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,fyniyerfltyfifi' 4.,

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This is to certify that the

dissertation entitled

AN EFFICIENT METHOD TO PREDICT‘
THE EFFECT OF DESIGN MODIFICATIONS
ON THE DYNAMICS OF STRUCTURES

presented by I

H. Metin N. Rizai

has been accepted towards fulfillment
ofthe requirements for ‘

Doctor of Philosophy (“Emmin Mechanicai Engineering

 

 

7
_ “/
<i\\[§:/é4g149 ::;’::\ LkéLqé¢7

 

Major professor

[Mme Juiy 22, 1983

 

MSU i: an Affirmativr Action/Equal Opportunity Institution 0-12771

 

AN EFFICIENT METHOD TO PREDICT
THE EFFECT OF DESIGN MODIFICATIONS
ON THE DYNAMICS OF STRUCTURES

BY

H. Metin N. Rizai

A DISSERTATION

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY

Department of Mechanical Engineering

1983

k.\~ \

/‘f-?* #0.? I}?

ABSTRACT
AN EFFICIENT METHOD TO PREDICT

THE EFFECT OF DESIGN MODIFICATIONS
ON THE DYNAMICS OF STRUCTURES

by

H. Metin N. Rizai

Static and dynamic analyses are important tools for the improvement
of structural systems. Consider an analysis where a number of different
changes may be synthesized to improve the dynamic and static behavior
of a system. Such an analysis usually depends on numerical optimiza-
tion schemes for systems with a large number of degrees of freedom.
These analyses become computationally burdensome for redesign study.

This thesis presents an efficient method to predict the effect of
design modifications on mechanical and structural systems with a large
number of degrees of freedom. The method minimizes a cost function
which includes natural frequencies, size of design change and the
static deflections. It uses a finite element preprocessor to find
derivatives of mass and stiffness matrices and computationally efficient

techniques to find eigenvalue and eigenvector derivatives.

DEDICATION

To mom and dad, Muriel and Andrew TenEyck.

ii

ACKNOWLEDGEMENTS

I particularly wish to express my gratitude to my major professor
and good friend Dr. James Bernard, for his guidance and constant encour-
agement as a teacher and a friend on this work and throughout my graduate
studies.

Special thanks to the members of my Ph.D. committee, Dr. Albert
'Andry, Dr. Ronald Rosenberg, Dr. William Symes and Dr. Clark Radcliffe
for their valuable suggestions which have improved the thesis measurably.

I would also like to thank the staff of the Case Center for Computer-
Aided Design at Michigan State University for financial support and for
the use of the computing facilities.

Finally many, many thanks to my parents Muriel and Andrew TenEyck
and the TenEyck family for their constant encouragement and loving

support.

111

TABLE OF CONTENTS

LIST OF TABLES

LIST OF FIGURES

CHAPTER I - INTRODUCTION

CHAPTER II - LITERATURE REVIEW

2.1 Design Optimization
2.2 Derivatives of Eigenvalues and Eigenvectors

CHAPTER III - DESIGN OPTIMIZATION WITH ONE DESIGN VARIABLE

Cost Function

Prediction of Eigenvalues and Eigenvectors
Example

.1 Initial Design

.2 The Design Change

The Cost Function

The Procedure

.5 Results

The Convergence of The Series

Restart

3
3
3.
3
3

TER IV - DESIGN OPTIMIZATION WITH MULTIPLE DESIGN
VARIABLES
The Cost Function
Calculation of Optimum Design Changes
.2.1 Linear Approximation
.2.2 Nonlinear Approximation
Restart
Example

J-‘L‘ E we:

The Design Change

The Cost Function

The Procedure

The Results

atic Deflection Function

Pit-3U)-
0
(1‘
m
H
O
O
U)
f?
t-rj
c
:1
O
f?
p.
O
D

xample with Static Deflection Function
.7.1 Results

iv

vi

vii

11

12
15
17
17
20
20
21
22
22
35

40
41
41
42
43
44
45
47
53
53
55
56
62
64
65
66

CHAPTER V - CONCLUSIONS
APPENDIX A
APPENDIX B

LIST OF REFERENCES

73
74

79

87

Table

Table

Table

Table

Table

Table

Table

Table

Table

3.1

3.2

3.3

4.3

4.4

4.5

LIST OF TABLES

Natural frequencies of the beam at h = .25 inches

Comparison of the predicted natural frequencies
and the frequencies from the eigenvalue solver at
h - .194 inches

Comparison of the predicted frequencies and the
frequencies from the eigenvalue solver at h - .1732
inches for the frequency band of 2000 Hz - 3000 Hz

Restart Results h . .1351 inches

Natural frequencies of the fixed-fixed beam
h . .25 inches

Comparison of the predicted natural frequencies via
linear approximation and the natural frequencies
via the eigenvalue solver at ef - -.2623
Comparison of the predicted natural frequencies
via nonlinear approximation and the natural
frequencies via the eigenvalue solver at e* - .2799
Design Change =‘Ekxf(ei, e5, e3)
Comparison of the predicted natural frequencies
via linear approximation including the static
deflection and the natural frequencies via the
eigenvalue solver at

* I * u - it a -

e1 2.1646 e2 2.3705 e3 3.2625
Comparison of the predicted natural frequencies
via nonlinear approximation including the static
deflection and the natural frequencies via the
eigenvalue solver at e* - .2964

Design Change -'E*xf(ef, e3, e3)

vi

19

28

34

38

49

57

59

67

69

Figure
Figure
Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

3.1
3.2
3.3

3.4

3.5

3.6

3.7

3.8

3.9

3.10

3.11

3.12

3.13

3.14

LIST OF FIGURES

Frequency content and size-of—change functions
Beam with legs fixed
Finite element representation of the beam

Comparison between the linear approximation and
the higher order Taylor series of the first
frequency of the beam

Comparison between the linear approximation
and the higher order Taylor series of the
second frequency of the beam

Comparison between the linear approximation
and the higher order Taylor series of the third
frequency of the beam

Comparison between the linear approximation
and the higher order Taylor series of the fourth
frequency of the beam

Comparison between the linear approximation
and the higher order Taylor series of the fifth
frequency of the beam

Predictions of the first eigenvalue at different
2 values

Predictions of the second eigenvalue at different
2 values

Predictions of the 6th frequency of the beam
at different thicknesses

Predictions of the 7th frequency of the beam
at different thicknesses

The effect of the restart on the predictions
for the sixth frequency

The effect of the restart on the predictions
for the seventh frequency

vii

13

18

18

23

24

25

26

27

30

31

32

33

36

37

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

4.1
4.2

4.3

4.4

4.5

4.6

4.7

4.8

A01

8.1

8.2

3.3

3.4

3.5

3.6

Schematic representation of the restart
Fixed-fixed beam

Finite element representation of the
fixed-fixed beam

Cubic polynomials that represent each half
of the fixed-fixed beam

Frequency content and size-of-change
functions for the example problem

Shape of the fixed-fixed beam after the
nonlinear approximation

Shape of the beam after the nonlinear
approximation including static deflections
in the cost function

Shape of the beam with nonlinear approximation
results with and without static deflection
functions in the analysis

Approximation of the derivative at xi
Comparison between the linear approximation and
higher order Taylor series of the eighth
frequency of the beam

Comparison between the linear approximation and
higher order Taylor series of the ninth frequency
of the beam

Comparison between the linear approximation and
higher order Taylor series of the tenth frequency
of the beam

Comparison between the linear approximation and
higher order Taylor series of the eleventh
frequency of the beam

Comparison between the linear approximation and
higher order Taylor series of the twelfth
frequency of the beam

Comparison between the linear approximation and

higher order Taylor series of the thirteenth
frequency of the beam

viii

46

48

48

51

54

61

71

72

74

79

80

81

82

83

84

Figure 3.7

Figure 8.8

Comparison between the linear approximation and
higher order Taylor series of the fourteenth
frequency of the beam

Comparison between the linear approximation and

higher order Taylor series of the fifteenth
frequency of the beam

ix

85

86

CHAPTER I

INTRODUCTION

Numerical optimization methods for specifying dynamic characteris-
tics of structures require eigenvalues and eigenvectors of the modified
system for each iteration. This can be a computationally burdensome
process for systems with a large number of degrees of freedom.

