Structural modification analysis using Rayleigh quotient iteration

Int. J. Mech. Sci. Vol. 32, No. 3, pp. 169 179, 1990 Printed in Great Britain.

STRUCTURAL

0020-7403/90 $3.00 + .00 © 1990 Pergamon Press pie

MODIFICATION QUOTIENT

ANALYSIS ITERATION

USING

RAYLEIGH

W. M. T o a n d D. J. EWlNS Department of Mechanical Engineering, Imperial College of Science, Technology and Medicine, London, U.K. Abstract--This paper highlights the application to structural dynamics of the sensitivity analysis methods developed by numerical analysts and presents a historical development of first- and higherorder eigenvalue and eigenvector sensitivities. Different formulae for an eigenvalue sensitivity are presented and it is shown that all of these are implicitly the same. A condition number is presented to give the limited bound of application for the first-order eigenvalue and eigenvector sensitivities. An alternative structural modification method based on Rayleigh Quotient Iteration is presented. A lumped spring-mass system with 7 degrees-of-freedom (7DoF) is used to show the applicability of the Rayleigh quotient iteration method. NOTATION [A], IdA] system matrix of the eigenvalue problem [A]{x} = 2{x} and the change in [A] [K], [M] system stiffness and mass matrices [dK], [dM] modifications in system stiffness and mass matrices {y,}, {~,}, ,~, left-hand and right-hand mass-normalized eigenvectors and eigenvalue of ith mode {x'}o, {~'},. orthonormalized eigenvectors of the original system, modified system .L.,, )~... eigenvalues of the original system, modified system 0~ design parameter of system matrices I1{ }11~ 2-norm (Euclidean norm) of a real vector, i.e. It{x} 112 ~- ({x}r{x}) a/2 I1[ ]11~ 2-norm of a real matrix i.e. II[A] 112= (max. eigenvalue of [A]T[A]) 1/2 }T, [ ]T transposes of a vector and a matrix

1. I N T R O D U C T I O N

Sensitivity analysis has been applied by several workers to the general eigenvalue problem [1-8] and, more specifically, to applications of structural modification in references [9-11]. In this area, both first- and higher-order eigenvalue and eigenvector sensitivities have been investigated with a view to predicting the dynamics of a modified structure from knowledge of its properties in an original, or unmodified, state. As the sensitivity analysis of a mechanical structure is based on a Taylor expansion of the eigenvalues and eigenvectors of the unmodified structure, and the computation of the higher-order terms of this series is difficult and time consuming, the effectiveness of this method is limited to small modifications. However, it is not easy to determine what is "small". In this paper, a condition number is presented to indicate how sensitive the eigenvalues and eigenvectors of a mechanical structure are to small modifications. The value of this condition number is used to determine a limit of applicability for the first-order eigenvalue and eigenvector sensitivities. The Rayleigh quotient provides a well-known procedure for the approximate evaluation of the eigenvalue of a structure. It is shown that under some conditions the Rayleigh quotient and first-order eigenvalue sensitivity are equivalent. Structural modification problems possess an inherent advantage in that some initial conditions are known and this information is very useful for an iteration method of analysis. A c c o r d i n g l y , a n iterative p r o c e d u r e u s i n g the Rayleigh q u o t i e n t w o u l d seem to offer a d v a n t a g e s for the s o l u t i o n of the m o d i f i c a t i o n a n a l y s i s p r o b l e m , especially in view of the c u b i c c o n v e r g e n c e p r o p e r t y of this m e t h o d . 2. THE O R I G I N A L P U R P O S E O F D E V E L O P I N G THE F I R S T - O R D E R P E R T U R B A T I O N F i r s t - o r d e r p e r t u r b a t i o n e s t i m a t e s (i.e. e i g e n v a l u e a n d e i g e n v e c t o r sensitivities) h a v e l o n g b e e n used b y n u m e r i c a l a n a l y s t s a n d scientists to investigate the stability of the e i g e n v a l u e p r o b l e m [ A ] { x } = 2{x}. T h e s e e s t i m a t e s h a v e the a d v a n t a g e t h a t they p r o v i d e a quick, 169

