Modal Analysis of A Robot Arm Using The Finite Element Analysis A PDF
Modal Analysis of A Robot Arm Using The Finite Element Analysis A PDF
Modal Analysis of A Robot Arm Using The Finite Element Analysis A PDF
Thesis/Dissertation Collections
6-1-1989
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SHASHANK C. KOLHATKAR
A Thesis Submitted in Partial Fulfillment
of the
Requirements for the Degree of
MASTER OF SCIENCE
in
Mechanical Engineering
Rochester Institute of Technology
Rochester, New York
June 1989
Approved by :
.......
Dr. Bhalchandra V. Karlekar
(Professor and Department Head)
Title
the Thesis
of
Shashank C.
Kolhatkar
my thesis
not
be
Dated
for
:
.
in
'Modal Analysis
whole
or
commercial
t/i/si
hereby
Library
in
use
of
part.
or
grant
A Robot
permission
R.I. T.
Any
of
to
to
reproduce
reproduction
profit.
Arm'
will
Abstract
The
objective
modal
UNIMATION five
was
NASTRAN
used
testing
the
size
its
free-free
and
first
large, it
error.
from
was
For
Assurance
discussed.
Hence
only for
of
of
Criterion
The
behavior.
This
structure.
The
element
method
patterns
of
mode
for
which
and
deflection
done
Also,
of
in
was
by
due
to
testing
could
be
are
seen
i i
supported
then
compared.
the
showed
showed
33%
of
seems
computational
technique
of
of
called
mode
showed
between
the
Modal
shapes
an
nonlinearities
discrepancies
modal
its
observation
shapes
mode
the
reflect
Due to
to test it in
possible
analyses
correlation
numerical
Measurement
frequencies
could
variation
NASTRAN.
instrumentation
were
the
natural
shapes.
probably
causes
analyses
was
experimental
was
not
the
display
with
shapes
mode
to
hammer.
and
tested
was
The
mode
one
was
both
animated
it
it
from
33%.
to
post-processor was
Kjaer's
&
element
Corporation's
Structural
accelerometer
structure
modes
1%
and
subsequently
using
results
most
comparison
displayed
the
Schwendler
pre-
Bruel
and
correspondence.
one-to-one
deviation
performed
condition.
thirteen
and
The finite
model.
MacNeai
analyzer,
of
weight
The
condition.
as
of
A PUMA
methods.
after
software
to
experimental
used
model
was
spectrum
and
using
the
shapes
STAR
including
was
mode
Systems'
work
robot was
performed
code.
modal
The
axis
discretize
to
animated
The
present
using theoretical
analysis
analysis
the
of
the
is
erratic
in
the
finite
in the two
analyses.
Acknowledgements
This
is dedicated to my
work
Chandrakant
Kolhatkar
it
encouragement
career.
will
done for
me.
Kaka
and
also
who are
with
influenced
my future
Thank
Dr.Torok
reviewing my
My
hardware
helping
out
me
Finally, my
diversions
of
the
past
and
special
when
two
me when
for seeing
My way
smile.
logical
be
gratitude
my
and
great
of
me
in my
day
need
them!
systematic
source
have
Maoshi
towards
thinking is very
smile
thanks
was
years
appreciate
Foley
and
used
with
and
of
much
approach
to
inspiration
in
career.
I really
work.
help
much
will
thanks to Dan
sincere
B&K
and
You
developing
you
there to
very
support
to see this
possible
Shri.
and
express
in-depth,
your
problems.
solve
you
patience
by
to
want
constant
whose
been
not
grateful
always
be
always
without
have
would
parents
to
to
our
so
David
own
got
valuable
spent with
will
you.
1 1
time
understanding.
all
the
Hathaway
help
for
with
always
into trouble.
all
working.
being
of
when
your
your
always
cherish
the
constant
memories
Table
of
Contents
Page
List
of
Tables
List
of
Figures
List
of
Symbols
1.
vii
viii
Introduction
1.1
General Concepts
Motivation
Modal
.2
.3
.4
.3.1
Testing
Applications
Modal
of
Testing
Range
of
Applications
of
Method
1
1
1
.5
1.6
.4.2
.4.3
10
Advantages
Limitations
Comparison
of
of
of
11
11
the Experimental
and
Finite Element
Modal Analysis
12
Mathematical Concepts
15
.6.1
Equations
of
of
Freedom System
1
.6.2
Equations
of
of
17
Freedom Systems
18
1.6.3
Decoupling
the Equations
Motion
21
1.6.4
Formulating
23
IV
of
Page
2.
MSC/NASTRAN
2.2
Finite Element
2.3
2.4
3.
Using
MSC/NASTRAN
28
28
With NASTRAN
Modeling
31
2.2.1
31
2.2.2
32
2.2.3
34
Dynamic Analyis
Using
NASTRAN
36
2.3.1
37
2.3.2
39
2.3.3
41
Dynamic Reduction
42
2.4.1
Guyan Reduction
43
2.4.2
45
2.5
Selection
of
48
2.6
Selection
of
49
3.2
Modal Analysis
and
Related instrumentation
51
51
3.1.1
Vibration Exciters
53
3.1.2
Hammer
55
3.1.3
Accelerometer
57
3.1.4
61
62
Curvefitting
3.2.1
Properties
3.2.2
SDOF
of
Curvefitting
65
68
Page
3.2.3
3.2.4
3.3
4.
Modal
Modeling
3.2.2. 1
68
3.2.2.2
71
MDOF
Curvefitting
3.2.3.1
Extension
3.2.3.2
General MDOF
73
of
Curvefitting
Autofitting
Testing
and
73
75
76
Procedure
77
Results
84
4.1
Background
4.2
Procedure to Create
4.3
Finite Element
4.4
Modeling
4.5
Results
of
1 03
4.6
Results
of
109
4.7
Comparison
4.8
Integration
and
Assumptions
Modeling
for Modal
of
of
and
the Robot
Testing
87
93
Testing
Experimental
Modal
84
102
and
and
Analytical Results
115
FEM
121
References
123
Appendixes
Appendix A
Appendix B
Appendix C
Modal
Appendix D
Appendix E
Appendix F
Testing Coordinate
VI
and
Testing
List
of
Tables
Page
Table
.1
Comparison
Modal
of
and
13
Testing
2.1
Comparison
4.1
Natural
of
Frequency
using NASTRAN
42
4.3
Natural Frequencies
Comparison
Analysis
and
of
50
106
obtained
from Modal
Testing
113
Modal
116
Testing
VII
List
of
Figures
Figure
.1
Page
and
Experimental
Modal Analysis
1
.2
Single Degree
14
of
Freedom System
1.3
Multiple Degree
2.1
3.1
Constructional Details
3.2a
Time
3.2b
Frequency
3.3
Constructional Details
3.4
Mounting Methods
3.5
Properties
3.6
3.7
Circle
3.8
Schematic
3.9
Analyzer
3.10
STARGATE
3.1 1
Driving Point
History
of
of
Freedom System
Data Decks
30
Hammer
of
56
of
56
the Pulse
of
56
Accelerometer
59
for Accelerometer
60
Modal Circle
of
of
and
19
Spectrum
Fitting
17
of
67
SDOF Curve
FRF Data
72
the Experimental
Setup
70
Fitting
Setup
for
Modal Test
Frequency
Domain
79
80
81
Measurement in the
Frequency
and
Time
83
Domains
PUMA UNIMATE Robot Arm
4.1
Picture
4.2
4.3
Various Views
4.4
Features
4.5
of a
of
of
86
85
the Robot
VIII
88
89
90
Figure
Rage
4.6
Various Views
4.7
Features
of
of
Applying
Loads
91
and
Boundary Conditions
92
4.8
Features
94
4.9
4.10
Isometric View
of
of
96
Intersecting
Cylinders
97
4.1 1
Front View
of
4.12
Front View
of
4.13
4.14
4.1 5
The
4.16a
The Experimental
Setup
110
4.16b
The Experimental
Setup
111
"STAR"
at
Intersecting Cylinders
100
Testing
Apppendix
A.1
Details
of
A.2
Details
of
the TRIA3
and
98
QUAD4 Elements
IX
101
104
105
List
[A]
Residue
Ajk
jktn
c,[C]
Damping
F(t), F(co)
External
FSW
Restoring
Mass
H(co), h(co)
Frequency
hjk(co)
Element in the
Tl
Damping
[I],
constant or
or applied
damping
force
Identity
response
the
system
spring
jth
function
row and
ktn
matrix
column of
[L]
Eigenvalue
or stiffness matrix
or mass matrix
force
N(t)
Generalized
modal
q(t)
Generalized
coordinate or generalized
Q(t)
Generalized
external
Laplace
co
[u], [u]r
matrix
matrix
Stiffness
u,
FRF
factor
k,[K]
Mass
matrix
normalized eigenvector
V-1
[M]
on
force induced in
m,
Symbols
of
Modal
matrix
force
variable
Frequency
of vibration
displacement
List
cor,
con
Natural
of
frequency
x(co)
Response
PFj]
Generalized
[]
'
]"
[
]"
Transpose
'
'
Inverse
'
Symbols (continued)
of
the system
the system to
of
excitation
of a matrix
of a matrix
Transpose
of
inverse
First derivative
of a
Second derivative
of a matrix
quantity
of a
quantity
DOF
Degree
FEM
FFT
FRF
Frequency
MDOF
Multiple Degree
SDOF
Single Degree
Im
Imaginary
Re
Real
of
w.r.t
time
w.r.t.time
Freedom
Response Function
of
of
Freedom
Freedom
XI
CHAPTER 1
INTRODUCTION
1.1
The
GENERAL CONCEPTS
of
study
with
or
structures
their
systems
Hence
take
place
forces, in the
system
under
frequencies.
some
when
forced
is
of
frequencies
of
and
to
bodies
of
engineering
in
vibrations
requires
vibrations
system
of
their
consideration
of
place
to
large
under
When
vibrate
the system,
is
oscillates
at
are
to
the
its
forced.
the
Free
action
are
of
condition
of
may
and
a
natural
dynamical
stiffness
major
with
external
is
excitation
of
its
design
to dynamic forces.
excitation
the
of
the
of
mass
excitation
the
oscillations
more
or
frequencies
coincides
one
exposed
at
and
under
properties
due
Natural
free
vibrates
structure
excitation
dangerously
of
configuration
vibration.
forced
frequency
All
forces.
applied
generally
frequencies
damping.
and
consideration
motion
oscillatory
subjected
absense
vibration
Natural
of
distribution
free
when
internal
system
are
are
vibrations
of
design
their
the
behavior.
oscillatory
There
influence
the
and
operation.
involves
vibrations
without
the
If
the
of
resonance
In
oscillatory,
frequency.
one
result.
forces is
the
natural
is encountered,
the
past,
failures
of
some
because
occured
frequencies is
All
of
vibrating
because
energy
the
damping
dissipated
by
friction
major
made
limiting
the
some
degree
and
other
internal
little
basis
cannot
natural
vibrations.
hence the
the
on
the
to
has very
and
damping
of
in
role
it
small,
be
can
is
the system,
of
frequencies
natural
of
to
damping
of
calculation
subject
are
the
have
airplanes
and
major
If the
resistances.
Thus,
resonance.
systems
the
of
of
buildings
be
influence
on
calculations
for
of
no
damping.
amplitude
oscillation
of
it
as
overlooked,
at
resonance.
The
number
motion
free
of
of
of
a system
particle
freedom.
A
of
Furthermore,
coordinate
degrees
may be
is
position
has
and
continuous
positions
to
motion
six
three
elastic
required
to
freedom
of
of
in
space
degrees
body
describe its
assumed to
be rigid,
dynamically
equivalent
Oscillatory
systems
For
linear
and
to one
can
be
systems
the
motion;
having finite
the
its
parts
classified
of
three
orientation.
infinite
hence
degrees
principle
the
the system. A
freedom,
an
may be
system
broadly
of
requires
describe
defining
angles
of
nonlinear.
general
body
rigid
coordinates
the degrees
called
undergoing
components
of
independent
number
it has
infinite
such
bodies
of
considered
of
as
to be
freedom. t19l
linear
superposition
and
holds,
there
and
is
developed
well
differential
equations
analysis
nonlinear
systems
systems
tend
of
All
apply.
amplitude
of
of
The idea
which
R.I.T.
of
structures.
use
computer
manufacturing
assembly
line
robot
of
robot
create
resonance
position.
weld
on
structure
are
high
for
The
precision
The
both
motors
in
to
it.
detrimental
effects
due
cause
undesired
known, the
area.
robot
of
of
might
natural
of
an
the
turning
in assembly
A
typical
frequencies
a
motor,
vibrations.
circuit
it
For
board,
its desired
throw a bead of
frequencies
operated
the
and
on
while
chip from
If
be
robots
natural
resonant
robot
could
of
frequency
in
automobile
accuracy.
the
and
There is
vibrations
welding
the
the
and
to
misplacement
example,
of
of
the
situated
accuracy
in the assembly
robot
role
If any
by
and
analysis
design
study.
function
precision
excited
using
in
modal
the
arm
case
especially
industries.
in
robot
for the
used
production.
six
another
an
due
components.
various
importance
is
could
As
of
structure
while
example,
introduced
also
are
Unimate
today,
critical
very
was
robots
requires
has four to
the
could
of
volumes
is
robots
Puma
laboratory
robotics
large
between
increasing
an
extensive
out
Nonlinearities
increasing
with
assuming
analysis
nonlinear
the
difficult to
and
of
is
become
to
known,
well
their
for
techniques
contrast,
less
are
connections
MOTIVATION
In
motion.
oscillation.
the complexity
1.2
of
handle
to
theory
mathematical
to
avoid
of
the
these
frequencies.
natural
Otherwise,
frequencies
Although in the
of
the
robot
problems
Modal
analysis
shapes
1.
In
of
Fourier
determination
or
be
could
of
as
used
and
deduced from
process
of
finite
the
data
to
modeled
by
and
elements.
and
the
and
and
numerically
are
residues.
and
polynomial
compared
associated
The
damping
Thus
residues.
Fast
obtained
then
the
many
a
using
is
implements
In this
method,
the
large
number
of
Appropriate
model
at
is
mode
be
can
it is
analysis.
and
into
structure
using
determine
frequencies
problems.
it
curves
mode
response
curvefitted
of
parameters
method
dividing
elastic
applied
for
model
experimentally
parameters
natural
measured
FRF to
of
modal
modal
vibration
are
equations
acquisition
element
approach
of
is
testing,
the
and
The
Functions
as
modal
structure
forms
series
values
or
analyzer.
resulting
known
coefficients
range.
frequencies
natural
frequencies
natural
excitation
the
Response
The
analysis
known
standard
shapes
modal
Transform
functions.
then
throughout
Frequency
defined
frequency
operating
the
shift
in the field.
with
locations
The
the
to
modified
vibrated
2.
from
be
could
purely academic, it
experimental
with
structure
away
present
was
actual
the
boundary
analyzed
using
theoretical
structure
is
theoretically
conditions
finite
are
element
This
analysis.
method
Damping
structure.
is
the
for
damping
simplicity
The accuracy
of
the
represents
it
times
many
results
real
is
depends
obtained
Modeling
structure.
In
the
present
determine the
here
made
correlated
to
by
corresponding
comparison
two
the
and
both
assumptions
The
chapters.
approaches
and
structure
mode
Modal
the
are
limitations
is
and
correlated
The
modes.
analysis.
model
experience
of
are
The
in
an
mention
must
not
exact
since
be
effort
is
are
in
the
are compared
to
frequencies
their
Criteria
used
of
shapes
results.
shapes
to
the
appproaches,
Assurance
assumed
in
approaches
mode
deviation
such
Hence,
components.
skills
mentioned
percentage
value
roles.
different
entirely
obtain
discussed in further
that
above
frequencies
natural
Using
structure.
the
work,
on
the
it is very difficult to
considered
not
of
its
and
structure
various
with
shapes
mode
In general,
model.
associated
and
the
of
property
frequencies
gives
values
by
direct
(MAC)
made
they
visual
technique
of
the fact
involve
many
as,
linear
have
stiffness
and
damping
properties.
2.
3.
The modeling
of
4.
The
of
errors.
process
boundary
are
assumed
conditions
measurement
of
is
the
to be very small.
not exact.
response
involves
some
1.3
MODAL TESTING
For
smooth
structure
in
many
excessive
that the
The
accelerometers
and
objective
or
of
modal
was
used with
on
an
If
vibration
one
few
of
of
the
phenomena
type
called
its
with
then
using
done
under
is
testing
and
testing
of
components
mathematical
In
or
analysis
other
the
of
using
it is
words,
structures
description
detailed
comprising
subsequent
Modal Testing.
known
measured
accurate
very
with
the
their
dynamic
STAR
software
behavior.
For
runs
analyzers.
and
testing
obtaining
oscillatory
is
vibrated
is
This
acquisition
is
response
conditions,
result.
involving
process
are
frequencies
vibrations
many
structure
spectrum
curvefitting techniques is
a
corresponding
will
data
response
structural
vibrations
There
controlled.
understand
study,
controlled
information
of
design
Hence it is important
(natural)
resonant
in the
practice.
experimental
excitation.
to
and
major
Resonant
discomfort.
and
vibrations
Vibrations is
applications.
monitored
the
methods
in
encountered
closely
noise
level be
minimum.
Experimental study
reliable
an
of
engineering
determine
to
structure.
In
operation
motion,
vibration
methods
most
safe
be kept to
must
limitation
create
and
t2l
IBM
PC/AT
and
is interfaced
Systems'
channel
with
FFT
the
analyzer.
spectrum
STAR
analyzer
using GPIB
impact
interfacing
the
structure
accelerometer to
The
driving
the
affect
tip
frequency
mounted
to
compared
response
the
on
mass
was
excite
are
to
applications
In
use.
transfer
the
and
to which the
the cases,
all
mathematical
The
structure
This
many
put
matrix.
model
the
of
1. Modal
function
applied
applications
testing
frequencies
and
theoretical
developed
For
used
stud
Thus
A
it
was
did
not
rubber
soft
in
the
at
lower
the
is
excitation
is
most
using
finite
finite
element
shapes
is
transfer
the
function
chapter
to
used
of
test
to
undertaken
function
of
response
of
the
frequency
modal
on
the
estimate.
or
any
exact
This
verify
structure
necessary
elements
model,
is very difficult to
the
by
as
the
modal
is
of
test
ratio
widely
This
analysis.
modal
from
co.
testing.
of
mode
results
represented
Some
4369
range.
There
resonances
obtain
B&K
the structure.
of
structure.
1.3.1
may be
to
used
accelerometer
structure.
the
of
the
of
was
with
structure
the
of
to
used
and
response
mass
characteristics
hammer
the
of
was
up the
pick
hammer
excitation
point.
insignificant
known
with
subsequently
accelerometer
A B&K 8202
software.
to
amount
model
the
analytical
of
can
natural
obtained
validate
other
the
model
method.
to
be
modified
by
damping
be
by
using
the
mathematical
Thus,
results.
further
finite
the
transient
measurements
causes
of
be
be
for
used
analysis.
The
insure
good
model
experimental
between
experimental
to
obtained
the
and
discrepancies
any
response
carefully
theory
the
then
can
model
harmonic
should
between
correlation
element
or
from
developed
model
them
so
be
can
that
easily
determined.
2. Modal
testing
can
be
Components
product.
tested
separately
developed
by
incorporated
used
their
into
the
In
substructuring
to
accurate
model
3. A
of
different
force
application
forces
by
caused
process
by
description
the
of
the
model
model
The
components
and
problems
process,
the
component
transfer
used
arise
is
forces
combined
functions
process
the
is very
structure
of
and
can
be
be
be
can
simplified.
be
may
effects.
A very
test
is that
modal
knowledge
desired
Here,
with
so
it is
via
but
a
the
direct
solution
is
the
mathematical
response
in
order
to
to the accuracy
essential
of
of
structure
sensitive
measurements
be
tested
measurements
whereby
are
the
possible.
not
structures
where
is
vibration
be
then
can
can
can
analysis.
incorporating
Situations
for the
of
assembled
model
study the
and
in this type
also
forces
mathematical
elements
causing
these
of
measurement
precise
structure
an
assembly
complex
required
determination.
dynamic
offered
is
an
Thus
improve the
modified
in
unreachable
assembly.
of
substructuring
substructuring.
broken
the
are
which
and
for
modal
that the
test.
