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CALCULATED STRUCTURAL RESPONSE USING

A "REDUCED" FINITE ELEMENT MODEL
presented by

Mark Norman Pickelmann

has been accepted towards fulfillment
of the requirements for

MASTER DEW degree in JIECHANIQAL
ENGINEERING

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CALCULATED STRUCTURAL RESPONSE USING

A "REDUCED“ FINITE ELEMENT MODEL

BY

Mark Norman Pickeleann

A THESIS

Submitted to

Michigan State University

in partial fulfilleent of the requireeents

for the degree of

MASTER OF SCIENCE

Department of Mechanical Engineering

1982

ABSTRACT

CALCULATED STRUCTURAL RESPONSE USING
A “REDUCED” FINITE ELEMENT MODEL .J
BY

Mark Norman Pickelmann

The use of finite element models for engineering
design has grown rapidly in the past few years. These
models are useful tools for predicting the behavior of
systems long before the system is actually constructed.
The resulting models, however, are often quite large,

requiring hours of computer time to use.

This thesis demonstrates that a finite element
model can be reduced for the purpose of calculating
structural response. This reduction is done
systematically so that the model is transformed into a
set of first order ordinary differential equations.
These equations are solved and used to calculate

frequency responses.

This reduction offers considerable time and cost
savings over computing the response directly from the

finite element model.

CALCULATED STRUCTURAL RESPONSE USING

A “REDUCED” FINITE ELEMENT MODEL

Submitted by:

Mark N. Pickelmann
M.S. Candidate

Approved by:

 

 

James E. Bernard
Associate Professor
Thesis Advisor

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Engineering

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5/x'7/527.
Date

"~/ 7 ~82/
Date

5-47-52
Date

To my loving wife Kristi

iii

AKNONLEDGEMENTS

I would like to thank Dr. James E. Bernard, my
major professor, for his help, friendship, and advice

over the last two years.

Also, I wish to thank Dr. Clark Radcliffe for all
of his help on the Oldsmobile project and for letting me

use his computer to print my thesis.

Many thanks to Don Hine, of Oldsmobile, for all

his help on the project.

Finally, I would especially like to thank my mom
and dad, Mary and Russell Pickelmann, for having me in

the first place.

iv

CHAPTER
CHAPTER
CHAPTER
CHAPTER

CHAPTER

TABLE OF CONTENTS

Flat-RES COCO-CC.ICIICCCCCIOIOICICOO-IOICCOOCI

1

VOU-FNN

APPENDIX A

INTRDMTIN 0....-ICC-OOICOIIIIIIOIOII

SMTLRK mm .COCIIOOOIOOOUIOICCIIC

UNCOUPLING OF MODAL EQUATIONS .........

FORCING FUNCTION ......................

fiSWSE Ckalm COCO-ICOIOIl-CIOCI

OLDSMOBILE PROJECT ....................

CONCLUSIONS

NOMENCLATURE

LISTmmRMES III-I....-I-...IIII-I...........-.

vi

15

22

26

8

44

LIST OF FIGURES

FIGURE 1 - FREQUENCY RESPONSE OF A UNDAMPED SYSTEM .. 11
FIGURE 2 - FREQUENCY RESPONSE OF A DAMPED SYSTEM .... 12
FIGURE 3 - FFT OF AN INPUT FORCE .................... 36

FIGURE 4 ‘ CALCULATED FREQUENCY RESPONSE ............ 37

vi

CHAPTER 1

INTRODUCTION

The use of finite element models for engineering
design has grown rapidly in the past few years. These
models are useful tools for predicting the behavior of

systems long before the system is actually constructed.

For complex structures, the finite element model
can become quite large, requiring a large computer for
the calculations. Even with the computing power of large
computers the finite element models require a great deal
of computing time. However, for the purpose of
calculating the frequency response of a structure in a
specified frequency range, the finite element model can

often be reduced so that the calculation can be done on a

PAGE 1

CHAPTER 1

minicomputer. This thesis is concerned with the process
of reducing a large model to a smaller one for the

purpose of such a frequency response calculation.