170

w.M. To and D. J. EWlNS

nonrigorous look at how eigenvalues and eigenvectors change when the elements of matrix [A] vary within the limits of permissible error. In most matrix computational textbooks, first-order perturbation theory is used to obtain the error bounds in computing eigenvalues and eigenvectors on a digital computer. By translating the above-mentioned feature to suit structural analysts, a first-order sensitivity analysis is used to determine the "order of importance" ranking of a structure's degrees of freedom for structural modification, for each mode of vibration. 3. HISTORICAL DEVELOPMENT OF FIRST (AND HIGHER-) ORDER SENSITIVITY In 1846, Jacobi [1] published a result on eigenvalue sensitivities for an eigenvalue problem I-A]{x} = 2{x}. Wilkinson [2] presented clear derivations for the first-order perturbation equation of an eigenvalue in terms of the normalized left-hand and right-hand eigenvectors. The equation presented in [-2] is:

=

{y,}" {x,}

(1)

In the 1960s, some methods were developed for sensitivity analysis of electronic networks and are notable for their nonreliance on eigenvectors in the eigenvalue sensitivity formula. Rosenbrock 1-3] and Reddy 1-4] developed a formula for eigenvalue sensitivities in terms of the matrix [A] and its eigenvalues. 0 2 , _ trace { a d j ( [ A ] - 2 , [ I ] ) [ ~ ] } c~

(2)

trace {adj([A] - 2/[I])}

If 2i is a simple eigenvalue of the matrix [A], then adj([A] - ).,[I]) can be expressed in terms of the corresponding left-hand and right-hand eigenvectors of 2,: adj([A] - 2,[1]) =

z,{x,} {y,}+

(3)

where z i is a constant. Therefore, trace {adj([a] - 2 ill])} = trace(r/{xi} {y,}T) = ~, {y,}T {Xi }.

(4)

Bodewig [12] showed that if [Q] is square and rank [Q] = 1 so that [Q] is a simple product of a column {x, } and a row {y, }+, i.e. [Q] = ~, {x, } {y, }v, and if [c~A/&t] is arbitrary, but square, then trace(I-Q] [~-aA] ) =

trace([~A ][Q])=zi{yi}T[~I{X,}.

(5)

By substituting equations (4) and (5) in equation (2), equation (1) is obtained. Thus, in the computation ofadj ([A] - 2, [I]), both left-hand and right-hand eigenvectors are implicitly computed, in view of equation (4). The trace of the adj([A] - ; h [ I ] ) , can also be expressed as: trace{adj([A] - 21[I])} = ~

()~, - 2i).

(6)

i,i¢:r

In structural analysis [A] is related to the stiffness matrix (usually written as [K]) and [ I ] is the unit mass matrix. If a point Single Degree of Freedom (SDoF) stiffness modification is made at location j, the numerator of equation (2) can be written as: trace

r

'

•

"

°" l(j,j)° il}

0

o

.

.

= A#(2,)

= the (j,j) cofactor of (I-A] - 2i(I]).

(7)

Structural modification analysis

171

The equation of motion for a linear system is: ([a]

-

hE/]){x}

= {f}.

(8)

It has been shown [13] that any frequency response function (FRF) of a grounded undamped structure can be described completely in terms of its poles or eigenvalues (squares of natural frequencies), its zeros (squares of anti-resonance frequencies), and a constant. Cjj 1-] r=l

where

Hjj(2) = Cjj = 2, = JJB~ =

1(9)

receptance of point measurement at location j static flexibility rth pole kth zero for Hjj.

