AGARD-AG-298

Volume 2

NORTH ATLANTIC TREATY ORGANIZATION

ADVISORY GROUP FOR AEROSPACE RESEARCH AND DEVELOPMENT

(ORGANISATION DU TRAITE DE L'ATLANTIQUE NORD)

AGARDograph No. 298

AGARD MANUAL
on
AEROELASnCITY IN AXIAL-FLOW TURBOMACHINES

VOLUME 2

STRUCTURAL DYNAMICS AND AEROELASTICITY

Edited by

Max RPlatzer
Department of the Navy
Naval Postgraduate School
Monterey, CA 93943-5100, USA
and
Franklin O.Carta
United Technologies Research Center
East Hartford, CT 06108, USA

This AGARDograph was prepared at the request of the Propulsion and Energetics Panel
and of the Structures and Materials Panel of AGARD.
 THE MISSION OF AGARD

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the fields of science and technology relating to aerospace for the following purposes:

— Recommending effective ways for the member nations to use their research and development capabilities for the
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and development (with particular regard to its military application);

— Continuously stimulating advances in the aerospace sciences relevant to strengthening the common defence posture;

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— Exchange of scientific and technical information;

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Published June 1988

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All Rights Reserved

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 REPORT DOCUMENTATION PAGE
1. Recipient's Reference 2. Originator's Reference 3. Further Reference 4. Security Classification
of Document
AGARD-AG-298 ISBN 92-835-0467-4 UNCLASSIFIED
Volume 2
5. Originator Advisory Group for Aerospace Research and Development
North Atlantic Treaty Organization
7 rue Ancelle, 92200 Neuilly sur Seine, France
6. Tide
AGARD MANUAL ON AEROELASTICITY IN AXIAL-FLOW TURBO-
MACHINES, VOLUME 2 - STRUCTURAL DYNAMICS AND AEROELASTICITY
7. Presented ai

8.Author(s)/Editor(s) 9. Date
Editors: MF.Platzer and F.O.Carta June 1988

10. Author's/Editor's Address 11. Pages

See flyleaf. 268

12. Distribution Statement This document is distributed in accordance with AGARD

policies and regulations, which are outlined on the
Outside Back Covers of all AGARD publications.
13. Keywords/Descriptors

Reviewing Flutter
Turbomachinery Blades
Structural dynamic analysis Discs
Aeroelasticity Rotors
Metal fatigue

14. Abstract

The first volume of this Manual reviewed the state of the art of unsteady turbomachinery
aerodynamics as required for the study of aeroelasticity in axial turbomachines. This second
volume aims to complete the review by presenting the state of the art of structural dynamics and of
aeroelasticity.

The eleven chapters in this second volume give an overview of the subject and reviews of the
structural dynamics characteristics and analysis methods applicable to single blades and bladed
assemblies.

The blade fatigue problem and its assessment methods, and life-time prediction are considered.
Aeroelastic topics covered include: the problem of blade-disc shroud aeroelastic coupling,
formulations and solutions for tuned and mistuned rotors, and instrumentation on test
procedures to perform a fan flutter test. The effect of stagnation temperature and pressure on
flutter is demonstrated and currently available forced vibration and flutter design methodology
is reviewed.

This AGARDograph was prepared at the request of the Propulsion and Energetics Panel and
of the Structures and Materials Panel of AGARD.
 PREFACE

The first volume of this Manual reviewed the state of the art of unsteady turbomachinery aerodynamics as required for
the study of aeroelasticity in axial turbomachines. It is the objective of the present second volume to complete the review by
presenting to state of the art of structural dynamic's and of aeroelasticity.

As pointed out in the preface to the first volume, further engine performance improvements and the avoidance of
expensive engine modifications required to overcome aerodynamic/aeroelastic stability problems will not depend only on
the continued systematic research in unsteady turbomachinery aeroelasticity. Rather, the need to transfer highly specialised
unsteady aerodynamic and aeroelastic information to the turbomachinery design community and the introduction of young
engineers to these disciplines suggested the compilation of a "Manual on Aeroelasticity in Turbomachines", similar to the
" AC ARD Manual on Aeroelasticity" for the aeroelastic design of flight vehicles, due to the lack of any textbook or other
comprehensive compendium on unsteady aerodynamics and aeroelasticity in turbomachines.

This conclusion was presented to and endorsed by the AGARD Propulsion and Energetics and Structures and
Materials Panels, the U.S. Office of Naval Research, the Naval Air Systems Command, and the Air Force Office of Scientific
Research. The support of these organizations is gratefully acknowledged. We are especially indebted to the late Dr Herbert
J.Mueller, Research Administrator and Chief Scientist of the Naval Air Systems Command, for his encouragement and
guidance during the initial phase of the project. Thanks are also due to Dr Gerhard Heiche and Mr George Derderian (Naval
Air Systems Command), Dr Albert Wood (Office of Naval Research), Dr Anthony Amos (Air Force Office of Scientific
Research) and Mr David Drane (AGARD) for their continuing interest and support.

In the present second volume, after an introduction and overview by Sisto, the structural dynamics characteristics and
analysis methods applicable to single blades and whole bladed assemblies are reviewed. Ewins first presents a chapter on
basic structural dynamics, followed by a chapter on individual blades, written together with RHenry, and concludes with a
chapter on bladed assemblies. This is followed by an exposition of the blade fatigue problem and its assessment methods,
written by Armstrong. The problem of life time prediction is reviewed by Labourdette, who also summarizes ONERA's
research in viscoplasticity and continuous damage mechanics. The remaining chapters are devoted to aeroelasticity. Carta
first introduces the reader to the problem of blade-disk shroud aeroelastic coupling. Crawley then presents detailed
aeroelastic formations and solutions for tuned and mistuned rotors. Special attention is given in this chapter to the effects of
mistuning. The sophisticated instrumentation and test procedures required to perform a fan flutter test are reviewed in
considerable detail by Stargardter. The effect of stagnation temperature and pressure on flutter is demonstrated by Jeffers,
who presents flutter boundaries obtained in a heavily instrumented fan rig as well as in a full-scale engine. In the concluding
chapter the currently available forced vibration and flutter design methodology is reviewed and put in perspective by Snyder
and Burns.

The editors are deeply indebted to the authors for their willingness to contribute their time and energies to this project
in spite of other pressing demands, Our thanks also go to the authors' employers for their support. Funding limitations made
it necessary to limit the number of contributors. Nevertheless, we hope that a fairly comprehensive and balanced coverage of
the field of turbomachinery unsteady aerodynamics and aeroelasticity was accomplished and that the two volumes on
unsteady turbomachinery aerodynamics and on turbomachinery structural dynamics and aeroelasticity will be found useful
as an introduction to this important special discipline and as a basis for future work.

Max F.Platzer and Franklin O.Carta, Editors

 STRUCTURES AND MATERIALS PANEL

Chairman: Professor Paolo Santini Deputy Chairman: Prof Dr-Ing. Hans Forsching
Dipartimento Aerospaziale Direktor der DFVLR Institut fur Aeroelastik
Universita dcgli Studi di Roma Bunscnstrassc 10
"La Sapienza" D-3400 Gottingen - Germany
ViaEudossiana, 16
00185 Roma — Italy

MEMBERS OF THE SUB -COMMITTEE ON AEROELASTICITY

Chairman: Prof Dr-Ing Hans F6rsching

Direktor der DFVLR Institut fur Aeroelastik
Bunscnstrasse 10
D-3400 Gottingen — Germany
SMP Members
DrL.Chesta —IT MrR,F.O'Connell-US
Dr R.Freymann — LU Prof. P.Santini — IT
Prof. V.Giavotto — IT MrC.W.Skingle-UK
Mr JB.de Jonge — NL MrD.C.Thorby-UK
Dr J J.Kacprzynski — CA Prof. AJF.Tovar de Lemos — PO
MrOJ.Maurer-US MrRJ.Zwaan —NL

PANEL EXECUTIVE

Mr Murray C.McConnell — UK

AGARD-OTAN From USA and Canada

7, rue Ancelle AGARD-NATO
92200 Neuilly sur Seine Arm: SMP Executive
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Tel:(1)4738 5790 Telex:610176
 ABSTRACT

The first volume of this Manual reviewed the state of the art of unsteady turbomachinery aerodynamics as required for
the study of aeroelasticity in axial turbomachines. This second volume aims to complete the review by presenting the state of
the art of structural dynamics and of acroclasticity.

The eleven chapters in this second volume give an overview of the subject and reviews of the structural dynamics
characteristics and analysis methods applicable to single blades and bladed assemblies.

The blade fatigue problem and its assessment methods, and life-time-prediction are considered. Aeroelastic topics
covered include: the problem of blade-disc shroud aeroelastic coupling, formulations and solutions for tuned and mistuned
rotors, and instrumentation on test procedures to perform a fan flutter test. The effect of stagnation temperature and
pressure on flutter is demonstrated and currently available forced vibration and flutter design methodology is reviewed.

RESUME

Le premier volume de ce manuel a examine 1'etat actuel des connaissances dans te domaine de 1'aerodynamique
instationnaire des turbomachines en vue de 1'etude de 1'aeroelasticite dans les turbomachines axiales. Ce deuxieme volume
vient completer cet examen, en presentant 1'etat actuel des connaissances dans les domaines de la dynamique des structures
et de 1'aeroelasticitc.

Les onze chapitres du present volume donnent un apercu du sujet, avec un examen des caracteristiques de la
dynamique des structures et des methodes d'analyse applicables aux aubes simples et aux ensembles munis d'aubes. Le
probleme de la fatigue des aubes est examine, ainsi que les m£thodes d'estimation de la fatigue et de la duree de vie des
aubes.

Parmi les questions aeroelastiques couvertes, nous signalerions, le probleme du couplage aeroelastique de 1'aube et le
talon du disque, les formulations et les solutions en ce qui concerne les rotors accordes et desaccordes, et I'instrumentation
necessaire et les procedures & suivre pour la realisation des essais de flottement des soufflantes. L'effet de la temperature et
de la pression d'arret sur le flottement est demontre et la methodologie de calcul de la vibration forcee et du flottement,
employe £ 1'heure actuel, est passee en revue.
 CONTENTS

Page

PREFACE iii

STRUCTURES AND MATERIALS PANEL iv

ABSTRACT v

XII. INTRODUCTION AND OVERVIEW

F.Sisto, Stevens Institute of Technology
Introduction 12-1
Overview 12-2

Xin. BASIC STRUCTURAL DYNAMICS

DJ.Ewins, Imperial College of Science and Technology
Introduction 13-1
Structural Dynamic Characteristics 13-1
Structural Dynamic Analysis Methods 13-4
Finite Element Modelling (by R.Henry) 13-8
Dynamic Analysis Methods for Structural Assemblies 13-11

XIV. STRUCTURAL DYNAMIC CHARACTERISTICS OF INDIVIDUAL BLADES

DJ.Ewins, Imperial College of Science and Technology, and
R.Henry, Institut National des Sciences Appliquees
Introduction 14-1
Vibration Properties of Uniform Beams and Plates 14-2
Factors Which Influence the Vibration Properties of Blades 14-5
Analysis Methods for Practical Blades 14-15

XV. STRUCTURAL DYNAMIC CHARACTERISTICS OF BLADED ASSEMBLIES

DJ.Ewins, Imperial College of Science and Technology
Introduction 15-1
Structural Dynamics Models for Bladed Assemblies 15-2
Basic Structural Dynamic Properties — Natrural Frequencies, Mode Shapes 15-5
Modal Properties for Various Bladed Assemblies 15-10
Dynamic Analysis for Practical Assemblies (by R.Henry) 15-15
Vibration Properties of Mistuned and Packeted Bladed Assemblies 15-21
Forced Vibration Response 15-31

XVI. FATIGUE AND ASSESSMENT METHODS OF BLADE VIBRATION

E.K. Armstrong, Rolls-Royce pic
Introduction 16-1
Character of Fatigue 16-1
Observations on Blade Fatigue 16-6
Factors Affecting Blade Amplitudes of Vibration • 16-9
Methods of Assessment 16-12
Amplitude Ratio Method 16-13
Stress Level Method 16-20
Comparison of Methods 16-26
Steps of Assessment and Test Conditions 16-28
Conclusions 16-31
Appendix I 16-33

XVH. LIFE TIME PREDICTION: SYNTHESIS OF ONERA'S RESEARCH IN VISCOPLASTICITY AND

CONTINUOUS DAMAGE MECHANICS, APPLIED TO ENGINE MATERIALS AND STRUCTURES
R.Labourdette, ONERA
Introduction 17-1
Behaviour of Macroscopic Volume Element 17-1
Damage Description 17-7
Structural Life Prediction at High Temperature Under Complex Loadings 17-12
Conclusions 17-15
 Page
XVni. AEROELASTIC COUPLING - AN ELEMENTARY APPROACH
F.O.Carta, United Technologies Research Center
Introduction 18-1
Nomenclature 18-2
System Mode shapes 18-3
Analysis 18-4
Results 18-8
Parametric Variations 18-12
Conclusion 18-15
Appendices 18-15
XIX. AEROELASTIC FORMULATIONS FOR TURBOMACHINES AND PROPELLERS
E.F.Crawley, Massachusetts Institute of Technology
Introduction 19-1
Formulation and Solution of the Aeroelastic Problem 19-3
Formulation for Multiple Section Degrees of Freedom 19-7
Formulation for Multiple Spanwise Blade Modes 19-8
Solutions for Sinusoidal Temporal Representations 19-10
Explicit Tune Dependent Formulation of Aerodynamic Forces 19-13
Trends in Aeroelastic Stability 19-15
Effects of Mistiming on Stability 19-18
Appendix A 19-23
Appendix B 19-24
XX. FAN FLUTTER TEST
H.Stargardter, Pratt and Whitney Aircraft
Introduction 20-1
Test Procedure for a Flutter Evaluation 20-1
Identification of Blade Vibration 20-2
Fan Flutter Test, A Case History 20-4
Summary 20-35
XXI. AEROELASTIC THERMAL EFFECTS
J.D. Jeffers, HI, Tampa, Florida
Introduction 21-1
Historical Background 21-1
Follow-on Research 21-4
Concluding Remarks 21-6
XXn. FORCED VIBRATION AND FLUTTER DESIGN METHODOLOGY
L.E.Snyder and D.W.Bums, General Motors Corporation
Introduction 22-1
Characteristics of Flutter and Forced Vibration 22-1
Forced Vibration Design 22-3
Flutter Design 22-17
Addendum to Program Listing for Unsteady Two-Dimensional Linearized
Subsonic Flow in Cascades, Volume I, Chapter 3, pages 3—34 to 3—30 A-l

ALPHABETICAL LISTING OF REFERENCES R-l

vu
 A0v 238- vIOL-Z
12-1

INTRODUCTION AND OVERVIEW

by
F. SISTO
Department of Mechanical Engineering
Stevens Institute of Technology
Hoboken, New Jersey 07030
USA

INTRODUCTION in Volume II) it is important to note the

gap between flutter prediction and stress
Background. With minor exceptions prediction. Although linear analysis of
Chapters 1 through 11 contained in the aerodynamically forced vibration will
earlier Volume I of this Manual have been yield a vibration amplitude, given some
devoted to an exposition of those types of specified law for mechanical damping, the
unsteady aerodynamics required for the same is not true for flutter, or self-
study of aeroelasticity in axial turbo- excited aeroelastic instability. Only the
machines. Referring once again to boundary between stability and instability
Collar's triangle of forces (see Chapter may be discriminated with linear theory.
1) it is clear that the present volume Thus the typical aeroelastic eigenvalue
dealing with structural dynamics, or the analysis will not yield a flutter stress
third side of the triangle connecting the prediction.
vertices labelled elastic forces and
inertial forces, completes the mnemonic of For the assessment of fatigue damage,
the aeroelastic triangle. or in general the severity of the flutter
vibration, it is currently necessary to
Structural dynamics, independently of employ experimental means. A system with
its role in aeroelasticity, has had a measured flutter behavior may be assessed
broad, vigorous development in the present for the accumulation of damage over a
century. This outgrowth of structural given operating cycle and redesigned to
mechanics has been marked, after the lower the damage level. Most likely, the
earliest work by Euler, Bernoulli and redesign will attempt to eliminate flutter
Lagrange and the subsequent practical con- entirely. With forced vibration the re-
tribution of those such as Timoshenko and duction of stress by redesign is quite
DenHartog, by the formalization of tensor feasible if the cause has been properly
notation, which is thought to be more com- diagnosed.
pact and therefore more general. Perhaps
the most recent and most useful develop-
ment has been the adaptation of the matrix In the future, application of non-
method to structural dynamics. With the linear analysis will improve the capa-
necessary substructuring and/or discreti- bility to predict blade stress during
zation of complex structures, the matrix flutter. This capability is now in an
method has brought to bear the powerful embryonic state with the first results
mathematics of linear algebra and the issuing from stall flutter theories em-
matrix calculus to the field of structural ploying nonlinear aerodynamics. The par-
dynamics. It is not surprising therefore, ticular theories are those which effec-
that for real system designs, (e.g., an tively display a hysteretic dependence of
elastic disk bladed with 4% tip thickness aerodynamic forces and moments on blade
airfoils and provided with butting part- displacement and velocity. In addition,
span shrouds) the matrix formulation of there may be amplitude dependence of
aeroelastic problems is not only useful, forces which is not linear in this
but increasingly needed. The widespread displacement and velocity.
adoption of the finite element method (for
fine-grain quantitative accuracy) coupled For single degree of freedom systems,
with modal analysis (for the development thus employing a single modal coordinate,
of robust computational and experimental the flutter amplitude prediction effec-
methods) characterize the present state of tively makes use of the aerodynamic work
structural dynamics. Vectorization of the concept. In turn this is an expression of
equations for efficient program execution the more general principle of harmonic
on super-computers is a reality and future balance expounded in nonlinear mechanics.
application of parallel processors seems How this capability may effectively be
certain. expanded to multiple degrees of freedom,
for particular formulations of the non-
Volume II, the present volume, begins linear aerodynamic forces, is currently an
with three chapters devoted exclusively to active area of analytical and computation-
subjects in structural dynamics. al investigation. Formulation of the ex-
Sufficient fundamental information there- plicit time dependence of the aerodynamic
fore is covered, with the completion of forces for arbitrary blade motions is a
Chapters 13, 14, and 15, to support some requirement first described in Chapter 19.
of the special aeroelastic studies pre- For complex geometry, nonlinear structures
sented in the remainder of the volume. and flows with sharp gradients, time do-
These topics range from fatigue and blade main solutions may be the solutions of
life prediction to aeroelastic coupling, choice. Simultaneous integration of un-
mistuning effects, experimentation and steady aerodynamic and structural dynamic
thermal effects. The Manual concludes subprograms by time marching will be
with the all-important implications for facilitated with modern computational
design. developments and allow the realistic
treatment of nonlinearity in both struc-
Stress Prediction. Referring to the ture and fluid models. Flutter stress
previouslistoftopics addressed in prediction will become increasingly possi-
Chapters 16 through 22 (the last chapter ble as these developments materialize.
12-2

OVERVIEW affected by geometry, boundary conditions

(end and shroud fixity), rotor rotation
Fundamentals. In Chapter 13, entitled and centrifugal and Coriolis forces.
Basic Structural Dynamics, the underlying
mechanism of blade vibration in axial Tables are presented showing the natu-
turbomachinea is reviewed briefly by David ural frequencies and modes shapes for the
Ewins. The concepts of structural dy- gravest modes in flapwise bending, edge-
namics begin with the establishment of wise bending, and torsional modes and for
continuous and later discrete models. The five different sets of end conditions.
distribution of inertial and elastic (or Second order effects due to rotatory in-
mass and stiffness) properties are ertia and shear deflection are elucidated
described, and also damping, thus deriving for typical geometry and material
a set of governing differential equations. properties.
This method of analysis concludes with
solutions of these partial differential Similar studies illustrating the ef-
equations. In the case of the free fect of aspect ratio using plate theory
vibration, the solution yields the modal are presented next for uniform geometry
properties: the eigenfunctions and the continuous plates. The modes are now best
characteristic frequencies. The forced described in terms of the location of the
vibration solution yields the actual dis- nodal lines.
tribution of displacements (strain) and
stresses Cor specified external When the uniform beam is modified to
excitations. It is not emphasized at this include linear taper or linear twist sig-
point that when the aerodynamic forces are nificant changes in natural frequencies
linearly dependent only on blade displace- are observed, although the mode shapes are
ments and velocities these forces may relatively unchanged for the tapered
effectively be taken on the left hand side blade. With the twisted blades the modes
resulting in the statement of an aero- become more complicated with flapwise and
elastic eigenvalue problem. edgewise components of deformation appear-
ing in the bending modes (i.e., flexural
Discrete system and continuous system modes are coupled bending-bendlng).
analyses are summarized by Professor Similar conclusions apply to plate-
Ewins. The finite element model is sum- type analysis of lower aspect ratio
marized in a separate section by Dr. R. geometries.
Henry who is also the co-author of Chapter
14 following. It is pointed out that the Discussions of root flexibility, sta-
energy method of equation derivation pre- tic deflection and rotor rotation follow.
sumed in Lagrange's Equation is capable of The limiting values of flexibility are
supplying the effects of centrifugal discussed qualitatively, the chief conclu-
stiffening, centrifugal softening and sion being that at some point of lowered
gyroscopics. root rigidity it is necessary to consider
the bladed-disc assembly (rather than the
This chapter concludes with a discus- blade alone) to properly identify the
sion of different methods of analyzing modal properties of that system due to the
structural assemblies, including most coupling between blades afforded by the
generally the bladed disk assembly with flexible disk.
interconnecting shrouds between adjacent
blades. In particular the concept of sub- Blade lean and pretwist in a centrifu-
structuring is presented along with the gal field and gas bending loads provide
use of modal coordinates for synthesizing sources of "static" deformations about
the overall system equations. For which vibrations may occur. The centrifu-
example, the blades may be represented by gal effect also changes the blade effec-
a very few modal degrees of freedom for tive stiffness, providing an increase for
inclusion in an analysis of a bladed disk vibration normal to the plane of rotation
assembly. Brief summaries then follow the and a small decrease for vibratory dis-
two methods of system couplings modal placements in the plane of rotation.
coupling and frequency response coupling. Professor Ewins notes that these effects
must be taken into account in order to
predict accurately the modal properties
under operating conditions which, for a
Beams and Plates. In Chapter 14 en- rotor blade, include the centrifugal
titled Structural Dynamic Characteristics effects attendant to rotation.
of Individual Blades, David Ewins and
R. Henry continue with the exposition of Noting that the finite element method
single blades treated alternatively as is most flexible and now in wide use, a
beams or plates. discussion of the beam-type elements and
plate shell elements then follows. Short
Beams, or beamlike blades, have defor- sections describe the ability of the FEN
mation functionally dependent solely on to formulate complex cross-section proper-
the radial coordinate whereas with plate- ties, root and platform flexibilities,
like structures the deformation will also shrouds, "static" loads (noted pre-
be dependent on the chordwise coordinate viously), Coriolis and centrifugal effect,
(i.e., the cross-section itself distorts). and the temperature effect on modulus of
For beams the aspect ratio is greater than elasticity.
for plate-like structures and in both
cases the thickness ratio is considerably The concluding section of this chapter
less than unity, at the blade tips is a description of the previous concepts
especially. as applied to three specific rotor config-
urations, by computation, and including
Initially, the vibration properties of some experimental verification. The first
uniform beams are described in terms of example is the analysis of a low pressure
natural frequency and mode shape and as turbine blade using a straight, tapered.
 12-3

twisted beam element and illustrating The computed results reveal the typi-
the effect of bending-bending-torslon cal feature (for each configuration) of
coupling. Inertia coupling due to non- 'single1 and 'double' modes of which the
coincidence of section centroids and latter is in the majority. Most of the
center of twist and the effect of rotation assemblies' modes occur in these pairs
are demonstrated in a series of examples. with similar shapes and identical
A number of practical conclusions are frequencies. This is consistent with the
drawn concerning the effects on natural identification of these combined modes by
frequency and mode shapes and the conse- the number of nodal diameters, n. The
quent need for using "refined" beam-type special nature of modes for n • 0 and
elements for aeroelastic studies. n • 1 is discussed as is the general
behavior as n becomes very large. With
The second example, the analysis of a the beam, plate, and finite element models
fan blade using three node triangular nodal circles may also appear. For an N-
plate elements, elucidates the untwist bladed disk, the maximum value for n is
effect which, along with accurate centri- 1/2N (or(N-l)/2 if N is odd) for the dis-
fugal stiffening, requires an iterative crete lumped parameter model. However,
solution procedure. Coriolis effects can for continuous disk or shroud models
be safely ignored for this particular pro- higher values of n are possible, but the
blem of mode definition. This is not blade or rim displacement for n > N/2 are
necessarily true for highly swept blades, Indistinguishable from the mode with N-n
or blades on a processing rotor. In any nodal diameters. This 'aliasing1 property
event, the nonlinear iterative solution is very significant and may allow an
procedure, for example, using an updated infeeding of energy from aerodynamic
Lagrangian formulation at each iteration, sources at frequencies associated with n
is recommended. nodal diameters, but with N-n nodal
diameters being measured. This important
The third example is a high pressure feature is fully discussed and
turbine blade modelled with thick shell elucidated.
elements. Dynamic analysis shows that the
rotation effect is weak for both frequency A number of configurations are ana-
and mode shape for this "thick" blade. lyzed, with an N = 36 bladed disk used to
The change in frequency can be adequately study parametrically the effects of disk
estimated from the nonrotating case using stiffness, root flexibility, stagger,
the method of Rayleigh Quotients. twist, shroud stiffness and shroud
connection
The final paragraphs of the chapter
consist of practical recommendations for Dr. Henry presents the results of
structural dynamic modelling of single analyzing a specific turbine rotor using
blades based on the findings of the pre- finite elements and assuming axisymmetry.
ceding studies and computations. The six lowest frequency modes are dis-
cussed as well as their variation with
rotational speed. The important point is
Blade-Disk Model. In Chapter 15, made that if cyclic symmetry must be
Structural Dynamics of Bladed Assemblies, assumed (e.g., due to a small N, and hence
the subject material of the previous two with blades which cannot be modelled as
chapters is brought together by David beams) then the mode shapes no longer dis-
Ewins for the treatment of practical play simple diameters and circles as nodal
systems. As in the previous two chapters lines.
there is also an important contribution by
Or. R. Henry in this chapter. The important characteristics of mis-
tuned assemblies is discussed next leading
to the important characteristics of mode
Initially, the configurations to be and frequency splitting. Each double mode
examined are described, ranging from un- with identical frequencies and mode shapes
shrouded identical blades to packetted in the tuned state splits into distinct
blades to blades that are not identical to modes with close natural frequencies and
each other. Disks are considered with modal shapes upon the introduction of
various degrees of flexibility up to com- mistuning. The very complex behavior of
plete rigidity. In the remainder of the mistuned assemblies is discussed including
chapter the natural frequencies and mode the effects of regular versus random
shapes are presented for these various mistuning, the effect of damping (may
assemblies and the major controlling introduce complex modes if the damping is
factors are described. The results of non-proportional) and the effect of lash-
analyzing and computing a number of repre- ing the blades in packets. The subject of
sentative cases are presented qualitative- mistuning as applied to self-excited sys-
ly in the text and in the form of figures tems is returned to in Chapter 19.
for more quantitative comparisons and
tests. The chapter closes with a discussion
of force vibration assuming two types of
Models which are evaluated by computa- forcing: single point harmonic excitation
tion are generally at three distinct and engine-order excitation. It is empha-
levels of abstraction: the lumped param- sized that a resonance can be obtained
eter model where full axisymmetry is only by excitation at the proper frequency
assumed; the beam and plate models of and with an appropriate spatial distribu-
blades with either axisymmetric, cyclical- tion (i.e., the sign of the work done on
ly symmetry (i.e., rotationally periodic the vibration will depend locally on the
structure) or full blade-to-blade phase between the force and the displace-
variation; and finally, the finite element ment and will be zero at the nodes). The
model in which axisymmetry, or at best detailed character of the response of a
cyclic symmetry with substructuring must number of computed cases is presented for
be assumed due to computing limitations. both types of excitation. It is empha-
12-4

sized that the aliasing phenomenon appears The emphasis is on jet engine experience;
most importantly in the engine order case. flight conditions and installation charac-
An (N-n)th engine order excitation and an teristics enter the discussion frequently.
nth engine-order excitation will be
equally effective in exciting modes with n A detailed derivation is then given of
diametral nodes and the vibration will the "amplitude ratio" method of forced vi-
occur at an N-n multiple of the rotation- bration assessment. The application of
al speed. The corresponding alias vibra- the method is discussed, including corre-
tion where the nth engine order excites lations supplied from cantilever beam
N-n nodal diameter modes is also demon- theory, different blade materials, fatigue
strated and discussed. This and other ex- capability, material damping, temperature,
tremely complex behavior is presented for statistical scatter, and the evolution of
a number of computed cases. The chapter design/development rules.
ends with a discussion and an estimate
of the increase in forced response on The "amplitude ratio" method is con-
the "worst", or "rogue", blade attribu- trasted with the stress level method of
table to a particular degree of mistuning. assessing blade vibration and fatigue
This is a fitting conclusion for the failures. This second method depends on a
third chapter on structural dynamics where detailed knowledge of the vibratory and
the emphasis is on forcing, not neces- steady stresses and has become increasing-
sarily or exclusively from aerodynamic ly used as the finite element method of
sources. It should be emphasized, as stress analysis has come into widespread
introduced briefly in Chapter 1 and dis- use. The application of the method, and
cussed more fully in Chapter 19, that the the special effects that can be handled
conclusions on the harmful effect of mis- (such as restraint of edge warping, cen-
tuning may be quite different in the case trifugal forces, etc.) is discussed at
of self-excited aeroelastic instabilities. length and practical examples are
presented.
Vibratory Blade Failure. Chapter 16, The two methods are compared and the
F a t i g u e a n d AssessmentMethods of Blade benefits of each are emphasized. Modern
Vibrations, by Keith Armstrong is a tho- practice in a large organization will be
rough treatise on the practical aspects of to use an efficient blend of the two. The
blade failure and its prevention. The influence on design and an historical
initial sections of this chapter are taken description, with many examples and prac-
up with many definitions related to metal tical observations, concludes this ex-
fatigue and the character of the tremely useful chapter on the blade vibra-
phenomenon. tion aspects of jet engine design and
operation. The successful developments of
Definition of alternating stress, the the past twenty years in the United
experimental means of testing metal sam- Kingdom are described and explicated in
ples and/or blades with a high number of the text of this chapter.
stress reversals,the establishment thereby
of S-N curves and the use of these data to High Temperature Material Behavior.
construct the modified Goodman diagram are Chapter H has a long tTtlV, Lifetime
all thoroughly discussed. The qualitative Prediction: Synthesis of ONERA's Research
nature of these diagrams is described, in- in Viscoplasticity and Continuous Damage
cluding the factors which affect them such Mechanics Applied to Engine Materials and
as notch sensitivity, mean stress level, Structures. Contrasted with the title,
(due to static loading as well as residual this short chapter by R. Labourdette deals
stresses) the presence of defects and the with the problems of high temperature tur-
propagation of fatigue cracks. bine blades: creep and fatigue. While in
the previous chapter temperature effects
This leads to a discussion of fracture are recognized but not dealt with in great
mechanics, particularly as applied to com- depth this is not so in the present
pressor/fan blades and turbine buckets. chapter; temperature effects are
The effect of grain size, loading history paramount. The work is a joint effort of
(leading to Miner's hypothesis for the ONERA and SNECMA.
accumulation of damage) surface treatment,
low cycle fatigue, fretting, erosion, Adopting a tensor notation, visco-
corrosion, and data scatter are discussed. plasticity of high temperature materials
The need for fatigue testing is empha- is formulated in the general thermodynamic
sized, the correspondence between sample framework of irreversible processes. This
or coupon testing versus blade testing branch of continuum mechanics is applied
is discussed and the methods of testing to the- cyclic straining of the prototyp-
are described. ical turbine blade beyond the elastic
limit. Creep represents the response to
A discussion follows of the factors loading over long periods of time, i.e.,
affecting amplitude of blade vibration, with a very low frequency of the loading
noting the previously discussed disparity cycle. For fatigue behavior the frequen-
between flutter and forced vibration inso- cies are much higher and the vibration
far as amplitude determination is thus defined is most appropriate to the
involved. The manner of stress measure- province of dynamic aeroelasticity. The
ment using strain gages is discussed and theory is also quite general and accounts
the practical diagnosis of vibration of for an interaction between creep and
both types in running compressors is pre- fatigue.
sented by examples. This broad ranging
mid-chapter exposition by Dr. Armstrong is The model developed for general visco-
one in which qualitative factors related plastic behavior of a metallic material is
to vane and blade vibration,its detection, then applied to the generation of a new
and developmental steps for its ameliora- theory for damage accumulation. This
tion are discussed in considerable detail. theory of damage on the microscopic scale
 12-5

leads to the all-important prediction of of most of the remaining chapters. The

crack initiation which is the starting principal value of Chapter 18 is that it
point for subsequent macroscopic treatment is a compact exposition of self-excited
of crack propagation under cyclic aeroelastic instability as it is actually
straining. The theory, based on continuum encountered in axial turbotnachines when
damage mechanics, is expressed in a finite dominated by strong structural coupling
element program EVPCYCL developed by Dr. between blades. Succeeding chapters build
Labourdette at ONERA. The program algo- on this exposition and may be contrasted
rithm integrates constitutive equations of and related to it for increased compre-
the developed theory, and predicts the hension and relevance of the material in
cycles to crack initiation for given load- those chapters. In this sense Chapter 18
ing and temperature histories. Comparison encapsulates the entire Manual by means
with experimental data shows that the of a cogent and historically signifi-
viscoplasticity model is an accurate tool cant example and its subsequent
for prediction of crack initiation. The generalizations.
method is being extended to anisotropic
materials (e.g., single crystal alloys) Mistuning Effects. In Chapter 19,
and more complex geometry. The appearance Aeroelastic Formulation and Trends for
of these very useful new results is anti- Tuned and Mistuned Rotors, the general
cipated with interest for later editions system approach to turbomachine aeroelas-
of this Manual. ticity is expounded by Edward Crawley.
The structural dynamic and aerodynamic
Systems Instability. Stemming from models are combined in a consistent manner
its central importance in the description to yield an aeroelastic model. The re-
of bladed-disk flutter, the coupled theory sults are discussed with major emphasis
introduced by Franklin O. Carta (1967) has on stability discrimination (flutter).
been summarized by him in Chapter 18, Explicit time domain aerodynamics is dis-
Aeroelastic Coupling - An Elementary cussed for use in predicting forced vibra-
Approach. The historical setting for the tion response. Considerable attention is
appearance of this type of flutter is out- given to the effects of mistuning and
lined briefly, noting that the bending/ algorithms for optimization of this effect
torsion coupling is augmented by part-span are presented and discussed.
shrouds typical of post-1960 turbofans.
This coupling of the bladed-shrouded-disk Noting that the available model formu-
is shown later to be critically lations are somewhat incomplete, particu-
destabilizing. larly with respect to aerodynamic
operators, the point is made that the time
The equations of motion of the aero- dependence can be expressed within the
elastic system, expressed in travelling time domain or the frequency domain. The
wave form, are combined with unsteady mathematical models are then developed
aerodynamic terms and the solution for the beginning with the representative section
flutter condition is discriminated by the having a single degree of freedom. A cas-
method of aerodynamic work integrated over cade of such blades is coupled only
the blade span. The intrablade phase through the moti6n-dependent force influ-
angle is fixed at 90 degrees by the struc- ence of one blade on another. By making
tural coupling, and, with isolated airfoil the assumption of sinusoidal time depen-
aerodynamics, the flutter instability is dence with a fixed interblade phase angle,
shown to be most susceptible in the modes a, the travelling wave modes of the aero-
having an intermediate number (i.e, 3, 4, elastic eigenvalue problem are determined.
or 5) of diametral nodes with forward tra- It is pointed out that this model, initi-
velling waves. The greater the degree of ally chosen for the aerodynamics
bending/torsion coupling the greater the formulation, is not unique. It is ade-
likelihood of coupled flutter instability. quate for tuned system SDOF analysis.
Parametric studies of shroud location con-
firm this trend and experiments with The aeroelastic eigenvalue problem can
research compressors lend further support be, and is, reformulated in terms of In-
to those general conclusions. dividual blade coordinates. When this is
done the aerodynamic influence coefficient
This chapter concludes with a survey matrix which results has a high degree of
of many recent studies employing compres- symmetry, depending on the degree of geo-
sible cascade aerodynamics and tuned and metric uniformity*. Each element is
alternatively mlstuned rotors. In this obtained from a discrete Fourier transform
section of the chapter Frank Carta shows of the complex force coefficient obtained
that the importance of the interblade from the conventional travelling wave
phase angle is undiminished in these sys- formulation. In this second form (i.e.,
tem mode instabilities. In fact, the num- individual blade coordinates) the aero-
ber of nodal diameters of a particular elastic equation demonstrates that a given
mode is a surrogate variable for the blade is strongly affected only by its
interblade phase angle; specifically a <• most immediate neighbors.
2nn/N where n is the number of nodal diam-
eters and N is the number of blades in the A third alternative formulation is in
row. This proxy relationship between n terms of the standing waves, or twin
and o is not emphasized elsewhere in orthogonal modes introduced earlier in
Volume II, although the groundwork for the Chapter 15 for the discussion of forced
importance of a in turbomachine aeroelas- vibration. The aeroelastic eigenvalue
ticity is laid down firmly in Volume I of problem expressed in this form allows the
this Manual.
Conclusions, drawn from this straight-
forward analysis of the fundamental aero- *For complete uniformity the matrix is
elastic problem of coupled systems, have circulant, the properties of which were
wide validity with respect to the material first exploited by Lane (1956).
12-6

use of experimentally determined standing the dynamic coupling, e.g. through the
wave structural modes. aerodynamic reactions, is usually not
strong enough to be of significance. The
These results are subsequently gen- effect of loading is speculated as being a
eralized for two degrees of freedom per possible source of flutter near stall, and
blade, e.g., coupled bending torsion, and the stability trends with reduced velocity
then to multiple degrees of freedom. In are discussed qualitatively, noting both
broad terms this amounts to replacing structural and aerodynamic implications of
single elements in the previous formu- the reduced frequency parameter.
lation by sub-matrices and sub-vectors.
Furthermore, modal coordinates are usually The remainder of the chapter Is
adopted for the multiple degree of freedom concerned with mistuning for stability
model. The transformation of the blade enhancement. Using a simplified example
aerodynamic forces is detailed in an for a supersonic fan with structural
appendix. In all these developments (mass) mistuning and restricted to tor-
Professor Crawley makes numerous qualita- sional motion, it is shown that optimal
tive comments concerning the properties of mistuning is more effective than alternate
the various matrices and describes under- mistuning which might intuitively be
lying physical characteristics of the chosen. The details of the optimal mis-
aeroelastic system. tuning patterns are then discussed, where
the optima have been obtained by mini-
Solution of the various problem formu- mizing a penalty function using nonlinear
lations are then exemplified by expressing programming techniques detailed elsewhere.
the aerodynamic forces in travelling-wave
form (sinusoidal motion) and restricting
the analysis to one or two degrees of Basically the optimum patterns are
freedom for a cascade of characteristic- 'almost' alternate mistuning, I.e., with
section blades. Traditional methods, such Important exceptions on one or two blades.
as the V-g and p-k methods are then dis- Practical considerations devolve upon the
cussed for solution, and subject to accuracy with which an intentional mis-
special conditions such as the very large tuning pattern can be effectuated due to
mass ratio of blade to air. I.e., the errors in measurement, manufacture, or
real part of the aeroelastic eigenvalue is selective assembly. Based on considera-
very close to the natural frequency in tion of this nature, alternate mistuning
yacuo. The character of the eigenvalue is more robust and may be the method of
plots is discussed giving some insight as choice. The detailed analysis of mistun-
to how these may be expected to differ ing patterns and their practicality in the
from fixed wing results. Naturally, the manufacturing phase forms a fitting con-
interblade phase angle is a key clusion to this extremely important
parameter. concept which is one of the dominant areas
of practical application in turbomachine
For the first time in the Manual the aeroelasticity today. More research on
need is described for obtaining the ex- mistuning may be expected to yield in-
plicit time dependence of the aerodynamic creasingly practical results.
forces for arbitrary blade motions. This
form of the aerodynamic operators, for
studying response to tip rubs or surge for Aeroelastic Research in Rotors. In
example, is theoretically derivable by a Chapter2"0"7FanFlutter Test,By Hans
complex inverse Fourier integral from the Stargardter, aeroelastic instability in a
frequency domain expression. In practice turbomachine is investigated experimental-
this is accomplished by using the so- ly. In this sense it epitomizes the
called Pads' approximation, whose deriva- purely experimental approach to the sub-
tion is outlined. ject of the Manual.
The analytical methods described up to Using an existing single stage fan
this point are applied next to a short research rig, a great deal of sophisti-
discussion of trends in aeroelastic sta- cated Instrumentation was installed for
bility of turbomachine rotors. Analyzing the measurement of steady and unsteady
a simplified system, it is demonstrated aerodynamic quantities as well as blade
that a necessary but not sufficient condi- deflection and deformation. (Deformation
tion for aeroelastic stability is that the measurements were limited to steady un-
blades be self-damped; i.e., the effect of twist and uncambering due to centrifugal
a blade's motion upon itself must be to effects.) In order to emphasize the range
contribute positive aerodynamic damping. and complexity of the instrumentation,
The unsteady interactions amongst or be- these systems are simply listed here;
tween blades in the cascade are destabi- their frequency response, accuracy, and
lizing for at least one possible inter- modes of operation are detailed in the
blade phase angle. This blade-to-blade text of the Chapter.
destabilizing influence is reduced by
mistuning, and is hence desirable. Instrumentation: Strain gages, laser
Mistuning, however, can never produce beams and blade-mounted mirrors, semicon-
stability when the self-damping is ductor pressure transducers, hot film
negative. With nonzero structural damping gages (probes and anemometers), time code
blades of larger (blade to air) mass ratio generator and revolution counter, wedge
are relatively more stable. probes, static taps and thermocouples.
Most unique was the large number of small
The effect of coupling is qualitative- mirrors fixed to the rotor blades in a
ly discussed and it is noted that kine- variety of radial and chordwise positions.
matic coupling (e.g., the presence of some By recording the position of reflected
bending displacements in a predominantly laser light beams it was posssible to re-
torsional natural mode) may be quite im- duce the data to yield blade deflection,
portant in determining stability whereas both steady and unsteady (i.e., mode
 12-7

shapes) as well as static untwist and terns are identified by spectral process-
uncamber. The acquisition of reliable ing of the experimental data. The
blade surface pressures and the recon- guidance that this result provides for the
struction of passage pressure distribution theoretical computational aeroelastician
near the housing are described. Of equal is profound. It will be interesting to
importance was the recording and elec- see in the future how this mutual stimula-
tronic reduction of these data. Sophisti- tion of theory and experiment will help to
cated enhancement and spectral processing define the most realistic and therefore
of the periodic signals was employed, and most useful aeroelastic model for a row of
analysis of the on-rotor data as a super- turbomachine blades which are always
position of backward and forward rotating mistuned to some extent.
harmonic waves, allowed comparison and
correlation with other data measured in Effects of Ambient Variables. In
absolute coordinates, i.e., relative to Chapter 21, Aeroelastic Thermal Effects,
the fixed housing of the fan. Phasing was by James D. Jeffers, II, the effect on
obtained using cross-spectral density flutter of variable inlet stagnation tem-
techniques. These overall data reduction perature and pressure is reported. These
methods are unique. As noted throughout experimentally derived results were ob-
the Manual, the phasing of certain param- tained in a sequence of tests beginning
eters along the cascade, e.g., the inter- with a heavily instrumented fan rig, pro-
blade phase angle, is of supreme impor- gressing on to a full-scale engine program
tance in understanding and describing and concluding with a full parametric
aeroelastic phenomena in turbomachines. flutter mapping of the engine under NASA
auspices.
Using this powerful system for the ac-
quisition of flutter-related data the ex- The 3-stage fan rig, operated under
perimental exploration of fan flutter led standard sea level static inlet conditions
to some interesting results. The flutter demonstrated a first stage stall flutter
region of the fan was mapped out. Aero- boundary referred to a representative
dynamic work computation led to the deter- span, similar to Figure 2 of Chapter 1.
mination of modal aerodynamic damping The mode was predominantly with a 5 nodal
values and are shown to be negative in diameter system mode, and classified in
flutter. The historic correlation of the region la of Figure 1 in Chapter 1. The
flutter region in incidence versus reduced important conclusion that the operating
velocity coordinates, and as depicted line of the fan component would not pene-
somewhat humorously in Figure 2 of Chapter trate the stall flutter region was a
1, is confirmed experimentally in the factor in clearing the engine for
final • figure of Chapter 20. A number of fabrication. Subsequently, simulated
other observations are made concerning the flight testing of the full engine which
steady pressure distribution, the passage included high Mach number conditions at
Mach number distribution, the localization altitude, uncovered a flutter condition.
of instability influence near the leading However, the mode was predominantly above
edge and the roles of uncamber and untwist shroud torsion at a slightly higher fre-
in flutter. quency corresponding to the next higher
natural mode. This important and unex-
Perhaps the most important contribu- pected result was then mapped in great
tion of the Chapter is the detailed infor- detail using the high altitude facility at
mation concerning blade frequency, vibra- NASA Lewis.
tion amplitude, and interblade phase
angle. For this type of flutter, identi-
fied as the subsonic/transonic flutter of In this program the imposition of
region I in Figure 1 of Chapter 1, the higher inlet stagnation temperature, and
frequencies of all blades are identical; to a lesser extent higher inlet stagnation
there is a single flutter freuqency, i.e., pressure, were found to be destabilizing.
"frequency entrainment" has taken place. Although the effect was quantitatively
Thus the interblade phase angle between measured for the particular flight article
any two blades is well-defined and remains of the test, no theoretical underlying
constant in time. However, the value of explanation was offered. Flutter has
the interblade angle varies from passage multiple-parameter dependence and when
to passage on the rotor and concomitantly, there is a validated analytical theory,
the vibration amplitudes vary from blade these dependencies are made explicit.
to blade (but not in time). This detailed However, the theory of unsteady stalled
experimental revelation of the flutter aerodynamics is still in a deficient
mode goes beyond the analytical or com- state, particularly with respect to Mach
putational description of the flutter number and Reynolds number dependence (see
mode in a mistuned stage. Mistuning, as Chapter 8). Thus, the results, reported
characterized by the assembled blade for the first time in Dr. Jeffers'
natural frequency distribution, is shown chapter, are quantitatively useful as the
to correlate with the blade amplitude at basis for a semi-empirical model for un-
flutter. Thus, it is suggested by Hans steady stalled aerodynamics.
Stargardter, that the analysis of flutter
in mistuned stages consider the following Perhaps of most importance was the
model: The blade amplitude pattern repre- experimental confirmation of the stall
sents "a family of spatial harmonics flutter mode as described in the previous
described by the superposition of a number chapter; the vibration at a single fre-
of rotating nodal diameter patterns, each quency in rotor-fixed coordinates is com-
characterized by a different number of posed of a number of different nodal
nodal diameters with different but uniform diameter mode shapes travelling in both
amplitudes and different but uniform phase forward and backward directions in these
indexing, with each pattern rotating at a coordinates. As noted in the previous
speed that results in the same flutter chapter, the blade-to-blade vibration am-
frequency u&" In Chapter 20 these pat- plitudes vary around the rotor circum-
12-8

ference as does the interblade phase Each of the five types of flutter are
angle. In essence, the present chapter then discussed: subsonic/transonic stall
demonstrated that the contribution of each flutter, unstalled supersonic flutter,
complex mode to the overall aeroelastic supersonic torsional flutter, choke
mode is dependent as well upon inlet and flutter, and supersonic stall flutter.
engine operating conditions. A reliable The characteristics with respect to unique
stall flutter prediction system in a prac- frequency and interblade phase angle, mode
tical sense remains somewhat elusive. shape and operative mechanism are dis-
cussed at length. Corrective actions that
Aeroelastic Design. The terminus of the may be taken by the designer, such as
M a n u a l I s C h a p t e r 22, entitled Forced lowering the reduced velocity for example,
Vibration and Flutter Design Methodology, are presented in the concluding sections
authored by Lynn Snyder and Donald Burns. of the chapter. in a sense this final
The focus here is on summarizing what is section of Chapter 22 of Volume II and of
known about aeroelasticity in axial turbo- the entire Manual, is the most immediately
machines and bringing it to bear on the useful to the operator or designer faced
process of design. The procedure may be with self-excited aeroelastic instability
iterative in the sense that a candidate in an axial turbomachine. Although the
design which fails flutter or fatigue cri- chapter is rife with practical explanation
teria is redesigned and then re-analyzed of vibration mechanisms and the means to
or re-tested. Whereas flutter is to be ameliorate or eliminate the resulting
avoided entirely, the criterion for forced fatigue damage, this is most particularly
vibration is to limit the stresses to some true of the final section. Although there
fraction of the endurance limit. These is always need for further research, this
two contrasting criteria have been noted final chapter indicates that much is known
previously in the Manual. Prediction is about aeroelastic problems in axial turbo-
based on empirical correlation and semi- machines and much can be done of a practi-
empiricism for certain phenomena while cal nature to eliminate its harmful
some types of flutter are assessed on a aspects. It is always nice to close on an
theoretical/analytical basis. optimistic note.
In order to limit forced vibration
response at the design stage it is neces-
sary to first identify the sources of ex-
citation by constructing a Campbell dia-
gram based on calculated natural modes and Acknowledgements. The author is pleased
on noting potential mechanical and aero- to acknowledge the support of the Naval
dynamic periodic forces. This is followed Air System Command, via the Naval Post-
by an assessment of stresses as the inter- graduate School, for supporting these par-
sections on the diagram in the running ticular publishing efforts. More Impor-
range. This assessment is based mostly on tantly, the editors and other contributors
empirical correlation for each mechanism, are to be congratulated for their applica-
e.g., rotor blade excitation in the tion to a difficult assignment, under
torsional mode from upstream struts. The sometimes trying circumstances. The
predicted stresses are then entered into a result has been a very valuable set of
modified Goodman diagram for several high reference volumes that are now available
stress points on the blade, taking into to the aerospace research and development
account stress concentration factors, community. AGARD is also to be commended
notch sensitivity, temperature distribu- for its foresight in undertaking the pro-
tion and scatter of fatigue data (a factor ject Initially and for underwriting
of safety amounting to three standard substantial production costs.
deviations typically is used). If a blade
is found deficient in this assessment, or
for that matter at a later stage in the
course of subsequent experimentation,
there are a number of corrective measures
that can be taken in the redesign effort.
These range from weakening the effective
source of excitation, to changing the
forcing and/or natural frequencies, to the
introduction of increased mechanical
damping. An example is given of the
assessment of a particular turbine rotor
row. This was found to have an adequate
vibratory stress margin, and hence, re-
design was not required.
In the discussion of flutter assess-
ment, five different types of mechanisms
are discussed by Dr. Snyder and Mr. Burns.
These correspond roughly to the regions in
Figure 1 of Chapter 1. Each of the five
phenomena are discussed with respect to
changes in five dominant flutter design
parameters: reduced velocity, Mach num-
ber, steady loading, mode shape, and pres-
sure or density level. In the last
parameter the effect of temperature is
subsumed in pressure and density through
the perfect gas law and the effect of
temperature is felt also in the Mach num-
ber through the acoustic velocity.
 A-l

ADDENDUM TO PROGRAM LISTING FOR

UNSTEADY TWO-DIMENSIONAL LINEARIZED SUBSONIC FLOW IN CASCADES
VOLUME I, CHAPTER 3, PAGES 3-24 TO 3-30

MODIFICATION TO LINSUB
By
D. S. Vhitchead

It has been found that progran LINSUB (pp. 3-24 through 3-30 in
Volume I) produces incorrect results in certain cases when the phase angle
is not within the recommended range. This la due to the condition used to
terminate the series, which terminates vhen one tern of the aeries becomes
small. In order to eliminate this behaviour, It is recommended that the
program should be modified so that the series is terminated vhen tvo
successive terms of the series become snail. The nodifications occur in
subroutine OSVK (pp. 3-26, 3-27) and are shown In the following tvo
extracts.
[Left Column, p. 3-27]
C ASSEMBLE MATRIX
C I(-M+1 IN PAPER) GIVES VORTEX POSITION
C J(-U1 IN PAPER) GIVES HATCHING POINT
C
30 CALL VAVB(IR.IV)
IF(Itf.EQ.l) GO TO 142
DO 131 I.l.NP
DO 131 J-l.NP
IF(ICHECK(I,J).E0.2) GO TO 131 Replace .EQ.l by .BQ.2.

[Right Column, p. 3-27]

C CHECK CONVERGENCE OP SERIES
C
X-TERHR*TERMR+TBRHI*TBR>tt
Y-KR(X>J)*KR(ItJ)*KI(I,J)*KX(X,J)
IP((X/Y).LE.1.0E-10) GO TO 111
ICHECK(I,J)-0 Replace 2 old lines
GO TO 131 by 7 new lines.
111 IP(ICHBCK(I,J).EO.l) GO TO 112
ICH£CK(I,J).l
GO TO 131
112 ICHECK(I,J)-2
ICOUNT-ICOUNT+1
 R-l

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R-4

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 R-5

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R-6

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 R-7

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R-8
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 R-9

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 13-1

BASIC STRUCTURAL DYNAMICS

by

D. J. EWINS
Imperial College of Science and Technology
Department of Mechanical Engineering
Exhibition Road, London SW7 2BX

INTRODUCTION parameters are introduced to the model.

The final analysis stage Is that for a
We now turn our attention to the forced vibration response and this task
other major aspect of this work: namely, requires the introduction of additional
the structural dynamics effects which com- information in the shape of the forcing
bine with the aero/fluid phenomena pre- function. Here, it can be seen that the
viously detailed in Volume I to constitute forced vibration response depends not only
the complete aeroelastic problem. The on the structure itself (as is the case
objectives of this present chapter are to for the modal properties) but also on the
provide the necessary introduction for the additional factors contained in the forc-
following three chapters which deal with ing or excitation function(s).
the purely structural dynamic aspects of
blade and bladed assembly vibration. Finally, In the section "Dynamic
These chapters will in turn be followed by Analysis Method for Structural Assemblies"
others describing methods which enable us we shall outline some of the special anal-
to combine both the aero-and the struc- ysis methods which are especially appro-
tural dynamic characteristics, thereby priate for dealing with the particular
describing the full aeroelastic phenomena. structural forms common in bladed
Throughout, we shall be examining the assemblies. It will be seen that in most
problem from the viewpoint of the analyst such cases it is essential that all the
wishing to predict the magnitude and blades in one row or stage be analyzed
nature of blade vibration in order to simultaneously in a single model and not
assess the damage incurred by the blades treated independently as individual
as a result. Thus, we shall be concerned blades. This often results in rather
with two complementary aspects: the vi- large models, especially for those stages
bration properties of a blade or bladed with many blades. Not only is this ex-
assembly and its response characteristics. pensive but since the blades are generally
Of course, these are closely related but (assumed to be) identical it is inef-
it is very important to establish at the ficient unless the multi-component and
outset that they are not one and the same repeated symmetry characteristics of the
thing. assembly are exploited. Methods to
achieve this are discussed in this con-
In this chapter, we shall be intro- cluding section.
ducing both the concepts applicable to the
structural dynamics aspects and the pro- STRUCTURAL DYNAMIC CHARACTERISTICS
perties of interest as well as summarizing
the appropriate analysis methods which are Modelling
available for deriving the required infor-
mation in specific cases. As it is The first task In any vibration anal-
intended to provide a summary applicable ysis is the definition of a suitable
to the most general cases, It will be mathematical model upon which the required
necessarily concise and readers requiring analysis can be based. In the present
a more exhaustive treatment are referred case, we seek to describe first the mass
to a suitable text. One the other hand, (or inertia) and stiffness properties of
readers already familiar with structural the structure in question, and then - if
dynamic analysis methods might only need possible - to include some consideration
to consult later sections of this chapter. of the mechanical damping effects. It is
Subsequent chapters will assume famili- usually the case that the mass and stiff-
arity with the contents of this one. ness effects can be much more readily
represented than can those for the damping
We shall begin by discussing the and so many dynamic analyses are performed
first task of the structural dynamicist including only these two primary struc-
-that of formulating a suitable mathema- tural elements. However, although it is
tical model. This amounts to specifying true that the damping often has a rela-
the distribution of mass and stiffness tively minor effect on the basic modal
and, if appropriate, the mechanical damp- properties (which are determined ignoring
ing effects as well (although this is damping), it must also be noted that it is
usually rather more difficult to do). A of particular importance in determining
separate section deals with the now wide- vibration response levels, especially
spread use of finite element methods for around resonance. Furthermore, it is
this modelling task. Following this found that one of the major contributions
modelling stage, a set of equations of of the aerodynamic effects to blade vibra-
motion can be developed and then solved in tion characteristics is in respect of the
different forms to determine the various contribution to the damping levels. Thus,
characteristics required. The first of due consideration must be given in our
these is the undamped free vibration studies of the structural dynamics to the
solution, yielding the fundamental vibra- consequences and methods of including
tion properties of natural frequencies and damping in the models.
mode shapes for the basic structure (the
'modal properties'). Modified values for Although the analysis methods which
these data can be derived if damping follow are general, and apply to all types
13-2

of structure, it ia appropriate and help- Continuous Models

ful if we confine our interpretation and
application of them to blade-and bladed If we wish to describe the vibration
assembly-like structures. (By bladed behaviour of a beam-like structure such as
assembly, we refer to a bladed disk, or to that sketched in Figure l(a), then it is
a set of stator vanes attached to an annu- clear that we need to be able to define
lar casing, or to a group of blades inter- the motion of all points of the structure
connected by a shroud band, as shown in at all times and thus that a varibale
Figure 1.) Before presenting the formal function of the type j£(y,t) would form the
analysis procedures in current usage, it basis of an analysis. If we extend this
should be noted that there are two funda- concept to a more complex configuration,
mentally different approaches to the taak such as the bladed disk in Figure Kb),
of describing the dynamics of a structure. then the variable function would need to
These approaches are embodied in the two become x(r,8,t). In this type of formula-
types of model which will be referred to tion, it is necessary to find expressions
as 'Continuous' and 'Discrete1. Although for the mass and stiffness distributions
the development of finite element methods which are also continuous functions of
has resulted in an emphasis on the latter position. These expressions are then used
type of modelling, both have a role to to determine inertia forces and stiffness
play and will be included in this review. forces (or kinetic energy and strain
energy) which are combined to develop
governing equations of motion for the
system. As will be seen later, these are
in the form of partial differential equa-
tions for which closed form analytic solu-
tions are only available under special and
rather restricting conditions. If these
conditions - which generally demand much
simpler and more uniform geometries than
are used in practice - are not applicable,
then approximate numerical solutions must
be sought. Because of this limitation,
and the non-optimal format of the equa-
x(y,t) tions thus generated for efficient numeri-
cal solution, the alternative procedure
employing discrete models such as the
finite element model is standard practice
for most practical applications. Never-
a) Single Blade theless, the continuous type of model does
have advantages and is particularly useful
for qualitative and comparative studies.

b) Unshrouded Bladed Disk c) Shrouded Bladed Disk

(e)

d) Stator Row e) Blade Packet

Figure 1. Basic Structural Configurations for Bladed Systems.

 13-3

Discrete Models example, we shall later refer to a lumped

mass-spring model for a bladed assembly In
The basis of the discrete model is which each blade is represented by just
the separation of the inertia and stiff- one or two masses: clearly, this is too
ness (and damping) effects so that the coarse a model to describe the deflection
model consists of a collection of separate pattern across the whole blade but it Is
components, each of which is either a mass adequate to describe the relative dis-
or a spring (or a damper) - see Figure placement from one blade to the next In
2(a). This process necessarily reduces the assembly.
the continuum of the previous form to a
finite set of discrete elements and so the
variable function which is used to Damping
describe the motion of the system now con-
sists of a finite set of individual It is also appropriate to mention
parameters of the form Xj(t), j-l,N, where here the question of Including damping
N is the order of the discrete model and effects In the model. In the present ap-
Is Its number of degrees of freedom. plication, any damping effects which are
(Note that a continuous system thus has an provided by the aero - or fluid dynamic
infinite number of degrees of freedom.) effects will be Introduced separately and
Describing the various forces or energy it is necessary to include here only those
terms in this case leads to a set of damping effects which are of a mechanical
linear ordinary differential equations origin - material hysteresis, friction In
which can be conveniently formulated in joints, etc. Unfortunately, it is not
matrix notation, thereby rendering them possible to introduce these effects in an
amenable to numerical solution. accurate way since almost all such mechan-
isms are non-linear and thus not easily
The development of a discrete model accommodated in our equations of motion.
for a given structure can be undertaken In Two types of damping model are used In an
various ways. In the limit, we can envis- attempt to simulate the energy dissipating
age the representation of each component effects of the actual mechanisms which ob-
of the structure by a large number of sep- tain, but neither is a 'correct1 model.
arate mass and spring elements following The first of these is the traditional vis-
closely the geometry of th'e structure. cous dashpot model, consisting of a damper
This is, In effect, the finite element ap- element which generates a force propor-
proach and results in a very large number tional to the relative velocity across its
of degrees of freedom and thus equations ends. The second is known as the 'struc-
of motion - see Section "Finite Element tural' or 'hysteretic' damper model and is
Modelling". Alternatively, it Is possible similar to the viscous one except that the
to devise rather more schematic models of damping rate is not constant (as- is the
much lower order which are capable of case for visous) but varies Inversely with
describing adequately the global behaviour the frequency of vibration. This alter-
of the model - i.e., its major mode shapes native model was developed as a result of
and their natural frequencies - but Is not the Inability of viscous damping to repre-
capable describing in detail what is hap- sent correctly the observed phenomenon
pening in every part of the system. For that energy dissipation mechanisms in most
structures are vibration amplitude-
dependent but not frequency-dependent.

(a) h
'x,(t) h(t) xh(t)
'x a
!
3 'h '-h
x '(t) H

a) Schematic Model

Figure 2. Discrete Model of Beam-Like Structure.

13-4

However, the drawback to the structural excitation. It is the former which leads
damping model is Its presupposition that to the description of the resonance phe-
vibration is taking place at a prescribed nomenon which is the forced vibration man-
frequency, or set of frequencies: a situ- ifestation of a natural frequency or mode
ation not strictly true for transient of vibration. However, it is most impor-
vibration. Having said that, use of one tant to establish that not all modes of
of these two models is almost essential if vibration will necessarily exhibit a
the equations of motion are to remain resonance under forced vibration condi-
linear: more accurate descriptions of the tions: the nature of the excitation func-
damping effects (and, to some degree, the tion is just as important in determining
stiffness effects) would lead to a non- resonance as are the modal properties of
linear model, the added complexity of the system itself.
which is almost certainly not practicable
in the present case. In the case of transient response
calculations, Interest Is very often con-
fined to what happens immediately after
Vibration Properties such an excitation has been applied. If
the combined effect of all the damping
Once the model has been established elements is negative, then vibration will
and used to generate a set of equations of tend to grow uncontrollably once started
motion, these are then analyzed for two by some transient disturbance: otherwise,
types of solution. The first of these is It will tend to die away.
the free-vibration solution, which yields
the intrinsic modal properties possessed
by the structure. Most important amongst Summary
these are the natural frequencies - those
frequencies at which vibration will take Thus, to summarize, the main features
place In the absence of any continuing ex- of a structural dynamic analysis are:
citation. Associated with each such
natural frequency Is a corresponding mode - the construction of a suitable mathe-
shape which describes how the displacement matical model;
of the system varies from point to point
across Its geometry, see Figure 2{b). It - the derivation of a set of governing
Is an important feature of these modal equations of motion;
properties that free vibration can take
place in any one of the 'normal' modes - the free vibration solution to yield
(natural frequency and mode shape combina- the intrinsic modal properties of the
tion) completely independently of all the structure; and
other modes.
- the forced vibration solution to
describe the actual displacements (and
When the system is undamped, the stresses, etc.) under some specified
modal properties are quite straight- excitation condtlons.
forward, and easy to interpret, but when
damping Is added, they become rather more
complex. The natural 'frequency* Itself
becomes complex with both an oscillatory
component (as for the undamped case) and a STRUCTURAL DYNAMIC ANALYSIS METHODS
decay component, the latter being due en-
tirely to the damping. The mode shapes Derivation of Equations of Motion for
can also become complex in that they Discrete Systems
describe not only the relative amplitudes
of vibration for the different positions
on the system but also introduce phase Having developed a discrete model for
differences. This means that when vibrat- a given structure, the derivation of its
ing in one such complex mode, each part of equations of motion Is a routine task. It
the system reaches Its maximum displace- should be noted at the outset that there
ment at a different instant to Its is no unique set of such equations. First
neighbour. In a classical or undamped of all, the variables chosen to describe
system normal mode, all parts reach their the system's behaviour (the coordinates)
maximum excursion simultaneously. are themselves not unique, but even given
a specific set of these variables, then we
may develop an infinity of equally valid
sets of equations of motion. The one
Response Characteristics thing all these different equation sets
share is the solution.
While the modal properties are an
essential part of the structural dynamic A valid set of equations can be de-
analysis of any system, they do not pro- rived by one of two methods, based respec-
vide the whole picture and in order to tively on equilibrium and energy
complete the analysis It is generally principles. In both cases, the starting
necessary to undertake also the second point is to assume that the system is in a
type of solution; namely, that for forced- state of general motion, with displace-
vibration response. At this stage in the ments, velocities and accelerations in all
analysis, it is necessary to introduce its coordinates, x, x, x*. Following this
some additional information in the form of assumption, It Is possible to define all
the excitation or forcing function (and it the forces which are acting upon each com-
is just this fact that makes the response ponent of the model and by applying equi-
solution fundamentally different from the librium conditions at all junctions
free-vibration analysis). This generally between components, we can derive a set of
falls into one of two types - steady, or governing equations - the equations of
continuous, excitation and transient motion. In the case of free vibration.
 13-5

with no externally applied forces, the and the corresponding patterns, or mode
equations contain only the system para- shapes (f}r , by solving the homogeneous
meters and the (unknown) displacement equations:
variables of each component.
([K] - ur2 [M]) {?}r =• (0) (5)
Alternatively, expressions for the
instantaneous energy levels - kinetic, (NOTE that this does not give a unique
stored and dissipated - can be derived answer for the displacement amplitudes of
from the same starting point and by apply- the individual variables, {x}. Because
ing the principle of conservation of the equations are homogeneous, only
energy, a similar and equally valid set of RELATIVE amplitudes are obtained and as a
equations can be derived. result It is appropriate to assign them a
different parameter, {41} , which reflects
In all cases, it is convenient to this indeterminancy of scale.)
write the equations in matrix form and
thus to describe the mass and stiffness The mode shapes have Important pro-
properties of the system in terms of a perties of orthogonality which, concisely
mass matrix (M] and a stiffness matrix stated, are:
tK]. A corresponding viscous damping
matrix [C], or structural damping matrix 0:
[H] may be defined as well, if the damping {*)rT [Ml mr (5a)
effects are included. The general form of
the equations of motion is then: and
IM] {x'} + [K] {x} + [C] {*} IK] (5b>
(1)
{x} =
Once again, it is important to note These can be grouped into the simple ma-
that these system matrices apply only when trix expressions:
the coordinate set is {x} and are not a
unique description of the system's charac- W T [M] [mr]
teristics. The system matrices must ex- (6)
T
hibit certain properties and in many cases [*J [K] lkr)
they will be symmetric matrices. Even if
the equations as derived are not sym- where the individual elements mr and kr
metric, they can often be rearranged so as are referred to as the 'modal masses' and
to become symmetric. However, under con- 'modal stiffnesses'. As mentioned
ditions where gyroscopic inertia effects earlier, the coordinates used are not the
are present, or where certain hydrodynamic only variables which can be used to
or aerodynamic effects apply, then some of describe the system's behaviour. We could
the system matrices may be non-symmetric, rewrite the equations of motion using a
with a consequent Increase in the complex- different set of coordinates, such as {p}
ity of the ensuing analysis and which are defined as
properties. We shall concentrate here on
the standard case where symmetry of the (p) = [»rl {x} (7)
system matrices is obtained.
Substituting into (2) and premultiplylng
by [?]T, leads to:
Free Vibration Solution for the
Undamped System [M] [t] {p} + [*]T [K] (P) - (0)
(8)
The basic system properties can be which reduces to:
derived from the equations of motion for
the undamped system in the absence of any [ror] (p) + [kr] {p} = {0} (9a)
excitation. For this case, the equations
of motion reduce to: It Is clear from this that each indi-
vidual equation contained in the complete
[M] {x} + [Kl {x} = {0} (2) set takes on the simple form:

mr pr + kr pr = 0 (9b)
and the method of solution is to assume
that simple harmonic motion is possible of leading directly to the solution that
the form:
<&f = kr/mr
(3)
It may be seen that the values for mr
and kr will vary according to the scaling
Substitution into the Equation (2) used for the mode shapes, and so these
shows that this is indeed a valid solution parameters are not unique. However, their
to the equations, provided that the fre- ratio - the square of the natural frequen-
quency is one of a finite set of specific cy - is unique. It is found convenient to
values (ur) and, further, that the dis- scale the modes in a particular way known
placements of different parts of the sys- as mass-normalization. This requires
tem conform to a specific pattern. The simply the rescaling of all the elements
specific values of frequency - the natural in {<>}r by /mr to give:
frequencies ur - are determined by solving
the determinantal equation: (10)
det| [K) - [M] (4)
13-6

This leads to a more covenlent form of the the damping Is relatively localized AND
orthogonality statement in (6): when the system has close natural frequen-
cies. This last characteristic la of
U1T IMJ Ul = til particular relevance here because axisym-
(11) metric bladed assemblies possess most of
U1T IK] [*] = Nr2J their modes In pairs with Identical or
very close natural frequencies. Such
modes may thus be particularly prone to a
Free Vibration Solution tor Damped high degree of complexity.
Systems
When damping is added to the system,
a similar type of result is obtained to
that described above but each of the modal Forced Vibration Solution
parameters becomes complex. The natural
frequencies become complex so that the
solution postulated in (2) takes the Although we are likely to be con-
form: cerned with response predictions for any
type of excitation, we shall show that
{x)est s {x}e~at eiwt (12) undertaking an analysis for the frequency
response properties (response to individ-
indicating not only oscillation (at fre- ual point harmonic excitation forces) per-
quency u) but also an exponential decay mits the extension to any more general
(rate = a ). Thus, free vibration of a excitation conditions. Thus, we shall
damped system also consists of a number of concentrate on this particular form of
independent modes, each of which has this response prediction.
type of complex natural frequency.
Similarly, the mode shapes may become The basic relationships are simply
complex, indicating phase differences be- stated: If the excitation forces can be
tween one point and the next, in addition represented by the simple harmonic func-
to the amplitude differences. tion
The exact form of the modal proper-
ties varies with the type of damping. For
viscous damping, the analysis becomes (ISa)
rather lengthy but a set of natural fre-
quencies and mode shapes are obtained in
the form of complex conjugate pairs: then it may reasonably be assumed that the
response will be similarly sinusoidal and
have the form:
fsr . 0I • *r* 1
;i ..... :.... ' (13)
L° • 8
*i
rJ
**r*J U5b)

These response and excitation func-

If structural damping is used, the tions are related by the frequency re-
algebra is simpler although the mathemati- sponse function matrix [H(u)] as follows:
cal rigour is less, and a set of complex
natural frequencies and mode shapes are
obtained:
{x}eiut {f}elwt (16)
[Xr2] 5 [V] (complex) (14)
Substituting (IS) into the basic
where equation of motion (1), first for the un-
damped case, gives:
Xr2

=• ([K] - ( (17)
In practice. It Is found that for
systems with relatively light damping (the which, while correct, is not very conven-
case for most applications to blade sys- ient for numerical application. It re-
tems), the inclusion of damping has quires the inversion of a large matrix at
virtually no effect on the oscillatory each frequency of interest and does not
component of the natural frequency and is readily permit the calculation of just one
only effective at introducing the decay or two of the frequency response elements
rate. If the damping is 'proportional' - in the matrix. Following the simplifying
i.e., is distributed in a similar way to effect of transforming the equations of
the mass and/or stiffness - then the modes motion by the mode shape matrix (Equations
of the damped system are not complex, and (8) and (9)), we find that if we premul-
are in fact identical to those of the sys- tiply both sides of Equation (17) by [$) ,
tem without damping. However, it is par- and introduce [$]U)'1-[11 , then it may
ticularly relevant to note here the be replaced by
conditions under which the mode shapes
become noticeably complex, and thus differ
significantly from the undamped system
properties. This is found to happen when [H(U)J (18a)
 13-7

which is both easier to compute and con- Those excitations which do not re-
siderably easier to interpret. Speci- spond to this approach, such as tran-
fically, it is possible to extract just sients, may be more amenable to a time do-
one element HJK(U) from the frequency main analysis. This is typically stated
response function matrix, which provides as follows: the time history response
the (harmonic) response at point j per x(t) caused by a force f(t) can be ex-
unit (harmonic) excitation at point k: pressed by the integral equation:

x(t) f(t')h(t-t')dt' (22)

(19b)
where h(t) is the unit impulse response
function. This, in turn, may be shown to
If damping is added, the expression be related to the frequency response func-
changes only slightly in form, becoming tion via another Fourier transform:
complex to reflect the additional influ-
ence. For structural damping, the matrix
equation becomes h(t) / (23)

[HUJ] Thus, the frequency responses provide

all the necessary information to evaluate
the forced response of a system to any
and the individual element: type of excitation.

•»« •« • (19b) Continuous System Analysis

It was mentioned earlier that at-
tempts to describe the dynamic behaviour
For viscous damping, the expressions are: of systems with continuously-distributed
mass and stiffness resulted In partial
differential equations which lacked con-
(20a) venient analytic solutions in the general
case. We shall summarize one particular
case here to illustrate the nature of the
and equations produced and to show some of the
potential advantages of this type of ap-
proach: for a more detailed development
(20b) of the analysis, reference should be made
to a standard vibration text, such as
Thompson ?1981) or Bishop & Johnson (1960).
or
r»k 1* r»k* Consider the case of bending of a
iur-s iu-Sr« beam, such as that shown In Figure l(a).
(20c) If we study an element of the beam while
it is undergoing free vibration (Figure
Where * denotes complex conjugate. 3), we find that the equation of equili-
brium can be stated as:
Lastly, it remains to deomonstrate
the applicability of these frequency re-
sponses to more general types of excita- ay (El PA - 0 (24)
tion than harmonic ones (even though these
latter may constitute the majority of
cases for bladed assemblies).

If the excitation is periodic, but

not harmonic, then It can generally be
described by a Fourier series:

.fn «n (21a)
x(y,t)
as also can the resulting response:
00

x(t) = S xnei">nt (21b)

n*--

Each of the constituent components,

fn and xn are then related by the cor-
responding frequency response function:

xn (21c) Figure 3. Continuous Model of Beam-Like

Structure.
13-8

If we- now assume simple harmonic FINITE ELEMENT MODELLING

motion, as before, but this time with the (contributed by R. Henry,
specific form x(y,t) - $(y) exp(iut), the Institut National des Sciences Appliquees)
this equation becomes:
So far, we have not explained how the
basic modelling and derivation of the
equations of motion is approached and it
(25) is now appropriate to address this task,
using finite element methods.

Solution of this equation requires the

specification of some boundary conditions The finite element method (FEM) Is a
and of the variation of the section pro- powerful numerical procedure for solving
perties along the length of the beam. The the mathematical problems of engineering
latter requirement means that the solution and physics and is particularly well
is difficult to obtain in a closed ana- suited for complex structures such as the
lytic form if the section properties vary bladed assemblies with which we are con-
along the beam, as is the case in most cerned here. The mathematical foundations
practical blades. However, for the cases of- the finite element method are unques-
which can be solved analytically (constant tionably an important area but are beyond
section properties), we find that our the scope of this text, where the tech-
assumption of simple harmonic vibration nique has to be considered simply as a
was a valid solution, provided that a cer- tool for a specific practical engineering
tain condlton is satisfied. This condi- problem. In the following paragraphs, we
tion varies with the boundary conditions, present a brief review of the method as
but Is usually similar to the specific it would be applied to analyze a bladed
example shown below for the case of a assembly including, where appropriate, the
cantilever beam of length L (clamped at effects of rotation - not easily accommo-
one end, free at the other ): dated In the preceding analyses. In the
finite element method, considerations of
energy are generally used In order to de-
velop the equations of motion and so this
1 + cost XL)cosh(XL) = 0 ; section will concentrate on that process:
(26) once the equations are available, they may
be used to study the structural dynamics
El as outlined in the preceding sections.

Clearly, solving this equation leads

to an Infinity of roots, Xr , or natural
frequencies, uj. , and each of these may be General Method for Analysis of
back-substituted to determine the corre- Rotating Structures
sponding mode shape, or 4>r(y) , such as
The analysis of rotating structures
has to take into account the centrifugal
4>r(y)= {(sinXry-sinhXry) effects in addition to the inertia and
(26) stiffness terms which are applicable to
all structures and, as a result, the po-
* (s!nXrL-sinhXrL) <cosxry-coshxry» tential and kinetic energies are a little
more complicated to derive than for non-
for the cantilever example. rotating structures. Assuming that the
material is homogeneous, continuous,
linearly elastic and undamped, a general
A forced vibration analysis may also procedure can be simply presented, as
be made using this type of model and leads follows.
to closed form expressions for individual
frequency response functions. As an ex-
ample, we quote the tip response function Expressions for the Strains and Stresses.
for the cantilever example: The strain vector {e} related t o t h e dls-
placement {x} of a typical point Is de-
rived according to the type of structure
sin(XL)cosh(XL)-cos(XL)sinh(XL) (one-, two-, or three-dimensional). In
HLL(u) I) matrix form, it reduces to:
ElX»(1+COS(XL)CDSh(XL)1
(27a)
UL) (28)
This can also be presented in a series
form:
where {eLl Is the linear classical strain
vector and {SNL^ a non-linear geometric
(4/ApL) strain vector representing the geometric
(27b) coupling due to rotation. Assuming the
ur"*- »'
validity of Hooke's law, the stress vector
is:
noting the infinite limit on the series.
Similar analyses may be made for the tor-
sional and axial vibration of beams and [D] {e} (29)
for two-dimensional bending of plates, re-
sulting each time in partial differential
equations of motion of various forms. The Strain Energy; Analytical Form.
general solution takes a different form in Integrating over the whole structure gives
each case, but there is a generic similar- an expression for the total strain energy
ity between all of them. (U):
 13-9

U •» 1/2 / U)T <o}dT Discretisation. The magnitudes of the

T continuous variables (such as displacement
(30a) {x}) at the nodes are denoted by a vector
= 1/2 / {e)T [D] U)dT {xe}, whose components are called the
nodal values or nodal degrees of freedom,
and these are to be determined. The con-
tinuous displacement {x} Is then approxi-
and substituting (28) in (30) leads to: mated over each element by a polynomial
expression (N] that Is defined using the
nodal values. A different polynomial is
UK UL,eL) + defined for each kind of element, but it
(30b) must be selected in such a way that conti-
U0 (eNL,eNL) nuity is maintained along the element
boundaries. The general form for the dis-
placements of an element e, is given by:
where UK, UG, and U0 are, respectively,
the classical linear elastic strain
energy, the strain energy due to the geo-
metric coupling (tension and bending) and {x} » [N] {xe} (33a)
a term of higher order that will be
neglected.
and in the same way for the velocity:
Absolute Velocity. Writing the absolute
velocity of a typical point M In a moving
coordinate system (O) linked to the axis {x} = [N] {xe> (33b)
of rotation of the structure rotating at
an angular velocity 0 , which gives:
and for the strains (28):
» —»
fl x OM (31)
dt {e} - [B] {xe} (33c)

and this can be expressed in term of the It must be noted that the matrix [B]
displacement {x} and velocity {x}. includes both the linear and the non-
linear parts of the strain definition
(Equation 28).
Kinetic Energy; Analytical Form.
Integrating over the whole component gives
the kinetic energy of the structure: Finite Element Expression for the
Potential Energy. Using equation <30a)
for a t y p i c a l element e and substituting
T - 1/2 /p V M -V M (32a) (33c) leads to:

2 U
(assuming a uniform mass density, p). (34)
{x
e>
Substituting (31) into (32a) leads
to: where [KL~] Is the element linear stiff-
ness matrix and [KQ~] the element geom-
etric stiffness matrix which Is linked to
T = TM(x2) + Tc(x,x) the initial stress vector {oo} due to the
(32b) centrifugal forces.
+ TS(X2) + Tp(x) +

where TM , Tc , Tg , and TF are respec-

nodes
tively the contributions to the kinetic
energy from the inertia or mass effect,
the coriolis effect, the supplementary
centrifugal effect and the centrifugal
forces effect respectively. To is a con-
stant term which cancels in later Beam Rate
calculations. Element Element

In most practical cases, the complex

geometry and non-classical boundary con-
ditions of real structures require U and T
to be evaluated using an approximate
numerical method, such as the finite ele-
ment method. Here, the structure is
divided into simple elements of finite Thick
size (the finite elements) that are con-
nected at certain points, called nodal Element
points or 'nodes', which are generally
(though not always) situated on the bound-
aries of the elements (Figure 4).
Figure 4. Basic Finite Elements.
13-10

Finite Element Expression for the Kinetic static centrifugal force distribution. F
Energy. Substituting (33a) and (33b) into can be any other static or dynamic excita-
the gene
general equation (32a) for an element tion force, such as the aerodynamic forces
e leads toi that will be detailed In other chapters.
However, we shall concentrate here on the
(35) basic case where rotating structures such
as blades or bladed disk assemblies are to
2Te - {ie}T [Me> <ie} be considered purely from a structural
dynamics standpoint and where such addi-
+ (xe>T [K8eJ {xe} {xe>T {Pce(Q2)} tional excitation forces are not
considered.

where (Me) Is the mass matrix of the ele-

ment e, [Gel the gyroscopic coriolis Static and Dynamic Solutions for Rotating
matrix, [K8eT the supplementary stiffness Structures
due2 to rotation which Is of the form
[n MGe] (where [MGe] Is called centrifugal
mass). (Fce(n2)} Is the vector of element The general form of Equation (38a)
nodal forces equivalent to the centrifugal applies to all rotating structures, what-
volumlc forces. ever is their physical shape.. The finite
element model Is able to Include and to
combine unldlmenslonal, two-or three-
Assembly. If the structure Is composed of dimensional components.
n elements, and remembering that energies
are scalar quantities, expressions for the
potential and kinetic energies of the Static Solution. It is of practical in-
whole structure can be derived: terest for the designer to know the stress
distribution due to centrifugal forces.
For this case, the equation of motion re-
n duces to:
» Z Ua ; T - £ Te (36) (38b)
e-1
[KL + KC(OO) - 02MG]{x}B - {Fc(fl2)>
The continuity of the nodal displace-
ments {*e} at the connecting nodes is now where OQ Is linked to {x}8 via Hooke's law
prescribed, and so U and T are expressed (29) and the strain-displacement relation
in terms of independent degrees of freedon (33c). Noting that (38b) is non-linear,
{x}, which are the unknowns of the since o0 Is unknown a priori, the solution
problem. needs an Iterative process, such as the
Newton-Raphson method. In practice, just
a few iterations (two or three) are suf-
ficient to obtain good estimates for the
Governing Equations of Motion. Applying centrifugal stresses In the finite ele-
Lagrange' equation: ments (at the center of gravity, or at
nodes) and the corresponding principal
_ stresses (o^, «2' 03)* A yield criterion
(37) such as von Mises can be used to determine
dt whether the elastic limit (OE) has been
reached:
to the expressions in (36) gives a set
of equations of motion for the whole —=
structure, including the effects of (39)
rotation. The general form of these equa- /2
tions Is:

[M]

- (Fc(a2) + F(t,x,x)} <38a) Dynamic Solution. In order to construct a

diagram whichiKows how the natural fre-
quencies vary with speed of rotation (a
Campbell diagram -the usual format for
Equation (38a) has the same general presenting the basic dynamic properties),
form as that quoted for discrete systems these frequencies and their associated
(see Equation 1) although here containing mode shapes have to be extracted assuming
additional terms due to the rotation that the displacement vector {x} is the
effects. It is important to note that the superposition of the static displacements
matrix [G] is not a viscous damping (x)3 and the dynamic displacement {x}d.
matrix, but the so-called coriolis matrix In other words, the system is assumed to
which is antisymmetric and which repre- vibrate about the equilibrium position
sents the gyroscopic effects. As to the determined by the static solution, (38b).
stiffness, some special terms are added In this case, the governing equations for
allowing for the centrifugal stiffening, the dynamic analysis are derived from
[KG], and the centrifugal softening due to (38a) and (38b) and reduce to:
the 2centrifugal mass effect, [MG]. (38c)
{Fc(n )} Is the nodal equivalent force
vector which Is the discrete form of the [M]{x} d +[G]{*} < j+[K L « G (o 0 )-fl 2 M G ]{x} <1 -{o)
 13-11

and the way to obtain the solution has their different types of vibration mode
been outlined in the section "Free Vibra- and response In Chapter 15, but it is
tion Solution for Damped Systems." appropriate here to review some of the
methods which are used for analyzing such
If the Coriolis effects can be ne- a system as a bladed assembly. The need
glected -and this is usually so for radial for special methods of analysis arises
or quasi radial blades -then [G] is disre- from the inefficiency of the direct solu-
garded and (38c) reduces to: tion of constructing a single model for
the whole structure without taking advan-
(38d) tage o'f Its high degree of symmetry.
[M] ) - QZMG] {x}d Also, it is sometimes found that different
types of model are appropriate for the
different parts of the assembly (blades,
which is the governing equation of motion disk, shroud) and if these components can
for a classical undamped system. The be treated independently, at least in a
motion is assumed to be simple harmonic first stage, a more optimal analysis may
and the method of solution has been de- be possible.
tailed in section "Free Vibration Solution
for the Undamped System." At this stage,
it should be noted that the finite element Thus, we shall outline some methods
approach may involve a very large number of analysis which respond to these re-
of degrees of freedom and, consequently, quirements: one which requires the dif-
the extraction of the eigenvalues and ferent components to be analyzed sepa-
eigenvectors can be a formidable task. In rately as a first stage - substructurlng -
this case, instead of solving the deter- and another which seeks to exploit the
rainantal equation (4), it is often more repetitive symmetry of a complete bladed
convenient to use special numerical disk - cyclic symmetry.
methods such as the Sturm Sequences
(Gupta, 1973) or the Simultaneous
Iterative Method (Jennings & Clint, 1970). Substructure Methods
The basic concept of substructuring
Conclusion is that a more efficient analysis of a
complex structural assembly can be ob-
The general theory developed in this tained if it is first considered in its
section can be adapted for the dynamic constituent component parts, or substruc-
analysis of any rotating structure. Some tures. By analyzing each of these as a
special treatment of the equations (38) separate system and determining its Indi-
can be outlined to particular systems, vidual vibration properties, we can (a)
i.e., axisymmetric or repetitive struc- use the most appropriate type of model for
tures, but the present generality is not each subsystem and (b) reduce the size of
affected. each model (if required) by retaining only
the primary or major vibration modes of
.the component. In this way, we can ensure
that the results from different types of
DYNAMIC ANALYSIS METHODS FOR STRUCTURAL analysis are presented in a standard for-
ASSEMBLIES mat and are thus suitable for combining in
an analytical equivalent of the physical
Introduction connections by which the actual assembly
Is formed.
Although there are a number of situa-
tions in which it is useful to study the
behaviour of a single blade at a time, in There are essentially two different
most practical applications the blades approaches to this procedure. The first
interact with each other and must be is that in which the modes of each compo-
treated as a complete assembly. Perhaps nent are first obtained by separate sub-
the most common of the bladed assembly system analyses and are then combined to
configurations is the complete bladed predict the modes of the complete
disk, Figure Ib, where the flexibility assembly. The second approach uses the
and inertia of the disk serve to couple frequency response properties and by de-
the vibration of the individual blades. A riving these characteristics for each com-
variant on this arrangement is where the ponent individually, and then combining
blades are additionally Interconnected them, it is possible to determine directly
through a shroud, which may be situated at the corresponding frequency responses for
the blade tips or at a midspan station, the complete assembly. Of course, since
Figure Ic. In some cases, the disk may we have seen from earlier sections that
be so stiff as to provide virtually no the frequency responses may be derived
coupling and it is then the shroud which from the modal properties, these two
takes on that role. In some special approaches are effectively different ways
cases, we may be concerned with a group of of organizing the same data and do not in-
blades spanning an arc of less than the volve any fundamentally different assump
complete 360°. This refers to a 'packet' tions. Where they do differ is in the
of blades in which the interconnection Is facility with which one or other type of
applied between just a few blades at a result is obtained: if the modes of the
time (Figure le). assembly are the essential end result,
then a modal coupling approach is more
All these assemblies of blades ex- appropriate but if it is the responses
hibit vibration characteristics which are which are ultimately of interest, then the
very much more complicated than are those response coupling approach is perhaps more
of the individual blade upon which the effective. The two approaches differ in
assembly is based. We shall be describing their sensitivity to errors or approxima-
tions introduced into the component
13-12

models, a point to be borne in mind when

condensing or reducing a large component
model in pursuit of computational
economy. (42)

Modal Coupling Methods

The first method of subsystem coup- - (0)
ling makes use of the modal properties of
each component and arranges the analysis
in such a way that the coordinates used to
set up the equations for the coupled
structure are modal coordinates and not
spatial coordinates, as is the case in which can then be solved for the modal
other methods. Further, this method properties of the complete structure.
allows for the Introduction of springs
and/or dashpots in between the subsystems The second modal method to be con-
at points where they are attached to each sidered is that used for most analytical
other. substructure analyses, and is based on
'constrained' and 'constraint* modes of
The general arrangement is shown in the subsystems. The analysis is outlined
Figure 5. The equations of motion for below, using the same notation as above.
subsystem A are: For subsystem A, with coordinates j at
junction points and coordinates a else-
where, we can write:
[MA] {XA} + [KA] (XA> - {fA>

which can be transformed Into modal coor-

dinates:

(43)

Now define a new coordinate vector by

Similarly for B.

