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Thesis/Dissertation Collections
6-1-1989
Modal analysis of a robot arm using the finite
element analysis and modal testing
Shashank C. Kolhatkar
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Recommended Citation
Kolhatkar, Shashank C., "Modal analysis of a robot arm using the finite element analysis and modal testing" (1989). Thesis. Rochester
Institute of Technology. Accessed from
Modal Analysis of a Robot Arm
Using The Finite Element Analysis
and Modal Testing
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. Richard G. Budynas (Advisor)
Dr. Joseph S. Torok
Dr. Wayne W. Walter
...
Dr. Bhalchandra V. Karlekar
(Professor and Department Head)
Title of the Thesis
'Modal Analysis of A Robot Arm'
I Shashank C. Kolhatkar hereby grant permission to
the Wallace Memorial Library of R.I.T. to reproduce
my thesis in whole or in part. Any reproduction will
not be for commercial use or profit.
Title of the Thesis : 'Modal Analysis of A Robot Arm'
I Shashank C. Kolhatkar
hereby
grant permission tothe Wallace Memorial
Library
of R.I. T. to reproducemy
thesisin
whole orin
part.Any
reproduction will not be for commercial use or profit.:
t/i/si
Abstract
The
objective ofthe
present work wasto
study
the
technique
ofmodal analysis
using
theoretical
and experimental methods.A
PUMA
UNIMATION five
axis robot was used as a model.The finite
elementanalysis was performed
using
MacNeai
Schwendler
Corporation's
NASTRAN
code.Intergraph's Rand Micas
pre- and post-processor wasused
to
discretize
the
model andsubsequently
to
display
the
animated mode shapes after
the
analysis wasdone
withNASTRAN.
The
modaltesting
was performedusing
Structural
Measurement
Systems'
STAR
software andBruel
&
Kjaer's
instrumentation
including
a spectrumanalyzer,
accelerometer andhammer.
Due
to
the
size and weight ofthe
structureit
was not possibleto
test
it in
its
free-free
condition.Hence
it
wastested
in
its
supported condition.The
resultsfrom
the
two
analyses werethen
compared.The
first
thirteen
modesfrom
both
the
analyses showed aone-to-one correspondence.
The
naturalfrequencies
showed adeviation
from
1%
to
33%.
Although
the
difference
of33%
seemslarge,
it
wasonly for
one mode which could reflect a computational error.For
most ofthe
modes,
the
variation wasless
than
10%.
The
comparison of mode shapes was
done
by
observation ofthe
displayed
animated mode shapes.Also,
atechnique
calledModal
Assurance
Criterion
for
numerical correlation of mode shapesis
discussed.
The
experimental mode shapes showed an erraticbehavior.
This
wasprobably
due
to
the
nonlinearitiesin
the
structure.
The
causes ofthe
discrepancies
between
the
finite
element method and modal
testing
arediscussed in
the text.
Similar
patterns ofdeflection
couldbe
seenin
the two
analyses.Acknowledgements
This
workis
dedicated
to
my
parentsSau.
Rohini
Kolhatkar
andShri.
Chandrakant
Kolhatkar
without whose constant support andencouragement
it
wouldhave
notbeen
possibleto
seethis
day
in
my
career.
I
will alwaysbe
gratefulto
youboth for everything
youhave
done
for
me.I
also wantto
expressmy
gratitudetowards
Maoshi
and
Kaka
who are alwaysthere to
help
me whenI
needthem!
Dr.
Budynas,
thank
youvery
muchfor
seeing
methrough
this
thesis
research with patience and a smile.
My
way
ofthinking
is
very
muchinfluenced
by
yourin-depth,
logical
and systematic approachto
solve problems.
You
willbe
a great source ofinspiration
in
developing
my future
career.Thank
youDr.Torok
andDr.
Walter for spending
your valuabletime
reviewing
my
work.I
really
appreciate yourbeing
so understanding.My
sincerethanks to
Dan
Foley
ofBruel
&
Kjaer for
allthe
help
withB&K
hardware
I
used andto
our ownDavid
Hathaway
for
alwayshelping
me out with a smile whenI
gotinto
trouble.
Finally,
my
specialthanks
to
allmy
friends
for
providing
constantdiversions
whenI
was working.I
will always cherishthe
memoriesTable
of
Contents
List
ofTables
List
ofFigures
Page
vii viiiList
ofSymbols
x1.
Introduction
1
1.1
General
Concepts
1
1
.2Motivation
3
1
.3Modal
Testing
6
1
.3.1Applications
ofModal
Testing
7
1
.4Finite Element Method
9
1.4.1
Range
ofApplications
ofthe
Finite Element
Method
10
1
.4.2Advantages
ofthe
Finite Element Method
1 1
1
.4.3Limitations
ofthe
Finite Element Method
1 1
1
.5Comparison
ofthe
Experimental
andFinite Element
Modal Analysis
12
1.6
Mathematical
Concepts
15
1
.6.1Equations
ofMotion for Single Degree
ofFreedom System
17
1
.6.2Equations
ofMotion for Linear Multiple Degree
of
Freedom
Systems
18
1.6.3
Decoupling
the
Equations
ofMotion
21
Page
2.
