Self-Excited Vibration
Self-Excited Vibration
Self-Excited Vibration
Self-Excited Vibration
Theory, Paradigms, and Research Methods
Wenjing Ding
Self-Excited Vibration
Theory, Paradigms, and Research Methods
AUTHOR:
Wenjing Ding Xuemin Zhang
Engineering Mechanics Department Dept. of Electrical Engineerging
Tsinghua University, Beijing China Tsinghua University
Email: dingwj@mail.tsinghua.edu.cn 100084 Beijing ,China
zhangxuemin@tsinghua.edu.cn
Ming Cao
Faculty of Mathematics and Natural Sciences
University of Groningen
The Netherlands
m.cao@rug.nl
ISBN 978-7-302-24296-3
Tsinghua University Press, Beijing
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Self-Excited Vibration
Theory, Paradigms, and Research Methods
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Preface
i
The first chapter briefly explains the main features of self-excited vibrations
and the remainder of the book is divided into three parts: the first, which consists of
chapters 2 to 5, describes a variety of qualitative and quantitative methods; and
the second is concerned with the detailed analyses of several types of self-excited
vibration in various engineering fields. The analysis results may be used to improve
the effectiveness of the practical design. Furthermore, a cross-fertilization of ideas
will evolve from the different self-excited vibration phenomena so that the common
excitation mechanism in different self-excited vibration systems and the effective
analysis techniques are summarized; the last part, namely, chapter 11, provides a
workable modeling routine for analyzing the unclear self-excited vibration pheno-
menon. In combination with the first part, this part constitutes a set of research
techniques to study all self-excited vibrations in mechanical systems.
The author also wishes to express his appreciation to Professor Shouwen Yu
for his recommendation on this book’s creation. In addition, the author wishes to
appreciate the valuable suggestions from Professor Haiyan Hu. Special thanks
are due to Review Ready Co. for patiently reviewing an early version and making
valuable suggestions. The author would like to thank Dr. Shichao Fan for producing
the computer-generated plots and equations. Last but not least, the author thanks
his daughter, Jinghua Ding, for extensive review and conscientious typing of the
manuscript.
Wenjing Ding
at Tsinghua University
2011.4
ii
Contents
Chapter 1 Introduction..................................................................................... 1
1.1 Main Features of Self-Excited Vibration ................................................ 1
1.1.1 Natural Vibration in Conservative Systems ................................ 1
1.1.2 Forced Vibration under Periodic Excitations .............................. 3
1.1.3 Parametric Vibration ................................................................... 6
1.1.4 Self-Excited Vibration ................................................................ 9
1.2 Conversion between Forced Vibration and Self-Excited Vibration......... 12
1.3 Excitation Mechanisms of Self-Excited Vibration ............................... 13
1.3.1 Energy Mechanism ................................................................... 13
1.3.2 Feedback Mechanism ............................................................... 15
1.4 A Classification of Self-Excited Vibration Systems ............................. 16
1.4.1 Discrete System ........................................................................ 17
1.4.2 Continuous System ................................................................... 17
1.4.3 Hybrid System .......................................................................... 18
1.5 Outline of the Book .............................................................................. 18
References ..................................................................................................... 20
iii
2.5.4 The Existence of a Limit Cycle ................................................ 40
2.6 Soft Excitation and Hard Excitation of Self-Excited Vibration ............ 42
2.6.1 Definition of Stability of Limit Cycle....................................... 43
2.6.2 Companion Relations................................................................ 43
2.6.3 Soft Excitation and Hard Excitation ......................................... 45
2.7 Self-Excited Vibration in Strongly Nonlinear Systems ........................ 46
2.7.1 Waveforms of Self-Excited Vibration ....................................... 46
2.7.2 Relaxation Vibration ................................................................. 47
2.7.3 Self-Excited Vibration in a Non-Smooth Dynamic System...... 49
2.8 Mapping Method and its Application ................................................... 52
2.8.1 Poincare Map ............................................................................ 52
2.8.2 Piecewise Linear System .......................................................... 55
2.8.3 Application of the Mapping Method......................................... 56
References ..................................................................................................... 58
iv
3.5.4 Influence of Circulatory Force on Stability of
Equilibrium Position................................................................. 78
References ..................................................................................................... 79
viii
10.1.2 Mathematical Model of the Heating Control System.......... 303
10.1.3 Time History of Temperature Variation............................... 305
10.1.4 Stable Limit Cycle in Phase Plane....................................... 306
10.1.5Amplitude and Frequency of Room
Temperature Derivation ....................................................... 307
10.1.6 An Excitation Mechanism of Self-Excited Oscillation ....... 308
10.2 Electrical Position Control System with Hysteresis ......................... 308
10.2.1 Principle Diagram ............................................................... 308
10.2.2 Equations of Position Control System with
Hysteresis Nonlinearity ....................................................... 310
10.2.3 Phase Diagram and Point Mapping ..................................... 311
10.2.4 Existence of Limit Cycle..................................................... 313
10.2.5 Critical Parameter Condition............................................... 314
10.3 Electrical Position Control System with Hysteresis
and Dead-Zone.................................................................................. 315
10.3.1 Equation of Motion ............................................................. 315
10.3.2 Phase Diagram and Point Mapping ..................................... 316
10.3.3 Existence and Stability of Limit Cycle................................ 318
10.3.4 Critical Parameter Condition............................................... 321
10.4 Hydraulic Position Control System .................................................. 322
10.4.1 Schematic Diagram of a Hydraulic Actuator....................... 322
10.4.2 Equations of Motion of Hydraulic Position
Control System.................................................................... 323
10.4.3 Linearized Mathematical Model ......................................... 325
10.4.4 Equilibrium Stability of Hydraulic Position
Control System .................................................................... 327
10.4.5 Amplitude and Frequency of Self-Excited Vibration .......... 328
10.4.6 Influence of Dead-Zone on Motion of Hydraulic
Position Control System...................................................... 330
10.4.7 Influence of Hysteresis and Dead-Zone on Motion
of Hydraulic Position Control System................................. 334
10.5 A Nonlinear Control System under Velocity Feedback
with Time Delay................................................................................ 338
References ................................................................................................... 344
ix
11.2.2 First Type of Extended Model ............................................. 352
11.2.3 Second Type of Extended Model......................................... 355
11.3 Mathematical Description of Motive Force...................................... 358
11.3.1 Integrate the Differential Equations of
Motion of Continuum ........................................................... 358
11.3.2 Use of the Nonholonomic Constraint Equations ................. 359
11.3.3 Establishing Equivalent Model of the Motive Force........... 360
11.3.4 Construct the Equivalent Oscillator of Motive Force.......... 361
11.3.5 Identification of Grey Box Model ....................................... 362
11.3.6 Constructing an Empiric Formula of the Motive Force....... 363
11.4 Establish Equations of Motion of Mechanical Systems ..................... 365
11.4.1 Application of Lagrange’s Equation of Motion................... 365
11.4.2 Application of Hamilton’s Principle.................................... 368
11.4.3 Hamilton’s Principle for Open Systems .............................. 372
11.5 Discretization of Mathematical Model of a Distributed
Parameter System ............................................................................. 374
11.5.1 Lumped Parameter Method ................................................. 374
11.5.2 Assumed-Modes Method..................................................... 376
11.5.3 Finite Element Method ........................................................ 379
11.6 Active Control for Suppressing Self-Excited Vibration....................... 380
11.6.1 Active Control of Flexible Rotor......................................... 381
11.6.2 Active Control of an Airfoil Section with Flutter................ 384
References ................................................................................................... 387
x
Chapter 1 Introduction
T U E, (1.1)
where T is the kinetic energy of the simple pendulum, U is the potential energy,
Chapter 1 Introduction
E is the total energy, which is an arbitrary constant. Obviously, the kinetic and
potential energies of the simple pendulum can be expressed as the following,
1 2 2
T ml T , U mgl (1 cosT ),
2
where m is the mass of the simple pendulum, g is the gravitational acceleration,
l is the length of the pendulum, T is the deviation angle, and T is the angular
velocity. If let T T 0, at the highest point of the motion, then,
T T 0 0 , U (T 0 ) E mgl (1 cosT 0 ) .
Using the trigonometric identity, we obtain
U 2mgl sin 2 (T / 2)
and
E 2mgl sin 2 (T 0 / 2) .
Expressing the kinetic energy as the difference between the total energy and the
potential energy yields
1 2 2
ml T 2mgl[sin 2 (T 0 2) sin 2 (T / 2)]
2
or
1
g
T 2 [sin 2 (T 0 / 2) sin 2 (T / 2)] 2 , (1.2)
l
from which we have
1
1 l
dt [sin 2 (T 0 / 2) sin 2 (T / 2)] 2 dT .
2 g
This equation may be integrated to get the analytical expression of the period W.
Actually, the motion is symmetrical, and the integral over T from T to T T 0
yields W / 4. Hence,
1
l T0
g ³0
W 2 [sin 2
(T 0 / 2) sin 2
(T / 2)] 2
dT ,
which is an elliptic integral of the first kind. This may be seen more clearly by
making the substitutions
sin(T / 2)
z , k sin(T 0 / 2) .
sin(T 0 / 2)
2
1.1 Main Features of Self-Excited Vibration
This yields
1
cos(T / 2) (1 k 2 z 2 ) 2
dz dT dT ,
2sin(T 0 / 2) 2k
from which we have
1
l 1
g ³0
W 4 [(1 z 2
)(1 k 2 2
z ) 2
]dz . (1.3)
in a power series[1]:
1
k 2 z 2 3k 4 z 4
(1 k 2 z 2 ) 2
1 ". (1.4)
2 8
Then, the expression for the period becomes
l § k 2 9k 4 ·
W 2 ¨1 "¸ . (1.5)
g © 4 64 ¹
If k is not too large, i.e., T 0 is less than / 2 , the expansion converges rapidly,
§T T3 ·
then, k # ¨ 0 0 ¸ , and the result is corrected to the fourth order, which is
© 2 48 ¹
l § 1 2 1 ·
W # 2 ¨1 T 0 T 04 "¸ . (1.6)
g © 16 3072 ¹
Therefore, we see that although the simple pendulum is not isochronous and
the solution of the differential Eq. (1.2), which describes the motion of the simple
pendulum, cannot be a harmonic function, it is nearly so for small amplitude of
vibration.
Vibrations that take place as a result of external excitation are called forced
vibrations. When the excitation is a periodic oscillatory, the system is forced to
vibrate at the excitation frequency. If the frequency of the excitation coincides
with one of the natural frequencies of the system, the condition of resonance is
3
Chapter 1 Introduction
x 2 E x Z 02 x A cos : t , (1.7)
where E c / 2m is the damping parameter and Z0 k / m is the natural frequency
in the absence of damping.
The solution for Eq. (1.7) consists of two parts: a complementary function
xc (t ), which is the solution of Eq. (1.7) with the right-hand side equal to zero, as
the homogeneous differential equation, plus a particular solution x p (t ), which
reproduces the right-hand side. The complementary solution is clearly the general
solution of the homogeneous differential equation, and it can be written as follows[2]:
xc (t ) e E t ª A1 exp
¬«
E 2 Z02 t A2 exp ¼
E 2 Z 02 t º» .
Since sin : t and cos : t are linearly independent functions, this equation can be
satisfied in general only if the coefficient of each term vanishes identically. From
the sin : t term, we have
2:E
tan G ,
Z02 : 2
so that
2:E Z02 : 2
sin G , cos G .
(Z 02 : 2 )2 4: 2 E 2 (Z02 : 2 )2 4: 2 E 2
4
1.1 Main Features of Self-Excited Vibration
A
D .
(Z : ) 4: 2 E 2
2
0
2 2
A
x p (t ) cos(: t G )
(Z : ) 4: 2 E 2
2
0
2 2
with
§ 2:E ·
G tan 1 ¨ 2 2 ¸
.
© Z0 : ¹
The general solution of a non-homogeneous linear differential equation is
x(t ) x p (t ) xc (t ) ,
where the term xc (t ) represents transient response. The terms contained in this
solution damp out with time because of the factor exp( E t ). The term x p (t )
represents the steady state response and contains all of the information when t is
much longer than 1/ E i.e.,
x(t ı 1 E ) x p (t ) . (1.8)
5
Chapter 1 Introduction
fact that the latter present not only periodic vibrations with the excitation frequency
: , but also periodic vibrations very often observed with circular frequencies : /2,
: /3 ," : /n , which are called subharmonic resonances, and periodic vibrations
with circular frequencies 2: , 3: ," n: , which are called superharmonic
resonances.
For linear vibration systems, the superposition method can be used to solve
the case of the simultaneous excitations caused by two harmonic forces with
frequencies :1 and : 2. The forced vibration contains only two components with
the circular frequencies :1 and : 2 . However, in the nonlinear case, completely
new types of vibrations may result and it is no longer possible to apply the
superposition method. Consider the undamped Duffing equation when there are
simultaneous excitations with two frequencies :1 and : 2. The forced vibration
now contains terms with circular frequencies : 2 r 2:1 and :1 r 2: 2 , which are
designated as combination frequencies. Such complicated phenomena have been
observed in various nonlinear electric and electronic circuits.
The above discussion is restricted to the vibration phenomena occurring in a
system with single degree of freedom, which has only single natural frequency and
a single mode of motion. In contrast, a system with n degrees of freedom has n
natural modes, which produce new physical phenomena such as internal resonance,
combinational resonances, saturation, and the nonexistence of responses to a
periodic excitation in the presence of positive damping. However, all of these lie
beyond the scope of this book.
In contrast with the external excitation in the forced vibration that appears as
inhomogeneities in the governing differential equation, the excitation in the
parametric vibration appears as the coefficients in the governing differential
equation. Thus, the equation is led to a differential equation with time-varying
coefficients. This type of excitation is called parametric excitation. Moreover, while
a small excitation in external excitations can produce a large response only when
the frequency of the excitations is close to one of the natural frequencies of the
system, a small parametric excitation can produce a large response if the frequency
of the excitation is close to one half of the natural frequencies of the system.
M. Faraday (1831) is considered the first person to recognize the phenomenon of
parametric resonance. He noted that the surface wave in a fluid-filled cylinder under
the influence of vertical excitations had twice the period of the excitation[3].
The problem of parametric resonance arises in many branches of physics and
engineering. One of the important problems is dynamic instability, which is the
response of mechanical and elastic systems to time-varying loads, especially
periodic loads.
6
1.1 Main Features of Self-Excited Vibration
Here, let us consider a simple example of parametrically excited systems, i.e., the
inverted pendulum illustrated in Fig. 1.1[4]. A particle is attached to the upper end
of a light rigid rod. When the lower end of the inverted pendulum is constrained
on a pivot to turn, this equilibrium position is obviously not stable. However, if
the rod, instead of being constrained to rotating about the fixed lower end, is
permitted to move with the lower end sliding freely on a vertical line, it is possible
to convert the inverted unstable equilibrium position into a stable one by applying
a vertical periodic force with proper amplitude and frequency to the lower end of
the rod. The motion is assumed to take place in the X-Y plane under the action of
the weight mg, the external applied force Fy, and the force Fx provided by the
constraint at point A, the end of the rod. The x-coordinate of point B is given by
x l sin T (1.9)
Since the mass of the rod is neglected, the equation of the motion of the system is
written as follows:
mx Fx (1.10)
Fy (t ) mg mA cos Z t . (1.13)
7
Chapter 1 Introduction
§g A ·
T ¨ cos Z t ¸T 0. (1.14)
©l l ¹
The previous equation is linear and the coefficient of T is a harmonic function to
time. Such an equation is known as Mathieu’s equation. Introduce the following
notations
4g 2A Zt
x T, G , H , W , (1.15)
lZ 2 lZ 2 2
and transform the time scale into a dimensionless one so that Eq. (1.14) is reduced
to the standard form of Mathieu’s equation,
8
1.1 Main Features of Self-Excited Vibration
Mathieu’s Eq. (1.16) produces the unbounded or the bounded solution depending
on the combination of parameters G and H. Thus, theG -H plane should contain the
regions of stability and instability as shown in Fig. 1.4[3], in which the shaded
areas are unstable. Obviously, the curves between the stable and the unstable regions
of the G -H plane are loci of transition values of the combination of parameters G
and H. Along these curves, at least one of the normal solutions is periodic, which
has the period Sor 2S. Fig. 1.4 is called the Strutt diagram, after J. Strutt (1928)
and B. van der Pol and J. Strutt (1928)[5].
Figure 1.4 Regions of Stability and Instability of the Solutions of the Mathieu’s
Equation[3]
Just as its name implies, self-excited vibration is caused by the interaction of all
elements inside a stationary dynamic system.
In engineering, there are various types of self-excited vibrations, such as whirl
of rotor, flutter of wing, shimmy of front wheel, and so on. Here, we introduce an
electronic oscillatory circuit shown in Fig. 1.5(a), which is a common assembly
in various electronic devices. The tunnel diode is a nonlinear element and its
9
Chapter 1 Introduction
I I 0 J (U U 0 ) G (U U 0 )3,
where Jand G are the experimental constants of the tunnel diode. This tunnel
diode behaves like an ordinary resister at the low and high voltages, but there is
an intermediate region that has negative resistance. Consequently, this leads to
amplification of small oscillations in the circuit when the working point is chosen
at A(U0, I0) in Fig. 1.5(b).
It is easy to write the following algebraic equation of the motion for the circuit:
I L I U I C 0
with
1 dU
L³
IL Udt , I C C .
dt
Using the theorem of circuits, we have the differential equation
dU 1
C I (U ) ³ Udt 0.
dt L
Differentiating the previous equation, we obtain
d 2U 1 dI U 1
2
U 0
dt C dt LC
or
d 2U 1 dU
2
ª¬ J 3G (U U 0 )2 º¼ Z02U 0.
dt C dt
Then, we introduce the resonant frequency of LC circuit, Z , and the dimensionless
coordinate of the variation of voltage, x, namely,
1 U U0
Z 02 , x .
LC U0
The previous differential equation can be written with respect to time t as follows:
J 3G CU 02
P , E .
C J
10
1.1 Main Features of Self-Excited Vibration
x H (1 E x 2 ) x x
0 (1.18)
with
P
H .
Z0
The above equation for the oscillator with nonlinear damping is named van der
Pol equation for it was originally formulated by Dutch physicist B. van der Pol
around 1920 to describe oscillations in a triode circuit[7].
Figure 1.6 shows the features of the time history of x (t ) for H 0.1 and E 1/3.
It is nearly a harmonic oscillation. When H is large enough, say, H 10, x (t ) will
become a jerky oscillation. The numerical results with different initial conditions
show that if the parameters of Eq. (1.18), H and E, are given, the amplitude of the
steady solution is a constant.
11
Chapter 1 Introduction
elements of the system. The self-excited vibration does so, while the forced and
parametric vibrations are exerted by external elements of the systeme. Thus, from
now on, we devote attention to the natural vibration in a conservative system and to
self-excited vibration. It is already known that the amplitude and the frequency of
the natural vibration in a conservative system vary continuously with the change of
the initial state of motion. Thus, if all of the parameters of the conservative system
are given, its amplitude and frequency may take arbitrary values in accordance
with the intensity of the initial disturbance, as discussed before. In contrast, if all
of the parameters of a self-excited vibration are given, the magnitude and the
frequency have their own constant values respectively, despite the intensity of its
initial disturbance.
So far, we have spread out the features of four types of the sustained periodic
motion. For the differences amang them are fundamental, they should be investigated
respectively.
12
1.3 Excitation Mechanisms of Self-Excited Vibration
13
Chapter 1 Introduction
the energy variation of the mechanical system is equal to the work done the
nonconservative forces. For the self-excited vibration system, it must be equal to
zero in a whole cycle. Let us choose a nonconservative system with a single degree
of freedom. Its governing equation may be written in the form
mx h( x, x ) f ( x) 0 , (1.19)
where m is the equivalent mass of the system, x the corresponding displacement,
f ( x) the potential force, and h( x, x ) the nonconservative force of the system.
With kinetic energy T mx 2 / 2, and potential energy U ³ f ( x)dx , Eq. (1.19)
may be written as follows:
For a periodic solution, we must have ED (a) EZ (a) , from which we can
determine the amplitude of the self-excited vibration. As a simple example, an
electronic oscillator circuit has been shown in Fig. 1.5(a). Its differential equation
of motion is Eq. (1.18), in which the function h( x, x ) H (1 x 2 ) x is not a
definite one. Therefore, it can produce self-excited oscillation, and the relevant
experiment results have verified the prediction. Hence, Eq. (1.20) becomes an
analytical description of the energy mechanism of the self-excited vibration.
14
1.3 Excitation Mechanisms of Self-Excited Vibration
15
Chapter 1 Introduction
16
1.4 A Classification of Self-Excited Vibration Systems
the terms containing explicit time t. Such differential equations are referred to as
autonomous equations and the self-excited vibration is just a periodic solution of
the autonomous equation. In fact, the periodic solution not only guarantees the
basic features of the self-excited vibration, but also provides benefits to study its
motion process.
The type of the differential equations of motion governing the dynamic systems
is basically related to the number of the variables describing the configuration of
the system. Therefore, to classify self-excited vibration systems according to the
number of generalized coordinates is meaningful. Generally, there are three types:
discrete system, continuous system, and hybrid system.
w g w , w ; , (1.21)
where w is the generalized coordinate vector with n dimensions, w and w are
the generalized velocity and acceleration vectors respectively, O is the variable
parameter vector, and g is the function of the variables w , w , and O.
Introducing state variable vector, i.e., x [ w T , w T ] , the second-order differ-
T
x f x; , (1.22)
where x is the state vector with 2n dimensions, O is the parameter vector with m
dimensions, and f (x; O) is the function of x and O. This equation is referred to as
the state equation of the dynamic system.
17
Chapter 1 Introduction
are a group of partial differential equations and this type of dynamic system is
referred to as a continuous system. If the considered object possesses remarkable
deformation, which should be taken into account, the resultant equations are always
a group of partial differential equations as those of the elastic pipe conveying
fluid studied in chapter 9.
The equations of the motion of a continuous system are established by means
of the mechanics of a continuous medium, including solid mechanics and fluid
mechanics, and the resultant equations may be written as operator form, namely,
L u; x, y , z; 0 , (1.23)
where L is the differential operator vector; u is the displacement vector of the
deformation body; x, y, and z are the coordinates of physical space; and O is the
variable parameter vector of the continuous system.
It is known that the equations governing the motion of a rigid body are ordinary
differential equations, and the equations governing the motions of deformable solid
and fluid are partial differential equations. Thus, if a dynamic system consists of
rigid and deformable bodies or (and) fluid, such as a rigid rotor with elastic axles,
or rigid body surrounded by fluid, such as an airfoil surrounded by air flow, the
differential equations of its motion consist of ordinary and partial differential
equations. This type of dynamic system is referred to as a hybrid system.
Obviously, differential equations that are simultaneously ordinary and partial
cannot be analytically solved as general cases. They are often transformed into a
group of approximate equations by various discretization methods such as Ritz-
Galerkin method and assumed-modes method. Alternatively, they may also be
directly solved by various numerical methods. The analytical researches about
self-excited vibration in this book are mainly concerned with studying ordinary
differential equations. Contents about numerical analysis are not discussed in this
book.
vibration, which has not been investigated before. The contents of each chapter
are arranged in the following manner.
Chapter 2 is devoted to the geometric approaches including the phase plane
method and point mapping method.
Chapter 3 examines the relationship between the stability of equilibrium of
autonomous systems and self-excited vibrations at first, and then introduces three
stability criteria, namely, Hurwitz criterion, geometric criterion, and quadratic form
criterion.
Chapter 4 provides four kinds of quantitative analysis methods for studying self-
excited vibrations, namely, Hopf bifurcation theorem with application, Lindstedt-
Poincare method, average method, and method of multiple scales.
Chapter 5 is concerned with control theory. The contents include Nyquist stability
criterion for a linear closed-loop system, Popov stability criterion for a nonlinear
closed-loop system, and describing function method for calculating amplitude and
frequency of self-excited vibrations in closed-loop control systems. In addition,
an optimal control algorithm based on a quadratic form index for a steady linear
system is introduced briefly.
Chapter 6 deals with the self-excited vibration caused by friction, namely, chatter
and hunting occurring in machinery with flexible elements. The relevant conclusions
are obtained by phase plane method and point mapping method. In addition, two
excitation mechanisms of friction leading to self-excited vibration are discussed
in detail.
Chapter 7 treats the shimmy of front wheel with pneumatic tire, whose
mathematical model is composed of the equation of motion of the front wheel
and the nonholonomic constraint equations of the tire. It is finally analyzed by
stability criteria for steady linear systems.
Chapter 8 is devoted to rotor whirls caused by fluid-film force and internal
damping respectively in a deformed rotation shaft. Some useful conclusions are
found by using stability criteria for steady linear systems.
Chapter 9 is concerned with the self-excited vibrations induced by fluid force,
in which there are vortex resonance of flexible structure, flutter of cantilevered
pipe conveying fluid, classical flutter of two-dimensional airfoil, stall flutter of
flexible structure, and fluid-elastic instability in an array of circular cylinders. To
acquire analytical results, the distributed fluid force is reduced to an equivalent
force to establish a simple expression or a mechanical model to describe the fluid
force.
Chapter 10 deals with some self-excited vibrations occurring in closed-loop
control systems. Heating control system, electrical position control system, and
hydraulic position control system are analyzed in detail. Various analysis methods,
namely, analytical integration, phase plane, point mapping, describing function
methods, and Popov stability criteria are used to determine self-excited vibrations.
The results show that self-excited vibrations in closed-loop systems may not be
only soft excitation but also hard excitation, and there exist more than one self-
19
Chapter 1 Introduction
References
[1] J B Marion. Classical Dynamics of Particles and Systems, Second Edition. New York:
Academic Press, 1970
[2] L Meirovitch. Elements of Vibration Analysis, Second Edition. New York: McGraw-Hill,
1986
[3] A H Nayfeh, D T Mook. Nonlinear Oscillations. New York: John Wiley & Sons, 1979
[4] J J Stoker. Nonlinear Vibrations in Mechanical and Electrical Systems. New York: John
Wiley & Sons, 1950
[5] B van der Pol, M J O Strut. On the Stability of the Solution of Mathieu’s Equation.
Philosophical Magazine, 1928. Ser. 5, 277, 18 38
[6] H D I Abarbanel, M I Rabinovich, M M Sushchik. Introduction to Nonlinear Dynamics
for Physicist. Singapool: World Scientific, 1993
[7] B van der Pol. The Nonlinear Theory of Electrical Oscillations. Proc. IRE, 1934, 22(9):
1051 1086
[8] J P Den Hartog. Mechanical Vibrations, Fourth Edition. New York: McGraw Hill, 1956
[9] P Hagedorn. Nonlinear Oscillations, Trans. By W Stadler. Oxford: Clarendon, 1982
20
Chapter 2 Geometrical Method
x f ( x, x ) 0, (2.1)
where x and x are the generalized coordinate and generalized velocity respectively,
and f is a function of variables x and x . Introducing a new variable
Chapter 2 Geometrical Method
y x , (2.2)
equation (2.1) will be separated into two first-order equations
x y , y f ( x, x ) (2.3)
Denoting x and y as Cartesian coordinate axes, we set up a coordinate plane
called a phase plane, which is shown in Fig. 2.1. Actually, it represents the state
space of second-order dynamic systems, in which the arbitrary point P(x, y) defines
the motion state of a dynamic system. Point P is referred to as phase point or state
point, and its coordinates x and y are state variables of the dynamic system.
Given the initial state, i.e., x(0) x0 and y(0) y0, the solutions x(t) and y(t) of
Eq. (2.3) may be represented with a curve in the phase plane. In fact, accom-
panying the time flow, the phase point P[ x(t ), y (t )] traces out a directed curve in
the plane called a phase path or a phase trajectory. A group of phase paths
starting from different initial state points are drawn in the x-y plane. This is called
phase plane, phase diagram, or phase portrait.
When we obtain the algebraic equation of the phase path, the path may be drawn
at once. In order to obtain the algebraic equation, we divide the second equation
of (2.3) by the first, which leads to
dy f ( x, y )
. (2.4)
dx y
Completing the integral of Eq. (2.4), we obtain an algebraic equation with two
arbitrary constants depending on the initial state of the dynamic system. Therefore,
corresponding to a pair of arbitrary constants, there is a phase path in the phase
plane.
The phase paths of Eq. (2.4) have two features as follows.
(1) If the right side of the equation is a single valued function, the phase paths
in the phase plane cannot intersect.
(2) In the upper half plane, y > 0, variable x monotonically increases, and the
phase path consistently elongates toward the right. In contrast, in the lower half
plane, y < 0, variable x monotonically decreases, and the phase path consistently
elongates toward the left.
22
2.2 Phase Diagrams of Conservative Systems
f ( x,0) 0 . (2.5)
The constant solutions of the above algebraic equation are a set of equilibrium
point coordinates of the system, which can be written as x0i, i 1, 2, " , Obviously,
they are a discrete set on the real axis x.
All of the equilibrium points and the phase paths in their neighborhood are the
main components of the phase diagram of dynamic systems.
Denoting x as the derivation angle of the simple pendulum, we have its differential
equation of the motion here
g
x Z 02 sin x
0, Z02 , (2.6)
l
where Z 0 is the angular frequency of the simple pendulum, l is the pendulum
length, and g is the gravitational acceleration. Introducing state variables x and y,
we obtain the differential equation of the phase path of the simple pendulum
dy Z 02 sin x
.
dx y
Given the initial state x0 and y0, we complete integration of the previous equation
and obtain the algebraic equation of a family of phase paths
y 2 2Z 02 (1 cos x) E0 , (2.7)
where E0 is the integrating constant depending on the initial state P0 ( x0 , y0 ) , i.e.,
23
Chapter 2 Geometrical Method
y(t) are periodic functions of time t, all these phase paths are closed curves. However,
they are not harmonic functions. Thus, the phase paths are not ellipsoids.
x f ( x) , (2.9)
where the force per unit mass f is independent of x . Define a function V(x),
dV ( x)
V ( x) ³ f ( x)dx or f ( x) . (2.10)
dx
Then V(x) is called potential energy function of the system. The equilibrium points
of the system (2.9) are given by
f ( x) V c( x) 0. (2.11)
By writing x y in Eq. (2.3), and integrating the following equation, which is
derived from Eq. (2.3)
dy f ( x)
, (2.12)
dx y
we obtain
1 2
y V ( x) E0 , (2.13)
2
where constant E0 is the total energy of the system. The algebraic equation of
phase paths may be derived from the previous equation, i.e.,
1
y r[2 E0 2V ( x)] 2 (2.14)
24
2.3 Phase Diagrams of Nonconservative Systems
For different values of E0, Fig. 2.3 shows the types of equilibrium points based
on three types of turning point of V(x): a local minimum always leading to a
center, a maximum to a saddle point, and a point of inflection to a cusp. They are
represented in Fig. 2.3(a), (b), and (c) respectively.
Figure 2.3 Three Types of Phase Diagram of Conservative Systems with One
Degree of Freedom
The construction can be thought of in this way: for a fixed E0, [2 E0 2V ( x)]
can be read out from the top frame for the range of x; when it is non-negative, y
can be calculated from Eq. (2.14) and a symmetrically placed pair of points is
inserted in the corresponding lower frame.
x cx kx 0, (2.15)
25
Chapter 2 Geometrical Method
where c is the viscous damping coefficient of one unit mass and k its spring stiffness.
The nature of the solutions of Eq. (2.15) depends on whether the roots of the
characteristic equation
O 2 cO k 0
are real and different, or complex or coincident and real. The roots are given by
1
1
O1 , O2 [ c r (c 2 4 k ) 2 ] ,
2
and the discriminant is
c 2 4k .
Besides, the general types of motion are determined by the previous parameters,
O, O, and . Here, we only consider the following three types of linear damped
vibrators.
1. Strong Damping ( ' ! 0 )
The solutions are given by
Figure 2.4 (a) Solution Curves for a Heavily Damped Vibrator; (b) Phase Diagram
x y, y kx cy .
Here, there is a single equilibrium point at x 0 and y 0, and the differential
26
2.3 Phase Diagrams of Nonconservative Systems
§ ct · ª1 1
º
x(t ) A exp ¨ ¸ cos « ( ) 2 t D » , (2.18)
© 2¹ ¬2 ¼
where A and D are arbitrary constants. A typical solution is shown in Fig. 2.5(a).
It represents a vibration with exponentially decreasing amplitude, which decays
more rapidly for large c. Plotted parametrically as before, its image on the phase
plane is shown in Fig. 2.5(b).
Figure 2.5 (a) Solution Curves of Weakly Damped Vibrator; (b) Phase Diagram
The equilibrium point at the origin is called a stable focus or a stable spiral.
3. Critical Damping ( ' 0)
In this case, 2O 2O c and the solution becomes
§ ct ·
x(t ) ( A Bt ) exp ¨ ¸ . (2.19)
© 2¹
27
Chapter 2 Geometrical Method
This solution resembles that of strong damping and its phase plane shows a
stable node.
Permutations of signs of the parameters c and k are possible.
(1) k < 0, c z 0 . In this case, O and O are real but have different signs. The
phase diagram shows a saddle point, as shown in Fig. 2.3.
(2) k > 0, c < 0. This is a negative damping case. Instead of being lost by the
equivalent of resistance or damper, energy is generated constantly in the system.
The node or focus is now unstable, as shown in Fig. 2.6, because a slight
disturbance from the equilibrium leads to the system being carried far from the
equilibrium state.
Now, returning to the general case, the second-order nonlinear autonomous system,
the equation of motion is
x f ( x, x ) 0. (2.1)
Assume that function f takes the form
f ( x, x ) h( x, x ) g ( x) , (2.20)
where h( x, x ) does not contain an additive function of x. Then, we have
x h( x, x ) g ( x) 0. (2.21)
In mechanical systems, function g(x) describes the restoring force of a spring.
If g(x) is of an appropriate type, we could expect a tendency to vibration, which is
modified by the presence of the term h( x, x ). Suppose that the kinetic energy T is
28
2.3 Phase Diagrams of Nonconservative Systems
equal to x 2 / 2. The potential energy function for the system (2.21) becomes
V ( x) ³ g ( x)d x .
Here, the total energy E is defined by
1 2
E T V x ³ g ( x)dx .
2
It is not a general constant.
The rate of the total energy may be found from the above equation and Eq. (2.21),
i.e.,
dE
x[ f ( x, x ) g ( x)] xh
( x, x ) . (2.22)
dt
Integrating it with respect to t from t 0 , to t W, we obtain
W
E (W ) E (W 0 ) ³ xh
( x, x )d t , (2.23)
W0
§ E x 2 ·
x H ¨1
¸ x x 0 , H ! 0 , E ! 0 , (2.24)
© 3 ¹
where Hand E are two characteristic constants of the vibrator. Obviously, in this
1
system, the negative damping occurs in strip | x | (3/ E ) 2 J 1 , as shown in region
I of Fig. 2.7, in which the phase paths depart from the equilibrium point; and the
positive damping occurs in the region | x | ! J 1, as shown in regions II and III, in
which the phase paths approach the equilibrium point.
29
Chapter 2 Geometrical Method
Figure 2.7 Negative Damping Region and Limit Cycle of Rayleigh Equation
(H 1, E 1)
x H (1 E x 2 ) x x
0, H ! 0, E ! 0 , (1.18)
where Hand E are two characteristic constants of the vibrator. In this system, the
1
negative damping occurs in strip | x | E 2 J 2 , as shown in region I of Fig. 2.8,
in which the phase paths depart from the equilibrium point; and the positive
damping occurs in the region | x | ! J 2 , in which the phase paths approach the
equilibrium point, as shown in regions II and III of Fig. 2.8.
Figure 2.8 Negative Damping Region and Limit Cycle of van der Pol Equation
(H 1, E 1)
30
2.3 Phase Diagrams of Nonconservative Systems
Differentiating the Rayleigh equation, i.e., (2.24) with respect to time t yields
x H (1 E x 2 )
x x 0.
By denoting x1 x , it may be written as follows
x1 H (1 E x12 ) x1 x1
0,
which is just a van der Pol equation, i.e., (1.18). Thus, we conclude that the solution
of a van der Pol equation is a first derivative of the solution of the corresponding
Rayleigh equation. In other words, the solution of the Rayleigh equation is the
integral of the solution of a corresponding van der Pol equation. Therefore, the
topological structures of their phase planes are very similar.
3. A special nonlinear autonomous equation is
x ( x 2 x 2 1) x x 0. (2.25)
Putting x y and using Eq. (2.20), we obtain
yh x, x ( x 2 y 2 1) y 2 .
31
Chapter 2 Geometrical Method
origin as well. The isolated closed phase path x 2 y 2 1 is called a limit cycle.
It is called ‘isolated’ because there is no other closed path in its neighborhood.
All phase paths approach the circle as t o f .
The limit cycle that the phase paths approach from both inside and outside is a
stable limit cycle, whereas, a limit cycle that the phase paths depart from both
inside and outside is an unstable one. Hence, Eq. (2.25) has a stable limit cycle,
as shown in Fig. 2.9.
x X ( x, y ) , y Y ( x, y ) , (2.26)
whose type considered in Section 2.1, i.e.,
x y , y f ( x, y ) ,
is a special case. Suppose that the equilibrium point to be studied has been moved
to the origin by a translation of axes, if necessary, so that X(0, 0) 0, and Y(0, 0) 0.
We can therefore write by Taylor expansion
wX wX wY wY
a (0,0) , b (0,0) , c (0, 0) , d (0,0) . (2.27)
wx wy wx wy
The linear approximation to Eq. (2.26) in the neighborhood of the origin is
defined as the system
x ax by , y cx dy . (2.28)
We expect that the solutions of Eq. (2.28) will be geometrically similar to those
of Eq. (2.26) near the origin, as fulfilled in most cases.
32
2.4 Classification of Equilibrium Points of Dynamic Systems
aO b
O 2 (a d )O (ad cb) 0 . (2.29)
c d O
This is called the characteristic equation. Here, we shall consider only the case
that the Eq. (2.29) has two different roots O1 and O2. Two linearly independent
families of the solutions are generated consequently by Eq. (2.28), corresponding
to O O1and O O2 respectively.
dy cx dy
. (2.30)
dx ax by
Let us introduce a vector variable
> x, y @ >u , v @
T T
x , u , (2.31)
and a nonsingular transformation matrix S that transforms x into u as follows
x Ax , (2.33)
where the coefficient matrix is
ªa b º
A «c d » . (2.34)
¬ ¼
Substituting Eq. (2.32) in Eq. (2.33), we obtain
dv O2 v
,
du O1 u
whose solutions are given by
O2
O1
v C |u| , (2.36)
where C is an arbitrary constant, u = 0 is also a phase path.
Plot the phase diagram with solution (2.36). Only two possible patterns emerge,
depending on whether O1 and O2 have the same or opposite signs. In case of same
signs, there is a node, as shown in Fig. 2.10(a), whose stability and hence the
direction of the arrows are determined from Eq. (2.36): if O1, O2 > 0, then, all
solutions are exponentially increasing, the node is unstable, and all arrows are
pointing away from the origin; if O1, O2 < 0, then the node is stable. In the latter
case, there is a saddle, as shown in Fig. 2.10(b).
Figure 2.10 O1, O2 Real, Non-Zero, O2 > O1 (a) O1and O2Have the Same Signs;
(b) O1 and O2 Have Opposite Signs
u D u E v, v E u D v. (2.37)
Set
z u iv .
Then, by Eq. (2.37), we have
z (D iE ) z
34
2.4 Classification of Equilibrium Points of Dynamic Systems
and by writing
z r (t )eiT (t ) ,
where r | z |, we obtain two equations in polar coordinates, i.e.,
r D r , T E,
whose solutions are
r r0 eD t , T T0 E t ,
where r0 r (0) , T 0 T (0) . The origin is therefore a stable focus if D< 0. It is an
unstable focus if D > 0, and is a center if D 0.
(3) Degenerate cases
Degenerate cases occur when there is a single repeated eigenvalue, and when an
eigenvalue is zero. These particular cases will not be discussed in detail.
O 2 pO q 0 (2.29)
with
p (a d ), q (ad bc) . (2.38)
Hence, we have
1 1 1
O1 , O2 p r 2
2 2
with
p 2 4q .
Various cases are displayed in Fig. 2.11, which is a bifurcation diagram of a
second-order linear system. The degenerate cases occur on 0 and q 0. Note
that the center constitutes a transition between the stable and the unstable focus,
and the existence of the center depends on particularly exact relations between
the coefficients of the system. It is a rather fragile feature. The consequence is
that even if the linear approximation of a nonlinear system predicts a center, the
equilibrium point, which could be a stable or an unstable focus, is not truly a center.
35
Chapter 2 Geometrical Method
If there exists a neighborhood of such equilibrium point that phase paths starting
from all points in the neighborhood ultimately approach, the equilibrium point is
known as an attractor. Both the stable node and the stable focus are examples of
attractor. Both the unstable node and the unstable focus are repellors, from which
all paths in the neighborhood depart.
If the eigenvalues of the linearized equation have non-zero real parts at the
equilibrium point, the point is said to be hyperbolic. Thus, the focus, node, and
saddle are hyperbolic, but the center is not.
x X ( x, y ), y Y ( x, y ) . (2.26)
36
2.5 The Existence of Limit Cycle of an Autonomous System
Let * be a smooth closed curve consisting of the ordinary point of Eq. (2.26).
Let Q be a point on * , as shown in Fig. 2.12. There is only one phase path
passing through Q. The path belongs to the family described by the equation of
the phase path
dy Y ( x, y )
. (2.39)
dx X ( x, y )
When time t ! 0 , the coordinates of the point Q( xQ , yQ ) will increase by x
and y respectively, where
x X ( xQ , yQ )t , y Y ( xQ , yQ )t .
Therefore, the vector T [ X , Y ]T is tangential to the phase path through the point
in the direction of increasing t. Its inclination can be measured by the angle I,
which is anticlockwise from the positive direction of the x axis to the direction
of T. So, we have
Y
tan I , (2.40)
X
where the value of I at one point has been decided and the value at the other points
is settled by requiring I to be a continuous function of position, except that I does
not, in general, have its original value on returning to its starting point after a full
cycle. The value may differ by 2Sn, where n is an integer.
In every case, the change in I must be a multiple of 2S, namely,
[I ]* 2I * , (2.41)
where I * is an integer, either positive or negative, or a zero. Described anticlock-
wise, I * is called the index of * with respect to the vector field (X, Y).
An algebraic representation of I * is obtained as follows. Suppose that the
37
Chapter 2 Geometrical Method
r ( s ) [ x( s ), y ( s )], s0 s s1 (2.42)
and s is a variable parameter. From (2.40)
d d §Y ·
tan I ¨ ¸,
ds ds © X ¹
after some reduction, we have
dI XY c YX c
,
ds X2 Y2
in which the prime denotes differentiation with respect to s. Then, from (2.41),
we have
1 s1 dI 1 s1 XY c YX c
2 ³s0 ds 2 ³s0 X 2 Y 2
I* ds ds .
1 XdY YdX
I* . (2.43)
2 X2 Y2
As r(s) [x(s), y(s)] describes * , R(s) [X, Y], regarded as a position vector
on a plane with axes X and Y, describes another curve * R . * R is closed, since R
returns to its original value after a complete cycle. From Eq. (2.41), * R encircles
the origin I * times. It is anticlockwise when I * is positive and clockwise when I *
is negative. It is illustrated in Fig. 2.13 for a particular case.
Figure 2.13 (a) Direction Vectors along the Closed Loop; (b) Closed Loop on
Plane [X, Y]
38
2.5 The Existence of Limit Cycle of an Autonomous System
The following are four theorems about the index of equilibrium point.
Theorem 1. Suppose that, on or inside * , X, Y, and their first derivatives are
continuous, and X and Y are not simultaneously zero. Then, I * is zero.
Proof: Applying Green’s theorem for plane curves to (2.43), we have
1 ª w X w Y º
2 ³³S R «¬ wX X 2 Y 2 wY X 2 Y 2 »¼
I* d XdY
If we already know the nature of the equilibrium point, the index is readily found
by simply drawing a figure and following the angle around. The following shows
the indices of the elementary types, as shown in Fig. 2.14.
(1) A saddle point. The change in I in a single circuit of the curve * surrounding
the saddle point is 2S, and the index is therefore 1.
(2) A center. The index is 1.
(3) A focus (stable or unstable). The index is 1.
(4) A node (stable or unstable). The index is 1.
39
Chapter 2 Geometrical Method
Here, without proof, we give a plausible theorem on which several results of this
section are based. For the proof, refer Andronov et al.[4].
Poincare-Bendixon Theorem. Let R be a closed bounded region consisting of
nonsingular points of 2 u 2 system x f ( x ) such that some positive half-path
H of the system lies entirely within R as t increases. Then H itself is a closed path,
or it approaches a closed path, or it terminates at an equilibrium point.
If we can isolate a region from which some path cannot escape, the theorem
describes what may happen to the path: roughly speaking, with some control of
its movement through the regular conditions based on the differential equations,
the path cannot wander about at random forever. The possible cases concluded in
the theorem are illustrated in Fig. 2.15.
In particular, the theorem implies that if R contains no equilibrium point, and
some H remains in R, then R must contain a periodic solution. The theorem can
be used in the following way. In Fig. 2.16, we can find two closed curves, 1 and
2, with 2 inside 1, so that all paths cross 1 point toward its interior and all paths
cross 2 point outward from it. No path that enters the annular region between 1
40
2.5 The Existence of Limit Cycle of an Autonomous System
and 2 can escape. The annular is therefore the region R for the theorem. Further,
if we can ensure that R has no equilibrium points in it, at least one closed path L
may exist somewhere in R according to the theorem. Evidently, L must wrap round
the inner curve as shown. Its index is 1 and it must therefore have an equilibrium
point inside; R contains no equilibrium points. For the same reason, there must
exist suitable equilibrium points interior to 2 for all this to be possible.
The same results are true if paths are all outward across 1 and inward across 2.
The Poincare-Bendixon theorem can be applied to obtaining two theorems
covering broad types of differential equations. The proof for these may be found
in mathematical textbooks[3].
Theorem 1. The differential equation
x h( x, x ) g ( x) 0 (2.44)
or the equivalent system
x y, y h ( x, y ) g ( x ) ,
41
Chapter 2 Geometrical Method
where h and g are continuous, has at least one periodic solution under the following
conditions:
1
(1) There exists a > 0 such that h(x, y) > 0 when ( x 2 y 2 ) 2 ! a ;
(2) h(0, 0) < 0, hence, h(x, y) < 0 in a neighborhood of the origin;
(3) g(0) 0, g(x) > 0 when x > 0, and g(x) < 0 when x < 0;
x
(4) G ( x) ³ 0
g (u )du o f , as x o f.
Let us consider the Rayleigh Eq. (2.24). Its equivalent system is
x y, y P (1 E y 2 ) y x .
In this case,
h x, y P (1 E y 2 ) y, g ( x) x.
Obviously, all required conditions of the theorem are satisfied so that there is a
limit cycle located in the neighborhood of the origin.
Theorem 2. The differential equation
x h( x) x g ( x) 0 (2.45)
has an unique periodic solution if h and g are continuous, and
x
(1) H ( x) ³ 0
h(u )du is an odd function;
(2) H ( x) is zero only at x 0, x a, or x a for some a > 0;
(3) H ( x) o f as x o f monotonically for x > a;
(4) g ( x) is an odd function, and g ( x) ! 0 for x > 0.
Let us consider the van der Pol Eq. (1.18). Its equivalent system is
x y, y H (1 E x 2 ) y x .
§E ·
H (1 E x 2 ) , H ( x) H ¨ x3 x ¸ , g ( x) x . So, all required
In this case, h( x)
© 3 ¹
conditions of the theorem are satisfied with a (3 / E )1/ 2 , and there is a limit
cycle located in the neighborhood of the origin.
42
2.6 Soft Excitation and Hard Excitation of Self-Excited Vibration
A limit cycle is called a stable limit cycle if all nearby solutions drift toward it,
and is called unstable if all nearby solutions recede from it. A semistable limit
cycle can also occur where the paths approach it on one side and recede from it
on the other side. It is a particular limit cycle where a stable limit cycle and an
unstable limit cycle coincide. Stable, unstable, and semistable limit cycles are
depicted in Fig. 2.17(a), (b), and (c) respectively. Obviously, the self-excited
vibrations occur only on the stable limit cycles. The state points on the unstable or
the semistable limit cycle must depart from it if they bear any small disturbance
and the corresponding motion cannot be maintained.
According to the geometry of phase planes shown in Fig. 2.17(a), (b), and (c),
there is a certain companion relation between the stability of the limit cycle and
the stability of the equilibrium point inside it. The companion relation originates
from the continuity of the direction field and the single-valued property of its
function.
Since the index of a limit cycle is equal to 1, a stable or an unstable or another
limit cycle should be enclosed within it. For inspection of the companion relation
between the limit cycle and the equilibrium point within it, we only consider the
first two cases here.
In the first case, a stable limit cycle requires that all nearby phase paths drift
toward it so that an unstable equilibrium point is enclosed within it. This is the
first type of companion relation between the limit cycle and the equilibrium point
inside it. As an example, the van der Pol equation, namely, Eq. (1.18), has an
unstable focus at the origin of the phase plane. Simultaneously, a stable limit
cycle is located in its neighborhood, as shown in Fig. 2.18, where all phase paths
are subject to the continuity of the direction field and the single-valued property
43
Chapter 2 Geometrical Method
of its function except the single isolated point, i.e., the equilibrium point, and the
isolated integral curve, i.e., the limit cycle.
In the second case, an unstable limit cycle requires that all nearby phase paths
departing from its interior elongate toward a stable equilibrium point. It means that
a stable equilibrium point is enclosed within it. Thus, we conclude that there is a
second type of companion relation between the limit cycle and the equilibrium
point within it, i.e., the former is unstable and the latter is stable. As an example, we
convert the sign of the damping term from negative to positive in Eq. (1.18), namely
x H (1 E x 2 ) x x
0, (2.46)
whose equilibrium point is a stable focus. According to the companion relation
mentioned above, there is an unstable limit cycle that encloses a stable focus.
Such expectation is illustrated in the phase diagram shown in Fig. 2.19.
44
2.6 Soft Excitation and Hard Excitation of Self-Excited Vibration
As we have seen, the unstable limit cycle cannot predict a true periodic motion
and it does not represent a self-excited vibration. However, we cannot say that the
self-excited vibration does not occur at all in the dynamic system with a stable
equilibrium point. Actually, there are some cases in which a stable limit cycle is
outside an unstable one. Such a case demonstrates a determinative companion
relation between a stable limit cycle and an unstable one. This is confirmed in the
following section.
Let us return to Chapter 1, where Eq. (1.19) was studied. Here, assume that the
motion is quasi-harmonic and is supported by work of the nonconservative force.
Here, we write an algebraic equation[2]
T
³ 0
h( x, x ) xdt 'ED (a) 'EZ (a) , (1.20)
in which 'ED is the lost energy due to damping, and 'EZ the energy filled by the
excitation effect during full vibration period T. For a periodic solution, we must
have 'ED ( a ) 'EZ (a ), from which we then determine the amplitude, as shown
in Fig. 2.20(a), (b), and (c).
Figure 2.20 Energy Diagram and Phase Diagram of Self-Excited Vibration Systems,
(a) Soft Self-Excitation; (b) Hard Self-Excitation; (c) Semi-Stable Limit Cycle
45
Chapter 2 Geometrical Method
therefore, all phase paths surrounding the closed path drift toward it and the closed
path becomes a stable limit cycle.
As an example, consider the van der Pol Eq. (1.18). The injected energy 'EZ is
found by the linear term H x ; the dissipated energy 'ED is calculated by the
nonlinear term HE x 2 x ; and their curves 'EZ (a ) , 'ED (a ) , and the phase diagram
of Eq. (1.18) are shown in Fig. 2.20(a).
In Fig. 2.20(b), 'ED (a ) and 'EZ (a) intersect at two points K1 and K2. There
are two limit cycles: the outer one K2 is stable and the inner one K1 is unstable.
The figure shows a companion relation between the two limit cycles. Here, as an
example, let us consider the following equation
x (0.8 4 x 2 1.6 x 4 ) x x 0.
In this system, the injected energy 'EZ (a) is found from the term 4x 2 x , and
the dissipated energy 'ED (a ) is found from the terms 0.8x and 1.6x 4 x . The curves
'E D ( a ), 'EZ (a ) , and the phase diagram of the system are similar to Fig. 2.20(b),
in which there are also two limit cycles — the outer one is stable and the inner one
is unstable.
In Fig. 2.20(c), the curve 'EZ (a) is tangential to the curve 'ED (a ) at point K,
which represents a semistable limit cycle. It is a special limit cycle because a
stable limit cycle coincides with the unstable one.
According to Fig. 2.20(b), if the initial disturbance is large enough, the initial
phase point is located out of the region surrounded by the unstable limit cycle
and the phase path tends asymptotically toward the stable limit cycle. This type of
self-excitation is referred to as hard excitation, whereas the other one, illustrated in
Fig. 2.20(a), is referred to as soft excitation in which very small initial disturbance
can produce the self-excited vibration.
x y x(1 x 2 y 2 ),
(2.47)
y x y (1 x 2 y 2 ).
In polar coordinates, the equation system becomes
r r (1 r 2 ), T 1 ,
and its equilibrium point is the origin of the phase plane and is unstable. It is
clear that all phase paths spiral around the origin in the clockwise direction. They
spiral outward for 0 < r < 1 since r ! 0 for 0 < r < 1; and they spiral inward for
r > 1 since r 0 for r > 1. Since r 0 on r 1, the unit circle describes a closed
phase path L, namely, a stable limit cycle, as shown in Fig. 2.21. Corresponding
to the limit cycle, the periodic solution departing from P0(cosT0, sinT0) on L is
represented as
x(t ) cos(T 0 t ) , y (t ) sin(T 0 t ) . (2.48)
There are many self-excited vibrations whose waveforms are far from the harmonic
function. Let us return to the Rayleigh Eq. (2.25)
x H (1 x 2 ) x x 0. (2.25)
If we choose a different order of magnitude for parameter H, the limit cycles with
47
Chapter 2 Geometrical Method
The general form of second-order differential equations, including van der Pol
equation and Rayleigh equation, may be written as follows
x H h( x, x ) Z 02 x 0, (2.49)
where H is a dimensionless parameter and h( x, x) is a general nonlinear function
of x and x. If H is small, the self-excited vibration system (2.49) is referred to as a
weakly nonlinear autonomous system and the corresponding self-excited vibration
is nearly a harmonic vibration. To study weakly nonlinear autonomous systems,
48
2.7 Self-Excited Vibration in Strongly Nonlinear Systems
The phase diagram of Eq. (2.50) may be exactly obtained by Liénard con-
struction[5]. Draw two lines x r P , which intersect the x axis at points B and C
respectively, as shown in Fig. 2.24. The phase paths consist of a series of circular
arcs with the centers C or B, depending on whether the phase point is in the
upper or the lower half-plane. In Fig. 2.24, a phase path is initiated at A. Draw a
circle clockwise with center C and radius CA until it meets the x axis at A1. Then,
switch the center to B and draw a half circle clockwise with a radius BA1 that
meets the x axis again at A2. Continue the process until the phase path intersects
the x axis between C and B. The motion ceases at the intersection because the
maximum possible friction force exceeds the force in the spring.
Compared with all the above diagrams, the one in Fig. 2.24 has two particular
structures: (1) there is an equilibrium zone, i.e., the interior of P - x - P . It is
a continuum, not an isolated point; (2) the phase path is non-smooth at the
intersecting point A1, A2, " This non-smoothness originates from the abrupt
direction conversion of the friction force.
2. Discontinuous point on the phase path
In mechanical clocks, mechanical vibrators (pendulum or balances) are used when
impulsive energy is supplied in certain states[2]. The motion of such mechanism
may be approximately described by the differential equation of a simple linear
vibrator with linear damping. The impulsive energy supply normally occurs twice
per full vibration. An extreme case with a fixed given increment 'EZ of the
kinetic energy at each impulse state has been particularly investigated. Next, let
us consider a system with linear damping with the equation of motion
50
2.7 Self-Excited Vibration in Strongly Nonlinear Systems
x [ x x
0. (2.52)
The solution is
1
x(t ) Ke[ t cos Z t , Z (1 [ 2 ) 2 . (2.53)
In Fig. 2.25, the phase diagram is composed of curves constructed in accordance
with solution (2.53). For a given 'EZ and [, only the amplitude A of the limit cycle
is computed here. Since x1 x2 0, the points 1 and 2 of Fig. 2.25 correspond to
t1 / 2Z , t2 /Z (2.54)
with
x1 KZ e [ t1 , x2 KZ e [ t2 . (2.55)
The mechanical damping energy withdrawn from the system in its passage from 1
to 2 is obtained as
1 2
'ED ( x1 x22 ) K 2Z 2 sinh([ / Z ) .
2
Figure 2.25 Limit Cycle of an Impulsively Excited System with Constant Energy
Increment
Since we must have 'EZ 'ED for a limit cycle, it follows that
1
1§ 'EZ ·2
K ¨ ¸ ,
Z © sinh([ / Z ) ¹
and eventually, the amplitude of the limit cycle is given by A x(ts), where ts is
the time when x vanishes. From
51
Chapter 2 Geometrical Method
we obtain
1 [
ts arctan .