V
[A] (44)

Combining these equations, we have

]
ft.:.?! !**{
f
IP '• IJ \PB $
(40)
T
o |*A
*A o] /£A1
1° *BTJ I«B]
but through the connections, we can relate
the forces to the responses, thus:

(41)
45'
where [Kcl is the stiffness matrix for the
connection and {XA} and {XB} are the geo-
metric coordinates of the attachment.
These, In turn, are related to the modal
coordinates by

(XA) = UAJ (PA) etc.(see Equation 7)

So, If there are no external forces

applied to the coupled structure, then we
have an equation of motion for the
assembly

Figure 5. Coupled Structure Analysis

Concepts.
 13-13
where Inverting each of these equations leads
to:
[•'V.'VT
[A)
" to-rr-J and then the application of compatibility
(48)

{Xc} • {XB} - {Xc} and equilibrium {fA> +

'constrained' modes of system (fB)° {fc)conditions as the common coor-
grounded at junction points. dinates generate a new equation:
'constraint' modes (deflections
of nonjunction coordinates for a {f c } - [H A U»-i{x A > + [ H B U ) ] - l { x B }
unit displacement in one
junction coordinate) = [H c (a>)J- 1 {x c )
(49)
Using these transformations, the
equation of motion for component A can be or
rewritten
[HCU)]-1

- (0)
which thereby permits direct derivation of
the frequency response function properties
of the assembled structure, described by
[Hc]. While the more representative sys-
tems consist of more numerous and complex
(45) configurations than this, the principle of
J the method is exactly that illustrated
[MA ] - [• here.

[M A J ] - [MAJ] Cyclic Symmetry Methods -

Axisymmetric Modelling
A similar expression can be written As stated earlier, It Is often bene-
for a second component B and the two sets ficial to seek a means of exploiting the
of equations combined, again using the repetitive symmetry of a complex bladed
commonality (compatibility) of .the junc- assembly. If the disk and blades for a
tion coordinates, i.e. (xA3 = xBJ). Thus, given assembly have similar stiffness,
we obtain: their motions are strongly coupled and
simple calculation of the individual
blade's properties would lead to false
results. Bladed disk assemblies can ef-
(46) fectively be classified into two groups
according to their structural dynamic
properties. In the first group are those
assemblies having an axisymmetric be-
haviour characterized by mode shapes
formed by simple diametral and circular
nodal lines (lines of zero displacement).
- CO' The structures in this group are usually
- {0} characterized by a large number of blades
mounted on a flexible disk, I.e., turbine
wheels of jet engines and eventually some
thick disks which have the blades con-
nected in a radial plane.
{0} The second group includes the rota-
tionally periodic structures such as
bladed disks or centrifugal impellers hav-
ing few blades with or without casing or
Frequency Response Coupling Methods shroud. This type of structure does not
exhibit purely axisymmetric mode shapes,
The basis of this method may be quite but includes some pseudo-axisymmetric and
simply stated again using the example in some repetitive modes showing "scalloped"
Figure 5: its implementation to practical patterns (Henry, 1980). In this case,
structures requires appropriate organi- classical axisymmetric theory can no
zation of the data from the various compo- longer be applied and one has to introduce
nent models. Turning to the example in the cyclic symmetry methods using expli-
Figure 5, we can see that the (harmonic) citly or Implicitly the wave propagation
response at the various coordinates of concept (Henry and Ferraris, 1984). This
interest on subsystem A can be related to group of structures is excluded from dis-
the (harmonic) forces which are assumed to cussion here.
act at the same points by:
(47a) In typical gas turbine assemblies, it
= [H A U)j {f A } ei«t is found experimentally that turbine
stages tend to have axisymmetric mode
and similarly for subsystem B: shapes; that is to say, the disk and blade
displacement follow a simple cosine fluc-
[HB(<o)J {f B } (47b) tuation both in time and in angular
position around the circumference. In
13-14

other words, the structure vibrates with a For the symmetric case, i.e., when
dominant n nodal diameter mode shape hav- the number of nodal diameters equals 0,
ing negligible contributions from other the coefficient N/2 Is replaced by N.
diametral orders. Thus, for example, the Thus, noting that Uo and To are Indepen-
displacement of the jth blade can be dent of n, the stiffness and mass matrices
written as: of the vibrating blading are those of a
single blade weighted by the factor N or
N/2 according to the value of n. The axi-
symmetric part of the bladed assembly is
(xn}j cos 2nn(j-l)
N
(50) modelled by axisymmetric finite elements
(thin and thick), U A < n) and TA(n) are de-
rived and the complete structure finite
where n and N are the numbers of nodal element energies U and T are obtained by
diameters and blades respectively. As N simple summation. Applying Lagrange's
Is generally large, the blades can be con- equation as described previously leads to
sidered to be a continuously distributed an individual governing equation for the
component attached to the rim of the disk. system (Equation 38a) for each value of n.
Thus, the blade array strain and kinetic Then, the static and dynamic solutions are
energies UB, TB, may be obtained from the performed as detailed in Equations 38b and
strain and kinetic energies of a single c. A practical application on a jet
blade (Uo, To), as follows: engine turbine stage Is reported In
Chapter 15.
N (51)
2
i i l>

figure It. T w i s t - - : n*d~ l - ' i n U . * -

 I-J -'•'

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I "' l"-'i.l i i,.; , ,t,r- , i ,,,.) ( . . . ( i j i t- i . J iu.-« | 1
"' I'' 'I'"' '-- 1 ' •' ' , ll • i I -MM r) | .- I .t 1 V* i V

••t ' . - . - . . l ' •••-•• • i ; ! ,,. • . . i- - • . • : ; , , . , [ : .. S I HIM 1 , • II'!- • 1-.' ] ,

: : . ' ' i . i i • •; t ., : I,,- ,.,,., T1 i.,,,1 ,.„!,, ,|
' I ' l I !-- I: , ..-.

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• •• ,,,,, Cl, m P ,,l H .,,,-, ,, t ) =• ' 1 1?\ --(••- ir - .-
"

i • 1 68 ihsa 16. 1 i no 0.2 . -\ C.. i • i , '.i

. 1133 3852 2 i .C j ."• '"i 7.0 - 10: i . •;
-, 4bi : 5248 1 .' . ; •i • • - • • 2,0 4809 \ .:

'• ' I. • , • 1.1 I - .•' V : .: 1 l.i I.i It--.

TABIJ: ~> TUP-BIKE BULDE FREQUENCIES AT ROCW TEMPERATURE

springs equivalent
to the root stiffness

'' . • ! ' • • - 1 "'. i'i- ido I 1 inn • •' 'v"

 14-1

STRUCTURAL DYNAMIC CHARACTERISTICS OF INDIVIDUAL BLADES

by

D. J. EWINS
Imperial College of Science & Technology
London, United Kingdom

R. HENRY
Institut National des Sciences Appliquees
Lyon, France

INTRODUCTION Outline. Before proceeding to a dis-

cussionofmethods for predicting the
Objective. The objective of this structural dynamic properties of an indi-
chapter is to provide the necessary intro- vidual turbomachine blade, it is appropri-
duction and grounding for a study of the ate to present a summary of the basic
vibration characteristics of individual nature of these characteristics and this
turbomachine blades. We are concerned can conveniently be done using simple
here with the structural dynamic proper- models of blades. Although the detailed
ties only and we shall concentrate on the geometry of a given blade will have a
two most important of these - the natural major influence in determining the exact
frequencies and the corresponding mode values for its natural frequencies and
shapes. Although some consideration will mode shapes, these properties will be
also be given in later chapters to the found to fall into one of a small number
damping properties, these cannot be treat- of categories, or families, which can be
ed in the same analytical way as is pos- related to those of rather simpler struc-
sible for predicting the natural frequen- tures than the blades themselves. Indeed,
cies and mode shapes. Knowledge of these we can learn much about the vibration pro-
two properties is a primary requirement perties of blades by studying first those
for a valid analysis of the aeroelastic of simple beams and plates. These will be
behaviour of blades and most of the struc- presented in the next section.
tural dynamicist's efforts are directed
towards their reliable identification and
prediction. Following this basic groundwork, the
influence of a number of relevant design
features is examined with a view to estab-
Having first outlined the basic char- lishing which parameters can have a marked
acteristics of blade vibration properties, effect on the actual values of natural
we shall then identify those design fea- frequency or mode shape. These factors
tures which have an influence on their includet
values. This is important since an appre-
ciation of these effects aids our under- - complex cross section properties
standing of how the various vibration - flexibility of root and/or shroud
problems and phenomena can arise in fixtures
service (often against our initial expec- - speed of rotation (of the blade in the
tations) and, more importantly, how they machine)
may be treated. - static loads
- temperature variations.
For these earlier sections of the
chapter we shall use relatively simple
models of blades but will then progress to Later sections deal with prediction
outline the methods which are available methods used for design calculations.
for the calculation of the structural dy- These are not intended to provide a com-
namic properties of real blades, as re- prehensive review of the very many differ-
quired for design purposes. Throughout ent methods or programs which are avail-
this chapter, we shall be dealing with in- able for this purpose but rather to
dividual blades only and it must be noted illustrate ways of dealing with a number
at the outset that in most applications of the difficulties often encountered in
the blades are grouped together in stages the process of predicting blade vibration
or assemblies and that this feature will properties by referring to a number of
also have an influence on their vibration differing case studies. Perhaps the most
properties. important point to be made is the need for
an appreciation of what to expect (from a
Indeed, in many cases, the natural calculation process) and of how to set
frequencies and mode shapes of a blade in about confirming or validating its
its operational environment (e.g., as one results. It is with this aspect in mind
of a set on a disk) may differ consider- that the early sections of the chapter are
ably from those calculated for the same presented.
blade as an isolated component. This
topic is dealt with in detail in the next
chapter (Structural Dynamic Characteris- The study and analysis of the blade
tics of Bladed Assemblies) but it can be vibration characteristics has led to the
noted here that it is usually convenient publication of a great many technical
in that more complex situation to relate papers, only a handful of which will be
the complete assembly's vibration proper- referenced here. However, there are a
ties to those of the single blade. Hence, number of review articles which provide
there is considerable advantage in under- useful surveys of different aspects of the
taking an individual blade analysis first, subject together with comprehensive lists
and that is the scope of the present of references, such as those by Rao (1973
chapter. and 1980) and Leissa (1981).
14-2

VIBRATION PROPERTIES OF UNIFORM BEAMS AND Plate-like characteristics are found

PLATES in those blades which have a lower aspect
ratio {typically, L/c<5> but are still
Basic Structural Models for Blades. 'thin' (d/c«l). The main difference
Althoughusually v e r y c o m p l e x I n t h e i r between these blades and the beam-like
geometric form, most turbomachine blades ones is that the cross section itself dis-
can generally be regarded as beam-like or torts during vibration with the result
plate-like in respect of their dynamic be- that the deflection(s) of any point on the
haviour. If we review the wide range of blade are now dependent on both its longi-
different blade configurations used in tudinal, and also its chordwise positions,
practice, we find that the vibration modes coordinates u and v.
of many of these can be well represented
by a one-dimensional or beam-like model The vibration characteristics of
(cross section remains undeformed during blades of both types can be studied using
vibration and deflection(s) are a function simple-profile beams and plates for which
of longitudinal position only - or radial, there is considerable data already in
in terms of the machine coordinates), existence (see, for example, Blevins 1979)
while others with different aspect ratios and for which additional calculations are
require the extra dimension provided by a easy to perform.
two-dimensional or plate-like model (for
which the cross section deforms and the Structural Dynamics of Simple Beams.
deflection is a function of both radial The methods of dynamic analysis for struc-
and chordwise positions); see Figure 1. In tural elements such as uniform beams have
both these cases, the vibration is effec- already been summarized in the previous
tively normal to the blade longitudinal chapter and so we need only present some
axis. There exist also a number of appropriate numerical results here.
special cases in which there is a signifi-
cant vibration displacement in the longi- Two sets of results are presented.
tudinal (or radial) direction as well as The first set -presented in Tables 1 and 2
the other two normal directions and the -give general non-dimensional natural fre-
dynamics of such blades are generally not quency and mode shape parameters for the
well represented by beam or plate models. bending and torsion modes of vibration of
These are usually very thick and/or very a uniform beam and are obtained from di-
short blades and must be described by a rect closed-form solution to the equations
full three-dimensional or solid model. of motion presented in the previous
However, such cases represent only a small chapter. Values for the vibration proper-
fraction of practical blades and the pre- ties of any specific beam can be deduced
dominance of the two former categories from these tabulated data by using the
provides the basis for classifying most appropriate scaling constant, as specified
blades as 'beam-like' or 'plate-like.1 in the table. The second set of results
-Tables 3 and 4, and Figure 2 -relate to a
specific beam model and have been computed
Beam-like blades are those which are using a simple finite element analysis,
essentially long and slender and so have a also summarized in the previous chapter.
high length-to-chord aspect ratio (L/c»l) Results for this particular case may also,
and a low thickness-to-chord ratio of course, be obtained from the non-
(d/c«l) - see Figure 1. During vibration, dimensional data in the first table and
the time-varying deflection(s) of any the small differences which can be ob-
point on the blade (x,y,z,9X,By,8Z) depend served in results derived from the two
only upon its longitudinal location (i.e., sources reflect the errors associated with
they are functions only of coordinate u). finite element modelling assumptions.
This is a consequence of an essential fea- However, this particular example is intro-
ture of beam-like vibration, namely that duced here in order to provide a reference
the cross section remains undeformed dur- case against which to compare the proper-
ing vibration. ties of variations on the basic model
later in the chapter.

DEFLECTION
z
edge

,y,z (w)

(a) GEOMETRY
(b)

Figure 1. Essential Features of Blade Vibration Analysis

 14-3

END CONDITIONS (ROOT-TIP)

MODS C-F P-F F-F C-P C-C
1st Bend 3.516 15.42 22.37 15.42 22.37
2nd Bend 22.03 49.96 61.67 49.96 61.67
3rd Bend 61.70 104.25 120.90 104.25 120.90
1st Tors 1.57 1.57 3.14 3.14 3.14
2nd Tors 4.71 4.71 6.28 6.28 6.26

VALUES OF X2

To obtain natural frequencies (Hz) of a specific rectangular-section

beam with dimensions: Length L, width w, thickness (in direction of
bending) t, and material properties: elastic modulus E, shear
modulus G, density p, polar moment of area Io, use the following
formulae:

bend * <E t2/12p) Hz

tors Hz

Table 1. Dimensionless Natural Frequency Parameters for Uniform Section Beams

In the tables and figures below, longitudinal mode (A) natural frequency,
natural frequencies and corresponding mode although this is usually so much higher
shapes are given for uniform rectangular than those of the bending or torsion modes
section beams with five different sets of that it is of relatively little interest
end conditions (the first of these at the in practice.
root - the second at the tip - see Figure
2):
(a) Clamped-Free (C-F) In fact, for each case considered,
(b) Pinned-Free (P-F) two results can be quoted for the natural
(c) Free-Free (F-F) frequencies of the bending modes, as shown
(d) Clamped-Pinned (C-P) in Table 3(b). These are found using (i)
(e) Clamped-Clamped (C-C) Bernoulli-Euler theory (B-E) and (ii)
Timoshenko theory (T) which respectively
exclude and include the second-order ef-
These represent different conditions fects of shear deflection (as well as
under which the blade might be analyzed or bending deflections) and rotatory inertia
tested: (a) considered to be rigidly (as well as translational inertia). From
clamped or grounded at the root; (b) these results it is seen how the second-
attached at the root but only restrained order terms become increasingly important
laterally - not in rotation; (c) suspended as the thickness of the beam relative to
as a free object; (d) and (e) with addi- its length becomes greater and the defor-
tional constraint at the tip, such as mation shape becomes more complex. In all
might be provided by interconnecting tip cases, the result of including the second-
shrouds. order terms is a lowering of the natural
frequencies.

The next set of results refers to a

specific example consisting of a beam of
length 250 mm, width (chord) 30 mm and
thickness 10 mm made from a material with
elastic modulus of 207 GN/m2 and density
y
7850 kg/m3. This beam has second moments TIP 'X
of area Ij - 0.250E-8 and I2 = 0.225E-7n>4
and a polar moment of area J - 0.7902E-8 (free)
ra4.
The natural frequencies for the first
few modes computed using a finite element
model of this beam are presented in Table
3(a) and the corresponding mode shapes
plotted in Figure 2. For each case, we
find the modes of vibration to be grouped
into three families: flapwise bending
(F), edgewise bending (E), and torsion
about the longitudinal axis (T) with the ROOT (clamped)
mode shape corresponding to each natural
frequency consisting of deflections in
only one of these three directions. Also
shown for reference is the first Figure 2. Mode Shapes of Uniform Beams
14-4

(a)
(b) CF

IB

2B
\ 2B

\
36
v/
1T
1T

Figure 2. Mode Shapes of Uniform Beams Figure 2. Mode Shapes of Uniform Beams
(a) free-free (b) clamped-free

PF (d) CP

IB 1B

2B 2B

3B \ 3B \
V/
Figure 2. Mode Shapes of Uniform Beams Figure 2. Mode Shapes of Uniform Beams
{c) pinned-free (d) clamped-pinned
 14-5

From these results it is seen that

the modes become rather more complex in
shape than for the beam model and are more
difficult to classify. Accordingly, it is
generally found that the most convenient
(B)
cc way of describing mode shapes of blades
such as these is by specifying the nodal
lines -loci of points which have zero de-
flection during vibration. In any event,
it is generally possible to follow the
1B evolution of each mode of the reference
case (30 ram wide) as the blade or plate
width is increased, thereby permitting the
tabulation shown below.

FACTORS WHICH INFLUENCE THE VIBRATION

2B PROPERTIES OF BLADES

Introduction. Real blades are gen-

era lly""~ratn"err~iore complex than the simple
beam and plate models which have been used
so far to introduce the basic types of vi-
bration characteristic. In this next
3B section, we shall seek to build on the
preceding groundwork by examining the in-

V fluence of a number of typical blade

design features which can have an influ-
ence on the structural dynamic properties
of interest here - primarily, the natural
frequencies and the corresponding mode
shapes. Although the vibration character-
istics of actual blades are similar to
1T those already discussed, the precise
natural frequency and mode shape data are
of vital importance to the analyst seeking
to predict a blade's vibration behaviour
under operating conditions. Hence, all
such influencing factors must be taken in-
to consideration at the design prediction
Figure 2. Node Shapes of Uniform Beams stage. while the finite-element based
(e) clamped-clamped prediction methods widely used today do
permit the inclusion of these various
effects, as will be illustrated in the
Structural Dynamics Properties of next section, it is appropriate first to
Uniform Plates. As the aspect ratio of extend the beam and plate models in order
the blade changes so that the chord width to explore some of the phenomena of par-
(c) approaches the length (L), the assump- ticular interest, as has been done by many
tion of beam-like behaviour becomes less workers including: Jacobsen (1938),
applicable and it becomes necessary to ex- Carnegie (1966), Montoya (1966), Rao
tend the model to that of a plate in order (1980), Irretier and Marenholtz (1981) and
to include the section deformation which Afolabi (1986).
will be present in such cases. Many stud-
ies have been reported on this approach to
blade vibration analysis, ranging from In this section we shall review a
early work by Dokainish and Rawtani (1973) number of the particular features of real
and MacBain (1975) to more recent work by blades, including:
Ramamurti and Kielb(1984). A survey of the
use of plate models for blade vibration
analysis is presented by Leissa (1980). complex geometry (including the non-
symmetric and non-uniform cross
In order to illustrate the particular sections which are found on tapered
characteristics which apply to plates, a and twisted blades and those designs
series of case studies are presented based for which 'simple* beam- or plate-
on a blade similar to that used like models are not adequate);
previously. This blade is 250 mm in
length, has a uniform rectangular cross
section of 10 mm thickness and width vary- attachment flexibility (the compli-
ing from 30 to 250 mm (aspect ratio L/C cations which arise at junctions
varying from 8 to 1) -see Figure 3. The between the blade and the disk and
material properties 9 are as used previously shroudband, or connecting blades);
(modulus = 207 x 10 N/m2, density = 7850
kg/m*) with a Poisson's ratio of 0.3, and
the blade is clamped at its root. Once special effects of operating condi-
again, the vibration characteristics for tions (including the quasi-static
this case are presented in tabular and load induced by gas pressure, cen-
graphical form with the fundamental trifugal and Coriolis effects caused
natural frequencies given in Table 4 and by rotation and by thermal effects
the mode shapes displayed in Figure 4. at extreme temperatures).
14-6

dimensions in mm

Figure 3. Uniform Plate Models

1st BENDING MODE

ROOT-TIP w-w C-F P-F C-P c-c

Freq 4.730041 1.875104 3.926602 3.926602 4.730041

X/L DISP STRESS DISP STRESS DISP STRESS DISP STRESS DISP STRESS

0.0 100 0 0 100 100 0 0 100 0 100

0.1 54 12 2 86 61 9 9 61 12 54
0.2 10 39 6 73 23 30 30 23 39 10
0.3 -27 69 14 59 -12 56 56 -12 69 -27
0.4 -52 92 23 46 -40 BO 80 -40 92 -52
0.5 -61 100 34 34 -58 96 96 -58 100 -61
0.6 -52 92 46 23 -66 100 100 -66 92 -52
0.7 -27 69 59 14 -62 91 91 -62 69 -27
0.8 10 39 73 6 -48 69 69 -48 39 10
0.9 54 12 86 2 -26 37 37 -26 12 54
1.0 100 0 100 0 0 0 0 0 0 100

2B BZKDXKS MOOB

ROOT-TIP F-F C-F P-F C-P C-C

Freq 7.853205 4.694091 7.068583 7.068583 7.853205

X/L DISP STRESS DISP STRESS DISP STRESS DISP STRESS DISP STRESS

0.0 100 0 0 100 100 0 0 100 0 100

0.1 23 30 9 52 30 26 26 30 30 23
0.2 -40 80 30 7 -29 72 72 -29 80 -40
0.3 -66 100 53 -32 -63 100 100 -63 100 -66
0.4 -48 69 68 -59 -60 88 88 -60 69 -48
0.5 0 0 71 -71 -26 38 38 -26 0 0
0.6 48 -69 59 -68 23 -28 -28 23 -69 48
0.7 66 -100 32 -53 61 -80 -80 61 -100 66
0.8 40 -80 -7 -30 70 -93 -93 70 -80 40
0.9 -23 -30 -52 -9 46 -61 -61 46 -30 -23
1.0 -100 0 -100 0 0 0 0 0 0 -100

Table 2. Mode Shapes for Uniform Beams

 14-7

BENDING MODI

ROOT-TIP F-F C-F P-F C-P C-C

Freq 10.995608 7.854757 10.210176 10.210176 10.995608

X/L DISP STRESS DISP STRESS DISP STRESS DISP STRESS DISP STRESS

0.0 100 0 0 100 100 0 0 100 0 100

0.1 -5 51 23 23 2 47 47 2 51 -5
0.2 -64 100 60 -39 -61 100 100 -61 100 -64
0.3 -40 58 76 -66 -51 76 76 -51 58 -40
0.4 33 -42 53 -47 12 -14 -14 12 -42 33
0.5 71 -93 2 2 66 -88 -88 66 -93 71
0.6 33 -42 -47 53 57 -77 -77 57 -42 33
0.7 -40 58 -66 76 -6 8 8 -6 58 -40
0.8 -64 100 -39 60 -63 85 85 -63 100 -64
0.9 -5 51 23 23 -60 82 82 -60 51 -5
1.0 100 0 100 0 0 0 0 0 0 100

1st TORSION MODS

ROOT-TIP F-F C-F C-C

Freq 3.141592654 1.570796327 3.141592654

X/L DISP STRESS DISP STRESS DISP STRESS DISP

0.0 100 -100 0 0 0 0

0.1 95 -95 16 -16 31 -31
0.2 81 -81 31 -31 59 -59
0.3 59 -59 45 -45 61 -81
0.4 31 -31 59 -59 95 -95
0.5 0 0 71 -71 100 -100
0.6 -31 31 81 -81 95 -95
0.7 -59 59 89 -89 81 -81
0.8 -81 81 95 -95 59 -59
0.9 -95 95 99 -99 31 -31
1 0 -100 100 100 -100 0 0

Data for each caa«

Heading : Dimensionless natural frequency parameters
Column 1 : Position as fraction of beam length
Cols 2, 4, etc: Relative displacement amplitudes
Cols 3, S, etc: Relative stress amplitudes

TABLE 2 MODE SHAPES FOR UNIFORM SECTION BEAMS

END CONDITIONS (ROOT-TIP)

MODS* C-F P-F F-F C-P C-C

IF 132.7 582.2 845.0 582.2 S45.0
2F 832.2 1891 2335 1891 2337
3F 2337 3972 4608 3977 4627
IE 398.2 398.2 2535 2535 2535
2E 2497 2497 7006 7012 7012
3E 7012 7012 13824
IT 1807 1807 3658 3658 3658
2T 5600 S600 7675 7675 7675
1A 5156 5156 10440 10440 10440

" See Mode Shape Figures for details.

TABLE 3(a) NATURAL FREQUENCIES FOR UNIFORM SECTION
BEAMS (He)
14-8
END CONDITIONS (ROOT-TIP)

MODE* C-F P-F F-F C-P C-C

IF (B-E) 132.7 582.2 845.0 582.2 845.0

(T) 132.6 581.1 843.4 579.6 838.3
(%diff) 0.08* 0.2% 0.2% 0.4% 0.9%

2F 832.2 1891. 2336. 1891. 2337.

827.8 1878. 2320. 1870. 2298.
0.5% 0.6% 0.7% 1.1% 1.7%

3F 2337. 3972. 4608. 3977. 4627.

2310. 3919. 4541. 3900. 4501.
1.2% 1.3% 1.5% 2.0% 2.8%

IE 398.2 398.2 2535. 2535. 2535.

395.4 395.4 2492. 2368. 2368.
0.7% 0.7% 1.7% 7.1% 7.1%

2E 2497. 2497. 7006. 7012. 7012.

2386. 2386. 6606. 6122. 6122.
4.7% 4.7% 6.1% 14.6% 14.6%

3E 7012. 7012. 13824.

6359. 6359. 12559.
10.3% 10.3% 10.1%

TABLE 3(b) NATURAL FREQUENCIES OF BEAM BENDING MODES (Hi)

WITH SECOND-ORDER EFFECTS INCLUDED

PLATE WIDTH (ASPECT RATIO)

Mode Beam 30(8.3:1) 62.5(4:1) 125(2:1) 250(1:1)

IF 132.7 134.1 134.8 135.9 137.0

2F 832.2 843.3 846.0 848.1 843.3
3F 2337. 2382. 2388. 2381. 2398.

IE 398.2 396.1 797.0 1430. 2208.

2S 2497. 2370. 4118.
IT 1807. 2196. 1095. 590.6 340.4
2T 5600. 6634. 3386. 1929. 1240.
3T 5963. 3714. 2830.

1A 5156. 5153. 5176. 5185. 5198.

_
TL1 3689. 1078.
TL2 - 5033. 2167.
-
F •= Flap; E=Edgo; T=Tor»ion; A=Axial; TL=Tramline

TABLE 4 NATURAL FREQUENCIES (Hz) OF CLAMPED-FREE PLATES

CLAMPED-FRES BEAM

UNIFORM TAPERED TWISTED

Mode No
1 132.7 F 192.4 F 151. S F E
2 398.2 E 576.7 E 443.3 E F
3 832.2 F 734.0 F 804.9 F E
4 1807. T 1769. F 1807. T
5 2337. F 2202. E 2279. FE
6 2497. E 3144. T 2390. EF
7 4617. F 3316. F 4600. F E
8 5156. A 5308. E 5156. A
9 5600. T 5493. F 5599. T
10 7012. E 6169. T 6731. EF

TABLE 5 NATURAL FREQUENCIES (Hi) OF TAPERED AND TWISTED

CLAMPED-FRBE BEAMS
 14-9

(B)
(b)

Figure 4. Mode Shapes of Uniform Plates

(a) bending mode; (b) twisting mode

Figure 5. Non-Uniform Blade and Beam Models

(a) tapered, twisted blade;
(b) tapered beam; (c) twisted beam
14-10

Complex Geometry. The first compli- the tapered case and (ii) that the fre-
cation to be considered is that arising quencies in the twisted case are less
from the complex geometry demanded by the noticeably affected but, in this case, the
aerodynamic performance requirements for mode shapes become more complex, many be-
the blade. In general, this results in a ing combinations of motion in both the
blade whose section varies along its lon- flapwise and edgewise directions simultan-
gitudinal axis in one or several respects eously. It can be seen that if the beam's
(as illustrated in Figure 5(al): symmetry were disturbed further by mis-
aligning the centroids of successive
- the section area varies (if the blade sections (up to now they have been thus
is tapered); aligned), then additional coupling would
result with all modes exhibiting some con-
- the centroid of each section may not tribution from each of the flap, edgewise,
lie on the same longitudinal axis as and torsion directions, although in many
the others; modes it is likely that just one of these
directions would predominate.
- the stagger, or principal axes, of
each section may vary in orientation
(if the blade is twisted). Similar considerations apply to
plate-like blades as well and reference
In addition to these features, the can be made to the recent studies by
cross section itself is unlikely to be Kielb et al. (1985) who have examined
symmetric, as was the case in the uniform the changes in natural frequencies and
beam and plate models discussed mode shapes for a rectangular uniform
previously. Work by Carnegie (1966), cross-section cantilever plate as it is
Montoya (1966), Rao (1980), Irretier et. progressively twisted more and more. The
al (1981), and others has demonstrated how purpose of that study was primarily to in-
the beam model can be extended and refined vestigate the difficulties encountered by
to include these various effects and the different methods of analysis, and to cat-
reader is referred to their publications alogue the various results which different
for full details. Also, some recent prediction methods produce. However, it
studies of plate models by Leissa et. al provides a convenient series of examples
(1986) have highlighted some of the diffi- for the present discussion and a selection
culties encountered when trying to include of results are included in Figure 8.
correctly the effects of twist on the vi-
bration properties of otherwise simple
cantilevered plates. Root Flexibility. In pursuit of an
accurateprediction of the natural fre-
quencies of a blade, careful consideration
In this section, it is proposed sim- must be given to the end or boundary con-
ply to illustrate some of the effects ditions which actually obtain in practice.
which can arise from the complexity of It is common to assume that the root pre-
blade geometry but without making any sents a grounded or cantilever base to the
attempt to provide a comprehensive cover- blade but this is often not at all realis-
age of the wide range of possibilities tic for actual designs. Clearly, the
which are dealt with in the literature. material of the root itself (whether of
We shall consider the changes that would fir-tree, pin or dovetail type) will pos-
be found in the previously-quoted vibra- sess some flexibility, as will the disk or
tion properties (for a uniform beam) if ring onto which the blade is mounted but
certain types of nonuniformity are intro- there will also be a 'junction flexi-
duced into the model. Two specific varia- bility* which is determined by the exact
tions on the clamped-free beam configu- conditions of contact that exist in the
ration are considered: (i) taper and (ii) joint. These various features all serve
twist. to reduce the rigidity of the boundary
condition(s) for the blade and thus to
In the first of these cases, the rec- lower its natural frequencies from those
tangular cross section is reduced linearly of the ideal clamped case.
from the root (where it is 30x10 mm) to
the tip (where, in the extreme case, it is
reduced to 15x5 mm), all other properties Whether or not the root flexibility
remaining unchanged: see Figure 5(b). effects will be significant depends on
several parameters but it is often found
In the second case, the cross section that the errors incurred by ignoring them
is kept constant (at 30x10 mm) but is pro- are greater than others in the prediction
gressively rotated (about the longitudinal process and than those associated with
or radial axis) so that the tip section is measurements. Thus, as the methods for
aligned with a stagger angle of up to 60" coping with the complex geometry and other
relative to that at the root of the blade, effects become more reliable, the effects
as shown in Figure 5(c). of root flexibility (or at the tip, where
similar comments apply to shroud attach-
ments) emerge as important factors and
Results for these two cases -which have been the subject of increasing atten-
may be taken to demonstrate the trends tion in recent years: Beglinger et al
caused by the common geometric features of (1976), Irretier (1981), Afolabi (1986),
tape and twist -are summarized in Table 5 for example.
and graphically in Figures 6 and 7. In
the table, an indication is given of the
direction(s) in which vibration is taking A clear illustration of the effect of
place. The essential results observed are root flexibility can be provided using the
(i) that the natural frequencies (but not simple beam model of a blade. Here, we
the mode shapes) are significantly af- suppose that the rigidity of the root end
fected by the reduced section present in fixing is less than complete and represent
 14-11

CLAMPED-FREE BEAM WITH TAPER OR TWIST

TAPER * TWIST **

50.% 33.% 25.% 17.% 0.% 15.° 30.° 45.° 60.°

192.4 166.6 156.4 147.6 133.8 133.8 137.2 143.0 151.5

596.7 499.7 469.3 442.7 398.2 400.9 409.0 423.0 443.3
734.0 771.7 788.2 803.7 832.2 830.4 825.2 816.6 804.9
1769. 1982. 2077. 2144. 1807. 1807. 1807. 1807. 1807.
2202. 2364. 2579. 2167. 2337. 2335. 2327. 2311. 2279.
3144. 2347. 2315. 2411. 2497. 2488. 2465. 2429. 2390.

% REDUCTION IN BOTH WIDTH ** TWIST OF TIP SECTION RELATIVE

AND THICKNESS AT TIP TO ROOT:
SECTION: LINEAR VARIATION ALONG
LINEAR VARIATION LENGTH
ALONG LENGTH

Figure 6. Natural Frequencies of Tapered and Twisted Beams

50% TAPER DATUM 60' TWIST

V y y

Mode 1 z

Mode 2 z

TAPER NO TAPER TWIST

NO TWIST NO TWIST NO TAPER

Figure 7. Typical Mode Shapes of Tapered and Twisted Beams

14-12

Figure 8. Natural Frequencies of Twisted Plates

Typical results from survey of different
analyses for first 4 modes of twisted plate
(1) 1st bend; (2) 1st torsion; (3) 2nd bend;
(4) 1st tramlines (after Kielb et al (1985))

T
* x-x
^ۤL
/xzT 7/

Figure 9. Effect of Root F l e x i b i l i t y on Beam Natural Frequency

(a) 1-coordinate f l e x i b i l i t y ; (b) 2-coordinate f l e x i b i l i t y
 14-13

Rigid Beam

150 —
Cantilever
Beam
(Hz) (b)

10" 10" X)1 10' 10* 10* 10*

k T (Nm/rad)

Figure 9. Root Flexibility Effects on Beam Natural Frequency

(a) 1-coordinate flexibility; (b) 2-coordinate flexibility

this effect by a boundary which prevents obtain a result such as that shown in
translational motion (i.e., which is rigid Figure 9(d), which clearly demonstrates
laterally) but which offers a finite the same basic trend as before.
stiffness kt to any rotation of the end of
the beam: see Figure 9(a). If the natur- Analysis of real blade designs tends
al frequency of the fundamental mode of to indicate that it is the rotational
vibration is calculated for a range of stiffness elements (or flexibilities)
values of 'root flexibility' (with k t which are most likely to be the determin-
varying between 0 and •»), then it is clear ing factors in root flexibility effects.
from Figure 9(b) that two regimes of vi- Furthermore, it is short or otherwise very
bration mode exist. First, for very low stiff blades which will be most affected
stiffness values, there is effectively a by the non-rigidity of actual blade fix-
rigid body mode with the inertia of the tures and in analyzing these, consider-
beam combining with the root flexibility ation of root flexibility is a primary
but involving very little bending of the requirement if the natural frequencies and
beam itself. Then, for much higher values the mode shapes are to be correctly
of kt (much less flexibility in the root), predicted.
the beam is found to approach the clamped-
free case which might normally be assumed. It should be noted that as consider-
However, unless the stiffness is very ation of the root flexibility effects be-
high, there may still be a noticeable re- comes more detailed, a point is reached
duction in the beam's fundamental natural where it is desirable to include the flex-
frequency from that of the fully grounded ibility of the disk also, and from there,
case and if this discrepancy is more than to incorporate the interblade coupling
2 or 3%, then it is necessary to include which is provided through the disk. At
the root flexibility in the analysis of this stage, it is necessary to review the
the blade vibration properties. whole analysis process and to employ a
much more complete model - of the entire
bladed assembly, or row. Such considera-
tions are the subject of the next chapter
In practice, of course, root flexi- and will not be considered further here.
bility effects are more complex than the However, in order to define a suitable
simple model presented here but the essen- 'boundary1 for the individual blade
tial results are generally similar to analysis, it is appropriate to include all
those shown. If we extend the previous those sections of the blade itself, plus
model to include flexibility in both the any additional flexibilities which are in-
rotation and translation directions troduced at the junction with the disk
(Figure 9(c)), and then consider ranges of (e.g., fir-tree 'branch') but not of the
values for both stiffness elements, we disk or annulus itself.
14-14

E f f e c t s of Rotation. There are a The type of plot just introduced - a

number of additional factors which apply Campbell Diagram - is a standard format
when the blade is in its operational en- for the presentation of blade (and bladed
vironment and which influence the struc- assembly) frequencies under operating con-
tural dynamics properties that are of in- ditions and is used even when there is no
terest to us here. Probably the most significant change in the blade's natural
important are those due to the centrifugal frequencies with running speed. It is
forces which are generated by the rotation convenient to superimpose on the basic
of those blades on moving rows. Very con- graph lines of different 'engine orders*
siderable forces are thus generated - for where the n" engine order line shows what
example, a 0.1 kg blade rotating at a mean frequencies are generated by a disturbance
radius of 0.3 m with a speed of 8,000 which occurs n times per revolution of the
rev/min will experience a radial force of stage. These, in turn, can help to iden-
more than 2 tons - and, as a result, tify potential regions of resonance -
these can have a marked influence on the speeds at which a known excitation source
vibration properties of some blades. will excite modes of the blade - an abil-
ity which takes on an even greater signif-
icance when dealing with bladed assembl ies
( next chapter) rather than individual
blades.
The first consequence of the centri-
fugal force (and, incidentally, the other Temperature Ef Cec\. Another effect
quasi-static bending forces which are gen- which is present under operating condi-
erated by the gas pressure on the surfaces tions, and which may have an influence on
of the blade) is to influence the geome- the vibration modes, is a variation in the
tric form of the blade. Those blades temperature of all or part of the blade.
which are heavily twisted or which have If there are any significant temperature
non-symmetric cross sections are liable variations, as there will often be in high
to suffer a change in their geometric form pressure turbine stages, then these may
under running conditions and this must be result in a change in the elastic modulus
duly taken into account in making the de- of the blade material with consequent
sign calculations of the vibration proper- effect on its natural frequencies. If the
ties (see the example in the next operating temperature profile is known,
section). along with the modulus-temperature charac-
teristics of the blade material, then it
is possible to account for this effect in
calculations of the blade's vibration
However, probably the most important modes with little difficulty.
consequence of the centrifugal load is the
stiffening effect it has on the blade -
'CF stiffening' -which, in turn, causes an
increase in the natural frequencies with
some corresponding changes in mode shapes.
Once again, it is possible to demonstrate
the essentials of this phenomenon using
simple beam-like models. Upon inspection
of the mechanics of the process, it can be
seen that there is a subtlety to the
effect in that the CF loads have a slight- (b) (a)
ly different influence on a blade mode
according to the direction of its vibra-
tion: relative to the plane of the
rotation. Figure 10 illustrates the basis
of this feature: Cor vibration normal to
the plane of rotation, (a) the (radial)
centrifugal force, which provides an addi-
tional restoring force and hence stiffens
the structure, acts purely along the blade
longitudinal axis, whereas for vibration
in the plane of rotation, (b) this same
restoring force has a (small) unstiffening
component because its line of action
(again radial) is not exactly in line with
the blade axis. Hence, the stagger atti-
tude of the blade on its disk will have a
small influence on exactly how much the CF
stiffening affects the blade vibration
modes.

A set of results is presented for the

simple beam model used earlier which is
now assumed to be clamped to the rim of a
rigid disk of 0.56 m outer radius and ro-
tating at speeds of up to 8000 rev/min.
The 'blade' is attached such that the flap
direction is in the plane of the disk - or
tangential, in terms of the disk coor- Figure 10. Centrifugal Stiffening (CP)
dinates. The results of these calcula- Effects
tions are shown in Figure ll{a) in a plot (1) axial vibration;
of natural frenjency vs speed of rotation. (2) tangential vibration
 14-15

1000 —

500 —

2000 4000 6OOO 8000

Speed (rev/min)

Figure 11. Campbell Diagram Showing Effects of CF Stiffening

on Blade Natural Frequencies

ANALYSIS METHODS FOR PRACTICAL BLADES Different glade Types. As indicated

earlier, "Tblades can generally be classi-
Introduction. In this section, we' fied into three types according to their
turn our attention to the analysis methods geometric characteristics such as length,
which can be used in order to calculate chord and thickness. If the length-
the specific values for the dynamic to-chord aspect ratio (L/c) is high
characteristics of individual blades (as -i.e., greater than about 5 -and the
described above) for the case of practical thickness is small compared with the
designs. It is first appropriate to length, then the blade may be considered
recall that the relative stiffness of the as a unidimensional structure and thus
disk to that of the blades is of impor- modelled as a beam. If the aspect ratio
tance in the mechanical modelling of ro- is low, but the thickness is still small
tating blading. If the disk and blade when compared with the other dimensions,
stiffnesses are comparable, then their then the blade is essentially two-
motions will be strongly coupled and dy- dimensional and should be modelled as a
namic analysis of the components indepen- thin plate or shell. If the blade has a
dently Is not sufficient to provide accu- low aspect ratio and is thick, it is a
rate estimates of the natural frequencies three-dimensional structure and must be
and mode shapes which will apply under nodelled as a thick shell.
operating conditions. However, if the
disk is very stiff, its influence can be
disregarded and just a single blade
modelled and analyzed for use in design Whatever blade type is considered,
predictions. In any event, the properties the natural frequencies and mode shapes,
of an individual blade are evaluated as a together with stress distributions, are
first step in most cases, even if the disk generally determined using the finite ele-
effects have to be added subsequently, as ment methort outlined in the previous
described in the next chapter. chapter. A parametric study of finite
14-16

element method modelling of all three finite element. These characteristics are
types of blade is reported by Hitchings et the area, the position of the centroid and
al, (1980). Other methods of analysis, that of the center of twist, the second
such as direct integration of the govern- moments of area and their principal
ing differential equations, the transfer values, the principal section axes (gives
matrix method, and others, are no longer the pretwist), the torsion constant due to
widely used; not because of a lack of pre- warping and some higher order moments of
cision but because their implementation area. Some of these quantities result
via standard computer programs is not from performing integrals of the forrni
straightforward.
yndS 0,1,..4
Blades Modelled by Beam Elements.
Generally, a beam type of model is well
suited to the analysis of low pressure The other parameters (torsion con-
turbine blades. This kind of blade may be stant, centre of twist) are quantities re-
pretwisted, have an asymmetrical varying lated to the torsion problem of airfoil
airfoil cross section and be mounted with cross sections, Ferraris (1982).
a stagger angle. Depending on its de-
tailed geometric characteristics, the As the blade profiles are usually de-
blade can execute either uncoupled bending fined by a set of points resulting from
or coupled bending-torsion vibrations, the aerodynamic design calculations, it is
latter occurring when the shear centre (or convenient to solve the torsion problem
centre of twist) does not coincide with and all the area integrals by using the
the centroid of the blade cross sections, boundary integral technique. A pre-
Saada (1974), Sokolnikoff (1956). Also, processor to the blade analysis program
bending-bending coupling exists due to the will furnish all the required cross sec-
pretwist and, under rotation conditions, tion properties, Ferraris et al (1983).
coupling occurs between longitudinal and Since the chord and pretwist tend to have
torsion motions (untwist due to centrifu- rather smooth variations along the blade
gal effects) and between longitudinal length, linear variations of the geometric
stress and bending (centrifugal stiffen- characteristics are assumed for the blade
ing). A procedure for deriving the cor- stiffness calculations.
responding beam element with two nodes and
six degrees of freedom per node is de-
tailed by Ferraris et al (1983) and repre- Root Flexibility and Platform Effect.
sents a finite element version of the Two approachesarepossibletotake
classical Montoya analysis, Montoya account of the blade platform and root
(1966). effects: the first consists of modelling
the root with a large number of finite
elements, while the second consists of
Blades Modelled by Plate or Shell using an equivalent but simple model.
Elements..This typeincludes fan blades
and some compressor blades. Doubly curved The first method is a priori more
thin shell elements and flat plate ele- precise but is much more costly because it
ments can be used to model blades in this increases considerably the number of de-
category. The only requirement is that grees of freedom in the model. The second
the longitudinal (membrane) stress and method is simpler and is most efficient
bending motions coupled by the centrifugal for three-dimensional turbine blades. It
effects are taken into account (geometric consists of neglecting the mass of the
and supplementary stiffness), but as the root (because of its relatively small
blades in rotating axial machines are al- motion), and deriving equivalent stiff-
most radial, the Coriolis effects can nesses calculated using simple formulas
often be disregarded. Some studies and (tension, bending, and shear) for the
checks show that the flat plate triangular strength of materials properties of an
element with three nodes and six degrees- equivalent parallelepiped element. These
of-freedom per node still remains compet- stiffnesses are then distributed to the
itive for the analysis of general thin convenient nodes. The platform is rela-
shells for practical engineering applica- tively thin and as such acts like a mass
tions in spite of extensive work done with that can also be neglected (see the sec-
refined shell elements, Clough (1980), tion "Three-Dimensional Blade: High
Olsen and Bearden (1979). Pressure Turbine Blade.*)

Blades Modelled Thick Tri-

Dimensional E l e m e n t s . A l l blades not in- Shroud Attachment. Where appro-
cluded in the two previous types must gen- priate, an integral shroud attachment can
erally be modelled as three-dimensional be modelled by equivalent stiffnesses,
structures, e.g., short and thick turbine masses, and inertias distributed on the
blades and some high pressure compressor convenient nodes. The restrictions on the
blades. A thick shell isoparametric ele- displacements are imposed by the limit
ment with 24 nodes is widely used for this conditions.
type of component, Trompette and Lalanne
(1974).
Static Loads. When using the finite
Accommodation of Special Effects. As element method, any inertial or external
a1ready noted in the previous section some surface loads must be expressed in terms
special feactures of turbomachine blades of equivalent nodal forces. The centrifu-
have to be accounted for. gal forces are derived from the expression
for the kinetic energy (see the previous
Complex Cross Section Properties. chapter), and the method of introducing
The f i r s t o f these requirementsConcerns the aerodynamic static forces as the pres-
the derivation of the geometric character- sure loads is detailed in later chapters
istics of the cross sections for the beam dedicated to aeroelasticity.
 14-17

Coriolis Effect - Centrifugal Stiffening. by the boundary integral technique, since

It is important to take into consideration they are defined by points (resulting from
the effects of rotation on the structure, aerodynamic design calculations). The
namely: the Coriolis effect and centri- geometric characteristics are: cross sec-
fugal stiffening. Their influence is re- tion area S, section centroid, G, princi-
ported along with the application to a fan pal moments of area Igr and lGn and their
blade in the following section. It must principal directions, leading to the pre-
be pointed out that provided the blade is twist angle, X . The center of twist
radial, the Coriolis effects are generally position T(£t,T)t) and the St. Venant tor-
negligible. On the other hand, the cen- sion coefficient JT are determined by
trifugal effects on the stiffness are of solving the torsion problem. Finally, the
great importance and must be introduced, remaining geometric characteristics for a
particularly for one-and two-dimensional complete bending-bending-torsion vibration
blades, as they can produce a significant of the blade at rest and in rotation are
stiffening effect on the frequencies and determined. They are of the form:
on the untwist in rotation.

Temperature Effect. As will be seen

in the following section, with an example
on a turbine blade, the effects of high 5U2+n2)dS;
temperatures -generally to reduce the
material stiffness -can be estimated prac-
tically by using Rayleigh's quotients / U2+n2)2ds
through the variation of stiffness due to s
changes in the value of Young's modulus.
In some circumstances, the temperature
effect can even overcome the centrifugal and are called higher order moments of
stiffening and result in a net reduction area in the following paragraphs.
of the frequencies at operating speeds.
The temperature static effects require the When all the cross section character-
use of a thermoelastic program that gives istics are known, a refined finite element
an initial thermal stress distribution. is derived considering a linear variation
of these characteristics along the mean
line of the element. The assembly is per-
REVIEW OF TOPICAL BLADE CHARACTERISTICS. formed using a global coordinate system
XY7, fixed in the rotor, where X is
Based on the above modelling and directed radially outward along the mean
'-.lysis methods, the calculation of three line of the blade, Z is colinear with the
•ifferent types of blade is now reported rotor axis of rotation and Y is chosen to
long with some experimental verifications provide a right handed orthogonal coor-
of the computed results. An example of a dinate system.
uni-dimensional blade (from a low-pressure
turbine) is used to illustrate the impor-
tance of the bending-bending-torsion coup- Dynamic Analysis; Coupling Influence.
ling both at rest and when rotating. The The low pressure turbine blade is modelled
two-dimensional example (a fan blade) in- with 12 beam elements. The tip shroud is
cludes a complete set of results: static modelled by additional masses and inertias
stress distribution and untwist due to and the blade is assumed clamped at the
rotation, influence of the geometric platform. The problem results in a model
stiffness on the static displacements and with 72 degrees of freedom for which the
on the frequencies related to the speed of first six natural frequencies are calcu-
rotation, Henry (1973), Henry and Lalanne lated at rest (n = 0 rev/min) and at the
(1974). The influence of the Coriolis the operating speed (ft = 5000 rev/min).
effect at operating speeds is also inves-
tigated. The last example (a high pres-
sure turbine blade) shows some of the In order to illustrate the various
practical difficulties of including the coupling Influences, three calculations
rotation, temperature, root and platform are performed on the twisted blade. The
effects, Trompette and Lalanne (1974). All first is made with the "refined coupling*
the applications presented use the finite model (noncoincident centroidand centre
element modelling methods summarized in of twist, and higher order moments of area
Chapter 13. noted: T # G, J^, J n , J * 0). The second
is made without the higher order inertia
terms, but with noncoinciding T and G.
Uni-Dimensional Blade. Beam Model of This case is referred to as "weak
an LP Turbine Blade. coupling," (T * G and J^, J n , J = 0).
Finally/ the blade is analyzed as a simple
Blade Finite Element Modelling. The blade twisted beam: T and G are coincident (T =
used for this example has an aspect ratio G) and J n, 0. This last case is
of just over 6 and a maximum angle of pre- referred to as "no coupling." It must be
twist of about 25° and this is suitable remembered that in all three cases, the
for a uni-dimensional model (Figure 12). pretwist and the varying cross sections
The finite element used is a straight are included in the model.
twisted beam element with two nodes and
six degrees of freedom per node, assigned Table 6(a) presents the results ob-
to the cross section centroid, (Figure tained for the blade at rest. The per-
13). The typical element is calculated in centages indicate the influence of the
a local coordinate system R(Gx?n). G? coupling terms relative to those for the
and Gn are the principal inertia axes of refined model calculations. Flapwise
the cross section S. The cross section bending modes are labelled F, edgewise
geometric characteristics are determined bending modes by E and torsion modes by T.
14-18

Figure 12. Low-Pressure Turbine Blade

Figure 13. Twisted Beam Finite Element

 14-19

Table 6(b) presents the corresponding re- Concluding Remarks.

sults obtained for the operating speed
while Figure 14 shows the mode shapes ob- (i) Disregarding the various coupling
tained at rest with the refined model (- terms gives overestimates for
.-), with the so-called "weak coupling" the bending frequencies and
(-x-) and with the refined model at underestimates for the torsion
operating speed ( + ). frequencies.
When examining the mode shapes pre- (ii) The influence of the coupling terms
sented in Figure 14 where the displacement is quite weak for those modes which
of the mean line of the blade are pro- are quasi-pure in bending or in
jected onto the YZ plane, (X, mean line torsion, but is significant on the
directed out of the plane), it can be seen highly coupled modes.
that the first four modes are respectively
IF, IE, IT, and 2FE modes. The highly- (iii) Introducing an incomplete set of
coupled bending-bending-torsion modes are coupling terms may be more disadvan-
numbers 5 and 6, and the coupling influ- tageous than calculating with an
ence is shown in Table 6(a) and (b) at uncoupled model.
rest and at the operating speed respec-
tively. The mode shapes are only slightly (iv) The lowest bending natural frequen-
modified by rotation but are very sensi- cies are sensitive to rotation
tive to the coupling terras, and particu- speed, but the mode shapes are only
larly so for the torsion modes. For slightly affected.
example, examining modes 5 and 6 for the
same scaling factor, (*x = I/ maximum tor- (v) Noting that the aerodynamic forces
sion angle), reveals a difference in the in aeroelasticity are closely re-
bending motions when the weak coupling lated to the frequencies and the
case is considered. When no coupling is mode shapes of the structures, it
considered, it becomes impossible to make becomes clear that it is very
any comparison, since the 5th mode important to use a refined finite
- coupled 3FE/2T - becomes 3FE uncoupled beam element for the uni-dimensional
and the 6th mode - coupled 2T/3FE blade model, particularly when the
- becomes a purely uncoupled 2T mode, with first bending and torsion modes are
a change in the order of natural highly coupled.
frequencies.

Mode Refined Weak % No % Experimental

Coupling Coupling diff Coupling diff (Clamped)
1 IF 113.4 115.2 +1.6 115.3+1.7 113
2 IB 455 460 +1.1 464 +2.0 455
3 IT 601 574 -4.5 581 -3.3 597
4 2FE 802 814 +1.5 811 +1.1 717
5 3FB-2T 1650 1707 +3.5 1839 +11.5 1647
6 2T-3FE 1983 1872 -5.6 1790 -9.8 1967

TABLE 6(a) INFLUENCE OF COUPLING EFFECTS (AT-REST)

Mode Refined Weak % No %

Coupling Coupling diff Coupling diff
1 165.7 167.1+0.8 167.2 +0.9
2 482 486 +0.8 491 +1.9
3 611 585 -4.3 589 -3.6
4 851 863 +1.4 861 +1.2
S 1697 1738 +2.4 1909 +12.5
6 2034 1939 -4.7 1814 -10.8

TABLE 6(b) INFLUENCE OF COUPLING EFFECTS

(AT OPERATING SPEED)
14-20

0x$1.5°/mm in y
in z

-1 1 y
Mode 1 -. 1 F Mode 2 » 1E

0X= 1 (scaling factor) ^#0.7'/mm

in y

1 y -1
Mode 3 • IT Mode A » 2 FE

(scaling z
_ 0X = 1 (scaling
factor) factor)

-1 1 V

Mode5 = 3 FE/2T coupled ModeG 2T/3FE

. Q = 0 refined model coupled
- + -.*Q = 5000 rev/mn refined model
1 = 0 weak coupling

Figure 14. Mode Shapes of LP Turbine Blade

Two-Dimensional Blade Plate Model of a Fan (Figure 15). The plate element used is
Blade. partly conforming: the displacements are
continuous across the element and at the
Blade Finite Element Modelling. This boundaries; the interpolation functions
blade is modelled with 80 three-node tri- are complete and the slopes are continuous
angular plate elements, with six degrees at the nodes, but not on the interelement
of freedom per node. The blade is assumed contours. Kirshhoff' assumptions are used:
to be clamped at the platform, leading to i.e., constant stress in membrane, linear-
some 300 effective degrees of freedom ly varying stress in bending.
 14-21

Figure 15. Finite Element Model for Fan Blade

a) 6r reduced b) o r % on the c) or°/0 on the

isodisplacement suction surface pressure surface
Figure 16. Static Isodisplacement and Stress for Fan Blade (at operating speed)
14-22

Stress Analysis; Displacements and Stress Let us consider the mean untwist if and the
Distribution maximum displacement 6 of the blade tip
section in terms of n. Firstly, these
At a typical operating speed, &, the quantities are calculated with the non-
steady displacements < and the Von Hises linear rotation effects included (KG * 0).
stress OVH are determined following the Secondly, the same quantities are calcu-
procedure described in the previous lated with KG neglected (KQ = 0 ) . In
chapter. Figure 17, the S and $ variations are pre-
sented in reduced form (6r = 6/<max; *r *
The reduced isodisplacement lines 6r as a */*max> wnere 5max and *max are Efie maxi-
percentage of the maximum are presented in mum values of 6 and 41 for the maximum
Figure 16(a) and one can see clearly the speed of rotation -non-linear effects in-
untwist <)> due to rotation since the zero cluded -and J!f is the non-dimensional
displacement line is located along the speed of rotation related to the first
blade axis. The stress distribution on natural frequency of the blade at rest (ftr
the suction and pressure surfaces of the = 8/foi)- It is clear that the non-linear
blade are also presented in terms of re- geometric stiffness Kg cannot be dis-
duced stress or in Figures 16(b) and (c). regarded, since neglecting the centrifugal
When examining these two figures, it can stiffening will lead to significantly
be seen that the maximum stress is not overestimated displacements. Remembering
located at the same point on the suction that the stresses are directly linked to
and pressure surfaces. This is due to the the displacements in finite element
fact that the stresses at a typical point modelling , errors in the design analysis
are the superposition of a membrane stress could follow, not only in the static
(constant across the thickness) and a cases, but also in the dynamic ones.
bending stress (of opposite sign on the
two surfaces).
These results are obtained when taking in- Dynamic Analysis Ten natural frequencies
to account the centrifugal stiffening and mode shapes are calculated for a range
offset but, noting that these calculations of values of the rotation speed and a
require an iterative procedure, it is of Campbell diagram is plotted and presented
some practical importance to examine the in Figure 18(a). It can be seen that the
extent of this effect. In other words, in influence of rotation is noticeable on the
view of the higher cost of such a non- bending frequencies but is relatively weak
linear calculation process, is it justifi- on the torsion modes. Some mode shapes -
able (by an improvement in the precision those of the second and third bending and
of the results) to include the rotation first and second torsion modes -are pre-
effect (KG, Kg) in the stiffness sented in Figure 18(b). The calculated
analysis?

error i 200 %> at Nr operating speed

6r(KG=

8r(KG)
S> r (KG)

Nr 2 Qr

Figure 17. Influence of the CF Stiffening (KG) on Static Deflection of Fan Blade
 14-23

nodal lines for the at-rest {fic » 0) and Comments

rotating (Slr = 1.5) conditions are com-
pared together with some experimental (i) In principle, a "two-dimensional"
results at rest. First, it is seen that blade dynamic analysis could be per-
good agreement exists at rest, and second, formed using the previously-presented
the calculated nortal lines in rotation bean elements but the designer must
show little modification when compared bear in mind that only the lowest
with those at rest. modes - those which are close to beam
modes - will be correctly predicted
All the preceding results have been ob- (e.g. the first four or five modes of
tained with the Coriolis effect neglected. the fan blade presented in this
In order to demonstrate the validity of section). Other more complex types
this assumption, the first ten natural of mode will not be reliably pre-
frequencies are re-calculated including dicted by a beam model.
the Coriolis effect for a speed corres-
ponding to ilr = 2, which is somewhat {ii) As was seen for the uni-dimensional
higher than the operating speed. The blades, centrifugal effects must be
ratios of the frequencies calculated with included in the FEW model of two-
Coriolis to the corresponding frequencies dimensional blades.
without Coriolis are in the range 0.998 to
1.0018: that is, ±0.28. As the computa- (iii) For classical blades which are
tion time is about four times higher for directed radially outwards, the
the with-Coriolis calculation, it can be Coriolis effect can be neglected.
concluded that the Coriolis effect can, However, in contrast to conventional
and indeed should, be disregarded in the blades, some new designs for thin
analysis of practical radial and quasi- and highly swept profiles give rise
radial blades. to large static bending and twisting
deformations and also to complex
vibratory characteristics. For

Figure 18a. Campbell Diagram for Fan Blade Natural Frequencies

14-24

these, estimating the static condi- Three-dimensional Blade; High Pressure

tions produced by the centrifugal Turbine Blade
loading using the geometrically non-
linear procedure (KG - (1 MG) might Blade Finite Element Modelling The high-
not be adequate, since large dis- pressure turbine blade shown in Figure
placements occur. A full non-linear 19(a) is modelled with nine 24-node isopa-
analysis, using an updated rametric thick shell elements, with three
Lagranqian formulation, or equiva- degrees of freedom per node.
lent, might be required. For the
same reasons, variations in the The blade root and the platform can also
blades natural frequencies with be modelled by isoparametric elements but
speed of rotation might now result this is quite cumbersome and increases
from the Coriolis effect and further significantly (and unnecessarily) the
investigation of this feature is total number of degrees of freedom in the
strongly recommended for this type model. It is generally more practical and
of blade.

Second bending (2F) First torsional (1T)

experiment
at rest
finite element
at rest
•—finite element
at operating
speed

Third bending (3F) Second torsional (2T)

Figure ISb. Mode Shapes (nodal lines) for Fan Blade Nodes
 14-25

convenient to choose a simpler way of tak- It is not appropriate to present again the
ing into account the root effect using an static analysis (for centrifugal stresses)
equivalent model obtained from mechanical that has been detailed in the preceding
considerations. First, noting that the section. Hence, in the following sec-
root motion is generally low, it is rea- tions, only the dynamic analysis is pre-
sonable to suppose that its inertia influ- sented and some practical considerations
ence on the kinetic energy may be concerning the rotation and temperature
neglected. The platform is thin and as effects are discussed.
such acts only as a mass (contributing
negligible stiffness) and so may also be
neglected. Secondly, the root is modelled
by an equivalent rectangular parallel- Dynamic Analysis The finite element model
epiped whose geometrical and mechanical results in 384 degrees of freedom. The
characteristics are defined as follows. first three natural frequencies and asso-
(Figure 19(b)): ciated mode shapes are calculated at rest
(0 - 0) and at room temperature, firstly
the dimensions are the mean dimensions with the blade clamped at the platform
of the fir tree (or other) root, (root effects neglected altogether) and
clamped at the first branch; and secondly including an equivalent root
iiodel. The natural frequencies are com-
the stiffness characteristics in the pared with experimental ones measured in
three directions are obtained from situ in the engine (see Table 7). Good
simple Strength of materials formulae agreement is obtained, provided that the
for bending, shear and longitudinal root is modelled, albeit with a relatively
effects. These stiffnesses are equally simple model.
distributed to the convenient nodes of
the blade base.

Freq. (Hz) EXP.f0* FE. % FE % FE + root %

Mode n- o base clamped Equiv. root CJ- 16000 rev/min

L 1368 1588 16.1 1370 0.2 1456 6.0

2 3133 3852 23.0 3354 7.0 3401 1.4
3 4661 5248 12.6 4755 2,0 4809 1.1

* mean values for several blades

TABLE 7 TURBINE BLADE FREQUENCIES AT ROOM TEMPERATURE

springs equivalent
to the root stiffness

19a. High Pressure Turbine Blade Figure 19b. Modelling of Root

14-26

The first mode is mainly a flap-wise bend- Estimation of the Rotation Effect on the
ing mode; the second one edgewise bending Frequencies. Provided that the at-rest
and the third one is predominantly natural frequencies are determined using
torsion. Still at room temperature, the the appropriate finite element mesh, the
blade properties are next calculated for a mass matrix [M] and the mode shapes («]
rotation speed of 16,000 rev/min. are known. At rest, the stiffness matrix
Examination of these results shows clearly [K] is the linear stiffness [Kr]. In
that for this case of a three-dimensional rotation, 2the stiffness becomes [KJ » IKL1
blade, the rotation effect on the natural + [KG] -n [MG], (see Chapter 13), and con-
frequencies is weak and almost negligible sequently it can be stated that:
on the mode shapes and this is likely to
apply to most other blades in this [AK] = IKG1 - 02[MG1 (2)
category.
where [KG], the geometric stiffness
In practice, it is often recommended that matrix, and [MQ], the centrifugal mass
the number of finite elements be increased matrix, are known at the end of the static
in order to improve the accuracy of the analysis.
analysis, but this will result in a higher
cost, especially if several different Using Equation (1) for a typical
speeds of rotation are required. However, mode, results in:
noting that the rotation may have little
or no influence on the results, a good
estimate of the natural frequencies at (3)
various non-zero rotation speeds (ft > 0)
can be obtained at low cost by using the
at-rest results together with the so- or fi<n)
called Rayleigh's Quotients.
The results presented in Table 9 for
a speed of 16,000 rev/min show very good
Use of the Ravleiqh's Quotients. It is agreement when compared with those ob-
well known that for a slight modification tained by a complete finite element calcu-
UK] or [AM] to the stiffness and mass lation, since the error incurred by the
matrices [K] and [M] of a structure, the Rayleigh's Quotient calculation is less
equation for free harmonic vibration: than or equal to 0.5%.

o>2[M] {x} - [K] {x} This procedure can be implemented

directly in the finite element program and
gives a first approximation to the natural will obviate the need for a complete
frequency variation (Af): re-calculation at several speeds of rota-
tion, thereby resulting in a considerable
if = Au/2n economy.
(1)
{x) T [AKHx}-<o 2 {x} T [AM] {x}) Estimation of the Platform Mass Effect.
PC) T [M] {x}) The platformmass h a s b e e n m o d e l l e d ap-
proximately and distributed equally to the
This expression is quite general and appropriate nodes of the blade base. The
applies to whatever is the cause of the resulting [AH) included in Equation (1)
structure modification, (rotation, mass of gives a less than IHz variation on the
the platform, temperature effect, etc.). first three natural frequencies, demon-
strating that the platform may be ne-
glected - at least in this case.

Freq. (Hz) FE £0(0) FE. Rayleigh

at room a -o £1 - 16000 rev/min a - 16000 rev/min
temp

1 1370 1456 1460 1 0.3%

2 3354 3401 3400 | 0.0%
3 4755 4809 4785 1 0.5%

TABLE 8 RAYLEICH'3 QUOTIENTS TOR ROTATION EBTBCT DZTERMXNATXOH

Freq (Hz)
t° f0 20° 200°C 400°C 600°C 700°C %
n - 16000

1 1456 1440 1401 1360 1334 -8.4

2 3401 3359 3259 3151 3084 -9.3
3 4809 4749 4605 4451 4355 -9.4

TABLK 9 TKKPERATORC INH1OTNCZ AT OPERATING SPIED

 14-27

Temperature Effect. Large temperature the choice of element depends pri-

variations modify the value of Young's marily upon the blade geometry. If
modulus, E. Remembering that the stiff- a beam model is chosen, the bending-
ness [KJ is proportional to E, then to a bending-torsion coupling must be
first approximation: included in the finite element theory;

the influence of rotation has to be

[AKJ = (AE/E)[K) (4) considered through both the geometric
stiffness and the centrifugal mass
effects. The Coriolis effect can be
and, finally, neglected if the blade is directed
radially outward. Por prop fan blades,
and other non-classical forms, further
(5) investigation is recommended before
ignoring this Coriolis effect. Also,
full non-linear analysis might be
or necessary for static behaviour in
rotation;
ft(R) = f0(fl)
some account may be taken of the
stiffening effects and temperature
In Table 9 the influence of tempera- effects by using Rayleigh's Quotients,
ture variations on the natural frequencies resulting in savings of time and cost,
at the operating speed n is presented, provided mode shapes are little modi-
when the temperature increases to a typi- fied by rotation;
cal turbine operating level, the natural
frequencies decrease and at 700°C it is considerable attention has to be paid
clear that this effect is more important to the root modelling and the limit
than rotation, since the frequencies at condition situation. Practically, at
operating speed and temperature are lower the operating speed, a rotating blade
than those at rest and at room temperature can be considered clamped if the root
(20°C). has a fir tree type fixation on the
disk; and
Conclusion - Practical Recommendations to a single blade calculation is appro-
the Designer. All these examples show priate only if the disk stiffness is
that the finite element modelling is an high compared with that of the blade.
efficient tool for practical rotating if the blade and the disk stiffnesses
blade vibration analysis. However, the are comparable, then the whole
designer has to remember that: assembly has to be modelled.
Appropriate methods for this case are
detailed in the next chapter.

ACKNOWLE DGEMENT

The authors gratefully acknowledge

the contributions to this chapter made by
Dr. M. Imregun, who provided many of the
computed data.
:ii. !; .. -. i. - i .-. Bl ado tMckel '»f -;| • • I :••'••• '

27. pact •?' < • . ! H L . I . I M i - ' ' --. !'>-•-.! niccu

 15-1

STRUCTURAL DYNAMIC CHARACTERISTICS OF BLADED ASSEMBLIES

by

D. J. EWINS
Imperial College of Science and Technology
Department of Mechanical Engineering
Exhibition Road, London SW7 2BX

INTRODUCTION configurations with which we shall be

dealing. There are several, although they
Background may be all classified into one of two dis-
tinct groups: (a) those which are circum-
It is clear from the preceding chap- ferentially or cyclically symmetric and
ter that considerable efforts have been (b) those which are not. The first group
expended In order to make accurate predic- comprises those assemblies in which all
tions for the essential structural dynamic the blades have the same interconnections,
properties of typical turbomachine blades. and include:
These components generally have quite com-
plex profiles and the calculations in- 1. unshrouded identical blades on a
volved are often lengthy since both the flexible disk (or drum);
natural frequencies and the mode shapes
are required with some accuracy: the fre- 2. continuously tip-shrouded blades on a
quencies because of their close relation- flexible disk;
ship with the rotation speed(s) oE the
machine and the mode shapes because of 3. raid-height-shrouded blades on a flex-
their direct influence on the aerodynamics ible disk;
of the working fluid. One other struc-
tural characteristic which is of impor- 4. shrouded blades on a rigid disk.
tance but which is far less accessible
than the mode shapes and frequencies is The second group contains those
the modal damping which derives from the assemblies which do not possess complete
various mechanical sources such as materi- circumferential uniformity and, for our
al hysteresis and slip at the mating sur- purposes here, include:
faces in joints. This property/ and
especially the latter effect, is very dif- 5. groups (or packets) of shrouded
ficult to predict and few attempts to do blades on a rigid disk;
so are reported. Our comments thus con-
centrate on the two primary properties of 6. shrouded blades on a flexible disk,
natural frequency and mode shape. where the shroud connections are dis-
continuous, resulting in a packet«d
In our pursuit of accuracy in these bladed disk; and
predictions, we have so far neglected one
feature which may - in certain cases - 7. assemblies where the blades are not
have a major influence on the actual identical to each other.
values of the vibration properties sought.
That factor is the coupling between one Although most current designs tend to
blade and its neighbours via the struc- fall in the first group, the second is
tural connections which inevitably exist, also of interest, partly because of the
either through the disk or annulus which possible advantages which these configu-
carries the blades, or through a shroud rations may enjoy and partly because im-
ring which may be incorporated to stiffen perfect or non-uniform assembly conditions
the whole assembly. Although we may have may (inadvertently) create them.
included root flexibility in our single
blade model (see Chapter 14), or ensured We have also included the implica-
the addition of the mass of a shroud tions of slight variations in the proper-
segment, these do not take into account ties of the blades in one row (7), since
the possibility that one blade will inter- this characteristic is likely to arise to
act with its neighbours and thereby influ- some degree in every practical assembly
ence the vibration modes of the whole set. and, also, because it sometimes has a very
In fact, as we shall see, the structural pronounced effect on the properties of
coupling between the blades comprising one interest.
stage will usually exert a significant in-
fluence on both the natural frequencies Objectives and Ojjtline of Chapjter
and the mode shapes and so it is important
that we extend our analysis to include it. Following the pattern set in the pre-
In most cases, determining the properties vious chapter, our main concern here will
of a single blade is a necessary stage in be to establish the essential features of
the structural dynamic analysis but is not the structural dynamic properties of
sufficient to describe the blades' vibra- bladed assemblies. We shall seek to
tion properties in their operational illustrate the patterns of both natural
environment: for that we must analyze the frequencies and mode shapes for various
complete assembly. assemblies and to identify the major con-
trolling factors in each case. Thus, it
is hoped that those engineers concerned
with the complete aeroelastic analysis of
Assembly Configurations such blading will be provided with both
insight into the types of vibration mode
Before developing our analysis possessed by bladed assemblies and to the
further, it is appropriate first to design parameters which are particularly
identify the various bladed assembly important for their exact determination.
15-2

Much of this chapter will make use of STRUCTURAL DYNAMICS MODELS FOR BLADED
simplified mathematical models of a bladed ASSEMBLIES
assembly since these readily permit the
detailed parametric studies necessary to Lumped Parameter Models
determine the patterns of behaviour. The
design analysis for an actual assembly In view of the inherent size and com-
will necessarily incorporate all the plexity of a typical bladed assembly, it
details of geometry and operating condi- is necessary to reduce the system model to
tions {as was the case for the single the most basic form which is appropriate
blade), but these calculations are very for the study in hand. Originally, this
expensive. As our primary interest here meant simplifying the usually complex geo-
is to understand the interaction between metry so that the blades could be repre-
the various blades, it is appropriate at sented by equivalent beams in order that
each stage to consider only those features the complete assembly could be modelled
which are essential and so a number of Armstrong (1955). Even then, it was
simplified models have been devised and necessary to assume full axisymmetry (so
will be used throughout this chapter. that the basic component consisting of a
Accordingly, the first discussion will be single blade/disk sector could be
concerned with the development of the modelled) and to use the repetitive nature
models themselves. The basic model, which of the actual assembly, together with some
is very simple, admits a single degree of knowledge of the anticipated results, in
freedom (or mode of vibration) for each order to extract a solution.
blade and includes a similarly simplified
representation of the disk and/or shroud Early attempts to study the effects
coupling. The resulting discrete lumped- of blade mistuning required a less re-
parameter (mass-spring) model permits strictive model and one which was proposed
rapid and cheap calculation of its vibra- for this purpose Dye and Henry (1969) (and
tion properties and is ideal for a para- derived from an earlier concept, Bishop
meter study but is of limited application and Johnson (I960)) now provides us with
as a design tool since the model para- the basis of a simplified, but nonetheless
meters can only be specified after a com- very useful, model for studying bladed
plete analysis has been made o 5 t h e blade assemblies of all types, (Ewins 1980),
and disk separately. A second group of Ewins and Han (1963), Griffin and Hoosac
models is used which is based on simple (1983). The model is a standard lumped-
beam and plate components. Although, once parameter mass-spring system of the type
again, not directly usable for design pre- introduced in Chapter 13 together with the
dictions, these models are more represent- basis for its analysis. Each blade is
ative of the actual assemblies than are modelled by just one or two degrees of
the first type and yet are still rela- freedom although later applications have
tively inexpensive to analyze. Both taken the concept rather further, Jones
models are suitable for studies of and MuszynshkR(1983), and the disc or
mistuned, as well as tuned, assemblies. shroud by equally simple spring and nass
The third and final type of model is that elements. Figure 1 shows the basic model
which contains few or no simplifications plus a number of variants which have
and is usually based on finite element evolved from it. It should be noted that
techniques. These are the models used for the model can he readily extended to in-
direct design predictions but are suf- clude damping elements as well, and some
ficiently expensive in computation time to applications have made use of this facil-
be inappropriate for exploratory studies ity albeit only in a rather qualitative
and are almost always limited to tuned way.
and/or cyclically symmetric assemblies.
All three types of models will be used in The model of this type is character-
this chapter. ized by a mass and stiffness matrix pair
such as the example which follows for
model (c) in Figure 1.

The studies reported here will con-

centrate on natural frequencies and mode
shapes but some consideration will be mi O
given to forced vibration response charac- IM] = 0 HH
teristics, it must be noted, however,
that these latter studies will be confined
to undamped or mechanically-damped systems
and will be based on assumed, nominal,
excitation forces. No attempt will be
made at this stage to analyze the actual
damping and excitation effects of the
\
working fluid. [KI = . . 0 0 0 -k< 0 0

Throughout the chapter, our main aim The main limitations of this model
is to demonstrate and to illustrate the are (1) its restriction to a range of fre-
essential structural vibration properties quencies around the blade mode(s) repre-
of various bladed assemblies. The calcu- sented, and (2) the difficulty of esta-
lation of their specific values in a prac- blishing suitable mass and stiffness
tical case is largely an exercise in values for a given design. The latter
numerical analysis and it is probably as, process is possible if the actual proper-
if not more, important to know what to ties of an individual blade and the disk
calculate and what to expect from the com- are known, Afolabi (1982), so that the
puted results as it is to know which cod- model can be used for parameter studies,
ing to use and how to use it. if not as a prediction tool.
 15-3

Once derived, analysis of the model where the six elements In the displacement
is straightforward, involving only the vector represent motion in the three
classical methods outlined in Chapter 13. translational and three rotational
In numerical solution for large assem- directions. 8 represents the position
blies, advantage can be taken of the around the disk and n the sinusoidal order
banded nature of the stiffness matrix and, of the assumed vibration pattern (and,
indeed, some studies have shown how a re- also, the number of "nodal diameters").
arrangement of the sequence of the coor- If we define a vector {Fn}d for the
dinates from that shown above can compact corresponding harmonic forces at the disk
the stiffness matrix even further to gain rim, and vectors {Xn}br, {Fn*br' (Xn}bt,
additional computational efficiency. The <Fn>bt' <Xn}g, tFn^s for the blade root,
model can be used just as easily for mis- blade tip and shroud respectively, then we
tuned assemblies as for the perfectly can relate all these parameters by apply-
tuned version and is capable of providing ing equilibrium and compatibility at the
natural frequencies, mode shapes and disk/blade and blade/shroud junctions.
forced response characteristics. The ad- Thus:
dition of damping terms to the model gen-
erally expands the computation effort by a <*n>d 5 {X }
n br ' <Xn}bt °- «n>s (2)
large amount and in view of the uncertain-
ty of their magnitude and distribution, and
such an extension is seldom made. It
should be noted, however, that it is in
the modelling - the definition of the (3)
appropriate parameter values - and not in
the analysis of that model'that the diffi-
culty lies. In addition, because of the assumed
harmonic vibration, the displacements and
forces for each component are related by
Beam and Plate Models its frequency response functions, thus:
A number of models based on beam and {Xn}d {Fn}d
plate components were developed prior to
the widespread availability of finite ele- where [an(u>] is the (receptance) frequen-
ment methods. Although essentially re- cy response function matrix for the disk
stricted- to uniform - or simple geometry - when vibrating in a cos n6 (or n nodal di-
components, these models are more repre- ameters) pattern. Similar expressions
sentative than those of the lumped para- apply for the shroud:
meter type but are still relatively inex-
pensive to use. One of the earliest lF n ) (5)
models of this type was devised for an
axisymmetric unshrouded bladed disk using and the blade:
the receptance (or frequency response)
coupling method, Armstrong (1955), and sub-
sequent developments have extended this f [B]rr I8lrt < p n>b,
type of model to shrouded assemblies,
Ewins and Cottney (1975), and to mistuned (6)
and packeted configurations, Ewins (1973),
Ewins and Imregun (1983). There are now <*n>bt [p] tr <F n >bt
two derivatives - one for cyclically sym-
metric configurations and the other for
the more general cases where blade- Combining (2) to (6) leads toi
to-blade variations exist. Figure 2 shows
the essential features of each type of (7)
model from which it can be seen that a de-
gree of non-uniformity (in component UBl
geometry) can be accommodated although not
to a great extent. Much the same comments
concerning derivation of the model para-
meters in a particular case apply to this (Ultr
type of model as well as to the previous
one although its ability to represent the
behaviour of actual turbomachine stages from which the determinantal equation
has been demonstrated, Ewins and Cottney
(1975). Uel rr
The method of analysis is similar in det (8)
the two types of model. Considering first
the fully symmetric or tuned case (all <lf» t t
blades identical), we can exploit the cir-
cular nature of the structure and the provides a method for finding the natural
known properties of the connecting disk frequencies and - by back substitution in-
(or shroud) and assume that the variation to the earlier equation - the mode shapes
in displacement around the assembly is es- of the bladed assembly. (All the elements
sentially sinusoidal. (The validity of in the matrices [anl, [01, and [in] are
this assumption will be borne out by later frequency-dependent, and the derivation of
results but its use may be likened to that their individual elements are detailed in
of assuming simple harmonic motion (in Ewins and Cottney (1975) and Ewins and
time)). He shall thus assume that the Imregun (1983).
motion at any point on the disk rim may be
expressed as Analysis of the other, more general
model follows similar lines except that it
(x(t)} cos ne cos ut (1) does not presuppose the coshe displace-
15-4

\
\
h h-
J
\
A A A f< , A A B

\
VA/V MAyv 1

VWH
h
-VW^ -^AAAA B.