Finite Element Analysis
Using
MSC/NASTRAN
28
2.1
MSC/NASTRAN
28
2.2
Finite
Element
Modeling
With
NASTRAN
31
2.2.1
Executive Control Deck
31
2.2.2
Case Control Deck
32
2.2.3
Bulk Data
Deck
34
2.3
Dynamic Analyis
Using
NASTRAN
36
2.3.1
Inverse
Power
Method
37
2.3.2
Given's Triadiagonal
Method
39
2.3.3
Modified Givens Method
41
2.4
Dynamic Reduction
42
2.4.1
Guyan Reduction
43
2.4.2
Generalized Dynamic Reduction
45
2.5
Selection
ofEigenvalue
Extraction
Method
48
2.6
Selection
ofDynamic Reduction Method
49
3.
Experimental Modal
Analysis
51
3.1
Modal Analysis
andRelated instrumentation
51
3.1.1
Vibration Exciters
53
3.1.2
Hammer
55
3.1.3
Accelerometer
57
3.1.4
Dual Channel FFT Analzer
61
3.2
Curvefitting
62
3.2.1
Properties
ofthe
Nyquist
Plot
65
3.3
3.2.2.
1
Peak
Amplitude
Method
3.2.2.2
Circle Fit Method
3.2.3
MDOF
Curvefitting
3.2.3.1
Extension
ofthe
SDOF
Method
3.2.3.2
General MDOF
Curvefitting
3.2.4
Autofitting
Modal
Testing
Procedure
Page
68
71
73
73
75
76
77
4.
Modeling
andResults
84
4.1
Background
andAssumptions
84
4.2
Procedure
to
Create
andAnalyze
the
NASTRAN Model
87
4.3
Finite Element
Modeling
ofthe
Robot
93
4.4
Modeling
for Modal
Testing
102
4.5
Results
ofthe
Finite Element Analysis
1
03
4.6
Results
ofExperimental Modal Analysis
109
4.7
Comparison
ofExperimental
andAnalytical Results
115
4.8
Integration
ofModal
Testing
andFEM
121
References
Appendixes
Appendix A
Appendix B
Appendix
C
Appendix D
Appendix E
Appendix F
123
MSC/NASTRAN
Cards Used
in
the
Modal Analysis
MSC/NASTRAN
Data File
Modal
Testing
Coordinate
andDisplay
Sequence Data
Mode
Shapes from
the
Finite Element Analysis
Mode
Shapes from Modal
Testing
List
ofTables
Table
Page
1
.1Comparison
ofthe
Finite Element
Modal
Analysis
andModal
Testing
13
2.1
Comparison
ofDynamic Reduction
Methods
50
4.1
Natural
Frequency
Results from Finite Element Analysis
using
NASTRAN
106
42
Natural
Frequencies
obtainedfrom
Modal
Testing
113
4.3
Comparison
ofthe
Results
from
the
Finite Element
List
ofFigures
Figure
Page
1
.1Procedures
for
the
Finite Element
andExperimental
Modal Analysis
14
1
.2Single Degree
ofFreedom System
1
7
1.3
Multiple Degree
ofFreedom System
19
2.1
NASTRAN
Punch
Cards
andData
Decks
30
3.1
Constructional Details
ofHammer
56
3.2a
Time
History
ofImpact Force
Pulse
56
3.2b
Frequency
Spectrum
ofthe
Pulse
56
3.3
Constructional Details
ofAccelerometer
59
3.4
Mounting
Methods
for Accelerometer
60
3.5
Properties
ofModal
Circle
67
3.6
Peak Amplitude Method
-SDOF Curve
Fitting
70
3.7
Circle
Fitting
ofFRF
Data
72
3.8
Schematic
ofthe
Experimental
Setup
for
aModal
Test
79
3.9
Analyzer
Setup
to
Measure FRF
in
the
Frequency
Domain
80
3.10
STARGATE
81
3.1
1
Driving
Point Measurement in
the
Frequency
andTime
Domains
83
4.1
Picture
ofaPUMA UNIMATE Robot
Arm
85
4.2
Graphic Model Of
the
Robot
86
4.3
Various Views
ofthe
Graphic Model
88
4.4
Features
ofthe
RAND
MICAS Preprocessor
89
Figure
4.6
Various Views
ofthe
Finite Element
Model
4.7
Features
ofin RAND MICAS
for
Applying
Loads
andBoundary
Conditions
4.8
Features
ofthe
RAND MICAS
Postprocessor
4.9
Finite Element
Mesh for
the
Base
4.10
Isometric View
ofthe Mesh
for Two
Intersecting
Cylinders
4.1 1
Front View
ofthe
Mesh
for
Two
Intersecting
Cylinders
4.12
Front
View
ofthe
Main
Arm
4.13
Finite Element Mesh
atthe
Main Arm
andForearm
Joint
4.14
RAND MICAS
Model
for Modal
Testing
4.1 5
The
"STAR"Model
for
the
Robot
Arm
4.16a
The Experimental
Setup
4.16b
The Experimental
Setup
Apppendix
A.1
Details
ofthe
Bar
Element
A.2
Details
ofthe
TRIA3
andQUAD4 Elements
Rage
91
92
94
96
97
98
100
101
104
105
110
111
List
ofSymbols
[A]
Residue
or modal constant matrixAjk
jktn element ofthe
residue matrixc,[C]
Damping
constant ordamping
matrixF(t),
F(co)
External
or appliedforce
onthe
systemFSW
Restoring
force
induced
in
aspring
M
Mass
normalized eigenvectorH(co),
h(co)
Frequency
responsefunction
matrixhjk(co)
Element
in
the