Z Z
Eventually, we have
1
1§ 'ED · 2 [ t s
A ¨ ¸ e cos Z ts .
Z © sinh([ / Z ) ¹
For small damping, [ 1 , we have Z | 1 , ts | [ , e[ | 1 , cos Z t s | 1 , and
2
sinh([ / Z ) | [ , with
1/ 2
§ 'E ·
A|¨ D ¸ .
© [ ¹
As the phase diagram in Fig. 2.25 shows that time histories of x (t ) have
discontinuous points at t1 and t2, the impulsively excited system belongs to the
non-smooth dynamic system.
Probably, the most basic tool for studying stability and bifurcations of periodic
orbits is the Poincare map, which was defined by H. Poincare in 1881. The idea
is quite simple: if * is a periodic orbit of the system
x f ( x) (a)
through the point x0, and 6 is a hyperplane perpendicular to * at x0, then for
any point x 6 sufficiently near x0, the solution of equation (a) through x at
t 0, an orbit curve )t ( x ) , will cross 6 again at a point P(x) near x0, as shown
in Fig. 2.26. The mapping x o P( x ) is called the Poincare map.
When 6 is a smooth surface, the Poincare map can also be defined through a
point x0 * , which is not a tangent to * at x0. In this case, surface 6 intersects
curve * transversally at x0. The next theorem establishes the existence and
continuity of the Poincare map P(x) and its first derivative P'(x)[7].
52
2.8 Mapping Method and its Application
Next, we cite some specific results for the Poincare map of planar systems. For
planar systems, if we translate the origin to the point x0 * 6 , the normal line
6 will be a line through the origin, as shown in Fig. 2.27. The point O * 6
divides the line into two open segments 6 and 6 when 6 lies entirely in the
exterior of * . Let s be the signed distance along 6 with s > 0 for points in 6
and s < 0 for points in 6 .
if d'(0) < 0, it follows that d(s) < 0 for s > 0 and d(s) > 0 for s < 0; i.e., the cycle
* is a stable limit cycle (see Fig. 2.27). Consequently, we have such corres-
ponding results: if P(0) 0 and P'(0) < 0, * is a stable limit cycle; and if P(0) 0
and P'(0) > 1, * is an unstable limit cycle. Thus, the stability of * is determined
by the derivative of the Poincare map.
Example 1. Let us return to system (2.47). In polar coordinates, it becomes
r r (1 r 2 ), T 1
with r(0) r0 and T T 0. The first equation can be solved either as a separable
differential equation or as a Bernoulli equation. The solution is given by
1
ª §1 · 2t º 2
r (t , r0 ) «1 ¨ 2 1¸ e » , T T (t ,T 0 ) .
«¬ © r0 ¹ »¼
If 6 is the ray T T0 through the origin, 6 is perpendicular to * and the
trajectory through the point (r0 ,T 0 ) 6 * at t 0 intersects the ray T T0
again at t 2, as shown in Fig. 2.28. The Poincare map is given by
1
ª §1 · 4 º 2
P(r0 ) «1 ¨ 2 1¸ e » .
¬« © r0 ¹ ¼»
Obviously, P(1) 1 corresponds to cycle * , and we have
3
ª §1 · º 2
Pc(r0 ) e 4 r03 «1 ¨ 2 1¸ e4 »
¬« © r0 ¹ ¼»
with Pc(1) e 4 1 . Hence, the limit cycle * shown in Fig. 2.28 is a stable one.
54
2.8 Mapping Method and its Application
There are some technical devices whose equations of motion contain the piecewise
linear function shown in Fig. 2.29. They are referred to as piecewise linear systems
and belong to a non-smooth dynamic system. It is known that the mapping method
is particularly suitable for analyzing their motion.
The procedure of the mapping method for analyzing a piecewise linear system
may be clearly explained by studying a position servo system with relay controller.
Its interior structure is illustrated by a block diagram shown in Fig. 2.30, in
which block A represents the controlled object and the servo motor, block B
represents the amplifier and the on-off element with dead-zone, and block C
represents the comparative operation of the desired position and the feedback
signal of the position servo system. After its modeling, the equations of motion
are eventually reduced to the following form
55
Chapter 2 Geometrical Method
ui and uo are the input and the output voltages of the on-off element, F(ui) describes
its input-output relationship, as shown in Fig. 2.29(f), and k1 and k2 are the
amplification coefficients of the servo motor and the controller of the position
servo system.
To apply the mapping method to analyze the position servo system with on-off
controller, there are four steps to follow.
Step 1: Divide the phase plane into three regions.
According to the piecewise linear function F(ui), the system Eq. (2.56) may be
respectively described as three forms in three different regions in the phase plane,
i.e.,
T
x x kuo , x ! G , (2.57)
T
x x 0, | x | G , (2.58)
T
x x kuo , x G , (2.59)
where uo is a constant voltage, namely, the output of the on-off element. Other
coefficients may be found from the Eq. (2.56), i.e.,
J k1 '
T , k , G , (2.60)
c c k2
where is the half width of dead-zone of F(ui).
According to Eqs. (2.57), (2.58), and (2.59), the phase plane is divided into
three regions by two lines L1 and L2, perpendicular to x axis, namely, x G and
x G. These regions are I, II, and III, as shown in Fig. 2.31.
56
2.8 Mapping Method and its Application
t1
§ 1 ·
t
y1 y0 e T
kuo ¨1 e T ¸ , (2.61)
© ¹
and the second Poincare map from P1 to P2, i.e.,
2G
y2 y1 , (2.62)
T
where t1 is the duration from P0 to P1, which may be determined by the equation
§ t1
·
kuo t1 T ( y0 kuo ) ¨1 e T ¸ 0. (2.63)
© ¹
Substituting (2.62) into (2.61), we obtain
t1
§ 1 ·
t
2G
y2 y0 e T
kuo ¨1 e T ¸ . (2.64)
© ¹ T
The equation system (2.63) and (2.64) describes the Poincare map from P0 to P2.
Step 4: Plot the Lamery diagram.
Denoting t1 as a variable parameter, we solve the simultaneous Eqs. (2.63) and
(2.64) by numerical computation, and draw a sequential function curve M. In
addition, plot a half-line L starting from origin O, whose equation is
L: y2 y0 . (2.65)
Here, make the y2 axis coincide with the y2 axis, as shown in Fig. 2.32. If the
57
Chapter 2 Geometrical Method
Obviously, the phase diagram accordant with the Lamery diagram in Fig. 2.32
is similar to the one shown in Fig. 2.20(b), which has two limit cycles. Since the
internal limit cycle is unstable, the equilibrium point of this position servo system
is stable. Consequently, the self-excited vibration can be caused by large disturbance
with its initial phase point out of the unstable limit cycle. Hence, the self-excited
vibration, occurring in the position servo system controlled by on-off element
with dead-zone, is produced by hard excitation.
References
[1] A A Andronov, E A Vitt, S E Khaiken. Theory of Oscillators. Oxford: Pergamon, 1966
[2] P Hagedorn. Nonlinear Oscillations. Oxford: Clarendon Press, 1981
[3] D W Jordan, P Smith. Nonlinear Ordinary Differential Equations, Third Edition. Oxford:
Oxford University Press, 1999
[4] A A Andronov, E A Leontorich, I I Gordon, A G Maier. Qualitative Theory of Second-
Order Dynamic Systems. New York: Wiley, 1973
[5] A H Nayfeh, D T Mook. Nonlinear Oscillations. New York: John Wiley & Sons, 1979
[6] J J Stoker. Nonlinear Vibrations. New York: Weley, 1950
[7] L Perko. Differential Equations and Dynamical Systems. New York: Springer-Verlag, 1991
[8] V I Arnold. Ordinary Differential Equations. Berlin: Springer-Verlag, 1992
58
Chapter 3 Stability Methods
Abstract: The phase plane is used not only to describe a solution to one
initial state, but also to describe a family of solutions to a number of initial
states. However, it is confined to applying to a second-order autonomous
equation only. Application to higher-order nonlinear equations is much
more difficult than higher-order linear equations. Fortunately, under certain
conditions, a nonlinear equation may be linearized in the neighborhood of
its equilibrium point, and the net results, the linear equation, can be applied
to examining the motion of the nonlinear system in the neighborhood of its
equilibrium point. This method has been widely used to predict the stability
of equilibrium and the existence of self-excited vibration in higher-order
nonlinear systems. Hence, the main material of this chapter is divided into five
sections: the first is concerned with the concept of stability of equilibrium of
an autonomous system, the second introduces the algebraic criterion of
stability, the third is arranged to deduce the geometric criterion of stability,
the fourth is devoted to constructing the parameter conditions that guarantee
stability of equilibrium of a higher-order linear system, and the fifth establishes
a quadratic form criterion of stability of equilibrium of a holonomic system
with multiple degrees of freedom.
Keywords: Lyapunov stability, linearized system, eigenvalue of system,
Hurwitz criterion, Mihairov curve, geometric criterion, homonomic system,
quadratic form criterion
x f ( x ), (3.1)
where x is an n-dimensional state variable vector, and f is an n-dimensional
function vector, i.e.,
f (x) 0. (3.3)
The above algebraic equation may have definite constant solutions, which are
all equilibrium positions of system (3.1). In general, each of them can be written
as a constant vector xe.
Suppose f (x) is an analytic function vector that may be expanded into a power
series in the neighborhood of the equilibrium position. Substituting the power
series, ignoring the higher-order power series, and retaining the linear terms, we
obtain the linear equation written in a vector form, namely,
x Ax , (3.4)
where A is an n-dimensional matrix and its elements are the partial derivatives of
f(x) with respect to x at the equilibrium position xe, i.e.,
ª wf º
A « wx » . (3.5)
¬ ¼ xe
In general, the linear Eq. (3.4) is called the first approximation equation of the
autonomous system (3.1).
At the end of the 19th Century, A.M. Lyapunov designated a type of stability of
motion of dynamic systems. The stability of equilibrium position is a particular
branch of stability of motion. Lyapunov’s definition assumes that the origin of
the coordinates is the equilibrium position and the stability of the equilibrium
position can be stated as follows[1, 2].
1. The equilibrium position is stable in the sense of Lyapunov if for any arbitrary
positive quantity H , there exists a positive quantity G such that the satisfaction of
the inequality
60
3.1 Stability of Equilibrium Position
A.M. Lyapunov verified that the three first approximation theorems give a simple
method to study the stability of equilibrium position for higher-order nonlinear
systems.
Theorem 1. If all eigenvalues of matrix A have negative real parts, the equili-
brium position of nonlinear system (3.1) is asymptotically stable.
Theorem 2. If at least one eigenvalue of matrix A has a positive real part, the
equilibrium position of nonlinear system (3.1) is unstable.
Theorem 3. If at least one eigenvalue of matrix A has a vanishing real part and
61
Chapter 3 Stability Methods
the other eigenvalues have negative real parts, it is impossible to determine the
stability of the equilibrium position of nonlinear system (3.1).
Nonlinear system (3.1) is called a critical stable system if its first approximation
Eq. (3.4) has at least one eigenvalue with vanishing real part and all other
eigenvalues have negative real parts. From the above three theorems, we conclude
that the stability of equilibrium position of nonlinear system, whose eigenvalues
do not have zero real part at all, may be determined by the first approximation
Eq. (3.4) and the stability problem of general nonlinear systems can be transformed
into one of its first-order approximation system.
Let us consider a solution of linear Eq. (3.4) given in exponential form[3, 4].
x (t ) e O t x0 , (3.9)
where x0 is the initial state.
Introducing solution (3.9) in Eq. (3.4), and dividing it by eOt , we obtain the
eigenvalue problem
O x0 Ax0
leading to the characteristic equation
det | A O I | 0 , (3.10)
where I is an n-dimensional identical matrix.
Expanding Eq. (3.10) into a high-degree algebraic equation, we obtain
The number of roots of an algebraic equation with negative real part is determined
by its all coefficients. To establish the relation is a well-known mathematical
problem first resolved by E. J. Routh and is named as Routh problem in honor of
his contribution. Later on, A. Hurwitz independently gave an answer to resolve
the problem by coefficient determinant. His method is called Hurwitz criterion
and is used to study various stability problems comprehensively.
First, we introduce the construction of Hurwitz determinant.
By arranging the coefficients of the algebraic Eq. (3.11) in accordance with
certain regulation, the Hurwitz main determinant 'n and all minors '1 ," , 'n 1
are listed below
a1 a0 0 0 " 0
a1 a0 0 a3 a2 a1 a0 " 0
a1 a0
'1 a1 , ' 2 , '3 a3 a2 a1 , ..., ' n a5 a4 a3 a2 " 0 .
a3 a2
a5 a4 a3 "
0 0 " an
(3.12)
63
Chapter 3 Stability Methods
Comparing the last two determinants, we find out the following relation
O 3 a1O 2 a2 O a3 0. (3.16)
According to criterion (3.14), if coefficients a1, a2, and a3 have the same sign, we
need to construct only one minor determinant
'2 a1a2 a3 .
Here, the coefficient condition for asymptotic stability of equilibrium of a
third-order autonomous system is
a1a2 a3 ! 0 . (3.17)
64
3.3 A Geometric Criterion for Stability of Equilibrium Position
According to criterion (3.14), if the coefficients a1, a2, a3, and a4 have the same
sign, we need to construct only one minor determinant
'3 a1a2 a3 a32 a12 a4 .
Here, the coefficient condition for asymptotic stability of equilibrium of a fourth-
order autonomous system is
a1a2 a3 a32 a12 a4 ! 0 . (3.19)
65
Chapter 3 Stability Methods
with
According to algebraic theory, the characteristic Eq. (3.11) has n roots O, O, " ,
On in complex domain and the polynomial D(O) may be expanded into a product
of n linear multipliers, i.e.,
D (O ) (O O1 )" (O On ) . (3.25)
Denoting O iZ, we have
If all coefficients of the algebraic Eq. (3.11) are real numbers, its complex roots
are all conjugate and the eigenvalues are symmetrically distributed about the
real axis of the complex plane, as shown in Fig. 3.3. The multiplier (iZ Ok ) in
expression (3.26) is a vector from point Ok to point iZ, which lies on the imaginary
axis of the complex plane.
If Ok is located in the left-half complex plane, with Z varying from f to f ,
the increment of argument of vector (iZ Ok ) , which is the angle included
between this vector and the abscissa, is equal to S. If Ok is located in the right-
half complex plane, with Z varying from f to f, the increment of argument
66
3.3 A Geometric Criterion for Stability of Equilibrium Position
As the Mihairov curve is symmetric with respect to the real axis of the complex
plane, with Z varying from 0 to f, the hodograph of vector D(iZ) should be
67
Chapter 3 Stability Methods
real parts is that the increment of the argument of the hodograph of vector D(iZ)
is equal to nS/2 with Z varying from 0 to f .
Now, let us return to the third-order linear autonomous system whose hodographs
of vector D(iZ) are shown in Fig. 3.2 (a), (b), (c), and (d). In case n 3, the
hodograph of D(iZ), as shown in Fig. 3.2(a), is equal to 3S/2 with Z varying
from 0 to f and the equilibrium of this system has asymptotic stability.
Meanwhile, the hodograph of D(iZ), as shown in Fig. 3.2(d), is equal to S/2
with Z varying from 0 to f and the equilibrium of this system is unstable.
Corresponding to the hodograph of D(iZ) shown in Fig. 3.2(b), the equilibrium
of the linear autonomous system has the first type of critical stability in case
a3 0, and the characteristic Eq. (3.16) possesses a zero root at the origin of the
complex plane. In addition, corresponding to the hodograph of D(iZ) shown in
Fig. 3.2(c), the equilibrium of this system has the second type of critical stability
and the characteristic Eq. (3.16) possesses a pair of conjugate roots on the imaginary
axis of the complex plane.
As shown in Fig. 3.2(c), the hodograph of D(iZ) of the third-order linear auto-
nomous system passes through the origin of the complex plane. Similarly, for a
higher-order linear autonomous system corresponding to the second type of
critical stability, the hodograph of D(iZ) also passes through the origin of the
complex plane, as shown in Fig. 3.4. Using this geometry, we can find out the
coefficient condition guaranteeing the second type of critical stability for higher-
order linear systems. Let us explain it in detail below.
68
3.3 A Geometric Criterion for Stability of Equilibrium Position
First, look at the hodograph of D(iZ) shown in Fig. 3.4. For Z Zc, both the
real part and the imaginary part of D(iZ) are equal to zero. Denoting O iZc in
the characteristic Eq. (3.11), we obtain two real equations
) (Z c ) 0, < (Zc ) 0,
in which polynomials ) and < are defined by expressions (3.22) and (3.23).
Denote E Zc2 to reduce their degree. If the positive n is odd and n 2l 1, we
obtain the equation system
f (E ) a0 E l a2 E l 1 a4 E l 2 " 0,
(3.30)
g (E ) a1 E l a3 E l 1 a5 E l 2 " 0.
If n is even and n 2l, we obtain another equation system
f (E ) a0 E l a2 E l 1 a4 E l 2 " 0,
(3.31)
g (E ) a1 E l 1 a3 E l 2 a5 E l 3 " 0.
According to algebraic theory, the necessary and sufficient condition guar-
anteeing the same roots for algebraic equation system (3.30) and (3.31) is that
their discriminant determinant R( f , g ) be vanishing, i.e.,
a0 0 0 " a1 0 0 "
a2 a0 0 " a3 a1 0 "
R( f , g ) a4 a2 a0 " a5 a3 a1 " 0.
" " " "
" " " " 2l u2l
Otherwise, equation systems (3.30) and (3.31) are coprime. After algebraic operation,
the above equation is arranged in the form
a1 a0 0 0 0 0 " 0
a3 a2 a1 a0 0 0 " 0
a5 a4 a3 a2 a1 a0 " 0 0. (3.32)
" " " "
0 0 0 " " an
In fact, if n is even, the determinant in the left side of the above equation is
just the Hurwitz determinant 'n . If n is odd, it is the maximum minor of the
Hurwitz determinant 'n 1 . Considering the relation in Eq. (3.13), two results
may be joined together and written as
'n 1 0. (3.33)
69
Chapter 3 Stability Methods
In the Eq. (3.13), denoting Hurwitz determinant 'n to be zero, we obtain two
coefficient equations, namely,
an 0 (3.34)
and
'n 1 0. (3.33)
Equation (3.34) states that a real root of characteristic Eq. (3.11) is located at the
origin of the complex plane, which means that the equilibrium position of the
dynamic system possesses the first type of critical stability. Similarly, Eq. (3.33)
states that a pair of conjugate roots of characteristic Eq. (3.11) lie on the imaginary
axis of the complex plane, which means that the equilibrium position of the
dynamic system possesses the second type of critical stability.
Obviously, Eqs. (3.34) and (3.33) define two surfaces with n 1 dimensions in
coefficient space R n. R n is constituted by n independent variables a1, a2, " , an,
whose n 1 groups of subspaces Sk, k 1, 2, " , n 1 are defined by the
inequalities 'k ! 0 . According to mathematical theory, these subspaces have a
geometrical property, that is, Sk is surrounded by Sk1, k 1, 2, " , n1. Hence,
we have
Sn 1 Sn 2 " S2 S1 . (3.35)
Thus, the stable region in coefficient space R is surrounded by two surfaces
n
O 2 a1O a2 0. (3.36)
In this case, equations of two surfaces w:1 and w: 2 corresponding to Eqs. (3.34)
and (3.33) are
a2 0 (3.37)
and
a1 0, (3.38)
whose stable region : and two boundary surfaces w:1 and w: 2 are shown in
Fig. 3.5.
Figure 3.5 The Stable Region and its Boundary Surfaces of a Second-Order Linear
Autonomous System
Dynamic systems always have some parameters that are variables in certain
71
Chapter 3 Stability Methods
Figure 3.6 The Stable Region and its Boundary Surfaces of a Third-Order Linear
Autonomous System
Here, consider the characteristic Eq. (3.11). Let us denote a0 1 and assume
that the remainder of its coefficients are independent variables so that they construct
an n-dimensional vector in R n, which is described in the form
[ P1 ," , P m ]T , Rm . (3.42)
Let us denote a function vector to describe the relation between the coefficient
vector a and parameter vector P, i.e.,
a G ( ), a R n , Rm , (3.43)
where G is an n-dimensional function vector, namely,
H1 ( ) 0. (3.45)
We define w:1c as the boundary surface of the stable region in the parameter
space Rm. If the parameter point of dynamic system lies on w:1c , the equilibrium
position possesses the first type of critical stability. Similarly, substituting
expression (3.43) in Eq. (3.33), which is satisfied by a set of coefficient points on
72
3.4 Parameter Condition for Stability of Equilibrium Position
H 2 ( ) 0. (3.46)
This defines w:2c as the boundary surface of the stable region in parameter space
Rm. If the parameter point of dynamic system lies on w:2c, the equilibrium position
possesses the second type of critical stability.
Here, let us consider a physical example of a stable region in the parameter
space of a vibration system with a single degree of freedom, whose equation of
motion is
mx cx kx 0, m ! 0 ,
where parameters m, c, and k are mass, damping, and stiffness coefficients of the
system and they are all positive variables. Consequently, the parameter space of
the system is a subspace in the three-dimensional space. The characteristic
equation of the system is (3.36) with the following coefficients
c k
a1 , a2 , m ! 0, c ! 0, k ! 0 .
m m
According to the above analysis, the equations of the boundary surfaces of the
stable region in the parameter spaces w:1c and w:2c are k 0 and c 0 respectively.
They are two orthogonal coordinate planes, and the stable region : in R3 is
surrounded by these two planes and plane m 0, as shown in Fig. 3.7.
Figure 3.7 A Stable Region of the Vibration System with Single Degree of Freedom
As we have seen, if the system parameters vary in the stable region of a dynamic
system, the stability of equilibrium position does not change and the behavior
73
Chapter 3 Stability Methods
> q1 " qn @ , q > q1 " qn @ , q > q1 " qn @ ,
T T T
q
ª m11 " m1n º
« », (3.48)
> f1 fn @ ,
T
f " M « " »
«¬ mn1 " mnn »¼
where q is the generalized coordinate vector; qi(i 1, 2, " , n) are the components
of q; q and q are the generalized velocity vector and the generalized acceleration
vector respectively; qi and qi (i 1, 2, " , n) are the components of q and q ;
f is the analytic function vector; fi (i 1, 2, " , n) are the components of f ; M is
the n-dimensional inertial matrix; and mij (i, j 1, 2, " , n) are the elements of M.
Denoting q q 0 in Eq. (3.47), we obtain the equilibrium equation of the
holonomic system, i.e.,
f ( q , 0) 0. (3.49)
Every solution of the equation, qk(k 1, 2, " ), is an equilibrium position of the
holonomic system. In general, there is more than one solution to the nonlinear
Eq. (3.49).
Denote square matrices B and C as the partial derivatives of f (q, q ) with
respect to q and q at an equilibrium position qk respectively, i.e.,
75
Chapter 3 Stability Methods
ª wf º ª wf º
« wq » B, « wq » C , (3.50)
¬ ¼ ( qk , 0 ) ¬ ¼ ( qk , 0 )
C D G, D DT, G G T,
(3.52)
B K E, K K T, E E T,
where D is the damping matrix, G the gyroscopic matrix, K the potential matrix,
and E the circulatory matrix. Substituting expression (3.52) in Eq. (3.51), we obtain
First, suppose that the initial value problem of linear Eq. (3.53) has a unique
solution
x Ae O t , (3.54)
where A is an n-dimensional complex vector, namely,
A [ A1 , " , An ]T ,
in which the components Ai (i 1, 2, " , n) of A are complex variables for the
systems with small damping.
Next, designate the elements of matrices M, D, G, K, and E as
¦ [m
s 1
rs O 2 (d rs g rs )O krs ers ] As 0, r 1, 2," , n . (3.55)
det | M O 2 ( D G )O K E | 0. (3.56)
It is known that the 2nth degree equation possesses 2n roots in complex domain.
If the dissipative force is small enough, all of the characteristic roots are complex
conjugate written as Os and Os* (s 1, 2, " , n). Corresponding to these eigen-
values, there are n pairs of conjugate eigenvectors,
where M( AA*), D( AA*), G( AA*), K( AA*), and E( AA*) are five quadratic forms
of variables Ds and Es, s 1, 2, " , n, i.e.,
1
M ( AA* ) M (D ) M ( E ) ¦¦ (mrsD rD s mrs E r E s ),
2 r s
1
D( AA* ) ¦¦ (drsD rD s drs E r E s ),
D(D ) D ( E )
2 r s
1
K ( AA* ) K (D ) K ( E ) ¦¦ (krsD rD s E r E s ),
2 r s
(3.59)
G ( AA* ) ¦¦ g rs (D r E s D s E r ),
r s
E ( AA* ) ¦¦ e
r s
rs (D r E s D s E r ).
The above expressions show that these quadratic forms respectively vary with
matrix elements mrs, drs, grs, krs, and ers with r, s 1, 2, " , n, and the quadratic
forms M( AA*), D( AA*), G( AA*), K( AA*), and E( AA*) represent the magnitude
of inertial force, dissipative force, gyroscopic force, potential force, and circulatory
force respectively. In general, M( AA*) is a positive definite quadratic form.
77
Chapter 3 Stability Methods
( D i G ) r ( D iG ) 2 4 M ( K iE )
O1,2 . (3.60)
2M
After algebraic operation in the complex domain, the previous expression is reduced
to the following form
( D B Z ) iG (1 B D / Z )
O1,2 (3.61)
2M
with
D 2 G 2 4 KM 1
Z2 (G 4 KM D 2 )2 4( DG 4ME )2 . (3.62)
2 2
According to expression (3.61), if D2 > Z 2, eigenvalues O1 and O2 possess
negative real parts, and the equilibrium position of the holonomic system (3.53) is
asymptotic stable.
Substituting expression (3.62) in (3.61), the necessary and sufficient condition
for asymptotic stability of equilibrium position of the holonomic system is found
and represented by an inequality of the quadratic forms depending on the system
parameters, i.e.,
4ME 2 D 2 K . (3.64)
78
References
In general, the quadratic form M is always positive, the inequality means that if
the circulatory force is large enough, the stable condition of equilibrium position
of holonomic systems will be violated. Thus, we conclude that the circulatory
force can cause the equilibrium position of holonomic systems without qyroscopic
force to lose stability.
References
[1] L Meirovitch. Methods of Analytical Dynamics. New York: McGraw-Hill, 1970
[2] L Meirovitch. Elements of Vibration Analysis. New York: McGraw-Hill, 1986
[3] L Perko. Differential Equations and Dynamical Systems. New York: Springer-Verlag, 1991
[4] D W Jordan, P Smith. Nonlinear Ordinary Differential Equations. Third Edition. Oxford:
Clarendon Press, 1999
[5] P Hagedorn. Nonlinear Oscillations. Oxford: Clarendon Press, 1981
[6] L Meirovitch. Introduction to Dynamics and Control. New York: John Wiley & Sons, 1985
[7] D R Merkin. Introduction to the Theory and Stability. New York: Springer, 1997
[8] B D Hassard, N D Kazarinoff, Y H Wan. Theory and Applications of Hopf Bifurcation.
Cambridge: Cambridge University Press, 1981
[9] J G Papastavridis. Analytical Mechanics. Oxford: Oxford University Press, 2002
[10] à à Çàíàæuñöè Å ½éêéëéìîé¾uëéìåéêãòàìåéäÌíÛÜñæãøÛñãã. ¿éåæ.
AH CCCP 1952. T. 86, B.1, Ìíë. 31 34
[I I Metelitsin. Problems of Stabilization by Gyroscopic Force. Doc. AS SSSR 1952 Vol.
81(1): 31 34 (in Russian)]
79
Chapter 4 Quantitative Methods
Abstract: Different from the geometry and the stability methods introduced
until now, which are all the qualitative methods, the methods arranged in
this chapter are quantitative methods that can calculate the magnitude of
amplitude and frequency of self-excited vibrations. In general, the analytical
solutions of the nonlinear differential equations cannot be found and the
dynamic problems of nonlinear systems are much more difficult to be solved,
in particular, the problems of high-dimensional nonlinear systems. Starting
from how to reduce the order of system equations, the contents of this chapter
are divided into five sections: the first is concerned with the basic concepts of
the center manifold from the local theory of ordinary differential equations; the
second is devoted to Hopf bifurcation theory and its application to calculating
amplitude and frequency of self-excited vibration; the last three sections
provide three analytical methods, Lindstedt-Poincare method, averaging
method, and the method of multiple scales, for seeking approximate solutions
of weakly nonlinear autonomous systems with single degree of freedom.
Keywords: nonlinear systems, flow, subspace, center manifold, bifurcation,
Hopf bifurcation, Linstedt-Poincare method, averaging method, method of
multiple scales
x f ( x) (4.1)
4.1 Center Manifold
x f ( x) 0 (4.3)
and f ( x ) satisfies the Lipschitz property, i.e.,
| f ( y) f ( x) | İ k | y x |
for some k f , then, there exists an interval 0 < t < T, in which Eq. (4.1) has a
unique solution x(t) satisfying the initial state (4.2).
The uniqueness of solution x(t) implies that the trajectories never cross each
other. Moreover, the trajectories vary smoothly over D except at singular points.
This leads to topological constraints on the trajectories in D and precludes complex
dynamical behavior in one- or two-dimensional systems. Consequently, a chaotic
behavior can occur only in three- or higher-dimensional autonomous systems.
As mentioned in Section 3.1, in the neighborhood of equilibrium point xe, the
nonlinear Eq. (4.1) can be linearized as
x Ax . (4.4)
According to the fundamental theorem of linear ordinary differential equations,
the solution to the initial state x(0) x0 associated with linear Eq. (4.4) is given by
x (t ) e At x0 . (4.5)
The mapping e At : R n o R n may be regarded as the description of the motion of
points x0 R n along trajectories of linear Eq. (4.4). The mapping is called the
flow of linear system (4.4).
An n-dimensional linear system has n linearly independent solutions. If the
eigenvalues of A are non-degenerate, it can be diagonalized and Eq. (4.5) can be
expanded into the scalar form
n
x (t ) ¦ c eO
k 1
k
kt
ek , (4.6)
The stability of linear systems has been studied with linear Eq. (4.4) in Chapter 3.
The equilibrium point xe is asymptotically stable if and only if Re(Ok) < 0 for each
eigenvalue Ok of A.
For nonlinear systems, the notion of local stability of equilibrium point xe can
be formalized with flow ) t :
Definition 1: The fixed point P(x xe) is a Lyapunov stable equilibrium of the
system for all neighborhoods U of P if there exists such a neighborhood U1
of P U that x )t ( x0 ) belongs to U for all times when x0 belongs to U1.
Furthermore, x converges uniformly to xe (with respect to t) as x0 approaches xe.
Definition 2: The fixed point P(x xe) is an asymptotically stable equilibrium
of the system, if and only if there exists such a neighborhood U of P that for all
x0 belongs to U1, we have
lim)t ( x0 ) P,
t of
) t (U ) ) s (U ) .
If xe is asymptotically stable for linear system (4.4), it will also be asymptotically
stable for the original nonlinear system (4.1) within a certain stable neighborhood
of the initial state x0. The importance of the notion of asymptotic stability arises
consequently. However, the linear problem (4.4) provides no information about
the size of the stable neighborhood around x0 for the nonlinear problem.
82
4.1 Center Manifold
each x0 U , there is an open interval I 0 R containing zero such that for all
x0 U and t I 0,
H D ) t ( x0 ) e At H ( x0 ) .
Near the origin, H maps the trajectories of nonlinear system (4.1) onto the trajectories
of linear system (4.4) and preserves the parameterization.
According to the Hartman-Grobman theorem, there is a one-to-one corres-
pondence between any qualitative change in the local nonlinear dynamics and
that in the concomitant linear dynamics.
The Hartman-Grobman theorem shows that near the hyperbolic equilibrium
point xe, nonlinear system (4.1) has the same qualitative structure as the linear
system (4.4). This result also paves way for local bifurcations whenever the
equilibrium loses hyperbolicity.
e At e k eOk t ek , (4.8)
and the set [e1, e2, " , en] is assumed to span Rn so that
n
x (0) ¦c e .
k 1
k k (4.9)
Equation (4.8) implies that the linear subspace of Rn generated by the eigenvectors
of A, which is called the eigenspaces EO of A, is invariant under the flow map etA,
i.e., if x (0) EO , then x (t ) EO for all t.
One way to divide this invariant subspace into three different invariant manifolds
is based on whether Re(Ok) is less than, equal to, or greater than zero:
1. Stable manifold Es is the subspace spanned by the eigenvectors {ek} with
Re(Ok ) 0 , namely, E s span{ek R n | ( A Ok I )ek 0 and Re(Ok ) 0};
2. Center manifold Ec is the subspace spanned by the eigenvectors {ek} with
Re(Ok ) 0, namely, E s span{ek R n | ( A Ok I )ek 0 and Re(Ok ) 0};
3. Unstable manifold Eu is the subspace spanned by the eigenvectors {ek} with
Re(Ok ) ! 0 , namely, Eu span{ek R n | ( A Ok I )ek 0 and Re(Ok ) ! 0}.
83
Chapter 4 Quantitative Methods
Corresponding to Es and Eu, the dynamics has the following simple asymptotic
property:
If x (t ) E s , then x (t ) o xe , as t o f;
If x (t ) Eu , then x (t ) o xe , as t o f .
Example 4.1 Consider linear system (4.4) with
ª0 1 0 º
A «1 0 0 » ,
« »
¬«0 0 2 ¼»
and O1 i, O2 i, O3 2, e1 [0,1, 0]T , e2 [1, 0, 0]T, and e3 [0, 0,1]T . The center
subspace Ec of the linear system is in the plane x1-x2. The stable subspace is on
axis x3. There is no unstable subspace for the system. These invariant subspaces
and some typical flows are shown in Fig. 4.1. Note that all solutions lie on the
cylinders x12 x22 c 2 .
V s , Re(Ok ) 0
°
Ok ®V c , Re(Ok ) 0 .
°V , Re(O ) ! 0
¯ u k
84
4.1 Center Manifold
x x, y y x2 .
Note that, for this system, the eigenspace Es corresponds to axis y, and Eu
corresponds to axis x. Eliminating t from the system, we obtain the equation of
the orbits
dy
x y x2 ,
dx
whose solution is
x2 C
y .
3 x
The Center Manifold Theorem states that the unstable manifold Wu is a tangent
to axis X(Eu) at the origin and is denoted by
x2
Wu : y .
3
85
Chapter 4 Quantitative Methods
Figure 4.3 Stable and Unstable Manifolds and Corresponding Invariant Subspace
x 2 Cx2 N 2 ( x1 , x2 ), (4.11)
where B is an nc u nc matrix with all eigenvalues on the imaginary axis, C is an
(ns nu ) u (ns nu ) matrix with all eigenvalues off the imaginary axis, and N1
and N2 denote the nonlinear terms. Here, ns dim(Ws ) , nc dim(Wc ) , and
nu dim(Wu ) .
A manifold Wc(xe, U) in a neighborhood U of the equilibrium point xe is said
to be a local central manifold for Eqs. (4.10) and (4.11) if
(1) Wc(xe, U) is invariant under the flow of Eqs. (4.10) and (4.11), and
(2) Wc(xe, U) is a group of function h(x1) x2 and is tangent to Ec at equilibrium
86
4.2 Hopf Bifurcation Method
wh( x1 )
h(0) 0, 0. (4.13)
wx1 x1 0
The local center manifold is then found by differentiating the relation x2 h(x1)
with respect to t, and by using Eqs. (4.10), (4.11), and (4.13), which leads to
Equation (4.14) is one of the local central manifolds in the neighborhood U of the
equilibrium point xe.
From Eqs. (4.10) and (4.12), the dynamics of the local central manifold are
then described by
x1 Bx1 N1 ( x1 , h( x1 )) . (4.15)
When the parameter point in parametem space goes across the boundary surface
of the stable region that is corresponding to the second type of critical stability, in
general, only two conjugate eigenvalues of linear system (4.4) cross the imaginary
axis of complex plane O and equation (4.15) is usually a two-dimension equation.
Thus, the central manifold at the bifurcation point causes a remarkable reduction
in dimensionality and therefore proves to be especially helpful in bifurcation
analysis of higher-dimensional systems.
In order to permit the occurrence of Hopf bifurcation, the first-order system (4.1)
87
Chapter 4 Quantitative Methods
must have dimension n ı 2. If the central manifold at the bifurcation point causes
the reduction of the dimension of Eq. (4.1), consider only the planar case n 2.
Here, Eq. (4.1) becomes
du
f (u , v; P ),
dt
(4.16)
dv
g (u , v; P ),
dt
where f and g are analytic functions of state variables u, v, and P.; Pis a variable
parameter of the system.
The critical point (u0(P), v0(P)), i.e., the equilibrium of Eq. (4.16), corresponds to
f (u0 ( P ), v0 ( P ); P ) 0,
(4.17)
g (u0 ( P ), v0 ( P ); P ) 0.
Its stability is determined by eigenvalues O1(P) and O2(P) of the matrix
ª f u (u0 , v0 ; P ) f v (u0 , v0 ; P ) º
A( P ) « g (u , v ; P ) g (u , v ; P ) » . (4.18)
¬ u 0 0 v 0 0 ¼
For a Hopf bifurcation, we have
O1 ( P ) O2* ( P ) D ( P ) iE ( P ). (4.19)
Assuming, without loss of generality, the bifurcation point is P= 0, we have
uˆ u u0 , vˆ v v0 , (4.21)
and expanding f and g in powers of û and v̂ , equation system (4.16) becomes
ª duˆ º
« dt » ªuˆ º ª F (uˆ , vˆ; P ) º
« » A( P ) « » « », (4.22)
« dvˆ » ¬ vˆ ¼ ¬G (uˆ , vˆ; P ) ¼
¬« dt ¼»
ˆ ˆ) as û and vˆ o 0, and are analytical functions
where F and G are O(uˆ 2 , vˆ 2 , uv
of û and v̂ .
To facilitate further discussion, let us suppose that A(P) has the following
88
4.2 Hopf Bifurcation Method
canonical form
ª D (P ) E (P )º
A( P ) « E (P ) D (P ) » . (4.23)
¬ ¼
Now, equation system (4.22) becomes
duˆ
D ( P )uˆ E ( P )vˆ F (uˆ, vˆ; P )
dt
. (4.24)
dvˆ
E ( P )vˆ D ( P )vˆ G (uˆ , vˆ; P )
dt
It shows that, at least in a small neighborhood of the bifurcation point, the origin
uˆ 0 , vˆ 0 in the plane uˆ -vˆ is a focus whose stability is determined by the sign
of D (P). Since D (0) 0, stability changes as P passes through zero.
To facilitate the discussion, let us introduce
dz
[D ( P ) iE ( P )]z N (z, z* ; P ) (4.26)
dt
with
1 2 1
N (z, z* ; P )
2
n1z n2 zz* n3 z* O (| z |3 ) (4.27)
2 2
To reduce equation system (4.26) to its normal form, introduce a near-identity
transformation
w z Q(z, z* ; P ) (4.28)
with the inverse
z w Q1 (w, w * ; P ) (4.29)
and
1 2 1
Q(z, z* ; P )
2
q1z q2 zz* q3 z* .
2 2
Differentiating equation (4.28) with respect to time t, we obtain
dw dz dz dz*
(q1z q2 z* ) (q2 z q3 z* ) ,
dt dt dt dt
which, by using Eqs. (4.26) and (4.29), becomes
89
Chapter 4 Quantitative Methods
dw 1 1
(D iE )w nˆ1 w 2 nˆ2 ww * nˆ3 w * O(| w |3 )
2
(4.30)
dt 2 2
with
Now, since E z 0 , we may choose q1, q2, and q3 in such a way that
dw
(D iE )w M (w, w * ; P ) (4.32)
dt
with
1 1
M (w, w * ; P )
2 3
m1 w 3 m2 w 2 w * m3 ww * m4 w * O(| w |4 ) .
3 3
In order to reduce Eq. (4.32) further, let us introduce another near-identity
transformation
[ w R (w, w * ; P ) (4.33)
with the inverse
w [ R1 ([, [* ; P ) , (4.34)
in which
1 1
R(w, w * ; P )
2 3
r1 w 3 r2 w 2 w * r3 ww * r4 w * .
3 3
Using Eqs. (4.33) and (4.34), we have
d[ dw 2 dw 2 dw *
(r1 w 2 2r2 ww * r3 w * ) (r2 w 2 2r3 ww * r4 w * ) ,
dt dt dt dt
which, with Eqs. (4.32) and (4.33), becomes
d[ 1 1
(D iE )[ mˆ 1[3 mˆ 2 [2 [* mˆ 3[[* mˆ 4 [* O (_ [ _4 )
2 3
(4.35)
dt 3 3
with
mˆ 1 m1 2(D iE )r1 , mˆ 2 m2 2D r2 ,
90
4.2 Hopf Bifurcation Method
mˆ 1 mˆ 3 mˆ 4 0 (4.36)
in the neighborhood of the bifurcation point P 0. However, r2 cannot be chosen
in any way to make mˆ 2 0 because D (0) 0. Consequently, choose r2 0. Given
mˆ 2 m2 J ( P ) iG ( P ) , (4.37)
and by using Eqs. (4.36) and (4.37), the system Eq. (4.35) becomes
d[
[D ( P ) iE ( P )][ [J ( P ) iG ( P )] | [ |2 [ O(| [ |4 ) , (4.38)
dt
where J (P) and G (P) are analytical functions of P. Equation (4.38) implies that
the nonlinear term N (z, z* ; P ) in Eq. (4.26) can be transformed to remove all
quadratic terms and all cubic terms except the term | [ |2 [. Note that | [ |2 [ is the
lowest-order nonlinear term with the same phase as [ and is, therefore, the most
dominant term producing resonance as | [ |o 0.
z1 D ( P ) z1 E ( P ) z2 h1 ( z1 , z2 ; P )
. (4.41)
z2 E ( P ) z1 D ( P ) z2 h2 ( z1 , z2 ; P )
where the third-degree functions h1 and h2 are respectively
h1 ( z1 , z2 ; P ) [J ( P ) z1 G ( P ) z2 ]( z12 z22 )
. (4.42)
h2 ( z1 , z2 ; P ) [G ( P ) z1 J ( P ) z2 ]( z12 z22 )
Let us introduce the polar coordinates (R, I) by
[ ReiI . (4.43)
91
Chapter 4 Quantitative Methods
R D ( P ) R J ( P ) R 3 (4.44)
and
I E ( P ) G ( P )R 2 . (4.45)
To benefit the discussion below, in which functions D (P), E(P), J (P), and G (P)
are expanded into power series, we neglect the high-degree terms and obtain two
approximate equations[4]
R c P R aR3 , (4.46)
I Z d P bR 2 , (4.47)
in which
dD ( P ) dE ( P )
c , d ,
dP P 0 dP P 0
Z E (0), (4.48)
1
a J (0) (h1,111 h1,122 h2,112 h2,222 )
16
1
[h1,12 (h1,11 h1,22 ) h2,12 (h2,11 h2,22 ) h1,11h2,11 h1,22 h2,22 ]
16Z
with
w 3 hi
hi , jkl , i, j , k , l 1, 2,
wz j wzk wzl
(4.49)
w 2 hi
hi , jk , i, j , k 1, 2.
wz j wzk
Equation (4.46) is independent of function I (t ) , and it is the equation of motion of
a one-dimensional autonomous system. Consequently, its equilibrium equation is
c P R aR3 0, (4.50)
which has two branches of critical solution, namely,
R 0 (4.51)
and
1
* § c P · 2
R ¨ ¸ . (4.52)
© a ¹
92
4.2 Hopf Bifurcation Method
To study the stability of the critical solutions (4.51) and (4.52), denote a function
h( R ) c P R aR 3 , (4.53)
whose derivative is
z F ( z, P ) A( P ) z H ( z , P ) O ( z 4 ), P R1 (4.55)
93
Chapter 4 Quantitative Methods
with
ªD ( P ) E ( P ) º
z [ z1 , z2 ]T , A( P ) « E (P ) D (P ) » , (4.56)
¬ ¼
O ( P ) D ( P ) r iE ( P ) ; and
dD ( P )
3. c z 0, a J (0) z 0 .
dP P 0
Then, at point P 0, the equilibrium of the system (4.55) is a focus, and Hopf
bifurcation occurs. If inequality acP < 0 holds good, a limit cycle occurs and its
stability is opposite to the stability of the equilibrium within it.
u1 u , u2 u (4.58a)
so that the equation (4.58) is written as the vector form
u1 ½ ª 0 1 º u1 ½ 0 ½
® ¾ « Z 2 ® ¾® ¾. (4.58b)
¯u2 ¿ ¬ 0 P »¼ ¯u2 ¿ ¯u12u2 ¿
According to the coefficient matrix A(P) of the linear term in above equation, we
obtain a pair of conjugate complex eigenvalues of the linearized equation of
system (4.58), i.e.,
94
4.2 Hopf Bifurcation Method
1
1
O (u ) D ( P ) iE ( P ) P i(Z 02 P / 4) 2 ,
2 (4.58c)
1
1
O2 (u ) D ( P ) iE ( P ) P i(Z 02 P / 4) 2 ,
2
Consequently, we have
P dD ( P ) 1
D (P ) , D (0) 0, c !0, (4.58d)
2 dP P 0
2
1
E ( P ) (Z 02 P / 4) 2 . (4.58e)
Next, performing near-identity transformation in the neighborhood of bifurcation
point P 0, we obtain the Poincare-Birhoff normal form of Eq. (4.58) and functions
h1(z1, z2, P) and h2(z1, z2, P) shown in expression (4.42). Then, according to
expression (4.48), we have
1
a J (P ) . (4.58f)
8
Eventually, substituting expression (4.58d) and (4.58f) in expression (4.52) and
using expression (4.58f), we obtain the approximate expressions of the amplitude
and the frequency of the stable periodic solution of the van der Pol Eq. (4.58), i.e.,
1
R* ( P ) 2P 2 (4.59)
and
1
§ P ·2
Z ( P ) E ( P ) Z0 ¨1 2 ¸ . (4.60)
© 4Z0 ¹
The phase diagram of the van der Pol Eq. (4.58) is shown in Fig. 4.5 and its
bifurcation diagram is shown in Fig. 4.6.
x Z 02 x H f ( x, x ) ,
(4.61)
in which His a small parameter, and f ( x, x ) is a nonlinear analytic function of x
and x . The generating linear system obtained by setting H 0 in Eq. (4.61) has
the period 2SZ 0. The nonlinear term H f ( x, x ) affects not only the amplitude but
also the period of the system. Hence, in the presence of the nonlinear term, it is
reasonable to expect that the system will not only have the period 2SZ0 but also
have the period 2S/Z, where Z is an unknown fundamental frequency Z (H)
depending on H, i.e., Z Z (H ) .
The essence of the Lindstedt-Poincare method is to produce periodic solutions
with every order of approximation to Eq. (4.61) by taking into account the fact
that the period of oscillation is affected by a nonlinear term. According to this
method, the solution of Eq. (4.61) is assumed in the form[5]
96
4.3 Lindstedt-Poincare Method
with the stipulation that the solution x(t) be periodic and of period 2S/Z, where
the fundamental frequency Z is given by
ª wf ( x , Z x c )
f ( x, Z xc) f ( x0 , Z0 x0c ) H « x1 0 0 0
«¬ wx
(4.65)
c
c wf ( x0 , Z0 x0 ) wf ( x0 , Z0 x0c ) º
x1 Z1 » H 2 ["] " ,
wx1c wZ »¼
Z 02 x0cc Z 02 x0 0,
Z 02 x1cc Z 02 x1 f ( x0 , Z 0 x0c ) 2Z 0Z1 x0cc , (4.66)
wf ( x0 , Z x0c ) wf ( x0 , Z 0 x0c )
Z 02 x2cc Z02 x2 x1 x1c
wx1 wx1c
wf ( x0 , Z 0 x0c )
Z1 (2Z 0Z 2 Z12 ) x0cc 2Z 0Z1 x1cc ".
wZ
Equation system (4.66) is solved recursively. However, we have an additional
task to determine the quantities Zi (i 1, 2, " ), which is accomplished by requiring
that each xi (t) (i 1, 2, " ) be periodic and of period 2S. The periodicity condition
has the mathematical form
97
Chapter 4 Quantitative Methods
Function xi may be periodic only in the absence of secular terms. To ensure that
xi is free of secular terms, we must prevent resonance. It requires that the right
sides of Eqs. (4.66) do not contain harmonic terms of W of unit frequency. This can
be guaranteed if the quantities Zi (i 1, 2, " ) are chosen to render the coefficients
of the harmonic terms of unit frequency equal to zero in the equation of xi (i 1,
2, " ). From the first equation of (4.66), we note that there is no danger of secular
terms in the case of x0 since the equation for x0 is homogeneous.
x H ( x 2 1) x Z 02 x
0, (4.68)
where H is a small parameter. Introducing dimensionless time W Z t, Eq. (4.68)
has the form
Z 2 xcc HZ ( x 2 1) xc Z 02 x 0 . (4.69)
The expansion of x(W ) and Z in power series with respect to H yielded
expressions (4.62) and (4.63). Completing quadratic operation, we obtain
x0cc x0 0,
Z1 cc 1 (4.71)
x1cc x1 2 x (1 x02 ) x0c ,
Z0 0 Z0
and so on. The general solution of the first equation of (4.71) is given by
98
4.3 Lindstedt-Poincare Method
we may only choose xc (0) 0 and satisfy this condition by setting x0c (0)
x1c (0) " 0 . Introducing B0 = 0 in the second equation of (4.71) yields
Z1 1 § 3 2· 1 3 1
x1cc x1 2 A0 cosW A0 ¨1 A0 A0 ¸ sin W A0
2
sin 3W . (4.73)
Z0 Z0 © 4 ¹ 4 Z0
For the equation to have a periodic solution, resonance must by avoided by
setting
§ 1 ·
A0Z 0, A0 ¨1 A02 ¸ 0 . (4.74)
© 4 ¹
These are equations defining A0 and Z1. As the trivial solution A0 0 is of no
interest, we have A0 2 and Z1 0 from (4.74). Hence,
2
x1cc x1 sin 3W (4.75)
Z0
has the general solution
1
x1 A1 cosW 1 B1 sin W sin 3W . (4.76)
4Z0
With x1c (0) 0 , we now obtain B1 3/(4Z 0 ). The value of A1 remains undeter-
mined for the moment. It is calculated in the next step of the iteration. However,
the intermediate calculations for the next step are omitted. Suffice it no note that
the subsequent equation is given by
5 3 3 A1 § Z 1 · A
x2cc x2 cos 5W cos 3W sin 3W ¨ 4 2 2 ¸
cosW 2 1 sin W .
4Z 2
0 2Z 2
0 Z0 © Z 0 4Z 0 ¹ Z0
(4.77)
The assumed periodicity of the solution requires Z 2 (16Z 0 )1 and A1 0, and
the limit cycle is now approximated by
3 1
x(W ) 2cosW H sin W H sin 3W . (4.78)
4Z 0 4Z 0
For the inverse transformation to the dimensional time t = W/Z, the relationship
§ 1 H2 ·
Z Z 0 ¨1 ¸ (4.79)
© 16 Z 02 ¹
is used. Since it is not possible to obtain an analytically accurate solution for
Eq. (4.68), such approximate formulas are extremely valuable. Figure 4.7 provides
99
Chapter 4 Quantitative Methods
Figure 4.7 Limit Cycles Corresponding to the Exact and the Approximate Solutions
100
4.4 An Averaging Method of Second-Order Autonomous System
x Z 0 a sin I . (4.81)
To determine the equation describing a(t) and E(t), we differentiate Eq. (4.80)
with respect to t and obtain
H
a f (a cos I , Z0 a sin I )sin I (4.86)
Z0
and
H
E f (a cos I , Z 0 a sin I ) cos I . (4.87)
Z0 a
Equations (4.80), (4.86), and (4.87) are exactly equivalent to Eq. (4.61) because
no approximation has been made yet.
For small H, a and E are also small. Hence, a and E vary much more slowly
with t than I Z 0t E. In other words, a and E hardly change during the period
2S/Z 0 of oscillation sinI and cosI. This enables us to average out the variables in
Eqs. (4.86) and (4.87). Averaging these equations over the period 2S/Z 0 and
taking constant a,E, a , and E , we obtain the following equations describing the
slow variations of a and E,
H 2S
a
2SZ 0 ³ 0
f (a cos I , Z0 a sin I )sin I dI (4.88)
and
H 2S
E ³ f (a cos I , Z 0 a sin I ) cos I dI . (4.89)
2SZ0 a 0
101
Chapter 4 Quantitative Methods
x H ( x 2 1) x Z 02 x 0. (4.90)
Here, the nonlinear function is
f ( x, x ) x x 3 (4.91)
so that Eqs. (4.88) and (4.89) become
Ha 2S 1 § 3 2 2·
S ³
a (sin 2 I Z02 a 2 sin 4 I )dI H a ¨1 Z 0 a ¸ , (4.92)
0 2 © 4 ¹
H
E
2
2 ³
(1 Z 02 a 2 sin 2 I )sin I cos I dI 0. (4.93)
0
The solution of Eq. (4.93) is E E 0, while the solution of Eq. (4.92) can be obtained
by separating variables. The result is
a02
a2 , (4.94)
3 2 2 § 3 2 2 · H t
Z0 a0 ¨1 Z0 a0 ¸ e
4 © 4 ¹
where a0 is the initial amplitude.
Equation (4.94) shows that the amplitude of the oscillation tends to be
as 2( 3Z0 )1 as long as it is different from zero, irrespective of magnitude of
the initial amplitude. Equation (4.92) shows that when a < as, a ! 0 , a tends to
increase. When a > as, a 0 , a tends to decrease. The value a as is a stable
amplitude, namely, the amplitude of the self-excited vibration.
In Fig. 4.8, the numerical solutions of Eq. (4.90) are compared with the
asymptotic results (4.94) for a0 > as and a0 < as, respectively.
102
4.5 Method of Multiple Scales for a Second-Order Autonomous System
In Fig. 4.8(a), the initial amplitude is less than that of the limit cycle, the damping
is initially negative and the amplitude increases until it reaches the limit cycle. In
contrast, the initial amplitude in Fig. 4.8(b) is greater than the amplitude of the limit
cycle, the damping is initially positive, and the amplitude decays until it reaches the
limit cycle. The corresponding trajectories in the phase plane are shown in Fig. 4.9.
Firstly, let us consider the weakly nonlinear quasi-harmonic system (4.61) again.
According to the method of multiple scales, the solution of this equation is
assumed in the form
103
Chapter 4 Quantitative Methods
d w w
H " D0 H D1 " , (4.97a)
dt wT0 wT1
2
d2 § w w ·
¨ H "¸ D02 2H D0 D1 H 2 ( D02 2 D0 D1 ) " , (4.97b)
dt 2 © wT0 wT1 ¹
and expanding in a power series of H , we obtain
f ( x, x ) f ( x0 , D0 x0 ) H [ f xc ( x0 , D0 x0 ) f xc ( x0 , D0 x0 )( D0 x1 D1 x0 )] O(H 2 ) .
(4.98)
Finally, by substituting (4.96), (4.97), and (4.98) in Eq. (4.61) and completing
an arrangement in accordance with the power of H , a comparison of the coefficients
yields
D02 x0 Z 02 x0 0, (4.99a)
D02 x1 Z 02 x1 2 D0 D1 x0 f ( x0 , D0 x0 ) , (4.99b)
D02 x2 Z 02 x2 2 D0 D1 x1 D12 x0 2 D0 D2 x0
f xc ( x0 , D0 x0 ) x1 f xc ( x0 , D0 x0 )( D0 x1 D1 x0 ). (4.99c)
104
4.5 Method of Multiple Scales for a Second-Order Autonomous System
D02 x1 Z 02 x1
2iZ0 D1 AeiZ0T0 cc f ( AeiZ0T0 cc,iZ 0 AeiZ0T0 cc). (4.102)
To get a periodic solution for the previous equation, the secular term must be
avoided by setting
Z0 2S Z 0
2S ³
2iZ0 D1 A f ( AeiZ0T0 cc,iZ 0 AeiZ0T0 cc)eiZ0T0 dT0 0. (4.103)
0
1 2S
i(D1a iaD1I )
2SZ0 ³ 0
f (a cos I , Z0 a sin I )(cos I i sin I ) dI .
Separating the real part and the imaginary part of the equation, we have
1 2S
D1a
2SZ 0 0³ f (a cos I1 Z 0 a sin I )sin I dI ,
(4.105)
1 2S
D1I
2SZ 0 a ³ 0
f (a cos I1 Z 0 a sin I ) cos I dI ,
which are two conditions to determine the solution of Eq. (4.102), x1(T0, T1, " ).
Then, substituting x0(T0, T1, " ) and x1(T0, T1, " ) in Eq. (4.99c) and using the
conditions for the absence of the secular term in the solution of Eq. (4.99c), we
obtain x1(T0, T1, " ).