This thesis presents an efficient method to determine the effect
of design modifications on the eigenvalues and eigenvectors of a system,
and uses this method to expedite design optimization. The next chapter
will review the literature in the areas of 1) numerical optimization
with dynamic constraints and 2) derivatives of natural frequencies
and mode shapes with respect to design changes. Chapter 3 presents an
efficient method to determine the effect of a single design modifica-
tion. The method improves the dynamic characteristics of structure
by removing natural frequencies from bands where excitations are likely
to occur. It illustrates the method with examples. Chapter 4 examines
design modification as a function of many possible changes and intro-
duces static deflection to the optimization. It illustrates the power
and the flexibility of the method with examples. Chapter 5 presents

conclusions and some remarks on the extension of this work.

 

__ ._. .i|il|l I TILII «Tl-11:1...

 

CHAPTER II

LITERATURE REVIEW
Dynamic response of structures in the absence of damping is typi-

cally described by a matrix equation of the form [1],

[ng + £ng - g (2.1)

where [M] and [K] are, respectively, the mass and stiffness matrices of
the structure. Their order n corresponds to the degrees of freedom of
the system. The vectors x, x and F represent the acceleration, dis-
placement and external loads respectively.

The mathematical model can be used to compute the natural fre-
quencies and the mode shapes of the system. Should these indicate that
the system is unsatisfactory, there is motivation to change the system.

This thesis develops and illustrates methods which are useful for
finding changes which improve the frequency spectrum of the system.
This review will discuss those areas of the literature which underpin
the methods developed here. The review begins with the literature
concerning optimization of the frequency spectra which result from
models of the form 2.1. Since optimization, particularly the techniques
to be developed here, makes use of derivatives of natural frequencies
and mode shapes with respect to design changes, a review of the litera-

ture in that area will also be included.

2.1 Design Optimization

Design optimization entails a search for the best set of design
variables satisfying an objective such as minimum weight or following
rules concerning the frequency spectrum. Such an optimization may be
thought of in either of two categories. The first category relies on
variational calculus to support a search for a function or functions
that will find minimum or maximum of some performance function subject
to prescribed constraints. These methods work with a continuous model
that is described by continuous functions of spatial coordinates. The
optimum.design is found through the solution of a system of equations
expressing the conditions for optimality.

Variational methods are well suited for fundamental optimization
studies. 0n the other hand, numerical methods employ iterative techniques
to seek a near-optimal design [2,3]. An initial guess is used as start-
ing point for a systematic search for better designs. The search is
terminated when criteria are satisfied that ensure the current design
is sufficiently close to the true optimum.

Frequency constraints are among the dynamic restrictions that have
been considered. Much of the research in dynamic constraints optimi-
zation applies equality or inequality constraints on the fundamental
frequency and specified frequency range of the structure. A good text
on numerical optimization is Applied Optimal Design by Haug and Arora
[4].

The computational methods with equality constraints allow less
flexibility of design than those with inequality constraints. Equality

constraints may be unnecessarily restrictive since an alternative and

more convenient design might be acceptable. But whether the problem
deals with equality or inequality constraints, the eigenvalue problem
consumes a major portion of the total computation effort. The process
may prohibitively be slow for systems with a large number of degrees of
freedom. Feng, Arora and Haug have presented a modal method which reduces
the size of the problem and thus limits the needed computer resources [5].
Starkey proposed a new method to modify an existing design to im-
prove the dynamic characteristics [6]. The approach was to select
design changes that improve the dynamic characteristics of structure
by removing natural frequencies from undesirable bands, for example,
ranges of frequency where excitation is likely to occur. This procedure
seeks to minimize a penalty function which becomes smaller as the design

improves. Starkey suggested a penalty function P(w,e) of the form

P(w,e) ‘ H(F(W). 3(6)) (2.2)

where w is n vector of natural frequencies, e is m vector that represents
design variables, F(w) is a function which is large when natural fre-
quencies are in undesirable ranges, S(e) is a function which becomes

large for undesirable size of variables and function E combines the effect
of the two functions to make up the penalty function. The best designs
are selected from local minima of the function P(w,e).

Starkey used linear approximation via first order Taylor series ex-
pansion to predict the effects of various design changes on natural fre-
quencies in the computation of the search for a minimum penalty function
[6]. He suggested that, for small systems, the eigenvalue problem might

be solved at every step of the optimization. But for the system with many

degrees of freedom, the solution of the eigenvalue problem is computa-
tionally burdensome since it requires 0(n3) calculations where n is the
number of degrees of freedom. On the other hand, the linear approximation
of the modified natural frequencies has a limited range of validity.

It is the goal of this thesis to proceed along a path laid out by
Starkey, i.e., to minimize a penalty function which is dependent on
several variables. But we proposed to attack large systems and large
changes. Thus, we seek, among other things, an improved method of
approximating eigenvalues as a function of design changes. The next

section discusses the literature in this area.

2.2 Derivatives of Eigenvalues and Eigenvectors
The eigenvalue problem associated with equation 2.1 may be written

88

[MJEUJUJ - [KJEU] (2.3)

where [U] is an nxn matrix whose columns are n eigenvectors of the
structure and [A] is an nxn diagonal matrix of the corresponding eigen-
values.

The determination of derivatives of eigenvalues with respect to a
system parameter goes back to the nineteenth century work of Jacobi
[7]. Wittrick presented an application of Jacobi's work to systems
with an infinite number of degrees of freedom [8]. Fox and Kapoor
showed the first derivative of an eigenvalue for discrete systems with

symmetric mass and stiffness matrices [9]. They also derived two

methods for determining an eigenvector derivative. Rogers has genera-
lized the second approach in reference 9 to include non-symmetric
matrices [10]. Rudisill and Bhatia used second derivatives of eigen-
values to find the role of change of the flutter velocity of an aircraft
structure with respect to structural parameters [11]. This work was
extended for general matrices by Rudisill [12]. Nelson proposed an
alternate procedure for calculating eigenvector derivatives of an
arbitrary nth order symmetric or non—symmetric systems [13]. This method
offered a significant computational improvement for sparse systems.
Whitesell presented a method to calculate the eigenvector derivatives
using only 0(n2) calculations independent of the sparsity of the con-
stituent matrices [14,15].

Since Whitesell's work is a key element of the work presented in
this thesis, it will be useful to present some details. Consider mass
and stiffness matrices which are functions of only one variable. Equation

2.3 can be written as follows
[M(e)][U(e)]D(e)] = [K(e)][U(e)] (2.4)

.where [M(e)] and [K(e)] are real symmetric on eeR, furthermore [M(0)]
is positive definite and [A(0)] are non-repeated. Assume that equation

2.4 can be expanded with power series:

on i as
( 2 ei([K](i) - 2 Iii’k) [u](k)))( z e1 11:11)) = o (2.5)
i=0 k-O 1=o —

where a superscript in parentheses denotes power series coefficient,

a superscript without parentheses denotes the ith power of that variable,
and subscript m specifies the mth eigenvalue and mth eigenvector. Also

in this analysis, (e) is dropped from each term for convenience. Collect-

ing terms in e in equation 2.5 yields

a i . i-J
z e1 z ([x](1‘J) - z A(i-J-k) [m]<k)) 0(J) - o (2.6)
1-0 J=0 kao m .2
Each coefficient of e1, 1 :_1, must vanish
i i-J
2 ([K](1_J) - 2 A(i-J-k) [m](k))u(J) = o (2.7)
J=0 k=0 “1 J2
Substracting the i - j term
i-l
([K] -. Ammo) utff’ -- - z (EKI‘H’
——- J=0
i-J
(i-J-k) (k) (J)
2 Am [M] ) Um (2.8)

kxo .—

Multiplying both sides of equation 2.8 by 0:, where T denotes the trans-
pose of the vector Um’ will result in zero on the left hand side of the

equation since U§([K] - Am[M]) is an eigenvalue problem. Therefore

1 i

i- -J
0 I U:( E (,[K](i-J) _ Z xii-J‘k) D1300)