jth row and ktn column ofFRF
matrixTl
Damping
factor
[I],
I
Identity
matrixi
V-1
k,[K]
Stiffness
or stiffness matrix[L]
Lower
triangular
factor
X
Eigenvalue
m,
[M]
Mass
or mass matrixN(t)
Generalized
modalforce
matrixq(t)
Generalized
coordinate or generalizeddisplacement
Q(t)
Generalized
externalforce
s
Laplace
variableu,
[u],
[u]r
Modal
matrix or mode shape matrixList
ofSymbols
(continued)
cor,
con
Natural
frequency
ofthe
systemx(co)
Response
ofthe
systemto
excitationPFj]
Generalized
modal coordinates matrix[ ]
'Transpose
of a matrix[
]"'Inverse
of a matrix[
]"
' '
Transpose
ofinverse
of a matrixFirst derivative
of aquantity
w.r.ttime
Second derivative
of aquantity
w.r.t.timeDOF
Degree
ofFreedom
FEM
Finite Element
Method
FFT
Fast
Fourier Transform
FRF
Frequency
Response
Function
MDOF
Multiple Degree
ofFreedom
SDOF
Single Degree
ofFreedom
Im
Imaginary
part of a complex numberCHAPTER
1
INTRODUCTION
1.1
GENERAL CONCEPTS
:The
study
of vibrationsinvolves
the
oscillatory
motion ofbodies
with or withoutthe
influence
of appliedforces.
All
engineering
structures and systems are subjected
to
vibrationsin
their
operation.
Hence
their
design
generally
requires consideration oftheir
oscillatory
behavior.
There
aretwo
general classes of vibrations-free
andforced.
Free
vibrationstake
place when a system oscillates underthe
action ofinternal
forces,
in
the
absense ofexternally
impressed
forces.
The
system underfree
vibration vibrates at one or more ofits
naturalfrequencies.
Natural
frequencies
are properties ofthe
dynamical
system of some configuration
due
to
its
mass and stiffnessdistribution
anddamping.
Natural
frequencies
are a majordesign
consideration when a structureis
exposedto
dynamic forces.
Vibration
that
takes
place underthe
excitation of externalforces
is
called
forced
vibration.When
the
excitationis
oscillatory,
the
system
is
forced
to
vibrate atthe
excitationfrequency.
If
the
frequency
of excitation coincides with one ofthe
naturalfrequencies
ofthe
system,
a condition of resonanceis
encountered,
of some
large
structures
like
bridges,
buildings
and airplaneshave
occured
because
of resonance.Thus,
the
calculation ofthe
naturalfrequencies
is
of majorimportance
in
the
study
of vibrations.All
vibrating
systems are subjectto
damping
to
somedegree
because
ofthe
energy
dissipated
by
friction
and otherinternal
resistances.
If
the
damping
is
small,
it
has
very
little
influence
onthe
naturalfrequencies
ofthe
system,
andhence
the
calculationsfor
the
naturalfrequencies
canbe
made onthe
basis
of nodamping.
However,
the
importance
ofdamping
cannotbe
overlooked,
asit
plays a major role
in
limiting
the
amplitude of oscillation atresonance.
The
number ofindependent
coordinates requiredto
describe
the
motion of a system
is
calledthe
degrees
offreedom
ofthe
system.A
free
particleundergoing
general motionin
spacehas
three
degrees
offreedom.
A
rigidbody
has
sixdegrees
offreedom,
three
components of position and
three
anglesdefining
its
orientation.Furthermore,
a continuous elasticbody
requires aninfinite
number of coordinate positionsto
describe
its
motion;
hence
it
has
infinite
degrees
offreedom.
However,
in
many
cases,
parts of suchbodies
may be
assumedto
be
rigid,
andthe
systemmay be
consideredto
be
dynamically
equivalentto
onehaving
finite degrees
offreedom.
t19l
Oscillatory
systems canbe
broadly
classified aslinear
andand
there
is
a welldeveloped
mathematicaltheory
to
handle
their
differential
equations of motion.In
contrast,
techniques
for
the
analysis of nonlinear systems areless
wellknown,
anddifficult
to
apply.
All
systemstend
to
become
nonlinear withincreasing
amplitude of oscillation.
Nonlinearities
are alsointroduced
due
to
the
complexity
of connectionsbetween
various components.1.2
MOTIVATION
:The idea
ofthis
work wasto
study
the
technique
of modal analysiswhich
is
assuming
anincreasing
importance
in
the
design
andanalysis of structures.