The F R F H jj(2) can also be obtained from equation (8). Hjy(2)

= x~(2) = the(j,j) cofactor of (EA] - 211]) det ([A] - 411])

(10)

since ~-~-/r,r,/r=lfi2r"

det ( [ A ] - 2 [ I ] ) = ,=,I~I( 4 - 2 , ) = ,=,fi ( 1 -

(11)

Using equations (9), (10) and (11), the (j,j)th cofactor of ([A] - 2 I-I]) can be expressed as:

c. the (j,j)th cofactor of ([A] - 2 [ I ] ) =

r=l

k=l n-1

t" k- 2) (12)

H JJk

k=l

Substituting equations (6) and (12) in equation (2), we obtain:

~i --

Cjj =

.-1

~

fi r=l

n--1

~r U (JJflk -- ~i) k=l

(13)

U JJflk fi ( 2 r k=l i,i¢:r

2i)

and this is the equation for a point SDoF stiffness modification obtained independently by Skingle [11]. The formula shown above for calculating the sensitivity of an eigenvalue to changes in the matrix (equation (13)) is implicitly the same as these shown in equations (1) and (2). This is to be expected if one begins with equation [A] {x } = 2 {x }. As the system properties do not vary under a harmonic excitation, the difference between equation (13) and equations (1) and (2) is the number of design parameters used to calculate the first-order eigenvalue sensitivity. Fox and Kapoor [5] considered the special case of symmetric stiffness [K] and mass [M] matrices but developed techniques applicable to more general cases. For eigenvalues their formula is:

{xi}

.L~-J)

{ .}

(14)

in which it is assumed that the eigenvectors are normalized such that

(xi}TEM]{xi}

= 1.

(15)

172

W . M . T o and D. J. EWINS

The first-order eigenvector sensitivity is: 1

T

_

+k=,,k*~/-" (2,

_

~M

2k){Xk}

OK

{X,} T

.

--2,

{Xk}.

(16)

Rogers [6] derived sensitivity formulae for eigenvalues and eigenvectors of a general problem and stated the need for two sets of normalization conditions for non-self-adjoint system. Vanhonacker [9] derived some formulae calculating the differential sensitivities of a mechanical structure subjected to parameter changes. The formulae obtained were based on an nDoF system under a sinusoidal excitation. Comparing his derivations with the equations shown above, one can observe that the differential sensitivity is a special case in the classical methods of sensitivity analysis. Many papers [7, 8] have presented the higher-order eigenvalue and eigenvector sensitivities. Wang et al. [10] have investigated the accuracy of structural modification by calculating the first- and second-order eigenvalue and eigenvector sensitivities and showed the divergence phenomenon for large structural modification. It has been shown [10] that including some higher-order terms does not always ensure a more accurate prediction. 4. C O N D I T I O N

NUMBERS

For an eigenvalue problem [A]{x'} = 2 {x'} Stewart [14] has shown that if {y'} and {x'} are orthonormalized left-hand and right-hand eigenvectors of [A] with simple eigenvalue 2, then II{y'} 112 is a condition number for 2 (this condition number was first observed by Wilkinson [2]). Let [U] e E" x(,- 1) be a matrix such that ({x'}, [U]) is unitary, then 11(2[I] -[u]T[A][U]) -1 I]2 is a condition number for the eigenvector {x'}. The inverse of I1( - • 0-1112 is a measure of the separation of 2 from its neighbours. In Stewarrs book, the following theorem is given: let 2 be a simple eigenvalue of [ A ] e E "×" with right-hand eigenvector {x'} and left-hand eigenvector {y'}. Suppose {x'} has been scaled so that rl {x'} 112 = 1 and {y'} has been scaled so that {y'}T{x'} = 1. Let [U]eR n×tn-1) be chosen so that ({x'}, [U]) is unitary and set

({x'}'[U])T[A]({x'}'[U])=[~

[u]T[A]{X'}T[A][U]][U] '

(17)