1.4
The
finite
element
whose
active
period
of
of
regions
stresses
often
structural
(the
described
be
can
structure)
finite
by
modeled
separate
in
chosen
finite
being
the
finite
The
method
that
of
composed
behavior
is
element
finite
systems
method
assembly
irregular
in
of
of
approach
the
is
by
directly
still
equations.
in
suitability
and
solution
complex
process
for
and
structures
The
is
behavior
differential
to
of
equation,
the
is
are
described
the
of
the
the
approximate
heterogeneous,
forms in
differential
formation
special
each
of
which
equation,
automation
loading
and
solution
advantages
in the ability to
and
into
the
applicable.
the
requires
total
functions
of
If the
structure
to
representing the
sets
approach
distinct
(the
behavior
the
which
These
structural
applied
subdivision
continuity
total
separate
method
its
an
its
short
relatively
when
functions
single
procedure
continuum
by
continuum.
algebraic
reside
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ensures
by
If
described
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region.
method,
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assumed
represents
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each
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element
of
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characterized
element
solution
set
form
is
for
pursued
analytically
elements)
numerical
analytical
of
analysis,
displacements in that
or
structure
an
time.
problems
is
method
I14l
of
the
the
equation
represent
conditions.
of
of
highly
The
finite
using
element
MacNeal-Schwendler
have
a good
Rand Micas,
is
which
Intergraph CAD
to scale
Corporation's
preprocessor or a
system.
meshed
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various
creates
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data file
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1.4.1
results
the
the
by
into
of
use
the finite
of
categories
For the
the
of
applications
solution
necessary
perhaps
thermal
problems
to
the
of
find
in
element
on
consists
finite
fluid
file
graphic
capability.
analysis
modal
mode
of
observation
loader program,
equilibrium
the
can
method
the
element
problems
shapes
by
converting
be divided
into three
distribution
for
mechanics,
it
10
known
in
for
the
is
solid
solution
to
of
majority
category.
mechanics,
given
necessary
solved.
equilibrium
into this
stress
or
to be
as
The
fall
method
problems
Similarly,
problem
problems.
displacements
temperature
loading.
takes the
using
time-independent
or
problems
elements
graphics.
depending
created
NASTRAN. Subsequently,
by
or
results
is
arm
robot
an
on
available
beam
which
its
using
the
and
solid,
readable
RANGE OF APPLICATIONS
Applications
broad
directly
program
of
translator
NASTRAN
Again, postprocessing
model
finite
MSC/NASTRAN does
of
generator.
element
plate,
finite
another
out
carried
purpose
general
version
model
is
structure
robot
element program
not
the
of
analysis
it is
distribution
or
mechanical
or
of
equilibrium
find
pressure,
temperature
or
velocity
distributions
under
steady-state
conditions.
2.
In the
second
fluid
solid
and
often
requires
of
vibration
the
are
category
These
mechanics.
the
of
so-called
determination
structures
or
are
of
eigenvalue
problems
whose
frequency
natural
problems
problems
solution
and
buckling
of
of
of
modes
slender
columns.
3.
In
the
third
propagation
composed
added
1.4.2
is
category
problems
of
of
of
time
is
is
irregular boundaries
Highly
2. Variable
be
the
various
material
or
of
advantages
can
element
be
the finite
modeled
properties
element
with
method.
this method.
throughout
the
continuum
modeled.
3. Element
size
number
and
of
nodes
can
be
refined
for
analysis
type.
4. Discontinuous and
5. Non-linearities
mixed
boundary
conditions can
be included in the
can
6. This
method
is very
7. This
method
can
1.4.3
or
This category
mechanics.
when
dependent
are
can
multitude
continuum
Following
.
the
cost
applied.
model.
effective.
be efficiently
LIMITATIONS
be
computer
1 1
implemented.
The limitations
of
finite
the
element
finite
with
2. Since
it
as
are
method
and
folllows
in modeling
experience
elements.
involves
many
the
approximations,
results
may
not
be
accurate.
3.
Exact
4.
Damping
5.
Problems
boundary
conditions
is very difficult to
properties
involving
are
of
COMPARISON
MODAL ANALYSIS
Table
modal
(1.1)
shows
analysis
difficult to
in
nonlinearities
computer
OF
represent
in the
material
or
model.
model.
very difficult to
6. The availability
1.5
are
solve
facilities
with
can
EXPERIMENTAL
this
be
AND
structural
method.
major
drawback.
FINITE
ELEMENT
the
comparison
of
methods.
12
the
experimental
and
analytical
Mass, damping
Actual mass,
distribution
and stiffness
properties of the
tested.
Structural geometry is
Very
represented in
tation
Surface
can
and
fine details.
internal geometry
Only
be modeled.
Large
number of
freedom
degrees
Material properties
of
the
The Figure
Modal
of
is
for
are
very
degrees
small.
tested.
to the best
be
Actual
assumption
boundary
approximate
conditions or
free-free
tested.
Testing
associated
accessible
of measurement
freedom
Actual
are assumed
conditions can
(1.1) t20'
mathematical
represen
structure.
conditions are
TABLE 1.1
structure are
geometry
surface
Number
of
or estimated.
modeled
coarse
and
testing.
are considered.
Boundary
the
stiffness of
damping
and
model
with
compares
Finite
or
the different
Element
Modal
Method
Model
it.
13
and
approaches
in
the
the
taken
by
formulation
subsequent
the
of
analysis
EXPERIMENTAL
FINITE ELEMENT
MODAL ANALYSIS
ANALYSIS
Dynamic
Discretize
Testing
Model
\/
\/
FRF
Mass, Stiffness,
Data
Damping
Matrices
\ /
\/
Curve
Solve Eigenvalue
Fit
Problem
\/
\/
Modal
Mo del
\/
\/
\/
Local
Forced
Vibration
Modifications
Response
Absorbers
\/
\/
Sensitivity
Animated
Analysis
Display
FIGURE 1.1
14
[20]
1.6
MATHEMATICAL MODELING
All
having
systems
Vibrations
vibrations.
the
frequencies
particular
by
frequencies
here that
solution
neglected
of
masses
be
correct
of
and
system
mathematical
parameters.
finite
vibration.
of
of
stiffness
are
the
motion
could
be
model
with
mainly the
minor
system
of
Equations
algebraic
of
motion
equations
and
15
eigenvalue
the
gives
It
namely
differential
an
natural
be
must
damping
of
stated
mass
and
controls
the
term
is
often
point
representation
of
and
can
the
stiffness
freedom
then
matrices.
discrete
as
lumped
are
of
not
be suitably
Many
will
is known.
system
considering
of
distribution
nonuniform
The
by
masses
of
frequencies.
finite.
reduced
equations
Damping
role.
with
every
to
and
To
these
properties
Hence the
natural
not
problem
the system.
of
number.
using linear
plays
rise
the
are
mechanics,
of
gives
eigenvalue
are
freedom
of
write
at
system.
that
to
simplification
shapes
calculation
unless
laws
place
properties
of
necessary
basic
the
of
of
elements
is
take
frequencies
material
free
undergo
excitations
Natural
transformation
damping
the
The degrees
the
The
mode
amplitude
in
various
frequencies
natural
diminishing
the
and
where
to
using
coordinate
The
system.
it
properties
external
the
of
law.
second
equations
This
system
elastic
any
frequencies
the
Newton's
stiffness
attributable
these
of
problem.
of
configurations
determine
and
without
frequencies
natural
motion
mass
reduced
to
expressed
the
complex
physical
systems can
be
freedom mathematical
is
simplification
necessary to
not
consider
studied
with
possible
in
two degree
or
one
simple
models.
be quantitatively
can
by
represented
these
larger
having
model
hence
case,
system
However
models.
every
the
of
it
such
might
be
degrees
of
the
of
of
number
of
freedom.
In
linear
equations
of
differential
motion
equations
freedom
of
involved
an
allow
with
from
(resulting
undamped
expression
transformation
second
constant
the
system,
of
these
solved
the
to
coupled,
order,
the
It
in
uncoupled
separated.
exists
form.
to
terms
system).
coordinates
Each
ordinary
the coupling
of
coordinates
principal
set
difficult
is
coordinate
principal
equations
coordinates
of
of
set
system
coefficients.
of
choice
But
which
Hence
is
equation can
if
carried
then be
independently.
According
of
is
obtained
dynamic
solve
for
multi-degree
to
Meirovitch
differential
coordinates
and
I11!,
equations
solving them
simultaneous
solution
of
vectors
multiplied
by
'the
using
the
natural
as
and
linear
coordinates
then
to
set
natural
expressing the
combination
is
uncoupling
transformation
independently
equations
of
process
whole
known
of
as
modal
Modal
Analysis.'
The
following
text
concisely
explains
16
the
formulation
of
equations
of
motion
uncoupling
for
single
the
differential
books. The
following
Meirovitch.
t12l
multi-degree-of-freedom
equations
system.
1.6.1
and
standard
explanation
for
and
systems
multi-degree
be found in
on
the discussion
is based
freedom
of
can
and
many text
by
given
SYSTEM
The
spring-mass-damper
model
of
generalization
basic
model
degree
mechanical
degree
of
to
single
further
derive
in
shown
complex
freedom
of
system
system.
of
the
Fig.
freedom
equations
It
is
also
system.
of
is
(1.1)
simple
simple
Consider this
motion
of
multiple
systems.
q(t)
>
F(t)
Fs(t)<r
>
FIGURE 1.2
Consider
q(t)
the
diagram
where
F(t)
is the
the
equilibrium
restoring force
and
damping force
damper
motion
body
displacement from
the resulting
and
free
F(0
Fd(tK
respectively.
Using
Newton's
is,
17
external
position.
Fs
force
and
F,j
and
are
second
law, the
equation
of
F(t)
where
Fs
hence,
is
constant
1.6.2
q( t)
q(t
+ c
second
q( t)
k q(
t)
Fd
=
of
cq( t)
F( t)
linear
order
(2)
ordinary
differential
section
consisting
dampers,
the
equation
with
coefficients.
of
as
presents
masses
shown
typical
mathematical
system.
Consider
mj
(i=1,..n)
m;
is drawn. [131
18
model
connected
masses
MULTIPLE DEGREE
multi-degree-of-freedom
with
(1)
mq(t)
and
OF FREEDOM SYSTEMS
This
But
Which
Fd(t)
-F8(t)-
body
linear
by
of
system
springs
diagrams
and
associated
c2
lJfjHJ}J>M>J>J.
>c2(q2-q1)
Since the
degrees
masses
motion
The
n.
7C3(q3-q2)4
takes place in
freedom
of
displacements
Applying
of
Newton's
ntrtWitiitttnit
k3(q3-q2)
k2(Q2-qi)
FIGURE 1.3
c3
niiirfiiimimnri
the
of
system
masses
law
second
with
coincides
coordinates
generalized
the
dimension,
one
the
representing
mj is denoted by qj(t)
to
write
the
number
differential
(i=1
of
the
,2,...n).
equation
of
motion,
q(t)
c.+1[q.+1(t)-qi(t)]+k.+1[q.+1(t)-qi(t)]
-c.[q.(t)
Where
can
be
m.q.(t)
Qj(t)
-qM(t)]
represents
the
-m.
-k.[q.(t)-q.
externally
(3)
q,(t)
impressed
force.
Equation
(3)
arranged as
-c.+lq.+1(t) +
(ci
c.+1)qi(t)-c.q..1(t)
-ki+1qi+1(t)^(kj
ki+1)qi(t)
k.q.l(t)
19
Q(t)
(4)
5ijmi
mij
5jj
i*j
h )
kjj=
cij
Cjj
cy
Where,
-ci
ky
kjj
of
qi.(t)
of
set
equations
for the
damping
and
arranged
in
[my]
Arranging
Eq.
(6)
can
the
mass,
the
be
i
i
matrix
and
of
set
stiffness
equations
of
as
where
second-order
i-1
n).
symmetric
Hence
(6)
,..,n
ordinary differential
qj(t) (i=1,2
are
form.
(5)
k..q.(t)]=Q.(t)
coordinates
damping
complete
written
coefficients
symmetric
i-2,i+2,....,n
1,2,
and
can
we
The mass,
can
be
simplify the
as,
the
[Cjj]
generalized
Qj(t), in the
{Qj(t)>
{q(M
now
[m]{q(t)}
-ki+1
now
ki+1
simultaneous
[m]
generalized
impressed forces,
{Qj(t)}
k,
-kj
Hence
c.jq.(t)
stiffness
the
notation
are
kjj
is
(6)
above
and
respectively.
X[rr>..
Eq.
ci+1
mjj, Cjj,
coefficients
motion
Cj
cy
kjj
-q
be
written
[c]{q(t)}
[c]
[ky]
coordinates,
column
in the
qj(t),
compact
20
and
(7)
generalized
matrices
(8)
{Q(t)}
[k]{q(t)}
[k]
{Q(t)}
form as,
(9)
These
the equations
are
in
system
the
1.6.3
motion
form.
matrix
damping,
mass,
of
and
The
stiffness
of
[m], [c],
matrices
matrices,
general
undamped
For simplicity
system.
of
mechanical
analysis,
[m]{q(t)}
is
(Q(t)}
where
generalized
column
we
they
symmetric
{q}
coordinates
and
(10)
ordinary
differential
to
may
generalized
any
represent
the
coupled
prior
motion
coordinates
coordinates
a set
equations
involve
of
(9)
freedom
damping
reduces
term
qj(t)
in
to,
whose
[m]
For
[k]
and
elements
elements
the
as
purpose
arbitrary,
vectors
of
this
of
that
except
The
constant.
n-dimensional
the
are
column
generalized
respectively.
represents
be decoupled
equations
the
are
multi-degree-of
neglect
forces.
their
and
{Q}
Thus Eq.
equations
matrix
the matrices
forces
and
[k]
(10)
impressed
externally
consider
matrices
freedom
[k]{q(t)}=qt)
discussion
are
of
respectively.
the
and
Consider
need
degree
multiple
to
are
of
with
mass
the
constant
and
final
using
21
can
be
second
order
These
coefficients.
stiffness
,2,....,n).
n)
linear
matrices
which
To facilitate this,
solution.
expressed
vFj(t) (j=1
(i=1,2
simultaneous
different
Linear theory
expressed
set
states
as
of
that
linear
combination
the
of
^(t).!13!
coordinates
Hence
consider
the
linear
transformation,
(q(t)}
in
which
can
is
[u]
(11)
constant
nonsingular
as
an
{q}. Since
[u]
{q(t)}
{q(t)}
that
vectors
transforming
the vector
matrix
{}
Eq.
of
into the
(11)
same
{}
and
Eqs.
(12)
transformation
{q}
(11)
the
and
matrix
sides
of
[u]
(10)
the
connects
acceleration
into Eq.
(12)
and
vectors
{}
velocity
and
[K]
where
[M]
to the
coordinates
and
and
(13)
{Q(t)}
Eq.(18) by
[u]T
results
in
(14)
an
*F:(t) defined by
[K]
[u]T
because
[m]
[u]T
and
[k]
(15)
[k] [u]
Moreover,
are symmetric.
(16)
(Q(t)}
n-dimensional
forces N:
are
[m] [u]
are symmetric
{N(t)J
is
[u]T
{q}.
yields
IM]W)}+[K]{T(t)}-{N(t)}
give
[u]{(t)}
[m][u]W)}+[k][u]mt)}
[M]
[u]
[u]{*(t)}
the
Substituting
operator
The
matrix.
square
be regarded
vector
So
[u] pp(t)}
associated
vector
with
the
whose
elements
generalized
22
are
the
coordinates
*Pj.
generalized
If
the
inertially
transformation
such
and
is
(11)
consist
to
M^(t)+K. y.(t)
Equations
have
(17)
linear
the
system
transformation
modal
representing the
of
matrix
of
This
exist.
modes
(or
that
as
particular
the
of
The
transformation
of
motion
(14)
differential
referred
shown
can
coordinates
with
23
[m]
[k]
and
is known
[u]
vectors
vFj(t)
procedure
by
equations
of
means
to as Modal Analysis.
that in the
be
solved.
modal
The
coordinates.
undamped
coordinates
previous
was
an
matrix
In the
of
[K]
(17)
diagonalizing
equations
and
Eq.
then
of
1.6.4
the
the
the type
the system.
simultaneous
it
of
Hence if
motion.
found,
of
consists
principal)
is generally
section
the
completely
be very easily
can
[u]
matrix
natural
system
modal
and
it
solving the
the
structure
since
natural
of
equations
matrix,
called
are
be
can
[M]
system
of
equations
then
j=1,2,...n
same
[u]
the
be
object
matrices
N.(t)
the
single-degree-of-freedom
does
then
independent
diagonal
independent
matrix
The
to
said
is diagonal,
[K]
uncoupled.
produce
is
(14)
system
hand if
other
only
of
transformation
represents a set of
then
elastically
because
simultaneously,
decouple
be
to
said
diagonal,
On the
uncoupled.
is
system
is
[M]
matrix
easily
the
absence
of
decoupled
modal
matrix
damping,
using
acting
as
transformation
eigenvalue
The free
whose
problem
vibration
solution
[k]{q(t)}
represents
'
'
(t)
j=i
(19),
(j=1,2,....,n)
motion
and
for
in
qj(t)
IJ
qj(t)
where
time
Uj
q(t)
homogeneous differential
where i= 1,2,..
is
(19)
.n
special
in
that
type
which
of
the
all
motion.
solution
the set of
of
coordinates
Physically,
same
this
the index
and
j,
dependence,
time
of
the
ratio
constant
between
during
the
two
any
motion.
coordinates
Mathematically
by,
expressed
(20)
,2,...,n)
is the
q:(t)
implies
Uj/(t)
to
reduces
that
so
motion
(j=1
which
problem
M31
simultaneous
synchronous
remains
q:(t),
of
eigenvalue
solved.
(18)
configuration
amplitude,
this type
be
must
vibration.
all
which
and
of
so-called
{0}
of
namely
execute
the general
the
modes
the
to
the
matrix,
vibration
free
directly
natural
set
modal
the type
of
j=i
the
with
leads
problem
of external
[m]{q(t)}
equations
associated
yields
In the absence
which
To determine the
matrix.
are
same
constant
for
all
amplitudes
and
the coordinates
the
following
equation
24
results
/(t)
qj(t).
/(t)
is
function
of
Inserting Eqs.
does
not
depend
/(t)l
+/(t) k J
u
J
j=i
Equations (21)
'
be
can
where
=0
i =1,2
(21)
'
j=i
in the form,
written
I k..
u.
(22)
/(t)
>
m.. u.
IJ
'
whereas
ratios
right
be
must
function, the
by X,
/(t)
(22)
Eq.
(22)
does
not
depend
to
equal
X/(t)
side
constant
the set
of
side
not
on
real
number.
depend
time,
Assuming
constant.
be
must
does
that
so
that
index i,
on
/(t)
Denoting
is
the
two
real
the constant
yields
(23)
Y
Consider
(k.
solution
/(t)
Xm..)
u.
of
satisfy the
equation
which
or
solutions,
exponential
form
(25)
into (23),
it
be
can
concluded
that s
S2
but
one
must
(26)
roots
negative
magnitude
in the
=0
has two
s-|
If A. is
(23)
(25)
solution
Eq.
est
Introducing
s2
(24)
=0
for
non
trivial
solution
V-A.
(27)
number, then
opposite
in
decreasing
s-j
sign.
and
and
In
the
25
S2
are
this
other
real
case
numbers, equal
Eq.
(23)
has
in
two
increasing exponentially
time.
with
motion,
and
(27)
or
s-|
however,
S2
is
negative
considered.
must
Letting
with
stable
be discarded
co2, where
+ico
co
is
(28)
of
Eq.
A2
ei(0t
A-,
positive
inconsistent
are
yields
/(t)
or
solutions,
so
real, Eq.
so
These
(28) becomes
e-'wt
(29)
/(t)
C is
where
(cot-<t>)
cos
an
$ its
motion,
every
arbitrary constant,
q:(t)
Eq.
