An accurate method for computing eigenvector derivatives for free-free structures

t

Pergamon

0045-7949(94)EO254-Y

1135-l

Cmpur~r.s & Srrucrurm Vol. 52. No. 6. pP. 143. 1994 Copyright 1994 Elsevier Science Ltd Printed-in Great Britain. All rights rcnerved cm457949/94 $7.00 + 0.00

AN ACCURATE METHOD FOR COMPUTING EIGENVECTOR DERIVATIVES FOR FREE-FREE STRUCTURES Zhong-sheng Liu,t Su-huan Chen$ and You-qun Zhao$

tDepartment

of Mechanics, Peking University, Beijing 100871, People’s Republic of China

IDepartment

of Mechanics, Jilin University of Technology, Changchun 130022,

People’s Republic of China (Received 7 June 1993)

Abstract-This paper presents an improvement to the truncated modal summation representation of eigenvector derivatives for free-free structures. In this method, the eigenvector derivative is assumed to be spanned by the available lower modes and the unavailable higher modes. The contribution due to the available modes to the eigenvector derivative is computed from the governing equation for the eigenvector derivative, while the contribution due to the unavailable modes is explicitly and accurately expressed using the available modes and the system matrices. An approach to the singularity of the stiffness matrix is presented. in which a shift in eigenvalues is introduced to avoid the singularity and accelerate the convergence speed. A numerical example is given to illustrate the effectiveness of the present method.

INTRODUCTION

Design sensitivity analysis, the study of changes in system response with respect to parameter variations, is of significance in a variety of engineering disciplines. Such disciplines include (I) system identification, (2) development of robust control systems, (3) use in gradient-based mathematical programming methods, (4) approximation of system response to a change in a system parameter, and (5) assessment of design changes on system performance [I]. Recent developments in this area were surveyed by Haftka and Adelman [2] and Baldwin and Hutton [3]. Derivatives of eigenvectors (vibration mode shapes) with respect to design variables are particularly useful in certain analysis and design applications; e.g. approximation of a new eigenvector due to a perturbation in a design variable; determining the effect of design changes on the dynamic behavior of a structure; and tailoring eigenvectors to minimize displacements at certain points on a structure. There are a number of efficient methods for calculating eigenvector derivatives. The different methods seek to overcome the practical difficulty of solving a singular matrix equation. Such methods include the finite-difference method, the modal method [4-IO], a modified modal method [I I] and Nelson’s method [12]. To obtain exact solutions, Nelson’s method appears to be more efficient than other approaches. The two improved approximate methods to the truncated modal summation representation of eigenvector derivatives was presented in a recent

work by Wang[ll]; the first method is the explicit method, in which a mode-acceleration type approach was used to obtain a static solution to approximate the contribution due to unavailable higher modes; the second is the implicit method. However, there are two problems to be further studied in Wang’s method: the first one is that it cannot deal with structures with rigid-body modes because it involves the inverse of stiffness matrix; the second one is that it is not able to given an accurate representation of the contribution due to the unavailable modes. This paper tries to extend Wang’s method in two senses: the first one is to extend Wang’s method to include the free-free structures; the second is to give an accurate expression of contribution due to the higher unavailable modes in terms of the available modes and the system matrices. A numerical example is given to illustrate the effectiveness of the present method. TECHNICAL

BACKGROUND

The eigenvalue problem for undamped systems in structural dynamics is (K - i, M)X, = 0

(1)

XfM.r, = 1,

(2)

where K and M are N x N real symmetric stiffness and mass matrices respectively, E+is an eigenvalue, X, is the associated eigenvector. Taking derivatives of eqns (I) and (2) with respect to a design variable b of

1135

1136

Zhong-sheng Liu et al.

concern, yields the following governing equation for eigenvector derivatives: d.u, (3)

A’db =&

first lower modes are calculated, while the other higher modes are truncated. Assume that the first L (L < N) modes are available with the other (N - L) modes truncated. An approximate eigenvector derivative can be written as

d”l, cL

(4)

(10)

db -,=,“+

where where c, is given by eqn (9). A,=K-1,M (c) Wang’s method [l I]

F, = -

$

- 2

M - 1, !$

_y, >

H

dM

E

--!x~_x.