Without loss of generality, we denote Z 0 1 , and the van der Pol equation is
x H (1 x 2 ) x x
0. (4.106)
Then, denote the initial conditions
x(0) a, x (0) 0, (4.107)
and expand the solution of Eq. (4.106) as a power series of H , i.e.,
105
Chapter 4 Quantitative Methods
Substituting the expression in Eq. (4.106) and carrying out the procedure of the
method of multiple scales, we obtain
D02 x0 x0 0 (4.109a)
and
D02 x1 x1 2 D0 D1 x0 (1 x02 ) D0 x0 . (4.109b)
Solving Eq. (4.109a) for x0(T0, T1), we obtain
1
A(T1 ) a (T1 )e iI (T1 ) . (4.111)
2
Substituting Eq. (4.111) in Eq. (4.109b), we obtain
2D1 A A A2 A* . (4.113)
Substituting (4.111) in (4.113) and separating the real part and the imaginary part,
we obtain
da a § a2 ·
¨1 ¸ (4.114)
dT1 2© 4¹
and
dI
0. (4.115)
dT1
Integrating Eqs. (4.114) and (4.115) and considering conditions (4.107), we
obtain
2
a(T1 ) 1
(4.116)
ª § 4 · T º 2
«1 ¨ 2 1¸ e 1 »
¬ © a0 ¹ ¼
and
I I0 . (4.117)
106
References
Eventually, the first-order approximate solution of the van der Pol Eq. (4.106)
with the initial conditions (4.107) is found, i.e.,
2
x(t ) 1
cos t . (4.118)
ª § 4 · H t º 2
«1 ¨ 2 1¸ e »
¬ © a0 ¹ ¼
References
[1] B K Shivamoggi. Nonlinear Dynamics and Chaotic Phenomena, An Introduction. Dordrecht:
Kluwer Academic Publisher, 1997
[2] L Perko.Differential Equations and Dynamical Systems. New York: Springer-Verlag, 1991
[3] A B Poore. On the Theory and Application of the Hopf-Fredrich Bifurcation Theory. Archive
for Rational Mechanics and Analysis, 1976, 60(4): 371 392
[4] J E Massden, M McCracken. The Hopf-Bifurcation and Its Applications. Springer-Verlag,
New York: 1976
[5] L Meirovitch. Elements of Vibration Analysis. New York: McGraw-Hill, 1986
[6] P Hagedorn. Nonlinear Oscillations. Oxford: Clarendon Press, 1982
[7] A H Nayfeh. Nonlinear Oscillations. New York: John Wiley & Sons, 1979
[8] P A Sturrock. Nonlinear Effects in Electron Plasmas. Proc. Roy. Soc. (London), A242:
277 299, 1957
107
Chapter 5 Analysis Method for Closed-Loop System
whole. When acted on by a given excitation, the system exhibits certain response.
In a generalized sense, dynamics is the study of this cause-effect relation. Excitation
is known as the input signal or simply the input, and the response is known as the
output signal or the output. The cause-and-effect relationship can be shown
schematically in the form of the block diagram in Fig. 5.1.
If the system is to exhibit some desired output, we must select certain input
and make the system subject to the selected input through a controller as shown
in Fig. 5.2. If the input is essentially predetermined and not influenced by the
output, the control system is an open-loop control system and the plant represents
the controlled object. However, in many cases, the input depends on the output
either inadvertently or intentionally. It is conceivable that unknown or unforeseen
factors may actually prevent a system from meeting the desired objectives. In such
case, we may wish to consider the actual output when implementing a change in
the input so that the objectives can be met. Such a system is depicted in the block
diagram in Fig. 5.3. In addition to the controller, there is a measuring device and
a comparing device. The measuring device senses the output and then feeds it to
the comparing device. The comparing device calculates the error, i.e., the difference
between the desired and the measured output. Based on this error, the controller
issues necessary input commands, termed the control effect, to reduce the error to
zero. The output signal of the controlled plant is called system state and the desired
output is called reference input.
109
Chapter 5 Analysis Method for Closed-Loop System
Any control system consists of some components and every component has its own
input and output, as shown in Fig. 5.4. Here, we just consider linear components
with lumped parameters, whose equations governing dynamic processes are linear
ordinary differential equations with constant coefficients.
xo (t ) Kxi (t ) , (5.1)
where constant K is called amplification coefficient. Such a component is
referred to as proportional component or linear ideal component.
2. Inertial component
The relationship between the output signal xo(t) and the input signal xi(t) of the
component is described by the equation
xo 2[ Txo xo
T 2 Kxi , [ (0,1) , (5.3)
where T is the time constant, [ the damping ratio, and K the amplification
coefficient. Such a component is referred to as oscillatory component.
4. Integral component
The relationship between the output signal xo(t) and the input signal xi(t) of the
110
5.1 Mathematical Model in Frequency Domain
xo Dxi xi , (5.5)
where D is the coefficient of the differential term. Such a component is referred
to as differential component.
6. Constant time delay component
The relationship between the output signal xo(t) and the input signal xi(t) of the
component is described by the equation
xo (t ) Kxi (t W ) , (5.6)
where W is called delay time and K is amplification coefficient. Such a component
is referred to as constant time delay component.
L[ f1 (t ) f 2 (t )] F1 ( s ) F2 ( s ),
(5.8)
L[ Kf (t )] KF ( s ).
In addition, Laplace transformations of the derivatives with definite order of a
function x(t) are given by
111
Chapter 5 Analysis Method for Closed-Loop System
L[ x ] sF ( s ) f (0),
x] s 2 F ( s ) sf (0) f (0),
L[
(5.9)
#
L[ x ] s F ( s ) s n 1 f (0) s n 2 f (0) " f ( n 1) (0).
( n) n
d
p .
dt
Assume that all initial values of xi(t), xo(t) and their relevant derivatives vanish,
namely,
N ( s ) a0 sn a1 sn 1 " an 1 s an ,
(5.12)
M ( s ) b0 sm b1 sm 1 " bm 1 s bm ,
where X o ( s ) and X i ( s ) are the Laplace transformation of xo (t ) and xi (t ).
Denote
M (s)
G(s) , (5.13)
N (s)
which is referred to as the transfer function corresponding to the given input-
output relationship described by Eq. (5.10). Transfer function is a property of a
system itself, independent of the magnitude and the nature of the input or
112
5.1 Mathematical Model in Frequency Domain
excitation[1].
Simultaneously, a transfer function includes units necessary to relate the input
and the output. However, it does not provide any information concerning the physical
structure of the system, and transfer functions of many physically different systems
can be identical.
According to Eqs. (5.1) (5.6), transfer functions of typical components are
derived easily and listed below.
1. Proportional component
G(s) K, (5.14)
2. Inertial component
K
G(s) , (5.15)
Ts 1
3. Oscillatory component
K
G(s) , (5.16)
T s 2[ Ts 1
2 2
4. Integral component
K
G(s) , (5.17)
s
5. Differential component
G(s) Ds 1 , (5.18)
6. Constant time delay component
G(s) Ke W s . (5.19)
Except for the constant time delay component, transfer functions of all above
typical components are rational fractions or polynomials.
In a closed-loop system, there are at least two signal channels, a forward channel
and a feedback channel, as shown in Fig. 5.3. The forward channel consists of the
plant, the controller, and the comparing device. Its transfer function is represented
by G(s). The feedback channel is usually the measuring device whose transfer
function is represented by H(s). As shown in Fig. 5.5, the block diagram of the
closed-loop system may be described by transfer functions.
113
Chapter 5 Analysis Method for Closed-Loop System
Here, let us consider the differential Eq. (5.10) describing the input-output relations
of a linear system with lumped parameters and assume that its input is harmonic,
namely,
d r iZ t
e (iZ ) r eiZ t , r 1, 2," , n , (5.22)
dt r
114
5.2 Nyquist Criterion
and inserting expressions (5.20), (5.21), and (5.22) in Eq. (5.10), we have
M (iZ )
G c(iZ ) . (5.23)
N (iZ )
Comparing functions (5.13) and (5.23) yields
xi (t ) A sin Z t , (5.25)
its output should be in the following form
K
G(s) (5.28)
T0 s T1 s 2 T2 s 1
3
and
K
G( s) . (5.29)
s (T0 s T1 s 1)
2
Their corresponding hodographs of G(iZ) are drawn in Fig. 5.7 (a) and (b)
respectively.
115
Chapter 5 Analysis Method for Closed-Loop System
Figure 5.6 Hodographs of G(iZ) (a) Proportional; (b) Inertial; (c) Oscillatory;
(d) Integral; (e) Differential; (f) Constant Time Delay Components
116
5.2 Nyquist Criterion
Y (s) G(s)
GC ( s ) . (5.30)
R( s) 1 G(s)
Hence, the characteristic polynomial of the closed-loop system is
D( s )
Q( s ) . (5.32)
N (s)
Using expression (5.30) and (5.31), we obtain
Q( s ) 1 G ( s ) .
Consequently, we have
Figure 5.9 Hodographs of Q(iZ) and G(iZ) of the Same Linear System
117
Chapter 5 Analysis Method for Closed-Loop System
According to expression (5.32) and the product formula of the vectors, we obtain
' Arg Q(iZ ) ' Arg D(iZ ) ' Arg N (iZ ). (5.34)
Z :0of Z :0of Z :0of
If l poles of G(s), i.e., zero points of N(s), have positive real parts, and (nl) zero
points of N(s) have negative real parts, we obtain
S S
' Arg N (iZ ) [(n l ) l ] (n 2l ) . (5.35)
Z :0 of 2 2
As mentioned in Chapter 3, if the argument increment of the hodograph of the
complex vector D(iZ) of the n-order closed-loop system is equal to nS/2 with Z
varying from 0 to f , all roots of its characteristic equation have negative real
parts and its equilibrium is asymptotically stable. Thus, the necessary and sufficient
condition of asymptotic stability of the equilibrium of the n-order closed-loop
system is
nS
' Arg D(iZ ) . (5.36)
Z :0 of 2
Substituting expressions (5.35) and (5.36) in (5.34), we obtain the necessary and
sufficient condition of asymptotic stability of the equilibrium of the n-order
closed-loop system, i.e.,
n S (n 2l )S
' Arg Q(iZ ) lS . (5.37)
Z :0 of 2 2
So, if l poles of G(s) have positive real parts, the necessary and sufficient
condition of asymptotic stability of a closed-loop system is that the argument
increment of the hodograph of Q(iZ) should be equal to lS with Z varying from
0 to f .
As shown in Fig. 5.9, we may conclude the following: If l poles of the transfer
function G(s) of an open-loop system have positive real parts, the necessary and
sufficient condition of asymptotic stability of the closed-loop system is that the
argument increment of the hodograph of G(iZ) around the critical point C(1, 0)
is equal to lS with Z varying from 0 to f . This is the Nyquist criterion.
As a particular case, the Nyquist criterion states that if all the poles of the transfer
function G(s) of an open-loop system have negative real parts, the necessary and
sufficient condition of asymptotic stability of equilibrium position of the closed-
loop system is that the argument increment of the hodograph of G(iZ) around the
critical point C(1, 0) is equal to zero with Z varying from 0 to f .
for linear autonomous closed-loop systems. Two illustrative examples are introduced
here.
Example 5.1 A second-order system shown in Fig. 5.8 has the open-loop transfer
function
K
G ( s) , [ ,T , K ! 0 , (5.38)
T 2 s 2 2[ Ts 1
where T, [, and K are inertial constant, damping ratio, and amplification coefficient
respectively.
The hodographs of G(iZ) for two cases, i.e., K < 1 and K > 1, are shown in
Fig. 5.10(a) and (b).
The transfer function of the open-loop system has one pole with positive real
part, i.e., l 1. According to the Nyquist stability criterion, the necessary and
sufficient condition is that the argument increment of the hodograph of G(iZ)
around C(1, 0) point be equal to S with Z varying from 0 to f . Hence, Fig. 5.10
shows that the equilibrium point of the closed-loop system corresponding to the
open-loop transfer function (5.38) is asymptotically stable if amplification coefficient
K > 1. In contrast, it is unstable if amplification coefficient K < 1.
Example 5.2 As shown in Fig. 5.8, a first order closed-loop system with constant
time delay has the open-loop transfer function
Ke W s
G(s) , K ,W , T ! 0 , (5.39)
Ts 1
where T, K, and W are inertial constant, amplification coefficient, and delay time
respectively.
Substituting s iZ in (5.39), we obtain
119
Chapter 5 Analysis Method for Closed-Loop System
and
K (sin WZ T Z cosWZ )
Q(Z ) . (5.42)
1 T 2Z 2
The hodograph of G(iZ) is depicted in Fig. 5.11.
T Z tan(WZ ) 0. (5.43)
Except the zero root, the transcendental Eq. (5.43) has numberless positive real
roots, i.e., Z1 , Z 2 " . In Fig. 5.11, the intersection A of the real axis and the
hodograph of G(iZ) corresponds to Z Z1 . The value of Z1 is determined by the
following expression
1§S § T ··
Z1 arctan ¨ ¸ ¸ . (5.44)
W ¨© 2 © W ¹¹
Substituting expression (5.44) in expression (5.41), we obtain
1 T 2Z12
K cr .
T Z1 sin WZ1 cosWZ1
120
5.3 A Frequency Criterion for Absolute Stability of a Nonlinear Closed-Loop System
In 1944, a small note was written by A. I. Lurie and V. I. Postnikov, in which the
directed method of Lyapunov was applied to stability analysis of motion of a
given automatic control system. Stability in large, i.e., stability for arbitrary initial
perturbation and stability under some conditions for arbitrary nonlinearity of
actuators, was considered. Such stability is called absolute stability[3]. In a
number of further works, A. I. Lurie developed concepts that he later introduced
in a monograph in 1951 and the results served as the starting point for further
121
Chapter 5 Analysis Method for Closed-Loop System
u f (V ) , (5.47b)
n
V ¦c x
j 1
j j , (5.47c)
f (V )
0 K f, (5.48)
V
and aD j, bD , and cj are constant coefficients.
Condition (5.48) means that the graph of M f (V) should lie between the V-axis
and line M KV in the V -M plane shown in Fig. 5.13 and the function M f (V)
may be arbitrary. In particular, it may have a form similar to the function shown
122
5.3 A Frequency Criterion for Absolute Stability of a Nonlinear Closed-Loop System
in Fig. 2.29(c) and (d). It can be seen that system (5.47) is obtained by closing
the open-loop system (5.47(a)) with an inserted nonlinear element (5.47(b)).
Denoting G(s) as the transfer function from input f (V) to output V of system
(5.47), G(s) may be derived from equation system (5.47) as
cT [c1 , c2 ," , cn ],
b [b1 , b2 ," , bn ],
T
(5.50)
A [aD j ]nun .
The basic theorems that were proved by V. M. Popov are outlined below. These
may be found in reference[7].
Depending on the position of the poles of the transfer function G(s), we
distinguish the non-critical cases, when all poles lie in the left half-plane of complex
variable s, and the critical cases, when some poles are on the imaginary axis of
the s plane.
Theorem 1 (for non-critical cases) Let the following conditions hold good.
(1) The nonlinear function f (V) satisfies (5.48);
(2) All poles of G(s) have negative real parts;
(3) There exists a real number H so that the frequency condition
1
Re[(1 iZH )G (iZ )] ı 0 (5.51)
K
holds good for all Z ı 0 . Then system (5.47) is absolutely stable.
Obviously, the poles of G(s) refer to the zeros of the polynomial in the
denominator of G(s), i.e., the characteristic roots of the open-loop system (5.47).
123
Chapter 5 Analysis Method for Closed-Loop System
1
P(Z ) HZ Q (Z ) ı 0 . (5.52)
K
We construct a modified frequency response G*(iZ), an image point defined
by the coordinate [P(Z), Z Q(Z)]. If we introduce a new plane defined by P1 P,
and Q1 Z Q, and construct the hodograph of the modified frequency response
G*(iZ) on this plane for Z ı 0 , condition (5.51) implies that there should exist a
1 § 1 ·
straight line P1 H Q1 0 through the point ¨ , 0 ¸ and lying to the left of
K © K ¹
the modified hodograph, as shown in Fig. 5.15.
dition (5.51) holds good for all Z > 0. Then, the system (5.47) is absolutely stable.
Theorem 3 (for critical cases with two zero poles) Let the following conditions
hold good.
124
5.3 A Frequency Criterion for Absolute Stability of a Nonlinear Closed-Loop System
(2) The transfer function G(s) has two zero poles, and the rest of its poles (for
n ! 2 ) have negative real parts;
(3) D lim s 2G ( s ) ! 0
s o0
d 2
U lim [ s G ( s )] ! 0
s o0 ds
lim S(Z ) 0
Z of
T1 x1 x1 x3 f ( x2 ),
T2 x2 x2 x1 , (5.54)
x3 x2 ,
where T1 > 0 and T2 > 0 are time constants and the function f (x2) satisfying
condition (5.48) is shown in Fig. 5.16.
125
Chapter 5 Analysis Method for Closed-Loop System
First, deduce the transfer function from input f(x2) to output x2. Then, let s
denote the differential operator of time. System (5.54) can be rewritten as
(T1 s 1) x1 x3 f ( x2 ),
(T2 s 1) x2 x1 ,
sx3 x2 .
Eliminating x1 and x3 from these equations yields
s
x2 f ( x2 ).
T1T2 s (T1 T2 ) s 2 s 1
3
s
G(s)
as 3 bs 2 s 1
with
a T1T2 , b T1 T2 . (5.55)
If we let
b!a, (5.56)
and apply Hurwitz criterion (3.14), then all the poles of the transfer function G(s)
have negative real parts and we can use Theorem 1 for this case.
Next, completing a simple operation, we obtain
Z 2 [(H b a)Z 2 (1 H )]
Re[(1 iZH )G (iZ )] .
(1 bZ 2 ) Z (1 aZ 2 ) 2
To ensure that condition (5.52) is satisfied for all Z ı 0 , the necessary and sufficient
condition is that H should satisfy H b a ı 0 and 1 H ı 0. Hence, we should have
a
İ H İ 1.
b
Such a H should exist due to inequality (5.56), and in terms of the given coefficients
expressed by equalities (5.55), we obtain
T1 T2 ! T1T2 .
The domain : of absolute stability in the parameter space for system (5.54)
is shown in Fig. 5.17. It is bounded by the lines T1 0, T2 0 and one of the
branches of the hyperbola
T1 T2 T1T2 . (5.57)
126
5.4 Describing Function Method
The describing function method notes that only the fundamental harmonic
component of the output is significant. Such assumption is usually valid since the
higher harmonic components in the output of a nonlinear element are often of
127
Chapter 5 Analysis Method for Closed-Loop System
| G (inZ ) |
1, n 2,3," . (5.58)
| G (iZ ) |
with
1 2S
2S ³ 0
Ak y (t )sin kZ t d(Z t ),
1 2S
2S ³ 0
Bk y (t ) cos kZ t d(Z t ), (5.61)
I1 arctan( A1 / B1 ),
1
Y1 ( A12 B12 ) 2 .
Thus, the describing function is given by
1
( A12 B12 ) 2 iI1
N ( A) e .
A
128
5.4 Describing Function Method
Obviously, A1, B1, Y1, and I1 are all the functions of the amplitude A of the
sinusoidal input. Let us denote
A1 B1
g ( A) , b( A) . (5.62)
A A
The describing function N(A) may be written as
129
Chapter 5 Analysis Method for Closed-Loop System
If self-excited vibration exists in a system, the amplitude and the frequency of the
output may be found from a graphical study of the describing function.
1 N ( A)G (iZ ) 0
or
1
G (iZ ) . (5.67)
N ( A)
If previous equation is satisfied, the system output will be a periodic motion,
namely, self-excited vibration. This situation of nonlinearity corresponds to the
case that the hodograph of G(iZ) passes through the critical point. According to
the Nyquist criterion, the critical point is just the point C(1, 0) in the complex
plane. Equation (5.67) means that the entire 1/N(A) becomes a locus of the
critical point in the describing function method.
To determine the amplitude and the frequency of self-excited vibration, plot
the 1/N(A) locus and the hodograph of G(iZ). If the 1/N(A) locus and the
hodograph intersect, the system output may exhibit self-excited vibration. Such
self-excited vibration is usually not sinusoidal, but it may be approximatly
sinusoidal. It is characterized by the values of A and Z corresponding to the
intersection of the 1/N(A) locus and the hodograph of G(iZ).
More accurate results of the describing function method may be obtained by
considering the higher harmonic components. This approach has been outlined in
reference [12].
130
5.4 Describing Function Method
Consider the case when a slight disturbance is given to the system operating at
point B2. Assume that the operating point is moved to point E on the 1/N(A)
locus. The hodograph G(iZ) in this case does not enclose the critical point E.
With the amplitude of the sinusoidal input to the nonlinear element decreasing,
the operating point moves toward point B2. If the operating point moves to point
F on the 1/N(A) locus, the hodograph of G(iZ) encloses the critical point F.
With the amplitude of the sinusoidal input to the nonlinear element increasing, the
operating point moves toward point B2. Thus, B2 possesses convergent charac-
teristics and the system operation at point B2 is stable. In other words, the limit
cycle at this point is stable.
By means of similar analysis, we conclude that the limit cycle at intersection
B1 is unstable.
nonlinearity as shown in Fig. 5.23. These 1/N(A) loci are straight lines parallel
to the real axis. The values of N(A) are obtained from expression (5.64).
From Fig. 5.23, we can see that the amplitude and the frequency of the limit
cycles are
A 0.27, Z 7, if ' 0.1;
A 0.42, Z 5.9, if ' 0.2;
A 0.57, Z 5.1, if ' 0.3.
Inspection of these values reveals that increasing the hysteresis width decreases
the frequency but increases the amplitude of the limit cycle, as expected.
Example 5.5 Consider the equation of the van der Pol vibrator
x H (1 x 2 ) x x
0. (1.18)
It may be written as
x H x x
H f ( x, x ), (5.68)
where the nonlinear function f ( x, x ) x 2 x .
We constitute the block diagram of system (5.68) as shown in Fig. 5.24.
132
5.5 Quadratic Optimal Control
x AZ cos Z t ,
y (t ) f ( x, x ) A3Z sin 2 Z t cos Z t .
Using expression (5.61), we obtain
A1 0, B1 A2Z / 4 .
Thus, N ( A) iA2Z /4 . According to Eq. (5.67), the characteristic equation of
system (5.68) is
A2 (iZ ) H
1 0.
4 (iZ ) 2 H (iZ ) 1
Solving the complex equation, we have A 2 and Z 1. This indicates the
existence of a limit cycle with amplitude 2 and frequency 1. This result is in
accordance with the first approximation by Lindsted-Poincare method in Chapter 4.
133
Chapter 5 Analysis Method for Closed-Loop System
x Ax Bu , (5.69)
in which
x [ x1 , x2 ," , xn ]T ,
u [u1 , u2 ," , um ]T ,
ª a11 ... a1n º
A « ... »,
« » (5.70)
«¬ an1 ... ann »¼
ª b11 ... b1m º
B « ... »,
« »
«¬bn1 ... bnm »¼
where x is the state vector, u the control vector, A the state matrix, and B the
control matrix.
The optimal control problem is to choose the control vector u U so that the
state vector x is transferred from an initial point x(0) x0 to a terminal point
x (T ) xT , and in general, xT 0 , at time T. The region U is called admissible
control region. If the transfer can be accomplished, the problem in quadratic
optimal control is to effect the transfer so that the performance index function
T
J ³ 0
( x T Qx + uT Ru)dt (5.71)
134
5.5 Quadratic Optimal Control
and
wL d § wL ·
¨ ¸ 0, (5.74)
wu dt © wu ¹
which give two equations
Qx AT O O 0 (5.75)
and
Ru B T O 0. (5.76)
Substituting O from Eq. (5.76) in Eq. (5.75) and making use of system Eq. (5.69)
yield
x Ax BR 1 B T O . (5.77)
Equation (5.75) shows that the function O and its derivative are linear combinations
of the state vector. Thus, lets us denote
O P (t ) x , (5.78)
and differentiate Owith respect to time t. Substituting Eq. (5.77) in the mathe-
matical expression of O , we obtain
O ( P PA PBR 1 B T P ) x . (5.79)
Substituting Eq. (5.78) and (5.79) in Eq. (5.75), we obtain
P PA AT P PBR 1 B T P Q 0. (5.80)
This equation is just the well-known Riccati differential equation, which is a
first-order nonlinear differential equation. E. Kalman investigated the behavior of
solution P(t) of Eq. (5.80) and pointed out that if A and B are constant matrices,
system (5.69) has controllability, Q and R are positive definite symmetrical
matrices, and control time T approaches infinity. Then, the solution matrix P
approaches a constant matrix, which is the solution matrix of the following Riccati
algebraic matrix equation
AT P AP PBR 1 B T P Q 0. (5.81)
Thus, if we choose the performance index (5.71) and solve the algebraic matrix
Eq. (5.81), we will obtain the symmetrical solution matrix P. Then, using Eq. (5.77)
and (5.78), we can obtain the optimal control law of linear system (5.69),
u* R 1 B T Px . (5.82)
135
Chapter 5 Analysis Method for Closed-Loop System
According to this optimal control law u*, we construct a control system that is
a steady linear closed-loop system and its block diagram is shown in Fig. 5.25.
The feedback signals are the linear combinations of the state variables. Thus, this
type of controller is referred to as state regulator.
Optimal state control requires measuring the signals of all state variables. However,
in practice, only some output signals which, in general, are the linear combinations
of the state variables may be measured. Therefore, we have
y Cx , (5.83)
where y is the output vector of system (5.69) and C is the output matrix.
The performance index function of optimal output control is constructed as
f
J1 ³ 0
( y T Qy uT Ru)dt . (5.84)
Comparing function (5.71) with (5.84), we can see that the mathematical
structures of the performance indexes J and J1 are the same and the integrands
are all quadratic forms of x and u. Thus, the optimizing results from index
function (5.85) are similar to those from Eq. (5.82). The correspondent Riccati
algebraic matrix equation is
According to this optimal output control law, we may construct a control system
that is still a linear autonomous closed-loop system. The block diagram is shown
in Fig. 5.26. In general, matrix C is not a square matrix, and the operation of
matrix C 1 is related to seeking the generalized inverse of the matrix.
As an illustrated example of the optimal output control, let us consider the equation
of the van der Pol vibrator, namely, Eq. (1.18), and write it in the form
z H (1 E z 2 ) z z
0. (1.18)
When H is equal to zero, it is the Hopf bifurcation point and Eq. (1.18) is reduced to
zz 0. (5.88)
With the state variables x1 z and x2 z , the state equation is written as
x Ax , x [ x1 x2 ]T (5.89)
with
ª 0 1º
A « 1 0 » . (5.90)
¬ ¼
If we make use of a force controller and a displacement sensor to construct the
control system, the state equation and the output equation of control system are
x Ax Bu , (5.91)
y Cx (5.92)
with
ª0 º
B «1 » , (5.93)
¬ ¼
C [1 0]. (5.94)
137
Chapter 5 Analysis Method for Closed-Loop System
Q 1, R r. (5.95)
Substituting matrices (5.90), (5.93), (5.94), and (5.95) in the Riccati algebraic
matrix Eq. (5.86) yields
2 p12 r 1 p122 1 0,
p11 p22 r 1 p12 p22 0,
1
2 p12 r p 2
22 0.
Solving the equation system, we obtain
1
ª 2 2
1
½º 2
p11 « 2r (1 r ) ®(r 1) 2 r ¾» ,
«¬ ¯ ¿»¼
1
ª 2 1
½º 2
p22 « 2r ®(r 1) 2 r ¾» , (5.96)
¬« ¯ ¿¼»
1
p12 (r 2 1) 2 r.
If r 1, we have
x 0.91x 1.414 x 0. (5.98)
138
References
The equation shows that the damping ratio and the natural frequency of the
controlled harmonic vibrator (5.88) are respectively [ 0.3826 and Z 1.1891.
If the term corresponding to u1* , the expression (5.97), is inserted in the
equation of the van der Pol vibrator, namely, Eq. (1.18), the Hopf-bifurcation
will not occur. Thus, the optimal output control can prevent the controlled system
from self-excited vibration.
References
[1] L Meirovitch. Introduction to Dynamics and Control. New York: John Wiley & Sons, 1985
[2] Katsuhiko Ogata. Modern Control Engineering, Second Edition. New Jersey: Prentice
Hall, 1990
[3] A I Lurie, V N Postnikov. On the Stability Theory of Control System. Sov. Appl. Math.,
1944, 8(3): 246 296
[4] A I Lurie. Some Nonlinear Problems in Automatic Control Theory (in Russian). M.
Gostakhizdat, 1951
[5] M A Aizerman, F R Gantmakher. Absolute Stability of Controlled System (in Russian).
M. AH. SSSR, 1963
[6] D R Merkin. Introduction to the Theory of Stability. New York: Springer-Verlag, 1997
[7] V M Popov. Hyperstability of Control System, New York: Springer-Verlag, 1973
[8] A Netushil. Theory of Automatic Control. Moscow: Mir Publishers, 1978
[9] R J Kochenburger. Limiting in Feedback Control Systems, Trans. AIEE Vol. 72, Part II,
1953
[10] E P Popov. On the Use of the Harmonic Linearization Method in Automatic Control
Theory. NACA TM 1406, 1957
[11] D P Atherton. Nonlinear Control Engineering Describing Function Analysis and Design.
New York: van Nostrand Reinhold, 1982
[12] Naumov. Philosophy of Nonlinear Control Systems. Moscow: Mir Publishers, 1990
[13] H L Trenteman, A A Stoorvogal, M Hautus. Control Theory for Linear System. London:
Springer-Verlag, 2001
[14] F L Lewis, V L Syrmos. Optimal Control, Second Edition. New York: John Wiley & Sons,
1995
139
Chapter 6 Stick-Slip Vibration
141
Chapter 6 Stick-Slip Vibration
The first recorded systematic study in the field of tribology was done by
Leonardo da Vinci (1591), who not only performed experimental studies of friction
but also introduced the coefficient of friction as the ratio between the friction
force and the normal force. His contributions to sliding friction, which formed
the basis for all the subsequent studies of sliding friction[2], were made about 200
years before the publication (1687) of Newton’s Principia.
The classical model of friction states that the friction force is proportional to
the load, opposes the motion, and is independent of the contact area. Though it
remained hidden in his notebook for centuries, it was already known to Leonardo
da Vinci. His friction model was rediscovered by G. Amontons (1699) and
developed by C. A. Coulomb (1785).
(1) Coulomb friction law
C. A. Coulomb investigated the influence of five main factors upon friction,
namely,
a) the nature of the materials in contact and their surface coatings,
b) the extent of the surface area,
c) the normal pressure (or force),
d) the length of time that the surface remained in the stationary contact,
e) the ambient conditions such as temperature, humidity, and even vacuum.
Coulomb summarized many of his results in the friction law
Fk Pk N , (6.1)
where Fk is the kinetic friction force, N the normal force, and P k the kinetic
friction coefficient. He found that the kinetic friction coefficient P k is usually
nearly independent of the contact area, and the surface roughness. As long as the
velocity is not too high or too low, it is also independent of the sliding velocity.
The Coulomb friction law can be used to study the stick-slip motion occurring
in multi-degree-freedom systems, as shown in the last section of this chapter.
However, it cannot be applied to studying the stick-slip motion in single-degree-
freedom systems, where the difference between the kinetic friction and the static
friction must be considered. Indeed, the static friction coefficient, P s, is higher
than the kinetic friction coefficient, P k. The maximum of the static friction force
is written as
Fs Ps N. (6.2)
In addition, the static friction force is time-dependent. The length of the rest
time that two solids are in contact affects the adhesion and consequently affects
the coefficient of the static friction.
(2) Stribeck curve
Neglecting the wearing process (since friction arises from the transfer of the
collective translational kinetic energy into the nearly random heat motion), the
friction force belongs to the dissipative force. Therefore, it is, in general, described
142
6.1 Mathematical Description of Friction Force
Figure 6.1 The Stribeck Curve Showing Friction as a Function of Velocity for
Low Velocity
143
Chapter 6 Stick-Slip Vibration
model of the friction force. In the next section, this geometric description of the
friction force has been used to for studying chatter phenomenon of mechanical
systems.
(3) Karnopp friction model
D. Karnopp proposed a force-balance model for one-dimensional motion with
a small velocity window. It is called Karnopp friction model and is sketched in
Fig. 6.2[3]. In a sense, the friction force Ff is always a function of the relative
velocity V of two contact bodies. A region of small velocity is defined as
'V V 'V . Outside this region, Ff is an arbitrary function of V. Inside the
small region surrounding V 0, we consider V to be zero. The finite region is
necessary for digital computation since an exact value of zero will not be computed.
Inside the V 0 region, Ff is determined by other forces in the system in such a way
that V remains in the region until the breakaway value of the force is reached.
dP
F Ff (6.3)
dt
with
P mV, (6.4)
where P is the momentum, F the net force on the mass from the rest of the
system, Ff the friction force, m the mass of body of motion, and V the velocity.
In the sticking region, according to the Newton’s law, Ff must mect the
requirement to V 0. For V 0 and P 0, we have
Ff Fstick F. (6.5)
In the sticking region, the momentum is constant and from Eq. (6.4), it is seen
that the velocity is also constant at some value between 'V and 'V . This means
that the mass is supported to be sticking and it really has a small velocity. In many
dynamic simulation studies, this small velocity does not affect the interpretation
144
6.2 Stick-Slip Motion
of the results, but it is easly possible to reduce the sticking velocity to zero by
modifying the relation between V and P.
A momentum range is defined as
145
Chapter 6 Stick-Slip Vibration
where m is the mass of the moving block of the model and k is the spring
coefficient. Ff (u V ) is described by the Stribeck curve shown in Fig. 6.1.
It is convenient to introduce a new variable x to replace u with the equation
1
x{u Ff (V ), (6.9)
k
which means that the position of the block is now measured from its equilibrium
position under the combined action of the spring force and the friction forces.
The differential equation of motion is reduced to
mx F ( x ) kx 0 (6.10)
with
F ( x ) Ff ( x V ) Ff (V ). (6.11)
Function F ( x ) will appear as that in Fig. 6.4 if V is not taken to be too large. It
is important that the slope of this curve is negative at the origin. Clearly, this
requirement can be fulfilled only if the friction force between the block and the
belt decreases numerically with V increasing.
146
6.2 Stick-Slip Motion
dy Z 2 F (Z y ) x
. (6.13)
dx y
Figure 6.5 shows the result of the application of the Liénard construction to
Eq. (6.13), in which the characteristic F ( x ) has the form indicated in Fig. 6.4.
Once any phase path of Eq. (6.13) touches the straight line segment P1P2, it
follows the straight line and moves from left to right until P1 is reached. Since on
this segment V z 0, the constant speed V is a solution of Eq. (6.13). Actually, the
segment P1P2 is also the phase path when the block is stuck to the conveyor belt.
From the point of mechanical view, it means that the friction force on the block
simply adjusts itself to the value of the applied external force, i.e., the value of
the spring force, as long as the critical value of the friction force is not exceeded.
Since the resultant force is zero, the system moves with a constant velocity.
To find the limit cycle approached by all other phase paths of Eq. (6.13) as
t o f , it is sufficient in the case shown in Fig. 6.5 to construct the phase path
that starts at P1 and to follow the limit cycle until it touches the segment P1P2 for
the first time. To guarantee the limit cycle touching the segment, point P2 must
lie far enough to the left, and this in turn requires that the critical value of the
friction force be not too small. From Eq. (6.13), we also see that the limit cycle L,
147
Chapter 6 Stick-Slip Vibration
departing from linearity, is a non-smooth closed curve as shown in Fig. 6.5. Thus,
particularly, when the spring coefficient k is very small, the self-excited vibration
will be of relaxation type.
Assuming that the static friction force is the same as the kinetic one, the friction
is subject to Coulomb friction law and the Stribeck curve is described as signal
function sgn x , as shown in Fig. 6.6(a), with the corresponding phase diagram
plotted by the Liénard construction and shown in Fig. 6.6(b).
Figure 6.6 Phase Plane of Mechanical Model Shown in Fig. 6.3 as Static Friction
is Equal to Kinetic Friction
It is not difficult to prove that the phase path departing from P1 is a circle L '
tangential to the straight line P1P2 at point P1. However, any mechanical system
has a certain dissipative force and the practical phase path departing from P1, as
shown in Fig. 6.6(b), should be a convergent spiral, not a circle limit cycle. Thus,
the stick-slip motion will not emerge.
The analysis has shown that the stick-slip motion cannot occur in the system
shown in Fig. 6.3 if the static friction force is exactly equal to the kinetic friction
force. Actually, the classical form of stick-slip motion may arise whenever the
static friction is markedly higher than the kinetic friction. Hence, we conclude
that the friction drop is an excitation effect for stick-slip motion in mechanical
systems, and it is named as first type of excitation effects for stick-slip motion.
Let x1 denote the relative displacement between the lumped mass and the
driving point T, i.e.,
x1 z Vt. (6.15)
Let us introduce the undamped natural frequency Z0 and the damping ratio [,
namely,
1
§ k ·2 c
Z0 ¨ ¸ , [ 1
, (6.16)
©m¹ 2
2(km)
and the dimensionless relative displacement x and dimensionless time W, namely,
x1Z 0
x , W Z 0 t. (6.17)
V
149
Chapter 6 Stick-Slip Vibration
By using expressions (6.15) (6.17), the motion equation of the model, i.e.,
Eq. (6.14), is reduced to the form
Fs P s N , Fk Pk N , (6.19)
where P s is the static friction coefficient, P k the kinetic friction coefficient, and N
the normal force on the contact surface.
'P Ps Pk . (6.20)
Moreover, let us introduce a dimensionless kinetic friction coefficient f and a
dimensionless friction drop d, namely,
Pk N 'P N
f , d . (6.21)
mZV mZV
The equation of motion of the mechanical model shown in Fig. 6.7 is reduced to
150
6.3 Hunting in Flexible Transmission Devices
ª cV º
k « z Vt F (0) 0, F (0) İ Fs . (6.23)
¬ k »¼
Using expressions (6.15) (6.17), (6.20), and (6.21), we can reduce the above
equation to the dimensionless form, namely,
xc 1 0, x ( f d )sgn( xc 1) ı 0, (6.24)
which governs the behavior of the mechanical model shown in Fig. 6.7 in the
stick stage.
A full cycle of the hunting motion consists of four stages: forward slip stage,
backward slip stage, stick stage, and breakaway stage. Their different phase path
equations are found by integrating the Eq. (6.22) with corresponding initial
conditions and algebraic Eq. (6.24).
1. Forward slip stage
The forward slip stage corresponds to the motion state of the lumped mass when
the direction of its velocity is the same as that of the velocity of the driving point
T. In this stage, the dimensionless Eq. (6.22) is
dy ( x f ) 2[ y
. (6.26)
dx y
Integrating the equation, we obtain the transcendental equation of the phase path
in the forward slip stage, i.e.,
1 ª º½
° y [ (x f ) »°
y 2 2[ ( x f ) y ( x f ) 2 C12 exp ®2[ (1 [ 2 ) 2 arctan «
»¾
,
« 1
° °
«¬ (1 [ ) ( x f ) »¼ ¿
2 2
¯
(6.27)
where C1 is an integral constant corresponding to the given initial state of the
system.
Obviously, the transcendental Eq. (6.27) shows that the phase paths in the
forward slip stage are a set of logarithmic spirals. In particular, when the damping
ratio is equal to zero, the transcendental Eq. (6.27) is reduced to the algebraic
151
Chapter 6 Stick-Slip Vibration
equation, i.e.,
y 2 ( x f )2 C12 . (6.28)
In this case, the phase paths in the forward slip stage degenerate into a set of
concentric circles.
2. Backward slip stage
The backward slip stage corresponds to the motion state of the lumped mass
when the direction of its velocity is opposite to that of the velocity of the driving
point T. Then, the dimensionless Eq. (6.22) takes the form
1 ª º½
° y [ (x f ) »°
y 2 2[ ( x f ) y ( x f ) 2 C22 exp ®2[ (1 [ 2 ) 2 arctan «
»¾
.
« 1
° °
«¬ (1 [ ) ( x f ) »¼ ¿
2 2
¯
(6.30)
Thus, the phase paths in the backward slip stage are also a set of logarithmic
spirals. In particular, when the damping ratio is equal to zero, the transcendental
Eq. (6.30) is reduced to the algebraic equation, i.e.,
y 2 ( x f )2 C22 . (6.31)
In this case, the phase paths in the backward slip stage are also a set of concentric
circles.
3. Stick stage
During the stick stage, the friction force acting on the lumped mass continuously
balances with the restoring force of the spring and it is never more than the static
friction force Fs all along. According to Eq. (6.24), the equations of the phase
paths in the stick stage are
y 1 0, x ( f d )sgn( y 1) ı 0. (6.32)
The equation shows that the phase path in the stick stage is a straight line L
parallel to the abscissa axis of the phase plane. If the phase paths touch it, they
will join in the straight line L.
4. Breakaway stage
The breakaway stage is only maintained for a very short duration when the static
152
6.3 Hunting in Flexible Transmission Devices
friction force on the lumped mass gets coverted to the kinetic friction force. In
this duration, the restoring force of the spring keeps is the magnitude to balance the
static friction force. When the stick state breaks, the lumped mass is accelerated
by means of the net force that is equal to the difference between the restoring
force of the spring and the kinetic friction force. As the damping force on the
lumped mass is usually small, its velocity variation in the breakaway stage can be
found by the theorem of kinetic energy. After considerable calculation, the velocity
variation in this stage is found, i.e.,
1
§ 2 f ·2 'F
'z z (t0 't ) z (t0 ) ¨1 ¸ 1
, (6.33)
© d ¹ 2
(km)
where t0 is the time when the sliding is impending and 't is the duration of the
breakaway stage, ' F is the difference between Fs and Fk.
Since the duration 't is very small and the velocity variation is finite, the
displacement variation of the lumped mass in the breakaway stage may be
neglected so that we get the approximate relations, i.e.,
153
Chapter 6 Stick-Slip Vibration
Second, let us recall the phase path in the stick stage, i.e., the segment P3 P4 on
the straight line L. If the phase path touches it, the phase path joins it at once.
Here, the lumped mass is stuck to the support surface.
Next, let us consider the outer limit cycle, which is a special phase path shown
in Fig. 6.9.
If a phase point is located at point P3(f d, 1) on the straight line L, the
friction force exerting on the lumped mass is equal to the static friction force Fs.
When the slip begins, the phase point rises up to the point G(f d, ad 1), which
represents the breakaway stage. Then, the phase point moves along the phase
path in the forward slip stage, which is a spiral toward the stable equilibrium point
P1, until it elongates to point H on the straight line L. The phase path continues
moving on the straight line L until it reaches point P3. In this way, a full cycle of
the periodic motion representing the hunting phenomenon is performed and the
q
closed phase path P 3GHP3 is a limit cycle L with two straight line segments,
HP3 and P3G, and one spiral GH q , as shown in Fig. 6.9. As the shape of the limit
cycle L is far from an ellipse, the hunting in a flexible transmission device belongs
to relaxation vibration.
Last, let us consider the general phase paths in the phase diagram describing the
hunting phenomenon.
Depending on the magnitude of the distance from the initial phase point P0 to
the equilibrium point P1, there are three types of phase paths.
1. If the initial phase point P0 is considerably far from the limit cycle L and
the equilibrium point P1, the phase path first touches the straight line L at point B
on the right of point P4, as shown in Fig. 6.10. Here, the static friction force is
converted to the kinetic friction force and the breakaway stage occurs. After the
breakaway stage, the phase point falls down to point G1 and then moves along
the phase path in the backward stage to point H1 in the stick zone P3P4. With the
phase path elongating along the straight line L to point P3, the phase path joins
the limit cycle L. Such a phase path is called the first type of phase paths.
154
6.3 Hunting in Flexible Transmission Devices
2. If the initial phase point P0c or P0cc is close enough to the limit cycle L, the
phase path first touches the straight line L at H c or H cc on the left of point P4, as
shown in Fig. 6.11. Then, the phase path elongates along the straight line L to
point P3 and the limit cycle L becomes the phase path. This is the second type of
phase paths. The phase diagram of Fig. 6.11 shows that all phase paths surrounding
L drift toward L and the limit cycle L is a stable limit cycle.
3. If the initial phase point P0 is very close to the equilibrium point P1, the
phase path cannot touch the straight line L. Eventually, the phase path spiral
approaches the stable focus P1, as shown in Fig. 6.12. This is the third type of
phase path. In this case, the hunting phenomenon cannot occur.
155
Chapter 6 Stick-Slip Vibration
Now, let us recall the companion relation between the stability of the equilibrium
point and the limit cycle explained in Chapter 2. Since the equilibrium point P1 is
a stable focus and the limit cycle L is a stable limit cycle, we conclude that there
is an unstable limit cycle between P1 and the limit cycle L. We call it L ', as
shown in Fig. 6.12. Obviously, L ' is a unstable limit cycle. Consequently, it is
a separatrix of two attractive regions of two attractors, P1 and L. Hence, if the
initial disturbance of the flexible transmission device is very small, the initial
phase point P0 is located within the unstable limit cycle L ', and the hunting
phenomenon will not occur. In such a case, the motion of the flexible transmission
device is smooth and steady. In contrast, if the initial disturbance of the flexible
transmission device is large enough, the initial phase point P0 is located outside
the unstable limit cycle L ', and the hunting phenomenon occurs. Therefore, hunting
in the flexible transmission device is caused by means of hard excitation.
y 2 2[ ( x f ) ( x f )2
1 2(a [ )d (1 a 2 2[ a)d 2
ª § · § ·º ½
° 2[ « arctan ¨ y [ ( x f ) ¸ arctan ¨ 1 (a [ )d ¸ » ° 0.
exp ® 1 « ¨ 1 ¸ ¨ 1 ¸» ¾
° (1 [ 2 ) 2 «¬ ¨ (1 [ 2 ) 2 ( x f ) ¸ ¨ (1 [ 2 ) 2 d ¸ » °
¯ © ¹ © ¹¼ ¿
(6.36)
Consider the mathematical condition describing the special phase path with the
critical parameters for energence of hunting in flexible transnission devices.
According to the phase diagram shown in Fig. 6.9, if the phase path departing
from point P3 touches the straight line L, the stable limit cycle L is certain to
emerge. Thus, the critical case in all phase diagrams for the hunting to occur is that
the special spiral that passes through points P3 and G should be tangential with
the straight line L at point H*(x*, 1), as shown in Fig. 6.13, where S denotes the
phase path in the forward stage. Obviously, the slope of the straight line L is equal
to zero. Thus, the mathematical condition describing the critical parameter system
is that the slope of the spiral curve S at point H* is equal to zero.
156
6.3 Hunting in Flexible Transmission Devices
The slope of the phase path in the forward stage is defined by Eq. (6.26) and
the mathematical condition describing the critical parameter system is written as
( x f ) 2[ y
0 (6.37)
y
in combination with Eq. (6.36). Substituting the coordinates of point H*(x*, 1)
into Eqs. (6.36) and (6.37) and eliminating the variable x*, we eventually obtain
the critical parameter equation for the occurrence of hunting, i.e.,
d 3 2 fd 2 ( f 2 1)d 2 f 0. (6.39)
157
Chapter 6 Stick-Slip Vibration
Let dcr denote the positive real root of the equation. In general, it is an implicit
function of the dimensionless kinetic friction coefficient f, d cr ( f ). Using expression
(6.21), we obtain the analytical expression of the critical speed of hunting, i.e.,
'P N
Vcr 1
. (6.40)
2
(mk ) d cr ( f )
With the normal load exerting on the contact surface of the lumped mass shown
in Fig. 6.7, i.e., N mg, expression (6.40) is reduced to
1
'P gm 2
Vcr 1
. (6.41)
2
K d cr ( f )
Based on the above expression and the computation results shown in Fig. 6.14,
a few useful conclusions about hunting in the flexible transmission device are
summarized below.
1. If the dimensionless friction drop d is more than its critical value dcr, which
decreases as the dimensionless friction coefficient f increases, the hunting pheno-
menon occurs in the flexible transmission device.
2. The hunting phenomenon is self-excited vibration due to the hard excitation.
As long as the flexible transmission device, whose system parameters are in the
hunting region, bears some disturbance large enough, the hunting phenomenon
occurs at once.
3. By increasing the dimensionless kinetic friction coefficient, or (and) decreasing
the spring stiffness, or (and) decreasing the driving speed of the flexible transmission
device, the hunting tendency of the flexible transmission device can be made to
rise.
4. With the viscous damping ratio increasing, hunting in the flexible transmission
device may reduce considerably and even vanish.
158
6.4 Asymmetric Dynamic Coupling Caused by Friction Force
A stage in theater is moved up and down by four vertical screw jacks located at
four corners of the platform. These screws are suspended to the concrete beams
of the theater structure. Taking into account the axial displacement of the screw,
a mechanical model with two degrees of freedom is proposed and shown in
Fig. 6.15(a). The generalized coordinates of the system consist of the screw angular
displacement and the axial displacement of the platform. Every driving nut is
operated by an electric motor. In order to give an intuitive equivalent kinematic
model, the screw jack mechanism can be thought of as a wedge mechanism. The
rotation movement of the nut is replaced by an equivalent horizontal movement
shown in Fig. 6.15(b).
159
Chapter 6 Stick-Slip Vibration
Neglecting the damping effect, the rotational motion equation of the screw is
J I kI I M, (6.42)
where I is the angular displacement of the screw, J the moment of inertia of the
rotating elements, kI the torsional stiffness of the screw, and M the torque exerted
on the screw by the nut.
The translational motion equation of the screw along its axis is
mu u ku u P mu g , (6.43)
where u is the axial displacement of the platform, mu the mass of the screw, ku
the stiffness of the axial equivalent of the screw and the support structure, P the
vertical force exerted on the nut by the screw, and g the gravity ational acceleration.
The kinematic relation between the vertical displacement of the nut uf , its
angular displacement I f, the angular displacement I , and the axial displacement
u is represented as
p
tan G ,
2r
where p is the pitch of the screw.
Using the Coulomb friction law to describe the friction force being exerted on
the nut, the expression of torque M is found by means of the equilibrium equation
of the nut, i.e.,
Pr (tan G P sgn Z r )
M (6.45)
1 P sgn Z r tan G
with
Introducing friction angular K arctan P , we rewrite the Eq. (6.45) in the form
M Pr tan(G K sgn Z r ). (6.48)
160
6.4 Asymmetric Dynamic Coupling Caused by Friction Force
Denoting m as the mass of the platform carried by the nut, the vertical load exerted
on the nut by the screw is
Equations (6.52) and (6.53) are linear ordinary differential equations with constant
coefficients. Therefore, the stability of the constant velocity motion of the screw
161
Chapter 6 Stick-Slip Vibration
jack mechanism in the upward and the downward stages depends on the sign of
the real part of all eigenvalues of the Eqs. (6.52) and (6.53) respectively. Here, the
characteristic equations in the upward and the downward stages are described in
a common form, i.e.,
AO 4 BO 2 C 0 (6.54)
with
A mJ , C kI ku ,
B kI m ku [ J mr 2 tan G tan(G K )], in the downward stage, (6.55)
B kI m ku [ J mr tan G tan(G K )], in the upward stage.
2
ª 1
º
B « § 4 AC · 2 »
O 2
1 r ¨1 2 ¸ . (6.56)
2A « © B ¹ »
¬ ¼
Applying the operation of complex numbers, we conclude that if O is a complex
number or a positive real number, at least one root in all roots of the Eq. (6.54)
has the positive real part. Consequently, for the stability of the system of (6.52)
and (6.53), O must be a negative real number and the condition for the stability
of the system of (6.52) and (6.53) is
AC 1
B2 4
and
B AC
! 0, ! 0.
A B2
AC
Moreover, since the last inequality ! 0 is always satisfied, the stability
B2
condition can be replaced by the following equivalent
B!0 (6.57)
and
AC 1
. (6.58)
B2 4
Introducing the equivalent moment of inertia
162
6.4 Asymmetric Dynamic Coupling Caused by Friction Force
kI m ku J e ! 0 (6.60)
and
J I ku Je
D and E , (6.62)
mkI J
D 2 E 2 2DE 4D 1 ! 0. (6.64)
In the logD-E plane shown in Fig. 6.16, each point represents a couple of
parameter values (D, E ). The couple of values (D, E ) that satisfy Eq. (6.63) are
represented by the area above the dot curve ē and the couple of values (D, E )
that satisfy Eq. (6.64) are represented by the area above the continuous curve Ē
and the area under the continuous curve Ĕ. Therefore, the couple of values (D, E )
that satisfy Eq. (6.63) at the same time are represented by the area above the
curve Ē. In short, the area above the curve Ē is a stable region in the logD -E plane.
163
Chapter 6 Stick-Slip Vibration
1 2 ª 1
º
D «1 ( E ) 2
». (6.65)
E E ¬
2
¼
If the axial stiffness ku of the screw is high enough, the value of parameter D is
also high so that the inequality is violated. Thus, the stick-slip motion should emerge
according to the above analysis. This has been observed in a newly constructed
theater platform.
The stick-slip motion has been found from the equation of motion of the screw
jack mechanism in the downward stage, i.e., Eq. (6.53), in which the friction
force is described by the Coulomb friction law and the friction drop is absent. It
means that there is another excitation effect of the stick-slip motion in this equation
of motion that is different from first type of excitation effets for stick-slip motion,
i. e., the friction drop.
Equation (6.53) is a non-homogeneous equation with constant non-homogeneous
terms. It is known that the equilibrium stability of a linear dynamic system depends
only on the signs of the real parts of all eigenvalues. Hence, we can neglect the
constant non-homogeneous terms in the equation and this is equivalent to translating
the origin of the system coordinates.
Then, let us introduce the following notations
ª a b º I½ ª kI 0 º I ½ 0 ½
« c d » ® ¾ « 0 ® ¾
ku »¼ ¯u ¿
® ¾. (6.67)
¬ ¼ ¯u ¿ ¬ ¯0 ¿
The inertial matrix is asymmetric when the friction angle K is not equal to zero.
This means that the asymmetric dynamic coupling is caused by the friction being
exerted on the contact surface between the nuts and the screws of the platform
system.
The characteristic equation is invariant in nonsingular linear coordinate transforms
for a given linear dynamic system. This means that the equilibrium stability of the
164
6.4 Asymmetric Dynamic Coupling Caused by Friction Force
ª a 0 º
x1 ½ ª k11 k12 º x1 ½ ª 0 e12 º x1 ½ 0 ½
« 0 d » ® ¾« ® ¾ ® ¾ ® ¾, (6.69)
¬ ¼ ¯ 2 ¿ ¬ k21
x k22 ¼» ¯ x2 ¿ ¬« e12 0 ¼» ¯ x2 ¿ ¯0 ¿
where
165
Chapter 6 Stick-Slip Vibration
asymmetric inertial coupling exerted by the friction force. without friction drop.
In true dynamic systems with two or more degrees of freedom of motion, previous
two kinds of excitation effects for stick-slip motion always coexits, so that the
stick-slip motion often occurs in the mechanical systems with higher degrees of
freedom of motion.
References
[1] B N J Person. Sliding Friction Physical Principles and Applications, Second Edition.
Berlin: Springer, 2000
[2] B Amstrong-Hélouvry, P Dupont, C C De Wit. A Survey of Models, Analysis Tools and
Compensation Methods for the Control of Machines with Friction. Automatica, 1994,
30(7): 1083 1138
[3] D Karnopp. Computer Simulation of Stick-Slip Friction in Mechanical Dynamic Systems.
ASME J. Dyn. Syst. Meas. Control, 1985, 107(1): 100 103
[4] F A Tariku, R J Rogers, Improved Dynamic Friction Models for Simulation of One-
Dimensional and Two-Dimensional Stick-Slip Motion. ASME J. Tribology, 2001, 123(4):
661 669
[5] B R Dudley, H W Swift. Frictional Relaxation Oscillations. Philosophical Magazine,
40(Series 7), 849 861, 1949
[6] J J Stoker. Nonlinear Vibrations in Mechanical and Electrical Systems. New York: John
Wiley & Sons, 1950
[7] Ding Wenjing, Fan Shichao, Lu Mingwan. A New Criterion for Occurrence of Stick-Slip
Motion in Drive Mechanism. Acta Mechanica Sinica, 2000, 16(3): 273 281
[8] P Gallina, M Giovagnoni. Design of a Screw Jack Mechanism to Avoid Self-Excited
Vibration. ASME J Dyn. Syst. Meas. Control, 2002, 124(3): 477 480
[9] L Meirovitch. Methods of Analytical Dynamics. New York: McGraw-Hill Book Company,
1970
166
Chapter 7 Dynamic Shimmy of Front Wheel
It is known that dynamic shimmy is induced by the tire force. Therefore, the first
step to study dynamic shimmy is to establish the analytical expression of the tire
force. A pneumatic tire is an anisotropic continuum, and such a complete model
is too complicated, therefore, the analytical researchs on dynamic shimmy become
impossible. Obviously, the tire force is a dynamic reaction force applied on the
tire-road interface when the tire is rolling on the road, and it is possible to
understand the tire force by the constraint theory in analytical mechanics.