(31)
[m]
and
[k]
and
determinant
expressed
where
(co2) is
called the
possessing
of
the harmonic
matrix
form
(31)
the
nontrivial
of
coefficients
Uj
associated
if
solution
with
and
only if the
This
vanishes.
matrices
can
be
in the form
(co2)
of
it
frequency
{u}
co2[m]
represents
is the
co
phase
coordinate
[k]{u}
(30)
ky
-co2mjj |
called
characteristic
in
general,
(32)
distinct
It is
an
roots
26
equation
referred
of
to
degree
as
(32) is
n
in
co2
characteristic
values or eigenvalues.
the
square
roots
frequencies
certain
{u}r
is
cor (r
nontrivial
The
and
are
{u}r
The
represent
roots are
quantities
Associated
{u}r
whose
the eigenvalue
elements
eigenvectors
they
natural
system
are
ujr
real
also
referred
to
natural
modes.
by
modes
is
numbers.
(33)
characteristic
multiplied
natural
1,2,....,n
and
that
such
as
so-called
be
can
shape of the
are
con2
frequency cor
every
problem
known
are
the
are
with
{u}r
physically the
eigenvectors
Hence the
of
co2[m]
vectors
eigenvectors.
l,n).
these
of
vector
solution
[k] {u}r
The
any
as
vectors
modal
or
vectors
Since these
constant.
arbitrary
amplitude
is
not!
Modal
or
vectors
factors. If
one
of
eigenvectors
the elements
value,
sense,
because it automatically
the
elements
of
elements
elements
natural
normalization.
convenience
the eigenvector
causes
'n-1'
remaining
called
of
relative
{u}r
displacement
is
certain
of
represent
and
is
modes
The
is devoid
of
by
adjustment
of
virtue
constant.
to
an
render
The
any
physical
27
an
of
in the
values
significance.
ratio
adjusting the
amplitudes
process
absolute
process
their
normalization
the
assigned
is
unique
for
is
mere
CHAPTER 2
FINITE ELEMENT ANALYSIS USING MSC/NASTRAN
2.1
MSC/NASTRAN
The
Finite
method
Element
to
used
Since then,
FEM
applications
in
for the
credit
With the
speed
in
of
software
the
oldest
National
Aeronautics
purpose
applications.
Normal
This
in
results!
most
and
method
of
its
of
Complex
eigenvalue
realizing
Much
now
shifted
(Although
of
exist
the
engineers
it
commercial
has
analysis
28
in
range
finite
developed
originally
Administration
Space
analysis
analysis
have
used
capabilities
Buckling
4'
the
and
many
analysis
are
still
).
analysis
modal
airplanes.!1
development
engineering.
gained
which
widely
and
approximate
technicians.
program
Some
Static
and
of
goes
MSC/NASTRAN
is
software
to
and
analysis
general
FEM
or
structures
way
field
packages
interpret the
One
of
newer
engineers
to
long
every
of
equations.
analyze
numerical
of analysis
needed
to
come
practically
advent
from
1950's
has
is
(FEM)
differential
advancement
user-friendly
work
Method
solve
importance
some
of
structural
(NASA).
element
by
It
capabilities
analysis
are
the
is
and
f15'
Direct
frequency
Transient
Static
and
Modal
complex
Modal
frequency
The
above
and
analysis
The
random
response
analysis
with
modes
cyclic
with
are
stiffness
eigenvalue analysis
analysis
analyses
differential
with
Modal transient
Normal
response
analysis
analysis
Static
random
cyclic
available
be
called
symmetry
symmetry
formats'
in 'rigid
upon
NASTRAN
of
by simply inputting
and
particular
number.
input
file
for
MSC/NASTRAN
consists
the
of
following
three
the
main
sections.
1. Executive
Control
Deck
functions
subroutine
NASTRAN
In the past,
was
when
shows
punch
cards.
over
MSC/NASTRAN. [11
defines the
physical
terminals
computer was
on
were
in the form
card.
input
and
to be solved
problem
and
typical
was
NASTRAN
equivalent
program
Although obsolete,
not
of
output
the
by
to
an
set
cards.
of
still
of
Each line
punch
input file.
consisting
literature
29
complete
'deck'
control
video
punched
called
user
program.
the input to a
data
provides
specific run.
NASTRAN
of
Fig.
various
refers
cards
(2.1
decks
of
was
)t1
and
to cards and
FIGURE 2.1
t1]
decks
as
input
file.
2.2
model
coordinate
the
type
of
boundary
model
conditions
by
connected
stress
identifies
as
elastic
of
thermal
analysis
run
shapes
2.2.1
points
for
points
of
modal
for
NASTRAN
stresses
for
card
static
card
of
etc.
elements
number
for
ratio,
mass
isotropic
and
produces
inside the
analysis
analysis.
31
grid
points
the same
about
stiffness,
The
property
properties
anisotropic
natural
the
use
can
density
or
in
identification
of
material
outputs
to
elasticity,
used
an
elements.
element
and
has
of
with
according
and
numbers
the
points
element
information
contains
Poisson's
expansion
of
density
Every
and
number
which
material
constants,
displacements,
grid
card
recovery
a
card
property
A
like
are:
orientation
etc.,
This
card.
and
thickness,
loads.
inputs
required
grid
properties
applied
the
connecting
connection
a
the
points
grid
stiffness,
material
the element.
property
or
mass
as
and
to
pointing
element
such
one
of
elements
element,
requires
number
card
system,
properties
NASTRAN
with
location
including
geometry
element
structure
the NASTRAN
of
sections
To
at
was
such
forces
and
such
coefficient
materials.
as
grid
and
frequencies
An
point
moments
and
mode
This deck
with
briefly.
in
the job
which
deck. Most
the
ID
card
part of a
be
to
Executive
control
of
Executive
deck
control
by
the
subroutines
carried
be
out can
to contain
fixed
the sequence
which
specified.
However,
subroutine
sequence
This
user
from
were
to
be
developed
required
in
high level
subroutines
formats
rigid
inputted
is
of
large
available
programming language in
identified
be
to
used
controlled
various
using
be
can
performed
of
the card,
by
SOLKn
Kn
where
is the
can
be
can
be done
format.
results
modified
This
or
card.
is
deck
cards.
In this
specified.
with
card
also
the
If this
2.2.2
This
NASTRAN
deck
saved
is
provides
program.
data
is
by
not
of
rigid
matrix
format
operations
option
by
card
of
used,
saving
of
means
intermediate
CHKPNT
or
by
used
is
the problem
one
minute.
over
control
user
selection.
number
CPU time to be
maximum
One
results
important
deck
the
provides
more
card
card.
previously
One
number.
ALTER
an
by inputting
using
RESTART
format
rigid
of
the
input
of
best
in
This
is
32
stated
and
output
of
the
'NASTRAN
any
used
at
form
of
name >
<
loads
<name>
analysis
the data
in
data
constraint
is
sets.
by
specified
The
particular
be
directive. The
control
case
to
set
is
selection
card
particular
type
SID
is the
SID
and
and
time
execution
general
where
of
is
the
deck."
Bulk data
the
identification
set
The
be
data to
of
number
following
included
in the
associated
example
with
this
shows
clearly:
CASE CONTROL:
SPC
BULK DATA
The SPC
of
24 in the
The
case
data.
It has
data
and
defined
can
set
control
be
24
having
to
matrices
selected
modal
extraction
Case
from
number.
algorithm
method
control
being
also
can
be done
defined
has
can
SPC, SPC1
problems,
analysis
cards
by
to
'n'
card,
the
by
EIGR
be
being
the appropriate
cards
selection
loads to
selected
MPC
temperature
of
by SPC
or
MPC
eigenvalue
control
33
or
be
structural
etc.
GRAV
n"
or
sets,
Static
Bulk data.
Constraints
geometric
any
constraint
sets,
"LOAD
by inputting
to the
contain
load
card
24.
of
not
select
input
ID
an
identification
For
case control
deck does
cards
direct
TITLE
card
prints
text
prints
top
on
sorted
the analysis
or
of
each
be
element
stresses
forces.
reaction
displacements
card
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shapes
at
the
deck
control
combination
the
the
This deck
information
element
extraction
analysis.
of
starts
file. A
mention
sets
cases
are
BEGIN
BULK
card
The
the
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output
of
static
be
can
defined
be
large
mode
the
case
analysis,
defined
in
with
individual
all
the
problem
the
made
cards
sorted
here
of
34
grid
can
can
and
for
eigenvalue
modal
thousands
be inputted in any
are
alphabetical
echoed
and
boundary
normal
contain
according to
cards
points
property,
the
all
contains
analysis
technique
in the deck
cards
sorts
static
and
including
material
and
reduction
By default,
must
geometry
for
loads
names.
constraint
linear
prints
DISPLACEMENT
important function
In
element
dynamic
because NASTRAN
card
cards
A
to
problems
definition.
cards print
SPCFORCES
nodes.
load
structural
applied
code.
ELFORCE
the
with
about
or
from
results
case control.
connectivity,
conditions,
of
and
and
VELOCITY
specified
Another
These
cases.
and
card
ELFORCE, SPCFORCE,
whereas
analysis
subcase
2.2.3
of
modal
loading
of
in the
subcases
velocities
resonances.
is
load
various
lines
in
used
forces,
and
The
specified.
STRESS
etc. cards.
DISPLACEMENTS
and
as
using STRESS,
printed
DISPLACEMENT, VELOCITY
the
of
bulk data
unsorted
can
page
in the
of
order
order
output
used
in
request
of
special
card
is
used
zero
or
very low
degrees
of
weight
Grid
point
the
start
element
e.g
cards
by
elements which
library
other
used
it
words
allow
PARAM, AUTOSPC
purges
freedom
of
the
with
unconnected
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cards
start
be any
might
properties
with
the GRID
with
All
of
These
The
card).
are
available
to
set
the
to apply
'P'
the letter
selecting
the
up
properties
local
number of
degrees
enables
using
structural
model
For
and
of
reduction
of
is
freedom in
the
solution
the
subsequent
35
The
etc.
property
cards.
elements,
defined in the
geometry.
cylindrical
analysis,
method
of
and
freedom
procedure
a system
of
cards
'MATi'
There
or
etc.
are
spherical
cards
modal
degrees
are
are
rectangular,
constraints.
number
Dynamic
element
eigenvalue extraction
reduction.
This
material
loads to the
static
to define the
elastic
systems.
grids
of
inputted
cards
The
card.
e.g.
geometric
used
cards which
the degrees
constrain
NASTRAN.
of
letter 'C.
coordinate
few.
in
stiffness
e.g.
cards.
for
automatically
information is inputted
the property
are
processing features,
vast
with
These
PARAM
number of
generator.
connected
in
to
are a
large
are
by
EIGR
in
which
reduced
problems
sections.
the
etc.
cards
DYNRED is
are
are
used
dynamic
the
total
to a selected
easily.
It
is
The NASTRAN
discussed in
2.3
cards
control
NASTRAN
the
the
executive
eigenvalue
motion
in the
for
modes
can
be
[M] is
stiffness
absence
the
mass
As
matrix.
of
of
an
problem.
or
given
undamped
as
(the
Mq
Consider the
system
eigenvalue
'SOL
3'
card
reduction
and
control.
Equations
be
expressed
can
the
The
[C]
in
is
damping
the
The
deformed
of
motion
brackets
are
be
can
can
solution
nontrivial
matrix
previous
problem
frequencies
natural
equations
Kq
(1)
equations
coordinates.
square
freedom
of
matrix,
The
by
in the case
specified
these
transformation
eigenvalues
This is inputted
explained
damping,
eigenvalue
Real
the
performs
as
[M]q+[C]q+[K]q
where,
are
degree
multiple
analysis
deck.
control
form
matrix
format
rigid
extraction
and
(A).
appendix
analysis or normal
of
case
In
in
in the
used
and
and
chapter,
decoupled
then
this
of
be
[K]
the
in
the
using
reduced
to
problem
associated
eigenvectors
shapes.
without
dispensed
external
with
excitation
are
for simplicity)
(2)
nontrivial
gives
solution
of
this
36
equation
in
exponential
form,
q
where
(3)
is
an
M
or
or
( K
co2
Substituting
K
This is
used
to
2.
3.
eicot
co2
co2
co2
available
1.
arbitrary
Substituting
motion.
e'<t
X)
X,
=
standard
extract
eicot
Eq.
is the
(2)
frequency
of
harmonic
gives,
(4)
(5)
0
(6)
gives
0
eigenvalue
eigenvalues
in MSC/NASTRAN
problem.
and
There
are
eigenvectors.
various
Three
techniques
such
methods
are,l7l
Givens triangularization
method.
2.3.1
in
co
and
this solution
+
constant
initial
guess
method
and
finds
is
the
an
t7l
iterative
method
eigenvalues
problem
has to be
in
its
which
starts
from
neighborhood.
placed
in the
To
following
form
(K-MA)u
resulting in
u
If (A.j, Uj)
an
(7)
0.
fixed
[K]'1
is
[M]
point
problem
(8)
u.
eigenvalue
and
associated
37
eigenvector,
then
the
substitution
in
of
vector
technique
Let un
be the
converges to
current
Wn+1
W
where
next
[K]-1
[L]T
side
estimate
Eq.(8)
of
would
used
to define
result
on
vector
iterative
an
Uj).
the
of
be
can
(Xj,
pair
eigenvector
iteration
first substituting un
by
obtained
The
Eq.(8)
where
is
which
hand
right
on
us
the
into
number.
Eq.(7)
after
The
iterations,
next
estimate
to obtain
[M] un
(9)
estimate
of
un is found by,
eigenvector
(10)
"n+i"^
m-1
Then
is the largest
Cn + i
where,
if
the
specified
value,
minimize
the
the
difference.
eigenvalue
whereas
drawbacks,
in
and
its
point.
is
which
Consider
(K-XM)u
un
it
that
accuracy
NASTRAN, in
process
is stopped,
The
gives
the
and
U2
else
Cn
are
slowly
slight
evaluated
u-j
is less than
iterations
eigenvector.
very
Hence
roots
-|.
parameter
the
converges
lost.
Wn +
between
difference
vector
in
element
gives
This
for
the
done to
lowest
has
procedure
close
modification
with
are
eigenvalues
is
respect
used
to
in
shift
again
(11)
38
and substitute
Xo is
or
[K
( Xo
called
hence
Xo
M]
p)
and
is the
(K
which
can
Xo
M)
be
substituted
Wn+1
pM
(12)
Eq.(12)
can
be
written
as,
=0
in
M)"1
=(K-Xo
eigenvalue.
shifted
Eq.(7)
equation,
Mun
(13)
W
"n+i-c^
where'
(14)
n+1
where
to the
are
again
un + i
shifted
few
this
structural
there.
this
is
A
it.
Also the
or
can
The
of
shift
within
Xo
matrices
be
can
also
specific
39
effort
this
easily.
convergence
point
point
p.
There
Triangular
is
to
required
encountered
method
is
at
can
One
at
the cost of
frequency
so
more advantage
changed
be
in
useful
have to be nonsingular,
not
be handled
shift point
rate
obtained
does
shift
converges
method.
less
mass
and
Cn+^
banded, hence
matrix
modes
the
this
considerably
stiffness
is that the
are
required
narrow
are
decomposition.
eigenvalues
2.3.2
of
in
noted
is
M)
banded,
to improve accuracy
matrix
Xo
stiffness
body
method
be
to
lot
analysis
(K
of
matrix
diagonalize
that is closest to
eigenvalue
points
decomposition
If
placed
range.
any
stage
additional
such
that
This
the
is very
method
for
eigenvalues
miss
this
the
of
any
also
positive
definite
it is
as
given
matrices
handled
are
symmetric.
However the
the
whereas
The
from
obtained
stiffness
steps
is
it
be
can
matrices
well.
problems.
small
hence
system,
Sparce
roots.
Dense
method.
methods
long
the
fast for
and
useful
not
all
possible
to
dynamic
matrix
mass
be
can
with
solved
easily
the
matrix
It finds
reduction
has to be
singular
as
are
as,
LCholesky decomposition
[M]
is done,
<15)
lower triangular
the eigenvalue
reduce
matrix
[L][L]T
is
[L]
where
2. To
the mass
of
matrix.
of
problem
Eq.
to
(6)
its
[L]"
[J
by,
given
[L]"1
I] [W]
0,
premultiply
Eq.
(6) by
and
substitute
and
form
standard
[K]
[L]T
[L]
[L]T
(16)
0.
-X[L]-1
or
[L]"1
-T
(17)
thus,
[L]_1
[L]"1
or
[L]"1'1
[K]
[K]
comparing this
0,
[L]"1'7
with
standard
(18)
0.
[l]){WJ
-X
the
[I]
(19)
0.
equation
eigenvalue
form,
[J-XI]
gives
{J}
3. The
W-X
{J}
matrix
extracted
method
[L]"1
using
to
[K]
is then
the
(2)
[L]"1-1
triadiagonalized
modified
determine
all
and
all
the
Q-R
algorithm.
the
eigenvalues
40
eigenvalues
(Q-R
of
algorithm
are
is
symmetric
triadiagonal
matrix.)
4. Eigenvectors
{W}
are
{u}
2.3.3
then found
are
[L]1-"1
eigenvectors
by solving
that
being
the
(MGIV)
shape
mass
As
method,
to the
similar
Givens
mode
(21)
nonsingular as
in
the
are
which
using
modification
eigenvalues
given
(W}.
for the
calculated
does
matrix
steps are as
decompose the
mass
have
not
to
be
follows,
matrix
using
Cholesky
decomposition.
XSM
optimizes
is
Eq.
[K
Xs
accuracy
(6)
M
if
[K+XSM]
[
of
the
number
the
is
specified
The
solution.
problem
is
by
matrix
the
in
program.
this
equation
formulated.
properly
It
Now,
as,
Xs ) M] {u}
for
(22)
positive
definite
positive
rearrange
[L]T
[L]
sides
(23)
0.
X'
by
-1
Xs)
and
substituting
gives
(L
LT)/(
Now premultiplying
by
Xs) ] {u}
[L]"1
and
(24)
0.
substituting
41
{W}
[L]T
{u}
gives,
{ [Lr1[M] [L]-1-T-X[l]}{W}
hence, {J}
[L]"1
Thus, {J} is
extracted
2.4
of
recovery
(i.e.
solution
since
Computationally,
of
of
equations
the
of
solving
square
or
cube
freedom).
Dynamic
of
assembly
reduced
to
of
eigenvalue
set
of
and
the
the
by
of
which
few
freedom. These
reduction.
They
are
the
selected
system
reduced
methods
two
response
schemes
are, ^
42
of
of
other
stage.
to
be
described
available
in
the
More so,
to
the
of
after
the
freedom
are
words
the
represented
can
and
degrees
applied
In
then
matrices
previously
quantities.
of
second
this
compared
degrees
ones.
are
in
and
important
proportionally
procedure
total
the
most
(number
problem
is
matrices
the
found
increases
matrices
of
is
of
assembly
Obviously,
etc.
expensive
of
recovery
size
assembled
are
system
frequencies
processes,
matrices)
very
reduction
dynamic
is
the physical
extraction
the
of
eigenvalues
matrices
a
characteristics
degrees
of
eigenvalues
solving
natural
three
displacements
step
of
obtain
following
assembled
this
assembly
cost
of
process
to
motion
and
the
all
solutions
stresses
and
basically
of
equations,
algorithm.!1]
It consists
shapes.
stage
is
analysis
equations
dynamic
some
(26)
DYNAMIC REDUCTION
mode
one
[M]
differential
(25)
[L]-1>T
then transformed to
dynamic
0.
solved
by
these
with
the
thereby saving
NASTRAN for
1.
2.
2.4.1
The
GUYAN REDUCTION
freedom, in
generalized
the
all
set
[%] (%)
T
where
[Kff] {u{}
for full
stands
(denoted
degrees
of
associated
keeping
Upon
the
discussion
are
the
of
is
represented
motion
is
degrees
the
all
containing
equations
given
by,
(27)
unreduced
equations.
subsets
used
o-set
of
set
into two
{Uf}
and
of
{uf}. The
by
[Fff]
partition
freedom
static
or
by {ua})
with
vector
following
dynamic
of
[8]
displacement
general
Condensation)
in the
are
solution
the
by
{u0}).
process
eliminated
of
characteristics
referred
from
structure
to
as
the
The
a-set
whereas,
those
the
the
equations,
same.
partitioning
f"al
{Uf}
'
the full
set
i-\
lUJ
of
(28)
equations
^a
H>a
"Kaa Kao
ua
<
Moo, \
in the
>+
partitioned
43
can
be
written
as
"f;
{ 1
^oj lUJ
form
fo_
(29)
Solving
{uo)
^oa} (ua}
{u0}
(30)
where,
{Goa}
K)
and
For
static
drop
hence
the
eliminate
the
analysis,
time derivatives
{u0}
of
of
back in
freedom.