Chapter 2 explains the formulation of a large
finite element model, and its transformation into a modal
model. The modal representation has a coupled
differential equation and associated mode shape for each
degree of freedom in the original finite element model.
Chapter 3 presents the rational for the reduction of the
modal model. Chapter 4 introduces the forcing functions
so that frequency response can be calculated, and the
calculation of the response from the reduced model is
presented in Chapter 5. Chapter 6 presents some details
of a project where structural responses were calculated
by this method. A summary of the assumptions used in the
analysis are reviewed in chapter 7 along with some of the

advantages to this method.

PAGE 2

CHAPTER 2

STRUCTURAL MODEL

The goal of the analysis discussed here is to
develop an analytical model of a structure which predicts
the response of that structure to forces of a given
frequency range. The starting point of the analysis is a
finite element model of the structure, which yields

equations of the form.

[MJ{N} + [C3{i} + llw [DJ{i} + [K]{X} = {F} exp(i w t)

Equation 1

PAGE 3

CHAPTER 2

Since our main interest is the calculation of
frequency response, the forcing function has been assumed
to be harmonic. However it could be any forcing function

provided it can be expressed as a Fourier series.

In the structural model developed here dissipative
forces arise from two different sources, viscous damping
and structural damping. The damping forces which are
proportional to velocity are classified as viscous
damping. Viscous damping occurs when molecules of a
viscous fluid rub together, causing a resistive friction
force that is proportional to, and opposing, the velocity

of an object moving through the fluid.

Damping forces which are proportional to
Displacement are classified as structural damping.
Structural damping may be viewed as a sliding friction
mechanism between molecular layers in a material. The
friction force is proportional to the deformation or
displacement from some equilibrium point with an
orientation opposite the relative velocity. Imagine a

rod made up of a bundle of axial fibers. The siding

PAGE 4

STRUCTURAL MODEL

friction force between each fiber and its neighbor will
increase as the rod is bent and the fibers are pinched
together. This pinching phenomena occurs in most
materials as the various molecular layers slide past one

another [1] [2].

A complex structure such as an automobile includes
several sources of dissipation. The shock absorber,
whose design mission is to provide damping, is closely
approximated by a viscous model. But important
dissipation occurs in mounting elements such as coil
springs and rubber mounts as well. Tests indicate that
the dissipation of a spring is most closely approximated
by a structural damping model. Tests done on rubber
mounts indicate a combination of viscous and structural
dissipation is needed to adequately model the

dissipation.

For the problems of concern here, we will assume
the structure is lightly damped, resulting in small but
non zero dissipation forces. Since the total dissipation

is small, the natural frequencies and mode shapes of the

PAGE 5

CHAPTER 2

structure can be determined from the mass and stiffness
matrices. But the amplitude of the forced response of

the structure depends on the damping as well.

In general, the matrices in Equation 1 are not
diagonal. Therefore the solution of one equation depends
on the solution of others and the system of equations is
said to be coupled. The size of the matrices depends on
the number of elements in the finite element model and
the number of degrees of freedom of each element. The
structure discussed as an example is modeled by 500
elements, each with six degrees of freedom, thus Equation
1 would include 3000 coupled equations. It is desirable
to simplify the model in such a way as to make the

response calculation more convenient.

The procedure which leads to a simplified model

begins with the equations of undamped free vibration.

PAGE 6

STRUCTURAL MODEL

[M] {I} + [K] (x; a {0} (2)
Hhere:

[M] and [K] are n x n matrices

Equation 2 is formulated from Equation 1 by neglecting
the damping matrices and setting the force vector to

zero. A solution for {X} may be found in the form

{X} B {A} exp(i w t) (3)

Using Equation 3 in Equation 2 results in

2

( -w [M] + [K] ) {A} exp(i w t) I {0} (4)

Rewriting Equation 4 defines the eigenvalue problem

[K] {A} = X [M] {A} (5)

CHAPTER 2

The solution of Equation 5 results in a set of n
eigenvalues xi. If these are distinct, as is the usual
case, there will be a corresponding unique set of n
eigenvectors {A}i. Since [M] and [K] are symmetric and

positive definite, both the eigenvalues and eigenvectors

are real. The eigenvectors are used to form two
transformation matrices [U] and [U]T where the
eigenvectors {A}i make up the columns of [U]. The

transformation matrix [U] is used to define a modal

coordinate Y

{X} 8 [U] (V) (6)

when the relationship from Equation 6 is substituted into

Equation 1, which is then premultipled by [UJI we get

[Mmlle + [CmJ{Y} + l/w [Dm]{Y} + [Km]{Y} a {Fm} (7)

This coordinate transformation uncouples the mass

and stiffness matrices, but in general does not uncouple

STRUCTURAL MODEL

the damping matrices [3]. Equation 7 is called the
"modal model“. The modal model is a set of n coupled
second order ordinary differential equations where n is
the dimension of Equation 1. Each coordinate Yi of the
modal model is associated with one natural frequency and

its corresponding mode shape or eigenvector.

