Practical application of modal analysis techniques

Rochester Institute of Technology

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3-1-1992

Practical application of modal analysis techniques

James Perconti

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PRACTICAL APPLICATION OF

MODAL ANALYSIS TECHNIQUES

James A. Perconti

A Thesis Submitted in Partial Fulfillment

of the Requirements for the Degree of

Master of Science

in

Mechanical Engineering

R. G. Budynas

(Thesis Advisor)

H. Ghoneim

W. W. Walter

c.

W. Haines

(Department Head)

Department of Mechanical Engineering

College of Engineering

Rochester Institute of Technology

Rochester, New York

ACKNOWLEDGEMENTS

I

would

like

to

dedicate

this

work

to

the

memory

of

my

father,

Thomas J.

Perconti

Sr. It

was

his

exampleandencouragementwhich prompted me

to

enter

the

engineering field

initially

and

later

gave me

the

ambition

to

continue

to

graduate

level.

I

would also

like

to

express great

thanks to

my

wife,

Kathy,

and

son,

Jamie,

who

have

ABSTRACT

A

procedure

is

presented

for

applying

modern modal analysis

tools

in

the

area ofstructural

dynamic

design

andanalysis

for

the

purpose of

minimizing

the

required

development

time

necessary

to

bring

a product

to the

marketplace.

Development

time

is

minimized

by

reducing

the

number of

design, build,

and

test

iterations

required

to

bring

aproduct concept

into

production.

This

is

done

by

using

available modalanalysis

tools

in

a coordinated manor

to

generate a good

understanding

of

the

behavior

of

the

design in

the

first

development

iteration. The

finite

element method and experimental modal analysis areused

for design

optimization and

design

verification.

In

addition,

the

finite

elementmethod

is

used

to

provide

information

which willensure good

quality

results

in

the

experimental

stage.

The

experimental results are

then

used,

with

the

aid of modal correlation

techniques,

to

refine

the

finite

element modeland

to

eliminate

inaccuracies

in

the

model.

The

refined

finite

element model

is

then

used

to

predict

the

impact

of

changes

to the

design

whichare required

in

order

to

fulfill

the

design's

performance

requirements.

A

modal

test

structure

is

analyzedand

tested,

and

the

results are

TABLE

OF

CONTENTS

List

of

Symbols

vi

List

of

Tables

viii

List

of

Figures

ix

1.0

Introduction

1

2.0 Background

5

3.0

Theory

10

3.1

The

Finite

Element Method

10

3.2

Experimental

Modal Analysis

21

3.3

Correlation

of

Finite Element

and

Experimental Results

28

4.0

Practical Application

of

Structural

Dynamics

Analysis

34

4.1 The Design Process

34

4.2 Application

of

the

Finite

Element Method

Using

MSC/NASTRAN

37

4.3

Application

of

Experimental Modal Analysis

Using

SMS/STAR

42

4.4

Application

of

Modal Correlation

Using

KIT-MAS

48

5.0

Problem Development

and

Solution

54

5.1

Problem Description

55

5.2 Finite

Element

Model

and

Results

57

5.3

Preparation

for

Experimental

Modal

Analysis

70

5.4 Experimental Modal Analysis

and

Results

76

5.5

Modal Correlation Results

89

6.0 Discussion

H6

7.0 Conclusion

121

References

123

Appendices

Appendix 1

List

of

Symbols

m

Mass for

single

degree

of

freedom

system

k

Stiffness for

single

degree

of

freedom

system

c

Damping

for

single

degree

of

freedom

system

X

Displacement

X

Velocity

X

Acceleration

t

Time

F(t)

Forcing

function

*0

Displacement

magnitude

CO

Natural

frequency

[M]

Mass

matrix

[C]

Damping

matrix

[K]

Stiffness

matrix

M

Displacement

vector

W

Velocity

vector

{*}

Acceleration

vector

im)

Forcing

function

vector

{*,}

Eigenvector

or mode shape vector

A

Eigenvalue

[D]

Dynamical

matrix

s

Laplace

variable

[X(s)}

Displacement

vector

in Laplace domain

[B(s)]

[(')]

Pk

pi

M

a

M

Wl

W

hi

Qk

mt

[I]

System

impedance

matrix

Transfer

matrix

Transfer function

between

points

i

and

j

kjh

Pole

Complex

conjugate of

kth

pole

Matrix

of residuesassociated with

k^

pole

Matrix

of residues

(complex conjugate)

Damping

coefficient

Complex

displacement

vectoror mode shape

Complex

mode shape

(complex conjugate)

Matrix

ofresidues associatedwith

ktn

pole

[rk\

=

2j[Ak

J

Matrix

of residues

(complex

conjugate)

[r*

J

=

2J[-\

J

Scaling

constant

Mass scaling factor for

im

mode shape

Identity

matrix

List

of

Tables

Table

Page

1

Natural

Frequencies,

FEM Results

60

2

Summary

of

Displacement Response

Data

75

3

Natural

Frequencies,

Experimental

and

FEM Results

79

4

MAC Matrix

102

5

Orthogonality

Matrix,

(FE

vs.

FE)

104

6

Orthogonality

Matrix,

(Exp.

vs.

Exp.)

105

7

Orthogonality

Matrix,

(Exp.

vs.

FE)

106

8

Orthogonality

Matrix,

(Exp.

vs.

Exp.),

Normalized Vectors

107

9

Orthogonality

Matrix,

(Exp.

vs.

FE),

Normalized

Vectors

108

10

Improved

MAC

Matrix

113

List

of

Figures

Figure

la

Single Degree

of

Freedom System

lb

Forces

Acting

on

Mass

(m)

2a

Two Degree

of

Freedom System

2b

Forces

Acting

on

Masses

3

Modal

Test Structure

4

NASTRAN

Model,

Undeformed Shape

5

Mode

1,

110.8 Hz

6

Mode

2,

137.1 Hz

7

Mode

3,

140.9 Hz

8

Mode

4,

245.2 Hz

9

Mode

5,

266.4

Hz

10

Mode

6,

366.0 Hz

11

Mode

7,

394.2 Hz

12

Mode

8,

522.8

Hz

13

Mode

9,

660.1 Hz

14

Possible Transducer Locations

15

X

Displacement

Response

vs.

Frequency

16

Y Displacement Response

vs.

Frequency

17

Z

Displacement

Response

vs.