AAAAr
X
-VWNA
h
Figure 1. Simple Lumped Parameter Models of Bladed Assemblies
(a) unshroudedf (b) tip-shroud; (c) mid-height shroud

(a)
(b)

Figure 2. Beam and Plate Models for Bladed Disks

(a) eyelie symme t ry; (b) qe ne ra1
 15-5

ment variation around the disk. If we let Finite Element Models

txi^d anc* ^Fi ^d ke tne harmonic displace-
ment and force vectors at position i One of the first published finite
around the disk, then we can write element models for a bladed assembly ap-
peared in 1971 Kirkhope and Wilson (1976),
and this has been followed by several
others extending the capability to more
}d and more complex designs. Undoubtedly,
there are many other studies and methods
(F)d (9) developed which remain unpublished. How-
ever, as the basic blade and disk are
modelled with increasing accuracy, the
scale of model required to represent all
the blades of an assembly individually re-
mains prohibitive and so, here in particu-
{xN)d lar, cyclic symmetry and substructure
methods must be exploited to the full.
where N is the number of blades. Thus, the simplest type of model is one
which assumes an axisymmetric structure,
and this is achieved by assigning negli-
(Note that the disk receptance matrix used gible circumferential stiffness to the
here [a(u)l is different to, and more gen- blading but otherwise taking uniform
eral than, that above for a specific modal inertial and elastic properties throughout
diameter pattern, [on(u)]. A typical sub- the assembly. This type of model is more
matrix of [a(u)j, such as [aij] relates appropriate for assemblies with many
the displacements at position i to a set blades (say, above 30) but becomes less
of forces applied at position j.) Using valid as the blade number is reduced.
the simple case of an unshrouded bladed
disk, we derive a single set of The next level of modelling is to
equations: construct a model of a single blade and
its corresponding disk (and shroud) sec-
tion. This is then considered as the
I 012' basic element in a cyclically repetitive
structure which may be analyzed as was the
<F 2 > {0} cyclically symmetric beam and plate model,
above.
[oN1][oN2] <FN>J (10)
Lastly, a more extensive sector model
can be developed, embracing several
blades, and again this can be treated as a
which can be used to derive a determinant- basic substructure, several of which are
al frequency equation, as before. The connected to form the whole assembly.
essential difference between this equa-
tion and the previous one, (7), is that in
the later version, no prior assumption is
made about the mode shapes and, also, BASIC STRUCTURAL DYNAMIC PROPERTIES -
every blade is individually represented so NATURAL FREQUENCIES, MODE SHAPES
that the resulting matrix is N times
larger than for the earlier case. Details All bladed assemblies with circumfer-
for derivation of the individual beam and ential symmetry exhibit certain well-
plate receptances may be found in Ewins, defined types of vibration mode and these
(1973) although a more general formu- can be illustrated using any of the
lation using the modal properties of the previously-discussed models. The most im-
individual components (if known) is out- portant feature is the existence of two
lined in Ewins and Imrequn (1983). types of mode - 'single' and 'double' - of
which the latter type represents the
majority. Most of a bladed assembly's
It should be noted that throughout vibration modes occur in pairs - double
the preceding analysis it has been assumed modes: these being two modes with the
there is complete coupling at the disk/ same natural frequency and similar mode
blade and blade/shroud junctions. This shapes. As with other structures where
may not be appropriate in all cases; for there are repeated natural frequencies, no
example, where inter-shroud slipping is unique mode shapes can be specified for
possible or where pinned blade roots are these modes. Rather, it is sufficient to
used. In such cases, the coordinates specify two suitably orthogonal shapes and
which are not coupled are simply omitted to note that when vibrating freely at that
from the coupling analysis. natural frequency the structure can assume
any form given by a linear combination of
the two specified shapes.
As noted, the analysis of this type
of model is based on frequency response In the present case, we find that for
methods (see Chapter 13) and, consequent- all the 'double' modes, suitable shapes
ly, the direct 'output' is itself a re- are provided by cos(ne) and sin(nB) cir-
sponse function. While this can be cumferential distributions of displacement
further analyzed to yield the natural fre- around the assembly. This means that at
quencies and mode shapes, because of its the corresponding natural frequency (say,
format it is sometimes difficult to ensure un), the assembly can vibrate in any com-
the identification of all the assembly's bination of these two patterns - i.e.,in a
modes. This feature is seldom a problem shape of the form cos(n6+$). What this
when analyzing the fully symmetric assem- means is that the natural frequency (or
bly but can present difficulties in the pair of identical natural frequencies) can
other, more general, type. be associated with a mode whose shape is
15-6

characterized by n "nodal diameters" all parts of the disk and the blades, in-
since the displacement is constrained to cluding not only the circumferential vari-
be zero along n equally-spaced diametral ation (as before) but also a radial dis-
lines, whatever the displacement 'shape' placement shape. When plotted as a dis-
is in a radial section, although the placement variation with radius, it is
orientation of these diameters will depend found that each family of modes is itself
on some additional external influence. associated with a number of nodal points
along the radial line - much as is found
for the vibration modes of a single beam -
Calculation of a typical assembly and these represent nodal 'circles' on the
using the simple lumped parameter model bladed assembly, in addition to the nodal
demonstrates this pattern clearly, as diameters already established by the cir-
shown in Figure 3 where the double modes cumferential variation. The nodal dia-
are identified. It is also clear from meter pattern used to identify each
this example that the assembly possesses a natural frequency on the plot is obtained
smaller number of 'single1 modes - each by analyzing the relative displacement of
with a single natural frequency and a each blade tip (or other selected refer-
unique mode shape - and that these fit in- ence location along the blade).
to the pattern set by the larger number of
double modes. The single modes correspond
to motion with all the blades having the It should also be noted from these
same amplitude of motion, either in phase results that each family of modes extends
with each other (0 nodal diameters) or out only up to a maximum diametral order of
of phase with their neighbours (N/2 nodal N/2 (or of (N-D/2, if N is odd). This is
diameters - only possible if N is even). to be expected with the discrete lumped
parameter model since higher diametral
orders simply could not exist but with the
In addition to the sketches of mode continuous disk (or shroud) model, vibra-
shapes which show the relative displace- tion of these components with higher nodal
ments of the various blades. Figure 3(b), diameter patterns is possible. However,
a second diagram is given in which each if we choose to describe the circum-
entry is obtained from a (discrete) ferential variation in amplitude by
Fourier analysis of the corresponding noting the relative displacement of some
column from the first set of results. In reference point of each blade, then we are
this format, the diametral order of each unable to resolve higher diametral orders
of the modes is clearly seen and readily than N/2 (or N-l/2 if N is odd). A dis-
facilitates the graphical presentation of placement shape in the disk of cos(nO)
the results shown in Figure 3(b). This (where n > N/2) which is described in
form of presentation, and where appro- terms of the rim or blade displacement at
priate, its tabular counterpart, will be N equal points around the disk will appear
used throughout the rest of this chapter. just as a mode shape in the form of
cos (N-n)6. In other words, n nodal dia-
meters and (N-n) nodal diameters are in-
From the example in Figure 3 and also distinguishable to a set of N blades - see
from a second one in Figure 4 for a more Figure 7. This phenomenon has implica-
complex model, another important charac- tions for a number of bladed assemblies,
teristic may be observed. As the number including all shrouded ones (where n nodal
of nodal diameters (n) increases, the diameters in the disk and (N-n) diameters
natural frequencies in each 'family' in the shroud will be directly compatible
approach asymptotically one of the 'blade since the two components are connected
cantilever frequencies' - the natural fre- only through the N discrete blades), and
quencies of an isolated blade with the those with small numbers of blades (where
disk attachment point (or root) grounded. n and (N-n) are both relatively low dia-
This behaviour is caused by the progres- metral orders).
sive stiffening of the disk as it adopts a
more complex shape, and it demonstrates
the relevance of performing an individual At the lower end of each family, the
blade analysis, as in Chapter 14, even 0-and 1-nodal diameter modes should also
when that blade is known to be part of a be noted for their special characteris-
coupled assembly. When the coupling is tics. These three modes (one single and
small, such as will apply with a very one double) differ from all the others
stiff disk, the same family characteris- with two or more nodal diameters because
tics are observed although, in this case, they involve a net motion of the center of
almost all the natural frequencies are the disk. Movement with a 0-nodal diameter
very close to the blade-alone values - as pattern involves axial or torsional disk
would be expected - Figure 5. movement while 1-nodal diameter displace-
ments indicate rocking of the disk about a
diameter or translation along a diameter.
Having established the main charac- All other diametral orders are 'balanced'
teristics using the very simple lumped and involve no motion of the disk center.
parameter model, we can confirm this be- The main practical implication of this
haviour using one of the beam and plate characteristic is that the 0-and 1-
models. Taking first the more general diameter modes will be influenced by the
model, a series of calculations are shown shaft and bearings which support the disk
in Figure 6 for a simple unshrouded 30- upon which the blades are mounted, while
bladed disk. From these results,we see the none of the other modes will be influenced
same general pattern - families of modes at all by these other parts of the system.
approaching the individual blade canti- This, in turn, means that the model should
lever frequencies - but now we observe be extended to include these components if
many more modes and, indeed, the full mode accurate estimates of the 0-and 1-diameter
shape for each one of these is more com- modes are to be determined. However, this
plex as it describes the displacement of extra complication is not often included.
 15-7

2OOO-

N
I

1000-
or
4>
t

6 8 10 12
Nodal Diameters

(b)
10 2O 0 10
Blade No. Fourier Component

IL I III 11 I I
r IP

II

Figure 3. Natural Frequencies and Mode Shapes for Bladed Disk

(Lumped Parameter Model of 24-Bladed Disk)
(a) natural frequencies; (b) typical mode shapes
15-8

Figure 4. Natural Frequencies and Mode Shapes for Bladed Disk

(Lumped Parameter Model of 33-Bladed Disk)

500

•400

300

200

100

8 TO 12
Nodal Diameters

Figure 5. Vibration Properties of 24-Bladed Disk: Stiff Disk

 15-9

o. oa
3 15°
30°
45a
90*
UNBLADED
DISC

5 10 15
No. Nodal Diameters (n)

Figure 6. Vibration Properties of 30-Bladed Disk

(Beam/Plate Model - Various Stagger Angles)

4 Nodal Diameter Shape

Blade
Displacement,

(a)
N=36Blades
n=4
CN-n)=32
32 Nodal Diameter Shape

(b) ,

360* 20° 40° 60° 80°90' Angle

iiit—,—i—i t i
36 8 Blade No.

Figure 7. Discrete Descriptions of Diametral Patterns

15-10

With the bladed assembly vibration Unshrouded Bladed Disk

modes clearly identified, the rationale
for the cyclic symmetry method of analysis (a) Datum Case
is not substantiated and we can use the
much simpler version of the beam and plate The datum system properties are shown
model (and those based on finite element in Figure 10 in the form of a natural fre-
models using the same approach) with some quency vs nodal diameters plot, and it is
confidence. This is illustrated in Figure against these results that the subsequent
8 for the 30-bladed example used above, cases will be compared.
first in the unshrouded configuration
(8(a)> and the second after the addition The properties illustrate clearly the
of a tip shroud, 8(b). The latter example influence of both the individual (canti-
serves to illustrate the way in which lever) blade properties and those of the
shrouding alters the (otherwise) predict- disk alone and, indeed, indicate how these
able pattern of modes and natural fre- separate component data might be used to
quencies and will be developed further in estimate the full assembly properties.
the parameter studies and practical
examples included in the next two (b) Disk Stiffness
sections.
The first influence studied is that
of the disk stiffness: keeping all other
Lastly, mention should be made of the dimensions unchanged, the disk thickness
various bladed assemblies not yet intro- is halved and doubled, resulting in the
duced here - including the cases with non- plots shown in Figure 11. These results
identical blades ('mistuned' assemblies) are understandable and show how the assem-
and those with deliberately non-symmetric bly modes tend towards those of the com-
configurations, such as blade packets. ponents as the disk stiffness increases
These will be discussed later, after a (so that the interblade coupling dimin-
thorough review of symmetric, or tuned, ishes), but how the full assembly analysis
assemblies has been completed. becomes more important as the disk becomes
more flexible.
(c) Root Flexibility
MODAL PROPERTIES FOR VARIOUS BLADED
ASSEMBLIES A related parameter - although quite
a separate effect - is root flexibility.
Introduction This is introduced to take account of the
loss of rigidity which occurs at the
In this section, we shall present a blade/disk junction because of the root
series of parameter studies to illustrate fixing used. Unlike disk flexibility, the
the major modal properties of a range of root effect does not influence the coup-
bladed disk assemblies. No attempt is ling but, in effect, changes the blade
made to encompass the very wide range of stiffness with a consequent influence on
configurations encountered in practical the assembly properties. It is noted from
machines but a selection of parameters is Figure 12 that once the flexibility
made in an attempt to highlight some of exceeds a certain level, its influence is
the more significant influences on the marked but below that threshold, its mag-
natural frequencies of a bladed assembly. nitude is less important.
(d) Blade Stagger
The characteristics presented are
based on a single reference or datum case In the datum case, the blades are
- a simplified 36-bladed testpiece. staggered at 30* and this has the result
However, it is not the properties of this that both flapwise and edgewise blade
which are the primary interest here: motions are coupled through the flexural
rather, we shall seek to illustrate how vibration of the disc. Figure 13 shows a
they are influenced by changes in the series of cases where the blade stagger is
system dimensions or configurations. increased from 0° to 90° and illustrates
the progressively greater influence ex-
erted by the disk on the flapwise blade
The system is studied in two basic motion.
configurations
(e) Blade Twist
- (a) unshrouded, and
- (b) shrouded A further series of calculations is
shown in Figure 14 for which the outer 30%
and within these groups, the influence of of the blade is twisted (staggered) rela-
various design features is explored, tive to the inner section. Here, it is
including clear how the higher families of modes are
significantly influenced while the lower
modes are relatively unaffected.
disk stiffness
blade root flexibility
blade stagger
blade twist Shrouded Bladed Disks
shroud stiffness
shroud connection (a) Datum Case
shroud position.
Here, the datum case is provided by
adding a "standard" shroud ring around the
Details of the datum case are given in blade tips of the 36-bladed disk used for
Figure 9. the previous studies. The results for the
 15-11

(a)

XX>0

500
./ . MEASURED
COMPUTED

100
2 5 10
NODAL DIAMETERS

5000

NATURAL FREQUENCIES OF
3O-BLADED DISC (SHROUDED)
too
NODAL DIAMETERS

Figure 8. Vibration Properties of 30-Bladed Disk

(Cyclic Symmetry Model)
(a) unshrouded; (b) shrouded
15-12

DISK radius 0.050 m (inner) NXTKRZJU, PROPERTIES

0.560 m (outer) (All components)
thickness 0.060 m

BLADE length 0.250 m Modulus - 207 GN/m2

width 0.030 m Density - 7850 kg/m3
thickness 0.010 m Poissons
stagger 30° (from axis) Ratio - 0.30
number 36
SHROUD thickness 0.010 m (radial)
depth 0.006 m (axial)

Figure 9. Details of Datum Case Bladed Disk Model

2000

1000

NODAL DIAMETERS

Figure 10. Natural Frequencies of Datum Case Unshrouded Bladed Disk

 15-13

2000

1000

NODAL DIAMETERS

Figure 11. Effect of Disk S t i f f n e s s on N a t u r a l Frequencies

disk thickness: (o) - 0.03m; (•) * 0 . 0 6 m j (i) - 0 , 0 9 m

2000

1000

NODAL DIAMETERS
F i g u r e 12. Effect of Root F l e x i b i l i t y on N a t u r a l Frequencies
(a) = 10~ 5 ; U) = 10"*; (o) * 10~ 3 ; (•) - 0 Nm/rad
15-14

NODAL DIAMETERS

F i g u r e 13. Effect of Blade Staaaer on Natural Frequencies

(o) - 0°; (•) - 30°; (V) - 45°; (O) - 60°; ( O ) - 90°

0 K) 18
NODAL DIAMETERS
Fiqure 14. Effect of Blade Twist- on Natural Frequencies
tip twist: (V)—15°; {•>- 0°; (o)-15°; (A)-30°; (0) -45°
 15-15

same frequency range as used before are results for other positions are shown
shown in Figure 15(a) and immediately show alongside those of the datum case.
rather more complex patterns of behaviour
than in any of the previous cases. The DYNAMIC ANALYSIS FOR PRACTICAL ASSEMBLIES
main difference is that the 1 natural fre- (contributed by R. Henry)
quencies in any one 'family no longer
necessarily increase steadily with the Dynamic Analysis of Low Pressure Turbine
number of nodal diametersi sometimes they Stage
do, following previous experience, but
sometimes not. The reason is that each Having explored the characteristics
mode labelled as "n nodal diameters" may of bladed assemblies in general, we shall
indeed have n diameters, but could also now present an example of a specific prac-
(or instead) have (N-n) diameters, or tical assembly analyzed by a finite ele-
(N+n) etc. (N being the number of blades). ment model, using the axisymmetric method
In all modes involving radial motion of outlined in Chapter 13 applied to a rotat-
the shroud, the lowest order pattern - n ing low pressure turbine stage of a modern
nodal diameters - is suppressed by the jet engine, Ferraris et al. (1973).
very high radial stiffness of the blades
and, as a result, it is the second order, Finite Element Modelling
(N-n), which dominates. When the shroud is
effectively dictating the coupling, as in Figure 20 shows the finite element
the case when the disk is relatively mesh used to calculate the natural fre-
stiff, then an increasing "n" involves a quencies and mode shapes of the above-
decreasing controlling shape, (N-n), hence mentioned turbine stage. The disk is
the falling natural frequencies seen in modelled mainly with thin shell axisym-
this plot. This conclusion is reinforced metric elements (nine elements for the
in Figure 15(b) where the same assembly is flanges and ten elements for the web),
analyzed admitting only the lowest order connected to the disk rim using junction
shroud modes to the model. This result elements (I to VI). The rim itself is
follows the same pattern as the earlier modelled with eight thick isoparametric
calculations - as expected - but is clear- elements. The blade is modelled with 16
ly an inadequate representation of the twisted, beam elements (see Chapter 14) and
actual system. continuity of displacements and slopes at
the blade root is ensured using two junc-
(b) Disk Stiffness tion elements (VII, VIII) placed on the
disk rim. The shroud element at the blade
A series of calculations is included tip is modelled by additional masses and
in Figure 16 where the disk is first made inertias in the three directions. The
much more flexible than the datum case, by disk is clamped as shown in Figure 20.
halving its thickness, and then much stif- The blade tip is free in the X and 2 di-
fer (double thickness). These results, rections, supported in Y and restrained in
though complex, are comprehensible. torsion. The whole model results in 288
Indeed, the latter case presents an indi- effective degrees of freedom.
cation of the typical characteristics for
an assembly of blades where virtually all Dynamic Analysis
the inter-blade coupling is through the
shroud band. The first six natural frequencies and
mode shapes are computed for the structure
(c) Shroud Stiffness vibrating with n = 0 to 5 nodal diameters
and at various speeds of rotation (1 . The
In the same vein, a series of calcu- results are presented using a dimension-
lations is shown in Figure 17 for two less frequency parameter fr(=f/fOl'» fQl
variants where the shroud is first less being the fundamental natural frorju'ncy
stiff and then more stiff than the refer- for 0 nodal diameters) and a dimension-
ence case, this effect being achieved by less speed Or = 0/fQl • The natural fre-
halving and doubling the radial thickness quencies related to n and 0 are presented
of the shroud band. in Figure 21(a) and (b) and the associated
mode shapes are arranged in families f^i,
f
(d) Shroud Attachments nli» •••• fnVl and presented in Figure
21(c) to <g).
In many assemblies, the connection
between blades and shroud is not fully in- Family I, Figure 21(c), is mainly a
tegral. Sometimes, there is a local flex- disk-rim vibration mode while the blades
ibility, similar to the root flexibility, move in the XZ plane following the rim
which relaxes the coupling provided by motion. It is seen that the natural fre-
this attachment and sometimes the connec- quencies of family I approach the first
tions are deliberately made only in cer- bending frequency of the clamped blade for
tain coordinates, or directions. In increasing values of nodal diameters but
Figure 18, we see the result of making the that the rotation effect results in a 30
blade/shroud attachment rigid in only 1, to 40% increase of the natural frequencies
or 2, or 3 of the 6 coordinates previously (Figures 21(a) and (b)).
included in the datum case. It is clear
how certain modes are greatly affected by Family II, Figure 21(d), shows a disk
these conditions while others are barely bending mode. Motion of the rim is
influenced at all. limited and the blade bends in the XZ
plane. It can be seen in Figure 21(d)
(e) Shroud Position that the rotation effect is significant on
the various n diameter modes as the asso-
Lastly, we show some typical effects ciated natural frequencies increase by
resulting from changing the radial loca- more than 20% in the range of rotation
tion of the shroud (previously, it was speeds studied (from rest to operating
at 100% blade length). In Figure 19, speed).
15-16

2000

1000

NODAL DIAMETERS

Figure 15. Natural Frequencies for Datum Case Shrouded Bladed Disk
(a) (•) full analysis
(b) (o) analysis omitting higher order terms

18
NODAL DIAMETERS

Figure 16. Effect of Disk Stiffness on Natural Frequencies

disk thicknessi (•) = 0.06m; (o) = 1.00m
 15-17

2000

1000

NODAL DIAMETERS

Figure 17. Effect of Shroud Stiffness on Natural Frequencies

shroud thickness: (o) •= 0.006m; (•) = 0.012m; (A) •= 0.024m

2000

1000

NODAL DIAMETERS

Figure 18. Influence of Blade/Shroud Connection on Natural Frequencies

tip coupling: (•) = 6 coords; (o) • 2 coords; (A) * 1 coord.
15-18

2000

I1OOO

NODAL DIAMETERS

Figure 19. Effect of Shroud Position on Natural Frequencies

shroud height: (•) = 100%; (o) «= 90%; (A) = 60% blade length

twisted
beam «t«mtnt* btad*

dtic rim

flanges

diic

Figure 20. Finite Element Model for Low Pressure Turbine Stage
 15-19

fr o flr=0.0
o Dr= 0.35
Qr=0.6 t«v

f
nVl

nlV

0 1 2 3 4 5 n

Figure 21a. Vibration Properties of a Low Pressure Turbine Stage

0 O.t 0.2 0.3 0.4 0.5 0.6 Qr

Figure 21b. Vibration Properties of a Low Pressure Turbine Stage
15-20

Figure 21c. Vibration Properties of a Low Pressure Stage

W)

Figure 21d. Vibration Properties of a Low Pressure Stage

 15-21
Family III, Figure 21(e), is mainly a VIBRATION PROPERTIES OF MISTUNED AND
blade vibration mode. Here again, the PACKETED BLADED ASSEMBLIES
natural frequencies tend to the first flap
bending frequencies of the cantilevered
blade as the number of nodal diameters Mistuned Assemblies
increases. The axisymmetric part of the
stage (disk plus rim) undergoes small am- (a) General
plitudes of vibration, except for n = 2
and n - 3 which show large motion for the All the examples presented so far
disk-rim and the disk respectively. The have related to 'tuned1 or symmetric
rotation effect results in a frequency assemblies where all the blades on one
increase of up to 8%. stage are identical to each other. In
practice, such uniformity is seldom
attained, either by chance or by design,
and so it is appropriate to consider next
Family IV mode shapes show mostly a the implications of a loss of complete
torsional motion of the blade with very cyclic symmetry for the vibration charac-
little motion of the disk. As n in- teristics, we shall first examine 'mis-
creases, the natural frequencies tend tuned1 assemblies, where small blade-
towards that of the first torsion mode of to-blade variations are admitted to the
the cantilevered blade. For this group of model, representing either random manu-
modes, it can be seen in Figure 21(d) that facturing variations or deliberate selec-
the rotation speed has a negligible tion of 'light1 or 'heavy1 blades as is
effect, never exceeding 1.5% for any value sometimes practiced during assembly.
of n. Later, we shall consider other, more
extreme, departures from symmetry.
Family V, presented in Figure 21(f), Using any of the assembly models
shows a disk-rim vibration with strongly which permit the individual representation
coupled second bending-torsion motion of of each blade (i.e., those models which do
the blade and, as n increases, a natural not presuppose cyclic symmetry), we find
frequency tending to the second torsion that a small amount of mistuning (i.e.,
mode of the cantilevered blade. Except blade variations of typically a few per-
for n - 2, for which the frequency in- cent) has a relatively minor effect on the
creases by about 3.5%, the rotation assembly natural frequencies but can in-
effects are relatively small. troduce potentially significant changes to
the mode shapes. This result is itself
important because of the central role
Family VI, Figure 21(g), is a highly played by the mode shapes in determining
coupled vibration mode with significant the blade response characteristics.
motion of the disk and rim, especially for
n = 1 and 4. The blade vibrates in a (b) Frequency Splitting
coupled bending-torsion mode and the fre-
quency tends to the second bending mode of Many of the essential characteristics
the cantilevered blade. For this family, of mistuned assemblies can be illustrated
the rotation effects result in a rather by the single example given in Figure 22
small increase in the frequencies (2.5% which relates to an unshrouded 24-bladed
for n = 1 to 5 and 5% for n = 0). disk used for a series of carefully-
controlled experiments. We see that
as a result of random mistuning of
Lastly, examination of Figures 21(a) the set of blades, each double mode
and 2Kb) shows clearly that the disk in- degenerates from the pair of identical
fluence is low for the families I, III, IV natural frequencies with pure n-nodal di-
and V. It can also be noted that the ameter mode shapes and 'splits' into two
families I, II .and, up to a point. III are distinct modes with close natural frequen-
affected by the centrifugal effects. cies and mode shapes which now contain
several diametral components. For those
double modes whose natural frequencies are
well separated from each other in the
tuned state - here, the low order modes
Comments with 2, 3, and 4 nodal diameters - the
mistuned assembly mode shapes retain a
The axisymmetric method is well strong resemblance to the original pure
suited to predicting the vibration be- nodal diameter patterns. For many of these
haviour of any rotating bladed assembly modes, the extra diametral components in-
where the number of blades is high, i.e., troduced by mistuning are relatively small
the actual structure must have an essen- compared with the 'parent' order.
tially axisymmetric behaviour. The re- However, as we progress through the family
sults obtained for this low pressure tur- of modes associated with one of the blade
bine stage example show that an adequate cantilever modes, we see that the contami-
model of such structures must consider all nation of the assembly mode shapes in-
the coupling effects due to the blade and creases to the point where some of them
disk geometry along with those due to ro- can no longer be associated with a partic-
tation. If the number of blades is low ular nodal diameter pattern at all. This
and the blades cannot be modelled by beams situation arises when the tuned system
(as is the case for many fan assemblies), natural frequencies are separated by the
the mode shapes are no longer axisymmetric same order of magnitude as the frequency
(simple diameters and/or circles for nodal split induced by the mistuning. This, in
lines), and the previous hypothesis is no turn, is related to the degree of blade
longer valid. In these cases, one has to mistune which can conveniently (though not
introduce methods using cyclic symmetry uniquely) be specified in terms of the in-
for rotationally periodic structures. dividual cantilever frequencies of the
15-22

Figure 21e. Vibration Properties of a Low Pressure Stage

n=0 z ,
n -s 1
A QS»^ blade tip
n =/ .^. ..*... 'Nj'"'""1*^
(f) n=3 *--4-
n =4o---o-
XXX
-S:T
"o,. °^*"" "io
n = 5 o —o- Wadetip^
n =4 de root

_ jtf
,"' °**^x0>
«x
^
ifis!^.
L. *-j*4K^3i?:I1=i jfcS. - ^* **"" ' "N ^ X

Figure 21f. Vibration Properties of a Low Pressure Stage

 15-23

(9)

Figure 21g. Vibration Properties of a Low Pressure Stage

6DIAS

0 0.01 O.O2
DEGREE OF DETUNE

Figure 22a. Frequency Splitting Effect Caused by Regular Mistuning

(a) separated modes; (b) close modes
15-24

frequencies of the actual blades used. As (d) Effects of Damping

a rough guide, a scatter of t x% in indi-
vidual blade cantilever frequencies will All the preceding discussion applies
generate frequency splits of up to x% in strictly to undamped structures, although
the various double modes although the very little change would result if
exact amount will depend on the specific proportional damping were introduced (see
distribution of the blades as well. Chapter 13). However, it Is possible that
in this particular case the actual mechan-
Single modes are only slightly ical (and/or aerodynamic) damping may have
affected by mistuning, the variations in a significant influence if, or because, it
natural frequency and mode shape always is distributed in a non-proportional way.
being of the same order as the blade mis- In practice, non-proportional damping will
tune itself. result if the actual source of damping is
localized and is not distributed in the
same way as the mass and/or stiffness of
(c) Patterns of Behaviour the structure; a situation likely to apply
to many bladed assemblies.
Although the vibration modes can al-
ways be computed for a given assembly mis- The significance of the distribution
tune configuration, such calculations are of damping lies in the fact that the modes
expensive and so it is useful to establish of non-proportionally damped systems may
any patterns which may relate specific be complex and, further, that systems with
mistuning arrangements with consequent close natural frequencies are known to be
changes from the tuned system datum pro- more prone to significant degrees of com-
perties. One such pattern is found to plexity in their mode shapes. All the
apply to those modes which exhibit a clear mode shapes illustrated so far have been
frequency splitting behaviour: i.e., real, with each blade vibrating exactly
those modes - usually with few nodal dia- in or out-of-phase with the others and
meters - which have well-separated natural thus reaching its maximum value at the
frequencies. It is found that in order same time as all the others. In a complex
for such an n-nodal diameter mode to split mode, adjacent blades can have any rela-
due to mistuning, the variation in indi- tive phase difference (and not just 0° or
vidual blade frequencies (the mistune 180°) and this can have a very marked
arrangement) must contain a component of effect on the aerodynamics.
order 2n. In order words, if the individ-
ual blade frequencies are given by
Blade Packets
The next assembly to be considered is
where that of a packet of blades, often a group
of up to 7 or 8 cantilevered blades con-
fo is the mean blade frequency, nected by a shroud band. Although analy-
fi is its individual cantilever sis of this configuration usually ignores
frequency, and the disk coupling (some examples do admit
61 is the position of blade i, disk or root flexibility, but not inter-
blade coupling), it is justified in the
then all modes with n nodal diameters will same way as was our earlier study of an
split and the extent of the frequency individual blade; the vibration properties
split will be closely related to the mag- for a complete (packeted) bladed disk are
nitude of xjn« Modes with other nodal related to those of the single canti-
diameter patterns will generally not be levered packet by these latter providing
split. This pattern ceases to apply when, asymptotic values and classification of
as mentioned previously, the natural fre- the various families of modes of the full
quencies of the (n-1)-, (n)-, and (n+1)- bladed system.
diameter modes of the tuned system are
already close together. Some examples of The basic characteristics can be
this effect as applied to a 30-bladed disk illustrated using a simple lumped para-
are shown in Figure 23 where both regimes meter model, such as that shown in Figure
are illustrated. 24(a). The natural frequencies and mode
shapes for a six-bladed packet of this
Another pattern which is observed type are shown in Figure 24(b) from which
with regular mistuning concerns the mode a general trend is seen: one fundamental
shapes. In all cases of mistuning, these mode with all blades moving together which
shapes are distorted from their regular is unaffected by the 'shroud' stiffness
cos(nd) form but in the case of cosine (or followed by a group of five modes whose
other periodic) mistuning, the distortion natural frequencies are grouped together
itself is also regular, and follows a and which are largely controlled by the
standard pattern. If the cosine mistuning shroud stiffness. Figure 24(c) shows a
is of order k, then the modes originally series of results based on the same very
identified by n nodal diameters will be simple model incorporating two, three, and
found to contain some component of (n+k), up to six blades in a single packet. The
(n+2k) . . . (n-k), (n-2k) . . . etc. trend of one fundamental in-phase mode
diameters. plus a group of (N-1) modes is clearly
seen throughout and it is also noted that
The significance of the various the basic two-blade packet yields the
diametral components present in each mode essentials of the vibration properties of
shape will be fully appreciated in the the other configurations with more
later discussion on forced vibration blades.
-each mode is susceptible to resonance
whose severity is determined by these The next level of model is one which
components. represents the blades and the shroud
 15-25

6.45

6.40

6.35

6.30

6.25

0.01 O02
DEGREE OF DETUC ^

Figure 22b. Frequency Splitting Effect Caused by Regular Histuning

(a) separated modes; (b) close modes

MISTUNED

f ¥
Figure 23. vibration Properties of a Mistuned 24-Bladed Disk
15-26

(a)
hx° h
|—\AAr- —vW

5.033 82.538 159.235 225.136 275.711 307.505

I I 1 1 f~l Pr-y. bv> A K A~^ k >T-K A A^

5.033 98.491 187.159 257.577 302.724

''ii [^ k A K ZtL > A x
|/-
5.033 121.196 225.136 294.123

s v
5.033 159.23O 275.719
m
5.O33 225.136

D
\

M = 1.O Kg
K=10'N/m
k«1C/N/m

Figure 24. Vibration Properties of a Simple 6-Blade Group

(a) model
(b) natural frequencies and their mode shapes
(c) various group sizes
 15-27

elements as beams, as shown in Figure One way of viewing a packeted assem-

25(a). Results from the analysis of a bly is as a special type of mistuned sys-
specific example of this type are shown in tem, the main differences from the pre-
Figure 25(b) where it is noticed that, in viously discussed cases being in the
fact, there are two separate sets of vi- regularity of the 'mistune' and also in
bration modes - the in-plane and out- its magnitude -likely to be considerably
of-plane modes. In this example, these greater than that due to blade manufactur-
two sets are totally independent of each ing tolerances. Such an approach provides
other but this is a direct result of the some insight into what narticular charac-
zero stagger of the blades. Examining teristics might be expected to apply to
each of the separate results shows that this type of bladed assembly since the
the same trend noticed in the simple 'mistune' applicable in this case has some
mass/spring model characteristics applies similarities with regular blade mistune
here also, with a single in-phase funda- mentioned earlier.
mental mode followed by a group of shroud-
controlled modes in a relatively narrow Another way of viewing regularly
frequency band. In this (more represent- packeted assemblies is to consider the
ative) model, the out-of-phase modes ex- subassembly formed by 1a single packet of
hibit clear and simple shapes by which the blades as a 'superblade so that the com-
various modes are identified -the second plete system consists of a disk with a
mode being a "rocking", the third a "U", (small) number of (identical) superblades.
the fourth "S", and so on, depending upon Since the number of 'blades' in this case
the number of blades included. would generally be small, the modes of the
system would not necessarily have simple
nodal diameter shapes (as is the case for
A study was made using a finite ele- a large number of blades) but would each
ment beam model to examine the signifi- include several prominent diametral com-
cance of blade number on the packet's ponents, although the diametral orders
vibration properties, Thomas and Belek present in each mode would be known a
(1977). As heralded by the very simple priori.
example quoted above, it was found that
the detailed analysis for a two-bladed
packet could form the basis for accurate Case Studies
prediction of the corresponding modes of
other packets containing any number of The calculations for the vibration
similar blades. Some results to illus- properties of a packeted bladed assembly
trate this are shown in Figure 25(c), are just as expensive and extensive as
still based on a set of blades with 0° those already encountered for mistuned
stagger to the plane of the 'disk'. systems. As a result, it is once again of
some importance to establish what patterns
or trends exist in these characteristics
and to seek ways of deducing or estimating
Finally, we consider the properties them without necessarily undertaking a
of a packet in which in-phase and out- 'full' analysis. In this section, we
of-phase motions are coupled by the blades shall present some results from a study
being staggered. In this case, all the made using both of the simple models of a
modes are coupled and contain some dis- bladed disk - the mass-spring representa-
placement in both in-plane and out- tion and the uniform beam and plate model
of-plane directions although, in many - in which the blades are interconnected
cases, each individual mode may be predom- in a variety of different packeting con-
inantly in one or the other directions. figurations. The latter studies are
An example derived from the earlier six- based on a simple 30-bladed disk shown in
bladed packet is shown in Figure 26. Figure 27.
The first series of results shown in
Figure 28 derive from the very simple
mass-spring type of model and include de-
Paeketed Bladed Assemblies tails of the natural frequencies and mode
shapes for a 36-bladed system with a
number of different packeting arrange-
Packeted Assembly Configurations ments. The results are summarized in the
graph of Figure 28 where the principal
Probably the most complicated bladed diametral component has been used to
assembly of all is the packeted bladed classify each mode (since there is gen-
disk in which all the previously-discussed erally more than one order present in the
elements are present - disk flexibility, mode shape). Also shown, on the graph
interblade coupling, and the lack of axis, are the natural frequencies of the
cyclic symmetry. The basic feature is a corresponding single packet of 'blades'
discontinuous shroud, there being gaps at with their roots grounded, so that the
a number of points depending on the spe- full packeted bladed disk frequencies can
cific packeting arrangement used. There be seen in relation to those of the canti-
are various possible configurations of levered blade packet. A pattern is clear-
this format, ranging from a simple case ly seen from these few examples leading us
where blades are coupled together in pairs to the possibility that the full packeted
("N/2 packets of two blades") to the most assembly natural frequencies can be
complex where blades are grouped in effectively deduced from (a) those of a
packets of different numbers spaced irreg- single blade packet together with (b)
ularly around the disk. Perhaps the most those of the continuously shrouded
likely arrangement to be encountered is assembly. Both of these two sets of cal-
one where there exist a number of identi- culations are very much easier to perform
cal packets ("P packets of p blades each") than those of the fully packeted
-referred to as 'regular packeting'. assembly.
15-28

OUT-OF PLANE IN PLANE

Figure 25. V i b r a t i o n Properties of a Blade Packet

(Unstaggered Blades)
( a ) model
( b ) mode shapes
 15-29

Family p-1 p-2 p-3 p-4 p-5 p-6

1A 261.18 241.19 235.34 232.76 231.21 230.48
504.21 365.44 308.95 280.83 290.58
707.92 518.62 417.80 375.04
808.04 639.04 526.43
857.89 723.72
876.32

IT 355.20 331.25 324.43 321.22 319.35 325.87

1494.11 1490.03 1470.67 1438.52 1403.76
1490.70 1487.33 1485.34 1480.34
1496.97 1493.96 1482.56
1497.81 1491.86
1492.52

2A 1618.95 1504.78 1468.23 1454.78 1454.19 1451.55

1628.08 1549.48 1502.28 1477.43 1489.32
1683.70 1586.36 1534.60 1535.88
1728.97 1619.48 1571.21
1766.04 1649.94
1818.31

2T 2181.24 1798.31 1758.64 1780.03 1777.26 1761.65

2050.23 1809.96 1798.37 1790.66 1766.02
2017.31 1814.51 1805.69 1798.30
2002.58 1816.69 1808.26
1994.26 1814.32
2038.49

3A 3506.94 3397.82 3335.34 3305.90 3288.75 3287.05

4380.69 3920.34 3760.86 3629.16 3575.05
4373.06 4204.97 3953.30 3869.36
4363.18 4045.29 4008.06
4356.75 4053.66
4344.77

Variation of cantilevered packet natural frequencies (Hz) with

number of blades in the packet (p)
A: Axial or out-of-plane
T: Tangential or in-plane (C)

Figure 25. Vibration Properties of a Blade Packet

(Unstaggered Blades)
(c) natural frequencies for various packet sizes
15-30

Figure 26. Mode Shapes for a Blade Packet of Staggered Blades

Figure 27. Packeted Bladed Disk Testpiece

 15-31

A second example is shown using the situation, each point on the rotating sys-
more representative beam and plate type of tem experiences the variations in steady
model, again referred to a 30-bladed disk. axial pressure or force as time-varying,
Using the general model of this type and thus responds by vibrating at a fre-
(i.e., one which does not presuppose quency or frequencies directly related to
cyclic symmetry), we can analyze any de- the speed of rotation. Such an excitation
sired packeting configuration and two not only has a characteristic frequency
specific cases are illustrated in the (an integer multiple of the rotation
results shown in Figure 29. Using the speed) but also has a characteristic shape
same form of presentation as before, we since it is applied simultaneously to all
show three sets of natural frequencies on points around the bladed assembly.
the same plot! (a) those for a single
packet of blades with roots cantilevered;
(b) those for the bladed disk when contin- In each case, it is necessary to in-
uously shrouded, and (c) those for the clude some form of damping and that which
packeted bladed disk. As before, with the is generally assumed is proportional,
simpler model, there is strong relation- thereby permitting most of the computa-
ship between the first two of these sets tional effort to be made on the basic un-
of results and the third, although in this damped system, introducing the damping
case they represent more of a trend than only at the very last stage. However, it
an exact pattern. Two different packeting should be noted at the outset that this is
arrangements produce the same essential a very crude approximation to the real
result. physical conditions.
It was mentioned earlier that the
principal diametral component of each mode
shape was used to identify that mode on
these plots, whereas in fact each such Review of Forced Response Analysis
shape has several significant components.
Once again, a pattern is observed which
connects the number of blades on the disk The basis for a general forced re-
and the number of blades in a packet with sponse analysis was presented in Chapter
the combinations of diametral orders which 13; explicitly, for the single point har-
appear together in the various modes. For monic excitation condition but also,
example, taking the case cited in Figure implicitly for the more complex situation
29, we find that each of the modes has a of engine-order excitation. A very impor-
shape which contains one of the following tant feature of all types of forced vibra-
sets of diametral components: tion is the double requirement to obtain a
resonance condition: namely, an excita-
(a) 0,5,10 or (b) 1,4,6,9,11,14 or tion at the appropriate frequency and with
(c) 2,3,7,8,12,13 the appropriate shape. The former is
self-evident but the latter condition is
A different set of diametral orders ap- more subtle and, indeed, plays a major
plies for other bladed assembly configura- role in the forced vibrations of interest
tions but the connection with the essent- here. In its simplest form, It is clear
ial parameters of the packeting is self- that a particular mode of vibration will
evident. not be excited into resonance, even at its
natural frequency, by an excitation force
which is applied at a nodal point of that
FORCED VIBRATION RESPONSE mode. This is a very simple example of
the excitation shape being incompatible
Scope of Response Analysis with the mode shape and we shall find
other morecomplex ones apply in this
Although the main objective of this study.
chapter has been to establish the modal
properties of the bladed systems of
interest, we shall include some brief con- We shall consider the implications of
sideration of the forced response behav- this aspect of forced vibration for the
iour as well. As mentioned in Chapter bladed assembly whose mode shapes, as we
13, such a response analysis requires the have already seen, are conveniently de-
introduction of additional information in scribed in terms of the diametral compo-
the form of the definition of a specific nents present in the circumferential dis-
forcing function to be considered plus the tribution of the disk and blades' dis-
inclusion of some damping terms. In this placement. Thus, we shall be looking at
application, both of these features in- or for the existence of similar diametral
volve assumptions or information which patterns in the excitation functions as a
extend beyond the structural dynamics measure of their potential ability to
aspects that are strictly the concern of excite resonant vibrations in the bladed
this chapter. assembly. The frequency of the exciting
force(s) is obviously of direct signifi-
We shall focus our attention in this cance as well, but the shape or distribu-
section on two specific excitation cases tion must not be overlooked.
and one specific assumption regarding the
damping effects. The first excitation
case is that of a single point harmonic Consider first the single point har-
excitation, such as Is used for most vi- monic excitation which will be assumed
bration test/measurement procedures. The (for the purpose of this discussion) to be
second, and major, case will be that of applied on a specific blade. The effec-
•engine-order' excitation: that which tive forcing function, as described in the
exists when a bladed disk rotates past (or governing equations of motion in Chapter
through) a steady flow pattern which is 13, will be a vector which only has one
nonunifonn around the annulus. In such a non-zero element so that
15-32

15
NODAL DIAMETERS

Figure 28. Vibration Properties of Packeted Bladed Disk (Simple Model)

MUMMR
Figure 29. Vibration Properties of Packeted Bladed Disk
(Beam/Plate Model)
(a) sinqle packet
(b) continuously shrouded
(c) packeted
 15-33

The total excitation experienced by

the entire assembly (since the forces are
exerted simultaneously on all blades) will
(11) consist of a full forcing vector (see
Chapter 13, Equation 1):
f cos(nflt-t-A)
{f} Fn cos(nQt+2A)
(14)
Provided that the point of application of
this force is not a node of any of the
bladed assembly's modes of vibration, then It can be seen that this complex excita-
it will be capable of exciting all modes. tion pattern - referred to as 'engine or-
The 'effectiveness1 of the excitation and der (EO) excitation1 - can be more simply
the strength of the resonant response at described by the sum of two harmonic com-
each natural frequency will depend partly ponents, both of the same n-diameter shape
upon the proximity of the exciting fre- and frequency (of n tines rotation speed)
quency to the natural frequencies of the but with a temporal phase difference of
various modes and partly on the relative 90".
amplitude of each mode shape at the drive
point. If any of these have a node at the
drive point then the amplitude of the re-
sponse will be zero but if the point is {f} - Fn {cos JA} cos(nut)
near an antinode, then a large response
will be generated at resonance. It is
certainly possible that any given excita-
tion point may well find itself on or near (15)
to a node for some modes and so such a
single point excitation cannot be guaran-
teed to excite all modes. This is impor-
tant if such an excitation is being Fn < {sin jA) • sin(nQt)
applied to measure the modes of a bladed
system: several such measurements may be
necessary before all the modes have been
sighted.
The engine-order type of excitation Engine Order Response Characteristics
is more complex but more important since
it constitutes a major source of steady From the above description of engine
forced vibration in most running turbo- order forcing, it is clear that such an
machines. As mentioned previously, we nth EO excitation can only excite those
shall consider here the excitation gener- modes which have an n-diameter component
ated by the rotation of a bladed system in their mode shape. For a tuned assem-
past a static pressure or force field, the bly, with all blades and interconnections
strength of which varies with angular pos- identical, this means that each EO excita-
ition around the machine - see Figure 30. tion will only generate response in se-
Such variations in the steady flow are lected modes - those with n nodal dia-
inevitable consequences of upstream meters. Even when the excitation is
obstructions - vanes, bearing supports - exactly at the frequency of an m-diameter
and other maldistributions in the fluid mode (n*m), the only response which will
flow. The rotation of the bladed assembly be generated will be the off-resonant com-
causes such variations in pressure to be ponent from the n-diameter modest there
experienced by the blades as time-varying will be zero response from the m-diameter
forces with a frequency or periodicity ones. In practice, of course, there are
based on the rotation speed. The effec- few assemblies with such pure nodal dia-
tive excitation can be prescribed by an meter modes and most systems, having some
analysis of the following form, here based degree of mistuning (or deliberate asym-
on a particular case. metry), will possess several modes which
have n-diameter components in their
Suppose the steady force has a varia- shapes. These will all be susceptible to
tion (or component in its variation) of some degree to excitation by the nEO
the formi forcing. However, the relative strengths
of the resonances thus caused will reflect
closely the magnitude of the n-diameter
f ( 6) » Fn cos n 6 (12) component in each mode. As was seen
earlier, most modes tend to have a primary
diametral component together with a number
then the force exerted on blade number j, of lesser components. Thus, we can pre-
located at 6j , where dict the general nature of engine order
forced vibration response charctertsticsj
a harmonic forcing generated at n times
rotation speed causing resonances in
several modes, the strength of each being
dependent on the n-diameter component in
that mode shape.
F<•* = Fn cos (nflt + ^J2i) (13) Lastly, we can see the possibility of
N the same type of allassing phenomenon al-
ready discussed in connection with the
15-34

mode shapes themselves. Because the ex- One characteristic of considerable

citation is applied to the assembly by a interest is the relative response levels
discrete number of forces (one per blade)* which will be encountered by a given
an (N-n)EO excitation will be equally ef- bladed assembly when it is (a) perfectly
fective at exciting n-diameter modes as tuned, (b) arbitrarily mistuned, and (c)
will an nEO type, although the frequency mistuned in the most disadvantageous way.
of the resulting vibration will be at {N- Many studies have addressed this question,
n) times the rotation speed. It will be focusing on the maximum possible increase
appreciated that with the above governing in response level, and a variety of
rules, the characteristics of engine-order results have emerged. If only the two
forced vibration can be quite complicated 1 modes of a double mode pair are involved
(i.e., only these two are coupled by
virtue of the damping and separation of
A convenient way to illustrate the their natural frequencies), then only a
essential characteristics of EO forced vi- modest increase in maximum response of up
bration is to use a version of the to 30% is forecast for the worst mistune
Campbell or interference diagram, as shown case, Ewins (1969). Once the various
in Figure 31. This Is a plot of frequency modes' natural frequencies become 'close',
versus rotation speed with, on the third then the interaction between them becomes
axis/ an indication of response level. very difficult to analyze - except by nu-
Shown on the diagram are lines depicting merical calculation - and the results for
the individual natural frequencies largely this case are founded mainly on empirical
independent of speed, and also the 'order observation, often based on many hundreds
lines' indicating the frequencies of prom- of case studies. One of the more compre-
inent excitation conditions. where the hensive of these. Griffin & Hoosac (1983),
nth EO line crosses the natural frequency shows response increases of up to 1001
'line1 of a mode of/containing n nodal under certain special, though plausible,
diameters, a resonance condition will conditions.
exist. Other intersections will produce
no significant response. The existence of At this stage, it becomes necessary
impure mode shapes, such as those of mis- to recall the assumptions which have been
tuned assemblies, results in several modes made concerning the damping, since de-
exhibiting a secondary level of resonant partures from these can result in signif-
response and this can also be indicated on icant differences in response charac-
the diagram. Lastly, the aliassing phe- teristics from those reported here. The
nomenon - where an nth EO excitation uncertainty of this Important parameter
excites modes with (N-n) modal diameters - means that although the trends indicated
is also demonstrated on this type of plot. by such results as those reported above
are valuable, it would be inappropriate to
draw too detailed a conclusion from them
unless the damping model were first
In order to illustrate the engine- validated.
order forced response characteristics of
bladed assemblies, a number of computed
case studies will be presented, all based ACKNOWLEDGEMENT
on the beam and plate type of model dis-
cussed earlier in this chapter. The first The author wishes to acknowledge,
example, in Figure 32, shows different with thanks, the assistance of Dr. M.
engine order excitations applied to a Imregun in obtaining many of the computed
tuned bladed disk, covering the same fre- data presented in this chapter.
quency range (although, of course, this
means different speed ranges). The
selectivity of the excitation in generat-
ing response only in a mode whose shape is
compatible with that of the engine-order
forcing is quite clear in this example.
Next, in Figure 33, just the 6EO forcing
Is applied to a slightly mistuned version
of the same assembly as before and now it
is clear that several modes are excited,
as anticipated, although not all to a
great extent. Some verification of these
response characteristics has been provided
by some carefully-controlled experiments,
Ewins (1976), an example from which is
shown in Figure 34.

In all the preceding examples, very

light damping has been introduced so that
the various modes with relatively close
natural frequencies are still clearly dis-
tinguishable. If a higher level of damp-
ing is assumed, then several of the
smaller close resonances merge together
and lose their separate identity, as can
be seen in Figure 35, taken from Bwins and
Rao (1976). In these circumstances, it
becomes almost impossible to 'anticipate'
reliably the form of the response curves,
although it IB always possible to calcu-
late the exact form as has been done here.
 15-35

STATIC PRESSURE

IDEAL
CONDITIONS
REAL
CONDITIONS

i
0° 180° 360'
CIRCUMFERENTIAL POSITION, 9

Figure 30. Variation of Working Fluid Pressure

40 36 32 28E 24E . 2OE. «E.17E16E 15E 14E 13E

NATURAL
FREQUEN-
CIES

10000 20OOO
SPEED (Rev/min)

Figure 31. Interference Diagram Showing EO Forced V i b r a t i o n Characteristics

15-36

37O 380 390

FREQUENCY [Hz I

Figure 32. Various EO E x c i t a t i o n s : Tuned Bla'lM ni*k

TO1

10'

10°

10"
300 350 400
FREQUENCY (Hz)

Figure 33. S EO excitation: Mistuned-Blade^i

 15-37

BLADE 2 61AOE 3 BLADE 4

4OO 4IO 4OO 4lO 400 4IO

4O
SIHESS

3O

2O

IO

380 39 O 380 39O 380

FREQLJEMCY(Hz)390
Figure 34. Measurements snd Calculations of SO E x c i t a t i o n Response

300 35O 4OO

FREQUENCY (Hz)

Figure 35. Calculated fX) Excitation of Ramped Assembly

 ••.•I LO KSSMfMT Mi.rn--:ns <> i-.i V ' K '•-•' ' - j i-NVM "'IN

by

i . K. -M-M-vn-'i in- :
! « _ < ! . I '--R'.'Yi':: ; •! • .
I •- 1 i • to

. ixin !'"•.
• • . , . . • • -. ..... • i in init La] rnn-n n.i . - i oas Hnwtjv*! , Fiqure 1 is . 1 M'" 1 iqrapti ni i
turbines in th 1930's, Mi-ii ,|,-y.-- | >->pmcnt .."i-jmrm-;>:'-.ii ..iLi.>-i "i 111,1 " ( ' • ! I-si liirf- w l Ii h
I i , .- 1 ..-, • i ; • n . . i > ! i .-if [ • • • • • '••-, J •; I I i f ' ' • L l l u s t r a t - . - ' : the • [ r v w i T y nt" the irohlen.
, t t h e bl • M L i. l 'i ,i ! •-• r--jf pei cent :ino - - - r
• ;, ;i failure; I hi ma joi i_-.:iu^c- w:i-: <)uo ' ;• i r (MM t > saiii that 1 1 i di'
! , ; . . » .,i • ,t i ., . . Tht-Ki l ,ii. i TII' r.i i I t i t '.-s hi?- -,, , • ! , ] f-FH .-,r metaJ Latiquo f •> i I u i •-• -'IH r.-r
,',1-n*-.' "' ' , • • - . ' - • imo f ' . K> ' rlurint] the r^rl y • x i s l r i-h.-n i 'n --.' i i ' i y "i- ; • i > i- ' i h r a t j i >n
: ••. , i . - • . • i,- i , r , ,r the ax M • ciiK' 1.1 i • i.iociaterl .-,«-!-• • > • ' .r i. i • i L v I'lfM-i 1 im-
i n pi •].--,' ,> T r i ,- • . "ii r i t ' j ' j 1 .in i t whirl, MTM would ' iv '" justified, it i--.
1 net i'1 :i:--.,-! ry apprC 1 i 3te
were used in the •• . i i -, iero "n i i i - ;. i ht-r.--ff.i-i-- »v i i. !>'••
t >i < i | i . < 1 l i l uri .'i ••>rni'. int.'t:' • - n f f i ' i t , t!t. j 111,1']". i' t'ltcUit'R r.inf r. .1 i i ii'; ' '«' f'lfi'iif. •'!
' , . > , , . 1 • . | -n, - r I M" i i ' L IC'fc i wll I i.''i I'f r i (i.Kl 'It '"" n1-nil-if w.'iii-h .ir>- encountered i-i i'"•
v e r y quick 1 y across the sjoct ion < •! ft-* ric-tu-'j I oixT.Mt i'in 'H I'T'n ' u rh i IL-S MI i ' N - -
..... ('••ivnl'. I '"i the "•'"•! i> 'i l t y - ' I <.-.T-,CS t h i s [ ; i . - i i - r i i-ril 9tepS -.v^iii't. ' ' i (-•-!! t'.'
i l t e r n a r i nt| stress is trie tli reel n .;n I i of .-,.-.•-,,..: the Levels - i vibration w h i rh a r t -
I f . •: i ! : i ,l i . i . ,i tin ' - I ' -I".., • • • . • " ! ] . I eS - ir <.r •si . . - M - I - I , t-n ensure a t.- r as p- is siblo i ti,--*r
i i ] i •: i - , , • , . i .. - M. i t i • anl amp II tU<1< . -\^ i, | H if Fa i lures "!i:-- t- fi f.n i^ cause ••irt i>
i ,. i -, i i , • ! • c:i • • - [,• • • 1 1 • • .-- ..i i.1 1 •"! SP t hi- -i i " i 1 1 - 1 7i".! . ;'li L.-: 1 •- t\n: :v,j r : • • •'-.<- Dl t h IS
t,|,-,ii-_- • • i i ' , , i 4 " leveJ '••t r.hf? steady ! r genl chanter.
;treas on rtic t .-ni., i M n<; part ot the se< -
tion ' r. i ..... . , md i ' m l 1 r<id. ure t a k < .•-
ic-e i. . thi : t •••-. -; ' i '-";• '.i!"1 ''••'" i '"-'
: , i ,• i ] . ,' i - t - n i i r ! i •• i. ' i < > m,- 1 < • ! KI t .
'I h,' | • r .1 . • I t FM> T i l ' .I I i ; ,|i' 111!

• ) • iy L • • i - , 7 , ' '•' f di lurr: ' ,1 .1 M .!•:<-• 1 n - > t impiirt a n < - > ' in nifCh.m i T.T 1 enqin ei
i ! way - i • .; • < • • i • l ••• in ""i .-tx I d I co M|-.I !-•.-,-- since 1.1"-'"1 r i t : ; ( , l i v : ol r ho r a i l w a y s .
Oj I m,-iTiy -t Kie: - • •• , ! [ > ' - r - - , : t t f nuch .^-(•ori- \ .-,,-•. \ :-., .in.- .-..f tru j u.;ii or i .-• I r.estlm] tnethori*
i , f I six.-jMf , --is '- H-- t 1 - -n 'it n- i.' l Lfniny .•ir-.1 '.juvi-* l<:»r>iTl frtw the eat'Iy
Li,.- blade ricochets l-.!i»- si at or -. w.'A i-.r -\. W('ihl<.-t [ c . l f t S Q ) who •*'ts in-
,-, , i ii i ,|h .; • i . . i , •< Ti is not volved in t:i-- inv>--:~r ! • n i i , . : i ' - •! the t .'i i 1 -
un,-. !'firn.:i tOI ill ''" ircn > r axles • >l t'.-i i 1 w t v wagons , -"'• >oi e
=;or r . , :„- f r a c L u t • - ' . i i 'iv ; i . ;; !in.-.-. i |-,.-it t 1 1 tt- I'l.K-t, I'.'i^ been
ifte t.ii- ! , i ! ' i r . - ^f an •j-.i i I y ';! .nf !"Ln: H! t:j .11.1 <<ur u r i i f , . - t .'1 indinq ol r t i * -
I . , :.-. i1 i- .-xtt-nt ot lni:i r.'-i.-i'i .t-i.-l - , ] • t - c i •• conl rr, 1 I i n'i tht-
l.-, i,-,.|. i' b - I I. rjoverneti '. in .1 v.--ry i:i/m i>r--i;i11.1^.. Mowi" i! /of, i"*!!!" k t t o w I « " ' l p i " ^t rh"
w.iv ) 'iv the flo; trin ar.d .-;ii.i<*i ti,j of t.ln- .-.•ui--iln 1 - I. y t":l •! nvif >• s i .i ! , ''in SO H , i - i • -in
, . ] ,,i i , . - ; , r i , r , ., Large < ' x r ,-nr ) by i_h>.' ;>nn<T:t , lt..*rt ",! i I I r.n i -• 1 v upon >n»p i i •• i 1
Mi: 1 H T i;i: j '„ I i :'i i - M 1 P I ' ' '. «'' t~ I M ' '-. of t'-ll-.' o,--it ,i -:.t ct 1'if.-.-; t rom •-.[•"••• i 1 1 "net nl 1 urn • i t
!, ' I'! I H . J . t yr>,'" f dt. pjut:' l_.--^t i m| , ni actu£ • i t i ,i,,i
1. ---.^t - , .11 '•oinni.iri'-'ni -^ . Tin - a in - i t ' I • SC~
! • j | > l i • - . ! • • i photntjCfjphs • ii i ..... •• 'in|.i es- l i o n i:; t « * i n'. rcHnr-' Mi-- I ictors wii i . ' i i
soi . • . ii iequeni r • • tli i u; ' VP 1 '- '^ f.i i 1 ' t : < • .itr..-c-r th." l ."'i ).-;u(.> -.t M'nnih 'it ,-i
.1 1 , no I i -.--.moii >; i , dm'1 , -i- • do'ilit. t ; , i tiiv iMriF«Kn--iit . Th-'s.- have ' • i " 1 ice •» lated
i . . . , ;. c - 1 1 1., I - . , I • - ; iluchance of MM nut fie I un.'i •-; iii iny aaKess.'nent conci r ( i i n , | t i •- ; i por-
f
i . release i l. i • yi- ol. i ntoi-m^t on. t nnce of a l«vei o) vi hrat i on •! i •' i • • - -

I . i I . . ' I . . . , ' ' ', , n > : | - . • - . : . : - t

. . - .1 • ' i : V ! .. i . 1 1 [•
I ,, 1 I , I •• I -, Mi I ..- I ,,-:i. [-.'• .', . iliq 1 , CyClO i'j'l.li;,.- IT .pe I I 1
 ,t_>-'r ! r I . ••, i l l .in i ,i '-i ; • [ . , . \ f - '-i,-|,-!.-, l,y will, -, 1 1 1 - i t . " -. ni.iMD.M- .1!" i ' reals ' • t.> i 1 •
HI.-I ! •• ,. . ! iuiVi-- ; . . ] , • • • i i -t t il i '|M'' .iti- i i i . - n Hn- re ! •-. • I arqei attei in
, , , . ,, , i . .• i •. . • • • ' ,' il-.'j1,'.' "i : L'ni Lai 1 i f,,. FI,|- ]uw ,'v 1' : , i i i . i' -it "..v H' 4
1. 1 , , , - . i- • ! . - i " • t .1 •••• ' i ----' '- - m i nna 1 l".u . v . l , , ; - . i. 11.-.- '-.- ,-ir t ••! it, Li Le is ">• >re "f tln. j
-,(..•(... " I ••! , . ! • • ! , .1 • I ,
w i Ui i ii t l i - - --..--ir-if . .-. t nt 2:1.
• -s- , . • , j , , - ! . - • . - .• -i ! 1 1 v ry , .]<M-"r-.-
:-T ' ipon t l . i - , - , . - ' | r . - i > t I '"-i i i ' . ; w < i r h • .....
! . • , , ! ..• • • , ..... - . 'l * -. , t i n prop* tiles
.1 . i :,, . i • , . - • . u - ' i • i . . . j u n c t i o n '"'f •'»"• . ••(. i.i |.:i j i t • .•!' i •. m whK.'li ' •(' cur. i 11

I,., , > | , i I ,: -1 .:,• .:,,•! Wl ! he ' l i l t fn Hi enni no«riiiq structures iv Mir- ;•>!'" 'I.I..H,'.TI ,t;
, | , , ' ! , 1 . 1 .. ,- • ,1 • .1 I I . ! > . .'.'.,-. i. • • -,,T is i. i ..•' I i nr: . M oci ' I T - : - wl.-ri two
i ... pl.Tf.fmn ii, ; f h c 1 am l»he • . r:om(>onf ol s ar«? Loaded Loqcther, CM w t i ' - n
r;,,. 1 , 1 , l , . • • i - i , ! i ! • • . - ' • - i t , . ,-. ••< i I -,i -
1 (
i |.,: -.!,.•!! "-VI"- '.f fnrc* •! -"nycni-nl-. .
i ', • • , : , -i pr • • < • • •:.•• ' M i i • i I , nf M M , Jei t he-s • :ondilir nr, t ••>• -.1 rlaci ^f t.lie
f : :
i ,i i . , , . , . . ,i i, i . - r , i , ! , . .; • - , . , - ,i >,-• i I i n i . Lnlei ' n " lf - '- • ' i ' ' i' •"!'"'" i-' l "' • ! " ' i ' > ' i '• • ' - 'nd
,i . i • -,.'.. i i , . i ' • j.- . i : nanuf act jri no i -, this i •-. Kir 'wn a3 i i ' < - t ; : n. - dan an? . rhese
._ | . • , • , ; l ' i • ' ' " ''I . ! • > ' , • • -i f
l : ;ituations • • * i :-t in t tu- bladinq nl rjas
I i r in i 11*1 il ' • " - " • •• 82, Heothnm i l "»\ . i . tuchi n e ? , w*-i- re the hlarlci ire • • inneei •••<•
• •-, : i . t , , ': . ' , , . . • ! , i V - - I J ! ! .--ft.;!!--'- in • : I ,i t n ,-int:'. I h^ ri";L - .L ' h-. onyine •-:! r . r - - in ' • ,
hoi -••:• i • •:••• M ! . 1 1 • ' •"•" i • " ' tri -I l i ' v , . , ] . , toLor hla i - - l i/ i !"i onto tl'.e disk j
p
, .. i i..\< iv -, ; raken t t o«n <•'•• '• i i in • it i v,itt': "iif < - r ( i \ i M - I intn • - . j > i t n j * -inr)
• . ) . Mi,, irint.-i- onrlK • - I " y a r i •(!-• 11-- statoi vanes.
: ! , . , , itory l-ostsj haterhoust! • ! »:•'-! I , Ahi-w,
t-|,,it. i »!.-• t.u. i'lut- .'iM-i'-ncr h of sofflQ nars
rial? r -i i. I.-'- i edue • • • • ' to 1/3 ol t"hi tar
1
-I : • ' >- , ( . • • • l n . t strength. A t t t - n r n - n t o J o t a i l dasiqi
ZIIK: ,inr i l-rt-U inq .^i^tin.|t>, Wrtterh . •
l -i . •; - i - . i?et ions, -i|n. i • it ( L H K I > , can h<- l|i to reduce th> • jorious
i ,-.,-..•, • .• i i , f i r, taticjm i'|lt.M L ii'--; • • t Eects.
. :-... : . • - . • ! , ••; i ven . ,.-r , to Hi •' i- i'i'i^.1 '• •'
,.--,.[ the . .-, .1- , i r t •Airi .it i i > n s I.--'. -'"--I
i . j irtc-R .<•: i .'I .•,"" i i,B ' . - - .. comply w i t.'i ' I'.-
I i.-'., . : ! I ! .- i - t - i , , . •••, will
.,r i - •• i ' i • nanuf acturinq M - I >M mces I tt t-hf rn:-i i»r i.t.y nf t»ny ' naei • - i
., i i, i - i. • , , . is , rln ,r i]aJ len in-., , - 1 1 i o n e , ' . • • ' • ' - Ls "i 1 1 1 - - t ri.u 1 1 ) , i s t P. ire
l, . , . , 1 1 , IL-I .1 ij • .1 ape and loi.-«i v^ri.-i- , t ,i constant w ] th ' i me , and - ' • ci -ns Ldi • .
!,,,!,•: ,n -,i t . • ncenlrat inn ire as ' i t •• ( j , , , , ],,-, - been given to w a y s in wt, i . • ! . t. ii i:-.
M l l c r radii, m-i rail inn; * - i j < V ' Ht i '.•knvss . niii]hi I--- icci mm. .»-|.-ti - - ' I i 'i any .-111,11 y s i;, to
i,. • , i i • H i i i - M i i t i f - r of t . l ,i.|'-s "it '" V I - • i . . l i .-I, hi .w .-i '.',-u-1 -at i • ,n in .Tim I i t in'!--1
• , . ,i . . . • i ...... •••," •• at' "li.-! .Mm -ii i ii- sano w i l l a f f e c t the i - a i <•!<!<; :-;t. r H T I M T t.. i-'f-t r.li-.-
!
I ,.,..,. I . ,l -It ! . i |l j . .I, , I i|,--M f Ill'l f.- W i l l >)<• d , - , . - , l •.'--, i / , . i i i,i;=t loads, >L sircrart struc-
• -,;•.,( tei i i ' mi ' s .' i l ii - • . ''"i* "ii =ih t hi;
1
antlcipntod, wh-.- 1 • ! !n- stress -n f . - i ! • i. Iy -iimpl.' in ' t,' "1 "t •-1.-H -,-! !• • lulal inn.
i n i t i ^ t t t o i poinl i ':- -it r octed hv 1 . 1 r i;" III. 1 ; cnri-jf-->L W.i.". , tli.'i'. t hi ICCUmuJ lt< '.
1
.ihil > i • i , ,1 fa tfira, s. in'- t-'i i 1 i;i t* !i frtl 1 •i,im,Ti-|r> ..-..in h<- expresaed i « r.Tin-> ui iho
l i t - . .1 i , - . ) HOT ,.-1 1 ' i '- I ' -; III.!' ] . . I I , I.''., - rn,r,):.---! - - t cycles applied, v i i v i ^ i e d Liy t.he
norm il riist-.ri hut i r.t ih,- !• "i-:ir i t h ' , M , . | ; ., to p!''-"!'"' 1 ' ' •' i lure -.r t tie - j i vii
i i •-,.• i i [ ... . Ujure 'i i '? t n k i - n L L >'>"> \>m\f stress level. Pur tin , this cumulative
( 14 n i w h i < - t how i t i ,i t h.-- v;i,'cat_.-!i- ir ,i HI,.-I:|I. r i i t - r . r v issumea •-'!•:* I ' liluru i fv-i
l i i . . n-i.ii. the hortesi ' • lonrn-r.t lil.o or lrttii|iif' w i l l i-.i.-t-ur wln-n clu:- •-. inirrinf i . <n - i |
. qi oup -i I- 1 blade: !•• :;f..?tl unrtir-r the various contri -".M ions ol rlamaqe •-.[iia 1 .s
r UK: it ions i . - .: fl ": i -mr t h a t Uiei • un i t y .
tiution approximate vrrv elrtt-tly I In,,
- K i t "i.-i I d i K t r i b u t ion. 11 i ^ 'i f • ot
i 11 ,-] i;, • ;. t < i J ' - : i i • • ' . ' H ,_•( iii!pnm~*n1 Lh.il.

•;. '- ..
O

(h)
•. h «rlu«fiCi iJilAi I I'llI'MfUM li.l^l I T V

I- l r ; u f (-• J . I'VI- I • i '1'"i' i.-|. u f . - nt

•-. t .-> n l.-ii ,', jri'1 um Ovi.i I ! t y
I ' t FI-4 KM: | i ' •_•',. { ISO nm Log -nbrth i 1 i L v Plol
• i --.-ii. 1 . - r ) . (44 f-'.^t i ipn.- .1 .
 "icks m a y also t , . we incipieni nea cracks
•jr Lhe ro .1 oi the notch, |'hi atrtuaJ
• • 1 :
i | • | i ! I «.- i I n1 ' • ' ' '" : [ | l i ' /I" • ' L'ract . . i l l '...- cU'pendenl
- • ' upon • . detai I of I r , , - for. igi !, ly ,, :
wel !
^ " s s i - / . - , veli - i ! ,' , and th. i .,, •
'I I , .I . . -,• les • | "'"" ' -'I ' • . ..... •' ' t tin rol i. E : > ! . - ! • if. only '••.
1 >;
U.i f a i I . • ' res 1
• • > • ti - . 1 i nq rorrectly amaqed b id<
->n ML- 1.,.^- , , - [ , , .,p,.r , ,,.,. . . . . . . 0, ,
' -
12 of , . i res ( i ' * r . ' t ; ,
ln
' •'• i Him i
:i
' ' •••-'
- • ' the ol hyr ' ' -nor* complex
' ' • ' " - ate : he i .u i ,ue ii imaae.
•' - ' ; ....... '• ten led specim -i • esl
1 :
' • "' ' ; • ' i • L t e r n a t ive mel hod was 1 ; r
noi considered . • r i j i,:-; . •' "i the i MUM i . | • i( , . ieai ch uoi
11
in Un- ". - ] -.• •!,.,.•, i , , . . . , . , ,;
1 ; ' 1 -^ 1.;,,,- . -.-,,•!, , | r , ,,, f -, . .
' •' • " • • ' ' ; - ' • • • •' " i i - i ' •- -. ; , ; i l . - j i r :
1
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• 1 •.!«; i nq may ni
1
•' ' ' ' - '• ' " ils a l u m l l i u m . • it m-
' • <" < teol. rhi • is I ... , , . - : . - t h<
- - . - X p , - ' ! i ,-•,, - i : •-.-.' r I f|( y. . , ,. (,,,_.,
' " '"' • ' ...... . ,i • •! v the n«i . 1
I"'"' i l y • ,, | , , , ..,. <n , , . , certain ispects ..f th« :.j.i.j,--
; :
' '!".. I -• , . ' . l . ' '' ' ' " - problnn i. . . . . , , , , , i r . ... and
h-C'i I " . J M' S ItrVDlop. i ! , i , , • i ,-. , > | ,,.;,
; " '-" " • i' ' *'- pn vion -, • - ! ,. ..,.,.,
• '• • '• i • ; "i imi '.-n , , , H
Can 1)1 ,.| .,,1,,..-. | r, - |,.- i , :i, , ... .
Made • : '")[ .1 •K, ' HIT i I K - ''"•'• ' • ' ' i Hurts. rhusi - . . i .- , ;
. i ' • -, ;," I i 1 i n ] i >"! IJHVl l - t i
"lvl ' ' • ' "' : - u|m i either t hdn v..r,| i t id. u ,, - - , -
- - inme M . '• iae • • the h iqh, -i i • ' : - : i t ' w i l J [ • « • ci --•• • ! - • • • In • i, , . PCI
i
,;.i., • .,-. ; - ' ' "ii in i ,..'•• •iny di t r < u
<M i i in ! : • • • M l .,,•• ,ii r 1 (.,., . • i t ac*? and
i ,. . ; I . live •.•i I ' i •• j in
1 '• ' 1 1 i r . i ' i •' - : . , • -
| • ,' f.- i, r . COl • •-. L , ;
•- ' : I. * LI .
!'..! i *• -'i 1 ! loss of ;-.r • : •• i i 1 i > .
i'J ' by
• ... . • • • . i ,. . I I . 'I I '. fractur. tae* •' rVit i«;ui
h
i n l • ' 1 .. ; i , • , . MI,. :_ , -. i •\ I I . ..-;;-
^s j vei v iisl incl ,ni\ M ,,-,.
1 i: :
i • • ' , ' . fir, iml • ' ' I • ,• <n.-iy t">'? ul <\-' I I 1 -
"' " i nee. ['hi - . , • j , ., ..... ,. , .. rf

n Ui" 1
-,..., ... - ,
, '- i •"'•• I y .-:i'ii».-i !, ,p:,, . . , . . ,, : . • ! , , , .
r tin: 1
1
• ! ••• l i I flu • " - • •• l im
•,u; 1 near cr; ' . Ln< i • -. i , , , - , - . . . . Una! ten
, w tI
ibli ;,i - ! , •. • - • i '" f i a c r u r e * me. rhis Li Cdusi Lh<
it ia] ii t , - •i: '.-, !
i ; . • i. t«.iunn • i . , . .. . i ,i . . • t r i , • -:
• ' ..... ' ' • • ' - propdyai • • : - in ,-» 1 1 ins u in-
11 !
in t,. these , r i 1 ' l -: "iu '-' : LK
-' mannei . n , •--:,..; ; , •. .[ ,,,,,, . ,..,, | | j f| .
| • - i . ' • • ) , ! 1
.. t , . i Iff • • i in r i| II ; • . life ol ' ' l ! " ' ; • • • " ' • i •! •• , -.1. not ional . being
: • , . .-lS-'30L-lHr>'tI W] t.h .;t , - , , . - . . , .-, i .. _ i, ,L ,
•" ' •" ' ' ' • •'" ' • • • • ' • le Cdt i t,- w i H, ..... , ri.-r
r
, , [ hi
i i, , [ 'i-ini,-,';, I ...... ft ! ' . i Li.-: i .[;,:. e I bhi . -• .
t>] ide ""•i ' I - ' in . I . : - . ! • - ,,f
. i '_• t • > 1 . , I i •.
' : •• ,:
l • • ,, • i , i. , . . , . , . , . . , , , . , ,
-: -
i I.T , . 1 ..; .
' !•<•• i ng i ii ! , • : _ • • i! i f t - mi i nal i • nr permits ' i . •< , . . . hanit:n
:
I ' ,>r|U , . . . . . _ |_
t a d nq iri.'iy *uffoi t ""' '"""-- I: ' " ipp led. • wevi r, *ith
'-• ' ' i n i f leant ni • • , , . , th< Lead i nq 1 he .:!, i f . - ..'. - ;,'! . . :-.,; i tUill " - I . • , •- LI
:
''"' l" !
il t(ju- : i r , . - i - , . ; t t, wh i • • ! «>l.ved 'Hi , : i ! • • v i hral ion, , i Ui
1|
1
' ' n i c k s , lependi upon MI " ' 1 ' ' • " ' • ; " '• n' •* POSH ii.,|.-', bul the ,- -
no t C h , ' • ' ' ods . 't
1 %
' ' • • i '• f i thi material having btjon h i < ] h Vlljr.-i! i i, . : I, . p, , , i , ,.
'"' incoi la i "'.ii A i , ) • ; wh i r i , act i-t-nteri .1 - i b - . i j r
" i i ' t l l ' " ' 1 • " • r ' ' lit n^i M I i. turi nq in--| htat 1
•- • , I«R. [ ri , T , ,,-,.. ,.., -,,.. '"-• ' • tique nitrleus. *> uhotoejra h it a
1
'" " - ' ' - ' ' -• Face •!- i. i t .in i UN t)]fldi in i :. i ..• -
DMxlUi ed I l, I M . 9. M, i :-; l;yj.--- ..[ f .-i t jfjue
cracking oxhibita oract. icall y no >luf-(-ility
l:nl !
l the t LIU l tens i ! • • t a i l u r o . Ehus, , -
!
" very . l i t r i,:ul t. to detect i he pi enc.
r
' ' ' "-' ' ' i " " ''.'-•'•-'•L- !' ipmr-nt in ,111 opnt, ,. n :
l 1 1
' " ' " 'Eeasor, rhe h r t i . .';i.,-d.iv st resa ' •. ' , - , , '
blade f .-i»• m ,-j i •;., .• • ^ i. : -,.- ^ 11,,-. ciack ' •
i > r . '[irnijti t - - quickly, anc .1-. -i • • • i •>.':" i ' ; i • 11 i i l
i« inusurfl t o f in.) ,:• • i n :on press •;
trot > ! - M.i-.j.' .-I.,.] ofoi I y "i '• ' -n< • \, I ,-, *,•-
1 ills. Wlieri the cracking nucleates in ( h e
bldde root f i n i n q , wh.,-r.> t h e ie,m t ,-,•
s t r e s s is lowor , thon Lh<- cracks ->••• - . \ . . «
L'' f>r'-.:.i.u],iM.- .-,r,d r;o cracks i n i I,, riiol •-,
f i n y h.- tound pi [ o r to E a i l u r e a .

trea
i-
In r : L 1 1 , - o r iri't Is U LJ tin
.-it ress n'cnLrut ions i Wh I t / h J , ,..:.-. i., ;•
1 t i n - pna LI- i uni ot MI.;- t , nut iei
I V[' I ' -'I i-'i aci u r t r/i.-,-s Due ti-
r.-;' Lque <L '[' i !•, .'in i urn i 'i i
Llie ;;t rt-ss V f l i.'l V t ! li I 'Jh , f lu-:i .•xpcr-
, >r 1 OI|,Jf
IU ides. f hot f.il i t- ci dfk ? (nj will
s I ,:. r t -H n-jmtj^r .->( s i -s .-mn.u-fnr- : v
simultaneously. i:? rt.-pr,-,-
 , . i •• i in F if) . \'-i I" i k - ' E i f >'-Mt Ar • .•; I i m.; ( h ) Wh.-it f ll r t hi-r ••••-.- pi ;i • .ii i i .'• -n
, I ' , • • . . - i ,. ,. • : • • . i• i w i- riuh i ' - i t ".I (•' ' I <>;, I , - w whnn ' i.. I" i I" i :,..-.( hits fUl ' r
hiqh alt' not ii , Lres ' I1 • tons/sqii I o ' revers i - ind Ls inLailwd? <.",:•- nu • • • - ,
wh i - :i cauiuij i i i I uri i d '< x hi'1 L'ycles n > ;.-.r v ' i •' r i i ••"• w i t ! Mui il :• :, i -|. :• i . , near I y
ll t ipli .' i' ' - w • • f i. A hi .-'.o.-,! . J i :-,!•- ll) :t. i; f". .

• j v lie ' i i , ! • ! ' • ! ' • : * i , • i- ii, aaaemhl y, w i <• h

niciny - i- i ! i f -. i : • - • . ii .--iii. .11. L. 'f . ; i < '-' '> How do ynu run th< .i , , ; .• r . • ; • • ' ,.
1
• !-• I -id i n ;. 1 1 i i i -, ' in- , L h ii. - i h i t ' l L i '. --t test t ,11 I, •.! • •<> r I y ; w.-i', H i ••••.•
: r l |.u ' .-., k ./-: in--. .1 r . ;,1 i ••-•-.• - I l l , 1li.-n iI : 1 1 .i.|f ,,i i ',., • „ . i t . ,,, , ., ] . , r i |, . , .,. r . - P , or
i: iir .si i i • - . . - 1 " i 'i 1 1 ' hey h,-i v. been c iuaed *,.i -" i. In:- - . u f i i ' l i i irfe i -•' ' . i i j ' i . ' Kernembxjr, tin. 1
[ .• :,. • . 1 tmpl it ud : • xamplfl i st ronq rn*ari t i r o w i J L v,.<iy :.'v ^t least 1 0 : i fui ,'
f l u t t e r cnnd i I ; .n , !0 •• 'hancie 1 • • -.11 ess Lev I, Kofi U'»7n > ,
•'• ! i ,'- i. • ',,.; i' ' • • < , . , [ - : . ) ,

H, - W " v ' ; , I • : i h j -i h i ••• .-..-\' • • . : •. • 'h >w > i f > y. 'h plan .. •-.. ries •: i .-.- . i ,••
i •' . !• t '-1«- n it ! I I . I •; | . ll.i'.,-, • r- • - i . • i - t • •-,!. I ',-,-• , j ! • - "• I • . i . . . - I i i i , i,.: i -
: i • i . • t i .,, i ,. i , . |,.,p.
I'i.p.l 1' IJit. , l I 'I ,'! .. i i ii r ] mi pi obi - - ' • ,nM In ili 1 .- tn t L \ .i
: • , M ! i. • . • 'I • : i •-•• , • -.y ""' that ' * - -
i! • •• ipletiot! .Li'.*;'.'
i ; , i] M,. • -;; . i . i, | [n iKtl. i-l :> I L.r^-
ir i i , • ! • . . • , , f, i |,|t- w.Ti vihl "ii ' ! • : . (••') Can you l• .my' i i r i , i w < < r r !,.vt i I •'• w i r h
ij • • • i r n.ir ion '- not ,.« a 11 y i. he , :• . ••>flly ;i -:'•!.- 1 1 n'li'i'.-v r t n • . • ; t . ,i
• \. tup l . - , Lhre* •! LI nit ?

' • , • - > * . i• i • i| •> ;•• i • i. .-,

l r is ft f"i\ nf :c - ' - i ry t , ,. : -; • jhll S Ii

the t :i; i , ! t refill" h it .-i ti 1 -i |i' . Two typ-
reason
• • • i I! i • i < i'.' i , I • • ; i r . , r. '
i in- r • • • • . • • ' t
nf t In- • i t idas i--t ,/i •.]•,.-
! | , -n .-.i • • • in an e i in. •I- CM L :.
est ahl Ish • ! .- hanqo in fir •i. i-v; whi -:\
Ftiriy r- :-;u I t frciii 'I., .i, pi -, ..-.i lit-
r • i . Hi i'i, tho of HE,.mi r i.
ou! sect inn o tve r ; ifi,.ny - > f t !«• i'i'.-jr;orn why
i I;- !",-.r i • ; . , " • p r ' > ; > i - rt i i •-, •t rni-- r>)nrtr' m.iy ho
• - ,<\ • • • < ' • : ' , diff EM f r;jfi t ho t i rpirc-^ whi f-h
.,[. obta ined E run -j aeries ' >f 1.-^', i ••
• ' - 1 1 111 ,; t ' • • t/[»•" L.it: iijii':- tt.*sct:. One
method i-! •.•rtviinj :.iut lati<;ur.- I.e'-it1-:
1« I I OHH A M { I tt AND
• -i • • .nip. - i ' - i . ! ' i «.-M-,'er^d in dot ^ i 1 in
r
A . T - r . t - 1 on') i i .'r.r,b}.

i .. - r . ' ,- prObJ .•"•• t --.nu J ns C' HI-, . ' t r i -

IM, ii.i*' iii- i.-.t progran is to '-"•-• C<-HI-
i r< . I ! • • • ! . ••• • • . ; - , ' i i M, - . ; , 1 1 ivi It i '- '-.t_.,n-
d.ii-ii -.-<-, t i c ' fot metal lurqlcaJ Laborai • • • .
LI:- si*; i" ! carried out ,.nd pi ^ieim-.-1. i ti
the; I :,r ,t an • - ! - 'h ,i'.|r.ifi'. whoi e [real
if ! . - n r i , i L i paid t ; . r h- ^ui-f.nn 1 t i rn :; h
i-'in-.i 1 1- 1 1 in , i.e., iri rror M n i ---h .-init ';i roas
rel iel i n • i , nu , :-•.-! 1 y I i'i7d ) , rtn-ri
s c a t t e r IS -.ih,-i 1 1 , ,-iruJ on-.' >r t.*>- i specimens
< >ril y n, ' I ' l l r ••• ! . - • ;t . - . - it each ".i r-':;.-i I t - v r - I
• . i ' - ' i i.-' i • ! > • > , . - ( • I ] , . . - t ' i , - - . [ • . . i f H _ - nt t'}'.' ti~ll
C u t ve at St r eSSeS ,i!.K.vt_' tut' Uil. i'l'-i'.' limit .
'!• '-..', ' v v t , Whu 1 6 t.-it.-' SUr I elt't: t i N lt;[j LS ti;>t:
to this hinh ?3i -ip..i,-ail, and where oossible
( • • • • • i du.-il streas L :'. nrt'MT.t t -. st.m," .; ; ;yrtM',
.=ir. wi 1 1 ("- t Id- .-,isi? WIT h I'.-nr'jino com-
pontinl ., then th< '.r.'it i . - r - will !n~- vr-ry
.s i rjn i l" i . - . i j f , . iri, t i i i •-. iv. ,-';•;.11 y l t < <••-1
.'it liMr.t 1 fo 4 hl.ii'l---^ il i?.irh s t r e s s
I ,",.'•• I . Rocaus* '• t l h.- 1 ,1 rg.! y - t r i /il i UN i M
l i f e -- •;,-!•/ hi; ! - -iMtl ,-il'm-i the- I.'K.'I lli-'il.
f !L: m,>,ir. I l l " i r i n i t i a l l y unknown f r-r ^
i | i v f n li:v-l Hi -.nhi-.-i- 1 1 >n .-im[il ini'io, it i :;
o f t e n .-i v - t y t. i " * t - .-nn^Mrfii rvi ctncl i _ - ) * ; t l y r e ;
t e . s t [jroijrrtPU Lo ^^tidbi ish tliy S-N u u t v e
I r.( h? ,i,U • f , - r i i om n t: .

These ;,! act i'. a L pioblt-ms Ct»n l)w

i 1 i M-t r 11 • • : ! IT, the f • :1 I "wi nrj <iui*r;r i nnn :
U.I Ut A d W i i
{a) l,-h it Should • ! • rh,. rtmpl i ru-i... of til..-
f i r^l i » - ^ l w.'i.-f k ii.iw I w<!qi" "f th^ pr''pt-i-
i i ' - " - .->!••• nnt kni"-wn l.o hot ler than <!l>- 3 0 » ?
'Hi i F, 1 ack o! kiT-iw I -di]^ m.iy hn duo l.o n now H! . Cr.icks liono.'! t.|. :-, t i t I un i mi
h 1 ,'ide :-,htipe , nt-w nanu f a c t uri rvj proces:;, on Completion of K
new m a r > - r i ,j 1
 'i he ' i ,( ,. al i .; s !

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from l hi '"" h' ' l ' " " ; ' ' '''""' ll
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' " ' " known. A ci,ai
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- < '•/. :• ah - i .< to; .„ • • re" ' " • • " ' ' " ' ; '' "-'•• <r ' • -"--'I ' , m ,,-., | •
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r r,.n, ltv WUh l r
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lop] IK; | i llt.;-. ,,[ Vr.!t j ,- | | ; jntr) ._

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UOWpaCl lii.l r ,-,,.! | j y
Pi'Hir.-^, l -.- i , , ,| ,. , ,,' f 13. Croquoncy V : ;>,,,,., Pll)fJS ,
( I'' 7 7 ) . w: l - V. nr.iM i t n t i<« C. nnent ,
1
'J Compressor ;.'<;r or fil.,,),..
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> iwai - • - i;'; ' ' - i" ICWL-I in it • spee'
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• •. • r> ; • ' • • • ' .. i , tin.- blarlns were '->.•.- i i-,-it- ions . M- the L Ull t o r - (Midi tlon |1
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inter- n u t ih u i •, > • • • •• , uid 111". n 111 «_• i - r t'Mi t r- m flu- ! • • -;- - '•'!• to
, ,- • , , i i -i • .,. t • . - , i ring coup! inrj r h,- .-d r - sonant1 type • • ' v i t r,,t inn. Since
!, i ! • ! . - - , i. , •• • • • • ' , md nroi'luci u - i ••" • ' ™b ! y , t I ut. i > • i- ; nvi i i ves • ri udic « han' :• in
. . > i i . i. • ; . . i - tut' • - i • idyn in i >l I h.-- '•! .MI ; i » i , th>
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: •• • m •- i • nit lv '• i I i 1 -' 1 '' '' ' < l:| ~" - -pj -im-'-ii is i !-• -L . r i ul • press
i .1 . .•.: -,•-:- h'.-l iiw ''••ii1-, f u l ! speed ' i - • u|.. i t ' s t n I \ •••'. i n '. IH-- ",i •• i i--- • ' • ' : • ' me •• • !
: •• .: i '• • i . , ; . . . :,, i. . i t , .•[• , ]T , i •,, i l M t i i incident. it IMP h n-i-i • i 1 >••
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the si q i i - > • ' ' >• • i - - 1 ' tin.-
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.: . , ! . • - - • t hi , i i t -.i! .1 I f r i - - i ] u « - M - ' . ; ' „ •-. in t I.--- i • ' i t . :i. ; [• , - ' , - ' , . i I t In-
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: ;. , .. ; ' With Hi*,1 : Iii I ' • IF'- i ' t
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!• . | . i - ' 1-1. t : rfq»H"ii:y v Spc-id P l o t s Engine f- Lt>. Frk:i|ii.-MH-y V 'Iin..- riot.-.

^ii-.li A i r r r . i f t t n t d k f rompr ,nc nt : I-;nq i nc llyiif-.---.it >.'h Cnmnri".s, >r Test
, :l Fin K'ltot HI .'id.- 0.060" Comporifrit K.in Fiolm H L nil? ,ind
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--*

CONVEX BURFACt
" Moot n,' M, yf MODE ,„ M/ |

CONCAVE SURFACE

M.OBI14IH4 IT MODI «0

'•'"i"1-1 12. -5urf.n-e - J t r a i n U i s t r t h u b i m - .