(J)
U (2.9)
— J-O k=0 —m

If the j-O term is separated from equation 2.9 and the k=0 term is carried

to the left hand side,

i
(1) T T (i) (i-k) (k)
,Am UmEMJUm - Umttxl - killm [M] )Um
i—J

1-1
+ 01% 2 <ExJ‘1‘” -
1‘- J=1 k=0

(J)
:13 ) (2.10)
Therefore equation 2.10 is the formulation of the power series coeffi-
cients for the mth eigenvalue where m-i,....n. The derivatives of eigen-
values with respect to a design change e can be calculated with a simple
substitution
1

i O

 

(2.11)
de

Equation 2.8 can be written in the following form to calculate deriva-

tives of the mth eigenvector [16],

1-1
0:1) - -<[KJ - Amtml)I X ([KJ‘i'J)
J30

i

-J
z . A11(11-.I-1c)[M](k))

U (2.12)
k-O m

(J)
32 ”i

where the superscript I denotes a generalized inverse of the singular

matrix ([K] s AmEMJ) such that

U:[M]([K] - AmEM])I - o and ([K] e AmEM])I[M]Um - o (2.13)

Whitesell [14]showed that the generalized inverse matrix in equation

2.12 can be computed as follows

([K] - meMI>I =- ([1] - EcIEMbcth - ‘i'mEMIrltm - [MJEGD

(2.14)
U UT
where [G] -‘=%}-£l-- , I = a unit matrix
U [M]U
.12 .JE
and
I - A + e
m m
The term Ci in the equation 2.12 can also be written as
i- l i- J
c1 - - z ( z 11‘1”J k) T[M](k))U(J) (2.15)
J-O k-0-—-

Thus the derivatives of eigenvectors with respect to a single design

change e, can be calculated with the following substitution.

d1 H

—J . 1' [1(1)
1 ° m

(2.16)
de --

10

In the solution for the derivative of eigenvalues and eigenvectors,
the only term that is candidate for 0(n3) calculations is the inverse
in equation 2.12. But if the eigenvalue problem for A and U is solved
with the inverse iteration method [17], the matrix ([K]—im[M]-1) is cal-
culated in the process. Thus the calculations of the derivatives of the
eigenvalue and the eigenvectors of the system with size n are 0(n2).

The next chapter illustrates the use of such a Taylor series in
support of the optimization procedure offered by Starkey. At this
stage, we will limit ourselves to one design change to facilitate the
use of equations 2.10 and 2.12. Later chapters will present a technique

to remove this restriction.

CHAPTER III

DESIGN OPTIMIZATION WITH ONE DESIGN VARIABLE

The traditional optimization methods for specifying dynamic
characteristics of structures have important limitations. In particular,
the methods which search for particular changes for natural frequencies
and/or mode shapes as a function of the size of a given change are unduly
restrictive since the designer must select an exact value for frequency
modification out of a large variety of acceptable modifications.

A less restrictive method that removes unwanted natural frequencies
from specified ranges was presented by Starkey [6]. The strength of
this procedure lies in the fact that the designer needs only to specify
what is not wanted, for example, a natural frequency occurring in a
certain frequency band. References [18] and [19], which illustrate the
effectiveness of the procedure, deal with problems in which a linear
approximation relating natural frequency to design change was adequate.

In this thesis, we will be concerned with economical methods for
the redesign of large structure via changes which may themselves be
large. The implications of the size of the system and the size of the
changes are these: We wish to deduce changes which will improve the
system with as few eigenvalue solutions as possible. Thus a major step
in this work is to improve upon the linear approximations used in
[18] and [19]. Of course, to be effective, whatever approximations are

used must be more economical than re-solving the eigenvalue problem.

11

12

This chapter presents an efficient method to determine the effect
of a single design modification. The proposed method is intended to
remove unwanted natural frequencies from the specified frequency ranges
in structures that have many degrees of freedom. The chapter includes
some examples which illustrate the procedure. Subsequent chapters will

deal with design modification as a function of many possible changes.

3.1 Cost Function

The optimal design is the best choice of the feasible designs. The
method presented in this chapter considers 'best' to be the design which
minimizes cost function C(w,e), which decreases as the design improves.

The form of C(w,e) to be used here was developed by Starkey [6],
cm» 4:607). S(e)) (3.1)

where w is an nxl vector of natural frequencies of the modified system,
n is the size of the system, F(w) is a function that is large when
natural frequencies are in the undesirable range, e is an mxl vector
of design variables, S(e) is a function that becomes large when design
variables begin to exceed prescribed limits. Figure 3.1 shows the
characteristics of the cost function.

The frequency content function, F(w), is largest near the center
of the undesirable frequency band since the center of the frequency
range is the most critical frequency. Its magnitude drops as natural
frequencies move away from the center of the function toward the edges

of the band.

13

‘F(w)

 

 

)4
(I)
i

[S(e)

 

 

C D

e

 

Figure 3.1 Frequency content and size-of—change functions.

14

The size of change function, S(e), becomes larger as the size of
the change increases. The total cost function is
k

C(w,e) I A S(e) + 2 Bi Fi(w) (3.2)
iIl

The parameter B weighs the relative importance of each critical fre-

i
quency band and A is a factor that weighs the importance of the size of
change.

To minimize C, it is necessary to relate the changes e to the
natural frequencies w. This can be done by reIevaluating the eigenvalue
problem as a function of the changes or by developing functions which

approximate the natural frequencies as a function of the change. For

example, one might use a Taylor series:

..2 .2.2 .ng

 

 

 

J 1 J 2 1 3
WiP W50 +~ de e + 2: 2 e + 3: 3 e + ... (3.3)
de de
The total cost function is then
k
C(e) I A S(e) + 2 B1 Fi(w(e)) (3.4)
iIl

and C can be minimized as a function of e.

Starkey minimized C(e) via the linear terms of the Taylor series.
Since we expect the w, e relationship to be fundamentally nonlinear, it
is clear that the range of usefulness of the linear series is limited.
In this chapter, we will show how the higher order terms can be used to

advantage.

15

3.2 Prediction of Eigenvalues and Eigenvectors

Numerical optimization is an iterative process which searches for
the best design. It requires eigenvalues and perhaps eigenvectors at
every iteration. But since each eigensolution requires 0(n3) calcula-
tions, recalculation of eigenvalues and eigenvectors computationally
burdensome.

In the method presented in this chapter, eigenvalues (square of
natural frequencies) and eigenvectors (mode shapes) of the modified
structure are approximated with a Taylor series including higher order
terms. The approximation is based in large part on theory developed
by Whitesell [14], which is summarized in some detail in Chapter 2. The

derivatives of eigenvalues have the form

1
1(1) [UT([K]<1)_ , x(1‘k’EMI‘k’)uk

 

[UM]Uk kIl
1-1 i-J _
+ U:( z ([K](i-J) - z 1(i-J-k)[M]<k))U;J)] (3.5)
_J.0 k=0 —
(i) 1 dixk (1) 1 diEK]
where; Ak .F-T , [k] '11" i ,
' de ° de
(1) 1 dint] (J) 1 dJ'Uk
[M] '1‘" 1 ’ Uk “3" J

16

and derivatives of eigenvectors have the form

1-1 i-J
(1) _ I (i-J) (i-J-k) (R) (J)
Uk -([K] - Ak[M]) J20([K] - 2 Ak [M] )Uk + C1 Uk
— 8 keno _.._
(3.6)
where;
1-1 i-J
.T-o k-O — —

([K] - Aktml>1 = ([11 - [G][M])([K] - ikrmi>'1<ExI - [M][G])

U UT

[Gj-lé-k 1:) +5
EEFMJEE

 

Equations 3.5 and 3.6 require derivatives of mass and stiffness
matrices with respect to the design variable. It is unwieldy to con-
struct mass and stiffness matrices with analytical functions in their
entries which are appropriate for the modeling of a wide variety of
potential design changes. Therefore, it is difficult to obtain deri-
vatives of these matrices via differentiation with respect to a design
variable. However, these derivatives can be approximated with finite
difference methods. For example, the first derivative of a mass matrix,
[M(e)], and a stiffness matrix, [K(e)], with respect to a design
variable e, can be approximated with the central finite difference

method as:

d[M(e)] = [M(e')] - [M(re')] (3 7)
de 28' .

l7

d[K(e)l [KQE'L] - [K(-e')]
de " 2e' (3‘8)

[K(e')], [M(e')] and [K(-e')], [K(-e')] are mass and stiffness matrices
that are perturbed by e' and -e' respectively. These matrices can be
constructed with a finite element preprocessor. Higher order derivatives
of mass and stiffness matrices can also be calculated via finite dif-
ferences. The central difference formulations for these derivatives
is given in Appendix A.

