THESES

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This is to certify that the
thesis entitled l

Deuuvituves of Frequency
Response Peaks

presented by

Raymond Brent Thompson

has been accepted towards fulfillment
of the requirements for

Mars—degree in .Meshanjm. Engineering

 

Major professor

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DERiVATIVEs
OF ‘
FREQUENCY RESPONSE

PEAKS

By

Raymond Brent Thompson

A THESIS

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

MASTER OF SCIENCE

Department of Mechanical Engineering

1983

ABSTRACT

DERIVATIVES
OF
FREQUENCY RESPONSE
PEAKS

By

Raymond Brent Thompson

When a system is excited at a natural frequency, the magnitude of
the response becomes large. This thesis concerns a method of redesign
to reduce the magnitude of the forced response at resonance.

The method uses derivatives of the forced response to compute a
first order Taylor series in the design change. This series can then be
used with standard minimization techniques to select the appropriate

design change to reduce the response at resonance.

ACKNOWLEDGEMENTS

I would like to express my appreciation to my major professor, Dr.
thmes Bernard. His expert guidance, sincere friendship and
encouragement greatly aided in this research.

Also, many thanks to Dr. Ronald Rosenberg and Dr. Brian Thompson
whose comments concerning this thesis are greatly appreciated.

I would especially like to thank three of my colleagues at Michigan
State University, Dr. tbhn Starkey (now a professor at Purdue
University), Matt Rizai and Guy Allen, who all leant a great deal of
advice, as well as friendship during my graduate career.

A special thanks goes to my parents, Clifton and Lorraine Thompson,
for their continuous loving support and understanding.

Also, I would like to thank the entire staff at the Case Center for
Computer-Aided Design at Michigan State University for the use of the
computer facilities, financial support and the friendship gained through

my association with it.

ii

LIST OF TABLES
LIST OF FIGURES

CHAPTER I
CHAPTER II

CHAPTER III

3.1
3.2

3.3
CHAPTER IV

4.1
4.2

CHAPTER V
CHAPTER VI
CHAPTER VII

TABLE OF CONTENTS

INTRODUCTION

MDDAL ANALYSIS OF DAMPED SYSTEMS: GENERAL CASE
GENERAL CASE WITH A SINOSOIDAL (EXPDNENTIAL)
FORCING FUNCTION

DERIVATIVE OF EIGENVALUES AND EIGENVECTORS

Derivatives of Eigenvalues

Derivatives of Damping Ratios and Undamped
Natural Frequencies

Derivatives of Eigenvectors

FREQUENCY RESPONSE DERIVATIVES

Frequency Response Derivatives
Variable Frequency Derivatives

A PROCEDURE TO REDUCE RESONANT RESPONSE

CONCLUSIONS

REFERENCES

iv

12

15
16

18
31
32

LIST OF TABLES

TABLE 5.1 Optimization Interaction Results b=l

TABLE 5.2 Optimization Interaction Results b=0.5

iv

27
29

FIGURE
FIGURE
FIGURE
FIGURE
FIGURE
FIGURE
FIGURE
FIGURE

4.1
4.2
5.1
5.2
5.3
5.4
5.5
5.6

LIST OF FIGURES

Effects of Constant Frequency Derivatives
Effects of Variable Frequency Derivatives

Ten Spring, Mass, Damper Example System
Freqency Response of the First Mass

Parameter Change Penalty Function
Consequenses of the Linear Assumption
Original System vs. Optimized System with b=1

Original System vs. Optimized System with b=0.5

l3
l4
T9
20
22
25
28
30

CHAPTER 1
INTRODUCTION

A system excited at or near one of its resonant can exibit large
responses. This thesis presents a method to compute the sensitivity of
this resonant response to changes in the system.

Chapter 2 presents some background information on modal analysis of
damped systems. Equations of motion will be presented. From these
equations an expression for the magnitude of the frequency response is
obtained. This expression is then used to facilitate the derivation of
the derivatives of the response at resouance.

Chapter 3 will review the derivation of the derivatives of the
eigenvalues, eigenvectors, undamped natural frequencies and the damping
ratios of a system.

Chapter 4 presents two methods for assesing the sensitivity of the
magnitude of the frequency response. The first method reviews of a
formulation which yields the sensitivity of the magnitude of the
response at any frequency. The second method presents an equation for
the magnitude of the response at a resonant frequency. It then finds
the sensitivity of this peak to design changes in the system.