In
simplified
Substituting
u0 do
not
out
Eq.(29), does
to
order
Eq.(30)
Eq.(29)
for
dynamic
and
eliminate
the
the
solution
equation,
{Goa} {ua}
Eq.
(33)
to
Eq.(30)
of
not
simplify
{u00}
entire
in
However
drop
in
term
{u00}
substituted
freedom.
of
Eq.(30)
of
be
can
procedure,
following
(30)
degrees
o-set
degrees
(32)
equation
substitution
o-set
(31)
F0-[M0a]ua-[M00]u0
analysis,
out
direct
[Koa]
-[K00r1
(33)
"I
{ua)
This
value
vibration
solution
for
the
contain
of
{Uf}
problem
of
the
o-set
is then
to
is
far less
substituted
a-set
problem
in
degrees
is found in terms
recovered
number
(34)
of
using
degrees
44
freedom
of
Eq.(33).
of
a-set
only.
After
DOFs, the
Since
these
freedom, they
can
the
the
solution
equations
be
solved
less
expensively
The
unaltered.
specified
or
The latter
degrees
selected
being
ASET1
cards
should
be
included
with
in
the
uniform
over
method
are
static
of
freedom
automatically
a-set
the
that
such
and
normally
the
can
are
be
the
good
of
both.
of
masses
a-set
within
or
freedom
of
are
points
obtained
and
user-
the ASET
with
large
with
eigenvalues
reasonably
be
can
degrees
distribution
the
system
combination
selected
grids
the
of
{ua}
set
user-specified
The
structure.
in
They
all
that
characteristics
or
NASTRAN. The
chosen
in
the
is
this
with
engineering
accuracy.
2.4.2
Generalized
dynamic
Reduction
Method.
approximate
the vibration
again
the
{Uf}
degrees
It
generalized
of
coordinates
selected
{ua}
coordinates
by inverse
iteration.
{u0}. The
and
{Ug},
and
the
Here
{ua},
vector,
remaining
neglected.
as
{uq}
But in
scalar
is
that,
were
thrown
generalized
points.
Hence
45
in
Guyan
away
physical
Reduction,
when
the
unreduced
and
the
vector
the
to
{u0} is
vector
displacement
reduced
Reduction
generalized
is
generalized
Guyan
the
of
was
t8l
modificaton
obtained
coordinates,
Generalized
{u00}
modes
The
the
uses
is divided into
eliminated.
eventually
contains
vector
is
reduction
{Uq}
displacement
vector
the
and
corresponding
stiffness
matrix
are
given
as
Kl
[K1
and
Again,
in Guyan
as
Reduction, {u0}
Ktt K,to
Ko, Kooj
(35)
be expressed in terms
can
of
{ua}
as,
( u0 }
{ Goa } { ua }
"u~
Gnn
oq
ot
(36)
A
where,
Thus
[ Got ]
static
degrees
{Goq},
of
the
properties
freedom
{ur}
contains
rigid
body
modes
eigenvectors
{uc}
are
One
of
more
are
approximate
To
and
and
are
in
preserved
the
evaluate
subdivided
{ur}
in the physical
transformation
{u|}.
and
The
matrix
set
{u|} is
to
define
{uc}. Here,
used
points where
rigidly
in
modal
motion
is
restrained
analysis
enforced.
when
the
approximate
calculated.
which
coordinates
are
free
when
approximate
calculated.
are
that
structure
coordinates
important
freedom
the
coordinates
the
eigenvectors
[ Kot ]
{ut}.
{u^}
the
the
are
of
is
{ut}
set
{u^}
]-1
[ K00
set
are
vibration
to be
free
modes.
considered
to
displace
Hence
46
is {uv},
containing
during
the
degrees
calculation
of
(37)
The
vibration
modes
are
calculated
by
solving
the
eigenvalue
problem,
[ Kvv
{uv}
is
XMVV ] {uv}
of
approximate
be
given
written
with
0.
{ub}
for
vectors
(38)
{uq}
and
vibration
removed.
modes,
If
[Ovq]
solution
of
is the
Eq.
matrix
(38)
can
by,
{uv}
Hence
{uf},
a set of
[<&vq] {uq}
the
modal
(39)
solution
of
the
entire
{uf}
vector
can
now
be
as,
~$~
ful
f*
1
ur
uc
{Uf}
>
$rv,
KJ
in
Writing Eq.(40)
{uf}
(40)
-O^.
compact
form,
<>
lu
Uq
K)
<
?
_
Ut
tj
1
0
^oq
{uq}
(41)
where,
(*.,)
Substituting
Locqj
Eq.
[<&oqJ {uq}
or,
[Goq]
Hence the
(41)
-
in Eq.
[QoqJ <uq}
[Ooq]
reduced
(36)
+
gives,
(42)
[Got] [Otq]
form
of
(43)
stiffness
47
and
mass
matrices
using
Eq.(36)
"O
oq
<
{uf}
qr
o,oq
iut
[KJ
3x,KooGoq
(44)
Ktto G
ot
^q^ooGoq
SYMMETRIC
.T
(45)
^T
m + m,0 Ga G; m; + G; Ko g01
+
and
(41)
Ktttt
HJ
{uq}
is
vector
by,
given
%?o
{Fa}
(46)
Ft+GotFo
where
{Ft} is
of
on
{ut}.
2.5
The
selection
relative
Modified
of
cost
and
Givens
Generally, they
singular.
The
most cases
the
Inverse
methods.
the
become
eigenvalue
methods
do
not
of
are
miss
any
results
quite
roots
is
fully
not
used
required.
similar
even
l9l
and
very
if the
of
method
is
less
compared
very
as
coupled
good.
method
and
the
and
Moreover,
of
the
accuracy
if
of
solution
GIV
Power
then
or
with
MGIV
obtaining the
Generalized
reduction,
Inverse
48
to
is
Hence in
cost
preferred.
reliable.
matrix
The
is
and
mass
other.
method
the
upon
Givens
each
depends
method
MGIV
Power
extraction
of
accuracy
costs
eigenvalues
reduction
an
the
method
dynamic
matrices
becomes
impractical.
important
for
Hence
criterion
which
important,
is
reduction
2.6
As
it
is
or
preferred
few
when
its
accuracy
the
Modified
is
Otherwise
method
of
cost
eigenvectors
good.
Givens
the
when
with
if
be
to
are
before,
there
two
are
MSC/NASTRAN,
with
Generalized
dynamic
degrees
200
exceeds
selection
reduction.
the two
many
and
depends
Generally
The
the
are
as
to
freedom
of
cost
upon
reasons
methods
expensive,
of
generalized
roots
very
dynamic
the
given
be
seen
is
of
favored
Guyan
preferred.
49
reduction
two
The
methods.
over
Guyan
comparison
in the
works
the
problem
exorbitant.
the
and
when
used
analysis
becomes
cost
reduction
in Table
be found,
hence it is
solution
methods
Reduction
are
dynamic
the
relative
dynamic
can
in
methods
reduction
Guyan
namely
not
used.
discussed
number
an
extracted
is
cost
available
is
solution
of
cases where
out
to
be
less
GUYAN REDUCTION
GENERALIZED DYNAMIC
REDUCTION
Accuracy
of modes
Relative Cost
Fair
Excellent
Lower if
accuracy
modes
Skill
required
for
of vibration
is required
Selection
of
good
excellent
to
accuracy is
required
A-set
Selection
are
No.
of
DYNRED
points
Labor intensive
Yes, if there
A-set
Troubles
Poor
many
points.
selection of
points
A-set
leads to
inaccurate
modes
Poor selection
parameters
of
DYNREU
leads to
erratic
missing modes,
results or excessive
cost
Diagnostic Aids
Sturm
None
sequence
indicates
missing
Number
of good
One-fourth to
one-half
number
of
Close Roots
No
May
loose
is too
50
modes
modes
problems
number of
some
small.
if NIRV
CHAPTER 3
3.1
Modal
characteristics
normally has
of
of
stiffness
matrices
modal
analyzer
to
in
motion
coupled
Modal
system.
measured
measurements
model
such
as
1. Excitation
of
the
made
of
measuring the
input force
measurement
the
three
of
the
Transducing
the
3.
Recording
and
about
at
removes
defined
analytically.
ratio
of
the
the
the force
points.
on
entire
of
the
Function
spectrum
mode
Fast Fourier
response
From
to
method
frequencies,
channel
and
from the
of
FRF
can
build
set
structure,
one
structure.
Modal
testing
steps:
structure
response
with
measured
force.
to a measurable form.
analyzing
modal
points
of
response
the mass
experimental
dual
choice
systematic
system
natural
(Frequency Response
measurement
primary
2.
information
parameters
is
analysis
an
system
by diagonalizing
testing is
modal
dynamic
the
freedom
of
and
between
response
consists
and
before,
of
degree
multiple
and
Transform
up
structure.
determining
of
determine the
system
process
coordinate
discussed
shapes
equations
particular
procedure
As
is
analysis
the
parameters.
51
response
to
extract
all
the
There
are
wave,
number
sweep
of
sine,
periodic,
Frequency Response
Where, X(co) is
that
of
complex
frequency
'co'. The
input
amplitude
between
output
output
type
input
and
is
model
used
the
consists
for
mode
modal
model.
degree
of
modal
in the
The
frequencies,
frequencies
and
of
F(co)
be
of
set
of
to
and
will
produce
the
Thus
modal
output
phase
FRF
the
independent
experimental
This
of
experimental
set
the associated
of
data,
The
52
referred
which
model
shapes
in the
of
any
response
in
terms
giving
modal
conditions
measurements
one
to as the
test may be
frequency
dynamic
FRF
mode
modal
single
structural
is
the
mathematical
conditions.
model
is
The
and
as
FRF
the
of
O(co).
system
An
input
from,
measured
measurement.
weak
by
measured
the
under
co
by |H(co)|,
shifted
the
output
frequency.
same
multiplied
will
result of an
showing
the
of
from
complexity
measurement
of
of
The
frequency
measurement.
system
normally
each
for
at
properties
constructed
represents
excitation.
interpretation
physical
input
the
sine
transient
or
between
ratio
force
motion
be
will
stepped
excitation.
the
sinusoidal
sinusoidal
random
structure:
the
of
represents
function
the
excite
is the transform
Eq.(a)
to
ways
animated
form.
Some
the
of
instrumentation
in
used
is
testing
modal
described
below.
3.1.1
The
VIBRATION EXCITERS
the
upon
consist
available
are
It
not
very
For
the
by
of
given
electromagnetic
However,
field
is
excitation
the
The
and
from
Pulse
of
exciters
Shakers
electrohydraulic.
the
rotation
of
force
of
an
at
be changed,
once
the
it
need
not
magnitude
of
is
excitation
Contacting
magnitude
cannot
operation
shaker,
to
attached
through
field.
magnitude
noncontacting
W Three types
limited
contacting
and
hammer.
vibration
since
types
is fixed,
this type
shaker
of
is
low frequencies.
electromagnetic
electromagnetic
is
at
two
excitation).
shakers.
depending
available,
structure)
held
force to the
measured
exciters
are
the
generate
But
electromagnetic
measurement.
efficient
to
provides
can
operation.
there
hand
variety
shaker
mass.
is in
verified
shaker
of
transient
variable
be
or
Mechanical,
mechanical
unbalanced
vibration
using
of
types
Mainly
pulse
obtained
typically
various
continuous
provides
normally
shaker
are
for applying
used
application.
(providing
(which
device
There
structure.
is
exciter
The
power
the
structure.
amplifier
power
force has to be
53
placed
coil
input
and
can
measured
is
be
by
in
varying
easily
some
to
the
controlled.
other
device,
the
since
vibration
impedence
of
the
structure
is
much
is just
inertia
smaller
than what
it
to
Hydraulic
for
higher
because
itself.
shaker
near
provide
But
by
be.
to
one
its
of
mass
natural
the
on
and
large, it is
and
the
But this
limitation
heavy
the
of
to
applied
sinusoidal, random or
is
There
structure.
is
these
of
can
This
can
the
of
conditions
They
of
are
natural
the
most
be
in
shakers
advantage
useful
in
they
size
the
of
very
can
operating
this type
of
vibration
the
laboratory.
are
and
and
simulating
test
the
is that
load
static
very
structure
although
greatest
apply
incorporating
power
usually large in
vibrations
it
hydraulic
of
use
excitation.
that
simultaneously.
make
amplitudes
range
drawback
to
used
force
appear
might
of
be determined.
cannot
actual
structure
amplitude
type of shaker.
electromagnets
shaker
be
the
range
exciters
frequency
the
excite
shaker can
the
of
used
provide
to
frequency
frequencies
widely
the
the
with
the force
of
the table,
excitation
operating
changes
of
enough
frequencies. This
periodic
coil
and
Because
force
the
of
working
The
main
complicated
and
expensive.
shaker
can
arrangement
structure.
in
one
should
provide
of
Drive
direction
to
be
excitation
drive
rods
and
taken
rod
are
so
must
nylon
flexible
only in
in
that
be
one
made
connectors
all
the
54
the
direction.
while
which
other
operating
Hence
attaching
are
proper
it to the
extremely
directions.
frequency
Also,
stiff
care
range
is
significantly
3.1.2
above
HAMMER
transient
or
random
impactor
with
pulse
sets
the details
of
above
of
its
function
The
impactor
the fundamental
head
the
and
is
combined system
Fundamental
The force
wave
which
as
spectrum
vibrations
frequency
pulse
shown
as
in
in
of
the
suspensions.
It
mass
of
Fig.
to
head
the combined
fundamental
an
excite
pendulum
shows
range
excited
and
into
energy
for
(3.1)I31
frequency
the
used
basically
tips
and
handle.
impart
The
is
used as a suspended
it has
of
be
can
contacting
the
system
structure
consisting
frequency
the
of
by,
/contact
Frequency
by
of
the
is
frequency
of
frequency
the
pulse
or
the
impactor,
the
the stiffness
55
of
the
tip
Tc,
range
impact
exciting
(up
the
to a
higher
covered
should
used.
frequency
in
spectrum
the pulse,
larger
effective
frequency
of
of
pulse
the duration
the
upon
pulse
This
portion
duration
ffness
impactor mass
(3.2b).
shorter
sti
(3.2a). A
and
head
of
hammer. The
of
cannot
given
Fig.
fc). The
case
structure.
Fig.
be
can
exciter
structure.
masses
frequency
delivered
of
the
of
different
construction
by it is mainly
stiffness.
excitation
of
hammer, in
transient type
or
frequencies. It
various range of
as
shaker
A hammer is
or
of
For
be
a
by
small.
HEAD
FIGURE 3.1
FfrjuJ
fit)
'e
'I
'
T.
^
i
J^
'
1ms
(a)
100
(b)
!mcact
Force Pulse
ana
-a)
Time History
(b;
Frequency Scectrum
FIGURE 3.2
u,
Hz
:ooo
10000
Ssectrum
t3l
stiffer
tip,
range
is
the
longer.
preferred,
range
to
it
magnitude
force is
determine
the
in the
crystal
used
of
direction
and
of
creates
all,
the
possibility
chance
might
3.1.3
of
to
available
is
tip
softer
limited
the
frequency
which
of
results
advantage
use.
for
It
measures
It
structure.
proportional
charge
force.
applied
gauge
the
develops
The
test depends
be
can
signal
is
charge
measured
from
this
is
problems
ACCELEROMETERS
the
in
Also,
from
signal
structure
all
in
offered
linear
by
inexpensive
structures.
57
this
the impacts
harder
with
the
structure
processing.
which
its
relatively
simple
hammer
the operator,
upon
magnitude.
the
of
overloading
be forced into
greatest
convenience
accurate
force
the same
the
structure
same
also
in
tips
metal
Generally
energy
delivered
impacts in
of
bounce
is
the
and
frequency
excited
of
tips,
plastic
which
across
magnitude
force
of
applied
force transducer is
more
all
the
and
respectively.
imparts
piezoelectric
rubber,
actual
when
are
shorter
interest.
of
simple
is
stiffnesses,
since
of
The tip
the
There
increasing
with
length
pulse
and
case
region.
method
There
is
gives
the
All
in
the
fairly
An accelerometer is
to
system
transducer
measurable form.
response
is to be measured.
crystal
certain
of
vibrates
the
The
crystal.
be
which
can
only
below
frequency
stiffness
on
the
it is
which
constructions
is
in
used
are
for
accelerometer
may
of
should
than
one
be
is
mounted
magnetic
mounting
and
well
of
whose
piezoelectric
The
Figure
the
exerts
of
the
shows
on
charge
be
accelerometer can
(k/m)1/2.
body
force
it to
converts
used
Hence
the
accelerometer
and
to
connection
(3.3)(31
When
'm'.
mass
then
mass
as
in
the
structure
wide
impose
mass
the
mass
of
of
is.
accelerometer
hand-held
and
response
the
followed
mounting.
the corresponding
stiffer
in
the
frequency
selecting
the
mass
the
the
on
should
be
Mounting
the
wax,
Fig.(3.4)[3l
response
of
structure
The
no
rule
more
the
of
connection,
connection
by
of
measurements.
structure.
The best
58
range
made
accelerometer
The
sensitivities.
contraints
frequency
of
Also,
application.
good
of
be
must
additional
for
range
mass,
particular
also
methods
structure
of
mass
frequency,
compromise
the
response
accelerometers.
avoided
tenth
accelerometer
stud-
as
with
is that the
thumb
seismic
crystal
available
Hence
the
on
consists
seismic
limited by the
mounted.
accelerometer
which
the
fundamental
sensitivity increases
reduced.
and
excitation,
crystal
Accelerometers
is
Basically, it
range
of
It is mounted
piezoelectric
its
the
convert
'k'
stiffness
the
under
to
used
is
given
cemented
shows
the
by
stud,
various
characteristics
(C)
/PiCo.i
>.a;
Construction
v.D)
(c) Construction
FIGURE 3.3
or
of
t3l
Cemented
stud
Thin layer
of
Hand
neid
wax
(t>)
Cemented
Wax
S,U\X
k Hertz
Accelerometer Attachment
of
Attacnment
f. a;
Methods
:b)
Freouencv Response
FIGURE 3.4
of
Different
Attacnments
t3]
by
obtained
be
should
the
perfectly
Accelerometers
as
well
3.1.4
are
Fast
channel
the
for
available
output
The
Function.
process, the
analog
process
1. The
excitation
and
input
sampling
rate
the
and
from
record
(weighted) by
and
analysis
of
weighted
window
end
of
all
the
of
the
record
sequence
the
other
fourier
is
and
to
the
of
to
the
transform
of
Frequency
simplified
FFT
used
channel
complicated
FRF.
the
sampled
and
The
signals
over
digitized
These
records.
determine
to
records
finite time.
The
the
frequency
be
multiplied
analysis.
sequence
may
each
mobility
follows^22':
or
lengths
continuous
as
sequences
the
of
the
filtered,
are
history
resolution
and
beginning
digital
time
fast
is
analyzer
H(co). One
signals
is
analyzer
mounted.
at
and
obtaining
response
is
rotational
signals
of
series
the
2. Each
in the
out
represent
range
Although
use
carried
(FFT)
displays
and
it
exciter
performs
accelerometer
which
freedom
of
the
signals
can
user
analog
give
to
analyzer
input
and
Response
on
measuring
Fourier Transform
connected
accelerometer.
3. The
surface
is
analyzer
for
also
the
the
of
axis
measure
the
to
normal
as
dual
the
The
accelerometers.
record
at
both the
records.
is transformed to the
61
frequency domain
as
complex
by
spectrum,
Transformation'. To
the
use
the spectral
estimate
4. An
complex
is
remove
of
noise
5. When the
products
complex
different
spectrum,
spectrum
is
output
a
and
of
and
then
of
different
conjugate
we
input
of
an
signal,
improve
and
and
one
force
and
response
together
force
and
response
are
and
with
number
is
spectrum
cross
by its
of
such
output.
The
spectrum.
phase
The
shift
quantities
needed
of
Modal
cross
between
coherent
autospectra
of
the
between the
exactly the
by
multiplied
representing the
magnitude
in the input
spectrum
spectrums.
the
get
averaging
complex
power
by multiplying
calculated
conjugate
independent
product
density
confidence.
autospectrum
the
Fourier
'Discrete
of
estimate.