’ ’ db

”

Equation (8) can be written as

(5)

(6) where

eigenvalue: Aiz ith xi = ith

w

and we assume that 1, < 1, < , . . , ~1,. The eigenvalue derivative is given by (7)

when L4N, li
BRIEF REVIEW OF MODAL METHODS FOR COMPUTING EIGENVECTOR DERIVATIVES

,=L+l

L+2,

2,

(13)

or Since the matrix A, is singular, the eigenvector derivatives can not be obtained from eqns (3) and (4) by standard techniques. However, several methods for computing eigenvector derivatives have been developed in the literature. They are all based on eqns (3) and (4).

s,,=

(14)

The first terms of ean . (14) \ I can be shown to be

(a) Exact modal method

,:,9x,=

For an N-degree-of-freedom structure, if all the modes are available, the eigenvector derivative can be expressed as a series expansion of the eigenvectors, i.e.

K-IF,.

I

(15)

Define y, = K-‘F,

(8)

(16) (17)

where Using these definitions, eqn (14) becomes (9)

s,, =y,-‘0.

(18)

Therefore, eqn (11) can be written as (b) Modal truncation method For a large-scale structure, there may be thousands of degrees of freedom. In this case only a few of the

(19)

1137

Computation of eigenvecctorderivatives THE PRESENT METHOD

In

Contribution due ta the unavailable modes

in Wang’s method, the contribution due to the unavailable modes, SR, is approximated by SRR, which is valid when I, G$AL+t (i d L). Then, we try to given an exact expression of SR only using the available modes and the system matrices. Equation (12) can be rewritten in compact form: s, = X,(/l,, - R,I)-‘X;F,,

order to obtain yk (k = 0, I,2,+. . ), the fol’low-

ing recursive steps can be used,

(20)

y*= K-*6 JJk= K-‘(My&,),

k 2 I.

(31)

The above recursive process can be terminated according to the accuracy requirement specified. If c is denoted as the accuracy requirement, the termination condition can be stated as

where

I&(r) - SR(Y- I)][2 6 L,

(32)

(33) and f is an identity matrix, the superscript T denotes

An qpraach

matrix transpose.

As can be seen from eqns (16) and (31), both W ang’s method and the method mentioned in eqn (31) fail to deal with the free-free structures because of singularity of the stiffness matrix. Now we try to transform the eigenproblem with a singular stiffness matrix into its equivalent eigenprobtem with a non-singular stiffness matrix, in the sense that these two eigenproblems have the same derivatives of eigenvalues and eigenvectors. Consider the following eigenvalue problem

Since A,< 1, (j > L -t l), matrix (A, - ll,I)-’ can be expanded into a convergent series:

(n, - A,])-’ = f A:/i,k-’ k=O substituting

(23)

eqn (23) into eqn f20) yields

tinfree-free

structure

(24)

(34) .: ,%K?;= 1,

It can be shown that (see Appendix)

(35)

Here p is a non-zero scalar parameter. It can be shown that (k=1,2,...),

(25)

xi= li

where

- fi

R{= x, , R N).

Substituting

(38)

is non-singular if P $4 From eqns (37) and (38) we have 2

eqn (25) into eqn (24). one has

(37)

_ d;i,

db -db

dxj -=db

where (29) (30)

dz, db’

Accordingly. the derivatives d_u,/db can be obtained from the derivatives d&/dB of the eigenproblem uf eqns (34) and (39, in which K is non-singular. In this sense, both Wang’s method and the method presented in eqn (31) can be applied to deal with the

Zhong-sheng Liu et al.

1138

free-free structures with rigid-body modes after a transformation from the eigenproblem of eqns (1) and (2) to the eigenproblem of eqns (34) and (35) is made. Effect of eigenvalue shift p on the convergent

speed

Noting that Li in eqn (23) should be replaced by I, (Ii = 1, - p) for the free-free structural eigenproblem, it can be seen that the convergent speed of the series eqn (23) depends on the eigenvalue shift p. Rewriting the series as

(A,,- X,1,-’ =&‘diag

k, = 2 x lo-‘, kz = 3 x IO-‘, k, = 7 x 103, k,=

13 x IO), k,=2

k,=l8x

x IO3

103, k,=22x

IO-‘, k,=8x

103,

k, = 22 x IO3(N/m).

This is a 10 degree-of-freedom system, N = 10, whose first 4 modes (contain one rigid-body mode) are computed with the eigenvalues as AL = diag(O.OOOOOO, 131.8465,653.6057,SSS.O055).

i Pi+*, . .1:

f pi+,, k=O

k=O

P”N 3 k=O

>

(41) where 2, 2, - P P’=~==

For the following four cases, the first four eigenvector derivatives are to be computed. Case I:

0 iZ*O (j>L+l).