Chapter 5 introduces an optimization scheme to reduce the magnitude
of a resonant peak. A penalty function formulation facilitates the
selection of an appropriate change to reduce the peak response. The use
of the technique is illustrated through an example.

Finally, concluding remarks summarize the thesis and discusses

future work.

CHAPTER 2

MODAL ANALYSIS OF DAMPED SYSTEMS:
GENERAL CASE WITH A SINOSOIDAL (EXPONENTIAL) FORCING FUNCTION

In order to lay the groundwork for the derivation of the
derivatives of the magnitudes at constant and variable frequencies, this
section reviews some of the fundamentals of modal analysis.

The equations of motion for a forced vibratory system with

n-degrees of freedom are

[m] {x} + [c] {x} + [k]{x} = {F(t)} (2.1)
where:
[m] = mass matrix {x} = Acceleration vector
[k] = Stiffness matrix {x} = Velocity vector
[c] = Viscous damping {x} = Displacement vector
matrix
{F(t)}

Force vector

t time

and the dot indicates differentiation with respect to time. In general,
we assume the mass, stiffness and damping matricies to the positive
definite and symmetric. In the case which will be discussed here {F(t)}

will be harmonic, i.e.,:

{F(t)} = {Eo}e(iwt)
where

{Fo} Magnitude of the Force

i square root of -1

w frequency of exitation

3
Our interest here is in the steady state response.
In general, the system of (2.1) cannot be uncoupled by using the
eigenvalues generated from the undamped system [I]. A first order
transformation of the form:

{;}
{x}

{y} =

 

may be used. This leads to a set of Zn symmetric first order ordinary

differential equations:
[MJ {9} + [m] {y} = {F(t)} (2.2)
where:

[M] = [0] [m] [K] = [-ml [0] {F(t)} = {0)
[m] [c] [0] [k] {F(t)}
This set of equations leads to a set of Zn eigenvalues (A1) and Zn

eigenvectors of the form:

{vi} = xi{u1} (2.3)
{U1}

where:

A1{ui} Eigenvectors corresponding to velocities of the n-degree of
freedom system.

{ui} = Eigenvector entries corresponding to the coordinates of the
n-degree of freedom system.
For non-repeated eigenvalues, the eigenvectors are [M] and [K]
weighted orthogonal and therefore can decouple equation (2.2) [2].
This decoupling can be accomplished by transforming to modal

coordinates, using the relation:

{y} = WM} (2.4)
where:
{q} = Modal coordinates

[U] Modal matrix (matrix of eigenvectors)

The pre-multiplication of equation (2.2) by [U]t (the transpose of the

modal matrix) yields:
EUJtEMJEUJth+EUJtEKJEUJrq1=EUJtmtn (2.5)

Equations 2.5 are uncoupled.
Equations 2.5 are often modified by normalizing the ith element of

[U]t{f(t)} with (UtMU)1i, the ith diagonal element of [U]t[M][U]. This

yields

[I]{q}-[AJ{qi= [U]t{fn(t)} (2.5)
where:

[I] = Identity matrix

[A] = Diagonal matrix of eigenvalues

{fn(t)} = The normalized force vector.

Since the forcing function has the form {f(t)}={Fo}e(th).

The particular solution will have the form

{q} = {A}e(iwt)
. . (2.7)
{q} = {A}iwe(lwt)
where:
{A} = Modal magnitude of the response vector.

Equation (2.6) can be re-written

(iw[I]-[A]){A}e(IWt) = [u]t{fn}e(iwt) (2.8)

Oi":

{iWEIJ-[A]}{A}=[U1t{fn} (2.9)

Pre-multiplication of equation (2.9) by (iw[I]-[)t])"1 yields an

expression for the modal magnitude
{A} = (iw[I]-[A])'1 [U]t{fn} (2.10)
Using equation (2.10) in equation (2.7) yields
{q}=(iw[I]-[1])-1[U]t{fn}e(iwt)

which is an expression for the modal response.

Since {y}=[U]{q}, we have

{Y}=[U](iwIIJ-IA])'1[U]t{fn} (2.11)
and

iY(t)}=LU](iWLI]-[A])'1[UJt{fn}e(th) (2.12)
where:

{Y} = Magnitude of response.

{Y(t)} = The response of the n-degree of freedom system with
displacements in the second n rows and velocities in the
first n rows.

The relationship between Y and w is the so-called frequency response.
Peaks on the frequency response plot indicate resonant frequencies of

the system. This occurs when w takes on the value of the imaginary

part of 11, causing Aj to become large.