Actually, in order to confine the motion of the rolling wheel, the road must
deliver a reaction force to the pneumatic tire, which is referred to as constraint
force in analytical mechanics and is called tire force in engineering. The tire
force is not normal to the tire-road interface when there is friction. In addition,
the tire force is a distributive force on the contact patch between the tire and the
road due to their deformation. The integration of the distributive force function
on the whole area of the contact patch is a resultant of the distributive force,
which is applied at the action center O' and denoted as F. Since the distribution
of the tire force is not homogeneous, point O' is usually different from the
geometric center O of the contact patch between the tire and the road. Take off a
front wheel from the moving vehicle and consider it as a free body shown in
Fig. 7.1. Point C is its mass center. According to classical mechanics, besides the
constraint force F, there are other forces: applied force Fa, applied moment Ma,
inertial force Fi, and inertial moment Mi of the front wheel. They are all applied
at point C when the wheel is rolling.
Let rc denote the special vector from point O' to mass center C of the front
wheel. We have
Mc rc u F , (7.1)
168
7.1 Physical Background of Tire Force
which is called constraint moment of the tire force, as shown in Fig. 7.1.
According to d’Alembert’s principle, for the free body shown in Fig. 7.1, there
are two equilibrium equations, i.e.,
F Fa Fi 0 (7.2)
and
Mc Ma Mi 0. (7.3)
These two equations demonstrate that the variations of the tire forces F and Mc
always follow the variations of the generalized applied forces Fa and M a , and
the generalized inertial forces Fi and M i . The natural regularity is so called passive
property of constraint force.
According to Eqs. (7.2) and (7.3), we conclude that all factors associated
with Fa , M a , Fi , and M i have a certain influence on the tire force F and M c .
Consequently, these factors may be broadly divided into four groups. The first
consists of a variety of physical parameters of the tire and the wheel disk, such as
the elastic modulus of the tire and the mass of front wheel. The second consists
of a variety of geometric parameters, such as their diameter and their width. The
third consists of a variety of kinematic parameters, such as the temporal values of
the generalized coordinates and the velocities. The last consists of a variety of
operation parameters, such as the magnitude and applied point of vertical load,
and the surface property of the road. Thus, finding the accurate expression of the
tire force is nearly impossible. Thus, we must focus on establishing its approximate
expressions, particularly on some components directly leading to dynamic shimmy
of the front wheel.
Let us recall the resultant F of the distributed tire force, whose applied point is
the action center O'. To define its components, SAE (Society of Autonotive
Engineers) has selected a reference frame OXYZ, in which the geometric center
O is denoted as the origin of the reference frame, axis X points to the direction of
the wheel heading, and axis Z points to the vertical direction, as shown in Fig. 7.2[3].
The slip angle D represents the difference between the direction of wheel heading
and the direction of wheel travel.
According to the reference frame OXYZ, the tire force F is decomposed into
three components, Fx, Fy, and Fz. They are called tractive force, lateral force, and
normal force respectively. Let r denote the spatial vector from the action center
O' to the geometric center O of the contact patch. The tire moment M is defined as
M r u F, (7.4)
whose components along the axes X, Y, and Z are Mx, My, and Mz and are
169
Chapter 7 Dynamic Shimmy of Front Wheel
the steer angle. The leading edge of the contact patch moves farther from the
wheel disk to maintain continuity of the tread, while the deflection at the rear is
limited by friction.
Figure 7.3 Tire Lateral Deflection Caused by Steer or Lateral Slip Gives Rise
to F and M
In accordance with the deformation geometry shown in Fig. 7.3, the curves
shown in Fig. 7.4 describe the typical cornering force obtained by a special rig.
For a small slip angle, the lateral force vs. slip angle relation is linear. As the
angle increases, it is bent down due to the sliding at the rear of the contact patch.
When the slip angle is large enough, the aligning moment falls off. Besides sliding,
there is another cause to make it bend down, namely, the sliding area in the contact
patch is continuously expanded as the slip angle increases so that the action
center of tire force progressively approaches the geometric center of the contact
patch. The angle at which the lateral force reaches its maximum varies with the
type of vehicles. For a racing car tire, the lateral force peaks at about 6e
, while for
[1]
passenger vehicles, the maximum force may occur at about 18e .
Figure 7.4 Typical Curves of Lateral Force and Alignment Moment for a Tire
It is convenient to present the tire test data in polynomial form. The following
expressions are usually used to study dynamic problems of various vehicles:
171
Chapter 7 Dynamic Shimmy of Front Wheel
and
where Dis the slip angle shown in Fig. 7.2. Constant a1, a2, b1, b2, and b3 are
found based on data obtained by considerable testing of the pneumatic tire and
by using the least square method, and each of them is a function of the vertical
force Fz.
Expressions (7.5) and (7.6) show that the lateral force and the aligning moment
are functions of the slip angle D. They are usually used to establish a nonlinear
mathematical model for studying the dynamic shimmy of a given front wheel
system. Meanwhile, many numerical results have been obtained. Though it seems
that the cornering force of a tire is only related to the slip angle, it is actually
relevant to many factors mentioned before. Therefore, it is necessary to find more
accurate expressions for the cornering force and such expressions are commonly
called the mathematical model for the cornering force of pneumatic tire[5].
172
7.1 Physical Background of Tire Force
The lateral deformation is a function of the traveled distance and the circum-
ferential coordinate. At the transition from the free range to the contact zone, a
kink may occur in the shape of the string and a shear force is required to maintain
173
Chapter 7 Dynamic Shimmy of Front Wheel
the kink. As the direction of sliding speed and the direction of shear force are
compatible with each other at the trailing edge but incompatible at the leading
edge, we conclude that a kink may arise only at the trailing edge of the contact line.
The condition of continuity of the slope at the leading edge leads to a first-order
differential equation for the deflection [ of the spring at the contact point, i.e.,
d[ [ dy dT
l2 T D l2I , (7.7)
ds l1 ds ds
where y is the lateral displacement of wheel center C, T the yaw angle of the
wheel symmetric plane, D the slip angle, I the path curvature, l1 the relaxation
length, l2 the half length of the contact line, and s the circumferential coordinate.
The above equation is a representative model deduced from the stretch spring
theory. After making a comparison investigation between the point contact theory
and the stretch spring theory, R. Collins concluded that both of them predict
dynamic shimmy with considerable success[79]. However, the simplicity of the
point contact theory offers noteworthy advantages in the analytical study of dynamic
shimmy. Therefore, more details about it are provided below.
For simplicity, assume that the lateral force during cornering is proportional to
the slip angle, i.e.,
Fy aD
where Fy is the lateral force, D the slip angle, and a the lateral slip-resistant
coefficient.
This expression was proposed by M. G. Brouheit in 1925 and named as slip law
of pneumatic tire during cornering.
2. Elastic body model with one deformation component
If the pneumatic tire is considered as an ideal elastic body, steering the rolling
tire causes the tread band’s lateral motion relative to the wheel rim, which produces
a lateral force, and we have
Fy k[
where Fy is the lateral force, [ the lateral deflection of the geometric center of the
contact patch of the tire, and k the lateral slip coefficient.
3. Elastic body model with two deformation components[10]
J. H. Greidanus developed a point contact theory of cornering force, in which
two deformation components, the lateral deflection and the torsional angle of
pneumatic tire, are taken into account. Besides, two assumptions restricting the
path of the geometric center of the contact patch are proposed, based on which
two differential equations presenting the point contact theory are found. The first
assumption is that the tangent of the path of the geometric center of the contact
patch is parallel to the tangent of the path of the mass center of the cornering
wheel, as shown in Fig. 7.6. The second assumption is that the curvatures of the
previous two path curves are equal to each other.
175
Chapter 7 Dynamic Shimmy of Front Wheel
y [ V T V M 0, (7.10)
where y is the lateral velocity of the mass center of the cornering wheel, [ the
deflection rate of the tire at the geometric center of the contact patch, T the yaw
angle of the cornering wheel, M the torsional angle of elastic deformation of the
tire about the vertical axis, and V the travel speed of the vehicle.
According to the second assumption, we write another first-order differential
equation
T M G V [ E V M 0, (7.11)
where T is the yaw angular velocity of the front wheel, M the torsional angular
velocity, G and E are two constant coefficients, and other signs are the same as
those in (7.10).
4. In addition to [ and M
M. B. Keldish considered another deformation component, namely, the inclination
angle of the pneumatic tire, which is caused by torsional deformation about heading
axis of the front wheel. Consequently, the differential Eq. (7.11) is modified as
follows.
T M G V [ E V M J V F 0, (7.12)
where F is the inclination angle shown in Fig. 7.2, J is the constant coefficient,
and other signs are the same as those in (7.11). Here, the differential equation
deduced from the first assumption is still Eq. (7.10).
When a tire contacts with the road surface, the soft rubber of the tread drapes
itself around the hard asperities of the road, most of the deflection occurs in the
tire, and the road is nearly undeformed. If the travel direction of a rolling tire
varies, the road surface makes a kinematical restriction to the motion of the tire.
The kinematical restriction is referred to as constraint in analytical mechanics.
The equation describing the kinematical restriction is called constraint equation.
If the variables in the constraint equation have both generalized coordinates and
velocities of the dynamic system, and the first-order differential equation is
unintegrable, the constraint is referred to as nonholonomic constraint and the
differential equation is called nonholonomic constraint equation.
Recall Eqs. (7.10) (7.12). Except the generalized coordinates y, [, M, and F,
the generalized velocities y and T are contained in them and these equations
are unintegrable. Therefore, they are nonholonomic constraint equations.
176
7.2 Point Contact Theory
The reaction force provided by the nonholonomic constraint Eqs. (7.10) (7.12)
just represents the cornering force being exerted on the tire.
Since the nonholonomic constraint equations cannot provide the explicit
expression of the cornering force as Eqs. (7.5), (7.6), (7.8), and (7.9), we need to
find an analytical expression of the potential energy of the pneumatic tire, through
which the differential equations of motion of the front wheel may be found by
the well-known Lagrange’s equation.
Since realization of the cornering motion of a tire is always is company with the
inclination angle, so that there are three cornering forces applied at the action
center of the contact patch, Fy, Mx, and Mz. With the deformation of the rolling
tire described by three components, [, M , and F, as shown in Eq. (7.12), we
denote the potential energy of the pneumatic tire as U([, M, F) and expand it into
a power series in the neighborhood of the equilibrium point ([0, M 0, F0), that is,
§ wU · § wU · § wU ·
U ([ , M , F ) U ([ 0 , M 0 , F 0 ) ¨ ¸ d[ ¨ ¸ dM ¨ ¸ dF
© w[ ¹0 © wM ¹0 © wF ¹0
1 ª§ w 2U · § w 2U · § w 2U · º
«¨ 2 ¸ (d[ ) 2 ¨ 2 ¸ (dM ) 2 ¨ 2 ¸ (dF )2 »
2 «¬© w[ ¹0 © wM ¹0 © wF ¹0 »¼
ª§ w 2U · § w 2U · § w 2U · º
«¨ ¸ d[ dM ¨ ¸ dM dF ¨ ¸ dF d[ » ". (7.13)
«¬© w[wM ¹0 © wMwF ¹0 © wFw[ ¹0 »¼
If the potential energy in the tire equilibrium state is assigned the value zero, namely,
§ wU · § wU · § wU ·
¨ ¸ ¨ ¸ ¨ F¸ 0. (7.15)
© w[ ¹0 © wM ¹0 © w ¹0
Substituting Eqs. (7.14) and (7.15) in expression (7.13), and neglecting all small
higher-order terms, we obtain an approximate expression of the potential energy of
the pneumatic tire in the neighborhood of the equilibrium point ([0, M 0, F0), i.e.,
1 ª§ w 2U · § w 2U · § w 2U · º
U ([ , M , F ) «¨ 2 ¸ (d[ ) ¨ 2 ¸ (dM ) ¨ F 2 ¸ (dF ) »
2 2 2
2 «¬© w[ ¹0 © wM ¹0 © w ¹0 »¼
§ w 2U · § w 2U · § w 2U ·
¨ ¸ d [ dM ¨ ¸ d M d F ¨ F ¸ dF d[ . (7.16)
© w[wM ¹0 © wM wF ¹0 © w w[ ¹0
177
Chapter 7 Dynamic Shimmy of Front Wheel
Since the elastic restoring force belongs to the potential force, the cornering
forces of the rolling tire, Fy, Mx, and Mz, are also the potential forces. According
to the theory of the potential field, the components of the cornering force are
equal to the derivatives of the potential energy with respect to the corresponding
deformation components, that is,
wU wU wU
Fy , Mz , Mx . (7.17)
w[ wM wF
In addition, let us designate the equilibrium point ([0, M 0, F0) as the origin of the
deformation coordinates, namely,
[0 M0 F0 0.
Consequently, we have the first approximate expression of deformation of the
rolling tire in the neighborhood of the equilibrium point, i.e.,
Fy a11[ a12 F ,
Mx a12[ a22 F ,
Mz a33M ,
in which a12 and a22 are approximately proportional to the vertical load N of the
178
7.3 Dynamic Shimmy of Front Wheel
Fy a[ V N F , (7.22)
Mx V N[ U N F , (7.23)
Mz bM , (7.24)
where a a11 , b a33 , V , and U are constant coefficients for the given tire.
According to analytical expressions (7.22) (7.24), we can write the analytical
expression of the potential energy of a rolling tire during cornering in the form
1
U (a[ 2 bM 2 U N F 2 2V N[ F ) . (7.25)
2
In the last section of this chapter, it will be used to establish the equations of
motion of the front wheel assembly by Lagrange’s equation.
Assume that a vehicle is moving along a straight line at a constant speed and the
direction of the kingpin axis of the front wheel is in its symmetric plane. Here, a
front wheel of the vehicle is considered, as shown in Fig. 7.7. This wheel is used
to study the dynamic shimmy of the front wheel, and it is named as isolated front
wheel model.
Now, let us consider only the one-degree-freedom motion of the rigid body,
i.e., the rotation about the kingpin axis of the front wheel assembly. The forces
associated with the rotation include the cornering force of the rolling tire, the
damping force exerted by the steering mechanism, and its own inertial force. The
179
Chapter 7 Dynamic Shimmy of Front Wheel
where T is the steering angle of the front wheel, J is the moment of inertia of the
front wheel assembly about the kingpin axis, h is the damping coefficient of the
steering mechanism, l is the mechanical trail, which is the distance from the
geometric center of contact patch of tire to the kingpin axis, and Fy and Mz are
the lateral force and the alignment moment respectively.
Next, use the length of travel path, i.e., s Vt, to replace time t. The equation
of the motion of the isolated front wheel model, Eq. (7.26)', is transformed into
the form
V 2 JT cc VhT c Fy l M z , (7.26)
where the prime represents the derivative of variable T with respect to length s.
Assume that the inclination angle of the tire vanishes. The expression of the
cornering force, (7.22) and (7.24), are reduced to the form
Fy a[, Mz bM
Substituting these expressions in (7.26), we obtain
180
7.3 Dynamic Shimmy of Front Wheel
Simultaneous Eqs. (7.28) (7.30) yield a closed equation system by which the
solutions of the generalized coordinates T, [ and M, may be found. Hence, it is a
simple mathematical model to study the dynamic shimmy of the front wheel.
The ordinary differential Eqs. (7.28) (7.30) are a linear autonomous equation
system. Consequently, the stability of the front wheel under steady rolling along
a straight line may be analyzed by using the Hurwitz criterion.
First, eliminating two coordinate variables from Eqs. (7.28) (7.30), we obtain
the characteristic equation of the isolated front wheel model, i.e.,
a0 V 2J, a1 V 2 J E Vh,
a2 V 2G J Vh E al 2 b, (7.32)
a3 VhG al E G lb,
2
a4 al E G b.
Next, we use the coefficient condition for the second type of critical stability
corresponding to fourth-order linear systems. When all coefficients shown in
expression (7.32) are positive, the critical coefficient condition of the isolated
front wheel model is reduced to
181
Chapter 7 Dynamic Shimmy of Front Wheel
To derive a qualitative conclusion about the relation between the stability of the
front wheel under steady rolling and the system parameters, we emphasize a special
case where the damping coefficient of the steering mechanism is small enough.
Let the damping coefficient vanish from Eq. (7.34). Consequently, we obtain
(G l E )( E JV 2 bl ) 0, (7.35)
E
l0 . (7.38)
G
Substituting this expression in equation (7.36), we obtain a new critical parameter
equation, i.e.,
l l0 . (7.39)
According to the critical parameter Eq. (7.37), we obtain the analytical expression
of the critical speed of dynamic shimmy of the front wheel, i.e.,
1
§ bl · 2
Vcr ¨ ¸ . (7.40)
© EJ ¹
Given all parameters in equation (7.37) except V and l, the Eq. (7.37) is a
parabola C in V-l plane, and the Eq. (7.39) is a line L perpendicular to axis l, as
shown in Fig. 7.8. It is obvious that curve C and line L intersect at point K, and
the l-V plane is divided into four regions, Ē, ē, Ĕ, and ĕ. Regions Ē and ĕ are
stable regions corresponding to high-speed rolling and low-speed rolling respectively,
and regions Ĕ and ē are unstable regions also corresponding to high-speed rolling
and low-speed rolling respectively.
182
7.3 Dynamic Shimmy of Front Wheel
In Fig. 7.8, the critical speed at the intersection K of curve C and line L is the
maximum speed Vm, and here the mechanical trail of the front is just equal to the
pneumatic trail of the tire. Substituting Eq. (7.38) and (7.39) in expression (7.40),
we have
1
§ b ·2
Vm ¨ ¸ . (7.43)
©G J ¹
This equation shows that if the tire parameter G or (and) the moment of inertia
of the front wheel J decrease, or (and) the tire parameter b increases, the maximum
of the critical speed of dynamic shimmy will increase. It helps to prevent the front
wheel from dynamic shimmy.
Reviewing Eq. (7.34) shows that if the damping coefficient of the steering
mechanism h rises, the critical speed of the front wheel will increase.
In addition, some nonlinear factors have an evident influence on dynamic
shimmy of the front wheel, such as the friction force in the steering mechanism,
the wheel-bearing clearance, and the saturation of the friction force on the contact
patch between the tire and the road surface.
In order to understand the effects of these nonlinear factors on dynamic shimmy,
J. T. Gordan performed a perturbation analysis on the nonlinear factors in an
183
Chapter 7 Dynamic Shimmy of Front Wheel
aircraft landing-gear that included terms of Coulomb friction between the oleo
struts and the freeplay in the torque links[12]. The method of multiple scales is
used to obtain the analytical expressions for the amplitude and the frequency of
the limit cycle, which are functions of the taxi speed. The analysis shows that
both stable and unstable limit cycles may exist for a taxi with speed above or
below a critical value and the stability of the limit cycle is determined by the sign
of a computed coefficient. If only freeplay is present, a stable limit cycle exists.
If both Coulomb friction and freeplay are present, a stable or unstable limit cycle
and a turning point may exist depending on the system parameter values. In such
a case, self-excited vibration occurs due to hard excitation. When this analysis
method was applied to a single wheel model, whose equation of motion is the same
as equation (7.26), the results obtained by perturbation analysis were in good
agreement with those obtained by direct numerical integration of the nonlinear
equation of motion of the front wheel.
According to the preceding analysis results found by the perturbation method,
the freeplay in the torque links can excite dynamic shimmy of the aircraft landing-
gear of aircraft, but the Coulomb friction between the oleo struts has opposite effect.
In the landing-gear shimmy test performed by D. T. Grossman, the landing-gear
was mounted to a taxi fixture designed to simulate the dynamic properties of
aircraft F-15[13]. The results of the test show that the shimmy speed of F-15 nose
landing-gear is clearly a function of the strut-torque-freeplay, namely, the shimmy
speed markedly decreases when the value of the strut-torque-freeplay increases.
Thus, the analysis results found by J. T. Gordan agree with the test results obtained
by D. T. Grossman.
a steering linkage; (2) the vehicle travels with a constant speed; and (3) the direction
of vehicle velocity keeps along a straight line.
Figure 7.9 A Shimmy Model Consisting of Two Front Wheels and One Front-
Axle Beam
There are five steps to establish the mathematical model of the front wheel system
shown in Fig. 7.9.
First, define the generalized coordinates describing the motion of the front
wheel system. Here, let T and I represent the steering angle of the front wheels
and the roll angle of the vehicle respectively, they are used to describe the attitude
motion of the front wheels, let [1 and [2 represent the deflection displacement of
the left and the right tires, and let M1 and M2 represent their torsional angles, they
are used to describe the deformation motion of the pneumatic tires.
Second, construct the analytical expressions of the kinetic and the potential
energies of the front wheel system. For simplicity, assume that the elastic defor-
mation displacement [1, [2, M1, and M2 are much smaller than the rotations of the
front wheel, T and I. Since the frequency of dynamic shimmy is usually very low,
all terms related to deformation velocity [1 , [2 , M1 and M 2 , may be eliminated
from the expression of the kinetic energy. Eventually, the approximate expression
of the kinetic energy of the front wheel system shown in Fig. 7.9 is written as
1§ V2 2 2 · JV TI
,
T ¨ 2 J 2 J1T J 2I ¸ (7.44)
2© r ¹ r
where J is the moment of inertia about the spin axis of the front wheel, J1 the
moment of inertia about the vertical axis of the front wheel system, J2 the
moment of inertia about the roll axis of the front wheel system, V the traveling
speed of the vehicle, and r the radius of the front wheel.
In addition, according to Eq. (7.25), the potential energy of the pneumatic tire
is written as
1
U [a([12 [ 22 ) 2V NI ([1 [ 2 ) 2 U NI 2 b(M12 M 22 )], (7.45)
2
185
Chapter 7 Dynamic Shimmy of Front Wheel
where a, b, V , and U are all the tire parameters obtained by the special test and N
is the vertical load on the two tires.
Third, establish the differential equation of motion of the front wheel system
by means of Lagrange’s equation.
According to the definition of the Lagrangian, we have
L T U . (7.46)
Inserting expressions (7.44) and (7.45) in the above expression, and substituting
it in Lagrange’s equations, we have
d wL wL d wL wL
QI , QT , (7.47)
dt wI wI dt wT wT
where QI and QT are generalized forces corresponding to the generalized coor-
dinates I and T respectively. As shown in Fig. 7.9, they are exerted on the front
wheel system by the steering mechanism of the vehicle.
Applying the virtual work principle, we find out the generalized forces
y1 B rI , y2 B rI . (7.51)
Substituting the expressions in Eqs. (7.10) and (7.12), we obtain four
nonholonomic constraint equations, i.e.,
T M1 G V [1 E V M1 J V I 0, (7.54)
186
7.4 Dynamic Shimmy of Front Wheel Coupled with Vehicle
T M 2 G V [ 2 E V M 2 J V I 0. (7.55)
Last, eliminating the tedious coordinates, we obtain the mathematical model of
the front wheel system. Here, we introduce two new generalized coordinates to
replace the four old deformation coordinates, i.e.,
[ [1 [ 2 , M M1 M 2 , (7.56)
where [ is the sum of lateral displacements of the geometric center of the contact
patch of the two tires, and M is the sum of the torsional angle of the elastic
deformation of the two tires.
Add Eqs. (7.52) and (7.53). Next, add Eqs. (7.54) and (7.55), and then combine
the resultant equations with the equations of motion (7.49) and (7.50). Finally,
we obtain the following equation system
[ V M 2rI 2V T 0, (7.59)
1 k2 2 N (V r U )
P , Q . (7.62)
V2 J 2V 2
They are linearly contained in Eq. (7.61).
Equation (7.61), with O iZ, indicates that there are a pair of conjugate
imaginary eigenvalues. Here, the steady rolling of the front wheel system is
critically stable, and we have
D(iZ ; P , Q) 0. (7.63)
Separating the real part and the imaginary part of the above equation, we have
P P1 (Z ) Q P2 (Z ) P3 (Z ) i[P Q1 (Z ) Q Q2 (Z ) Q3 (Z )] 0. (7.64)
Consequently, we obtain two independent equations, i.e.,
P P1 (Z ) Q P2 (Z ) P3 (Z ) 0,
(7.65)
P Q1 (Z ) Q Q2 (Z ) Q3 (Z ) 0.
Solving these two equations, we obtain
P2 (Z )Q3 (Z ) P3 (Z )Q2 (Z )
P ,
P1 (Z )Q2 (Z ) P2 (Z )Q1 (Z )
(7.66)
P1 (Z )Q3 (Z ) P3 (Z )Q1 (Z )
Q .
P1 (Z )Q2 (Z ) P2 (Z )Q1 (Z )
(3) Let Zbe a parameter variable in equation (7.66). By numerical calculation,
we obtain a couple of data (P,Q). Draw the curve with a set of points (Pi,Qi), i 1,
2, " , in the P -Q plane, as shown in Fig. 7.10. This curve is called D-decomposition
curve. Its two branches CB p and DEp divide the P -Q plane into four domains,
Ē, ē, Ĕ, and ĕ. If the variation of the parameter point (P,Q) is maintained in the
same domain in Fig. 7.10, the number of eigenvalues with positive real parts does
not vary. If the variation of the parameter point (P,Q) enters another domain as
shown in Fig. 7.10, the number of eigenvalues with positive real parts varies as the
parameter point (P,Q) crosses over the D-decomposition curve. The calculation
result shows that domains Ē and ĕ are stable while domains ē and Ĕ are unstable.
(4) Dividing the second equation of (7.62) by the first one yields a linear
equation, i.e.,
Q kP 0 (7.67)
with
k2 2 NV r U
k . (7.68)
J2
188
7.4 Dynamic Shimmy of Front Wheel Coupled with Vehicle
Equation (7.67) defines a straight line L in the P-Q plane, as shown in Fig. 7.10.
In general, line L intersects with two branches of the D-decomposition curve BC p
p
and DE at points P2 and P1. Denote V1 and V2 as the vehicle speeds at P1 and P2
respectively. If the vehicle speed increases from the initial velocity V0 to a higher
velocity, the corresponding point P0 moves along the straight line L toward the
origin O in the P -Q plane. Since domains I and IV are stable, when the vehicle
speed remains in the region (0,V1 ) or (V2 , f), the steady rolling of the front
wheel system is stable and dynamic shimmy does not occur. It is known that
most road vehicles have two stable regions. One has low speed and the other has
high speed. Hence, this result qualitatively agrees with the practical observation.
Many effective methods to prevent the front wheel from dynamic shimmy have
been proposed in the previous sections. Besides these methods, we suggest a new
way here. Let us take a look at Fig. 7.10, in which the stable domain I is directly
connected with another stable domain ĕ at the intersection A. In order to make
the straight line L pass through intersection A, the geometric condition may be
met by adjusting system parameters. If the system parameters satisfy the
mathematical condition, with the vehicle speeding up from a lower velocity to a
higher velocity, point P on the straight line L does not pass through any unstable
domain in the P-Q plane, and dynamic shimmy does not occur to the front wheel
system of this vehicle.
Thus, in the design stage, we should carefully adjust the position of the
intersection A of two branches of the D-decomposition curve and the slope of the
straight line L so that the straight line L may just pass through the intersection A
of two curves BC p and DE p , as shown in Fig. 7.10. This method is an optimal
scheme to prevent the front wheel from dynamic shimmy.
189
Chapter 7 Dynamic Shimmy of Front Wheel
References
[1] T R Kane, G K Man. Characterization of Wheel-Roadway Intersection for Recreational
Vehicles. SAE Trans. Sect. 1, 790181, 1979
[2] H B Pacejka, R S Sharp. Shear Force Development by Pneumatic Tire in Steady-State
Conditions. A Review of Modeling Aspects. Vehicle System Dynamics, 1991, 20(2):
121 176
[3] J R Ellis. Vehicle Handling Dynamics Mechanical Engineering. London: Publications
Limited, 1994
[4] T D Gillespie. Fundamental of Vehicle Dynamics. Warrendale PA USA: SAE Inc., 1992
[5] R van der Valk, H B Pacejka. Analysis of a Civil Aircraft Main Gear Shimmy Failure.
Vehicle System Dynamics, 1993, 22(2): 97 121
[6] H Pacejka. The Wheel Shimmy Phenomenon. Ph. D. Thesis Dec. 1966, Delft Technical
Institute Holland
[7] R Collins, R Black. Tire Parameter for Landing-Gear Shimmy Studies. J. Aircraft, 1969,
6(3): 252 258
[8] R Collins. Theories on the Mechanics of Tire and Their Applications to Shimmy Analysis. J.
Aircraft, 8(4): 271 277
[9] R Collins. Frequency Response of Tire Using the Point Contact Theory. J. Aircraft, 1972,
9(6): 427 432
[10] ÙÃÈàäçÛëå,ûïîïÛàÝ. ¿ãèÛçãåÛ ÈàÞéæéèéçöðÌãìíàç.ÇéìåÝÛ.
Ïãâ-ÇÛ í.Æãí. 1967
[Y I Neimark I Afifaev. Dynamics in Nonholonomic Systems. Moscow Phys. -Math. Lit.
1967 (in Russian).]
[11] L Meirovitch. Methods of Analytical Dynamics. New York: McGraw-Hill, 1970
[12] J G Gordon. Perturbation Analysis of Nonlinear Wheel-Shimmy. J. Aircraft, 2002, 39(2):
305 317
[13] D T Grossman. F-15 Nose Landing Gear Shimmy, Taxi Test, and Correlative Analysis,
SAE Trans. 801239, 1980
190
Chapter 8 Rotor Whirl
Abstract: In the past three decades, researches in rotor dynamics met with
considerable success in modeling the structural dynamics of flexible rotors
and in developing analysis techniques for the bearings supporting these rotors.
As a result of these substantial technical advances, new high-performance
turbomachinery was designed, developed, and put into service to operate at
higher speed and higher energy density level. At the same time, instable
rotor vibrations in compressors and turbines occurred more frequently and
caused severe failures. The frequency leading to whirling instability is usually
near one of the shaft critical speeds and can be caused by many factors,
including hydrodynamic bearings, seals, internal damping, aerodynamic cross
coupling, and torsional coupling. Instability due to fluid film forces and the
internal damping of shafts often leads to self-excited vibration. This is
introduced in this chapter in detail. The contents associated with rotor whirl
are divided into six sections: the first emphasizes the construction of the
mechanical model of the rotor in planar whirl; the second introduces the
analytical expressions of the fluid-film forces of bearings and seals; the third
treats the stability problem of planar whirl rotors with nonlinear and linear
models; the fourth is devoted to establishing the analytical expressions of
the internal force of rotation shafts; the fifth turns attention to instability
caused by the internal damping of the rotor system; and the last discusses
the cause and the prevention of the rotor whirl.
Keywords: planar whirl, nonsynchronous whirl, oil-film force, Musynska model,
threshold speed, Hopf-bifurcation, oil whip, internal damping, circulatory force
of papers on this subject have been published. In order to better understand the
rotor operation, we must turn to the basic theory of mechanics.
A variety of rotor whirls occur in many rotating machines. They are classified
with different evaluation standards, including the physical property of the force
leading to the rotor whirl, the level of the whirl frequency in comparison with the
rotating speed of the rotor, the motion pattern of the whirling, and so forth.
1. It is known that rotor whirls may be caused by different motive forces, such
as the centrifugal force due to its mass unbalance, the oil-film force on the lubricated
journals, the work-fluid force applied on the disk of the rotor, the internal damping
force in the rotor shaft, and the rubbing friction applied on the tip of the rotor.
Rotor whirls caused by different motive forces are usually studied and reduced to
different types respectively. We emphasize two types of rotor whirls, i.e., the
rotor whirl caused by the oil-film force and the rotor whirl excited by the internal
damping force.
2. Taking the rotation speed of the rotor as the reference standard, all rotor whirls
may be broadly divided into two types, synchronous whirls and nonsynchronous
whirls. The whirl frequency of synchronous whirls is constantly equal to the
rotation speed of the rotor, but the whirl frequency of nonsynchronous whirls is
not the same as the rotation speed of the rotor. As addressed by H. H. Jeffcott,
synchronous whirls are usually excited by the mass unbalance of the rotor. of
which the studied results are arranged into the vibration theory. Since synchronous
whirls belong to forced vibration.
Nonsynchronous whirl may be further grouped into three classes as follows.
(1) Supersynchronous whirl due to shaft misalignment, whose frequency is
usually twice the shaft speed. Actually, this is also a kind of forced vibration;
(2) Subsynchronous whirl due to cyclic variation of parameters. In general, it
is excited by loose bearing-housings or shaft rubs;
(3) Nonsynchronous whirl due to instability of the rotor’s steady rotation whose
speed reaches certain threshold speed. The rotor’s unstable rotation gradually
evolves into a periodic motion. In general, its frequency is not the same as the
rotation speed of the rotor. Thus, it is called nonsynchronous whirl and it is the
main subject of this chapter.
3. According to the motion pattern of the rotor whirls, they are classified as
cone whirls and planar whirls. Cone whirl is a kind of space motions of a rigid
body and will be discussed in Chapter 11. A rotor of cone whirl has at least four
degrees of freedom of motion. With regard to planar whirl, the rotor is a rigid body
under planar motion. In particular, if it rotates with a constant speed, it has only
two degrees of freedom of motion. For convenience of analytical investigation,
the planar whirl of rotor is the main subject of this chapter.
192
8.1 Mechanical Model of Rotor in Planar Whirl
m
r Kr F, (8.1)
J M.
where r is the displacement vector of the mass center of the rotor, r the acceleration
vector of the mass center, Z the angular velocity vector, F all external forces
exerted on the rotor except for restoring forces, M the moment of the external
forces about the mass center, m the mass of the rotor, J the inertia matrix about
the mass center, and K the stiffness matrix of the rotating shaft.
In general, finding the analytical solution of equation system (8.1) is almost
impossible. The equation system is usually solved by numerical computation. To
acquire concise conclusions about the rotor whirl by qualitative analysis, we
present some simplified assumptions and construct the mechanical models, in
which the planar motion rotor is usually a favorable model for analytical study of
the rotor whirl.
Rotor whirl has been the subject of the continuing development of rotors. A
mechanical model with two degrees of freedom of motion was first used by W. A.
Rankine in 1869 to explain the critical speed behavior of rotor-bearing systems.
The system model consists of a rigid mass with an elastic spring whirling in a
circular orbit, as shown in Fig. 8.1. W. A. Rankine applied Newton’s second law
incorrectly in a rotating coordinate system and predicted that rotating machines
would never be able to exceed their first critical speed.
Since a perfectly balanced rotor never occurs in real machines and since it is
the rotation unbalance that excites the most commonly observed type of vibration
(synchronous) in rotating machines, the rotation balance is an essential ingredient
of one of the most useful models for rotor dynamics analysis. The model is called
Jeffcott rotor, as shown in Fig. 8.2, and is named after the English scientist H. H.
Jeffcott who used the model for the first time in 1919 to analyze the response of
193
Chapter 8 Rotor Whirl
Here, we consider another mechanical model of rotor simpler than the Jeffcott
rotor, where the centrifugal force from rotating unbalance and the elastic
deformation of the rotating shaft are ignored. It is a symmetrically rigid body
supported by external forces applied on the left and the right journals, as shown
in Fig. 8.3. The external forces exerted on the two journals are also symmetric.
The motor will maintain the planar motion when the transient response excited
by the external disturbance has been damped by the natural damping, which
always exists in real rotors.
It is known that the planar motion of a rigid body has only three degrees of
freedom of motion. If we consider the stability of the rotation with constant speed,
the degree of freedom of rotation is eliminated and the mechanical model shown
in Fig. 8.3 is reduced to a particle moving in a plane. The differential equations
of its motion are consequently reduced to the form
mx cx Fx ( x, y, x , y , t ),
(8.2)
my cy Fy ( x, y, x, y , t ),
where x and y are displacement components of the mass center of the rotor, x ,
y , x , and y are components of their velocity and acceleration values, m is the
194
8.2 Fluid-Film Force
mass of the rotor, c is the external damping coefficient, and Fx and Fy are the
components of external force except external damping force.
As mentioned before, the mechanical model shown in Fig. 8.3 is simpler than
the Jeffcott rotor, and equation system (8.2) is the simplest mathematical model of
the rotor whirl. As long as the external force components, Fx and Fy, are replaced
by the components of the oil-film force, this model will be used to study the
rotor’s oil whirl.
adhering to the walls, and since the clearance between the walls of the bearing
is very small in comparison with their radii of curvature, we may follow O.
Reynolds to adopt the following model: the flow, assumed to be laminar, fills the
region 0 < x < h(y, z, t) defined in terms of an orthonormal frame of the reference
OXYZ, and the pressure does not vary within the thickness of the film. In other
words, pressure p p(y, z, t) is independent of x, and we shall make a similar
assumption regarding the density U U( y, z, t) in the case where the lubricant is
a compressible fluid, gas-filled bearing. With u, v, and w denoting the components
of the fluid velocity depending on x, y, z, and t, the equation representing the
principle of mass conservation can be written as[3]
wU wU u wU v wU w
0.
wt wx wy wz
Or, integrating with respect to x from 0 to h yields
h
wU w
wt wy ³ U vdx wwz ³ U wdx U ©¨§ u u wwhy v wwhz ¹¸·
0
h
0
h
x h
Uu x 0
0
with
h h
qy ³ 0
vdx, qz ³ 0
wd x (8.3)
and
wU h wU q y wU qz § wh wh wh ·
U ¨u u v ¸ Uu x 0.
wt wy wz © wz wz wt ¹ x h
0
wU h wU q y wU qz
0. (8.4)
wt wy wz
Or, in case of incompressible fluid, we have
wh wq y wqz
0. (8.5)
wt wy wz
On the other hand, components Wy and Wz of the shearing stresses per unit area
parallel to the plane OYZ are obtained from the viscosity law
wv ww
Wy P , and W z P ,
wx wx
where P is the coefficient of viscosity.
197
Chapter 8 Rotor Whirl
With the assumption that inertial effects can be ignored, we may write down
the equilibrium condition for a volume element of sides dx, dy, and dz within the
oil film,
wp w § wv · wp w § ww ·
¨ P ¸ 0, and ¨P ¸ 0. (8.6)
wy wx © wx ¹ wz wx © wx ¹
Since p is independent of x and we assume that the same is true of P, these
equations can be easily integrated. Taking the adherence condition into account,
the velocities of the fluid at points (0, y, z) and (h, y, z) are the same as those of
the points on the walls with which they coincide. With y and z components of
these velocities denoted by U1, V1, and U2, V2, we obtain
x 2 wp ª P h wp º
Pv « (U 2 U1 ) x PU1 ,
2 wy ¬ h 2 wy »¼
(8.7)
x 2 wp ª P h wp º
Pw « (V2 V1 ) x PV1 .
2 wz ¬ h 2 wz »¼
Hence, we deduce by (8.3) and obtain
h3 wp U1 U 2 h3 wp V1 V2
qy h, q z h.
12P wy 2 12P wz 2
Finally, using (8.4), we obtain
w § h3 wp · w § h3 wp · w
¨U ¸ ¨U ¸ 6 [ U (U1 U 2 )h]
wy © P wy ¹ wz © P wz ¹ wy
w wh U
6 [ U (V1 V2 )h] 12 .
wz wt
Let us simplify this equation with U U1 U2, V V1 V2, and make U 0 by a
suitable choice of axes. Accordingly, in the case of impermeable walls and an
incompressible lubricant, assuming V and P to be constant, we finally obtain the
Reynolds’ equation.
w § 3 wp · w § 3 wp · wh wh
¨h ¸ ¨h ¸ 6 PV 12P . (8.8)
wy © wy ¹ wz © wz ¹ wy wt
It can be further simplified for the stationary case. By omitting the last term, the
Reynolds’ equation for the stationary case is established, i.e.,
w § 3 wp · w § 3 wp · wh
¨h ¸ ¨h ¸ 6PV . (8.9)
wy © wy ¹ wz © wz ¹ wy
198
8.2 Fluid-Film Force
In the case where the shaft turns in the bearings of the circular cross section, we
assume that the axis of the shaft always remains parallel to that of the bearing in
the common direction Z. In a plane normal to this axis, the internal cross section
of the bearing is a circle of center O and radius R1, while the internal cross
section of the shaft is a circle of center A and radius R < R1, as shown in Fig. 8.6.
We refer the plane of this cross section to the orthonormal axes A[K where AK is
Figure 8.6 Geometry of the Journal and the Circular Bearing of the Rotor
199
Chapter 8 Rotor Whirl
where Z is the angular velocity of the shaft, and \ is the angular velocity of the
journal whirling.
For a plain journal bearing, with two ends z L 2 and z L 2 open to the
atmosphere, and with an uncavitated oil film, the boundary condition is
p (T , L 2) p(T , L 2) Pa (8.13)
and
P(0, z ) P(2S, z ) P0 , (8.14)
where Pa is the atmosphere pressure, and P0 is the supply pressure to the bearing.
With realistic boundary conditions (8.13) and (8.14), closed-form solutions to
Eq. (8.8) in the function form have not been obtained yet, except for two special
cases, namely, the long bearing case and the short bearing case.
We first seek to define the operating conditions in the stationary mode, i.e., the
mode in which the eccentricity H and the angle between AK and the upward
vertical axis are constant. The Reynolds’ Eq. (8.8) is reduced to Eq. (8.9) in such
a case.
In the long bearing case, where L D is large, the pressure distribution around
the bearing is invariant along the length of the bearing; simultaneously, it is
usually assumed that p does not depend on z and the second term in Eq. (8.12) is
of negligible magnitude compared to the first term. The function expression of
pressure p (T , H ) has been found for this case.
In the short bearing case, with L D 1 , the pressure gradient in the direction
parallel to the axis of the shaft is important. Here, assuming that p does not
depend on y, the Reynolds' equation is reduced to
w2 p 6PV dh
.
wz 2 h3 dy
It can be easily integrated with the boundary condition p 0 at z r L 2 , which
gives
dh § 2 L2 · 3
p 3PV ¨ z ¸h . (8.15)
dy © 4¹
This formula represents fairly well what happens in the region with positive
pressure, i.e., the region where the bearing clearance h decreases in the direction
of the shaft rotation. A satisfactory approximation is to adopt formula (8.15) in
the region 0 T S, and take p 0 in the region S T 2S , as shown in Fig. 8.7,
which is called the Gumbel condition. The condition describes the fact that the
actual pressure in the lubricant usually does not drop more than the atmospheric
pressure. The bearing, whose boundary pressure satisfies the Gumbel condition, is
called cavitated bearing[4]. We can easily deduce this pressure exerted by the
200
8.2 Fluid-Film Force
S
L
PVL3 SH
P[ ³³
0
2
L pR sin T dT dx
4c 2 3
,
2
1 H 2 2
S
L
PVL SH 23
PK ³³0
2
L
2
pR cosT dT dx
c 2 (1 H 2 )2
.
The terms due to the viscous shearing stress in the lubricant are of the order of
c/R relatively to P[ and PK . Thus, the load-bearing property of the bearing is
essentially due to the pressure of the lubricant. Denoting P as the proportion of
the weight of the shaft supported by the bearing concerned, in the stationary state,
we have
1
1
SPVL3 H ª§ 16 · 2 º 2
( P[2 PK ) «¨ S2 1¸ H 1»
2 2
P (8.16)
4c (1 H 2 ) 2
2
¬© ¹ ¼
and
P[ S 1
tan\ (1 H 2 ) 2 . (8.17)
PK 4H
Taking load weight P as a variable, we solve the simultaneous Eqs. (8.16) and
(8.17) and obtain the locus of the equilibrium position of the journal for a
cavitated short bearing, as shown in Fig. 8.8[4].
A dimensionless parameter referred to as Sommerfeld’s number is defined by
P DLN § D ·
2
S ¨ ¸ , (8.18)
P ©c¹
201
Chapter 8 Rotor Whirl
SH ¨ ¸
© D¹
It shows that the eccentricity of the shaft supported on the cavitated short bearing
has a direct relationship with the Sommerfeld’s number.
Figure 8.8 Locus of Journal Equilibrium Positions for a Cavitated Short Bearing
If the rotor bearing system is unstable (oil whirl) or if imbalance is added to the
rotor, the journal may execute orbits about the static operating points, as shown
in Fig. 8.9. In order to analyze the motion, the incremental variations of the fluid
film force on the journal must be expressed in terms of functions of the relative
displacement of the journal and the velocity measured from the static operating
point, and the incremental force functions may be linearized when the orbit is
small enough. The idea to represent the incremental variations of the fluid film
force applied on the journal by means of stiffness and damping coefficients was
suggested by A. Stodola (1925)[3, 5].
The linear expressions of the components of the fluid film force are deduced
from the analytical solution of the Reynolds’ Eq. (8.12), and two components of
the fluid film force along the axes A[ and AK are written in the form
202
8.2 Fluid-Film Force
with
[ 'e, K e'\ ,
(8.21)
[ e, K e\ ,
1
8(1 H 2 ) S(1 H 2 ) 2
kKK Q (H ), kK[ Q(H ),
1 H2 H
(8.22)
S(1 2H 2 )
k[K 1
Q (H ), k[[ 4Q(H ),
H (1 H )
2 2
1
Q(H ) [S2 (16 S2 )H 2 ] 2 , (8.23)
where x and y are components of the relative displacement of the journal center
from the stationary operating point (equilibrium point), and x and y are the
velocity components in frame AXY. The dimensionless stiffness and the damping
coefficients of the cavitated short bearing are respectively
203
Chapter 8 Rotor Whirl
k xx 4[2S2 (16 S2 )H 2 ]Q 3 (H ),
S>S2 2S2H 2 (16 S2 )H 4 @Q 3 (H )
k xy 1
,
H (1 H ) 2 2
In general, kxx and kyy are called direct stiffness coefficients of the fluid film;
kxy and kyx are called cross-coupled stiffness coefficients of the fluid film; cxx and
cyy are called direct damping coefficients; and cxy and cyx are called cross-coupled
damping coefficients.
A concentric shaft steady rotating inside a bearing or a seal clearance drags the
surrounding fluid into rotation. After a transient process, the fluid exhibits a very
regular pattern of motion: the angular velocity of the fluid layer next to the shaft
is the same as its rotation speed : , while the fluid layer next to the house of the
bearing or the seal has zero velocity. The fluid average velocity varies for various
types of bearing and seal. Therefore, it is reasonable to introduce a coefficient W,
which is called the fluid average circumferential velocity ratio, to describe the
motion of the fluid. Obviously, introducing coefficient Was a representative of
the circumferential flow is certainly a strong simplification of the complex flow.
Actually, when the shaft is displaced from its concentric position inside the bearing
or the seal, the fluid average circumferential velocity decreases. Inside one cycle of
the fluid rotation, the pattern changes for half cycle, and the fluid, in average, flows
down to the smallest gap created by the eccentrically located shaft. Numerical
calculations of the flow pattern have confirmed this qualitative prediction.
In fact, the flow field of the fluid film is intimately related to the eccentricity
of the rotating shaft. In a slightly eccentric shaft, an average circumferential
204
8.2 Fluid-Film Force
velocity per cycle can still be associated with the fluid circumferential motion,
whose magnitude is smaller than that in a concentric shaft. There exists a certain
value of eccentricity, for which the circumferential flow periodic pattern partially
disappears and the backward circumferential flow, which is called secondary
flow, appears, as shown in Fig. 8.10. If the backward circumferential flow exists,
the model of the fluid film force presented below is not valid. Fortunately, oil
whirl and oil whip do not occur when the shaft’s eccentricity is high.
Figure 8.10 Flow Pattern for Eccentric Shaft inside a Bearing with Relatively
Large Clearance and a Comparison between Fluid Average Circumferential Velocity
Ratio for a Concentric and Eccentric Shaft
According to the flow field of the fluid film shown in Fig. 8.10, the magnitudes
of the fluid average circumferential velocity ratio can be obtained and depicted in
Fig. 8.11, where, for a concentric shaft, the curve shows that the fluid average
circumferential velocity ratio is a decreasing function of the eccentricity starting
at a value W 0 and reaching zero at high eccentricity.
205
Chapter 8 Rotor Whirl
z zr eiW: t , (8.28)
where z x iy, and x and y are the coordinate components of the journal center.
In the fixed reference, the fluid-film force will have the form
F k0 z d 0 ( z iW: z ) m f (
z 2iW: z W 2 : 2 z ).
° Fx °½ ª k0 W 2 : 2 m f W: d 0 º x½
® ¾ « »® ¾
«¬ W: d 0 k0 W : m f »¼ ¯ y ¿
2 2
°¯ Fy °¿
ª d0 2W: m f º x ½ ª m f 0 º x½
« » ® ¾« » ® ¾. (8.30)
¬ 2W : m f d 0
¼¯ ¿ ¬
y 0 m
f ¼ ¯ y¿
206
8.2 Fluid-Film Force
appearing on the main diagonal of the stiffness matrix is now modified by the
centripetal fluid inertia force that carries a negative sign. In experimental testing,
by applying the perturbation method, the character of the fluid force described by
Eq. (8.30) was fully confirmed[7]. An important conclusion relates to the cross-
coupled stiffness coefficient, which is the most important component affecting the
stability of the rotor.
For relatively large bearing clearance to radius ratio, the fluid inertia force
becomes significant and modifies the damping and the stiffness matrices con-
siderably. However, in journal bearings, the clearance to radius ratio is very small
and we may neglect the influence of mass mf on the fluid film force. Hence, the
following expressions of the fluid film forces are often used to analyze the stability
of the rotor motion.
Fx d 0 x k0 x W: d 0 y,
(8.31)
Fy d 0 y k0 y W: d 0 x.
The fluid film in the seal clearance is dragged into rotation by the rotating shaft.
Therefore, the flow field in the annular space is very similar to that in the journal
bearing and the seal force is also the function of the displacement and the velocity
of the rotor at the seal location. Consequently, the components of the seal force
along x and y axes are usually represented in the form[8]
where k xxs and k yys are direct stiffness coefficients of the seal, k xys and k yxs are the
cross-coupled stiffness coefficients, cxxs and csyy are the direct damping coefficients
and cxys and csyx are the cross-coupled damping coefficients.
From the data available to date, it appears that the linear interpretation of the
force coefficients, as previously described for journal bearings, should be used
for most seals under the assumption of small motions about the centered position.
The configuration of plain seals is geometrically similar to that of plain journal
bearings but has large c/R ratios of 0.01 as compared to that of bearings for
which c/R is of the order of 0.001. Seals customarily operate in the turbulent
regime both axially and circumferentially, and have a substantial direct stiffness
at a centered zero eccentricity position. Furthermore, seals are nominally designed
to operate in a centered position, while the operating eccentricity of journal bearings
varies with the running speed and the load. In addition, the identification work on
207
Chapter 8 Rotor Whirl
dynamic bearings has had the objective of validating the dynamic coefficients
versus the eccentricity relationship. Due to the general similarity between bearings
and seals, the technique developed for the coefficient identification of bearings
may be applied for seals.
mu Fu u , v, u , v, S ,
(8.33)
mv Fv u , v, u , v, S ,
PZ LD § D ·
2
S ¨ ¸ , (8.18)
2SP © c ¹
where the force components Fu and Fv consist of the oil-film force exerting on
the journal, and the Sommerfeld number S is defined in the foregoing except that
the number of revolutions per second N is replaced by Z /2S.
208
8.3 Oil Whirl and Oil Whip
S
xcc f x x, y, xc, y c, S ,
Z2 (8.35)
S
y cc f y x, y, xc, y c, S ,
Z2
where the prime represents the derivative with respect to the dimensionless time W.
To apply the Hopf bifurcation theorem, it is necessary to convert Eq. (8.35) to
a first-order system. Denoting
x x1 , x x2 , y x3 , y x4 (8.36)
and
x [ x 1 , x2 , x3 , x4 ]T ,
F [ x2 , SZ 2 f x , x4 , SZ 2 f y ]T . (8.37)
x F x,Z , S . (8.38)
Obviously, this equation is a nonlinear vector equation. It shows that the rigid
rotor supported on the hydrodynamic bearings is a fourth-order dynamic system.
In Eq. (8.38), dimensionless rotating speed Z takes on the role of the variable
parameter in the Hopf bifurcation theorem.
The equilibrium point of the rotor center xe may be determined by solving
simultaneous Eqs. (8.35) and (8.12). Furthermore, the stability of the equilibrium
position may be determined in terms of the Jacobian matrix of F with respect to
the state vector x, i.e.,
ª 0 1 0 0 º
« k xy cxy »»
A(Z )
ª wF ( x , Z ) º « k xx cxx
, (8.39)
« wx » « 0 1 »
¬ ¼ xe 0 0
« »
«¬ k yx c yx k yy c yy »¼
with
209
Chapter 8 Rotor Whirl
S wf x S wf x
k xx , k xy ,
Z 2 wx1 Z 2 wx3
S wf y S wf y
k yx 2 , k yy 2 ,
Z wx1 Z wx3
(8.40)
S wf S wf
cxx 2 x, cxy 2 x,
Z wx2 Z wx4
S wf y S wf y
c yx 2 , c yy 2 .
Z wx2 Z wx4
The characteristic equation of the rotor is obtained with matrix A Z , i.e.,
The stable condition of the steady rotation of the rigid rotor may be established
by using Hurwitz criterion and written as
Z Z0 , (8.42)
where Z 0 , the dimensionless threshold speed of the oil whirl of the rigid rotor, is
a function of H e only. The stability borderline Z 0 f (H e ) is shown in Fig. 8.12.
210
8.3 Oil Whirl and Oil Whip
For H e ! 0.8 , the rotor is always stable, while for H e 0.8 , there is a bifurcation
point in the parameter plane H e - Z .
The dimensionless whirl frequency Z 0 can be easily found from the characteristic
Eq. (8.41) by numerical calculation.
As the threshold speed is exceeded, a pair of eigenvalues crosses into the right
half plane and the rotor becomes unstable. Let us designate D (Z ) iE (Z ) as the
eigenvalue of matrix A(Z ) . This is a continuous extension of the imaginary
eigenvalue iE (Z 0 ), which corresponds to the stability borderline Z 0 f (H e ) .
Derivative (dD / dZ )Z0 is also required. For calculation, it is necessary to consider
the relationship among the parameters Z , S, and H e . A change in the rotor speed
Z alters the Sommerfeld number S and hence the corresponding equilibrium
point. Therefore, a new system parameter V must be introduced. It is independent
of the rotor speed and is constant for a given rotor system with constant lubricant
viscosity. Let us define
S LD 3 P
V . (8.43)
Z 1
2 2
8( Pmc) c
A series of operating curves for different values of V is shown in Fig. 8.12, where
each curve illustrates the relationship between the rotor speed and the corres-
ponding equilibrium point. Derivative (dD dZ )Z0 is calculated from Eq. (8.41) and
tabulated in Table 1, in which D , E and G are the functions of P in Eq. (4.38).
As summarized below, it has been shown that the system of Eq. (8.38) possesses
the following properties:
(1) F ( x, Z ) is differentiable in a neighborhood of ( x , Z ) ( xe , Z0 ) ;
(2) The locus of equilibrium points is determined by the Sommerfeld number S;
(3) Matrix A(Z0 ) has a pair of conjugates with pure imaginary eigenvalues
riE (Z 0 ) for each H e 0.8 . The remaining two eigenvalues have negative real parts;
(4) (dD / dZ )Z 0 ! 0 for each H e 0.8 .
Therefore, the condition of the Hopf bifurcation theorem is satisfied and the
existence of a periodic solution of the system with small amplitude at speed close
to the threshold speed has been proved.
211
Chapter 8 Rotor Whirl
The remaining work is to determine the direction of bifurcation and the stability
of the periodic orbit. This information is deduced by calculating the sign of the
quantity G c(Z0 ), which may be determined by algebraic formulas derived by A.
B. Poore[10]. To use the formulas, the following need to be calculated.
(1) The left and the right eigenvectors for the eigenvalue iE (Z 0 ) of A(Z0 ) ,
(2) The elements of matrices Fxx and Fxxx,
(3) The inverse of matrices A(Z0 ) and [ A(Z0 ) 2iE (Z 0 ) I ] .
The calculated values of G c(Z 0 ) are given in Table 1. The results indicate that
there are three separate regions of parameter plane H e - Z , as shown in Fig. 8.12,
including:
(1) Region I(0 H e 0.14) and Region Ĕ (0.74 H e 0.8). In these regions,
G c(Z 0 ) 0 , and therefore, subcritical bifurcation occurs for Z Z 0 , and the
bifurcation periodic orbit is unstable, and
(2) Region ē( 0.14 H e 0.74 ). It is the largest region with G c(Z0 ) ! 0 , and
supercritical bifurcation occurs for Z ! Z 0. The bifurcation periodic orbit is stable.
With the value of system parameter V 0.2, a numerical calculation is carried
out. The behavior of the rotor is shown in Fig. 8.13, in which each plot represents
the motion of the rotor center. For rotor speed Z below the threshold speed Z0, only
the equilibrium solution is found, as shown in Fig. 8.13(a). However, for rotor speed
Z immediately above the threshold speed, a stable amplitude whirl orbit appears,
independent of the initial condition, as shown in Fig. 8.13(b) and (c). Although the
amplitude of the orbit increases continuously as the rotor speed increases, the orbit
remains independent of the initial condition, as shown in Fig. 8.13(d) and (e), before
becoming completely unstable at speed of 2.0, as shown in Fig. 8.13(f).
By using the short bearing theory, a similar analysis was published by P. Hollis
and D. L. Taylor[11]. J. C. Deepak and S. T. Noah provided experimental evidence
that verified the existence of the Hopf bifurcation of a single disk rotor supported
on a short journal bearing[12].