For , iov/''li>piri».'nr- w o t h , ' li< j -i.-v-r i f.y if <t itr<iin :.!<-:iui.ie P o s i t i o n tin I '.pr<i-ni.ion
:
m<;vi;;nri .".l -51. iv.'s;; 'i n y !,.• ..."tpt.i''S5'?'.l OS n
"l)> 4 i ,.''Mi l.rin,- i-niiiir.Tn • • • " , by coiiipn ri ,in t'li--
1
• •v-.'.r.- ir.u'l - i l ^rn.il iii'i 'jtC'.-ssi wLlh U.r: Tin- iiiimh-vr ."i I •|."i1.n'|r.-.s wliii:!i .'ire ii-'.'id
,i I i - r n . T i - i iv| i-:-i iuf..ini,-.i s.l re :•;:-• fnvn c\irvo un •'! blade nay bu rcMitr i cf.'.'d by the ai<:e
'C^i' havin.i .=ippl i.'j'l i.h'.'- t-v-irrorr i'i..;flri i.if' I. hi: bl.'jili; ^irul tine lead Out win?
ai.-r.iivioirionr.s. Normal ly 3 or 1 strain
i:ji3un,o IO'";at i ons art 1 used pat -SLiK'jii1, rinrl 4
bUirles in ft 'ir.,iqr- w i l ! \'K? i'is t rum.; n fart l..i
1
'I'yfi ..-'I I v,~i! iir-«; »f 11 t'-rneit i'l'i ^Liv.'HS 'il Low Lor bUi-'l'" ro blade.- variation .irirl
wliii.-h "i-tv b-^ nfii'il --ii"'' •;(.-! deim ni|jort«-.'l. 'i'-iit'i'-1 tviilur.-. l-kiw.'ver, I.Vn:: probli'Oi •->!•
.sli.'inu.)l. P,TV;,>V ' I O V 6 i iJoes provide i.iio .jtUi'H"' f.'ii I. iirr. i •> mi.url'. r'jducf'l by thfl .'idop-
i r»l I ov/ii-,(| t.ibl" •>"! s.-'if.'-' l e v o l s or rilr.';r- ti.jn- Kof. f ( i S t V f i ) - of thin f i l m ..|.:iuije!3
n .:>!•. i in st r«SB . ririun.- 37. It i« s t a t e d C h a t i;hin f i l m
'I.TtuQOS o f f < , - r hi'ihvM- < ; u < i l i t v ' , Lowijr cost,
i/Tiprovi;..;- tt-Kt snrv i.v.:ib i 1. i t.y , .'.nd ,1 do* Im-
S/M-'K "il.'I'KKf.'AI I Nli. ''TK3SS HANCil-: [.H'r.'VOini?nL in rhi> sl.riin l<;v^l t licit i---.ni !><•
in?-in jrisd . T'IC- ufinn I .*n ramie: y t t r a i n naii'ies
z'Hriti L (Po.ih-l'cak) Lh/:j-.| in £rfil at -ippi:r>x iinnf" 1. y h a l f tlit1 Hmp.l Luudfl
'I'i'C'.-ss.'.ry i.o pro..!uco an .aei'oCoil U'lilurc-,
A l u m i n i u m 'MLny 1.1,1100 whor..-art t e a t s w i t h thin fil'ti i|riu';(.-s h<ivy
Tit;nuu--i M.loy 2'j,i)00 oorai l:.f.;:(! mea-iuronii.'rit:5 oC s t r o i n Ic-volr, (in
1 n tiiu E a i l u i f r t'linpl it .1.1 "3 .

TV: a b i l i t y of the sir.'iiri nn.i-j<; ro

11
u'.S'.'y -il l.r i r.ul •;•-; the low lovi.'l ol withsi.ind hiijh l.i-veis ol r, I I c r n a f i ng
rlioiii; 3li'".'!a!jt*$ f ' l I ' l l ' 1 L n f l ' . i i - n i j t i 01 ,'l:i,|i^:|rj s t r a i n , wi'. I ilso c o n t r o l the p o s i t i o n i n g
an I surCdCi-.- i.inpt'r.'S .'Cl. ions . of the i I.M 11' i is s. The roqi.;i romi':nt:.i d r f t , tiiitf.
it shuuld bis .'i Dl.fi io m'.-ni;un-- « si qriiC leant
Kofi. ! I'JVll) r:xprv:;:i^s tlie opinion ri.--;[)(jri:;;i"; in mare than ono n.v.'.c, fiut it its
tdl:'l.|ue si.r-l'nnf.l' ir. l i m i t ' . - d , then it must
•-jii'"lin-- ..'f3'Tit-i f >ri'_-n t: ^ is I-MISI- t:V"il uri t^'! l.'V 'mi; b'^ usf'.i in ,=int i<: i i>atii-<l MHJAS r
d thf*
bench f.i.qriiv;, whoro lit.?. pfli'i s .ire s-.i!j- ''liuhcist a) t f r n r i t in'i t i t r r i i n . V.'ith '( ,-.r 4
}i}(-l ml to v i b r . t t s j i y l.'\,i'l I • < induce f j i l - l|rtlll|l!S, U!. to 1! IllOfJlJS C.'UI l-.-.l1! i 1 '/ Lie
nri.-." 'L'ost''. i^'3 ^.-.ri-iutl cut on inslL'u- covonvd. Kidure jl:l 1. rum Poti'irson (l'J7S>
ni.T,(-«,l l.i I .'i 11 .:• s fo cvnl nati.- the ftesivin, shows i-.htr po:;i f. i.o-is s e l e c t e d tor ^ ;;ncill
turbine blade. I'he det'iilod i n v (.•:;-. ti-
ij.-ir f. i.-i I ly si.i't-11 a s ':tl conil-. t ions, i.i-:., ri.Titi.on, :;HI; l a t e r , e;enterecl on H'u: 7f.li
iir>.'ir-iim .•!•. i Is -iii5 imori ro r.ji ••...' the ten- mode. Hoc .=11.1 Si": of the Snd 11 olode -)\.7.r;, in
pv>r.iruri: l'.') p r o v i d e hoi f - i t i<]ii.-' i..'ist i n<i. this insr.Tici!, only ona i|<:i'jqo pr>r hi ado
Pr'iMi tin.- t ' ' M t i •!•.!, it is | :.>'.;;; i lilt- tu CI'.IIN- could l/r; ac
1
li.irc tin i . j - s u l t s wij.li .ml- i c i patv;c: propoi.'-
i i.-:'; Tor tdii [jarful, is a i:':tr i;i I , mu'i a l a r > to
n o m p l D t o ..i tatii'ii.iii l i m i t - .-|j arjran for us<; !
iy riij test Ceil ihfot AOIIH for e.ai;li
i i-i l->l.-ii-|n .I'-ir.ii-ssm'-.'.Ml , i . « * . , oiirvo ' ' ' 2 ' o|: mode in turn the r t j l a t i v u s e n s i t i v i t y of
I-if in r.-! Hi. tho '>-)i.icio pn:; i t i <mt; may be ost.lh I i nhoil,
,-in.:l rr-1 a i-i.-.i to the level of I: ho lii'i lies-it
C«r»mlcwer»u» Thin Rim Strain 0*9*1 , i I f o r n a t i n:( a t c e a a Of. l.'he- h l n r j o .

This f'l.il-.i, coupled w i t h ti knov/1 orlqo

of the stress distribution ovor the sur-
l;.:ic«i of the bU\cie makes It :>o.v.i! i b 1 e to
obtain n m t i o between the c r i t i c a l point
vibratory stress, to tho apfiaronl- s t r e s s
•scnsijil by the slC'Vin gauge (~-^) . This
Th i n r ibn sl'.rosn distriljution w i l l have boon ustal)-
:", t r» \ n I lishud by one of toe inothodr. iliscuased
above. OanCoirth (.li»75) uses thi.s rTorrn of
r a t i o t c j ofitablish f.hn "scope limits" Cor
I»BO durinci live s t r a i n (jauqo r-.estlnf], to
asDo:;::, the s o v o r i t y of <\ v i b r a t i o n .

u'hero is the stress r.inyo

ts
si reng for tli«
point: under the
fliven operating conditions.

«« I 4 t t k l . r o
i:; the s t r e s s r.itrio above
Hud wi.ll bo mode and speed

f'i(H.irre 3 f t . Typical Turbine Blade Strain

!jriijrji5 I n s t a l l a t i o n .
 16-1

FATIGUE AND ASSESSMENT METHODS OF BLADE VIBRATION

by
E. K. ARMSTRONG
ROLLS-ROYCE pic.
Bristol

INTRODUCTION

Ever since the initial running of qas However, Figure 1 is a photograph of a

turbines in the 1930's, their development compressor after a major failure which
progress has been handicapped by failures illustrates the gravity of the problem.
of the blading. In a large percentage of
these failures the major cause was due to Indeed, it can be said that if the
metal fatigue. These fatigue failures be- problem of metal fatigue failure did not
came of prime importance during the early exist, then the study of blade vibration
development of the axial flow compressor, and the associated aeroelasticity phenom-
in place of the centrifugal units which ena would barely be justified. It is
were used in the early aero engines. In a therefore necessary to appreciate the
fatigue failure a component suffers the major factors controlling the fatigue of
development of a crack, which propagates blading which are encountered in the
very quickly across the section of the actual operation of gas turbines and the
component. In the inajority of cases this practical steps which can be taken to
alternating stress is the direct result of assess the levels of vibration which are
the vibration of the blades, compressor or present, to ensure as far as possible that
turbine, at a significant amplitude. As blade failures due to this cause are
the fatigue crack progresses across the minimized. This is the purpose of this
blade section, the level of the steady present chapter.
stress on the remaining part of the sec-
tion increases, and final fracture takes
place as this stress rises above the CHARACTER OF FATIGUE
tensile strength of the material,
The problem of metal fatigue has been
obviously, the failure of a blade in of importance in mechanical engineering
this way - especially in an axial compres- since the first days of the railways. In
sor of many stages -can create much secon- fact some of the material testing methods
dary damage, as the released portion of used today are developed from the early
the blade ricochets amongst the stators work of A. Wo'hler (c.1860) who was in-
and high speed rotor blades. It is not volved in the investigations of the fail-
uncommon for all the blading of a compres- ures of axles of railway wagons, Koore
sor to be fractured or severely damaged (1927). since that time much has been
after the failure of an early stage rotor achieved to aid our understanding of the
blade. The extent of this secondary reasons and factors controlling the
damage is both governed (in a very complex problem. However, our knowledge of the
way) by the design and spacing of the capability of a material, and so the com-
blading, and (to a large extent) by the ponent, has still to rely upon empirical
impact and mechanical properties of the data obtained from special "metallurgical
bladinq. type" fatigue testing, or actual fatigue
tests on components. The aim of this sec-
Published photographs of the compres- tion is to introduce the factors which
sors subsequent to this type of failure affect the fatigue strength of a
are not common, due, no doubt to the component. These have to be accommodated
understandable reluctance of manufacturers in any assessment concerning the impor-
to release this type of information. tance of a level of vibration of a blade.

MVMI1D UNDIM

HtWUlOOH •jt
.00 * OH LOB •
Jiofl I'M

O, C I C l l l

Figure 1. Major Compressor Failure Due To

Secondary Damage After Blade
Fatigue Fracture. Figure 2. High Cycle Fatigue Properties
16-2

Alternating Stress Influence of Steady Stress

Wohler developed a testing machine, In an operating environment it is

in which a cylindrical type specimen was rare for a component to be subject to an
rotated about its axis while subjected to alternating stress alone, and so experi-
a static bending moment. mental studies have established the influ-
ence of an applied steady stress in con-
These tests showed that failures junction with the applied alternating
could be obtained when the alternating stress.
stress levels (± value) were between 1/3
and 1/2 of the ultimate tensile 6 strength A number of these empirical studies
of the material for lives of 10 to 10' were correlated by Goodman, Moore (1927).
cycles. With higher alternating stress He showed that for a given life there was
levels, the test specimen broke at •.re- an approximately linear relationship of
duced number of reversals. It is still the peak tensile 'stress on the specimen as
the standard practice today to present a the steady stress was reduced from the
material's ability to withstand applied tensile stress to the tensile component of
alternating stress on a diagram where the the purely applied alternating stress.
alternating stress is plotted against the
number of reversals to failure at this It is now common practice to use a
stress. Because of the high number of re- slightly modified version of Goodman's
versals involved, it is normal to plot the original correlation. This is shown in
number of reversals on a logarithmic Fig. 3. The Modified Goodman line con-
scale. These diagrams are known as the nects the value of the endurance stress
S-N curves. A set of test data, for a for a high number of reversals (>105) to
standard titanium alloy (Ti -6A1 -4V), the tensile strength, or creep strength in
which is used for compressor blading, is the ease of high temperature applications.
plotted in Figure 2. The tensile strength As Passey (1976) points out, the Modified
of this alloy is 990 MN/m2. Thus, the Goodman diagram is pessimistic for some
ratio of alternating stress to tensile materials, but is on the optimistic side
strength for an endurance of 10" cycles is when specimens are notched.
0.51.
Because of the ease with which the
Further data which shows the influ- diagram is constructed, coupled with the
ence of, manufacture method, microstruc- expense and difficulty in obtaining data
ture, and section size, for different from material tests under combined load-
materials can be obtained from text books ing, the Modified Goodman diagram tends to
and reference manuals, Heywood (1962), be used in design and assessment work, in
E.S.O.U. (1983). the absence of actual test data.

For some materials, e.g., steel

alloys, it is found that failure does not Notch Sensitivity
occur for lives above 10' cycles, if the
applied alternating stress is below a cer- The standard S-N properties of a
tain value. This threshold stress value material are obtained from tests on speci-
is known as the fatigue limit of the mens which are designed to give a constant
material. For a life of a specified num- stress over the test length. To overcome
ber of reversals 10N, the corresponding scatter in the test results, it is neces-
stress is called the endurance stress for sary to use highly polished surfaces for
10N cycles. For some alloys the fatigue the test section, and so remove the
characteristics are found to have two effects of surface finish irregularities.
slopes. At high alternating stress levels It was also found that the fatigue results
the gradient is steeper with a sudden kink were also dependent on the specimen size,
or 'knee1 in the S-N curve at the transi- the larger specimens giving lower results
tion point as the curve becomes a flatter (although work by Kelly (1971) has shown
line for low alternating stress levels. that some of these effects may be due to
The actual shape is very dependent upon residual stress concentrations). In
the material type, its condition, and addition, it was established that all
temperature. materials did not have their properties
changed in the same way with a common
stress concentration.
This gave rise to the concept of
"Notch Sensitivity" of the material.
The definition normally used in the
fatigue of materials for notch sensitivity

Where q = notch sensitivity index

Kf • the fatigue strength reduction
OKIUTnia •!««•• TIK8H.I (TMMTH factor actually determined,
•TIAPT (TRIM
i.e., ratio of fatigue strength
of unnotched specimen to that
of notched specimen
Kt = the theoretical stress concen-
tration factor
Figure 3. Modified Goodman Diagram.
 16-3

The reason for subtracting 1 is to Fracture Mechanics and Fatigue

provide a scale for q which goes from zero
(no notch effect, i.e., K f =l) to unity In the early development of linear
(full theoretical effect Kf=Kt). elastic fracture mechanics, G. R. Irwin
studied the stress field in front of a
The value of q is dependent on the growing crack in an elastic material. He
alloy strength, grain size, specimen size showed that the stresses died away propor-
and degree of stress concentration -e.g., tional to the inverse square root of the
notch root radius -but it does enable distance along a radius vector from the
materials to be compared for their ability tip of the crack. He found that the pro-
to withstand notches. Typical values for portionality was a function of the applied
blade materials are given below, but the stress and the square root of the crack
values relevant for a particular component length, and termed this 'the stress inten-
will depend upon the above factors and so sity factor', K. Irwin (1957). For a crack
its method of manufacture and component of length 2a and an applied remote tensile
heat treatment. stress o , for a through crack in an infi-
nite plate the stress intensity factor is
Typical q values are: given byi

- for aluminum and titanium alloys 3/rra"

q • 0.5
The subscript I relates to the mode of
- for steel and nickel-base alloy opening of the crack, i.e., tensile in
q • 0.35 this case.
In the application of fracture me-
Low Cycle Fatigue (L.C.F.) chanics to fatigue, the rate of crack
growth under cyclic loading is related to
In an operating engine, a large num- the value of the cyclic change in the
ber of components are subjected to a pro- stress intensity factor at the crack tip.
gressively increasing stress, as the The resistance of a material to crack
engine accelerates from idle to full growth is established experimentally. For
power. Throughout the life of the engine these tests, a thick specimen known as a
these components will therefore experience compact tension specimen as shown in Fig.
many cycles of this repeated stress. With 4 is usually used to provide uniform con-
components like disks, shafts, and casings ditions at the crack front and the speci-
which are highly stressed, a fatigue crack men is subjected to an oscillating tension
can be initiated at a relatively low num- between two positive values (°min, °max).
ber of reversals - say 3 x 10* - if the Thus the stress intensity at the tip of
applied stress is high enough. the crack will vary between Kmin and
Kmax. Since both are dependent on the
Because of the seriousness of a fail- crack length, so will their difference.
ure of this class of component, much work Thus, A K = °max(l- -^j^ fOa) . Normally
has been done since the early 1960's, to
apply the analysis of Fracture Mechanics °min is kept small. During the test, the
to the propagation of a fatigue crack or length of the crack is measured either
the development of cracking from a defect. optically or by an electrical method and
It is currently not feasible to apply is plotted against the number of cycles.
fracture mechanics analysis to the problem From the slope of this plot it is possible
of blade failure caused by high cycle to obtain the rate of crack growth, da/dN.
fatigue. But a brief introduction to the
topic is given for completeness. It is
possible from its application to material For fatigue cracking it was proposed by
tests, to obtain a better understanding of Paris et al. (1963), that the rate of
the influence of geometry, size, and crack growth per cycle could be a function
stress concentration in the results of of the power of the stress intensity
fatigue test specimen, Burdekin (1981). factor range, under the cyclic loading.

CRACK
OAOWTH (IATC

I CUMUCNT I/O

Figure 4. Compact Tension Test Piece Figure 5. Range Stress Intensity Factor.
Dimensions (mm).
16-4

two major factors. In general the unknown

Thus § = C(*)« fluctuating nature of blade vibration am-
plitude is often so great that the evalua-
resulting in plots as shown in Fig. 5. tion of da/dN is unreliable, also the
operating stress field is not sufficiently
At very low values of AK there is a constant in the crack zone to be consis-
threshold below which the crack appears tent with the underlying analysis of frac-
not to develop, while in the central zone ture mechanics. In addition, the bulk of
most materials exhibit a fairly linear the blade life is associated with crack
relationship. At high values of AK , the initiation which cannot normally be dealt
crack propagation rate becomes high as AK with by a fracture mechanics approach.
approaches the critical values *MC, which Once a crack is established then the re-
is termed the fracture toughness of the maining life is short because the high
material. With this value the propagation frequency of stressing propagates the
of the crack is extremely rapid to event- crack very quickly.
ual failure.

Application of Fracture Mechanics Residual Stress and Surface Treatments

Apart from an improved understanding It has already been explained that
of 'metallurgical type1 fatigue test data, the fatigue strength is influenced by the
and the characterization and comparison of presence of a steady stress. However,
the fatigue properties of materials, the this steady stress need not be due to an
main application of fracture mechanics has external force system, but may arise from
been to the analysis of low cycle fatigue an internal system of stresses. One sig-
(L.C.F.) life of disk and other similarly nificant cause is residual stresses re-
stressed components. If the fracture face sultant from the method of manufacture,
of a component which has undergone L.C.F. either forging or machining. In some
testing is examined under high levels of instances these stresses result from
magnification, for example with an elec- cold setting of blading to correct
tron beam microscope, it is possible to stagger errors, or forging manufacture
identify striations on the surface. distortions. Obviously, the value of
Laboratory test piece specimen analysis these stresses is not known, but their
has shown that each striation is associ- variability must contribute to the scatter
ated with a cycle of fatigue loading. It in fatigue strength of blades.
is therefore possible to obtain values of
da/dN by measurement from the fracture The surface finish of a component has
face of the disk. This information, to- a significant influence on the fatigue
gether with the crack propagation rate of strength, because the vibratory stresses
the alloy, enables the operating stress will be highest at the surface. The sus-
level to be determined, together with an ceptibility to fine polishing scratches
estimate of the number of cycles since will depend upon the notch sensitivity of
crack initiation. Such data are valuable the material, and the thickness of the
evidence when seeking the conditions which local blade section, for example at the
have given rise to a failure. The appli- trailing edge. In most cases, these sur-
cation of fracture mechanics to this type face effects can be mitigated by super-
of component also enables crack propaga- imposing a residual compressive stress in
tion rates to be established with confi- the surface and this can be achieved by a
dence under the stresses caused by L.C.F. cold work of the surface by controlled
conditions, which then permits component shot peening, or vapor blasting, for
lifing techniques to be adopted. However, example. Pig. 6 shows the improvements in
the necessary conditions for fracture fatigue properties obtained by this tech-
mechanics to be applied do not exist in nique, Metal Improvement Co (1980), Hanson
the case of compressor and turbine (1971).
blading. This is because of the following
Other surface processes like plating,
annodic treatments, or antierosion or oxi-
dation coatings may reduce the properties,
by the creation of a residual tensile
OAMAU MtSTHtM •!••«• I W • stress, or by the fact that the
coating itself lacks ductility and will
cause micro cracking which then propagates
110 into the parent material. The influence
of these effects is best established by
IN
carrying out fatigue tests on actual com-
ponents with the treatment applied.

Material Structure
I Illllj I I ihllll l I il.llll l I ll.llll
In most materials which are used for
CYCLIl TO rULUM blading, the fatigue properties are depen-
dent upon the grain structure of the
metal, which is determined by the way the
component is made. A forging, in which
the aerofoil is forged to size from an ex-
Figure 6. Shot Pee n ing as a Means of truded bar stock, provides possibly the
Overcoming Prior Fatigue Damage best grain flow and with the optimum heat
with 4340 Steel Tested in treatment, the finest grain size, to ob-
Rotating Bending. tain the highest fatigue properties for
 16-5

aterial. A titanium blade made by with a higher number of reversals to fail-

this method will have superior fatigue ure then there is a larger scatter in
properties by some 20% above a similar life. For low cycle fatigue at say 10*
blade machined from conventional bar cycles the scatter in life is more of the
stock. Also, within the same aerofoil, order of 2:1.
the fatigue properties will vary, depen-
dent upon the degree of forging work done Fretting Fatigue
at various stations. Thus, the properties
of the material around the junction of an One major problem which occurs in
aerofoil and a snubber will be different engineering structures is the phenomenon
than those in the aerofoil midway between known as fretting. It occurs when twd
the platform and the snubber. components are loaded together, and when
the interface is subjected to an oscillat-
ing shear type of force or movement.
The importance of process control, of Under theso conditions the surfaces of the
both temperatures and degree of working, interface become marked or damaged, and
at the various stages in manufacturing is this is known as fretting damage. These
given in the section on process control of situations exist in the blading of gas
titanium alloys p. 82, Meetham (1981), turbines, where the blades are connected
Figure 7 showing the difference in grain onto the rest of the engine structure,
size between standard and premium quality e.g., rotor blade fixing onto the diski
of Ti 6AL 4V is taken from Meetham stator vane outer fixing into casings and
(1981J. the inner ends of variable stator vanes.
Laboratory tests, Waterhouse (1981) , shows
that the fatigue strength of some mate-
rials can be reduced to 1/3 of the stan-
Scatter in Fatigjje dard strength. Attention to detail design
and antifretting coatings, Waterhouse
In previous sections, a number of (1981), can help to reduce these serious
reasons for scatter in fatigue properties effects.
have been given. However, to these must be
added the geometric variations between
blades which are made to comply with the Variation of Stress _Lev_e_l
same drawing. These differences will
arise due to the manufacturing tolerances In the majority of engineering appli-
on thickness, chordal length, blade cations, levels of alternating stress are
length, aerofoil shape and local varia- not constant with time, and so considera-
tions in streaa concentration areas like tion has been given to ways in which this
fillet radii, and trailing edge thickness. might be accommodated in any analysis to
Thus, if a large number of blades are vi- establish how a variation in amplitude
brated in the same mode and at the same will affect the fatigue strength. For the
level of vibration, then there will be a analysis of gust loads, on aircraft struc-
scatter in time to failure. As might be tures, Miner (1945), suggested a relative-
anticipated, where the stress at the crack ly simple method of damage accummulation.
initiation point is affected by a large His concept was, that the accumulated
number of factors, the failure lives fall damage can be expressed in terms of the
into a log normal distribution, i.e., a number of cycles applied, divided by the
normal distribution of the logarithm of number to produce failure at the given
blade life. Figure 8 is taken from Hunt stress level. Further, this cumulative
(1972) which shows that the scatter in damage theory assumes that failure from
life from the shortest to longest life of fatigue will occur when the summation of
a group of 44 blades tested under the same the various contributions of damage equals
conditions is 200:1 and that their distri- unity.
bution approximates very closely to a log
normal distribution. It is a feature of
fatigue properties of components, that

•• » 1 I I • W

(M
STANDARD QUAUTV PREMIUM QUALITY

Figure 7. Typical Macrostructure of

Standard and Premium Quality
Ti 6-4 Forging Stock (350 mm Figure 8. Log Normal Probability Plot
Diameter). (44 Blades in Fatigue).
16-6

nicks may also have incipient tear cracks

Thus I# « 1
at the root of the notch. The actual
shape and type of crack will be dependent
when n = number of cycles applied at upon the detail of the foreign body as
stress S well as its size, velocity, and the tan-
gential speed of the rotor blade. Only by
N = number of cycles for a specimen fatigue testing correctly damaged blades
to fail at stress S alone can the loss of properties be correctly
determined.
In Chapter 12 of Sires (1959), Miner
considered some of the other more complex
proposals to integrate the fatigue damage. OBSERVATIONS ON BLADE FATIGUE
However, the more extended specimen test-
ing needed for the alternative method was During the running of research com-
not considered Justified. pressors, or in the early development of
aero engines, fatigue cracking or failure
Indeed even today Miner's simple pro- of blading may occur.
posal , given above, is widely used for the
usual blade materials -aluminium, titan-
ium, and steel. This is because the This experience over the years has
resonable accuracy given by the method allowed certain aspects of the blade
does not justify employing a much more fatigue problem to be appreciated and
complex analysis. techniques developed, in addition to those
covered in the previous section which
Corrosion, Erosion, and Damage can be of value in the task of overcoming
these fatigue failures. Those associated
The blade surfaces of a gas turbine with fatigue rather than amplitude assess-
are often operating in a severe ments will be covered in this section.
environment. Because of the high, air or
gas, speeds over their surface any dirt or
grit in the air will scour the surface and
may damage any protective coating or Fracture Characteristics
treatment. The corrosion which will
follow will cause a loss of properties by The fracture face of a fatigue fail-
two mechanisms. There is the stress rais- ure has a very distinct and unique
ing effect due to the geometry of the cor- appearance. The fatigue zone has a rela-
rosion pit, and also there may be a detri- tively smooth appearance compared with the
mental change in the chemistry of the near crystalline texture of the final ten-
material on the exposed surface, with sile fracture zone. This is because the
possible preferential attack along the fatigue crack propagates in a transgran-
grain boundaries. Losses in properties ular manner in a series of very small in-
due to these effects must be considered crements; each increment notionally being
when determining the service life of associated with a stress cycle. In the
blading. case of low cycle fatigue with the higher
stress levels, this is indeed the case,
In the case of local damage to blade and the analysis of electron microscope
aerofoils as a consequence of small ob- examinations permits fracture mechanics
jects (metal or stones) being ingested in- techniques to be applied. However, with
to compressors, the blading may suffer the more complex amplitude histories in-
significant nicking on the leading edges. volved with blade vibration, similar de-
The loss in fatigue strength which results ductions are not possible, but the short
from these nicks, depends upon the notch periods of high vibration do produce the
sensitivity of the material having been concoidal marking which are centered about
subjected to the manufacturing and heat the fatigue nucleus. A photograph of a
treatment processes. In some cases these fracture face of titanium blading is re-
produced in Fig. 9. This type of fatigue
cracking exhibits practically no ductility
until the final tensile failure. Thus, it
is very difficult to detect the presence
of the crack development in an operating
compressor. The high steady stress in the
blade form also causes the crack to
propagate quickly, and as a consequence it
is unusual to find cracks in compressor
rotor blade aerofoils before one blade
fails. When the cracking nucleates in the
blade root fixing, where the mean steady
stress is lower, then the cracks are slow
to propagate and so cracks in the roots
may be found prior to failures.

Influence of Alternating Stress Level

In engineering components it is the
stress concentrations present which locate
the positions of the fatigue nuclei. When
Figure 9. Typical Fracture Faces Due to the stress level is very high, then exper-
Fatigue of Titanium Compressor ience shows that fatigue cracking will
Blades. start at a number of sites apparently
simultaneously. A good example is repro-
 16-7

duced in Fig. 10 taken from Armstrong (ti) What further test program should you
(1966a) where a shaft was subjected to follow when the first blade has run for
high alternating stress ± 40 tons/sqin 10' reversals and is unfailed? The neces-
which caused failure in 5 x 104 cycles and sary test time with 100 Hz blade is nearly
multiple cracks were found. A bladed disk 30 hours,
may be considered to be an assembly, with
many similar stress concentrations in (c) How do you run the second test if the
the blading. If it is found that a lot of first test failed early; was it a weak
fatigue crack systems are present, then it blade at the bottom end of the scatter, or
is most likely that they have been caused was the amplitude too high? Remember, the
by a high amplitude -for example a strong mean life will vary by at least 10:1 for a
flutter condition. 20% change in stress level, Koff (1978),
Armstrong (1966b).

However, if only one blade is cracked (d) How do you plan a series of fatigue
or broken then it is likely to have re- tests to solve a development or manufac-
sulted from a moderate amplitude, maybe a turing problem and be able to fix a
resonance. Of course, it may be that be- complet ion date?
cause of the spread in natural fre-
quencies, only one blade was vibrating. (e) Can you do anything worthwhile with
Such a situation is not usually the case. only a small number of blades; for
example, three or four?

Determination of Properties

It is often necessary to establish

the fatigue strength of a blade. Two typ-
ical reasons are (1) to provide data for
the assessment of the amplitudes of vibra-
tion measured in an engine, or (2) to
establish the change in properties which
may result from the adoption of a, dif-
ferent method of manufacture. The previ-
ous section covers many of the reasons why
the fatigue properties of the blade may be
expected to differ from the figures which
are obtained from a series of basic
"metallurgical type" fatigue tests. One
method of carrying out fatigue tests U) F O R W A R D IANO
on components is covered in detail in
Armstrong (1966b).

However, the problem remains concern-

ing how the test program is to be con-
trolled. As explained above it is stan-
dard practice for metallurgical laboratory
tests to be carried out and presented in
the form of an S-N diagram. where great
attention is paid to the surface finish
condition, i.e., mirror finish and stress
relief in a vacuum, Kelly (1970), then
scatter is small, and one or two specimens
only need be tested at each stress level
required to determine the shape of the s-N
curve at stresses above the fatigue limit.
However, where the surface finish is not fbl CENTRE IANO
to this high standard, and where possible
residual stress is present to some degree,
as will be the case with engine com-
ponents, then the scatter will be very
significant, and it is necessary to test
at least 3 to 4 blades at each stress
level. Because of the large variation in
life - say lOjl - and also the fact that
the mean life is initially unknown for a
given level of vibration amplitude, it is
often a very time consuming and costly rig
test program to establish the S-N curve
for blade components.

These practical problems can be

illustrated in the following questions:
(el H E A N W A M D *ANO
(a) What should be the amplitude of the
first test where knowledge of the proper-
ties are not known to better than 20-30%?
This lack of knowledge may be due to a new Figure 10. Cracks Beneath Stiffening Bands
blade shape, new manufacturing process, on Completion of Endurance
new material. Test.
16-8

Some of these problems can be avoided ysis based on Miner fatigue damage simula-
by the adoption of an incremental method. tion, and with the statistical tests,
Hunt (1972), Armstrong (1966b), Armstrong Students 't1 and 'F test1, see "Blade
(1967). The purpose of the test is not Fatigue Capability," a more formal assess-
to provide data which replaces the basic ment of the results may be obtained.
metallurgical type of S-N curve, but to Because a valid result is obtained from
provide an effective engineering answer to each specimen, an informative result can
the questions; how do these two standards be obtained with as few as three
of blade compare in fatigue properties? specimens. The limited duration for each
What is the scatter and mean fatigue test also allows the test sequence to be
strength of this blade for use in asses- planned, which is often a useful advantage
sing the level of vibration in an engine? when faced with an engine development
Remember, the blades in an engine do not problem.
vibrate at constant amplitude, and are
only subjected to relatively short periods A comparison between constant-
of vibration during the engine life, and amplitude and incremental tests is hard to
so the results of an S-N curve cannot be form, as it depends entirely upon the pur-
applied directly. This is dealt with more pose of the test.
fully in the section "Amplified Ratio
Method." If the purpose of the tests is to
establish a "full" knowledge of the
In an incremental fatigue test such fatigue properties of the blade, then per-
as that illustrated in Figure 11, the am- haps the constant amplitude tests are the
plitude of vibration is increased regular- best. This will involve testing over the
ly every half hour. The starting level of full S-N curves, or say from 10* to lo"
vibration is standardized for a particular reversals, although to cover 100 hours for
material so that results will be more a frequency of 3 kHz the number of rever-
comparable. A value of about one third sals would need to be 1 x 10'. A number
the failure amplitude is normal, and en- of specimens will be needed to establish
sures that time is not wasted at very low the degree of scatter. This type of test-
ineffective amplitudes, but also provides ing is hardly, if ever, required other
ample accommodation for specimens with than for research purposes. Also, in
very low fatigue strength. The time axis doing this type of testing, certain data
was selected in hours rather than number like the slope of the S-N curve for the
of cycles, as engine life is in hours material are reassessed. One criticism of
rather than reversals, and so the stan- the incremental test is that the number of
dardized test method will be equally rele- reversals during an amplitude increment is
vant to blades of all frequencies. During low. For 300 Hz these are 5.4 x 10s re-
the test each blade is stepped through the versals in half an hour which of course
increments of increased amplitude each could be extended at the expense of a
lasting for 30 minutes. longer testing time. However, the testing
procedure integrated in an assessment
The incremental steps are fixed at a method, Armstrong (1966a and 1966b), seems
constant value. If the amplitudes are to provide a very satisfactory method.
measured in 'af (see the section "Mechan-
ical Aspects of af"), then this is either
0.5 ft/sec or 0.2 mHz. At some amplitude
level, the blade will fail, and the time
in the increment is recorded together with
the level and site of fatigue crack on the
component.
The major advantage of the above
method is that all specimens are taken
through the same test cycle until failure
occurs. By plotting the results in a
simple form. Figure 12, a ready appraisal
of the relative strengths of two groups of
blades can be obtained. By using an anal-

Figure 11. Incremental Test Programme/ - Figure 12. Comparison of Two Test Groups
Fatigue Failures (Example©). Tested by Incremental Method.
 16-9

Strength and Life The format is similar to the conven-

tional Campbell interference diagram, with
axes of frequency and rotational speed.
Problems in fatigue generally fall However, in the figures, the brightness
into one of two groups. They are either indicates the amplitude of vibration, and
those dominated by problems of low cycle is usually proportioned to the logarithm
fatigue and repeated high stress levels, of the amplitude. To obtain this form of
or those of vibration and high cycle presentation, of the response from a blade
fatigue. Two of the features of fatigue strain gauge, the amplitude signal from a
which contribute to this identification of frequency analyzer is used to modulate the
two problem classes, are the shape of the intensity of an oscilloscope. The y axis
S-N curve of materials, and the distribu- deflection of the oscilloscope is driven
tion scatter. from the signal from the analyzer propor-
tional to the analysis frequency while the
In the case of the repeated high x axis deflection of the oscilloscope is
steady stress, the level of applied stress controlled by a signal proportional to the
is relatively well-known, and so with rotational speed of the engine as derived
knowledge of the S-N curve and the reduced from the tacho signal. During a slow
scatter with high stress, the life of the acceleration, the display is photographed,
order 10* is also known. A change in and at 10% increments in speed, the
material processing which changes this frequency calibration marks are applied,
life by a factor of 3 is very relevant. thus producing the array of spots which
However, in the case of high cycle are used for reference purposes.
fatigue, the levels of vibration are not
known or measured to anything like the Fig. 13 shows the result of a single
same accuracy as the L.C.F. stress levels strain gauge attached to an early stage
and typically a factor of 2 is relevant. compressor rotor blade. There is suffi-
Thus, it is more meaningful to quote the cient turbulence in the air stream to
results of a comparison of fatigue proper- cause the blade to respond at its natural
ties between two groups in terms of stress frequencies, and to give a strain gauge
level, rather than in terms of mean life. signal above the background noise level.
Remember that 20% in stress is equivalent The responses of the first flexural (IF),
to a factor of at least 10:1 in life. second flexural (2F), and first torsional
mode (IT) are clearly seen. Multiple
lines in the 2F zone are associated with
the effects of blade-disk coupling. For
FACTORS AFFECTING BLADE AMPLITUDES OF the flexural modes, the rise in natural
VIBRATION frequency with centrifugal stiffening is
clearly seen; this is absent on the IT
A method of assessing the severity of mode, thus confirming the modal
blade vibration must make allowance in identification.
some way for the factors which control the
vibration. These factors can be summa- The sloping lines of various intensi-
rized as the mode of vibration, and the ties are the strain responses of the ro-
form of excitation. The mode shape influ- tating blade, due to the air flow not
ences the fatigue strength and its being uniform round the compressor
scatter, while the excitation controls the annulus. In fact, in this engine there
level of vibration and the variation dur- are 3 intake vanes and this is the reason
ing the flight life of the engine. why the third sloping line labelled 3 E.G.
Excitation can be divided into two major for the third engine order is stronger
groups. Self excitation or flutter, which than the others. The brightness of the
is due to the interplay between the
blade's vibratory motion, and the result-
ant change in aerodynamics forces on the
blade, which is the subject of the earlier
chapters of this handbook. Flutter is a
serious problem, because the amplitudes of
vibration are limited by nonlinear
effects. The other class of vibration is
the larger group, and a number of sources
of excitation can be grouped together, be-
cause the response is due to the forced
resonant characteristics of a single or
multi-degree-of-freedom system.

Identification of Modes and Excitation

The identification of the modes of

vibration of a blade, or bladed assembly,
must be derived from the observed frequen-
cy of vibration, and a knowledge of the
calculated values or experimental frequen-
cies obtained on a static test for fre-
quency determination of the individual
blade. Modern signal frequency analyzers
enable the results from engine or compres-
sor strain gauge tests to be presented in
a compact and readily appreciated form. Figure 13. Frequency V Speed Plots Engine
Figures 13-16 are taken from Armstrong with Venturi Intake Component:
(1977). '0' Compressor Rotor Blade.
16-10

second engine order line is stronger The one marked 3D is so identified since
towards 100% than lower in the speed it is resonant with the 3 engine order
range. While the excitation strength is (EO) excitation. Notice also that the 2
expected to rise with engine power, an EO signal is stronger than the 1, 3, and 4
additional reason is the proximity of the EO components; this is because the engine
excitation to the IF natural frequency was run behind an aircraft intake with a
line. It will be seen that there is no bifurcated inlet.
resonance of the IF mode between 70%
of full speed-flight-idle, and 100%. In Because of the difficulty of accu-.
fact, as a result of this form of testing, rately predicting whether flutter will be
a minimum frequency limit was imposed on present on a compressor rotor blade,
the IF mode to ensure that a second engine strain gauge testing is usually carried
order resonance did not occur on a blade out to establish if flutter is in the
with a particularly low IF natural operating range. Fig. 15 shows an ana-
frequency. In the region below 60% engine lyzed strain gauge signal from a rotor
speed, and at frequencies between 200 Hz blade having part-span shrouds. For this
and 800 Hz, excitation lines additional to test the engine speed was progressively
the engine orders can be seen. These are increased, and the frequency data are pre-
due to the presence of rotating stall sented against time rather than engine
cells. speed, as is normal for a Campbell
diagram.
Fig. 14 is a reproduction of a fre-
quency analysis from a strain gauge on a In the early part of the record, the
fan rotor blade. The fan blade has a natural frequencies of the blades can be
snubber or part-span shroud. In this par- easily distinguished from the engine order
ticular engine test, the blades were excitations. At the flutter condition it
assembled with a clearance between the will be seen that the strain gauge signal,
snubber contacting faces of 0.060 inches. at the natural frequency, is greatly in-
Under the influence of the centrifugal creased. This is, of course, characteris-
field, the blades untwist, until the tic of self excitation, and unmistakably
inter-snubber gap closes, and then the different from the response due to a
snuhber ring acts as a ring coupling the forced resonant type of vibration. Since
blades together, and producing assembly, the flutter involves a periodic change in
rather than blade modes. the aerodynamics of the blading, the re-
sultant variations in pressure can be de-
The characteristics of the above tected on the casing. In Fig. 16 also
effects can be readily seen from Figure reproduced is a record of a pressure pick-
14. At speeds below 60% full speed the up installed in the casing at the time of
cantilever natural frequencies IF, 2F, IT, the flutter incident. It can be seen that
3F can be seen. Between 60% to 70% full the two signals are coincident in time.
speed, a signal which has intermittent
components over the whole frequency range However, the difference in the fre-
is evident. Experience has shown that quency of the signals is because the
this type of signal is usually present strain signal is taken from the rotating
when two surfaces are in intermittent blade, and the pressure pick-up is on the
contact. Above the 70% speed condition a casing. By using these two frequencies,
new set of natural frequencies are it is possible to establish the number of
present. These are the natural frequen- lobes in the rotating pattern and the
cies of the bladed assembly, all the speed of rotation.
blades being coupled with the ring formed
by the snubbers. The modes are character-
ized by the presence of nodal diameters.

*ffl
-'::-i:^:;
^MwRLsi
.S-i 'i^K^^W'1
Figure 14. Frequency V Speed Plots Engine Figure 15. & 16. Frequency V Time Plots
with Aircraft Intake Component: Engine Research Compressor Test
1st Fan Rotor Blade 0.060" Component Fan Rotor Blade and
Snubber Gap. Casing Pressure PU.
 16-11

Amplitude - Time Variations

1 11
Amplitudes of blade vibration at a v
Loss Factor (n) Log Decrement (8)
steady engine condition are seldom con-
stant with time, and are only so when at Mode Loq Decrement (6)
the peak of a forced resonant condition of
high amplitude or a serious flutter IF .04 - .08
condition. Examples of stress against
time can be found in early references. IT .02 - .03
Carter (1957), or more recently in
Cardinale (1980), from which Figure 17 and 2F .005 - .015
18 have been taken. These show responses
in conditions of turbulent flow and rotat- 2T .01 - .03
ing stall, while Figure 18 shows condi-
tions of stress at the strain gauge in a 1-2S .005 - .01
flutter condition.

Variations of amplitude with engine It is also stated that data of axial

speed will occur, due to the change in dovetail designs appear to fall in the
aerodynamic conditions, e.g., rotating above range. When it is remembered that a
stall and resonance. Typical results can reduction of 20% in stress amplitude re-
be seen in Fig. 19 and is reproduced from sults in at leaat an increase of more than
Armstrong (1960) and Fig. 20 is reproduced 10:1 in life, then the potential life at
from Cardinale (1980). It will be seen stresses less than half the peak amplitude
that, in both cases, large amplitudes of will be between 103 to 10$ times longer
vibration exist over a relatively small than that associated with the peak stress
increment of speed range, as a result of level. Therefore, because of the high 0
resonances between the natural frequencies of blading, it is only necessary to con-
and specific engine orders of excitation. sider the response at their natural fre-
The shape of these amplitude speed curves quencies, and not the responses at of f-
is controlled by the total aerodynamic and resonant conditions.
mechanical damping in the system. Typical
values from published papers, e.g.. Hunt
[1972), Armstrong (1967), give values of Q
of about 30 which correspond to log dec of
0.1. However, values for high modes are
given in Cardinale (1980) for integrally
bladed assemblies.

PUCUTUM COMFHCB«Oft BPUB

Figure 17. Induced Flow Vibrations

Figure 19. Engine Order Vibration,

Figure 20. Engine Order Excitation Due to

Vanes.
Figure 18. Self-Excited Vibration,
16-12

Characteristics of Types of Vibration greater use and ability to help diagnose

the cause of failure, it was realized that
The assessment of the degree of vi- by their early application, it should be
bration of a blade or vane, in terms of it possible to detect potential failures and
causing a fatigue failure, must be made on their source of excitation. This would
the evidence from one or more blades of a then enable early modifications to be made
stage, and possibly from a limited amount prior to any failures, and thus to avoid
of engine running. As previous sections their attendant disruptions to engine de-
have shown, the vibration can be due 'to velopment programs. However, this aspect
one of many sources of excitation. of the work places new requirements on the
However, the consequences in terms of con- assessment method which is adopted.
sistency of the stress level is very de-
pendent on the particular type of excita-
tion under consideration. The Table, Assessment Requirements
Figure 21, has been drawn together from
Danforth (1975, 1974) Armstrong (1967,
1960), Cardinale (1980). The data in this The prime requirement of any assess-
table provide a good indication of the ment method must be to establish whether a
dependence of the amplitude of vibration particular stage of blading will give a
on the operating conditions of the engine, satisfactory service life. As the intro-
but it must not be taken as exclusive of duction to Cardinale (1980) states, "the
additional factors, nor new types of prob- evaluation of aeromechanical behaviour
lems which will become evident as the rat- must consider practical operational
ing and performance of compressors and effects and sensitivities, including air-
turbine designs are extended. The section craft maneuver and flight transition dis-
"Design Assessment" addresses the problem tortion, and the integrated effects of a
of how an engine test series should be number of other variables, including vari-
conducted. able geometry, bleed, power extraction,
operating line, and other engine and inlet
transient conditions, such as those asso-
METHODS OF ASSESSMENT ciated with environmental and weapon
delivery gas ingestion. The long range
Strain gauges were first used on com- effects of deteriorations, foreign object
pressor blading in the late 1940's, see damage, airfoil erosion and potential con-
discussion Carter (1957), to establish the trol malfunctions also need to be
reason for the fatigue failure of blading. addressed. Predictions of vibratory re-
This early work set the obvious style for sponses, fundamental mode instability mar-
investigations of this sort and the strain gins, and surge-induced stresses are not
gauges were positioned at the sites of yet adequate to eliminate the need for
fatigue nucleation. Under these condi- experimental validation of these effects.
tions of high alternating stress, and with Overall experience, guided by the aero-
the very early strain gauges, the instru- mechanical fundamentals, serves to estab-
mentation life was very short. However, lish systematic design verification proce-
as the quality of the strain gauges and dures with considerations given to the
their installations improved, their use total engine system."
became more commonplace. With this

IHOIHB/ftlJlClUn
coHBincm ROMTIBQ ITALL MR) EMI Ml OKPBH VM1ATICTI

IVUIATXHa LIMB
Tbrottk* •ur.UcbopJ Bigh Opeiitln? Lift* •Igh Operating l>ia*
* C«a*ff«ll]r Aggravate*
* tfc«C bttrMf TraOBiont ItedUeO Marf ftU
Aff*et*4 lt«g» Depend* on
etatac ftehoftaU Betting « MXO Mfllgn Hay
Halfenotion Hay M4*o* taplitod** tapwt-
c«wo rlvtu* dant o* •efaadula
141* *p**d Variation Hay a*Milt la Longer Via*
Hay Can** Longer Kuanlng on K**ooanc*
Tin* in hulifcud* •*«•)».
HBTQUtlOU
Aircraft lauka
•> May 10cr*M* HargiB Can Caw** L*r«« Mtpll- iBUrodiMtion of Hay lUault in Largo
CUco»C*t**U«L tudM on S«9«*at of •tator ttakon
Likely to Reave* •tator Vaa«
ladial Hug in
Oc«ataa 3rd BagiM
CIMI HIM Ordar
• Magnitude Donend on
Hiftat Condition,
t.f., Incidence, low.
rjjfltr mt
Intake ToBvorotnco ••doetion of ftugie |p«*d R«ng* CoatEollad by IlfOdUOtlon Of *fi*M t*BOO*Jk oa At Migk «•• Duo to Drop
Corr*ot*4 BpMd. la lUtural frogjacoeUi wit* 1r**p*r«tac«.

Xaltt rr««««*« BeAaotloo of MAcgla tMtnaM ID Aaplitod* »oul»I* ZMn«M to ftjvllt*1*.

Hodtt of vlbMtio* ir or IT vltb Canti- tew)toaM Ht«h wd«s iMludlaa Hat*

lever Bledift* HftdO
be* Di«o*t*M «itb
•laded AioMtUoo
AnplitOde Varl*tiM Inpttltivo for tare* BiBiiar Max Jk^tltoto on Cl««*loal •vaooaiuw rook* taparatod DU* to
Vnry Baddon IncroMO All UaOu tor • Oondl- riiMt«at«a oa r**k •toady M P**k FroittMoy SomttM
la Aevlltode -Uh ttoa. Oftoa Hlok 'fi' Valw*i
•aglno Condition, to* >M«OM in CnM*ot«Jt
Flitter
Difference* BotVMH Very Variable for CoMi«t*nt totvooA >1*4M COMiltMt CQMlaUOt Vary VnriaMo
r tut tax
BlBiUr for *li •laJoa
io-laus

Figure 21. Character and Changes of Compressor Blade Vibration.

 16-13

In addition to the primary task of B. The surface finish and the finishing
assessing a particular design, it is also process, e.g., shot peened.
important to be able to relate the results
of a particular investigation with similar C. The actual local profile, e.g., any
data from other engines or rig test local undercutting or thinning.
results. This comparison across a field
of experience then allows a feedback to D. Variations in the manufacture process
take place into the design processes for heat treatment.
new engines. Ideally, this type of infor-
mation should be in a form which will be E. Method of manufacture whether it be
of use at the earliest project stage, and forged, machined, or degree of cold
often before the details of the mechanical setting.
design are finalized. Thus, there are the
two important aspects of the aeromechan- F. Level of steady stress in blading.
ical behaviour to be considered in the
assessment method; the excitation and the Each group will contain variations in
response in terms of ability to withstand the values of the factors, and so they
the vibration. will create some form of distribution as
indicated in the Figure 22. It is evident
Aerodynamic and Mechanical that when the two distributions are well
separated, i.e., the upper end of the
There is no doubt that the factors amplitude distribution is well below the
which are relevant in blade vibration low end of the fatigue capability, then
fatigue problems can be divided into two failure will not occur. It will be
major groups. This is illustrated in equally obvious that failure is certain to
Figure 22 which is taken from Armstrong occur when the fatigue capability is below
1980. The degree of vibration in the the engine amplitude distribution. The
operating environment is controlled by a most difficult problems are when the two
number of factors of which perhaps the distributions just overlap as indicated in
following are the most important, for a Figure 22.
particular condition of resonance.
The desire to separate the problems
A. Basic blade aerodynamics and nominal associated with blade fatigue capability
flow conditions. from the level of vibration in the engine,
has resulted in a method of assessment be-
B. Individual blade geometry, stagger ing developed which is an alternative to
angle, aerofoil shape, and thickness. the more direct technique of stress
measurement.
C. The mode shape of the blade which is in
resonance.
Stress Method and Amplitude Ratio
D. Aerodynamic factors controlling the Techniques
balance between excitation and damping.
The two methods which have been de-
E. The mechanical damping. veloped have much in common, and both are
effective in satisfying the basic require-
F. The flight condition and the mode of ment of evaluating the seriousness of a
operation of engine and aircraft. particular vibration incident. Unfortu-
nately, the degree of expertise invested
G. The consistency between engines and in each method, and the extensive exper-
aircraft. ience gained and succcess with its opera-
tion, rather precludes the possibiity of a
While these, and others control the serious assessment of the benefits and
level of the vibration, another group disadvantages of the alternative method.
establish the ability of the blade to In the following sections each method will
withstand the vibration, for example: be introduced and a brief comparison will
be attempted.
A. The material of the blade which will
control the fatigue strength. AMPLITUDE RATIO METHOD

The amplitude ratio method of assess-

ment has been developed by the Bristol
Division of Rolls-Royce, and is reported
in Armstrong (1966b, 1967), and is a de-
velopment of the early work on assessment
of blade vibration work which was reported
in Blackwell (1958). In this work the
level of vibration is determined in terms
of the product of tip amplitude of vibra-
MOMl <» WJUfO* (NOT MAI, tTMHl
tion of the blade (a), and the frequency
M AMOIMI 'ATIOWI CAPABILIT*
of vibration (f). As will be shown, this
cow mot »ACTOm
1. BIAM **MPf • •TAOQC*
com not r AC Tout product 'af1 is a very useful measure of
1 MATBMM.
the fatigue strength of the blade. It had
5: ."=« {HS^} 1.
1.
tUIVACI riNtftM
ACIkML LOCAL MKVM.I
been shown earlier in Pearson (1953) and
4. ttCCnAMtCAL OAMPtttg *. MAI TMJATMUT

Parry (1954) that the product 'af1 was al-

.. PL*., co-mo."""*1""
so a measure of the aerodynamic excita-
tion, assuming that quasi-static condi-
tions exist (i.e., zero frequency param-
eter) and also that mechanical damping
maybe neglected in comparison with the
Figure 22. Factor Influencing Failures. aerodynamic forces, This analysis is
repeated below.
16-14

Thus, while these aerodynamic assump- From the vector diagram

tions are not valid in absolute detail,
the approach does provide the ability to
relate the degree of excitation to other
similar experience without introducing the
detailed geometry of the blading. A de- (V0+vwsinpt)cosa0-x sinO
cision can then be made concerning the
strength of the excitation, and whether it
will have to be reduced, or whether the
blade can withstand it. It is also of U-(V0-t-vwsinpt)sina0+x cos 6------2
assistance in relating the measured level
of vibration to the fatigue strength of
the blading, and so predicting the likeli-
hood of blade failure in service. Expanding the L.H.S. and using the approx-
imation for small quantities
Aerodynamic Excitation and 'af'
cos4oi=l Svx6oj " 0
This analysis under quasi-static con-
ditions, i.e., zero frequency parameter,
is based on that given in Pearson (1953) we get
and Parry (1954) for wake excitation of
compressor blading.
Cascade Notation

Now from 1 & 2, for steady state

conditions, i.e., vw = x = Sv -

V0sina0+u-
Therefore, using equations 1 to 6, we
obtain
Vector Diagram

vwsinptcosao-x sine-

- vwsinptsinao+x cose 8

Multiplying 7 by cosai and 8 by sinaj and

adding we get
where
Vo absolute gas inlet velocity (steady) 6v=vwsinpt
Vi gas relative inlet velocity (steady)
Multiplying 0 by cosai and 7 by sino^ and
V0+vwsinpt absolute gas inlet velocity subtracting we get
with wake component
U blade speed
Now with e the force on the blade
F lift force on blade in direction of
vibration
x = a<usin(a>t+ E)
» - !*£ ««i + H 8vi and >
**
Therefore,
= vibrational velocity of blade
OQ absolute gas inlet angle -^)- |L8in(e-B1)]

01 relative gas inlet angle

C staqger angle
inclination of vibration to (J
direction Fo+Ax-Bvwsinpt
«v,6ai increments due to vibration
 16-15

The vibrational energy gained per The above analysis also holds in the
second by the blade is W with frequency case of flutter, but with vw » 0 . Thus,
for energy to be fed into the vibration to
ID overcome the mechanical damping, the term
2n ~jAa2u)2 must be positive. Hence A must
0 = mechanical damping work done per be negative but2 again the term is propor-
second. We have: tional to (af) .
In the next section it will be shown
that the product (af) is also a very use-
W=f /(-Fx ful and general method for specifying the
intensity of a vibration from a mechanical
aspect.

A - f-r xsinpt-D Mechanical Aspects of af

With Perhaps the first reference to veloc-
ity being an important criterion for the
x=dujsin( ut+e) assessment of vibration stress, was given
in Appendix 1 of the paper by H. G. Yates
2 ir (1948). He proposed the following general
proposition based on reasoning from a di-
W=-A~-/'!'a2<L>2sin2Ut+E)dt mensional type of analysis.
"Mechanical vibrating systems, having
ll geometrical similarity and constructed
-t-B-j vw /uausin( o)t+E)sinptdt-D of the same materials, when vibrating
freely in the same mode with equal
linear velocities, will suffer the
The value of the first term, which is a same vibrational stresses."
damping term, is independent of the value
of c whereas the second excitation term He then extends his reasoned argument
will be a maximum when u = p (i.e., on to the case of transverse vibration of a
resonance) and when e » 0. Therefore, cantilever beam. However, this can be
confirmed by mathematical analysis which
,.,2JL is given in full in Appendix I.
4" The analysis follows the standard
Assuming that the mechanical damping D type, e.g., Timoshenko (1937) for a beam
is small and can be neglected, we have for of constant section. The assumption is
a steady state vibration (i.e., when made that the beam is vibrating harmon-
M = 0 ) and for no increase in energy in ically with time and that the applied load
the vibration. intensity on the beam is due to the re-
versed mass inertias as a result of the
or af beam's motion. The constants in this
"f =f
A f*
2w general solution are determined by the end
conditions for the beam -fixed and free
Thus, the product of (amplitude of -for a cantilever. This, therefore,
vibration) times (frequency) is propor- yields the frequency relationship equa-
tional to the velocity strength of the tion, the roots of which define the
wake. natural frequencies. Using this frequency
equation in conjunction with the expres-
Clearly the expression: sions for the bending moment 1 at the root
section, and the amplitude 'a at the free
end of the cantilever, it is possible to
•«-*£- derive the following expression, which is
valid for all flexural frequencies since
the frequency relationship equation was
used in the analysis.

af
is not a precise prediction of the re-
sponse of the vibrating blade, but as was Where a = the tip amplitude of vibration
pointed out in Pearson (1953) and Parry
(1954) and in the Blackwell contribution f - the natural frequency of the
the discussion of Carter (1957) and flexural mode
Blackwell (1958) the product of af is a
very convenient method of assessing the k - radius of gyration
relative strength of the aerodynamic exci-
tation from the results of a strain gauge y = distance of the highest stress
test on a compressor or turbine. fiber from the neutral axis
a = stress on the fiber
E • Young's modulus
m = mass per unit volume
16-16

This is the important relationship The analysis given above applies to

for cantilevers, relating the product of the flexural modes only, since a similar
the (tip amplitude) times (frequency) with exact analysis does not exist for non-
the material properties of Young's modulus circular sections vibrating in torsional
E , mass density m and the alternating modes. However, Yates (1948) refers to
stress o at the root section. It has simple calculations for non-circular sec-
been shown by Blackwell (1958) and Yates tions, and states that the ratio of maxi-
(1948) k/y is reasonably constant for mum shear stress to maximum linear
similar section shapes. Thus, if the velocity is nearly equal to Gm
stress a is taken as the value for an where G is shear modulus. This is
endurance life of say 10' reversals for a borne out by correlations of fatigue tests
particular material, then all cantilevers on blades in the torsional modes, where
in that material will fail at the corre- the amplitude is measured at the leading
sponding fixed value of af irrespective edge. The early work reported by Backwell
of the length, breadth, and thickness of (1958) shows that the failing values of
the beam. af in torsion are very similar to those
Typical values of f=g* for blade type in flexure - see Figure 23 which is
reproduced from this reference.
materials are given in the table.
Young" s Endurance Whilst the relationships developed
Modulus Density Stress above for flexural vibration of uniform
cantilever beams are exact, the benefits
Alloy E GPa m Mg/m3 o MPa which are gained by using af as a measure
of the severity of vibration from a me-
Aluminium 72.4 2.71 145 10.35 chanical aspect, derive from the ability
to compare the performance and fatigue
Steel 214 7.83 591 14.44 properties of bladinq covering a wide
range of designs, sizes, and methods of
Titanium 113 4.43 550 24.58 manufacture. It is not intended, as
Passey (1976) comments, to be an alterna-
Nickel 214 7.86 340 8.29 tive to stress as a criterion of fatigue
Based of material, but rather as an indicator of
a the ability of the whole blade to with-
The relative values of •3rl~ in this table
stand vibration. This must, and does, in-
illustrate the ability of blading in the volve all the mechanical aspects of the
various materials to withstand vibration blade.
as a cantilever. It will be noted that
whilst the endurance stress of steel is
some four times that of aluminium, the The Measurement of af
actual performance as blading will be only
some 40% better. This is because of the A strain gauge positioned on a vi-
high Young's modulus and the higher brating blade in an engine, will indicate
density of the steel material. the strains due to all the modes which may
be excited. Each mode will of course be
The lower values of E and m for identifiable by the frequency content of
titanium enable the titanium blading to the signal. As reported in Armstrong
demonstrate its far greater superiority in (1960) it is possible to find a position
a vibration environment. on a blade which provides almost equal
response in a number of modes. The unit
Similar relative values obtained from of response for this comparison is strain
fatigue tests on blades are taken from gauoe output per unit tip af , and so to a
Armstrong (1960). The values being for first approximation the modal output from
107 reversals. the strain gauge is a correct indication
of the seriousness of the vibration.
Aluminium 5.5 ft/sec Initially, the position of the strain
Steel 6.5 ft/sec gauge was determined experimentally on a
Titanium 11.0 ft/sec static rig, where the blade was vibrated
Glass fiber laminates 12.0 to 13.0 ft/sec. in each mode in turn, for the modes of
interest, and the corresponding gauge out-
put and tip amplitude measured. Changes
to the strain gauge position normally re-
sulted in one being satisfactory for 3 or
4 modes. If more modes are required, then
an additional position can also be used.
In these calibrations, it is important to
measure the tip amplitude in an identical
way to that which is used during the
fatigue testing on the blade. Normally,
this is the leading edge, but when a nodal
line is close to this edge then it is more
reliable to measure the amplitudes at the
trailing edge for that particular mode.

Whilst the above methods relied upon

experimentally determined positions of the
strain gauge to cover a number of modes,
it is now possible to predict the likely
Figure 23. Compressor Blade Fatigue sites and orientations of the strain
Strength for Different gauges by working with data from the blade
Materials and Blade Designs. design detail using Finite Element (F.E.)
 16-17

analysis methods. It is expected that an A. Incremental and Constant Amplitude

empirical calibration will be performed to
calibrate the strain gauges prior to en-
gine build but the finite element analysis It was explained before that signifi-
route will minimize the experimental work cant benefits can be gained by using the
required. incremental fatigue test techniques when
carrying out test work, to establish the
The use of tip af , as a measure of fatigue capability of a blade standard.
the importance of the intensity of vibra- However, it is necessary to convert the
tion, has also led to the development of results into a form which can be compared
methods other than the use of strain with the amplitudes measured in the
gauges to measure blade vibration. One of engine. The method which has been adopted
these, which is particularly useful where is to convert the experimental fatigue re-
slip rings cannot be fitted to the shaft, sults to constant amplitude data. This
is the F.M. (frequency modulated) grid. then allows the effect of scatter to be
allowed for easily because, as explained
In this method - which is fully de- before, the scatter properties of a blade
scribed in Eccles (1962) and Raby (1970) - population form a normal distribution of
a small magnet is inserted in the tip of a the log life at a constant amplitude or
rotor blade. In the casing, above the alternating stress.
track of this magnet, is fixed an accu-
rately pitched zig-zag conductor, and the
near axial portions of this conductor are The conversion from the incremental
spaced at 2 degree intervals. As the tests to the life at a specific amplitude
blade rotates at a uniform speed, a series is done by the use of Miner's fatigue
of electrical impulses is generated in the damage summation. Thus for r steps in
conductor. Because of the uniform speed the incremental test:
and the accurate pitch of the bars of the
conductor, the frequency of the impulses
will be constant. If, however, the blade x=r n
is also vibrating while it is rotating,
then the frequency of the impulses will x
x= 1
vary. By the use of pulse shaping tech-
niques and filtering the signal it is pos-
sible to frequency demodulate the signal
and to produce a signal which is propor- represents a criterion for failure.
tional to the alternating velocity of the
magnet in the blade tip. As the signal is
proportional to the component of the Where N x is the life (number of cycles)
blade's vibrating velocity at right angles at the amplitude Sx and is the time
to the bars of the conductor, it is neces- increment (number of cycles) durino the
sary to calibrate the direction and ampli- fatique test at the amplitude Sx . It
tude of the magnet for the blade's modes has been found that metallurgical
of vibration relative to the blade's lead- constant-amplitude fatigue test data for
ing edge amplitude. blade materials when presented as a plot
of log (stress) against log (number of re-
As the signal generated is a versals) approximates well to a straight
frequency-modulated one, the signal line. Thus, the relationship:
strength -which is dependent upon the mag-
net to grid clearance -is unimportant, and SNd = constant
so the technique is not dependent on blade
tip clearance. Normally, 2 or 3 grids per holds and obtain
stage can be incorporated. The tempera-
ture limit is that of the materials used
to bond the conductor into the casing.
SN°=

The tip af method of assessment also

allows ready use to be made of the data Thus, by carrying out a small number
obtained from measurements of the blade of constant amplitude fatigue tests, it is
tip displacements as reported in Raby possible to obtain a value for the index
(1970) and Koff (1978) from probes mounted d for the material. From the above equa-
on the casing. tion it is possible to obtain values of N
and log N for a specific constant ampli-
tude of vibration.
Blade Fatigue Capability
As explained before, the fatigue pro- B. Scatter in Properties
perties of a blade depend upon many fac-
tors, and in general these are not known One of the difficulties in the
very accurately. So the Amplitude Ratio assessment of blade fatigue problems is
method relies upon a comparison betwen the allowing for the scatter in properties.
measured af in the engine and the fatigue However, this can be achieved in the
capability of the blade as determined by following way. From the results of a
test. To predict the strength of the small number of blades tested in the rele-
blade in the engine, it is necessary to vant mode, it is possible to establish the
apply a number of correction factors to mean of the log life of the group at a
the fatigue tests, and these will be specified amplitude.
covered in this section. The majority of
this section is taken from Armstrong
(1966b).
16-18

Because there are only a small number .(n-l)52

in the sample, their mean value is likely
to be different from that of the popula- X*
tion from which they have been taken. By *2
using a statistical technique, it is
possible to estimate within a 95% confi-
dence level, a lower limit on the mean of The values of x are obtained from
the whole group. This lower limit is statistical tables, found in most statis-
given by tics books, which are based on the
Biometrika Tables for Statisticians,
to Vol. 1, Table 8.

where t is the Students t value C. Effect of Temperature

for a chosen confidence level.
o is the best estimate of the To allow for the effect of tempera-
standard deviation of the population ture on the failing amplitudes of blades,
(see below) and Armstrong (1966b) proposed that test data
from metallurgical specimens from rotating
n is the number in the sample. bend tests conducted over the temperature
range may be used. This information would
allow a correction to be applied to the
For 95% confidence level the values mean property, and also the index 'd1 de-
of t for different sample sizes are: fining the slope of the log S vs log N
curve. In applying this it is assumed
that these changes apply directly to the
material in blade form.
4.3 3.2 2.8 2.6
In the absence of more detailed in-
With a knowledge of the mean of total formation on the actual variation of the
population, and an estimate of the stan- blade's operating temperature with engine
dard deviation, it is possible to predict speed, it is assumed that the metal tem-
the lowest log life in a large number of perature will vary on a speed squared law.
blades. Armstrong (1966b) considered the Therefore we have
lowest one in 10,000 which is 3.72o below T = 20 •*• N2 (TT - 20)
the population mean. These relationships R
are shown graphically in Figure 24.
In the previous paragraph, a value of Where ..R a actual rev/min
the standard deviation has to be used. An maximum rev/min
accurate value will not be obtained solely
from a small sample size, but it is not T *> metal temperature
unreasonable to consider that the standard
deviation o is common for a similar de- TT = metal temperature at maximum rev/min
sign of blade in a certain material.
Thus, it is possible to obtain a good
estimate of a from an overall considera- D. Amplitude for Given Life
tion of all the blades tested. Hunt
(1975) has done this and Figure 25 is re-
produced showing how the values of a tend
to definite values for each of the three
examples of typical blade materials. The
bounds for each of the trends are the chi-
square distribution for n-1 degrees of
freedom, and are given by:

t 4 • • .0 TOTAL. MOOT KUM4 TltTID

LOO to tm TO »«.u«i «i c.o «f riiMC

Figure 25. Best Estimate of Standard

Figure 24. Statistical Determination Deviation.
of 1 in 10,000 Level.
 16-19

With the application of the foregoing F. 100 Hour Amplitude Curves

sections, it is possible to predict the
failing amplitude of the weakest blade in It is now possible, with the steady
a large population in the absence of stresses applicable for a particular en-
steady stresses.' The effect of steady gine speed, to obtain from the Goodman
stress can be allowed for by the use of a type diagram the amplitude of vibration.
Goodman type diagram, if the equivalent to af , for the 100 hour life, If this is
the alternating stress axis can be completed for all speeds, a series of
established. To do this it is necessary curves can be drawn, one for each mode of
to 'use' a realistic life and Armstrong vibration under consideration. A typical
(1967) explained that a value of 100 hours curve is presented by Armstrong (1966b) in
was chosen for this purpose as this is a Figure 26 for four modes of vibration.
mean on a logarithmic scale between 10
hours and 1,000 hours. It must be noted These curves can be used directly in
that the material properties are such, comparing the measured af , from an engine
that a 20% change in stress level is worth test, in order to assess the seriousness
a factor of at least 10 to 1 on life, and of the vibration.
the time to be considered is the time at
maximum vibration amplitude and not the Amplitude Ratio and Rules
total blade life.
It has been explained before that an
informative way to assess the importance
E. Steady Stress Correction of a vibration amplitude, which may cause
a high cycle fatigue problem, is to form a
In allowing for steady stress comparison between the measured amplitude
effects, Armstrong (1966b) used a Goodman and that which will cause failure. This
type of correction and so: therefore gave rise to the concept of an
amplitude ratio which is defined as:
a«.
S
x .£S
0 u Amplitude ratio % =
a, Maximum measured amplitude x 100
where So is the allowable alternating Amplitude for 100 hour life
stress or af at zero mean stress:
A survey of the observed amplitude
Ss is the allowable alternating ratios, as determined above, was compared
stress or af at steady stress and o S,,
u is with the service experience of the blad-
the effective ultimate stress at the ing, and is reported by Armstrong (1966b
operating temperature. and 1968). From the results of this cor-
relation it was possible to propose the
It is assumed in their analysis that following design/development rules.
the steady stress varies with rotor speed
as follows

Where St is the direct centrifugal stress;

Sx is the total bending stress
(allowing for the restoring centrifugal
moment),

and X

Where M xv amd Mxx are the uncorrected

and corrected gas bending moments respec-
tively for a radial blade.

The positions on the blade for which

these steady stresses are evaluated, are
those at the positions of the fatigue
crack in the mode in question. If the
position of cracking varies from blade to
blade, in the batch of the fatigue tested
blades, then the most severe steady stress
is taken. |rj „ TMCBA.MI • AT MM BL BTATIC

As discussed in the next section with

the possibility of the results of finite
element analysis being available today, to
establish the value of steady stresses,
these results may be more applicable than
the relationships given above.
Figure 26. Typical 100 Hour Life Curves.
16-20

A. If the amplitude ratio in any mode is perience, and so only the basic aspects of
found to be greater than 100%, . then the methods, which are published in the
adequate engine restrictions must be technical press may be reviewed here.
imposed until satisfactory engine
modifications have been incorporated.
Service evidence has shown that short Steady Stress Distributions
life failures had taken place with
amplitude ratios greater than 100%. A knowledge of the steady stress dis-
tribution is equally vital to both the
B. When the amplitude ratio lies between Amplitude Ratio Method as it is to the
50% and 100% then failures during Stress Level Method, because the level of
long service use may be expected. In steady stress, through the Goodman or
these cases, long term rectifications equivalent type of diagram, establishes
and improvements should be prepared. the level of vibration which may be
permitted. It is discussed here because
C. For amplitude ratios less than 50%, some aspects are also applicable to the
the vibration may be considered to be study of the alternating stress
acceptable for the full service use. distributions.
Modern finite element analysis
These rules have proved to be satis- methods permit the stress distributions
factory for the last 20 years, Armstrong over the whole of the blade surface to be
1980, but it is emphasized that to obtain predicted. However, to obtain a suffi-
a thorough assessment it is necessary to cient accuracy does demand a fairly small
carry out the strain qauge survey at all grid size, and for the/modelling of the
flight conditions. See also the section blade platform and root fixing zones,
"Flight Testing" further below. solid or brick elements will be required.
As is reported by Koff (1978), it is pos-
sible to confirm these calculated values
Foreign Object Damage (F.O.D.) and either by strain gauge measurements or by
Ex Engine Blades photoelastic tests and analysis. The ob-
vious advantage of the finite element
During engine service use, the blad- analysis is that it is possible to perform
ing suffers foreign object damage (F.O.D.) the calculations for various combinations
or in some cases the surface of the blades of the steady state force systems which
deteriorate due to corrosion and erosion. might be present.
This results in the problem of establish-
ing what should be the limits of accep- In order to appreciate the results of
tance for this form of damage, and in the the detailed finite element analysis, it
case of blades showing corrosion and is beneficial to understand the physical
erosion on engine overhaul, to determine representations of the force systems which
if they are acceptable for engine rebuild. are present, and also the reasons for the
stress/strain distributions which will be
The amplitude ratio method provides a observed in practical blading. The steady
technique for readily answering -these state forces may be considered, for canti-
queries. Fatigue tests are carried out on lever blading, to be caused by four force
the defective blades, in the modes which systems.
exhibit cracking where the damaqe is most
severe. Or alternatively in those modes A. Centrifugal forces acting on the blade
with the highest amplitude ratios. From sections. In the absence of high de-
these results it is possible to establish grees of twist if the centers of mass
a new 100 hours life for this standard of of each section lie on a radial line,
bladinq. With the originally measured then these forces will be a radial
bench or flight amplitude data it is pos- force givinq rise to an average P/A
sible to create a new assessment which can type stress; at sections remote from
be called an Effective Amplitude Ratio. the fixing.
Obviously, the same rules for acceptance
can be used as for the new blades. B. Because of the variation in stagger
angle of the blade from root to tip,
the centrifugal body forces acting on
STRESS LEVEL METHOD the leading and trailinq edge zones
of the blade will not be normal to the
The stress level method of assessing blade sections. Thus, there will he a
the severity of a blade vibration is ba- component of force, in addition to the
sically very straightforward, and follows radial force, which will act in the
directly from the strain gauge investiga- tangential plane and cause a twisting
ton work of the early days of jet engine moment which will alter the blading
development. In essence, the alternating stagger anqle.
stress at the most critical oart of the
blade is measured by the use of strain C. The aerodynamic forces on the blade
gauges, and then this level is compared sections. There will be both forces
with the material properties. The prob- which cause bending of the blade about
lems which arise are a consequence of the its two principal axes, and also an
complex stress distributions of current aerodynamic twisting moment.
high performance blading and the lack of
sufficient knowledge of the material D. If, as is the usual practice for large
properties. However, these problems have and medium size blades, design causes
been overcome, and a number of companies the centers of mass of the blade
emnloy these techniques very successfully sections to lie off a radial line
in their assessment methods. However, its through the root section of the blade,
successful operation does rely upon a good then the centrifugal forces will
background of practical knowledge and ex- also result in bending moments
 16-21

about the blades' principal axes. It is Blades with High Rate of Stagger Change
customary to design the blade shape so
that the combined gas bending moment, and when a blade has a high rate of
the centrifugal bending moments, provide change of stagger along the blade span,
the optimum stress condition. Note the and when it is subjected to a torque load
gas bending moment will be a function of the twist distortion is accompanied by
the flight condition, and so the optimum tensile stresses in the leading and trail-
arrangement will depend upon the aircraft ing edge zones of the blade. In order to
duty. provide the equilibrium of forces normal
to th£ aerofoil section, a stress of op-
The stress distribution within the posite sign and reduced magnitude is set
blade aerofoil which would be obtained by up over the central zones of the blade.
the application of the elementary Euler
beam theory, under the action of the above
forces, will be modified by a number Blades with Root Camber and High Degree of
effects. These modifying factors are dis- Stagger Change
cussed by Danforth (1975) and Shorr
(1961).
Shorr (1961) explained that when a
Warping Stress due to End Restraint high rate of change of stagger was present
with highly cambered sections, then the
If a thin rectangular beam is rigidly three force systems, - radial load, tor-
fixed at its end, then under the action of sional twist, and bending moment about the
a torque, the Saint Venant shear stress least moment of inertia - are all coupled
distribution (which would be obtained in a with their corresponding displacement
free beam) is modified. The edges of the strains.
section undergo a bending type of distri-
bution as in Figure 27 from Danforth
(1975). For a thin rectangular section the
bending stress can be 2.9 times the shear
stress.
Effect of Partial Chord Root Fixing \
It is unusual in the design of the
root fixing for the root to be the same
chordal width as the aerofoil at the sta-
tion above the platform. Thus, the lead-
k
ing and trailing edge zones of the blade
will not be fully supported, and the
stress distribution as determined by a
completely rigid fixing will be modified.
Figure 28 from Danforth (1975) shows this
effect of axial width reduction. This
effect would be much more severe in the
case of blades which are carried in a cir-
cumferential slot, where the axial width
of the portion below the platform may be I ..
only 1/3 or less of the aerofoil chord.

B) mmimfm uimmitM-^im.

PCKOtMTCMOW

fIG 27 - ILLUSTRATIVE ELEMENTARY END EFFECT

STRESS BENDING AT HOOT OF CANTILEVER BAA IN
TORSION

Figure 28. Illustrative End Effects Stress Figure 29. End Effect Stresses for
Blade Root in Spanwise Pull Illustrative Fan Blade Root
Compared to Nominal Pull/Area Section (A) Spanwise Pull, (B)
Stress. Moment, and (C) Torque Load.
16-22

Danforth (1975) provides the stress

distribution around the root section (Turn OAM «*»mu"we«T» ««t ut»
periphery, for a fan blade design, as TO VCRV* THE PIKlTI-lllutNT MOOCl
shown in Figure 29. Where the loading is,
in turn, that of a pure pull load -the
radial component of the centrifugal field
-a pure bending moment about the section
axis with the least second moment of area,
and a pure torque. The stress magnitudes
are normalized with respect to their
respective beam theory counterparts.

Results for Typical Fan Blade

The actual stress distribution for a

particular blade will depend upon its de-
tail geometry and blade/disk root design,
but the values may be obtained by the ap- Figure 30. Fan Blade Stress Analysis -
plication of finite element analysis, Comparison of Finite-Element
photoelastic test, or an approximate nu- and Strain Gage Measurements
merical solution following the work of at Airfoil Root.
Zbirohowski-Koscia (1967). Koff (1978)
provides a comparison of the strain dis-
tribution at an aerofoil root of a fan Alternating Stress Distributions
blade, determined both by strain gauge
test and by finite element analysis. The stress distributions of a rectan-
These results are reproduced in Figure 30. gular section cantilever beam when vibrat-
It will be seen that the character of the ing in the flexural modes are relatively
distributions is of the same form of those straightforward, and in Figure 31 are pre-
given in Figure 29. sented the variations of stress for three
different taper ratios. However, as in
the case of the steady stress distribution
the distributions become more complex,
when actual blade geometries are
considered. In the case of vibration, as
opposed to steady loading, the situation
is more involved. As for each mode of vi-
bration, the modal shape, and so the fre-
quency, are dependent upon the elastic
distortion of the blades under the action
of major inertia force systems. Thus, on

MOOt

Figure 31. Deflection and Stress Distributions

for Tapered Cantilevered Beams.
 16-23

a fan blade, which has at its root, both In Figure 32 are presented the re-
high camber, and a rapid rate of change of sults for the IF, 2f, 3F, and IT modes of
stagger at sections above the platform, the first stage rotor blade of an LP com-
then in the IF mode, which is predominant- pressor. It will be seen that, for the IF
ly bending about the axis of minimum mode, the maximum strain on the concave
second moment of area, the stress distri- form is in the central zone of the aero-
bution will be similar to that of Fig. 29. foil, as would be anticipated from Fig. 29
These coupling effects, primarily due to for the bending moment about the minimum
end effects, will therefore result in axis. This is also true of the root
motion in the torsional and edgewise coor- stations for the higher modes, but the
dinates being present in the mode. These peak strains in these roodes are found in
additional distortions together with the areas nearer the blade tips. Also in Fig.
associated inertia forces result in a 32 are the strain distributions for the IT
change in the natural frequency, from the mode, which shows the peak strain to be
'Euler beam' frequency value which would present in the leading and trail ing edges,
neglect these effects. and not in the areas of maximum aerofoi1
thickness as one would expect from tor-
Experiment a j^Deterra i_n_a t ion of S t r a in sional strains of a beam wi th long thin
Distributions sections. The reason was discussed in the
section "Blades with High Rate of Stagger
As reported in Passey (1976), a Change."
visual impression of the strain distribu-
tions of a blade may be obtained by carry- These alternating strain distribu-
ing out an incremental vibration test, tions will of course be altered to some
with the aerofoil surface coated by a degree under the action of the centrifugal
strain sensitive brittle lacquer. This field. The extent of the change will
then gives a series of strain contours/and depend upon the mode, the stagger angle,
so the strain distribution. the hub to tip ratio of the -stage, and the
blade aspect ratio, but the general
pattern will remain.

convex
1F MODE S2 Hi 2F MODE ?B4 Hi IT MODE 420 Hi SF MODE •!• Mi

C O N C A V K >UHfACC
IP MODE 82 Hi 2F M O D E 246 Hi >T MODE 430 Hi 3F M O O t B I B Hi

Figure 32. Surface Strain Distributions.

16-24

It is of course possible to obtain an which are necessary to derive the allow-

indication of the strain distribution by able values for comparison with the
the use of a matrix of strain gauges, but stress, as determined from the strain
the accuracy and degree of appreciation gauge measurement. In the work of
will depend upon the number of gauges banforth (1975), Peterson (1978), Passey
employed. (1976) use is made of a Modified Goodman
type of diagram or stress range diagram.
Fig. 36 and its description is taken from
Calculation of Strain Distribution Danforth (1975) and represent the typical
form. "Curve 'Ci1 represents the upper
It is also possible to calculate bound of alternating stress for unlimited
these strain distributions, from a finite life, vs mean stress as given for statis-
element analysis of the blade, provided tically "minimum" properties. Curve 'Ci_'
that the element size is sufficiently reflects undamaged material manufactured
small, and that the blade fixing is according to prescribed blade surface pre-
modelled accurately enough. Such an in- paration, and without concentration.
vestigation of a turbine blade has been Curve 'Co' represents the blade fatigue
reported in Peterson (1978) for the as- strength In terms of the nominal stress in
sessment of the levels of blade vibration the presence of a notch, (whether of de-
measured by strain gauge testing. Figure sign geometry, foreign-object damage
33 is taksn from that paper, and shows the induced, or a notch-equivalent degradation
finite element model which employed three- induced by instrumentation surface prepar-
dimensional isoparametric elements, which ation for test vehicles). The level and
are formulated either as solid or thin shape of curve 'C21 relative to 'Ci/, de-
shell versions, to allow the representa- pends upon the notch concentration - the
tion of thin airfoils and attachments with higher the concentration, the lower and
the same element configuration. The re- more concave is curve 'C2'« Curve 'C31 is
sult in Figure 34 shows the steady state in a sense the limiting case of the curve
effective stress distribution. In Figure 'Cj' family. It represents the upper
35 is presented the strain of the 7th mode bound of alternating stress, consistent
which was in resonance with the nozzle with crack propagation avoidance, a level
guide vane order. of significance for blade durability in
the presence of extreme P.O.D.".

Material Properties
One of the major considerations, in
the stress level method of assessment, is
the knowledge of the material properties,

£ MM.tl MAI4MUW •THAI.

O M N O T I K MMHWIM BTM.IH

Figure 33. Turbine Blade Finite - Element Figure 35. Vibratory Strain (Plus Three
Model Standard Deviations) on Tur-
bine Blade Pressure Surface
(Micro-Strain X 10 ).

• MOUftAIICt LIMIT COHTOUIU

a, K r > t
MOPAQATIOM TNM9HO10

Ci

MAN ITMM -

•WCTIOM .U«fACt

figure 34. Turbine Blade Steady-State Figure 36. Schematic of Typical Stress
Effective Stress kN/cmz. Range Diagram.
 16-25

For development work, the severity of a Strain Gauge Position and Operation
measured stress may be expressed as a
"percentage endurance", by comparing the
measured alternating stress with the The number of gauges which are used
alternating endurance stress from curve on a blade may be restricted by the size
•C^' having applied the correct mean of the Marie and the lead out wire
steady stress. arrangements. Normally 3 or 4 atrain
gauge locations are used per stage, and 4
blades in a stage will be instrumented to
Typical values of alternating stress allow for blade to blade variation and
which may be used are seldom reported, gauge failure. However, the problem of
although Passey •''1976} does provide the gauge failure is much reduced by the adop-
following table of safe levels of alter- tion- Koff (1978)- of thin film gauges
nating stress. Figure 37. It is stated that thin film
gauges offer higher quality, lower cost,
improved test survivability, and a 60% im-
SAFE ALTERNATING STRESS RANGE provement in the strain level that can be
measured. The usual ceramic strain gauges
Material (Peak-Peak)Ib/sq in fail at approximately half the amplitude
necessary to produce an aerofoil failure,
Aluminium Alloy 10,000 whereas tests with thin film gauges have
Titanium Alloy 25,000 permitted measurements of strain levels up
Steel 25,000 to the failure amplitude.
Nickel-Iron Alloys 35,000

The ability of the strain gauge to

Passey attributes the low level of withstand high levels of alternating
these stresses to the influence of damage strain, will also control the positioning
and surface imperfections. of the gauges. The requirements are, that
it should be able to measure a significant
Koff (1973) expresses the opinion response in more than one mode, but if its
"that the actual fatigue strength of fatigue strength is limited, then it must
engine components is best evaluated by not be used in anticipated areas of the
bench testing, where the parts are sub- highest alternating strain. With 3 or 4
jected to vibratory load to induce fail- gauges, up to 8 modes can easily be
ure." Tests are carried out on instru- covered. Figure 38 from Peterson (1978)
mented blades to evaluate the design, shows the positions selected for a small
material, and manufacturing process under turbine blade. The detailed investi-
partially simulated conditions, i.e., gation, see later, centered on the 7th
heating coils are used to raise the tem- mode. Because of the small blade size, in
perature to provide hot fatigue testing. this instance, only one gauge per blade
From the tasting, it is possible to com- could be accommodated.
pare the results with anticipated proper-
ties tor the parent material, and also to
complete a fatique limit diagram for use By rig test calibrations for each
in blade assessment, i.e., curve *C2 l of mode in turn the relative sensitivity of
Figure 36. the gauge positions may be established,
and related to the level of the highest
alternating stress on the blade.

This data, coupled with a knowledge

of the stress distribution over the sur-
face of the blade makes it possible to
obtain a ratio between the critical point
vibratory stress , to the apparent stress
sensed by the strain gauge This
Figure 37. Ceramic Versus Thin Film stress distribution will have been estab-
Strain Gages. lished by one of the methods discussed
above. Danforth (1975) uses this form of
ratio to establish the "scope limits" for
use during live strain gauge testing, to
assess the severity of a vibration.