A
Puma
Unimate
robot arm situatedin
the
R.I.T.
roboticslaboratory
was usedfor
the
case study.There
is
anextensive use of robots
today,
especially
in
the
automobile andcomputer
manufacturing
industries.
The
function
of robots onthe
assembly
line
requires ahigh
precision andaccuracy
whileturning
outlarge
volumes of production.The
role of vibrationsin
assembly
robots
is
very
criticalfor
both
precision and accuracy.A
typical
robot
has four
to
six motorsin
it.
If any
ofthe
naturalfrequencies
of
the
robot structureis
excitedby
the
frequency
of amotor,
it
could create
detrimental
effectsdue
to
resonant vibrations.For
example,
whileusing
a robotin
the
assembly
of a circuitboard,
a resonance could cause a misplacement of achip
from
its
desired
position.
As
anotherexample,
awelding
robot mightthrow
abead
ofweld on an undesired area.
If
the
naturalfrequencies
ofthe
frequencies.
Otherwise,
the
structure couldbe
modifiedto
shiftthe
natural
frequencies
away
from
the
operating
frequency
range.Although
in
the
presentstudy
the
interest
in
the
naturalfrequencies
of
the
robot waspurely
academic,
it
couldbe
used as a modelfor
actual problems
in
the
field.
Modal
analysis ordetermination
of naturalfrequencies
and modeshapes of
the
structure canbe done
in
two
different
ways-1.
In
experimental modal analysis or modaltesting,
a structureis
vibrated with a
known
excitation andthe
response atmany
locations
throughout
the
structureis
measuredusing
aFast
Fourier
Transform
analyzer.The
experimentally
obtainedFrequency
Response
Functions
are curvefittedusing
polynomialfunctions.
The
resulting
equations of curves arethen
comparedwith standard series
forms
ofFRF
to
determine
the
associatedcoefficients
known
as modal parameters and residues.The
modeshapes and values of natural
frequencies
anddamping
canbe
deduced
from
the
modal parameters and residues.Thus
it
is
a process ofdata
acquisition and analysis.2.
The
finite
element methodnumerically
implements
atheoretical
approach
to
vibration problems.In
this
method,
the
structureis
modeled
by
dividing
it
into
alarge
number oftheoretically
defined
elastic elements.Appropriate
boundary
conditions areanalysis.
This
method givesfrequencies
and mode shapes ofthe
structure.
Damping
is
aproperty
ofthe
structure andits
valuehas
to
be
inputted
in
the
model.In
general,it
is
very
difficult
to
estimate
the
damping
associated with various components.Hence,
for
simplicity
many
times
it
is
not consideredin
the
analysis.The
accuracy
of results obtaineddepends
onhow
closethe
modelrepresents
the
real structure.Modeling
skills and experience ofthe
analystplay
very
important
roles.In
the
presentwork,
the
above mentioned approaches are usedto
determine
the
naturalfrequencies
and mode shapes ofthe
robot armstructure.
Using
two
entirely
different
appproaches, an effortis
made
here
to
obtain correlated results.The
frequencies
arecorrelated
by
the
percentagedeviation
in
their
valuesin
the
corresponding
modes.The
mode shapes are comparedby
direct
visualcomparison and
the
Modal
Assurance
Criteria
(MAC)
technique
discussed
in
further
chapters.A
mention mustbe
made ofthe
fact
that
both
approaches are not exact sincethey
involve
many
assumptions and
limitations
suchas,
1
.The
structureis
assumedto
have
linear
stiffness anddamping
properties.
2.
The resulting
displacements
are assumedto
be
very
small.3.
The modeling
ofboundary
conditionsis
not exact.4.
The
process of measurement ofthe
responseinvolves
some1.3
MODAL
TESTING
:t2l
For
smooth and safe operation ofany
machine,
the
vibrationsin
the
structure mustbe
kept
to
a minimum.Vibrations
is
a majordesign
limitation
in
many
engineering
applications.Resonant
vibrationscreate excessive
motion,
noise anddiscomfort.
Hence it is important
that the
vibrationlevel be
monitored and controlled.There
are afew
methods
to
determine
the
resonant(natural)
frequencies
of astructure.
Experimental
study
of structural vibrationsis
one ofthe
most reliable methods
to
understandmany
vibration phenomenaencountered
in
practice.In
an experimentalstudy,
a structureis
vibrated with aknown
excitation.
The
corresponding
responseis
then
measuredusing
accelerometers and spectrum analyzers.
If
testing
is
done
underclosely
controlledconditions,
very
accurate anddetailed
information
will result.This
type
oftesting
comprising
the
response
data
acquisition andits
subsequent analysisusing
curvefitting
techniques
is
calledModal
Testing.
In
otherwords,
it is
a process
involving
testing
components or structures withthe
objective of
obtaining
a mathematicaldescription
oftheir
dynamic
oroscillatory
behavior.