Frequency

18

STAR

Model,

Undeformed

Shape

19a

Mode

1,

Experimental

Results

19b

Mode

1,

FEM

Results

Figure

Page

20a

Mode

2,

Experimental Results

81

20b

Mode

2,

FEM

Results

81

21a

Mode

3,

Experimental Results

82

21b

Mode

3,

FEM Results

82

22a

Mode

4,

Experimental Results

83

22b

Mode

4,

FEM Results

83

23a

Mode

5,

Experimental Results

84

23b

Mode

5,

FEM

Results

84

24a

Mode

6,

Experimental Results

85

24b

Mode

6,

FEM

Results

85

25a

Mode

7,

Experimental

Results

86

25b

Mode

7,

FEM

Results

86

26a

Mode

8,

Experimental

Results

87

26b

Mode

8,

FEM

Results

87

27a

Mode

9,

Experimental Results

88

27b

Mode

9,

FEM Results

88

28a

Mode

1,

FE

and

Experimental

Mode

Shapes

91

28b

Mode

1,

Mode

Shape Difference

91

29a

Mode

2,

FE

and

Experimental Mode Shapes

92

29b

Mode

2,

Mode

Shape Difference

92

30a

Mode

3,

FE

and

Experimental

Mode Shapes

93

30b

Mode

3,

Mode Shape

Difference

93

31a

Mode

4,

FE

and

Experimental

Mode

Shapes

94

Figure

Page

32a

Mode

5,

FE

and

Experimental Mode Shapes

95

32b

Mode

5,

Mode Shape Difference

95

33a

Mode

6,

FE

and

Experimental Mode Shapes

96

33b

Mode

6,

Mode Shape Difference

96

34a

Mode

7,

FE

and

Experimental

Mode

Shapes

97

34b

Mode

7,

Mode

Shape

Difference

97

35a

Mode

8,

FE

and

Experimental Mode

Shapes

98

35b

Mode

8,

Mode

Shape

Difference

98

36a

Mode

9,

FE

and

Experimental Mode

Shapes

99

36b

Mode

9,

Mode Shape Difference

99

37

MDI Vector

1,

Difference in

Flexibility

Estimates

100

38

MDI Vector

2,

Reciprocal

of

Flexibility

Difference

101

39

MDI Vector

3,

1.0

Minus CO-MAC

101

40a

Mode

2,

FE

and

Improved Experimental Mode Shapes

111

40b

Mode

2,

Mode Shape Difference

111

41a

Mode

3,

FE

and

Improved Experimental Mode Shapes

112

1.0 Introduction

Due

to

the

competitivenature of

the

world

economy,

manufacturers are now

being

forced

to

develop

their

products

in

amuch shorter

time

period

than

was

the

case several years ago.

Development

cycles which

took

as muchas

five

to

seven years are now

being

shortened

to

as

little

as

two

years.

In

addition,

product

quality,

and

reliability

requirements

have

increased

and acceptable unit

manufacturing

costs

have decreased.

In

short,

engineering

teams

must

develop

better

designs

than

in

the

past

but

spend

less

time

doing

it.

Companies

whichare able

to

do

this

can capture a generous market share

by

beating

their

competitors

to the

market place and

keep

this

market share

by

introducing

improvements

to their

product

lines

periodically.

In

the

past,

product

development

programs

have

often

been

multi-phased

in

order

to

give

design

engineers

the

opportunity

to

refine

their

designs iteratively.

A

multi-phased program

may have involved

the

design

and

fabrication

of

breadboards,

feasibility

models,

engineering

modelsand

finally

production prototypes of a

product.

Each

of

these

phases

involved

design,

procurement,

fabrication,

assembly

and

testing

with

the

later

phases also

including

the

design

and

fabrication

of

production

tooling.

In

order

to

decrease

the

development

time

for

a

product,

development

programs now often

have only

two

phases.

In

the

first

phase

fully

functional

engineering

models are

designed,

built

and

tested.

Based

on what

is

learned from

the

engineering models,

the

designs

are modified where

necessary,

frozen

and released

for

production.

In

order

for

this

to

be

successful,

the

engineering

models must

be

quiterefinedso

that the

risk

associated with

taking

this

design

into

production

is

minimized.

Therefore,

the

design

engineer

is

required

to

productionmodel

in

the

first design iteration. To do

this,

the

engineermustuse all

the tools

available

to

him

to

ensure

this

happens.

In

today's

engineering

environment

this

means analytical

modeling,

usually

by

computer,

design

optimization,

and

then

verification

by

testing

the

performance of

the

resulting

design. The

resultsof

testing

can

then

be

correlated

to the

analytical model

to

determine

what

design

modificationsare

necessary

to

meet

the

required

performance.

This

thesis

deals

with

this

concept

in

the

areaof mechanical

engineering,

specifically in

the

areaof structural

dynamics.

The

structural

dynamics

analyst

has

some

very

powerful

tools

available

for

design

and

design

verification of structural systems.

Finite

elementanalysis

is

a

widely

acceptedmethod

for

analysis of

design

concepts and

design

optimization.

This

approach

is

particularly

powerful

for analyzing

conceptsand

designs

during

early design

stagesof a

development

program when no

hardware

is

available

for

testing.

However,

for

complex

structures,

inaccuracies

in

the

mathematical model

caused

by

difficulties in modelling details

such as

boundary

conditions or material nonlinearities can produce

deceptive finite

element results.

Therefore,

experimental modalanalysis

techniques

are used

to

verify

the

dynamic

properties of

a

design

once

hardware is

available

for

testing.

Differences

which exist

between

the

FEM

and experimental results will

identify

performancecharacteristicsof

the

design

whichwere not predicted

by

the

FEM

results.

These

can

be

classified

into

two

categories: attributes which

adversely

effect

the

performanceof

the

design,

and

those

which

do

not.

If

the

dynamic behavior

does

not produceperformance

compromises,

then

the

engineer

does

not need

to

investigate any further

and

the

design

can

be

considered adequate.

However,

if

the

performance

is

compromised,

of

the

system.

The difficult

part

is

determining

whataspectsof

the

design

need

to

be

changed.

If

reliable experimental

data is

obtained,

the

experimentaland analytical results can

then

be

correlated

to

determine inaccuracies

in

the

finite

element model.

Methods

of

performing

this

correlation

have

currently been

a research

topic

in dynamics

analysis and

commercially

available

tools

are now

becoming

available

for

this

purpose.

Using

the

information

obtained

through

correlation

techniques,

the

finite

element model can

be

modified

to

more

accurately

represent

the

actual

design

and

then

used

to

predict what modifications are

necessary

to

eliminate

design

problems.