The steps described above are usually done by the
finite element programs on large computers. The output
from the finite element program would be the
transformation matrix [U], the diagonal modal mass matrix
[Mm], the diagonal modal stiffness matrix [Km], and the

damping matrices [Cm] and [Dm].

The damping matrices can be thought of as a
coupling by which energy can flow from one mode to
another. The damping can then be thought of as an input
force. This can be seen by rewriting Equation 7 in the

form

[Mm] {Y} + [Km] {Y} . {Fm} - [Cm] {9} - 1/» [Dm] {9} (8)

PAGE 9

CHAPTER 2

The fact that the {Fm} are harmonic dictates that the
velocities {Y} are also harmonic at the same frequencies.
Thus, it is convenient to think of the right hand side

of Equation 8 expressed as {Fm}‘exp(iwt).

[Mm] {Y} + [Km] {Y} a {Foi‘oxpti w t) (9)

The solution to Equation 9 is shown in Figure 1.

Of course, in order to calculate a response the
damping must be included in the left hand side of
Equation 8. But the introduction of this small amount of
damping will limit the peak amplitude but will not
drastically alter the basic chatacter of the frequency
response as shown in Figure 2. Thus if the frequency of
the force is near a natural frequency, the response of
that mode will be large. By knowing the frequency range
over which the forcing function is active, the modes

which heavily participate in the response can be

PAGE 10

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STRUCTURAL MODEL

identified. Those which do not participate are
eliminated by dropping the associated modal coordinate
from Equation 7 and the mode shape vector from the

transformation matrix [U].

The number of modes has been reduced by detemining
which modes fall significantly outside of the frequency
range of the forcing function. A rule which is often
used is to keep modes whose associated natural frequency
is less than twice the maximum frequency of the force

[4].

Since the number of modes which have meaningful

participation in the response may be a great deal smaller

than the number of degrees of freedom, the size of

Equation 7 and the transformation matrix [U] can often be
substantially reduced. In our previous example there
were 500 elements each with six degrees of freedom
resulting in 3000 equations. If, for example, only 100
of the 3000 natural frequencies are determined to
meaningfully participate in the response, we can reduce
the size of the matrices from 3000 x 3000 to 100 x 100

without significant loss in accuracy.

PAGE 13

CHAPTER 2

At this point the problem has been substantially
reduced. But the equations are still coupled in the
damping matrices and thus the response calculation is not

in a convenient form. needs further attention.

PAGE 14

CHAPTER 3

UNCOUPLING OF MODAL EQUATIONS

Chapter 2 showed that the n degree of freedom
finite element model could be cast in the form of the
modal model of Equation 7 and that the modal model could
be reduced by eliminating the modes whose participation
in the response was determined to be insignificant. The
reduced modal model is then a set of k coupled second
order differential equations. The fact that k<<n
facilitates the solution. However, since the equations
are still coupled in the damping matrices, they are not
in a convenient form for solution. In this chapter a
coordinate transformation will be introduced to uncouple

the modal equations.

In some cases the modal equations are uncoupled by

neglecting the off diagonal terms in the damping

PAGE 15

CHAPTER 3

matrices. This assumes that the off diagonal terms have
a small effect on the response of the system. To
uncouple the modal equations without having to make this
assumption, a more general approach will be taken. To do

this the reduced form of Equation 7 is written.