3.2
CURVE FITTING
Curve
fitting
measurements
is
at
the
Response
numerical
techniques
frequency, damping
obtained,
this
in
which
an
analytical
to
are
individual
for
the
62
structure
curvefitted
shapes.
expression
expression
the
modal
extract
mode
for every
on
points
Functions
and
analytical
phase
second
various
Frequency
process
Test
taken.
are
Frequency Response
matching
FRF
expressed
is
it
is
in
as
such
fitting
curve
closely
The
sophisticated
with
parameters
Basically,
after
is
Function
found.
its
If
series
form, the
coefficients
Thus,
parameters.
deduced
from
squares
method
Generally,
domain.
equations
and
harmonic
[M] {q}
general
When
best
the
however,
of
fit
with
fit the
usually
a
as
well.
The
in
the
methods
of
directly
least
function.
frequency
to
available
modal
the
use
polynomial
response
algorithms
be
can
do
the
curvefitting
given
[coq {q}
be
MDOF
system
with
hysteretic
damping
as,
{ f}
eiut
(1)
expressed
in the form,
ei03t
(2)
back
substituted
Equation
into
The
problem.
eigenvalue
for
are
can
solution
motion
[K] {q}
damping
and
to the
related
Ewins. M
by
excitation
be
can
algorithms
algorithms
are,
explained
q(t)
find
The
The
shapes
curvefitting
the
of
terms
various
mode
The
There
best
the
to
most
curvefitting
are
it.
of
rtn
(1),
this
eigenvalue can
be
leads
to
written
as
complex
\2
<o2
cor is the
where
the
rtn
mode.
properties
(3)
irir)
natural
The
frequency
and
eigenvectors,
[u],
by,
given
[u]T
[M] [u]
[u]T[K+
orthogonality
icoC] [u]
case
(4)
[mr]
of
(5)
[Kr]
harmonic
excitation
63
and
response,
the governing
equations
of
[K
motion
+ icoC-
The direct
(q)
[K
eicot
eicDt
{ f}
(6)
be expressed
can
icoC
is
[h(co)]
its
constituting
matrix
M] {q}
as,
o^M]"1
as
{f}
[h(co)] {f}
where
co2
solution
are given
h;|<(co)
(7)
"N
the
response
can
be
NH
receptance
model.
given
for
matrix
general
element
the
in
system
this
FRF
as
^
MO)
(8)
Vk7
fm
m=1,N ;
=0;
*k
-1'
[K+icoC
Let
[<E>
-co2M]
[h(co)]
the
represent
(mass-normalization
Premultiplying
both
(9)
is
sides
by
mass-normalized
eigenvector
for
eigenvectors).
[<D]T
and
[0]T
[K
or
from Eq.
icoC
co2
-
M] [C>]
(3) following
process
scaling
postmultiplying
by [&],
(10)
[h(co)f [O]
relationship
can
be obtained,
-1
[X;
co2]
[d)] [h()]
(11)
[<D]
[X'
that is
Eq.
(12)
[h(co)J
is
N. Individual
an
(12)
[O]
expression
receptances
for the
can
be
receptance
represented
64
matrix
as
of
the
order
N X
hikN
f
r
In Eq. (13),
point and
(Oj)r
(a>k)r
-co2+
gives the
2-~^
is
(Ajj<)r
form
series
proximity
and
Multiple Degree
Nyquist
the
or
of
the
adjacent
Nyquist
of
Freedom
the
part
does
not
explicitly
must
be
added
particular
to the
of
used
resonance
is
FRF's.
in
used
There
are
upon
the
depending
modes.
is
divided
FRF
the
of
for
used
plane
into
is
give
the
on
are
the
curvefitted
well
to
according
or
whether
is
plot
of
of
frequencies,
the values
In
separated
65
of
vs.
the
Since this
plot
the
Response Function.
values
curve.
methods
curvefitting.
plot
Frequency
by identifying
points
the
It
(MDOF) curvefiting
further
are
Argand
or
Residue. This is
Function.
of
curvefitting
or
plot
imaginary
to
of
magnitude
circle
Response
3.2.1
The
from
techniques
whether
Modal Constant
as the
Frequency
coupling
2.
(14)
curvefitting techniques
of
Single Degree
two
as
ITlrC02
the
of
1.
The
near
written
the resonance
near
-2+
referred to
types
main
be
also
2-
of
deducing information
two
shape amplitude
(Aik)r
rTl
where,
can
V
=
mode
the amplitude
gives
hik^
J
irirco2
co2
real
this
information
frequency corresponding
those
points
close
resonance
are
clustered
displaying
the
important
(3.5)l3]
shows
together.
resonance
For light
traces out
This
of
damping,
line
vertical
h(co)
is
is
in
region
of
an
some
some
plot
exhibits
the
resonance
the
as
effecive
the receptance
resonance known
through
as
some
frequency
detail.
or
Frequency
symmetry
sweeps
about
The
(k
m)
(15)
(coC)
i (coC)
(16)
(k
The
m)
right
hand
side
imaginary
parts
as
Re(h)
co2
-
of
(coC)
Eq.(16)
can
co2m
be
separated
Im(h)
into
its
real
coC
=
(coC)'
(k
and
it follows
(k-
-co2m)
x2
lm
is the
co2m)
and
(17)
that
(Re)'
which
given as,
m)+
(k
FRF
from 0 to
'
Fig.
the properties
of
frequency.
co
of
way
follows.
proved as
mode r
it
displays
also
Nyquist
the
complete circle
be
can
a single
passing
Thus
of
(18)
2coC
2coC
equation
<
circle.
66
m(h)
Im(h)
FIGURE 3.5
From
equation
(15)
of a single mode,
(19)
co
<(1-
(f-)
+mr)
can
be
written:
(20)
<>2>.
(90
tan
or,
Now,
the
-y)
consider
tan
tan
tan
(9/2)
two points
frequency
natural
From this
(6/2)
(92/2)
(1
cor.
(1
-(co/co,)2)
(21)
/r\T
on
-(co/,)
below
can
be
and
said
co2
above
that
)/ t|r
(22)
(1
expression,
-(co2/cor)
the
)/ t\f
damping factor
of
the mode
can
be
written
as
ft"?-*?)
For the
case
{co2
of
tan(62/
2)
(23)
tan(e/2))}
above
67
expression
becomes
2 (co
^r
When
co)
]
jov
(tan(e2/ 2)
points
and
TV
tan(61 /2))|
is
light,
not
(26)
points
one
at
technique
near
natural
to
response
curvefitting.
above
coupled.
curvefitting
3.2.2.1
This
This
is
from
is
are
used.
is
other
the
In
single
is easily
be found if the
mode can
the
or
here, is
is curvefitted,
resonance
that
the
response
is completely dominated
closest.
In
other
modes
adjacent
simplest
valid
mode
type
only if the
event
of
of
is
by
neglected
curvefitting
very
are
the
of
that
mode
the contribution
words,
modes
having
far
close
SDOF
in
method.
The
or
they
apart
modes,
MDOF
method
any
assumption
resonance
assumptions
not
which
frequency
whose
the
by
of
known.
are
The
time.
structure
are
damping
SDOF CURVEFITTING
further
(24)
eq.
points,
power
It is
power
(25)
3.2.2
half
the
are
damping
nr
half
(24)
to,
simplifies
If the
"
applied
to
Frequency Response
68
Functions
of
the
structure
It
the
uses
the modes
where
simplest
1. Individual resonances
first
The
noted.
in
amplitude
particular
or
modes
mode
Fig.
separated
to
and
in
FRF
plot of the
to
corresponding
gives
is
and
curvefitting
very damped.
not
the
fastest
below:
outlined
frequency
that
mode.
well
approach
The procedure is
method.
are
the
natural
(3.5) graphically
magnitude
the
frequency,
shows
the
maximum
cor,
peak
are
that
of
amplitude
curvefitting.
is
the
Then
points
on
are
and
co-j
damping
the
either
amplitude
side
ratio
CO
points
to
establish
curve
noted.
power
of
the
mode.
of
gives
amplitude,
the two
half
mode.
also gives
3. As
From Eq.(26),
<27>
Eq.(14)
in
(Aik)r
hik(co)=X
mode
(28)
2 Tir
X.
For the
and
-co
Cr
in
shown
|h|,
power
damping
Tlr=-V^
and
in
This bandwidth
frequency band
maximum
"h/V2"
peak.
question,
given
(29)
neglecting
the
contribution
from
other
as,
A-
(3)
<rir
In
other
words, the
Ar
modal
constant
can
be
written
as
(31)
|h|co2Tir
69
Thus the
modal
mode
shapes
of
(A^),..
constant
FIGURE 3.6
Although this
limitations
damping
that
mode.
single
correctly
Firstly,
depends entirely
inaccuracy
a
it.
Measurements
point
which
measured.
the
on
of
to
simple
implement, there
of
accuracy
one
point,
the
modal
affecting the
mode
when
with
the
Secondly,
modes,
under
modes
other
are
serious
constants
maximum
amplitude
or
of
determines the
other
of
combined
of
contribution
useful
is very
method
to
maximum
this
method
even
consideration.
are
methods
well
70
amplitude
entirely
though
they
Hence this
separated.
effectively to
used.
give
may
are
be
the
somewhat
method
good
not
neglects
However,
a
Hence
it
is really
can
curvefit.
be
3.2.2.2
This
is
peak
amplitude
slightly
vs.
imaginary
plot
for
made
that
the
the
for
modal
effects
Frequency
small
than
method
curvefitting
uses
traces
mode
extract
frequency
It
part of
single
to
sophisticated
method.
real
circle
more
which
is
the
plot
of
circle.
Again,
parameters.
due to
adjacent
frequency
range.
is found
curvefit
modes
This is
assumption
is
independent
of
an
are
for this
shown
by expressing
W-l^Z
<4
s=i
J.
w2
-
Vs
^
co2
small
(32)
:^2
hrco2
-i^U
co2
frequency
rOange
in the vicinity
computed
damping
plot
m.sco2
Damping
of
of
mode,
mode
r, the
can
be
follows.
as
of a mode
is
given
as
CO2
(34)
^-V1
where
The
9-j
sides
of
circle
90
number
manually
2. A
circle
entire
1.A fixed
92
natural
by
the
fitting is
of
represent
points
carried
are
frequency.
out
as
selected
The
follows:
on
selection
be automatically
or
operator.
is fitted to the
points
71
squares
method.
The
selected
points
angles
between
points
(from
are
each
co=0
of
to co=o)
the
increments along
the
natural
given
to the centre
connected
frequency
forming
successive
that
of
Then,
circle
points.
mode.
of
This
the
circle
by calculating
maximum
be
can
the
is found
The
and
seen
rate
gives
in the figure
below. I41
Im
FIGURE 3.7
3.
is
Damping
on
of
modal
frequency
given
by
The
set of points.
4. The
using Eq.
estimated
two sides
resonance.
mean
constants
and
co
of
the
set
Thus, damping is
gives
the
determined
are
diameter
substituting
value
circle.
damping
knowing
of
points
calculated
and
for
each
ratio.
the
damping,
The diameter
of
natural
the circle is
A.
jk
Di^k
or
Thus,
the
(35)
A.^Dia^co2^
72
be determined.
3.2.3
MDOF CURVEFITTING
where
is
structure
is
system
affected
by
combined
methods
for the
curvefitted
be
can
of
range
modes
m-jto
A..
the
range
represents
whereas
Equation
the
the
frequency
m-j to
above
term,
The
can
the
of
number
close
Thus,
are
the
various
domain.
be
written
are
in
series
form
they
(1/Kjk)R,
of
represents
superscript
73
m-j
as
and
modes
represent
behavior
be further divided
the range
term,
(1/co2Mjk)R,
The
equation
outside
since
stiffness-like
modes.
(36)
m2.
Hence,
co2M]k
modes
terms,
residual
the
k"
msco2
of
effect
of
response
There
the
-co2+
as
can
when
functions.
estimated.
some
co2
st^
account
m2
wherein,
frequency
or
resonance.
polynomial
expression
lk
the
used,
using
resonances
of
are
analytical
called
algorithms
of
modes,
for
suitable
very
of
coupling
3.2.3.1
The
effect
close
more
are
modes
damped.
highly
MDOF curvefitting
adjacent
is
there
not
in
high
mass
stands
the
They
m2.
on
each
Equation
frequency
effects
for
the
of
are
side
(36)
modes,
the
low
residuals.
m.
(Aik)s
<*,,.>.
w>
co2
ir)rco2
co2
2+
J
The SDOF
is
side
for
constant
if
analysis,
good
These
the
magnitude
for
frequency
magnitude
results
for
all
data
of
the
of
in
analysis.
(in
h:k(co)
gives
Eq.
the
the
be
be
can
the
the
constant.
analysis
at
as
cor,
each
the
available,
calculated
from
values
from
nri2)
the
at
value
in
calculated
to
m-|
MDOF
resonance,
is
value
range
FRF
the
(hjk(co)). Then
by
Subtracting
the
to
SDOF
around
(37)
in
coefficients
trial
FRF
on
assumed
points
measured
modes
adjacent
from
FRF denoted
the
SDOF
the
value
measured
of
which
of
the
set
be
not
found
of
the
be
could
Consider
it may
Jk
But
range.
can
then
)k
bracketed term
frequency
small
estimates
estimates
follows
(Aj|^)r,
term,
second
with
(37)
"ls4
mode
the
under
consideration.
rn
(A;J
k's
h>)
)kv
'
s=m
Thus,
found
by
co2
co2
good
,
(s*r
,
estimate
+ m
.cor
's
for the
parameters
can
be
for any
repeated
mode
to
co2
co2^
|k
coefficient
procedure
<Aik)r
1
2
obtain
an
(Aj^)r
of
SDOF
better
irirco2
the mode
analysis.
estimates
74
co2
-
of
(38)
r can
The
the
be
same
modal
3.2.3.2
in
measurements
works
minimizing the
on
that determined
and
the
independent
an
case
error
of
coupled
modes.
between the
theoretically.
The
to curvefitting
approach
t5l
measured
individual
The
method
FRF
of
value
measured
whereas
hi=s=m
In
the
g)2,...
analytical
"Hi
measured
given
at
theoretical
and
jk
the
are
(A,k)i
coefficients
These
unknowns.
are
(A;k)2,.. co^
the
modal
values
for
frequency
which
is
considered.
is
as
=
<c
E,=
make
frequency
(40)
-M
(41)
i ef I
the
natural
frequency
measurement
is
interest
of
more
analysis
is
consideration
under
is
is then
general,
considered.
given
higher
given
lower
given
the
The
weighted
closer
modes
weight
weight.
and
error
to
should
unknowns
be
for every
the
mode
The
total
error
in
the
by,
(42)
IW1E1
To find the
error
Jk
M;^
and
M?
co2
each
61
To
K"
iiyo|
expression,
Ki|<
t|2>...,
parameters.
error
<39>
TL?Jr--TT
o/ +
co^
is
value
in
Eq.(39),
minimum.
This
the condition
is
75
achieved
is that the
weighted
by differentiating
the
total
error
them
equating
equations
3.2.4
testing
of
structure.
location
these
manually is
modal
peaks
the
the
table.
existing
long
measurements
modal
final
natural
table.
then
This
These
animated
These
data
is
they
for
single
used
mode
the
all
parameters.
freedom
at a
displacements
shapes
for
the
of
all
the
entire
modes
In this method,
measurement
at
all
the
using SDOF
or
entered
repetitive
values
output
in
of
modal
into
process
and
AUTOFIT
an
on
computes
all
the
all
the
damping
are
modal
resonant
frequency bands
are
test,
modal
the
SMS
the
with
process!
painful
performs
these
using
frequencies and
measurements.
hence
linear,
measurement
These bands
modes.
parameters.
The
method.
extract
The
every
and
curvefitted
computer
be
simultaneous
unknowns.
to a different degree
the
give
of
the
available
to
structure.
and
each
very
for
iterative
part
curvefitted
locations
are
final
the
corresponds
on
Fitting
of
set
and
unknown
be
must
each
not
may
some
using
As
measurement
each
solved
generated
solved
software.
different
generates
be
to
respect
user-friendly feature
measurements
with
This
then
can
AUTOFITTING
Autofitting is
(42)
zero.
equations
have to be
might
Each
to
which
simultaneous
Eq.
by
given
into
further
mode shapes.
76
are
averaged
the
frequency
processing
or
out
for
and
for
all
the
damping
display
of
3.3
This
describes the
section
SMS STAR
test using
test
There
software.
2.
Setting
up the
experiment and
3.
Setting
up the
computer
Preparation
In
an
or
support
first
degrees
The
they
be
of
the
structure
complete
in
stages
modal
modal
actual
structure
But in
of
is
way
of
all
by
absorbed
some
it from flexible
in
the
of
freedom in
all
the energy
the structure
to
or
springs
its
and
point on
a
is
place
condition.
supported
structure
which
suspended
not
cases
is tested
to
identify
it
it
on
The
the
structure
can
be tested.
points
connected
easiest
of
or
preparation
degrees
six
measurements
free-free
analysis,
gives
structure
taking
analyzer
modal
suspend
the
and
spectrum
number
are
to
the
in
coordinates
the
the
ground.
possible
step
could
of
excite
foam, hence
the
three
are
the structure
the structure
of
to
applied
not
of
experimental
condition
is
perform
Preparation
its
to
procedure
1.
1.
by
lines.
in
which
the
points
The
points
connection
are
are
lines
determined
computer.
77
numbered
so
can
as
be
to
according to
written.
enter
The
them into
2. Setting
Using
up
the experiment
hammer to
the structure,
excite
hammer is connected to
channel
conditioning
The
amplifier.
response
is
amplifier.
Appropriate
and
accelerometer
the
amplifiers.
instrumentation
Next,
and
response
are
as
of
of
in
its
gives
point
Setting
is
good
The
to
up
pick
the
transducer
are
set to
on
set
the
schematically
for
conditioning
charts
setup
modal
test.
the excitation
measure
is
analyzer
position
upon
up the
modal
test.
Fig.
modal
to
a project,
as
(3.1
undertaken.
number
set
up
the
the
of
0)I211
Various
driving
point
point
point
is the
are
hammer
tested
and
is
one
antiresonances
locations
and
driving
driving
and
response
in
by
finally
in
point.
taking
most
of
driving
resonances
accelerometer
and
is the
Choice
that the
such
whereas
domain. Then
test. A good
Function.
computer
measurements
perform
of
frequency
modal
Response
Taking
open
experimental
in the
seen
maximum
is decided
calibration
is
using
hammer force
shows
analyzer
(3.9).
structure
Frequency
changing the
3.
FRF
Fig.
the
analyzer
using
signal
the
(3.8)
again
the
of
used
the
of
of
the structure is
survey
B,
channel
Figure
and
spectrum
accelerometer
according to their
used
shown
the
of
sensitivities
excitation
to
connected
measurements
important
shows
and
graphically
lengthy
the
stage
procedure
to
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points
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are
are
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by
display
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the
freedom
of
response
(3.11)
shows
domain.
and
'Modal
Peaks'
checks
all
averaged
identify
out
driving
point
functions
and
measurements
frequency
also
basic
The
and
allows
response
in
the
all
same
structure
are
mathematical
mode
can
be
All these
model
the
of
shapes
printed
by
steps
the
can
structure
82
for
then
saved,
the
of
it
FRF's
easy
range
peaks
the
all
command
part
to
tested.
in
the
polynomial
rest
the
of
peaks
the
in
parameters
are
displayed
and
be
results table.
be
time
the
using
modal
structural
and
This
The
After
can
from
are
frequency
repeated
the
Fig.
to
makes
curvefitted.
all
point.
transferred
This
the
all
For this
frequency
imaginary
in the
of
executed.
function.
curvefitted,
processing
simulation.
is
process
finished
driving
of
curvefitted
are
damping
further
then
Autofit
animated
is
plots
the
using
with
the
measurements.
are
the coordinates
measurements
STAR
and
measurement
the
in
are
response
performed.
subsequently
the
the
measurements
measurement
extracted.
is
all
all
acquired
every
After
measurements
for
are
location
measurement
and
Having
sequence.
the fixed
measurement
command
the
at
freedom,
of
coordinate
typical
saved.
excitation
measurements
recorded
Every
computer
The
is
are
of
the structure,
of
entered
degree
point
SMS STAR
modification
carried
out
is determined.
or
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3
CHAPTER 4
MODELING AND RESULTS
4.1
The
objective
arm
robot
in the
find
and
experimentally
software
Fig. (4.1)
techniques.
is
different
to
an
each
infinite
arm
of
fixed
the
that
is horizontalO
locked,
every
assembly
in
Also, it is
are
to
and
of
and
STAR
System's
means
that
there
are
five
oriented.