In

order

to

demonstrate

the

capabilities of

currently

available

tools

in

the

areaofstructural

dynamics,

a

test

structure was chosen

to

be

analyzed

using

finite

element analysis as well as experimental modal analysis.

This

structure,

described

in

detail in later

chapters,

was chosen

to

illustrate

the

difficulties in measuring

highly

coupled vibrational modes which were

known

to

be

present.

It

was

thought

that the

coupledmodes would

be

a morestringent

test

of

the techniques

described

in

this thesis

as well as

for

the

computational

tools

used

to

carry

out

these

The

choiceof which

tools

would

be

used

to

carry

out

this

investigation

was

made

primarily

through

current

availability

at

Rochester

Institute

of

Technology.

MacNeal-Schwendler Corporation's MSC/NASTRAN for finite

element analysis

and

Structural Measurement

Systems'

SMS/STAR

for

experimental modal analysis were

both

available at

RIT.

In

addition, the

KIT-MAS

package

by

Kensinger

Integrated Technologies

Corp.,

used

for

performing

modal

correlation,

was

currently

planned

for installation

at

RJT. The

KIT-MAS

program

is

designed

to

use results

from

the

MSC/NASTRAN

and

SMS/STAR

packages.

The

goalsof

the

work

described in

this thesis

were

threefold.

First

the

KIT-MAS

package

had

to

be

installed,

and understood.

The

operation of

the

KIT-MAS

program was

learned

and

demonstrated

by

analyzing

a simple rectangular plate.

The

second goal was

to

analyze a more complex

three

dimensional

problem and

obtain good correlation

between

the

finite

element and experimental modal

analysis.

Finally,

by

performing

the

sequence of steps

described

above,

the

practical application of available

tools

used

to

minimize

the

required

design

time

with respect

to

a structural

dynamics

design

problem would

be

demonstrated

and

documented.

Although

the

work

described

here

wouldnot

be

used

to

meet

the

requirements of a

2.0

Background

The

power of

the

digital

computer

has

made possible

the

creation of

many

analysis

tools

which

simplify

the

analysisof

many engineering

problems which would

have

been nearly

impossible

to

solve withoutcomputers.

Through

the

use of

numericalmethods and

the

computers on which

they

are

implemented,

complex

problemswhich

may

have

taken

years

to

solve,

now can

be

solved

in only

minutes or

hours.

The finite

elementmethod

is

an excellent example of one of

these tools.

By

describing

a complexstructure

mathematically,

the

behavior

of

the

structure can

be

simulated

by

the

digital

computer.

The design

engineer can

then

determine

whether

the

subject structure will

adequately fulfill

the

intended

need even

before

it is

built.

Through

the

use of

engineering

tools,

an engineer can

quickly

completea

design

which

has

a

high

probability

of

fulfilling

its

expectations,

verify

the

performance of

the

design

after

it is

built,

and gain

information

about

how

the

design may be further

improved. The

alternative

is

anexercise

in

trial

and error and

involves

building

many

versions ofa

design in

order

to

seewhich works

best.

Clearly,

the

former

technique

is

the

only

means

by

which

the

engineerand

the

company he

works

for

will

be

successful.

This

section

is intended

to

give some

background information

as

to the

originof some of

these tools

and

the

capabilities which

they

possess.

The

fundamental

concept on which

the

finite

elementmethod

is based is

the

idea

of

representing

a

geometrically

complex

domain

as a collectionofsimple

subdomains.

The

subdomains,

or

finite

elements,

can

be easily described

mathematically

and

the

full

complex

domain

can

be

described

by

assembling

the

finite

elements.

The

origin of

this

concept

is

unknown

but it

was recorded

that

polygon of a

finitely

large

numberofsides.

The

evolutionof

this

concept

into

matrix mathematics occurred

through the

1940's,

50's

and

60's

and

there

is

a

large

amountof

literature

_whichreviews

this

_{evolution as well as}

the

basic

theory

of

the

method. * J

The

implementation

of

the

technique

into

acomputerprogram was

sponsored

by

the

National Aeronautics

and

Space

Administration

(NASA)

during

the

late 1960's

andwas released

into

public

domain in

1969

through the

Computer

Software Management

and

Information

Center

(COSMIC).

This

versionof

NASTRAN

(NAsa

STRuctural

ANalysis)

is

called

COSMIC/NASTRAN.

Another

proprietary

version of

NASTRAN

wasreleased

in

1971

and

is

developed

and

maintained

by

the

MacNeal-Schwendler

Corporation

(MSC).

MSC/NASTRAN

became

the

acceptedversion

due

to

its

wide usage and advanced

features. MSC has

continually

enhanced

the

program and

is

now

up

to

version

67.

Since

the

release of

these

finite

element

programs,

many

companies

have

released versions of

finite

elementsoftware

for

useon

every

computer

from

mainframes

to

microcomputers

making FEM

the

most

widely

accepted analysis

technique

for

structural statics and

dynamics.

MSC/NASTRAN

was chosen

for

the

work

described here due

to

its

availability

at

Rochester

Institute

of

Technology,

and

its

compatibility

withanother

software package which will

be described

shortly.

The

program was used

to

mathematically determine

the

naturalor modal

frequencies

andassociated mode shapes

for

the

structure

in

question.

In

addition,

the

response of

the

structurewas

determined for

given

inputs in

order

to

choose

the

best

position

to

locate

the

transducer

for

the

experimentalportionof

this

work.

Since

the

finite

element

model providesagreat

deal

of

information

about

dynamic

behavior

of

the

structure,

it

was alsoused

to

determine

where

to

locate

measurement points.

These

provide good results without

having

to

repeat experimental work.

Although

only

the

vibrationalanalysis capabilitiesof

the

program were used

for

this

work,

the

program

also

includes

statics

analysis,

buckling

analysis,

heat

transfer analysis,

aeroelasticity

andsomeoptimization.

Origins

of modern experimental modalanalysis

techniques

can

be

traced to

the

early 1960's

and

the

development

of

the

tracking

filter for

obtaining

frequency

response

function

measurements.^]

Obtaining frequency

response

functions

from

a structure

is

fundamental

to

modal analysis.

The

modalproperties of a structure are obtained

from

the

frequency

response

functions

as will

be described in

the

next section.

Although

the

tracking

filter

made

these

measurements

possible,

the

filters

could

only be

used

to

measureanarrow

band,

slowly

varying

signal.

This

resulted

in

the

almostexclusiveuse of swept sine

inputs

for

determining

the

frequency

response of a system.