For
modaltesting,
Structural
Measurement
Systems'
STAR
softwarewas used with a
Bruel & Kjaer 2032 dual
channelFFT
analyzer.STAR
using
GPIB
interfacing
software.A
B&K
8202
hammer
was usedto
impact
the
structure withknown
excitation and aB&K
4369
accelerometer
to
subsequently
pickup
the
response ofthe
structure.The
accelerometer was mounted onthe
structure with a stud atthe
driving
point.For
this
problem,
the
mass ofthe
accelerometer wasinsignificant
comparedto
mass ofthe
structure.Thus
it
did
not affectthe
response characteristics ofthe
structure.A
soft rubbertip
ofthe
hammer
was usedto
excite resonancesin
the
lower
frequency
range.1.3.1
APPLICATIONS OF MODAL TESTING
:There
aremany
applicationsto
whichthe
resultsfrom
a modaltest
may be
putto
use.In
allthe
cases,
a modaltest
is
undertakento
obtain a mathematical model represented
by
the
transfer
function
matrix.
The
transfer
function
is
a ratio ofthe
response ofthe
structure and
the
applied excitation as afunction
offrequency
co.This
theory
is
discussed
further
in
the
chapter on modaltesting.
Some
ofthe
applications ofthis technique
arelisted
below.
W
1.
Modal
testing
is
mostwidely
usedto
verify
the
naturalfrequencies
and mode shapes of a structure obtainedby
atheoretical
analysis.This
is
necessary
to
validatethe
modeldeveloped
using
finite
elements orany
other analytical method.For
afinite
elementmodel,
the
exact amount ofdamping
to
be
usedis very difficult
to
estimate.This
model canbe
modifiedby
using
the
mathematical modeldeveloped
from
the
experimentalresults.
Thus,
the
finite
element model canthen
be
usedfor
further
transient
orharmonic
response analysis.The
measurements should
be
carefully
obtainedto
insure
goodcorrelation
between
theory
andthe
experimental model sothat
causes of
any
discrepancies
between
them
canbe
easily
determined.
2.
Modal
testing
canbe
usedfor
substructuring
of an assembledproduct.
Components
which are unreachablein
anassembly
canbe
separately
tested
andtheir
precise mathematical model canbe
developed
by
substructuring.The
components canthen
be
incorporated
into
the
assembly.Thus
complex structures canbe
broken
into
their
basic
elements and problems canbe
simplified.In
the
substructuring
process,
the
componenttested
may
be
modifiedto
improve
the
structure andstudy
the
effects.A
very
accurate modelis
requiredin
this type
of analysis.3.
A
different
application alsoincorporating
the
modaltest
is
that
of
force
determination.
Situations
arise whereknowledge
ofthe
dynamic
forces
causing
vibrationis
desired
but
direct
measurement of
these
forces
is
not possible.Here,
a solutionis
offered
by
a processwhereby
measurements ofthe
responsecaused
by
the
forces
are combined with a mathematicaldescription
oftransfer
functions
ofthe
structurein
orderto
deduce
the
forces.
This
processis very
sensitiveto
the
accuracy
of
the
model usedfor
the
structure and soit is
essentialthat the
1.4
FINITE ELEMENT METHOD
:I14l
The
finite
element methodis
an analytical numerical procedurewhose active
development
has
been
pursuedfor
arelatively
shortperiod of
time.
The
basic
concept ofthe
method,
when appliedto
problems of structuralanalysis,
is
that
a continuum(the
total
structure)
canbe
modeledanalytically
by
its
subdivisioninto
regions
(the
finite
elements)
in
each of whichthe
behavior
is
described
by
a separate set of assumedfunctions
representing
the
stresses or
displacements
in
that
region.These
sets offunctions
areoften chosen
in
aform
that
ensurescontinuity
ofthe
described
behavior
throughout
the
complete continuum.If
the
behavior
ofthe
structure
is
characterizedby
a singledifferential
equation,
the
finite
element method represents an approachto
the
approximate solution ofthat
equation.If
the
total
structureis
heterogeneous,
being
composed ofmany
separate structuralforms
in
each of whichthe
behavior
is
described
by
adistinct
differential
equation,
the
finite
element approachis
stilldirectly
applicable.The
finite
element method requiresthe
formation
and solution ofsystems of algebraic equations.
The
special advantages ofthe
method reside
in
its
suitability
for
automation ofthe
equationassembly
and solution process andin
the
ability
to
representhighly
The
finite
element analysis ofthe
robot structureis
carried outusing
MacNeal-Schwendler
Corporation's
general purposefinite
element program
NASTRAN. The R.I.T.
version ofMSC/NASTRAN
does
not
have
a good preprocessor or a model generator.R.I.T. does have
Rand
Micas,
whichis
anotherfinite
element program available on anIntergraph
CAD
system.A
graphic model ofthe
robot armis
createdto
scale meshed with variousplate,
solid,
andbeam
elementsusing
Rand Micas.
Rand Micas has
atranslator
whichtakes the
graphicfile
and creates adata file
directly
readableby
NASTRAN.
Subsequently,
analysis
is
done
in
NASTRAN
using
its
modal analysis capability.Again,
postprocessing
ofthe
results or observation of mode shapesis done in
Rand Micas
by
the
use of aloader
program,
by
converting
the
numerical resultsinto
graphics.1.4.1
RANGE
OF
APPLICATIONS
:Applications
ofthe
finite
element method canbe
divided
into
three
broad
categoriesdepending
onthe
nature ofthe
problemto
be
solved.1.