The goal of this thesis is to be able to deduce changes in the
magnitude of the resonant response of the system which may result from
changes in the system. This will be done through differentiation of the
magnitude of the response at a resonant frequency with respect to a
system change.

The next chapter discusses the techniques involved in obtaining

these derivatives.

CHAPTER 3
DERIVATIVE OF EIGENVALUES AND EIGENVECTORS
3.1 Derivatives of Eigenvalues

An important step in the derivation of the derivative of the
frequency response is the ability to find the derivative of the
eigenvalues and eigenvectors of the system.

This section is based on a paper published by Rogers on
derivatives of eigenvectors and eigenvalues [3].

Consider the homogeneous set of equations of the form:
[M]{y}+[K]{y}={0} (3.1)
Assume the solution

{y}={Uj}e(XJt)

where:

{Uj}=jth eigenvector of the system.
j=jth mode

Substitution placed into equation (3.1) leads to:
Aj[M]{Uj}+[K]{Uj}=O (3.2)

If the [M] and [K] matricies are symmetric, pre-multiplying

through by {Uj}t produces the Rayleigh Quotient.

Ajiujittnltuji + {ujittklruj1=o (3.3)

Taking the partial derivative of equatin (3.3) with respect to some

parameter e yields

Aj.e {Uj}tIM]{Uj} + Ajtuj1t,e [M]{Uj} + Aj{Uj}t[M],e {Uj}+
AjiujitiMJIUj},e + {Uj1t.e LMJ{Uj} + {Uj}t[M],e {uj)+
{Uj}t[K]{Uj},e=0 (3.4a)

where the comma indicates differentiation with reSpect to e. Collecting

terms yields,

xj.e {UthLMJIUj} + {Uj}t.e(xj[M]{Uj} + LKJIUj}) +
{Uj1t(xj[MJ.e + [K].e){U1} + (AjtujitEMJ + {UjitEKJ){Uj1.e
= o (3.4b)

In view of equation (3.2) and the fact that for symmetric [M] and [K]
equation (3.2) is also valid for the transpose Aj[M]t{Uj}+[K]t{Uj},

equation (3.4b) reduces to:
A3.e=-({Uj}t[A3[M1.e+[K].e]{Uj}/({{Uj}t[M]{uj}) (3,5)

Equation (3.5) is an expression for the derivative of an eigenvalue with

a desired parameter change.
3.2 Derivatives of Damping Ratios and Undamped Natural Frequencies

To find the derivative of the frequency response, it will be
necessary to find the derivatives of the damping ratios (cj) and the
undamped natural frequencies (mj) of the system. These derivatives can

be obtained through term by term differentiation of the eigenvalues.

The eigenvalue can be written as:

*1=-Cjw1+iwj(1-CJ2>1/2 (3.6)
Take the partial of Aj with respect to a parameter e:

Ajse = ‘ (Cjwj)ae + (ij(1-Cj2)1/2 )se (3'7)

Equate the real and imaginary parts on each side of (3.7)

-Re(1je) = cj,e wj + wj,e Cj (3.8)

Im(Aje) = (l-cj2)1/2,e wj + wj,e (1-c32)1/2 ' (3.9)
where:

Re(1j) = The real part of Aj.

Im(Aj) = The imaginary part of A3.

cj and wj can be determined from the eigenvalue.

Equations 3.8 and 3.9 yield cj,e and wj,e
-(1-Cj2)1/2((1-Cj2)1/2 R6(kj).e - Cj1m(kj).e)/wj (3-10)

Cj.e

"1’9 (‘91 mi Reilj):e + wj(1-Cj2)1/2 Im(xj).e)/mj (3.11)

3.3 Derivatives of Eigenvectors

To find the derivative of the frequency response, it will also be
necessary to find the derivative of the eigenvectors. This derivative
can be obtained by taking the partial derivative of equation (3.2) with

respect to e.