This

was a

very

time

consuming

procedure

in

which

the

experimentalist

had

to

construct

the

frequency

response

function

from

the

input

and output signals

he/she

measured.

In

the

early 1970's

the

advent of

the

digital

FFT

analyzer

greatly

increased

the types

of excitationsources which could

be

used

to

measure

the

frequency

response

functions

ofasystem.

Since

the

frequency

response

is

the

Fourier

transform

of

the

output of a system

divided

by

the

Fourier

transform

of

the

input

of

the system,

the

FFT

analyzer couldgive

the

frequency

response

function

directly

as

long

as

the

input

andoutput are

Fourier

transformable.

Virtually

any

physically

possible

input

satisfies

this

restriction.

This

greatly

reduced

the time

required

to

perform modal measurements and

lead

to

the

more modern modal

analysis

techniques

used

today.

The final

piece of

the

modern modalanalysis

test

all

the

computational

tasks

required

to

extract

the

modal parameters

from

the

frequency

response

functions

and

display

the

results graphically.

The

packageused

for

this

work was

Structural Measurement Systems STAR

_{system used with an}

HP

FFT

analyzer.

Again,

this

packagewas chosen

due

to

its availability

and

its

compatibility

with anothersoftware package.

Once

the

finite

element analysis and experimentalmodalanalysis are

completed

for

a

structure,

the

natural

frequencies

and mode shapes

from

the two

sourcescan

be

compared.

Unfortunately,

there

usually

are

differences

between

the

two

setsofresults.

Although

the

frequency

differences

are

easily

noted,

they

don't

givemuch

information

about

why

the

differences

exist.

Much

more

information

exists

in

the

mode shapes

but

since

these

are motions of

the

analyzed

structure,

again

it is difficult

to

determine

the

cause of

the

differences

just

by looking

at

the

modeshapes.

Because

of

these

difficulties,

correlation

techniques

have been

developed

and software packages written

to

perform correlations

between

the

finite

elementand modal results.

Work

at

Rochester Institute

of

Technology

by

Krebs

has

produced acorrelation program called

Modal.

P]

The Modal

programworks

withresults

from

the

NASTRAN

and

STAR

programsand computes

the

mode shape

difference

as well as

the

orthogonality between

the two

sets of results.

Kensinger Integrated Technologies

Corp.'s

KIT-MAS

program was used

to

perform

the

correlation

in

the

work

described here.

The KIT-MAS

program

inputs

the

results

from

NASTRAN

and

STAR,

performs

any necessary

coordinate

transformations,

reduces

the

data

to

a common set of

degrees

of

freedom

and

then

performs

any

requested correlations.

KIT-MAS

performs severalcorrelation

difference,

andmodal

difference

index. Some

of

the techniques

were

found

to

be

more useful

than

othersas will

be

discussed

in

later

sections of

this thesis.

Difference

vectors

resulting from

the

difference

correlation

technique

can

be

transferred

back

to the

STAR

program

for display.

By

using

apackage such as

KIT-MAS,

one can

hopefully

determine

the

causes of poor correlationand

modify

the

finite

element model

to

obtain

better

correlation.

Once

this

is

done,

the

finite

elementmodel

better

represents

the

real world structureandcan

be

used

to

determine

whatchanges need

to

be

made

to the

design

in

order

to

eliminate

design

3.0

Theory

3.1

The Finite Element Method

The finite

element method

has

become

a

very

popular method

for

solving

complex problems

for

which

it is

notpractical

to

derive

a closed

form

solution.

Through

the

use of

the

finite

element

method,

geometrically

complex

domains

are

broken up

into

simple subdomains or

finite

elements.

The finite

elements are

connected

to

each other at points called node or grid points.

These

are

the

points at

which

the

problem solution will

be

calculated.

The

propertiesof

the

elements are

described

by

approximate

functions

called

interpolation functions. The

interpolation function

used

for

any

particular

finite

element

is

based

on

the

geometry

of

the

element,

the

number ofnodes and

the

loading

condition

the

element

is

to

model.

For

example,

the

interpolation function

used

to

describe

tension

or compression

in

a

simple,

one

dimensional bar

element gives

displacement

asa

linear

function

of position

along

the

bar. Elements

used

to

modelmore

complex

geometry

and

loading

conditions

have higher

order

interpolation

functions.

The

interpolation functions

approximate

the

response

(displacement,

for example)

at a given grid point

based

on

the

value at

the

othergridpoints

in

the

element.

Once

the

elementsare assembled

into

the

original

domain,

the

resultantsystem of

equations are solved

simultaneously

to

yield

the

overallproblemsolution.

Although

this

has

been

a

very

simplified

discussion

of

the

general

theory

of

the

finite

element

In

order

to

understand

the technique

with which

the

finite

element method

determines

the

natural

frequencies

and mode shapes ofa

structure,

let

us

first

consider some simple

dynamical

systems.

The first step is

to

write

the

equations of

motion

for

the

system.

Figure la

showsasingle

degree

of

freedom

system.

In

order

to

formulate

the

equationsof

motion, the

forces

whichacton

the

mass

(m)

must

be

determined. Figure

lb

shows

the

forces

whichacton

the

mass

(m)

when

it is

displaced from

its

rest position.

In

the

figure

the

mass

has

been displaced

to the

right

(positive

x)

and

it is

assumed

that the

displacement,

velocity

and acceleration

are all positive.

>

F(t]

Figure la: Single Degree

of

Freedom

System

kx

ex

>

F(t)

By

applying Newton's

second

law

of motion

for

the mass,

we obtain

the

familiar

second order

linear

ordinary

differential

equation

mx+cx

+kx

=F(t)

/]\

The

solution

to

equation

(1)

willgive

the

position of

the

mass asa

function

of

time

for

all

time

t.

However,

since

in

modal analysis we are

only

interested

in

finding

the

resonance

frequencies

and mode shapes

for

the

structure,

weneed

only

consider

the

transient

solution

to the

problem.

This is

given

by

the

solution

to

mx+cx+

kx

=

0.

(2)

Next,

negligible

damping

is

assumed

(c

=

0). This

is

done

to

simplify

the

problem

and

is

validsince we are

only

interested

in

the

natural

frequency

and mode shape

information for

the

problem.

Once

the

solution

is

found for

the

undamped

case,

the

effect of

damping

can

be

determined

as shown

in

reference

[5].