Let IdA] e ~"×" be given and let =ll[dh]ll2, and

rt=llfU]W[h]W{x'}lJ2,

~= JI[u]T[dA]{x'}II2

# = I1(;-[I] - [u]T[A][U]) -' II;'.

Then if,

7(q (# _ +e)2~) < ~1,

(18) (19)

an eigenvalue Am of [A + dA] with eigenvector {x' }m can be predicted accurately by firstorder perturbation theory. This theorem is modified to deal with the generalized eigenvalue problem [K] {x'} = 2[M]{x'}. The system matrix [A] is substituted by the matrix [ M ] - X [ K ] if [M] is invertible, the perturbation matrix [dA] is approximated by [M + d M ] - a ( [ d K ] - 2[dM]) if [ M + dM] is invertible. 5. R A Y L E I G H

QUOTIENT

The Rayleigh quotient provides an approximation of the eigenvalues based on trial vectors that are often approximate eigenvectors of a self-adjoint system. Assume that the eigenvector of a modified system {x' }m is approximated by the eigenvector of the original system {X'}o, then:

{X'}oT[K + dK] {X'}o & =

+ TM]

(20)

Structural modificationanalysis

173

where 2R is called the Rayleigh quotient. For a stiffness modification, the Rayleigh quotient is expressed as: 1 ;..

=

+

'~'~t[dK] v,

{X'}o.

(21)

Comparing the results obtained by Rayleigh quotient (21) with the results obtained by perturbation theory (14), it is noticed that the Rayleigh quotient and first-order perturbation theory yield identical results for a self-adjoint system when there is no mass perturbation. For a mass modification, the Rayleigh quotient can be expanded to a power series: 1 &

+ {. ,}o EM3{x,}o ( - ;-o{X'}o EdMl{x'}o)

=

+ 2o ({x'}x°[dM]{x'}°)2 - ). ( { x ' ~ [ d M ] { x ' } ° ) 3 + . . .

(22)

Hence, the first-order term in the Rayleigh quotient is the first-order eigenvalue perturbation ((62i/6~)A~). The second-order term is one of the mass perturbation terms which appear in the second-order eigenvalue sensitivity. However, most of the mass perturbation terms for the second-order eigenvalue sensitivity are not accounted for in the Rayleigh quotient expression. Some higher-order mass perturbations terms are present because mass terms appear in the denominator of the Rayleigh quotient. 6. RAYLE1GH QUOTIENT ITERATION If {X'}o is an approximate eigenvector for the modified system, then 2R (as defined above in (20)) is a reasonable choice for the corresponding eigenvalue. On the other hand, if 2r is an approximate eigenvalue, inverse iteration theory shows that the solution to ([K + d K ] - 2 r i m + dm]){X'}m = [ m + dm]{x'}o will almost always be a good approximate eigenvector. Combining these two ideas in the natural way gives rise to Rayleigh quotient iteration. For a self-adjoint system [K + dK] is symmetric [M + d M ] is symmetric and positive definite, and the Rayleigh quotient iteration is expressed as follows [15]: {X'}o given, II {X'}oN2 = 1. For k = 0, 1. . . . {x'}~[K + dK] {x'}k , ~t dM] " Solve ([K + dK3 - 2 k [ M + d M ] ) { z } k + 1 = [ M + d M 3 { X ' } k for {Z}k+l

{z}k+l {x'}k+, -II{z} +1112' 7. NUMERICAL EXAMPLES In order to evaluate the effectiveness of the first-order sensitivity and the Rayleigh quotient iteration methods, a modification study was made on a lumped spring-mass model with 7DoF shown in Fig. 1. Figure 2 shows the point frequency response function for this model when the excitation was applied on point 4 (damping loss factors 0.001 were used to produce the FRF). For several parameter changes at different locations of the model the first-order sensitivity and the Rayleigh quotient iteration have been applied to the original data to predict the consequent changes to the system's natural frequencies. The results are compared with the exact solution obtained by complete reanalysis. In Fig. 3(a-c), the shifts of eigenvalues are plotted with respect to a mass change at point 6. From these figures, it is noticed that the Rayleigh quotient iteration method gives accurate results for modes 1 to 6 although some numerical difficulties are observed for

174

W . M . T o and D. J. EWINS

M i = lkg

i= 1,2,...,7

K 1 ffi LF~N/m KT.= 2E4 N/m

FIG. 1. A lumped spring-mass model with 7 degrees-of-freedom.