The first category
consists ofthe
problemsknown
as equilibriumproblems or
time-independent
problems.The
majority
ofapplications of
the
finite
element methodfall
into
this
category.For
the
solution of equilibrium problemsin
solidmechanics,
it is
necessary
to
find
the
displacements
or stressdistribution
orperhaps
the
temperature
distribution
for
a given mechanical orthermal
loading.
Similarly,
for
the
solution of equilibriumvelocity
ortemperature
distributions
under steady-stateconditions.
2.
In
the
secondcategory
arethe
so-called eigenvalue problems ofsolid and
fluid
mechanics.These
are problems whose solutionoften requires
the
determination
of naturalfrequency
and modesof vibration of structures or problems of
buckling
of slendercolumns.
3.
In
the
third
category
is
the
multitude oftime
dependent
orpropagation problems of continuum mechanics.
This
category
is
composed of
the
problemsthat
result whenthe
time
dimension
is
addedto
the
problems ofthe
first
two
kinds.
1.4.2
ADVANTAGES OF
THE
FINITE
ELEMENT
METHOD
:Following
arethe
various advantages ofthe
finite
element method.1
.Highly
irregular boundaries
canbe
modeled withthis
method.2.
Variable
material or element propertiesthroughout
the
continuum canbe
modeled.3.
Element
size and number of nodes canbe
refinedfor
analysistype.
4.
Discontinuous
and mixedboundary
conditions canbe
applied.5.
Non-linearities
canbe included in
the
model.6.
This
methodis very
cost effective.7.
This
method canbe efficiently
computerimplemented.
The
limitations
ofthe
finite
element method are asfolllows
1.
User
musthave
extensiveknowledge
and experiencein
modeling
withfinite
elements.2.
Since
it
involves
many
approximations,
the
resultsmay
notbe
accurate.
3.
Exact
boundary
conditions aredifficult
to
representin
the
model.4.
Damping
is very difficult
to
model.5.
Problems
involving
nonlinearitiesin
material or structuralproperties are
very
difficult
to
solve withthis
method.6.
The availability
of computerfacilities
canbe
a majordrawback.
1.5
COMPARISON
OF
EXPERIMENTAL
AND
FINITE
ELEMENT
MODAL ANALYSIS
:Table
(1.1)
showsthe
comparison ofthe
experimental and analyticalFINITE ELEMENT
ANALYSIS
EXPERIMENTAL
MODAL
ANALYSIS
Mass,
damping
and stiffnessdistribution
properties ofthe
structure are assumed.
Actual
mass,
damping
andstiffness of
the
structure aretested.
Structural
geometry
is
represented
in
fine details.
Very
coarsegeometry
representation
ofthe
structure.Surface
andinternal
geometry
can
be
modeled.Only
surfaceis
accessiblefor
testing.
Large
number ofdegrees
offreedom
are considered.Number
of measurementdegrees
of
freedom
arevery
small.Material
properties are assumedor estimated.
Actual
material properties aretested.
Boundary
conditions canbe
modeled
to the best
assumptionActual
boundary
conditions orapproximate
free-free
conditions aretested.
TABLE
1.1
:COMPARISON
OF
FE AND
EXPERIMENTAL MODAL ANALYSIS
The Figure
(1.1)
t20'
comparesthe
different
approachestaken
by
the
Modal
Testing
andFinite
Element
Method
in
the
formulation
of amathematical model or
Modal
Model
andthe
subsequent analysisEXPERIMENTAL
FINITE
ELEMENT
MODAL ANALYSIS
ANALYSIS
Dynamic
Discretize
Testing
Model
\
/\/
FRF
Mass, Stiffness,
Data
Damping
Matrices
\
/
\/Curve
Fit
Solve Eigenvalue
Problem
\
/ \/
Modal
Mo del
\ /
\ /
\/
Local
Forced
Vibration
Modifications
Response
Absorbers
\ /
\
/Sensitivity
Animated
Analysis
Display
FIGURE
1.1
:PROCEDURES
FOR THE FINITE ELEMENT AND
1.6
MATHEMATICAL
MODELING
:All
systemshaving
mass and elastic properties undergofree
vibrations.Vibrations
withoutany
external excitationstake
place atthe
naturalfrequencies
ofthe
system.Natural
frequencies
arethe
frequencies
attributableto
various material properties andparticular
configurations
ofthe
elements ofthat
system.To
determine
these
frequencies
it
is
necessary
to
write equations ofmotion of
the
systemusing
basic
laws
ofmechanics,
namely
Newton's
secondlaw.
The
simplification ofthese
differential
equationsby
coordinatetransformation
gives riseto
an eigenvalueproblem.
The
solution ofthe
eigenvalue problem givesthe
naturalfrequencies
andthe
mode shapes ofthe
system.It
mustbe
statedhere
that
naturalfrequencies
aremainly
the
properties of mass andstiffness where
damping
plays a minor role.Damping
controlsthe
diminishing
amplitude of vibration.Hence
the
damping
term
is
oftenneglected
in
the
calculation of naturalfrequencies.
The degrees
offreedom
of a system with a nonuniformdistribution
of masses and stiffness are notfinite.