Assuming

the

motion

is

harmonic,

then the

solution

to

equation

(2)

can

be

given

by

x(t)

=

x0ei'

(3)

Substitution

of equation

(3)

back into

equation

(2)

with

damping

equal

to

zero

yields

-ma)2x0eim+kx0eiat=0

^

which

has

the

solution

a=

The quantity

co

is

the

undampednatural

frequency

of

the

system.

For

the

single

degree

of

freedom

(SDOF)

system

there

is only

one natural

frequency

(or

resonance).

Mode

shape

information for

the

SDOF

system yieldsnouseful

information

since

the

mode shape

describes

the

relativemotion of

degrees

of

freedom

within

the

systemand

this

system

has only

one

degree

of

freedom.

The

procedure

for solving

amultiple

degree

of

freedom

system

is

the

same as

the

single

degree

of

freedom

system.

Figure 2a

shows a

two

degree

of

freedom

system made

up

of

two

masseseachable

to

move

linearly. Figure 2b

shows

the

forces

whichact on

the

masses when

displaced from

their

rest positions.

Figure 2a: Two Degree

of

Freedom System

kixl'

ciV

m,

F,(t)

F,(t)

As

before,

the

equation of motion can

be

written

for

each mass

by

applying Newtons

second

law

ofmotion.

This

yields

n\x\

-i-^

+

Cz

)ix

-c2x2

+

(^

+

*2 )xl

-k2x2

=

Fl (t)

mjXj

-CjXj

+

c2x2

-hjX^

+

k2x2

=

F2 (t)

This

systemofequationscan

be

written

in

matrix

form

as

follows:

(6)

m,

0

m-,

(Cl+c2)

-c7

{k.+k,)

-k2

-k.

(7)

This

procedurecan

be

used

for

any

numberof

degrees

of

freedom

to

obtain

the

equations of motion of a

system;

one equation

for

each

degree

of

freedom.

The

system ofequations can

then

be

written as

the

general matrix equation

[M\{X}

+

[Q{X}

+

[K\{X}

=

{F(t)}

(8)

where

[M]

[Q

[*]

{*}

{*}

{F(t)}

is

the

symmetric system mass matrix

is

the

symmetric system

damping

matrix

is

the

symmetric system stiffness matrix

is

the

system

displacement

vector

is

the

system

velocity

vector

is

the

system acceleration vector

is

the

fencing

function

As

before,

the transient

solution

to the

systemof equations

is

used

to

find

the

natural

frequencies

(eigenvalues)

and mode shapes

(eigenvectors)

for

the

problem

and

the

damping

is

assumed

to

be

zero.

Setting

the

forces

and

damping

to

zero

[m]{x}

+

[K]{x}

=

{o}

Once

again,

if

the

motion

is

harmonic,

then the

solution

to

equation

(9)

is

{X}

={X0}eu*

where

0)

(9)

(10)

Substituting

this

matrixequation

back into

equation

(9)

yields

-O)2[m]{x0}

+

[k]{xq}

=

{0}

(n)

By

premultiplying both

sides

by

the

inverse

of

the

stiffnessmatrix weget

-o)2[k]-1[m]{xq}+{x0}

=

{o}

(

(12)

which can

by

rewritten as

[Z)]{^0}

=

A{^0})

(13)

[D]

=

[K]'l[M]

(14)

(15)

Equation

(13)

is

the

classical representation of aneigenvalue problemwhere

the

eigenvalues are represented

by

A,

and eigenvectors arerepresented

by

the

displacement

vector

{Xq}.

The

solution

to this

equation will yield

the

same number

ofeigenvalue-eigenvectorpairs as

there

are

degrees

of

freedom in

the

system.

The

eigenvaluesgive

the

natural

frequencies

of

the

systemand

the

eigenvectors

give

the

corresponding

modeshapes.

Once

the

eigenvalues andeigenvectors

have been

using

modal summation

techniques.

The

effectof

damping

can also

be included

in

this

analysis.

A

thorough

discussion

of

these techniques

can

be found

in

reference

[5].

Although

there

are

many

techniques

for solving

the

eigenvalueproblem

stated

above,

the

easiest

to

understand usesequation

(11)

rewritten as

follows:

[K-G)fM]{xi}

=

{0}t

(16)

where

{Xj}

is

the

mode shape

corresponding

to

coj.

If

the

displacement

vector

is

zero,

we

have

the trivial

solution.

Nontrivial

solutionsare obtained

by

forcing

the

condition

that

\K-(D2M

=0

(17)

which gives one equation

for

each eigenvalue of

the

system.

The corresponding

eigenvectors can

then

be found

by

substituting

the

eigenvalues

back

into

equation

(16)

and

solving

for

each eigenvector.

The

technique

for

determining

the

solution

to

an eigenvalue problem

is

straightforward,

however due

to the

large

amount of

computations required

to

evaluate

the

determinant in

equation

(17)

the

method

is

not practical

for

use with

finite

element models.

Other

methods

implemented

on a

digital

computer will save

time

and cost when

solving

this type

ofproblem.

Version

66

of

MSC/NASTRAN

gives

the

user sixchoices ofmethods

for

the

solutionof

eigenvalue problems.

Five

of

these

methodsextractrealeigenvalues

from

equation

(16).

These

include

the

Inverse, Givens,

Modified

Givins,

Householder

and

Modified

Householder

methods.

The

sixth

method,

called

the

Lanczos

method,

uses a

block

shifted

iteration

algorithm

to

solve

the

eigenvalue problem.

Each

of

The Inverse

(INV)

method obtainseigensolutions

by

iterations

based

on

the

equation

[*-A,.Af]{jr,+1}

=

[M

]{*,}

^

(18)

where

Af

is

anestimateof

the

eigenvalue.

[6]

This

method

is best

suited

to

large

problemswith sparse matriceswhere

only

a

few

eigenvectorsare

desired.

The Givens

(GIV)

and

Householder

(HOU)

methods

first decompose

the

mass matrixas

[M]

=

[L][L]T

f

(19)

where

[L]

is

a

lower

triangular

matrix.

By

premultiplying

equation

(16) by

[L]"land

substituting for

[M]

using

equation

(19)

the

following

is

obtained:

[Lf[*]{X}

-a>\L\-\L][L]T

{X}

=

{0}

.

(20)

By

using

the transform

$}=[lY[x)

t

(21)

equation

(20)

can

be

writtenas

[J-<D2I]{X}

=

{0}

?

(22)

where [jMlVIkIlT1

(23)

and

[7]

is

the

identity

matrix.

The

[J]

matrix

is

then

transformed to tridiagonal

form

eigenvalues and eigenvectors.