To illustrate the power of the procedures presented here, it will

be instructive to consider an example.

3.3 Example

Consider the horizontal beam of Figure 3.2. The legs are twice
as thick as the horizontal part of the beam. The design variable is
the thickness h of the horizontal beam. The objective is to remove

any natural frequency from the frequency range of 400 Hz - 580 Hz.

3.3.1 Initial Design

The initial design is modeled with the six beam finite elements
(four horizontal, two vertical) via the ANSYS finite element preprocessor.
Each beam element has two nodes with three degrees of freedom (two
translation and one rotation) at each node [20]. Figure 3.3 shows the
elements and nodes of the beam. Table 3.1 lists the fifteen natural
frequencies that correspond to the fifteen degrees of freedom. These

were obtained by solving

18

2.4'.’ A

 

ALL

.25"

 

 

 

 

)*|

\\\\\\ \

Figure 3.2 Beam with legs fixed.

a .g
, m S (a) (4) L (s) (2)

Figure 3.3 Finite element representation of the beam.
Numbers in parenthesis represent the element

number.

0.50

19

Table 3.1 Natural frequencies of the beam at hI.25 inches

Beam Frequencies (Hz)
1. 88
2. 245
3. 490
4. 918
5. 1520
6. 2323
7. 2782
8. 5197
9. 7721

10. 10260

11. 14054

12. 19394

13. 20694

14. 44783

15. 44921

20
([K] - {DIME = o (3.9)

where [K] and [M] are the stiffness and mass matrices respectively, w1

is the ith natural frequency of the system and U1 is the associated

mode shape.

3.3.2 The Design Change

The only admissible design change is assumed to be the thickness h
of the horizontal part of the beam. The derivatives of the eigenvalues
and eigenvectors with respect to h are calculated using equations 3.5
and 3.6. The derivatives of the mass and stiffness matrices were
obtained via the ANSYS preprocessor and the central difference method

as discussed in section 3.2.

3.3.3 The Cost Function

The penalty function C(e) can be minimized with routinely available
software [21]. In this example, there is one undesirable frequency band
which is in the band 400 Hz - 580 Hz. A frequency content penalty that
meets our needs is

m
F(w) I Z (l - cos(21r(wJ - p)/(u-p)) (3.10)

J=l
where p and u are lower and upper bounds of the frequency band respec-
tively, m is the number of modes in the finite element model and wj is

the jth modified natural frequency.

21

We assume that the size of change of the penalty function is

quadratic in e

S(e) a e2 (3.11)

and the linear combination of equations 3.10 and 3.11 with scaling
factors A and B at l and 100, respectively, becomes the total cost

function as indicated in equation 3.2

3.3.4 The Procedure

An optimal design for this example is sought according to the
following procedure.

1. Construct the stiffness and mass matrices for the orignial
design via a finite element preprocessor.

2. Extract the dynamic characteristics of the system, that is,
the natural frequencies and mode shapes.

3. Calculate mass and stiffness matrix derivatives as shown in
Appendix A.

4. Calculate eigenvalue and eigenvector derivatives using
equations 3.5 and 3.6.

5. Input the scaling parameters A and B in equation 3.2.

6. Minimize the total cost function C(e) as it is defined by
equation 3.4.

7. Verify the results by solving eigenvalue problem for the

modified model with the proposed design change.

22

3.3.5 Results

The algorithm used for this example approximates the modified fre-
quency with 9 terms in the Taylor series. Figures 3.4 through 3.8
illustrate the approximation. The figures show clearly that the first
five frequencies of the example do not have linear relationships with
the design variable.

The results of the minimization are shown in Table 3.2. The
optimal thickness of the beam found was to be .056 inches less than the
original .25 inch design, and the corresponding frequencies are out of
the unwanted frequency band. Note that the approximated frequencies
for the new thickness agree with the eigensolution of the new design,
indicating that the series has satisfactorily replaced the re-solution

of the eigenvalue problem.

3.4 The Convergence of the Series

The eigenvalues and eigenvectors of a modified system are approxi-
mated with the Taylor series. This approximation is accurate only
within the radius of convergence of the series. For example, consider

the following 2x2 matrix

1 Z
..[z 2]

for this case, the eigenvalues are

A - 321+ %- /—2__ (3°13)

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28

Table 3.2 Comparison of the predicted natural
frequencies and the frequencies from the eigenvalue
solver at hI.194 inches

PREDICTED ANSYS
FREQUENCIES EIGENVALUE
(Hz) SOLVER (Hz)
1. 69.5 69.5
2. 192.6 192.6
3. 383.1 383.2
4. 721.6 721.8

5. 1193.1 1193.5

29

where z is variable. The eigenvalues are complex and the matrix A has

a double root at
z I :32; i where i I [:1- (3.15)

The magnitude of this 2 value is the radius of convergence of the Taylor
series about zIO. This simple example illustrates that the radius of
convergence of the series may depend on complex values of the design
parameter. Figures 3.9 and 3.10 are plots of 12 and 12 for real values
of 2.

Of course, in a large complicated problem, this radius is not at
all obvious and may, in fact, have to be dealt with via rather ad-hoc
methods. In section 3.3, for example, the series had a large enough
radius of convergence to cover frequencies of the problem. But, in
the same example, if the unwanted frequency range is set to 2000 Hz -
3000 Hz, then the frequency penalty includes sixth and seventh frequencies.
Figures 3.11 and 3.12 indicate that the radius of convergence of the
Taylor series for these two frequencies is smaller than for the first
five frequencies. Therefore, the usefulness of the Taylor series
approximation of these frequencies will be limited.

To illustrate the consequences of this limitation, consider Table
3.3, which presents the results of an attempt to drive frequencies out

of the range of 2000 to 3000 Hz. The predicted frequencies in Table 3.3

show that the frequencies are removed from range. However, when these

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34

Table 3.3 Comparison of the predicted frequencies
and the frequencies from the eigenvalue solver
at hI.1732 inches for the frequency band of 2000 Hz-3000 Hz

PREDICTED ‘ANSYS
FREQUENCIES EIGENVALUE
(Hz) SOLVER (Hz)
1. 63.8 63.8
2. 176.9 177.0
3. 351.6 351.9
4. 663.1 663.7
5. 1096.3 1097.2
*6. 1905.7 *1759.2
87. 3172.3 *2885.7
8. 5537.2 5539.2
9. 8065.3 8067.6
10. 10780.0 10782.9
11. 14771.0 14776.2
12. 20875.0 20882.6
13. 21777.0 21784.9
14. 47545.0 47555.0
15. 47669.0 47678.7

35

results are compared with the eigensolution at the indicated thickness,
it is obvious that the approximations of the sixth and seventh frequen-
cies are inaccurate at the proposed thickness, and that, in fact, the
seventh frequency has not been moved from the unwanted frequency range.
The next section presents a method to deal with the series con-

vergence problem.

3.5 Restart

Restart is a re-evaluation procedure used when resolution of the
eigenvalue problem indicates that the Taylor series did not converge
for a frequency of interest. In short, the fact that the eigenvalue
problem has been redone to check the accuracy of the series allows new
derivatives to be calculated to expand about the new operating point.

The example from Section 3.4, with an objective of a renoving
frequencies from the 2000 Hz - 3000 Hz range, yielded a design which,
according to the approximation, drove all frequencies out of the unde-
sirable range. But re-evaluation of the eigenvalue problem at the new
design point indicated that the series did not faithfully represent the
eigenvalues at the indicated minimum. Figures 3.13 and 3.14 verify
that the next step in the optimization, which is an expansion about the
new design point based on the eigenvalue check solution, results in
locally accurate solutions. With this new starting point the optimizer
found a design change that met the objective. In this case, re-solution
indicated that the series remained in an accurate range. (See Table 3.4).