When the rotating speed of a rotor is much lower than its fundamental frequency,
no elastic vibration mode is excited and the differential equation of the motion of
the rigid rotor may be used to study the oil whirl problems. J. W. Lund took a rigid
rotor as the mechanical model to study the stability problem of rotor-bearing systems
in 1965, by which the whirl frequency and the threshold speed of the rotor-bearing
system are found with the expressions of the dimensionless stiffness and the damping
coefficients of the short bearing[13]. The calculation results are plotted in Fig. 8.14.
It demonstrates that the ratio between the whirl frequency and the threshold
speed, W 0 , is a function of the equilibrium eccentricity H e . In normal conditions,
the ratio of the whirl frequency and the threshold speed is higher than one half.
However, this result is not well consistent with the practical observation. It is
212
8.3 Oil Whirl and Oil Whip
Figure 8.14 The Ratio of Whirl Frequency and Threshold Speed versus
Eccentricity of Rotor
213
Chapter 8 Rotor Whirl
obvious that the deficiency is caused by the error of the expressions of the fluid
film force. To acquire a rational result, let us replace the theoretical expression of
the fluid-film force by the Muszynska model, i.e., Eq. (8.30). Consequently,
substituting expression (8.30) in the equations of motion of the rigid rotor (8.2),
we obtain the differential equation system, i.e.,
x d 0 x 2W: m f y (k0 m f W 2 : 2 ) x W: d 0 y
(m m f ) 0,
(8.44)
y d 0 y 2W: m f x (k0 m f W : ) y W: d 0 x
(m m f ) 2 2
0,
where m is the half mass of the rotor, : the rotating speed, and W the fluid
average circumferential velocity ratio.
The above equations show that the stability of the symmetric rigid rotor supported
on the fluid-film bearings may be roughly determined by the linear model of a
fourth-order dynamic system, in which there are inertial force, damping force,
gyroscopic force, potential force, and circulatory force. The steady rotation of the
rigid rotor may lose stability under certain parameter conditions, which may be
found by Hurwitz criterion.
The characteristic equation of the linear system (8.44) may be found by using
the traditional method, i.e.,
a0 O 4 a1O 3 a2 O 2 a3 O a4 0 (8.45)
with
a0 (m m f )2 , a1 2d 0 (m m f ),
a2 d 02 2(m m f ) (k0 m f W 2 : 2 ) 4m f W 2 : 2 , (8.46)
a3 2d 0 (k0 m f W : ), a4
2 2
dW : .
2 2
0
2
According to the Hurwitz criterion, determine the coefficient conditions for the
second type of critical stability of system (8.44), which consists of five inequalities
a0 ! 0, a1 ! 0, a2 ! 0, a3 ! 0, a4 ! 0,
and an algebraic equation
a1a2 a3 a0 a32 a12 a4 0. (8.47)
The expressions (8.46) show that the above inequalities are all satisfied
naturally. Therefore, the algebraic Eq. (8.47) is the sole coefficient condition for
the second type of critical stability of the rigid rotor supported on the fluid film
bearings. Then, substitute expressions (8.46) in Eq. (8.47). After some tedious
operation, we obtain the sole parameter condition for the second type of critical
stability, i.e.,
(d 02 4m 2f W 2 : 2 )(k0 mW 2 : 2 ) 0,
214
8.3 Oil Whirl and Oil Whip
in which the first factor is always positive, and the critical parameter equation of
the rigid rotor becomes
k0 mW 2 : 2 0.
The threshold speed of oil whirl of the rigid rotor is also found consequently, i.e.,
1
1 § k0 · 2
: . (8.48)
W ¨© m ¸¹
It is obvious that when the rigid rotor operates under the threshold speed,
characteristic Eq. (8.45) possesses a pair of conjugate imaginary roots, namely,
riZ . Substituting them in Eq. (8.45) yields a complex equation, i.e.,
Z W: . (8.50)
The oil whirl frequency Z is just equal to the average angular velocity W: of the
fluid film in bearings. The curve in Fig. 8.11 illustrates that the average angular
velocity of the fluid film is related to the rotor eccentricity and the average
circumferential velocity ratio W is slightly lower than one half if the eccentricity
is not very high. Thus, the oil whirl frequency of the rigid rotor is also slightly
lower than one half of the threshold speed and the oil whirl of the rotor is usually
called half-speed whirl.
For studying the influence of shaft elasticity on the oil whirl of a rotor, a symmetric
rotor model consists of a centric rigid disk and the left-hand and the right-hand
shafts without a mass, whose two journals are supported on the fluid film bearings
respectively, as shown in Fig. 8.15. The centers of two bearings are O and O'.
The origin of the static frame of reference, OXYZ, is located at O. Let us denote
215
Chapter 8 Rotor Whirl
mxc k ( xc x) 0,
(8.51)
myc k ( yc y ) 0,
where m is the half mass of the rotor, and k is the stiffness coefficient of left-hand
or right-hand flexible shafts.
Applying the approximate expression (8.31) of the fluid film force, the equation
of motion of two elastic shafts is reduced to
k ( xc x) k0 x W: d 0 y d 0 x 0,
(8.52)
k ( yc y ) k0 y W: d 0 x d 0 y 0.
Let us substitute Eqs. (8.52) in Eq. (8.51). After necessary operations, we obtain
the differential equations of motion of the flexible rotor supported on the fluid
film bearings, i.e.,
216
8.3 Oil Whirl and Oil Whip
b0 m 2 d 02 , b1 2m 2 d 0 (k0 k ),
b2 m2 [(k0 k ) 2 W 2 : 2 d 02 ] 2md 02 k ,
b3 2md 0 k (k 2k0 ), (8.55)
b4 d k 2mk (k k0 k W : d ),
2
0
2 2
0
2 2
0
2
b5 2d 0 k0 k 2 , b6 k 2 (k02 W 2 : 2 d ). 2
0
According to the Hurwitz criterion, the coefficient condition for the second
type of critical stability for the system (8.53) can be determined. It consists of the
following inequalities
b0 ! 0, b1 ! 0, b2 ! 0, b3 ! 0, b4 ! 0, b5 ! 0, b6 ! 0, ' 3 ! 0 ,
and the algebraic equation
'5 0 (8.56)
Expressions (8.55) show that the above inequalities are all satisfied naturally.
Taking into account the geometrical property of Hurwitz inequalities: ' K ! 0,
K 1, 2," , i.e., Eq. (3.35), we conclude that the algebraic Eq. (8.56) is the sole
coefficient condition for the second type of critical stability for the flexible rotor
supported on the fluid film bearings.
It is known that when the flexible rotor operates under the threshold speed, the
characteristic Eq. (8.54) possesses a pair of conjugate imaginary roots, namely,
riZ . Substituting them in Eq. (8.54) yields a complex equation, i.e.,
217
Chapter 8 Rotor Whirl
1
1 ª k0 k º 2
: « » . (8.61)
W ¬ m( k0 k ) ¼
After comparing the expression (8.60) with (8.61), we may conclude that the
threshold speed is directly related to the oil-whirl frequency of the flexible rotor
supported on the fluid film bearings, namely,
Z W: . (8.62)
Thus, the oil whirl of the flexible rotor is also a half speed whirl.
Summarizing all the above, we obtain the following conclusions.
1. The oil-whirl frequency is slightly lower than one half of the threshold speed
of the rotor.
2. Inspecting expression (8.60), we find that the oil-whirl frequency of the
flexible rotor is equal to the natural frequency of a rigid rotor supported on two
series springs, whose stiffness coefficients are equal to the oil-film stiffness k0 and
the shaft stiffness k respectively.
3. If the shaft stiffness k is comparable to the direct stiffness coefficient of
hydrodynamic bearing k0, the threshold speed of the rotor is remarkably lower
than that of the rigid rotor, and the influence of shaft elasticity on the threshold
speed must be considered.
To understand the influence of the external damping on the oil whirl of rotor, Z.
L. Guo and R. G. Kirk considered a symmetric flexible rotor, which was supported
on the oil-film bearings, exerting an external damping force on the disk and
perfectly balanced[14]. Let us denote O1 and O1c as the centers of two bearings,
whose midpoint O is the origin of the static frame OXYZ. Axis Z lies along the
line O1O1c and axis Y is an upward vertical, as shown in Fig. 8.16(a).
Let us assume that the rotor operates under the planar whirl. Let us denote
O1 j ( x, y , l ) and O1cj ( x, y , l ) as the centers of two journals and C ( xc , yc ,0) as the
center of the rotor, as shown in Fig. 8.16(b). The equations of motion of the rotor
become
mxc cxc k ( xc x) 0,
(8.63)
myc cy c k ( yc y ) 0,
where m is the mass of the rotor, c the external damping coefficient, and k the
sum of stiffness coefficients of the left-hand and the right-hand shafts. Using the
linear mathematical model of the fluid film force, i.e., Eq. (8.25), the differential
equations of motion of the elastic shaft are
k ( xc x) k xx x k xy y cxx x cxy y 0,
(8.64)
k ( yc y ) k yx x k yy y c yx x c yy y 0,
where kzx, kxy, kyx, and kyy are stiffness coefficients of the fluid film force, and cxx,
cxy, cyx, and cyy are its damping coefficients.
When the system parameters make the rotation with a constant speed into the
second type of critical stability, the displacement components of the rotor center
and the journal centers, i.e., xc, yc, x, and y, are harmonic functions of time t.
After damping the transient response of the rotor, we have
[ xc yc x y ] [ xc 0 yc 0 x0 y0 ]eiZ t , (8.65)
where xc0, yc0, x0, and y0 are the initial values of xc, yc, x, and y. Substituting the
expression in Eqs. (8.63) and (8.64), and eliminating xc0 and yc0, we obtain
mkZ 2 ickZ
x0 (k xx icxxZ ) x0 (k xy icxyZ ) y0 ,
k mZ 2 icZ (8.66)
mkZ 2 ickZ
y0 (k yx ic yxZ ) x0 (k yy ic yyZ ) y0 .
k mZ 2 icZ
Using the following expression to define the equivalent stiffness keq and the
equivalent damping ceq,
mkZ 2 ikcZ
keq iceqZ , (8.67)
k mZ 2 icZ
we obtain
(k mZ 2 )mkZ 2 c 2 kZ 2
keq (8.68)
(k mZ 2 ) c 2Z 2
and
219
Chapter 8 Rotor Whirl
c 2 k
ceq . (8.69)
(k mZ 2 ) c 2Z 2
Substituting Eqs. (8.68) and (8.69) in Eqs. (8.66) and eliminating x0 and y0, we
have
Solving Eqs. (8.70) yields the formulas for calculating the whirl frequency Z
and the threshold speed : of the oil whirl, i.e.,
(keq k xx )(keq k yy ) k xy k yx
Z 2 (8.71)
(ceq cxx )(ceq c yy ) cxy c yx
and
P Bkeq [k m(Z : )]mk (Z : ) 2 c 2 k (Z : )2
: . (8.72)
3
c [k m(Z : )]2 c 2 (Z : ) 2
Simultaneously, we obtain the formulas to calculate keq and ceq
P BR3ceq ck 2
, (8.74)
c3 [k m(Z : )]2 c 2 (Z : ) 2
where P is the dynamic viscosity coefficient of the lubricant, B the width of the
bearing, c the clearance of the bearing, and R the radius of the bearing.
If the external damping coefficient c is equal to zero, the above analysis will
be simplified and become the Lund stability method where only equivalent
stiffness is presented.
In order to recognize the stability pattern of the model shown in Fig. 8.17,
given the system parameters of a flexible rotor with external damping, namely,
m 50 kg, k 13.72 u 106 N / m, R 25 mm, B 30 mm, c 2.5 u 103 mm, P
1.92 u 102 (N s) / m 2 we may find all eigenvalues using Eqs. (8.63) and (8.64).
The results are shown in Fig. 8.18, i.e., the plot of system damping versus rotor
speed curves of a given rotor system with different intensity of external damping.
In this plot, Z n2 k / m , and D is the minimum value of the negative real part in
all eigenvalues. It can be seen that under the circumstance without external damping,
namely, c 0, the system will get into an upper region of stability from the lower
region of instability with the rotor speed increasing. When there exists external
220
8.3 Oil Whirl and Oil Whip
damping, the system will normally have two speed thresholds that evolve a stability
pattern as one region of instability, A, sandwiched between two regions of stability,
B1 and B2, whose top region of stability is up-boundary free. With the increase of
the external damping value, the width of the unstable region will decrease. When
external damping is increased to a certain value, the middle region of instability
will disappear, which means that the system will always be stable over the entire
running speed range. These results basically agree with the analytical work done
by S. H. Crandall in 1996[15].
Figure 8.18 is a stability chart with a curve of threshold speed versus external
damping of the system shown in Fig. 8.16. This chart demonstrates that external
damping is very helpful to improve the system stability at the low-speed region.
221
Chapter 8 Rotor Whirl
has been found[15, 16]. The theoretical result of an isoviscous standard solution of
the Reynolds’ equation predicts an increment of something like 10% 40% in the
stability threshold. While some increase in stability threshold is due to the use of
the fluid boundary condition, temperature effect in the fluid may also increase
the theoretical stability threshold to some extent. J. K. Wang and M. M. Khonsari
studied the influence of the inlet oil temperature on the instability threshold of
rotor bearing systems and pointed out that the inlet temperature can either improve
or deteriorate the instability threshold speed of rotor bearing systems, depending on
the operating point. Their conclusion has been verified by some experimental data[17].
1. When the shaft starts rotating with a slowly increasing rotation speed,
synchronous lateral vibrations with minor amplitude are observed all along the
rotor axis, as shown in Fig. 8.19. These vibrations are caused by the inertia forces of
unbalance of the rotor. At low rotation speed, these vibrations are stable. Shortly
222
8.3 Oil Whirl and Oil Whip
after an impulse perturbation of the rotor causes a transient process, the same
vibration pattern is reestablished.
2. At higher rotation speed, usually below the first balance resonance speed,
the forced synchronous vibration is the only regime of motion. Along with it, oil
whirl appears, as shown in Fig. 8.19. Oil whirl is the rotor’s lateral forward
processional subharmonic vibration around the bearing center at the frequency
close to, and usually smaller than, half of the rotation speed. In this range of
rotation speed, the rotor behaves as a rigid body. The amplitude of oil whirl is
usually much higher than that of synchronous vibration. It is, however, limited
by the bearing clearance and the fluid nonlinear forces. With rotation speed
increasing, the pattern of the vibration remains stable. The half-speed oil whirl
follows the increasing rotation speed and maintains the speed ratio with it. The
vibration amplitude remains nearly constant and is usually high. In the considered
range of rotation speed, the bearing fluid dynamic effect clearly dominates. The
forced synchronous vibration represents a small fraction of vibration response, as
the spectrum cascade indicates in Fig. 8.19.
3. When the increasing rotation speed approaches the first balance resonance,
i.e., the first natural frequency of the rotor, the oil whirl suddenly becomes
unstable, disappears, and is suppressed and replaced by increasing synchronous
vibrations. The forced vibration dominates and reaches the highest amplitude at
resonance frequency corresponding to the mass and the stiffness properties of the
rotor.
4. At the speed above the first balance resonance speed, the synchronous
forced vibration decays. Again, the bearing-fluid forces come back into action.
With the rotation speed increasing, shortly after the first balance resonance, the
oil whirl occurs again, and the previous pattern repeats. The width of the rotation
speed region, in which the synchronous vibrations dominate, depends directly on
the amount of rotor unbalance. The higher unbalance, the wider this region.
5. When the rotation speed approaches double the rotor first balance resonance
speed, the frequency of the half-speed oil whirl reaches the first unbalance
resonance speed, i.e., the first natural frequency of the rotor. The oil whirl pattern is
replaced by oil whip, which is a lateral forward processional subharmonic vibration
of the rotor. Oil whip has a constant frequency. Independent of the increasing
rotation speed, the oil whip frequency remains close to the first natural frequency
of the rotor. In this range of high rotation speed, the shaft cannot be considered
rigid. Its flexibility, i.e., the additional degrees of freedom, causes the rotor
bearing system to be closely coupled. The rotor parameters, in particular, its mass
and its stiffness, become the dominant dynamic factors. The amplitude of the oil
whip journal vibration is limited by the bearing clearance. However, the shaft
vibration may become very high, as the shaft vibrates at its natural frequency,
i.e., in the resonant condition caused by the interaction of the oil whirl of the
rotor-bearing system and the fundamental frequency resonance of the unbalanced
rotor.
223
Chapter 8 Rotor Whirl
Figure 8.20 Journal Orbits of Periodic Motions with Different Rotating Speeds
(a) Synchronous Motion ( V 10 ); (b) Period 5 Motion ( V 25 )[20]
Figure 8.21 Chaotic Motions Computed by the Oil-Film Force Model of Present
Method ( V 48 )[20]
symmetrically supported by two identical fluid film bearings. They established the
equations of motion in polar coordinates and implemented the Hopf bifurcation
theory to determine the threshold speed : of rotor stability[21]. When the running
speed : is close to but below : , a closed curve of the center of the journal
exists in the polar coordinates plane. If the center of the journal is released inside
the closed curve, the orbit of the center of the journal tends asymptotically to
approach the steady equilibrium point. If the center of the journal is released outside
the closed curve, the orbit of the center of the journal tends asymptotically to
approach the clearance circle and the motion becomes stable. Therefore, the closed
curve is called stability envelope.
Secondly, after crossing the stability threshold and with the speed gradually
decreasing, the whirl disappears at running speed that is below the threshold speed.
The phenomenon is repeatable and occurs even though all other system parameters
including oil viscosity remain unchanged. J. K. Wang and M. M. Khonsari provide
an analytical explanation for the hysteresis phenomenon by means of the previous
225
Chapter 8 Rotor Whirl
mechanical model and the Hopf bifurcation analysis[22]. The hysteresis phenomenon
can be predicted thus: if the bifurcation is subcritical, the hysteresis phenomenon
is detectable and the hysteresis loop is determined by the subcritical bifurcation
profile; if the bifurcation is supercritical, the hysteresis phenomenon does not exist.
This conclusion is consistent with the experimental result presented in paper[22].
The idealized Hook’s Law is taken as a straight-line variation of stress with strain.
In the actual case of alternating stresses, there is a small derivation from this
linear law, as shown in Fig. 8.22[23]. Thus, a fiber of a rotating shaft experiencing
the alternating tension and compression moves on the ellipse from the maximum
strain position A1 to the maximum negative strain position A3 through the zero
stress position B1 and the zero strain position A2, and completes the cycle through
B2 and A4. The derivation from Hook’s Law is exaggerated in the figure and
always runs through in the clockwise direction, as shown in Fig. 8.22.
226
8.4 Internal Damping in Deformed Rotation Shaft
Here, consider the effect of the internal force of the deformed rotating shaft
from the point of view of energy.
In this figure, the stress and the strain in the rotating shaft originate from
applied force caused by motive power and reacting force coming from the loads of
the machine. From point A3 to point A1, the direction of stress and the direction
of strain are coincident and the applied force makes positive work for the rotor.
From point A1 to point A3, the direction of stress and the direction of strain are
contrary and the applied force makes negative work for the rotor. In a cycle of
rotation, the total amount of work made by applied force, which eventually comes
from motive power of the machine, is proportional to the area of the hysteresis
loop in Fig. 8.22. Thus, the internal damping of the rotating shaft provides enough
energy sources for the stationary periodic motion, which is just a self-excited
vibration and occurs in the rotor supported on the bearings without fluid film.
In order to establish the analytical expression of the internal force of the deformed
rotation shaft, it is necessary to describe its geometric and physical features in
details.
The shaft under the undeformed state is a cylinder of revolution of radius r
supported by two circular bearings with centers O1 and O1c at a distance of 2l
apart. Let OXYZ denote the orthonormal frame of the reference in which the axis
OZ lies along the line O1O1c , O is the midpoint of the line and OY is the upward
vertical axis, as shown in Fig. 8.23(a) and (b).
The shaft carries at its center a rigidly attached disk of mass m and the whole
shaft revolves with a uniform angular velocity represented by the rotation vector
227
Chapter 8 Rotor Whirl
: k. Under the effect of inertial forces, the shaft can undergo deformation, which
we assume to be flexural without any torsion or elongation.
The motion of the planar whirl can be described completely with the help of the
coordinates x and y of mass center C, the point of intersection of the plane OXY
and the neutral axis of the shaft, which is the line joining the centers of the circular
section. The trajectory of C is a planar curve, which is the image of the axis O1O1c
under deformation. With x and y as coordinates of C in the plane OXY, we write
y
U2 x2 y 2 , M arc tan , (8.75)
x
where M is the angle of the precession and describes the whirling, as shown in
Fig. 8.23(b).
The bent shaft exerts on the disk a force T, which lies in the plane OXY by
symmetry, with components Tx and Ty.
Neglecting the mass of the shaft, it is clear that T/2 represents the force exerted
by the bearings. With M denoting the moment at C of the forces exerted through
the cross section z 0 by the material elements of the shaft that is contained in
the half space z ! 0 but acting on the material elements in the other half space
z < 0, we can write
JJJJG T
M CO1 u 0
2
or
1 l l
M ( yTx xTy )k Ty i Tx j , (8.76)
2 2 2
where i, j, and k are unit vectors along X, Y, and Z axes respectively.
If S denotes the cross section of the shaft in the plane z = 0, we can express M
by
M ³³ (
S
uV k )d[ dK , (8.77)
where V is the component of the constraint force per unit area in the direction
normal to the cross section. Vector is shown in Fig. 8.23(b).
According to Eq. (8.76), the components of M on OZ are small because of the
small deflection of the shaft. Comparing Eq. (8.76) and Eq. (8.77), we obtain
2 2
l ³³ l ³³
Tx [V d[ dK , Ty KV d[ dK. (8.78)
S S
228
8.4 Internal Damping in Deformed Rotation Shaft
where H denotes the rate of elongation, and H the partial derivative of H with
respect to time t. We could calculate H on the basis of linear approximation.
Terms other than EH in Eq. (8.79) involve small perturbations only, but can cause
the hysteresis loop in plane (H, V ), as shown in Fig. 8.22(b).
For a fiber intersecting the frame of reference CUV plane at point P of
coordinates u and v with respect to the CUV plane, as shown in Fig. 8.23(b), in
which the axis CU is in the direction OC, the extension is H u R , where R is
the radius of the neutral line at C. In accordance with the elementary theory on
the beam bending, the deflection G ( z ) along the neutral line is represented in
each interval (l, 0) and (0, l) by polynomials of degree 3 in z whose coefficients
can be simply obtained by expressing the kinematic condition G (rl ) 0 , the static
d2
conditions, the zero moment at the bearings 2 G (rl ) 0 , and finally the boundary
dx
d
conditions G (0) U , G (0) 0 . Thus, we have
dz
U 3U
G ( z) 3
( z l )3 ( z l ), z (l ,0),
2l 2l
and in particular, as the radius of the curvature at C has the value R l 2 / 3 U , it
follows that
3u U
H . (8.80)
l2
According to Eqs. (8.78) (8.80), calculate the contributions to Tx and Ty from
three terms in Eq. (8.79) respectively[3].
1. Contribution from EH
From the analytical expressions of the internal force of the circular beam, i.e.,
Tx kx, Ty ky (8.81)
with
k 6 EIl 3 , I Sr 4 4.
2. Contribution from f (H , H )
To calculate the rate of strain H , under the usual assumption of the beam theory,
we have to consider the point P as being rigidly bound to the cross section S.
Since the angular velocity of S relative to the CUV plane is : M , we have
u (: M )v,
and by Eq. (8.80), we have
229
Chapter 8 Rotor Whirl
3U 3u U
H 2
(: M )v 2 .
l l
The result, after some calculation, is that we find the contribution from f (H , H)
to Tx and Ty, i.e.,
Tx xF1 ( U , U , : M ) yF2 ( U , U , : M ),
(8.82)
Ty xF2 ( U , U , : M ) yF1 ( U , U , : M )
with
1
2l 5 h ( h 2 ] 12 ) 2 § U ·
Fj
27 U 4 ³ h
d] 1 ³ 1
( h 2 ] 12 ) 2
] j f ¨]1,
© U
] 1 (: M )] 2 ¸ d] 2 ,
¹ (8.83)
h 3r U l 2 , j =1,2.
If we only consider a particular form of f (H , H) , i.e., F sgn H , namely,
8F r 3 U U
F1 ,
3U l 1
[( U U ) (: M ) ] 2 2 2
(8.84)
8F r 3 : M
F2 .
3U l 1
[( U U ) (: M ) ]
2
2 2
Hence, we have
8F r 3 x : y
xF1 yF2 1
,
3l
[( x : y ) ( y : x) ]
2 2 2
(8.85)
8F r 3
y : x
xF2 yF1 1
.
3l
[( x : y ) ( y : x) ]
2 2 2
F1 1 , F2
b UU b(: M ) (8.86)
with
b 6 HIl 3 , I Sr 4 4.
Therefore, we have
230
8.4 Internal Damping in Deformed Rotation Shaft
8F r 3 x : y
Tx kx b( x : y ) 1
,
3l
[( x : y ) ( y : x) ]
2 2 2
(8.89)
8F r 3
y : x
Ty ky b( y : x) 1
,
3l
[( x : y ) ( y : x) ]
2 2 2
in which the first and the second terms are linear, but the third is nonlinear.
Let us denote (x0, y0) as the equilibrium position of the disk center when the shaft
remains at constant rotation speed. Let us deduce the linear parts of the nonlinear
terms of Eqs. (8.89) in the neighborhood of point (x0, y0). After some calculation,
we obtain the linearized expressions of the internal force increment of the
deformed rotating shaft, i.e.,
with
8 F r 3 x0 y0 8F r 3 x02 8F r 3 y02
a1 , a2 , a3 , U02 x02 y02 .
3l :U03 3l :U03 3l :U0 3
ª a2 b a1 º ª k : a1 : (b a2 ) º
D « a , and C « : (b a ) k : a » .
¬ 1 a3 b »¼ ¬ 3 1 ¼
231
Chapter 8 Rotor Whirl
Obviously, D is a symmetric matrix and the first term on the right side of (8.91)
represents the damping force. C is an asymmetric matrix and may be separated
into a symmetric matrix and a skew-symmetric matrix. The symmetric one represents
the potential force while the skew-symmetric one represents the circulatory force.
Thus, we conclude that the internal force of the deformed rotating shaft contains
three components, namely, the potential restoring force, the damping force, and
the circulatory force.
xi yj.
232
8.5 Rotor Whirl Excited by Internal Damping
ve : k u : yi : xj. (8.93)
According to the relative motion principle of kinematics of classical mechanics,
the relative velocity vr of disk center C measured in the carrier frame of reference
CUVW is
vr v ve .
Substituting Eq. (8.92) and Eq. (8.93) in the previous equation, we obtain the
analytical expression of the relative velocity of geometric center C, i.e.,
vr ( x : y )i ( y : x) j . (8.94)
Next, let us propose the basic assumptions to establish the simple model of
internal damping force in the deformed rotating shaft: (1) the average strain rate
of the rotating shaft is proportional to the relative velocity of the disk center; and
(2) the average stress in the rotating shaft is directly related to the strain rate
under the condition of small deformation.
Last, summarizing the previous two assumptions and the relative velocity
expression (8.94), we obtain an approximate expression of the internal damping
force in the rotating shaft, i.e.,
F d ( x : y )i d ( y : x) j ,
whose two components in the static frame Oxyz are
In general, rotors are endowed with a certain unbalance themselves, which has to be
taken into account, because synchronous whirl is usually excited by the unbalance.
233
Chapter 8 Rotor Whirl
Here, consider the planar whirl of a vertical Jeffcott rotor with unbalance, as
shown in Fig. 8.24(a), whose mechanical model is illustrated in Fig. 8.24(b). The
origin of the static frame of the reference OXYZ is located at the midpoint of the
bearing centers O1 (0,0, l ) and O1c(0,0, l ). At the same time, let C(x, y, 0) and
G(x1, y1, 0) denote the geometric center and the mass center of the disk
respectively, as shown in Fig. 8.24(b). Then, we have
mx1 cx kx Fx ,
(8.97)
my1 cy ky Fy ,
where m is the mass of rotor, c the external damping coefficient, and k the
flexural stiffness coefficient of the rotating shaft. Differentiating expressions
(8.96) twice with respect to time t, we have
x1
x e: 2 cos : t ,
y1
y e: 2 sin : t.
Substituting these expressions and (8.95) in Eqs. (8.97), we obtain the equation
of motion of the rotor model shown in Fig. 8.24, i.e.,
234
8.5 Rotor Whirl Excited by Internal Damping
mx (c d ) x kx : dy e: 2 cos : t ,
(8.98)
my (c d ) y ky : dx e: 2 sin : t.
Next, let us introduce the system parameters
c d k e
ne , ni , Z n2 , e0 ,
2m 2m m m
where ne is the external damping parameter of the rotor, ni the internal damping
parameter, Z n the natural frequency, and e0 the unbalance parameter of the rotor.
Hence, the equations of motion of the rotor model are written as
x 2(ne ni ) x Z n2 x 2ni : y
e0 : 2 cos : t ,
(8.99)
y 2(ne ni ) y Z n2 y 2ni : x
e0 : 2 sin : t.
With the complex variable z x iy, joining Eqs. (8.99) yields
z 2(ne ni ) z (Z n2 i2ni : ) z
e0 : 2 ei: t , (8.100)
whose nonhomogeneous solution, which describes the steady rotation of the rotor,
may be written as
z Aei: t ,
where A is the complex amplitude of the steady rotation of the rotor.
Actually, the steady rotation is just the synchronous whirl of the flexible rotor
excited by its unbalance. Substituting the above expression in Eq. (8.100), we
obtain
e0 : 2 (Z n2 : 2 2: ne i)
A .
(Z n2 : 2 ) 2 4ne2 : 2
After some elementary operation, we find the analytical expressions of the
amplitude and the phase of the synchronous whirl excited by unbalance, i.e.,
1
e0 : ª§ : 2 · 4ne2 : 2 º
2 2 2
|A| « ¨ 1 ¸ » ,
Z n2 «© Z n2 ¹ Z n4 »
¬ ¼ (8.101)
§ 2n : ·
T arc tan ¨ 2 e 2 ¸ .
© Zn : ¹
Furthermore, the components of the displacement of the geometric center of the
rotor are also found, i.e.,
x | A | cos(: t T ), y | A | sin(: t T ).
235
Chapter 8 Rotor Whirl
If the mass unbalance of the Jeffcott rotor shown in Fig. 8.24 vanishes, we have
e0 0 , and thus, the differential equations of motion of the rotor model (8.100)
are autonomous, namely,
z Be E iZ , (8.103)
where B is the complex amplitude determined by the initial condition of the rotor.
Substituting (8.103) in Eq. (8.102) and separating the real part and the imaginary
part, we obtain
E 2 : 2 2(ne ni ) E Z n2 0 (8.104)
and
( E ne ni ): niZ 0. (8.105)
Let us assign values to the rotor parameters ne , ni , and Z n , and denote the
rotation speed : as a variable parameter. Next, let us solve the simultaneous
Eqs. (8.104) and (8.105), and finally find out the functions E (: ) and Z (: ) . It is
known that if E (: ) 0 , the steady rotation of the rotor is asymptotically stable.
In contrast, if E (: ) ! 0 , the steady rotation of the rotor is unstable. Thus, the
236
8.6 Cause and Prevention of Rotor Whirl
E (: ) 0. (8.106)
Here, solution (8.103) is a periodic function of time t. The rotation speed is just
the critical speed :cr , at which the Hopf bifurcation of the previous nonlinear
equations of motion of the flexible rotor emerges and the rotor whirl is impending.
Thus, substituting the critical parameter condition (8.106) in Eq. (8.104) yields
the expression of the frequency of the rotor whirl, i.e.,
Z Z n . (8.107)
Then, substituting (8.106) in Eq. (8.105) yields the expression of the threshold
speed of the rotor whirl, i.e.,
§ ne ·
:cr ¨1 ¸ Z n . (8.108)
© ni ¹
Now, let us summarize the results of the analysis.
1. The whirl of the flexible rotor supported on the dry bearings can be excited
by the internal damping force in the rotating shaft, which is generated due to
nonelastic deformation. Obviously, it is an internal force of the rotor system and
the whirl caused by the internal damping force belongs to self-excited vibration.
2. The frequency of this whirl is still equal to the natural frequency of the flexible
rotor, but the critical rotation speed of the flexible rotor is more than its natural
frequency as the whirl impends. Therefore, this whirl is a supersynchronous whirl.
3. If the external damping parameter ne increases or (and) the internal damping
parameter ni decreases, the critical rotation speed of the flexible rotor will raise.
Therefore, the supersynchronous whirl will be delayed remarkably.
At the end of this chapter, we pay attention to the flexible rotors supported on
the hydrodynamic bearings, which always suffer two kinds of circulatory forces:
one is generated by the oil-film force and the other is generated by the internal
damping force of the deformed rotating shaft. Thus, the super-synchronous whirl
occurring in such rotors is more easily excited. In other words, their critical
rotation speed should be lower than that of the rotors suffering only one kind of
circulatory force.
In addition, other forces induced by the fluid flow in the radial gaps of rotors,
such as the clearance excitation force, often cause the rotors to whirl[25]. Due to the
complication of the motive force, such problems have not been perfectly solved yet.
shaft. The main conclusion may be obtained by the quadratic form criterion of
equilibrium stability for the holonomic system. In addition, we introduce some
effective techniques to prevent rotors from whirling. The contents of this section
are expanded in three steps: (1) demonstrate a common mathematical structure of
the differential equations of the motion governing rotor whirls caused by the
fluid-film force and the internal damping force of the rotating shaft; (2) point out
the common motive force included in the equations of the motion governing these
whirls; and (3) introduce some effective techniques to prevent rotors from whirling.
Let us separate the complex Eq. (8.102), which governs the supersynchronous
whirl caused by the internal damping force of the rotating shaft, into two real
equations of motion. Eventually, two real equations are described in the vector
form, i.e.,
ª1 0 º
x ½ ª ne ni 0 º x ½ ªZ n2 0 º x ½ ª 0 2ni : º x ½ 0½
® ¾ « » ® ¾ « 2»® ¾
« ® ¾ ® ¾.
«0 1 »
¬ ¼ ¯ y¿ ¬ 0 ne ni ¼ ¯ y ¿ ¬ 0 Z n ¼ ¯ y ¿ ¬ 2ni : 0 »¼ ¯ y ¿ ¯0¿
Recall the differential equation system (8.44), which governs the oil whirl of the
rigid rotor. Denote m f 0 in the equation system to neglect the influence of the
mass of the fluid film on the motion of the rotor, and supply the external damping
force cx and cy . Then, we obtain the equations of motion governing the oil whirl
of the rigid rotor, i.e.,
mx (c d 0 ) x k0 x W d 0 : y 0,
(8.109)
my (c d 0 ) y k0 y W d 0 : x 0.
c d0 k0
ne , no , and Z n2 ,
m m m
the equation system (8.109) may be described as
ª1 0 º
x ½ ª ne n0 0 º x ½ ªZ n2 0 º x ½ ª 0 2n0 : º x ½ 0 ½
«0 1 » ® ¾« » ® ¾« 2»® ¾
« ® ¾ ® ¾.
¬ ¼ ¯ y¿ ¬ 0 ne n0 ¼ ¯ y ¿ ¬ 0 Z n ¼ ¯ y ¿ ¬ 2n0 : 0 »¼ ¯ y ¿ ¯0 ¿
(8.110)
Obviously, if the internal damping parameter of rotating shaft ni is replaced
by the damping parameter of oil film n0 , the equation of motion (8.102) will be
238
8.6 Cause and Prevention of Rotor Whirl
transformed into the equation of motion (8.110). This means that the mathematical
structure of the equations of motion, which respectively governs the supersyn-
chronous whirl caused by internal damping of the rotating shaft and the oil whirl
caused by the fluid film force, are all the same.
Let us look through Eqs. (8.102) and (8.110). In both, the first term represents the
inertial force of the rotor, the second term represents the damping force, the third
term represents the potential force, and the last term represents the circulatory force.
Thus, the rotor systems described by equations of motion (8.102) or (8.110) are
regarded as the linear holonomic system without gyroscopic force. The quadratic
form criterion of the equilibrium stability for the special system is reduced to
4ME 2 D 2 K , (3.69)
where M, K, D, and E are the quadratic forms calculated by the inertial, the
potential, the damping, and the circulatory matrices in Eqs. (8.102) and (8.110),
whose magnitudes respectively represent the intensity of the inertial, the potential,
the damping, and the circulatory forces.
In general, the quadratic forms M and K are positive definite functions.
Consequently, inequality (3.69) is violated only due to the large magnitude of the
quadratic form E, and we conclude that the circulatory force is a common cause
of rotor whirl.
The analysis shows that all circulatory forces are caused by the shear stress in
the rotating continuum, such as the fluid film in the hydrodynamic bearings and
the rotating shaft made of the viscoelastic materials. The shear stress is always
proportional to the viscosity and the strain rate of the continuum. In fact, the
physical background has been shown in the analytical expressions of the
circulatory force in Eqs. (8.44) and (8.98) respectively, namely, W d 0 : and d : ,
in which both d0 and d are the viscous-damping parameters of the oil film and the
rotation shaft espectively.
The rotor whirl may bring about severe plant troubles and to prevent it is always
an important task in the design of any high-speed rotor. It is obvious that the
increase of the threshold speed of rotor instability is the most effective way to
prevent the rotor from whirling.
From the previous analyses, we have obtained the approximate expressions of
the threshold speed of the oil whirl and the threshold speed of the sypersynchronous
239
Chapter 8 Rotor Whirl
whirl caused by internal damping of the rotating shaft respectively, i.e., Eq. (8.48)
and Eq. (8.108). According to these expressions, increasing the stiffness coefficients
of the fluid film and the rotating shaft, or (and) decreasing the rotor mass, or (and)
raising the external damping coefficient are all helpful to lift up the threshold
speed of rotor whirl and benefit suppressing the rotor whirl. A squeeze-film
damper installation supporting a fluid-film bearing is shown in Fig. 8.25. It is an
effective way to increase the external damping and has been successfully employed
in many types of machinery.
In addition, the equations show that the threshold speed of oil whirl is inversely
proportional to the circumferential flow average velocity ratio W. Thus, decreasing
its magnitude is also an effective way to lift up the threshold speed of oil whirl of
the rotor. According to this principle, an anti-swirl technique to inject the lubricant
into the hydrodynamic bearings has been proposed and used to guide the design
of the high-performance bearings.
Thus, active control technique is considered the most effective method to prevent
the rotor from whirling. This will be introduced in the last chapter of this book.
References
[1] H H Jeffcott. The lateral Vibration of Loaded Shaft in the Neighborhood of a Whirling
Speed: The Effect of Want of Balance. Philosophical Magazine, 1919, Ser. 6, 37, 304
[2] J M Vance. Rotordynamics of Turbomachinery. New York: John Wiley & Sons, 1958
240
References
241
Chapter 8 Rotor Whirl
[23] A L Kimball. Jr., Internal Friction Theory of Shaft Whirling. Phys. Rev., 1923, [2] 21, 703
[24] S H Crandall. Physical Explanations of the Destabilizing Effect of Damping in Rotating
Parts. NASA Conference Pub. 2133. Rotordynamic Instability Problems in High-Performance
Turbomachinery. Proceedings of a Workshop Held at Texas A&M University, Texas,
May 12 14, 1980, pp. 369 382
[25] B Leie, H J Thomas. Self-Excited Rotor Whirl due to Tip-Seal Leakage Factors Bid. pp.
303 316
242
Chapter 9 Self-Excited Vibrations from Interaction
of Structures and Fluid
with the flow. The periodicity of the wake of a cylinder was associated with the
vortex formation by H. Benard in 1908 and with the formation of a stable street
of staggered vortices by von Karman in 1912. In engineering, the large amplitude
vibrations induced in flexible structures by the vortex shedding are of great
practical significance because of their destructive effect on bridges, antenna,
cables, and heat exchangers of power plants and nuclear reactors.
As a fluid flows toward the leading edge of a bluff cylinder, the pressure in the
fluid rises from the free stream pressure to the stagnation pressure. The high fluid
pressure near the leading edge impels the developing boundary layers about both
sides of the cylinder. However, the pressure forces are not sufficient to force the
boundary layers around the back side of the bluff cylinder at high Reynolds
numbers. Near the widest section of the cylinders, the boundary layers separate
from each side of the cylinder surface and from two free shear layers that tail aft
in the flow. These two free shear layers bound the wake. Since the innermost
portion of the layers are in contact with the free stream, the free shear layers tend
to roll up into discrete, swirling vortices. A regular pattern of the vortices is
formed in the wake that interacts with the cylinder motion and is the source of
the effects called vortex-induced vibration.
Any structure with a sufficient bluff tailing edge sheds the vortices in a
subsonic flow. The vortex streets tend to be very similar regardless of the tripping
structure. As an example, the major regimes of the vortex shedding from a circular
cylinder adapted from J. H. Lienhard are shown in Fig. 9.1[1]. At very low Reynolds
numbers based on cylinder diameter, the flow does not separate. As Re increases,
a pair of fixed vortices is formed immediately behind the cylinder. As Re further
increases, the vortices elongate until one of them breaks away and a periodic wake
and a staggered vortex street is formed. Up to an Re of approximately 150, the
vortex street is laminar. At an Re of 300, the street is turbulent and it degenerates
into a fully turbulent flow beyond approximately 50 diameters down the stream
of the cylinder. The Re range from 300 to approximately 3 u 105 has been called
subcritical because it occurs prior to the onset of the turbulent boundary layer at
an Re of approximately 3 u 105 , depending on the free stream turbulence and the
surface roughness. In the subcritical Re range, the shedding occurs at a well-defined
frequency. At the transition Re, the flow separation point moves backward, the
vortex shedding is disorganized, and the cylinder drag drops sharply. At the higher
supercritical Re range, the vortex street reestablishes itself.
A periodic force on the bluff cylinder is generated as the vortices are alternately
shed from each side of its surface. The oscillating pressure fields and the net force
on a circular cylinder are shown in Fig. 9.2 for a portion of the shedding cycle[2].
According to Hopf bifurcation theory of continuous systems, C. P. Jackson
calculated the critical Reynolds number, Re*, of the onset of the vortex shedding
244
9.1 Vortex Resonance in Flexible Structures
Figure 9.2 A Sequence of Simultaneous Surface Pressure Fields and Wake Forms
at Re 112,000[2]
in steady flow past circular cylinder and other section cylinders by the finite
element method[3]. With Reynolds number varying, the transition to the periodic
behavior corresponds to a Hopf bifurcation point where a pair of complex
eigenvalues of Jacobian matrix of the flow equations, i.e., Navier-Stokes equations,
crosses the imaginary axis. Thus, the Karman vortex street is a self-excited periodic
motion in autonomous continuous systems. Corresponding to the critical Reynolds,
Re*, the pure imaginary eigenvalue iZ is the nondimensional frequency of the
245
Chapter 9 Self-Excited Vibrations from Interaction of Structures and Fluid
In 1879, V. Strouhal found that the Aeolian tones, which are the resonance of the
wire excited by the vortex shedding in the wind, were proportional to the wind
speed divided by the wire diameter. Therefore, the Aeolian tones are equal to the
predominate frequency fs of the vortex shedding. The phenomenon may be
described by the formula
S tU
fs , (9.1)
D
where U is the stream velocity approaching the cylinder, D is the cylinder width,
and St is the proportional constant named as Strouhal number in honor of his
contribution of the discovery.
As mentioned before, by using the finite element method, C. P. Jackson has
demonstrated that the purely imaginary eigenvalues riZ at the bifurcation point
are just the angular frequency of the vortex shedding, i.e., Z s 2Sf s . Thus, the
vortex shedding phenomena have been interpreted perfectly.
A large quantity of experimental data show that the Strouhal number is a
geometrical function of the cylinder and the Reynolds number for low Mach number
flows[6]. At the transition Reynolds numbers, the shedding frequency is defined
in terms of the predominate frequency of a broad band of shedding frequencies.
The relationship between the Strouhal number and the Reynolds number for the
circular cylinder is shown in Fig. 9.3. According to this figure, the Strouhal number
of the circular cylinder in the Reynolds number range from 2 u 102 to 2 u 105 is
Figure 9.3 Strouhal Number – Reynolds Number Relationship for Circular Cylinders[6]
246
9.1 Vortex Resonance in Flexible Structures
equal to 0.2. For noncircular cylinders, the Strouhal number has been measured in
terms of wind tunnel experiments. Some of them are listed in Table 1 as
reference data for engineering designers.
247
Chapter 9 Self-Excited Vibrations from Interaction of Structures and Fluid
Vortex shedding also occurs in the group of a few cylinders and in larger arrays
of cylinders whose Strouhal numbers characterize the peak in the turbulence
spectrum within the array. In very closely spaced tube arrays with tube center to
tube center space less than about 1.5 diameters, the distinct frequency associated
with the shedding degenerates into a broad-band turbulence. The Strouhal numbers
for the flow within arrays of cylinders of heat exchangers have been compiled to
make Fig. 9.4 and Fig. 9.5.
248
9.1 Vortex Resonance in Flexible Structures
Cylinder vibration perpendicular to the fluid flow, at or near the shedding frequency,
can increase the vortex strength and force the frequency of the vortex shedding to
change from the stationary cylinder shedding frequency to the cylinder vibration
frequency. This phenomenon is called the frequency lock-in effect of the vortex
shedding. The lock-in effect can also be produced if the vibration frequency of
the cylinder is equal to a multiple or a submultiple of the shedding frequency.
This lock-in or synchronization effect was first documented by R. E. Bishop and
A. Y. Hassan[7]. The lock-in band is the range of flow velocities over which the
vortex shedding frequency synchronizes with the frequency of the vibrating cylinder.
If the periodic force on the cylinder is a forward effect, the lock-in effect of the
frequency of the vortex shedding generated by the cylinder is a feedback effect
and the dynamic system combined with the fluid flow and the cylinder is a
closed-loop system.
According to the previous discussion, the vortex-induced vibration of a flexible
structure is just like an internal resonance of two modes of a dynamic system.
Based on the experiments of a flow over a cylinder, R. E. Bishop and A. P. Hassan
suggested that the fluid dynamic behavior of the self-induced vortex shedding
might be modeled with a simple nonlinear oscillator[7]. This idea was first applied
by R. T. Hartlen and I. G. Currie to develop a wake-oscillator model in which a
generalized Liénard’s equation is used to simulate the fluctuating pressure on the
bluff body[8]. R. A. Skop and O. M. Griffin modified the model by adding a cubic
nonlinearity and presented a method for parameter estimation, which yields very
good agreement with experimental data[9]. The equations of the model are
described below[10, 11].
x E x x
aU 2 cL ,
J (9.2)
cL G UcL cL3 U 2 cL bx ,
U
where x is the dimensionless displacement of cylinder, cL is the instantaneous lift
coefficient, which may be considered as a generalized coordinate to describe the
alternating wake, U is the dimensionless wind speed, E is the damping factor, a
and b are the interaction parameters, and G and J are the fitted wake oscillator
parameters.
Let us denote the generalized vector as
x [cL cL xc xc ]T .
The linearized state equation of system (9.2) may be written in the vector form
x A U x (9.3)
with
249
Chapter 9 Self-Excited Vibrations from Interaction of Structures and Fluid
ª 0 1 0 0 º
« U 2 GU 0 b »
A U « ». (9.4)
« 0 0 0 1 »
«¬ aU 2 0 1 E »¼
Our following investigation is based on the variable wind speed with other
parameters held fixed. The stability of the steady state can be determined by finding
the wind speed U0 at which state matrix A(U) has a pair of purely imaginary
eigenvalues riZ0 . For U close to U0, the eigenvalues of A(U) can be written
as D (U ) r iZ (U ) , O3(U), and O4(U), where D (U0) 0 and Z (U0) Z0, which is
assumed to be positive. The stability of the steady state can be determined from
the sign of the derivative of D c(U ) with respective to U, namely, D c(U 0 ) , and the
sign of the real part of O3(U0) and O4(U0). The stability of the bifurcating periodic
solution can be determined from the sign of D c(U 0 ) and the real parts of the
eigenvalues of A(U0) other than riZ0 . This D c(U 0 ) may be found from an algebraic
function established with the eigenvectors of the fourth-order system (9.3), as
introduced in the reference[11]. If D c(U 0 ) 0 and the eigenvalues other than riZ 0
have negative real parts, the periodic solution is stable near the bifurcation point.
On the other hand, the periodic solution is unstable near the bifurcation point
if D c(U 0 ) 0 or if there is an eigenvalue of A(U0) with a positive real part.
The classification of various bifurcation diagrams is based on the numerical
values a 0.002, b 0.4, J 2/3, and all values of G > 0 and E > 0. This numerical
values of a, b, and J are the same ones used by R. T. Hartlen and I. G. Currie and
correspond to the system parameters used to fit the existing experiments[8]. The
parameter plane G -E is divided into eight regions: Ē1, Ē2, ē1, ē2, ē3, ē4, Ĕ, and
ĕ, as shown in Fig. 9.6. The bifurcation diagrams corresponding to various
250
9.1 Vortex Resonance in Flexible Structures
parameter couples (G i, E i) picked out from these eight regions are respectively
plotted in Fig. 9.7, which shows the stability of various periodic solutions for the
positive wind speed.
The curve of the relationship between the amplitude of the periodic solution
and the wind speed is referred to as response diagram, as shown in Fig. 9.8. The
experimentally observed double amplitude response corresponds to the existence
251
Chapter 9 Self-Excited Vibrations from Interaction of Structures and Fluid
of two stable oscillations. The existence of two stable oscillations occurs for certain
wind speed, as shown in Fig. 9.8(a), (b), (c), (f), and (g), which basically agrees
with experimental data obtained by D. M. Griffin and R. A. Skop[12]. Consequently,
the hysteresis phenomenon, which occurs whenever G and E are in regions ē1
252
9.1 Vortex Resonance in Flexible Structures
and ē4, can be illustrated by Fig. 9.8(c) and (f) due to the existence of certain
range of the wind speed. Two stable periodic solutions of equation system (9.2)
exist in this range.
In order to predict the amplitude magnitude of the flexible structures in the vortex
resonance condition, R. H. Scalan and I. Goswami proposed a simple model, which
is a nonlinear oscillator undergoing harmonic excitation[13]. The differential equation
of motion is given as
UU 2 D ª x x 2 x x
x 2[Z n x Z n2 x
Y
« 1 ( : ) Y2 ( : ) J1 ( : )
m ¬ U D2 U D
x º
J 2 (: ) cos 2Z s t » , (9.5)
D ¼
where x is the cylinder displacement, [ the damping ratio of structure, Z n the
natural frequency of the structure, D the cylinder width, m the mass of the structure,
Uthe density of the fluid, Z s the dominant frequency of vortex shedding, U the
flow velocity, and the reduced frequency, i.e.,
Zn D
: . (9.6)
U
The right side of Eq. (9.5) consists of a representation of various aeroelastic
effects witnessed in the dynamics of a sprung cylinder. The Y1 and Y2 terms are linear
and nonlinear aeroelastic damping terms, which lead to the commonly observed
self-excited and self-limited character of the model. The J1 term is an aeroelastic
stiffness term that provides for a shift in the mechanical response frequency from
zero-wind velocity frequency Zn. The J2 term is a parameter stiffness coupling
between the wake and the cylinder, which excites the body at twice the frequency
of the vortex shedding. It is considered as the representative of the principal
contribution of the wake oscillator.
To inspect the reliability of the simple model (9.5) of the vortex-induced
vibration, an experimental setup was designed and located in a low turbulent tunnel.
The bluff body specimen was a rigid circular cylinder. The additional mechanical
damping in the setup was provided electomagnetically[14]. The experimental data
consists of time histories of the cylinder vibration and the velocity fluctuations in
the upper shear layer. The general character of the cylinder response, as monitored
by the spectrum analyzer, shows that the spectrum of the transverse oscillations
of the cylinder contains two distinct frequencies: the frequency of the vortex
shedding fs and the natural frequency of the structure fn.
253
Chapter 9 Self-Excited Vibrations from Interaction of Structures and Fluid
Using the numerical algorithm for parameter estimation, parameters Y1, Y2, J1,
and J2 are obtained through the range of the reduced frequency and damping.
The parameter curves of Y1 (: ), Y2 (: ), J1 (: ), and J 2 (: ) for several levels of
mechanical damping are plotted. According to these experimental data and the
model Eq. (9.5), the predicted amplitude may be found and shown as in Fig. 9.9,
in which we use a dimensionless parameter V U / f n D as the abscissa. It is called
reduced velocity. Obviously, the prediction deviation from the experimental data is
small. Therefore, the mathematical model proposed by R. H. Scalan and J. Goswami
has been verified and can be used in engineering design.
Figure 9.9 Dimensionless Amplitude versus Reduced Velocity for several >@
From the above investigation, we can see that substantial reduction of the amplitude
of the vortex resonance can be achieved by modifying the structure as follows.
1. Increase damping. If the mass or the internal damping of the structure can be
increased, resonance vibration will diminish. Damping increment can be achieved
by (1) permitting scraping between the structural elements; (2) using composite
materials, such as concrete, instead of welded steel; (3) incorporating materials
with high internal damping, such as wood, rubber, and sand; or (4) using external
254
9.2 Flutter in Cantilevered Pipe Conveying Fluid
dampers or bumpers.
2. Avoid resonance. If the reduced velocity (U / f n D ) is kept below 1.0,
resonances with the vortex shedding through the third harmonic are avoided. This
is ordinarily accomplished by stiffening the structure with guide wire or braces to
increase the natural frequency.
3. Change cross section. If the structure can be streamlined for a marine pier
design, the flow will not be separated from the section and the vortex shedding
will be eliminated. Effective streamlining requires a taper ratio of at least 6 to 1
or an included angle not bigger than 8e 10e .
The linear mathematical model to study the flutter problem of cantilevered pipes
conveying fluid is comprised of its linear differential equation and the corresponding
boundary equations.
255
Chapter 9 Self-Excited Vibrations from Interaction of Structures and Fluid
Figure 9.10 shows a vertical cantilevered pipe that has transverse deflection
w( z , t ) from its equilibrium position. A fluid with density U flows at pressure p
and constant velocity U through the internal pipe cross section of area A. The
length of the pipe is l, the modulus of elasticity of the pipe is E, and its area
moment of inertia is I. Consider the small elements cut from the pipe in Fig. 9.11.
The fluid element in 9.11(a) has been extracted from the pipe element in Fig. 9.11(b)
for clarity. As the fluid flows through the deflecting pipe, it is accelerated because
of the changing curvature of the pipe and the lateral vibration of the pipeline.
This acceleration is opposed by the vertical component of fluid pressure applied
to the fluid element and the normal force F per unit length applied on the fluid
element by the pipe walls. A balance of forces on the fluid element, in the X
direction for small deformation, gives
2
w2w §w w·
F Md M d ¨ U ¸ w, (9.7)
wz 2 © wt wz¹
where M d U A is the conveying fluid of mass per unit length.
Figure 9.11 Forces and Moment exerted on the Elements of the Fluid and the Pipe
256
9.2 Flutter in Cantilevered Pipe Conveying Fluid
The pressure gradient in the fluid along the length of the pipe is opposed by the
shear stress of fluid friction against the pipe walls. For a constant flow velocity,
summing the forces parallel to the pipe axis in Fig. 9.11(b) gives
wp
A qS 0, (9.8)
wz
where S is the inner perimeter of the pipe, and q is the shear stress on the internal
surface of the pipe.
The equations of motion of the pipe element are derived from Fig. 9.11(b).
Summing the forces parallel to the pipe axis gives
wT w2w
qS Q 2 0, (9.9)
wz wz
where T is the longitudinal tension in the pipe, and Q is the transverse shear force
in the pipe.
The forces on the element of the pipe normal to the pipe axis accelerate the
pipe element in the X direction. For small deformation, we have
wQ w2 w w2 w
T 2 F m , (9.10)
wz wz wt 2
where m is the mass per unit length of the empty pipe.
The transverse shear force Q in the pipe is related to the bending moment M in
the pipe and the pipe deformation by
wM w3 w
Q EI . (9.11)
wz wz 3
Equations (9.7), (9.10), and (9.11) may be combined to eliminate the variables
F and Q, which yields
2
w4 w w2 w §w w· w2w
EI ( pA T ) M d ¨ U ¸ w m 0. (9.12)
wz 4 wz 2 © wt wz ¹ wt 2
w3 w
Since Q is proportional to , the third term on the left side of Eq. (9.9) is of
wz 3
the order w2 and can be neglected for small deformation analysis. Here, the shear
stress q may be eliminated from Eqs. (9.8) and (9.9), which gives
w ( pA T )
0. (9.13)
wz
This equation implies that ( pA T) is independent of the position along the span
of the pipe. At the pipe end where z l, the tension in the pipe is zero and the
257
Chapter 9 Self-Excited Vibrations from Interaction of Structures and Fluid
w4 w w2 w w2w w2w
EI M dU 2 2 2M dU ( M d m) 2 0. (9.15)
wz 4
wz wzwt wt
The boundary conditions for a cantilevered pipe clamped at z 0 and free at
z l are
ww
w(0, t ) (0, t ) 0,
wz
(9.16)
w3 w w2w
(l , t ) (l , t ) 0.
wz 3 wz 2
Let us introduce the following nondimensional variables and parameters
1
w z § EI · 2 t
K , [ , W ¨ ¸ 2,
l l © Md m ¹ l (9.17)
1
§ Md · 2 Md
v ¨ EI ¸ Ul , E .