[MmJ(Y} + [Cm]{Y} + l/w [DmJ{Y} + [Km]{Y} = {Fm} (10)

The matrices of Equation 10 are of dimension k. Before
this set 'of equations can be uncoupled the l/w which
multiplies the structural damping matrix must be
eliminated. This is done by assuming a solution for {Y}

of the formESJ

{Y} a {B} exp(i w t) (11)

Taking the first derivative with respect to time yields

{Y} = i w {B} exp(i w t) (12)

PAGE 16

UNCOUPLING OF MODAL EQUATIONS

substituting Equation 11 into Equation 12 yields

{9} - i w {Y} (13)

Following the procedure of [5], Equation 13 is then

substituted into Equation 10 which results in

[Mm]{Y} f [CeltYI + ( [Km] + i [DI] ){Y} ‘ {Fm} (14)

Next, consider the homogenous solution i.e.,

{Fm} - {0}. The following solution for {Y} is assumed

{Y} = {C} exp(i t) (15)

PAGE 17

CHAPTER 3

Taking the first and second derivatives with respect to
time of Equation 15 and substituting them into Equation

14 will result in

( izrnaz + x [Cm] + {[Km] + i [Dell ) {C} - {0} (16)

Equation 16 has a nontrivial solution only if the
determinant of the coefficient matrix is zero. This
results in an algebraic equation of order 2k in X (k
being the dimension of the matrices). This equation will
result in a set of 2k x solutions. with each eigenvalue
xi there is an associated eigenvector {C}i, and both are
complex. In the case where the system is modeled with
viscous damping only, the eigenvalues and eigenvectors
would appear as complex conjugates, but with structural
damping in the system the pairs are rotated so they are
no longer conjugates [7]. This means that the
eigenvalues and eigenvectors cannot be used to solve the
transient problem but the solution must be of the form of
Equation 15 £63. The forced vibration problem however

can be solved by uncoupling Equation 14.

PAGE 18

UNCOUPLING OF MODAL EQUATIONS

The eigenvectors can be used to form the k x 2k
rectangular modal matrix [V]. But since, this modal
matrix [VJ cannot be used as a transformation matrix of

the form

(V) 8 [V] {Z} (17)

to obtain a solution to the nonhomogeneous problem. The
reason is that there are 2k modes and consequently 2k
coordinates 21, and only k coordinates Yi' This
difficulty can be overcome by introducing a set of.
auxiliary variables and converting the set of k second
order ordinary differential equations into an equivalent
set of 2k first order ordinary differential equations
known as Hamilton's Canonical Equations (8]. The
auxiliary variables are the modal velocities {Y}. The
modal coordinate v and the modal velocities § define a

set of new variables P in the following way

-—PAGE 19

CHAPTER 3

P. a i i - i,k
(18)

Pi+k 3 Yi 1 I 1,k

P is then used into Equation 14 to formulate the

following:

\

{P} + {P} =

[03 [Mm] -[Mm] [OJ {0}
Mm] [Cm] [OJ ([KmlfiIDmJ) {Fm}

Equation 19

which can be written as

(up: {5} + [Kp] {P} a {Fp} (20)

PAGE 20

UNCOUPLING OF MODAL EQUATIONS

Setting {Fp} - {0} defines the homogeneous problem

which is similar to Equation 2 and can be solved in a
similar fashion. The result is a set of 2k complex
eigenvalues and a corresponding set of 2k eigenvectors.
As before, the eigenvectors are used as the columns of
T

the transformation matrix [V] and as the rows of [V].

Equation 21 defines a new coordinate Z

{P} = [V] {Z} (21)

which is substituted into Equation 20 and the result is

premultiplied by [V]: leading to

[M2] {2} + [K2] {2} a {F2} (22)

This coordinate transformation uncouples both the
mass and stiffness matrices. Equation 22 is now a set of

2k uncoupled ordinary first order differential equations.

PAGE 21

CHAPTER 4

FORCING FUNCTION

In the preceeding chapters a finite element model
was reduced first to a coupled second order modal model
and then to a first order uncoupled modal model, which is
the left hand side of Equation 22. Before Equation 22
can be solved the right hand side of the equation must be
defined. Two ways of doing this will be discussed in

this chapter.

In order to calculate a response to a forcing
function first the function must be defined. In the
analysis here the forcing function will first be defined
for the structure and then undergo the same
transformations as the finite element model. The forces
which act on the structure can be either measured or

analytically determined forces.

PAGE 22

 

FORCING FUNCTION

Analytical functions can be any periodic forcing
function limited only by the frequency restrictions used
to reduce the equations. They could be obtained by
simulating components of the system and calculating the
forces which would be transmitted to the structure, or
the frequency range of interest may be spanned for the
purposes of computing frequency response. Once the
functions have been calculated or defined they are

transformed into the frequency domain.