The

physical eigenvectors

(mode

shapes)

are

recovered

by

performing

the

inverse

ofGivens'

transformation

and

the

inverse

of

the

transformation

given

in

equation

(21). The

mass matrix must

be

non-singular

for

this

method

to

be

successful since

the

decomposition

of

the

mass matrix or

the

inversion

of

the

resulting lower

triangular

matrix will

fail for

this

case.

The Modified Givens

(MGIV)

and

Modified

Householder

(MHOU)

methods use a similar method

but

decompose

the

shiftedmatrixas

["J:

[2s:

+

Ajm]

=

[l][l]t

_

(24)

Note

that

equation

(16)

can

be

rewrittenas

[k

+

Xsm-(X

+

Xs)m]{x}

=

{0}

^

(25)

where

k=(p.

Multiplying by

_-1/(A,+XS)and substitution

from

equation

(24)

yields

M

-LLT

(X

+

Xs)

W

=

{o}

(26)

Premultiplication

by

[L]"l

and

the

substitution

{x>}

=

[Lf{x}

A

+

/l-(30)

s

The

rest of

the

procedure

is

the

same as

Givens

or

Householder

methods.

MGIV

and

MHOU

methods allow

the

massmatrix

to

be

singular

if

there

are

compensating

terms

in

the

stiffness matrix.

The

GIV,

HOU,

MGIV,

and

MHOU

methodsare

best

suited

to

small

problems,

or

for

large

problemswhere

many

eigenvectors are

required,

after

dynamic

reduction

(described

later in

this

section) has

been

performed.

The Lanczos

method uses a

block

shifted

iteration

algorithm.

Sets

of vectors

obtained

by

a recursive

form

are used

to

reduce

the

problem

to

areduced

block

triangular

form.

The

eigensolutionsare computed

in

the

reduced

basis

and

then

back

transformed to the

original

basis. This

is

the

most modernmethod and should

be

considered

for large

problems.

Since

this

method

takes

advantageof

sparsity

of

the

input

matrices,

it is

mosteconomical when used without

dynamic

reduction.

As

was mentioned

above,

the

solution

to the

eigenvalue problem

in dynamics

gives as

many

mode shapes and

corresponding

natural

frequencies

as

there

are

degrees

of

freedom

in

the

model.

Generally,

when

building

a

finite

element

model,

hundreds

or even

thousands

of

degrees

of

freedom

arecreated

in

order

to

describe

complex geometries

for

statics analysis.

However,

determining

all

these

modes of

vibration

is

notpractical nor

is

this

information

needed

in

most

engineering

problems.

Usually

the

lowest frequencies

are of most

interest

and

the

modes of

vibration

below

aspecified

frequency

aresolved

for. The implication is

that the

finite

element model contains more

detail

than

is

needed

to

describe

the

dynamic

dynamics

analysis,

a

technique

called

dynamic

reduction

is

used

to

reduce

the

complexity

of

the

model.

Dynamic

Reduction

reduces

the

numberof

degrees

of

freedom in

the

model prior

to

computation ofeigensolutions

in

order

to

make

eigenvalue extraction

less

costly.

Using

dynamic

reduction provides a cost savings as

compared

to

analyzing

the

full

model or

constructing

smaller

models,

yet retains

high accuracy for

the

frequency

range of

interest.

MSC/NASTRAN

offers

two

methods of

dynamic

reduction.

Although

notgiven

here,

a complete

derivation

of

these

methods

is

given

in

the

program

documentation.

[8],[4]

The

first

method,

called

Guyan Reduction

or

Static

Condensation,

involves

choosing

a reduced set of

degrees

of

freedom for

which

the

solution will

be

calculated.

By

reducing

the

numberof equations

the

problem

is

more

easily

solved while most of

the

accuracy

is

maintained.

However,

the

choice of which

degrees

of

freedom

to

retain can effect

accuracy

and

the

use of

this

method

is

not advised

for inexperienced

users.

Generalized Dynamic Reduction

(GDR)

involves producing

aset of

displacement

vectors which are"rich"

in

the

lowest

eigenvectors of

the

system.

The

program uses

an

inverse

iteration

method

to

generate

the

displacement

vectors which are

themselves

a

linear

combination of

the true

eigenvectors.

The displacement

vectors

areused

to

define

atransformation

between

the

physical

displacement degrees

of

freedom

and a set of modal coordinates.

The

eigenvalue problem

is

then

defined in

the

reduced vector space and

the

solution

is

found using

aneigenvalue extraction

routine.

This

method eliminates

the

need

to

pick which

degrees

of

freedom

will

be

used

to

find

the

solution and

is

usually

more accurate

than

Guyan Reduction.

In

summary,

it is

possible

to

determine

the

natural

frequencies

and

corresponding

mode shapes of a

generally

complexstructure

by

treating

it

as a

problem

down into

manageablepiecescalled

finite

elements.

The

finite

elements

are connected at grid

points,

where

the

mass

is

concentrated.

In

doing

this

it is

easy

to

describe

the

distribution

of massand stiffness

by

using

matrices.

By

assembling

the

elements

into

the

original

structure,

the

matrix equation which

describes

the

dynamic

motion of

the

structure

is

approximated.

Solution

of

this

equation

by

eigenvalue

extraction,

yields

the

natural

frequencies

andmode shapes

for

the

structure.

32 Experimental Modal Analysis

Experimental

modal analysis

is

the

process of

determining

the

dynamic

properties ofastructure

by

exciting

the

structure

in

acontrolledmanor and

measuring its

response.

The

response and excitation signals are analyzed

by

an

FFT

analyzer

in

order

to

determine

the transfer

function

which

defines

the

interaction

between

the

excitation

input

pointand

the

response point.

This is done

at enough

points on

the

structure

to

map

out

the

dynamic

properties,

or modal

parameters,

of

the

entire structure.

The

transfer

functions,

or

frequency

response

functions,

contain

the

required

information

needed

to

determine

the

modalparametersof

the

structure.

The

modal parameters

frequency,

damping

and modeshapeare all

contained

in

the

functional

expression

for

the transfer

function

and

therefore

can

be

determined

by fitting

this

expression

to the

experimental

data.

Finally,

a computer

is

used

to

perform

the

requiredcurve

fitting

to the

frequency

response

functions.

Resulting

mode shapesare

displayed

by

the

computer.