The steps of the algorithm that are explained in this chapter can

be summarized as follows:

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38

Table 3.4 Restart Results
hI.1351 inches

PREDICTED ANSYS
FREQUENCIES EIGENVALUE
(Hz) SOLVER (Hz)
1. 50.2 50.3
2. 139.6 139.6
3. 277.1 277.2
4. 523.6 523.7
5. 865.5 865.8
6. 1392.6 1393.0
7. 3061.9 3061.3
8. 5856.7 5855.4
9. 8389.1 8387.8
10. 11124.0 11122.5
11. 15179.0 15178.6
12. 22101.0 22097.1
13. 22788.0 22785.4
14. 49235.0 49228.8
15. 49349.0 49342.8

39

- Generate the Finite Element Model of the original design

- Obtain the eigensolution of the model

- Assess the results

- Set up the cost function

- Calculate derivatives of mass and stiffness matrices via the
finite element preprocessor and finite difference methods

- Calculate derivatives for eigenvalues and eigenvectors for the
natural frequencies of interest

- Execute the minimization algorithm to find a new design

- Solve the eigenvalue problem at the new design point

- Compare the solution of the eigenvalue problem with the

series approximation

Restart if necessary.

This chapter dealt with design optimization using one design
variable and a cost penalty which is a simple function of the size of
the design change. The next chapter deals with the multi-variate

design optimization and introduces static deflection to the cost function.

CHAPTER IV

DESIGN OPTIMIZATION WITH MULTIPLE DESIGN VARIABLES

This chapter presents an efficient method for predicting the
effects of design modifications which may be dependent on several
variables. The particular problem of interest here is to find an ime
proved design, as indicated by eigenvalues and eigenvectors and by
static deflection. We assume that the finite element analysis finds
the eigenvalues and eigenvectors of the initial design, and that this
analysis indicates a need to improve the system.

Additional eigenvalue analyses for optimization purposes are com-
putationally burdensome process for systems with a large number of
degrees of freedom. Thus the procedure presented here seeks to avoid
resolution of the eigenvalue problem. To avoid unnecessary resolutions,
the eigenvalues and eigenvectors of the modified system are approxi-
mated using the Taylor series expansion about the original design
point. Since the series will require only 0(n2) calculations, as
opposed to 0(n3) for an eigenvalue problem, the optimization procedure
requires only 0(n2) calculations to find the improved design.

Initially, the design analysis for several variables will be
developed without static deflection as part of the design criteria.
Then in section 4.5 static deflection function will be added to the cost
function. The following sections explain these steps in detail. The

power of the method will be illustrated with examples.

40

41

4.1 The Cost Function

As in previous chapters, the cost function is
C(w,e) foo). S(e)) (4.1)

where w is an nxl vector of natural frequencies of the modified system,
F(w) is a function that is large when natural frequencies are in an
undesirable range, e is an mxl vector of design variables, S(e) is a
function that becomes large when design variables begin to exceed
prescribed limits. The methods given in chapter 3 allow efficient
minimization of such penalties as a function of one change. This
chapter presents an efficient minimization procedure which includes

many changes.

4.2 Calculation of Optimum Design Changes

The calculation of optimum changes require a search for the minimum
specified cost function. The key issue in the total cost function is
determining eigenvalues and eigenvectors of a modified system for the
frequency content function. This can be done in two steps, 1) a
linear approximation of the eigenvalues of interest, a procedure which
can handle multiple design variables while using only 0(n2) calcula-
tions, and 2) nonlinear approximation which improves the results of the
linear calculations. In the first step, the design variables are deter-
mined based on the assumption that the rates of change of eigenvalues
are constant. For the second step, the ratios of the design variables

with respect to each other are kept constant and a new variable that

42

scales the magnitude of the variables is determined. In this case, the
Taylor series, including higher order terms, is used to estimate the
eigenvalues of interest. The next two sections explain the steps in

detail.

4.2.1 Linear Approximation

The penalty function can be computed as a function of the modified
eigenvalues. As a first step, the approximations of the eigenvalues are
based on a linear expansion, therefore, the predictions of the natural

frequencies of a modified system are

m 3w2
wz - 6:04. 2 gig, (4.2)
JIl J

where w and w are the modified and the original ith frequencies,

1 10
2
respectively, and awi is the rate of change of the ith frequency with
8e
J

respect to the jth design variable. For distinct eigenvalues, the

derivative can be computed as follows [9],

 

 

2
aw
i _ T 8[K] 2 3[M]
eJ U1( aeJ ‘ “I aeJ )Ui (4'3)

T
where 11$[M]U_i- 1

This procedure has 0(n2) calculations, where n is the number of degrees
of freedom. The first derivatives of stiffness and mass matrices can
be approximated via the central difference method as explained in
chapter 3. Therefore, the final solution becomes a combination of the
design variables that best removes the unwanted natural frequencies

under the constraint of the size-of-change penalties.

43

4.2.2 Nonlinear Approximation

If design variables have a linear relationship with the natural
frequencies of the structure, the linear approximation will give the
correct solution. But since the natural frequencies are, in fact, non-
linear functions of the design variables, the linear results have a
limited range of validity. The use of higher order terms can lead to
accuracy over a much wider range of changes. The method that is presented
here uses the Taylor series including the higher order terms to improve
the linear results.

An initial set of design variables e£* is found using linear appro-
ximations of the natural frequencies. This is followed by a nonlinear

optimization using the methods of chapter 3. The single design variable

for the nonlinear optimization is E; where
(4.4)

The natural frequencies of the system are now a function of the scale
variable 3. Since 3 scales the change along the path indicated by the
linear analysis, this may be viewed as a steepest descent procedure.
Equations 3.5 and 3.6 yield the higher order terms in the Taylor series
for approximation of the modified natural frequencies and mode shape

vectors with respect to E: In particular,

 

 

61w2 d2w2 d3w2
w2Iw2+—1;+—1, 1E2+—1, i;3+... (4.5)
i 10 _ 2. .2 3o -3

de de de

_.*
The solution is a set of design variables eeJ which minimize the
total cost function under the constraint that the design variables

retain the ratio to each other that was indicated by the linear

44

calculations. Thus there remain two questions, namely 1) does the

Taylor series yield an adequate representation of the penalty along

* k

the path indicated by the e , and 2) did the eJ

J indicate a path that

led to a minimum of C(w,e).
To answer the first question, it is necessary to find the eigen-
_.*
solution of the system modified by the changes eeJ, a procedure which

requires 0(n3) calculations. If the eigenvalues and eigenvectors at

*
J

then it is apparent that the series representation inadequate. In this

28 do not closely match the values predicted by the series expansion,

case it is necessary to use the restart procedure. If the eigensolution
_.*

at ee matches the predicted eigenvalues, it remains necessary to verify

J

_.*
that the ee

J indicate a minimum penalty. This also invalues the restart

procedure.

4.3 Restart

This procedure entails finding the optimum design variables by
minimizing the cost function based on linear approximations of the eigen-
values. Then, these variables are scaled according to the minimization
procedure of the total cost function with the estimation of the eigen-
values via the Taylor series including the higher order terms. This
process assumes constant ratios of the design variables and improves
the optimization of the variables by scaling. But this enhancement may

be inaccurate if the direction that is assumed constant is incorrect.
*

In this case, the optimization requires a new ratios of eJ.

45

Thus for the initial step of the restart the eigenvalues become

2
2 2(" *) + I; W“ A (4 6)

w . W 83 _ e o
k J J J31 BeJ J

_.*
If the linear optimization about eek returns the solution AekIO, this

minimizes C(w,e). Consider, on the other hand, the

*
k

case wherein one or more of the Ae

indicates that 28
I *
k are non-zero, say they are Aek.

Then ratio of Aek must be established, and nonlinear optimization
is again used. During the nonlinear approximation analysis, the new

design variable size function becomes
*
e ‘+ e Ae (4.7)

where Z is now the single nonlinear design variable.

Figure 4.1 presents a pictorial explanation of the restart proce-
dure in the case of an incorrect direction for the search of an optimum
result with one variable. Again, this can be viewed as optimization via
steepest descents.

The next section illustrates the power and the flexibility of the

analysis with an example problem.