2o

ia the stress
Where oacp range diagram
endurance stress for the
critical point under the
given operating conditions.
•UCTIOH tIM FHI1IUM IiOl
W1TI HO'I MMHIIOHl AM in MCHU
9 «"OH. »•".* MUM MM**. OH* V-.U11
ML* - « « ' W i n . 1 0 •(• M.1OI
is the stress ratio as above
and will be mode and speed
dependent.
Figure 38. Typical Turbine Blade Strain
Gauge Installation.
16-26

is an experience - derived The predicted maximum strain was ob-

factor greater than one, im- tained by using a scale factor of 3 stan-
plying the presence of a dard deviations above the mean. This
blade in the stage more resulted in the principal strain distribu-
active than the one directly tion on the pressure surface shown in
observed. Figure 35. It shows a peak level at a
position removed from a measurement point,
and some 4 times the highest peak
Ke is a mode frequency dependent measured.
function to allow for fre-
quency response of equipment. The test data was presented also on a
modified Goodman diagram, and is repro-
The factor 2 is dependent upon duced in Fig. 40 where it will be seen
whether the oscilloscope is calibrated, that the points all lie below the nominal
using 'peak-to-peak' signals or not. endurance line. However, the fact that
the critical points may not be represented
in the measured data was highlighted by
Correlations and Criteria For Failure the inclusion of Fig. 41. This plots all
the calculated points from the finite ele-
One of the major problems in the ment analysis nodes, with the alternating
assessment of blade vibration test data is levels scaled to correspond to the empiri-
the variability of the information between cal data. It will be seen that two points
blades and strain gauges. The test data with high steady strains are above the
presented in Peterson (1978) illustrate endurance line.
this point. During this turbine test, 6
strain measurements were made for the 9 COMPARISON OF METHODS
strain gauge positions. Figure 38. If all
the blades had been vibrating at a common It is important in discussing the
amplitude, then the scale factors to apply pros and cons of the two methods of
for the calculated stress pattern, which assessment which have been reviewed, to
is necessary to provide a common level at identify the purpose of the investigation.
the point of maximum stress, would have The prime purpose in this handbook is to
been the same. The mean and standard de- be able to assess those situations where
viation of these 54 scale factors were failure is principally due to hiqh cycle
calculated, and then applied to the strain
at the measurement position. These are
compared with the measured data in Figure
39.

The reasons which are given as con-

tributing to the variation in the scaling
factors for the nine gauges and six time
points are:
A. Blade to blade variations in dimen-
sions, material, properties, tempera-
tures, loading, damping, etc.
B. Variations in engine conditions and
excitation levels in the operating
range, including acceleration rates,
pressures, temperatures.
C. Gauge position and wiring routing
influences. Figure 40. Modified Goodman Diagram for
Peak Strains Measured in
Engine Test.

MAKMMH
5 "•
i n.

^T

lit • -•9
4

Figure 39. Ranges of Measured and Figure 41. Modified Goodman Diagram
Analytical Strain Data. Including Analytical Data.
 16-27

fatigue, i.e., if there were no vibration Fatique and Amplitude Scatter

the component would not fail. If this pur-
pose is accepted, then it automatically
rules out that portion of the modified
Goodman diagram where the magnitude of the There can be no doubt that the big-
applied steady stress is comparable with gest problem which has to be overcome by
the ultimate tensile strength. The ro- the adopted assessment method, is the
peated application of these steady flight scatter in fatigue capability of blades,
flightstresses would cause a failure which and the variation of amplitudes between
is basically an L.C.F. type of failure. blades in the fleet of engines to be
Under these conditions of high steady assessed. Many factors contribute to the
stress the failure will be aggravated by scatter in the fatigue strength of
the presence of high cycle vibration and blading, and it would appear to be very
so the problem will not be corrected by difficult to allow for all these factors
reduced aerodynamic vibration - it is effectively by correcting the results of
essentially a steady stress problem, which laboratory type fatigue tests which miqht
is identified by stress analysis and not be obtained from special specimens, both
strain gauge and vibration testing. notched and un-notched. As inferred by
Koff (1978), even for the stress level
Company Organization and Size method, fatigue testing of representative
blades is the optimum way to establish the
There can be no doubt that both fatigue capability of the blade. From
methods have been developed to provide these tests, it has been shown that a sat-
accurate and practical tools for assess- isfactory allowance for the fatigue
ment purposes. However, each method does scatter can be wade.
require significant expertise and experi-
ence and background data. Much of this
data, especially the material test data
for the stress level method, is not avail- Equally difficult to accommodate is
able in the open literature, and can only the scatter in amplitude between blades
be obtained by extensive testing, which is within a stage, and within a group of
usually only practicable in large engines. The stress level method uses a
companies. Equally, the amplitude ratio special factor Kv to account for the anti-
method nay require a high content of ex- cipated variations. However, its value is
perimental rig testing on components. The obtained from experience, and so is dif-
facilities for these tests will require ficult to obtain initially. By using the
moderate investment, as long as the number maximum modal amplitude from a small num-
of components to be reviewed is not ber of instrumented blades, together with
large. a survey of service experience, it has
been possible with the amplitude ratio
method to set up assessment rules which
Which of the two methods is adopted have not needed to be altered during some
will depend to a large extent upon the way 20 years of use Armstrong (1966b) to
in which the technical departments in the Armstrong (1980). One advantage of this
company interact, and also the engineering method is that it does not involve any
style which has been adopted by the experience factors other than the inter-
company. The 'af aspects of the ampli- pretation of likelihood of lonq term
tude ratio method enable some aspects of failures, when the amplitude ratios are
the aerodynamic excitation to be assessed between 50% and 100%.
independently of the mechanical stress
analysis/fatigue considerations of the
problem. This will be of prime interest
to the aerodynamic desiqn departments,
while the development groups will be more
familiar with stress levels and material Non-Cantilever Assemblies
properties.
One failing of the amplitude ratio
method is when the blades form part of an
Defect Investigation assembly, e.g., a part span bladed disk
design. In these circumstances it is not
practicable to carry out fatigue tests in
When it is necessary to investigate the correct modes. For these conditions,
the cause of a fatigue crack, defect or the method adopted compares closely with
failure, then it must be the correct pro- the stress level methodf although the
cedure to ensure that a strain gauge is fatigue strength of the section can be
positioned at the crack location. Care derived from instrumented blade fatique
must be taken to be sure that the gauge is tests in a mode which approximates to that
applied to the relevant surface, if the of the assembly mode of interest. Under
crack is in a thin section like a trailing conditions when the anticipated amplitude
edge. In these investigations, it will be ratio is becoming high, it is possible to
prudent to use other strain gauge posi- carry out a calibrating fatigue test on a
tions in addition and to cross-calibrate whole bladed assembly, in a facility simi-
the gauges for all possible modes so that lar to the rotating fatigue test in the
should the gauges at the crack location whirligig facility, Koff (1978).
fail, then the testing may continue by
monitoring the alternative gauges. In
this aspect the methods are very similar,
especially if the amplitude of the leading Of course, the assessment of complete
edge is used as a reference. assemblies by the stress level method
should provide few problems above those
encountered for cantilever blading.
16-28

Ex Engine and Damaged Blades Design Assessment

The consequences of surface damage on It is now possible with modern finite
blading after service engine running can element analysis to predict the natural
be readily assessed from the amplitude frequencies of cantilever blades, bladed
ratio method using the effective amplitude assemblies, and bladed disks including the
ratio. For the stress ratio method, if necessary allowance for temperature and
the fatigue properties of ex engine blades centrifugal effect. Thus, it is possible
are compared to new blades by fatigue at the design stage to construct a
tests, then they can easily be incor Campbell diagram of frequency against en-
porated into the method. Otherwise, it is gine speed. This diagram will be com-
difficult to see how they can be assessed pleted by adding the major sources of ex-
by the stress level method, because of the citations due to non-uniform variations of
difficulty of reproducing the damage on flow around the blading annulus, e.g.,
laboratory type fatigue test specimens. structural struts, intake distortions,
numbers of blades in adjacent rows.
These same comments also apply to the
necessary approval process which has to be Types of Vibration
cleared before a repair technique can be
used for damaged blades. Blade vibration phenomena fall into
two major categories. That due to self
excitation, and that due to forced reso-
Optimum Method nance type of excitation. The methods of
accommodating each by the assessment
No doubt the optimum technique is a method is quite different.
blend between the two methods. Calibra-
tion of the strain gauges to a movement of
the blade's tip section, allows the Self Excitation
severity of the aerodynamic excitation to
be quantified. The application of this For the cases of self excitation or
calibration may also reduce one of the instability, i.e., subsonic stall, super-
sources of scatter when deriving the de- sonic stall, supersonic shock, conditions
gree of vibration of the blades. The of choke, disk shroud system subsonic
fatigue strength of blading could be mea- stall, etc., it is generally accepted,
sured by fatigue tests on instrumented Cardinale (1980), Danforth (1975),
blades, and after corrections for the Armstrong (1960), Danforth (1967), that
environment, scatter and steady stresses, sufficient operational margin must always
the results could be compared with strain be provided.
gauge measurements from engine test. In
this way bladed assemblies could be Methods of calculating the conditions
handled, and the benefits of both general for this type of self excitation are pro-
methods obtained. vided in other chapters of this handbook
but, because of the difficulty of perform-
ing these calculations, it will always be
STEPS OF ASSESSMENT AMD TEST CONDITIONS necessary to verify that sufficient opera-
tional margin is provided. At the design
stage, in addition to the prediction for
The assessment of blade vibration the nominal conditions, it will also be
must be a continuous process throughout necessary to allow for other additional
the life of an engine, from the project effects. For instance intake distortion
stage right through to the evaluation of and intake air density under flight condi-
engine blading during service use. At the tions, Figure 42 and Figure 43, are taken
project stage, the general vibration from Halliwell (1978) as examples.
characteristics of the blading should be
considered when the scantlings of the
major engine components, compressors and
turbines, are being sized. Detail adjust-
ment can then be made during the design
phase. The design predictions must then
be verified as early as possible during
compressor and turbine rig testing. Full
•MAIM HO fUTTW Al
engine development then follows, includ- MAI VCKD MAOMD
ing, tests in altitude facilities and
flight testing where appropriate, as well
as bench testing. • - ^ ^ ^ TMtT DATA

The above sequence of testing provides

the ability to obtain the earliest indica-
tion possible of any likely major
problems. This is advantageous in two
ways. Not only does it give the best pro-
tection against possible component
failure, but the early knowledge that a
new component or change in operation is
required minimizes the quantity of compo-
nents which will be scrapped or need modi-
fication. By minimizing the cost of the Figure 42. Correlation of Integrated
alterations in this way, it enables the Distortion Parameter with
correct change to be adopted without risk Average Flutter Onset Speed
of compromise. For Different Intake Types.
 16-29

Forced Response and Resonance Conditions For obstructions downstream of a ro-

tor blade, Armstrong (1967) suggests that
The location of possible resonant if the obstruction in the air stream can
conditions can be obtained from the pre- be approximated to an aerofoil, then it is
dicted Campbell diagrams. However diagrams possible to estimate the velocity varia-
will indicate many resonances, especially tions of the flow by employing a potential
if the harmonics, as well as the basic flow calculation for the equivalent stan-
frequencies generated from obstructions in dard shape. It was found that the best
the air stream are included. The problem correlation was obtained by a comparison
exists as to which of these resonances may of the measured 'af' of the resonant blade
be accepted in the design, and to identify and the relevant harmonic of the
those which should be removed, either by excitation. This correlation is repro-
modification to the blade to change the duced in 1Figure 44 from which an antici-
frequency or by adopting a different num- pated 'af can be obtained from the
ber of obstructions, e.g., struts across magnitude of the flow disturbance.
the air stream.
Rotor to Stator Resonance
Currently, the ability of theoretical
methods to predict the amplitudes of these With close pitching of rotor to
resonances is not very good, and so re- stator blade rows in a compressor, there
course has to be made to experience based is a possibility of the flow disturbance
design rules. Armstrong (1960), Danforth arising from the blades in the adjacent
(1967) recommended that resonances with blade row. Because of the relatively high
the lowest 4 modes, and a recognized cir- numbers of blades in a row, the modes of
cumferential disturbance should not exist vibration which will be excited are nor-
in the high portion of the speed range. mally the higher modes. Cardinale (1980)
Armstrong (1960) suggests that the lower discusses this problem and considers de-
limit of this speed band should extend signing the appropriate adjacent blade
down to the idle zone, and also that this rows with the required aspect ratio to
rule should apply to two stages adjacent avoid the rotor's or stator's first two
to the source of excitation. It is also stripe (l-2s) panel mode resonance with
stressed in Danforth (1967) that a "two its adjacent row's passing frequency.
per rev" resonance in the idle to maximum This is illustrated in Figure 45 from the
speed range should not be accepted, as reference. Based on their current solidity
some degree of this excitation pattern trends, they found the criterion to be
will always be present. satisfied when the ratio of blade to
stator aspect ratio was approximately 0.6.
The presence of low engine orders is
especially relevant when considering the
early stages of a compressor, and the pos-
sible excitation due to intake distortion
patterns. Armstrong (1965) reports on
some early work which was carried out to
predict the levels of fan blade vibration
from aircraft intake distortion patterns.
With today's improved analysis procedures,
it is expected that this type of predic-
tion could be much improved.
Danforth (1974) outlines the type of
analysis which will be roquirad to predict
fully the levels of vibration from distor-
tion patterns. However, there is little
published data showing the accuracy
attained by theso more complex and
thorough analyses. Figure 44. Comparison of Downstream
Excitation.

01 M M 0* Or M 0» 1C II
WTAKI •TAOttAIION OtNWTV CL»/M*)

Figure 43. Effect of Intake Air Density

on Supersonic Flutter Onset -
Accumulated Test Data For
-30 MPa Stress. Figure 45. Schemes for Frequency Tuning.
16-30

Rotating Stall and Separated Flow at least a small number of blades.

Usually, 4 blades are instrumented an
Until the mid 1960's, there was a identical way. This will then allow for
significant problem of compressor blade differences in manufacturing tolerances,
failure due to unsteady rotating stall or which may affect the mechanical damping,
separated flow type of excitation. etc., to be covered in a general way. One
Armstrong (1960) in the Design Rules for factor which should be specifically
Blade Vibration the following is proposed, accounted for is the influence of blade
"so that the general level of vibration frequency variations on the amplitudes of
and the lower levels of random type of vi- vibration. Research work has shown that
bration can he accommodated, the compres- the largest and smallest amplitudes of a
sor blades should be stressed to withstand blade row are often predicted to occur on
a minimum vibration of ± 2.0 ft/sec.". either of the blades, with the highest or
lowest frequencies. It is therefore ad-
With the aluminium (and to a lesser visable to select these blades and to in-
extent with the steel) alloys which were clude thorn in trhe smaller number to be
used for blades, this anticipated level of instrumented.
vibration, together with the steady
stresses would be very significant, and in
some cases would not provide an acceptable Rig Compressor Testing
blade design. Two things have eased the
problem. The first is the wider use of Aerodynamic rig tests provide an
the titanium alloys with their inherent ideal opportunity to establish the vibra-
ability to withstand higher levels of vi- tion characteristics of the blading over
bration, thereby making it easier to com- the whole performance map of the unit. In
ply with the above design rule. The this way it will be possible to establish
second major factor is the wider use of the zones of self excitation in relation-
variable geometry within the compressor. ship to the anticipated work line. In
This rasuits in a reduced speed range zone this connection, it is worth reminding the
with rotating stall, or separated flow, reader that it is often the practice with
and often the excitation is of reduced large compressor tests to throttle the in-
intensity. take to reduce the power requirements. As
we have seen, this will have the effect of
It is, however, a prudent design rule increasing the onset speed for flutter.
to adopt wherever possible, to enable
unanticipated minor excitations to be With units incorporating variable
withstood. stator vanes, it is normal to carry out
testing with the full range of variable
angle settings, to establish the extent of
Design Verification rotating stall zones, and any possible
flutter conditions on the following rows.
In the design process, many assump- This information is essential to set sat-
tions have to be made in order to idealize isfactory operating control laws for the
the problem until it becomes one which is vanes, so that a satisfactory margin from
tractable. It is therefore necessary these flutter conditions is determined.
throughout the engine development process,
to carry out testing to ensure that the A number of the techniques employed
behaviour of the parts is in acceptable in this type of testing with variable
agreement with the predicted characteris- stator geometry are covered in Cardinale
tics of the Idealized systems. (1980).

Mechanical Aspects Engine Development Tests

Testing therefore has to establish If design verification is to be
the following characteristics of the basic applied satisfactorily, it is essential
components. that engine components are instrumented in
engine use as early as possible. Priority
A. The natural frequencies are as pre- should be given to the first and last two
dicted with a representative type of rows of compressors, as well as to stages
fixing. adjacent to variable geometry stators.
This testing will confirm, on the correct
B. The fatigue strength of the blade is as engine component parts, the indications
anticipated. obtained during compressor testing. This
tasting, however, will be representative
C. The natural frequencies in the operat- of engine use, as it will be operated over
ing machine are not unexpectedly influ- the correct work lines. It is normal
enced by touching platforms, shrouds, practice to carry out recordings over slow
spacer constraints, etc. accelerations and decelerations at the
rate of approximately 2,000 rpm/min
D. The combined bladed disks assembly fre- Armstrong (1960), Armstrong (1967), and in
quencies are as predicted. addition, slam accelerations and decelera-
tion should also be recorded. This type
The levels of vibration for a given of testing is included in the table Figure
set of component characteristics will be 21 of conditions to be surveyed and in-
dependent upon the aerodynamic test con- cludes data from Danforth (1975).
ditions, and the mode of operation of the
unit which will be briefly discussed As part of the engine development
below. However, these amplitudes may also program, tests should always be carried
be influenced by factors which cannot be out with the engine fitted behind an air-
readily allowed for by calculation, and so craft intake. This will confirm whether
it is necessary to obtain test data from any additional forcing excitations are
 16-31

present in the intake flow. The bench turbines, the level of vibration may be
test program should also include the con- sufficiently high to cause fatigue
sequences of likely malfunction of the failure. It is normal for these high-
engine control system; examples being; cycle fatigue cracks to propagate very
sudden opening of reheat nozzle, mal oper- quickly and as there is no prior elonga-
ation of variable vanes or blow off valve, tion of the material the final fracture
operation of deicing air. takes place without warning. The fatigue
strength of a component is dependent upon
Strain gauge testing should also be many factors, e.g., material charac-
employed during any testing in engine al- teristics, manufacturing methods which
titude test facilities, as this will pro- determine the material structure and re-
vide an early anticipation of the influ- sidual stress, applied steady stress,
ence of the worst flight conditions. As stress concentration and surface
Fig. 43 would imply, high _ intake densi- condition. The materials which are used
ties, i.e., high aircraft speeds* at low conventionally for gas turbine blading ex-
levels, nay cause a worsening of any flut- hibit a high degree of scatter in their
ter condition. fatigue strength and this characteristic,
combined with the variability of manufac-
ture of components, creates one of the
Flight Testing major problems in the assessment of blade
vibration levels.
A series of flight tests will be
essential to ensure that service failures The blading of a gas turbine is con-
can be avoided. Normally it will be the tinuously subjected to forces which will
early stage blading which will be instru- cause it to vibrate. The most difficult
mented because they will be affected most vibration situation to assess is that
by the special flight conditions. These which may cause a long-term service fail-
conditions will result from the aircraft ure to occur. Under these conditions, ex-
operations which cannot be simulated in perience shows that it will 'be necessary
the altitude test plant. Typically, they to carry out fatigue tests on the compo-
will be: nent to obtain a sufficiently accurate
measure of the blade's fatigue strength.
A. Intake conditions during the take-off A high fatigue strength requires attention
phase for supersonic aircraft. to detail design and also to ensure that a
good distribution is obtained for the
B. Operation of single and dual engines in applied steady stress as well as the al-
installations where the intake of one ternating stress caused by the vibration.
will affect the second. The use of the af - tip amplitude times
frequency - techniques helps to verify
C. Aircraft incidence. that a high fatigue strength has been
obtained.
D. Aircraft yaw.
It is expected that it will always be
E. High speed 'g' turns and spirals. necessary to confirm by engine testing
that a design is satisfactory from a vi-
F. Approach and -landing conditions; bration point of view. Blades can be sub-
reverse thrust. jected to two major classes of excitation
-self excitation, e.g., flutter, or forced
G. Firing of any armament. resonant vibration. In the case of self
excitation the published data recommend
H. Operation of reheat. that a margin of operation from the onset
of flutter be available throughout the
I. Aircraft intake operation for total operation of the engine. Self exci-
supersonic flight. tation can be distinguished from forced
resonance response by detailed analysis of
The testing should include any the strain gauge signal from the vibrating
special aircraft maneuvers as listed above component. For flutter, the response will
but recordings of deceleration and accel- be at the natural frequency of the compo-
erations should also be made during high nent or assembly and is not dependent upon
and low speed flight at a range of. alti- a forcing function being present. With a
tudes throughout the flight envelope. forced resonant condition the essential
Finally, it is prudent to carry out re- forcing frequencies are likely to be seen
cording throughout normal flights or away from resonance and the response of
sortie patterns. the vibrating component will increase as
the forcing frequency coincides with the
Normally in this work the early stage natural frequency.
blading will be instrumented. However, be-
cause of the knowledge from previous test- As it is generally not possible to
ing, if specific engine orders are operate engines without forced resonances
generated, which may cause problems on being present in the running range, it is
later stages, then on subsequent flights necessary to ensure that the levels of vi-
these too should be included for bration will not give rise to an unaccept-
verification. able incidence of fatigue failure during
the engine working life. This can be
achieved by the measurement of the maximum
CONCLUSIONS alternating stress present on the blade
and then forming a comparison with the
Fatigue is the failure mode of a fatigue properties of the material. tn
metal which has been subjected to a large this process it is necessary to allow for
number of applications of stress. Under the material scatter, the influence of
conditions of high-frequency vibration, steady stresses and temperature, stress
which is typical for the blading of gas concentrations and the degradation of
16-32

properties due to erosion, corrosion, and The 2:1 range of amplitudes between
fretting. A good data bank in conjunction those which are acceptable for long life
with experience of application is required and those which may result in a dangerous-
to ensure all factors are accommodated ly short life ensures that it will always
correctly. An alternative method of be necessary to measure the operating am-
assessment measures the level of vibration plitudes in an engine in order to assess
achieved in engine operation for each of them accurately. It is difficult to see
the normal modes of vibration of the com- how predictive methods can become a prac-
ponent and then compares this level with tical alternative due to the largely un-
an acceptable one which is derived from known factors which must be considered to
fatigue tests on the component. achieve the amplitude accuracy. The fac-
Experience has shown that, when measured tors which have to be considered include
amplitudes are double those which are sat- mechanical damping, all sources of excita-
isfactory for service then failures with tion including variations in aircraft in-
short lives are likely to take place. The take distortions over the full flight
advantage of the method lies in the abil- envelope, the influence of blade to blade
ity to allow for the variation of the coupling due to variation in natural
fatigue strength of components and can frequencies. However, the role of the
easily be extended to include surface predictive methods must be to ensure that
deterioration effects, e.g., erosion and designs are not considered which would
corrosion. Because the vibration is mea- generate extremely short life failures.
sured in the amplitudes of the normal The aeroelastic work will also indicate
modes rather than the maximum stress, one the best way in which a design can be
strain gauge, or some other convenient modified, to reduce the level of excita-
method of vibration measurement, can be tion, should an unacceptable level be mea-
used to cover a number of modes and this sured during the engine design verifica-
is an advantage in the testing for design tion testing.
verification. A disadvantage is that the
method is not directly applicable to vi-
bration involving an assembly mode because
of difficulty of carrying out the neces-
sary fatigue tests and calibrations.
 16-33
APPENDIX I For a beam of uniform cross section,
El is constant and we have
THE DERIVATION OF THE 'af RELATIONSHIP
FOR CANTILEVERS
where
The vibration velocity can be used as
a criterion for the assessment of vibra- b2 . «
tion stress. The proposition, which is mA
outlined in a general way by H.G. Yates
(1948), can be developed analytically for
a cantilever beam of constant cross sec- If we assume that the beam is
tion when it is vibrating in its flexural vibrating at a natural frequency <o and
mode. The analysis follows a standard so, with y=Xsinut where X is the mode
type of text, e.g., Timoshenko (1937). shape and is a function of x
d*X
A.I

By using the notation

A.2

it can easily be verified that sinpx ,

cospx , sinhpx , and coshpx are
solutions of A.I and thus that the
general solution can be of the form
X= cj(cospx+coshpx)+c2(cospx-coshpx)
+C3< sinpx-t-sinhpx)
+C4(sinpx-sinhpx) - - - - - - - A.3

With the usual assumptions that the cross with the values of ci. , c2 , 03 , and c4
sectional dimensions are small compared being determined by the particular end
with the length of the beam, and that it conditions for the beam. In the case of a
is vibrating in one of its principal cantilever beam for the fixed end
planes of flexure, then the following will
define the deflection curve. x-0 , x=0 and = 0 (i.e., deflection
and slope zero);
for the free end
, 3Y
where El is the flexural rigidity x»L, 5^7=0 and ^3=0 (i.e., bending
M is the bending moment at any moment and shearforce are zero).
cross section
Thus at x-0
Differentiating twice:

+C4P(cospx-coshpx) A. 4
~ dx ="°
with x=0 dX c3
dx' dx Therefore

The last equation is for a bar sub- -C4P2(sinpx+sinhpx) A. 5

ject to a distributed load of intensity w.
This load can be due to the inertia load
of the vibrating beam itself; the inten-
sity being equal to the reversed mass
inertias of the cross section. Therefore,
with m = mass per unit volume, A the cross -C4p3(cospx-t-coshpx) A.6
sectional area:
16-34

relationship A.7

Squaring both sides we have

Oo-c 2 p 2 (cospL+coshpL)-C4P 2 (sinpL+sinhpL) clsinzpLsinh2pL
coszpL+cosh2pL+2cospLcoshpL

= (sinpL+sinhpL)
°2 (cospL+coshpL) 4c^ainzpLainh2p
A.10
coszpL+coshzpL-2
From A.6
by using the frequency relationship A.8.
Thus, the following relationships between
stress and tip amplitude will be valid for
-C4 < cospL+coshpL) any of the flexural natural frequencies.
0=sin pL-8inh2pL-T-cos2pL+cosh2pL
2
From A.8 and A.7 cos2pLcosh2pL=l
+2cospLcoshpL

Remembering sin29+cos2e=l
H-sinh2pL A.11
2 2
and cosh e-sinh e>l A.7
we have cospLcoshpL=-l - - - - - - - A.8
l-cos2pL_-Bin2pL
cos'pL
This is the frequency relationship
which determines the natural frequencies From A.10 and A.7
of a cantilever beam in flexure, the first
six roots of which are:
?_4cjsin2pLsinh2pL 4cpain2pL^ coa*pL
P4L p5L * -sinzpL-sinhzpL~-ainzpL-f l-co82pL.
coszpL
4.730 7.853 10.996 14.137 17.279
_ _ ,13
The frequencies can be obtained using the z z z
definition of p from equ. A.2 -sin pLcos pL+sin pL

A.14

etc. for higher roots.

Hence using A. 9

We now proceed to obtain the rela- "x-o - EJP2a

tionship between the bending moment and
stress at the fixed end x=0 for the From normal bending theory the stress
amplitude of vibration at the free end o on a section of 2nd moment of area
x-L l=Ak2 where k* radius of gyration and
y is the distance of stressed fiber from
from A. 5 the neutral axis then with A. 14

yEp2a = yEaw
For half total tip amplitude a at x»L
from A. 3 as cj-»C3"0 Therefore af
2 ( cospL-coshpL) +c4 ( sinpL-sinhpL)

(ainpL-ainhpL from A.6

(cospL-fcoshpL

, [-28inpLainhpLl
*• cospL-f-coshpL
 17-1

LIFE TIME PREDICTION: SYNTHESIS OF ONERA'S RESEARCH

IN VISCOPLASTICITY AND CONTINUOUS DAMAGE MECHANICS,
APPLIED TO ENGINE MATERIALS AND STRUCTURES
BY
R. LABOURDETTE
Office National d'Etudes et de Recherches Aerospatiales
BP 72 - 92322 CHATILLON CEDEX, France

INTRODUCTION describe now the above mentioned steps of

the life prediction method. Most of them
Improvements in the metallurgy of are described in more detail in the cited
superalloy materials make it possible to references.
use them at higher temperatures. This is
of great importance in aeronautics, where It must be mentioned that all the re-
gas turbines are widely used for propul- sults obtained here were obtained in the
sion. In addition, new cooling techniques course of a very close and fruitful coop-
for turbine blades lead to increased tur- eration between ONERA and SNECMA.
bine inlet temperatures and therefore
better turbine efficiency.
BEHAVIOR OF MACROSCOPIC VOLUME ELEMENT
However, it follows from this increase
in temperature that more severe thermo- General Remarks
mechanical problems are encountered when
dealing with the design and life predic- Two main therraodynamical concepts are
tion of turbine blades (and disks). generally considered when describing the
mechanical behavior of materials, Odquiat
Roughly speaking, two phenomena are to and Hult (1962).
be described for such a prediction: creep
and fatigue, with particular emphasis on A. - the present state of the material
low cycle fatigue. depends on the present values and
past history of observable vari-
This part of the chapter deals with ables only (such as: total
the method of life prediction developed at strain, temperature...)
ONERA whose main steps are:
B. - or it depends only on the pre-
- Behavior of the macroscopic volume sent values of both observable
element. and internal variables.
- Damage accumulation under creep-fatigue
conditions. All the developments made at
- Numerical methods of stress and damage O.N.E.R.A. are derived from the second
calculation in complex structures one, and assume the existence of a thermo-
undergoing thermomechanical loading, dynamic potential (the free energy for
i.e., submitted to generalized viaco- example) from which the relations between
plaatic behavior. the state variables and thermodynamic
forces are defined; it is also considered
Although it was developed for gas tur- that dissipative potentials, associated
bines, the method presented here can be with the generalized normality rule, allow
applied to any kind of structure under- the a priori verification of the second
going thermomechanical loading. We principle, Mandel (1962).

efasKc

Figure 1. "Hardening" Effect due to Straining.

17-2

In the context of plasticity and vis- In order to describe this kind of be-
coplasticity, such general concepts have haviour, it is worthwhile to use a state
been applied by many workers, giving rise equation written ass
to a coherent tool, especially for the
classical flow rules, Halphen and Nguyen P («> ep, cp) = o (2)
(1975), Sidoroff (1975).
(for the one-dimensional case).
In the next sections we shall derive
and present a model able to describe com- A product of power functions gives good
plex behavioral effects such ast cyclic results over a wide range for the primary
hardening or softening, time recovery, creep as well as for the tensile teat or
aging and strain memory. relaxation teat, Lemaitre (1971).
Description of the Model of Viacoplastic KK
Behavior Cp
C (3)

Strain hardening This three-coefficient relation la easily

generalized to the three-dimensional form
The inelastic straining of a material if we assume an iaotropic hardening
almost always results In hardening. This hypothesis. With the Von-Mises criterion,
is perceptible macroscopically, e.g. after we can say that:
a plastic flow under tensile stressing:
the apparent yield strength increases and
the material exhibits a greater resistance •1/n
(4)
to a subsequent plastic flow (Fig. 1).
The decrease in the strain rate during the
primary phase of creep tests also results where p is given by (1) and is the
from work-hardening. We thus associate an deviatorlc stress tensor.
increase in the inelastic strain with an
increase in the density of the disloca- The viacoplastic strain occurs at con-
tions, which lose their mobility by piling stant volume and we have:
up on obstacles or by forming cells.
(5)
The cumulated plastic strain p is
thus a natural hardening parameter. It is
defined by: This elementary theory of viscoplasticity
with isotropic hardening is valid in the
case of nearly proportional monotonic
P ' (f 'pi j ) (1) loadings.

The hypothesis of strain hardening For other kinds of loadings, inter-

(SH) is more accurate than the time har- nal stress must be introduced in the
dening one (TH) as can be seen from a two- equations.
level creep test. Figure 2 shows, sche-
matically, what should be the results of a
two level creep test according, respec- The various internal stresses
tively, to the SH and TH hypothesis. In
that sort of test, the higher level (92) The idea of internal stress is not
follows the lower one (0}). Experimental new. It was introduced in the thirties by
data confirms, generally, the SH hypo- Orowanto provide a better way of express-
thesis, Rabotnov (1969), Larson and ing the observed macroscopic behavior, in
Storakers (1978). particular, In polycrystal and multiphase
materials. In these materials, the expo-
In order to describe this kind of be- nent in the secondary creep phase law
haviour, it is worthwhile to use a state (Norton's law), for example, is very high:
equation written as:

f (a, 6p, ep) (2)

ff\ tnin ff:

Figure 2. Time and Strain Hardening Visualized Schematically by Two-Level Creep Tests.
 17-3

threshold k also depends on the size of

(6) the precipitates, because of the various
bypasses of shear forces making the dislo-
Among other things, introducing an cations move across the precipitates. Let
Internal stress o^ makes it possible to us note immediately that a macroscopic
reduce this exponent greatly, to values model has been developed to express the
of 2, 3, or 4, as justified by physical variations in this internal stress, in-
theory. The internal stress can be mea- duced by temperature changes (partial dis-
sured in the secondary phase of creep by solution of the medium-size precipitates,
successive unloadings, modifying the law • followed by reprecipitation of a finer
as follows, Culie et al. (1982). phase), Chabache and Cailletaud (1979).

(7)
This model has been shown to describe,
very accurately, the additional hardening
induced by overheating periods for a the IN
The internal stress may be a scalar or 100 alloy in the range of 900-1000 C.
a tensor depending on the type of interac-
tion considered. On the macroscopic level, - R is the variation in Orowan's stress
four types of internal stresses can be induced by a plastic strain. It la di-
introduced, corresponding to an additive rectly related to the increase in the dis-
decomposition of the applied stress. In location density, but may also depend on
3-D form we can write: the dislocation configuration, e.g. creat-
ion of dislocation cells, size, and fine-
ness of the cell, etc.
j(<r-X)-k-R-R-Kp (8)
- R*, sometimes called RSQL or the draO
stress, is included to describe the har-
which, under pure tensile stress, reduces dening as seen by the atoms or particles
to: in solution. This hardening slows down
the movement of the dislocation by a drag
a - X + k + R R* - phenomenon.
where (9)
o - «i + Kp 1/n - The last term, - Kp1/n la the vis-
cous stress itself' (viscous friction)
- X la a second-order tensor, called the which can be approximated initially by a
back stress or rest stress correaponding power function. Of course, equation (8)
to long-distance interactions: intergran- can be rewritten In the ordinary form
ular stress induced by the nonhomogeneous (Di
plastic strains from one grain to the
other, interactions between dislocations - k - R -R
and precipitates, as exact calculations K (10)
have shown on the precipitate scale. Carry
and Strudel (1978). Many models have been developed to
describe the variations in the internal
- k is Orowan's isotropic (or scalar) stresses X, R, R*.
stress. It corresponds to the initial
yield strength of the material and, among
other things, depends on the volume frac- Constitutive equations for
tion of precipitates and the initial den- viscoplasticity
sity of the dislocations. This flow
The law of viscoplasticity (10) de-
rives from a viscoplastic potential of the
form, Chaboche (1977):
n+1
* K .Jfo-Xl - k - R - R*
* = n-t-1 {
K (11)
Using expression (10) for the modulus p
of the plastic strain rate, we again find:

_ = 3 v• o' - X' (12)

So I J l.o - X)

J(o-X) designates, for example, the

second invariant of the deviator o'-X'
of o-X, for a Von Mises type material.

This model brings in a flow threshold,

given by ke • k + R' + R . Figure 3 indi-
cates schematically the actual domain of
elasticity centered at X, of radius ke, in
Figure 3. Elastic Domain and Visco- the deviator plane of the stress state.
plastiticy Equipotentials. The surface.of equal dissipation (or of
equal rate p) is found by similarity.
17-4

dX c a dep - cx|dep| (15)

It can be noted that, in the limit
case of a very slightly viscous material, The nonlinearity introduced by the re-
(n being very large) the present theory call term la thus not the same during a
must be replaced by a time-independent flow under tensile or under compressive
theory of plasticity which will not be loading: the relation Is nonunique be-
described here. tween X and en and the concavity of the
stress-strain curve is correctly repro-
duced, even in the limiting case of plas-
Non-linear kinematic hardening ticity (Pig. 4b). At each half cycle,
beginning with epo, Xo, the kinematic
Let us first take R = R* • 0. The on- model (15) is integrated explicitly:
ly hardening effect is kinematic: a
translation of the surfaces in the stress
space. This type of hardening is prepon- X(ep) - va + (16)
derant in many situations, and, con-
sequently, must be accurately described if
one wishes to predict the cyclic behaviour and the stress is then expressed by:
of a material.
.1/n
The simplest kinematic model is X(ep) + vk + vKp (17)
linear, and was originally developed by
Prager (1949). It reads:
where v = * 1 gives the direction of the
2 flow. Figure 3b shows this stress decom-
C.Er
position schematically.
or (13) The kinematic model gives the shape of
the hysteresis loop, but also the relation
between the amplitudes at the stabilized
dX cycle (here stabilization is very fast for
a periodic symmetrical loading).
This linear kinematic model has two dis-
advantages: 1) The streaa-atrain curve is
poorly expressed In the limit case of a.tanh (c *&) +k+Kp1/n (18)
time-independent plasticity. 2) It cannot
deacribe the controlled-stress ratchet ef-
fect or the effect of the mean stress For the viacoplastic material, they depend
relaxation, under controlled cyclic on the rate.
strain. Figure 4a shows how the model
stabilizes at the first cycle. This model of viscoplasticity, using
Introducing a recall term to express only the five temperature-dependent coef-
an evanescent plastic strain memory effect ficients n, K, k, a, and C, already gives
brings a significant improvement, a very good approximation of the cyclic
Armstrong and Frederick (1966). behaviour, as illustrated for IN 100 alloy
in the Figures 5 and 6. In a rather
limited strain domain «0.5%) the model
dX = j c a dep - cX dp; X[o) (14) correctly reproduces the cyclic curves,
the hysteresis loop and the effect of the
loading rate or hold time in cyclic creep
Let us clearly note the essential differ- tests.
ence between the two plastic strain in-
crease terms dcB and dp. Under tensile-
compress ive leasing, for example, we
have:

Figure 4a. Immediate Stabilization of the Linear Kinematic Model

(Time-Independent Scheme).
Figure 4b. Nonlinear Kinematic Modem: Stress Decomposition.
 17-5

Note that the cycles stabilize in the

nonlinear kinematic model only if the
loading is symmetrical (zero mean stress).
Qualitatively, the model thus describes (MPi)
the ratchet effects (nonzero Ornean* under
tensile-compressive loading as well as,
for example, the cyclic torsion super-
imposed on a constant tensile streas, and
also describes the effects of the mean 300-
stress relation under controlled strain
loading (nonzero
The valid strain range is widened and
the quantitative description of the rat-
chet effects is improved when we super-
impose several models of the same types as
follows, Krempl (1977):
m
E X
1 *
(19)
dx
k " 3 ck ak de
p - ckxk dp

One of the models can be linear, for

example:

- f CdE p (20) 900'C

For proportional loading, the work-

hardening model is still Integrated ex-
plicitly. Figure 7 illustrates the model-
ing possibilities.

Isotropic hardening
In order to obtain a better fit of the Figure 5. The Model of Viscoplasticity
monotonic hardening curve, and of the cyc- Applied to the IN 100 Alloy:
lic softening or hardening ones, it is Stabilized Hysteresis Loop.
necessary to describe an isotropic harden-
ing, that is to introduce an equation for
R evolution. A simple form of such an
equation, similar to the kinematic case, R(P) - R8(l-exp(-bp)) (22)
is:
Coefficients b and R8 depend on the
dR •> b' (Rs-R)-dp temperature. Tensile-compressive stress
(21) is now expressed in the form:
R(o) = o
.1/n
This internal stress varies as a func- X (ep) * vk -t- vR(p) vK-p (23)
tion of the cumulated plastic strain p .
After a certain number of cycles (less aa For a controlled strain amplitude (alter-
the strain amplitude increases) it stabil- nating, for simplicity), we have at each
izes at the value Rg . This is necessary, cycle:
or else the only possible stabilized cycle
would be elastic. We integrate to get: 0M = XM(Aep) + k + R(p) (24)

where epM la the plastic strain rate for

the maximum stress. In this loading con-
figuration EpM is not very different from
the total controlled strain rate; in any
case, it can be considered to be approxi-
mately the same at each cycle. Further-
more, XL is little different from a
AE
Th C p/2. Applying relation (24) to each
cycle, to the stablized cycle and to the
first cycle, considering the approxima-
. i,-IS or Ml tions we have mentioned, we get:
• i- 165 or ISOt
coniroUfd dapitctmtnt
1,0
• i. lOOar JtOi 3j_^MO- , -i- . ^-bp) {25)

o.i O.t
where OM and o^ are the stress peaks
s o
Figure 6. Modelization of the Cyclic
Curves for the IN 200/Alloy in the stabilized cycle and in the first
at 1000°C. cycle.
17-6

/ (a, AJ, e e - T, aj) (30)

For a theory with time (or rate) effects

the generalized normality is expressed by:
0
jEto' 102500 .2500
i
6 . .12- (31)
hound P 30
We see that the intrinaig dissipation la
necessarily positive if * la convex, pos-
itive and equal to zero at the origin
(o = AJ « 0)i

Dl 20 (32)
too To fit the viscoplasticity law with
nonlinear kinematic hardening and isotro-
pic hardening into this thermodynamic
framework, we simply use the following
expressions for the two potentials:

Figure 7. Stress-Plastic Strain Curve pf = ^ A:E e «E e + J cao:a+w(p) (33)

Obtained by Superimposing
Three Kinematic Variables. (34)

Thermodynamlcal aspects *m K . G-(3/4a)X:X+(l/3)c2ao:a]tVI'1

* n+1 K
The present approach has been deve-
loped in a general thermodynamic framework where G expresses the elastic range:
of irreversible processes with internal
variables, Germain (1973), Sidoroff G » J(o-x) - R - k £ 0 (35)
(1975), Halphen and Nguyen (1975).
Let us summarize this theory briefly assuming here that R > 0 . We get the
as it applies to small quasistatic trans- associated variables from the first
formations, using T for the temperature, potential:
ee for the internal variable that describe
the current atate of the volume element. caa
P f e - *"«' X " P
Two potentials are used: a thermo-
dynamic potential, e.g. the free energy: (36)

» = *(Ee, T, Oj) (26) 3a = W'(p)

The total strain la divided into ther- The hypothesis of normal dlssipativity
moelaatic strain ee and plastic strain E - leads successively to:
ee + Ep. The second potential controls
the diaalpation:

T
» (27)
(37)

The variables EO, T, a appearing in this

potential are considered as parameters. 3X -P 2a
The Clausius-Duhem inequality, expressing
the second principle, leads to: Prom the last relation we get:
(28) (38)
x = 4 ca& » !• caED - cXp

which is identical to (14) in the non-

where a is the stress tensor, c la the linear kinematic model. We note that
entropy, p is the density and AJ are the p is actually the cumulated plastic
thermodynamic forces associated with the strain. Using (34) and (35), we get the
internal variables cu . plastic strain rate modulus by:
G-r(3/4a)XtX-(l/3)c2aaia .n
- q grad T £ o (29) (39)
3G K
2
and, using X • j C a a, we find the
This must be positive because of the
second principle. This is automatically relation (10) again exactly, with R* - 0 .
the case if we adopt the hypothesis of
generalized normality, using the potential
<> obtained from 4> by a Legendre-Fenschel
transformation on the variables
 17-7
DAMAGE DESCRIPTION The Continuous Damage Approach:
One-Dimensional Aspects
General Remarks
Mechanistic concept of a macroscopic
In the previous paragraph, we have internal variable.
seen that a model with internal variables
is able to describe the macroscopic me- Before any damage theory can be de-
chanical behaviour of a metallic material veloped, it is necessary to define what we
undergoing viscoplastic straining. Yet, mean by the ultimate stage of the damage
during this straining, and particularly processes. Current Continuous Damage
under cyclic conditions. microscopical Mechanics assumes that this final stage
phenomena take place, such as dislocation corresponds to the macroscopic crack
motion, which lead to macroscopical de- initiation, that is the presence of a
fects of significant size classically material discontinuity, sufficiently large
known as cracka. as regards the microscopic heterogeneities
(grains, subgrains...). In such a case,
Since the presence of cracka can mark- the main macroscopic crack is assumed to
edly reduce the atrength of a structure, be developed through several grains, in
it is obvious that the prediction of crack order to show a sufficient macroscopic
initiation is of paramount importance in homogeneity, in size, geometry and direc-
engineering deaign. For inatance, the tion, leading to a possible treatment
assessment of the structural integrity of through the Fracture Mechanics concepts
a turbine disk must be achieved before any (see the schematic illustration in Figure
usage of the engine, since the burst of 8). Let us remark that, contrary to the
such a disk can lead to very serious usual definitions, the present one can be
damage to the passengers and the airframe, approximately associated with the breaking
and even to complete loss of the up of a uniaxial specimen subjected to
aircraft. tension-compression.
The classical way to predict crack The theory is supported by the physi-
initiation was to use the so-called "para- cal idea that crack initiation is preceded
metric failure relations." Since the by a progressive internal deterioration of
original idea and work of Kachanov (1958) the material, which induces a losa of
and Rabotnov (1969) on the concept of strength in terms of strain as well a* In
"macroscopic damage," we, at ONERA, have terms of remaining life. Kachanov (1958)
contributed to develop an alternate way of first proposed to relate these two aspects
prediction known as "Continuous Damage through a macroscopic damage variable,
Mechanics." introduced with the effective stress
concept. The phenomenologlcal idea of a
The objectives of this paragraph are damage parameter D taking into account, on
to briefly introduce the theory, review a macroscopic scale, the microacopic
its specific advantages compared to the deteriorations (voids, decohesions, micro-
classical approach, illustrate it by the cracks,...) Is generally accepted, Krerapl
results obtained for some materials and (1977), with the following limiting
finally open the discussion on its values: D a o for the initially un-
applicability and possible future stressed material, D • Dc at failure, that
developments.
'*<»rv • Micro* j Micro • i~i9crv —
slip •crick \pnatgilion\-crielt \proptgition
btndt initiation i \inititlion'.
0.01 0.1 ;
CUtiictl
crtek ittitntion

Prntnt dtfinition
of trtck miliilian

„ _ _ _t . »-' 1 Frteturt
j1 mtcfitnict

Surttet

Gnini Frtcturt
mtchtnict
idnliittion
Microscopic
initinioia

i ont man
microscopic trie*

Figure 8. Schematic Illustration of a Macroscopic Crack Initiation Concept.

17-8

is at the macroscopic crack initiation 1/m

time (in several works the critical value 1 - (At /Ac p )
Dc is taken as 1). The effective stress *a
concept is uaed to describe the effect of
damage on the strain behaviour: a damaged Figure 10 shows the example of IN 100
volume of material under the applied refractory alloy tested at 1000'C, at 5
stress a shows the same strain response as Hz, under completely reversed stress
the undamaged one submitted to the effec- control. Let us underline the high non-
tive stress (Fig. 9). linearity of damage evolution and the
stress dependency.
e - o/U-D) Using similar procedures, directly de-
duced from the initial propositions of
Kachanov one can obtain the creep damage
evolution during creep tests (Lemaitre
and Chaboche (1975), (Fig. 11). Eq. (1)
tM corresponds to the
Kachanov. Actually,
initial theory of
the microscopic
damage process can give little loss of
D.O effective area before crack initiation,
t.Ht especially under fatigue loading:
gives rise to very small values of the D
this
parameter until a large fraction of life
is consumed, which induces highly non-
Figure 9. The Effective Stress Concept. linear damage evolution curves. However,
it must be emphasized that small values of
D can lead to large reductions in life:
Under this definition D represents a loss using the present approach, the damage is
of effective area taking into account no longer proportional to N/Np.
decohesions and local stress concentra-
tions through homogeneization concepts,
Duvaut (1976).
Depending on the concerned areas, dif- D
ferent macroscopic measures of damage have
been uaed in the past: in addition to 0.5
denaity changes or electric resistivity
measurements, Cailletaud et al (1980), one
can distinguish between:
- measures of the remaining life as used
in creep, Woodford (1973), or in
fatigue, Kommers (1945).
- measures of the reduction in fatigue 0.5 I N/Nr
limits, Bui Quoc et al (1971), which
need many tests to define the damage
evolution curves,
- measures of the stress-strain behav- Figure. 10. Fatigue Damage Evolution
iour, which are now retained because Of Curves as Measured Through
their easier applicability in each teat Plastic Strain Range and
(independently of the others). They Effective Stress Concept:
can be obtained in terras of elastic IN 100 Alloy, 1000'C, 5 Hz
strain (Youno's modulus), Lemaitre and Freguency, Load.
Dufailly (1977), plastic strain range,
Chaboche (1974), or stress range. The lost area interpretation can be
considered under many situations but con-
stitutes a first approximation only. It
In the particular case of stress con- applies mostly to the ultimate crack
trolled tests, measurements of plastic micropropagation stage of the tests.
strain range evolution, lead to damage Future improvements could be obtained by
measurements through the classical Hanson- introduction of two damage parameters
Coffin equation for the undamaged stabil- corresponding to the micro-initiation and
ized stress-strain behaviour and the ef- macropropagation stages, as in the aimpli-
fective stress concept for the last part •fied approach of the Double Linear
of the test. According to this hypo- Cumulative Damage Rule, Manson et al
thesis: (1965).

Lf Aq im (40) Development of damage constitutive

"P K(!-D) J equations
Stabilized conditions correspond to:
Let us begin with an important remark:
1
these eguations have to be developed in
- Uo/K) " where is differential form: an expression of D aa
r
B a function of time, for example, consti-
tutes only a response of the.material to a
determined from a number of cyclic tests. particular forcing parameter. All equa-
The damage follows from, Chaboche (1974), tions using the consumed potential aa a
damage parameter are thua eliminated.
 17-9

Second general remark: if nonlinear

cumulative effects are needed, these equa-
tions must have unseparable variables in D
terms of damage and the chosen forcing IH100 lOOO'C
parameter, Krempl (1977), Duvaut (1976), .SO
Cailletaud et al (1980), Woodford (1980),
Kommers (1945), Bui Ouoc et al (1971),
Lemaitre and Dufailly (1977), Chaboche
(1974). In other words, the damage
response functions have to be different
under different loading conditions, Bui
Ouoc et al (1971).

Three types of damage evolutions can

be considered: as a function of stress in
the static plastic failure (or in fa-
tigue); as a function of time for the
creep processes (or for corrosion or ir-
radiation processes), and as a function of
cycles for the fatigue processes. Each of
them has to be identified by some specific SO-
tests, independently of the others, lead-
ing to the determination of the corre-
sponding differential damage equations.
Their one-dimensional isothermal form is, 25
Lemaitre and Chaboche (1975).
dD Fj. (*»OL,DI, . . . )d<J
l

dDj F2(«,a,D2, .. . )dt (42) 0,5 i i/ie

F3($,a,D3, .. . )dN

Figure 11. Creep Damage Evolution Curves

Here $ denotes the chosen forcing vari- as Measured Through Strain
ables: stress or strain or plastic Rate and Effective Stress
strain, and a represents the internal Concept: IN 100, 1000°C.,
variables, describing for example the har- AU2GN, 130°C.
dening state of the material.
Application to the Refractory Alloy IN 100
When several processes act simultan-
eously, the interaction effect has to be Creep-fatigue interaction
determined through special tests: for
example, the combination of creep and The creep and fatigue damage were in-
fatigue effects could be treated by intro- dependently characterized from pure creep
ducing coupling terms in eq. (42) and pure fatigue (high frequency) teats,
through failure measurements and damage
dD2 »t>2'D3'- •• )dt measurements.
(43)
dD3 P3(»,a,D2,D3,. .. )dN Equations of damage, Lemaitre and
Chaboche (1975), Manson et al (1965),
Chaboche et al (1978), can be explicitly
Although some microphysical studies Integrated (see Figs. 10, 11). This leads
use such coupling parameters, moat of the to the possibility of description of se-
present day cumulative damage theories are quence effects aa in the case of two level
based on the simplifying hypothesis that fatique tests (see Fig. 12). As already
damage variables (here D2 and D3) are of mentioned by several workers, Bui Ouoc et
similar nature and interact in an additive al. (1971), Lemaitre and Dufailly (1977),
manner, which corresponds to the special Chaboche (1977), Lemaitre and Chaboche
forms: (1975), Manson et al (1965), theme se-
quence effects are consistent with the
stress dependency. Figure 13 illustrates
how the linear Palmgree-Miner rule does
(44) not hold in such cases.
dD3 • F3(4>,a,D2
Two typea of creep-fatigue situation*
can be considered. First, the two level
As shown in several applications, the teats where a portion of life is spent in
nonlinearities of the interaction pro- fatigue (reap, creep), the remaining life
ceases can effectively be described, be- is being measured under creep (reap,
cause of the different nonlinearities in fatigue). For IN 100, such testing leads
functions f 2 and f 3 . Under this hypo- to nonlinear effects with life summations
thesis only one damage variable has to greater than 1 (reap, smaller than 1), be-
be considered for the general case. cause the measured fatigue damage rate
Combination of eqs. (42) leads to: (Fig. 10) is much smaller than the creep
damage rate (Fig. 11) when they are nor-
malized by the total life. This has been
dO + F 2 U,o,D)dt experimentally checked and compared quite
(45) satisfactorily with damage equations (Ptg.
* F3(»,o,D)dN 14).
17-10

T'930'C
ffu • 439 Moo
T'/OOO'C
'6/6 &Ot'«H ..
a aat "tu os

O.S l

Figure 12. Prediction of Two-Stress-Level-Fatigue Tests - IN 100 Allow.

The second case, more important for Stress-strain behaviour of the damaoa
the applications, appears when creep and material
fatigue act simultaneously during each
cycle: low frequency stress controlled or In the case of high temperature viaco-
strain controlled cyclic testa, cyclic plastic behaviour of some refractory
creep, cyclic tests with hold times under alloys, particular strain rate equations
strain control. In such cases the deve- have been developed, using isotropic and
loped damage equations, Chaboche et al nonlinear kinematic hardening rules,
(1978), are numerically integrated, the Chaboche et al (1978), consistent with a
nonlinear interaction effect being repro- general thermodynaraical framework. For IN
duced through the different damage rates 100 alloy, the one-dimensional isothermal
under creep and under fatigue for a given equations are:
damage state. This is supported by the
physical idea that creep cavities nucleate Jo-xJ-R \n . , ,
early in the life and accelerate the nu- '—jr1 J ' sign(o-x) (46)
cleation and the propagation of fatigue
microcracka.
x = cF(pHaep-x|Ep|-b|xfsign(x) (47)
Figure 15 shows that such a theory
predicts nonlinear creep-fatigue inter-
action with a stress range dependency: F(p) = l+(l-
the lower the stress, the greater the (48)
interaction that is the greater the life
reduction by comparison with pure fatigue. P-
Such predictions are consistent with ex-
perimental results reported in the liter-
ature, Krempl and Walker (1968), and agree where p and X are isotropic and
fairly well with results of cyclic creep kinematic internal variables and n, K,
tests performed with IN 100. For this C, a, b, m, B, » are coefficients.
material the Low-Cycle High Temperature
fatigue, with or without hold timea, is
predictable from damage eguations deter-
mined under pure creep and pure fatigue ,. non imiw cuffluinnt
conditions, as shown in Figure 16. Let us t
c/lc / pridieti«ni
emphasize that the predictions shown in
Figures 15 and 16 have been made with
equations whose constants have been deter-
mined only from pure creep and pure
fatigue tests.

OS lin«i» N
cumulation .>

On.20S.5MPt ^
0.5
a. 0.965
K.15

Figure 14. Creep Rupture after Pure

Fatigue Cycling; IN 100,
Figure 13. Schematic Explanation of Non- 1000'C. (0 teats,
Linear Cumulative Effects From predictions).
the Damage Evolution Curves.
 17-11

AcC/.) soo'c

linear cumulation
'0.5

0.4

10* 10* 10*

.JM) • *"• '•'•»>• «»•* eomrog

0 tfffmO
1
(n-JBO • t«,.JO»
aa- •N

0.6
\^^ tOOO'C

0.4-

Figure 15. Creep-Fatigue Interaction for 5Hi

Cyclic Creep Tests (IN 100,
lOOO'C.). 0.2-

This formulation bringa together con- to' to*

cepts and equationa proposed by several
workera and contains many descriptive pos- Figure 16. Prediction of Strain Controlled
sibilities of nonlinear hardening, creep Cyclic Tests on IN 100, from
and relaxation, Bauschinger effects, with the Measured Stabilized Loops
the strain rate equation (46) and the non- and the Non-Linear Creep-
linear kinematic hardening (47), stabil- Fatigue Damage Equations.
ized cyclic behaviour, cyclic hardening or
softening through isotropic hardening de-
pendence (48),time softening (high temper- Introduction of the coupling with
ature recovery affects) induced by the damage effect is easily done by the effec-
last term in eq. (47). tive stress, replacing o by o/l-D . Con-
sistency in the hardening rule needs also
Moreover, some microstructural harden- the modification of X by X/l-D .
ing effects induced by temperature changes
can be described through the introduction This makes possible to describe terti-
of additional internal variables, Chaboche ary creep effects as shown for IN 100 in
and Cailletaud (1979). Fig. 17.

prediction!
x o. tent

INtOO.T.IOOO'C

0 *•
10 20 40 60

Figure 17. Prediction of Creep Curves and Creep-Fatigue through Coupled

Viscoplastic and Damage Eguations IN 100, 1000°C.).
17-12

STRUCTURAL LIFE PREDICITON AT HIGH In some isothermal problems, K is then

TEMPERATURE, UNDER COMPLEX LOADINGS, triangularized and assembled only once;
CHABOCHE AND "CAILLETAUD T1984') for the time dependent temperature cases,
we use a linear interpolation of the in-
Structure Analysis in viscoplasticity verse of K between two exact values, in
order to minimize the computation cost.
The object of this paragraph is to
present the "structural" aspect of life
time prediction, i.e., the numerical Viscoplastic algorithm
methods and tools which have been devel-
oped at o.N.E.R.A. to integrate the As already seen, we can write the
results described in the two previous constitutive equations as:
sections.
F(o,aj,T)
This has resulted in a finite element
code named 'EVPCYCL", suitable for treat- (52)
ing two-dimensional structures under plane g(o,aj,T)
stress, plane strain, or axisymmetricai
cases.
where the a- are scalar or tensorial vari-
This code exhibits a great versatility ables describing the hardening state of
concerning the choice of constitutive the material, and T is the temperature.
equations; for instance, one can choose to It follows that a step by step integration
describe isotropic hardening or kinemati- process is required; several options are
cal hardening or both, etc.... Time de- offered in EVPCVCL:
pendent temperature fields are also
accepted. We describe now some features - completely explicit
of this code. (Euler, second order)

- semi-implicit
EVPCYCL Finite Element Code (Euler-Cauchy), Chaboche (1978 )

Finite element aspects

The most often used is the second
This part of the code is quite clas- order one, developed at ONERA, Savalle and
sical: the unknowns are the nodal dis- Culie (1978). In this method we perform
placements (q), which are linked to the an automatic computation of the time inte-
displacements (u) and strains (e) within gration step before the time increment, by
the elements through the matrices N and B, using the first and second derivatives of
giving: EP and 3i which are explicit at time t,
since, if y designates the stage variable
N. q vector (ep and a*) one can writei
(49)
B. q
y(t) » Y(y(t),t)
The total strain Ls the sum of three
terms: (5->)

ee : elastic strain

ep : viscoplastic strain
One feature of the chosen algorithm is to
EJ.^ : thermal strain (= 06) increment only the state variables c,
aj which leads to:
so that Hooke's law leads to:
pt tp)t.4t t. at*
2
o = De - Dep - Do9.n (50)
where a is the thermal expansion (54
coefficient.
Jt+it .at
8 the temperature,

Jl unit tensor in space The first order terms are obtained

after expressing equilibrium at time t,
(51) giving the knowledge of e p ; the
The last two terms can be considered second order terms are obtained by solving
as an initial stress, so that one can
write:
Kq F + F, {551
K.q (51)
where F depends on T.
where K is the classical stiffness
matrix (depending on the elastic constants
of the material and not on the visco- We can then obtain the third order
plastic behaviour). term in ep and aj , which allows the opti-
mal determination of the time increment,
by assessing that the third order term in
F and F0 are the nodal forces the Taylor expansion must be negligible
equivalent to external loads and initial compared to the sum of the first and
stress. second order terms.
 17-13

The time increment is obtained from Results

eguation (56):
For the stresses, results are shown in
Figures 19,20, and 21 for the two tempera-
(t) + y(t).-*|| (56) tures. At the lower temperature, the
material is only slightly viscous and
n being a precision factor of roughly stress redistribution is essentially
5.10~z. Although this method is not plastic; in contrast, at 650°C the vis-
rigorously established, it leads to a re- cosity is present, so that we observe vis-
markably safe algorithm. In addition, it coplastic flow during the hold time.
has the advantage of needing only the
approximate incrementation of ep and a« For the 550"C case, two computations
which avoids the cumulation of errors ana were made, one using a complete model
gives good stability to the algorithm describing the behaviour continuously from
(although, by nature, an explicit algo- the first cycle until obtaining the stabi-
rithm is less stable than an implicit lized one, and a second one where only the
one). model describing the stabilized cycle la
used. Figures 19 and 20 show the great
gualitative difference between these two
Example of an Application; Biaxial cases. In particular, the zone under high
Fatigue Disk stress being much more extended in the
first case. This would lead to large
The geometry and mesh are indicated in differences in predicting life time.
Figure 18, the elements used are tri-
angles, the problem is axisymmetrical. Finally, a comparison was made with a
corresponding test where the increase in
Loading conditiona the diameter of the bore was measured as
.04 mm. The computed values were .04 and
The temperature is assumed as uniform .07 mm respectively at the external sur-
and constant. Two cases were studied! face and in the plane of symmetry which
shows good agreement.
- 550°C, low rotation speed i
1500 r.p.m
high rotation speedt
27700 r.p.m/10 a
- 650"C, low rotation speed :
1500 r.p.m
high rotation speedt
24000 r.p.m/90 a

Figure 18. Mesh for F.E.M. Analysis.

Figure 19. 550»C. - Stress Distribution with Complete Behaviour Described.

17-14

<sso

Figure 20. T - S50*C. - Stress Distribution with Behaviour of Stabilized Cycle.

Figure 21. T - 650*C. - Stress Distribution (Complete Behaviour)

Life time prediction of turbine blades tions, Manson (1954), Coffin (1954), for
fatigue, and the Kachanov-Rabotnov equa-
The general method previously de- tion for creep, Rabotnov (1969).
scribed, i.e.:
The computed crack initiation numbers
- structure analysis under viscoplastic of cycles were compared (Fig. 22) to the
behaviour experimental ones obtained through tests
on actual blades with thermal and centri-
- damage cumulation for creep-fatigue in- fugal loadings (see Pig. 23 for the de-
teraction has been applied to the case scription of the experimental set-up).
of two turbine blades with different
cooling ducts, but made of the same All this work is fully described by
material, the IN 100 refractory allot. Policella and Culie (1981). It can be
shown, from Figure 22, that the predic-
In this case, the structure analysis tions fall reasonably in the experimental
was performed using Bernoulli's kinemati- scatter zone, the number of cycles to
cal hypothesis for beams, Chaboche and initiation being determined by extrapolat-
Culie (1980); the damage behaviour was ing the curves of crack length versus num-
described using the Hanson-Coffin equa- ber of cycles toward zero length.
 17-15

CONCLUSIONS For the time being, extensions of this

approach are in progress concerning the
The Continuous Damage Mechanics has following items:
been proved to be an accurate tool for
prediction of crack initiation. - description of viscoplastic behaviour
and damage of materials exhibiting a
It must be noted that it is a complex high initial anisotropy, like single
tool, either in its numerical implementa- crystal alloys;
tion on computers, or concerning the ex-
perimental determination of behaviour and - numerical treatment of fully three-
damage models, particularly for multi- dimensional structures and behaviour,
dimensional situations. Yet, this com- it must be noted that this requires
plexity allows one to treat very difficult very powerful computers.
problems, where, for instance, the
materials undergo microstructurol changes
during the period of loading, as already
pointed out.

Figure 22. Comparisons of Predicted Number of Cycles to the Test Results,

Fore*
Control Dyrumormtar
Air outlet
Inductort

Figure 23. Description of the Experimental Set-Up.

IS-I

ANALYSIS

Two-Dimensional sp^r ion Coefficients

The unsteady aerodynamic ilieuiy

t vn-dimensional Uuii AH lu; 1 «xe'_ut ii'itf
hnnnnnic motiuii in vei Licdi Urfiiilaiion Aiid''yr
t w i s t has been e»l«iii.iv«iy investigated by a
numbei ui. dulliwis (TUwudcn sen 1935, Scar! an 311^
1931, Bisplinfihoff, A s h l e y . *nH Ha1 fntan
and -ill not be <*1«rti«s*>rf ir> any great
detail in t h i s chapter. Howevrr. fnr r l.in r. in
the #-11511 lug rt^rivarinn, 11 i r. expedient '.u
L
desr r1 be b r i p f l v rhf j>hvsi'"al ;V^ LPIFI ht;t IIR
<-on^idpr*>H .nid 10 define (he itooenclaiuie to I"'
use-d In Th

; . -
-

Figure 5 Airtoil section notation showing both

undellecied and deflected Matin

... , •
. ; : i . •i
, • .
1

: ' ' . .
,
., . • ,
FiqL»e d Rubber wheel deformation . . . . .

! i. t be 0.11 Jy i K'li'Is , > mimbfi • • ' \, • •>,.

ii j otdei ••-• ihrar iors^ .1 r . rn •
iii l)i»lh engine ,'ind U'" i • if iti pi •
1
The Elrtui iov.-'i o.ic.hi • • . <ei »'.
:
. . .-- .'«t « : : , . ' . ! i L v u L l y tiigli lu • • . • • : the
Sit ft * per a* !»A" i Ai'ir.' ot ' '.'- ".ii'( -,;..-,ILH . AM t'i!i('l,v
to r e l a t e the.s*-' vibrations • th« LaJ ';-,iiiei ••i, K . .>;i i n,, . . • - , . jj lls
r t o rotrHinfl ; r _all laii-.j. j r t i K « l - . . ., ','.:...,: aeiwtJ iini •• ' - . H f" i
vtbrari->n«, AfT*r, nn n r v * - ^ on 01 m»jit 1
i' • • :
• v : ' . i i..- . : .. ; ' • - . • , . , ! lue to
I lie engLiie npcr ^t inp i in^ . iinhsf rnifnt Ans I vs 1 •: ol ^iriiii.;. 'ili-l r.--jFs f!!i ! ' . • • WlSli i f^f-is ' • ' "*1'..
rhM'-:«* ' liif •; re liases iif.reaic(J I bni i hr i-th^prvrd i 01 -••;i»:ii|. I*.-, i t It; v .-,(|., i •-t.,i ' •-: i (iijyj v ( I'iVj i !>.i ,ii i
f i f((!iprir; , H- .j{ ideie irisiatj i 1 i tie.-- r f v r r r l A rrr "•- : isolated diituiJ ;i' zorc inr-1 dpni**; > • • ing in
v i r n t i m j . - HI! i ,-• i.^ij ! F ._-"ji.i_'i!i_ : ' .; ;jj 1 he r r > " . pd an lncompt*»tiible, iwn dimrns J i>i'i« I ' M,. . •.- .
L
J!U '->nFJ nj'ti I'lii dcsr ri iirn prrv i nns!y . f 1
r h & ^ e iuan 11 K;;, tnay br tr-'i i r < p M ;,, ML- ' nnn nl
Stni IR and
Thf i n i t . i n i O!'KI_I ui (lie-
t o oxplorf- rhf muter I y ing uni(.)jai]iBfu ot r lii.i
uis Labi li t y .ind TO shciv r tin! mill*-! VHI i.a i it
coii-ji t lutii ul ai t Llow and L-.vior grnnimry rhi s
i:oi.ipl«<J oscillation way -japabic ot c x t f.ic t ing
*>n*rgy from the aiistt^ani in stit ti^ienl ^uauLilies
rf> prfiditf P nn i.tn? rahle v i bi dtoiy i:njtir>ji. A
further o b j f c r i vp was t o make the analysis
yuL t jc icni ly jj^nrra 1 ro ppi-nl t t ts us* ui th
arrody namir anri/or structural dynamic
icr., anrf nl r i m s r p l y , to provide the
w i t h a loot for flurrrr fr^p op
 - 2^8 - VOL- 2. 18-1

AEROELASTIC COUPLING - AN ELEMENTARY APPROACH

by Franklin 0. Carta
United Technologies Research Center
Bast Hartford, CT 06108
USA

INTRODUCTION
In the Introduction and Overview of Volume 1, Within Units, the Increase in torsional
the vide diversity of flutter and vibration frequency vas a viable fix to this problem. It vas
instabilities of turbomachinery blade rovs faced by generally accomplished by Increasing the thickness
the designer have been described. Common to all of of the blade, vhlch had the effect of Increasing
these Instabilities are tvo necessary requirements engine weight. Early designs vere based on steam
for them to occur: 1) an available energy supply turbine technology, in vhlch thick blades vere the
(I.e., the moving air stream) and 2) a zero or norm, and the percentage Increase In bulk vas not
negatively damped system. In the noat elementary prohibitive to the overall system operation.
sense, the vord "system" refers to the combination Hovever, as higher performance vas required of the
of the blading and the alrstream. Taken alone, the engine, thin blades of higher aspect ratio vere
blades have positive damping. The Inclusion of the found to provide this extra performance margin, and
alrstream will either increase or decrease the unfortunately, both of these blade characteristics
damping of the system, and In the case of a vere found to place the operating point securely
negatively damped system, vill lead to within the flutter region of Pig. 1. Further
self-excitation or flutter. discussions of these early problems can be found In
Shannon (1945), Armstrong and Stevenson (1960), and
It should be noted that the emphasis on Slsto and Nl (1970).
self-excited phenomena in this chapter does not
imply that this Is the only important vibratory The introduction of part-span shrouds (called
problem faced by the engine designer. Indeed, snubbers or clappers in Great Britain) provided an
there are several chapters In this Manual (Chapters additional constraint that dramatically raised both
9, 16, 17, 19, 20, 22) that are concerned with the torsional and bending frequencies of the blades
forced vibration, resonant response, and fatigue of vlthout materially affecting the overall velght,
engine structures. Nevertheless, most or all of and in some instances permitted the use of thinner
these phenomena involve the close coupling and (and hence lighter) hardvare. The need for this
Interaction of aerodynamics and structures, and for configuration vas driven by the introduction of the
this reason, the present topic vas chosen to fan engine, vhich required one or more stages of
introduce the reader to the subject of coupled extra long blades at the compressor inlet to
aeroelaticity in axial flov turbomachines, in provide an annulus of air to bypass the central
preparation for the chapters that follow. core of the engine. This solved the stall flutter
problem, but introduced a more Insidious problem,
It is historically appropriate to initiate initially termed "non-integral order flutter",
this chapter with a brief reviev of the evolution vhich vas Impossible to predict vith the available
in engine design that led to the occurrence of empirical tools. The term "non-Integral order" vas
coupled flutter, and to Its subsequent analysis and chosen because the flutter, vhlch Involved the
prediction by the energy method. Prior to the coupling of bending and torsion modes, did not
early 1960s the observed flutter of turbonachinery occur exclusively at the intersections of the
blading vas usually a slngle-degree-of-freedom engine order lines and the natural frequency curves
instability associated with high blade loading, and of the Campbell diagram (cf. Pig. 3 of the
vas invariably called "stall flutter" (cf. Chapter Introduction to Vol. 1). The problem vas further
7). Although several multiblade unsteady exacerbated by the relative supersonic speeds at
aerodynamic theories existed (Lane and Vang 1954, vhlch the blade tips operated.
Sis to 1932, and Vhitehead 1960), they vere all
based on linear potential flov of an incompressible
fluid past Infinitely thin flat plate airfoils. FLUTTER
Clearly, none vere applicable by virtue of the BOUNDARY
nonlinear nature of the governing aerodynamics.
Hence, empiricism vas the only tool available to
the designer.
The empiricism took the form of a plot of
reduced velocity (U/btu) vs. Incidence angle
(Fig. 1) in vhlch the lover left portion of the
plot vas flutter free and represented "goodness",
while the upper right portion, above the curved
boundary, vas a region of progressively increasing
torsional stress, leading ultimately to blade
failure (cf. Pig. 2 of the Introduction of Vol. 1, UNSTABLE
and Pig. 2 of Chapter 7). If a blade design
yielded an operating condition at "1", within the
flutter regime, the designer could apply a series
of corrections that vould lead to a reduction In STABLE
angle of attack (which vould yield a lover pressure
rise and vas therefore unfavorable), or a reduction
in velocity (having the sane unfavorable effect on
performance), or an increase in torsional
frequency, u., vhich vould also decrease ' the INCIDENCE ANGLE, o
value of the ordlnate of the operating point to "2"
In Pig. 1.
Figure 1 Schematic stall flutter map.
18-2

At this time (in the early 1960s) there vere

no applicable aerodynamic theories capable of
modeling the complex flov field in the tip region
of the nev fan jet geometry, shown schematically in
Fig. 2. Here the axial velocity was subsonic, but
Its vector sum vlth the vheel speed yielded a
supersonic relative speed that placed the leading
edge Mach vaves ahead of the leading edge locus of
the blade rov. Thus the relatively simple theory
of Lane (1957) for supersonic through-flow vas
Inapplicable. Furthermore, the incompressible
theories cited above vere Inappropriate for
compressible flov, and vere far too complicated for
routine computations on existing computer hardvare,
and hence could not be used even for trend studies.
The present chapter will reviev the first
published vork to provide a means for Identifying
the phenomenon and for predicting its behavior
(Carta 1967). Although the initial paper relied on
unsteady aerodynamic theories for isolated airfoils
in an incompressible flov, the fundamental
principle vas sound, and its later use vlth the
supersonic cascade theory of Verdon (1973) and
subsequent aerodynamic theories vas shovn to be
accurate as a predictive design tool by engine
manufacturers (Mikolajczak et al 1975, Ballivell
1975, 1980). DIRECTION OF
ROTATION
The object of this chapter is to introduce the
reader to the concept of the coupled, multiblade
flutter instability. The fundamental aeroelastic
equations linking aerodynamic forces and moments Figure 2 Supersonic cascade- with aubaonlc toadlng
with elastic blade deformations vil be manipulated
to yield a prediction of the vork per cycle of
coupled motion, leading to a stability prediction.
Although superceded by more modem approaches and
techniques this will serve to highlight the need to
understand both the aerodynamic and the structural stiffness, Ib/ft
contributions to the phenomenon. The remainder of
this tvo volume vork has been devoted to satisfying kinetic energy, ft-lb
this need.
lift, Ib (positive upvard), or
NOMENCLATURE lift function

a dimenslonless distance of pivot axis of m mass, Ib-sec /ft

the mldchord, in semichords
M moment, ft-lb (positive nose up), or
A lift function moment function, or Mach number
A amplitude of nth vave number of blades
b semi chord, ft N number of vaves, or number of nodal
diameters
B moment function
radius, ft
c damping, Ib-sec/ft
peripheral distance along rim, ft
ccr critical damping, Ib-sec/ft
s - s'/S dimenslonless peripheral distance along
C(k) Theodorsen function rim
f frequency, cps S peripheral vave length, ft
F.a damping force, Ib t time, sec
P(k) real part of Theodorsen function T period of damped motion
G(k) imaginary part of Theodorsen function U velocity relative to moving blade,
ft/sec
h' bending deflection, ft (positive
downward) V vork, ft-lb
h - h'/b dimenslonless bending deflection, x displacement, ft
in semichords
a tvlst angle, rad, or incidence angle,
deg (positive nose up)

k - boVU reduced frequency parameter OLg chordal stagger angle, deg

kc - 2kH/(M2-l) compressible reduced Y damping ratio
frequency
 18-3

5 logarithmic decrement, or blade physical embodiment of the •Igensolutiona of th*

deflection, ft system, and It can b* ahovn, using standard
structural dynamical techniques, that th* syat«a
T) dimenslonless spanvise station frequency for each mod* is primarily a function of
th* physical distribution of th* ayatea maaa and
0 phase angle betveen bending and stiffness and is only slightly affected by th*
torsional motions, rad rotation of th* system. Thus th* system
frequencies do not necessarily coincide vlth
X normalized vork term integral multiples of the cotor speed, and In
fact, such coincidences of frequency are avoided
li real part of eigenvalue for the lover frequencies if possible.
v imaginary part of eigenvalue
p air density, Ib-sec/ft4
T gap, ft
o Interblade phase angle
u frequency, rad/sec
Subscripts
ax axial
B bending term
C coupling term
d damped value
0) TWO NODAL DIAMETER PATTERN
h due to bending
I imaginary part

0 coot radius or
frequency
P pitching term
R real part
t torsional
tan tangential
T tip value
TOT total b) THREE NODAL DIAMETER PATTERN
due to pitch
Superscripts
(~) amplitude or average over Figure 3 Typical diametric nod* configurations.
one cycle
(•),(•*)first and second derivatives
with respect to time
A graphic depiction of thes* coupled diak
node* can b* found in th* "rubber wheel"
SYSTEM MODE SHAPES experiment performed by Stargardter (1966) in
vhleh a flexible nultiblad* rotor, vlth integral
The vibratory mod* shapes vhich can exist on pact-span ring, vaa spun over a range* of
a rotor consisting of a flexible blade-disk-ihroud rotational speeds and subjected to integral order
system are veil knovn to structural dynamiclsts in excitations with air jeta. The deformation mod*
the turbomachinery field and are discussed in shapes vere exaggerated relative to k "real*
detail in chapter 15 of this volume. Although rotor, but left no doubt about the phyiics of the
both concentric and diametric modes can occur, the problem and the key role played by the part span
latter are the only system modes which are of shroud in coupling the bending and torsion mode*.
interest in the present chapter. These diaaetric Figure 4, taken from the vork that led to th*
modes are characterised by node lines lying along paper, shovs the flexible wheel in plan view, and
the diameters of the wheel and having a constant tvo other edgevlae vleva of tvo- and three-
angular spacing. Thus, for example, a two-nodal nodal diameter vibrations. Of necessity, tbaae
diameter nod* vould have tvo nod* lines are integral order modes because of the use of aa
intersecting normally at the center of the disk, excitation source that was fixed in apace.
and a three-nodal diameter mod* vould have thrt* However, they differ from the nonintegral order
nod* lines Intersecting at the disk center vlth an flutter mode* only in that they are stationary in
angular spacing of 60 deg betveen adjacent node apace while the nonintegral modes are traveling
lines (see Fig. 3). Theae diametric modes ar* the wave*.
18-4

ANALYSIS
Two-Dimeniional Section Coefficients

The unsteady aerodynamic theory of a

two-dinensional thin airfoil executing simple
harmonic notion in vertical translation and/or
twist has been extensively investigatad by a
number of authors (Th*odors*n 1935, Seanlan and
Rosenbaum 1951, Bisplinghoff, Ashley, and Halfman
1955) and vlll not be discussed in any great
detail in this chapter. However, for clarity in
the ensuing derivation, it is expedient to
describe briefly tht physical system being
considered and to define the nomenclature to be
used in the analysis.

NOTE: QUANTITIES POSITIVE AS SHOWN

UN DEFLECT?

Ufa

Figure 5 Airfoil eection notation showing both

undetected and deflected blade.

Figure 5 is a schematic representation of a

(b> Two Nodal Diar.eter Three llodal Diameter two-dimensional airfoil section displaced in both
VibratLon Vibration vertical translation (normal to the chord) and
twist. The effects of translation of the airfoil
parallel to the chord are of second order
(Vhtteneac 1960) and have been neglected herein.
The complex. t ine-dependent uns teady lift and
Figure 4 Rubber wheel deformation, moment per unit span are given by

.3,2 h'
iL- * A-a (1)
In the early 1960s, a number of instances of
nonintegral order vibrations at high sirtss
occurred in both engine and test rig compressor
rotors. The stress levels reached in a number of LBh — * B. (2)
cases were sufficiently high to severely liuit the
safe operating range of the compressor. Attempts
to relate these vibrations to the stall flutter where B^, and Baa represent the
phenomenon or to rotating stall failed, largely standard' unsteady aerodynamic coefficients -- lift
because the vibratons often occurred on or near due to bending, lift due to tvist, moment due to
the engine operating line. Subsequent analysis of bending, and Moment due to tvist, respectively.
these these cases revealed that the observed For exampie. if Theodorsen's theory (1935) for an
frequencies of these instabilities correlated well isolated airfoil at ztro incidence oscillating in
with the predicted frequencies of the coupled an incompressible, two-dimensional flow is used,
blade-disfc-shroud motion described previously. these quantities may be rewritten in the for* of
Sailg and Wasserman (1942) as
The initial object of the 1967 analysis vas
to explore the underlying mechanism of this
Instability and to shov that under certain
conditions of airflow and rotor geometry this
coupled oscillation was capable of extracting
*n«rgy from the alrstream in sufficient quantities
to produce an unstable vibratory motion. A
further objective was to make the analysis
sufficiently general to permit its use vlth
advanced aerodynamic and/or s t ructural dynamic
theories, and ultimately, to provide the designer - M -
vlth a tool for flutter free operation.
 18-5
vhere. in turn, H, and a
an M „. The line integrals over one cycle of action are
tabulated in both Scan Ian and Smilg. (Mote that equivalent to an integration ovar the range
in these sources the positive liftP and vertical 0 <w < 2r; after the indicated integrations
translation are both directed downward which in equation <6) are perforned and the equation is
accounts for the negative right-hand side of simplified, the total work done on the system is
equation (l).) Appropriate coefficients for other given by
aerodynamic conditions nay be inserted for A.,
Aa, flh, and &„, and it shown in Mikolajcxak {Anlh2 * ,)si.n 9
TOT
(1975) and Halliwell (1975) that this leads to an
accurate prediction of the instability boundary. * «hl)coa (7)
At present, though, the development will be based
on the coefficients A h , A a , BH, Ba, and
consequently will be quite general. In this equation, lh« quantities A,_ and fl^
represent the damping in bending and the damping
It Is well known from unsteady aerodynamic in pitch, respectively. For an isolated airfoil
theory that the forces and moments acting on an oscillating at zero incidence in- an incompressible
oscillating airfoil are not in phase vith the flow, both of these damping terns will be negative
motions producing these forces and moments. A and hence will contribute to the stability of the
convenient representation of this phenomenon Is system.
obtained on writing the unsteady coefficients in
complex form as AR . AhR * iAhJ1 etc. and The sign of the cross-coupling term in
the time dependent displacements as equation (7) (the tern enclosed by the square
brackets and multiplied by the product oh )
is strongly dependent on the phase angle between
tie1 cos we * ih ain the notions, 9 , In the usual classical flutter
analysis, the phase angle remains an unknown until
il ^X the end of the calculation, at which time It may
be evaluated as an output quantity. For the
* a cos (u« + 6) + ia sin. t wt «• S) configuration presently under consideration,
however, the physical constraint of the structure
on the node shape fixes 0 to be a specific Input
where, in general, it has been assumed that the quantity, as will be shown in the next section.
torsional notion leads the bending motion by a This quant i ry w i thin the square brackets Is
phase angle, fl. in this equation, h . h'/b Is dominant in specifying regions of unstable
the dinensionless bending displacenemt, and h and operation.
a are the dimcnsionless amplitudes of the notion
in bending and torsion, respectively. Relations Between Blade Motions
and Disk Deformation
Two-Dimensional Vork Per Cycle
It is assumed that a given rotor system
The differential work done by the aerodynamic consists of a set of flexible blades uniformly
forces and moments in the course of this motion Is distributed on the periphery of a flexible,
obtained by computing the product of the In-phase rotating disk. To determine the phase relations
components of force and differential vertical between the components of blade vibration,
displacenent and of moment and differential twist. consideration is given to the portion of the blade
Accordingly, the vork done p«r cycle of notion in which Is in thai Immediate neighborhood of the disk
each node is obtained by integrating the rln. The nomenclature for a compressor blade rov
fferential work in each mode over one cycle. is illustrated in Fig. 6, in vhich the blades havm
The total vork done per cycle of coupled motion is been schematically represented as a serlts of __
flat
given by the sum plates oriented at a chordal stagger angle o UH
relative to the line connecting the leading edges
of all blades.
(5)

OF
where the minus sign is required because L and h
are defined to be positive in opposite directions.
It Is Important to note that In equation <5),
positive work Implies Instability since them
equations represent work done by the air forces on
tht system.

To compute these integrals. L and H_ ar«

obtained from equations (1) andK<2), ?he real
parts of equations (4) are differentiated, and
thesa quantities are substituted into equation (S)
to yield

U
TOT " -'"»>4u2{h"dJ[Ahilh coa <* - A hI S sin we Figure 6 Ca*cad* geometry,
A
* aR* cosd* + fl) - A OI 5ain(«t + 8) ]sin u>td(ut> The disk deformation, which Is primarily In
the axial direction, is denoted by £ (s') at the
in
* "* * disk rim, where s' is the peripheral distance
along the rim, measured from a diametric node
- B al o «U (ut + 6) B)d(ue <6) point. 11 can be assumed that for small
amplitudes of vibration thm rim mode will be
sinusoidal and given by the formula
18-6

ein (2»»'/S) 2»a (8) The amplitudes of the periodic functions must be
equal, and therefore
where a is the dinensionless peripheral distance
measured in units of the wavelength, S, and where h - (13)
«CH
6 ax is the amplitude of the peripheral wave
at the disk rim.
The blade embedded in the rim of the 2 if
compressor disk may nov be represented by the (14)
intersection of a line segment (i.e., the blade)
and a portion of the disk deformation curve, vlth
the angle of Intersection betveen the line segment Hence equations (11) and (12) become
and the tangent to the curve equal toOCH. This
is shown in Fig. 7, In vhich both a deflected and coi ut • -ain 2ns
an undeflected rim are depicted.
(15)
coa (ut » 9) • coa 2"3
TANGENT TO DISK
DEFORMATION CURVE'
In order to satisfy this set simultaneously, one
possible solution for the phase angle is 8• - w/2
(i.e., the bending motion leads the torsional
notion by 90 deg). This result can also be
obtained intuitively (at least for the magnitude
of the phase angle) from the fact that at the
nodal points in the disk rim the blade will
experience maximum twist with no normal
displacement, whereas at the antinodes in the disk
rim the blade will have no twist, but will
experience maximum normal displacement, as shovn

DIRECTION OF
V UNOEFLECTED
fxsie TOACF
OISK TRACE
in Fig. 8.