2OO 0

----1

-200 60

J E o~ E

Doto from INERT X4F4

-I10 008

Frequency

I 637"4

(Hz)

FiG. 2. A point F R F of the analytical model.

mode 7. Figure 3(c) shows that mode 7 converges to mode 6 if the mass change is greater than 60%. The first-order sensitivity method is a linear approximation based on an infinitesimal change of a parameter, and it is seen that the accuracy of prediction based on this parameter decreases with increasing magnitude of the mass change. Figure 3(d) shows the condition number (LHS of inequality (25)) against the percentage of mass change. Modes 3 and 4 are close modes and for these the first-order sensitivity is applicable only for small mass changes ( ~ 2.5%). Modes 5, 6 and 7 are also ill-conditioned for the mass perturbation at point 6. The accuracy of the mode shapes predicted by the two techniques is assessed by using the Modal Assurance Criterion (MAC) [16]. Tables 1 and 2 contain the MAC values between eigenvectors from the first-order sensitivity analysis and the exact solution and between the Rayleigh quotient iteration and the exact solution. Examining the tables, it is seen that the Rayleigh quotient iteration yields consistently better correlation with the exact solution except when the mass change is greater than 60%, mode 7 converges to mode 6. In Fig. 4(a c) the shifts of eigenvalues are plotted for the case of a stiffness change made between points 3 and 4. It is seen that the Rayleigh quotient iteration method gives accurate prediction for all modes. The condition number against the percentage of stiffness modification change is shown in Fig. 4(d). From this figure, it is noticed that modes 2, 3, 5 and 6 are ill-conditioned for this stiffness modification. The limit of application for the first-order perturbation theory for all modes is approximately equal to 4%. The MAC values between eigenvectors from those two techniques and those from the exact solution are shown in Tables 3 and 4. From the results, it is observed that the Rayleigh quotient iteration again yields better results even when the magnitude of stiffness modification is large. 8. P O S S I B L E

USE OF THESE

MODIFICATION

TOOLS

An analytical or Finite Element model of a complex structure normally has a large number of degrees-of-freedom and design variables. The primary analysis yields some eigenvalues in a given frequency range, together with the corresponding eigenvectors.

32O0O

33000

la)

50~0

lb)

286(i0

[~- 30000

0

0

15ooo,

20~00 '

> Z 31000 t.t2

< > Z t~

•

25000

•

i

',

'

I

•

I

40 60 % OF MASS CHANGE

I

.....

•

~

80

" I

80

I

.... •, o - - - , ~ . . _ _ . . ~ _

~-z.-.Z--"

40 60 % OF MASS CHANGE

' 'I

•, .

.....

Key :- Exact'solution' (EX) l s t - o r d e r sensitivity (SE) Rayleigh quotient iter. (R1)

R1 3 SE4 R14

-

~"~,-" --.~-

_

_..°_.

100

l

~

EX 3

RI2

--+'-

100

RI 1 EX 2 SE 2

EX I SE 1

--"~-'-a'...... •o , - -

-'-D'..... ' # ' -

~

0

0.00 : 0

0.:15

(d)

E

Z 0.50 ©

0.75

45000

~o 55000 ~

t.tl

75000'

85000'

"

'a"

~l !1

" ,'

I

/

"

I /

5

'

""

80

I

",o.

'

1

15

'r e

,

~

Key ~ :- Condition number (C)

I0 % OF MASS CHANGE

f

40 60 % OF MASS CHANGE

I

~ ' ~ ' ~ " : " ' " g

,;t

lii !;;

ii;

J i:

i

20

"1"

~ ~ ' ; t g ~ Z Z " . . . . . . ....

................ •.,,:::.g ................. ~ . ....................

FiG. 3. ( a - c ) E i g e n v a l u e s shift w i t h m a s s c h a n g e at l o c a t i o n 6. (d) C o n d i t i o n n u m b e r s v a r y w i t h m a s s c h a n g e at l o c a t i o n 6.