© ¹ Md m
The Eqs. (9.15) and (9.16) may be written in the dimensionless form
w 4K 2 w K
2 1
w 2K w 2K
v 2 E 2
v 0 (9.18)
w[ 4
w[ 2
w[wW wW 2
and
wK
K (0,W ) (0,W ) 0,
w[
(9.19)
w 2K w 3K
(1,W ) (1,W ) 0.
w[ 2 w[ 3
Obviously, the flutter of the cantilevered pipes conveying fluid is a two-point
boundary value problem of the linear partial differential equations.
The critical parameter condition makes Eqs. (9.18) and (9.19) possess the periodic
solution, but it is not deduced from Hurwitz criterion, and the procedure is different
258
9.2 Flutter in Cantilevered Pipe Conveying Fluid
from the analysis about the threshold speed of the rotor whirl. Consequently, it is
necessary to explain it in detail.
Here, we introduce the undefined deflection W([ ) and the undefined frequency
Z. The periodic solution of Eqs. (9.18) and (9.19) is written as
K ([ ,W ) Re[W ([ )eiZW ] . (9.20)
Substituting this equation in Eqs. (9.18) and (9.19), we obtain the ordinary
differential equation describing function W([ ), i.e.,
1
d 4W d 2W dW
v2 i2 E 2 vZ Z 2W 0 (9.21)
d[ 4
d[ 2
d[
with the boundary condition
W (0) 0, W c(0) 0,
(9.22)
W cc(1) 0, W ccc(1) 0.
In general, the solution of the linear homogeneous Eq. (9.21) is
W ([ ) BeiD[ . (9.23)
Substituting the expression in Eq. (9.21), we obtain the characteristic equation
1
D 4 v 2D 2 2 E 2 vZD Z 2 0. (9.24)
Denoting D D Dand D as four different eigenvalues, the previous equation
may be written as
(D D1 )(D D 2 )(D D 3 )(D D 4 ) 0. (9.25)
Comparing each coefficient of all terms with the same degree of D in Eqs. (9.24)
and (9.25), we obtain four algebraic equations
D1 D 2 D 3 D 4 0, (a)
D1D 2 D1D 3 D1D 4 D 2D 3 D 2D 4 D 3D 4 v ,2
(b)
1
D1D 2D 3 D1D 2D 4 D1D 3D 4 D 2D 3D 4 2 E vZ ,
2
(c)
D1D 2D 3D 4 Z 2 . (d)
The Eq. (a) shows that four eigenvalues D1, D2, D3, and D4 are not independent
and each one depends on the other three. Therefore, denoting a, b, and c as three
independent variables, the relationship among them is established, i.e.,
D1 a b c , D 2 a b c ,
(9.26)
D 3 a b c, D 4 a b c.
259
Chapter 9 Self-Excited Vibrations from Interaction of Structures and Fluid
2(a 2 b 2 c 2 ) v 2 ,
1
4abc E 2 vZ , (9.27)
a b c 2a 2 b 2 2b 2 c 2 2c 2 a 2
4 4 4
Z .2
According to the number of the sign variations of all coefficients in Eq. (9.24),
we can conclude that this equation has one positive real root and three negative real
roots. Thus, the general solution of the ordinary differential Eq. (9.21) is written as
ª 4 iD [ º
W ([ ) Re « ¦ C j e j » , (9.28)
¬j 1 ¼
where C1 and C2 are conjugate, and C3 and C4 are conjugate, namely,
B1 B3 0,
(D1 D 2 ) B1 (D 3 D 4 ) B3 0,
(D cos D1 D cos D 2 ) B1 (D12 sin D1 D 22 sin D 2 ) B2
2
1
2
2
(9.30)
(D 32 cos D 3 D 42 cos D 4 ) B3 (D 32 sin D 3 D 42 sin D 4 ) B4 0,
(D13 cos D1 D 23 cos D 2 ) B1 (D13 sin D1 D 23 sin D 2 ) B2
(D 33 cos D 3 D 43 cos D 4 ) B3 (D 33 sin D 3 D 43 sin D 4 ) B4 0.
If the above equation system has a nontrivial solution, its coefficient determinant
must vanish, i.e.,
1 0 1 0
D1 D 2 0 D3 D 4 0
det 2 0.
D1 cosD1 D 22 cosD 2 D12 sin D1 D 22 sin D 2 D 32 cosD 3 D 42 cosD 4 D 32 sin D 3 D 42 sin D 4
D13 cosD1 D 23 cosD 2 D13 sin D1 D 23 sin D 2 D 33 cosD 3 D 43 cosD 4 D 33 sin D 3 D 43 sin D 4
(9.31)
Substituting (9.26) in this equation, we obtain an algebraic equation, i.e.,
D(a, b, c) 0.
In general, as a parameter condition, the differential Eq. (9.18) has no periodic
solution. In this case, a, b, cand Dj ( j 1, 2, 3, 4) are complex and the above
260
9.2 Flutter in Cantilevered Pipe Conveying Fluid
equation is a complex equation. Separating its real and imaginary parts, we have
Figure 9.12 Critical Flow Velocity and Flutter Frequency for Onset of Instability
in a Cantilever Pipe as a Function of Mass Ratio[17]
The dynamics of cantilevered pipes conveying fluid have been studied quite
extensively by many investigators. Most of the early theoretical work on this
problem has been carried out within the framework of linear models, as introduced
above. With the developments of the theory on nonlinear dynamic systems, much
attention has been paid to the study of nonlinear models.
261
Chapter 9 Self-Excited Vibrations from Interaction of Structures and Fluid
Figure 9.13 Comparison of Theory and Experimental Data for Onset of Instability
of Fluid-Conveying Pipe[17]
First, let us consider the resistance to the fluid motion. It is proportional to the
square of the fluid velocity U and to the length of the pipe l with a constant of
proportion D. The fluid velocity U is a new variable that can satisfy a nonlinear
integro-differential equation deduced from fluid dynamics. Consider the second-
order terms of the elastic deformation of the pipe. The equation of motion of the
pipe conveying fluid becomes a nonlinear differential equation. J. Rousselet and
G. Herrmann assumed that the pipe center line is inextensible[18]. They constructed
the equation of motion of the cantilevered pipe conveying fluid. Here, we omit
the detailed procedure deriving these equations and merely outline the derivation
and define the system parameters.
Let us denote s as the arc length along the deformed pipe, and u and w as the
displacements of the center line in the X and the Z directions. The signs of other
variables and parameters are the same as that of Eq. (9.17). Applying the theory
of elastic beam with large deformation and the theory of visco-fluid dynamics,
we construct the equation of motion of the flexible pipe conveying fluid and the
equation of fluid in the pipe, i.e.,
w 2K w 2K 2 w K
2
w 4K w 2K ª dv 2º w 2K
2v E v « E (1 [ ) 2 v E v » 2 E v
wW 2 w[wW w[ 2 w[ 4 w[ 2 ¬ dW ¼ w[wW
(9.33)
and
1§ wK w K w] w 2] ·
2
dv
E 2D v E D v 2 E 2 ³ ¨ ¸ d[ 0. (9.34)
dW 0 w[ wW 2
© w[ wW 2 ¹
262
9.2 Flutter in Cantilevered Pipe Conveying Fluid
w 2K w 2K 2 w K
2
w 4K
2v E v 0 (9.37)
wW 2 w[wW w[ 2 w[ 4
and
dv
2vJ v 0
dW
with
D
J .
E
Obviously, Eq. (9.37) is the same as Eq. (9.18). In addition, the boundary
condition for Eq. (9.37) is Eqs. (9.19). Let
wK wK
q1 2v E , q2 K , (9.38)
wW w[
263
Chapter 9 Self-Excited Vibrations from Interaction of Structures and Fluid
wq
Lq (9.39)
wW
with
ª q1 º ª0 v 2 (<)cc (<)cccc º
q « », L « », (9.40)
¬ q2 ¼ «¬1 2v E (<)c »¼
and the boundary condition (9.19) can be written as
OW ([ ) LW ([ ) (9.42)
with boundary condition
1 cosh Z cos Z 0.
When v > 0 but small, all the eigenvalues are in the left-half plane. Instability can
occur if for a given E, v (i.e., the flow rate) is increased to a critical value at
which an eigenvalue or several eigenvalues cross into the right-half plane. We
merely postulate that the instability occurs if for some value of v vcr > 0, there
is a pair of complex conjugate eigenvalues that cross the imaginary axis from
the left-half to the right-half complex plane. Furthermore, it is assumed that, for
0 < v < vcr, all other eigenvalues remain in the left-half plane.
More precisely, there exists a pair of complex conjugate eigenvalues O and O * ,
O O (v, E ) , such that
O (vcr , E ) iZ 0 , Z0 z 0,
d Re(O ) (9.43)
z 0.
dv v vcr
264
9.2 Flutter in Cantilevered Pipe Conveying Fluid
In Fig. 9.14, numerical solutions for the behavior of the three lowest eigenvalues
as a function of v are shown for the case when E 2 0.6. This figure shows that
the loci of the eigenvalues of second mode cross the imaginary axis at point P0.
According to the condition (9.43), the flow velocity at point P0, the flutter velocity
of the pipe conveying fluid, is the critical flow velocity.
The critical flow velocity vcr, at which Hopf bifurcation occurs, depends on
parameter E in a complicated manner. A graph of vcr vs. E 2 has been obtained by
the numerical calculation and is plotted in Fig. 9.12(a). According to the sign of
dRe(O)/dv at point v vcr, we can distinguish the subcritical Hopf bifurcation
from the supercritical Hopf bifurcation. The computing results are shown in a graph
of Dcr vs. E 2 , which is plotted in Fig. 9.15. For D > 0.71, the Hopf bifurcation
is supercritical, while for D < 0.71, the Hopf bifurcation can be subcritical or
supercritical depending on the value of parameter E 2 . It is known that only the
supercritical Hopf bifurcation generates self-excited vibration. Thus, if the parameter
couple ( E 2 ,D ) is located in the supercritical region in Fig. 9.15, the flutter of the
cantilevered pipe conveying fluid occurs.
First, let us discuss the physical mechanism generating the flutter in the cantilevered
pipe conveying fluid based on the basic principle of classical mechanics. As
mentioned before, self-excited vibration cannot occur in a conservative system. It
265
Chapter 9 Self-Excited Vibrations from Interaction of Structures and Fluid
Figure 9.15 Supercritical and Subcritical Hopf Bifurcation Regions in Plane ( E 2 ,D ) [18]
occurs only in a nonconservative system. In fact, the pipe conveying fluid has mass
flow through its inlet and outlet. It is an open system, not a conservative system
unless the flow velocity of the fluid in the pipe vanishes. It is known that the
restoring force of the elastic pipe is a conservative force. In this case, the Coriolis
force and the centrifugal force of the fluid flow belong to nonconservative force,
and they can excite the flutter in the cantilevered pipe conveying fluid.
Second, let us check the energy feeding into the cantilevered pipe conveying
fluid. Review the forces in Eq. (9.18). It is obvious that all of them are internal
forces exerted in the dynamic system. Therefore, they must cause their reactive
forces to be applied on the fluid flow. As there is a relative velocity on the
interface of the pipe and the fluid flow, the sum of the virtual work done by the
Coriolis and the centrifugal forces of the fluid flow is not equal to zero. They can
transfer the kinetic energy of the fluid flow to the motion of lower order modes
of vibration of the cantilevered pipe.
Third, study the mechanical nature of the Coriolis and the centrifugal forces in
the cantilevered pipe conveying fluid.
The pipe conveying fluid with stationary supports is usually not subject to any
nonholonomic constraint. Thus, it is a holonomic system with indefinite degree
of freedom of motion. The stability theory of the holonomic system, which has
been introduced in Chapter 3, indicates that the forces in the holonomic system
are classified into four types, namely, (1) the potential force or the conservative
force; (2) the dissipative force; and (3) the gyroscopic force; and (4) the circulatory
force. Among them, only the negative dissipative force and the circulatory force
can excite the self-excited vibrations in dynamic systems.
P. J. Holmes studied the behavior of a pipe conveying fluid[20] by numerical
calculation and the flutter phenomenon in the pipe was found with a discrete
model comprising of the first and second modes of the cantilevered pipe under
certain parameter conditions. However, the flutter phenomenon cannot be found
266
9.2 Flutter in Cantilevered Pipe Conveying Fluid
with the model with only the first mode even if the divergence instability occurs
in it when the flow velocity of the fluid is high enough. A rational interpretation
for such an analysis is that the flutter occurring in the cantilevered pipe conveying
fluid is excited by the circulatory force, which exists only in a system with two or
more degrees of freedom of motion. It cannot exist in a system with one degree
of freedom of motion.
In addition, according to S. S. Chen’s analysis[21], the role of the Coriolis force
applied on the cantilevered pipe conveying fluid always contains the component
of the dissipative force. This is the reason that all modes of the cantilevered pipe
conveying fluid are damped and the flutter phenomenon does not occur if the flow
velocity is low enough. Now, let us review Eq. (9.18), in which the magnitudes
of the second and the third terms, corresponding to the centrifugal force and
Coriolis force, increase with the flow velocity parameter v. Therefore, if the flow
velocity is high enough, the exciting effect of the centrifugal force and the Coriolis
force exceeds the damping effect in the fluid and the cantilevered pipe conveying
fluid, and the flutter phenomenon occurs at once.
Finally, we may find an effective means to prevent the cantilevered pipe
conveying fluid from flutter.
Various supports at the free end of the cantilevered pipe conveying fluid offer
a variety of boundary conditions for its differential equation of motion[22]. In
accordance with them, a variety of reactive forces exerted by the supports directly
affect the deformation of the pipe conveying fluid. The deformation of the
cantilevered pipe turns back to adjust the Coriolis force and the centrifugal force
exerted on the pipe wall. Thus, the support schemes at two ends of the pipes
conveying fluid have a very important effect. In fact, except the cantilevered pipe,
the flutter phenomenon cannot take place in the pipe supported at both ends,
either simply supported or fixed, even if the flow velocity is considerably high. If
a spring support is placed at the position close to the free end of the cantilevered
pipe conveying fluid, its dynamic behavior has a great variation. When the spring
stiffness is very small, the flutter phenomenon still occurs at a certain flow velocity.
However, the critical flow velocity at the onset of the flutter will gradually rise as
the spring stiffness increases. If the spring stiffness is raised to a value high
enough, the flutter phenomenon in the cantilevered pipe conveying fluid cannot
take place anyway.
Apart from the support scheme at the free end, the behavior of the cantilevered
pipe conveying fluid is closely associated with three nondimensional parameters,
namely, D, E, and v. According to the analysis results shown in Fig. 9.12 and
Fig. 9.15, increasing the mass ratio E 2 or decreasing the slenderness-damping
parameter D can also raise the critical flow velocity at the onset of the flutter. Con-
sequently, these means can effectively prevent the cantilevered pipe conveying
fluid from flutter.
267
Chapter 9 Self-Excited Vibrations from Interaction of Structures and Fluid
Consider a long cantilever wing with a straight elastic axis perpendicular to the
fuselage, which is assumed to be fixed in space. The wing deformation can be
measured by a deflection h, which is positive downward, and a rotation D about
the elastic axis, which is positive with the leading edge up. The chordwise distortion
is neglected. A frame of reference is chosen as shown in Fig. 9.16, with the Y axis
coinciding with the elastic axis. Let xD denote the distance between the center of
the mass and the elastic axis at any section. It is positive if the center of the mass
lies behind the elastic axis. Let c denote the chord length and x0 denote the distance
of the elastic axis aft the leading edgy. In a steady flow with speed U, the wing will
have some elastic deformation. To consider the free motion of the wing following
an initial disturbance, let h and D be the deviations from the equilibrium state, and let
the inertia, elastic, and aerodynamic forces also correspond to the deviations from the
steady-state values. Then, for small disturbances, the principle of superposition
holds good and we have the following equations of motion[23].
268
9.3 Classical Flutter in Two-Dimensional Airfoil
w2 § w2h · w2h w 2D
¨ EI ¸ m mxD FL 0,
wy 2 © wy 2 ¹ wt 2 wt 2
(9.44)
w § wD · w 2D w2h
¨ GJ
¸ D 2
I mxD MD 0,
wy © wy ¹ wt wy 2
where EI and GJ are the bending rigidity and the torsional rigidity of the wing, m
and ID are the mass and the mass inertia respectively of the wing section at y about
the elastic axis per length along the span, and FL and MD are the aerodynamic lift
and the moment per unit span respectively. Under the quasi-steady assumption,
the aerodynamic lift FL and the moment MD per unit span can be written as
1 1 § x ·
FL UU 2 cCL , M D UU 2 c 2 ¨ Cm 0 CL ¸ , (9.45)
2 2 © c ¹
where CL is the lift coefficient, Cm the moment coefficient, c the chord of wing,
x0 the distance between the elastic axis and its leading edge, U the density of the
air flow, and U the velocity of the air flow.
According to Joukowsky’s theorem, the lift coefficient CL of a two-dimensional
thin airfoil in a steady flow of an incompressible fluid is
dC L ª 1 dh 1 § 3 · dD º
CL
dD «D U dt U ¨ 4 c x0 ¸ dt » , (9.46)
¬ © ¹ ¼
and its moment coefficient Cm is
Sc dD 1
Cm CL . (9.47)
8U dt 4
269
Chapter 9 Self-Excited Vibrations from Interaction of Structures and Fluid
w2 § w2h · w2h w 2D
¨ EI ¸ m mxD
wy 2 © wy 2 ¹ wt 2 wt 2
UU 2 dCL ª 1 wh 1 § 3 · wD º
c
dD «D U wt U ¨ 4 c x0 ¸ wt » 0,
2 ¬ © ¹ ¼
w § wD · wD2
w h
2
¨ GJ ¸ ID 2 mxD 2
wy © wy ¹ wt wt
UU 2 2 Sc wD § x0 1 · dCL ª 1 wh 1 § 3 · wD º ½
+ c ®
w
¨ ¸
D «D U wt U ¨ 4 c x0 ¸ wt » ¾ 0.
2 ¯ 8U t © c 4 ¹ d ¬ © ¹ ¼¿
(9.48)
The boundary conditions are
wh
h D 0 at y 0,
wy
(9.49)
w2h w3h wD
0 at y l,
wy 2 wy 3 wy
where l is the semispan of the wing.
If xD and U are zero, Eqs. (9.48) would be reduced to two independent equations,
one for h and one for D. The terms involving xD and U indicate the inertia and the
aerodynamic coupling.
Recently, E. H. Dowell and D. Tang investigated the flutter problem of a
high-aspect-ratio wing by a wind tunnel test of an aeroelastic model and a time
simulation of a mathematical model, which consists of three equations of motion
governing flapping bending, chordwise bending, and torsion motion respectively[24].
These equations are similar to Eqs. (9.44), and the lift and the moment are
obtained from the original ONERA stall aerodynamic model. The results of the
theory and the experiments are in good agreement for static aeroelastic response,
the onset of the flutter, and the amplitude and the frequency of the flutter.
Therefore, the Eq. (9.48) may be used to predict the onset of the flutter.
Since Eqs. (9.48) are linear equations with constant coefficients, the solution can
be written in the usual form
O t
h Af ( y )eOt , D BI ( y )e ,
270
9.3 Classical Flutter in Two-Dimensional Airfoil
where O, f ( y ), and I ( y ) are determined by Eqs. (9.48) and (9.49), and A and
B are determined by the initial conditions. Since the solution of a differential
equation continuously depends on the coefficients of the equation, and since the
coefficients of Eqs. (9.48) vary continuously with U, constant O will also vary
continuously with U. In general, O is a complex number. Let O p iq. When p is
positive, the amplitude of the motion will increase with time. When p is negative,
the amplitude will decrease. If p is negative at U1 and positive at U2, U2 > U1,
then there exists at least one value U, say U0, between U1 and U2, at which p
would vanish. At U0, O is purely imaginary, physically corresponding to a
simple-harmonic motion. Such a flow velocity separates the flow velocity range
in its neighborhood into two regions. One is p < 0, where the motion is damped
and stable, and the other is p > 0, where the amplitude increases with time and is
unstable. Assume that the motion at the onset of flutter is simple harmonic and
can be represented as
ª d2 § d2 f · dC U c º
A « 2 ¨ EI 2 ¸ mZ 2 f L UiZ f »
¬ dy © dy ¹ dD 2 ¼
ª d C U c d C U c 2
§ 3 x · º
B « mxD Z 2I L U 2I L ¨ 0 ¸ UiZI » 0,
¬ dD 2 dD 2 © 4 c ¹ ¼
ª dC U c § x 1· º ° d § dI ·
2
A « mxD Z 2 f L U ¨ 0 ¸ iZ f » B ® ¨ GJ ¸
¬ dD 2 © C 4¹ ¼ ¯° dy © dy ¹
UU 2 c 2 § x01 · dCL UUc3 ª S § x0 1 ·§ 3 x0 · dCL º °½
ID Z 2I
¨ c 4 ¸ dD I « 8 ¨ c 4 ¸¨ 4 c ¸ dD » iZI ¾ 0.
2 © ¹ 2 ¬ © ¹© ¹ ¼ °¿
(9.51)
The boundary condition is given by Eqs. (9.49), provided that h(y, t) and D (y, t)
are replaced by f(y) and I ( y ) .
As a first approximation to the critical flow velocity and frequency, the Galerkin
method may be used (see Section 11.5). Assume that the functions f ( y ) and
I ( y ) are known and real valued. Multiplying the first equations of Eqs. (9.51)
by f ( y ) dy and the second by I ( y ) dy, and integrating them for 0 to l, we obtain
2 2
l§ d2 y · l § dI ·
a11 ³0 ¨© dy 2 ¸¹ dy, a22
EI ³0 GJ ¨© dy ¸¹ dy,
U dCL l U dCL l § x0 1 · 2 2
2 dD ³ 0 2 dD ³ 0 ¨© c 4 ¸¹
b12 cf I dy, b22 c I dy ,
l l
³ mf ³ mxD f I dy,
2
c11 dy , c12 c21
0 0
(9.53)
l U dC L l
³ I I dy , d11 ³ cf
2 2
c22 dy,
0 D 2 dD 0
U dC L
§ 3 x0 ·
l U dC L l § x0 1 ·
³c
¨ 4 c ¸ f I dy, d 21 2 dD ³c ¨ c 4 ¸ f I dy ,
2 2
d12
2 dD
0
© ¹ 0
© ¹
U ª§ 3 x ·§ x 1 · dC S º
l
d 22 ³ «¨ 0 ¸¨ 0 ¸ L » c3I 2 dy.
2 0 ¬© 4 c ¹© c 4 ¹ dD 8 ¼
LU 4 MU 2 N 0 (9.57)
With
272
9.3 Classical Flutter in Two-Dimensional Airfoil
L D2 B1C2 D2 A1 ,
M B1C2 D1 B1C1 D2 B12 E2 2 D1 D2 A, (9.58)
N B1C1 D1 B E1 D A1 .
2
1
2
1
The equations of motion of classical flutter for a long wing in the steady flow,
i.e., Eqs. (9.44), are established by the Largrange equation. In these equations,
both the lift force and the aerodynamic moment are nonconservative forces. If
they are eliminated, the other terms in Eqs. (9.44) are combined into the equations
of motion of a conservative system. It is known that self-excited vibration cannot
occur in a conservative system and the classical flutter of the wing is excited by
the lift force and the aerodynamic moment. According to the reduction principle
of the force system acting on a rigid body, any complicated force system
containing distributed areodynamic force, can be reduced to a resultant acting on
a particular point, namely, the center of the force system. In this case, the resultant
is a concentrated force exerted on the aerodynamic center. Thus, we conclude
that the classical flutter is excited by the aerodynamic force exerted on the wing,
in which the lift force originates from the velocity circulation of the flow in the
boundary layer of the flow over the wing surface. According to Joukowsky’s
theorem, the lift force on the wing in the steady flow is described as
FL UU * ,
where * is the circulation of the stead flow over a two-dimensional airfoil, U
the density of the flow, and U the velocity of the flow.
Let us recall the supporting force of the oil film in a journal bearing. It also
originates from the circulation of the oil flow around the journal. Thus, both the
oil whirl of rotor and the classical flutter of the wing have essentially the same
excitation mechanism. Since the oil whirl of rotor is excited by the circulating
force, as mentioned in the previous chapter, the motive force of the classical
flutter of the wing is also the circulatory force.
273
Chapter 9 Self-Excited Vibrations from Interaction of Structures and Fluid
Now, we make a qualitative study about the classical flutter of the wing, for
which the continuous model shown in Fig. 9.16 is replaced by a simpler discrete
model with flexible motion and torsional motion coupled with each other. This
model with two degrees of freedom of motion can transfer the lift force to the
circulatory force that is a necessary mechanical condition for the occurrence of
the self-excited vibration. In accordance with this model, B. H. K. Lee, L. Liu,
and K. W. Chung have used two coupled ordinary differential equations to study
Hopf bifurcation of an airfoil placed in a subsonic airflow. The aerodynamic
forces are represented by other first-order ordinary differential equations, and the
nonlinear restoring forces of the springs in the pitch and the plunge degree of
freedom are given by two terms respectively. The Hopf bifurcation is found by
numerical calculation[25]. In addition, E. F. Sheta, et al performed a numerical
analysis of the classical flutter of wing and predicted a limit cycle oscillation of
an aerodynamic system with combined structural and aerodynamic nonlinearities.
Corresponding wind tunnel measurements were also carried out[26]. In addition,
an aeroelastic computation based on a discrete model with two degrees of
freedom predicted the amplitude and the frequency of the classical flutter in very
close agreement with the experimental data of the wind tunnel testing. Thus, it is
demonstrated that the discrete model is credible to study the classical flutter of
the wing.
In order to find effective schemes to prevent the classical flutter of the wing, it is
necessary to establish the relationship between the critical speed of the classical
flutter and the parameters of the wing.
First, we construct a discrete model of a two-dimensional airfoil shown in
Fig. 9.17. It is a rigid body supported on a flexible spring and a torsional spring
at the elastic center E, which is defined in one of the following ways. E is a point
on the wing where a vertical force causes only a vertical displacement and no
rotation occurs; E is also a point of the wing that does not displace itself if the
wing is subject to a pure torque leading to the rotation of the section. These two
properties of E always go together as shown by Maxwell’s theorem of reciprocity.
Let A and G be the aerodynamic center and the mass center of the section of the
wing respectively. Let us assume that A, E, and G are located at the midchord
line of the wing.
Let kh be the up-and-down spring constant and kD the torsional spring constant
per unit length of the wing. Let FL be the lift force acting at the aerodynamic
center A. The lift force is the vertical component of the resultant of the distributed
pressure on the surface of the wing.
274
9.3 Classical Flutter in Two-Dimensional Airfoil
mh kh (h bD ) FL ,
(9.60)
ID D kD D kh b(h bD ) FL a,
where h is the plunge of the airfoil, D the pitch of the aeroplane, m the mass of
per unit length of the wing, ID the inertial moment of per unit length of the wing,
b the distance between the elastic center and the mass center, and a the distance
between the aerodynamic center and the mass center.
The lift force per unit length of the wing is described by the first equation of
(9.45). According to Joukowsky’s theorem, the lift coefficient CL is approximately
proportional to the small angle of the attack under the moving state, i.e.,
(D h / U ) . The proportional constant is denoted by CLD , which is somewhat
smaller than 2S. Therefore, we have
1 § h ·
FL UU 2 cCLD ¨ D ¸ .
2 © U¹
1 1
d UUcCLD , f UU 2 cCLD .
2 2
The equations of motion, i.e., Eqs. (9.61), may be considered as a discrete model
of the two-dimensional airfoil to study the classical flutter.
Next, we deduce the analytical expression of the critical speed of the classical
flutter of the wing.
275
Chapter 9 Self-Excited Vibrations from Interaction of Structures and Fluid
The equations of motion, i.e., Eqs. (9.61), are a linear ordinary differential
system with constant coefficients. Therefore, we use Hurwitz criterion to determine
the parameter condition corresponding to the second type of critical stability of
the model of the airfoil.
The characteristic equation of the model is deduced from Eqs. (9.61), i.e.,
a0 ! 0, a1 ! 0.
Thus, according to Hurwitz criterion, the necessary and sufficient condition for
the second type of critical stability of the dynamic system (9.61) consists of three
inequalities
a2 ! 0, a3 ! 0, a4 ! 0, (9.64)
and an algebraic equation
§ I ·
kD ¨ b 2 D ¸ kh ! af ,
© m¹
kD ! (a b)bkh , (9.66)
kD ! (a b) f .
Then, in Eq. (9.62), let us denote O = iZ, and separate the real part and the
imaginary part of the complex equation. This yields two algebraic equations, i.e.,
276
9.4 Stall Flutter in Flexible Structure
It is not difficult to find the flutter frequency Z0 in the onset state of the flutter
from the Eq. (b). The analytical expression of the flutter frequency is
1
ª kD a b bkh º 2
Z0 « » . (9.67)
¬ ID ¼
Substituting this expression in Eq. (a), after some considerable operations, we
obtain the analytical expression of the critical speed of the classical flutter, i.e.,
1
ª 2bkh º 2
U cr « D »
. (9.68)
¬ U cCL ¼
Last, we summarize the study and outline the effective schemes to prevent the
classical flutter of the wing.
Combining the analytical expression (9.68) with three inequalities (9.66), we
can find some schemes to prevent the wing from the classical flutter by raising
the critical speed Ucr. The following four ways are these schemes.
1. Increasing the spring coefficient kh and kD;
2. Raising the geometric parameter b. This scheme may be realized by shortening
the distance from the elastic center to the aerodynamic center and elongating the
distance from the mass center to the aerodynamic center;
3. Widening the chord length c;
4. Decreasing the density of the air flow by varying the flying altitude.
aeronautic terminology for aerodynamic instability of the wing with large angle
of attack, while galloping is a term used by civil engineers for aeroelastic instability
of bluff structures in wind and currents. Nowadays, galloping has become an
important subject to prevent violent vibration of flexible structures in intense
wind. This section aims at qualitatively analyzing the motion and the excitation
mechanism of the stall flutter of bluff structures. There are four subsections,
namely, aerodynamic force exciting stall flutter; a mathematical model for studying
stall flutter; critical speed and hysteresis phenomenon of stall flutter; and some
features of stall flutter and prevention of stall flutter.
Stall flutter is usually induced by the separated flow whose flow field is extremely
complicated[28]. Therefore, the aerodynamic forces exerting on the structure under
stall flutter are often determined by a model test in a wind tunnel.
The aerodynamic force per unit length of a cylinder with bluff section in a
planar steady flow F is comprised of aerodynamic lift FL and drag FD, as shown in
Fig. 9.19. The direction of drag FD is opposite to the one of the flow velocity U.
According to steady flow theory, the magnitudes of the lift and the drag of a
cylinder are represented as
1 1
FL UU 2 HCL , FD UU 2 HCD , (9.69)
2 2
where H is the width of the cylinder, CL the lift coefficient of the cylinder, CD the
278
9.4 Stall Flutter in Flexible Structure
drag coefficient of the cylinder, Uthe density of the fluid, and U the velocity of
the steady flow.
A great quantity of test results shows that the lift coefficient and the drag
coefficient of the cylinder are functions of the angle of attack, as shown in
Fig. 9.20.
§ y · y
D arctan ¨ ¸| . (9.70)
©U ¹ U
In addition, according to the geometry shown in Fig. 9.19, the lateral force Fy is
1
Fy FL cos D FD sin D UU 2 HC y , (9.71)
2
in which Cy is called the lateral force coefficient. Comparing Eq. (9.69) with
Eq. (9.71), we obtain
279
Chapter 9 Self-Excited Vibrations from Interaction of Structures and Fluid
A typical curve describing the relationship between the lateral force coefficient
and the attack angle is shown in Fig. 9.21. The lateral force exerted on the
cylinder with bluff section may be approximately represented by a third degree
function, i.e.,
Fy aD bD 3 , (9.73)
where m is the mass of the cylinder, c the damping coefficient of the dashpot,
and k the stiffness coefficient of the spring. If the model is thought of as a true
flexible structure, these parameters will represent the inertia, the damping, and
the stiffness coefficients respectively.
280
9.4 Stall Flutter in Flexible Structure
§ a b ·
my ¨ c 3 y 2 ¸ y ky 0. (9.75)
© U U ¹
As the damping coefficient c is always small, the term in the bracket of the
equation is negative under certain extent of wind speed, namely,
a b 2
c y 0. (9.76)
U U3
In this case, the fluid flow will feed energy to the flexible structure. Previous
inequality means that the energy dissipated by the internal damping is less than
the one acquired from the fluid flow, the flexible structure vibrates under the
negative damping state and the galloping phenomenon will occur. Consequently,
inequality (9.76) demonstrates the potential of galloping in the flexible structure
with bluff section.
Cy AD BD 3 CD 5 DD 7 , (9.77)
1 ª y § y ·
3
§ y ·
5
§ y · º
7
Fy UU 2 H « A B ¨ ¸ C ¨ ¸ D ¨ ¸ » , (9.78)
2 ¬« U ©U ¹ ©U ¹ © U ¹ ¼»
where H is the width of the test specimen. Let us denote a set of nomenclatures,
281
Chapter 9 Self-Excited Vibrations from Interaction of Structures and Fluid
1/ 2
y §k·
Y , Zn ¨ ¸ , W Znt,
H ©m¹
dY d 2Y c
Yc , Y cc , E , (9.79)
dW dW 2
2mZ n
UH 2L U
n , P nA, V ,
2m Zn H
where V is called reduced velocity and L is the length of the square prism. By
substituting expression (9.78) in Eq. (9.74) and considering expressions (9.79),
the equation of motion of the square prism is reduced to a dimensionless form by
dividing through by kH and it becomes
ª§ 2E · § B · c3 § C · c5 § D · c7 º
Y cc Y nA «¨ V ¸Y c ¨ ¸Y ¨ 3 ¸
Y ¨ 5 ¸
Y ». (9.80)
¬© nA ¹ © AV ¹ © AV ¹ © AV ¹ ¼
It is the mathematical model of the experimental rig, as shown in Fig. 9.22, for
our study of the galloping of flexible structures.
In Eq. (9.80), both the dimensionless parameter n, which is equal to the half of
the ratio of displaced air mass to prism mass, and the dimensionless parameter E,
which gives the testing rig damping in the absence of aerodynamic force, are
very small, of the order of 10 –3 in the wind tunnel test. Therefore, Eq. (9.80) is a
weakly nonlinear equation that is written as
Y cc Y P f (Y c), P nA, (9.81)
282
9.4 Stall Flutter in Flexible Structure
§ 2E · § B · c3 § C · c5 § D · c7
f (Y c) ¨ V ¸Y c ¨ ¸ (Y ) ¨ 3 ¸
(Y ) ¨ 5 ¸
(Y ) . (9.82)
© P ¹ © AV ¹ © AV ¹ © AV ¹
According to the concept of the average method, the approximate solution of
autonomous Eq. (9.81) is described as
Y Ys cos(W I ), Y c Ys sin(W I ), (9.83)
where Ys is the amplitude of the approximate solution and I is the phase shift.
For P 0, the solution of Eq. (9.81) is a harmonic function. It is assumed that the
solution for P z 0 but small can be described as a series of expansion in powers
of P, i.e.,
1 dYs2
P f [Ys sin(W I )]Ys sin(W I ).
2 dW
According to Eq. (9.81), it is clear that Ys varies slowly with W. Thus, we have
dYs2 P 2
S³
f (Ys sin J )Ys sin J dJ , (9.84)
dW 0
where J W I. Inserting the function f (Y c), namely, Eq. (9.82), into the above
expression and integrating the resulting equation, we obtain
dYs2 ª§ 2 E · 2 3 § B · 4 5 § C · 6 35 § D · 8 º
nA «¨ V ¸ Ys ¨ ¸ Ys ¨ ¸ Ys ¨ ¸ Ys .
dW ¬© nA ¹ 4 © AV ¹ 8 © AV 3 ¹ 64 © AV 5 ¹ »¼
Denoting R Ys2 , the equation is written as
dR
(a bR cR 2 dR3 ) R F ( R) (9.85)
dW
with
283
Chapter 9 Self-Excited Vibrations from Interaction of Structures and Fluid
The stationary vibrations correspond to dR/dW 0, and the amplitudes are therefore
given by R 0, and the real positive roots of the algebraic equation
a bR cR 2 dR 3 0. (9.87)
In the phase plane (Y, Y' ), R 0, the equilibrium point, is a singular point at the
origin known as a focus. The positive roots Ri of Eq. (9.87) define the closed
1
trajectories and the concentric circles with radius Ysi ( Ri ) 2 , i 1, 2, 3. The stability
of equilibrium and limit cycles is determined by investigating the tendency of the
disturbed motions after a small disturbance GR. The pertinent function is dGR / dW ,
and from Eq. (9.85), it is given by
dGR
F c( Ri )R, i 1, 2,3
dW R Ri
with
§ 2E ·
F c(0) nA ¨ V ¸. (9.88)
© nA ¹
Obviously, if E < 0, the focus at the origin is unstable; if E > 0 and the reduced
velocity is small enough, it is stable. Consequently, the previous equation yields
a critical reduced wind velocity, i.e.,
2E
Vcr . (9.89)
nA
If V > Vcr, Eq. (9.87) has at least a positive real root R1. In this case, there is a
stable limit cycle in the phase plane, as shown in Fig. 9.24(a). If Eq. (9.87) has
three positive real roots, namely, R1, R2, and R3, and R1 < R2 < R3, as shown in
Fig. 9.24(b), there are three limit cycles in the phase plane. The limit cycles
corresponding to R1 and R3 are stable, while the limit cycle corresponding to R2
is unstable.
For a square prism with a given mass and damping parameters, namely, n and
E, the choice among these possibilities depends on the reduced speed parameter
V. This is an example of bifurcation theory in nonlinear dynamics. Coefficients a,
b, c, and d all depend on the change of the system from one to another. The
dependence of the system on the parameter is usually found by numerical
computation. Here, it is convenient to use a (Ys ,V ) diagram, with the stationary
amplitude Ys plotted against the wind-reduced velocity V directly, as shown in
284
9.4 Stall Flutter in Flexible Structure
Fig. 9.25. The figure shows that the system has oscillation hysteresis. If V < Vcr,
Ys = 0 and the phase plane has only one stable focus. At V Vcr, the system forks
into a form whose phase plane has a stable limit cycle, like Fig. 9.24(a). Between
V1 and V2, however, the phase plane becomes Fig. 9.24(b). The lower limit cycle
Ys1 is reached from rest. For V > V2, the form of the phase plane is like
Fig. 9.24(b) with the amplitude of the stable limit cycle, Ys3, increasing with V. If
the wind speed is then decreased to a value between V1 and V2 while the prism is
still oscillating, the upper limit cycle is reached. Until for V < V1, the amplitude
drops to that of the lower cycle again. Therefore, a hysteresis loop in V-Ys plane
is formed. An appropriate experiment was carried out in a low-speed wind tunnel.
285
Chapter 9 Self-Excited Vibrations from Interaction of Structures and Fluid
The test section was 104 inches long, 36 inches wide, and 27 inches high. A one-
inch square tested prism was mounted vertically at the center of the principal cross
section. Damping was measured by obtaining decay curves, and the system value
of E was 1.07 u 103. Additional damping was provided by two electromagnetic
eddy current dampers. The measurement of stationary amplitude as a function of
wind velocity was made at four damping levels and the results are plotted in
Fig. 9.26. Also shown are the theoretical curves obtained by the average method
above. The associated parameters of these curves correspond to the polynomial
approximation to the Cy(D) curve shown in Fig. 9.23. The figure shows that the
topological behavior predicted by the theory is observed experimentally. Oscillation
hysteresis is present, and the amplitude and the range of the wind speed for two
stable limit cycles are quite accurately predicted.
direction of wind on the earth surface may arbitrarily vary, the angle of attack to
the civil structures may get large magnitude easily. Therefore, the stall phenomenon
often occurs in civil engineering. With high-strength materials being extensively
used and the flexibility of the civil structures increasing greatly, galloping has
become an important topic.
According to the excitation mechanism of galloping, as mentioned above, there
are at least three features:
(1) Since civil structures always have bluff section across the span, the stream
velocity separating the boundary layer is much lower than the velocity of the
streamlined body like wings. Hence, galloping of civil structures with high
flexibility usually occurs at lower flow velocity. On the other hand, since the
internal damping of the flexible structures is often very small, the negative
damping force required by galloping is also small. Consequently, the wind speed
for exciting galloping is much smaller than the critical flow velocity for inducing
the classical flutter of wings.
(2) Since the critical wind speed for onset of galloping is very low, the aerody-
namic force under galloping condition is much smaller than the elastic restorative
force of the structures. Consequently, the aerodynamic force cannot dramatically
change the stiffness of the flexible structures; the galloping frequency is always
close to the natural frequency of the excited mode of the flexible structures.
(3) For most civil structures, the frequency of the flexural mode and the frequency
of the torsional vibration mode are sufficiently separated, and these structures
usually have two symmetric planes. Therefore, the center line of the mass is close
to the center line of elasticity, and the flexural mode and the torsional mode are
often coupled very weakly. In this case, we may use a one-dimensional model to
study the galloping of flexible structures in civil engineering.
Obviously, if the distance between the mass center and the elastic center in some
structures is large enough, particularly, the flexural mode frequency is sufficiently
close to the torsional mode frequency, it is necessary to use a mechanical model
with two degrees of freedom to study the galloping of flexible structures. Some
structures, such as the bridge made of steel, is curved either in elevation or in
plane, so that torsion involves some lateral sway. Sametimes the bending flexible
strdeture couples in some torsion, in the case the vibration modes posses of three
components: the vertical deflection; the lateral deflection; and the torsion
identifiable at any point along the line of section center across the span. Thus, we
should use a mechanical model with three degrees of freedom to study the
galloping of the structures[30].
According to the analysis above, the galloping or the stall flutter can be pre-
vented by ensuring that the critical wind speed is more than the actual wind speed
in the locality. This task may be accomplished by the following technical schemes.
(1) Reform aerodynamic contours of structures
If the slope of the aerodynamic force coefficient of structures is stable, as
shown in Fig. 9.27, the flexible structure with bluff cross section is stable. For
287
Chapter 9 Self-Excited Vibrations from Interaction of Structures and Fluid
example, the unstable section of an ice-coated power line can be changed into a
stable circular section by melting the ice with resistance heating. The rectangles
that have narrow face into the wind are more stable than the rectangles that have
broad face into the wind if the wind direction always varies in a small range.
Cylinder arrays in the form of closely packed tube banks, which are subject to
cross-flow, are found in many types of heat exchangers, boilers, and steam
generators. Cross-flow vibrations of cylinder arrays are caused by one or more of
following mechanisms: (1) turbulent excitation, namely, the randomly varying
pressure on the surfaces of the cylinders generally produces relatively low
amplitude cylinder vibration; (2) acoustic excitation; (3) vortex-induced vibration;
and (4) fluid elastic instability: at certain flow velocity, fluid energy may be fed
288
9.5 Fluid-Elastic Instability in Array of Circular Cylinders
into cylinders and result in large cylinder vibrations. The first two types of
vibrations are forced vibrations beyond the topic of this book. The third has been
investigated before. The last will be discussed in this section. It has been known
for over four decades that this type of cross-flow may cause violent vibrations of
arrays and it is a dominant reason for the tube failure in heat exchangers. Thus, it
is not surprising that there has been a great deal of effort devoted to developing
the design criteria to prevent this vibration from occurring. This section emphasizes
the study of the excitation mechanism of the fluid-elastic instability in arrays of
circular cylinders. The contents are divided into five subsections, namely, the
fluid-elastic instability phenomena, the fluid forces depending on the motion of
the circular cylinders, the analysis of the flow-induced vibration, the expressions
predicting the critical flow velocity, and the prevention of fluid-elastic instability.
One of the fluid-elastic instability properties is that once the critical flow velocity
is exceeded, the vibration amplitude increases very rapidly with the flow velocity.
Figure 9.28(a) shows the response of an array of metallic tubes to the water flow.
The initial hump is attributable to the vortex-induced vibrations[21]. Figure 9.28(b)
shows the response of an array of plastic tubes to the air flow. In water, the onset
of instability tends to coincide with the vortex resonance with the tube natural
frequency. In air, the onset of instability generally occurs well above the vortex
resonance. The maximum amplitude is usually limited by clashing against the
adjacent tubes, as shown in Fig. 9.29. The tubes generally vibrate in the oval orbits
289
Chapter 9 Self-Excited Vibrations from Interaction of Structures and Fluid
and somewhat synchronize with neighboring tubes. The orbits vary in shape from
nearly straight lines to circles.
290
9.5 Fluid-Elastic Instability in Array of Circular Cylinders
fluid forces. Other fluid excitation forces, which are not mentioned, are independent
of the structural motion.
The cylinder array with n cylinders is usually placed with a simple geometric
pattern. Several schemes are shown in Fig. 9.30.
n
°ª w 2 uk wuk º ª w 2 vk wv º °½
gj ¦ ® «D jk D c D cc u » «V V cjk k V ccjk vk » ¾,
wt wt wt wt
2 jk jk k jk 2
j 1°¯¬ ¼ ¬ ¼ °¿
(9.90)
n
°ª w 2 uk wuk º ª w 2 vk wv º °½
hj ¦ ® «W jk W c W cc u » « E E cjk k E ccjk vk » ¾ ,
wt wt wt wt
2 jk jk k jk 2
k 1°¯¬ ¼ ¬ ¼ °¿
291
Chapter 9 Self-Excited Vibrations from Interaction of Structures and Fluid
where Djk, V jk, W jk, and E jk are the elements of the added mass matrix, D cjk , V cjk ,
W cjk , and E cjk are the elements of the damping matrix, and D ccjk , V ccjk , W ccjk , and E ccjk
are the elements of the stiffness matrix.
In addition, there are also motion-independent fluid forces acting on the surface
of the cylinders, which are not related to the flow-induced vibration of circular
cylinders.
An array of n cylinders is subject to a cross flow, as shown in Fig. 9.30. The axes
of the cylinders are parallel to the Z axis and the flow is parallel to the X axis. The
equations for the jth cylinder in the X direction and the Y direction are
w 4u j wu j w 2u j
EjI j cj mj gj,
wz 4 wt wt 2
j 1, 2," , (9.91)
w4v j wv j w 2v j
EjI j cj mj hj ,
wz 4 wt wt 2
where EjIj is the flexible rigidity, cj is the structural damping coefficient, mj is the
cylinder mass per unit length, uj and vj are the displacements in the X and the Y
directions, and gj and hj are the forces per unit length in the X and the Y
directions, as given in Eqs. (9.90). Substituting Eqs. (9.90) into (9.91), we obtain
w 4u j wu j w 2u j n
§ w 2u w2v ·
EjI j cj mj ¦ ¨ D jk 2k V jk 2k ¸
wz 4 wt wt 2 k 1© wt wt ¹
n
§ wu wv · n
¦ ¨ D cjk k V cjk k ¸ ¦ D ccjk uk V ccjk vk 0,
k 1© wt wt ¹ k 1
(9.92)
w vj
4
wv j w 2v j n
§ w 2u w2v ·
Ej I j 4 cj m j 2 ¦ ¨W jk 2k E jk 2k ¸
wz wt wt k 1© wt wt ¹
n
§ wu wv · n
¦ ¨W cjk k E cjk k ¸ ¦ W ccjk uk E ccjk vk 0.
k 1© wt wt ¹ k 1
Let us denote the mth orthonormal function Im as the mth modal function for
the circular cylinder, and let
f
u j ( z, t ) ¦a
m 1
jm (t )Im ( z ), (9.93)
292
9.5 Fluid-Elastic Instability in Array of Circular Cylinders
f
v j ( z, t ) ¦b m 1
jm (t )Im ( z ). (9.94)
With the assumption that all cylinders are identical, have the same natural
frequencies, and are damping in the X and the Y directions, using Eqs. (9.92),
(9.93), and (9.94) and neglecting the modal coupling due to the dependence of
the fluid force coefficients on the flow velocity yields
¦ D
n
1
ajm 2] jmZ jm a jm Z 2jm a jm a V jkmbkm
jkm km
mj k 1
¦ D c
n n
¦ D cc a V ccjkmbkm 0,
1 1
c
jkm akm V jkm bkm jkm km
mj k 1 mj k 1
(9.95)
¦
n
1
bjm 2] jmZ jmb jm Z 2jmb jm W jkm akm E jkmbkm
mj k 1
¦ W '
n n
¦ W '' a E ccjkmbkm 0
1 1
jkm akm E cjkmbkm jkm km
mj k 1 mj k 1
with
1 L 1 L 1 L
L ³0 L ³0 L ³0
D jkm D jk Im2 dz , D cjkm D cjkIm2 dz , D ccjkm D ccjkIm2 dz ,
1 L 1 L 1 L
L ³0 L ³0 L ³0
V jkm V jk Im2 dz , V cjkm V cjk Im2 dz , V ccjkm V ccjk Im2 dz ,
(9.96)
1 L 1 L 1 L
L ³0 L ³0 L ³0
W jkm W jkIm2 dz , W 'jkm W cjkIm2 dz , W ''jkm W ccjk Im2 dz ,
1 L 1 L 1 L
L ³0 L ³0 L ³0
E jkm E jk Im2 dz , E cjkm E cjkIm2 dz , E ccjkm E ccjk Im2 dz ,
where Zjm and ]jm are respectively the natural frequency and modal damping
ratio of the jth cylinder in the mth vibration mode in vacuum.
Equations (9.95) may be written in the matrix form
CQ KQ
Q 0
where Q is a 2n-dimension vector, namely, Q [a11 , b11 , a12 , b12 ,"]T , and C and
K are constant matrices.
For a cylinder array in the cross flow, matrices C and K are not necessarily
symmetric. Therefore, they can be separated into a symmetric matrix and a skew-
symmetric matrix respectively. Hence, we conclude that the fluid forces on the
oscillating cylinder in the cross flow include the gyroscopic force and the
circulatory force.
293
Chapter 9 Self-Excited Vibrations from Interaction of Structures and Fluid
with
1
§ 1 ·2
¨ 1 S3 J 1V D11cc ¸
2
Zf Z1 ¨ ¸ ,
¨¨ 1 D11J 1 ¸¸
1
© ¹ (9.98)
1
] 1 3 J 1V 2D11c
]f 2S ,
1
ª § 1 ·º
« J 1 11 ¨
D J D c
2
1 1 V 11 ¸ »
© S
3 1
¬ ¹¼
294
9.5 Fluid-Elastic Instability in Array of Circular Cylinders
1
§ 2S3] 1 · 2
Vcr ¨ ¸ . (9.99)
© r1D11c ¹
where xj and yj are the longitudinal and the transverse displacements of the jth
cylinder from its equilibrium position, m is the mass per length including the
added mass of the jth cylinder, ]jx and ]jy represent the damping factor, Zjx and
Zjy indicate the natural frequency, and Fjx and Fjy are the fluid-dependent forces
per unit length of the jth cylinder, whose linearized expressions may be represented
in the following form.
Figure 9.32 Tube Row Model (Dampers Parallel to Springs not Shown)
1
Fjx UU 2 [k xx ( x j x j 1 ) k xx ( x j 1 x j ) k xy ( y j y j 1 ) k xy ( y j 1 y j )],
4
1
Fjy UU 2 [k yx ( x j x j 1 ) k yx ( x j 1 x j ) k yy ( y j y j 1 ) k yy ( y j 1 y j )],
4
(9.101)
where U is the fluid density, U is the flow velocity its value is determined with
295
Chapter 9 Self-Excited Vibrations from Interaction of Structures and Fluid
the minimum gap between cylinders, and kxx, kxy, kyx, and kyy are the elements of
the stiffness matrix of the fluid dependent force.
Substituting Eqs. (9.101) in (9.100), we obtain the linear equations of motion
of the jth cylinder parallel and perpendicular to the stream, i.e.,
1
mxj 2m] jxZ jx x j k jx x j UU 2 [k xx ( x j 1 x j 1 2 x j ) k xy ( yi 1 y j 1 )],
4
1
my j 2m] jyZ jy y j k jy y j UU 2 [k yx ( x j 1 x j 1 ) k yy (2 y j yi 1 y j 1 )].
4
(9.102)
The stability analysis of these equations is considerably simplified by using an
assumed mode of cylinder motion. As shown in Fig. 9.29, the cylinders tend to
vibrate in the semisynchronized oval orbits. This mode of cylinder-to-cylinder
vibration in Fig. 9.29 is[32]
x j 1 x j 1 , y j 1 y j 1 . (9.103)
Another simplifying assumption is that the stiffness coefficients kxx and kyy
may be neglected, for they serve only to shift the natural frequency of the
cylinder and experiments have shown that the shift of the natural frequency is
very small.
With these assumptions, the equations of motion of the jth cylinder, i.e.,
Eqs. (9.102), are reduced to
1
mxj 2m] jxZ jx x j k jx x j UU 2 k yx y j 1 ,
2 (9.104)
1
my j 1 2m] j 1, yZ j 1, y y j 1 k j 1, y y j 1 UU 2 k xy x j .
2
Let us denote the solutions of these equations as
1
(mO 2 2] jx mZ jx O k jx )(mO 2 2m] j 1, yZ j 1, y O k j 1, y ) UU 4 k xy k yx 0.
4
(9.106)
On expanding the equation, we obtain a four-degree algebraic equation
O 4 a1O 3 a2 O 2 a3O a4 0.
296
9.5 Fluid-Elastic Instability in Array of Circular Cylinders
¨¨ ¸
1 1
© ] jx f jx ] j 1, y f j 1, y ¸¹
2SD( f jx f j 1, y ) 2 (k xy k yx ) 4
1
ª§ f f jx ·
2
§ ] j 1, y ] jx · º4
«
u ¨ j 1, y
¸ 4 ¨¨ ¸¸ (] jx f jx ] j 1, y f j 1, y ) » .
«¨© f jx f j 1, y ¸¹ f
© j 1, y f jx ¹ »
¬ ¼
(9.109)
If the damping in the X and the Y directions are the same and relatively small, i.e.,
]jx ] j+1, y ], and the natural frequencies in the X and the Y directions are close
but not equal, namely, f x f y 1 O(] ), then the critical flow velocity falls to
1
1
2 S § m · ª«§ º4
2
U cr 32 2 fx ·
1 ¨ ¸ ¨ 1 ¸ 4] 2
» . (9.110)
1
4 ©
U D 2 ¹ «¨© f y ¸¹ »
( fx fy ) D 2
(k xy k yx ) ¬ ¼
This expression predicts that the critical flow velocity for onset of instability
increases with damping and detuning between cylinders. Both effects have been
seen in the experimental data.
If all cylinders are identical in frequency and damping, fx fy f and ]x ]y ],
then Eq. (9.109) is reduced to a more simple expression, which was derived by
H. J. Connors first[34], i.e.,
1
U cr ª m(2S] ) º 2
C« » (9.111)
¬ UD ¼
2
fD
with
1
2(2S) 2
C 1
.
4
(k xy k yx )
297
Chapter 9 Self-Excited Vibrations from Interaction of Structures and Fluid
Although expressions (9.99), (9.109), (9.110), and (9.111) have determined the
relationship between the critical flow velocity of fluid-elastic instability and the
system parameters, a variety of fluid-force coefficients must be found with accuracy
to carry out the effective prediction of the fluid-elastic instability. Under such
circumstances, the prediction is successfully made only by testing many special
rigs. Figure (9.33) represents a compilation of a great quantity of experimental data,
in which the solid curves were found by S. J. Price and M. P. Paidousis[35]. They
considered a single flexible cylinder surrounded by rigid cylinders. The second
equation of (9.102) with yj+1 yj1 xj+1 xj1 0 is the equation of motion in the
transverse direction for the jth cylinder with all other cylinders stationary. It yields
1
my 2m] yZ y y k y y UU 2 k yy y. (9.112)
2
Figure 9.33 Theory for Instability of Tube Array in Comparison with Data; the Price
and Paidoussis (1986) theory is based on a single flexible tube surrounded by fixed tubes
If the motion is harmonic, the displacement is 90° out of phase with the
velocity, namely,
y Ay sin Z t , y AyZ cos Z t. (9.113)
298
References
where D is the cylinder diameter and U is the flow velocity. Let Fy denote the
fluid force acting in phase with y . Consequently, we have
1
Fy UU 2 k yy Ay sin[Z (t W )].