(A).eIM]+Aj[M].e+[KJ,e){Uj}+(Aj[M]+[K]){uj},e = o (3.12)

10

Since the eigenvectors are independent, derivative of an eigenvector can

be written as a linear combination of the eigenvectors

2n
{Uj},e = Z ajk {Uk} (3.13)

If equation (3.13) is substituted into equation (3.12) and

equation (3.12) is pre-multiplied through by {Ug}t (g¢j) then we have

{Ug}t(xj,e[M] + xj[M],e + [K],e){Uj} +
2n
{Ug}t(lj[MJ + [K1) 2 1 ajkIUk} = 0 (3.14)

Observe that, for non repeated eigenvalues, the orthogonality relation

{Ug}t[M]{Uj}={Ug}t[K]{Uj}=O for g¢j leads to

-lj.e{Ug}t[M]{Uj} + {Ug}t([K],e - 1j[M],e){Uj} +

 

ajgtu91t([K]-Aj[MJ){ugi (3.15)
a. = _ {Uk}t([K].e - XJEMJ.E){UJI (3,15)
3k {Uk}t([K] - Aj[M]){Uk}

This is an expression for all of the coefficients, except k=j. In order
to obtain the k=j coefficient, assume that the largest element in the
jth eigenvector has been normalized to 1. Then denote this largest
element of the jth eigenvector by Ugj, where the normalization of the
eigenvector should be the same before and after the increment in

parameter. This means that:

ll

Ugj=1
and
Zn
Ugjse =2 ajk ng = O (3.17)
k=1
2n
3“ ”9i = “LENS?"
or
2h
311 = -k§1, Egg ng (3.18)

Equation (3.16), (3.18) along with equation (3.13), gives an expression

for the derivative of the eigenvectors.

12

CHAPTER 4

FREQUENCY RESPONSE DERIVATIVES

Consider the case wherein a change in the frequency response of a
system is desired which produces a lower frequency response at a given
frequency. This is illustrated by Figure 4.1, which compares the
frequency response of a system before and after some change to the
system. The peak response of the average system at 1.8 rad/sec has been
changed by AYC, from about 2.8 down to 0.6. However, note that the
resonant peak itself has only slightly decreased, from about 2.8 down to
about 2.5.

Now consider the case wherein a change in the peak response may be
desired. This is illustrated by Figure 4.2, where the response at the
second peak has been reduced by AYV, from about 2.8 down to 0.8.

Observe that, while the magnitude of the peak itself has been reduced,
the magnitude of the response at 1.8 rad/sec has again only been reduced
to 0.6.

This chapter will present derivatives of the frequency resonse
appropriate for each of these cases. First, the derivative of the
frequency response at a constant frequency will be obtained This will be

followed by the derivation of the derivative of the resonant response.

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15

4.1 Constant Frequency Derivatives

The derivative of the frequency response at a constant frequency
can be obtained by differenting the equation for the modal magnitude
(equation 2.9). The chain rule can then be applied to obtain the
constant frequency derivative. Recalling the equation for the modal

magnitude,
{A}=(iw[I]-[A])'1[U]t{fn} . (4.1a)

or in component form:

2n
A3 = Z Ujk fnk (4.1a)
k=1 (iw-Aj)

Differentiating Aj with respect to parameters that appear in the "ass,

stiffness or damping matrices gives,

2% 2n
A°,e =
J k=

t t
(U )jk:e fnk + (U )kl fnk Aj, e
1 l=1 (iw-kj) (1m-)j)2

 

 

t t t t
((U )ji.e Mlk Ukj + (U )jl Uik.e Uki + (U )1) Mik Ukj.e) (U )jk ink

 

((Ut)jl Uik Ukj) (iw-Aj)
(4.2)

or since [M] is symmetric,
2n 2n

t t
Aj,e = I (U )jk,e fnk + (U )kl fnk 13,2 -
k=l l 1 (iw-Aj) (1w-)j)2

 

 

((Ut)jl Mik,e Ukj + 2(ut)j1 M1k ukj,e) (ut)jk fnk
((Ut)j] Mlk ukj)(iw-1j) (4.3)

 

16
Using the relation {Y}=[U]{A} and applying the chain rule yields
{Y},e=[U],e{A} + [U]{A},e (4.4)
This derivation is presented in Reference 4.
4.2 Variable Frequency Derivatives

We have considered the derivative of the frequency response at a
given frequency. This derivation is valid at any specified frequency,
whether or not it corresponds to a resonant point. '

Now we wish to consider the derivative of the magnitude of a
resonant point, i.e., a point where the forcing frequency matches the
imaginary part of one of the eigenvalues. This is different in
character from the previous derivation because our frequency of interest
changes as the eigenvalue of interest responds to system changes.