This

section

describes

the

As

was

described in

section

3.1,

the

differential

equations of motion

for

a

multiple

degree

of

freedom

system can

be

expressed

by

the

matrix equation

[M]{X}

+

[C]{X}

+

[K]{X}

=

{F}

(31)

By

taking

the

Laplace

transform

of

both

sidesof equation

(31)

and

assuming

all

initial

conditionsare zero we obtain

(s2[M]+s[c]

+

[K]){x(s)}

=

{F(s)}

(32)

Define

the

system

impedance

matrix

[B(s)]

=

S2[M]

+

s[C]+[K]

(33)

such

that

[5(*)]{X(s)}

=

{F(s)}

(34)

Premultiplying

by

the

inverse

of

the

impedance

matrix yields

{x(s)}

=

[B(s)]~l{F(s)}

(35)

Defining

the transfer

matrixas

[H(s)]

=

[B(s)Y1

(36)

gives

{X(5)}=[H(j)]{F(j)}

(37)

Notice

that the transfer

matrix gives

the

response of

the

system

for

a given

input

to

X^s)

X2(s)

= .Xn(*).

hn(s)

hn(s)

*2l(*)

A22(5)

hnl(s)

*u(0

*(0

F2(-0

(38)

Each

elementof

the

transfer

matrix

is

a

transfer

function

which

describes

the

response atonepointof

the

system

for

a given

input

at anotherpoint

in

the

system.

Each

of

the

transfer

functions is defined in

terms

of

the

Laplace

variable

(s)

and

is

a

complex valued

function.

One

of

the transfer

functions

can

be

expressed as

follows:

K(s)

=

Fj(s)

(39)

The

transfer

function

hu(s)

defines

the

response at point

i,X[(s),

for

an

input

excitation

Fi(s)

at point

j. Notice

that

by

using

an

FFT

analyzerand

measuring

a

disturbance

at point

j

and

the

response at point

i,

this transfer

function

can

be

measuredexperimentally.

Likewise,

all of

the

elements of

the transfer

matrix can

be determined

experimentally.

The

key

to the

use of modal analysis

using

transfer

functions is

to

express

the

transfer

matrix

in

terms

of

the

modalparameters:

frequency,

damping

and mode

shape.

The

following

derivation

provides

this

expression.

Recall from

equation

(33)

that

the

elements of

the

system matrixarequadratic

functions

of

the

Laplace

variable

(s).

The

transfer

matrix

is

the

inverse

of

the

systemmatrix.

The inverse

of

a matrix

is

given

by

[AT

=

det[A]

*ti[A]

(40)

adM

=

[c,f

(41)

and

Qj

is

the

cofactor of

a{\

in

the

matrix

[A]. Since

the transfer

matrix

is

the

inverse

of

the

system

matrix,

the

elementsof

the transfer

matrix are ratios of

polynomials.

The

numerator ofeach elementwill

be

acofactor of

a{\

and will

have

order

2n-2

where n

is

the

number of

degrees

of

freedom in

the

system.

The

denominator

of each will

be

the

determinant

of

the

system matrix and will

have

order

2n. An

elementof

the transfer

matrix can

by

writtenas:

h

""

+b2s

"~1+"'+b2n-ls+b2n-2

^

det[B(s)]

(42)

The determinant

of

the

system matrix

is

the

characteristic equation

for

the

system

and can

be

expressed as

the

product of

its

roots.

The

roots are called polesof

the

transfer

function.

When

the

system

is

subcritically

damped,

the

poles are complex

numbers and occur

in

complexconjugate pairs.

Equation

(42)

can

be

rewritten as:

_

b

tj

2n~2

+b2s2""14-

+b

2n_xs+

b

2n_2

"

A(S-Pn){s-Pn}"iS-Pl)(S

'Pi)

(43)

whereA

is

a

constant,

and/?^ is

the

km

pole.

If

the

roots of

the

determinant

[B(s)]

are

distinct,

then

the transfer

matrix can

be

rewritten

in

partial

fraction form

as:

[ff(-)]=X

k=i

s-pk

s-pk

(44)

where

[^4^1

is

the

matrix of residues

for

the

km

pole.

At

the

locations (s

=p])

structure.

Each

complex conjugatepair ofpoles

is

associatedwith a resonance or

mode ofvibrationof

the

structure.

They

are given as:

Pk=-ok

+

\(ok

Pl=-Vk-]k

(45)

where

o^ is

the

damping

coefficient,

and

cojj

is

the

natural

frequency.

The

modal

vectorsor modeshapes

(u^)

aresolutions

to the

homogeneous

equation

[B(P*)]M

=

{0}

(46)

It has been

shown

[9]

that

when

they

are

defined in

this

way,

the transfer

matrix can

be

rewritten as

fr(0]

= *=i

{*}{"*}

_,

{*}{*}

s-pk

S-Pk

(47)

where

the

mode shape vectors

u^

are complex valued and u

is

the

complex

conjugate.

Note

that

in

equation

(47)

the transfer

matrix

has

been

expressed

in

terms

of all

the

modal parameters.

It is

important

to

note

that

each row and column

of

the

numerators of matrices of equation

(47)

contains

the

same vectormultiplied

by

a component of

itself.

This

is

easy

to

see

by

performing

the

outerproductof

the

vectors

in

the

numeratoras

is

shown

in

equation

(48).

This is

the

most significant premiseofmodal

testing.

Only

onerow or column of

the

transfer

matrix needs

to

be

measured

in

order

to

determine

all

the

modal

parameters ofastructureas

long

as

the

following

assumptions aremet:

1.

The

motion

is linear

and

is described

by

the

linear

second-order

equations.

2.

The symmetry

of motion

property

or

reciprocity property is

valid

(B

and

H

matricesare symmetric).

3.

No

more

than

one modeexists at each pole

location

of

the

system

transfer

matrix.

4.

Modes

are

defined in

a global sense

(mode

shapes are

defined for

all

degrees

of

freedom

of

the

systemand

their

frequency

and

damping

don't very significantly from

one part of

the

structure

to

another).

Now

that the transfer

matrix

has

been

expressed

in

terms

of

the

modal

parameters,

curve

fitting

can

be

used

to

determine

the

modal parameters

from

experimental

data.

However,

the

STAR

package uses a

slightly

different form

of

equation

(44)

to

determine

the

modalproperties of a system.

This

equation

is

given

by:

[#(*)]

=X

*=

2](s~pk)

2j(s-p*k)

(49)

where

hi

rk=rlk+ir2k

(50)

In

this

form

the

amplitude of

the

residues

is

more

directly

related

to the

impulse

response

function. The

STAR

program offers several

different

methods of curve

fitting

for different

situations.

An

exampleof one of

these

is

the

single

degree

of

freedom

polynomialcurve

fitting

algorithm.