ROTATION
NOTE: U DENOTES UPPER SURFACE
L DENOTES LOWER SURFACE

Rgurt 7 Blade deformation notation for disk deformation

at Mad* root showing both undeflected and I f / ) > 1 *fi
olUNOEFLECTED ROTOR
deflected blade.
DIRECTION OF
ROTATION
Figure 7 shovs that the blade twist, o , is
given by the alope of the disk deformation curve,
and upon differentiating equation (8), the result
is
b) DEFLECTED ROTOR

tan
-,M, cos 2Rs

coa 2ns (9) c) BENDING AND TWIST DISTRIBUTIONS

for a sufficiently small angle, a . The figure

also shows that the dlmensionless normal Figure 8 Torsion and banding motlona caused by
deflection is given by coupled blade-dlek-ahroud Interaction.

h'/b • •* coa a_.,

CT Thus far the development has been carried
b out at the disk rim. (Actually, the disk
participation in the coupled motion is usually
,... >in 2"s (10) quite small, and It will be seen later that the
major contribution to the coupling betveen
bending and torsion often derives from the
presence of a part-span shroud.) In considering
where use has been made of equation (8). any arbitrary outboard station on the blade span,
Equations (10) and (9) will nov be set equal to it can be assumed that the same phase angle,
the real parts of h and a , respectively, from S- -*/2 vill exist betveen the bending_ and
equations (4) with the result torsion motions. However, the amplitude h vill
be different from that given by equation (13) by
an amount obtained from the vector sum of the
blade deformation relative to the disk rim in the
h coa ut • ** cot a-.,
vn sin 2»s (11) axial and tangential directions, and the
b amplitude a will be different from that given in
equation (14) by an amount equal to the blade
tvist relative to the root position. Use vas
made of an existing structural dynamics computer
(12) program to determine the characteristic vibration
modes and frequencies of a typical rotor system
consisting of disk, blades and part-span shroud.
(As stated earlier, these procedures are
discussed in detail in Chapter IS.) The computed
 18-7

r titles relevant to the present study are

o* *** tan' th* axial and tangential
deformations of the blade at the given radial
- .a
station, measured relative to the original blade
position in the plane of the undeformed disk, and
& , the total tvist distribution. Thus, at any
arbitrary radial station outobard of the rim, the
1. EL.
bending amplitude vill be' computed from the KI ' T (18)
formula (see Fig. 9).
Idr
<$
(16) It vas stated earlier that the stability of
the system vas related to the algebraic sign of
°CM - °CH the vork expression; I.e., positive aerodynamic
vork implies instability and negative aerodynamic
vork implies stability. As shown in equation
(18), the aerodynamic vork done on the system is
a direct function of the squares and products of
the oscillatory amplitudes vhlch are ordinarily
obtained from a numerical solution of the
characteristic equation for the vibratory system.
It is veil knovn from elementary vibration theory
that the results of such a calculation are given
in the form of relative amplitudes rather than
absolute amplitudes. Therefore, the aerodynamic
vork can only be calculated on a relative basis
and in its present form It cannot be used to
predict either the absolute stability level of a
particular configuration or the relative
stability levels betveen tvo configurations.
Since one of the objects of the original study
vas to devise a prediction technique that vould
permit the evaluation of alternative rotor
designs from the standpoint of system stability,
the theoretical development vas necessarily
'tan extended to overcome this deficiency.
In addition to the relative amplitudes of
motion, the structural dynamics solution provides
Figure 9 Blade deformation notation outboard of blade average kinetic energy of vibration of the entire
root showing both undeflected and deflected blade-disk system, based on the relative
amplitudes of motion. It is shown in Appendix 1
blade. that for a simple, linear, spring-mass-dashpot
system, the ratio of damping vork per cycle to
Stability of Blade-Disk-Shroud System average kinetic energy is proportional to the
logarithmic decrement of the system, vhlch is
The value of the phase angle, 8, -v/2, may Independent of the absolute amplitudes of the
nov be substituted into equation (7) and the system. A comparable ratio of the aerodynamic
resulting two-dimensional aerodynamic vork per vork done per cycle on the entire
cycle at each spanvise station reduces to blade-dlsk-shroud system to the average kinetic
energy of vibration of the system may be made and
this vill be equal to the logarithmic decrement
of the system. First, however, the normalised
"TOT bV [ - k 2 B hR >° h aerodynamic vork per cycle obtained for one blade
In equation (18) must be multiplied by the number
(17) of blades on the entire disk, n, and then by the
proportionality factor, 1/4, from equation (26)
in Appendix 1. The result is
where k-U*yi) is the reduced frequency parameter.
(The use of the combinations k A.-,
k2*OR» ^"hRi ""I "2Bai in the original 1967 6 -- (19)
paper vas dictated by convenience and by this
availability of the aerodynamic coefficients
in this form.)
where Kg/hf is the average kinetic energy
of the system, also normalized vlth respect to
Equation (17) is an expression for the tXj. , and where a positive value of $
tvo-dlmensional aerodynamic vork per cycle at any represents stable operation. Equation (19)
arbitrary span station, say at radius r, as yields an absolute measure of system stability
measured from the engine centerllne. The vork vhlch is Independent of relative amplitudes.
done on the entire blade Is obtained by Hence the results obtained may be used to
Integrating equation (17) over the blade span, evaluate both the absolute stability of a
from the root, at to the tip, at r - particular configuration and the relative
r_ and after normalising0' the deformations, h stability between tvo or more configurations.
and 5 , with_ respect to the tip bending
deflection, h^, this gives
18-8

RESULTS the overall system vibration modes encompass 2

through 8 nodal diameters. Figures 10 and 11
Calculation Procedure contain the spanvlse variations of bending
deflection and blade twist, both normalized vlth
To illustrate the use of this theory, the respect to the bending deflection at the blade
stability characteristics of a typical rotor vere tip for the specific nodal diameter under
investigated and the results are presented below. consideration. Figure 10 shows that as the
The input quantities for use in equation (18) number of nodal diameters Increases the bending
vere obtained both from experiments and mode shape undergoes a consistent change In vhlch
analytical studies conducted on an actual rotor the deformation of the tip region, outboard of
at a given rpm. These data consisted ofi (a) the the shroud, increases relative to the Inboard
geometric parameters for the configuration, region.
a
CH' b > rO' rT* *"d n' ^the ste«dv-st*t« aero- In Fig. 11, hovever, it is seen that the
dynamic parameters, p and U| and (c) for each torsional content of the vibration (relative to
prescribed disk nodal diameter pattern and blade the tip bending) first 'increases and then
mode, the relative amplitudes of the blade decreases vith increasing number of nodal
deformation components, orax,3'tan, and a , the diameters. This suggests a variable amount of
average system kinetic energy, Kg, and the coupling betveen bending and torsion as the
system natural frequency, WQ. For each nodal system mode changes from one nodal diameter to
diameter, equation (16) vas used to calculate h another. Furthermore, the major change In tvist
from _ and 5. ln
both n distribution for each curve occurs over the
"tan' *"" °° and a vere portion of the blade inboard of the deformed
then normalized vlth respect to the tip bending shroud, vhlch Imposes a tvlsting moment on the
deflection, fL, and plotted in Figs. 10 and 11 blade as a result of this deformation. Bence,
as functions of the dinensionless spanvlse the presence of a deformed shroud produces the
variable n -(r-r0)/(r_-r0). In both of these coupling betveen blade tvlat and bending, and the
figures the shroud location at 17- 0.653 is degree of this coupling la modified by the
Indicated by short tic-marks on each curve. diametric modal pattern.
The stability analysis of the rotor system
for any given nodal pattern vaa performed by
calculating the spanvise variation of the reduced
frequency parameter, k, and using this to obtain
the spanvlse variations in the unsteady
2 2
aerodynamic
2
coefficients, k A.T>I
T » k A/,
*"»
k B««»
k B^f vhlch are functions of k. These
quantities vere then inserted into equation (18)
together vlth the normalized mode shapes and
other spanvlse variables and integrated
numerically. Finally, the logarithmic decrement,
5 , vas computed from equation (19). The
relevant details of this analysis are described
In the subsequent sections of this chapter.
Structural Dynamics of Typical Rotor System
The dynamic system chosen for analysis in
this .chapter consists of a blade-dlsk-shroud
configuration in vhich the blade oscillates in
its 'tlrst bending and first torsion modes, and 0.2 0.4 0.6 OB
SPANWISE STATION, T)

Figure 11 Spanwise variation of twist distribution

normalized with reaped to tip bending
deflection for each nodal diameter.

A very revealing and Informative plot is

shown In Pig. 12, in vhich only the tip value of
the normalised deformation ratio (a/n), has been
plotted as a function of frequency. This
figure Indicates a very strong variation of
(a/hL)T vith the number of nodal diameters
(I.e., vlth natural frequency); it is relatively
small at both small and large nodal diameters and
reaches a maximum value at approximately 4 or S
nodal diameters. Thus, it appears that
torsion-bending coupling is a maximum for
intermediate nodal diameters, vlth a
predominantly bending motion occurring at either
extreme. This increase in coupling for
Intermediate nodal diameters may be regarded as
either a relative increase In blade tvist or a
•-Q2 relative decrease in blade bending. The absolute
Q2 0.4 06 0.8 deformations are unimportant since the stability
SPANWISE STATION. 17 equations (18) or (19) are expressed solely In
terms_ of the normalized deformations h/n_
and a/n.
Figure 10 Spanwlao variation of bending deflection
normalized with respect to tip bending
deflection for each nodal diameter.
 18-9

0.15
TIP DEFLECTION RATIO. <a/h" T ) T

£ O
0.10
5
C

O
o
o
8 O gS 5

o
O
o O
O

STABLE
NOTML UNSTABLE 3
al

DtAMCTEItS o
BLADE

4 5 e T S O
O
i

500 -O05
FREQUENCY, f-Cp* MODAL
DIAMCTEKS
Figure 12 Variation of blade tip deflection ratio with /"
frequency. II 4 S • T •
-0.10
aoo 350 400 45O 500
Stability Analysis Using Isolated Airfoil Theory FREQUENCY, f - cps
Figure 13 Variation of logarithmic decrement with
The stability of the system vaa originally frequency uatng Isolated airfoil theory.
determined using equations (18) and (19) and
employing unsteady isolated airfoil theory as
described in Appendix 2. The logarithmic LOCUS OF POINTS FOR WHOM 8 - 0 (THEORY)
decrement, 5 , vas calculated for each nodal CONFIGURATIONS WHICH FLUTTERED
diameter (2 through 8) at the resonant CONFIGURATIONS WHICH DID NOT FLUTTER
frequencies appropriate for each case. Results DENOTES REDESIGN OF UNSTABLE
of these stabilitity calculations are found in CONFIGURATION TO OBTAIN STABUTY
Pig. 13. The natural frequency in each case Is
denoted by the circled point. System stability
is Indicated by positive values of 5 and
instability is indicated by negative values of
5. It is seen from this figure that the system
is stable for the 2, 6, 7, and B-nodal-diameter
modes and Is unstable for the 3, 4, and
5-nodal-diameter modes, vith minimum stability
(i.e., maximum instability) occurring at four
nodal diameters. A comparison of Pig. 13 vlth
Pig. 12 reveals a rather strong correlation
betveen maximum system Instability and maximum
torsion-bending coupling, represented in Fig. 12
Ity the maximum values of (a/h_)_ at four
nodal diameters. Similar results vere obtained
for a number cf rotor configurations vhich vere
analyzed using these procedures. Thus it vas
tentatively concluded that the greater the degree
of coupling betveen torsion and bending in a
shrouded rotor, the greater the likelihood of a
flutter instability.
0 02 0.4 0.6 08 1.0
Comparison Betveen Theory and Experiment
DEFLECTION RATIO, (a/HT)T (ARBITRARY SCALE)
The theoretical procedure for an Isolated
airfoil at zero incidence described in a Figure 14 Stability boundary — comparison between
previous section vas modified slightly (by the theory and experiment.
engine development groups) for use in evaluating
various rotor configurations. An iterative
procedure vas developed vhich produced the rotor
parameter values for the neutrally stable condi- of Fig. IA may also be confirmed by
tion, 6 - 0, in vhich the incompressible interpolating Pigs. 12 and 13 to 6 - 0, through
Isolated airfoil theory of Theodorsen (1935) vas fictitious curves passed through the circled
used at lov speeds, and the supersonic isolated eigensolutlons. This vas done for several
airfoil theory of Garrick and Rublnov (1946) vas configurations in the report (Carta 1966) that
used at high speeds. A number of rotors vere formed the basis for the 1967 paper.
considered in this study, and In each case,
values of blade tip deflection ratio, In Pig. 14 the region beneath and to the
(a /h*T)T, and reduced velocity at the blade left of the curve represents stable operation,
tip, (U/bwt)T, vere obtained for the and the region above and to the right of the
condition of zero logarithmic decrement, 5- 0. curve represents unstable operation. Super-
The locus of points so obtained yielded a narrov imposed on this curve are the results of a
band of scattered points through vhlch a faired number of engine and rotor tests. The solid
curve vas dravn. This formed a single zero circular symbols represent configurations vhich
damping stability boundary valid for all fluttered and the open triangular symbols
configurations, as shovn in Fig. 14. It is represent stable configurations. It is seen
interesting to note that the theoretical curve that the engine experience And the theoretical
18-10

prediction vere in good agreement Cor these

early cases. Furthermore, tvo of the unstable Pb2 O2 k2 Ahi
ht H
2
configurations vere redesigned to yield stable (25)
rotors, as Indicated by the arrovs connecting kz Aht
tvo pairs of points. Hence, it vas felt that
the theory had some merit in predicting as veil
as explaining the occurrence of coupled 2 2 2
blade-disk-shroud flutter. Hovever, it vas also Pb U (k^a, - k BhR'hB)a G
(26)
felt that the agreement achieved through the use BT (Pb' U* k< Ahl h')
of incompressible, isolated airfoil theory on a
high speed, multiblade systen vas tenuous at
best, and a major effort vas continued by a
number of researchers to extend the aerodynamic W. Pb 2 U2 k2 BaT a2
(27)
model to include more realistic effects. Before
these modern developments are discussed, a brief
reviev of the energy distribution among the
modes vill be made, and a portion of a
parametric study of coupling effects vill be
examined. It vas stated earlier that the quantity
W-_ vas alvays a negative, non-zero number,
Spanvise Variations in the Damping ana hence implied a stable condition for this
and Coupling Terms parameter at the blade tip. By definition the
value of XB at the tip vill be +1.0 since at
In the previous section on Stability this point V. is divided by Itself. Bence a
Analysis, the major emphasis vaa placed on the positive value of any of the normalized vork
variation in stability parameter vlth changes in terms vill indicate a stable tendency (in
frequency (i.e., number of nodal diameters). In contrast to a negative value of the absolute
this section, some brief considerations vill be vork parameter) and a negative value of any of
given to the Individual terms in the vork the normalized vork terms vill Indicate an
equation and their spanvise variations as they unstable tendency.
are affected by changes in frequency.
From Fig. 13 the most unstable condition
For convenience the expression for the vork vas a 4 nodal diameter vibration, vhile the most
per cycle at each spanvise station, equation stable vaa an 6 nodal diameter vibration. The
(17), vill be revrltten as quantities XB, X-, and X_ are plotted
versus dimenslonless span station for these tvo
(20) conditions in Pig. 15. An examination of this
TOT plot provides a great deal of insight into the
mechanism Involved in this flutter phenomenon.
In both panels, the distributed vork is
vhere negligible Inboard of the mldspan, and is
HB - n2 Pb2 O2 k2 Ahl C2 (21) significant only over the outer 1/3 of the span.
The normalised vork in bending, XB, Is alvays
positive (stabilizing), and reaches its normal
is the local tvo-dimenslonal vork due to value of 1.0 at the tip. In both Instances the
bending, normalized coupling vork, \_, is negative
(destabilizing), and the normalized vork in
pitch Xp, is of second order and can be
WP - -n2 Ob2 O2 (k2 A^ - k2 BhR)a S (22) and can bi Ignored. For the 4 nodal diameter
case, the bending and coupling vork both reach
approximately the same value at the tip, but the
Is the local tvo-dlmensional vork due to distribution over the span is such that the
coupling and destabilising Integrated vork in coupling is
greater in magnitude that the stabilizing
integrated vork in bending. This leads to a net
2
pb2 U2 k2 B OI o (23) instability of the system In this mode. (A
similar situation, vlth less obvious differences
betveen bending and coupling distributions, is
is the local tvo-dlmenslonal vork due to pitch. found for the less unstable 3 nodal diameter
Before these quantities can be usefully mode in Fig. 31 of Carta (1966).) In contrast,
investigated, they must be normalised in such a the spanvise distributions for the 6 nodal
vay that meaningful comparisons betveen various diameter mode shov that the stabilising vork in
cases may be made. In the course of this study, bending is significantly greater in magnitude
it vas found that the tip value of the local than the coupling vork over the entire span, and
vork due to bending, accordingly, the system is stable at this
condition.

*BT (Pb2 U2 k2 AhJ h2)TIp (24) Use of Advanced Aerodynamic Theories

and Typical Mode Shapes
vaa negative and non-zero for all values of These results In Fig. 15 vere for a
frequency In all cases considered. (It vill be so-called first family spanvlse mode. More
recalled that negative vork Implies stability typical higher order family modes, having
since this represents vork done by the air on additional circumferential nodes lines, vere
the blade.) Therefore, in vlev of the negative examined independently by Hlkolajczak, et al
definite behavior of VnT, it vas decided to (1973) and by Ballivell (1975), for state of the
normalize all three local vork terms, VB, art compressors of the mid 1970s. Ballivell
Vc, and Vp, vith respect to this tip value used the aerodynamic theory of Nagashlma and
of W_ for the specific number of ncdal Vhltehead (1974) and obtained the spanvlse
diameters being considered. The normalized distribution of vork per cycle shown in Fig. 16
quantities are defined belov. for the second family modes. This confirms the
 18-11

8-NODAL DIA. VIBRATION

STABLE
UNSTABLE

-1

J I
0 0.2 0.4 0.6 0.6 1.0 0 0.2 0.4 0.6 0.8 1.0

DIMENSIONLESS SPANWISE STATION, rj

Figure IS Variation of normalized work terms with span.

16
CROSS-
COUPLING ,
TERM
12

CROSS-
COUPLING
TERM

PURE
TORSION
TERM

-4 STABLE

-8 PURE
BENDING \
TERM
I
0.7 0.8 0.9 1.0
-2
OIMENSIONLESS SPANWISE STATION. r\
3 4
DIMATERAL MODE NUMBER, N
Figure 16 Fen unsteady work component* et
100 per cent speed four-diameter
eecond family mode (from Halllwell Figure 17 Fen unsteady work components et 100 per
1975). cent epeed variation with eecond family
mode number (from Hafllwell 1976).
distributions of vork shorn in Fig. 15. Note
that Halliveil's results are dimensional, and typical compressor designs. The vork by
are Inverted relative to Fig. 15. He further Hlkolajczak and his co-authors employed several
computed the integrated vork per cycle for each aerodynamic theories, depending on the local
component of the stability equation, and the aerodynamic conditions. For the supersonic
results are shown in Fig. 17 as a function of relative flov (vlth subsonic axial Mach number)
nodal diameter number, again for the second use vas made of Verdon (1973), and for subsonic
family mode. Here again the coupling term is flovs Smith's theory (1971) vas used. In
destabilizing, and opposite to the bending term. addition, cambered thin airfoils vere treated
An overall system instability for the 4, 5, and using an extension of Vhltehead's early vork
6 nodal diameter forvard traveling vaves vas (1960) or the analysis of Slsto and Ni (1974).
predicted, and vas confirmed as 4 nodal
diameters for the test compressor. Three radial modes vere examined In this
vork, vhich concentrated on tvo compressor
The vork by Hlkolajczak et al (1975) rotors. Rotor A vas designed specifically to be
concentrated on the overall aerodynamic damping susceptible to an (installed supersonic flutter
of several compressor designs. This vas in its second radial mode. It experienced
preceded by the cascade study of Snyder and flutter at 13,290 rpm in its second mode vith a
Connerford (1974) vhlch also examined the 4 nodal diameter vibrational pattern. The
18-12

4.0
ROTO* FLUTTERED IN SECOND MODE
1 AT FOUR NODAL DIAMETERS
4.0

MOOil
2.0
2.0

1*8
STABLI STABLE
K 0
UNSTMLf
l.p
IQO «
UNSTABLE

•2.0 -2.0

•4.0
•4.0 0.2 0.3 0.4 O.S 0.6
0.2 0.3 0.4 O.S 0.6
REDUCED FREQUENCY. K
REDUCED FREQUENCY. K

Figure 18 Damping predictions for PaWA research rotor Figure 1ft Second mode damping n e function of rotor speed.
et flutter speed.
predicted aerodynamic damping for the three
•odes, plotted as a function of reduced
frequency, is found In Fig. 18. This clearly
shovs the second node to be the least stable
mode, although the S and 6 nodal diameter
patterns appear to be theoretically more
unstable than the 4 nodal diameter pattern. The
sensitivity of predicted stability to rpm for
Rotor A is shown in the prediction of Fig. 19
for the second radial mode. Rotor B (a
NASA 1800 fps design) vas specifically designed
to be flutter free over its performance range.
It vas tested successfully vlth no flutter up to
12,464 rpm. The predicted aerodynamic damping
for the first three radial modes vas positive
for all nodal diameters, as shown in Fig. 20.
In a summary of these and several other rotor
designs, the Hlkolajczak paper shovs that the
use of the analytical prediction techniques
described here vere consistently conservative
(at least up to the date of publication) and
generally capable of predicting the correct
flutter mode, vhen it occurred. This is shown
in Pig. 21, vhere the predicted minimum
aerodynamic damping for each of several rotors 0.9 0.8 0.7 0.» 1.1
Is plotted horizontally for the first three REDUCED FREQUENCY.*
radial modes. The tabulation at the right of
this figure briefly describes each rotor and
indicates the observed flutter mode vhen it
occurred. It should be noted that the estimated Figure 20 Damping prediction for NASA 1SM fpa
mechanical damping of a typical rotor system rotor et 100H speed.
(the sum of material and frictlonal damping) vas
approximately 0.03. Thus, it can be seen from
Pig. 21 that whenever flutter vas observed, the
analysis predicted a level of negative
aerodynamic damping vhich vas comparable to this
expected level of mechanical damping of the
SOLID SYMBOLS INDICATE FLUTTER NUMBER
rotor.
O MOOE1 O WOOES 0 MODE) noTOR °*

PARAMETRIC VARIATIONS D O • 1 NONE THREE

<1 > 1 2 NONE TWO
Bffect of Shroud Location O > i1 J NONE NONE
In the original 1966 report a parametric 1i 0 • 4 ONE TWO
study vas made of the effect of part span shroud O 0 c 5 ONE NONE
location on system stability. The results a <x. « TWO NONE
already discussed vere for the standard
configuration having a part span shroud at the 06 0.04 0.02 -0.02 -0.04 -0.06
65.3X span station. Additional locations of
SOX, 60.5X, 69.IX, and BOX span vere also
examined. As before, this study vas constrained Figure 21 Damping analysis aa an unatalled supersonic
by the use of isolated flat plate aerodynamic flutter prediction tool.
theory, but it is believed that the physical
principles involved are sound and that the
relative changes In predicted stability are
correct.
 18-13

The most important finding of this study

vas that the stability of the system decreases
rapidly as the shroud is moved outboard. This
is shown in Fig. 22 in which a number ol
normally stable conditions for the basic
configuration became quite unstable as the
shroud vaa moved outboard. This unstable
tendency is caused by the associated increase in
coupling betveen the torsion and bending nodes,
shown In Fig. 23, In vhlch a number of curves of
the normalized tvist distribution have been
plotted as a function of spanvlse station for
three values of nodal diameter number (., 6, and
8) and five shroud locations. For each nodal
diameter, curves have been plotted for the
shroud location at the 50*. 60.5X, 65.3X, 69.U,
and 80S span station. The position of the
shroud has been indicated by a short tic mark on
each curve.
The primary effect to be noted In Fig. 23
Is the increase in coupling (i.e., the Increase
in normalized tvist value) within each nodal
diameter plot as shroud location is moved
outboard. This increase in coupling is caused
by tvo factors. First, as the shroud Is moved
outboard, a larger portion of the blade (the
inboard portion) Is subjected to the oscillating
tvlsting moment of the shroud at resonance in
the system mode vibration, and a smaller portion
of the blade (the remainder of the blade
outboard of the shroud) is driven at an
off-resonance condition. (Actually, for the 6
and 8 nodal diameter vibrations vith an SOX
shroud position it appears that the outboard
portion may also be at or near a resonance
condition, but it Is felt that this Is an
Isolated phenomenon and is not Important in
general.) Second, as the shroud Is moved
outboard, the portion of the blade outboard of
the shroud becomes sitffer in bending;
consequently, the tip bending deflection, vhich
j_s_the normalizing factor in the denominator of
o/hT, becomes rela 11vely smaller, and
therefore the entire level of the curve is
raised. In effect, the torsional motion has
Increased at the expense of the bending motion.
50 60
Finally, it is obvious that as the shroud SHROUD LOCATION, PERCENT SPAN
Is moved inboard the system stability increases
— at least for the type of coupled flutter
instability being considered herein. Movever, Figure 22 Effact of shroud location on Mobility
another effect of moving the shroud Inboard is parameter.
to Increase the cantilever length of the blade
portion outboard of the shroud, vhlch reduces
both the cantilever bending and cantilever
torsion frequencies of this part of the blade.

4 DOOM. DMMCTtM • NOML MAMETtM a NOBU. OUMETCM

2 030

0 0.2 04 06 08 10 0 O2 0.4 OA OS LO 0 01 M QA OH 1.0

DtUCNSIONLESS SPANWISE STATION, r,

Figure 23 Eftact of shroud location on normalized twlvt distribution.

18-14

This reduction In frequency at a fixed value of CASCADE A eiaOLATED AIRFOIL

resultant velocity into the stage produces an
Increase in reduced velocity, UVbcj, and the MOTIi CIRCLED POINTt REFMtENT
• •HCNEMENT OF M* RELATIVE TO G am 0 A •• •
blade aystea outboard of the shroud may become ADJACENT POINT ON CURVE
susceptible to either a tocslonal stall flutter, t*
a bending buffet, or both. Clearly in any
design procedure a coapro»l8« must be made
betveen a flutter-free configuration relative to
the coupled flutter phenomenon, and a
flutter-free configuration relative to either
torsional stall flutter or bending buffet.
Interblade Phase Angle
In nost of the early vork described above
the interblade phase angle vas necessarily a
quantity fixed to the configuration under
examination, and not usually subjected to
parametric scrutiny in a sensitivity analysis.
By virtue of its definition it vas inextricably
tied to the fixed number of blades in the rotor,
n, and to the number of disturbances over the
rotor circumference, N, by the formula
•1.0 1.0
a - -2»/(n/H) • -2i N/n (26)

Figure 24 Moment coeffldenta due to pitching

vhere the minus sign is associated vith a mottona of cascade A and Isolated
backyard traveling wave relative to the rotor, airfoil.
and vhere N is also the number of nodal
diameters that foras the basis of this chapter. Infinite cascade of theoretical airfoils, the
Note that in general the forvard traveling vave points and their connecting curves represent a
is associated vith system instability (Halllvell continuum of valid and realizable solutions.
1975). Thus, any interblade phase angle can be
represented by this plot. It vill be shovn
Physically, the interblade phase angle is a presently, however, that for a rotor vith a
measure of the phase lag or lead of adjacent finite number of blades, a similar closed
blades, and has been the subject of several diagram is generated by the stability analysis
experlaental studies at lov susbonlc speeds of of a coupled motion, in vhich the only valid
its important effect on aerodynamic damping solutions are for the specific values of
(Carta and St. Hllaire 1980, Carta 1983), Interblade phase angle that satisfy equation
primarily in cascade. An analysis of the effect (28). Under these circumstances, a variation in
of varying a for a single degree of freedom a implies a corresponding change in number of
pitching notion of a supersonic cascade of thin blades, and the effect on stability vill be
blades operating in a subsonic axial flov vas profound.
presented by Verdon and McCune (197S). It Is
veil known (e.g. Carta 1983) that the stability An illustration of the assertion that the
of an airfoil executing a pure pitching motion number of blades has a strong effect on system
depends only on the sign of the imaginary part stability through the Interblade phase angle is
of the pitching moment. (Note that in this case, discussed in the papers by Kaza, et al (1987a,
vith h - 0, equation (17) reduces to the form 1987b) vhlch deal vlth single rotation propfans.
Both papers deal vith thin, flexible, lov aspect
o o
ratio blades susceptible to large, nonlinear
IT k* B, (29) deflections in a strongly three-dimensional
»TOT flov. To further complicate natters, the blades
have large sweep and tvist vhich couples blade
bending and torsional motions within each blade,
vhere the last term is a consequence of using and their flexibility and proximity to one
equation (2).) If ¥_ is positive, then the another engenders an aerodynamic coupling
vork per cycle is also positive, vhlch similar to that caused by part span supports.
represents an unstable condition. Figure 24 is In addition, a flexible hub also contributes to
a phase plane plot of the complex moment the systea coupling.
coefficient, taken from, Verdon and McCune (1975)
for this condition. The open points connected An analysis of the stability of the basic
by curves represent Increments of A o - 30 propfan, denoted by SR3C-X2, vas performed for
deg relative to the specified values of o - 0, an eight blade, a four blade, and a single blade
ff/2, ir, 3»/2. Each curve is for a different configuration. The results are presented In
compressible reduced frequency, kc - 2kH/(H*-l), Pig. 25 (taken from the 1987a paper), vhlch is a
from 0.5 to 2.0. Corresponding values of the root locus plot of the complex eigenvalues,
complex moment coefficient for an isolated flat consisting of the real part, p (proportional to
plate airfoil in a supersonic flov, computed damping), and the Imaginary part, •'(proportion-
froa the theory of Garrick and Rublnov (1946), al to frequency). Thus, in this phase plane
are represented by the solid symbols. Tvo plot the stable/unstable boundary is at p- 0,
concepts are revealed by this figure. The vith flutter occurring for a positive real part.
first, vhich is obvious, is that Isolated The single blade ays ten has a single eigenvalue,
airfoil theory Is inadequate to predict the located veil within the stable region of the
extent of the unstable region for this single phase plane. The four blade system has four
degree of freedom oscillation. The second, elgensolutions, represented by Interblade phase
vhich is also obvious, but vhlch has subtle angles 90 deg apart. This configuration is also
implications, is that the Interblade phase angle fully stable. However, the eight blade system,
has a significant effect on system stability. vlth eight elgensolutlons spaced 360°/8 . 45 deg
For the analysis described here, vhlch vas an
 18-15

apart, surrounds the tvo other system solutions,

and borders on the unstable region forc-180 deg
and 225 deg. In this instance it vould appear
that an Increase In the number of blades has
Intensified the Interblade coupling, possibly 320 INTER-BLADE PHASE ANGLE
through the cascading effmc.t, and has caused a
deterioration ot the system stability. (The
prediction is shown to be in good agreement vlth 310
theory in Pig. 15 of this paper.)

An extension to a mistuned case vas the 300

subject of the Kaza 1987b paper. Once again the
SH3C-X2 rotor vas analyzed, together vlth a
rotor designated the SR3C-3. Both of these vere 290
tuned rotors, as in the 1987a paper. In
addition, a deliberately mistuned rotor, the 260
SR3C-X2/SR3C-3, vas analyzed. It vas modeled as 28 -24 -20 -16 -12 -8 -4
an idealized alternately mis tuned rotor having REAL PART OF EIGENVALUE, *i/2n, Hz
four identical blade pairs vlth tvo different
blades In each pair, one from the SR3C-X2 rotor,
and one from the SR3C-3 rotor. The analysis Ftgure 25 Root locus plot of the mode with least damping:
yielded the plots shown in Fig. 26, taken from Mi0.50, Q-6080 rpm, SR3C-X2 rotor.
the 1987b paper. In this case all three rotors
have eight eigensolutions. The SR3C-X2 (circled
points) vas already discussed in the previous
paragraph. It vas an unstable rotor, vith a O EUC-X2
neasured flutter condition that coincided vith
the real part of the eigenvalues equal to zero. Q S8JC X2/SIX-J
Conversely, the SR3C-3 (triangular points) was __ _ „ —IHTEMUUC PHASE MtLC. 0,
a stable rotor during the experiments, and
3101— JO0'"
yielded eigensolutions comfortably avay from the
stable/unstable boundary. The aerodynamic
coupling of the mistuned rotor (square points)
appears to be gone, and the eigensolutions are
divided into tvo nested groups, vith the high
frequency group near the center of the SR3C-X2
eigensolutions, and the lov frequency group in
the vicinity of the SR3C-3 solutions.
The previous paragraphs serve to point out
the complications associated vith nmltiblade
systeit coupling, and the need for accurate
modeling of the Interblade phase angle effects.
They also Introduce the reader to the concept of
mistuning as it affects system stability,
although the discussion is restricted to Ideal HEM. PMT OF EIGENVALUE, - - . Hi
paired mistiming. A detailed study of arbitrary 2*

or random mistuning is beyond the scope of the

present chapter, and the reader Is referred to Figure 26 Calculated root locus plot of the lowest
the several papers by Kaza and Kielb (1982, damped mode for the SR3C-X2, SR3C-3, and
1984, 1985), Kielb and Kaza (1983, 1984),
Bendiksen (1984), and to the chapter that SR3C-X2/SR3C-3 rotors at 6320 rpm and 0.528
follovs. freeatream mach number.

CONCLUSION
APPENDIX 1
It vas shown in the original 1967 paper
Logarithmic Decrement for Simple Linear System
that the energy method, using unsteady isolated
airfoil theory, and applied to actual multiblade
The simple aprlng-masx-dashpot system is
rotors, yielded results that vere remarkably governed by the linear differential equation
accurate. This fortuitous agreement vas
sufficiently encouraging to foster a continuing
development of the technique and its constituent (30)
aerodynamic and structural dynamic components.
As shown in these tvo volumes and In the several
citations to advanced analyses, current practice vhere (see Scanlan and Rosenbaum 1951)
has gone veil beyond the relatively simplistic
viev of this early paper. The aerodynamic input u> " /K/B * undanped natural frequency
nov encompasses multiblade systems subjected to
compressible flows, and structures are modeled
to include nonlinearities and mis tuning. Y • c/ccr - damping retio (31)
Nevertheless, the paper has served its purpose
well. In its original fora it set the stage for c • 2nu - critical damping
the continual Improvement of engine flutter
prediction methods, and in this Manual it The damping force alvays opposes the velocity
provides the reader vith a vehicle for and is given by
coordinating the separate disciplines vhlch,
together, represent the modern approach to
flutter prediction of turbomachlnery blade rovs. P - -ex • ~ Y c x - -
18-16

Hence the differential work done by the damper APPENDIX 2

la
Unsteady, Incompressible, Potential Plow Theory
for Isolated Airfoil
dW - F d dx - -2Yn««tdx - -2lfm«I 2 dC The appropriate unsteady aerodynamic
coefficients for Incompressible, invlscld,
potential flow past a two-dimensional Isolated
Therefore, the work done per cycle of motion is flat plate airfoil have been previously
mentioned in equation (3). On page 399 of
Scanlan and Rosenbauo (1951), the coefficients
W - -2Ymu j> x2dt (32) L^, La , H^, and Ma are related to the
Theodorsen (1935) circulation function C(k) •
At any Instant of time the kinetic energy P(k) + IG(k), by the equation
due to the notion is

L 1 i[
,
'-"; 2C(k)] - -y C(k)
and hence the average kinetic energy over one
cycle of the motion Is
h "2
i (37)

(33 3^ i_
2T > M
°"7 " k
In these equations the reduced frequency, k, is
where T - 2tr&v, is the period of the damped based on the freestream velocity component, U,
motion. It can be shown (cf. Scanlan) that the parallel to the flat plate airfoil.
damped frequency is given byw. -uiv \-"T. After C(k) and equation (37) are
substituted into equations (3), multiplied
through by k and separated into real and
If the ratio of the work per cycle to imaginary parts, the required coefficients in
average kinetic energy for the same cycle is equation (IB) are given by
taken, then from equations (32) and (33),
• -2kP
(34)
2
k A0& - - -2F * 2kfl (38)

On page 58 of Church (1957), the logarithmic

decrement of a damped motion is given by

- lo g (A n /A n+1 ) - — - (35)
a

and A and A , are the amplitudes of two

consecutive waves. A comparison of equations
(34) and (35) yields

(36)

which is the required result.

 - MOL-Z 19-1

AEROELASTIC FORMULATION FOR

TUNED AND HISTUNED ROTORS

EDWARD F. CRAWLEY
Gas Turbine Laboratory
Massachusetts Institute of Technology
Cambridge, Massachusetts

INTRODUCTION

In previous chapters, the analytic mechanical disturbances such as FOD

tools necessary to approach the problem of impact, blade loss, rubs, etc. The third
aeroelastic analysis have been presented. challenge facing the working aeroelas-
In the terminology of Bisplinghoff and tician is that all the required analytic
Ashley (1962), three operators, Inertial, tools to progress in an orderly and rig-
Structural, and Aerodynamic, are needed in orous manner from the starting point to
the appropriate form. The current state the end point are not available within the
of the art techniques for determining the state of the art. For example, a three-
aerodynamic operators, which are contri- dimensional, heavily loaded, large shock
buted by the unsteady aerodynamicist, have motion unsteady aerodynamic operator for
been presented in Chapters 2 through 7. the analysis of transonic fan aeroelas-
The inertia! and structural operators, ticity dimply does not exist as of this
which together form the structural dynamic writing. Therefore existing tools, exper-
model have been reviewed in chapters 12 imental data and empirical rules must be
through 14. combined to yield an appropriate engineer-
ing solution to the aeroelastic problem.
The task of the aeroelastic analysis
is to combine the formulations of the
structural dynamic and unsteady aerody- To illustrate these three problems,
namic model in a consistent manner, to varying start points, various goals, and
solve the resulting aeroelastic model for unavailability of analysis tools, consider
the desired results (e.g., stability, the very general flow chart for aeroelas-
forced vibration), and to interpret those tic analysis shown in Figure 1. The fig-
results for both qualitative trends, and ure is largely self-explanatory, especial-
quantitative detail. This task of formu- ly in view of the discussion in earlier
lation of the aeroelastic problem and chapters, but presents a consistent
interpretation of the results will be the strategy for combining and extending those
subject of this chapter. topics. What is important to note are the
start points, end points, and limitations
that prevent full implementation of the
Specifically, the topics to be ad- charted procedure.
dressed are: the formulation of the aero-
elastic problem, including a summary of
the relations necessary to transform vari- Essentially, three starting points are
ous diverse structural and aerodynamic available, either a structural model of
models to a consistent notation; a brief the blade alone, of the nonrotating blade-
review of the solution techniques disk assembly or the rotating blade-
applicable; the trends in aeroelastic disk assembly. In each case, assumed
stability for tuned rotors; and the modes, calculated eigenmodes, or measured
effects of mistuning on stability. eipenmodes are possible forms of the
starting data. After inclusion of thermal
and shaft/rotor support effects, the first
possible end point is reached, the rotat-
In order to understand the motivation ing natural frequencies, which can be used
for a lengthy discussion of aeroelastic in traditional Campbell diagram analysis
formulations, one must appreciate the of forced vibration. It is reasonable to
challenges and dilemmas faced by the work- say that all of the analytic tools neces-
ing aeroelastician. First, the starting sary to reach this point on the flowchart
point of the analysis can vary. Typical are reasonably well developed, and the
starting points can include experimentally temporal dependence of the motion can be
or analytically determined mode shapes of expressed either in the time or frequency
the entire blade-disk assembly, mode domain. As soon as the next step in the
shapes of individual blades, or the pro- chart is taken, the inclusion of the homo-
perties of a simple typical section. geneous unsteady aerodynamic forces, two
Secondly, the objective or end point of limitations appear. First, as has been
the analysis may vary. Most often in discussed in Chapters 2 through 7, aerody-
current practice, a simple assessment of namic operators do not exist for all flow
the stability of the turbomachinery stage regimes, and secondly, intrinsic to the
is desired. Increasingly, however, the development of these operators is the
full forced vibration response to aerody- assumption of sinusoidal motion of the
namic disturbances is of interest. In blade row. If the ultimate end point is
principle, the ultimate objective is to only the flutter behavior, the assumption
develop a completely coupled, time accu- of sinusoidal behavior is not limiting, as
rate dynamic and aerodynamic model which well known techniques exist for assessing
can be used in such diverse analysis as stability even under the assumption of the
stall and surge loading, and analysis of sinusoidal motion.
19-2

STARTIHC INTERMEDIATE STEPS AND RESULTS END POINTS

Typical section
Blade assumed modes Blade Structural
Blade calc. modes Dynamic Hodel
Blade raeas. nodes
+ disk elastic coupling
+ shroud coupling
+ mistuning

Assumed twin modes

Blade-disk calc. nodes Non-Rotating Blade -
Blade-disk meas. modes Disk Dynamics

T
+ rotational/centrifugal effects
(stiffening, untwist, etc.)

Rotating blade-disk
modes from calculation Rotating Blade -
or measurement Disk Dynamics

+ thermal-elastic effects
+ shaft elastic support effects
(gyroscopic & centrifugal)

Rotating, Hot Blade - Coupled critical

Disk - Shaft Dynamics speeds, natural
frequencies for
Campbell diagram
+ unsteady homogeneous
aerodynamic model

I
Homogeneous Aeroelastic
Blade - Disk - Shaft Hodel Flutter
Response
I

+ unsteady aero disturbance

model (stall, surge, blade
passage. Inlet distortion)
+ unsteady nonhomogeneous
aerodynamic gust
response function

Complete Rotor Aerodynamic

1
Aeroelastic Model Forced
Response
+ mechanical disturbance
model (FOD.BHQD Impact,
blade loss, rubs, etc.)

1
Return to start
Complete Rotor Aeroelaatlc-
Nechanlcal Model
Aerodynamic/
' Structural
for design or Forced
optimization Response

Figure 1. Flowchart for Aeroelaatic Analysis

 19-3

At the next step, however, the addi- section of the ith blade is given as
tion of the unsteady aerodynamic distur-
bances and unsteady aerodynamic "gust11
response function, even fewer analytic
tools are available, and the assumption
of sinusoidal motion becomes limiting.
Techniques will be presented below to
transform the aerodynamic influences de- where mi is the generalized mass, t>t its
rived in the frequency domain, back to the natural frequency, q^ its displacement,
time domain.
f? the motion dependent aerodynamic
Of course, the complete model would forces, and f, the aerodynamic disturbance
include the capability to couple the forces acting on the ith blade. When
structural dynamic, aeroelastic and modelling a typical section, the general-
mechanical disturbance models to produce a ized mass and force traditionally have
complete, time accurate model of the units of mass per span and force per span.
turbomachine aeromechanical response. The assembly of N structurally
However, due to lack of the proper uncoupled blades would then be governed
analytic tools, this, is probably not pos- by
sible at the current time. Ultimately,
iteration takes place over this entire
procedure, either in the form of heuristic
design or formal optimization.

Over the past decade, as the state of

the art of aeroelastic analysis has pro- where equation (2) represents N
gressed, a number of different formula- 'independent* equations, which will be
tions of the aeroelastic problems have recoupled by the motion dependent aerody-
evolved. These have included travelling
wave formulation, individual blade formu- namic forces fm . In its most general
lations, and standing mode formulations, form, the motion dependent force can be
Kielb and Kaza (1983), Crawley and Hall written as
(1985), Dugundji and Bundas (1984). These
formulations have been applied to single
and two degree of freedom typical section
models, and to blade modal models,
Srinivasan (1980), Bendiksen and Friedmann
(1981), Srinivasan and Fabunmi (1984). In (3)
some models the effect of disk and shroud
elastic coupling has also been included,
Kielb and Kaza (1984). There has been etc.
some doubt as to whether these various
formulations are equivalent, and as to
which is is most appropriate. One of the with /* qj
objectives of this chapter is to review
and summarize these formulations in a con- y
sistent notation for single blade degree l+l 'o
of freedom analysis, and to show that they
are mathematically equivalent. This does *,_! •*«••
not imply that in a given situation one
may not be preferred over another due to
its ease of application or insight con- where, of course, fo , f+i , f_i depend on
tributed, but merely that simple simi- the Mach No., reduced frequency, and geom-
larity transforms are available to trans- etry of the blade and cascade. Equation
form easily from one formulation to (3) expresses in a very general way the
another. The direct extension of the one dependence of the force acting on the ith
degree of freedom formulation to multiple blade due to its motion and the motion of
section or blade modal degrees of freedom its neighbors, and on the time history of
is also demonstrated. those motions through lags due to shed
vorticity and finite speed of sound.
In the next section the mathematical These lag effects are explicitly repre-
formulations and transformations which sented by the augmented state variables
allow coupling of the various existing
analytic tools along the lines of the
flowchart of Fig. 1 will be presented. Unfortunately, within the state of the
art, the aerodynamic operators are not
available in the very general form of eq.
FORMULATION AND SOLUTION OF THE (3). In fact, they are derived for a very
AEROELASTIC PROBLEM specific temporal and spatial motion
pattern: sinusoidal in time and fixed
Basic Relationships interblade phase along the cascade in
space. The kinematic relationship between
At the foundation of the aeroelastic these travelling wave coordinates and the
analysis of turbomachines and propellers displacement of the ith blade is
are three fundamental relationships: a
structural dynamic model of the bladed
disk; a kinematic relationship between
various expressions for blade motion; and
an unsteady aerodynamic model of aerody-
namic forces. The most general possible
model of the single degree of freedom
aeroelastic response of a typical blade
19-4

where q Bn is the amplitude of the

travelling wave of interblade phase Sn [
8n = 2irn/N. The sum in n can be taken as
where the structural damping factor g has
n = O. 1. 2,---. N-1 been added. The second is the kinematic
relationship between individualand
or. equlvalently for N an odd number of blades, traveling wave blade motion (eq. 5)

or. for N an even number of blades. (10)

n o- - I,- •• .-1.0.1. •••.
where the last relation simply assumes
sinusoidal motion of the individual
since for a rotor of N blades there are N blades.
possible interblade phase angles, and
small negative angles are equivalent to The third is the relationship between
large positive ones travelling wave motion and unsteady aero-
dynamic forces, supplied by the aero-
dynamicist (eq. 8)

(11)

It will be convenient to rewrite equation

(4) as These three fundamental relationships
can be combined to yield the governing
aeroelastic equations in several ways.
(5) First, the equations can be expressed in
terms of interblade phase angle "modes".
This requires transformation of the struc-
tural dynamic equation (9) to interblade
phase coordinates. Second, the equations
fe E
o.o • • " o,N-l can be expressed in terms of individual
• blade displacements. This requires trans-
where [E]S •
•
* (6) formation of the aerodynamic forces, eq.
• •
(11) to individual blade coordinates. And
Vl.o' ' " Vl.H-l third, the equations can be expressed in
terms of standing modes of the bladed
disk, such as sine and cosine modes, or
The aerodynamic forces per span are structural eigenmodes. This requires
usually derived assuming that the blades transformation of both the dynamic equa-
are undergoing the travelling wave motion tions and aerodynamic forces. Each of
of eq. (4),Whitehead (1966), Smith (1972). those approaches has some value, as will
Adamczyk and Goldstein (1978). Under this be discussed.
assumption the forces per span acting on
the zeroth blade undergoing the nth
travelling wave, constant interblade phase Travelling Have Formulation
angle motion of eg. (4) can be expressed
as The aeroelastic eigenvalue problem was
first formulated in travelling wave coor-
(7) dinates, that is in those coordinates for
which the aerodynamic forces are derived,
Whitehead (1966). In order to derive the
equations in traveling wave coordinates,
where qg n is the amplitude of the nth equations (10) and (11) are substituted
travelling wave pattern, and the complex into equation (9), giving:
force coefficient due to en is ign .
The force on the itn blade due to the
superposition of all the interblade phase
angle waves is (12)

premultiplying by E~*, and cancelling the

(8) time variation exp(jut) gives

At this point the three fundamental (13)

equations of the aeroelastic problem are
at hand and will be repeated for clarity.
The first is the dynamic governing equa-
tion of motion, as would be derived by the
structural dynamicist (eq. 2).
 19-5
Equation (13) now represents the aerodynamic forces must be transformed by
formulation of the aeroelastic problem in substituting equation (10) into equation
terms of travelling wave coordinates. It (11), yielding
has the advantage of using the aerodynamic
force coefficients in exactly the form in
which they are derived. Furthermore, if
the blades have a single degree of freedom (17)
and if the blades are uniform in mass and
stiffness such that where

The flutter equation is found by sub-

(14) stituting into equation (9)
and [Xw^l+Jgj) J - nwJ(

then equation (13) becomes

U\ O8)

which is the aeroelastic equation in terms

of individual blade coordinates. The
ffpb' principal advantage of this formulation is
that it is expressed in a coordinate sys-
tem which is a natural one for the
which is the governing homogeneous equa- structure. Thus, if any complicating fea-
tion for single degree of freedom flutter tures are added to the structure, such as
for a perfectly tuned rotor. Note that disk elastic coupling, shroud elastic
the separate equations in equation (15) coupling, blade nonuniforraity or mis-
are completely- uncoupled. This implies tuning, or multiple blade degrees of
that fora tuned rotor, the travelling wave freedom. This is a simpler starting point
coordinates are the normal aeroelastic for the resulting model than the
eigenmodes, and the eigenvalues associated travelling wave form.
with each mode are directly related to the
unsteady aerodynamic coefficients for that Another advantage of this formulation
interblade phase angle. is that although the aerodynamic coeffi-
cients must be transformed into the [L]
matrix form, the aerodynamic coefficients
mu2(l+Jg) as they appear in the [Ll matrix give tre-
(16) mendous insight into the unsteady aerody-
namic interactions in a cascade. Each
term in the aerodynamic influence matrix
(l+ffpb« ~/m)
p
(L] has a unique physical significance
(Fig. 2). The term in the first row and
the second column, for example, designates
the force acting on the first blade due to
It is an advantageous coincidence that the motion of the second blade. By the
the kinematic assumption of constant symmetry of the rotor, assuming that the
interbalde phas travelling wave coordi- blades are geometrically identical, this
nates made by the unsteady aerodynamicist must be the same as the force felt by the
eventually turn out to be the eigenmodes second blade due to the motion of the
of the aeroelastic problem for a tuned third. Likewise, each term on the dia-
rotor. The disadvantage of this formula- gonal represents the force felt by a blade
tion is that it requires transforming the due to its own motion. The [L] matrix has
structural model to travelling wave coor- the form in which there are only N inde-
dinates, in effect forcing the structural pendent complex terms, and the entries of
representation into a form chosen for its each column are the same, with each column
convenience in the unsteady aerodynamic permuted one row relative to the adjacent
problem. Although not inconvenient for columns.
tuned rotors, transformation of the
governing equations to this form makes it
very difficult to interpret the aeroelas-
tic response of mistuned rotors with non-
uniform blades, and difficult to explic- 'Lo
itly include the effects of shroud and 4
disk elastic coupling, Crawley and Hall (19)
(1985), Kielb and Kaza (1984). Although
the representation of the aerodynamic -2 4-3
forces in this form obscures the real
physical dependence of forces on specific
blade motions, Szechenyi et al (1984),
much more insight into these aspects is The most significant term in [L] is the
gained by examining the equations formu- diagonal term Lo, which expresses the
lated in terms of individual blade force acting on any given blade due to its
coordinates. own motion, in effect the blade self-
stiffness and self-damping. It has been
shown that this is the only term in the
Individual Blade Formulation influence coefficient matrix which can
provide a net stabilizing influencing on
In order to formulate the problem in the rotor, Crawley and Hall (1985),
terms of individual blade coordinates, the Szechenyi et al (1984).
19-6

Mathematically, the individual blade

aerodynamic forces LK are related to the
travelling wave forces through a complex
Fourier transform relationship

N-1
(20) _ O

and a) average offset represents blade's influence on Itself. L Q

N-l
2
(21)

Equation (20) shows that Lk is just

the Ktn coefficient of the discrete
Fourier series representation of ign given
in equation (21). So, for example, if a b) first harmonic represents neighboring blade influence,
plot of the aerodynamic coefficients ver- L, andUN.,
sus 0 is dominantly the first harmonic of
6 and an average offset, this implies that
[L] is almost tridiagonal, and the physi-
cal interpretation is that only the two
adjacent blades to a given blade and the
blade itself have any direct effect on the
"•'-. , "\ .
blade (Figure 2). If the plot of t0n vs. 6
has higher harmonics in 0, then the influ-
ence of more distant blades is relatively
more important. "2 - l-N-2
c) second harmonic represents influence of blades two
stations away, L 2and L N.2
Standing Mode Formulation
When the starting point of the aero-
elastic formulation is a set of calculated Figure 2. Graphical Relationship Between
or experimentally measured standing Aerodynamic Forces in Inter-
structural eigenmodes of the bladed disk blade Phase and Complex
assembly, it is desirable to formulate the Influence Coefficient Form
aeroelastic problem in terms of these
modal coordinates, Brooker and Halliwell
(1984), Crawley (1983).
If the rotor is tuned, then there will co
be pairs of repeated structural eigen-
values. In this case, there is not a
unique representation of the eigenvectors. <cn [P] (24)
Two natural ways to represent the mode
shapes are by forward and backward travel- *sn c2
ing waves, or by sine and cosine standing
waves, Dugundji and Bundas (1984).
Expressing the motion of the rotor in
terms of sine and cosine modes, also known
as twin orthogonal modes or raultiblade
coordinates, gives the representation
C C
0.0 0.1 8 0.1 C 0.2 8 0.2' ' '
'1,0 .1 C1.2 81,2
GL;2»0* . • * • • ** •
* (25)
i.2,...!tl for N odd, (22)
n o i.2.«««,j for N even,
a _ 2H1
fl. a -B- for N odd

C a cos 2TJ= and 8ln

which still allows arbitraryu time depen-

dence of the motion. If the motion is
assumed to be oscillatory, the displace-
ment is Since equation (23) expresses the
relationship between the individual and
sine/cosine modal coordinates, the aero-
q
l • ^eo * 5 ^co'O (23) elastic formulation can be transferred to
these coordinates by simply substituting
equation (24) into 1 equation (18) and
which can be written premultiplying by P" giving
 19-7

Note that the left hand side of equa-

tion (30) will now be uncoupled, since ^
are the structural normal modes, but these
modes will be aerodynamically coupled by
the terms on the right hand side.
(26) The advantage of this formulation is
that the starting point is the set of
blade disk normal modes, which can incor-
porate all forms of blade, disk, and
shroud elastic coupling. The disadvantage
is that the aerodynamic forces in the form
in which they appear in equation (30), and
the resulting flutter eigenvectors may be
difficult to interpret physically.
The process of deriving equation (15)
from equation (8) is a similarity trans-
form, in which eigenvalues are preserved.
FORMULATION FOR MULTIPLE SECTION DEGREES
The unique aspect of the pure sine and OF FREEDOM
cosine standing modes is that a pair of
like nodal diameter modes can be directly So far the various formulations for
superimposed to form a traveling wave single blade degree of freeedom flutter
mode. By comparison of equation (10) end have been outlined. However, it is often
equation (24), it can be seen that desirable to include multiple degrees of
freedom for each blade in the aeroelastic
model, Bendiksen and Friedroann (1980),
Kielb and Kaza (1984).
q
cn
[Elf1!?] (27) For such a model, such as a bending-
torsion coupled typical section analysis,
the equations presented above are still
valid, but must be generalized appropri-
If the rotor is mistuned, or contains ately. This generalization process essen-
coupled bending torsion motion of the tially consists of letting each scalar
blades, it is no longer simple to relate quantity in the equations (9), (10), and
the standing and travelling waves, but it (11) take on a sub-matrix nature. The
is still straightforward to relate the three fundamental relations for one degree
standing blade-disk modes to the indi- of freedom system are summarized here
vidual blade deflections. The itn blade again. The dynamic equation of equilibrium
deflection is given as is

N-1
(28a)

the kinematic relationship between stand

which can be written ing and travelling waves is

(28b)
do)
where 41 is the matrix whose columns are
the traditional structural modes, and qn
are the coordinates of those modes. The dependence of the aerodynamic force on
Comparison of equation (24) and equation motion is
(29) show that for perfect sine and cosine
twin orthogonal modes that P matrix is
just a special case of the normal modal
vector matrix $ under the assumption of
sinusoidal motion.

M (29)
If each section is allowed a translational
and pitching degree of freedom, then the
and substitution into equation (18) and generalized coordinate sub-matrix map to
premultiplication by <(>T gives the aero-
elastic formulation in terms of arbitrary
blade-disk modal coordinates
(31)

(30) where h is the translation of the section,

and a is the pitch. The other terms in
the equation (9), therefore, map as
follows
19-8

(38)
(32)
S/b I/b' N-I

(33)

where equation (38) includes the effects

of impinging wakes of velocity ugn and of
periodicity 8n being convected into the
cascade. Equation (38) can be written

(39)

In this typical section analysis all

of the generalized mass and force terms N-1
are defined on a per unit span basis.
The pitch motion is defined about the
elastic axis, such that the stiffness,
sub-matrix in equation (33) is diagonal, provided E has the definition of equation
but the inertia matrix is populated. Note (37), q gn has the definition of equation
that the usual (unfortunate) aeroelastic (35), and * is defined as
convention for positive signs has been
used (Figure 3).In modifying the kinematic
relationship (equation (10)) the traveling
wave coordinates also take on two coor-
dinates for each interblade phase angle
•r *• and (40)
n n .C
(35) With these relationships, the bending
torsion aeroelastic problem has the same
notation as the single degree of freedom
problem and all the transformations de-
veloped above can be employed. The aero-
elastic problem can be formulated in terms
The E matrix is now fully populated by of travelling modes, individual blade
sub-matrix blocks deflections and standing modes of bending-
torsion deflection.

FORMULATION FOR MULTIPLE SPANWISE BLADE

•Bk.< (36) MODES
In order to gain a more accurate model
of the aeroelastic behavior of a turbo-
machine component, it is necessary to in-
so that the E matrix has the form tegrate the unsteady aerodynamic forces
over the entire span. Whether two-dimen-
sional strip theory operators (Chapter
3) or a full three-dimensional model is
used (Chapters 4 and 5) will depend on
>E
0.0 ° E the availability and refinement of such
0. operators. The inclusion of Spanwise
0 integration of aerodynamic forces in the
S.c E
0.
aeroelastic formulation is a straightfor-
E
1.0 ° (37) ward extension of the results of the last
section. The governing dynamic equation
0 E for the i *> 0,1,..., N-1 blades is now
i.c

Finally, the aerodynamic forces and with m=l.2.•••.!! for every 1=0.1.*".N-1
moments now depend on translation and
pitch, so that
 19-9

where the generalized displacements and

generalized forces of the i fc " blade are
now represented at N/2 spanwise stations

w With p=1.2,- —,P for every i-0.1.•••,N-1

l
V* (42)
where the modal mass, modal stiffness, and
modal force associated with these P modes
are

Note that the mass matrix of equation (41)

now has units of mass, rather than mass
per span, and the other matrices have been
redimensioned accordingly. The formula-
tion of equation (41) still assumes that
shroudless blades are rigidly fixed to a
stiff disk, such that no structural coup-
ling exists between blades.
Rather than solve the coupled
<Vt = (49)

structural-aerodynanic problem, the usual

procedure is to solve equation (41) for
the structural normal modes of the ifc^ With these definitions the left hand side
blade, (i.e., with fi set to zero), by of equation (46) is completely uncoupled
solving and the mapping of the multiple spanwise
blade mode problem to the simple single
degree of freedom problem of equation (9),
(43) (10), and (11) is possible.
XxX NxN For the displacements, the generalized
displacements map to the blade modal
coordinates
The result of theth structural eigenvalue
problem for the i blade is a set of M
natural frequencies and mode shapes.

h /b (P) (50)
"l
Vs
p = 1,2,--,H (44)

for the inertia term the inertia maps to

"M/2 the modal inertia

and an associated set of blade modal

coordinates a .
(51)
In the aeroelastic problem, only a few
of the blade modes are generally of
interest. Let the number of modes of
interest be P, so that the displacement of
the ith blade is expressed in terms of P
modes and the stiffness terms map to the modal
stiffness

(45)
(52)

and, finally, the blade force maps to

and upon substitution into equation (41) the blade modal forces
19-10
When written in the notation of equation
(53), the forces acting on the blade modes
of the N blades in terms of the motion of
the modes of the individual blades is
(53) given by equation (54). Note that the form
assumes that aerodynamic strip theory has
been used. The transformation matrix T is
used to change the order of notation for
blade degrees of freedom from that used
for the structural problem (inner loop on
the blade DoP) to that used in the aerody-
The proper transformation of the blade namic problem (inner loop on the cascade-
aerodynamic forces acting on' the blade wise coordinate).
modes is somewhat complex. Careful atten-
tion must be paid to keeping track of
effects at the same spanwise location
around the rotor versus effects along the
blade.

<Vi-o tfl uo tE]

2 2
ttpb u

\f\ I=W-1

G0~
(54)

'OB'-i
IvU \fl 1=N
<V> uo
where the transformation =[T]
[T] Is defined by :

inner loop Inner loop over-

over 1 blades a blade stations
of the 1™ blade

If the aerodynamic forces were derived To this point all the necessary trans-
from a three-dimensional aerodynamic model formations and formulations have been
which assumed a travelling wave pattern of rigorously developed to express the spa-
an assumed blade mode shape, then the tial (i.e., spanwise and circumferential)
aerodynamic forces are dependencies of the aeroelastic
formulation. However, the entire formula-
tion to the point, except for the basic
equations (3), (4), and (9) have assumed
temporally sinusoidal motion. This is due
to the assumptions inherent in the deriva-
tion of the aerodynamic operators. In the
next section, solution techniques for the
sinusoidal formulation will be presented,
and in the following section, an approxi-
mate transformation to an explicit time
where the £3.0 matrix is the representa- accurate formulation will be discussed.
tion of travelling wave three-dimensional
unsteady aerodynamic forces due to travel-
ling wave motion (Chapters 4 and 5). SOLUTIONS FOR SINUSOIDAL TEMPORAL
REPRESENTATIONS

It may be desirable to express the Under the assumption that the aerody-
aeroelastic equations of motion of a com- namic operators are only available for
plete rotor in terms of both spanwise sinusoidal motion, the steps remaining
blade modes and coupled blade-disk circum- after formulation of the aeroelastic pro-
ferential modes. In this case the formu- blems are its proper nondimensionalization
lation for blade modes of his section can and solution for stability and forced
be coupled with the formulation for stand- response. For reference, the dynamic
ing blade-disk modes given above to yield equation of equilibrium, assuming sinu-
the governing equations of motion. soidal motion is
 19-11

(59)

(56)

- BpU2
Generalized forces on the ith blade due to
the nth travelling displacement wave
pattern and the wake forced vibration
HP.
terms are
This will be referred to as the c t formu-
lation for the aerodynamic forces.
(57) If the homogeneous aerodynamic force due
to translation and moment due to pitch are
examined in the Ct form, they are

This will be referred to as the I formula-

tion of aerodynamic forces. Note that the
forces are nondimensionalized in time by (60)
the square of the frequency of
oscillation, and therefore have the form
of virtual inertias. A second common form
of the aerodynamic operators is (Chapter (61)
III)

= ffpUc L In contrast to the I formulation, the

aerodynamic forces in the cj formulation
(58) are nondimensionalized in time by the
square of free stream velocity in the case
of the moment (eq. 61), and by the velo-
city an_d frequency, in the case of the
in which q (unfortunately) stands for the force,Teq. 60). Therefore these terms
translational velocity, and o for the appear in the equations of motion as vir-
pitch angle (Fig. 3 ) . I f the assumption tual stiffness and damping like terms,
of sinusoidal motion is made, and the respectively. Note that in the t form
coordinates are assumed to be the transla- there is an explicit frequency dependence
tional and pitch displacements,, and wake but no explicit dynamic pressure depen-
velocity amplitude, then equation (58) can dence, whereas in the Cjj form, there is
be manipulted to have the form of equation explicit dynamic pressure dependence, and
(57). the explicit frequency dependence is dif-
ferent from that in the former. Thus in
comparing reduced frequency dependence of
the nondimensional aerodynamic forces, one
must keep in mind that the form of the
nondimensionalization impacts the apparent
trend as the reduced frequency is varied.
Of course, as always, one must pay close
attention to the sign convention for posi-
tive moment and displacement, and for the
chord location which is used for the coor-
dinate system reference. A summary of
these conventions for the £ and c t forms
is given in Appendix B. The nondimension-
alization and solution techniques will be
developed for the simple single degree of
freedom equations (9), (10), and (11),
since it was shown above that the problems
Figure 3a. Notation Convention for 2-dof with multiple blade degrees of freedom
Model in the 1 Force Notation were simply extensions of the one DoP per
blade formulations.

Continuing with only the 1 form, combi-

nations of eq. (9), (10), and (11) give
the aeroelastic problem formulated in

y&
gust velocity
individual blade coordinates as

(62)

Figure 3b: Notation Convention for the 2-dof Model

In the C Force Notation [Chapter 3]
19-12

Division by the blade mass of a sec- where g is interpreted as the degree of

tion of the nominal blade gives us the structural damping necessary to provide
nondimensional form of the problem neutral (oscillatory) dynamic behavior.
The corresponding velocity is then

(67)

(63)

For a N degree of freedom system this

where e< and 5^ are the fractional mass will produce N points on the V-g diagram,
and stiffness mistuning of the ith blade, as shown in Figure 4. By choosing various
0 is the nondirnensional eigenfrequency, values of k, families of curves of re-
n = U/UK , and ii is the section mass quired damping can be plotted. The sta-
density ratio bility boundary is then defined as the
velocity at which the required damping
exceeds the structural damping actually
present in the rotor.
(64) Unlike in aeroelastic analysis of air-
craft, a key simplification of this pro-
cess can usually be -made for gas turbines.
Since the mass ratio is usually large (u»
10), the aerodynamic forces are very small
which premultiplies all of the aerodynamic compared to the inertial and elastic
terras in the governing equations. Note forces acting on the blade, that is */u«
that in the form of equation (64) the mis- 1. Therefore the oscillatory component of
tuning or nonuniforml'ty effects appear the aeroelastic eigenvalue is usually very
explicitly in the formulation. close to the reference frequency, implying
that the reduced frequency for all of the
Equation (63) is of the form of a eigenvalues is very close to the reduced
traditional aeroelastic eigenvalue problem frequency associated with the natural
used to determine the stability of the frequency.
system. The task is to solve for the com-
plex eigenvalues of equation (63). The This relative weakness of the aerody-
eigenvalues in general will have a nega- namic terms leads to treating equation
tive real part or a positive real part, (63) as a standard eigenvalue problem.
indicating mode stability or instability, That is, a reference value of the reduced
respectively. The contradiction present frequency is calculated based on the
in the formulation is, of course, that the structural frequencies at speed but in
system eigenvalues are either exponen- vacuum. The aeroelastic eigenvalues are
tially damped or unstable, but in general all then calculated and used as is, since
not purely sinusoidal, while the aerody- little difference between aeroelastic and
namic forces were derived assuming pure in vacuum frequency is present.
sinusoidal motion. Furthermore, these
aerodynamic terms depend implicitly on the
reduced frequency, but the actual fre-
quency of oscillation is not known until
after the eigenvalues are determined.
The traditional solution to this
problem is the so-called V-g method, in
which the structural damping is assumed X - Points derived for a
uniform, and treated as a free parameter, single value ol k
Bisplinghoff and Ashley (1966). Rewriting
equation (63) under these assumptions
g
V for instability
I

(65)
where

For a fixed reduced frequency kR , the

eigenvalue problem is then solved for the
complex eigenvalues , and for each the
frequency of oscillation and. damping
factor are calculated

n » {••(*) r (66)
Figure 4: V-g Representation of System Stability
 19-13

If more accuracy is desired, then two EXPLICIT TIME DEPENDENT FORMULATION OP

approaches are available. In an iterative AERODYNAMIC FORCES
approach, after the first calculation, the
reduced frequency is modified based upon While sinusoidal representation of
the calculated oscillatory component of motion is adequate for stability analysis,
the most critical aeroelastic eigenvalue. it is sometimes desirable to express the
This iteration is then continued until the aeroelastic equations of motion with ex-
reduced frequency assumed in determining plicit time dependence of the unsteady
the aerodynamic coefficients, and the cal- •aerodynamic terms. Examples of when this
culated reduced frequency of the most might be needed are when the excitation or
critical eigenvalues converge. This pro- response is expected to differ from a
cedure resembles the traditional p-k sinusoidal behavior. Such non-sinusoidal
method of aeroelastic analysis. behavior occurs in certain forced vibra-
tion phenomena, such as impacting or
A second procedure which eliminates mechanical rubs, and time unsteady aero
the need for this iteration is based on disturbances, such as rotating stall and
expanding the explicit functional depen- surge. Furthermore, whenever time march-
dence of L on k. If the aeroelastic coef- ing calculations are to be done, it will
ficients are locally fit by a least be necessary to have the aerodynamic
squares procedure to an expression of the forces in a time domain representation.
form
Unfortunately, the unsteady aerody-
namic operators have been derived assuming
(68) sinusoidal behavior in time and travelling
wave constant interblade phase angle in
space. In the special transformations
above, a complex inverse discrete Fourier
transform (eq. 17) was used to remove the
Substitution into equation (63) gives a restriction of assumed travelling waves,
new eigenvalue problem and to express the aerodynamic forces in
terms of the individual blade motions.
The resulting form was

which can be rewritten as a standard

eigenvalue problem and solved directly for
the aeroelastic eigenvalues.

The results of these formulations are

aeroelastic eigenvalues which can be
plotted in the complex plane. If the
traditional complex s-plane interpretation
is desired, then the plot must be of stable unstable

(70)
R»(8)
as shown in Figure Sa for a single value
of reduced frequency k. If a range of k is Figure 5a: Complex s-Plane Interpretation of
plotted, the root locus of the individual Aeroelastic Eigenvalues for a Single k
eigenvalues plot out as curves originating
at (0 + lj ) in the case of no structural
damping. Instability is then defined to
occur as the first root crosses into the
right half planes (Fig. 5b).
There remains in all this analysis the
contradiction that the system behavior is
non-oscillatory, while the aero forces
were derived for oscillatory behavior.
Where accuracy is most needed, at the
point of neutral stability, the behavior
is truly oscillatory, so the aerodynamic
forces are exact. Common sense would
dictate that for lightly damped and mar-
ginally unstable systems, the stability
margin would approximate the true damping
ratio of the system. This, in fact, has
been shown to be the case, but a proof re-
quires the expression of the aerodynamic
forces in time explicit form, Dugundji and 0 Re(s)
Bundas (1984). An approximate scheme for
this time accurate representation will be Figure 5b: Complex s-Plane Interpretation of
shown in the next section. Eigenvalue Root Loci for Increasing k
19-14

The vectors of y^ are augmented

(71) states, related to qi by
(ft) =

fop
where each column of L was identical, and
shifted down one row relative to its
neighbor. Thus all the diagonal terras are
L0 , the blade's aerodynamic force on
itself, the first diagonal below the prin- In other words, the y^ variable is a
cipal is LI , the effect of the adjacent first order lag of time constant g(, on
blade downstream, etc. (eq. 19). The the rate of change of the displacement q<.
elements of the matrix L are of course The time constants are the same for all
complex and functions of the reduced fre- the nominally identical blades. Such
quency k. The restriction of sinusoidal approximations ,are motivated by their suc-
temporal behavior was therefore still cess in approximately unsteady aerodynamic
present. forces in external flows and cascades.
In principle, a complex inverse In order to evaluate the unknown con-
Fourier integral in the reduced frequency stants in C2» Cjf Cfj, G0, Gj etc., equa-
parameter k, allowing k to range from zero tions (73) and (74) are expanded to
to infinity, could be taken of the ele- examine the forces acting on the zeroth
ments of L in order to explicitly trans- blade. Equation (73) gives
form them to the time domain. In
practice, the frequency dependence of the
L terms is either expressed as a very com-
plicated expression of k, or, if L is
found through computational techniques,
never written as an analytic function of
k. Thus approximate transform techniques
from the frequency to time domain must be
used.
The most popular approximate transform
technique for unsteady aerodynamic forces
involves the so-called Pade approximation
of exponential lags in the aerodynamic (75)
forces, Edwards et al. (1979). In order
to prepare the aerodynamic coefficients
for this approximation procedure, it is
necessary to convert the coefficients to a . b • for
form in which the frequency does not U l !fql and
appear explicitly in the nondimension-
alization
Assuming pure sinusoidal motion

(77)
(72)
where [C]
then substitution into equations (75) and
The CL form of the coefficients is (76), and combining the two, the force on
similar, but not identical to the c t form. the zeroth blade can be written
Now a general approximation to the time
dependent form of the aerodynamic forces
is Introduced

(f ^ - UpU2

NM'VKK (78)

If equation (72) is expanded in a

manner similar to equation (78), then the
where C2 i C} and C0 are real circulant force on the zeroth blade is
matrices of the same form as L (i.e., only
N unknowns, all columns identical but
shifted). The matrices C2 • C\ i and C0
represent the inertial, damping, and
stiffness effects of the aerodynamics.
The matrices Go, Gj, G2, etc. are sparse
real circulant matrices with only one en-
try per column. They contain the impact *1
of the relative lags in the aerodynamics
on the blade forces. G<. for example, (79)
contains the coefficient which expresses
the lagged forces of the i+j blade on
the i blade.
 19-15

By comparing equations (78) and (79) Stabilizing and Destabilizing Influences

term by term, the following relations are in Cascades.
apparent
Simply from examination of the stabil-
ity eigenvalue problem, certain stabiliz-
ing and destabilizing effects can be iden-
tified for a single degree of freedom
flutter model. The nondimenaional form of
the stability problem, equation (63), is

-J
(82)
where the C's, G's, and g's are real con-
stants to be determined, and CLr is a
complex function of k. in which g is the structural damping £,
and e the stiffness and mass nonuni-
All that remains is for the real un- formity, and L, the complex aerodynamic
knowns to be determined by a fitting pro- influence coefficients of the form
cedure, such as a least squares fit to CLr
versus the reduced frequency k for each
value of the index r. Such experience in
fitting sometimes produces an adequate fit
using the single lag pole shown. This is
true for the case of an incompressible
Hi-i
cascade, Dugundji and Bundas (1984). More (83)
accuracy is attained by introducing a
second set of poles g' and associated con-
stants G*. The classic Jones approximation
to the Theodorsen function is an example
of this kind of two pole fit, Bisplinghoff
and Ashley (1962).
Once the aerodynamic constants have In order to identify the stabilizing
been determined, the governing equation of and destabilizing influences, we simplify
equilibrium, equation (9),and the time do- the problem by allowing the blades to be
main expression for the aerodynamic forces uniform in stiffness and structural
can be combined into a single expression. damping. The governing equations for one
degree of freedom per blade flutter are
then

DpU2 (84)

(81)

The remaining parameters in the pro-

blem are the structural damping g, the
where mass mistuning et , the mass ratio u , and
aerodynamic coefficients Lo through LN .
0.1.«".H-1 Each of these terms somehow influences the
eigenvalues n .
If a similar procedure is used to The complex eigenvalues of equation
represent the unsteady wake or gust re- (84) form a pattern in the s-plane, with s
sponse function, then a complete time « jft, as shown in Figure 6. This pattern
accurate time domain representation of the can be considered to have a centroid, and
aeroelastic behavior can be achieved. the eigenvalues are distributed about this
centroid.
TRENDS IN AEROELASTIC STABILITY The location of the centroid is criti-
cal to the stability. If the centroid is
As with many engineering analyses, in the right half plane, then by defini-
there are certain dominant trends in the tion some eigenvalues will be in the right
analysis of the aeroelastic stability of half plane, and the system will be
turbomachine rotors. Some, such as the unstable. Thus, to assure system stabil-
role of the mass ratio or the importance ity the centroid must be in the left half
of blade mistuning can be determined sim- plane. Returning to equation (84), it has
ply from careful examination of the been shown, Crawley and Hall (1985), that
governing equations. Others require solu- the only terms which can exert a net
tions for ranges of parameters to deter- stabilizing influence on the rotor are the
mine overall trends. In this section four structural damping g, and the term L0,
trends will be addressed: the stabilizing which expresses the aerodynamic force felt
and destabilizing Influences in a cascade, on the blade due to its own motion. To
and the critical role of the blade self- show the importance of this term, consider
damping; the effects of bending-torsion the problem of equation (84). Making use
coupling; the real rotor effects of load- of the matrix property that the sum of the
ing, three-dimensionality and stall; and eigenvalues of a matrix equals the trace
the differences in analysis of actual of the matrix, we have the following rela-
rotors and "rubber" designs. tionship for the sum of the eigenvaluest
19-16

That is, in the absence of structural

damping, the real part of the centroid,
<s>, depends on the imaginary part of L0 ,
and the structural damping g . The
imaginary part of the centroid depends on
the real part of LQ and the mean value of
the mistuning.
In the absence of unsteady aerodynamic
forces, the reference blade vibrates at
at the nondimensional eigenfrequency
Q m aR s i. in the presence of aerody- The location of the eigenvalues in the •
namic forces, which are small compared to s-plane can be considered to be distri-
the elastic and inertial moments, ft will buted around the centroid. Recalling that
still be nearly equal to flR . The eigen- the system will be unstable if any eigen-
frequency can be expressed as a sum of its value is in the right half plane, the ob-
reference value and a perturbation from jective is to assure that the least stable
the reference value 8R . eigenvalue is as far to the left as
possible. If the rotor is unstable,
increases in stability can be achieved
(86) either by moving the centroid to the left,
or by reducing the size of the distribu-
tion about the centroid, which pulls the
The last step in eq. (86) is due to nR rightmost eigenvalue to the left.
being unity (see Eq. 68). Hence the
eigenvalues of eq. (85) can be expanded as
Interpreted in this light, eq. (88) is
an important result. It shows that in the
absence of structural damping the centroid
= 1-23 (87) of the eigenvalues lies in the left plane
if and only if Im{L0 ) is less than zero.
Since a necessary condition for aero-
.elastic stability of the rotor is that the
For convenience, let s = Qj. Substitu- centroid of the eigenvalues lies in the
tion of eq. (87) into eq. (85) yields that left half plane, it can be deduced that a
the centroid of the eigenvalues (Fig. 6) necessary but not sufficient condition for
is given approximately by stability is that im(Lfj ) be less than
zero. This is equivalent to the condition
that the blades be self damped.
«e<s>
(88)
The location of the centroid is set by
•»-i*hr -T the average value of the mass (and stiff-
ness) of the blades, the structural damp-
ing, and the blade self damping term. The
distribution of the eigenvalues about the
centroid is controlled by the nonuni-
Im (t )
formity in the mass and stiffness and by
• lirT'C
t
the off-diagonal terms in the aerodynamic
influence coefficient matrix equation (83)
(i.e., the unsteady cascade influences in
A A 1
the aerodynamics).
\
•% A
Note that any amount of off-diagonal
4 A EIGENVALUES
aerodynamic Influence, that is any un-
steady aerodynamics effects due to neigh-
A ' ^ CENTROID boring blades, will distribute the eigen-
values about the centroid, and therefore
move some of the eigenvalues to the right,
^ I destabilizing the cascade. Thus, unsteady
* A*1
aerodynamic Interactions amongst the
blades in a cascade are destabilizing.
* * i
I

STABILITY MARGIN--^ \
CONSTRAINT i The distribution pattern of eigen-
i values about the centroid is influenced by
i the pattern of stiffness and mass mistun-
i ing of the blades, but the location of the
i
centroid is not Influenced by the pattern
i of mistuning so long as the average value
i
i
i
is zero. Thus, the effect of mistuning
is to reduce the influence of the blade to
blade aerodynamic coupling and move the
Rets) less stable eigenvalues toward the
centroid. Note that no amount of mistun-
ing will cause the centroid to move in a
stabilizing direction and no amount of
Figure 6. s-Plane Interpretation of mistuning can increase the stability mar-
Eigenvalues Showing Centroid gin of the rotor beyond that given by the
and Stability Margin Constraint blade self damping.
 19-17

Finally, the importance of the mass limit to the potential effectiveness of

ratio and structural damping can be seen mistuning is the centroid of the eigen-
for a one degree of freedom flutter by values of the tuned rotor.
examining equations (84) and (88). It is
clear that all of the aerodynamic influ- While these four trends are rigously
ences are scaled by the mass density true Cor single degree o£ freedom por
ratio. In particular, if a necessary blade flutter, they are generally appli-
stability criterion is that the centroid cable to any turboraachine in which the
of the eigenvalues is in the left-half flutter dominantly involves a single de-
plane then for stability gree of freedom per blade. This is gen-
erally true of solid metallic blades. In
the case of hollow or composite blades
with significant bending torsion coupling,
Se<s> i 0 more judgment should be used in interpret-
(89) ing these stablizing and destablizing
cascade influences.
J
Bending-Torsion Coupling
Several authors have investigated the
A similar relationship is derived from impact of modeling cascade flutter as a
the previous tuned rotor analysis in which classical bending-torsion coupled problem.
a sufficient condition for stability of a In order to not confuse issues, two dis-
tuned rotor was that tinct mechanisms of bending-torsion
coupling must be distinguished:
A. Single mode coupling -this occurs when
a single torsional mode has some
-8 (90) translational component, or a single
bending mode has some torsional
component. Although its origin may be
dynamic, this is essentially a kine-
for the largest positive value of the matic coupling. It may be due to the
aerodynamic coefficient 4gn . In each of root not being supported along a line
these cases, the relative contribution of normal to the elastic axis (i.e.,
the aerodynamic component and structural structural sweep), the presence of an
damping is scaled by the mass density offset between the elastic axis and
ratio u . center of mass, the presence of aniso-
tropic materials or fibers, or the
In the limiting case of no structural presence of shrouds at tip or mid-
damping, the stability boundary is inde- span;
pendent of the mass ratio, since even a
small amount of destabilizing aerodynamic B. Dynamic coupling between two modes -
influence will cause the rotor to go which is the case when two independent
unstable. However, in the presence of a modes dynamically interact to cause a
fixed nonzero structural damping ratio, classic bending-torsion like coales-
the mass ratio sets the magnitude of cence flutter.
destabilizing aerodynamic effect which can
be tolerated before the system becomes
unstable. If the rotor speed is increased
past the reduced velocity corresponding to
neutral aerodynamic stability for a fixed
structural frequency and damping, a rotor
blade with a larger mass ratio will be
more stable than a rotor with a smaller
mass ratio, as shown in Figure 7. The mass
ratio of course can be changed by either
changing the gas density, or by a change
in the blade material. 7 for
Instability
The stabilizing and destabilizing (small u) r;for
effects for a single degree of freedom
flutter model can be summarized as instability
follows: / (large ji)
1. In the absence of structural damping, T unstable 1/
the blade must be self damped, so that the \S...
centroid of the eigenvalues lies in the
left half plane. Jet— w
2. In the presence of structural damping,
blades of larger mass ratio are relatively neutral aerodynamic
more stable than those of smaller mass
ratio for the same damping g. stability point

3. The cascade unsteady aerodynamic

influences are destabilizing. large n
4. Structural mistuning does not change
the location of the centroid, but can
rearrange the eigenvalues to increase the Rgure 7: Sensitivity of Rotor Stability to Mass Ratio
stability of the least stable root. The for Single Degree of Freedom Flutter
19-18

In gas turbine blading, the first or

single mode kinematic coupling can be very uV dynamic pressure
important. The presence of shrouds or the ~ structurml stiffness
distribution of mass will often cause a
vibrational mode not to be pure torsion or
pure bending, and this mixture must be
taken into account. Often the presence of Generally, these two influences on
some bending in a dominantly torsional aeroelastic stability are opposing, that
mode will exert a stabilizing influence. is, the flow is more unsteady at high k,
low V, but the dynamic pressure is greater
However, in gas turbine blading, the at low k, high V.
dynamic bending-torsion coalescence
coupling is more important when the struc- Further, when aeroelastic trends such
tural modes are very close in frequency, as the V-g diagram of Figure 8 are plotted
or when blades are hollow or composite, versus V, attention must be paid to
The lesser importance of this dynamic whether the trends are at constant Mach
interaction between modes can again be number, or if the Mach number changes with
traced to the large mass ratios usually V. The former, which is the traditional V
found in turbine components, The aerody- -g diagram, is useful in the analysis of
namic forces are simply not strong enough rubber engines, i.e., in considering de-
to significantly shift the structural sign trades, since V can be changed at
frequencies, unless they are already in constant Mach number and temperature only
close proximity. Thus the aeroelastic by changing b and u. These are parameters
instability found in most gas turbine com- which are only variable in design. For
ponents is not the classic bending-torsion analysis of an actual component, a V-M-g
coalescence flutter, but instead a cascade diagram is required, i.e., one in which
induced blade-to-blade interaction Mach number changes in proportion to V, as
flutter. determined by the operating line of the
rotor on the performance map. Such dia-
grams then represent the performance of a
The Effects of Loading and Three- given component running on an operating
Dimensionality line.
Actual turbomachine components work in When plotting both V -g and V -M-g
a complex heavily loaded, three- diagrams, it is useful to plot the per-
dimensional flow environment which is not formance of a tuned rotor, and one with a
easily modeled in unsteady aerodynamic nominal degree of mistune as in Fig. 8.
models. The aerodynamic loading and In limiting case a mistuned rotor asymp-
associated turning of the flow impact the totically approaches the stability associ-
aeroelastic problem in at least two ways. ated with the blade self damping.
First, the presence of the loading can
push the blading to a near stall
condition. In this heavily loaded EFFECTS OF MISTUNING ON STABILITY
condition, the additional load per unit of
incidence is known from quasi-steady All rotors are, by the nature of the
analysis and cascade experiments to manufacturing process, mistuned to some
diminish. Translated into an unsteady degree. Here mistuning is defined as a
aerodynamic effect, this implies that the distribution in the frequencies of the
forces on the blade for a unit of motion blades in the cascade. It has been sug-
diminish. These forces per motion of the gested that the level of mistuning be
blade on itself are expressed in the LQ deliberately increased to further augment
unsteady aerodynamic coefficient, which the stability margin of the rotor, and
was shown to be pivotal to system stabil- that this mistuning be introduced in
ity in the discussion above. Thus, any specific patterns.
slight reduction in the Iro(L0 ) the blade
self damping term, might lead to flutter. STABILITY DIAGRAM
This could be the origin of heavily loaded -005r-
Mach number • 1.317
flutter occurring near the stall line.
-0.04 -
A second effect of the turning is to
introduce swirl into the flow. This swirl
vastly complicates the downstream flow,
and couples the acoustic, vorticity and
pressure fields. Even current three- Optimally miitunid
rotor
dimensional aerodynamic analyses do not
take this into account. The implications
on the unsteady aerodynamic forces of this
swirling downstream flow have yet to be
considered.