2
Substituting this equation in Eq. (9.112) and using expression (9.115), we obtain
the equation of motion, i.e.,
ª 1 º
my « 2m] yZ y UU 2 k yyZ 1 sin(Z DU 1 ) » y k y y 0. (9.116)
¬ 2 ¼
This equation possesses instability provided that the quantity in the square
brackets passes through zero so that the net damping passes through zero and
becomes negative. With the bracket term set to zero and Z assumed to be Zy,
instability is predicted at a reduced flow velocity Vcr, which is just a positive real
root of the equation
§1· 4m] y U
V 2 sin ¨ ¸ , V . (9.117)
©V ¹ k yy U D 2 ZyD
Owing to the periodicity of the trigonometrical function, this equation predicts
multiple ranges of instability if stiffness coefficient kyy is negative. S. J. Price and
M. P. Paidousis found that kyy is negative in some cases. The solid curves in
Fig. 9.33 show their result for a single flexible cylinder in comparison with the
result for cylinder arrays with all cylinders flexible. Although theoretical foundation
is not perfect enough, numerous experimental data measured by many investigators
are densely distributed in the neighborhood of the theoretical curves. Thus, this
long and narrow band constructed from experimental data can be used as a
prediction foundation, and Fig. (9.33) has been extensively referenced in many
literature[36,37]. Particularly, since the aeroelastic instability in an array of circular
cylinders in the cross flow is a very complicated problem and a perfect theory
has not been constituted yet, Fig. 9.33 becomes a very useful implementation for
design engineers. In practice, the rational values of various parameters of an
array of circular cylinders in the cross flow may be determined in this way.
References
[1] J H Lienhart. Synopsis of Lift, Drag, and Vortex Frequency Data for Rigid Circular
Cylinders. Washington State University, College of Engineering, Research Division Bulletin
300, 1966
[2] H Drescher. Mussung der auf querongestromte Zylinder ausgeubten seitlich veranderten
Drucke, Zeitschrift für Flugwissensschaft, 1956, 4, 17 21
299
Chapter 9 Self-Excited Vibrations from Interaction of Structures and Fluid
[3] C P Jackson. A Finite-Element Study of the Onset of Vortex Shedding in Flow Past
Variously Shaped Bodies. J. Fluid Mech., 1987, 182
[4] P M Gresho, S T Chan, R L Lee, C D Upson. A Modified Finite-Element Method for
Solving the Time-Dependent Incompressible Novier-Stokes Equations. Part 2 Applications,
Int. J. Numer. Math. Fluids, 1984, 4, 619
[5] C Friehe. Vortex-Shedding from Cylinders at Low Reynolds Numbers. J. Fluid Mech.,
1980, 100, 237
[6] R D Blevins. Flow-Induced Vibration, Second Edition. New York: Van Nostrand Reinhold
Company, 1990
[7] R E Bishop, A Y Hassan. The Lift and Drag Forces on a Circular Cylinder Oscillating in
a Flowing Fluid. Proc. of the Royal Society, 1964, London, Series A, 277, 51 75
[8] R T Hartlen, I G Currie. Lift-Oscillator Model of Vortex-Induced Vibration. Proc. ASCE,
J. Engrg. Mech., 1970, 96, 577 591
[9] R A Skop, O M Griffin. A Model for the Vortex-Excited Resonant Response of Bluff
Cylinder. J. Sound Vibration, 1973, 27, 225 233
[10] A B Poore. On the Theory and Application of the Hopf-Friedrichs Bifurcation Theory.
Arch. Rational Mech. and Anal., 1976, 60(4): 371 393
[11] A B Poore, A R Al Rawi. The Dynamic Behavior of the Hartlen-Currie Wake Oscillator
Model, Wind Engineering Proceedings of the International Conference, ed. J. E. Cermark,
pp 1073 1083, Oxford, Pergamon Press, 1980
[12] D M Griffin, R A Skop, G H Koopmann. The Vortex-Excited Response of Circular
Cylinders. J. Sound Vibration, 1973, 31, 235 249
[13] I Goswami, R H Scanlan, N P Jones. Vortex-Induced Vibration of Circular Cylinders, II,
New Model. J. Engrg. Mech. ASCE, 1993, 119(11): 2288 2302
[14] I Goswami, R H Scanlan, N P Jones. Vortex-Induced Vibration of Circular Cylinders, I.
Experimental Data, Ibid, 1993, 119(11): 2270 2287
[15] M P Paidoussis. Flow-Induced Instability of Cylindrical Structures. Applied Mechanics
Review, 1987, 40, 175
[16] M P Paidoussis, N T Issid. Dynamic Stability of Pipes Conveying Fluid. J. Sound
Vibration, 1974, 33(3): 267 294
[17] R W Gregory, M P Paidoussis. Unstable Oscillation of Tubular Cantilevers Conveying
Fluid, I. Theory, II. Experiments. Proc of the Royal Society, London, 1966, Series A, 293,
521 542
[18] A K Bajaj, P R Sethna, T S Lundgren. Hopf Bifurcation Phenomena in Tubes Carrying a
Fluid. SIAM J. Appl. Math., 1980, 39(2): 213 230
[19] J E Marsden, M McCraken. The Hopf Bifurcation and Its Applications. New York:
Springer-Verlag, 1976
[20] P J Holmes. Bifurcations to Divergence and Flutter in Flow-Induced Oscillations: A
Finite Dimensional Analysis. J. Sound Vibration, 1977, 53(4): 471 503
[21] S S Chen. Flow-Induced Vibration of Circular Cylindrical Structures. Washington:
Hemisphere Publishing Cor., 1987
[22] J D Jin. Stability and Chaotic Motion of a Restrained Pipe Conveying Fluid. J. Sound
Vibration, 1997, 208(3): 427 439
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[24] D Tang, E H Dowell. Experimental and Theoretical Study on Aeroelastic Response of
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[25] B H K Lee, L Liu, K W Chung. Airfoil Motion in Subsonic Flow with Strong Cubic
Nonlinear Restoring Forces. J. Sound Vibration, 2005, 28(2): 699 717
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Investigation of Limit Cycle Oscillations of Nonlinear Aeroelastic Systems. J. of Aircraft,
2002, 39(1): 133 141
[27] J P Den Hartog. Mechanical Vibrations, Fourth Edition. New York: McGraw-Hill, 1956
[28] K B M Q Zaman, D J Mckinzie, C L Rumsey. A Natural Low-Frequency Oscillation of
the Flow Over an Airfoil near Stalling Conditions. J. Fluid Med., 1989, 202, 403 442
[29] G V Parkinson, J D Smith. The Square Prism as an Aeroelastic Nonlinear Oscillator.
Quarterly J. of Mech. and Appl. Math., 1964, 17, 225 239
[30] R H Scanlan. The Action of Flexible Bridges Under Wind, I: Flutter theory. J. Sound
Vibration, 1978, 60(2): 187 199
[31] S S Chen. Instability Mechanism and Criteria of a Group of Circular Cylinders Subjected
to Cross Flow, Part I: Theory. J. of Vibration, Acoustics, Stress, and Reliability in Design,
1983, 105(1): 51 58
[32] R D Blevins. Flow-Induced Vibration. New York: Van Nostrand Reinhold Company,
1977
[33] L Meirovitch. Methods of Analytical Dynamics. New York: McGraw-Hill, 1970
[34] H J Connors. Fluid Elastic Vibration of Tube Arrays Excited by Cross Flow. Paper
Presented at the Symposium on Flow Induced Vibration in Heat Exchangers, 1970,
ASME Winter Annual Meeting
[35] S J Price, M P Paidoussis. A Single Flexible Cylinder Analysis for the Fluid Elastic
Instability of an Array of Flexible Cylinders in Cross-Flow. J. Fluid Engineering, 1986,
108(2): 193 199
[36] H Tanaka, S Takahara. Fluid Elastic Vibration of Tube Array in Cross-Flow. J. Sound
Vibration, 1974, 77(1): 19 37
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Subjected to Liquid Cross-Flow. J. Sound Vibration, 1981, 78(3): 355 381
301
Chapter 10 Self-Excited Oscillations in Feedback
Control System
Figure 10.1 A Heating Control System (a) Structural Drawing (b) Block Diagram
governing the temperature variation in the closed room. The equation is obtained
from the conservation law of heat energy in physics and its mathematical description
may be reduced to
ChT KT W, (10.1)
where T is the room temperature, Ch is the heat capacity of the room, K is the
dissipative coefficient of the heat, and W is the heat power of the furnace.
Since the variation of the room temperature is very slow, the dynamic process
in the controller may be ignored. Thus, the on-off relay controlling the room
temperature is considered as a nonlinear inertialess element, whose input-output
characteristic is shown in Fig. 10.2, where W0 is a constant power and 'uc is the
half width of hysteresis.
According to the block diagram shown in Fig. 10.1(b), the activation voltage
of the controller is
K W0
D , E , (10.4)
Ch Ch
and substitute Eq. (10.2) and (10.3) in Eq. (10.1). Two differential equations of
304
10.1 Heating Control System
T DT E , if T ! 0, T T r 'T c ; (10.5)
2. Halting heat stage
T DT 0, if T 0, T ! T r 'T c ; (10.6)
where 'T c is the maximum temperature deviation, whose magnitude is determined
by the formula
'uc
'T c . (10.7)
K1
Hence, the mathematical model of the heating control system with on-off relay
consists of Eqs. (10.5) and (10.6).
If the initial temperature T0 is lower than the reference temperature Tr, the system
is in the heating stage. Here, integrating Eq. (10.5), we obtain its analytical
solution, i.e.,
T T 0 eD t . (10.9)
According to Eqs. (10.8) and (10.9), draw time histories in the heating stage
and the halting heat stage respectively in Fig. 10.3(a) and (b).
305
Chapter 10 Self-Excited Oscillations in Feedback Control System
Joining the curves in Fig. 10.3(a) and (b) together, depict the entire time history
of the room temperature under the controlled condition. Figure 10.4(a) shows the
case that the initial temperature T0 is lower than the reference temperature Tr,
while Fig. 10.4(b) shows the case that the initial temperature T0 is higher than the
reference temperature Tr.
Now, differentiating Eqs. (10.8) and (10.9) with respect to time t, we obtain the
first-order derivative of the room temperature in the heating stage
306
10.1 Heating Control System
The phase diagram of Fig. 10.5 shows that the amplitude of the room temperature
variation is equal to 'T c . According to expression (10.7), it is proportional to the
width of hysteresis in the input-output characteristic shown in Fig. 10.2. The
period of the self-excited oscillation in the heating control system consists of the
duration t1 from A to B and the duration t2 from C to D in the phase plane of
Fig. 10.5.
Furthermore, making use of expressions (10.8) and (10.9), we obtain the analytical
expressions of t1 and t2, respectively, i.e.,
Ch ª W0 K (T r 'T c ) º
t1 ln « » (10.14)
K ¬W0 K (T r 'T c ) ¼
and
Ch ªT r 'T c º
t2 ln « ». (10.15)
K ¬ T r 'T c ¼
By summing these two expressions, the expression of the period of the self-
excited oscillation in the heating control system is obtained, i.e.,
Let us return to Fig. 10.4 and Fig. 10.5. The curve section from A to B on the time
history and the phase path describe the temperature variation of the controlled room.
From A to E, the room temperature T is lower than the reference temperature Tr
and the heating effort is the negative feedback, while from E to B, the room
temperature T is higher than the reference temperature Tr and the heating effort is
the positive feedback. This positive feedback is introduced by the hysteresis
nonlinearity in the input-output characteristic shown in Fig. 10.2. It is known that
the negative feedback benefits the stability of the feedback system, while the
positive feedback may make the equilibrium position lose stability. Taking a look
at the time history shown in Fig. 10.4, the alternative negative and positive
feedbacks with nearly the same intensity force the heating control system to work
in the self-excited oscillation state.
Thus, we conclude that there are alternative negative and positive feedbacks in
the heating control system controlled by on-off relay with hysteresis. Moreover,
it is an excitation mechanism leading to self-excited oscillation.
A simple position control system is shown in Fig. 10.6[1]. The electrical connections
are drawn by the solid lines and the mechanical connections are drawn by the dashed
lines. With a position command dialed into the system, the input potentiometer
converts this mechanical motion into an electric signal, Vc, a representative of the
command. A gear connection on the output shaft transfers the output angle To to
a shaft and the shaft feeds the sensed output Tf back. The feedback shaft is
308
10.2 Electrical Position Control System with Hysteresis
Frictions and backlashes always exist in the interfaces of the moving parts of
the position control systems. For analytical investigation, these physical factors
are usually described with the piecelinear functions and are called the static
characteristic of nonlinear systems. Depending on the structure of the mechanical
system, the influence of the friction and the backlash between the sliding surfaces
on the motion of the system is often simplified as a hysteresis nonlinearity, or a
dead-zone nonlinearity, or the combination of hysteresis and dead-zone, as shown
in Fig. 10.7(a), (b) and (c).
In general, the nonlinear factors are described by static characteristic and are
embedded in the closed loop of the electrical position control system. Furthermore,
309
Chapter 10 Self-Excited Oscillations in Feedback Control System
the transfer function is used to describe the dynamic process. Here, the electrical
position control system is represented with a block diagram, as shown in Fig. 10.8.
According to the block diagram shown in Fig. 10.8, the equations of motion of
all elements in the electrical position control system are established and combined
in a mathematical model.
First, lets us establish the differential equation of motion of the dc motor and
its inertial load, i.e.,
M K1u1 , (10.18)
where u1 is the output voltage of the amplifier and K1 is the amplification co-
efficient of the motor. Here, we directly write the equation of the amplifier, i.e.,
u1 K 2 'T , (10.19)
where K 2 is the amplification coefficient and 'T is the derivation between the
desired angle T c and the sensed output angle T f , namely,
'T Tc T f . (10.20)
Next, make use of the hysteresis characteristic to describe all nonlinear factors
in the electrical position control system. According to Fig. 10.7(a), we have the
piecelinear function
Tf F1 (T ,T) (10.21)
310
10.2 Electrical Position Control System with Hysteresis
with
T ', if T ! 0
F1 (T ,T) ®
¯T ', if T , (10.22)
T 0,
if | F1 (T ,T ) T | '
where ' is the half width of the hysteresis.
In accordance with the motion in the self-excited vibration, with T c 0 , we
introduce the following parameters
J K1K 2 t
T , K , W .
c c T
(10.23)
T F1 (T ,T)
x , z .
' '
Then, with the dimensionless time W, Eqs. (10.17) (10.21) are simplified as
xc y, y c D z y, z c f ( x, y, z ), (10.24)
where the prime represents the derivative with respect to the dimensionless time W,
f ( x, y, z ) is the analytical function, and D is the system parameter, namely,
D TK. (10.25)
At last, since the function f ( x, y, z ) is a piecelinear function, it is described in
some regions in the phase space, namely,
H z x 1, if y 0
(10.26)
H zx 1, if y ! 0,
f ( x, y, z ) 0, if | z x | 1 (10.27)
where H and H are two half-planes in the phase space.
Now, the mathematical model of the electrical position control system has been
established. It consists of Eqs. (10.24), (10.26), and (10.27). All these equations
are linear ones and the electrical position control system shown in Fig. 10.6 is a
piecelinear dynamic system.
Since the system equations are linear differential equations, all phase paths of the
position control system with hysteresis may be directly obtained by their analytical
integration.
First, the phase space (x, y, z) is divided into four subspaces by half-planes H1
311
Chapter 10 Self-Excited Oscillations in Feedback Control System
and H2 and the horizontal coordinate plane y 0, as shown in Fig. 10.9. The
half-planes H1 and H2 intersect with the horizontal coordinate plane y 0. The
intersection lines are L1 and L2, whose equations are described in the form
L1 : z x 1, y 0, (10.28)
L2 : z x 1, y 0. (10.29)
Next, let P1 denote the initial point of the phase path at line L1. Then, line L1
becomes a set of the initial points of all phase paths. As the initial points depart
from line L1, after duration time W1, they reach at point P2 in the half-plane H2, as
shown in Fig. 10.9. According to Eq. (10.27), the phase path set P q
1 P2 is in a
parallel plane set perpendicular to the axis Z. A curve S1 in the half-plane H2 is
constructed by the point set of P2. Integrating the differential Eqs. (10.24), and
taking account Eq. (10.27), we obtain the parameter equations of the curve S1, i.e.,
2(1 e W1 ) 2
y2 , z2 . (10.30)
e W1 W 1 1 D (1 W 1 eW )
1
Eliminating the duration time W1 from the previous equations, we obtain mapping
expressions from P1 to P2, i.e.,
§ y D z1 · y2 D z1
M 1 : exp ¨ 2 ¸ , z2 z1 . (10.31)
© D z1 ¹ D z1
At last, choose the point P2 on the curve S1 as the initial point of the phase path
departing from S1 to point P3 on the line L2, as shown in Fig. 10.9. According to
q
Eq. (10.26), the phase path P2 P3 is in the half-plane H2. Integrating the differential
Eq. (10.24) and taking account Eq. (10.26), we obtain the mapping expression
312
10.2 Electrical Position Control System with Hysteresis
¯ ¬« z2 (4D 1) 2 (4D 1) 2 ¼» ¿°
(10.32)
It is clear that system Eqs. (10.24), (10.26), and (10.27) are all symmetric about
the origin O of the phase space. Therefore, the limit cycle of this system, if it
exists, is also symmetric about O.
Assuming that P1 on L1 and P3 on L2 lie in the limit cycle, their coordinates
must satisfy the following equations
x3 x1 , y3 y1 , z3 z1 .
Considering that points P1 and P3 are constrained on lines L1 and L3, the necessary
condition for the occurrence of the limit cycle is reduced to the sole equation
z3 z1 , (10.33)
which is utilized to determine the condition of the occurrence of the limit cycle.
Given the magnitude of parameter D, selecting a group of values of y2 and
using Eqs. (10.31) and (10.32) to calculate the corresponding coordinates z1 and
z3, we obtain two sets of points (y2, z1) and (y2, z3). Then, by using all of them,
draw the curve G1 ( z1 , y2 ) 0 corresponding to Eq. (10.31) and the curve
G2 ( y2 , z3 ) 0 corresponding to Eq. (10.32), as shown in Fig. 10.10. This is called
Lamery diagram. If the parameter D is large enough, these two curves intersect at
point Q, which corresponds to a limit cycle in the phase plane. The directions of
313
Chapter 10 Self-Excited Oscillations in Feedback Control System
the arrows on the staggered lines show the convergent processes of the phase
paths toward the limit cycle. Obviously, the limit cycle is a stable one and the
self-excited vibration occurs in the electrical position control system. According
to the coordinates of the intersection Q, y 2 and z1 , the amplitude of the limit
cycle may be found. Then, by calculating the duration time W1 and W2, the period
of the self-excited vibration is determined.
Let us review Fig. 10.11. The critical condition for the occurrence of the limit
cycle is that the point P1 is at a special point N1, which is the intersection of line
L1 and axis x. According to Eqs. (10.31) and (10.32), draw the mapping curves
from P1 to P2 and from P2 to P3 , whose equations are G1 ( z1 , y 2 ) 0 and
G2 ( z2 , y3 ) 0 respectively. When parameter D is more than 3.04, they intersect
in the region | z1 | ! 1, and when parameter D is less than 3.04, they intersect in
the region | z1 | 1 , as shown in Fig. 10.12(a) and (b). Under the critical parameter
condition D 3.04, they intersect on the line | z1 | 1 , as shown in Fig. 10.12(c),
and the limit cycle reaches the minimum one. Thus, substituting (10.23) in (10.25)
and denoting D 3.04, we obtain the critical parameter condition for the occurrence
of the self-excited vibration in system (10.24), i.e.,
K1 K 2 J 3.04c 2 , (10.34)
in which the parameters K1, K2, J, and c are the same as those in expression (10.23).
In order to reduce the occurring tendency of the self-excited vibration in the
electrical position control system, we may raise the viscous-damping coefficient
of the system, decrease the moment of inertia of the dc motor and the load, and
lower the amplification coefficients of the electric circuit.
314
10.3 Electrical Position Control System with Hysteresis and Dead-Zone
'
J , (10.35)
H
where H is the half width of the dead-zone and ' is the half width of the
hysteresis.
Taking into account the dead-zone nonlinearity shown in Fig. 10.7(c), the
equations of motion, i.e., Eqs. (10.24), are transformed into the form
x y, y D z y, z f1 ( x, y, z ), (10.36)
where f1 ( x, y, z ) is still a piecelinear function and is represented in six regions
respectively, namely,
315
Chapter 10 Self-Excited Oscillations in Feedback Control System
z x 1 J 1 0, if y ! 0, (10.37)
°
°z x 1 J
1
0, if y ! 0, (10.38)
f1 ( x, y, z ) ®
°z x 1 J
1
0, if y 0, (10.39)
°¯ z x 1 J 1 0, if y 0, (10.40)
and
f1 ( x, y, z ) 0, if z ! 0, 1 J 1 x z 1 J 1 ,
(10.41)
f1 ( x, y, z ) 0, if z 0, 1 J 1 x z 1 J 1.
These equations show that the electrical position control system with the hysteresis
and the dead-zone nonlinearity is still a piecelinear dynamic system.
First, according to Eqs. (10.37) (10.41), construct four half-planes H1, H2, H3,
and H4 in combination with the coordinate plane z 0. They divide the phase
space (x, y, z) into six regions, as shown in Fig. 10.13. The equations of these
four half-planes are
H1 : z x 1 J 1 , y İ 0,
H 2 : z x 1 J 1 , y İ 0,
(10.42)
H 3 : z x 1 J 1 , y ı 0,
H 4 : z x 1 J 1 , y ı 0.
316
10.3 Electrical Position Control System with Hysteresis and Dead-Zone
These four half-planes intersect with two coordinate planes, y 0 and z 0, and
yield eight half-straight lines, L1 L8, as shown in Fig. 10.13. The equations of
these lines are
L1 : zx 1 J 1 , y 0, z İ 0,
L2 : zx 1 J 1 , y 0, z İ 0,
L3 : zx 1 J 1 , z 0, y ı 0,
L4 : zx 1 J 1 , z 0, y ı 0,
(10.43)
L5 : zx 1 J 1 , y 0, z ı 0,
L6 : zx 1 J 1 , y 0, z ı 0,
L7 : zx 1 J 1 , z 0, y İ 0,
L8 : zx 1 J 1 , z 0, y İ 0.
Next, taking into account Eqs. (10.41), we integrate Eqs. (10.36) and obtain
the mapping expressions from the point P1(x1, y1, z1) on the line L1 to the point
P2(x2, y2, z2) in the half-plane H2, namely,
2 2(1 eW1 )
z1 , y2 . (10.44)
D (1 W eW1 ) eW1 W 1 1
where W 1 is the duration from P1 to P2.
The set of point P2 constructs a curve S1 in the plane H2, as shown in Fig. 10.13.
At last, consider a phase path departing from L1, which intersects with S1, L3,
L4, and L5 at P2, P3, P4, and P5 sequentially. The phase path is divided into four
sections, namely, P q q q q
1 P2 , P2 P3 , P3 P4 , and P4 P5 , as shown in Fig. 10.13. Taking
into account Eqs. (10.41), (10.38), and (10.39), lets us integrate Eqs. (10.36) and
obtain four mapping expressions in accordance with P q q q q
1 P2 , P2 P3 , P3 P4 , and P4 P5 ,
respectively, i.e.,
§ y 2· y2 D z1
M 1 : exp ¨ 2 ¸ , z2 z1 , (10.45)
© D z1 ¹ D z1
1 ª 1 § 2 y2 z 2 S ·º
M 2 : y3 ( y22 z2 y2 D z22 ) 2 exp « ¨¨ arctan ¸ » , (10.46)
«¬ 4D 1 © z2 4D 1 ¸¹ »¼
M 3 : y4 y3 2J 1 , (10.47)
1 ª 1 § S 1 ·º
M 4 : z5 E1 y4 , E1 exp « ¨ arctan ¸» . (10.48)
D ¬ 4D 1 © 4D 1 ¹ ¼
317
Chapter 10 Self-Excited Oscillations in Feedback Control System
W1 ln ª¬D z1 ( y2 D z1 )1 º¼ ,
2 ª 2 y2 z 2 Sº
W2 «arctan »,
4D 1 «¬ z2 4D 1 2 »¼
(10.49)
W 3 ln ª¬ y3 ( y3 2J 1 )1 º¼ ,
2 §S 1 ·
W4 ¨ arctan ¸.
4D 1 © 2 4D 1 ¹
M M 1 M 2 M 3 M 4 . (10.50)
Since the system equations, namely, Eqs. (10.36) (10.41), are symmetric about
the origin O of the phase space, if expression (10.50) can yield the result
z5 z1 , (10.51)
the existence of a limit cycle in the phase space is confirmed, as shown in Fig. 10.14.
318
10.3 Electrical Position Control System with Hysteresis and Dead-Zone
y2 Kz2 . (10.52)
Solving the equation systems (10.45) and (10.52), we obtain an explicit expression
of function z1(K), i.e.,
2
z1 K . (10.53)
ªK § K ·º
D« ln ¨1 ¸ »
¬ D © D ¹¼
2 1
ª 1 § 1 2K S ·º ½
° ( K K D ) 2
exp « ¨ arctan ¸» °
° ¬ 4D 1 © 4D 1 ¹ ¼ 1 °
z5 ( K ) 2 E1 ® ¾ (10.54)
° § K· J°
K D ln ¨1 ¸
°
¯ © D¹ °
¿
with
1 ª 1 § S 1 ·º
E1 exp « ¨ arctan ¸» . (10.55)
D ¬ 4D 1 © 4D 1 ¹ ¼
According to expressions (10.54) and (10.55), the following inequalities are
verified
d( z1 ) 2K
1. ! 0, K [D , 0] ,
dK ª § K· º
( K 2) «D ln ¨1 ¸ K »
¬ © D¹ ¼
2 E1
2. lim ( z1 z5 ) !0,
K of J
3. lim( z1 z5 ) f.
K o0
These inequalities confirm that the geometric patterns of all curves of –z1(K)
and z5(K), i.e., curves C1 and C2, are classified into three types, as shown in
Fig. 10.15(a), (b), and (c). In figure (a), since there are two intersections of C1
and C2, there are two limit cycles in the phase space. In the figure (b), the curve
C1 is tangential to the curve C2 and there is only one limit cycle. In the figure (c),
the curve F1 does not intersect with the curve C2. Thus, there is no limit cycle.
Then, draw the staggered curve in the neighborhood of the intersection of
curves C1 and C2 in Fig. 10.15. The Lamery diagrams of the electrical position
319
Chapter 10 Self-Excited Oscillations in Feedback Control System
control systems are constructed. According to the directions of the arrows on the
staggered lines, we may conclude that (1) the limit cycle L1 corresponding to
the intersection Q1 is unstable; (2) the limit cycle L2 corresponding to the
intersection Q2 is stable; and (3) the limit cycle L corresponding to the tangent
point Q0 is semistable.
320
10.3 Electrical Position Control System with Hysteresis and Dead-Zone
First, substituting the analytical expressions of coordinates z1 and z5, i.e., Eqs. (10.53)
and (10.54), in the necessary condition to form the limit cycle, Eq. (10.51), we
obtain a transcendental equation of parameters D, J , and K, i.e.,
J I (D , K ),
in which
ªK § K ·º ª 1 § 1 S ·º
D« ln ¨ 1 ¸ » exp « ¨ arctan ¸»
¬D © D ¹¼ ¬ 4D 1 © 4D 1 ¹ ¼
I (D , K ) .
1
ª 1 § 1 2K 1 ·º
( K K D ) exp «
2 2
¨ arctan arctan S ¸» D
¬ 4D 1 © 4D 1 4D 1 ¹¼
(10.56)
Next, given a group of values for parameter K, by numerical calculation, we
obtain a set of relative curves between parameters D and J, corresponding to
different values of parameter K, as shown in Fig. 10.17. The figure shows that
these curves have an envelop E that is the boundary of the stable region in the D-J
plane. This envelop E possesses an asymptote, namely, a straight line D 3.04, and
the analytical result is entirely consistent with the previous conclusion.
At last, we conclude the main influence of the dead-zone of the static characteristic
on the motion in the electrical position system as follows.
1. The dead-zone of static characteristic causes the position control system to
possess a stable equilibrium interval. If the self-excited vibration occurs in this
system, it is induced by hard excitation. In this case, there are two limit cycles in
the phase diagram, as shown in Fig. 10.16(a). The inner one is unstable and the
outer one is stable.
2. According to the configuration of the boundary of the stable region in
321
Chapter 10 Self-Excited Oscillations in Feedback Control System
the D -J plane, i.e., the envelop E, as shown in Fig. 10.17, if the width of the dead-
zone increases, the critical value of the parameter J decreases. Here, the critical
value of parameter D, which guarantees the stability of the equilibrium of the
electrical position control system, increases as well. Thus, enlarging the width of
the dead-zone of the static characteristic is of benefit to prevent the electrical
position control system from self-excited vibration.
Fig. 10.18 shows a simplified sketch of an early hydraulic piston control system
to control elevators[4]. The input link drives the lever to rotate and the lever drives
the slide valve along the direction of the input link. When the input link is displaced
to the right, the oil under constant pressure flows from the center valve up into the
left cylinder chamber. This operation causes the piston to move the inertial load
to the right. As the piston moves to the right, it expels the oil in the right cylinder
chamber down through the right return valve, which opens when the slide valve
moves to the right. The oil thus returns to the main supply. Simultaneously, the
piston moves to the right, and the lever rotates around the terminal point of the
input link and forces the slide valve to move to the right. The displacement is just
a negative feedback effect and the hydraulic position control system shown in
Fig. 10.18 is actually a closed-loop control system. Input motion to the left makes
the load move to the left in a similar manner.
Hence, the controlled motion of the hydraulic position control system may be
described with a block diagram, as shown in Fig. 10.19.
As the working medium, the oil in a hydraulic actuator with higher pressure always
displays remarkable compressibility, and it can delay the motion transmission
and decrease the stability of the system. Hence, the compressibility of the oil is
an important factor to be considered when we establish the equations of motion
of the hydraulic position control system operating in high-pressure conditions.
First, establish the equation relative to the displacement of the valve stem, the
flow rate of the oil, and the pressure difference between the cylinder chamber
and the main supply. This is normally found by some testing method.
In order to describe the relationship between the flow rate of the oil and the
relative displacement of the valve stem about the valve port, we install the hydraulic
actuator on a special testing rig. In the experiment, the pressure difference between
the cylinder chamber and the main supply is maintained to be constant by this rig.
After measuring the pressure difference and the relative displacement of the valve
stem and the valve port repeatedly, we obtain a large number of testing data to
draw the relation curve of the oil flow rate and the relative displacement between
the valve stem and the valve port. The curve is called flow rate curve of the slide
valve. It is shown in Fig. 10.20(a) and (b) for the slide valve without or with
overlap respectively.
Then, change the pressure of the main supply repeatedly and obtain a group of
flow rate curves shown in Fig. 10.20. These curves fit the nonlinear function
Q1 Q1 ( E , P0 P1 ), if E ! 0, (10.57)
323
Chapter 10 Self-Excited Oscillations in Feedback Control System
where Q1 is the flow rate of the oil up into the left chamber, E is the relative
displacement between the valve stem and the valve port, P0 is the pressure of the
main supply, and P1 is the pressure in the left chamber.
Since the geometry of the right channel is the same as the one of the left
channel, we have
Q2 Q2 ( E , P2 P3 ), if E 0, (10.58)
where Q2 is the flow rate of the oil up into the right chamber, P2 is the pressure in
the right chamber, and P3 is the atmospheric pressure.
Second, define the bulk elasticity modulus of the oil, i.e.,
V 'P
N ,
'V
where V is the volume filled up with the oil, 'P is the variation of the oil
pressure, and 'V is the variation of V caused by 'P . In this expression, replacing
'P
by the derivative of P with respect to time V yields
'V
VdP
N . (10.59)
dV
Obviously, the mass of the oil filling up V may be described as M UV , where
U is the density of the oil. Differentiate the expression with respect to time t and
designate number 1 and 2 as the indexes of the variables in the left and the right
chambers of the hydraulic cylinder. We obtain
M 1 U Q1 , M 2 U Q2 .
Substituting the above expression in Eqs. (10.60) yields
V1 U V2 U
Q1 V1 , Q2 V2 . (10.61)
U U
The equivalent length of the cylinder is defined as
V1 V2 V0
L0 ,
S0
where V0 is the total volume of the oil channel from the valve port to the cylinder
324
10.4 Hydraulic Position Control System
and S0 is the effective area of the piston. Simultaneously, through let x1 denote
the displacement coordinate of the piston, and its zero point is corresponding to a
particular configuration, namely the piston just passes the center of the hydraulic
cylinder, so that we have
§ L0 · § L0 ·
V1 ¨ 2 x1 ¸ S0 , V2 ¨ 2 x1 ¸ S0 .
© ¹ © ¹
Substituting these expressions in Eqs. (10.61) and considering the expression
(10.59), we obtain
S0 ( L0 2 x1 ) S0 ( L0 2 x1 )
Q1 P1 S0 x1 , Q2 P2 S0 x1 . (10.62)
2N 2N
Next, write the equation of motion for the assembly of the piston and the
inertial load, i.e.,
mx1 S0 ( P1 P2 ) R(t ), (10.63)
where m is the equivalent mass of the assembly directly connected with the
piston of the hydraulic actuator and R(t) is the external force exerted on the
assembly.
Establish the equation of motion for the feedback lever. Neglecting its inertia
and considering that the pivot is usually positioned at the midpoint of the lever as
shown in Fig. 10.18, we have
x2 2 x1 x0 , (10.64)
where x2 is the displacement of the valve stem and x0 is the displacement of the
input link. As the valve port is integrated with the piston body, the relative
displacement of the valve port about the valve stem is
E x1 x2 .
Substituting Eq. (10.64) in the above equation, we obtain the equation of motion
of the inertialess feedback lever, i.e.,
E x0 x1 . (10.65)
At last, combine the equation system, namely, (10.57), (10.58), (10.62), (10.63),
and (10.65), with the nonlinear mathematical model of the hydraulic position
control system shown in Fig. 10.20. Since Eqs. (10.57), (10.58), and (10.62) are
nonlinear, the linearization operation on them is necessary.
Q10 0, Q20 0, E0 0,
Q10 0, Q20 0, E0 0,
P0 P3 R0 P0 P3 R0 (10.66)
P10 , P20 , x10 x00 .
2 S0 2 S0
Designate Gq1, Gq2, GE, Gp1, Gp2, and Gx1 as the variations of the variables Q1, Q2,
E, P1, P2, and X1 with respect to their stationary values. Therefore, we have
On the other hand, summing two equations of (10.62), considering Q10 = Q20 = 0,
and neglecting the higher-order small terms, we obtain
S0 L0 S x
Gq1 Gq2 2S0 x1 Gp1 Gp 2 0 1 ( P1 P2 ) . (10.70)
2N N
326
10.4 Hydraulic Position Control System
In addition, the flow rates Q1 and Q2 mainly depend on the velocity of the piston
x1 . Thus, there is an approximate relation Q1 | Q 2 , and by means of Eqs. (10.57)
and (10.58), we have
P0 P1 P2 P3
or
P1 P2 P0 P3 ,
whose right side is a constant. Substituting the above equation in Eq. (10.70) and
denoting Gp Gp1 Gp2 , the last term vanishes and Eq. (10.70) is reduced to
S0 L0
Gq1 Gq2 2 S0 x1 Gp , (10.71)
N
where G p is the derivative of the pressure difference between the oil in the left
and the right chambers of the hydraulic cylinder from the stationary value.
Eliminating the variable (Gq1 Gq2) from Eqs. (10.70) and (10.71) yields
Ve Cp
Gp Gp Ce GE S0 Gx1 , (10.72)
2N 2
where Ve is the effective volume of the hydraulic cylinder, i.e., S0L0.
At last, substituting (10.67) in Eq. (10.63) and (10.65), we obtain
mG
x1 S0 Gp (10.73)
and
GE Gx1 . (10.74)
Equations (10.72), (10.73), and (10.74) are combined into a linearized mathe-
matical model of the hydraulic position control system shown in Fig. 10.18.
Here, consider the hydraulic position control system without overlap, whose flow
rate curve is shown in Fig. 10.20(a). Below is the stability analysis of the sole
isolated equilibrium point of the system.
First, eliminating the variables Gp and GE from Eqs. (10.72) (10.74), we obtain
the system equation
a0 G
x1 a1G
x1 a2 Gx1 a3Gx1 0 (10.75)
with
a0 mL0 2 N , a1 C p m S0 , a2 2 S0 , a3 2Ce . (10.76)
327
Chapter 10 Self-Excited Oscillations in Feedback Control System
a0 O 3 a1O 2 a2 O a3 0.
Next, according to Hurwitz criterion, for asymptotic stability of the system
equilibrium, the necessary and sufficient condition is that all coefficients, a0, a1,
a2, and a3, are positive and the minor Hurwitz determinant ' 2 ! 0. Actually,
expressions (10.76) show that all coefficients a0, a1, a2, and a3 are positive, and
the necessary and sufficient condition of asymptotic stability of equilibrium of
hydraulic position control system is ' 2 ! 0. Substituting expressions (10.76) in
' 2 ! 0 , we obtain the sole condition of its asymptotic stability, i.e.,
2C p N L0 Ce ! 0 . (10.77)
At last, take into account the necessary condition under which the self-excited
vibration occurs. The system equilibrium has the second type of critical stability.
In this case, the system parameters satisfy the equation ' 2 0. Hence, substituting
the expressions of (10.70) in ' 2 0 , we obtain the critical flow rate gain
Ce* 2C p N L0 . (10.78)
If the flow rate gain of the hydraulic position control system Ce is smaller than Ce* ,
namely,
Ce Ce* 2C p N L0 , (10.79)
the equilibrium position of the hydraulic position control system without overlap
is stable and the self-excited vibration does not take place in this system.
If inequality (10.79) does not hold good, self-excited vibration will occur in the
hydraulic position control system. Here, the amplitude and the frequency may be
found by the describing function method[5, 6].
The nonlinearity in the hydraulic position control system is mainly a flow rate
saturation caused by the limited width of the valve port. As long as the oil flow
reaches the maximum rate Qm, with the displacement of the valve stem about the
port increasing, the flow rate remains constant, as shown in Fig. 10.20. For
approximate analysis, a piecelinear function F(E) is used to describe the
saturation phenomena, as shown in Fig. 10.21, in which Em is the half width of
the valve port and Qm is the maximum of flow rate. With Qm Ce Em, the slope of
the linear segment in the neighborhood of the origin O is equal to 1.
328
10.4 Hydraulic Position Control System
Figure 10.22 Block Diagram of the Transfer Function of the Hydraulic Position
Control System
The block diagram of the hydraulic position control system is shown in Fig. 10.22,
whose transfer function G(s) is found from Eqs. (10.72) (10.74), i.e.,
Ce
G(s) . (10.80)
ª § L Cp · º
s « ms ¨ 0 s ¸ S0 »
¬ © 4N 2 S0 ¹ ¼
According to the piecelinear function F(e) shown in Fig. 10.21, its describing
function is
N ( A) 1, if A Em ,
½ 1
2° § Em · Em § Em · °
2 2 (10.81)
N ( A) ®arcsin ¨
S° ¸ ¨1 2 ¸ ¾ , if A ! Em .
© A¹ A © A ¹ °
¯ ¿
Then, draw the locus of [ N ( A)]1 , which lies on the negative real axis in the
complex plane and ranges in the interval (1, f) , as shown in Fig. 10.23.
Denoting s iZ in expression (10.80), we obtain the frequency response G(iZ)
of the linear part of the hydraulic position control system and the hodograph of G
(iZ) is drawn in Fig. 10.23.
Depending on the system parameters, the hodograph of G(iZ) with respect to
the locus of [ N ( A)]1 may be classified into three types.
The first type is that the hodograph of G(iZ) intersects with the locus of
[ N ( A)]1 at point K1 (b,0), b ! 1 , i.e., curve I in Fig. 10.23. In this case, there
329
Chapter 10 Self-Excited Oscillations in Feedback Control System
is a limit cycle in the phase space and the self-excited vibration occurs in the
hydraulic position control system. Designate A and Z as the amplitude and the
frequency respectively of the self-excited vibration. They can be determined by
solving the following equation system
N ( A ) b 1
and
Re G iZ b.
The second type is that the hodograph of G(iZ) and the locus of [ N ( A)]1 are
separated. The hodograph is curve II in Fig. 10.23 and there is no limit cycle in
the phase space. Consequently, the self-excited vibration does not occur in the
hydraulic position control system.
The third type is that the hodograph of G(iZ) is connected with the locus of
[ N ( A)]1 at the critical point C(1, 0) which determines a critical parameter
condition though it is approximate.
If the hydraulic actuator has overlap between the valve stem and the valve base,
and their relative displacement is less than half width of the overlap, the valve port
still closes, and consequently, the flow rate is equal to zero. Here, the hydraulic
position control system has a stable equilibrium interval (H, H), as shown in
Fig. 10.20(b). Thus, if the self-excited vibration occurs, it must be hard excited.
330
10.4 Hydraulic Position Control System
Consequently, there are two limit cycles in the phase space. The internal one is
unstable and the outer one is stable.
Popov criterion may be used to determine the necessary and sufficient condition
for the occurrence of the self-excited vibration in the hydraulic position control
system with overlap between the valve stem and the valve base[7, 8]. The block
diagram of the system is shown in Fig. 10.24 and the transfer function G(s) is
still the expression (10.80). Denoting s iZ, we obtain the frequency response, i.e.,
P Z Ǭ 0
S Z ¸ Z » ,
2S0 ¬«© 4N ¹ 4S02 ¼»
1
(10.82)
· ª§ mL0 2 · C p m 2 º
2 2 2
Ce § mL0 2
Q Z Z S « S Z Z » .
Z ¨© 4 N 0¸ ¨ 0
¹ ¬«© 4N
¸
¹ 4 S02 ¼»
331
Chapter 10 Self-Excited Oscillations in Feedback Control System
If the hodograph of G* (iZ ) intersects with the real axis of the complex plane
at point B( K 01 , 0) , we can draw an oblique line L passing through the point B.
The hodograph of G* (iZ ) lies to the right of the line L, as shown in Fig. 10.25.
According to the theorem 2 of Popov criterion, as long as the flow rate curve Q(E)
lies within the region between the axis E and the oblique line L', whose slope is
equal to K0, as shown in Fig. 10.26(a), the hydraulic position control system
possesses absolute stability. In contrast, if the flow rate curve Q(E) intersects
with the oblique line L', as shown in Fig. 10.26(b), the hydraulic position control
system is unstable.
Figure 10.26 Stable and Unstable Flow Rate Curves Judged by Popov Criterion
0, if E İ H ,
°
° § H H Em2 ·
° S¨ ¨ arcsin 1 ¸ if H İ E İ Em ,
N ( A) ® © A A A2 ¹¸
°
° § Em H Em Em2 H H2 ·
°S ¨¨ arcsin arcsin 1 1 ¸ if E ı Em .
A A A A 2
A A2 ¸¹
¯ ©
(10.83)
Here, the block diagram of the hydraulic position control system is the same as
that in Fig. 10.22 except that the nonlinear function F(E) is replaced by F1(E)
shown in Fig. 10.27.
332
10.4 Hydraulic Position Control System
In order to determine the amplitude and the frequency of the self-excited vibration,
we may draw the locus [ N ( A)]1 in the complex plane according to expressions
(10.83). The locus lies on the negative real axis, as shown in Fig. 10.28. Then,
according to expression (10.80), we draw the hodograph of G(iZ) in Fig. 10.28.
If it intersects with the locus of [ N ( A)]1 at point K(b, 0), there are two limit
cycles in the phase space, whose amplitudes are A1 and A2 respectively. Two
amplitudes can be found from the transcendental equation
N ( A) b 1 . (10.84)
The limit cycle with the smaller amplitude A1 is unstable, and the limit cycle
with the larger amplitude A2 is stable and describes the self-excited vibration.
The influence of the dead-zone in the flow rate curve of the hydraulic actuator
on the motion of the position control system can be summarized as follows.
333
Chapter 10 Self-Excited Oscillations in Feedback Control System
If there are friction and backlash in the linkage of the hydraulic position control
system, the flow rate curve of the hydraulic actuator possesses the hysteresis, the
saturation and the dead-zone. It may be described by a piecelinear function
F2(E), as shown in Fig. 10.29, whose slope of the inclined segment is equal to
one. Therefore, the transfer function of the linear part is still expression (10.80).
However, the describing function of the nonlinear function F2(E) is a complex
function with real variable A, i.e.,
N ( A) g ( A) ib( A)
with
1ª cd c Ed d Ed
g (A) arcsin arcsin arcsin arcsin
S «¬ A A A A
cd (c d ) 2 c E d (c E d ) 2 d d2 Ed E 2d 2 º
1 2
1 2
1 2 1 2 »,
A A A A A A A A »¼
2dc(1 E )
b( A) , if A ! c d ;
SA2
334
10.4 Hydraulic Position Control System
1 ªS d (1 E ) ½ d Ed
g ( A) arcsin ®1 ¾ arcsin arcsin
S «¬ ¯ A ¿ A A
d (1 E ) ½ ª d (1 E ) º
2
d d2 Ed E 2d 2 º
®1 ¾ 1 «1 1 1 »,
¯ A ¿ ¬ A »¼ A A2 A A2 »
¼
2d § d ·
b( A) ¨1 ¸ (1 E ), if d İ A İ c d ;
SA © A ¹
g ( A) 0, b( A) 0, if A İ d ;
(10.85)
where c Qm Ce , and E and d are the constants depending on the width of the
hysteresis and the dead-zone.
Here, the block diagram of the hydraulic position control system is the same as
that in Fig. 10.22 except that the nonlinear function F(E) is replaced by F2(E)
shown in Fig. 10.29.
To calculate the amplitude and the frequency of the self-excited vibration, we
select a particular case, i.e., H ' . Consequently, we have E 0. Then, according
to expression (10.85), draw the locus of [ N ( A)]1 in the complex plane, as
shown in Fig. 10.30. Besides, according to expression (10.80), draw the hodographs
of the frequency response G(iZ), as shown in Fig. 10.30.
335
Chapter 10 Self-Excited Oscillations in Feedback Control System
Figure 10.31 Phase Portrait Mapped into Two-Dimensional Space for Hydraulic
Position Control System with Four Limit Cycles
336
10.4 Hydraulic Position Control System
F2(E) shown in Fig. 10.29, and the latter is caused by the excessively large gain
of the linear part and the saturation of the flow rate curve of the hydraulic actuator.
As the open gain of the control system is large enough, the hodograph of G(iZ)
intersects with the locus of [ N ( A)]1 at points K1c and K 2c . In this case, the
hodograph is the third type curve shown in Fig. 10.30, namely, curve III. There
are two limit cycles in the phase space and they are compressed into the two-
dimensional phase plane as shown in Fig. 10.32. The internal limit cycle L1
corresponding to K1c is unstable, while the outside limit cycle L2 corresponding
to K 2c is stable.
Figure 10.32 Phase Portrait Mapped into Two-Dimensional Space for Hydraulic
Position Control System with Two Limit Cycles
because the divergent motion is limited due to the saturation of the nonlinear
element. Eventually, a steady periodic motion occurs in the closed-loop system.
Owing to the time delay in both the controller and the actuator, the actual
control force is modeled in the form
g (t ) J 0 x (t W ) , (10.87)
where J 0 is the feedback gain and W is the constant delay time.
338
10.5 A Nonlinear Control System under Velocity Feedback with Time Delay
c J0
] , J . (10.88)
2 mk mk
Then, substituting expression (10.87) and (10.88) in Eq. (10.86), we obtain
x(t ) 2] x x(t ) P x3 (t ) J x (t W )
0. (10.89)
The equation implicates various dynamic characteristics, but we only emphasize
its periodic motion, namely, the self-excited vibration.
Here, the analysis is confined to the case of small damping, weak nonlinearity,
and weak feedback. Thus, we have
0 H 1, ]ˆ O 1 , Pˆ O 1 , Jˆ O 1 .
Now, solve the Eq. (10.89) by the method of multiple scales. Since the frequency
of a bifurcation motion is unknown, we denote
Z 2 1 HV , (10.91)
where V O (1) is the detuning frequency. Thus, Eq. (10.89) is written as
339
Chapter 10 Self-Excited Oscillations in Feedback Control System
1
A(T1 ) a (T1 )eiE (T1 ) . (10.96)
2
Then, substituting Eq. (10.95) in Eq. (10.94b), we obtain
3P 2 2
2] J cos ZW 0, Z 2 1 JZ sin ZW a 0. (10.101)
4
Eliminating cos ZW and sin ZW in the previous equations yields
2
§ 2 3P a 2 ·
¨Z 1 ¸ (4] J )Z
2 2 2
0. (10.102)
© 4 ¹
Noting that | J |ı 2] , we have two branches of the solution, i.e.,
1
ª 4
2 º
2
« 3P Z 1 B Z J 4] » .
2 2
a1,2 (10.103)
¬ ¼
Solving the first equation in (10.101) for Z gives the frequency corresponding
to each branch of the solution, i.e.,
340
10.5 A Nonlinear Control System under Velocity Feedback with Time Delay
1 § 2] ·
° arccos ¨ ¸, a a1 ,
1 §] · °W ©|J |¹
Z arccos ¨ ¸ ® (10.104)
W ©J ¹ ° 1 ª 2S arccos § 2] · º , a a .
°W « ¨ ¸»
© | J | ¹¼
2
¯ ¬
Now, let us consider a system with dimensionless parameters: ] 0, J 0.5,
and P 0.1. Substituting these parameters in Eqs. (10.104) and (10.103) yields
1
° 4 ª§ S ·2 § S · º ½°
2
a1 ® «¨ ¸ 0.5 ¨ ¸ 1» ¾ ,
¯° 0.3 ¬«© 2W ¹ © 2W ¹ ¼» ¿°
1
(10.105)
° 4 ª§ 3S ·2 § S · º ½°
2
a2 ® «¨ ¸ 0.5 ¨ ¸ 1» ¾ ,
¯° 0.3 «¬© 2W ¹ © 2W ¹ »¼ ¿°
Figure 10.35 presents two trajectories of the system with time delay W 1 . The
phase trajectory initiating from motion x(t ) 1 t 2.5t 2 , t (1,0) approaches
the asymptotically stable equilibrium as shown in Fig. 10.35(a). In this case, the
self-excited vibration cannot occur in system (10.92), while the phase trajectory
initiating from motion x(t ) 10 10t 25t 2 , t (1,0) approaches an asymptotic
stable limit cycle with frequency Z 4.713 , as shown in Fig. 10.35(b). In this
case, the self-excited vibration occurs in system (10.92). The numerical result
coincides very well with the approximate solution given by Eqs. (10.103) and
(10.104), where a 17.73 and Z 4.713 .
341
Chapter 10 Self-Excited Oscillations in Feedback Control System
At the end of this chapter, let us briefly discuss some structural conditions for
the occurrence of the self-excited vibration in the closed-loop system.
First, the locus of [ N ( A)]1 of a single-valued nonlinear function, which is
similar to the configuration shown in Fig. 10.21 or Fig. 10.27, lies on the negative
real axis of the complex plane, as shown in Fig. 10.36. In this case, since the
hodograph G(iZ) can intersect with it, the locus of [ N ( A)]1 must pass through
the negative real axis of the complex plane and enter into the second quarter, as
shown in Fig. 10.36. Thus, the order number of the differential equation of motion
of the closed-loop system is not less than three. Actually, Eq. (10.75), the differential
equation of motion of the hydraulic position control system with the occurrence
of the self-excited vibration, can satisfy the structural condition and its order number
is equal to three.
Figure 10.36 Explaining the Minimum Order of the Nonlinear System without
hysteresis-loop for occurrence of self-excited vibration
Second, the locus of [ N ( A)]1 of the nonlinear function with hysteresis, which
is similar to the configuration shown in Fig. 10.29, lies in the third quarter of the
342
10.5 A Nonlinear Control System under Velocity Feedback with Time Delay
complex plane, as shown in Fig. 10.37. Since the hodograph G(iZ) can intersect
with it, the locus of [ N ( A)]1 must enter the third quarter of the complex plane,
as shown in Fig. 10.37, and the order number of differential equation of motion
of the closed-loop system is not less than two. Actually, the order number of the
equation of motion of the electric position control system is just equal to two.
Thus, the structure condition for the occurrence of the self-excited vibration can
be satisfied.
Figure 10.37 Explaining the Minimum Order of the Nonlinear System without
Hysteresis-Loop for Occurrence of Self-Excited Vibration
Third, if the constant time delay component is contained in the first-order closed-
loop system, whose hodograph of the frequency response is shown in Fig. 5.11,
when the gain of the linear part is large enough, the critical point C(1, 0) will be
surrounded by the hodograph of G(iZ), as shown in Fig. 10.38. Therefore, the
stability of the equilibrium position is lost and the self-excited vibration occurs.
In the same way, the self-excited vibration, occurring in the nonlinear control
343
Chapter 10 Self-Excited Oscillations in Feedback Control System
system under velocity feedback with time delay, is caused by delay time that is
large enough. This has been studied in detail in the previous section. Actually,
common closed-loop systems in engineering fields usually have a variety of
many factors leading to time delay, and the gain of the linear part must be strictly
restricted to a level low enough. Otherwise, the self-excited vibration will occur
in the closed-loop system.
At last, the differential equations of motion of the heating control system,
Eqs. (10.5) and (10.6), are first-order differential equations, whose hodograph of
the frequency response G(iZ) is restricted in the fourth quarter of the complex
plane. Consequently, it cannot intersect with the locus [ N ( A)]1 of the nonlinear
function. However, the self-excited oscillation occurs in it and the cause of the
paradox may be that the filtering condition (5.58) is not satisfied. In this case, the
describing function method cannot correctly predict the self-excited vibration in
the closed-loop system. In fact, the components of higher harmonics in on-off
control system always are larger, so that the describing fauction method does not
make correct result in this case.
References
[1] P Emanuel, E Leff. Introduction to Feedback Control Systems. New York: McGraw-Hill,
1979
[2] W J Ding. Second-Order System without Restoring Force with Dead Zone and Hysteresis
Loop. Journal of Tsinghua University, 1979, 19(4): 69 76 (in Chinese)
[3] M Jelali, A Kroll. Hydraulic Servo-Systems Modeling, Identification, and Control. London:
Springer, 2003
[4] R K Thomasson. Stability of Jack-Type Power Controls. Aircraft Engineering, 1962, Vol.
34, No. 401
[5] E P Popov. On the Use of the Harmonic Linearization Method in Automatic Control
Theory. NACA TM 1406, 1957
[6] D P Atherton. Nonlinear Control Engineering. New York: Van Nostrand Reinhold, 1982
[7] V M Popov. Hyperstability of Control System. New York: Springer-Verlag, 1973
[8] D R Merkin. Introduction to the Theory of Stability, New York: Springer, 1997
[9] H Y Hu, Z H Wang. Dynamics of Controlled Mechanical Systems with Delayed Feedback.
Berlin: Springer, 2002
344
Chapter 11 Modeling and Control
Abstract: This chapter presents a common procedure for studying the unclear
self-excited oscillation phenomena in physics and engineering. Since the
main method used to analyze the nonlinear autonomous equations governing
the self-excited oscillation have been introduced in the first part of this book
from Chapter 2 to Chapter 5, the remaining task of studying the unclear
self-excited oscillation problems is mainly to establish a mathematical
model that can be used to solve all proposed problems. In the second part of
this book, from Chapter 6 to Chapter 10, we analyzed five classes of self-
excited vibrations in engineering, including the stick-slip vibration, the
shimmy of the front wheels, the whirl of rotors, the fluid-induced self-excited
vibrations, and the self-excited vibration occurring in control systems. Since
system modeling always occupies an extremely important position, we
construct a feasible modeling procedure for solving new self-excited oscillation
problems by summarizing the accumulated experience in the second part of
the book. This is comprised of the following five steps.
1. Find out the excitation mechanism leading to the self-excited oscillation;
2. Determine the mechanical model, whose parts all have remarkable
influence on the self-excited oscillation;
3. Construct the mathematical expressions of the motive force;
4. Establish the equations of motion governing the self-excited oscillation;
5. Discretize the mathematical model of the distributed parameter systems
with the occurrence of the self-excited oscillation.
In this chapter, we will introduce each step in detail.
The essential goal to study the self-excited oscillation is to control it
efficiently. It is known that the application of passive damping to any
oscillation system is a reliable means to reduce the oscillatory intensity. In
particular, the active damping is usually superior to the passive damping.
Thus, we will briefly explain the active control of the self-excited oscillation
in the last section of this chapter.
Keywords: energy mechanism, feedback mechanism, minimal model, extended
model, motive force, Lagrange’s equation, Hamilton’s principle, lumped
parameter system, distributed parameter system, active control
Chapter 11 Modeling and Control
'ED ( A) 'EZ ( A) .
The equation shows that the work done by the nonconservative force is equal to
zero in a whole period. Thus, for a dynamic system with the occurrence of the
self-excited oscillation, the work done by the nonconservative force is positive
during some time and is negative during the other time in a full cycle.
All analysis results given in the second part of this book show that the negative
damping force and the circulatory force can do positive work to increase the
mechanical energy of the dynamic system. Therefore, we conclude that mechanical
systems with the occurrence of the self-excited oscillation must possess the
negative damping force or (and) the circulatory force. In fact, the chatter and the
stall flutter may be excited by the negative damping force, while the oil whirl and the
classical flutter are induced by the circulatory force. For simplicity, we combine
the negative damping force and the circulatory force together and refer to them
as generalized negative damping. Now, we can say that the self-excited vibration
in a mechanical system is induced by the generalized negative damping.
346
11.1 Excitation Mechanism of Self-Excited Oscillation
Here, let us recall the analysis about the heating control system, whose self-excited
oscillation is induced by the alternative positive and negative feedbacks. The
positive feedback makes the room temperature increase while the negative
feedback makes the room temperature decrease. Thus, the room temperature is
maintained in a steady oscillation with small amplitude.