4.4 Example
In this section, the method is applied to the design of a fixed-

fixed beam (see Figure 4.2). The design variable is the height of the

46

 

(b
a
(O.
on

 

 

Starting point

Linear approximation

Nonlinear approximation

Linear approximation

Restart with nonlinear approximation

bWNl-‘O

Figure. 4.1 Schematic Representation Of The Restart.

47

top of the beam. The objective is to remove natural frequencies from the
frequency band of 450 Hz - 550 Hz. The size of design change is included

in the cost function.

4.4.1 Initial Design

The fixed-fixed beam is modeled with ten beam finite elements
using ANSYS finite element preprocessor. The beam elements each have
two nodes and each node has three degrees of freedom (two translation
and one rotation). Figure 4.3 shows the element and the nodes of the
beam. The eigensolution from.ANSYS finite element processor found
the frequencies which correspond to twenty-seven degrees of freedom
(Table 4.1).

To facilitate the redesign of the shape of the top of the beam,
we write the height of the beam, h, as a function of four cubic para-
metric equations. Figure 4.4 shows the four parametric equations that
model each half of the fixed-fixed beam [22]. The redesigned beam will
remain symmetric about the mid point.

The parameter t ranges from 0 to l on each of the plots. The
first two plots, B and B

1 2’

have non-zero coordinate only at tIO for B1 and tIl for B2. B3 and B4,

on the other hand, have zero coordinate at each end and non-zero slope

have zero slope at each end point, and

3 and t=l for 34'

The design variable h, which is the top surface of the beam, may

only at tIO for B

be written for each half of the beam as

48

50 Lb.

 

 

2 i .
g .25'
It 2 4.” egg. T-

Figure 4.2 Fixed-Fixed Beam.

 

Figure 4.3 Finite Element RepreSentation Of The Fixed-Fixed
Beam.

49

Table 4.1 Natural frequencies of the fixed-fixed
beam at hI.25 inches

FREQUENCY (Hz)

2. 248
3. 487
4. 806
5. 1208
6. 1695
7. 2271
8. 2937
9. 3651
10. 4226
11. 4867
12. 5099
13. 7172
14. 8557
15. 8683
16. 10462
17. 12529
18. 13098
19. 14842

20. 17195

50

Table 4.1 Continued

FREQUENCY (Hz)

21. 17953
22. 19096
23. 23205
24. 28874
25. 34798
26. 40446

27. 44757

51

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52

e e

l 3 2 2 3 2
h ho + 76(2t - 3t 4' 1) + EEK-21: + 3t )
e3 3 2 3 2
+3-(t - 2t + t) + e4(t - t ) (4.8)

The variables e1, e2, e3 and e4 are the design variables which determine
the top surface of the beam. The design parameters are scaled such that
elIl corresponds to a 20% change in height at the end of the beam, and
e2Il corresponds to a 20% change in height at the mid point of the beam.
The slope scaling is such that e3Il corresponds to a maximum of 15%
change in height at an interior point fairly near the ends of the beam

(t ..%9. To ensure slope continuity at the mid point, e4 is set to

zero. Thus for the left hand side of the beam, h may be written as

e e
l 2 l 3 3 3 2 2
h'h0+('2‘0"1'0'+353)t + (70e1+2‘092‘333)t
1 e1

and for the nonlinear approximation, h may be written

* *
e e
- 1 2 I * 3 3 * 2 * 2
h h0+e[(‘1'6"'1'6+3'e3)t +("'2'6‘31'3‘23)t
*
1 * e1] 410
+(333)t+-2_O- (°)

*
where eis are the values of the design which result from the linear

approximation.

53

4.4.2 The Design Change
The approximation of the natural frequencies and mode shapes of
the modified design requires the first derivative of mass and stiffness

matrices with respect to the e and the high order derivatives of the

i
mass and the stiffness matrices with respect to :2 They are computed
via the central finite difference method as explained in section 3.2

using ANSYS finite element preprocessor.

4.4.3 The Cost Function
The cost function is seperated into two functions. First, the
frequency content function is represented by the following mathematical

form,

In

F(w) I Z (l - cos(21r(wi - p)/(u - p))) (4.11)
JIl

where p and u are lower and upper bounds of the frequency band, respec-

tively, m is the number of modes in the finite element model and W1

is the ith natural frequency. The w1 are found by the linear approxi-

mation at each design change. The design change size functions are

quadratic functions of e where iIl,2,3. That is,

i
3 2
S(e) I 2 e1 (4.12)
i=1

Figure 4.5 summarizes the cost functions for this example.

54

(F(w)

IOOO 4-

 

 

4:50 487 550 W (H2)

 

 

 

 

Figure 4.5 Frequency Content And Size-Of-Change Functions For
The Example Problem.

55

4.4.4 The Procedure

The procedure for this example is as follows:

1. Construct the Finite element model

2. Extract the natural frequencies and mode shapes of the
initial design

3. Calculate the first derivative of mass and stiffness matrices

with respect to each design variable e at the initial design via the

J
ANSYS finite element preprocessor

4. Calculate the first derivative of the natural frequencies
with respect to each design variable eJ

5. Minimize the total cost function in equation 4.1 using the
linear approximation for the frequencies. This yields e:

6. Formulate the new design variable E.by defining the design
change in terms of e: and 3 according to equation 4.4

7. Calculate the higher derivatives of mass and stiffness
matrices with respect to 3 via the central finite difference method
using finite element preprocessor

8. Calculate the higher derivatives of the natural frequencies
and the eigenvectors with respect to 2; according to equations 3.5
and 3.6

9. Minimize the total cost function to find 3

10. Resolve the eigenvalue problem at 26:
11. Repeat steps 1-5 with the modified design
12. If the eigensolution matches the predicted results from step

10 and if AekIO at the completion of step 11, then the procedure is

completed

56

13. If AekIO, use the restart procedure
14. Do a restart until AekIO, which indicates the minimum solution

for the given cost function.

4.4.5 The Results

Table 4.2 shows the results obtained by linear approximation. It
also shows the natural frequencies that are computed with the suggested
design changes via eigenvalue analysis. The table indicates that the
predicted natural frequencies deviate considerably from the correct
solutions. In fact, the proposed design failed to remove the natural
frequency from the specified frequency range of 450 Hz - 550 Hz.

Table 4.3 shows results after the nonlinear optimization. It
scales down the change considerably from the linear-based analysis.

This proposed design does not have any frequencies in the specified
frequency range of 450 Hz - 550 Hz. The table indicates that the eigen-
value analysis with this proposed design matches the predicted natural
frequencies.

Further linear approximation with the modified design as an
operating point results AekIO. This means that the path that is taken
with the linear approximation lead to the minimum cost function.
Figure 4.6 shows the shape of the beam with the new changes.

The analysis and figure 4.6 indicate that the design change has
thinned the beam. Thus, while the design objective with regard to
frequency spectrum has been met, the designer might well be concerned

with the static strength of the new design.

57

Table 4.2 Comparison of the predicted
natural frequencies via linear approximation
and the natural frequencies via the
eigenvalue solver at

*- 262 * 2 * 1 3012

MODE FREQUENCY VIA (Hz) FREQUENCY VIA (Hz)

LINEAR APPROXIMATION EIGENVALUE SOLVER
1. 63 66
2. 151 170
3. 279 324
4. 441 522
5. 643 770
6. 890 1068
7. 1188 1419
8. 1570 1848
9. 1896 2277
10. 2519 3043
11. 3138 3755
12. 3706 4457
13. 4394 4541
14. 4431 5478
15. 5273 6567
16. 6224 7845
17. 7208 9066

18. 7620 9221

58

Table 4.2 Continued

MODE FREQUENCY VIA (Hz) FREQUENCY VIA (Hz)

LINEAR APPROXIMATION EIGENVALUE SOLVER
19. 8770 10276
20. 8886 11461
21. 13266 13347
22. 18043 18075
23. 23205 23205
24. 28769 28732
25. 34584 34483
26. 40049 39839
27. 44629 44581

59

Table 4.3 Comparison of the predicted
natural frequencies via nonlinear approximation

and the natural frequencies via the eigenvalue solver at

_a _a * * *
e I.2799 Design Change-e f(e1,e2,e3)

MODE FREQUENCY VIA Hz FREQUENCY VIA Hz
NONLINEAR APPROXIMATION EIGENVALUE SOLVER

1. 83 83

2. 226 226
3. 441 441
4. 727 727
5. 1086 1086
6. 1522 1522
7. 2038 2038
8. 2641 2641
9. 3275 3275
10. 4278 4278
11. .4366 4366
12. 5306 5306
13. 6436 6437
14. 7784 7784
15. 8660 8660
16. 9369 9369
17. 11208 11209
18. 13150 13150

19. 13253 13254

60

Table 4.3 Continued

MODE FREQUENCY VIA Hz FREQUENCY VIA Hz
NONLINEAR APPROXIMATION EIGENVALUE SOLVER

20. 15255' 15256
21. 16975. 16977
22. 17980 17980
23. 23205 23205
24. 28842. 28842
25. 34732. 34732
26. 40323 40323
27. 44718 44718

61

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62

One way to codify the loss in load carrying capacity is to calculate
the deflection of the new and the original designs under a static load.
For example, calculations would indicate that under centerpoint loading
the new design deflects about 302 more at the center than the original
design.