Trends with Reduced Velocity Blade tell damptno,

In considering aeroelastic trends with Reduced Velocity at which rotor

wot optimiied
reduced velocity, or reduced frequency, it
must be remembered that those parameters
are in fact used in two ways: first, as a
general measure of the unsteadiness of the Figure 8. V-g Stability Diagram of a
flow, and, secondly, as a way to non- Tuned and Optimally Mistuned
dimensionalize the relative strength of Rotor (5*0.002), with the
the structural and dynamic pressure Limiting Case of the Blade Self
forces Damping
 19-19

In considering the effect of mistuning At this point, the optimization state-

on stability, one roust understand the ment has be»n completely specified. The
mechanisms of mistuning; that is, what cost function to be minimized is a measure
physical effects cause a change in of the level of mass mistuning to be
stability. The auxiliary questions are introduced into the rotor while the con-
then: how much mistuning must be present straints are that the rotor meet minimum
to consider a rotor mistuned; how much stability requirements. The independent
change in stability margin can be achieved variables are the individual mass mistun-
by mistuning; what is the optimal pattern ing of the blades or stiffness and the
of mistuning; and what are the limitations governing system equation is equation
to mistuning for stability augmentation. (63). This problem can be solved using
appropriate numerical optimization
When deliberately introducing mistun- techniques.
ing for stability, one would like to pick
an arrangement of mistuning which provides
a large increase in stability for a given To illustrate the results of optimal
level of structural mistuning. It has mistuning, consider a specific high bypass
been suggested, for example, that alter- ratio shroudless fan. The aeroelastic
nate mistuning may be nearly optimal in behavior is modelled using a typical sec-
increasing the stability of shroudless tion analysis by assuming a single tor-
fans. In this first section, an appropri- sional degree of freedom per blade. At
ate criterion for optimal mistuning will this typical section, the relative Mach
be defined and typical optimal mistuning number, M, is 1.317; the reduced
patterns examined. In the next section, frequency, k, is 0.495; the solidity, is
the mechanisms and limitations of mistun- 1.404; the mass ratio, is 182; and the
ing will be discussed. nondimensional radius of gyration, r, is
0.4731. Because the computational dif-
The selection of a definition of an ficulty of the optimization problem rises
optimal mistuning pattern is of course quickly as the number of blades increases,
necessarily subjective. One must deter- the number of blades of the fan was taken
mine how to weigh the unlike quantities of to be 12, 13, or 14.
stability, mistuning level, and forced
response of the rotor. One choice is to
implement the level of mistuning as a cost The unsteady aerodynamic model used is
function to be minimized, and the desired the supersonic linearized model of
level of stability of the least stable Adamczyk and Goldstein (1978). The aero-
eigenmode as a constraint. Hence, the dynamic influence coefficients found from
optimal mistuning problem can be posed as this model are shown in Figure 9. Note
a constrained optimization problem. that Im(Lo ) is less than zero indicating
that the blades are self-damped. Hence,
The cost function which represents the although the tuned rotor is unstable, mis-
level of mistuning in the rotor should of tuning may stabilize the rotor as previ-
course strongly penalize large amounts of ously discussed. Figure 9 also shows that
mistuning in any single blade. The cost the neighboring blades and the blade it-
function used by Crawley and Hall (1985) self exert the dominant forces on a given
is given by blade. Therefore, the aerodynamic in-
fluence coefficient matrix is strongly
banded.
(90a)

1.0
where n is in general some positive
integer. In particular, n was chosen to Q1
be 4. Since this cost strongly penalizes —K
large amounts of mistuning in any single 1
blade, no blade mistuning becomes exces-
sively larger than that of any other
ioc "» 0
1 2 3 4 S 6 7 S 9 10 II 1*2 13 M

blade. That is to say that there will be «£ -as

no "rouge blades" in the optimal mistuning
pattern. -1.0
8
The designer of a fan might wish to
specify that a fan have at its operating i 1.0
point some minimum stability level. Hence
the stability requirements are simply that | as •
the damping ratio of every eigenmode of
the mistuned fan be greater than some s
minimum damping ratio. This is expressed d o
5 1 Z 3 4 S 6 T 8 9 10 II 12 13 14
symbolically as 1

-as Blodt Numbtf.K

-1.0 .
-ct - r 2 o (91)

where c< is the damping ratio of the Figure 9. Unsteady Aerodynamic Moment
eigenvalue, and J, is the desired stability Coefficients Showing the Influ-
margin. This requirement is shown graph- ence of the i-th Blade on the
ically in Figure 6. 14th Blade in a 14-Bladed Rotor
19-20

Figure 10 shows the eigenvalues of the

14-bladed tuned rotor in nondimenslonal
form. Note that four of the 14 eigen- * °T
values lie in the right half plane and are
therefore unstable. Since the blades are COST EFFECTIVENESS OF
self-damped, the centroid of the eigen- MISTUNING
values lies in the left half plane.
-OPTIMAL MISTUMNG /

Next the rotor is optimally mistuned -ALTERNATE MISTUNINOJ

by numerically finding a mistuning pattern -OPTIMAL MISTUNING 13 BLADES
which minimizes the cost function and -OPTIMAL MISTUNING 12 BLADES
satisfies all the constraints. Figure 11
shows the cost of the optimal mistuning
pattern versus the desired amount of sta-
bility margin for the 12, 13, and 14-
bladed cases. Also shown is the cost of
alternate mistuning for the 14-bladed
case. Two important points are clearly
Illustrated. First, although it has been
previously thought that alternate mistun-
ing may be nearly optimal (in the sense
that a small amount of mistuning is
necessary) this is not the case. For a
desired damping ratio of 0.002, alternate
mistuning requires nearly twice the level
of mistuning as optimal mistuning.
Second, it appears that the number of
blades on the rotor is unimportant when -0.006 -0.004 -0.002 0.002
optimally mistuning the rotor, and also
that the optimal cost for 12, 13, and 14- Stability Margin,
bladed rotor are very similar.

Figure 11. Cost of Mistuning for 12, 13,

and 14-Bladed Rotors

l.04r
EIGENVALUES OF TUNED ROTOR
DAMPING RATIO { • -0-00602 IJ04 - EIGENVALUES OF OPTIMALLY MISTUNED ROTOR
1.03 "
DAMPING RATIO 1 *0

IX>3
2 1-02 309- OT
3 ^..0-^ 2 1-02
o o
uT i.oi / \ 309-
^
r> i.oi
^ 1.00
' \ o«
^ 257«o \
\
* °»
UI \
0 5 '•«» /
w \ 2
0.99 \
X
«\ \
ui 257-4, \
v*
^^ ^V V*
tO
Q9I*
i

.,
^^
v % \
\ n\ 154' I \ 206-^
C O98
%
d
/ § ase
^ *.^
^ ' ^^
-»e
Ov
^
/
^Qf

103* a.
£ o.9T Interblade phase ' \ \\
angle | 0.97 / \ \\
z Stable *- -».Uns
< 0.96
3
<
/
/

0.96 - tuned inter blade phate

^\
x,"'; '
'•*•<
103-
2
angle attocioted with
eigenvalue
0.95 - 0.95
^^-e>Unttable

O.94 1 1 1 1 1 O.94 1 1 1 1 1
-0.05-004 -OX)3 -O02 -0.0) -OOO 0.01 -0.05 -004 -009 -0.02 -0.01 -0.00 0.01
REAL PART OF EIGENVALUE , RE ( S) REAL PART OF EIGENVALUE, RE (S)

Figure 12. Eigenvalues of Optimally Tuned

Rotor _
Figure 10. Eigenvalues of a Tuned Rotor C-0
 19-21

Mechanisms and Limitations of Mistuning Finally, there is a certain fine structure

to the mistuning pattern, The details of
Some insight into why mistuning is this structure depend on the details of
effective can be gained by examining the the minimization, and it is difficult to
eigenvalues of the tuned and mistuned predict what this structure will look like
rotors in the complex plane. Figures 10, without actually performing the numerical
12, and 13 show the eigenvalues for the optimization.
tuned case and the c =• 0.0 and the c =
0.002 optimally mistuned cases for the 14- Unfortunately, the strict optimal mis-
bladed rotor. As mistuning is introduced tuning pattern is sensitive to errors in
the eigenvalues are "pushed" to the left implementation. Although the designer may
as much as necessary to satisfy the specify a certain mistuning pattern, the
constraints. manufacturing process may place limits on
the tolerances which can actually be
The optimal mistuning patterns found achieved. Hence, it is necessary to con-
in the optimization procedure for the 14- sider the sensitivity of a given mistune
bladed rotor are shown in Figure 14. pattern to errors in implementation. For
Beginning with the ? = -0.005, the pattern instance, if one wishes to implement an
of mistuning is "almost alternate" optimal mistune pattern on an actual
mistuning. The odd numbered blades have rotor, the actual mistuning pattern which
little or no change from their nominal is implemented will be given by
mass. As the stability margin is
increased, the nearly alternate blade mis-
tunings become more and more apparent.
Upon examination of a number of opti- {92}
mal mistuning patterns such as these, cer- ^specified* **
tain characteristic trends become
apparent. An almost alternate pattern is
evident which serves to reduce the domi-
nant influence of the neighboring blades. where ej is the error in mistuning the
This almost alternate mistuning pattern, rotor. The stability of this actual pat-
however, is usually broken at one or two tern may be significantly less than the
points around the rotor. It is thought one desired, depending on the errors
that these breaks disrupt the communica- introduced.
tion of longer "wavelength" forces, that
is, the smaller but nonzero influence co-
efficients from non-neighboring blades. To investigate this problem, errors
were introduced into the optimally mis-
tuned 14-bladed rotor with a stability
margin of 0.002. The procedure was to
compute the worst case arrangement of the
error and then assess the degradation in
stability due to that case.
1.04
EIGENVALUES OF OPTIMALLY MISTUNED ROTOR
DAMPING RATIO f -0.002
I .03
M
5 1.02 —

MASS MISTUNING VECTORS OF OPTIMALLY

jjj I.OI 309- MISTUNED ROTOR

5 t v
, *Q

SUJ
I. 00 -

.
V*
°,
U.0.99
25
/1 i
5 0.98 / \ 206y^
a. / \ 1 \ 6»»
1 \ 1 1
/ \ 1 1
K 0.97
V:
^

Z / \
/ '\ IB4-V
< 0.96
tuned interblade phase X, ,'e
angle associated with ~C3*
eigenvalue •Va.oo2
0.95
o.ooi
o.o
Stable •«— — >.Un -o.ooi
-0.002
0.94 1 1 1 1 2 3 4 S 6 T 8 9 10 II 12 13 14^ -a oca
-a05 -0.04 -0.03 -0.02 -0.01 -OJOO Ofll -o. oo*
BLADE NUMBER -0.008
REAL PART OF EIGENVALUE. RE(S)

Figure 13. Eigenvalues of Optimally Tuned

Rotor _ Figure 14. Optimum Mistuning Patterns of
C - 0.002 14-Bladed Rotor
19-22

For an RMS scatter of 1 percent in can be shown that for the first increment
mass mistuning, it was found that the of mass mistuning of blades with a single
stability was reduced from 0.002 to degree of freedom, no change in stability
-0.00317 (Figures 15 and 16). The opti- occurs. Thus, on average, a rotor must
mally mistuned rotor is extremely sensi- have several percent mistuning before it
tive to errors in mistuning. begins to exhibit the behavior of a
mistuned rotor.
Hence we have seen that even though
the optimal mistuning is the best possible
mistuning pattern in one sense, that is, Beyond the first few percent in mis-
it requires the lowest level of raistuning tuning, the trend enters an approximately
to achieve a desired level of stability, linear region of sensitivity, that is,
it is clearly not practical to implement a linearly increasing stability with in-
pattern of mistuning which requires very creasing mistuning. Beyond this region,
close tolerances on the natural frequen- one moves into a region of diminishing
cies of the blades. As an alternative, returns. Eventually, the asymptotic limit
consider the case of alternate mistuning. of stability, the centroid of the eigen-
As was shown earlier, this mistune pattern values, is approached and the level of
is not nearly as effective as the optimal mistuning required per increase In sta-
mistuning in terms of required levels of bility rises sharply.
raistuning. However, the pattern is not as
susceptible to errors in implementation as This idealized trend can be used to
the optimal mistuning pattern. The same explain the sensitivity of the optimum
sensititivy analysis was applied to an mistuning patterns. figure 9 shows that
alternately mistuned rotor with a perfect- the optimum cost curve has a very shallow
ly mistuned stability margin of 0.00171. slope in the region of c = 0.002. This
For a 1 percent RMS scatter in mass implies that a small amount of mistuning,
mistuning, the stability margin was re- if introduced correctly, can greatly in-
duced from 0.00171 to 0.00047 as shown in crease the stability of the rotor. But
Figure 16. Therefore, although alternate for the same reason, small errors in mis-
mistuning is not as cost effective as tuning can cause large decreases in
optimal mistuning, it is clearly much more stability. On the other hand, alternate
robust to errors in implementation. mistuning is relatively insensitive to
errors in mistuning but is not nearly
Some insight into this difference in optimal. Thus there is a clear design
sensitivity can be gained by examining the trade-off between the level of mistuning
trends shown in Figure 16. These trends and the robustness of the design.
can be divided into three regions. For
the first few percent of mistuning intro-
duced into the tuned rotor, very little
change in stability occurs. In fact, it

SENSITIVITY OF EIGENVALUES TO
ERRORS IN MISTUNING

--- PERFECTLY MISTUNEO .0 002

— I % RMS WORST CASE ERROR

308*

SENSITIVITY OF STABILITY
MARGIN TO EfWORS IN * ni t
MISTUNE
287' I
— Opting Miiurtng
_._ Alrcrnot. Uithmlug
U— I RMiKilon li> Jiowiit, $»-
1

y ^
^ '<»• Ml%«i.l.«.
Irw
lMill<M Tnndi
^''AifnBtOliC
0.97 MOST SENSITIVE f f- ,' Limit 10
EIGENVALUE f ,.' Mltnwng
I",/
1— **
O96 ^^f*
/ &
0.95 litial /X'V 002
uniilivc *^r /
*v**~^r/ ^s^^1' '»
O94 I I J
f 1 • 1 0 \ 1 l
-0.05-0.04 -0.03 -0.02 -0.01 0.00 0.01 -0005 -0.004 -0.002 0 0002 0.004 0006 0008
REAL PART OF EIGENVALUE. RE ( S) Stability Margin.(

Figure 15. Sensitivity of Optimally Mis-

tuned Eigenvalues to Errors in Figure 16. Sensitivity of Stability
Mistuning Margin to Errors in Mistuning
 19-23

Summary Comments
For travelling rove coordinate*:
In this chapter an attempt has been transform equation 1 using equation 3
made to outline a complete and generalized
formulation for the aeroelastic problem
and its solution. This includes informa- ..Zrr-i-l
tion necessary conventions of the aerody- (AS)
namic and structural dynamic operators.
The most important lesaon to be learned ffpb*
from this review is that in the case of
linear analysis, all of these analyses are
equivalent, and the practicing engineer
should use the one which gives the most
insight into a particular problem. For standing mode coordinates (•In/cos):

In addition, a brief review of the transform «qn. 4 for <ii and prowl tlply by [P]-1
most common trends in stability analysis
was conducted: the destabilizing influ-
ences in cascades, the influence of kine-
matic vs. dynamic bending torsion
coupling, the effects of mistuning, and
the yet largely unmodelled effects of V «n
(A6)
three-dimensionality.

Much of what has been presented in

this chapter, except for the treatment of
explicit time dependent motion in a
cascade, exists in fragments distributed For standing node coordinates (general):
throughout the literature. But, here an
*]*
attempt has been made to unite all of transform eqn. 4 for q. and prenultlply by [4>]
this material in a common formulation, and
reference it to the remaining chapters of
this Manual.
-*>2[4>]T[!l][0]{q;} * [<t>]T[K][0>]{q;} = (A7)

Other useful transforoatlon relationships:

APPENDIX A: SUMMARY OF TRANSFORMATION RELATIONSHIPS

Basic relationships (for single degree of nSl nSl l]«J /

freedom per blade):

Structural dynamics: m
V
(Al)
co
Unsteady aerodynamics:

«p if,
r r
\ 1\ *\
(A2)

.O C0.1 S0.1 %.2S0.2

C
Kinematics: . C1.181.1C1.2S1.2
1.0

-
S.o
{A3}

For blade coordinates: where cos 2D and 8 m sin

transform equation 2 using equation 3

(A4) [P]T[P>[D]= N/2

19-24

•here n = 0 at the leading edge. 1 at the trailing

O
edge, and X a uc/U. The new coefficients are:

(B2)
E
E
c .0 • • • o,»-i ] p _ eJTT~
*
• »•' ~ 2HkC
[E] •

s -!• ,--^,J^-J.^

" £
1 0 0 0 0 0 0 • • •

O j - g j O 0 0 0 • • •
0 0 O j - j J O O ' - ' (B3b)
[E] *[PJ
BT
?]
Force notation used with the t notation
0 0 ® J jj 0 0 • • •
0 i M 0 0 0 0 • • •
The forces and noments are assumed to be
acting at the elastic axis (see figure 3a). and
APPENDIX B: FORCE AKD MOMENT NOTATION the displacements and wake velocities are,

Force notation usually used with the Cg notation

h o h-e^wt : displacement of the reference axis
The forces and moments are Initially defined a =~a*e lut : angular displacement
to be acting at the leading edge (see figure 3b), w s w*e : wake velocity of the reference axis
and the displacements and wake velocities are:
And the forces and moments are written as:

: leading edge velocity

1
^.J* - angular displacement
w-ej : wake velocity at leading edge
(B4b)

The forces and moments acting at the leading By comparison of the two convention*.
edge are:

(Bla)
V-
(Bib)

If the axis of pitch Is shifted to a point TJC

behind the leading edge, the coefficients about c x 2b
this axis (designated by subscript T)) are derived
by considering the transforoatlon in coordinates and with these conventions:
and forces:
 20-1

FAN PLOTTER TEST

by
HANS STARGARDTER
United Technologies Corporation
Pratt & Whitney Aircraft
East Hartford, Connecticut 06108
INTRODUCTION o Resonant Vibration
The object of this chapter is to Campbell Diagram
describe an aeroelastic investigation of
fan flutter. Discussion includes test Avoid known resonant stimuli
procedures for flutter evaluation, data
acquisition and reduction, safety, and o Stability
instrumentation. Some data contents are
included, but are not the primary concern Supersonic unstalled flutter
of this document. Initially, a complete
sequence of preparation, testing and Transonic stall flutter
analysis of an ideal fan flutter test is
presented. This is followed by identi- Other flutter
fication of dangerous blade vibrations, a
case history of a NASA/Pratt & Whitney Test Preparation
Subsonic/Transonic Flutter Study, and a
discussion and analysis of the program o Blade Selection and Acceptance
results. Lastly, a summary relating this Criteria
teat to the overall discussion of fan
flutter testing is presented, it will be Dimensional verification
noted that many of the steps in the ideal
test situation have been omitted in the Frequency checks
actual case history. Most of these
differences are in the test preparation Instrumentation
area because the caae history used a
previously tested design in which much of o Strain Gages
the preliminary work and safety screening
had already been accomplished. Select blades for instrumentation
based on measured frequencies. Place
TEST PROCEDURE FOR A FLUTTER EVALUATION strain gages at locations where strain
is high on modes of vibration
What be measured? anticipated in test (root leading edge
is normally good location).
o Flutter Boundaries
o Number of Instrumented Blades
Locations on compressor map, intensity
gradients, stress gradients Strain gage of high and low frequency
blades
o Response in Flutter
o Mirrors
Frequencies, coherence. modulation,
phase, mode shapes, amplitude varia- Select locations
tions from blade to blade
Decide installation technique
o Blade Running Position
Recognize limitation that optical path
Steady untwist, camber changes, un- must be clear
steady mode shapes, variations from
blade to blade, variations with o Kulites
speed, temperature, pressure
Mount on blade and/or case; concen-
o Aerodynamic - Steady Performance trate on leading edge region where
moat of the flutter action will be.
o Aerodynamic - Unsteady Performance
o Hot Films
Incidence, pressure and velocity
fluctuation, in-passage fluctuations, Blade-mounted hot film good only for
airfoil surface fluctuation qualitative data. Amplitude cannot be
calibrated.
o Rig Safety
o Include Steady State Performance
Vibration, critical speed, tempera- Instruments
tures, tip ruba, sudden changes in
performance or signals o Safety
Pretest Analysis Determine steady state safety limits
o Blade Steady Stress Determine transient state safety
limits
Goodman Diagram
Test Plan
Check for steady-state stresses
o Define number of test points required
Check for safe steady-state deflec- to get satisfactory data for flutter
tions boundaries, resonances, surge, etc.
20-2

o Establish flutter boundaries dangerous conditiona require instanta-

neous reaction, while other situations
allow more time for control. A descrip-
o Establish safe operating regime tion of various types of blade vibrations
associated with excessive amplitudes
o Obtain most important information follows, along with the required test
first; flutter tests can end abruptly control actions.

Test Procedure o Flutter

o Safety and exploring new regimes Flutter is identified on the monitor

scope as a sinusoidal signal usually
During first acceleration, overshoot on all blades but isolated to one
all steady points by a small margin stage. It is probably the moat
to assure a safe condition on all dangerous aeroelastic event, often
blades, including those not instru- associated with rapid increases in
mented . amplitudes. The observed frequency of
flutter is not an integral multiple of
o Accuracy the engine rotational frequency. Safe
test stand practice requires backing
Verify accuracy of test stand obser- off the flutter boundary quickly by
vations by careful playback. reversing the operating changes that
caused penetration into the flutter
o Fatigue Damage boundary. Hysteresis associated with
the flutter boundary is rare.
To assess fatigue damage keep records
of time above fatigue limits, esti- Figure 1 shows typical unfiltered
mate amplitude and number of cycles strain gage responses associated with
above limits, visually inspect blades a high-speed torsional condition.
using borescopee, if necessary, This figure shows a coherent flutter
inspect often for foreign object response, i.e., all blades vibrating
damage, i.e., nicks, gouges, erosion, at the same frequency with the inter-
tip rub burrs and temperature dis- blade phase angle fixed. Rarely flut-
coloration. ter may occur in a leas coherent man-
ner with variable amplitude and vari-
Data Acquisition able phase, each blade vibrating at
its own natural frequency.
o Frequency Response
o Resonance
Recovery of wave forms requires an
order of magnitude margin above Resonance ie excited when the frequen-
highest anticipated blade frequency. cy of periodic aerodynamic forces
If sampling is used watch for loss of (wakes) matches the blade's natural
frequency spectrum. frequency. Some sources of resonant
vibration are inlet distortion,
o Determine calibrations for amplitude. structural struts, instrumentation,
burner cans or nozzles. The blade
o Determine calibrations for phasing response frequency is an integral
(spectral and fast Fourier Transform). multiple of the engine rotation.
Use a common clock on all recorders
by paralleling a prime signal from Resonant vibration can be recognized
the strain gages. on the test monitor by constant
amplitude sinusoidal wave forms that
Strain Gage Signals appear to stand still when the scope
sweeps are triggered by the engine
Each aeroelastic phenomenon has a per-revolution signal. For example, a
characteristic signature or wave form two per-revolution signal displays two
that can be identified on the oscil- full waves on the scope; a six per-
loscope during the test or on playback. revolution shows six full waves. See
The major signals are identified below. Figure 2.

o Flutter - coherent While the only safe reaction to

flutter is to back off, resonant
o Resonant vibration vibration can be avoided by either
acceleration or deceleration of the
Buffeting (also called separated flow test vehicle. How far to stay away
vibration, bending flutter) from a dangerous resonance depends on
the instrumentation. Enough margin
o Rotating stall must be allowed to keep non-
instrumented high and low frequency
o Surge or atall blades out of danger. Usually 5 to 10
percent change in speed in either
o Tip rub direction is sufficient. However, in
the case of a new machine it is
o Bad signals (faulty slip ring, prudent to accelerate to a slightly
breaking gages, etc.) higher speed than the intended, new
data point and then back down to the
IDENTIFICATION OF BLADE VIBRATION intended design speed. This will
assure the absence of resonant vibra-
Rapid identification of blade vibra- tion of non-instrumented high frequen-
tion signals observed on the test stand cy blades that may be excited to
is essential. Certain potentially dangerous amplitudes.
 20-3

HIGH SPEED TORSIONAL FLUTTER

TIP M1REL = 1.6

Pressure
ratio

10 11 12 13 14 15 16
VVVT\A/YVYVYVV\
Corr airflow ~ioibs/sec

Figure 1. High Speed Torsional Flutter

BLADE RESONANT VIBRATION OSCILLOGRAPH

FROM STRAIN GAGE
4E primary signal with weaker 2E

Strain gage signal

Signal with small component

of higher frequency

Speed signal I/revolution

Time code

Figure 2. Blade Resonant Vibration Oscillograph

20-4

o Buffeting frequency variation around the rotor.

Buffeting, also called separated flow Instrumentation and Data Aequaition
vibration or bending flutter, is an System
irregular motion of the blades
excited by turbulence in the flow Instrumentation provided full docu-
field. It usually shows on the scope mentation of both steady-state and non-
as a first mode response with steady aerodynamics and for rotor struc-
amplitude varying randomly. Buf- tural behavior in and out of flutter.
feting is associated with high The location of instrumentation is shown
incidence angles and often occurs in Figures 6 and 7. Table V lists the
toward the stall line or preceding instrumentation and readout systems.
flutter. It ie alao common on front
stages of multistage compressors at A fully computerized steady-state
part speed. data acquisition system was used during
testing. Data were transmitted to a
This vibration is not necessarily computer located in the control room and
dangerous because penetration into then to an automatic data reduction
the buffeting regime normally in- computer which performed preliminary data
creases slowly. However, occasion- reduction and returned the results to the
ally a threatening situation occurs control room for use directing the test
when the buffeting shifts rapidly program. Positioning and readout of all
into true torsional coherent flutter. the traverse probes at the rotor inlet,
Typical responses to this condition rotor exit, and stator exit were con-
are shown in Figure 3. trolled by the rig automatic traverse
system.
o Rotating Stall
Unsteady data recording systems for
Rotating stall, although often men- the Kulites, hot films, and strain gages
tioned as a cause of blade vibration, recorded a common 1:1 speed signal,
is rarely a threat. Blade frequen- common time code, and a common strain
cies are usually too high for one- gage signal. Before the start of test-
sector excitations, and multi-sector ing, a common sine wave and white noise
excitations provide only transient signal were recorded on all system
weak stimuli. Therefore, the required channels in order to calibrate frequency
excitation-response conditions do not and phase response.
often exist for a rotating stall to
cause blade vibration. Data from the blade mirror system
were recorded on still photographs, movie
FAN FLUTTER TEST, A CASE HISTORY film and video tape.
Fan Rig Description Steady-State Aerodynamic Instrumentation
A stall flutter test was conducted Hedge probes measured total pres-
on a single stage fan research rig. The sure, static pressure, and air angle.
fan rig combined moderate tip speed with Combination probes measured total pres-
a high pressure ratio, high flow rate sure, static pressure, air angle, and
per unit annulus area, and good total temperature. Wall taps were used
efficiency. The 3.6 blade aspect ratio for measuring wall static pressures, and
was aeroelastically aggressive even with total pressure rakes were used for
a single partspan shroud. The stage, measuring stator exit total pressures.
with its 81.8 cm (32.21 in.) rotor tip
diameter, was large enough to permit The pressures sensed by the probes,
good definition of the unsteady flow, fixed rakes, and static taps were
blade deflections, and mode shapes measured by transducers and recorded in
during flutter. millivolts by the automatic data acquisi-
tion system. The accuracy of the pres-
The rig is schematically represented sure measurement was ±0.1 percent of
in Figure 4. Stage design parameters are fullscale value. All temperatures were
listed in Table I, and rotor blade measured with Chromel-Alumel, type K
specifications, in Table II. thermocouples connected to reference
junctions attached to uniform temperature
Blade Inspection reference blocks located in the test
cell. Temperature elements were cali-
Blade leading and trailing edge brated for Mach numbers over their full
angles were determined. The maximum, operating range. The thermocouple leads
minimum, and average edge angles at were calibrated for each temperature
fourteen span locations are compared with element. Overall rms temperature ac-
design values in Tables III and IV. curacy wae estimated to be ±0.56K
(1.0'R).
The second mode bending frequency of
the isolated blades was considered to be Compreaoor speed was neasured using
representative of blade flutter frequen- an impulse-type pickup, which counted
cies. Therefore, the second mode frequen- time. The data were recorded through a
cy with clamped root and unrestrained frequency-to-DC converter. Accuracy was
shrouds was measured for each of the ±1 rpm.
thirty-two rotor blades. The minimum
measured frequency was 241 Hz; the Airflow was measured with an
maximum. 249 Hz. The blade positions in orifice calibrated to International
the rotor (Figure 5) were selected to Standards Organization/DIS 5167
minimize differences in frequency between Standards. Accuracy of the airflow
adjacent blades and provide a smooth measurement was within one percent.
 20-5

TABLE I TABLE II

TS22 FAN STAGE DESIGN PARAMETERS TS22 BLADE DESCRIPTION

Aerodynamic Corrected Design Speed 11,042 rpn

Airfoil Series Multiple Circular Arc
Pressure ratio Aspect Ratio 3.6
Rotor 1.702 Taper Ratio 1.5
Stage 1.67 Tip Speed 472.4 m/sec
(1550 ft/sec)
Ad1abat1c Efficiency
Rotor 0.871 Root Dimeter
Stage 0.838 Inlet 26.2 en
(10.3 1n.)
Corrected Flo* 95.56 kg/sec
(210.67 Ion/sec) Tip Diameter
Inlet 81.7 cm
Specific FloM 202.78 (kg/sec/m*) (32.2 In.)
(annulus at rotor Inlet) (41.53 Ibm/sec/ft*)
Exit 79.7 en
Geometric (31.4 In.)
Rotor Tip Diameter 0.8178m Beta !*(«)
(2.7 ft) Root 54.999 deg
T1P 27.0399 deg
Number of Blades 32
Beta 1* Suction Surface^)
Hub Solidity 2.60 Root 48.503 deg
Tip 25.398 deg
Tip Solidity 1.315
Chord Length
Hub/Tip Ratio 0.32 Root 7.47 cm
(rotor leading edge) (2.94 1n.)
Partspan Shroud Location 62 Tip 10.42 cm
(percent span from hub) (4.10 1n.)
Notes:
(a) Beta 1* Is the leading-edge metal angle. U *. the angle between the
tangent to the mean camber line and the meridional direction.
(b) Leading-edge metal angle ba'sed on suction surface.

Type of vibration Characteristics Wave form

Resonance Multiple of engine order
Sinusoidal
Individual blade response
occasionally full stage response
MAAAAAA,
Flutter Not multiple of engine order
Sinusoidal

Usually ill blades at tame frequency

MAAAAM,
Buffet Not multiple of engine order Random amplitude

Individual blade response usually first

bending node

Figure 3. Identification of Blade Vibration

20-6

. .

)20 MIRRORS LOCATED AT (3)« INNER WALL STATIC

7 RADIAL. ^ O S i T i O N S PRESSURE TAPS 4 ID ANO OO WALL STATIC
PRCSSUC PRJC35UR.C TAPS
@ T O T A L . PRESSURE. STATIC
70 STRAIN GAQCS LOCATED ON V A R I A T I O N IN
PRESSURE AND f LOW ANG Ll
10 DIFFERENT BLADES
1» SLAOE MOUNTED 3WIOOI P A O B E S T R A W E R S E O LOCAL MASS FLOE FLOW
HOT FILMS TO T RADIAL POSITION* I vj)

1 STRAIN GAGE PER BLADE • v ] HOT FILM PROBES

T)l COMBINATION PHOBIS TMAVCHSIO TO TRAVERSED TO
32 MICH RESPONSE PRISSURC T R A D I A L POSITIONS
J R A O t A L POSITIONS
TRANSDUCERS ONI, DIFFERENT
SLAOCS LOCATED TO OBTAIN 3 DUAL TOTAL TIMPf MATUME AND
MEASUREMENTS AT T RADIAL TOTAL PRUSURE PROBES WITH
KICL MtAOEO SENSORS LOCATED AT ft 2 WEDGE PROBES T R A V E R S E D
POSITIONS TO 7 RADIAL POSITIONS
RADIAL POSITIONS. THESE AMf
LOCATCD APPROXIMATKLY 130°
'I HOT FILM PROBES T R A V E R S E D Ar>»*r IN A ClNCUMFEAINTIALLV. j l O KULlTE
TO 7 RADIAL POSITIONS MOTATABLC T R A V f R S i RiNQ THAT SINSORS
CAN BK POSITIONED TO PROVtOf AT
TOTAL AND ITATi LIAST 11 T O T A L PRIMURS AND 10 STATIC
TOTAL TIMPf RATUM. FLOW ANOLl TOTAL TIMPi»ATuRE REAOiNOC PRISSURt
ON ONE COMBINATION PROBE ACROSS A ONE ID STATOR VANf TAP*
T R A V E R S E D TO 1 R A C I A L POSITIONS SAP AND TWO 00 STATOR VANE
OAPS
3 WCDOE PROBES T R A V E R S E D TO )* OUTER WALL STATIC
7 RADIAL POSITIONS TAPS

AOQITIONAL INSTRUMENTATION

GEARBOX ROTOR ROTATIVE 3 iMPULSI T V P E PICKUPS

SMtQ '
tNLBT DUCT CALlBRATCOORIFlCi
FLOW RATI
ROTOR I N L E T E WALL STATIC PRESSURE
TOTAL PRESSURE TAPS LOCATED IN TH«
PLENUM CHAMBER
TOTAL TCMPIRATURf
E BASE WIRE CMROMBL ALUMt L
THERMOCOUPLES LOCATIO IN THE
PLENUM CHAMBER

Figure 4. Schematic Diagram of TS22 Rig

 20-7

TABLE III

LEADING EDGE ANGLE

BLADE INSPECTION RESULTS

Design
Percent (a) Value Minus
Span Minimum Maximum Average Value
0.08 47054- 47040' 48012' 47050 OOQ4'
0.12 46010' 45024' 460Q' 45044 0026'
0.22 42054' 42015' 43012- 42042 0012'
0.32 40° 54 i 39028' 39046- 39037 0030'
43 3/058- 37018' 37040' 37029 0029'
52 36052' 3508' 1 370Q' 36029 0023'
55 3704- 360 SO 3700' 36055 009-
58 37010- 36010' 36028' 36018 0052'
62 35026' 350Q' 3508' 1 3502' 0024'
66 34036- 33030'1 34054 34015 0021'
72 320Q' 31048 31052' 3105' 0°55'
82 28050' 28024' 28044- 30029 0014'
92 25024' 25041 25050' 25029 -0005'
0.99 23010' 22056' 2304' 22058 0012'
Note: U)Percent Span From Hub

AXIAL DIRECTION

LEADING
EDGE
ANGLE

TABLE IV

TRAILING EDGE ANGLE

BLADE INSPECTION RESULTS

Design
Percent(a) Value Minus
__.Spa_n__ Minimum Maximum Average Value
0.08 96050' 95050' 9602' 95054 0056'
0.12 87° 10' 86014' 86042 ' 86028 0°42'
0.22 69056' 690Q- 70016' 69° 37 0°29'
0.32 58°28' 57058' 58034' 58014 0°14'
0.42 49«8 ' 48°42' 49°0' 48°40 0°18'
0.52 44034. 43044. 44040' 4407- 0027'
0.55 43042 ' 43024' 43030' 43028 0014'
0.58 43014' 42022' 42046' 42037 0037'
0.62 410Q' 4004' 40028' 40015 0°48'
0.66 39042' 4904' 39052' 39025 0017'
0.72 350 28' 35026' 35038' 35032 004'
0.82 30028' 30014' 30048' 30029 Q01«
0.92 24024' 24024' 25°0' 2 4°41 -OP 17'
0.99 21058' 21038' 21056' 2P44 0014'
a
Note: ( )Percent span from hub
20-8

20 MIRRORS

NOTE
ALL BLADES HAD ONE
ASMT STRAIN SAGES

Figure 5. Circumferential Location of Blade Instrumentation

:-:_ ..
IL .|, !
._ + rr
.... . i
_ ._ J ... i
.i
'
4 - -H 1 « f • MAM «D. i

'7777/7%
...|... *_
h.^.* .
. .^4. —
r f * -

j\\ n
I) m rti m
\\\\\
flf
., MrtlMMMNaMMt

0 «MA n*ne RU
» *u» MMMIB KUUTI

Figure 6. Circumferential Schematic View of TS22 Compressor Instrumentation

 20-9

LOCATION OF KUUTE TRANSDUCERS

ON BLADE SURFACE

Blade Percent Probe Location Percent . Chord) b)

No. SpanTa) Pressure Surface SUC1 .Ion Sur ace
2 76.4 5 IS 25 40 65 90 5 25
3 76.4 5 25 5 IS 25 40 65 90
4 86.3 S 15 25 40 65 90 5 25
5 86.3 5 25 5 15 25 40 65 90
6 86.3 50
Note: (a) Percent Span Fran Hub
(b) Percent Chord Fran Leading Edge

RADIUS
78.4% SPAN

RADIUS
86.3% SPAN

Figure 7. Installation Locations for Blade-Mounted Kulite Pressure Transducers

PLENUM CHAMBER
/
STILL CAMERA
SCREEN

MOVIE
CAMERA j_

TELEVISION

USER
LASER IN PORT
BEAM SPLITTER
Figure 8. Schematic of TS22-X204 Laser Configuration
20-10

TABLE V
INSTRUMENTATION AND READOUT EQUIPMENT
Non-Steady Instruments Recorded
32 Strain gages
24 Hot films - blade mounted
4 Stationary hot film probes
10 Wall Kulltes
3j2 Blade mounted Kulltes
102 Sensors
Recorders
1 70 channel multiplex
2 9 channels Sangano
3 11 channels Sangamo
4 12 channels strain gage console
5 4 channels strain gage console
106 channels

Each of the five recorders had strain gage 3 1n parallel as a common signal to
permit time correlations between any of the 102 sensors.

TABLE VI
HIGH RESPONSE INSTRUMENTATION SPECIFICATIONS
Kullte - Model XCQL-8V-80B
Rated Pressure: 17.24 N/onZ (25 lbf/ln.2)
Sensitivity: 3.8 x 10* mV/N/m* (2.62 mV/lbf/ln.*)
Temperature Compensation 278K to 422K (40°F to 300°F)
Acceleration Sensitivity:
Traverse 0.00004* Full Scale Gage
Perpendicular: 0.0002X kHz
Natural Frequency 230 kHz
Non-Linearity and Hysteresis: +0.75X full scale maximum
Kulltc =.Model LQL5-080-25S
Rated Pressure: 17.24 N/oj2 (25 Ibf/ln.*)
Sensitivity 3.8 x lO-* HV/N/riZ (2.62 mV/lbf/1n.2)
Temperature Compensation: 278K to 422K (40°F to 300°F)
Acceleration Sensitivity:
Transverse: 0.00008X Full Scale Gage per g
Perpendicular: 0.00042 Full Scale Gage per g
Natural Frequency: 125 kHz
Quartz Hot Film
Thermo-systems model 1210-60
0.0154 cm (0.006 In.) quartz rod with platinum sensor deposited 0.203 on
(0.080 In.) between posts
Temperature coefficient of resistance » 0.0026 ohm/ohm-*
Frequency Response at 91.44 m/sec (300 ft/sec): 200 kHz
 20-11

TABLE VII
TS22 NASA fLUTTB TEST
TEST HATX11

Percent Rotor
Run Speed Point Unsteady Percent Corrected Pressure
Muaber Code Hunter Record Speed Oeslqn flax Ritlo
Snikedowi
AH Hlrrort
001 70 01 20-27 70 72.8 1.228 UldB Open OUcrurgt

" -• -- 28 70 -- -- doting Discharge Valve, Transient Into

Flutter

001 70 OS 43 70 66.9 1.2722 Check Point Wide Open Discharge:

-- « — 64T -• -- -- Trintlent
003 CI 01 71 63 65.6 1.1776 uldt Open Dticturge
003 63 03 76-84 63 53.4 1.2374 Stress Level Fluctutttng. SfukedoMi
Coopleti
Perfor<unc4
All Mirrors
003 73 01 SS-94 73 74.7 1.2587 Wide Open Discharge
003 73 01 100-108 73 60.0 1.1117 rUitmua Flutter
1 Uatt User
4 ROMS Mirrors (Above Shroud)
004 70 03 128-135 70 59.8 1.1004 70t UM Flutter Point
004 67 01 136-143 67 68.5 1.18M Ulde Open Discharge
004 7S 01 176-183 75 75.3 1.2840 Hide Open Otsctwrje
004 7S 04 195-202 75 60.3 1.3169 Hulnji Flutter
005 73 08 220 70 56.5 1.2978
All Mirrors
3 Witt Laser
007 66 01 at 66 54.5 1.26 Nulaui flutter
007 60 01 242-249 SS 52.0 1.1530 Mtr Surie (Rotating Still)
007 as 02 279-286 'as as. e 1.3792 wide Open Discharge
- - - 287 - " - Transient To Surge
007 85 0$ 268-29$ as 75.1 1.4662 Near Surge

i l l '

BLADE MIRROR LOCATION z 86 } » : f

BLADE Z 76 ? v v
NUMBER PERCENT SPAN (») PERCENT CHORD (b) u
UI
4

ee
2
3
86.3
86.3
50
50 S 66 p 1? SHROUD
£ y
4 66.0 50 M I? V
S 66.0 25 | 55 —
9 95.2 5 25 50 70
86.3 5 25 50 70
76.4 5 25 SO 47 '/ v
66.0 S 25
S5.0 S 25 50
47.0 5 25
38.0 S 38 V
20.0 5

Note: (a) Percent Span From Hub TIP

20

HUB

Figure 9. Mirror Installation Locations

20-12

Structural Instrumentation loscope beneath the output window was

photographed simultaneously with the
Blade deflections were measured by light spots to correlate the time code
a mirror system. Steady state and and rotor speed signals with the mirror
unsteady deflections were recorded using deflection.
the techniques described in Stargardter
(1977) and Stargardter (1979). One
strain gage was placed on each blade to Strain Gage on Blades
measure unsteady stresses and to the on-
set of flutter. To ensure safety and to measure the
response of all blades in flutter, one
Optical Mirror System strain gage was installed on each rotor
blade immediately above the shroud at the
Rotor blade deflections were mea- midchord location. This location was
sured both in and out of flutter. During selected because it is sensitive to the
stable operation these deflections were second coupled mode expected in flutter.
due to centrifugal and aerodynamic Signals from these gages were transmitted
forces. When in flutter, vibrational through a slipring and recorded on
mode shapes had to be determined. A magnetic tape.
patented system Stargardter (Patent
4080823) of optical mirrors and reflected The frequency response of the strain
laser light was used to measure rotor gage data was limited by the bandwidth of
blade surface angle changes from which the data recording system. The accuracy
blade deflections were determined. of any given strain gage was statistic-
ally determined to be approximately ±5
The optical mirror system consisted percent, representing one standard devia-
of an array of mirrors installed on the tion for typical data.
blades, a laser light source, and a
readout and recording system, shown Unsteady Aerodynamic Instrumentation
schematically in Figure 8. The laser
light was split into several beams, each The instrumentation for measuring
directed to the radial location of one unsteady aerodynamic effects included:
of the mirrors. Variable beam splitters 1) Kulites on the case over the rotor
were adjusted to provide equal intensity blade tips, 2) an array of Kulite high
of all the light beams. Each splitter response pressure transducers on the
was mounted on a five-axis adjustment rotor blades, 3) traversing hot-film
system to facilitate accurate aiming of probes at the inlet and exit locations of
the light beams. The laser light and the rotor, and 4) an array of hot-film
mirror system for measuring the instan- anemometers on the rotor blades.
taneous blade surface positions used a
604.5 cm (238.0 in.) optical path up- The fan case Kulites and the hot-
stream of the test stage inlet. This film probe outputs were recorded on two
path was long enough to provide light wide band FM tape recorders having a
spot deflections to detect blade motion capacity to 40 kHz center frequencies and
of 0.1 degree rotation in bending or 40 kHz output filters. In addition to
torsion. As the instrumented blade the data signals, 60:1 and 1:1 speed
rotated through the beams, the mirrors signals were recorded along with 1 kHz
reflected the beams back to the readout IRIG B Format Time Code Signal.
system to separate points that had been
selected to avoid pattern interference Data from all blade-mounted Kulites
during flutter. were recorded on a constant bandwidth
frequency-division multiplex system,
The mirrors were attached to the which provided a frequency response of2
blades with epoxy cement. Additional 2 kHz with a resolution of 690 N/ra
details of the mirror system and (14.4 lbf/ft2) and an error no greater
mounting techniques are given in than il dB. Gach of the 12 tracks on the
Stargardter (1977). Twenty mirrors were system recorded six data signals plus a
installed on one blade to provide full time code signal. One data channel on
coverage of the blade in the region each track was used to record a reference
above the midspan shroud and along the strain-gage signal for phase determina-
leading edge below the shroud. Addition- tion. The hot-film anemometer data were
al mirrors were mounted on other blades. also recorded on this multiplex system.
The complete array of mirrors is shown in
Figure 9. Circumferential locations of Case and Blade Mounted Kulites
instrumented blades are shown in Figure
5. Thirty-two Kulite, high-response
pressure transducers were distributed
Two windows were installed in the over the pressure and suction surfaces of
inlet section walls of the test facility: four blades (Figure 7) instrumented in
one to provide laser light entry; the pairs such that the instrumentation
other to provide a screen for receiving locations on the suction surface of one
the reflected light beams. The interior blade matched those on the pressure sur-
of the inlet was frosted in selected face of the adjacent blade across the
locations to prevent secondary reflec- flow channel. Transducer specifications
tions from interfering with the laser are shown in Table VI. The data from
beam signal. these transducers provided detailed
mapping of the pressure fluctuations
The optical data were recorded with during flutter.
a movie camera, and video tape. The
cameras were placed to receive the Signals from the rotating Kulite
maximum scattered light when the beams transducers were amplified before pas-
were on the output window. A oscil- sing through the slipring to the record-
 20-13
ing device by an amplifier package which VII. Flutter boundary was defined as the
rotated with the rotor assembly. flow 2 at which a vibratory stress of *2068
N/cm (t3000 lbf/in.2) was attained.
The rotating Kulite pressure trans-
ducers were calibrated both before and
after mounting on the blade surfaces. Data Reduction Procedures
The accuracy of the calibration facility
was 0.1 percent full scale over a range Parameters calculated for all data
of zero to 345 N/cm2 (550 lbf/in.2). points:
Hot Film Anemometers o Overall stage performance
Radially traversing hot-film probes, o Blade element performance
two located at the inlet and exit of the
rotor, respectively, measured fluctua- o Blade untwist and uncamber
tions in inlet and exit flow during
stable and flutter operation. The hot- o Flutter frequency on all blades
film probes were calibrated at ten
different flow velocities. Hot-film o Blade stress level on all blades
anemometers were located on the rotor
blades to characterize the flow over the o Vibratory mode
blade surfaces in and out of flutter and
during transition from stable flow to o Pressure contours over the blade tips
flutter. The sensors were oriented with
their length tangent to the case and o Incidence angle at seven radial
perpendicular to the rig axis. stations
The frequency response of the o Reduced velocity for seven radial
probes and the associated data acquisi- stations
tion system was 40 kHz with a resolution
of 1.0 percent of the mean flow More extensive data reduction was
velocity. The dynamic accuracy of the performed for six of the data points.
probes was 1 dB for axial Mach numbers This reduction provided:
below 0.4. The accuracy at higher Mach
numbers was less because of a loss in o Analysis of the mirror data for
sensor linearity caused by flow compres- amplitudes and phase relations on one
sibility. blade correlated with strain gage
signals and other non-steady signals
Twenty hot-film sensors were in- to define the blade and stage mode
stalled on four blades above the shroud shape and its relationship with the
at positions corresponding to those of instantaneous aerodynamics.
the blade-mounted Kulite pressure trans-
ducers. o Average steady pressure distribution
at the wall for two blade passages.
The film with its polyimide backing
was mounted on a 0.041 cm (0.016 in.) o Unsteady pressure distribution at the
Kapton film substrate to minimize heat wall for two blade passages.
transfer to the blades. The grid was
oriented in the direction of flow with o Amplitude and phase angles of all
the leadwires routed off the trailing fluctuating signals from rotor-mounted
edge to avoid an additional turbulence sensors — both amplitude and phase
source. angle were determined relative to the
reference signals from the No. 3
The sensors were calibrated to iden- blade.
tify strain induced errors, but their
nonlinear response and the difficulty in The blade centrifugal untwist and
simulating the test situation in the uncamber resulting from centrifugal and
laboratory made calibration to obtain gas bending loads were determined
quantitative data unrealistic. There- directly from the blade deflection data
fore, their function was limited to obtained with the otpical mirror system.
qualitative characterization of the Images from twenty of the twenty-six
flutter. However, comparing the flutter blade mounted mirrors were used to
response from one data point to another determine blade movement.
gave useful data on suction surface
nonsteady flow characteristics. These The steady pressure contours in-
sensors and associated data acquisition dicated that flutter may alter the blade
system provided a frequency response to 2 passage pressure distribution.
kHz.
The contour maps of unsteady
Tests pressure amplitude revealed regions of
high amplitudes near the leading edge,
Overall aerodynamic performance and lower amplitudes near the trailing edge,
high response aerodynamic and aeroelastic and nodes near the midchord position.
data were obtained at all operating
points, and surge points were determined
at several speeds between 63 and 85 Steady Deflections
percent of design.
Steady deflections were determined
Testing involved mapping the extent for speeds from 25 to 85 percent of
of the flutter boundary by taking data at design. Photographs taken at selected
operating conditions both in and out of speeds in nine percent increments were
flutter. Test points are listed in Table used to determine steady blade movement.
20-14

The vertical position of each spot on orders, m , of blade vibration where

each photograph was measured. Using the 1 <m < N .
idle spot positions as a baseline, the
movement of the spots for any speed was
scaled. This movement on the screen was
then converted to angle of twist and Pm(x,y,t) - I ?,„„ (x)exp } (1)
change in bending slope.
Unsteady Data Reduction when n , the number of full waves per
intrablade passage, is added to the
The reduction of the high frequency fractional number represented by m
response data from the hot-film probes, waves around the full circumference.
wall-mounted Kulites, blade-mounted hot- u is the flutter frequency common to all
film sensors, blade-mounted Kulites and phenomena in the rotating system. The
blade-mounted strain gages was done by unsteady periodicity condition defines
signal enhancement, signal phasing from the wave number.
rotating instruments, and signal phasing
between rotating and stationary instru-
ments. smn • " s 2nn (2)
Signal Enhancement where the interblade phase angle,
The signal enhancement, a time
domain technique, extracted or enhanced (3)
particular frequency components from a
broadband signal. The technique in-
volved averaging numerous time segments and blade spacing, s <* 2trr
of a broadband signal, the start of each
segment being triggered by a reference where r is radius
signal. Each successive time segment was
summed and averaged in a storage memory.
The result was an enhancement or hence *.„ - ro * Nn '
reinforcement of those components that
were synchronous with the triggering In the stationary system, the coordinates
signal and suppression of components that x' and y' are related to their
were not. rotating counterparts by
Phasing of Signals from Rotating Instru- x = x1
mentation (4)
y « y 1 + flrt
Phase information between the
reference strain-gage signal on the where 0 is the rotor speed. Equation
Ho. 3 blade and all other strain-gage (1) then becomes
and rotating Kulite and hot-film signals
was produced at the flutter frequency, Pm<x' , V' , t)
using cross spectral density techniques. (5)
The analysis range fpr the task extended
to 2 kHz. An 800 element spectral reso-
lution was selected. To produce each
final plot, 128 sweeps from the analyzer
were averaged. The analysis was con- This is in the form of waves having wave-
ducted from zero to 2000 Hz. The ana- length 2ir/Bmn , moving at velocity
lyzer filter bandwidth yielded a spectral (flr + -/Bmn) • Tne frequency us mea-
resolution of about 3.75 Hz. All result- sured by a stationary probe is the
ing phase angles were corrected for product of the wave number mn and the
errors introduced by the signal condi- wave velocity. Hence
tioning and recording systems.
Bmnnr flNn + am (6)
Phasing of Signals Between Rotating and
Stationary Instrumentation where flm is a multiple of shaft speed,
and nNn is a positive or negative
Phasing between signals from the multiple of blade passing speed. There-
rotating blades and the stationary wall fore, the single flutter frequency, u ,
instrumentation was determined using a in the rotating system becomes a
variation or extension of the cross- spectrum when detected by a stationary
spectral density technique. sensor. The observed frequencies, us ,
are spaced at multiples of shaft speed,
The vibration of the individual 0 . The index m identifies the
blades in a stage in flutter is fully associated harmonic wave component of
defined by the sum of a finite series of blade vibration and the index n
circumferential harmonic waves where the identifies the added number of waves
number of component waves equals the within a passage between adjacent
number of blades, N , in the stage. The blades.
associated unsteady pressure, p , at a
particular axial coordinate, x , varies Because flutter was seen at many
with tangential coordinate, y , and time, frequencies by the case-mounted Kulites,
t, is described by the sum of an infinite phase information could not be produced
series of forward and backward rotating directly. Instead, an aliasing tech-
harmonic waves having all integer numbers nique was used. By selection of the
of cycles around the circumference of the
stage. This function is epxressed in
terms of sets of responses to individual
 20-15

sampling rate to equal the rotational Blade mode shape was determined by
frequency, both the rotating and analysis of the laser optics mirror data.
stationary transducer flutter signals Blade deflection amplitudes were
were transformed to a new coordinate determined from the mirror data. For the
system in which a single flutter blades without mirrors, the deflection
frequency existed for both sets of amplitudes and relative phases were
signals. Phasing of the signals in determined through correlation of the
question could then be performed. The strain gage data and the mirror data.
one-per-revolution speed pip was used as
the sampling rate command Typical still photographs of the
mirror data in and out of flutter are
Two different procedures were used shown in Figure 10. The difference in
to produce phase information. One width of the same spot in the two images
procedure was to allow all the flutter is proportional to the torsional
components in the stationary signal to amplitude and the difference in height la
be aliased. The other procedure was to proportional to the axial component of
isolate individual spectral components the bending slope.
of the flutter with a narrow filter
before aliasing. This latter technique The 16mm film record of the
extracted a single nodal diameter signal reflected laser beams was digitized using
plus its harmonics at multiples of the a Spatial Data Systems Scanner. The
rotor speed. In all cases, the sampling measurement accuracy was better than
command was properly conditioned to ±0.00254 cm (0.001 in.). The data were
allow the rotor to be in a selected stored on magnetic tape for computer
orientation before a data sample was processing. A fast Fourier transform was
taken. Corrections to the final phase used to convert the data from the time
angle were included for influences of domain to the frequency domain. This
all signal conditioning and the aliasing procedure allowed the calculation of
process. power spectral densities and cross-
spectral densities to determine amplitude
and phase angles for the different
Strain Gages mirrors.
Each of the 32 blades was Instru- Case Mounted Kulites
mented with one dynamic strain gage
located near the maximum thickness point The case-mounted Kulites were used
above shroud at 64 percent span. The to obtain nodal diameter patterns present
stage flutter response was obtained from in the rotor system during flutter,
the strain-gage signals consisting of contour maps of the pressure distribu-
amplitude, frequency, and phase. The tions over the blade tips during stable
amplitude and frequency characteristics operation, contour maps of the unsteady
of the individual blades were obtained pressures during operation with flutter,
from power spectral density (PSD) plots and contours of the real and imaginary
from 0 to 2 kHz. Phases relative to the components of the unsteady pressure and
gage on the No. 3 blade were obtained by relative phase during flutter.
using the cross spectral analysis tech-
nique described above. The strains were The nodal diameter patterns in the
of the form Sveiut where the complex rotor system during flutter operation
number, S v , representing the strain in were determined through Fourier analysis
the No. 3 blade, defined phase as well as of the signals from the case-mounted
amplitude. The Sv numbers, where l£v<N , Kulite pressure transducers.
may be represented by the finite summa-
tion The contour maps of pressure
distributions over the blade tips were
obtained from wall Kulite and wall static
12irmu (7) tap data. The one-per-revolution speed
°m signal was the reference signal used in
the enhancement. Data from 512 rotor
where is the amplitude of a series revolutions were averaged to produce the
of patterns having numbers of lobes, m , final plots. The enhancements were
where l<m<N , rotating with respect to timed to allow a selected group of
the disk~a<? speed u/m . From the known blades to occupy a desired orientation
amplitude and phase of each strain gage, relative to the wall Kulites. These
Sr , the complex coefficients, m , of the enhancement techniques produced a signal-
series in Equation (7) may be determined to-noise improvement factor of about
by mathematical inversion to give the 22.6
strength of the mtn modal component or
spatial harmonic and its phasing with Plots of pressure versus time were
respect to all other components. digitized to obtain an array of pres-
sures representing the variation from the
The broadband and flutter frequency mean at the specific axial location. A
amplitudes for all strain gages and minimum of ten samples per blade gap was
rotating Kulites were plotted versus time digitized. The time location of each
to help establish the stability of the pressure sample was translated into a
data during the two-minute steady-state rotating frame, with the leading edge of
records. The plots were also used as a the No. 2 blades used as the zero
cross-check with the power spectral reference. The wall mean static pressure
density curves to help identify possible for each axial location was added to the
errors in engineering unit conversions. local variation to obtain the steady
state pressures. The array of local
Mirrors static pressures was input into a contour
plotting package, which linearly inter-
20-16

polates in space to find specified levels (i.e., Fourier decomposition).

of pressure. The lines of constant pres-
sure were normalized as percentages of o The net aerodynamic energy of a node
the maximum local steady-state static Is the algebraic sum of the aero-
pressure sampled, and contour maps of the dynamic energy associated with each
constant percentages of pressure were harmonic response.
machine plotted.
o The susceptibility of a rotor vibra-
When the contour maps at the blade tion mode to flutter is a function of
tips were plotted, the pressure field the stability of the individual
with respect to the blade leading edges harmonic responses.
was observed to be shifted about three
degrees tangentially in the direction of An aerodynamic damping exists for
rotation, corresponding to a time delay each harmonic response. This damping is
of about 30 microseconds. However, this defined by the log decrement parameter,
shift, which was nearly independent of which is proportional to aerodymamic work
rotor speed, did not appear in the divided by kinetic energy of the harmonic
nonsteady pressure plots obtained from
the same data. The shift is, therefore, 5 aero
believed to have resulted from the data
reduction procedure used to obtain the
steady-state plots. Although the source
was not found, the location of the For positive values of the aero-
blades was evident from the plots. Each dynamic damping parameter, the energy
steady pressure plot in the report has flow is from the structure to the flow
been corrected to place the blades in stream and in the reverse direction for
the proper positions. negative values.
Blade-Mounted Kulites The aerodynamic work per cycle done
by each of the individual harmonics is
Blade-mounted Kulites provided un- computed by integration.
steady pressure amplitude and phase
distribution for both the pressure and 2* dh
suction surfaces of the airfoil at two w_ d(ut)db
radial positions. Amplitudes were
determined from the power spectral
density for each signal over a frequency where:
range of 0 to 2 kHz. The power spectral
density data were confirmed by backup &pm * pressure jump across the airfoil
plots of amplitude against time during from the mth harmonic and is of the
the two-minute test period. Cross- form 4Pra • Pm (x,yp,t)-Pm (x,ys,t)
spectral density functions were used to
determine the phasing of the pressure where:
signals relative to the strain-gage
signal from No. 3 blade. yp and ys are blade surface coordi-
nates on the pressure and suction sur-
Blade-Mounted Hot-Film Sensor faces at axial location x
Blade-mounted hot-film sensors hm • deflection normal to the airfoil
provided air velocity measurements on surface of the mth harmonic
the blade suction and pressure surfaces
at two radial positions. The flutter (ut) = position angle during the vibra-
response from these sensors consisted of tion cycle
frequency, amplitude, and phase. Am-
plitude and frequency were obtained from b = chordwise location
the power spectral density for each
signal over a frequency range of 0 to 2 u = flutter frequency
kHz. The measured amplitude had a
repeatability of ±20 percent, making it t - time
possible to relate the data from one
point to another. The strain gage on the The pressure at the airfoil tip at
No. 3 blade was used as the reference for any axial location is the summation of
determining phase angle. the distribution resulting from indi-
vidual blade harmonic motions.
Blade-Work Interaction Calculation
N
The flutter characteristics of the P(x,y,t) Pm(x,y,t,)
test rotor and the types of data obtained
allowed for an evaluation of a theory of
energy transfer that takes place during The pressure of each harmonic is
flutter. The assumptions associated with defined by a Fourier series described
this theory are: above .
o Self-excited vibrations occur in a i(ut + B y)
bladed rotor when the energy supplied Pm(x,y,t) mrr
by the air stream exceeds the energy
dissipated through the structural
damping associated with that mode. These pressure waves are translated
into the stationary system using the
o The complex rotor vibration mode can relationships given above.
be defined aa a summation of simple
circumferential harmonic responses
 20-17

Pm(x',y',t) surge, and the points of flutter initia-

tion as well as the surge were
(m + Nn)Q)t + Bmny'] determined. Data along the 70 percent
Pmn<*'>« speed line are shown in Figure 11.
The data points in Figure 11 ware
The values of Pmn at frequencies taken during a slow transient in which
(""ran = (ra «• Nn)0) are obtained by pro- equilibrium conditions were not fully
cessing the casing wall Kulite signals established. Furthermore, the pressure
through a wave analyzer. The frequency ratios were obtained from arithmetic
band of the recorded data was 40 kHz averages of a limited number of rake
which at the maximum rotational test readings. As noted on Figure 11, flutter
speed of the rotor permitted ten was first indicated by the hot-film gages
harmonics (i.e., -10<n<+10) to be deter- at about 67 percent of design flow, and
mined. the first indication of flutter on the
strain gages appeared at about 63 percent
Since no direct measurement was made of design flow. Surge occurred at about
of the mode shape at the blade tip, a 56 percent of design flow.
NASTRAN analysis was used to predict the
deflections. A composite map for all test speeds
is shown in Figure 12. These data were
The predicted mode shapes were obtained at fully stabilized conditions
scaled and phased in accordance with the and represent mass-weighted average
measured strain components to define the performance. Flutter occurred at speeds
tip motion, hm, required to calculate the between 63 and 75 percent of design. The
energy transfer in three dominant flutter boundary shown in Figure 12
harmonic components. The NASTRAN mode represents a blade vibratory stress level
shape energy levels were scaled in of ±2068 N/cm2 (3000 lbf/in.2) as
proportion with the strain component am- measured on the strain gage just above
plitude squared to determine the kinetic the shroud. Surge was encountered before
energy level of the individual harmonic flutter at speeds below 63 or above 75
responses. percent of design.
The kinetic energy per cycle of the Structural Deformations
individual harmonics was computed by
integration: Steady-state structural deformations
of the blades were determined from data
from the optical mirror system. Unsteady
Em = ~ hm)didb deformations during flutter were deter-
mined from the optical mirror system, the
strain gages, and by analysis of the high
where: response pressure data.
u • airfoil material density Steady-State Deformations
T = airfoil thickness
Local steady-state untwist at from
fm , gm , hm *> spatial components of 35 to 85 percent of the design speed are
deflection in the mth harmonic shown in Figure 13 for the 95 percent
span location. Above 25 percent speed
t = spanwise location coordinate the amount of untwist varied as the
b = chordwise location coordinate square of the rotational speed, the
o> = flutter frequency midspan shroud was not seated, and the
untwist varied in an unpredictable
PROGRAM RESULTS manner.
The following conclusions are based The distribution of untwist along
on test results: the span at 73 percent speed is shown in
Figure 14. Approximately equal amounts
o Deviations from uniform phase angle of untwist occurred above and below the
from blade-to-blade, previously at- shroud, which was at the 62 percent
tributed to insignificant anomalies in blade span and constrained untwist to
the data, are important, being indica- near zero at this location. Untwist was
tive of a complex flutter characteris- essentially a function of speed only.
tic. The effects of gas loading were
negligible. As shown in Figure 15, the
o Flutter alters the passage steady- variation of untwist with flow at 75
pressure pattern only slightly, as percent speed was less than 0.1 degree
shown in the steady-pressure contours. for a flow change from 70 to 59 kg/sec
(155 to 130 Ibm/sec), corresponding to a
o Work input is concentrated near the blade tip D-factor increase of 0.1764.
leading edge, as shown in the un-
steady-pressure contours. Measured untwist as a function of
chordal position is shown in Figure 16
o Local supersonic flow is required in for 73 percent speed. Uncambering was
order for this flutter to occur. significant at all stations above the
shroud, exceeding 0.3 degrees near the
Test Matrix blade tip.
Tests were run over a range of Figure 17 shows both the measured
corrected speeds from 54 to 85 percent of untwist and predicted by NASTRAN ana-
design. A transient was run at each of lysis for rotor speed at 75 percent
several speeds from wide-open-throttle to speed. The measured uncambering at the
20-18

tip was slightly higher than predicted. which is 2 times the ratio of available
However, the deformations in this damping to critical damping, since this
region, where the airfoil was very thin, number represents the percentage rise or
were sensitive to the actual airfoil decay of the signal, a negative of the
thickness, and slight variations within logarithmic decrement represents an
specified tolerance might have been unstable or flutter condition. Complex
sufficient to cause the observed discrep- pressures used in the stability calcula-
ancies. tions are listed in Table IX and plotted
in Figures 21, 22 and 23 for th upper and
Unsteady Deformations lower surfaces of the airfoil. The
chordwise position of the pressures is
Previous to this program, fan flut- the same as for the wall mounted Kulltes
ter had been visualized as a sinusoidal, from which the data was obtained.
circularly traveling wave superimposed on
the rotor, forming a single multinodal Table IX shows that the fifth
pattern, each rotor blade deflecting harmonic was the principal source of
sinusoidally in sequence as the wave instability at 70 percent speed. The
traveled around the rotor, Bendat and seventh harmonic was marginally unstable,
Piersall (1971). the ninth, marginally stable. The
results suggest that the effect of
Such a wave was characterized by asymmetries, or "mistuning," on the
concentric ring nodes and traveling system in flutter is to couple secondary
nodal diameters or diametral lines of modes into the instability. This is an
zero deflection. Figure 18 shows such a important result, clearly demonstrating
system with two ring nodes and three that any future flutter analysis that is
nodal diameters. This pattern is to be correlated against test data for a
referred to as a vibration in the second mistuned bladed disk system must be
mode with three nodal diameters. On a capable of handling several spatial
rotating stage, the radial lines travel harmonics.
either forward or backward, and adjacent
blades experience a relative time delay The present analysis is not capable
or phase difference (interblade phase of explaining the mechanism that deter-
angle) as the wave passes. With such a mines what patterns will occur or what
concept, all blades are assumed to their relative indexing will be. How-
flutter at the same frequency and ampli- ever, the mechanism probably relates to
tude, with uniform phase angles between the mistuning of the stage, which results
adjacent blades. from small dimensional differences among
these airfoils. These airfoils had been
The results of the current program deliberately grouped by frequency when
revealed a different picture: All the rotor was assembled, see Figure 24.
blades fluttered at the same frequency, And it may be significant that the group
but not at the same amplitude and of airfoilds with the highest flutter
interblade phase angles were not equal. amplitudes were those that individually
Typical amplitudes and phase angles had natural vibratory frequencies equal
observed during the program are shown in to the average frequency for the blade
Figures 19 and 20, respectively. These set. It may also be significant that
data were obtained from the strain-gage only forward traveling waves (traveling
measurements. Amplitudes in Figure 19 in the same direction as the rotor) were
are expressed in terms of measured observed.
stress. The patterns shown represent a
family of spatial harmonics described by Pressure Distributions
the superposition of a number of rotating
nodal diameter patterns, each charac- Increasing rotor speed on a given
terized by a different number of nodal operating line resulted in a strengthen-
diameters with different but uniform ing of the expansion waves and normal
amplitudes and different but uniform shock and a rearward shift of the shock.
phase indexing, with each pattern Moving up a speed line to higher loading
rotating at a speed that results in the and incidence shifted the shock forwards
sane flutter frequency. towards the leading edge. Crossing the
flutter boundary produced little change
The detailed definition of the although the normal shock appeared to
amplitude and phase for each nodal have spread, which is probably indicative
diameter pattern was determined from wall of shock oscillation.
Kulite data. A result of this analysis
is presented in Table VIII. As shown in Steady-State Pressure Distributions
the table the fifth nodal diameter pat-
tern had the strongest signal at 67 At 63 percent outside of flutter,
percent speed. The seventh nodal dia- data from the case-mounted Kulites
meter pattern was strongest at 73 percent showed that high loading occurred at the
speed, and the eighth was slightly leading edge and that the flow was
stronger than the others at 75 percent subsonic (Figure 25). Moving up to a
speed. high operating line Into flutter is
shown in Figure 26.
To further study the complex mode
shapes of the rotor and blading, At 67 percent speed on the low
stability calculations were made for the operating line, expansion waves occurred
fifth, seventh, and ninth nodal patterns behind the leading edge, culminating in
at 70 percent design speed. These a shock at about 15 percent chord (Figure
patterns represented two strong signals 27). At the flutter boundary at 67%
and one weaker signal. The results of speed, the shock appeared to be a gradual
these calculations are given in Table IX compression, which may be indicative of
in terms of the logarithmic decrement an oscillating shock (Figure 28).
 20-19

FLUTTER NO FLUTTER

figure 10. Typical Laser M i r r o r Results for Operation

at 67 Percent Speed in and Out of F l u t t e r

RECORD 133 O TRANSIENT DATA POINTS

D FULL DATA POINTS
1.30 - 9 ""°°-^
r 5 /^^
RECORD 220 Y^
— x ^ ^.^RECORD 43
1.28
H^
o
,— TVN
5 1.26 - v

S«
TRANSIENT DATA POINTS \
2 1.24 - 1. WIDE OPEN ^ ^RECORD 25
QC 2. OUT OF FLU HER
*"
1.22
3. FIRST FLUTTER INDICATION ON HOT FILMS
FIRST FLUTTER INDICATION ON STRAIN GAGES
a
'^T

P\
- 4.
5. a UTTER
i
ib i i
l.?Q 1 1 1
50 55 60 6S 70 75 SO

PERCENT DESIGN FLOW

Figure 11. I d e n t i f i c a t i o n of Data Points at 70 Percent Speed,

.Including Transient from open Discharge into Surge
20-20

I INDICATES NCAA SURGE SURGE

LIMIT

1.3

70ft
87«
63%
54*

tO 70 80
PERCENT OF CORRECTED OESION f LOW

Figure 12. TS22 Performance Map Showing Test Points

in Relationship to Flutter Boundary

15

1.4

1J

1.2

1.1

10

OJ

0.8

0.7

0.8

04

0.4

04 STARTING
REFERENCE SPEED
FOR MEASUREMENT
aa

o.i

o
10 40 so JO 1

ROTOR SPEED. PERCENT OF DESIGN

Figure 13. Measured Untwist for TS22 Fan Blade as a Function

of Rotor Speed at 95 Percent Span
 20-21

A 5% CHORD FROM LEADING EDGE

Q 25% CHORD
O SO* CHORD
0.8 Q 70% CHORD

QUESTIONABLE
DATA POINT

0.5

03

0.1

&0 60 «0 100
TIP

PERCENT SPAN

Figure 14. Measured Untwist for TS22 Fan Blade at 73 Percent

Speed Relative to Untwist at 25.4 Percent Speed

5% CHORD
JJ <-*(- 7S% CHORD

60 n TO 7»
CORRECTED FLOW

Figure 15. Measured Untwist for TS22 Pan Blade as a

Function of Flow Rate at 75 Percent Speed
20-22

95,2%
SPAN FROM MUl

0.7

0.8

1
0.4

0.1

0 1O 20 40 SO 60 70 80 90 100
L.e. T.t.
•CHORD. PERCENT

Figure 16. Measured Untwist for TS22 Fan Blade as a

Function of Chord at 73 Percent Speed

100

10
RADIUS' 88% SPAN
OJ

04

0.4
100

RADIUS'7T%SPAN

0.4 -

0 40 «0 100
UAOING PERCENT CHORD TRAILING
IDGt fOGE

Figure 17. Measured Untwist for TS22 and Predicted by NASTRAN

Analysis for Rotor Speed at 75 Percent Speed
 20-23

TABLE VIII
UNSTEADY HALL PRESSURE AMPLITUDES FOR INDIVIDUAL
NODAL DIAMETER PATTERNS FUNDAMENTAL MOOES ONLY
(NO HARMONICS)
Relative Power Spectral Density
Nodal Percent Chord
Diameters -55.4 -15.1 -3.6^ ZZ.Z 34.5 47.5 73.4 99.3 141.4
67 Percent Speed
2
3 30
4 40 45 24 25 33 34
5 65 58 50 100 140 220
6 65 60 60 SO 75 70
7 65 70 75 70 85 80
8 100 140 80 80 85 60
9 65 100 45 35 28 20
10 30
73 Percent Speed

2
3 26 24 25
4 40 36 24 57 41
5 9 25 96 54 78 100 140 170
6 10 32 92 59 86 74 150 125
7 24 84 96 68 165 240 180 340 270 280
8 35 77 130 165 150 140 160 150 120
9 110 88 75 44 35 38
10 26 125 80 34 31 24
75 Percent Speed
2
3 18 1?
4 25 20
5 50 100 55 39 45 60
6 50 80 78 70 70 100
7 120 70 50 45 45
8 140 UO 78 55 75
9 120 85 45 35
10
TABLE IX

COMPUTED DAMPING IN DOMINANT HARMONICS

AT 70 PERCENT SPEED
Log
Harmonic Decrement
5

Complex pressures used In damping calculation normalized to 1600 N/ra2

(0.232 lbf/1n.2).

Percent Upper Upper Lower Lower

Chord Real Imaginary Real Imaginary

-3.4 0.039 -0.091 0.022 -0.022

9.4 0.573 -0.681 0.056 -0.134
22.2 0.504 -1.000 -0.254 -0.060
34.6 0.095 -0.125 -0.069 -0.060 5 Nodal Diameters
47.5 0.030 -0.246 -0.086 -0.142
73.4 0.138 -0.086 -0.108 -0.086
99.3 0.198 -0.026 -0.228 -0.039
-3.4 -0.065 0.017 0 0.043
9.4 -0.250 0.190 0.039 0.026
22.2 -0.177 0.384 -0.091 0.241 7 Nodal Diameters
34.6 -0.091 0.056 -0.121 0.052
47.5 -0.095 0.060 -0.129 0.052
73.4 -0.129 -0.004 -0.112 0.004
99.3 -0.112 -0.052 -0.147 -0.043
-3.4 -0.112 -0.099 0.009 -0.009
9.4 -0.030 -0.259 -0.069 -0.134
22.2 -0.181 -0.134 -0.129 -0.091
34.6 0.017 -0.034 -0.043 -0.043 9 Nodal Diameters
47.5 -0.030 -0.172 -0.052 -0.164
73.4 -0.043 0.056 -0.017 0.060
99.3 0.043 -0.194 0.022 -0.233
20-24

Figure 18. Three Nodal Diameter Pattern Second Mode - Previous Theory Predicted
the Presence of only one Nodal Diameter Pattern at Any Time

- S
O f* 3

si
r,
J 4 B t 10
wm 12 14 1-
BLADE NUMBER
It 30 33 34 24 M 3D H

Figure 19. Blade Flutter Amplititude for TS22 Rotor at 67 Percent

Speed from Strain-Gage Measurements

oo

0°00

12 IS
•LAOENUMftEft

Figure 20. Blade Flutter Phase Angles for TS22 Rotor at

67 Percent Speed from Strain-Gage Measurements
 20-25

O MM'ApVT-U*f<tA
Q IUO. PAMT - UffUt fcJ)t»>ACI
RIAL FAJir - LOWIfl
A IMG »A«T - LONIH CURFMI

Figure 21. Complex Pressures Used in Damping Calculation,

Five Nodal Diameters

NORMALIZED TO -0.23] fSI

MOflMALIZED TO 1«OON/MJ 1-0.233 LIF/IN2)

<to>
a
UJ
N

i
ec
o

-0.1

O REAL FART - UPPER SURFACE

O IMG. FART - UFP£R SURFACE

Q REAL FART - LOWER lURPACe

A IMC. FART - LOWER SURFACE

I | , |
0 40 60 100
L.I. T.S.
PERCENT CHORD

Figure 22. Complex Pressures Used in Damping Calculations,

Seven Nodal Diameters
20-26

REAL FART - UPPER SURFACE

IMC. FART - UPPER SURFACE
REAL FART - LOWER SURFACE
IMC. PART - LOWER SURFACE

NORMALIZED TO -'600N/MI(-0.r»L8FflNzl

cc
q -01

-0.2

-0.3
0 30 40 60 80 100
L.E. T.E.
PERCENT CHORD

Figure 23. Complex Pressures Used in Damping Calculations,

Nine Nodal Diameters

280

00

248 o o
oo o
246 ooo ooo
-oooooo
oo oo
o o
242 oo ooo

240
1 4 • 12 IS 20 14 28 32

BLADE HUM8ER (CLOCKWISE AFT LOOKING FORWARD)

Figure 24. Distribution of Natural Second Mode Vibration

Frequencies of Blades in Assembled Rotor
 20-27
% MAXIMUM PRESSURE
CWIVe CUAVC
LRBK. milt
0.9900001*02

0* 790QOOC*C2
0. 7

0*3IOOOOC*02

MAXIMUM STATIC

•9.63 N/cm2 II 3.96 iW/in.Z

ROTATION BLADE 4 BLADE 1

-zn.oo -20.00 -16.00 -iz.oo-e.oaJ*.oo o.oo i.ao e.oo

TBNGEMTIflL IDEGl

Figure 25. Steady-State Pressure Contours at Blade Tip at 63 Percent

Speed Outside of Flutter on a Low Operating Line

% MAXIMUM PRESSURE
oant eiMvc
LMO. VHLUt
O.llOOOOt'OJ
0. MBBOflt^K
O.T»OOuOt>0*
o.7jooooe»e»
o.isoooot ot

10.18 N/cm2 IK.75 lbf/in.2]

-».oo -20.00 -le.oo -12.00 .i.ofl -li-oo o'.oo «.oo a.oo

TflNGENTlflL tOEG)

Figure 26. Steady-State Pressure Contours at Blade Tip at 63 Percent

Speed Inside of Flutter on a High Operating Line
20-28

ft MAXIMUM PRESSURE
cunvf cwvc
uwo. mi*
2 O.iSOOCM'OZ
} C.880000f>01
4 O.T90OTOE»Oi
s o.72oacoe*o2
e o.8soooce*02
7 o.saooooc«c2
g o.si<nooe*oz
MAXIMUM STATIC PI»»UPja
9.76 N/cm<14.14 IM/ln.1)

-28.00 -2U.OO -20.00 -18.00 -12.00 -8.00 U.OO 0.00

T f l N G E N T I P L fDEG)

Figure 27. Steady-State Pressure Contours at Blade Tip at 67 Percent

Speed Outside of Flutter on a Low Operating Line

% MAXIMUM PRESSURR

MAXIMUM STATIC FRBSIURI

10.38 N/cm 115.00 Ibf/ln.2)

Figure 28. Steady-State Pressure Contours at Blade Tip at 67 Percent

Speed Inside of Flutter on a High Operating Line
 20-29
At 70 percent speed outside the Velocity Fluctuations
flutter boundary, supersonic Mach number
expansion at the leading edge was more Upstream and Downstream Velocity Fluctua-
clearly evident, and the normal shock tions
moved rearward to the 20 percent chord
position (Figure 29). At the flutter Enhanced wave forms from the hot-
boundary, the shock moved forward, very film probes ahead of and behind the rotor
close to the leading edge (Figure 30). are shown in Figure 36 for two test
Near surge the leading edge expansions points at 73 percent speed: One at wide
appeared to be weaker, but the passage open discharge, the other in the flutter
shock appeared stronger (Figure 31). region. Because these signals were not
calibrated for amplitude, the magnitudes
of fluctuation are not known. For the
At 73 percent speed on a low open discharge condition, the inlet
operating line the shock moved further signal at the blade tip showed a velocity
rearward to about the 30 percent chord fluctuation of blade passing frequency
position. Moving into flutter the that was caused by the passage of
principal loading remained at the expansion and shock waves emanating from
leading edge with the data showing the blades. There was no defined pattern
considerable smearing of the normal at the inlet near the shroud and at the
shock. Essentially identical trends blade root where the relative inlet
occurred at 75 percent speed. velocity was subsonic. At the rotor
exit, a well defined blade wake pattern
At 85 percent speed, surge occurred existed for all three radial positions.
before flutter. On the low operating The inlet probe patterns in flutter were
line significant reacceleration occurred similar to those for the nonflutter
behind the shock and the compression condition. Behind the rotor at the hub,
process was far from optimum, with the pattern was also similar to that for
negative lift occurring on the aft wide open throttle. However, at the near
portion of the blade. Moving up the shroud and tip exit position, the blade
operating line resulted in a high Mach wakes were not as well defined as for the
number with strong leading edge expan- nonflutter condition. The tip pattern
sion and a strong detached bow shock. had some random fluctuations at other
Operating near surge produced little than blade passing frequency, but did not
change in this pattern. show a significant fluctuation at flutter
frequency.
Unsteady Pressure Distributions Blade Surface Unsteady Velocities
Unsteady pressure data were reduced Unsteady velocities and phase
to contours of unsteady pressure ampli- angles were determined from the hot-film
tude and contours of the real and gages mounted on the rotor blades. Data
imaginary components of the unsteady for a flutter condition at 67 percent
pressure to provide relative phasing speed are shown on Figure 37. The arrow
information. A typical plot is presented length in this plot represents the
in Figure 32. To interpret these plots amplitude of the unsteady velocity
it should be noted that the real and relative to the maximum fluctuation
imaginary components represent the observed for that test point. The
instantaneous unsteady pressures at two direction of the arrow indicates the
time phases separated by 90 degrees. phase angle referenced to the strain-
Hence, the square root of the sum of the gage signal from the No. 3 blade. The
squares of the real and imaginary ampli- major fluctuations of velocity occurred
tudes shown in Figures 33 and 34 is equal on the forward part of the airfoil, but
to the amplitudes shown in Figure 32, and some significant fluctuations also occur-
the relative phase angle of the unsteady red at midchord and near the trailing
pressure is equal to the arctangent of edge.
the ratio of the real and imaginary
components. Reduced Velocity Versus Incidence Angle
The data showed high unsteady pres- Empirical correlations of reduced
sures near the leading e dge (back to velocity versus incidence angle have
approximately the 25 percent chord been used extensively as a stall flutter
position), relatively low values near the criterion. The range of design types
trailing edge, and minimum amplitude near over which any specific correlation will
midchord. Similar trends were evident in accurately predict flutter boundaries,
the blade unsteady surface pressures however, is questionable. Existing cor-
measured by the blade-mounted Kulite (see relations were based on measured air
Figure 35). The arrow lengths in this angles, but blade metal angles were
plot represent unsteady amplitudes and usually taken as the calculated metal
the directions represent phase angle as angle at design speed. In this program,
referenced to the strain-gage signal from actual metal angles were measured.
the No. 3 blade. As shown, significant Figure 38 presents a plot of reduced
unsteady pressure amplitudes were con- velocity versus measured incidence
fined to the leading edge portion of the angle. Incidence angles were based on
airfoils. the blade leading-edge mean-line metal
angle. The reduced velocity parameter,
These results clearly indicate that V/bu , is the ratio of the relative
the major portion of the action was inlet velocity, VI 1 , to the product of
concentrated in the first quarter of the the blade half-hord, b , and the
airfoil, implying that future flutter rotational flutter frequency, u , in
research should concentrate on the aero- radians per second.
dynamics near the leading edge.
20-30

% MAXIMUM PRESSURE
CU8VE CIUIVE
LABEL VALUE
2 0.930000E»OZ
S 0.86000aE*02
H O.?90000t>02
5 0.7200001*02
6 0.8500BDE.OZ
7 O.S80000E*02
8 o.sioooae«02
MAXIMUM STATIC PRCBSUP1E
9J3 N/cm] 114.25 IM/M,2)

-12.00 -8.00 -U.OO O'.OO u'.OO 8'. 00 12.00 ;e.oo

TANGENT IRL (DEG)

Figure 29. Steady-State Pressure Contours at Blade Tip at 70 Percent

Speed Outside of Flutter on a Low Operating Line

% MAXIMUM PRESSURE
onm cum
i tun VH.UI

e.iioogoc>et
g.«uoooc««
to n.mooot^a
ti o.taamtxa
II O.tWOODt^i
MAXIMUM STATIC FRESSUflE

1 1 .87 Hlon2 (T7.20 IB«/ln.*)

-'n.eo -ii.ra -is.oo -o.oo -<.oo o.oo i.oo 11.00

TRNGENTIRL (DEC)

Figure 30. Steady-State Pressure Contours at Blade Tip at 70 Percent

Speed Inside of Flutter on a High Operating Line
 20-31

» MAXIMUM PRESSURE
cum* cum
uun. VKLUC
o.uaooofo*

0.7MOttE»et
o.7«coot'0»
O.I9QOOOC*8t
T O.KOOBOE*8i
I 0.(10000C«Ot
MAXIMUM STATIC PPltSSURt
10.4)N/an2l16.11 let/In.3)

-21.00 •2V.OO -20.00 -11.00 -12.00-1.00 0 00 H.OO 1.00

TRNGENTIRL (DEC)

Figure 31. Steady-State Pressure Contours at Blade Tip

at 70 Percent Speed Near Surge

* MAXIMUM PRESSURE

MAXIMUM STATIC PRESSURE

0.308 N/cm2 (0.447 Ibffin.*)

-tf.M -K.N -10.00

Figure 32. Unsteady Pressure Amplitude Contours for

TS22 Rotor in Flutter at 73 Percent Speed
20-32

K MAXIMUM PRESSURE

MAXIMUM STATIC PRESSURE

Figure 33. Real Component of Unsteady Pressure

in Flutter at 73 Percent Speed

«MAXIMUM PRESSURE

I l.ll

MAXIMUM STATIC PRESSURE

0.155 N/cm 10.225161/ifl^

Figure 34. Imaginary Component of Unsteady Pressure

in Flutter at 73 Percent Speed
 20-33

PRESSURE SURFACE

SUCTION SURFACE

VECTOR: 100 N/cm2 1145 lbl/.n.2l PHASCTO S/0 1

l«Cf

180-r -lo-

270*

' ^X 4-
**»

-25 *

BLADES BLADE 4 BLADE! BLADE2

Figure 35. Blade Mounted Kulite Unsteady Pressure Amplitude

and Phase Obtained in Flutter at 67 Percent Speed

BLADE TIP BLADE TIP

BEHIND ROTOR BEHIND ROTOR

BEFORE ROTOR BEFORE ROTOR

NEAR SHROUD NEAR SHROUD

BEHIND ROTOR BEHIND ROTUM

BEFORE ROTO4I

BLADE ROOT BLADE ROOT

BEHIND ROTOR BEHIND ROTOR

-

BEFORE ROTOR
BEFORE ROTOR

Figure 36. Signal Enhanced Wave Forms of Hot Film Probes

at 73 Percent Speed (Noncalibrated Amplitudes)
20-34

90*
PRESSURE SURFACE

SUCTION SURFACE

AMPLITUDES ARE RELATIVE TO 180--*-

MAXIMUM OBSERVED FLUCTUATION

270*
i ROTATION r™«.._«-.