Recalling the values of the component terms of the eigenvalue from
equation 3.6:

1,- = 4...». +1 wj(l-c.2)1/2

J J J
Assume the ith mode is at resonance. The equation for the ith component
of the modal magnitude can be written as:
2n

t
A,- = i (U )ik ink (4.6)
k=1 ciwi

 

and the other 2n-1 modal magnitudes have the form

2n 1:
k=1 (iw1(1-c12)1/2-Xj) (4.7)

 

l7

Differentiating equations 4.6 and 4.7 with respect to parameters
that appear in the mass, stiffness or damping produces the expressions
for the rate of change of the equation of the peak. This

differentiation yields (since [M] is symmetric),

 

 

 

 

 

 

 

 

A1,e = i ) (Ut)1k.e ink - (Ut)ik ka Ci.e - (Ut)ik ka Wi.e -
(4.10)
t t t
(2(U )il Mlk Uki.e + (U )il Mik.e Uki) (U )ik fnk
t
((U )il ”1k Uki)i ‘1“)
and
Zn 2n 2 1 2 .
Aj,e = 2 Z (UtIJk:e fnk + (Ut)jk fnk{(Wi :1/(1-c1 ) / )c1.e 1)
k=1 l=1 (1(1-C12)1/241-XJ) (1(1_C12)1/2wi -Aj)2
t t t
- (2(U )jl Mlk Ukj.e + (U )5] Uik.e Ug1)(U )jk fnk +
t _ 2 1 2 -
((U )3“) Mlk Ukj)j(1(lcl° )/w1 A3)
(4.11)
2 1 2
(Ut)jk fnk Aj,e - (Ut)jk fnk (i(l-c1 ) I wi,e )
(1(1-212)1/2 mi -xj)2 (1(1-212)1/2 m1 -1j)2

Equations 4.11 and 4.12 can be used in conjunction with 4.4 to
obtain the derivative of the peak of a frequency response with respect

to a change in the system.

CHAPTER 5

A PROCEDURE TO REDUCE RESONANT RESPONSE

This section presents an example of a procedure to lower a resonant
response of a system. In particular, an optimization technique will be
used to obtain changes in a system to reduce the magnitude of a peak of
the frequency response.

The system to be considered here, which is shown in Figure 5.1, is
made up of ten identical masses, springs and dampers. The system is

subjected to an axial load on the end mass of the form,
{F(t)}={Fo}exp(lwt) . (5.1)

The frequency response plot of the first mass (m1) of this ten mass
system is shown in Figure 5.2.

Consider the following optimization scheme. Assume the design of
the system can be changed. However, a penalty will be assesed for the
change.

P1 = ((ei'eoiI/eoiIZ (5.2)

"OM a
H

1
where,

eoi=The original value of the design parameter.’
ei=The value of the design parameter after the design change.
m=The number of design variables (ei) being changed.

The ideas underlying this penalty scheme are given in Reference 5.

18

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21

Another penalty will be assessed for the magnitude of the peak,

namely
P2=|Yn|A|Yo| (5.3)
where,

|Yn|=The magnitude of the frequency response plot after a design
change.

|Yo|=The magnitude of the frequency response plot before a design
change.

In this section a weighted sum of the two penalties will be

minimized:
P = bPI + 22 (5.4)

In this way, a lower peak will be derived while limiting the changes in
the parameters to reasonable levels.

This penalty function is initially equal to one, since with no
change in the system P1 equals zero. As e1 is changed, P1 increases as
shown in Figure 5.3. This increase is weighted by b. The larger the
value of b the greater the penalty for changing e01.

In order to minimize the penalty function P, we need

m
dP = Z (aP/ae1)de1=0 (5.5)
i=1

Thus,

3P/8e1=0 (5.6)

OI“

3P2/aei + b aPl/ae1=0 (5.7)

22

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23

P2 and P1 are differentiated according to their expressions given in

equations 5.2 and 5.3.
8P1/3e1=2((e1-e01)/e012) (5.8)
aP2/ae1=(a|Yn|/ae1)/|Yo| (5.9)

When these two expressions are substututed into equation 5.7 an

expression for the minimum e1 (eimin) can be obtained,

e1m1=eoi = ((alYnl/ae1)e02)/2b|Yo|) (5.10)

All of the terms on the right hand side of equation 5.10 are known
except for the derivative of the magnitude e1. This term can be found
by considering the equation for the magnitude of the response |Y|,

|Y| = (Rem2 + name)”2 (5.11)
where

Re(Y)=The real part of the frequency response Y.
Im(y)=The imaginary part of the frequency response Y.