This
could
give
rise
be
of
is
no
this
of
is
that all
relative
84
structure.
changes.
fundamental
assumed
configuration.
the forearm
at
in
In this
vertical
as
the moving
motion
Since for
Theoretically
frequencies
combination.
that there is
Puma
The
R.I.T.
which
number
assumed
both
using
robot
distribution
robots
involved
modeling
PUMA
could
mass
line
Element Analysis
Finite
configurations
particular
explicitly
so
it
infinite
be
corresponding
position
of
number
could
the
robot,
which
configuration,
there
are
axis
about
axes
mode
corresponding
Measurement
details
shows
five
and
analyzer.
the
explains
chapter
Unimation
B&K FFT
with
Structural
using
model
frequencies
natural
shapes.
This
its
is to
problem
current
typical
ail
the
axes
position,
the
main
which
shown
parts
in
of
Fig. (4.2).
the
between them.
robot
WAIST
(JOINT
1)
SHOULDER
JOINT 2)
FIGURE 4.1
cc
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4.2
drawing
geometric
fig. (4.2)
Intergraph
The
programs
various
Fig. (4.5)
the
and
profile
and
mesh
material
boundary
options
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stiffness.
from
expected
about the
bending
small
mode
oscillations
theory.
classical
shaft
connecting the
the shaft
the
of
base
Hence
not
be
can
this
oscillations
for
mode, the
and
is
horizontal
arm
vertical
pure
of
and
bending
are
arm.
torsion
of
Mode
the
of
the
main
ninth
is
this
For
the
of
sixth
local
is
main
Mode
arm
number
only.
is
again
some
108
but the
and
all
bending
pure
slight
of
the
of
the
and
mode of
in the
bending
other components
bending
of
main
is
other components
the
after
vertical
bending
combined
is
fourth
the
shaft
eight shows
it
modes
of
mode shows
bending
shafts.
Also,
is surprising in that it
Again
second
excite.
bending
second
All the
arm.
in
bending
in the
oscillation,
mode.
of
shafts.
cylinder
example,
there is local
The
arm.
under
at
the forearm
torsion
mode a
mode
structure
mode
bending
two different
displaying different
starts
and
the
as
vertical
comes
be
oscillation
requires
number
stationary.
The tenth
second
main
entire
mode
The
the
In the fifth
torsion of the
stationary.
bending
components.
cylinders,
cylinder.
structure
and
the
of
structure
cylinder
vertical
mode
of
can
displays torsional
mode
a combined
excite
various
arm
shows
as
to the forearm. It is a
arm
well
entire
to
main
mode
structure
stationary.
bending
main
the
possible
the
main
mode,
mode
as
slight
This
seen.
mode,
arm
mode,
although
easily
second
of
of
The
bending
are
the main
It
be
must
arise
noted
As
basically
there
which
trully
an
consists
theoretical
of
is
additional
of
is
mode as can
forearm.
They
are
stiffness
second
be
of
two
adjacent
each
This
is
all
is
This is
the
modes
of
limitation
major
mode
in
This
is
components.
bending
the
main
arm
number
the
and
both
of
modes
not
since
model,
complex
very
in
other
seen
third
and
between
mathematical
mode
bending
structure
that point.
at
analysis.
torsion
the
touch
surfaces
eleventh
and
motion
theoretical
linear
bending
a complex
in
The
models.
two
in
nonlinearities
conflicting
the
as
that
point,
a
into consideration
taken
it
soon
there
oscillation,
is
there
whenever
surfaces.
this
at
the
of
components.
The
above
discussion
It
requires
robot
be
must
modes
require
less energy to
careful
4.6
attention
characteristics
it
that although
stressed
quite
in
generally
practice.
They
excite.
any
the higher
all
excite
damped
modes
Hence
vibration
are
structure
modes.
Usually
most
should
the
and
robot.
b)
It
seem
show
can
the
be
Also, the
first three
designed
with
modes.
Figures (4.16a
might
of
higher
structure.
the
summarizes
actual
seen
experimental
109
top
fcr
co
-i-
ui
Ql
X
UJ
UJ
CO
Ui.
CC-:
cr
E:
0.
r-
UJ
CO
_l
<
z
Ul
cc
Ul
D.
X
UJ
UJ
X
H
JO
CO
T.
UJ
cc
cover
the
of
driving
point
in
point
the
main
arm
was
point
survey
was
mounted with
was
modal
structure
This
system
were
second
0.5
all
the
should
Hence
modes
global
screen.
local
were
a
redefined
hence
measurements,
(C)
it
frequency
modal
gives
geometric
driving
were
averages
is
system.
of
in the
structure
new
the
on
points
coordinate
systems
systems.
The
It
was
0-800
Hz.
range
The
determine
0-400
of
analyzer
to
measured
the
Hz
correct
frequency
up to
values
for
frequencies
measured.
the
shows
process
range
frequency
to
coordinate
in the
spaced.
very closely
be correctly
the
all
cylindrical
frequency
smaller
The curvefitting
Appendix
maximum
accelerometer
structure
rectangular
experimental
for the
are
much
the
first test,
the
new
done for
for
on
was
out
In
the
test
Table (4.2)
the
in
the points
Hz.
the
performed
and
measure
on
defined
proved
were
up
set
that gives
point
driving
accurately.
directions.
correct
be the
to
chosen
initial
an
a stud.
model
were
After
test.
on
was
radial
selected
63
number
for the
removed
the
model
performed
analysis
coordinates
of
the
damping
and
table and
robot
arm
112
for
results
the
on
display
modal
obtained
from
robot
arm.
sequence
table
testing.
Results
Mode
TABLE 4.2
(Freq&Damp)
Freq.
Table
ROBOT. PRJ
(Hz)
Damp
(%)
13.266
4.726
21.136
3.528
35.957
3.942
39.708
2.224
52.201
1.702
66.347
2.134
107.390
1.738
117.405
1.293
151.286
3.742
10
199.078
11
205.468
1.985
12
213.083
1.055
13
224.702
661.696m
620.581m
Curvefitting
function.
The
curvefitted
coupled
measurements
modes
the
using
modes
were
The
test.
(E)
animated
characteristics
are
seen
is very
structure
from
ail
of each
mode
shapes
are
few
This
points
the
be
will
vibrations
resonance.
above
the
The
200 Hz,
wherein
the
the
Some
in the
and
experimental
the
There
possibly
lack
for
reasons
of
points.
neighboring
structure
The
used.
oscillate
oscillations.
the
with
response
components
their
modes
considering the
harmonic
get
and
analysis
of
are
section.
modes
modes
The higher
modes.
damped
zoom
in
shapes
mode
structure
To find the
phase
methods
next
damping
of
out
bending
realistic
behavior in
from the
torsional
some
nonlinearities
erratic
and
percent
spaced.
more
Closely
freedom
of
obtained
Pure
the modes.
all
giving
erratic
moving
to
discussed in the
good
an
curvefitting
correlation
is
show
shapes
were
separate
degree
shapes
it is very hard
and
other
show
due
is
improper
The
mode
polynomial
technique.
multi
independent
mode
in nearly
complex
freedom
using the
the
the
using
distinctly
of
This is probably
all.
at
degree
gives
done
was
were
curvefitted
bending
not
which
single
Appendix
technique.
modal
the
of
it
since
out,
above
correct
analysis
measurements
damping
means
thus
of
be
114
the
reducing
200 Hz
natural
in the FFT
can
of
damping
for
the
are
effect
of
of
very closely
of
should
a
modes
amplitudes
frequencies
analyzer
taken
the
and
that
partly
frequency
values
is higher
much
modes
be used,
smaller
frequency
4.8
range.
COMPARISON
RESULTS
Table
modal
first thirteen
of mode
than
the
that,
structural
which
frequencies.
animated
show erratic
very ideal,
means
it
Some
of
in
the
is
10%,
trend
no
have
that the
said
good
and
they
good
NASTRAN
mode
two
However,
not show a
shapes
9 (34%). Other
is
which
explained
modes
consistently
concerned.
does
to
other
large deviation in
number
really be
there
are
from
obtained
with
both
is
or
correlate
visual
those
obtained
the
higher
obtained
well
as
inspection
correlation
and
of
or
models
any
positive
none
lower
quite
between
obtained
experimentally
since
the
be
are
shapes
they do
a number of reasons
between
to
can
1%
finite
and
the frequencies
of
most
and mode
cannot
that
experimentally
mode
modes
(18%)
frequencies
natural
There is
models.
variation
frequencies
natural
experimentally.
be
ANALYTICAL
experimental
be seen, that
variation
The
from
between
tendency
Hence,
and
analytically
as
shows
is
variation
reasoning.
negative,
models
AND
the
of
modes
can
number
The large
reasonable.
the
It
analysis.
frequencies
of
comparison
match
far
the
shows
for the
obtained
and
EXPERIMENTAL
(4.3)
element
OF
two
not
for
models.
include any
not
getting
Some
of
follows.
115
damping
a good
the
values.
There
could
reasons
can
be
cited
as
Mode#
NASTRAN RESULTS
PERCENTAGE
FREQUENCY (Hz)
FREOUENCYfhW
DIFFERENCE
1.
12.546
13.266
5.427 %
2.
20.094
21.136
4.929 %
3.
23.654
35.957
34.226 %
4.
37.989
39.708
4.329 %
5.
58.622
52.201
-12.30%
6.
71.853
66.347
-8.298
7.
101.155
107.390
5.805 %
8.
114.958
117.405
2.084 %
9.
1 24.208
151.286
17.898%
10.
185.405
1 99.078
6.868 %
11.
203.095
205.468
-1.154%
12.
21 1
213.083
-0.062
13.
215.696
224.702
4.007 %
.762
TABLE 4.3
: COMPARISON
1.
The
it
and
is
there
where
components.
at
resonance.
other
to
or
Structural
is
suitable
making
differential
the
nonlinearities
the
be
analysis,
very
well
theoretical
3.
in
used
the
simplifications
the
stiffness
in the
account
like
for
matching
results.
could
The
by
for
the
treat
these
to
performed
nonlinear
into
take
gap
elements
available
other
good
each
mode
is
analysis
In
of
to
incorrect
algorithm
handled
due
with
phase
damping.
not
may be
points
theory.
reason
not
the
of
doubled
of
out
eliminate
does
the
and
lack
experimental
modal
to
are
the
major
adjacent
performance
of
of
accurately known,
really
moving
curvefitting
analysis.
Incorrect curvefitting
having
It
equations.
be
predicted
theoretical
nonlinearities
and
are
effective
elements
nonlinear
cannot
no
not
some
may
points
distortions
Also,
nonlinearities.
others
lead to
nonlinearities
There
shapes.
of
of
Also, in the
between
not
the
known
not
in the form
damping
are
predict
is
methods.
motion
damping
of
Amplitudes
that
could
structural
relative
theoreticaly
and
This
values
resonance.
damped
highly
is
Since the
difficult
structure
2.
structure
friction
it
cannot
actual
damping
structural
account
words,
in
well.
any
of
the
of
This
modal
could
between
correlation
in
NASTRAN
aspects
very
terms
by
the
models.
be
stated
robot
as
one
structure
117
more
has
reason
some
weak
for not
points
like
the
two
small
resonances
or
component
element
in the
Poor
mode
measurements
shapes
exciting the
in
the
always
the
after
structure
same
of
the
minimize
the
error
point
is
But
in
results.
for
animated
mode
displacement
The
prime
models
grids
condition
problems.
them.
be
for
this
in
condition
of
to
erratic
conditions
while
the
test,
be
no
10,
was
error
it
at
the grid
type
such
is
other
problems,
methods
118
in
difficult to
correlation
of
considerably.
drastically
point
was
comparison
visual
points
should
which
could
of
very
is
the
of
comparison
the two
rather
not
that count.
on
correlation
of
driving
error
This is
measurement
could
obtained
Hence in large
show
location.
correctly chosen,
this
There
created
This
Insufficient
attributed
to
prime
same
averaging
not
given
shapes.
values
should
on
are
reasons
above
which
is
structure
the
of
exact
and
The
the
One
the
at
used
operator
and
hits
algorithm
shapes.
number
finite
with
direction
with
applied.
techniques
cause
curvefitting.
particular
obtained
be
local
to
rise
to
appropriate
should
mode
could
shapes
an
these
give
points
confined
mode
account
on
questionable correlation
4.
the
behavior,
into
available
week
vibrations
in
For this
modes
information
of
seen
analysis.
local
takes
modes
is
as
These
shafts.
that
are
satisfy
of
models.
the
two
matching
in
larger
only frequencies is
done. One
the methods
of
properties
the
of
It
experimentally.
finite
element
FRF
the
with
the two
good
can
it
where
from
analysis.
more
effective
One
noted
of
more
the two
For
on
X-axis.
If
passing
through
correlate
they
Probably
mode
the
on
the
measure
correlation.
for
should
trends
it
the
is
be
good
not
for
This is
observed.
for
possible
in
a
simpler
FRF
an
complicated
plot
problems
degrees
If both the
axes.
all
it
same
of
freedom
eigenvectors
in
are
and
be
would
at
can
method
by plotting
is graphically
problems
simple
correlation,
frequencies,
directly
inconsistency
synthesize
corresponding
The
the
used.
normalized,
origin.
and
from
plot
compared
any
be
can
structurally
used
of
be
can
analytical
45
to the
straight
be
Vs.
line
to
applied
measured
most
shapes
and
FRF
test. Then
however,
the X and Y
are
natural
frequencies
larger
displacements
straight
particular
generally
on
It
model.
modal
theoretically
an
synthesize
the response
compare
obtained
be easily feasible to
methods
normalized, then
lie
or
will
models
the
of
comparison,
method
the
plotting
to
in the
obtained
be
models
possible
analysis
method
models
two
is
is to
of correlation
widely
used
method
for
numerical
least
deviation
squares
MAC is mathematically
expressed
119
comparison
(MAC)'J61
from
by
It
of
provides
straight
the formula
line
mac(P,x)
where
from
|2
(ox).(d,p):
'p'
subscript
from
the
represents
matrix
in
when
calculated
for
Ox,
model
resulting
modes
will
that for
be unity
whereas
Practically, it is
but
close
In
correlation
the
the
two
points.
of
Criterion'
about
were
The
comparison.
symmetric
displacements
MAC
be
will
to
and
shapes.
an
attempt
writing
initially
close
ail
zero.
different
correlation
identical
made
FORTRAN
to
be
Thus it
can
be
terms
mode
compute
program.
enough
After
doing
the
there
requires
entire
finite
120
were
should
shapes,
the
'Modal
However,
few
zero.
reasonably
information
very
two
will
respectively indicate
created,
analysis
the
points
the diagonal
exactly
was
and
element
when
measured
the finite
matching
to
When
get
zero
of
MAC
square
models.
However,
=1.
the
all
Thus
MAC
mode
by
for
models.
in the two
modes
off-diagonal
unity
were
models
available
models
to
of
value
displacement
calculated
form
the
identical,
possible
problem
robot
Assurance
the
all
is
for the
experimental
ideal correlation,
not
close
values
of
stands
displacement
the
or
This
the
all
value
the
the value
and
seen
easily
the
different,
are
different
the
equal
in
in
quantity
(0p). (0p);
'x'
and
and
) (
predicted
mode.
analytical
scalar
(0x).
model
measured
points
matching
for
stands
finite element
the
value
when
was
not
Hence the
matching
almost
identical
models
element
analysis
and
for
modal
testing, it
the
testing
some
very difficult to
was
Therefore,
again.
correlation
FORTRAN
back to
go
although
it
analysis,
did
an
was
MAC
matrix
do
to
made
great
yield
of
attempt
not
programs
recreate
The
results.
included in
are
4.8
In
design
the
the
structure
analytical
and
assumptions
the
design
of
are
and
also
results.
Next,
also
detailed
model
to
more
its
mathematical
preliminary
location
finite
with
account
the
of
driving
the
determine
to
driving
model
element
detailed
and
modal
and
model
is
measured
helps
121
in
modal
for
modal
and
element
flexibilities.
other.
of
combined
in
The
results.
can
be
model
is
model
for
locations
test
will
modal
test
yield
is
test
is
then
correlated.
run,
which
model
can
This
understanding
are
lot
created
The finite
for the
point
each
element
appropriate
survey.
point
be
dynamical
that the
so
of
possible
finite
both
using
they involve
could
best
the
transducers
compared
adjusted
and
if they
But
yield
could
analyzed
The
by
verified
performed.
it
response
and
since
that
disciplines
two
independently
much
simplifications.
important
very
properties
These
methods.
very
is
for dynamic
completely accurate,
follows:
constructed
it
structures,
verified
work
structure
as
good
not
stage,
constructed
exciters
and
and
analysis
of
experimental
and
them
of
be
should
entirely different
Both
analysis
makes
the
is
be
the
system
completely.
is
satisfactory,
further
model
like
processing
response
simulation
response
can
be
and
rerun,
When
the
final
performed
changes
This
is
requires
modal
very
correct
final
finite
test
verify
that
both
not
in
the
analysis
of
element
always
analysis
to
possible
is
forced
dynamic
easier
to
changes.
should
the
be
design
modeled.
in
is
or
the
model
modal
to
if
see
If it
for any
used
any hardware
require
configuration
procedure
It
to
element
completed,
lengthy
expertise
of
is
be
can
studied.
modifications
changes
not
is
model
it
and
The finite
it does
design
accurately
analysis.
because
the
were
any design
or
the
of
dynamic
structural
since
the
of
is complete
improved.
modify
response
and
yields
very
122
good
and
follow
ideally
results.
structures.
It
experimental
all
the
steps
speaking, this is
References
Richard L. Burden
(Boston
Prindle,
Hertfordshire, England
Kjaer), pp 2-5.
3
Ibid., pp 95-106.
Ibid., pp 157-168
Ibid., pp
Ibid., pp 222-226
Gockel, M. A.,
Theory
and
Practice.
174-177
ed.
Corporation), pp 4.2-4
8
Ibid., pp 4.1-1
Ibid.,
p.
(Letchworth,
MacNeal Schwendler
4.2-9
-4.1-10
5.1-2
10 MacNeal, R. H.,
ed.
Schwendler
MacNeal
Corporation)
12 Ibid, pp 145-148
13 Ibid, pp 157-164
123
York
15
p.
16 Ibid.
p.
17 Ibid, pp 13-20
18 Ibid.
19
p.
193,236
and
Bruel &
of
(Bruel &
with
Applications.
Inc.)
Structures. (Minneapolis
Kit-Mas
Kjaer)
22 Structural Testing
Vibration
Prentice-Hall
Corporation
of
Structural Measurement
Systems)
Kjaer)
124
APPENDIX A
MSC/NASTRAN CARDS USED IN THE MODAL ANALYSIS
The
bar
straight
is
element
prismatic
line
element
i.e. the
element,
element are
and
area
connection
PBAR
two
connecting
cross
of
and
card
not
for this
card
property
is
does
ssection
in the
It
nodes.
image.
card
CBAR CARD
EE>
CBAR
pro
GA
GB
X3
X2
GO
XI,
123
PA
PB
VIA
V2A
V1B
V3A
V2B
V3B
+23
In the CBAR
identification
by
connected
One
of
the BAR
degrees
with
of
BAR
rigid
the
of
freedom
element,
element
X-| X2
and
PB
are
be fixed
arms.
and
the
X3
axis
the
at
can
pin
and
are
the
are
property
grid
points
the components of a
ends
flags for
the
of
can
element.
be defined
ends
and
be
offset
from
by inputting
the
grid
as
B. Six
This is done
in WA's
GB
orientation
of
neutral
has
GA
identifies
and
can
connecting
offset vectors
The bar
element.
represent
respectively.
which
PA
PID
and
pinned connections.
the
EID
numbers
point
at
vector
card
In
points
the components
WB's.
extensional,
torsional, bending,
and
transverse
shear
card.