(42)

When p is chosen to be close to li, the convergent speed of the series eqn (41) will be faster. However, p cannot be chosen to be equal to Ai, in order to avoid the singularity of matrix R (R = K - PM). Assume that the first L, eigenvector derivatives are to be computed. In order to save computational effort needed for the factorization of matrix R (K = K - PM), it is better to choose a common /J for all the first L, eigenvectors. In this case, only one matrix K is needed to be [L][U] factorized and that [L][U] result can be used for computing all the first L, eigenvector derivatives. To achieve a faster average convergent speed for all the first L, eigenvector derivatives, p can be determined as

#A, (1 ,
(43)

NUMERICAL EXAMPLE

Consider a free-free mass-spring system, shown in Fig. 1, with parameters given by M,=l,

Mz=2,

M,=3,

M4=4,

MS=5

Mb=6,

M,=7

MS = 8,

MS = 9,

M,, = 10 (kg)

i=lO,

3=. db

l
”

Case II: dA4, z=O

l,
2 Case III: 1

dM, -= db,

0

Case IV:

dM -= d&

1 i=3,7 o iz3,7

dk. -&=O

l
We choose p = 400.00. The first four eigenvector derivatives are computed using the present method and the modal truncation method eqn (13) respectively. The results are summarized in Table 1. For the sake of comparison, the exact derivatives of eigenvectors is also included in Table 1. The eigenvector derivative errors, defined as the difference between the exact eigenvector derivatives and the ones obtained by the specific methods, are also listed in Table 1. It can be seen from Table 1 that even though both two methods used the same first four modes in mode-superposition, the present method gives better accuracy than the modal truncation method; for example, the errors of dx3/db, are reduced to about lO-6-1O-s from the errors, due to the modal

Fig. I. Free-free mass-spring system.

1139

Computation of eigenvector derivatives Table I. Comparison of eigenvector derivatives d.x,lb,

d.v, lb,

d.u, lb,

d.u,lb,

Exact solution O.l225819E-02 0.1225819E - 02 0.1225818E - 02 O.l225818E-02 O.l225818E-02 O.l225818E-02 0.1225818E - 02 0.1225818E - 02 0.1225818E - 02 0.1225818E - 02

0.3804057E - 03 0.1203598E - 03 -0.3562213E-03 -0.7030792E - 03 -0.96375008 - 03 -0.30758168 - 02 -0.31571328 -02 -0.3062302E - 02 -0.2280956E - 02 -0.1803152E - 02

-0.1519378E -01 -0.786878lE - 02 0.2561268E - 02 0.6641470E - 02 0.735404OE- 02 -0.2676820E - 02 -0.42298588 - 02 -0.54545818 - 02 -0.6968820E - 02 -0.4876142E - 02

-0.7996594E -0.5345302E 0.1083526E 0.1204437E 0.22769458 0.5785313E 0.4392139E 0.1933754E -0.6745522E -0.7413485E -

02 02 02 02 02 02 02 02 02 02

Present method The first term retained

The first two terms retained

O.l225818E-02 O.l225818E-02 0.1225818E-02 O.l225818E-02 0.1225818E-02 0.1225818E - 02 0.1225818E - 02 0.1225818E - 02 0.1225818E - 02 0.1225818E - 02

0.3804515E 0.1203487E -0.3562552E -0.7030859E -0.9637352E -0.3075746E -0.315713lE-02 -0.306236OE -0.228099OE -O.l803112E-02

03 03 03 03 03 02 02 02

-O.l519417E-001 -0.786871 IE - 02 0.2561530E - 02 0.6641530E - 02 0.7353939E - 02 -0.2677317E - 02 -0.422986OE - 02 -0.5454157E - 02 -0.6968585E - 02 -0.4876415E - 02

-0.7998007E - 02 -0.5344953E - 02 -0.10825OOE - 02 0.1204625E - 02 0.2276491E - 02 05783397E - 02 0.4392138E - 02 O.l93539lE-02 -0.6744639E - 02 -0.741452lE -02

Errors due to the present method 0.7279229E 0.5882094E 0.372638OE0.3794781E 0.3971607E 0.1983125E 0.2562202E 0.2084486E 0.8314625E 0.6466290E -

09 09 09 09 09 09 09 09 10 10

0.4578212E - 07 0.1114992E - 07 0.3389186E - 07 0.6657062E - 08 O.l481269E-07 0.6959976E - 07 0.1215539E - 08 0.5789823E - 07 0.3340192E - 07 0.3975138E - 07