\jifr -25
-16
-S

BLADE » BLADE 21 BLADE 20

BLADE 21
/

Figure 37. Blade Mounted Hot Film Unsteady Velocity Amplitude

and Phase Obtained in Flutter at 67 Percent Speed

STABILITY PLOT BASED ON CONDITIONS

AT 77 PERCENT SPAN FROM HUB PERCENT SPEED

O 80
1.9 l-
Q 63
O 87
& 70
1* -
6 73
X n
V 85
I
1
„
(f

1.8

UNSTABLE

STABLE

1.3 1 1 ' ' .' ' I 1 1 -1 -1 -1

7 B 9

MEAN INCIDENCE IINCM], DECREES

Figure 38. Observed TS22 Flutter Boundary Correlation of

Reduced Velocity as a Function of Incidence
 20-35
Figure 38 shows that flutter occur- tion. Test execution includes the test-
red at high incidence angles only over a ing procedure as well as data acquisi-
limited range of reduced velocity values, tion, reduction and analysis. Safety
with flutter-free operation being ob- procedures are outlined. Included is a
tained at reduced velocities both above description of the on-test appearance of
and below those at which flutter was several aeroelastic phenomena and an
achieved for a given incidence. A pos- assessment of corrective actions to
sible explanation is that locally super- prevent fatigue failures due to over-
sonic flow may be required for flutter stress.
and that this was not achieved at low
rotor speeds and velocity ratios, even at ACKNOWLEDGMENT
high incidence. At very high speeds and
velocity ratios, the incidence was too Th« author gratefully acknowledges
low even at surge to support flutter. the assistance of the following individ-
uals in the performance of the program!
SUMMARY Robert A. Arnoldi, and Ron-Ho Ni for
providing technical guidance, Arthur R.
This chapter describes the pro- Guerette for providing instrumentation
cedure of conducting a fan flutter test. and data reduction, William N. Dalton,
The experience of an actual test is used III, for reducing the laser mirror data
as an example. Test preparation includes and performing the analysis, and Dr. John
a test plan, selection of blades, and J. Adamczyk for providing technical
description of measurement instrumenta- direction.
 21-1

AEROELASTIC THERMAL EFFECTS

by
JAMBS D. JEFFERS, III
9321 Forest Hills Drive
Tampa, Florida 33612

INTRODUCTION The fan has three rotating stages, each

with partspan shrouds, as shown in figure
The adverse effect of increasing tempera- 1. The original configuration of the
ture on the stability of turbomachinery first stage fan blade, which is located
airfoils has long been recognized but re- immediately behind a variable inlet guide
mains today one that is not fully vane, had a shroud mounted aft of the air-
understood. The quantitative effect on foil midchord at approximately 65% span.
the reduced frequency parameter, k-wc/V,
which has been experimentally and analyt- Upon completion of the intitial test, air-
ically shown to be one of the most influ- foil fatigue failures were discovered in
ential stability parameters, is readily several first stage blades. Materials
calculated. The effects of increasing laboratory tests indicated that the
temperature on the parameters that com- failures were high-cycle fatigue cracks
prise reduced frequency, i.e., the fre- caused by a high amplitude response in a
quency of the unsteady airfoil motion and predominantly torsional mode of vibra-
relative flow velocity, are well known. tion. Both the nature of the failure and
Unfortunately, the resulting effect on the aerodynamic environment of the air-
aeroelastic stability, particularly for foil at the time of failure suggested that
•stall" flutter, is not. the first stage had experienced sub-sonic
stall flutter.
This inability to completely assess ther-
mal effects, among others, was dramati- Initial Flutter Analysis
cally underscored by an occurrence of sub-
sonic stall flutter in the first stage fan The design system used to predict the sub-
rotor of the FlOO turbofan engine early in sonic stall flutter margin of the FlOO
its development program. The severity of first stage consisted of a stability
the problem prompted an intensive experi- boundary that is a function of two vari-
mental test program and analysis to first ables (Figure 2). One of the variables is
determine a solution and, further, to reduced velocity, Vr - V/c», which is the
assess the design techniques used to inverse of the reduced frequency. The
avoid the problem. As a result, a decade second is a normalised incidence
of extensive analytical and experimental parameter, Bif, which is defined as
research has been conducted at the
Government Products Division of Pratt & (1)
Whitney Aircraft, the United Technologies
Research Center, and the NASA Lewis The flow angles, 8 , in this expression
Research Center. The following is a sum- are determined from nonrotating two-di-
mary of some of the findings of these mensional cascade loss correlations which
investigations. generate "loss buckets*. %\ represents
the actual inlet flow angle, 6min is the
HISTORICAL BACKGROUND flow angle at minimum loss, and BI is the
flow angle at a specified loss on the
In late September of 1972, a prototype stalled side. Thus, &if is a measure of
P100 engine was tested at simulated high- the degree of stalling.
Mach-number flight conditions in prepara-
tion for military qualification tests
(Jeffers 4 Meece, 1975). The FlOO engine
is a twin-spool, high thrust-to-weight
ratio, turbofan engine developed for the LEGEND
F-15 and F-16 fighter aircraft.
FLUTTER BOUNOARKS
• ENVELOPE

2.0

Q '•»
u

1.0

1.5 2.0 2J 3.0

NORMALIZED INCIDENCE, /i,,

Figure 2. Torsional Stall Flutter

Figure 1. FlOO Fan Cross-Section. Experience
21-2

The two-dimensional bi-variate experience number flight. For flight inlet condi-
limit was generated by enveloping the tions, the normalized incidences of the
available stall flutter experience at a first stage were calculated to be the same
reference span to create a "conservative" as those in the rig at part corrected
stall flutter boundary. speed on the normal operating line.
However, the increased temperatures did
Extensive rig component testing was con- increase the reduced velocities (decrease
ducted early in the development program to the reduced frequencies) such that the
evaluate the aerodynamic design and flutter region on a fan map was projected
structural durability of the FlOO fan. to enlarge (Figure 5). The enlargement
The testing included heavily instrumented was small, and therefore the engine was
fan rigs that were operated at ambient in- predicted to be flutter-free throughout
let conditions in sea level facilities at the flight envelope. Obviously, the
both normal and off-design engine operat- blade failures experienced durinq flight
ing conditions. Instrumentation included testing indicated that the prediction was
standard steady state temperature and in error.
pressure probes at stator leading edges
and strain gages placed at strategic loca- Engine Test Results
tions as dictated by modeling and
laboratory tests. A full-scale engine test program was
initiated to investigate the blade
During off-design testing of an FlOO fan failures. Strain gages were placed on the
rig, a response at a non-integral order first stage fan as dictated by laboratory
frequency was observed in the first staqe tests that reproduced the failure.
rotor at low corrected speed and high Aerodynamic instrumentation was sparse due
pressure ratios, well away from the normal to the difficulty of the twin-spool enqine
operating line (Figure 3). The flutter environment. The engine was placed in an
frequency was very near that calculated altitude test facility capable of simulat-
for a 5 nodal diameter, blade-shroud-disk ing flight inlet conditions in excess of
system mode, but phasing of the strain Mach 3.0 and 70,000 ft.
gage responses proved inconclusive. In
the 5 nodal diameter system mode, the
blade motion is calculated to be comprised
primarily of bendinq motion with a high BOUMMftY
degree of shroud coupling. At these near
surge operating conditions, the first ENOXC CONOmONS
stage experiences very high incidences, ON OP LINE f
100 STALL FUCTK*
and consequently the response was desig- (800 Hi)
nated as stall flutter.
Correlation of the rig stall flutter data
on the empirical experience curve in-
dicated that it was indeed a conservative no CONDITIONS
operating limit (Figure 4). The rig data NEW SUM
formed a "nose curve" similar to the cor-
relations of previous Pratt £ Whitney
Aircraft flutter data considerably beyond
the experience boundary. Because many of
the variables not included in a two- 24 U M
dimensional flutter were reasoned to not NORMALIZED INCIDENCE, £„
appreciably change, the rig boundary was
used to assess the stability of the FlOO
first stage fan at the elevated tempera-
tures and pressures of high Mach Figure 4. Prediction of First-Stage Fan
Blade Stability at Engine
Flight Conditions.

,SLTO
SLTO
R» STALL FLUTTER
BOUNDMTT (BOO Hi)
RIO STALL FLUTTER
BOUNDARY (800 Ml)

TOTAL CORRECTED FLOW

TOTAL CORRECTED FLOW

Figure 5. Predicted Stall Flutter

Figure 3. FlOO Rig Stall Flutter Boundary at Engine Flight
Fan Map. Conditions on Fan Map.
 21-3

The flutter region was probed by two basic The FlOO engine data determined the first
paths, as illustrated in three formats in stage durability problem wasr in fact the
Figure 6. Initially, inlet conditions result of an aeroelastic instability that
were set at a maximum safe experience was a function of stalled incidence. The
point based on endurance testing (point engine flutter exhibited a marked dif-
A). Inlet temperature was then steadily ference from that observed in the com-
increased to simulate a high Mach number ponent rig tests; i.e., the flutter
excursion into the failure region (point response changed from an 800-Hz, coupled
B). Above a certain Mach number, the bending-torsion system mode to a 1000-Ht,
engine is scheduled such that the fan cor- above-shroud torsion mode (Figure 7).
rected speed decreases with increasing Because phasing of the strain gage data
temperature to avoid aircraft inlet again proved inconclusive, the engine
Instabilities. On a fan map, the fan flutter mode could have been either a
approaches lower corrected flows and pres- high-nodal-diameter, second coupled mode
sure ratios as temperature or flight Mach or a low-nodal-diameter, third coupled
number is increased. During this tempera- mode.
ture excursion, the first stage fan exper-
iences increasingly higher normalized
incidences but only slightly decreasing
reduced velocities. Temperature was in-
creased until stability was encountered.
The second basic path employed suppressed
speed excursions to flutter onset (points
A to C, Figure 6). Speed was decreased at
constant temperature by circumventing the
engine control system. The fan followed
an off-schedule operating line at in-
creasing higher pressure ratios as speed
and corrected flow were reduced. As a
result, flutter was generally encountered
at higher normalized incidences but lower
reduced velocities than on the tempera-
ture transients.

0 2 4 ( I 10 12
ROTOR SPEED, rpm (Thouwmii)

Figure 7. Rig vs. Engine Stall Flutter

Frequency Response.
TOTAL INLET TEMPERATURE, Tt2

CKONt SPOD SCHEDULE

fr

TOTAL CORRECTED FLOW

•T

TOTAL INLET TEMPERATURE, Ttt - deg F

NORMALIZED INCIDENCE. />„

Figure 6. Engine Operation to Figure 8. Stall Flutter Boundaries Prom

Encounter Flutter. Engine Tests
21-4

The data confirmed the destabilizing The redesigned first stage blade was
effect of increased temperature, but in demonstrated to be free of flutter
addition demonstrated that increasing throughout the aircraft flight envelope in
pressure is destabilizing, a fact not pre- tests of the FlOO engine in altitude test
viously documented. This pressure depen- facilities and in F-15 and F-16 aircraft.
dence was readily seen when the engine Concern remained, however, for the FlOO
stall flutter boundaries were presented on stall flutter problem had revealed grave
a speed-versus-inlet-temperature plot inadequacies in the state of the art of
(Figure 8). Subsequent testing of candi- stall flutter prediction.
date redesigns substantiated the pressure
effect as illustrated in a correlation of FOLLOW-ON RESEARCH
pressure-versus-temperature flutter bound-
aries (Figure 9). As pressure was In recent years, the emphasis on light-
increased, flutter occurred at lower weight, high performance engine designs
temperatures. has resulted in flutter problems similar
to the one encountered in the FlOO engine.
When plotted on the empirical Vr-versus- Efforts have intensified in academia,
6lf correlation, the engine flutter was industry and government to develop accu-
shown to occur at considerably lower re- rate and reliable flutter prediction sys-
duced velocities and normalized incidence tems that go beyond the limitations of the
than previous stall flutter experience empirical method.
(Figure 10). In fact, the engine Vr-
versus-Bif boundary would have predicted Carta (1967) proposed the aerodynamic
the first stage fan to have fluttered on damping approach to flutter prediction in
the normal operating line at low tempera- which the unsteady aerodynamic work over
ture and pressure. Nevertheless, the one oscillatory cycle is normalized by
general stability trends represented by four times the system kinetic energy to
the empirical correlation were used to obtain the logarithmic decrement commonly
redesign a flutter-free first stage fan. denoted as <aero> The total system log-
The thickness-to-chord ratio in the air- arithmic decrement is the sum of <3aero and
foil tip region was increased to increase the mechanical logarithmic decrement,
the torsional frequency from 1000 Hz to 6mecn, which is always positive or
1240 Hz and thereby decrease the reduced dissipative. Thus, instability occurs
velocity. The airfoil thickness increase only when Saero is negative and of a mag-
also improved the loss characteristics re- nitude such that «mech is overcome. The
sulting in lower calculated normalized unsteady aerodynamic work for the entire
incidence at a given operating condition. blade is obtained by summing the contribu-
A finite element analysis (NASTRAN) of the tions from two-dimensional strips of con-
steady-state aerodynamic loading on the stant airfoil cross section along the
blade also indicated that the destabiliz- blade span in an attempt to account for
ing effects of increased pressure were, at three-dimensional effects.
least in part, due to increased blade
deflections and consequently the cascade The aerodynamic damping approach requires
geometry. The increased stiffness in the an unsteady aerodynamics model to predict
tip region created by the thickness in- the unsteady lift and moment responses to
crease minimized the cascade geometry the unsteady motion, which is assumed to
changes and the accompanying increase in be that of the system natural modes.
normalized incidence. Unsteady aerodynamic modeling has inves-
tigated two areas for the subsonic stall
flutter problem. Some investigators

CONFIGURATION ®\ ®

no CONDITIONS
- ON OP. UNE

if
AUBCMT INLET

M ii u
NORMALIZED INCIDENCE. />„
TOTAL INLET TEMPERATURE, Tt2 - deg F

Figure 10. Empirical Stall Flutter

Figure 9. Inlet Pressure vs. Temperature Reanalysis of FlOO Rotor 1
at Flutter Inception. at Rig Conditions.
 21-5
have probed the effects of high loading NASA Flutter Data
due to incidence and blade shape for
attached flow (Atassi & Akai, 1978, Verdon The objectives of the FlOO engine tests at
& Caspar, 1982), while others have studied NASA Lewis were to further define the par-
the effects of flow separation (Sisto & ametric effects of inlet temperature and
Perumal, 1974, Chi, 1980, Chi & pressure, determine flutter responses as a
Srinivasan, 1984). The Chi model, which function of operating conditions, and to
assumes that the flow separates at a fixed generate a data bank for future research.
point along the blade chord, was very The instrumentation included strain gages
successful in predicting the relative on all fan rotors and stators and total
stabilities of the two flutter modes ob- temperature and pressure probes at the
served during FlOO first stage fan tests leading edges of all stators. In addi-
at the disparate temperature and pressure tion, a traversing flow angle probe was
inlet conditions. Further research may placed behind the inlet guide vane to more
establish a means of determining the sepa- accurately define the inlet flow angles to
ration point as a function of the steady- the first stage fan. The more atypical
state operating conditions. However, al- instrumentation included high-response
though some of the results have been pressure probes and optical sensors
encouraging, no fully substantiated model mounted in the engine case directly over
has as yet been developed. the tip of the first stage rotor. A high-
response wake probe was also placed down-
A semiempirlcal unsteady aerodynamics stream of rotor 1. These sensors recorded
model for stall flutter {Jeffers, et. the unsteady airfoil motion and the aero-
al., 1978) was evaluated using FlOO dynamic responses to the blade motion dur-
engine data from tests at the NASA Lewis ing stall flutter.
Research Center. The model was moderate-
ly successful in that the above-shroud A broad range' of inlet temperatures and
torsion mode was calculated to be the pressures as well as fan operating condi-
least stable and the destabilizing effect tions were surveyed to generate stall
of increasing inlet pressure was satis- flutter data. The stability trends
factorily represented. However, other associated with changing inlet temperature
inadequacies, most notably the inability and pressure were in agreement with those
of the model to fully account for the observed in the earlier FlOO tests
effects of inlet temperature, led to the (Mehalic, Dicus & Kurkov, 1977). Of most
conclusion that a conservative design sys- interest were the flutter responses mea-
tem was possible only within certain sured in the stationary reference frame.
limits. More importantly, the aero As reported by Kurkov (1978, 1980,
elastic data used to assess the model 1981) and Jeffers, et. al., (1978), the
yielded valuable information concerning
the stall flutter phenomena.

Steady Slats Point No. 125

Flutter Frequency: 8.448E (1078 Hz)

14

Figure 11. Static Pressure Transducer Frequencies During Flutter Indicate Presence
of Several Nodal Diameter Modes and Wave Directions.
21-6

stationary sensors in combination with Thus, stall flutter occurs in complex

the rotating strain measurements showed modes whose contributors are determined by
that although stall flutter occurred at a aerodynamic and structural asymmetries.
single frequency relative to the rotating Although mistuning has been shown to be
reference frame, the flutter mode shape stabilizing and a conservative design
was comprised of several nodal diameter system could be based on tuned aero-
patterns traveling in both the forward and mechanical systems, a comprehensive flut-
backward directions with respect to disk ter prediction system capable of verify-
rotation. In the stationary reference ing an unsteady aerodynamics model for
frame, the frequency of the contributing stall flutter would have to account for
mode was displaced from the non-integral mistuning effects.
flutter frequency by positive or negative
multiples of engine order depending upon CONCLUDING REMARKS
nodal diameter and wave direction (Figure
11). The blade-to-blade maximum amplitude An encouraging amount of progress has been
response varied considerably around the made over the past decade in the under-
circumference of the disk similar to that standing of the stall flutter phenomenon.
observed for forced excitation of mistuned The parametric effects of inlet flow, par-
blade-disk assemblies (Ewins, 1974). ticularly temperature and pressure, and of
Further, the modal contributions and the engine operating conditions have been
resulting blade-to-blade amplitude and extensively demonstrated in experiment.
phase angle responses were found to vary Analytical aerodynamic and structural
with inlet and engine operating modeling has become more ambitious and
conditions. sophisticated with, in many cases, promis-
ing results. However, no fully substan-
tiated stall flutter prediction system
has as yet been developed.
 22-1

FORCED VIBRATION AND FLUTTER DESIGN METHODOLOGY

by
Lynn E. Snyder and Donald U. Burns
Allison Gas Turbine Division
General Motors Corporation
Indianapolis, Indiana 46206

INTRODUCTION under forced vibration design the areas

of source definition, types of com-
Prevention of high cycle fatigue in ponents , vibratory mode shape definitions
turbomachinery components is the aim of and basic steps in design for adequate
the structural designer. High cycle high cycle fatigue life will he present-
fatigue considerations account for a ed. For clarification a forced vibration
significant percentage of development and design example will be shown using a high
operational costs of a gas turbine en- performance turbine blade/disk component.
gine. In development, costly time delays Finally, types of flutter, dominant flut-
and redesign efforts may be incurred due ter parameters, and flutter procedures
to high cycle fatigue failures of com- and design parameters will be discussed.
ponents. Decreased reliability, short- The overall emphasis of this chapter is
ened time between overhauls, and in- on application to initial design of
creased need for spares may be associated blades, disks and vanes of aeroelastic
with high cycle fatigue failures. These criteria to prevent high cycle fatigue
also add to the costs of operation of gas failures.
turbine engines. Based on the accumulat-
ed knowledge of the cause of high cycle CHARACTERISTICS OF FLUTTER AND FORCED
fatigue, empirical and analytical design VIBRATION
tools to aid the designer have been and
continue to be developed. Proper appli- The classification of the two types of
cation of these design aids leads to the vibration which can cause high cycle
ultimate goal of eliminating high cycle fatigue failures is delineated by the
fatigue from gas turbine engines through relationship between the component dis-
judicious design of turbomachinery com- placement and the forces acting on the
ponents. component. Forced vibration is defined
as an externally excited oscillating
This chapter will cover the aeroelastic motion where there are forces acting on
principles and considerations of design- the component which are independent of
ing blades, disks and vanes to avoid high the displacement. Where the nature of
cycle fatigue failures. Two types of the forces acting on the component are
vibration that can cause high cycle functions of the displacement, velocity
fatigue, flutter and forced vibration, or acceleration of the component, and
will first be defined and the basic
governing equations discussed. Next,

UNSTEADY
FORCES
STRUCTURAL FUNCTION
DYNAMIC UNSTEADY FORCE*
PROPERTIES F (BLADE MOTION)

STRUCTURAL AERODYNAMIC
DAMPING
DAMPING UNSTEADY FORCE:
F (BLADE MOTION)

Figure 1. Primary Elements of Flutter and Forced Vibration,

22-2

these forces feed energy into the system, For either forced vibration or flutter,
the self-induced oscillations are clas- the response (equilibrium amplitude) of
sified as flutter. Therefore to avoid the component is equal to the work done
forced vibration and flutter through by the component. For forced vibration,
design requires an accurate knowledge of the equilibrium amplitude is reached when
the forces and the dynamic properties the work done on the component by the
of the structural component involved. external forcing function is equal to the
work done by the structural damping force
A simplified view of. the forces and the and by the aerodynamic damping force.
dynamic characteristics of the structural This work balanced is expressed as:
component are shown in Figure 1. The
basic equation of motion shown combines WORK/CYCLE) WORK/CYCLE) (1)
IN OUT
the structural dynamic properties on the
left side of the equation with the
forcing function on the right. The WORK/CYCLE)FORCING (2)
dynamic properties of the component are
based upon the mass (M) and stiffness (K) FUNCTION
matrices of the system from which natural
undamped frequencies (>o) and mode shapes = WORK/CYCLE)
STRUCTURAL
are determined. The force required to
move the component in each mode shape is DAMPING
dependent on the structural damping (C)
of the system. WORK/CYCLE)
AERODYNAMIC
Definition of the forcing function is DAMPING
divided to distinguish between external
and self-induced forces. External forc- For flutter, equilibrium is reached when
ing functions which are independent of the work on the component by the self-
component displacement can be generated induced force, aerodynamic damping, is
by such things as air flow nonuniformi- equal to and opposite in sign to the work
ties or by mechanical mechanisms such as done by the component by the structural
rub. The aerodynamic force which is damping force. This is expressed as:
created as a result of the component's
displacement is classified as a self- WORK/CYCLE) (3)
AERODYNAMIC
induced force called aerodynamic damping.
This self-induced force may either be DAMPING
stabilizing (positive aerodynamic damp-
ing) or destabilizing (negative aero- s. WORK/CYCLE)
dynamic damping).
DAMPING

Structural dynamic* analyses Unsteady aerodynamic analytet

Prediction of natural modm and frequencies Provide time-dependent preuure
of structure, Including effecti of distribution du* to airfoil motion and/or
nonoxl symmetry flow nonunlfoimity

L Aeroelaitlc analyses

Forced response prediction analyiti

Flutter analyui
Goal i Avoidance In design phase
of aeroelastic instability
and limiting vibratory stress
due to resonance

Structural damping model Distortion analysis

Quantify sources of Inherent damping Prediction of itrengm of excitation source

nich at material and Interface friction such o» Inlet distortion, upstream raw waltM
Design ruin for optimum application of Attenuation of strength with distance from
damping materials source

Figure 2. Elements of an Aeroelasticity Design Analysis.

 22-3

The key elements of an analytical design external forces acting on the blade, disk
system for aeroelastic response pre- or vane component. The type of component
diction is shown in Figure 2. This geometry can be tailored to limit or
system can be used to predict steady lesson the effects of these forces
state (equilibrium) response of turbo- through displacement limitation, fre-
machinery components to forced response quency tuning, mode selection and/or
or flutter with the ultimate goal being damping control. Accurate calculation of
elimination of HCF failure in the design the undamped natural frequencies and mode
phase. The basic elements of the shapes is required to effect an accept-
equation of motion are shown here with able geometry for forced response. These
the structural dynamic properties on the areas will be discussed and an example of
left side and the forcing function on the the basic steps in forced vibration
right. The structural dynamics analyses design will be presented in this section.
used presently are based on finite
element techniques and are able to
accurately predict natural modes and Sources of Unsteady Forces
frequencies of blade, disk and vane
structures. Structural damping is defined The most common aerodynamic sources of
by qualifying the various sources of forced vibration are shown in Table 1.
damping such as material and interface Aerodynamic sources due to structural
friction. Recently structural damping blockages to the flow are mainly due to
has been measured by Srinivasan (1981) the upstream or downstream airfoil rows.
and Jay (1983). Damping materials have Upstream vanes and struts create a
been identified for optimum application periodic unsteady flow field for down-
to various designs to improve flutter stream rotating blade rows. Likewise the
characteristics and/or reduce forced viscous flowfield of rotating blade rows
vibration responsiveness. D.I.G. Jones creates a periodic unsteady flowfield for
(1979) gives an extensive list of efforts downstream stationary vanes and struts.
to increase/add structural damping to Generally, vanes, struts and blades are
components. Prediction of the strength equally spaced circumferentially but if
of the forcing function due to aerodynamic they are nonuniform in a) circumferential
disturbances is also required. Research location b) shape (i.e., thickness,
to acquire data and model such distur- camber, trailing edge thickness, chord)
bances to provide an experimentally or c) setting angle, for example, then
verified analytical prediction system has the unsteady downstream flowfield will
been carried out by Callus (1982a, contain harmonics of the pattern which
1982b). Research to develop unsteady may coincide in the operating speed range
aerodynamic analyses to calculate the of the engine with a natural frequency of
time-dependent pressure distribution due a downstream airfoil structure.
to airfoil motion and/or flow uniformity
has been conducted by Smith (1971), Downstream vanes and struts can also
Caruthers (1980). All of the elements create a periodic unsteady flow field for
shown are necessary to adequately design upstream rotating blade rows. Likewise
components for forced vibration or flut- potential flow effects of rotating blade
ter considerations. rows create a periodic unsteady flowfield
for upstream vanes and struts.
The preceding definitions and equations
form the basis for the design systems Asymmetry in the stationary flowpath can
used for preventing high cycle fatigue of cause unsteady forces on rotating (rotor)
gas turbine blades, disks and vanes. airfoils. Examples of flowpath asymmetry
These design systems are largely centered are a) rotor off center, b) non-circular
on defining the sources and/or mechanism case and c) rotor case tip treatment.
of forcing function generation and
accurately predicting the aeroelastic Circumferential inlet flow distortion can
properties of the component. The success be a source of unsteady forces on rotating
of a design system is directly dependent blade rows. A non-uniform inlet flow con-
on how well it can define these elements dition creates unsteady forces on the ro-
of forced vibration and flutter. Use of tating (rotor) airfoils. The strength and
empirical relationships are still re- harmonic content of the forcing function
quired as a substitute for exact defini- produced will be dependent on the magni-
tions of some elements. Estimates of tude of the velocity/pressure/temperature
values of certain elements based on defect and the radial and circumferential
experience are needed. These approx- extent of the distortion.
imations compromise the ability of the
designer to completely avoid high cycle
fatigue of blades, disks and vanes but Table 1.
are used to prevent most high cycle
fatigue problems. As more exact defini- Sources of Unsteady Forces in Rotating
tions of these elements are obtained Turbomachinery Structures.
through experimental and analytical
approaches, the designer will be able to o Aerodynamic sources
more adequately attain the goal of a Upstream vanes/struts (blades)
elimination of high cycle fatigue failure <> Downstream vanes/struts (blades)
in turbomachinery components. o Asymmetry in flowpath geometry
0
Circumferential inlet flow distortion
Additional references on aeroelasticity (pressure, temperature, velocity)
are Scanlan (1951), Bisplinghoff (1955, o Rotating stall
1962) , and Fung (1955). a Local bleed extraction
FORCED VIBRATION DESIGN 0
Mechanical Sources
0
0
Gear tooth meshes
Forced vibration is the result of Rub
22-4

Circumferential inlet flow distortion Gear tooth (NT = number of teeth) meshing
taking the form of velocity, pressure or can be a source of such translation
temperature variations at the inlet to motion of the rotor system (fTR = NT x
the compressor or turbine can induce high fN). Therefore whenever the blade fre-
sinusoidal forces through the length of quency matches the order frequency due to
the compressor or turbine. Crpsswinds or the number of gear teeth plus or minus
ducting at the compressor inlet may one, excitation is possible (fc * <N T x
produce distortion patterns of low order fN) + fN or fc (NT N.
harmonic content. Combustor cans, because
of the variations in operation, may
produce temperature patterns of low order Considering the case of torsional vibra-
harmonic content. Even annular combustors tion of a rigid rotor, all blades experi-
may produce velocity/temperature patterns ence the same in-phase excitation forces
of low order harmonic content which are at any instant, independent of whether
due to circumferential flow variations. the rotor is turning. Each blade will
resonate when its natural frequency (fc)
equals the rotor torsional vibrating fre-
Rotating stall zones are another source quency.
of aerodynamic blockage which can produce
high response in blade, disk and vane TR (5)
components. Stall zones are formed when
some blades reach a stall condition Again, gear tooth meshing can be a source
before others in a the row. A zone(s) of of such £torsional motion of the rotor
retarded flow is formed which due to system ( TR - NT * *$)• Therefore when-
variations of angle of attack on either ever the blade frequency matches the
side of the zone begins to rotate other frequency due to the number of gear
opposite the rotor rotation direction. teeth, excitation is possible.
This speed of rotation has been observed
to be less than the rotor speed. Thus, Rub, as a source of forcing function, can
the zone(s) alternately stalls and un- produce high response in components.
stalls the blades as it rotates. The Contact of a rotor blade tip with the
number, magnitude and extent of the zones stationary casing locally, may cause an
and the relative speed between zone initial strain 'spike" of the blade
rotation and blade, disk or vanes followed by strain decay in a natural
determines the magnitude and frequencies mode. At its worst the rub excitation
of the forcing function available for frequency will be equal to a blade
component excitation. High stresses natural frequency. Causes of contact may
observed with this source of excitation be related to rotor unbalance response,
can lead to quick failure of turbo- ovalizing of the case, casing vibration
raachinery components. characterized by relative blade to case
radial motion, casing droop, and non-
Local bleed extraction, where air flow is uniform blade tip grind.
not removed uniformly around the case
circumference, may produce unsteady
forcing functions which may excite Types of Turbomachinery Blading
natural modes of blade and disk com-
ponents. Stages upstream and downstream There are may types of turbomachine
of the bleed locations have been observed blades and vanes. Table 2 is a partial
to respond to harmonics of the number of list of the types of blades and vanes.
bleed ports. Each of these descriptors have a definite
impact upon the dynamic properties of the
components. They describe some aspect of
Tooth meshing of a gear that is hard the component design from how it is sup-
mounted on the same shaft is a common ported, general shape, structural geo-
mechanical source of blade forced metry, material, to its aerodynamic de-
vibration excitation. Rotor blade sign.
failure is possible when the rotor system
is excited in a natural mode in which Some examples of turbine blade and disk
there is high vibratory stress at the geometries are presented in Figure 3. As
blade root. The mechanism of this exci- shown, blades may be integrally cast with
tation can be illustrated by examining blades or may be separate and have at-
two examples. tachments at the blade root. The dif-
ferences in dynamic characteristics of
Consider a rotating rigid rotor system each of the blades must be accurately
vibrating in a fixed plane, the instanta- considered during the design of testing
neous direction of acceleration that is phases.
applied to each blade root differs from
blade to blade for a total variation of Table 2.
360» around the rotor. Each blade will
resonate when its natural frequency (fc) Types of Turbomachinery Blades and Vanes
equals either the sum or differenceand of
the rotor translation frequency (fjR) Blades Vanes
the rotational frequency (CN)>
Shrouded/shroudless Cantilevered/
f f {4) Axial/circumferential inner banded
fc ' TR ± N attachment High/low
This equation defines phase equality be- Stiff/flexible disk aspect ratio
tween the vibrating blade and the forcing High/low aspect ratio Solid/hollow
function dynamics. If the rotor was not High/low speed Metal/ceramic
rotating only those blades which were Solid/hollow Compressor/
normal to the plane of translation would Fixed/variable turbine
be resonant due to common phase equality.
 22-5

INTEGRAL CAST • FIRST STAGE

PACKET, INTEGRAL CAST - THIRD STAGE

AIRCOOLED - HRST STAGE

PAIRED BLADES -
SECOND STAGE

Z SHROUD - FOURTH STAGE

Figure 3. Turbine Blade Configurations,

BLADES VANES
• VANE ALONE*
• STIFF DISK Oft CIRCUMFERENTIAL aa^-mx -_*»aaKtt»j
A / ',
ATTACHMENT*

• VANE/SHROUD ASSEMBLY*

HSK/BLADE/SHROUD** * MODES 1,2, 3

NB
* * NODAL DIAMETERS 0.1, 2 _.
"2
FOR MODES 1.2,3..__

NB •- NUMBER OF BLADES

Figure 4. Types of Blade and Vane Vibratory Modes.

22-6

Metal/ceramic/ High/low first stage compressor blade is at the

composite pressure ratio bottom (hub) end of the airfoil. A vari-
High/low hub-to-tip ety of mode shapes characterized by node
radius ratio lines are identified as either basically
Compressor/turbine bending (B), torsion (T), edgewise (BW),
High/low pressure ratio lyre (L> or chordwise bending, or complex
(C). The finite element method used to
Natural Modes calculate the frequency and mode shapes
shows the excellent accuracy possible by
Natural modes and frequencies of the com- analytical means. This ability to predict
ponents are defined by the physical geo- natural modes is necessary in order to
metry of the component. These natural have accurate forced response and flutter
modes are described by the location of design systems.
node lines (zero motion) and general mode
deflection. Generic types of natural when several blades or vanes are tied
modes are shown in Figure 4. together and/or are a part of a flexible
disk, the combined dynamic properties of
The first, lowest frequency, mode of a the components couple to produce ad-
beam-like component is that mode which ditional modes called system modes {see
has no nodes present on its unfixed Figure 4). Packets of blades or vanes
surface. This is illustrated by the tied together have assembly modes in
stiff disk and vane alone modes. For a which combined bending/twisting of all
blade or vane Eixed at one end the motion blades take place at one natural fre-
is one of bending from side to side of quency Flexible disk natural mode
the whole structure. The fixed blade or shapes are characterized by line of zero
vane at both ends bends like a bow string mot ion across the diameter called nodal
in its first mode. For both types of diameters (ND). These may couple with
fixity the second, third, etc. modes the blade natural modes to produce system
become more complex with node lines ap- modes with elements of motion from each
pearing on the blade or vane. component's natural mode but at a new
natural frequency. Circumferential node
Actual holographic and calculated mode lines also may describe higher frequency
shapes for an unshrouded compressor blade natural modes of blade/disk systems.
are shown in Figure 5. The fixity of this

21 IT 1IW 3B

CALC TI5T CALC CALC TEST CALC TEST CALC TEST

39O 4O2 1768 21OO 2186 3323 3057 4541 4853

2T 3T 4B 4T IL

CALC TEST CALC TEST CALC TEST CALC TEST CALC TEST
5224 5473 8088 B610 8779 9214 HB84 mu 12866 13515

2L 3L 58 3T

rej S£
° E
q 2
2 n
aM

CALC TEST CALC TEST CALC TIST CALC TEST CALC TEST
13600 NOT 14115 14166 16018 15688 16233 17145 17741 17914
rouNo

Figure 5. Frequency and Mode Shape Correlation

 22-7

Holographic photographs (see Figure 6) of hollow blade is a low aspect ratio (length
a blade/disk system illustrate the rela- to width) design, with twenty-two (22)
tionship between nodal diameter pattern airfoils in the stage. A nickel alloy
and mode number. Three nodal diameter which has good structural properties at
patterns are shown with the 3ND pattern high temperature and stress conditions is
family expanded for the first three used in this design.
modes. The second mode is characterized
by one circumferential node while the Step one calls for an identification of
third mode has two. possible sources of excitation (forcing
function) while step two requires the
Ten Steps of Forced Vibration Design definition of the operating speed ranges
the component will experience. For the
The ten basic steps in designing to example turbine the possible sources and
prevent high cycle fatigue due to forced speed ranges are shown in Figure 9.
response are listed in Figure 7. These Several sources of aerodynamic excitation
steps involve evaluating the environment exist and are listed. Two upstream and
in which the component must operate two downstream sources have been identi-
(steps 1, 2, 5), predicting the aero- fied. Each of these sources creates a
elastic characteristics of the component periodic forcing function relative to the
(steps 3-8), investigating possible de- rotating second stage blade/disk com-
sign changes (step 9), and finally the ponent. The relevant content of these
actual measurement of the dynamic re- forcing functions will be the harmonics
sponse of the component in the engine associated with the second stage blade
environment (step 10). These steps will passing these stationary sources. The
be illustrated by examining the design of frequency of the forcing function is
a second stage gasifier turbine blade/ dependent upon the rotating speed of the
disk component. The choice of a turbine second stage blade. The speeds of pos-
instead of a compressor component was sible steady state operation are between
arbitrary since the steps are the same idle and design. Any resonance occuring
for each. below idle would be in a transient speed
range implying lower chance of accumu-
The example blade is an aircooled design lating enough cycles for failure or main-
incorporating the features listed in taining high enough response to produce
Figure 8. It is a shroudless blade which failure.
is integrally bonded to the disk. The

2677 Hi
MODE 1, 3ND

CIRCUMFERENTIAL
NODE

2356 Hz 6858 Hz 2985 Hz

MODE 1. 2ND MODE 2, 3ND MODE 1. 4NO

CIRCUMFERENTIAL
NODES

HD-NODAL DIAMETER 13457 HZ

MODE 3. 3ND

Figure 6. Mode Number and Nodal Diameter Pattern,

22-8

STEP 1 IDENTIFY POSSIBLE SOURCES OF EXCITATION

STEP 2 DETERMINE OPERATING SPEED RANGES
STEP 3 CALCULATE NATURAL FREQUENCIES
STEP 4 CONSTRUCT RESONANCE DIAGRAM
STEP 5 DETERMINE RESPONSE AMPLITUDES
STEP 6 CALCULATE STRESS DISTRIBUTION
STEP 7 CONSTRUCT MODIFIED GOODMAN DIAGRAM
STEP 8 DETERMINE HIGH CYCLE FATIGUE (HCF) LIFE (FINITE
OR INFINITE)
STEP 9 REDESIGN IF HCF LIFE IS NOT INFINITE
STEP 10 CONDUCT STRAIN GAGED RIG/ENGINE TEST TO VERIFY
PREDICTED RESPONSE AMPLITUDES

Figure 7. Summary of Basic Steps in Designing to Prevent High

Cycle Fatigue Created by Forced Vibration.

COOLING AIR OUT

* FEATURES
• 3 CHANNEL SINGLE-PASS COOLING
• METERED COOLING AIR
• TIP DISCHARGE
• WALL-TO-WALL PIN RNS

COOLING AIR IN METERING TUBE

Figure 8. Gas Generator Second Stage Blade Cooling Geometry,

 22-9

Figure 9 also notes that the spacer and To determine if the natural frequency of
disk have been designed to be in constant a blade coincides with the frequency of a
contact throughout the engine operating source, resonant condition, a resonance
conditions. This contact limits the disk diagram is constructed (step 4). A
flexibility and eliminates the disk resonance diagram relates frequency to
participation in the assembly modes. rotational speed as shown in Figure 11.
Therefore accurate prediction of natural Since the forcing function frequency is
modes, step 3, can be made for this type dependent on rotational speed, lines of
of design by modeling only the blade concurrent frequencies can be drawn for
geometry and fixing the blade at the various harmonics (i.e., 1,2,3,.... 10,
proper radial location. ... 13 ,... 19,. .21... sine waves per revo-
revolution of the rotor) of engine speed
The natural modes of a blade as compli- for which sources exist. Placing of the
cated as this example, can be calculated calculated natural frequencies on the
using finite element techniques developed diagram with the lines of concurrent
especially for rotating turbomachinery frequencies, engine order lines, of the
components. The natural frquencies of known sources, identifies possible reso-
the blade have been calculated using a nant conditions of a component natural
model constructed with triangular plate frequency coinciding with a forcing func-
elements. The elements have been used to tion, source frequency. In this example,
simulate the hollow airfoil, platform, a dropping of natural frequency with
and stalk geometry as shown in Figure 10. rotor speed indicates that temperature
The stiffness and mass matrices formed by effects are dominant over centrifugal
these elements are solved to compute the stiffening within the operating speed
natural frequencies. The more elements range.
used, the closer to the actual blade is
the mathematical model. The possible resonant conditions are
identified by intersection of natural
This method of calculation is shown to b« frequencies and order lines which occur
accurate by the comparison of frequencies within or near to the steady state
and mode shapes of test holograms with operating speed range. The strongest
those of the finite element model. This expected aerodynamic sources of excita-
comparison is for zero RPM and room tem- tion are those immediately upstream (in
perature conditions. Additional calcula- front of) and downstream (behind) the
tions are made for various temperature blade. The amount of response (step 5),
and rotational speeds to determine depends not only on the strength of the
natural modes at the actual operating forcing function, but also upon the
conditions. aeroelastic characteristics of the com-
ponent. For this example, the dynamic

DESIGN SPEED=48,500 RPM

IDLE SPEED*32,000 RPM
DISK BOUNDARY CONDITION = SPACER AND DISK
IN CONSTANT CONTACT

SOURCES:
13 1ST STG VANES
19 2ND STG VANES
21 LP1 VANES
5 EXIT STRUTS
(NOT SHOWN)

Figure 9. Gas Generator Turbine General Arrangement

22-10

FINITE ELEMENT MODEL 7050 HZ

MODE1
7018 HZ
(,4%)

13279 HZ 19970 HZ

MODE 3
20225 HZ

Figure 10. Calculation of Natural Frequency for Second Stage Blade.

• POSSIBLE RESONANCES
O RESONANCES OF MAIN CONCERN
IDLE DESIGN
20
21E (LP1V)

19E (HP2V)
15

13E {HP1VJ

10
10 E(2X STRUTS)

10 20 30 40
ROTOR SPEED-RPMX 10 *3

Figure 11. Resonance Diagram for Second Stage Blade.

 22-11

characteristics for the first torsional The response of turbine blade modes to
mode (IT), mode shape and damping are the turbine downstream vane source of
significant in determining possible re- excitation has been shown to be related
sponse amplitudes. to the variation of static pressure in
front of the vane source. Figure 13. iP/Q
The response of modes of blades due to is a calculated value based on the aoro-
aerodynamic sources of excitation have dynamically predicted static pressure
been empirically defined based on experi- field created by the presence of the vane
ence. This empiricism groups typical in the airflow stream. This pressure
blades by common mode shape, damping, field is dependent upon aerodynamic
type of source, and distance from the characteristics (velocity triangles, mass
source to correlate with response experi- flow, etc.) and vane cross-sectional
ence. The use of an empirical method for geometry. Plotting iP/Q versus a nor-
estimating response is due to a current malized value of axial gap allows the
inability to adequately predict the designer to space (gap) the blade-vane
strength of the forcing functions pro- row to avoid a larqe forcing function. A
duced and the damping present in the gas specif ied lirnit would be based upon
turbine environment. turbine blade experience to date and
represents the maximum value of fiP/Q that
For example, the response of first tor- is considered acceptable. The second
sional modes of turbine blades due to an stage blade range (based on build toler-
upstream vane source might be empirically ances ) indicates an acceptable value of
defined as in Figure 12. A plot of axial fiP/0 and thus indicates that a low
gap/vane axial chord (rate of decay of response due to the third vane row is
forcing function) versus the vane overall expected.
total to static expansion ratio (forcing
function) may allow the designer to pick Calculation of dynamic stress distribu-
the combination of variables that will tion, step 6, is necessary for determin-
ensure a viable design. A range of blade ing locations of maximum vibratory stress
gap/chord values is defined for the for high cycle fatigue assessment and as
second stage based upon build up toler- an aid in the placement of strain gages
ances from engine to engine of the to measure strain due to blade motion
various rotor components. This range during engine operation. A number of
indicates that a maximum dynamic stress these gages placed in various positions
of six to ten thousand psi would be on the airfoil can be used to qualify the
expected for the first torsional mode relative responses of the blade at each
coincidence with the upstream vane order location for each natural mode. A
line. distribution of stress for each mode is
thus identified.

6 789101214 MAXIMUM FIRST TORSION

MODE STRESS LEVEL ±KSI

GAS GENERATOR SECOND

CD

VANE OVERALL TOTAL STATIC EXPANSION RATIO

Figure 12. Prediction of Response due to Second Stage Vane

22-12

MAXIMUM STATIC PRESSURE • MINIMUM STATIC PRESSURE

" DYNAMIC HEAD

HIGH STRESS
AP
/Q REGION
SPECIFIED LIMIT

SECOND STAGE BLADE

RANGE

J L-J l 1 I I I
DOWNSTREAM AXIAL GAP/VANE TANGENTIAL SPACING

Figure 13. Prediction of Response due to Third Stage Vane

CALC
1ST STG. TURBINE BLADE BENCH
X MAX
• HOLLOW
• AIR COOLED PRESS SUCTION
• IMPINGEMENT TUBE T.E. T.E.
• DIFFUSION BONDED TO WHEEL TIP

PRESSURE STRAIN
. r GAGES

\ HUB L

i
-1 +1 -1 +1
S CRO MN FIRST BEND
w [il
j
5

j HUB
INSTRUMENTED BLADE -1 +! -1
FIRST TORSION
TIP

HUB L
FINITE ELEMENT -1 +1 -i +
MODEL SECOND BEND
REtAMP. RELAMP.
Figure 14. calculation of Dynamic Stress Distribution for
First Stage Blade—Trailing Edge.
 22-13

An example of a first stage turbine blade Parameters which can affect the distribu-
stress distribution is shown in Figures tion of mean fatigue strength and, thus
14 and IS. The analytical results are allowable vibratory stress, are notch
based upon finite element calculations factor, data scatter, and temperature.
and ahow good correlation with test data. These are illustrated in Figure 17. The
The results of the bench test are limited fatigue notch factor is related to a
to the number of and direction of the stress concentration factor which is a
gages, while the analytical results cover ratio of the maximum steady stress to the
all locations and directions. Use of average steady stress of a particular
analytical stress distributions to pick a geometry (notches, fillets, holes). The
limited number of gage locations on the relationship between fatigue notch factor
blade helps to ensure the best coverage (KKj), and stress concentration factor
of all modes of concern. It should be ( t), is dependent upon the notch sensi-
noted that one gage location may not tivity of the material. The one-sigma
cover all modes of concern since maximum scatter, which is obtained from test,
locations vary with mode shape. data, accounts for variations in mean
fatigue strength due to compositional
To determine the allowable vibratory changes of the material and processing
stress for various locations on the blade differences from piece to piece. A minus
a diagram relating the vibratory and three sigma (-30) value of fatigue
steady state stress field is used, step strength used accounts for 99.865 percent
7. A typical modified Goodman diagram ia of all pieces having a fatigue strength
shown in Figure 16. Material properties greater than this value. The temperature
are normally obtained through testing at affects both the ultimate and fatigue
several temperatures with smooth bar strengths. The example shows a dip in
samples, no notches or f i1lets. From fatigue strength with temperature which
these properties mean ultimate strength is characteristic of some alloys used in
at zero vibratory stress and mean fatigue turbine blades.
strength at 107 cycles (or infinite life)
of vibration at zero steady stress are
placed on the diagram. A straight line
is drawn between these two values which,
for most materials/ is a conservative
mean fatigue strength as a function of
loading (steady stress).
CALC
BENCH
1ST STG. TURBINE BLADE X MAX
• HOLLOW
• AIR COOLED
• IMPINGEMENT TUBE
CROWN TIP LE.
• DIFFUSION BONDED TO WHEEL

PRESSURE CUES
HUB
+1 -I
\
; CRI))VN FIRST BEND
ti S
t-
TIP

/
II
(
1 g
ia »—.

, HUB
-1 -1
INSTRUMENTED BLADE
FIRST TORSION
TIP

HUB
FINITE ELEMENT
-1 -t-1 -1
SECOND BEND
MODEL
RELAMP. REL.AMP.

Figure 15. Calculation of Dynamic Stress Distribution for

First Stage Blade—Leading Edge and Crown.
22-14

SMOOTH BAR, 10 ' CYCLES. 800'F, MEAN FATIGUE STRENGTH

NOTCHED, 10 7 CYCLES, 800'F, MEAN FATIGUE STRENGTH

Kf=1.25
t NOTCHED, 10 7 CYCLES, 800'F, -StrFATOUE STRENGTH

ALLOWABLE
VIBRATORY STRESS
MARGIN N. 800'F,
ULTIMATE STRENGTH
800'F, MEAN
ULTIMATE STRENGTH
STEADY STRESS

Figure 16. Typical Modified Goodman Diagram.

FATIGUE NOTCH FACTOR, Kf,

ACCOUNTS FOR NOTCHES 2.0
(FILLETS, NICKS, FDD, HOLES)
1.0
141 2.0 3.0
Kt, STRESS CONCENTRATION
FACTOR
-3 <r ACCOUNTS EXAMPLE ON PREVIOUS CHART
FOR DATA SCATTER ASSUMES T - 7%
.'. -3' - 21% DEGRADATION

TEMPERATURE AFFECTS
ULTIMATE AND FATIGUE
STRENGTH
TEMPERATURE, 'F

Figure 17. Parameters which Affect the Allowable Vibratory Stress

 22-15

Returning to Figure 16, degradation of the stress values for the two locations of
line representing the mean distribution of concern, yields maximum allowable stresses
fatigue strength is made for notches, of 15.3 and 14.3 ksi. These values of
scatter and temperature. Where notches or maximum allowable vibratory stress are
fillets exist, it is necessary to degrade above what is expected based upon evalua-
the mean fatigue strength by the notch tion of the upstream and down stream vane
fatigue factor (Kf) . Lowering of both row sources. Thus, the high cycle fatigue
the mean fatigue and ultimate strengths life is predicted to be infinite and no
for -3a scatter effects is made. Tempera- redesign is required.
ture effects which also affect both fa-
tigue and ultimate strengths was included Redesign, step 9, was not necessary in
initially in establishing their values. this example, but if it had been, design
The lowest line now represents the distri- changes such as those listed in Table 3
bution of fatigue strength versus steady would have been reviewed. Redesign
stress at temperature for a specified considerations fail under the categories
notch factor and for which 99.865 percent of changes to the source, component geo-
of blades produced will have a greater metry, fixity, damping, material and pos-
strength. sible maximum amplitude. Changing the
proximity of sources may lower the forc-
This diagram is now entered at the steady ing function strength and thus blade re-
stress value based on the resonance speed sponse. The frequency of resonance may
and specific component location of con- be moved to occur outside the operating
cern. The maximum allowable vibratory range by changing the number of sources
stress level for infinite life', 10' and thus forcing function frequency.
cycles in this case, is then read. The Geometry changes to the sources may be
difference between the maximum allowable made to lower the disturbance factor
vibratory stress level and the predicted (i.e., fiP/0).
vibratory stress level is called the
vibratory stress margin. Table 3
The modified Goodman diagram for the Typical forced vibration redesign
example second stage blade is based on considerations
MAR-M2465 material properties at 1300»F,
Figure 18. The mean ultimate and the « Proximity of sources (gap/chord,
mean fatigue strengths are 142 and 31
thousand pounds per square inch (Ksi) o Number of sources (resonance speed)
respectively. The one-eigma scatter for
each strength is 6 and 2.2 ksi, respec- « Geometry of sources (lower disturbance)
tively. The stress concentration factor
(Kt) for the hub fillet radius is 1.36. o Geometry of resonant piece (stiffness
For this material and fillet radius, this and mass distributions )
gives a notch factor (Kf) of 1.18. De-
grading the mean strengths by the respec- « Boundary conditions (type of fixity)
tive factors and entering the steady
o Increase system damping (coating,
fixity)
* Amplitude limitation (shroud gap)

40 0
Increase fatigue strength (geometry,
material, temperature)

CO
a*
•
eo MEAN (Kf=1.0) MAR M246 at 1300'F
CO 30
HUB, CCTE-15.3 KSI
CO MAX ALLOWABLE DYNAMIC STRESS
HUB, CROWN-14.3 KSI
20 MEAN (Kf =1.18)
OB
CD •3<r(Kf»1.1B)

10
42.000 RPM
- HUB, CCTE 42.000 RPM
HUB, CROWN

60 80 100
STEADY STRESS--KSI

Figure 18. Modified Goodman Diagram for Second Stage Blade.

22-16

Component geometry changes such as varia- To maintain or increase fatigue strength

tions in thickness or chord distribution material or processing changes may be
which change stiffness and mass distribu- required. Choosing materials which in-
tions can be used to raise or lower crease corrosion resistance, decrease
natural frequencies and thus move reso- notch sensitivity or increase tempera-
nances above or below operating speed ture capability can help the designer to
limits. Geometry modifications such as attain the necessary high cycle fatigue
changing of fillet sizes or local thick- strength. Processes such as airfoil
ening can be used to lower notch effects coating, peening, brazing, grinding, heat
and thus increase allowable vibratory treating all affect fatigue strength and
stress. should be considered in the design pro-
cess.
Changes to boundary conditions to raise
or reduce fixity may be used to raise and For the example chosen, testing to verify
lower natural frequencies, again to move the predicted response amplitudes, step
resonances above or below operatng speed 10, was accomplished using a gasifier
limits. Some examples of fixity changes turbine rig with the instrumentation con-
which affect blades and vanes include figuration shown in figure 19. The
disk rim restraint aa In our example, amount and location of the vibratory
slot and tang restraint of vanes, blades instrumentation to cover the rotating
and vanes cast rigidly to support struc- components is shown. The testing prior
ture, use of Z shrouds for turbine to rig running included determining fre-
blades, pinned retentions for compressor quencies, mode shapes, stress distribu-
blades and packeting of blades and vanes tions and fatigue strengths for the
together. These changes may also in- second stage blade. Note that in the
crease or decrease damping by modifying absense of the low pressure turbine, th*
the frictional losses. Damping may also third stage vane row was not present.
be increased by application of a selected
coating to the blade or vane surface. The vibratory responses of the second
stage blade obtained during the gasifier
Amplitude limitation has been used mainly test are shown on the predicted resonance
in turbines to reduce the amount of diagram in figure 20. The two strain
response. Reducing gaps between adjacent gage locations of crown and concave
shrouds will limit motion in modes where trailing edge are denoted by the circle
the adjacent blades are not in phase when and triangle symbols respectively. The
at resonance. Application of a wear percent gage sensitivity for the first
resistant coating to the shroud faces is torsion and first bend modes are bracket-
usually required to limit material loss ed directly below the symbols. The
due to wear. maximum vibratory allowables for these
locations are ±14.3 and ±15.3 ksi.

"Ul

5 STRUTS

VIBRATORY STRESS INSTRUMENTATION

RRST STAGE BLADE S GAGES
16 HUB, CCTE
46 HUB.CVTE
SECOND STAGE BLADE 3 GAGES
1 HUB, CROWN
2§ HUB,CCTE
SPACER DISK 2 GAGES (WEB-RIM FILET)
1@0'C1RCUMFEREMT1ALLY
16 30'CIRCUMFERENT1ALLY

Figure 19. Gasifier Test Configuration.

 22-17

Maximum measured stresses were ±6.3 to FLUTTER DESIGN

±8.8 ksi for the IT mode and upstream
vane resonance. This compares to a Designing to avoid flutter is an important
ratioed maximum IT response of ±13.3 ksi. part of the aeroelastic design process for
The predicted response was estimated to aircraft engine fan and compressor blades.
be ±6 -10 ksi. The design shows a Although there is much more to be learned
minimum vibratory margin of 7.5% or ±1 about turbo-machinery flutter, there are
ksi using the ±13.3 ksi as a maximum basic principles that have been developed.
response. In this section these principles will be
presented. This presentation will contain
Based on these measured responses this a discussion of the types of fan/compres-
design should exhibit infinite high cycle sor flutter and the dominant design para-
fatigue life. Additional instrumented meters associated with each. A definition
tests during actual engine running will of the ideal flutter design system and the
assure that adequate HCF margin is overall flutter design procedure will also
present by better defining the vibratory be included. Finally, a detailed review
response distribution (scatter) among of five types of fan/compressor flutter
blades. and empirical and analytical design sys-
tems for each will be presented.
This completes the discussion of forced
response. Basic sources of excitation, Description of Flutter
component geometries, natural mode shapes
and frequenc ies have been presented. The designer is interested in predicting
Fundamental steps in designing for forced the onset of flutter rather than pre-
response included discussion of sources, dicting a specific vibratory response
environment, resonance and Goodman dia- level as in forced vibration. As dis-
grams, calculation of natural frequen- cussed earlier, the blade vibration pre-
cies, mode shapes and stress distribu- sent during flutter is not caused by an
tions, determination of response ampli- unsteady external force but Instead by the
tudes and high cycle fatigue life, re- fact that the blade is absorbing energy
design considerations, and strain gage from the flow around the blade.
testing to verify the design.

IDLE DESIGN
20
21E (LP1V)
LOCATION CCTE CROWN
SYMBOL A »
15 ALLOWABLE 15.3 KSI 14.3 KSI 19E (HP2V)

o 8.8 (13.3 KSI). 13E

i 10
10E (2X STRUTS)
IB S/G SENS. (100) ±6KSI (6.9 KSI)

20 30
ROTOR SPEED-RPMX 10 "3

Figure 20. Resonance Diagram for Second Stage — Gasifier Test Configuration
22-18

Once any random excitation causes a small The criterion for stability requires that
vibration of the blade, if the blade the unsteady aerodynamic work/cycle re-
aerodynamic damping is negative, the main positive {i.e., system is not
blade will absorb energy from the air- absorbing energy). The unsteady aero-
stream as the blade vibrates. If the dynamic work/cycle is the integral over
energy absorbed from the airstream is one vibratory cycle of the product of the
greater than that dissipated by the in-phase components of unsteady force
structural damping, the blade vibratory (pressure times area) and unsteady dis-
amplitude will increase with time until placement.
an energy balance is attained. Random
excitation is always present at low (h0exp{l«tH (8)
levels in the turbomachinery environment.
Thus predicting the onset of flutter 211
entails predicting the aeroelaatic condi- h_exp{i<ot}d(ut) (9)
tions that exist when the absorbed energy
due to negative aerodynamic damping VIBRATORY
RESULTANT UNSTEADY
equals the dissipated energy due to DISPLACEMENT
structural damping at the equilibrium FORCE
vibratory stress level.
In-phase Components
For most blade/disk/shroud systems,
structural damping (i.e., frictional Thus positive aerodynamic damping is
damping , material damping, etc.) Is not related to the aerodynamic characteris-
large. Therefore, the stability (design) tics of the flow field (unsteady forces)
criterion essentially becomes positive and vibratory mode shape (displacement).
aerodynamic damping. Aerodynamic damping
is proportional to the nondlmensional Dependence upon the flow field Is noted
ratio of unsteady aerodynamic work/cycle by the names given to five types of
to the average kinetic energy of the fan/compressor flutter which have been
blade/disk/shroud system. observed and reported during the last
thirty-five years. These five are
SAERO -AERODYNAMIC DAMPING presented in Figure 21 on a compressor
performance map. Each of these types of
flutter is characterized by a distinct
UNSTEADY AERODYNAMIC WORK aerodynamic flow field condition. Each
(6) of these types will be discussed later
~ BLADE/DISK/SHROUD KINETIC ENERGY
with respect to avoidance of flutter In
design. References for this section are
Carta (1966) and Snyder (1974).

(NB)(ROOT-TO-TIP INTEGRATED UNSTEADY WORK/CYCLE) (7)

(4) (AVERAGE KINETIC ENERGY of B-D-S-SYSTEM)

SUPERSONIC
STALL
aUTTER
SURGE LINE / / . A100TYPE
o£>ffu& SUPERSONIC
SUBSON1C/TRANSONIC
FLUTTER
STALL
FLUTTER

CLASSICAL
UNSTAUED
ce SUPERSONIC
o_
FLUTTER

100%

CORRECTED MASS aOW RATE

Figure 21. Types of Fan/Compressor Flutter,

 22-19

Dominant Design Parameters fined as the ratio of relative inlet

velocity to the product of blade vibra-
Since the distinct flutter regimes are tory frequency and blade semi-chord
identified by the type of flow present, length.
it is not surprising that four of the
five dominant flutter design parameters v
involve aerodynamic terms. The five are Reduced velocity Fu (10)
as follows:
1. Reduced velocity Decreasing this parameter is stabilizing.
In general, the reduced velocity para-
2. Mach number meter is between 1 and 5 at the flutter
stability boundary. Examination of the
3. Blade loading parameter unsteady flow equations for flow through
a cascade of airfoils shows that the
4. Static pressure/density unsteady flow terms are important for
these levels of reduced velocity.
5. Vibratory mode shape
Blade loading parameters have also been
The fifth parameter, vibratory mode shape, used in correlations of flutter data.
Is necessary since the vibratory displace- Incidence or non-dimensional incidence,
ment directly affects the magnitude and pressure ratio, diffusion factor and
sign of the unsteady aerodynamic work per margin to choke are parameters which have
cycle. All five design parameters are been used to describe the blade loading.
pertinent to each type of flutter and are Blade shape geometry descriptors such as
important elements of the flutter design leading edge radius, maximum thickness/
system. chord ratio and the maximum thickness
location also indicate the magnitude and
The first two dominant design parameters distribution of aerodynamic loading.
are dimensionless and appear in the
governing equations for unsteady flow over Either static density or pressure can be
a vibrating airfoil. Mach number is a used as a design parameter if static
parameter which describes the nature of temperature is held constant and the flow
the unsteady flow field whether it be sub- of a perfect gas, p » PRT, is being con-
sonic, transonic or supersonic. It has sidered. The primary effect of changing
been seen that Mach number and reduced air density (or pressure) is a propor-
velocity are key parameters in correlat- tional change in unsteady aerodynamic
ing flutter data and developing empirical work/cycle and therefore in aerodynamic
design systems. Reduced velocity is de- damping.

CONSTANT
N AND WV*

AERODYNAMIC oMODEl
DAMPING, Q MODE 2
A MODE 3
'AERO

'LEAST STABLE
NODAL DIAMETER
AND WAVE DIRECTION
ACCURATE PREDICTION OF
• AERODYNAMIC DAMPING
• CORRESPONDING FLUTTER MODE SHAPE

FOR EACH WAVE DIRECTION OF EACH NODAL DIAMETER

OF EACH MODE

Figure 22. Definition of Ideal Flutter Design System.

22-20

Increasing the gas density is stabilizing of the airfoil about the minimum polar
if aerodynamic damping is positive. moment of inertia axis. Some cases of
Likewise, increasing the gas density is flutter have been encountered in a chord-
destabilizing if aerodynamic damping is wise bending with node lines nearly per-
negative. An indirect effect of changing pendicular to the airfoil chord. Since
gas density is that of changing the flut- blade modes generally contain chordwise
ter mode shape which is a weak function bending, bending and torsion motions,'the
of mass ratio. Thus aerodynamic damping modes can best be described in terms of a
is also a function of density through the generalized mode shape where motion per-
effect of air density on the flutter mode pendicular to the mean line is expressed
shape. Aerodynamic damping is also a as a function of radial and chordwise
function of density through the effect of position.
density on the Reynolds number and the
effect of Reynolds number on the unsteady References for this section are Pines
flow field. (1958) and Theodeorsen (1935).
The final dominant design parameter is vi-
bratory mode shape. The unsteady aerody- Design System and steps
namic work per cycle of blade motion is a
function of both the unsteady surface This dependence of flutter or vibratory
pressure created by the blade's motion in mode shape is illustrated in Figure 22 in
the air flow and the vibratory mode shape. the definition of the ideal flutter
Thus, since the blade unsteady surface design system, an experimentally verified
pressure distribution is also a function analytical prediction system. Classical
of the blade mode shape (motion), the supersonic unstalled flutter is one type
aerodynamic damping is a strong function of flutter for which such a design system
of the vibratory mode shape. Vibratory exists. The analysis which is a part of
mode shape may be described as pure bend- an ideal flutter design system considers
ing or torsion of the airfoil or a coup- the mode shape and frequency for each
ling of bending and torsion. Rigid body nodal diameter of each mode. It also
bending or translation of the airfoil is considers both forward and backward
displacement of the airfoil perpendicular traveling wave directions, Campbell
to the minimum moment of inertia axis. (1924). There is a least stable nodal
Rigid body torsion or pitching is rotation

STEP 1 - AERODYNAMIC DESIGN AND ANALYSIS

DESIGN ANALYSIS
• PRESSURE RATIO/STAGE • RELATIVE VELOCITY AND MACH NUMBER
• SPECIFIC FLOW • BLADE CAMBER AND CHORD
.... • 1NCIDENCE, Df, MARGIN TO CHOKE
~ 'inlet & exit -»• • STATIC AIR TEMPERATURE
• FLOWPATH TYPE • STATIC AIR DENSITY AND PRESSURE
• BLADE ASPECT RATIO
• THICKNESS/CHORD RATIO
• BLADE SERIES
• TAPER RATIO
STEP 2 - BLADE STRUCTURAL DESIGN AND ANALYSIS
DESIGN ANALYSIS
• BLADE MATERIAL w • STEADY STRESS
• NO. OF SHROUDS "*" • BLADE/<DISK/SHROUD) VIBRATORY MODE SHAPES
AND FREQUENCIES

STEPS- FLUTTER ANALYSIS AND REDESIGN

ANALYSIS REDESIGN
• BLADE STABILITY MARGIN -*> • CHANGE ASPECT RATIO
OR
• CHANGE BLADE THICKNESS
OR
•CHANGE TAPER RAT 10
OR
• CHANGE SHROUD LOCATION

Figure 23. Flutter Design Procedure.

 22-21
diameter and wave direction for each mode If the blade is predicted to exhibit flut-
and of these there is a least stable mode ter within the desired operating range, a
(i.e., mode 2) for the structure. The redesign effort must be initiated.
design system must accurately calculate Changes are usually in the form of geome-
the aerodynamic damping and corresponding try modifications which not only affect
flutter mode shape for each of these the system modal characteristics (dis-
modes in order to predict the stability placement and frequency) but also the flow
of the blade. Stability is determined by characteristics (velocity and blade load-
maintaining positive total damping (i.e. ing). These changes are aimed at obtain-
above dashed line) for all modes and all ing a design which exhibits stability,
nodal diameters and wave directions. positive total damping, throughout the
Furthermore, the aerodynamic damping must engine operating environment.
be determined at the least stable fan/
compressor steady state aerodynamic Detailed Review, Types of Flutter and
operating point. The impact of the Design
steady state aerodynamic operating point
will be discussed in more detail in later The types of flutter will now be dis-
sections. cussed with respect to the dominant de-
sign parameters and the available analyt-
The three steps in the flutter design ical tools. The purpose of this discus-
procedure are outlined in Figure 23. The sion is to demonstrate how to avoid fan/
first step is to perform the aerodynamic compressor flutter through judicious de-
design of the blade and to obtain the sign. Five types of flutter and the
pertinent aerodynamic parameters that location of their boundaries on a
have an impact on aerodynamic damping. compressor map are shown in Figure 21.
This should be accomplished at the most The order of discussion will be subsonic/
critical points within the predicted transonic stall flutter, classical un-
operating envelope of the engine. The stalled supersonic flutter, AlOO super-
definition of the modal displacement and sonic flutter, choke flutter and super-
frequencies of the blade/disk/shroud sonic stall flutter.
system is the next step. This can be
done by conducting a structural dynamic The first documented type of turbo-
analysis using a finite element model of machinery flutter, subsonic/transonic
the blade/disk/shroud system to determine stall, was first reported almost at the
the modal frequencies and mode shapes of same time that the performance of the
the natural modes of the system (i.e. ND first axial compressor was reported. At
» 0, 1, 2,...for modes 1, 2, 3...). The first, this type of flutter was confused
final step is to conduct the analysis to with rotating stall. Characteristics of
combine the steady state aerodynamics and the subsonic/transonic stall flutter
dynamics results and conduct a Clutter vibratory stress response are non-
analysis (unsteady aerodynamics analysis integral order, sporadic amplitude with
plus stability analysis). Depending on time, stress holds or increases with in-
the type of flutter this analysis may creasing stage loading and blades vibrate
entail an actual calculation of the aero- at different frequencies and amplitudes
dynamic damping (e.g. classical super- in same mode, whether bending, torsion or
sonic unstalled flutter) for each mode or coupled modes.
may involve an empirical correlation of
flutter data (stalled flutter) using the
dominant design parameters.

SURGE LINE
SURGE LINE
O

ec
g FLUTTER BOUNDARY
ut
CO OPERATING
LINE

50% 60% OPERATING LINE

V*
BLADE LOADING PARAMETER. INCIDENCE

(a)

Figure 24. Subsonic/Transonic Stall Flutter Design System Using

Reduced Velocity and Blade Loading Parameter.
22-22

All five dominant flutter design para- Based on this empirical design system, a
meters are needed to describe subsonic/ blade design may be stabilized by lowering
transonic stall flutter (S/TSF). By ita the blade loading parameter. This may be
very name, S/TSF is dependent on Mach accomplished by modifying the position of
number. The shape of the flutter boundary the operating line as shown in Figure 25a.
on the compressor map shows its dependence This change may be made through reschedul-
on a blade loading parameter such as inci- ing staters or re-twisting the airfoil.
dence or diffusion factor. The simplest Likewise, by changing the blade shape, the
S/TSF design system is a correlation of position of the flutter boundary may be
flutter and no flutter data on a plot of moved. Such changes to shape would in-
reduced velocity versus a blade loading clude leading edge radius, recamber of
parameter such as incidence (see Figure leading edge, blade thickness and maximum
24). Experience has shown that with such thickness location. All of the above
a correlation with parameters chosen at a changes demonstrate the effect of lowering
representative spanwise location it is the blade loading parameters (i.e. dif-
possible to separate most of the Clutter fusion factor, incidence) to increase the
and no flutter data with a curved line. stability of the blade in the operating
This curve is then called the flutter environment (Figure 25b).
boundary. The relationship between points
A and C on the flutter boundary on the Another way of avoiding a potential flut-
compressor map and the same points on the ter problem suggested by this empirical
design system flutter boundary are shown. design system is to lower the reduced
This example shows that S/TSF flutter can velocity. This is roost commonly done by
prevent acceleration along the operating increasing the product of semi-chord timeA
line of the compressor. For the case frequency, but . Increasing the chord or
where the boundary falls between the op- lowering blade thickening, adding part
erating line and surge line the flutter span shrouds (also called snubbers,
boundary can become a limiting character- dampers, bumpers and clappers) or changing
istic at the compressor performance if taper ratio have been used. Use of com-
distortion, increased density or tempera- posite materials have been made to change
ture or changes in the operating line the material modulus/density ratio to in-
occur. The design goal is to have all crease frequency. The effects of increas-
points on the surge line be below the ing bu are shown graphically in Figure 26.
flutter boundary with an adequate margin. The change shows up as a relocation of the
operating and surge lines in the correlat-
ion plot (Figure 26a) while it Is a shift
in the flutter boundary on the performance
map.

SURGE LINE
SURGE LINE

FLUTTER BOUNDARY

OPERATING LINE

BLADE LOADING PARAMETER, INCIDENCE

(a)

Figure 25. Stabilizing Effect of Lowering Blade Loading Parameter.

 22-23

SURGE LINE SURGE LINE

bo
FLUTTER BOUNDARY

*/*
OPERATING
OS.
Q-
LINE

_N •50% 60%
tf OPERATING LIRE
wW BLADE LOADING PARAMETER, INCIDENCE

(b) (a)

Figure 26. Stabilizing Effect of Increasing the Product bu .

SURGE LINE
SURGE LINE

b»
FLUTTER BOUNDARY

OPERATING LINE

BLADE LOADING PARAMETER, INCIDENCE

(a)

Figure 27. Stabilizing Effect of Lowering Static Pressure/Density

at Constant Static Temperature
22-24

Blade inlet static density (or static Sufficient flutter margin must be designed
pressure) changes may occur as the air- into a new compressor or fan such that
craft changes altitude and/or flight speed flutter will not be encountered under any
or as the engine changes speed. As dis- aircraft operating point.
cussed earlier aerodynamic damping is
proportional to static density. If the Vibratory mode shape is a dominant S/TSF
aerodynamic damping is positive, in- design parameter. For a given reduced
creases in static density are further sta- velocity a bending mode is much more
bilizing. If the aerodynamic damping is stable than a torsional mode (Figure 30a)
negative, decreases in static density are with node-line located at mid-chord. This
stabilizing. The latter is shown in implies the need of the designer to evalu-
Figure 27 on both the compressor map and ate the S/TSF flutter margins of both
the S/TSF design system as shifts in flut- bending and torsion modes. If bend-ing
ter boundary. Changes in aircraft alti- and torsion modes are coupled by the
tude and/or flight speed also affect blade presence of a flexible disk or part span
inlet static temperature. However, to or tip shroud, the ratio of bending to
properly predict the independent effects torsional motion and the phase angle be-
of density and temperature changes, they tween them must be considered in the flut-
should be considered independently. After ter analysis. This is illustrated in
they are considered independently, the two Figure 30b.
effects can be combined. References for subsonic/transonic stall
flutter are Shannon (1945), Graham (1965),
Changes in blade inlet static temperature Huppert (1954), Pearson (1953), Sisto
affect the relationship between Mach num- (1953, 1967, 1972, 1974), Schnittger
ber and velocity. If Mach number is held (1954, 1955), Carter (1955a, 1955b),
constant and static temperature is de- Armstrong (1960), Rowe (1955), Halfman
creased, velocity is decreased and, there- (1951), and Jeffers (1975).
fore, reduced velocity is reduced. Thus,
reducing static temperature and holding Classical unstalled supersonic flutter
static density will be stabilizing. The (USF) is a design concern if a significant
effect of such a change is shown on both portion of the blade has supersonic rela-
the compressor map and the S/TSF design tive inlet flow. The term unstalled is
system plot in Figure 28. used because USF is encountered at the
lowest corrected speed when the stage is
In gas turbine engine applications, tem- operating at the lowest pressure ratio.
perature and density changes generally oc- Classical is used because of its similar-
cur simutaneously. Such is the case as ity to classical aircraft wing flutter.
aircraft flight speed is changed. As air- The stress boundary is very steep with
craft flight speed is increased, the blade respect to speed as shown in Figure 21,
inlet static temperature Increases, cor- thus, preventing higher speed operation.
rected speed drops if there is a mechani- The stress level does not usually fluct-
cal speed limiter and blade inlet static uate with time and all blades vibrate at a
density increases. These effects can common frequency unlike S/TSF. Experience
cause a S/TSF boundary to move nearer to to date has been predominately in torsion-
the compressor operating region, while at al modes but has occurred in coupled
the same time causing the engine operating bending/torsion modes or chordwise bending
point to move closer to the S/TSF region. modes. Four dominant design parameters
This is illustrated in Figure 29. are used to describe USF. They are re-
duced velocity, Mach number, vibratory
mode shape, and static pressure/density.

SURGE LINE
SURGE LINE

o
5
FLUTTER BOUNDARY

OPERATING
LINE

OPERATING LINE
Lv_^
WV5" BLADE LOAD ING PARAMETER, INCIDENCE

(b) (a)

Figure 28. Stabilizing Effect of Reducing Inlet Static Temperature

at Constant Static Density.
 22-25

Sea Level
Sea Level Static
Flutter Boundary

o
5
t/1
1/1 OPERATING Sea Level
LINE Static
100%
\^>
80% 90% •Sea Level
Ram
wVT BLADE LOADING PARAMETER. INCIDENCE
r SEA LEVEL STATIC DESIGN POINT ' SEA LEVEL, HIGH FLIGHT MACH NUMBER
OPERATING POINT (RAM CONDITION)
(b) (a)

Figure 29. Consideration of Subsonic/Transonic Stall Flutter in Fan/Front

Compressor Stages at Sea Level Ram Conditions.

FIRST
BENDING
MODE

FIRST
TORSION
MODE

INCIDENCE INCIDENCE

SHROUDLESS BLADES SHROUDED BLADES

(b) (a)

Figure 30. Effect of Mode Shape on Subsonic/Transonic Stall Flutter.

22-26

The simplest classical unstalled super- is shown on both the empirical and ana-
sonic flutter design system consists of lytical design systems in Figure 33. In
plotting available classical USF data on a each case three points are shown: The
plot of reduced velocity versus inlet Mach original flutter point, the same operating
number and drawing a curve (flutter bound- point after the decrease in static dens-
ary) which best separates the flutter and ity, and a new flutter free operating
no flutter data points (Figure 31). The point at higher rotor speed.
flutter data points should be about this
curve, while the no flutter data points Since reducing inlet static temperature at
should be below the line. The design sys- constant corrected rotor speed causes the
tem can be applied to new designs by cal- mechanical rotor speed and, hence, blade
culating the parameters reduced velocity inlet relative velocity to decrease, the
and Mach number for points along the com- effect of reducing inlet static tempera-
pressor operating line and then plotting ture at constant static density is stabi-
the operating line on the design system lizing for classical USF. The result of
plot. No classical USF is predicted if such a change is to move the flutter
the operating line is below the flutter boundary to a higher speed. This is Illu-
boundary. strated in Figure 34 on both empirical and
analytical design system.
As in the case of subsonic/transonic stall
flutter, increasing the product bo is sta- References for classical unstalled super-
bilizing for classical supersonic un- sonic flutter are Snyder (1972, 1974),
stalled flutter. The effect of increasing Mikolajczak (1975), Garrick (1946),
bu> is to push the flutter boundary to Whitehead (1960), Smith (1971), Verdon
higher operating speeds. This is illu- (1973, 1977), Brix (1974), Caruthers
strated in Figure 32. The slope of the (1976), Nagaahima (1974), Goldstein
"new" operating line on the design plot is (1975), Ni (1975), Fleeter (1976),
inversely proportional to boo. For success- Adamczyk (1979), and Halliwell (1976).
ful designs, the flutter boundary is be-
yond the highest expected operating speed. A third type of fan/compressor flutter,
which has been identified, is A100 type
Classical unstalled supersonic flutter is supersonic flutter, Troha (1976). This
the one type of flutter for which a rea- was identified as a torsional mode flutter
sonably accurate analytical design system of a shroudless blade. The flutter bound-
exists. This analytical design system ary for this type of flutter is unlike the
parallels the ideal flutter design system. other types of flutter, indicating that
The existing analytical design system con- the unsteady aerodynamics of this type of
tains a blade-disk-shroud vibrational flutter are unique. Looking at Figure 21,
analysis, an unsteady, flat plate, cascade a moderate pressure ratio at constant cor-
analysis, and an aerodynamic damping cal- rected speed is destabilizing, while at
culation. The result is the capability to sufficiently higher pressure ratio the ef-
calculate the aerodynamic damping for each fect of the same change is stabilizing.
mode (and nodal diameter if necessary) of However, though unique in boundary it is
a compressor blade/disk assembly. A typi- very similar to USF. The outer portion of
cal plot of the resulting data is shown in the blade is supersonic. The stress
Figure 22. boundary is steep. All blades vibrate at
the same frequency and interblade phase
The effect on classical USF of lowering angle. The reduced velocity/inlet Mach
static density at constant static tempera- number empirical method also predicts this
ture is stabilizing since aerodynamic instability. Varying of bu , static
damping is proportional to blade inlet pressure/density and inlet static tempera-
static density. This stabilizing effect ture produces similar effects as those ob-
served with USF.

OPERATING
LINE V FLUTTER
Of
UJ bo>
Q£
in
i/t STABLE
LU
OH
0. OPERATING
t LINE
V0"-90% 95% 100%
Ur 1.0 INLET MACH NUMBER

(a)
Classical Unstalled Supersonic Flutter Design
System Using Reduced Velocity and Mach Number.
 22-27

o OPERATING
5 LINE V
^^^^«
bo>
Crt
in
UJ
ce.
a.
"NEW" OPERATING LINE
V?~-90% 95% 100% 1
LO INLET MACH NUMBER
8
(b) (a)

Figure 32. Stabilizing Effect of Increasing the Product bu.

(b) (a)

OPERATING
LINE V
^^^•^
bw
Ul
cn
OPERATING
LINE
90% 95% 100%
WVF 1.0 INLET MACH NUMBER
S
(0

AERO D<u

Figure 33. Stabilizing Effect of Lowering Static Pressure/Density

at Constant Static Temperature.
22-28

Choke flutter received its name from the the mode is generally first bending. An
close proximity of the flutter boundary analytical design approach which deter-
and the choke operating region of the com- mines the unsteady aerodynamic force,
pressor. This boundary can be encoun- aerodynamic damping, as a function of
tered during part speed operation. Figure interblade phase angle has been developed
21. Blades are usually operating at nega- for the designer by Adamczyk (1981). The
tive incidences in the transonic flow authors use two dimensional actautor disk
regime. In this near choke condition, in- theory in which flow separation is repre-
passage shocks with associated flow sepa- sented through rotor loss and deviation-
ration are thought to influence the aero- angle correlations. The analysis is for
elastic characteristics and thus the blade the fundamental mode bending of shroudless
stability. Recently, modal aerodynamic blades. Based on experimental data,
solution codes which analytically predict Rugger! (1974), blade stability is in-
aerodynamic danping as in the ideal flut- creased by increases in bu and reduction
ter design system and the USF analytical in blade loading. The presence of strong
design system have been developed for shocks is indicated to have an effect on
choke flutter. The crucial element in this type of flutter, Goldstein (1977).
these codes is the development of the This effect is one of destabilizing for.
transonic unsteady aerodynamic programs. both bending and torsional motions and as
Improvement in this area is presently such may be expected to lower the back
underway and will benefit the designer in pressure at which this flutter first
predicting the occurance of choke flutter. occurs. Flow separation was observed to
Experimental data has been cocrelated much exist, Riffel (1980), for a cascade of
like S/TSF data as a function of reduced airfoils representing the airfoils exhib-
velocity and incidence. Reducing solidity iting flutter above 105% speed in Ruqgeri
has been found to increase stability due (1974).
to increases in incidence. As with S/TSF
lowering reduced velocity, static This concludes the discussion of forced
pressure/density and inlet static tempera- vibration and flutter design methodology.
ture are stabilizing effects. Choke flut- Design principles have been presented to
ter has been observed in both bending and aid the designer of turbomachinery in
torsional modes. understanding the mechanisms involved and
in properly evaluating the crucial compo-
References for choke flutter are Carter nents of turbomachinery. Effective appli-
(1953, 1957), Schneider (1980), and Jutras cation of the design steps for both forced
(1982) vibration and flutter are necessary to
limit the occurrences of HCF failure in
The last type of flutter to be discussed new turbine engine designs. Research is
will be supersonic stall flutter. The continuing at this time to define and
position of this flutter boundary on the model the unsteady flow fields and forces
compressor map is suggested by the title. present during forced vibration and flut-
This flutter is like unstalled supersonic ter. As knowledge is acquired, experi-
flutter in that all blades vibrate at a mental and theoretical, and combined to
common frequncy. Experience indicates that develop better analytical predictions
tools, the possibility of eliminating high
cycle fatigue from turbine engines in de-
sign is increased and costs decreased.

(b)
(a)

o OPERATING
5 LINE

"NEW"OPERATING LINE
90% 95% 100%
10 INLET MACH NUMBER

(c)

AERO
, :^_fc I t^MECH

Figure 34. Stabilizing Effect of Reducing Inlet Static Pressure.

 A-l

ADDENDUM TO PROGRAM LISTING FOR

UNSTEADY TWO-DIMENSIONAL LIN EARIZED SUBSONIC FLQW „,
VOLUME I, CHAPTER 3, PAGES 3-24 TO 3-30

MODIFICATION TO LINSUB

By

D. S. Uhitehead

P
?' 3'24
is not within the r e c o e d r a S ^\^ l '**** _ hen then "hase
the condltl
terminate the series, which terThSL, £ ° ° »sed to
small. In order to eli.i2te th^ h h ?"" °"? term °f the serles bec°"es
1 that the
program should
successive termsbeofmodified ^ St
the «,ori!> L he series
series iss terminated
^"TTT when two
eC Smal
subroutine Dm (pp 5 26 3 27? ?) andT i' The
are shotfn in •
odi
"«tion8 occur in
extracts. ' ^ following two
[Left Column, p. 3-27]
C ASSEMBLE MATRIX
C I(=M+1 IN PAPER) GIVES VORTEX POSITION
C J(=L+1 IN PAPER) GIVES MATCHING POINT
c
30 CALL WAVE(IR,IW)
IF(IV.EQ.l) GO TO 142
DO 131 1=1,NP
DO 131 J=1,NP
IF(ICHECK(I,J).EQ.2) GO TO 131
Replace .EQ.l by .EQ.2.

[Right Column, p. 3-27]

C CHECK CONVERGENCE OF SERIES

X=TERMR*TERMR+TERMI*TERMI
Y=KR(I,J)*KR(l,j)+KI(I,j)*KI(I,J)
IF((X/Y).LE.1.0E-10) GO TO 111
ICHECK(I,J)=0
GO TO 131 Replace 2 old lines
HI IF(ICHECK(I,J).EQ.l) GO TO 112 by 7 new lines.
ICHECK(I,J)=1
GO TO 131
112 ICHECK(I,J)=2
ICOUNT=ICOUNT+1
•
 R-l

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AOAMCZYK, a., STEVENS, W.. and JUTRAS, R. 1981 Subsonic Stall Flutter of High Speed Fans. ASME
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MOABI. H.D. 1982 Vibration of Mistuned Bladed Disc Assemblies. Ph. D. Thesis, Department of Mechanical
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- rrsa. ss7
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May 26, 1924.

-
29 September - 30 October 1980

„ «..*.—. ^. -
Inst. Mech. Engrs. 180 (31).
°f

- SS.-SW.fi
c .
dum No. M. 181, November 1953.

-
its

en,ent. These Publication ONERA

*•
.
Science Meeting, Paper No

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r
 R-3

asr
"
—
'•-—•-
—
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- — - -•- — - — — -—•
-
' ^

""
• ""
. 1,73

at int. Conference on

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(Ewins, M
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These 3 cycle, Univ.

-ass-
SSSSM--
««.«..,«,..-.--"».«•»»
—: rr - •: rr -.
Engineering for Power, Vol. iu«,
 R-4

GARRICK, I.E. and RUBINOW, S.I. 1946 Flutter and Oscillating Air Force Calculations for an Airfoil i
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GUPTA, K.K. 1973 On a Combined Sturm Sequence and Inverse Iteration Technique for Eigenproblem Solution
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D 1975 Fan
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iy /Df pp. 39— 63»

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R L Lm vibration
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•industry\^
yol * ^^o',£l*
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HinJI^^iabmtriS^cS'SsL""6- ***** °fEnvi™*al a*"""' !«*» Sy-nposium on

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 R-5

JEFFERS, J.D. and MEECE, C.E. 1975 FlOO Fan Stall Flutter Problem Review and Solution. J. Aircraft 12,
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 R-6

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 R-7

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 R-9

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sss
 REPORT DOCUMENTATION PAGE

1. Recipient's Reference 2. Originator's Reference 3. Further Reference 4. Security Classification

of Document
AGARD-AG-298 ISBN 92-835-0467-4 UNCLASSIFIED
Volume 2
5. Originator Advisory Group for Aerospace Research and Development
North Atlantic Treaty Organization
7 rue Ancelle, 92200 Neuilly sur Seine, France
6. Title AGARD MANUAL ON AEROELASTICITY IN AXIAL-FLOW TURBO-
MACHINES, VOLUME 2 - STRUCTURAL DYNAMICS AND AEROELASTICITY

7. Presented at

9. Date
8. Author(s)/Editor(s)
Editors: M.F.Platzer and F.O.Carta June 1988

11. Pages
10. Author's/Editor's Address

See flyleaf. 268

12. Distribution Statement This document is distributed in accordance with AGARD

policies and regulations, which are outlined on the
Outside Back Covers of all AGARD publications.
13. Keywords/Descriptors

Reviewing Flutter
Turbomachinery Blades
Structural dynamic analysis Discs
Aeroelasticity Rotors
Metal fatigue

14. Abstract

The first volume of this Manual reviewed the state of the art of unsteady turbomachinery
aerodynamics as required for the study of aeroelasticity in axial turbomachines. This second
volume aims to complete the review by presenting the state of the art of structural dynamics and ot
aeroelasticity.
The eleven chapters in this second volume give an overview of the subject and reviews of the
structural dynamics characteristics and analysis methods applicable to single blades and bladed
assemblies.
The blade fatigue problem and its assessment methods, and life-time prediction are considered.
Aeroelastic topics covered include: the problem of blade-disc shroud aeroelastic coupling,
formulations and solutions for tuned and mistuned rotors, and instrumentation on test
procedures to perform a fan flutter test. The effect of stagnation temperature and pressure on
flutter is demonstrated and currently available forced vibration and flutter design methodology
is reviewed.
This AGARDograph was prepared at the request of the Propulsion and Energetics Panel and
of the Structures and Materials Panel of AGARD.
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