PID
numbers.
moment
MID
and
'A'
of
is the
area
inertia
J
respectively.
the
are
section.
and
is the torsional
PBAR property
and
material
I-,,
and
property
of cross
about
input in the
are
l2
and
axes
identification
l12
the
represent
product
inertia
of
constant.
PBAR CARD
PBAR
pro
MTO
11
12
NSM
PBAR
123
C 1
C2
K 1
K2
D 1
D2
E 1
F 1
E2
F2
+23
112
unit
flexibility
stresses
the
in
or
stiffnesses.
shear
point
length
is
at
in the
the
of
per
section
and
unit
of
unit
area.
their
from
In
points.
that
per
default
Output
grid
cross
C, D, E
element.
By
zero.
coordinates
the
mass
mass
length. It is defined
K-|
K2
and
value
is
NASTRAN
the
BAR
F fields.
Up
mass
the transverse
infinite
generally
element,
point.
are
as
or
that
consists
stresses
recovered
the
at
of
any
by inputting
The figure
(A.1)t1
81
shows
the
BAR
There
four
are
different
plate
or
shell
and
other
the
names
number
stress
TRIA3
suggest,
the
model
QUAD4
used
to
at
elements,
e.g.
transition
between
These
shows
card
and
triangular
to
can
bending
CTRIA3 CARD
places
meshing
elements
combined
since
they
give
where
of
As
in
the
differ from
points
connected
this
it
better
is
work
with
as
unequal
membrane
quadrilateral
center
flat
shell
is the PSHELL
elements
are
as
The
given
In
and
elements
use
of
grid
QUAD4
circle
figure
The
(A.2)t1
or
81
property
connection
below.
or
points.
membrane
elements.
card.
TRIA3
of
or
The
noded
elements are
to
number
properties.
their
respectively.
TRIA3
results.
having bending
four
is
only
feasible
not
in
numbers
nodes
eight
triangular plate or
for these
They
QUAD4
and
and
grid
in
available
are
behave
elements
card
six
presented
edges
and
and
of
number
noded
QUAD8 have
hence they
TRIA3
only
three
structure
are used,
preferred
are
the
of
is
QUAD8.
and
points.
recovery
elements
card
CTRIA3
EID
pro
Gl
G2
G3
XX
ABC
XX
T 1
T2
T3
G 1
G2
G3
XXX
+BC
CQUAD4 CARD
CQUAD4
EID
pro
G4
ABC
XX
T 1
T2
T3
T4
XX
+BC
The
shell
the thickness
value
the
0 is
surface.
the
Each
materials.
element
not
card
property
degrees
values
T is
of
of
is
PSHELL CARD
variable
thickness.
specified
and
connection
thickness
material
element
freedom.
not
have
can
elements
property
is
taken
orientation
elastically
Rotation
provided.
is then
card,
about
This is
number
grid
by
G. If the
it has to be input in
as
uniform
angle
connected
the
specified
normal
for
over
the
anisotropic
to
five
of
to
the
surface
the
six
of
PSHELL
pro
MIDI
MID2
TS/T
MID3
121 /T
NSM
BCD
Z 1
Z2
MID4
+CD
In the
MIDI
property card,
for
numbers
membrane
by inputting
leaving
unit
the
it blank.
length,
normalized
shear
is
are
fourth
the distances
material
membrane
to
take
plane.
ratio
of
of
coupling.
warping
Also the
up to 20
point
results
i.e.
are
fairly
and skewness
shell
four
grids
accurate
do
for
Format
: DYNRED
by
per
is the
thickness and T is
mass
are
not
and
lie
elements
and
input
for
conditioned
well
up to 45.
Z-|
recovered.
be
can
inertia
Rj=TS/T
is to be
elements
selected
deselected
bending
shear
MID4
be
can
or
inertia.
stress
number
when
these
nonstructural
where
The
of
Whereas
respectively.
number
TS is the
NSM is the
a
of
identification
material
normalized
moment
where
identification
bending
some
is the
bending
thickness,
Any
identification
Rj=12*I/T^
Z2
behavior.
shear
the
are
behavior
bending
and
appropriate
where
MID2
and
in
the
with
same
aspect
This
is
case
control
parameters.
Here,
data
Use
card.
Reduction
is
the
to
used
select
is the identification
n,
of
card
this
card
method
applied
of
number
that
means
dynamic
to
reduction
DYNRED bulk
Generalized
the
reduce
Dynamic
assembled
matrices.
2.
card
data
needed
card
image is
is
perform
as
shown,
the
generalized
and
dynamic
it defines the
reduction.
The
DYNRED CARD
1
DYNRED
SID
FMAX
NIRV
NIT
NQDES
ron?
DYNRED
I
Where,
identification
SID
is the
FMAX
set
number
of
makes
sure
that
NIRV
is the
number
of
initial
NIT
is the
number
of
iterations.
IDIR
is
integer
random
NODES is the
used
card
to
random
select
vectors.
starting
point
to
generate
initial
vectors.
number
is blank, the
3. METHOD
interest. This
of
generalized
automatic
coordinates
selection
is
used).
to be
used.
(If it
Format
METHOD
This
is
case
extraction
parameters.
EIGR
an
EIGR
card :
This
bulk
card
Here
(inputted
card
parameters
4.
control
for the
data
method
card
real
eigenvalue
identification
deck).
data
number
The
data
the
image is
needed
as shown
to
of
necessary
are
card
the
select
set
bulk
the
defines
The
eigenvalue analysis.
is the
in
to
used
card.
perform
real
here
EIGR CARD
EIGR
METHOD
SID
Fl
F2
NE
ND
ABC
EIGR
NORM
is the
set
+BC
SID
identification
specifies
extraction
methods
number.
one of
to
the three
be
F1
F2
Tnis
in
|\E
specifies
which
the
estimates
the
the
frequency
eigenvectors
number
of
The
used.
will
roots
of
is
method
'INV, 'GIV
range
be
eigenvalue
or
interest
inputted
'MGIV.
or
calculated.
in the
frequency
range
specified.
ISD
number
of
roots
of
Inverse Power
range
Method
and
desired
number of eigenvectors
for Givens
Method.
E
It is the
mass
NORM
specifies
Following
methods can
MASS
MAX
is the
is the
It
should
be
normalizes to
5. QSET1
q-set
and
unit
noted
must
of
must
have
of
eigenvectors.
generalized
maximum
of
unit value
identification
point
parameter.
used
unit value
component number
matrix
freedom
be
normalizes to
grid
the eigenvalues
mass
to
normalizes
POINT
G
orthogonality test
number
for NORM
displacement.
the component defined.
for NORM
or
the
be
positive
mass
are
POINT
POINT
mass.
definite.
Hence
all
method
the degrees
all
the
of
properties.
card :
are
the
generalized
component
coordinates
alternate
synthesis.
manually
formats
QSET1 CARD
modal
of
coordinates
or
used
automatically.
the QSET1
for dynamic
bulk data
card
places
Following
card.
reduction
are
these
the
QSET1
Gl
G2
G3
G6
G5
GA
G7
QSET1
ABC
G8
G9
-tc-
+BC
ALTERNATE FORMAT
QSETl
"THRU"
GID1
GD2
QSET1
where,
is the
component
are
numbers
grid
identification
Gj, GID;
5. ASET1
or
grid
points.
scalar
It
are
numbers
point
when
be
must
scalar
identification
point
identification
zero
if
point
points.
number.
card :
card
Alternately,
by
as
ASET1 CARD
of
by
used
freedom to be
the
number
specifying
automatically
is
defines
it
freedom. Again
user
number
the
individual
software.
The
placed
of
in the
selection can
points
card
or
it
image is
degrees
be done
could
as
set.
analysis
independent
be
shown
It
by
of
the
done
below.
ASET1
ASET1
ABC
-ETC-
+BC
ALTERNATE FORMAT
ASETl
"THRU"
EDI
E>2
ASET1
where,
is the
G, ID1, ID2
Please
at
fatal
exist,
are
note
sequence
be
component
ID1
least
error
the grid
that
if
Any
collectively
of
are
identification
point
scalar
later
ID2
degree
results.
or
the
through
one
number
format
is
not required
freedom in the
points
produces
implied
warning
in
all
used,
numbers.
points
in
the
a-set
the
THRU
message
but
that
is
do
or
not
otherwise
ignored.
6. SPOINT
card :
of
SPOINT CARD
card
defines
this card is
scalar
shown
points
below.
of
the
structural
model.
SPOINT
ID
ID
ID
ID
ID
ID
ID
ID
SPOINT
ALTERNATE FORM
SPOINT
roi
"THRU1
ID2
XXXX X
SPOINT
where,
scalar
point
other
structural,
used
to
define
multipoint
are
must
of
identification
scalar
and
scalar
constraint
connected.
degrees
are
numbers
fluid
points
equations,
unique
Primarily,
appearing
but to
be
must
points.
numbers.
in
which
this
single
no
scalar
with
all
card
is
point
or
elements
freedom
in the
{uq}
set
card.
are
scalar
points.
Hence
they
APPENDIX B
ID
SHANK.
TIME 30
THESIS
SOL 3
CEND
SNORMAL
TITLE=
ROBOT ARM
MODES
ANALYSIS
SUBTITLE=
MODAL ANALYSIS
LABEL = M0DE#
DISP (PRINT, PUNCH )= ALL
SPC=1
DYNRED=20
METHOD
30
BEGIN BULK
PARAM,GRDPNT,0
,15,,,+EIl
+EI1,MAX
YES
MAT1
1.03+7
3.872+6
MAT1
2
1
1.02+7
3.835+6
GRID
GRID
GRID
GRID
2
3
4
4.94975
.33
.33
2.512-4
.000265
4.94975
123456
2.22-16
123456
4.94975-4.94975
123456
1.11-16-7.
123456
7.
123456
GRID
5
6
-4.94975-4.94975
GRID
-4.94975
GRID
-2.22-16
GRID
4.94975
4.94975
2.
2.
2.
GRID
10
7.
2.22-16
2.
GRID
GRID
11
12
4.94975-4.94975
GRID
-4.94975-4.94975
GRID
13
14
15
2.
2.
2.
2.
-2.22-16
7.
123456
GRID
16
-4.94975
4.94975
123456
GRID
17
-3.18
18
-2.2486
GRID
GRID
GRID
123456
-7.
4.94975
7.
1.11-16-7.
-7.
2.
-2.2486
19
20
21
5.55-17-3.18
GRID
22
GRID
23
-2.50-16
3.18
GRID
24
-2.2486
2.2486
GRID
25
26
27
-3.18
GRID
GRID
GRID
GRID
GRID
GRID
28
GRID
29
30
GRID
GRID
31
GRID
32
GRID
33
2.2486
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34
GRID
35
GRID
36
GRID
37
GRID
38
2.
2.
3.18
1.67-16
2.
2.2486
2.2486
2.
2.
2.
-2.2486
22.5
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5.55-17-3.18
2.2486
-2.2486
22.5
22.5
22.5
3.18
1.67-16
22.5
2.2486
2.2486
22.5
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3.18
22.5
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2.2486
22.5
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1.
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i
4.94974-4.94974
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39
40
41
4.94974
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GRID
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5.09
3.59917
43
44
3.59917
GRID
45
5.09
GRID
GRID
GRID
GRID
GRID
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46
3.59917 -3.59917
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GRID
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GRID
GRID
GRID
GRID
GRID
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3.59917
5.09
3.59917
47
48
49
50
51
52
53
54
55
56
-5.09
3.59917 -3.59917
3.18
3.18
3.18
2.2486
2.2486
2.2486
2.2486
2.2486
2.2486
3.18
3.18
57
58
59
60
3.18
2.2486
2.2486
2.2486
2.2486
2.2486
2.2486
1.
1.
2.
2.
2.
2.
2.
2.
2.
2.
7.125
12.25
17.375
7.125
12.25
17.375
7.125
12.25
17.375
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12.25
17.375
61
3.18
7.125
GRID
62
3.18
12.25
GRID
63
64
GRID
GRID
GRID
GRID
65
66
3.18
2.2486
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2.2486
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2.2486
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GRID
GRID
GRID
GRID
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68
69
70
71
72
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1.59
2.75396-
-1.59
17.375
7.125
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24.09
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22.926
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2.75396
1.59
1.59
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1.59
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79
1.59
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2. 75396
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76
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32
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83
84
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36
37
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38
89
90
GRID
31
6.3
92
6.3
3.18
3.18
24.09
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25.68
23.93
24.09
27.27
28.434
28.86
28.434
27.27
-1.59
23.434
93
94
6.3
1.59
28.434
6.3
2.75396
27.27
95
6.3
3.18
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96
6.3
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24.09
GRID
97
3.7
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98
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28.86
GRID
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99
100
101
3.7
102
7.8
4.74
GRID
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GRID
GRID
GRID
GRID
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
GRID
118
6.3
GRID
GRID
119
120
6.3
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28.86
19.68
4.5
25.68
31.68
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27.27
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28.434
28.86
1.59
28.434
2.75396 27.27
3.18
25.68
2.75396 24.09
1.59
22.926
22.5
-1.59
22.926
-2.75396
24.09
1.59
28.434
2.75396 27.27
3.18
25.68
1.59
22.926
22.5
-1.59
22.926
-2.75396
24.09
GRID
121
6.3
-3.18
25.68
GRID
122
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GRID
GRID
GRID
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123
124
125
126
127
1.59
1.59
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GRID
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128
129
130
-2.78-17
7.8
7.8
4.74
4.74
4.74
4.74
4.74
4.74
4.74
4.74
4.74
4.74
-3.7
-3.7
-3.7
6.3
-2.75396
1.59
-1.59
1.59
1.59
1.59
1.59
1.59
2.75396
3.18
2.75396-
-2.75396
2.75396-3.18
131
132
2.75396-
GRID
GRID
GRID
133
134
135
2.75396
2.75396
2.75396
GRID
136
2.75396
GRID
137
2.75396
-2.75396
2.75396-
-1.59
1.59
2.75396
3.18
2.75396
28.434
28.86
28.434
27.27
25.68
24.09
25.68
27.27
28.434
28.86
28.434
27.27
25.68
24.09
27.27
GRID
138
3.18
-2.75396
GRID
3.18
3.18
-1.59
GRID
139
140
GRID
GRID
141
142
3.18
6.3
-2.75396
GRID
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143
144
3.18
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GRID
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145
146
2.75396
1.59
GRID
147
3.18
3.18
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149
3.18
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151
152
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153
-1.59
24.09
22.926
22.926
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25.68
27.27
28.434
28.86
28.434
GRID
154
-2.75396
27.27
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GRID
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155
-1.59
156
157
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2.75396
1.59
28.434
28.86
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25.68
27.27
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GRTD
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159
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161
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168
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172
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187
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191
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193
194
195
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7.8
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199
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202
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203
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27.27
28.434
28.86
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24.09
-3.18
25.68
3.18
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27.3894
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25.68
25.68
24.2937
25.68
22.2611
21.6968
6.41667
-12.3587
8.33333
-12.3587
-9.6565
21.1324
20.5681
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20.0037
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29.0989
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29.6632
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30.2276
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30.7919
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31.3563
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29.0989
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GRID
205
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208
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211
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213
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216
11.8
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30.2276
30.7919
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218
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21.5968
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11.8
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219
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11.8
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223
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226
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228
229
230
231
232
233
234
235
236
237
238
239
240
9.8
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-4.25217
-9.6565
29.6632
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30.2276
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30.7919
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21.6968
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21.1324
-17.0111
20.5681
28.1272
-16.5
23.1261
-16.5
28.2339
-4.25217
9.8
9.8
9.8
11.8
11.8
11.8
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11.8
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11.8
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21.1324
20.5681
-18.0345
-18.7457
23.7063
24.5814
-18.7457
25.68
26.7786
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27.6537
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23.2328
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-17.0111
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23.1261
28.1272
28.2339
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24.5806
25.6793
27.6535
241
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243
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26.7782
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244
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23.2328
245
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23.1261
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246
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7.8
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248
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28.2339
28.1272
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23.706
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25.6796
26.7784
27.6536
23.2328
28.6664
22.6936
256
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11.8
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28.6664
9.8
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28.6664
258
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259
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260
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22.6936
26.8775
261
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27.5557
-17.6975
25.68
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23.8043
24.4825
31.5965
31.5965
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19.7635
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19.7635
28.5181
25.68
GRID
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251
252
253
254
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-18.7455
-18.7458
-18.0346
-17.0111
-14.4293
-18.3487
GRID
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263
264
265
266
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268
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269
7.8
11.8
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270
7.8
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7.8
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271
272
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273
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22.6936
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22.8419
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28.6382
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275
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23.5181
GRID
276
11.8
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25.68
277
11.8
-1.55
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GRID
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279
280
11.8
11.8
281
7.3
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282
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283
7.8
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284
7.8
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285
286
287
288
7.8
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289
290
291
292
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294
7.8
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7.8
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295
296
297
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298
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301
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299
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302
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303
9.8
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304
305
306
9.8
7.8
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-17.6063
26.1383
25.2217
307
7.8
-17.3468
24.8332
308
7.8
-16.9583
24.5737
309
310
7.8
-15.8753
24.6584
7.8
-15.4341
25.1343
-15.5392
23.6641
7.8
7.8
7.8
7.8
-12.9999
22.9844
-15.4784
26.3047
GRID
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311
312
313
314
315
7.8
-18.1761
25.8872
26.4585
GRID
316
7.8
-17.6907
27.0903
317
7.8
-16.9847
27.4569
318
7.8
-18.1758
24.9014
-17.6902
24.2696
7.8
-16.9847
23.9033
7.8
-14.7651
GRID
319
320
321
7.8
27.4861
322
7.8
-15.9737
GRID
23.6592
323
7.8
-17.0872
25.6138
324
7.8
-16.5
25.68
325
7.8
-16.5008
11.8
25.033
26.8775
GRID
326
-16.5
327
11.8
-16.9583
26.7863
GRID
328
11.8
-17.3468
26.5268
329
11.8
-17.6063
26.1383
11.8
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25.68
11.8
-17.6063
25.2217
11.8
-17.3468
24.8332
11.8
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24.5737
11.8
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24.4825
11.8
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24.6584
11.8
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25.1343
11.8
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26.3047
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25.8775
GRID
GRID
GRID
GRID
GRID
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GRID
GRID
GRID
GRID
GRID
GRID
GRID
GRID
GRID
GRID
GRID
GRID
GRID
GRID
GRID
GRID
330
GRID
331
GRID
332
GRID
333
GRID
GRID
334
GRID
336
GRID
337
GRID
338
335
7.8
7.8
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25.68
-.4
22.7218
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23.6884
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25.68
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27.6716
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23.4062
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25.68
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27.9538
2.25
2.25
23.124
25.68
28.236
28.68
22.68
4.723
28.748
7.196
4.723
28.816
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7.196
22.612
22.544
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26.5268
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26.7863
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28.6382
19.7635
31.5965
GRID
339
GRID
GRID
GRID
GRID
GRID
GRID
GRID
GRID
GRID
GRID
340
341
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9.8
9.8
9.8
9.8
9.8
9.8
9.8
9.8
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
9.8
9.8
9.8
11.8
11.8
11.8
11.8
9.8
9.8
9.8
9.8
11.8
358
-16.9583
-17.3468
-17.6063
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24.5737
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24.4825
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26.3047
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24.2937
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22.9844
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23.6641
25.8872
24.2937
22.9844
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27.5557
27.5557
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359
360
361
362
363
9.8
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11.8
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9.8
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11.8
9.8
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364
365
7.6
7.6
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366
367
368
369
370
371
372
373
374
375
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23.4711
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4.1
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4.1
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4.1
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5.85
5.85
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377
GRID
378
4.1
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GRID
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379
380
381
4.1
5.85
5.85
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382
4.1
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383
384
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5.85
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385
386
387
5.85
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GRID
GRID
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GRID
GRID
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388
389
390
25.68
25.68
23.8043
23.8043
23.4711
25.7211
27.9711
24.7211
25.7211
26.7211
24.0961
27.3461
27.9711
27.9711
-19.5441
7.6
7.6
7.6
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376
26.7363
26.5268
26.1383
25.68
25.2217
24.8332
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-13.4071
-13.5057
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4.1
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23.4711
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26.7211
27.3461
24.0961
26.7211
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25.7211
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15.3878
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391
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178
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139
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152
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162
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165
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154
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3
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162
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181
182
183
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185
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221
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223
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211
215
216
217
212
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204
205
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198
199
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215
216
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226
213
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189
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191
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192
193
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249
239
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CQUAD4
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195
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214
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253
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236
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232
233
251
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4
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198
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200
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205
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207
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208
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255
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199
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214
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219
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227
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233
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237
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238
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15
55
424
CTRIA3
449
12
422
19
CTRIA3
450
12
425
21
41
CTRIA3
451
423
17
45
C0NM2
452
12
263
C0NM2
^53
.007874
307874
822631
853143
1.