20

I

20

'

20

- "--~ -'-

[email protected]

-----o---...... • , ..... ---'~--

1013

-....4--'-~-•"---'~-'~ ---4----D-..... o- .... --'~-

...... o- .....

-'-~'-

EX 5

C6 C7

CA (:5

CI C2 (:3

RI5 EX 6 SE 6 RI6 EX 7 SE 7 Pd7

SE 5

©

~q

176

W . M . T o and D. J. EWINS

TABLE l. M A C VALUES FOR THE MODIFIED EIGENVE('TORS BETWEEN FIlE [ST ORDER SENSITIVITY ANALYSIS AND THE EXACT SOLUTION [~ORTHE MASS MOI)lI~l(,~.rION AT LOCATION 6 60% P e r t u r b a t i o n

Exact

1

2

1 2 3 4 5 6 7

1.000 0.004 0.006 0.000 0.016 0.000 0.004

0.003 1.000 0.003 0.000 0.007 0.000 0.002

Predicted l i s t - o r d e r sensitivity) 3 4 5 0.002 0.(X)O 0.959 0.031 0.019 0.000 0.004

0.002 0.003 0.035 0.964 0.003 0.000 0.000

0.005 0.001 0.000 0.(X)0 0.812 0.157 0.012

6

7

0.002 0.001 0.004 0.000 0.094 0.834 0.082

0.021 0.011 0.018 0.001 0.06 l 0.000 0.938

6

7

0.003 0.003 0.008 0.000 0.130 0.667 0.203

0.037 0.022 0.035 0.001 0.101 0.000 0.821

100% P e r t u r b a t i o n

Exact

1

2

1 2 3 4 5 6 7

1.000 0.012 0.016 0.001 0.034 0.000 0.005

0.008 0.999 0.007 0.000 0.016 0.000 0.003

Predicted (1 st-order sensitivity) 3 4 5 (1.(X14 0.001 0.855 0.111 0.049 0.000 0.005

0.005 0.009 0.097 0.879 0.025 0,(X10 0.001

0.007 0.000 0.001 0.000 0.625 0.312 0.(X)9

TABLE 2. M A C VALUES FOR THE MODIFIED EIC,ENVECTORS BETWEFN THE RA'*[.EIGH QUOTIENT ITERATION AND THE EXACT SOLUTION I-OR THE MASS MODlt-I('ATION AT LOC,\TION 6 60% P e r t u r b a t i o n

Exact

1

"~

1 2 3 4 5 6 7

1.000 0.004 0.006 0.000 0.016 0.000 0.004

0.004 1.(X)0 0.002 0.000 0.007 0.000 0.002

Predicted (Rayleigh quotient iteration) 3 4 5 0.006 0.002 1.000 0.000 0.010 0.000 0.003

0.000 0.000 0.(X10 I .(X)0 0.001 0.000 0.000

0.016 0,007 0.010 0.001 1.000 0.000 0.006

6

7

0.000 0.000 0.000 0.000 0.000 1.000 0.000

0.004 0.002 0.003 0.000 0.006 0.000 1.000

6

7

0.000 0.000 0.000 0.000 0.000 1.000 0.000

0.000 0.000 0.000 0.000 0.000 1.000 0.000

100% Perturbation

Exact

!

2

1 2 3 4 5 6 7

1.000 0,014 0.017 0.001 0.034 0.000 0.005

0.014 1.000 0.007 0.000 0.014 0.000 0.002

Predicted (Rayleigh quotient iterationl 3 4 5 0.017 0.007 1.000 0.000 0.017 0.000 0.003

0.001 0.000 0.000 1.000 0.001 0.000 0.000

(/.034 0.014 0.017 0.001 1.000 0.000 0.005

. . . . .