The
representation will notbe
correct unlessthe
motion ofevery
point ofthe
systemis known.
This
system couldbe
reducedby
considering
adiscrete
mathematical model with masses and stiffness as
lumped
parameters.
Thereby,
the
infinite degrees
offreedom
are reducedto
a
finite
number.Equations
of motion canthen
be suitably
expressedphysical systems can
be
representedby
simple one ortwo
degree
offreedom
mathematical models.The
dynamic
behavior
ofthe
system canbe
quantitatively
studied withthese
models.However
such asimplification
is
not possiblein
every
case,hence
it
mightbe
necessary
to
consider a modelhaving
larger
number ofdegrees
offreedom.
In
alinear
multi-degree offreedom
dynamic
systemthe
set ofequations of motion obtained
is
secondorder,
coupled,
ordinary
differential
equations with constant coefficients.It
is
difficult
to
solve
these
equationssimultaneously because
ofthe
coupling
terms
involved
(resulting
from
the
choice ofthe
coordinate system).But
for
an undampedsystem,
a set of principal coordinates exists whichallow expression of
these
equationsin
uncoupledform.
Hence
if
transformation
of coordinatesto
the
principal coordinatesis
carriedout, then these
equations canbe
separated.Each
equation canthen
be
solved
independently.
According
to
Meirovitch
I11!,
'the
whole process ofuncoupling
a set ofdifferential
equationsusing
transformation
to
natural coordinates andsolving
them
independently
andthen
expressing
the
solution of simultaneous equations as a
linear
combination of modalvectors multiplied
by
the
natural coordinatesis
known
asModal
Analysis.'
of motion
for
single and multi-degree-of-freedom systems anduncoupling
the
differential
equationsfor
multi-degree offreedom
system.
The
theory
is
very
standard and canbe
found
in
many
text
books. The
following
explanationis based
onthe
discussion
givenby
Meirovitch.
t12l
1.6.1
EQUATIONS
OF MOTION FOR SINGLE DEGREE OF FREEDOM
SYSTEM
:The
spring-mass-damper system shownin
Fig.
(1.1)
is
a simplemodel of a complex mechanical system.
It
is
also a simplegeneralization of a single
degree
offreedom
system.Consider
this
basic
modelto
further
derive
the
equations of motion of multipledegree
offreedom
systems.q(t)
I
>
mF(t)
Fs(t)<r
Fd(tK
>F(0
FIGURE
1.2
:SINGLE DEGREE OF FREEDOM SYSTEM
Consider
afree
body
diagram
whereF(t)
is
the
externalforce
andq(t)
the
displacement
from
the
equilibrium position.Fs
andF,j
arethe
resulting
restoring force
anddamping
force induced
in
the
spring
and
damper
respectively.Using
Newton's
secondlaw,
the
equation ofF(t)
-F8(t)-Fd(t)
=mq(t)
(1)
where
q
representsthe
time
derivative
ofthe
quantity
'q'.But
Fs
=k
q(t)
andFd
=cq(
t)
hence,
mq(
t)
+ cq(
t)
+k
q(
t)
=F(
t)
(2)
Which
is
a second orderlinear
ordinary
differential
equation withconstant coefficients.
1.6.2
EQUATIONS OF MOTION FOR
LINEAR
MULTIPLE
DEGREE
OF FREEDOM SYSTEMS
:This
section presents a mathematical model of amulti-degree-of-freedom system.
Consider
alinear
systemconsisting
of n massesmj
(i=1,..n)
connectedby
springs anddampers,
as shownin
the
figure (2.2).
Free
body
diagrams
associated withthe
typical
masses m;is drawn.
[131
Q
Q
c2Q
O
c3Q
Q
lJfjHJ}J>M>J>J. niiirfiiimimnri ntrtWitiitttnitk3(q3-q2)
7C3(q3-q2)4k2(Q2-qi)
>c2(q2-q1)
FIGURE 1.3
:MULTIPLE
DEGREE OF FREEDOM SYSTEM
Since
the
motiontakes
placein
onedimension,
q,
the total
number ofdegrees
offreedom
ofthe
system coincides withthe
number ofmasses n.
The
generalized coordinatesrepresenting
the
displacements
ofthe
massesmj
is denoted
by
qj(t)
(i=1
,2,...n).Applying
Newton's
secondlaw
to
writethe
differential
equation ofmotion,
q(t)
+c.+1[q.+1(t)-qi(t)]+k.+1[q.+1(t)-qi(t)]
-c.[q.(t) -qM(t)] -k.[q.(t)-q. -m.
q,(t)
(3)
Where
Qj(t)
representsthe
externally
impressed
force.
Equation
(3)
can
be
arranged asm.q.(t)
-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)
=
Q(t)
Next
using
double
indices
for
simplicity,
mij
=5ijmi
(
5jj
=0
i*j
=1
h )
cij
=kjj=
0
j
=1,2,
i-2,i+2,....,n
Cjj
=-q
kjj
=-kj
j
=i
-1
(5)
cy
=Cj
+ci+1
ky
=k,
+ki+1
j
=i
cy
=-ci
+1
kjj
=-ki+1
j
=i
+1
Where,
mjj,
Cjj,
andkjj
arethe
mass,
damping
and stiffnesscoefficients respectively.