Differentiating the magnitude of the response |Y| with respect to e1

 

 

yields,
a(v( = Re(Y)(8Re(Y)/8ei) + Im(Y)(aRe(Y)/aei) (5.12)
361 [VI

The derivative of the real and imaginary parts of the peak equation

are given by (4.4).

24

Once the minimized parameter eimin is obtained the new magnitude

|Yn| can be estimated with a first order expansion:

m
|Ynl=|Y0| + I (alYl/aei)AEi (5-13)
i=1
where

Ae=ei~eoi

Since only a first order Taylor's series is being used, the
procedure discussed above may be inaccurate for large changes. This can
be seen more clearly in Figure 5.4, which illustrates the first order‘
expansion of |Y| vs. e about e=1. The linear expansion follows the
straight line (line 1) of Figure 5.4. To deal with the inaccuracies
resulting from large changes in e, after solving for the change in e,
the eigenvalue roblem should be re-solved to obtain |Y| an ay/ae. The
optimization procedure can then be restarted using the new e values and
the process to obtain eimin can be done again. The procedure is
complete when the linear expansion for |Y| is satisfactorily close to
the solution of the eigenvalue problem.

The example to be addressed here will obtain changes in a ten
spring, mass, damper system to reduce the magnitude of a peak of the
frequency response plot. Initially, each mass, spring and damper will
have a value of 1., 1., and .1, respectively as shown in Figure 5.1. To
illustrate the minimization process the design variables that will be
optimized will be the ten dampers and ten springs. In this example the

changes in the dampers will be assumed to be proportional to the changes

25

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26

in the springs, i.e., Aci=Aki/5. In practice the design variables that
can be altered are based on engineering judgement and the constraints of
the system.

Table 5.1 Summarizes the first example. In this case, the change
penalty is weighted by b=1. The columns of the table show the value of
each of the ten dampers and springs after each iteration. These values
are followed by the magnitude and predicted penalty function values,
predicted using the first order Taylor series, and the new magnitudes
and penalties obtained after resolving the eigenvalue problem.

Figure 5.5 compares the frequency response of the original system
and the final system. As the table indicates, the peak has dropped from
84 to about 72. Note that the frequency response at the frequency of
the original peak has dropped far below 72, to about 30. Thus, if the
minimization was performed at a constant frequency rather than traking
the peak amplitude, a deceptively low value would have been calculated
which is not at all descriptive of the peak magnitude.

Table 5.2 summarizes a closely related example, this time with
b=0.5. In this case, it took five interactions to converge to within
one percent. Figure 5.6 compares the frequency response of the original
system and the final system. Clearly the frequency of the peak has
changed, with the magnitude of the peak of the peak reduced from about
84 down to about 66. The magnitude of the frequency response at the
initial resonant frequency now has a magnitude of approximately 13,
again illustrating that the magnitude at the initial resonant frequency
has been changed by an amount much different than the change in the peak

values.

27

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CHAPTER 6

CONCLUSIONS

This thesis presents a formulation which can determine the
sensitivity of the resonant response of the vibratory system to changes
in the system. The thesis then illustrated through an example, that
this sensitivity could be used to determine changes which are useful in
lowering the peak response.

Since the sensitivity is only computed to the first order the
optimization scheme which was used to obtain the desired design changes
is iterative. The number of iteractions involved in this proCedure
could be reduced if a more sophisticated optimization technique was
employed. This technique would consider higher order derivatives in
order to facilitate the determination of an improved design.

A point that was not considered here is the effect of design
changes in a system on other resonant peaks of a system. This could be
of importance when the resonant frequency of a system are closely
spaced. In this case, the optimization technique should be extended to

include all the peaks of interest.

31

REFERENCES

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

Caughey, T. K., and O'Kelly, M. E. 41, "Classical Normal Modes in
Damped Linear Dynamic Systems,” (burnal of Applied Mechanics, Vol.
32, pp. 583-588, 1965.

Rogers, L. C., "Derivatives of Eigenvalues and Eigenvectors,“
Technical Mote, AIAA (burnal, Vol. 8, No. 5, pp. 943-994, May
1970.

Chrostowski, 41 0., Evensen, D. A., and Hasselman, T. K., "Model
Verification of Mixed Dynamic Systems,“ (burnal of Mechanical
Design, Vol. 100 pp. 266-273, April 1978.

Starkey, (L M. and Bernard, .1 E., “Optimal Redesign Based on Modal

Data," "Proceedings," First International Modal Analysis
Conference, Orlando, Florida, November 1982.

32