-

,

100

EX3 SE 3

~ m

(b)

31000

m~ 31500

rr 3 2 0 0 0 [.., ,q-

32500

33000

33500

(a)

~ 20

-

I

~

~

~

~

~ , 40 60 % OF STIFFNESS C H A N G E

-

80

,

--.t---.---.-~-

100

-

--',t-o

....w - -

- "q" ---O'-

-.-D.I .... ' ~ - -

RI4

RI3 EX4 SE4

(d)

0.00

Z O 0.25

0-50

0.75

1.00

(c)

0

0

I

; , l

i

t

"

! ! ! /i

80

I

- "*- -.-O-.....' * " - --.t,- -.-D-..... , t . - "-~- -

20

A

--41,--

-.-as..-

---'4"--

---'Ill--

. . . . . •g , .....

~

100

:l.---::zEZ~

~ K e y :- Condition , N u m b e r (C) !

10 15 % O F STIFFNESS C H A N G E

:

/

l

I

I

I I I I

/" I t

;t /

5

!

...........

...........

40 60 % O F STIFFNESS C H A N G E

:

I

20

- - ..--...'7..--.....-.,4,

. _ .._.,,.._,,~,,,.,,,.~...-".~-.'1~" .....

-'-D-.....~'-"

FiG. 4. (a-c) Eigenvalues shift with stiffness c h a n g e b e t w e e n points 3 a n d 4. (d) C o n d i t i o n n u m b e r s v a r y with stiffness c h a n g e b e t w e e n points 3 a n d 4.

.....

80

"-''''''¢1''~''~'r4~'--~'''-'~

40 60 % OF STIFFNESS C H A N G E

.t._._,~.,--t

20

6OOOO

I

SE 2 RI2

RI1 bO EX2 ~ 90000

5000

,

......*"..... ---'~--

--'¢"-.-o.-

lOO000

70000

,

Key :- Exact solution (EX) 1 st-order sensitivity (SE) Rayleigh quotient iter. (RI)

-'-D'EX1 I ......tt, .... SE 1

[- 10000

t"4

~3 Z

,~ ~5ooo

0

> Z ua

2oo0o

25000

(26

C4

C1

RI5 EX 6 SE 6 RI6 EX 7 SE 7 RI7

EX 5 SE 5

0

O

E

e-

178

W . M . To and D. J. EWlNS

TABLE 3. MAC VALUESFOR THE MODIFIED EIGENVECTORS BETWEEN THE 1ST-ORDER SENSITIVITYAND THE EXACT SOLUTION FOR THE STIFFNESS MODIFICATION BETWEEN LOCATIONS 3 AND 4 100% Perturbation

Predicted (lst-order sensitivity) Exact

1

2

3

4

5

6

7

1 2 3 4 5 6 7

0.979 0,014 0.000 0.001 0.000 0.002 0,004

0.017 0.909 0.001 0.017 0.001 0.020 0.035

0.000 0.001 0.994 0.004 0.000 0,000 0.000

0,000 0.023 0.005 0.964 0.000 0.003 0.005

0.001 0.016 0.000 0.004 0.255 0.532 0.192

0.001 0.014 0,000 0.003 0.697 0.125 0.161

0.000 0.005 0,000 0.001 0.004 0.506 0.483

TABLE 4. MAC VALUESFOR THE MODIFIED EIGENVECTORS BETWEENTHE RAYLEIGHQUOTIENT ITERATIONAND THE EXACT SOLUTION FOR THE STIFFNESS MODIFICATION BETWEENLOCATIONS 3 AND 4 100% Perturbation

Predicted (Rayleigh quotient iteration) Exact

1

2

3

4

5

6

7

1

1.000

0.000

0.000

0.000

0.000

0.000

0.000

2 3 4 5 6 7

0.000 0.000 0.000 0.000 0.000 0.000

1.000 0,000 0.000 0.000 0.000 0.000

0.000 1.000 0.000 0.000 0.000 0.000

0,000 0.000 1.000 0,000 0.000 0.000

0.000 0,000 0.000 1.000 0.000 0.000

0.000 0.000

0.000 0.000

0.000

0.000

0.000 1.000 0.000

0,000 0.000 1,000

Usually, the model needs to be progressively modified during an automated optimum design process. The first-order eigenvalue sensitivity ranks possible structural modification sites in their order of effectiveness at influencing each particular mode. The Rayleigh quotient iteration produces a more accurate prediction. The condition number monitors the behaviour of the modes in structural modification. By combining these three tools, an iterative program can be written to calculate the optimum condition for a structure specified by the user. 9. S U M M A R Y