In general, feedback control systems are constructed on the negative feedback
rationale and the equilibrium position is always asymptotically stable. The
occurrence of the positive feedback is a necessary condition to induce the self-
excited oscillation. The positive feedback comes into being in various systems
with the occurrence of the self-excited oscillation. For example, the positive
feedback in the hydraulic position system, which was studied in the previous
chapter, is caused by the delay action of the information in the system elements.
However, the realization of the positive feedback requires the transferred information
to possess the delay action long enough. Therefore, there are certain structural and
parameter conditions for the occurrence of the self-excited oscillation in dynamic
systems. The condition is discussed here.
If the feedback information is a harmonic function, the time delay action is
directly turned into a phase lag. When the phase lag reaches S, the negative
feedback is entirely transformed into the positive one. As the phase lag is more
that S/2, the component of the positive feedback will be contained in the feedback
information. By Fourier analysis, the arbitrary function of time may be transformed
to its frequency spectrum, in which if the component of the positive feedback is
superior to that of the negative feedback, the closed-loop system possesses the
features of the positive feedback and the equilibrium is unstable. By completing
the analysis about some physical examples, this view point will become clearer.
Now, we consider a typical closed-loop system, the hydraulic position control
system studied in the last chapter, whose block diagram is depicted in Fig. 11.1(a).
Here, the frequency response of the open loop is represented as CeG(iZ) with
G(i0) 1. Its hodograph is drawn in Fig. 11.1(b). If the amplification coefficient
Ce is equal to the critical value, the hodograph of CeG(iZ) passes through the
critical point C(1, 0), that is, curve I. With the amplification coefficient
increasing, the hodograph of CeG(iZ) will envelop the critical point C, namely,
curve II, on which there is an intersection P called the phase crossover point, as
shown in Fig. 11.1(b). It defines a system parameter Zc known as the phase
crossover frequency[1].
Let us consider a harmonic component of the negative feedback x1 with
x1 sin Z c t . According to the definition of the frequency response of the open
loop CeG(iZ), the harmonic response in the forward channel of the closed system
can be found, i.e.,
Previous discussion shows that if the time delay action due to the inertia, the
damping and other factors in the mechanical system is large enough, the negative
feedback will be converted into the positive feedback.
Here, let us consider a simple mechanical system studied in Chapter 6. The
block diagram of the system is depicted in Fig. 11.2, in which there are two
information channels, the forward channel and the feedback channel. The
information in the forward channel is the friction force, while the information in
the feedback channel is the relative velocity between the lump mass and the
translating belt, which causes the transition from static friction to kinematic
friction and vice versa. Actually, the sharp drop of the friction resistance leads to
the emergence of the limit cycle in the phase plane shown in Fig. 6.5. In accordance
with it, the chatter phenomenon occurs in some flexible transmission devices
with friction.
348
11.1 Excitation Mechanism of Self-Excited Oscillation
Now, we may conclude that the positive feedback information is usually hidden
in the mechanical systems. For example, in the mechanical model shown in
Fig. 11.2, it occurs on the interface of two contact bodies.
With regard to the time delay in both controllers and actuators, we consider
a nonlinear control system under velocity feedback with time delay shown in
Fig. 10.33 again. When the perturbed motion is near the equilibrium x xe , the
differential equation of motion of the system of Eq. (10.89) is linearized as a
linear delay differential equation, i.e.,
J seW s
G( s) .
s 2] s 1
2
Figure 11.3 Block Diagram of a Feedback Control System with Time Delay
As pointed out above, the feedback mechanism leading to the chatter has been
represented in the block diagram in Fig. 11.2. The feedback mechanism that causes
the self-excited oscillation in various mechanical systems may also be explained
by the corresponding block diagrams[1].
Let us recall a mechanical system with the shimmy occurring in the front
wheel as discussed in Chapter 7. The block diagram is depicted in Fig. 11.4. The
information in the forward channel is the tire force, and the information in the
feedback channel is the relative motion between the tire and the road, which
governs the change of the tire force. In fact, both the inertia of the front wheel
349
Chapter 11 Modeling and Control
and the deformation of the tire cause the delay action of the transferred information
so that the positive feedback may take place in this dynamic system.
Figure 11.4 The Block Diagram describing the Shimmy of the Front Wheel
For the shimmy duration, we carefully inspect the energy variation in the front
wheel system. When the front wheel moves in the neighborhood of the equilibrium
point, the positive work done by the tire force is more than the negative work
done by the dissipative force and the motion of the front wheel is divergent.
According to the viewpoint of the system science, the front wheel just operates in
the positive feedback state. In contrast, when the motion of the front wheel is far
from the equilibrium point, the positive work done by the tire force is less than
the negative work done by the dissipative force and the motion of the front wheel
is convergent, it means that the front wheel just operates in the negative feedback
state.
It becomes clear that the energy mechanism does not come into conflict with
the feedback mechanism. The difference between them originates from different
view points only. The energy mechanism only focuses on the supplement and
dissipation of the energy in the system, while the feedback mechanism only
emphasizes the interaction of the related parts in the system. Either of them does
not repel the other. If we bring two classes of excitation mechanisms together,
we may understand the self-excited oscillations more comprehensively and more
deeply. This will be helpful in constructing the model with self-excited oscillation.
350
11.2 Determine the Extent of a Mechanical Model
Figure 11.5 The Block Diagram describing the Oil Whirl of the Rotor
351
Chapter 11 Modeling and Control
Actually, most of the mechanical models used in the investigations from Chapter
6 to Chapter 9 are the minimal models. Their analytical results consistently show
that the cause of the self-excited vibration and the influence of system parameters
on the critical speed at the onset of the self-excited vibration may be determined.
Therefore, we may conclude that the minimal model is available for the qualitative
analysis of the self-excited vibration. However, the minimal model still has some
defects since the coupled effects of the other elements are neglected. Therefore,
the critical speed determined by them, such as Eqs. (6.41), (7.40), (8.62), and (9.68),
is not accurate enough to predict the accuracy speed at the very onset of the
self-excited vibration. Hence, establishing the extended model of the self-excited
vibration system is necessary to satisfy some special requirements. In general, the
added element, which is directly connected to the vibrating body or the exciting
body, has some important effect on the motive force leading to the self-excited
vibration. Thus, there are two types of extended models. In the first type, the
added element is directly connected to the vibrating body. In the second type, the
added element is directly connected to the exciting body.
Provided that the added element is directly connected to the vibrating body in a
minimal model, the added element will exert an added force on the vibrating
body. The added force is usually related to the relative displacement and the
relative velocity between them. It contains two components, the elastic restoring
force and the damping force. As a result, the motive force applied on the interface
between the vibrating body and the exciting body varies due to the response of
the vibrating body to the added force, and the behavior of the extended model is
different from that of the original minimal model. Eventually, the effect of the
added element shows in the critical parameter condition that induces the onset of
the self-excited vibration.
Now, we consider an example used to confirm the possibility of quenching of
the galloping in ice-coated power lines by dynamic dampers. The galloping in
the power lines has been observed to vibrate with great amplitude and very low
frequency when an intense transverse wind is blowing. To suppress the violent
self-excited vibration, some dynamic dampers are often hung on the power line.
The dynamic dampers can provide stronger damping force to suppress the
galloping. In this case, the power line is the vibrating body, the transverse wind
(air flow) is the exciting body, and the dynamic damper is just the added element.
As mentioned above, the minimal model consists of a segment of the
ice-coated power line and the surrounding air flow, and the extended model is
composed of the minimal model and a dynamic damper connected to the power
line. The block diagram of the extended model is depicted in Fig. 11.6.
352
11.2 Determine the Extent of a Mechanical Model
Figure 11.6 The Block Diagram describing Galloping of Ice-Coated Power Line
The aerodynamic force exerted on the ice-coated power line has been shown in
Eq. (9.73). It can be written as
Fy ax1 bx13 ,
where x1 is the relative velocity between the vibrating body and the exciting body,
and a and b are the aerodynamic parameters describing the air damping and the
structure damping.
According to the block diagram of the extended model shown in Fig. 11.6, the
differential equations of motion are written as
m1
x1 ax1 bx13 k1 x1 c2 ( x1 x2 ) k2 ( x1 x2 ) 0,
(11.1)
m2
x2 c2 ( x2 x1 ) k2 ( x2 x1 ) 0,
where x1 is the relative displacement between the vibrating body and the exciting
body, x2 is the displacement of the dynamic damper, m1 and k1 are the reduced
mass and the reduced stiffness coefficient respectively based on the fundamental
mode of the ice-coated power line, and m2, c2, and k2 are the mass, damping, and
stiffness coefficients of the dynamic damper.
Now, introduce the following dimensionless parameters
m2 x1 x2
P , u1 , u2 ,
m1 l l
1 1
§ k1 · 2 § k2 · 2
z u1 u2 , Z1 ¨ ¸ , Z2 ¨ ¸ ,
© m1 ¹ © m2 ¹
Z2 bZ12
: , W Z1t , D ,
Z1 a
a c2
[1 1
, [2 1
,
2 2
2(m1k1 ) 2(m2 k2 )
353
Chapter 11 Modeling and Control
in which l is the span of the power line between two neighboring towers and t is
the time.
Then, the cubic term in Eq. (11.1), bx13 , is replaced by its describing function
so that the differential equations of motion are linearized. After some tedious
operation[2], two algebraic equations are eventually obtained from the complex
equation. The first equation is used to calculate the frequency of the galloping
and is written in the form
a0 : 4 , a2 : 4 (1 P ) 2: 2 (2[ 23 1),
a4 : 2 (1 P )(4[ 22 1) : 2 1.
Using this equation, we obtain the frequency of the galloping Z . Then, the second
equation is used to determine the amplitude of the galloping, and its approximate
expression can be found[2], i.e.,
1
° 4 ª P[ 2 :Z 4 º ½° 2
A ® 2 «
1 2 »¾
. (11.3)
¯° 3DZ ¬ [1 (a b ) ¼ ¿°
2
In general, the dimensionless parameters P and [ are not large, and the resonance
condition : 1 is utilized to determine the parameters of the dynamic damper[3].
In this case, expression (11.3) may be reduced to the form
1
ª 4 § P ·º 2
A « ¨1 ¸» . (11.4)
«¬ 3D © [1[ 2 ¹ »¼
The expression shows that if both the damping parameter of the power line, [1,
and the damping parameter of the dynamic damper, [2, are small and the mass
ratio P is large enough, the galloping amplitude of the power line A is remarkably
reduced by the added element, namely, the dynamic damper. In contrast, if [1
and [2 are large enough or P is very small, the galloping amplitude of the power
line A cannot be remarkably reduced by the dynamic damper. In this case, the
added element cannot exert a force that is large enough to change the motion of
the vibrating body. Hence, the effect of the added element on the motion of the
vibrating body may be neglected. In other words, if the added element is removed,
the motion of the original system is not remarkably changed and assembling the
added element to the vibrating body is ineffective. Hence, what is done above
provides a technique to establish an extended model of the dynamic system whether
the self-excited vibration occurs or not.
354
11.2 Determine the Extent of a Mechanical Model
Similar to the vibrating body in 11.2.2, if the added element is directly connected
to the exciting body in a minimal model, it will exert an added force on the
exciting body. This added force is usually related to the relative displacement
and the relative velocity between them. Thus, it contains two components, the
elastic restoring force and the damping force. As a result, the motive force applied
on the interface between the vibrating body and the exciting body varies due to
the response of the exciting body to the added force, and the behavior of the
extended model is different from that of the original minimal model. Eventually, the
effect of the added element shows in the critical parameter condition that induces
the self-excited vibration.
Now, let us consider another example that is used to inspect the effect of
foundations with the additional damping on the stability boundary of the rotor-
hydrodynamic bearing-support system. It was thought that a rotor system supported
in the fixed geometry hydrodynamic bearings would be unstable to pass through
the unstable region. This has hindered the further development of the high-speed
flexible shaft machinery applying this type of bearing. In order to solve the
problem, the displacement of the rotor foundation is taken into account. In fact, if
the elastic displacement of the rotor foundation is large enough, the dissipated
energy through the foundation has an important effect on the stability of the rotor
with a constant speed. In this case, a simple model to study the rotor whirl should
contain the bearing houses, which are supported on the flexible foundations, as
shown in Fig. 11.7(a). Here, the bearing house is the added element connected to
the exciting body, which is the oil film between the journal and the bearing house.
The block diagram of the extended model is depicted in Fig. 11.7(b). Fortunately,
this model just is the one that was established by Z. L. Guo and R. G. Kirk to
study the effect of external damping on the threshold speed of rotors[4].
355
Chapter 11 Modeling and Control
Provided the symmetric motion of the rotor, namely, the motion of the centers
of the journals and the bearing houses on the left and the right ends, maintains
synchronous planar motions all along, the rotor system with 10th degree of freedom
may be reduced to a system with 6th degree of freedom, whose equations of
motion are described in vector-matrix form, i.e.,
Cx Kx
Mx 0
with
T
xª¬ xd yd x j yj xs ys º¼ ,
ªm º
« m 0 »
« »
« mj »
M « »,
« ms »
« 0 ms »
« »
«¬ ms »¼
ª0 0 0 0 0 0 º
«0 0 0 0 0 0 »
« »
«0 0 cxx cxy cxx cxy »
C « »,
«0 0 c yx c yy c yx c yy »
«0 0 cxx cxy cs cxx cxy »
« »
¬«0 0 c yx c yy c yx cs c yy ¼»
ªk 0 k 0 0 0 º
«0 k 0 k 0 0 »
« »
« k 0 k k xx k xy k xx k xy »
K « »,
«0 k k yx k k yy k yx k yy »
«0 0 k xx k xy ks k xx k xy »
« »
«¬0 0 k yx k yy k yx ks k yy »¼
356
11.2 Determine the Extent of a Mechanical Model
k 13.72 u 106 N/m, the diameter of the journal bearing D 50 mm with the
width B 30 mm, the clearance ratio \ 2.5 u 103 , and the dynamic viscosity of
the oil P 1.92 u 102 N S / m 2 . Then, the eigenvalues of the system can be
calculated by using MATLAB, and the stability boundary of the steady rotation
of the rotor can be represented by the threshold speed of the whirl :cr and can be
drawn on the parameter plane cs - : . With the dimensionless support stiffness
k s / k 0.1 , the stability boundary is shown in Fig. 11.8(a), in which A is the
stable region and B is the unstable region. The figure shows that if the support
damping coefficient cs is not large enough, there are up to five threshold speeds
to form a regional pattern of stability, including three stable and two unstable
threshold speeds. Each stable threshold speed corresponds to a stable periodic
motion of the steady rotation of the rotor, i.e., the rotor whirl. Such stability
mapping shows that if the support damping coefficient is large enough, the first
several regions of instability are reduced or eradicated. Therefore, the magnitude
of the support damping has a strong effect on the first several lower threshold
speeds. However, it only has little effect on the top threshold speed at the back,
which is determined by the portion of the journal mass. Another stability
mapping, ks / k 1.0 , is drawn in Fig. 11.8(b). It is easy to see that when the
support stiffness increases, the stable region becomes narrow, which means that a
small external stiffness is good for broadening the range of the optimum external
damping.
Now, we can conclude that decreasing the support stiffness and (or) increasing
the support damping can raise the threshold speed of the oil whirl of the rotor. If
the support stiffness of the rotor is small enough, and (or) the support damping is
large enough, the mass of the bearing house should be included in the mechanical
model of the rotor system. Here, it is an added element connected to the exciting
body.
When the added body enters the extended model, the analysis of the model is
357
Chapter 11 Modeling and Control
much more difficult than that of the minimal model. Thus, the modeling of the
self-excited vibration system should start from the minimal model. If the critical
parameter condition has no essential variation with the addition of the added
element, we should adopt the minimal model.
358
11.3 Mathematical Description of Motive Force
the classical flutter of two-dimensional airfoil and the fluid film force in the oil
whirl of the rotor. Let us introduce them below.
1. Lift force on two-dimensional airfoils
Under the quasi-steady flow assumption, the lift force on the two-dimensional
airfoil in an ideal fluid has been found by fluid dynamics. According to
Joukowsky’s theorem, the lift force is proportional to the sine function of the
angle of attack, namely[5],
1
FL UU 2 cCL , CL 2S sin D ,
2
where FL is the lift force per unit span, U the density of local air flow, U the
velocity of the steady flow, c the chord length of the wing, CL the lift coefficient,
and D the angle of attack of the airfoil.
The testing data of the wing model in the wind tunnel shows that the derivative
dCL dD is approximately equal to 6.0. Therefore, the above expression has
sufficient accuracy for qualitative investigation of various engineering problems
and it has been successfully used to study the classical flutter of the wing.
2. Fluid film force on the surface of the circular journal
The distribution of the oil pressure around the journal surface is an interesting
subject in fluid dynamics. The Reynolds’ equation governing the pressure of the
oil film is derived from the basic equation of the viscous fluid, i.e.,
Navier-Stokes equation, by introducing a number of simplifying assumptions[6].
The closed form solutions of the oil film force have been found for two notable
cases, the long bearing and the short bearing, as shown in Eqs. (8.16) and (8.19).
After applying the general linearization routine, the components of the fluid film
force in the neighborhood of the equilibrium position of the journal may be
reduced to the linear expression of the displacement and the velocity of the
center of the journal, i.e., Eq. (8.25). This linear expression has been utilized to
find the threshold speed of the oil whirl of the rotor. Simultaneously, it also
provides a theoretical foundation to establish the equivalent model of the fluid
film force, namely, the well-known Muszynska model.
359
Chapter 11 Modeling and Control
pneumatic tire. However, it belongs to the constraint force. Thus, use of the
nonholonomic constraint equations of the rolling tire to substitute the analytical
expressions of the motive force exerted on the rolling wheel is an available
method. In fact, this method has been successfully utilized to establish the
mathematical model of the front-wheel assembly with the occurrence of shimmy,
as introduced in Chapter 7. The result is qualitatively consistent with a large
number of practical observations.
where x and y are the components of the displacement of the rotor center with
respect to the stator center, x and y are the relative velocity between them, x
and y are the relative acceleration between them, and kxx, kxy, kyx, kyy, cxx, cxy, cyx,
cyy, mxx, mxy, myx, and myy are the fluid dynamic coefficients of the fluid film force.
The entire testing data is obtained on a special testing rig, in which the stator is
hung by the elastic elements. Therefore, the equations of motion of the stator are
360
11.3 Mathematical Description of Motive Force
M s
xs ½ ° Fxc °½ ° f x °½
® ¾ ® c¾ ® ¾, (11.6)
¯ M s
ys ¿ °¯ Fy ¿° ¯° f y °¿
where M s is the stator mass, xs and ys are the components of the acceleration of
the stator, f x and f y are the components of the input excitation forces, and Fxc
and Fyc are the components of the fluid film force. Substituting Eq. (11.5) in
(11.6), we obtain
° f x M s
xs °½ ª k xx k xy º x ½ ª cxx cxy º x ½ ª mxx mxy º x½
® ¾ «k ® ¾ ® ¾ ® ¾ . (11.7)
°¯ f y M s
ys °¿ ¬ yx k yy »¼ ¯ y ¿ «¬c yx c yy ¼» ¯ y ¿ «¬ m yx m yy ¼» ¯
y¿
° Fx M s Ax ½° ª H xx H xy º X ½
® ¾ « ® ¾
H yy »¼ ¯ Y ¿
(11.8)
¯° Fy M s Ax ¿° ¬ H yx
with
where Fx, Fy, Ax, Ay, X, and Y are the Fourier transformation results of the
variables f x , f y , xs , ys , x, and y, respectively.
The exciting forces, fx and fy, are usually harmonic forces. Under steady state
condition, the data of fx, fy, xs , ys , x and y may be accurately measured by
special apparatus. Through Fourier transformation, Fx, Fy, Ax, Ay, X, and Y, which
are all complex functions of the excitation frequency Z, are found. In general, the
parameter estimation algorithm is always deduced from the least square principle
and is applied to finding the frequency response functions Hxx, Hxy, Hyx, and Hyy
in expression (11.9).
It is obvious that the stiffness and the damping coefficients of the fluid film
force may be calculated with expressions (11.9)[9]. In fact, the direct damping
coefficients, cxx and cyy, and the cross-coupled damping coefficients, cxy and cyx,
may be directly found from the imaginary part of the frequency response functions.
If the directly added mass term is included in the stiffness term, which is called
dynamic stiffness, the direct and cross-coupled dynamic stiffness coefficients may
be directly found from the real part of the frequency response functions as well.
If the exciting body is a nonsteady flow in space, the motive force cannot be
361
Chapter 11 Modeling and Control
described with an explicit function because the relationship between the force
and the state variables of the flow field is overcomplicated. Here, an available
method is that the exciting body is replaced by an equivalent oscillator that
provides a periodic force exerted on the vibrating body. The coefficients
contained in the oscillator model may be found by the testing data of the physical
model in the wind tunnel.
As mentioned before, the wake oscillator to study the vortex resonance in the
flexible structure proposed by R. E. Bishop and A. Y. Hassan is an equivalent
model of the motive force that induces the vortex resonance[10]. The differential
equation governing the motive force is given in Eq. (9.2), in which the
instantaneous lift coefficient CL is denoted as a generalized coordinate to
describe the variation of the motive force exerted on the vibrating body, though it
is essentially different from the displacement components of the vibrating body
(flexible structure) or the exciting body (nonsteady flow). The undetermined
coefficients in the oscillator model may be found by processing a great number
of testing data and using the parameter estimation algorithm.
If the quantitative relationship between the output and the input variables of a
dynamic system cannot be found by means of the basic law and the equation of
its motion cannot be established with the theoretical method, the dynamic system
may be considered as a grey box in system science. In fact, mathematical models
of many complicated control systems have been established by identification of a
grey box. Such a technique may be utilized to establish the mathematical model
for mechanical systems with self-excited vibration.
Now, we introduce an example, namely, a dynamic system composed of
machinery, cutting tool, and workpiece. As the cutting is going, it is a extremely
complicated system. Conseuently, its differential equations of motion have not
been found so far. However, its grey box model has been established with the
identification technique in frequency domain by I. E. Minis et al., which has been
used to predict the chatter in turning[11, 12].
The interaction between the machinery-tool system and the cutting process
may be described by a closed-loop system consisting of two blocks as shown in
Fig. 11.9. The dynamics in the machinery-tool system and the formation of the
workpeice is represented by a block describing the relation between the cutting
force f and the relative displacement x of the machinery-tool system, and the
dynamics of the cutting process of the workpiece is represented by another block
describing the deformation and the fracture processes of the workpiece.
The frequency curves of these two open-loop dynamic systems may be directly
obtained by simultaneously measuring the input and the output signals. They can
be transformed from time domain into frequency domain. Then, select two lower
362
11.3 Mathematical Description of Motive Force
order fractionals as the frequency response functions, whose curve matches the
data points obtained from special experiments. All undetermined coefficients in
the two selected frequency functions are usually found by the least square
method and eventually, the grey box model in frequency domain is established
by the identification technique.
In this modeling, the linear assumption cannot but be utilized to describe the
dynamics of the machinery cutting tool system and the cutting process of the
workpiece, which can be represented by the frequency responses of the cutting
force f and the relative displacement x, namely, F(iZ) and X(iZ), and their
frequency response functions G(iZ) and H(iZ) are frequency response matrices
of the machinery tool system and the cutting process respectively. The dynamics
of the system composed of the machinery and the cutting tool is approximately
described in the form
main variables that are intimately relevant to the motive force may be known by
understanding the excitation mechanism of the self-vibration formation, An
available way is to collect a variety of testing data of the motive force under
different conditions by special testing on the physical model and selecting a trial
function matching the testing data. With the undetermined coefficients found by
the least square method, the curve of the function is usually close to the points
obtained from the special testing. The resultant function is just the empiric
formula of the motive force that induces the self-exciting vibration. In the early
studies, empiric formulas were utilized in practice to describe the motive force to
study the self-excited vibration.
Now, let us recall an empiric formula introduced in Chapter 9. It represents the
motive force of the galloping in the square cylinder with flexible supports in the
wind tunnel. The first step to establish the empiric formula is to perform a large
number of model experiments in the wind tunnel and collect a great deal of
testing data, as shown in Fig. 9.23. Then, select the angle of attack as the variable
related to the lift force of the square cylinder and use a seven-degree polynomial
of the angle of attack as a trial function, as shown in Eq. (9.77), in which all
undetermined coefficients are calculated with the least square method. Consequently,
the empiric formula of the lift force to study the galloping in the flexible square
cylinder is established. At last, the amplitudes of the steady response of the
square cylinder with various wind velocities are calculated by using the empiric
formula. As shown in Fig. 9.26, the predicted curve of the amplitude-velocity
characteristic is very consistent with the testing data and the accuracy of the
empiric formula, expression (9.77), is verified successively.
A variety of experimental researches have demonstrated that the friction does
not depend only on the instantaneous sliding velocity Q but also on the whole
sliding history. The friction force should be written as
F P (Q , I ) N ,
where P is the friction coefficient, N is the normal load, and I is the average age
of the micro-contacts, which grows while the material creeps under normal load.
Recently, A. Cochard et al proposed a simple function to represent the friction
coefficient, i.e., [13]
§Q · § V1I ·
P (Q , I ) P 0 A ln ¨ ¸ B ln ¨1 ¸, (11.10)
© V0 ¹ © D0 ¹
in which
dI QI
1 , (11.11)
dt D0
where P0, A, B, D0, V0, and V1 are constants calculated by the testing data.
364
11.4 Establish Equations of Motion of Mechanical Systems
365
Chapter 11 Modeling and Control
d wL wL wR
Q (q, q ) , (11.12)
dt wq T wq T wq T
in which
> q1 " qn @ ,
T
q
q > q1 " qn @ ,
T
>Q1 " Qn @ ,
T
Q
L(q, q ) T (q, q ) U (q ),
where q is the generalized coordinate vector, q the generalized velocity vector,
Q the nonconservative force vector, L(q, q ) the Lagrangian, T (q, q ) the kinetic
energy function, U(q) the potential energy function, and R (q, q ) the dissipative
function.
The detailed routine to use Lagrange’s equation to establish the equations of
motion for the dynamic system is introduced with a meaningful example.
The sketch of the mechanical model of a vertical symmetric flexible rotor,
which is used to study the cone whirl phenomenon, is comprised of a slender
rotor and two identical segments of massless rotating shafts supported in dry
bearings. It is depicted in Fig. 11.11(a).
Figure 11.11 A Mechanical Model of the Flexible Rotor Supported on the Dry
Bearings
366
11.4 Establish Equations of Motion of Mechanical Systems
frame with the constant speed : , namely, OXYZ, denote the undisturbed motor
of the rotor, and let O'X'Y'Z' denote a frame fixed at the rotor with X', Y', and Z'
representing a set of principal axes and O' representing the mass center of the
frame. As the generalized coordinates, x and y are the coordinates of the mass
center of the rotor along with the axes X and Y, and D and E are the rotations
about axis X' and axis Y' respectively, as shown in Fig. 11.11(b).
Next, calculate the potential energy and the kinetic energy of the rotor. Denote
l as the half length of the rotor, P1(x1, y1, l) and P2(x2, y2, l) as the centers of the
upper and lower tops. Consequently, we have
x1 x l E , x2 x lE ,
y1 y lD , y2 y lD .
Compared with the undisturbed motion, the cone whirl motion is very small if
we consider only the motion of the rotor in the initial stage. Therefore, all
higher-order terms of the state variables are neglected and the equations of the
cone whirl become a linear equation system consequently. In accordance with it,
the potential energy of the rotor is described in the quadratic form
1 T 1
U r1 Kr1 r2T Kr2 ,
2 2
in which
> x1 y1 D @ , r2 > x2 y2 D @ ,
T T
r1 E E
ª k kl / 2 0 0 º
« kl / 2 kl 2 / 3 0 0 »»
K « ,
« 0 0 k kl / 2 »
« »
¬ 0 0 kl / 2 kl 2 / 3¼
1 1 1
T sin E E 2 : 2 ) J1: 2 .
m( x 2 y 2 ) J (D 2 2DE
2 2 2
The dissipative forces in the rotor system may be reduced to the linear viscous
forces, and the dissipative function is written as
1 T
R (r1 Dr1 r2T Dr2 ),
2
in which
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Chapter 11 Modeling and Control
E ª¬ x2 E y 2 D º¼
T T
r1 ª¬ x1 y1 D º¼ , r2
ªd 2K k K kl 0 0 º
« K kl (ds 2K kl 2 ) / 3
2
0 0 »,
D « »
« 0 0 d 2K k K kl »
« »
¬ 0 0 K kl (ds 2K kl ) / 3¼
2 2
where d is the external damping coefficient of the rotor system, s is the distance
from the external damping force to the mass center of the rotor, and K is the
internal damping coefficient. In addition, let GW denote the virtual work of the
nonconservative forces except the dissipatic forces in the rotor system. It is done
by the generalized forces, i.e., Qx, Qy, QD, and QE, on the corresponding virtual
displacements, i.e., Gx, Gy, GD, and GE. Hence, we have
GW Qx Gx Qy Gy QD GD QE GE , (11.13)
from which the expressions of the generalized forces Qx, Qy, QD, and QE are
obtained.
At last, join the expressions of T (q, q ) and U (q) together and obtain the
expression of Lagrangian L(q, q ) . Then, substituting L(q, q ) , R(q, q ) and the
generalized forces Qx, Qy, QD, and QE into Lagrange’s equation, we eventually
get the equations of the rotor systems in the matrix form
ªm 0 0 x ½ ª d 2K k
0 º K kl 0 0 º x ½
« » ° ° « K kl (ds 2K kl ) / 3 J J1 : »» °° y °°
0 » °E ° «
2 2
J 0
«0 0
® ¾ «
«0 0 m 0 » °
y° 0 0 d 2K k K kl » ®°D ¾°
« » « »
¬0 0 0 J ¼ ¯°D ¿° «¬ 0 J J1 : K kl (ds 2 2K kl 2 ) / 3»¼ ¯° E ¿°
ª 2k kl 2K kl K kl : º x ½ Qx ½
« kl 2
K kl : 2K kl 2 : / 3»» °° E °° ° °
2kl / 3 °QE °
« ® ¾ ® ¾. (11.14)
« 2K kl K kl : 2k kl »° y° ° Qy °
« »
¬ K kl : 2K kl : / 3 kl
2
2kl 2 / 3 ¼ ¯°D ¿° °¯QD °¿
This equation may be used to find the stability condition for the stationary rotation
of the rotor system.
Hamilton’s principle: The actual path in the configuration space renders the
t2
value of the definite integral I ³ t1
Ldt , which is stationary with respect to any
arbitrary path between two instants t1 and t2 provided that the path variations
368
11.4 Establish Equations of Motion of Mechanical Systems
or rewritten as
t2
³ t1
(GT GU GW )dt 0, (11.16)
where GT is the variation of the kinetic energy of the system, GU is the variation
of the potential energy, and GW is the virtual work done by the nonconservative
force applied on the system. The extended Hamilton’s principle can lead to the
equations of motion of systems with the self-excited vibration. As a typical
example, the finite element equation of motion of a laminated composite plate is
introduced below.
A laminated composite plate with a piezoelectric sensor and an actuator layer
is depicted in Fig. 11.12. The routine to establish the equations of motion includes
three steps.
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Chapter 11 Modeling and Control
1. Find expressions of the strain-energy, the kinetic energy, and the virtual
work done by the nonconservative forces[17, 18].
Consider only the response of the flexural vibration and ignore the in-plane
displacements. The displacements and the strains are represented in the vector form
u x ½ ª0 z 0º T
° ° « » ª ww ww º
u «0 0 z » , Q [ w, E x , E y ] « w, wx , wy » ,
T
®u y ¾
°u ° ¬ ¼
¯ z¿ ¬«1 0 0¼»
wE y wE y
T T
ª w2 w w2 w w2w º ª wE x wE º
« 2 , , 2 » , H «z ,z ,z z x» z ,
¬ wx wy 2 wxwy ¼ ¬ wx wy wx wy ¼
where z is the half-thickness of each layer, w is the flexural displacement, Ex and
Ey are rotations of the normal to undeformed middle surface respectively in the
X-Z and Y-Z planes, and H and are the strain vectors of each layer.
The strain energy of each layer is
1
2 ³Vk
Uk [H1 , H 2 , H 3 ][V 1 , V 2 , W 12 ]T dVk ,
where Vk is the volume of the kth layer and and Uk is its strain energy. The
potential energy for each layer is summed up in the z direction and the total strain
energy of an element in the laminated plate is
1
U
2 ³ A
T D0 dA , (11.17)
where A is the area of the surface perpendicular to the z direction and D0 is the
flexural stiffness matrix of an anisotropic plate. The kinetic energy T of an
anisotropic plate is
1 1
T
2 ³V
U u T u dV
2 ³ A
Q T RQdA , (11.18)
where R is the inertia matrix of the anisotropic plate. The virtual work done by
the nonconservative forces f in an anisotropic plate is
GW G ³ v T f dA. (11.19)
A
370
11.4 Establish Equations of Motion of Mechanical Systems
x [ w1 E x1 E y1 " w4 E x4 E y4 ]T , (11.20)
ww j ww j
where wj, j 1, 2, 3, and 4 is the normal displacement, E xj and E yj ,
dy wx
j 1, 2, 3, and 4 are the rotations about the X and the Y axes respectively.
w½ N ([ ,K ) ½
° ° ° °
Q ®Ex ¾ ® H x ([ ,K ) ¾ x , (11.21)
°E ° ° °
¯ y¿ ¯ H y ([ ,K ) ¿
where N, H x , and H y are the interpolation function matrices for w, Ex, and Ey
respectively, whose components are four-degree polynomials and only have 12
terms with 12 undetermined constants.
3. Assemble the equation of motion for the entire system.
Substituting the energy and the work terms in Eq. (11.16) leads to the variational
equation. To derive the discrete equations of motion of the entire system, the
displacement Q and the curvature are expressed in terms of nodal variables via
the shape functions using the four-node, 12-degree-of-freedom quadrilateral plate
bending element with one electrical degree of freedom, namely,
Q Q x , x,
where Q and are the interpolation function matrices of Q and , respectively.
Substituting the discretized expression of the displacement and the curvature
into the variational equation and assembling the equations of motion for the
entire system in terms of nodal variables, we have
Mx Kx Q (11.22)
with
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Chapter 11 Modeling and Control
M Ms M p , K Ks K p , (11.23)
¦ ³ ³ Q R Q | J | d[ dK ,
1 1
Ms T
s
1 1
¦ ³ ³ Q R Q | J | d[ dK ,
1 1
Mp T
p
1 1
¦ ³ ³ D | J | d[ dK ,
1 1
Ks T
s
1 1
¦ ³ ³ D | J | d[ dK ,
1 1
Kp T
p
1 1
¦ ³ ³ Q f | J | d[ dK ,
1 1
Q T
1 1
where M and K are the inertia and the stiffness matrices, J is the Jacobian matrix,
Q is the generalized nonconservative force vector, f is the nonconservative force
vector, and the subscripts s and p represent the main structure and the
piezoelectric material respectively.
372
11.4 Establish Equations of Motion of Mechanical Systems
At the considered instant, if the open volume Ro(t) coincides with the closed
volume Rc(t), the general transport theorem is
d D
dt ³³³R0 (t ) Dt ³³³Rc ( t )
(<)dV (<)dV ³³³ (<)(U u) nds . (11.24)
Bo ( t )
Since the notation D (<) D(t ) is employed for the closed volume, it is
permissible to write
D D
Dt ³³³Rc ( t ) Dt ³³³Ro ( t )
(<)dV (<)dV . (11.25)
D
Dt ³³³Rc (t )
GL GW U (u r )dV 0. (11.26)
d
GL GW ³³
dt ³³³Rc ( t )
U (u Gr )(U u) ndS U (u Gr )dV 0. (11.27)
Bo ( t )
Integrating the equation with respect to time from t1 to t2, with the prescribed
configuration, we obtain the Hamilton’s principle for a system with a varying
mass
t2 t2
G ³ Ldt ³ GWdt ³³ U (u Gr )(U u ) ndS 0. (11.28)
t1 t1 Bo t
This general principle may be applied to pipes conveying fluid. The system is
open, with the control volume boundary coinciding with the exterior surface of
the pipe and the inlet and the exit of the pipe, as shown in Fig. 11.15[20]. Si(t) and
Se(t) are the open boundary at the inlet and the exit of the pipe. The pipe inlet is
fixed, and there is no virtual work contribution from the force and the moment of
the reaction. The fluid velocity at the pipe exit is u R U , where R is the
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Chapter 11 Modeling and Control
velocity of the pipe at its exit end, and the normal velocity relative to the control
surface Se(t) is (u U ) n U . Thus, the last term in Eq. (11.28) becomes
where Md is the mass of per unit length of the fluid. Therefore, we have
The equation is useful in deriving the equation of motion and the boundary
condition for pipes conveying fluid. Particularly, for straight pipes conveying
fluid, the deduced equation of motion is the same as that of Eq. (9.15) and the
deduced boundary condition is also the same as that of Eq. (9.16).
The vibratory bodies in the physical world usually have distributed properties,
such as mass and stiffness. In many cases, the mass distribution and the stiffness
distribution are highly nonuniform. For such systems, it may be more feasible to
construct the discrete mathematical model directly.
In the lumped parameter method[21], as its name suggests, the parameters are
lumped at discrete points of the system. In particular, the length of the system is
divided into small increments and the distributed mass within the increments is
lumped at either the geometric center or the mass center if higher accuracy is
desired. Then, any two lumped masses are assumed to be connected by a massless
374
11.5 Discretization of Mathematical Model of a Distributed Parameter System
spring with the stiffness equivalent to the stiffness of the segment between the
two masses. Consistent with this, the continuous displacements, w(x, t), is replaced
by the discrete displacement wi(x,t), where i identifies the lumped mass mi. The
net result is a discrete model of the type encountered before.
The Jeffcot rotor is a typical lumped parameter model. Its elastic shaft is a
bending beam that is simply supported at both ends and a disk is located at the
central position of the beam span, as shown in Fig. 11.16(a). Let P denote a
transverse force applied on the central lumped mass m and let G denote the
deflection of the point of application of P. By using the Euler-beam theory, the
displacement of the beam is simply
Pl 2 x Px3
w( x) ,
16 EI 12 EI
where EI is the flexural stiffness of the uniform beam and l is its span. Therefore,
we have
Pl 3
G ,
48 EI
and the equivalent spring coefficient is
P 48EI
k .
G l3
Hence, the lumped parameter model may be drawn in Fig. 11.16(a), which is
exactly the same as Fig. 8.2.
The assumed-modes method begins with the discretization of the boundary value
problem of the continuum, albeit in an implicit manner. It provides us with the
opportunity to formulate the discretized system response of the motive forces,
rather than merely the free response. Therefore, it is available to construct the
discrete mode of the elastic body with the occurrence of the self-excited vibration
occurring.
The assumed-modes method aims at deriving equations of motion by discretizing
first and then making use of Lagrange’s equations. For simplicity, we only introduce
the discretization of the elastic body with one dimension.
Consider an elastic body and its approximate displacement w(x, t) is described
by a finite series, namely,
n
w( x, t ) ¦I ( x)q (t ) ,
i 1
i i (11.31)
where Ii(x), i 1, 2, " , n, are the known trial functions, namely, the assumed
mode functions, and qi(t) are the known generalized coordinates. Assuming that
there is no lumped mass at the boundaries, we can discretize the kinetic energy as
1 n n
T (t ) ¦¦ mij qi q j
2i1 j1
(11.32)
with
l
mij m ji ³ m( x)I ( x)I ( x)dx,
0 i j i, j 1, 2," , n, (11.33)
where m(x) is the distribution function of the mass of the elastic body and l is its
span. As an example, for a nonuniform beam in bending fixed at x = 0 and
supported by a spring of stiffness k at x l, the potential energy has the form
2
1 l ª w 2 w( x1t ) º 1 2
2 ³0
V EI ( x ) « » dx kw (l , t ). (11.34)
¬ wx 2
¼ 2
By inserting Eq. (11.31), the discretized potential energy can be written as
1 n n
V (t ) ¦¦ kij qi (t )q j (t )
2i1 j1
(11.35)
with
d 2Ii d I j
2
l
kij k ji ³ 0
EI ( x)
dx 2 dx 2
dx kIi (l )I j (l ), i j 1, 2," , n. (11.36)
376
11.5 Discretization of Mathematical Model of a Distributed Parameter System
Finally, let f(x, t) be a distributed nonconservative force, and discretize the virtual
work as
n
W (t ) ¦ Q (t )q (t )
i 1
i i (11.37)
with
l
Qi (t ) ³ 0
f ( x, t )Ii ( x)dx, i 1, 2," , n. (11.38)
¦ m q (t ) ¦ k q (t )
j 1
ij j
j 1
ij j Qi (t ), i 1, 2," , n. (11.39)
Mq Kq Q, (11.40)
where M and K are the inertia and the stiffness matrices, respectively, and q and
Q are the generalized coordinate and the generalized force vectors, respectively.
The routine shows that the assumed-modes method consists of two steps. The
first is to find the kinetic energy of the elastic body, the potential energy of the
conservative force, and the virtual work done by the nonconservative force. The
second is to apply Lagrange’s equation to deduce the differential equations of
motion of the vibratory elastic body.
Now, we introduce an example to explain the application of the assumed-modes
method.
The cantilevered pipe conveying fluid is hung vertically and supported at the
free end by a nonlinear spring, as shown in Fig. 11.17[22]. The restoring force of
the nonlinear spring is
f ( K1 w K 3 w3 )G( z l ),
where w( z ) is the flexural function of the pipe, K1 and K3 are the spring
constants, and G( z l ) is the Dirac function. Consider a fixed frame OZY, as
shown in Fig. 11.17. The differential equation of motion of the pipe conveying
fluid is written as
w4 w 2 w w
2
w2 w w2 w
EI U AU 2 U AU m ( K1 w K 3 w3 )( z l ) 0,
wz 4 wz 2 wzwt wt 2
(11.41)
where the coefficients in the equation are the same as those in Eq. (9.15). By
377
Chapter 11 Modeling and Control
Figure 11.17 A Cantilevered Pipe Conveying Fluid and Supported at the Free End
w 4K 2 w K
2 1
w 2K w 2K
v 2 E 2
v (k1K k3K 3 )G[ 1 0 . (11.43)
w[ 4 w[ 2 w[wW wW 2
Neglecting all higher-order modes and only retaining the first two modes, we have
378
11.5 Discretization of Mathematical Model of a Distributed Parameter System
x Ax F ( x ) (11.46)
with
> q1 q2 @ ,
T
x q2 q1
ª0 0 1 0º
«0 0 0 1 »»
A « ,
« a1 a2 a3 a4 »
« »
¬ b1 b2 b3 b4 ¼
>0 F4 @ ,
T
F ( x) 0 F3 (11.47)
where O1 and O2 are the first two eigenvalues of the uniform beam. Other
parameters are the same as those in Eqs. (11.42) and (11.45).
Here, let us recall the mechanical model of the flexible rotor shown in Fig. 11.10,
whose differential equation of motion has been found. It is Eq. (11.14).
Now, we introduce a state vector with eight dimensions[23], i.e.,
x E
T
x ª¬ x E y D y D º¼ .
ª 0 0 1 0 0 0 0 0 º
« »
« 0 0 0 1 0 0 0 0 »
« »
« 2 k kl ( d 2K k ) K kl 2K k : K kl : »
0 0
« m m m m m m »
« »
« kl 2 kl 2
K kl ( ds 2
2K kl 2
) K kl : 2K kl 2
: J »
0 1 1
« J 3J J J J 3J J »
« »
« 0 0 0 0 0 0 1 0 »
« »
« 0 0 0 0 0 0 0 1 »
« 2K k : K kl : 2 k kl ( d 2K k ) K kl »
« 0 0 »
« m m m m m m »
« »
« K kl: 2K kl :
2
§ J1 · kl 2kl 2 K kl (ds 2 2K kl 2 ) »
0 ¨1 ¸
«¬ J 3J © J ¹ J 3 J J 3 J »¼
(11.49)
According to the state equation of the flexible rotor, Eq. (11.48), we obtain the
characteristic equation, i.e.,
det | O I A | 0 . (11.50)
Here, consider a flexible rotor with the following parameters: m 0.3 kg,
J1 6.132 u 104 kg m 2 , J 4.38 kg m 2 , l 0.07 m, s 0.02 m, k 60 kg s ,
K 0.01 s, d 0.06 kg s 1 , : 120 s 1 . Using the numerical method to solve
the characteristic Eq. (11.50), we obtain all eigenvalues, i.e.,
O1 0.4623, O2 0.0606, O3,4 3.716 r i 3.757,
O5,6 0.0275 r i 111.65, O7,8 11.745 r i 123.59.
The result shows that the real parts of three eigenvalues, namely, O2, O3, and O4,
are positive. Thus, the steady rotation of the flexible rotor is unstable when it
operates under the given rotation speed : . Actually, there is saturation nonlinearity
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Chapter 11 Modeling and Control
in a real rotor system, such as the limiting of the energy supply. Consequently,
the unstable rotation is usually transformed into the self-excited vibration, the
rotor whirl[24]. As mentioned before, active control is the most effective way to
suppress the self-excited vibration in many cases. Let us expand it here.
A control system of the flexible rotor shown in Fig. 11.18 is comprised of a
flexible rotor, two sensors A1 and A2, a signal amplifier D1, an operating
computer C, a power amplifier D2, and two actuators B1 and B2[25]. Let us have a
look at the operation routine to seek the optimal output control law.
First, by adding the control force exerted on the actuators, the state equation of
the controlled flexible rotor is transformed into the form
x Ax Bu, (11.51)
where u is the control vector and B is the control matrix.
According to the geometry of the configuration of the controlled rotor, we have
T
ª0 0 1 b 0 0 0 0 º
B «0 0 0 0 0 0 1 b » , (11.52)
¬ ¼
in which b is a constant that is obtained based on the structure of the control
system.
Similarly, according to the geometry of the configuration of the controlled
rotor, we establish the output equation of the control system, i.e.,
y Cx (11.53)
with
ª0 0 1 h 0 0 0 0 º
C «0 0 0 0 0 0 1 h » , (11.54)
¬ ¼
where y is the output variable of the control system and h is a constant that is
found based on the structure of the control system.
382
11.6 Active Control for Suppressing Self-Excited Vibration
u Ky, (11.55)
where K is a constant gain matrix, whose elements are all found by the special
algorithm explained in the following.
Next, combine Eqs. (11.51) and (11.53) with Eq. (11.55) and eliminate the
variables y and u. We obtain the state equation of the controlled rotor, i.e.,
x ( A BKC ) x. (11.56)
This shows that the controlled flexible rotor is a steady linear dynamic system.
Consequently, make use of an integral functional of the quadratic form of the
output vector y and the control vector u as the performance index, namely,
f
J ³ 0
( y T Qy uT Ru)dt (11.57)
with
ª q11 0º ª r11 0º
Q «0 , R .
¬ q22 »¼ «0
¬ r22 »¼
Then, using the optimal output control algorithm of the steady linear system,
which has been introduced in Chapter 5, we establish the Riccati algebraic
Eq. (5.87), and apply the numerical method to find the Riccati matrix P*. Further,
inserting P* in expression (5.88), we obtain the optimal output control law, i.e.,
u* R 1 B T P * x . (11.58)
Eventually, substituting u in Eq. (11.51), we obtain the state equation of the
*
x ( A BR 1 B T P * ) x . (11.59)
To calculate the elements of the gain matrix K, we compare Eq. (11.56) with
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Chapter 11 Modeling and Control
Eq. (11.59) and obtain a linear algebraic equation in the matrix form
R-1 B T P * KC 0. (11.60)
As long as all elements of the gain matrix K are found by means of the numerical
method, the design calculation of the output control of the flexible rotor is
completed.
At last, according to Eq. (11.59), we establish the characteristic equation of the
controlled flexible rotor, i.e.,
det | O I A BR 1 B T P * | 0. (11.61)
Here, given a set of the system parameters, namely, b 1, h 1, q11 q22 1,
r11 r22 1 , and the remainder of them are the same as those in the previous
calculation of the eigenvalues of the uncontrolled rotor, we compute all eigenvalues
of Eq. (11.61) and obtain
The past decade has witnessed an increasing interest in control theory and the
actuator technique[21]. Active flutter suppression is facing several challenges,
such as development of new concepts and actuators with high efficiency and
reliability, the design of robust controllers, and the integration of sensors, controllers,
and actuators. Among them, the challenge due to flap actuators is most important.
The ultrasonic motor is a kind of promising flap actuators for flutter suppression
of light wings, featuring not only small size and light weight but also large output
torque, quick response, and direct mechanical transmission. Recently, H. Y. Hu
and M. L. Yu performed a research on robust flutter control of an airfoil through
an ultrasonic motor[27, 28]. The studied airservoelastic system is a two-dimensional
airfoil section of NACA0012 with a flap servo as shown in Fig. 11.20 and is
subject to a steady flow at speed U as shown in Fig. 11.21, in which h is the
plunge along a guide rail, Dthe pitch around a hinge, and Ethe flap of control
surface. A linear spring is attached to the elastic axis of the airfoil to provide
plunge stiffness kh, and a pair of linear springs is installed to offer the pitch
stiffness kD. All length symbols ab, cb, b, xD, and xE are shown in Fig. 11.21.
384
11.6 Active Control for Suppressing Self-Excited Vibration
in which f a is the noncircular part including the structural variables only, i.e.,
ª S Sab bT1 º
Ma U b « Sab S (1/ 8 a ) 2b 2T13 »» ,
2« 2 2
«¬0 0 T18 S »¼
U
qa Pqa S1qs US2 qs , P diag [0.0045 0.3]. (11.69)
n
Let us denote qw [qsT qsT qaT ] R8 . Then, substituting Eqs. (11.65), (11.67),
and (11.69) in Eq. (11.63), we obtain the state equation of the airfoil shown in
Fig. 11.21, i.e.,
qw Aw qw Bw E c , (11.70)
in which the input command E c is applied to the ultrasonic motor.
The flap angle of the control surface, E (t ), is just the output angle of the
ultrasonic motor provided the output torque of the ultrasonic motor is more than
the aerodynamic torque ME on the control surface, In this case, the control surface
386
References
driven by the ultrasonic motor looks like a rigid body with a single degree of
freedom, and we have
I E E cE E k E E k0 E c , (11.71)
in which the parameters I E , cE , k E , and k0 are all identified from the measured
frequency response function.
Given the input command Ec, substituting E determined in the previous equation
in the first two Equations in (11.62) can greatly simplify the analysis of the entire
system. When the ultrasonic motor is under the self-lock condition due to
E c { 0, E { 0. In this case, the entire system becomes a dynamic system with
two degrees of freedom.
While many uncertainties exist in control systems, there are mainly two types
of uncertainties: one is the friction in the guide of the testing model and the other
is the deviation between the real controlled system and the mathematical model
described by Eqs. (11.70) and (11.71). J. Doyle proposed a concept of Psynthesis
to design robust controllers[30]. With help of the MATLAB tool box, the robust
controller for flutter suppression of the airservoelastic system shown in Fig. 11.20
is constructed and used to control the system subject to a steady flow in a
wind tunnel, whose system parameters are set as L 0.3 m, b 0.1 m, a 0.5,
U 1.225 kg m3, m 1.85 kg, ID 3.142 u103 kg m 2 , ShD 0.0309 kg m, ShE
kh m kD ID
8.608 u 104 kg m, SDE 1.215 u 104 kg m, Z h 5.9 Hz, ZD
2S 2S
kE I E
5.9 Hz, Z E 61.34 Hz.
2S
With the testing model, the acquired study results show that when the
controller does not work, the computing value of the critical flutter speed from
Eq. (11.70) is 22.3 m / s and the testing value of the critical flutter speed in wind
tunnel is 22.4 m / s ; when the robust controller is working, the critical flutter
speed measured in the wind tunnel test reaches 27.7 m / s . This means that the
critical flutter speed is raised up to 23.6%. Hence, this research project provides
reliable evidence that the robust control of an airfoil with the ultrasonic motor
can effectively suppress the flutter.
References
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1985
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387
Chapter 11 Modeling and Control
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389
Subject Index
1 1.1.4
self-excited vibration van der Pol equation
forced vibration
1.2
1.1 vortex resonance
natural vibration vortex shedding
conservative system Karman vortex
parametric vibration
non-autonomous system 1.3.1
nonconservative force
1.1.1 positive definite
generalized coordinates stable equilibrium
elliptic integral negative definite
unstable equilibrium
1.1.2 quasi-harmonic
homogeneous
non-homogeneous 1.3.2
primary resonance generalized dynamic system
feedback
main resonance
closed loop
sub harmonic resonance
Harkevich
super harmonic resonance
combination frequency
1.4.2
natural mode
operator
internal resonance
differential operator
combinational resonance
1.4.3
1.1.3 hybrid system
Faraday, M. Ritz-Galerkin method
parametric resonance assumed-modes method
Mathieu’s equation
Strutt diagram 2
Strutt, J. phase plane method
van der Pol point mapping method
Subject Index
3 limit cycle
Hurwitz criterion hard excitation
3.5 2.1
quadratic form criterion Poincare, H.
state space
4 phase point
Hopf bifurcation theorem phase path
Lindstedt-Poincare method phase trajectory
average method
method of multiple scales 2.2.2
turning point
5 center
Nyquist stability criterion saddle
Popov stability criterion
describing function method 2.3.1
optimal control characteristic equation
coincident
6 discriminant
chatter node
focus
6.3 spiral
hunting
2.3.2
7 restoring force
nonholonomic constraint Rayleigh vibrator
van der Pol vibrator
8
Rayleigh equation
fluid film force
van der Pol equation
internal damping
2.4.1
9
Taylor expansion
flexible structure
nonsingular transformation
flutter
cantilevered pipe
classical flutter 2.4.3
airfoil attractor
stall flutter repellor
fluid-elastic instability
2.5.1
10 index
time delay
2.6
2 soft excitation
phase diagram
391
Subject Index
2.7.2 3.3.1
relaxation vibration Mihairov curve
2.7.3 3.3.2
non-smooth hodograph
Coulumb friction
equilibrium zone 3.3.4
Liénard construction coprime
continuum
impulsive energy 3.4
coefficient space
2.8.1 parameter space
Poincare map
hyperplane 3.4.1
transversally subspace
open segment stable region
Bernoulli equation boundary surface
3.4.3
2.8.2
bifurcation
block diagram
node-saddle bifurcation
dead-zone
Hopf-bifurcation
sequential function
Hopf, E
Lamery diagram
Andronov, A. A.
3 3.5
stability holonomic system
algebraic criterion Lagrange’s equation
geometric criterion
3.5.1
3.1 inertial force
Lyapunov, A. M. dissipative force
gyroscopic force
3.2.1 conservative force
eigenvalue problem circulatory force
characteristic equation
characteristic value 3.5.2
eigenvalue quadratic form
positive definite
3.2.3
Routh, E. J. 4.1.1
Routh problem smooth function
Hurwitz, A. family
criterion of Liénard-Chipart trajectory
392
Subject Index
4.1.2 4.4
Hartman-Grobman theorem Krylov, N. M.
flow Bogoliubov, N. N.
hyperbolic flow
hyperbolic linear system 4.5
invariant Sturrock, P. A.
hyperbolic fixed point Nayfeh, A. H.
open subset
open interval 4.5.1
parameterization differential operator
concomitant
hyperbolicity 5
transfer function
4.1.3 frequency domain
center manifold theorem frequency criterion
eigenspace optimal algorithm
invariant subspace
invariant manifold 5.1.1
stable manifold input
center manifold response
unstable manifold output
center subspace controller
stable subspace controlled object
unstable subspace error
input command
4.2 control effect
canonical form desired output
reference input
4.2.1
Poincare-Birkhoff normal form 5.1.2
component
4.2.2 proportional component
Poincare-Androvov-Hopf theorem amplification coefficient
inertial component
4.3 time constant
secular term oscillatory component
393
Subject Index
394
Subject Index
395
Subject Index
396
Subject Index
9.2.1 9.3.2
transverse deflection Galerkin method
397
Subject Index
9.3.3 9.5.4
aerodynamic center reduced velocity
velocity circulation
circulation 9.5.5
Lee, B. H. K. Price, S. J.
Liu, L. Paidousis, M. P.
Chung, K. W.
Sheta, E. F. 10.1.2
dissipative coefficient
9.3.4
elastic center
10.2
Maxwell’s theorem
reciprocity position control
aerodynamic center command
midchord line
plunge 10.2.2
pitch half-plane
9.4 10.4
aeroelastic instability hydraulic actuator
9.4.1 10.4.2
angle of attack compressibility
lateral force coefficient
attack angle 11.1.1
generalized negative damping
9.4.2
Parkinson, G. V. 11.1.2
Smith, J. D. phase lag
reduced velocity Fourier analysis
dissipative force
frequency spectrum
phase crossover
9.5
feedback channel
cylinder arrays
turbulent excitation forward channel
acoustic excitation
fluid-elastic instability 11.2.1
minimal model
9.5.2 motive force
added mass vibrating body
fluid stiffness force exciting body
9.5.3 11.2.2
cross flow dynamic damper
398
Subject Index
399