Test data can be obtained by measuring the forces
which would be transmitted to the structure. This
testing can be carried out in two ways: 1) measuring the
forces on a prototype of the structure or 2) measuring
forces from components which transmit forces to the
structure being studied. If the second approach is
chosen care must be taken to assure that the boundary
conditions of the components are the same as in the total
system. Once the forces have been measured the Fast
Fourier Transform (FFT) can be taken to put the forces

into the frequency domain.

PAGE 23

CHAPTER 4

At this point the forces which act on the
structure are defined in the frequency domain. A check
must be made see that they fall within the original
specified frequency range. Forces having large
magnitudes and oscillation frequencies outside of the
specifed range would violate the original assumption
which was used to reduce the modal model. If the
assumption is not valid then the calculated response will

be inaccurate.

Each of the force time histories has been broken
up into discreet frequencies. At a given frequency w

there is an amplitude {D} and a phase angle I.

{F} I {D} e(i Q) (23)

PAGE 24

FORCING FUNCTION

Letting exp(i I) 8 Cos(Fee) + i Sin(Fee) we have

(P) I {R} + i {I} (24)
{R} i {D} Cos(§)

{I} I {D} Sin(C)

{F} is then premultiplied by the reduced w)T which

results in {Fm}. {Fm} is then used to create the {Fp}
vector which is premultipled by [VJT resulting in (F2).

Now that {F2} is defined Equation 22 can be solved.

PAGE 25

CHAPTER 5

RESPONSE CALCULATION

Thus far a finite element model has been reduced
to a set of uncoupled first order ordinary differential
equations. Chapter 4 defined the forcing function for
these equations. This chapter will discuss the solution

of the uncoupled equations.

Since the equations represented by Equation 22 are
uncoupled, each can be solved independently. If the

equations to be solved are written in the form

3
N
f
r
N
ll

. . . . f. exp (i w t) (25)
J J J J J

PAGE 26

RESPONSE CALCULATION

Following the procedure given in [9], the solution can be

expressed as

2. = f. exp( i w t )l[ k. +i w m.) (26)
J J J J

With all of the 2k zj known, Equation 21 can be
used to find all 2k pj. The definition of P in Equation
18 is then used to find the k second order modal
coordinates yj. The reduced form of Equation 6 can then
be used to find the response at any point in the original

finite element model.

These steps are repeated for each discreet
frequency in the forcing function. The result is a
complex amplitude for each frequency. The magnitude of
the response is found by taking the magnitude of the
complex number. The phase angle with respect to the
forcing function can be computed based on the real and
imaginary parts of the amplitude. A time response can be
computed from the frequency response by taking the

inverse Fourier Transform.

PAGE 27

The main goal of the analysis was to find a simple
and fast way to calcualte predicted structural response
using a large finite element model. This was done by
reducing the finite element model to a small set of
complex modes. The response was then calculated by
summing the modes. The next chapter will give some

highlights of a project in which this analysis was used.

PAGE 28

CHAPTER 6

OLDSMOBILE PROJECT

In the preceeding chapters it was shown that a
finite element model could be reduced to a set of
uncoupled first order ordinary differential equations.
These equations were then solved for the response to a
given forcing input. This chapter gives some of the

details of a project in which this analysis was used.

Currently the Albert H. Case Center for Computer
Aided Design at Michigan State University is involved in
a joint project with the Oldsmobile Division of General
Motors. One goal of this project is to limit the forces

transmitted from the engine to the passenger compartment

PAGE 29

CHAPTER 6

of an automobile. As part of this project a method was
developed to quickly and inexpensively calculate the
response of an automobile to a given set of forces from

the engine.

The finite element model for this project was
created by General Motors Engineering Staff, using the
finite element program Nastran. The model consisted of
408 nodes, each of which was allowed six degrees of
freedom, yielding 2448 equations to be solved for
frequencies and mode shapes. The Nastran program solves
this second order eigenvalue problem and uses a post
processor to create the modal model. Of the 2448 modes
found, 60 fell within the frequency range of interest.
The diagonal [Mm] and [Km] matrices and the 60 x 60 tCmJ
and [Dm] matrices as well as the 2448 x 60 transformation

matrix [U] were sent to the Case Center.