First-order eigenvalue sensitivities are very useful to rank the order of importance for the sites during structural modification. First-order eigenvector sensitivities are sometimes impossible to compute because to do so requires the complete set of right-hand eigenvectors for a self-adjoint system. Calculations of higher-order eigenvalue and eigenvector sensitivities are difficult and expensive. Summation of a truncated eigensystem sensitivities series does not converge rapidly to give satisfactory results. A condition number has been presented to provide information about the sensitivities of the eigenvalues and eigenvectors of the generalized eigenvalue problem [K] {x} = 2[m] {x} to small perturbations of [K] and/or [M]. A Rayleigh quotient iteration method has been presented to compute the modified eigenvalues and their associated eigenvectors of a large analytical model. It has long been known that if the first prediction of an eigenvalue by using the unperturbed eigenveetor is closer to the modified one than its neighbour eigenvalues, this iteration method converges globally and the convergence is ultimately cubic. Acknowledgement The authors gratefully acknowledge the financial support of the Croucher Foundation in Hong Kong.

Structural modification analysis

179

REFERENCES 1. C. G. J. JACOBI, "Uber ein leichtes Verfahren die in der Theorie der Saecularstoerungen vorkommenden Gleichungen numerisch aufzuloesen", Z. Reine Angew. Math. 30, 51 (1846). 2. J. H. WILKINSON, The Algebraic Eioenvalue Problem, pp. 62-109. Oxford University Press, London (1963). 3. H. H. ROSENBROCK, Sensitivity of an eigenvalue to changes in the matrix. Electron. Lett. 1, 278 (1965). 4. D. C. REDD¥, Sensitivity of an eigenvalue of a multivariable control system. Electron. Lett. 2, 446 (1966). 5. R. L. Fox and M. P. KAPOOR, Rates of changes of eigenvalues and eigenvectors. AIAA J. 6, 2426 (1968). 6. L. C. ROGERS, Derivatives of eigenvalues and eigenvectors. AIAA J. 8, 943 (1970). 7. C. S. RUDISlLL, Derivatives of eigenvalues and eigenvectors for a general matrix. AIAA J. 12, 721 (1974). 8. H. V. BELLE, Higher order sensitivities in structural systems. AIAA J. 20, 286 (1982). 9. P. VANHONACKER, Differential and difference sensitivities of natural frequencies and mode shapes of mechanical structure. AIAA J. 18, 1511 (1980). 10. J. WANe, W. HEYLEN and P. SAS,Accuracy of structural modification techniques. Proc. 5th Int. Modal Analysis Conference, pp. 65-71 (1987). 11, G. W. SK]NGLE, Resonance and anti-resonance sensitivities for SDoF mass and stiffness modification. Imperial College, Dynamics Section, Int. Rep. No. 8802 (Jan. 1988). 12. E. BODEWlG, Matrix Calculus, pp. 35-36. North Holland, Amsterdam (1959). 13. W. J. DUNCAN, Mechanical Admittances and their Applications to Oscillation Problems. HMSO, London (1947). 14. G. W. STEWART, Introduction to Matrix Computations, pp. 289-307. Academic Press, New York (1973). 15. G. H. Goeue and C. F. VAN LOAN, Matrix Computations, pp. 317-318. North Oxford Academic, Oxford (1983). 16. R. J. ALLEMANGand D. L. BROWN, A correlation coefficient for modal vector analysis. Proc. 1st Int. Modal Analysis Conference, pp. 110-115 (1983).

I~ 32:3-8