49
425
:*!!
993348
5
-.6
.568576-.
313
63
62
61
CBAR
434
411
407
15
15
CBAR
380
372
373
47
-3.25-17
8.33-17
C0NM2
C0NM2
C0NM2
C0NM2
PSHELL
PSHELL
PSHELL
PSHELL
PSHELL
PSHELL
PSHELL
PSHELL
PSHELL
PSHELL
PSHELL
PSHELL
PBAR
PBAR
PBAR
ENDDATA
454
201
455
204
31312
.02624
456
20b
.02624
457
410
.007874
3
4
1
1
5
6
1
1
1
1
1
10
11
12
1.
1.
.5
.25
1.
1.
1.
2
1
1
1
1
1.
1
1
1
1
1
1
1
2
.11
.35
1.
.25
1.
1.
1.
1
1
2
1.
1.
1.
.25
1.
25.
1.
15
1.
.225
.175
1
2
13
14
.25
.0625
25.
1.
3.255-4
25.
1.
.000326
.8333
.8333
.8333
.8333
.8333
.8333
.8333
.8333
.8333
.8333
.8333
.8333
25.
1.
1.
APPENDIX C
MODAL TESTING COORDINATE AND DISPLAY SEQUENCE DATA
Coordinates
Point
Table
crd
ROBOT. PRJ
#1
crd
#2
crd
#3
Componer
7.000
270.000
0.00
CYLINDER
7.000
315.000
0.00
CYLINDER
7.000
0.00
0.00
CYLINDER
7.000
45.000
0.00
CYLINDER
7.000
90.000
0.00
CYLINDER
7.000
135.000
0.00
CYLINDER
7.000
180.000
0.00
CYLINDER
7.000
225.000
0.00
CYLINDER
7.000
270.000
2.000
CYLINDER
10
7.000
315.000
2.000
CYLINDER
11
7.000
0.00
2.000
CYLINDER
12
7.000
45.000
2.000
CYLINDER
13
7.000
90.000
2.000
CYLINDER
14
7.000
135.000
2.000
CYLINDER
15
7.000
180.000
2.000
CYLINDER
16
7.000
225.000
2.000
CYLINDER
17
3.180
270.000
2.000
CYLINDER
18
3.180
315.000
2.000
CYLINDER
19
3.180
0.00
2.000
CYLINDER
20
3.180
45.000
2.000
CYLINDER
21
3.180
90.000
2.000
CYLINDER
22
3.180
135.000
2.000
CYLINDER
23
3.180
180.000
2.000
CYLINDER
24
3.180
225.000
2.000
CYLINDER
25
3.180
270.000
12.250
CYLINDER
26
3.180
315.000
12.250
CYLINDER
27
3.180
0.00
12.250
CYLINDER.
28
3.180
45.000
12.250
CYLINDER
Coordinates
Point
Table
Crd
ROBOT. PRJ
#1
29
3..180
30
31
3.
32
Crd
#2
Crd
43
Componei
90.000
12.250
CYLINDER
.180
135.000
12.250
CYLINDER
.180
180.000
12.250
CYLINDER
.180
225.000
12.250
CYLINDER
33
.180
270.000
22.500
CYLINDER
34
.180
315.000
22.500
CYLINDER
35
.180
0.00
22.500
CYLINDER
36
.180
45.000
22.500
CYLINDER
37
.180
90.000
22.500
CYLINDER
38
135.000
22.500
CYLINDER
39
180.000
22.500
CYLINDER
40
.180
225.000
22.500
CYLINDER
41
.180
180.000
0.00
HCYL
42
.180
135.000
0.00
HCYL
43
.180
90.000
0.00
HCYL
44
.180
45.000
0.00
HCYL
45
.180
0.00
0.00
HCYL
46
.180
315.000
0.00
HCYL
47
3.
.180
270.000
0.00
HCYL
48
3,
.180
225.000
0.00
HCYL
49
3.
135.000
520.000m
HCYL
50
3,
.180
90.000
520.000m
HCYL
51
3,
.180
45.000
520.000m
HCYL
52
3,
.180
0.00
520.000m
HCYL
53
3,
315.000
520.000m
HCYL
54
3.
.180
270.000
520.000m
HCYL
55
3.
,180
225.000
520.000m
HCYL
56
.180
.180
.180
.180
,180
135.000
1.450
HCYL
Coordinates Table
^oinc
crd
ROBOT. PRJ
#i
Crd
#2
Crd
#3
Componen-
57
3.180
90.000
1.450
HCYL
58
3.180
45.000
1.450
HCYL
59
3.180
0.00
1.450
HCYL
60
3.180
315.000
1.450
HCYL
61
3.180
270.000
1.450
HCYL
62
3.180
225.000
1.450
HCYL
63
3.180
90.000
3.700
HCYL
64
3.180
45.000
3.700
HCYL
65
3.180
0.00
3.700
HCYL
66
3.180
315.000
3.700
HCYL
67
3.180
270.000
3.700
HCYL
68
3.180
135.000
5.950
HCYL
69
3.180
90.000
5.950
HCYL
70
3.180
45.000
5.950
HCYL
71
3.180
0.00
5.950
HCYL
72
3.180
315.000
5.950
HCYL
73
3.180
270.000
5.950
HCYL
74
3.180
225.000
5.950
HCYL
75
3.180
135.000
6.880
HCYL
76
3.180
90.000
6.880
HCYL
77
3.180
45.000
6.880
HCYL
78
3.180
0.00
6.880
HCYL
79
3.180
315.000
6.880
HCYL
80
3.180
270.000
6.880
HCYL
81
3.180
225.000
6.880
HCYL
82
3.180
180.000
10.000
HCYL
83
3.180
135.000
10.000
HCYL
34
3.180
90.000
10.000
HCYL
Coordinates Table
fomt
Crd
ROBOT. PRJ
#1
Crd
#2
crd
#3
Component
85
3.180
45.000
10.000
HCYL
86
3.180
0.00
10.000
HCYL
87
3.180
315.000
10.000
HCYL
88
3.180
270.000
10.000
HCYL
89
3.180
225.000
10.000
HCYL
90
4.615
180.000
10.000
HCYL
91
4.615
135.000
10.000
HCYL
92
4.615
90.000
10.000
HCYL
93
4.615
45.000
10.000
HCYL
94
4.615
0.00
10.000
HCYL
95
4.615
315.000
10.000
HCYL
96
4.615
270.000
10.000
HCYL
97
4.615
225.000
10.000
HCYL
98
7.800
8.000
31.680
MAIN
99
7.800
5.330
31.680
MAIN
100
7.800
2.660
31.680
MAIN
101
7.800
0.00
31.680
MAIN
102
7.800
-1.550
31.350
MAIN
103
7.800
-5.150
30.600
MAIN
104
7.800
-8.750
29.850
MAIN
105
7.800
-12.350
29.090
MAIN
106
7.800
-12.350
26.810
MAIN
107
7.800
-12.350
24.540
MAIN
108
7.800
-12.350
22.260
MAIN
109
7.800
-8.750
21.500
MAIN
110
7.800
-5.150
20.750
MAIN
111
7.800
-1.550
20.000
MAIN
112
7.800
0.00
19.680
MAIN
Coordinates
Point
Table
Crd
ROBOT. PRJ
#1
Crd
#2
Crd
#3
Component
113
7.800
2.660
19.680
MAIN
114
7.800
5.330
19.680
MAIN
115
7.800
8.000
19.680
MAIN
116
7.800
9.990
23.510
MAIN
117
7.800
9.990
27.840
MAIN
118
4.615
180.000
11.500
HCYL
119
4.615
135.000
11.500
HCYL
120
4.615
90.000
11.500
HCYL
121
4.615
45.000
11.500
HCYL
122
4.615
0.00
11.500
HCYL
123
4.615
315.000
11.500
HCYL
124
4.615
270.000
11.500
HCYL
125
4.615
225.000
11.500
HCYL
126
7.800
7.320
27.840
MAIN
127
7.800
7.320
23.510
MAIN
128
7.800
-8.750
27.070
MAIN
129
7.800
-8.750
24.280
MAIN
130
11.800
-1.550
31.350
MAIN
131
11.800
-5.150
30.600
MAIN
132
11.800
-8.750
29.850
MAIN
11.800
-12.350
29.090
133
MAIN
11.800
-14.420
28.660
134
MAIN
11.800
-17.010
28.120
MAIN
11.800
-18.750
26.780
MAIN
11.800
-18.750
24.580
MAIN
11.800
-17.010
23.230
MAIN
11.800
-14.420
22.690
MAIN
11.800
-12.350
22.260
MAIN
135
136
137
138
139
140
Coordinates
roim:
Table
Crd
ROBOT. PRJ
#1
Crd
#2
Crd
#3
Component
141
11.800
-8.750
21.500
MAIN
142
11.800
-5.150
20.750
MAIN
143
11.800
-1.550
20.000
MAIN
144
11.800
-1.550
22.840
MAIN
145
11.800
-1.550
25.680
MAIN
146
11.800
-1.550
28.520
MAIN
147
7.800
-13.080
28.970
MAIN
148
7.800
-16.580
30.720
MAIN
149
7.800
-20.080
28.970
MAIN
150
7.800
-19.480
22.880
MAIN
151
7.800
-18.880
16.800
MAIN
152
7.800
-18.280
10.720
MAIN
153
7.800
-17.780
9.800
MAIN
154
7.800
-17.710
7.550
MAIN
155
7.800
-15.460
7.550
MAIN
156
7.800
-15.380
9.800
MAIN
157
7.800
-14.880
10.720
MAIN
158
7.800
-14
.300
16.630
MAIN
159
7.800
-13.720
22.540
MAIN
160
7.800
-17.010
28.120
MAIN
161
7.800
-18.750
26.780
MAIN
162
7.800
-18.750
24.580
MAIN
163
7.800
-17.000
23.230
MAIN
164
4.300
-13.080
28.970
MAIN
165
4.300
-16.580
30.720
MAIN
166
4.300
-20.080
28.970
MAIN
167
4.300
-19.480
22.880
MAIN
168
4.300
-18.880
16.800
MAIN
Display
Line
l
Sequence
Table
Lift
Pen
X
2
3
17
25
33
33
12
13
17
14
25
15
33
16
10
17
18
18
26
19
34
20
21
11
22
19
23
27
24
35
25
26
12
27
20
28
16
24
32
25
10
11
End
17
start Pt
6
7
ROBOT. PRJ
40
Pt
Display
Line
Sequence
Table
Pen
Lift
ROBOT. PRJ
29
28
30
36
31
Pt
Start
13
33
21
34
29
35
37
X
37
14
38
22
39
30
40
38
41
23
43
31
44
39
45
24
48
32
49
40
50
52
16
47
51
7
15
42
46
Pt
32
36
End
41
39
49
48
55
53
39
54
55
38
56
56
62
Display
Line
Sequence
Table
Lift Pen
57
59
start
35
75
66
35
X
68
X
90
118
118
74
41
39
35
76
82
77
90
78
118
79
89
97
90
72
75
82
81
82
70
73
74
34
65
71
67
36
68
63
69
End
33
62
67
pt
37
63
60
64
ROBOT. PRJ
40
58
61
42
80
49
81
56
82
63
83
68
34
75
125
Pt
Display
Line
Sequence
Table
Lift
Pen
ROBOT. PRJ
Start
85
83
86
91
87
119
88
56
89
37
90
68
91
43
92
50
93
57
94
63
95
69
96
76
97
84
98
92
99
120
100
44
101
51
102
58
103
64
70
104
77
105
85
106
93
107
121
108
109
110
111
112
45
52
59
65
Pt
End
Pt
Display
Line
Sequence
Table
Lift
Pen
ROBOT. PRJ
Start
113
71
114
78
115
86
116
94
117
122
118
46
119
53
120
60
121
66
122
72
123
79
124
87
125
95
126
123
127
47
128
54
129
61
130
67
73
131
80
132
88
133
96
134
124
135
136
137
138
139
140
48
55
62
67
74
Pt
End
Pt
Display
Line
Sequence
Table
Lift
Pen
ROBOT. PRJ
start
141
81
142
89
143
97
144
125
145
33
147
74
X
149
150
98
105
147
152
160
153
159
154
108
X
126
157
114
X
100
159
121
160
99
161
114
163
164
117
166
168
120
126
165
167
113
119
162
163
99
156
158
117
98
151
155
End
62
146
148
Pt
120
127
127
Pt
Display
Line
Sequence
Table
Lift Pen
169
170
Start
111
125
103
124
176
110
X
104
178
128
179
109
180
128
182
124
183
129
184
107
X
187
196
133
105
194
195
132
104
192
193
131
103
190
191
130
102
188
189
130
130
186
129
106
181
185
End
102
175
177
Pt
123
173
174
ROBOT. PRJ
116
171
172
143
111
146
Pt
Display
Line
197
Sequence
Table
Pen
Lift
X
108
162
137
163
138
147
164
164
218
219
148
165
220
221
149
166
222
223
150
167
224
159
147
216
217
161
136
214
215
160
135
212
213
139
159
210
211
134
147
208
209
Pt
140
206
207
End
109
204
205
Pt
141
202
203
Start
110
200
201
ROBOT. PRJ
142
198
199
176
Line
225
Lift
Pen
X
226
227
158
159
147
164
167
176
246
247
157
176
244
245
156
175
242
243
155
174
240
241
154
173
238
239
153
172
236
237
152
171
234
235
151
170
232
233
Start Pt
169
230
231
ROBOT. PRJ
168
228
229
168
175
248
249
169
174
250
251
160
149
252
End
Pt
Display
Line
253
Sequence
Table
Lift
Pen
X
254
255
Start
163
158
151
157
152
258
259
ROBOT. PRJ
150
256
257
145
260
177
261
120
Pt
End
Pt
APPENDIX D
MODE SHAPES FROM THE FINITE ELEMENT ANALYSIS
<7
t=
^3
^
rf3tr~~'
^^^^
>'
T-.
'
,.-.
'
^7
^f
i*
^T
^f-
IT
a
\.
rifely/ ,7
Mffltr
to
UJ
~>4-
-IJL&.
TTTT
II
III
n~
-^-f
^^
~vfc-~~'l
II,
!l\i
I
-r^r.
-rf
I-*
ae
r*^
I
y*-
wbb^:\
APPENDIX E
MODE SHAPES FROM MODAL
TESTING
Trace
#1(13.266
Hz)
Mode 4
Frequency
Damping
13.27
Hz
4.73
x-
Trace
Mode
#2(21.136
Hz
Frequency
21.14
Hz
Damping
3.53
X'
Trace a
#3(35.957
Mode
HZ)
Frequency
35.96
Hz
Dampina
.94
X'
Trace
Mode
#4(39.708
HZ)
Frequency
39.71
HZ
Damping
2.22
X'
Trace a
Mode #
#5(52.201
Frequency
52.20
HZ
Damping
1.70
Hz)
X"
Trace a
#6(66.347
Mode
Hz)
Frequency
66.35
HZ
Damping
2.13
X'
Trace
Mode
Frequency
Damping
#7(107.390
Hz)
107.39
1.74
HZ
X'
Trace a
#8(117.405
Mode
Frequency
117.41
HZ
Damping
1.29
HZ)
X"
Trace a
#9(151.286
Mode
Frequency
151.29
HZ
Damping
3.74
HZ)
--TN
x-
Trace a
#10(199.078
Mode
10
Frequency
199.08
Hz
Damping
661.70m
HZ)
X'
Trace
Mode
#11(205
468
HZ)
11
Frequency
205.47
Dampina
1.98
Hz
X'
Trace
Mode
#12(213
083
HZ)
12
Frequency
213.08
Damping
1.06
Hz
X'
Trace
Mode
#13(224.702
Hz)
13
Frequency
224.70
Hz
Damping
620.58m
Z_
X'
APPENDIX F
run
the
Run
normal. for
2.
Run
MAC
Run
where
assign
expl.asc
assign
normal. out
column.
assign
3.
program,
for
for005 ( expmental
for006
forOlO
(file
feal.f06 for015
normal. out
for020
assign
column. out
for006
where
containing
corresponding nodes)
(nastran output file)
assign
for
file)
where
match.dat
assign
mac.
output
assign
column. out
assign
mac
out
for005
for006
Name
:
Shashank Kolhatkar
Description of the program
This program normalizes
from
obtained
*
using
the
experimental
unit
displacements
the
modal
analysis
displacement
maximum
scheme
A
ex
is
the
modal
array
normalized
big
variable
real
do
30
shapes
determine
big
,exl(144,3)
1, 13
(5, A) ( (
ex(l,l)
biggest
big, fact
ex<
j)
i = 1,144
if (abs(ex( i,2) )
j^l
,4)
,i
,144)
20
big
then
.gt.big)
ex(i,2)
endif
if (abs(ex( i
big
,3
) )
.gt.big;
then
ex(i,3)
endif
if (abs(
big
ex(
,4
) )
gt
.big)
then
ex(i,4)
endif
20
enddo
fact
do
110
l/(abs(big) )
i
110
es(i2)
*>
fact
exl(i,2)
ex(i,3)
exl(i,3)
ex(i,4)
fact
fact
(6,M
enddo
enddo
stop
end
1,144
=
write
30
exl(i.l)
write
( 6,*)
obtained
displacement
or
the
i,j,m
read
do
mode
to
none
ex(144,4)
integer
values
from
testing
exl
implicit
displacement
of
ex(
1 )
(exl( 1
j l, 3 )
=
values
factor
in
the
t
array
*
*
Name
Shashank Kolhatkar
Description
This
results
the
*
data
IDF
program
and
finite
for
the
corresponding
no.
of
finite
from
data
collects
file
grid
IDE
the
file
results
element
grids
experimental
and
together
in
puts
column
form
model
element
XF,
XE,
file.
results
implicit
none
,YE(144)
,XF(426)
,YF(4
26)
,ZF(426)
,ZE(144)
,G(66)
DO
1,66
READ
*
F(I)
(10, A)
G(I)
,
f(i),g(i)
(6, A)
write
ENDDO
DO
1,13
DO
1,426
READ
*
(15, *)
write
(6, A)
IDF(I)
XF( I)
,XF(I)
,YF(
I)
,YF(I)
,ZF(
,ZF(I)
I)
ENDDO
DO
1,144
READ
a
write
(20, A)
(6, *)
IDE(I)
XE( I) , YE( I)
,XE(I)
,YE(I)
,ZE(
,ZE(I)
I)
ENDDO
DO
WRITE
ENDDO
ENDDO
STOP
END
1,66
(6, A)
XF(g(I) )
,YF(g(I)
,ZF(g(I)
,XE(f
(I) )
,YE(f
(I)),ZE(f (I)
Name
*
Shashank Kolhatkar
Description
This
and
A
A
*
fin
ex
mac
program
them
displacement
displacement
values
value
Modal
implicit
values
the
of
the
calculates
arranges
in
the
values
MAC
of
form.
matrix
analysis
none
real
f in(66
real
mac(
13
,3,13)
,13
ex(66,3,13)
,numl
,den2
,den3
,suml,sum2
,sum3
integer m,i,j,e,f
do
10
do
10
1,13
=
1,66
(5, A)
read
10
(f in( i
,m)
j=l
,3
do
200
do
f
100
den2
den3
do i
suml
+
numl
sum2
den2
sum3
ex(
,m)
j=l
,3
1,13
0
.
0.
1,66
-t-
numl
( f in ( i
den2
den3
1
+
f ) **2 ) + ( f in ( i
(ex(i,l,e)**2)
den3 + sum3
100
suml/
(den2Aden3 )
enddo
500
f=l,13
write
format
99
enddo
stop
end
,2,e)
f ) **2 ) + ( f in ( i
f ) **2 )
(ex(i,2,e)**2)-t-(ex(i,3,e)**2)
enddo
do
)*ex( i
sum2
mac(f,e)
,f
suml
enddo
500
1,13
numl
200
enddo
(6,*) (mac(f
(E9.3)
,e)
,e=l
,13)