The

method

fits

the

following

equation

to the

experimentally

measured

frequency

response

functions

for

each

individual

mode:

(<rk

+

ork-(o2+2}(Tka))

(51)

The

first

term

in

equation

(51)

is

equivalent

to

equation

(49)

for

a single modeand

the

other

terms

are

included

to

compensate

for

the

effect of other modes.

Equation

(51)

is fit

to

the

measured

data in

a

least

squares error sense

to

determine

the

modal parameters

frequency,

damping

and complex residue.

Mode

shapesare

obtained

from

the

residue matrixas was shown

previously

using

the

equation

[rk]

=

Qk

{"*}{"*}

}

(52)

where

Qfc

is

an

arbitrary scaling

constant.

Other

curve

fitting

methodsare also

provided

to

obtain

better

curve

fitting

results

for

cases such as

high

modal

coupling

between

modes.

This

section

has

shown

the

mathematical

basis behind

experimentalmodal

analysis

techniques.

In

order

to

gain an appreciation

for

the

level

ofcomputation

involved in

this

technique,

one needs

to

examine equation

(51).

As

was

previously

generate

the

all

the

elementsof

the

complete matrix.

By fitting

equation

(51)

to

a

measured

frequency

response

function,

the

frequency, damping

and one element of

the

complex residue matrix

is

obtained.

In

order

to

obtainall

the

residues,

the

equation

has

to

be fit

to

a

frequency

response

function for every degree

of

freedom

which

is

measured

in

the

system.

In

addition,

this

needs

to

be

repeated

for

each

modeofvibration since equation

(51)

is

a single mode model.

For

a

relatively

small

model of

100

degrees

of

freedom

and

10

modes which are of

interest,

the

algorithm

needs

to

fit

the

equation

1000

times,

and

this

doesn't include processing

the

FFTs

needed

to

obtain

the

frequency

response

functions.

With

this

in

mind,

it is

easy

to

see

why

the

adventof

the

FFT

analyzer and

the

processing

power of

the

microcomputer

has

made

this technique

a reality.

3.3

Correlation

of

Finite Element

and

Experimental Results

As has been

shown

in

the

previous

sections,

the

natural

frequencies

and

associatedmode shapes of a

dynamical

system can

be

determined analytically

as

well as experimentally.

Although

this

providesa check of either

technique,

it does

notprovide

information

as

to

where

discrepancies

exist

between

the two

sets of

results.

If differences

exist

between

the two

solution

sets,

one needs

to

determine

which solution provides a

better description

of

the

dynamic

properties of

the

system

being

studied.

This

information

is

needed

in

order

to

providea

basis for

determining

the

adequacy

of a

design

or

to

support

engineering

decisions

required

to

implement

design

changes needed

to

ensure

the

systemmeets

the

design

requirements.

In

addition,

if

the

reasons

for

the

differing

resultscan

be determined

predict

the

impact

ofpossible

design

changes.

Needless

to

say,

better

techniques

than

comparing

natural

frequencies

or

observing

different

versions ofaparticular

modeshape areneeded

for

determining

differences

between

experimental and

analyticalresults.

Some

of

the techniques

which

have

become

popular

for

this

purpose are

described

in

this

section.

In

order

to

make comparisons

between

mode shape

vectors, the

vectorsmust

first be

referenced

to the

same coordinate system and

the

geometry

of

the

vectors

must

be

matched.

The

Kensinger Integrated

Technologies'

Modal Analysis System

(KIT-MAS)

program offers a coordinate

transformation

utility

to

perform

the

required

transformation.

This

is

not

necessary if

the

analyticaland experimental

models were

both

set

up

using

the

same coordinate system.

After both

vectors are

in

the

samecoordinate

system,

commonpointsof

the two

vectors are

determined.

Since

the

finite

element model

usually has

many

more points

than the

experimental

model,

points

in

the

finite

elementmodel without counterparts

in

the

experimental

model are eliminated.

The

mode shape

is

then

represented

by

the

remaining

points.

Points

do

not

have

to

be

coincident

to

be

matched and

the

KIT-MAS

program

provides

the

user selectable

tolerance

which

determines

the

maximum

distance

the

points can

be

separated and still

be

matched.

Once

this

step

is

completed,

the

mode

shapevectors can

be

compared

both mathematically

aswellas

visually

by

animating

the

mode shape.

Since

the

mode shapes are now represented

by

the

same

degrees

of

freedom,

it is

much easier

to

make comparisons

between

the

two

models

to

determine

where

differences

exist.

The

KIT-MAS

program offers several mathematical

techniques

for

first

method

is

called

Modal

Assurance Criteria

(MAC).

The MAC

provides a

measure

for

determining

whatextent

two

mode shapes arecorrelated.

The

MAC is

calculated

using

the

following

formula:

MAC({Z}A{Z}B)=|

|Wa'Wb|2

[{'K-WaIwI-Wb]

(53)

The MAC

can

take

value

between

zeroandonewhere one

indicates

exact

correlation.

The

closer

the

MAC

is

to

a valueof

one,

the

more similar

the two

vectors are.

The KIT-MAS

program calculates

the

MAC between

all

the

vectors

in

the two

sets and uses

this

information

to

determine

whichmode shape vectors are

related.

The

user

then

has

the

option

to

ehminate vectors which

have

not

been

correlated

to

a counterpart

in

the

other

data

set.

Rigid

body

modes produced

by

finite

element programsare an example of vectors which would

be

eliminated.

Once

this

calculation

is

completed,

each mode shape

in

the

finite

element solution

has

been

paired with

its

counterpart

in

the

experimentalsolution and

the

relative

similarity between

them

has

been

computed

through the

MAC.

Calculating

the

difference

vector

between

two

mode shapesprovidesan

excellent qualitative measure of

the

level

of correlation

between

the two

vectors as

well as

insight

as

to

where

the

vectors

differ.

Once

vectors arescaled

in

the

same

manor,

the

vector

difference

is

calculated

using

the

equation

{*y}Difrercnce=WA~{-y}B

.

(54)

When

the

difference

vector

is

animated,

it

is easy

to

see

differences between

the two

the

vectors.

The

KIT-MAS

program provides

the

difference

calculation,

as well as

the

originalcorrelatedvectors.

These

can

be

animated

by

the

SMS/STAR

program.

The orthogonality

check

is

anothermethod

the

KIT-MAS

program uses

to

determine

the

degree

of correlation

between

eigenvectors.

The

eigenvectors

found

in

the

solution