To deal with these concerns, the next section considers static
deflection as a design criterion. It seeks for an improved design

indicated by static deflection as well as eigenvalues and eigenvectors.

4.5 Static Deflection Function

The changes which produce a desirable frequency spectrum may so
weaken the structure that it will not withstand expected static loads.
The static deflection penalty to be presented here is meant to prevent
the formulation of such a structure.

The static deflection cost function, D(ds), reflects the relative
desirability of all values that static deflections can accrue. That
is, reasonable values of static deflection should be associated with a
small cost and less desirable ones should have larger cost.

Since it is an goal here to limit the number of calculations, we
will use the solution to the vibration problem to determine the static
deflections. Then the economical series approximation for eigenvalues
and eigenvectors can be used to determine expected changes in static

deflection as a function of changes in the system.

63

The determination of the static deflections of a modified system

is given by the following method. The system is defined by

[MJg + [K]x 1 E (4.13)

where

[M] I mass matrix (nxn)

[K] I stiffness matrix (nxn)

F. I force vector of static loads (nxl)
g. I acceleration vector (nxl)

x. I displacement vector (nxl)

If the eigenvalues are distinct and non-zero, equation 4.13 can be

uncoupled by the following transformation

5 I [U]y (4.14)
[where y is an nxl generalized coordinate vector and U is the nxn modal
matrix with columns which are eigenvectors. Assume the eigenvectors

are normalized such that [U]T[M][U] I [I], and premultiply by UT. Equa-

tion 4.13 becomes

[1]): + £sz = In?“ F (4.15)

where [A] is a diagonal nxn matrix of eigenvalues.

64

In the static case, that is, y=0, equation 4.15 becomes

E111 = EuIT F (4.16)
01'

z = [AJ'IEUJT 3 (4.17)

where [A]1 is the inverse of the diagonal matrix [A].

The generalized coordinate vector y can be obtained from equation
4.17 for the modified system. The entries of U]-1 and [U] are the
reciprocal of the modified eigenvalues and the modified eigenvectors for

a given design change. The static deflection can then be found from 4.14.

4.6 Total Cost Function
A total cost function can be written as a linear combination of

the size, frequency content and static deflection functions:

k r m
C(w,e,d ) I Z A.F (w) +- X B S (e) +. 2 E D (d ) (4.18)
s J=l J J J31 J J J=1 J J s

where k is the number of unwanted frequency ranges, r is number of design
variables, m.is the number of sections of the structure where static
deflections are of concern, dS is the static deflection of a point on

the structure under a set of prescribed static loads and D(ds) is a
function that becomes large for undesirable static deflections of the
system. Coefficients of each function weight the relative importance

of each frequency, size and static deflection.

65

As indicated in chapter 3, the frequency content function is a

function of design variables,

w1 I f(eJ) J I 1,2,...m (4.19)

where wi is modified ith natural frequency and the m eJ

variables. The static deflection is also a function of design variables

are design

eJ as indicated in equations 4.17 and 4.14. Therefore, the total cost

function becomes a function of the design variables.

k r
C(e ) I Z A F (w(e )) + 2 B S (e
J J31 J J J ng J J J
m .
+ J51 EJ(DJ(eJ)> (NO)

4.7 Example with Static Deflection Function

In this section, the beam example from section 4.4 is considered
to illustrate an application of the static deflection penalty. In
this case, the static deflection penalty is included in the cost func-
tion. In particular, a linear static deflection function for the mid
point of the beam under mid-point loading is used. The linear equation

is

(4.21)

*
where Z is a scale factor, dS is the static deflection of the middle

66

node of the original design under a give external load at that node,

and d is the static deflection of the middle node for the various
designs. The total cost function is the linear combination of the

three functions of equation 4.20. For this example, the scaling factors

A, B and Z are set to 1000, l and 125, respectively.

4.7.1 Results

The results of this example show the behavior similar to the last
example. The linear approximation eigenvalue results deviate from
the eigensolution of the modified design (Table 4.4). However, the
nonlinear approximation procedure with constant ratios of design
variables yields deign that does not have an frequencies in the range
of 450 Hz - 550 Hz and the predicted natural frequencies agree with
the eigensolution of the design (Table 4.5). Figure 4.7 shows the
modified beam after the nonlinear approximations. A further linear
approximation with the modified design as an operating point yields
AekIO, which indicates that the given set of design changes for this
example lead to the minimum cost function.

Although this example is similar to the previous one, the analysis
synthesized a remarkably different design modification. In this case,
this modified beam has mid-point a static deflection only about 13%
higher than the original design, whereas, the previous example lead to
a static deflection about 30% more than the original design. Figure
4.8 shows the final design for each case. Clearly, the particular
deflection chosen here lead to increased thickness in key areas of the

beam.

67

Table 4.4 Comparison of the predicted natural
frequencies via linear approximation including the
static deflection and the natural freequencies via

the eigenvalue solver at

s * *
e =2.1646 e I-2.3705 e I-3.2625

1 2 3
MODE FREQUENCY VIA (Hz) FREQUENCY VIA (Hz)
LINEAR APPROXIMATION EIGENVALUE SOLVER

1. 82. 71

2. 176 177
3. 300 326
4. .442 504
5. 612 733
6. 685' 1055
7. 758 1294.
8. 823 1659
9. 1088 1984
10. 1494 2640
11. 1686 3384
12. 2209 4164
13. 2915 4829
14. 3213 5078
15. 3690 5983
16. 4196 7053
17. 4663 8897

18. 4705 9842

Table 4.4 Continued

MODE

19.

20.

21.

22.

23.

24.

25.

26.

27.

68

FREQUENCY VIA (Hz)
LINEAR APPROXIMATION

4834

9361

13517
18178
23205
28610
34255
39442

44377

FREQUENCY VIA (Hz)
EIGENVALUE SOLVER

10002
10153
13662
18180
23205
23611
34087

38895

44281 .

69

Table 4.5 Comparison of the predicted natural
frequencies via nonlinear approximation including the
static deflection and the natural frequencies via the

._*
eigenvalue solver at e I.2964

_* * z *
Design ChangeIe x f (e1,e2,e3)

MODE FREQUENCY VIA (Hz) FREQUENCY VIA (Hz)
NONLINEAR APPROXIMATION EIGENVALUE SOLVER

1. 87 87

2. ‘ 230 230
3. 442 442
4. 721 721
5. 1072 1072.
6. 1496 1496
7. 1989 1989
8. 2596 2596
9. 3208 3209
10. 4278 4279
11. 4383 4383.
12. 5232 5233
13. 6329‘ 6331
14. 7639 7640
15. 8831 8831
16. 9175 9176
17. 10957° 10959

18. 12920 12921

Table 4.5 Continued

MODE

19.
20.
21.
22.
23.
24.
25.
26.

27.

7O

FREQUENCY VIA (Hz)
NONLINEAR APPROXIMATION EIGENVALUE SOLVER

13233
146801
16262
18022
23205
28793
34626
40120

44637

FREQUENCY VIA (Hz)

13233
14681
16263
18022.
23205
28794
34627
40121

44638

71

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CHAPTER V

CONCLUSIONS

Design modifications to improve the frequency spectrum.of large
systems often requires many solutions of large eigenvalue problems.
Thus, this has been computationally inefficient. This thesis develops
a more efficient approach to the problem.