The reduced modal matrices were then used to make

up the two first order [Mp] and IKp] matrices. Initially

PAGE 30

OLDSMOBILE PROJECT

the Case Center’s minicomputer, a thirty two bit Prime
750 with one megabyte of memory, was used to solve the
120 by 120 first order eigenvalue problem. It took the
computer four hours to compute the eigenvalues and
eigenvectors. The resulting eigenvectors were used as
the transformation matrix [V]. But the resulting [M2]
and [K2] matrices were not diagonal. Since the same
software had been sucessful in solving smaller problems,
the indication was that the trouble was in the size of

the problem.

The size of the problem can affect the accuracy of
the solution because computers use floating-point

arithmetic '5103- This means that after each operation in
floating-point arithmetic the result is rounded off to a
fixed number of digits. The resulting number is then an
approximation of the actual result. The larger the size
of the problem the greater the number of operations
required for the solution. This, along with the relative
size of the numbers in the problem, can cause error due
to round off, and have detrimental effects on the

accuracy of the solution.

PAGE 31

CHAPTER 6

With single precision, the number of digits kept
after each operation on the Prime 750 is seven digits.
More digits were needed for accuracy in this problem.
The number of digits kept can be increased to fourteen by
in double precision. However the library subroutines we
used to solve the eigenvalue problem were not available

using double precision.

The round off problem was solved through the use
of a Control Data Corporation Cyber 170 series model 750,
at the Michigan State University Computer Center. Single
precision on the Cyber is fourteen digits and the same
library subroutines were available. The problem took
less than fifteen minutes to compute and the resulting

eigenvectors produced diagonal [M2] and [K2] matrices.

With the first order eigenvalue problem properly
solved, the diagonalized [M2] and [K2] matrices and the
transformation matrix [V] formed from the eigenvectors
were put up on the minicomputer where the response would

be computed.

PAGE 32

OLDSMOBILE PROJECT

Hith the equations uncoupled the next thing needed
was the force input to the model. In the Oldsmobile
project the forces that drive the finite element model
came from two sources. These were test data taken from
an engine and forces derived from a rigid body simulation
of the engine. The testing phase of the project is well
under way and producing results. The analytical phase is
still under development and as such no results have yet

been calculated using this type of data.

The tests fall into three categories: 1) engine

in a prototype vehicle, 2) engine on a rigid test stand,

and 3) engine in a test buck ( the test buck having the

front suspension and part of the structure of the
automobile). Data was taken in a prototype vehicle to
establish a base line for later tests on the test stand.
This also gave the opportunity to collect some vehicle
response data for correlation purposes. In each case an
engine was run and the time histories of the forces which

the engine transmitted were recorded on a PM tape

PAGE 33

CHAPTER 6

recorder. This consisted of recording forces in the X, Y
and 2 directions of each of six mounts, yielding eighteen
forces. These forces are then postprocessed using a
Hewlett Packard (HP 5423A) Structural Analyzer. The
FFT’s of the the forces are then transferred to the
minicomputer and transformed so they can be used in

Equation 26.

The frequency response corresponding to each set
of forces can be calculated in approximantly three

minutes on the Prime 750 minicomputer.

The force data measured from the prototype vehicle

was "sad t0 calculate a response. This response could

then be compared to the response data taken from the same
vehicle. Figure 3 is the FFT of one of the eighteen
forces which came from the testing. In this case it is
the 2 direction of mount number one while the engine was

running at 750 RPM. This and the other seventeen forces

PAGE 34

OLDSMOBILE PROJECT

were then used as input to calculate a response. In
Figure 4, this response is compared to the measured

response of the vehicle.

Figure 4 indicates that the model did not exactly
predict the measured response. There are many possible
reasons for the response difference, the most important
one being that the results are limited by the original
finite element model. The measured data presented in
Figure 4 was measured from a prototype vehicle, while the

finite element model is of a production car which has

been structurally up-dated from the prototype stage.

PAGE 35

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PAGE

CHAPTER 7

CONCLUSIONS

In the preceeding chapters an analysis was
developed whereby a finite element model of a complex
structure could be reduced to a set of uncoupled ordinary
first order differential equations. These equations
could then be solved and the response of the structure
calculated. In the last chapter the analysis was put to

use and the results compared to the measured data.