Hence
the
complete set of equations of motion ofthe
system can nowbe
written asX[rr>..
qi.(t)
+c.jq.(t)
+k..q.(t)]=Q.(t)
wherei-1
,..,n(6)
Eq.
(6)
is
a set of n simultaneous second-orderordinary differential
equations
for
the
generalized coordinatesqj(t)
(i=1,2
n).The
mass,
damping
and stiffness coefficients are symmetric and canbe
arranged
in
the
symmetric matrixform.
Hence
we cansimplify
the
above notation
as,
[my]
=[m]
[Cjj]
=[c]
[ky]
=[k]
(7)
Arranging
the
generalizedcoordinates,
qj(t),
and generalizedimpressed
forces,
Qj(t),
in
the
column matrices{Qj(t)}
-{q(M
{Qj(t)>
={Q(t)}
(8)
Eq.
(6)
can nowbe
writtenin
the
compactform
as,
[m]{q(t)}
+[c]{q(t)}
+[k]{q(t)}
=These
arethe
equations
of motion of a multipledegree
offreedom
system
in
the
matrixform.
The
matrices[m],
[c],
and[k]
arethe
mass,
damping,
and stiffnessmatrices,
respectively.1.6.3
DECOUPLING
THE
EQUATIONS
OF
MOTION
:Consider
a general undamped mechanical multi-degree-offreedom
system.For
simplicity
ofanalysis,
we neglectthe
damping
term
in
the
equations of motionhere
onwards.Then
Eq.
(9)
reducesto,
[m]{q(t)}
+[k]{q(t)}=qt)
(10)
where
(Q(t)}
is
a column matrix whose elements arethe
ngeneralized
externally
impressed
forces.
For
the
purpose ofthis
discussion
considerthe
matrices[m]
and[k]
asarbitrary,
exceptthat
they
are symmetric andtheir
elements constant.The
columnmatrices
{q}
and{Q}
represent n-dimensional vectors of generalizedcoordinates and
forces
respectively.Thus Eq.
(10)
represents a set of n simultaneouslinear
second orderordinary
differential
equations with constant coefficients.These
equations
may
involve
coupled mass and stiffness matrices whichneed
to
be
decoupled
priorto
the
final
solution.To
facilitate
this,
the
equations of motion are expressedusing
adifferent
set ofgeneralized coordinates
vFj(t)
(j=1
,2,....,n).Linear
theory
statesthat
combination of
the
coordinates ^(t).!13!Hence
considerthe
linear
transformation,
(q(t)}
=[u]
pp(t)}
(11)
in
which[u]
is
a constant nonsingular square matrix.The
matrix[u]
can
be
regarded as an operatortransforming
the
vector{}
into
the
vector
{q}.
Since
[u]
is
constant, the
time
derivatives
ofEq.
(11)
give{q(t)}
=[u]{*(t)}
{q(t)}
=[u]{(t)}
(12)
So
that
the
sametransformation
matrix[u]
connectsthe
velocity
vectors
{}
and{q}
andthe
acceleration vectors{}
and{q}.
Substituting
Eqs.
(11)
and(12)
into
Eq.
(10)
yields[m][u]W)}+[k][u]mt)}
={Q(t)}
(13)
Next,
premultiplying
both
sides ofEq.(18)
by
[u]T resultsin
IM]W)}+[K]{T(t)}-{N(t)}
(14)
where
[M]
and[K]
arethe
mass and stiffness matricescorresponding
to the
coordinates*F:(t)
defined
by
[M]
= [u]T[m]
[u]
[K]
= [u]T[k] [u]
(15)
and are symmetric
because
[m]
and[k]
are symmetric.Moreover,
{N(t)J
= [u]T(Q(t)}
(16)
is
an n-dimensional vector whose elements arethe
generalizedIf
the
matrix[M]
is
diagonal,
then
system(14)
is
saidto
be
inertially
uncoupled.On
the
otherhand
if
[K]
is
diagonal,
then
the
systemis
saidto
be
elastically
uncoupled.The
object ofthe
transformation
(11)
is
to
producediagonal
matrices[M]
and[K]
simultaneously,
because
only
then
does
the
systemcompletely
decouple
and consist ofindependent
equations of motion.Hence
if
such a
transformation
matrix[u]
canbe
found,
then
Eq.
(14)
represents a set of n
independent
equations ofthe type
M^(t)+K.
y.(t)
=N.(t)
j=1,2,...n
(17)
Equations
(17)
have
the
same structure asthat
of an undampedsingle-degree-of-freedom system and can
be
very
easily
solved.A
linear
transformation
matrix[u]
diagonalizing
[m]
and[k]
simultaneously does
indeed
exist.This
particular matrix[u]
is known
as
the
modalmatrix,
sinceit
consists ofthe
modal vectorsrepresenting
the
natural modes ofthe
system.The
coordinatesvFj(t)
are called natural
(or
principal)
coordinates.The
procedure ofsolving
the
system of simultaneousdifferential
equationsby
meansof