The procedure developed here uses a finite element preprocessor and
series approximation to develop an approximation for eigenvalues and
eigenvectors as a function of design changes. Thus, the iterations which
lead to the optimal redesign take only 0(n2) calculations. Examples
indicate that the approximations are useful over a wide range of design
changes.

Future work should include applications of this method to systems
with damping and the investigation of the possibility for use of a more
sophisticated choice of direction of the change vector in restart appli-

cations.

73

APPENDICES

APPENDIX A

The central finite difference formulations for the various deriva-
tives involve values of the function on both sides of the x value at
which the derivative of the function is desired. By utilizing the
appropriate Taylor series expansions, one can obtain expressions for

derivatives as follows:

3: *(X)

 

HI-I 9; givl

 

 

 

 

 

Figure A.1 Approximation of the derivative at x‘

74

75

The Taylor series for a function y=f(x) at (x1+Ax) expanded about x1 is

.Y;(Ax)2 yi"(AX)3
y(xi + AX) . yi + yi(AX) + 2: + 3, + ... (A.1)

 

where yi is the ordinate corresponding to x and (x1+Ax) is in the

1
region of convergence. The function at (xi—Ax) is similarly given by

y;<4x)2 y;"<4x>3
y(xi - Ax} . yi - yi<Ax) + 2: - 3,. + ... (A.2)

 

Using only the first 3 terms of each expansion, we can obtain an expres-
sion for y; by substracting equation (A.2) from (A.1), yielding

y(xi + Ax) - y(x1 - Ax)

Vi ' 2(Ax) ' (A'3)

 

Equation (A.3) is called the first central-difference approximation of

y' at x Graphically, the approximation represents the slope of the

1.
dashed line in Figure A.1. The actual derivative is represented by

the solid line drawn tangent to the curve at x If we add equations

1.
(A.1) and (A.2), the second derivative can be approximated as

y(xi + Ax) - 2yi(Ax) + yi(xi - Ax)

 

y" a (A.4)
i (Ax)2
The four terms on the right hand side of each equations (A.1) and (A.2)

can be used to obtain an expression for the third derivative. Sub-

stracting equation (A.2) from equation (A.1) yields

76

2 1" 3
y1 .(Ax)
Yi(xi + Ax) - yi(xi - AX) - 2yi(AX) + 3: (A.5)

 

If we expand the Taylor series about x to obtain expressions for

i
yIf(x) at (xi+2Ax) and (xi-2Ax), respectively, we obtain

y;(2Ax)2 y;"(2Ax)3

 

y(xi + 2Ax) = y1 + yi(2Ax) + 2, + 3, (A.6)
n 2 I”
Y (24x) Y1 (24X)
y(xi - 2Ax) - yi - yi(2Ax) + ——£,— - T— (A.7)

Substracting equation A.7 from equation A.6, and using just the 4 terms

of each expansion shown, gives
y (x + 2Ax) - y (x - 2Ax) I 4y'(Ax) +I-8--y"'(Ax)3 (A 8)
i i i i i 3 i '

The simultaneous solution of equations (A.5) and (A.8) yields

"' yi(xi + 2Ax) - 2yi(xi + Ax) + 2yi(xi - Ax) - yi(xi - 2Ax)

3'1

 

““03 (A.9)

Equation (A.9) gives the central difference difference expression for
the third derivative of y at xi.
Successively higher derivatives can be obtained by this method,
but, since they require the solution of increasingly larger number of
simultaneous equations, the process becomes quite tedious. Derivations

for the higher derivatives are accomplished with much greater facility

and far less labor by using difference, averaging and derivative

77

operators [ ]. Central-difference expressions, however, are more
accurate than either forward or backward difference expressions. The
following is a summary of the differentiation formulas which may be
obtained from Taylor series expansions.

. _ y1+1 ' yi-l
yi 2(Ax)

 

" . y1+1 ' 2V1 + yi-l

 

 

 

y
i (Ax)2
y - 2y + 2y _ - y _
yin . 1+2 1+1 3 i l i 2 (A.10)
2(Ax)
y.... . y1+2 ' 4y1+1 + 6Y1 ’ 4y1-1 + y1-2
1 (Ax)4
where
y1+1 ' y1(x1 + A“)
y1+2 ' y1(x1 + ZAX)

yi-l ' y1(x1 ' AX)

yi-2 ' yi(xi ‘ 2Ax)

Similarly the derivatives of mass and stiffness matrices may be for-
mulated via central difference method. The following is a summary of
the derivatives of mass matrix [M] and stiffness matrix [K] with

respect to a design variable e.

' [M]1+1 ‘ [MJi-l
[M] a 2(Ae)

 

78

" _ [M]i+1 ' ZEM11 + [MJi-l

[M]
we)2

 

... . [M]1+2 - ZEMJi+1 + 2EM]1_1 - [M]i_2

 

[M]
2(Ae)3

I!!!

[M] . [u]i+2 - 4[M]1+1 + etuji - 4EM11_1 + [M11_2

(Ae)4

 

where

[M]i+l - Mi(ei + Ae)
[M]i+2 = M1(e1 + 2Ae)
[MJi-l = Mi(ei - Ae)

[MJi-Z I Mi(ei - 2Ae)

' [K]1+1 ' [KJi-l
[K3 ‘ 2(Ae)

 

EX)" g E§]1+1 ' ZEKji + [KJi-l

 

(Ae)2

[K]... . [K]1+2 - 2EK11+1 + 2EK]1_l - [K]i_2

 

2(Ae)3

HM a [K]i+2 ' 4EK11+1 + GEKJi - AEKji-l + [KJi-Z

[K]
(Ae)4

 

where
E“1+1 . Ki(ei + Ae)
[K]1+2 - Ki(e + 2Ae)

[K]i_l = K1(ei - Ae)

[KJi-Z = Ki(ei - 2Ae)

APPENDIX B

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REFERENCES

10.

ll.

12.

13.

REFERENCES

L. Meirovitch, Analytical Methods in Vibrations, MacMillan Company,
New York, 1967.

A.V. Fiacco and G.P. McCormick, Nonlinear Programming: Sequential
Unconstrained Minimization Techniques, John Wiley, New York, 1968.

F. Moses, "0ptimum.Structural Design Using Linear Programming,"
Journal of the Structural Division, ASCE 90, No. 8T6, 89-104
(December 1974).

E.J. Haug, and J.S. Arora, Applied Optimal Design, John Wiley and
Sons, New York, 1979.

T-T. Feng, J.S. Arora and E.J. Haug Jr., "Optimal Structural Design
Under Dynamic Loads," Int. Jr. for Num. Methods in Engr. V01. 11,
39-52 (1977).

J.M. Starkey, "Redesign Techniques for Improved Structural Dynamics,"
Ph.D. Dissertation, Mech. Engr. Dept., Michigan State University,
1982.

C.G.J. Jacobi, "Uber ein Leichtes Vergahren die in der Theorie der
Sacularstorunger Vorkommenden Gleichungen numerisch aufzulosen,"
Crelle's Journal, Vol. 30, pp. 51-95, 1846.

W.H. Wittirick, "Rates of Change of Eigenvalues with Reference to
Buckling and Vibration Problems," Journal of the Royal Aeronautical
Society, Vol. 66, pp. 590-591, Sept. 1962.

R.L. Fox and M.P. Kapoor, "Rates of Change of Eigenvalues and Eigen-
vectors," AIAA Journal, Vol. 6 No. 12, pp. 2426-2429, Dec. 1968.

L.C. Rogers, "Derivatives of Eigenvalues and Eigenvector," AIAA
Journal, Vol. 8, pp. 943-944, May 1970.

C.S. Rudisill and K.G. Bhatia, "Second Derivatives of the Flutter
Velocity and the Optimization of Aircraft Structures," AIAA
Journal, Vol. 10, No. 12, pp. 1569-1572, Dec. 1972.

C.S. Rudisill, "Derivatives of Eigenvalues and Eigenvectors for a
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