The analysis is based on four assumptions, 1) the
finite element model is an accurate model of the
structure, 2) the damping forces which occur in the
structure are small, 3) the frequency of oscillation of

the forcing function are known to be in a given range, 4)

PAGE 38

CONCLUSIONS

modes whose natural frequencies are not near the range of
the forcing frequencies do not significantly affect the

response of the structure.

The goal of this analysis is to facilitate the
calculation of the response of a structure based on a
finite element model of the structure. The analysis met
this goal offering considerable cost savings over
computing the response directly from the finite element

model.

PAGE 39

APPENDIX A

WHENCLATURE

PAGE 40

[H]
{i}
cc:

ID]

{i}
[K]
{X}

{F}

{A}
[U]
[U]
[Mm]

{Y}

[Cm]

c?)

NOMENCLATURE

Matrix of inertia coefficients (mass matrix)
The acceleration vector

Matrix of viscous damping coefficients
Matrix of structural damping coefficients
The frequency of oscillation

The velocity vector

Matrix of stiffness coefficients

The displacement vector

The force vector

F—‘I

Time

The exponential function

w2 is the eigenvalue and the square of the
undamped natural frequency

The associated eigenvector

The transformation matrix of eigenvectors
The transpose of the [U] matrix

The diagonal modal mass matrix IUJTIM] [U]
The second order modal acceleration vector
The coupled modal viscous damping matrix
[UJTIC] run

The second order modal velocity vector

PAGE 41

APPENDIX A

[Dm] I The coupled modal structural damping matrix
[UJTED] run

[KmJ a The diagonal modal stiffness matrix [U]T[KJ [U]

(V) a The second order modal coordinate vector

{Fm} - The modal force vector [UJT{F} exp (i w t)

{B} I A solution vector

x 8 The first order eigenvalue

{C} a The first order eigenvector

[V] I The first order transformation matrix made
up of the first order eigenvectors

[V] 8 The transpose of the [V] matrix

{P} - The first order transformation coordinate

{P} 8 The first order transformation velocity

[Mp] - The first order mass matrix

[Kp] I The first order stiffness matrix

{Pp} = The first order force vector

[M2] = The diagonal first order modal mass matrix
rvnTran [V]

{I} = The first order modal velocity

[K2] 8 The diagonal first order modal stiffness matrix
[VJTEKpJ [V]

{Z}

The first order modal coordinate

PAGE 42

{Fz}

{D}

{R}

{I}

I The

- The

- The

NOMENCLATURE

first order modal force vector [VJT{Fp}

phase angel with respect to time
amplitude of the force
real part of the force

imaginary part of the force

PAGE 43

LIST OF REFERENCES

PAGE 44

1)

2)

3)

4)

5)

6)

LIST OF REFERENCES

Kimball A. L. and Lovell D. E., Internal Friction
in Solids, Physical Review Vol. 30 December, 1927

Bishop R. E. D., The General Theory of Hysteretic
Dasping, The Aeronautical Quarterly February, 1956

pp 60 - 70.

Meirovitch L., "Analytical Methods in Vibrations“,

MacMillan Co. London, 1967 pp 100 - 104.

Swanson Analysis Systems Inc., 'Ansys Users Manual”,

Houston, Pennsylvania.

Bishop R. E. D., The Treateent of Daaping Force:
in Vibration Theory, Journal of the Royal
Aeronautical Society, Vol. 59 November 1955

pp 738 —742.

Myklestad N. O. The Concept of Coeplex Daeping,

Journal of Applied Mechanics, Vol. 19 1952.

PAGE 45

LIST OF REFERENCES

7) Richardson M. and Potter R., Uiscoa: 9: Structural

G)

9)

Danping in Nodal Analysis, 46th Shock and Vibration

Symposium, October 1975.

Frazer R. A., Duncan H. J. and Collar A. R.,
"Elementary Matrices", Cambridge University Press,
New York, New York, 1957 p 289.

Wylie C. Ray, "Advanced Engineering Mathematics“,

McGraw - Hill Book Co., 1975 Sec. 1.6.

10) Stewart B. R., ”Introduction to Matrix

Computations“, Academic Press New York,

New York 1973.

PAGE 46