0.3882797E 0.7057204E 0.2618074E 0.5996576E 0.1005943E 0.4969540E 0.2203560E 0.4240165E 0.2350202E 0.2727491E -

06 07 06 07 06 06 08 06 06 06

0.1413128E - 05 0.3492888E - 06 0.1025890E - 05 0.1880765E - 06 0.4535942E - 06 O.l916079E-05 0.1526058E - 08 0.36370458 - 05 0.8825915E - 06 0.10362798 - 05

01 02 02 02 02 02 02 02 02 02

-0.8096096E - 02 -0.5337156E - 02 0.1004307E - 02 0.1240906E - 02 0.22585888 - 02 0.5442920E - 02 0.43 19469E - 02 0.219392lE -02 -0.6111395E-02 -0.79568828 - 02

Modal truncation method O.l225818E-02 O.l225818E-02 O.l225818E-02 0.1225818E-02 O.l225818E-02 O.l225818E-02 O.l225818E-02 O.l225818E-02 O.l225818E-02 O.l225818E-02

0.3623228E - 03 0.12121488 -03 -0.3416690E - 03 -0.6955 l86E - 03 -0.9663628E - 03 -0.3146005E - 02 -0.3174641E-02 -0.3010396E - 02 -0.2131782E - 02 -0.3929005E - 02

-0.1536428E -0.78569918 0.2697634E 0.6707154E 0.7324933E -0.3290989E -0.4367928E -0.4991978E -0.5775657E -0.5892856E -

Errors due to the modal truncation method 0.7279229E - 09 0.5882094E - 09 0.3726380E - 09 0.3794781E - 09 0.3971607E - 09 O.l983125E-09 0.2562202E - 09 0.2084486E - 09 0.8314625E - 10 06466290E - IO

0.180829lE - 04 0.85495758 - 06 0.1455228E - 04 0.7560624E - 05 0.2612791E - 05 0.701893lE - 04 O.l750813E-04 0.5190590E - 04 0.1491738E - 03 0.1258533E - 03

0.1705028E 0.1 I79079E 0.1363658E 0.6568355E 0.2910728E 0.6141695E 0.1380702E 0.4626032E 0.1193163E 0.1016713E -

03 04 03 04 04 03 03 03 02 02

0.9950190F 0.81467948 0.79218988 0.3646931E 0.1835615E 0.3423934E 0.7267067E 0.260167OE0.6341269E 0.54339688 -

04 05 04 04 04 03 04 03 03 03

1140

Zhong-sheng Liuetal. TableI.(conr.) d.u, lb2

d?cz lb2

dx,lb2

hlbz

Exactsolution 0.0000000E+00 0.0000000E+00 0.0000000E+00 0.0000000E+00 0.OOOOOOOE+00 0.OOOOOOOE+OO 0.0OOOOOOE+00 0.0OOOOOOE+00 0.0000000E+00 0.0000000E+00

O.l04697OE-03 0.9964797E-04 -0.3021744E-04 -0.31434068-04 -0.30006258-04 -0.5036025E-05 -0.2183580E-05 O.l380944E-07 0.5133699E-05 0.6161452E-05

-O.l119506E-02 -0.7822084E-03 0.2342975E-03 0.3961585E-03 0.4054442E-03 -0,1646745E-03 -0,1797774E-03 -0.14465588-03 0.67190588-04 O.l168196E-03

-0.84767798-03 -0.1244043E -03 -0.4818472E-03 0.3011062E-04 0.2314532E-03 0.4100710E-03 0.3388942E-03 0.2172983E -03 -0.2280359E-03 -0.32.54429E-03

Present method The first term retained -0,1837137E-17 -0.18333968-17 0.1001193E-18 O.l054883E-18 O.l087873E-18 O.l064364E-18 O.l057914E-18 O.l054519E-18 O.l059390E-18 O.l061455E-18

The first two termsretained O.l047022E-03 0.9965268E-04 -0.3023217E-04 -0.3143518E-04 -0.2999875E-04 -0.5034482E-05 -0.2184104E-05 0.1211361E-07 0.5133551E-05 0.6162009E-05

-O.l119481E-02 -0.7821921E-03 0.2342382E-03 0.3961543E-03 0.4054748E-03 -0,1646680E-03 -0,1797795E-03 -O.l446628E-03 0.6718993E-04 O.l168219E-03

-0.84818828-03 -0,1246818E-03 -0.4807252E-03 0.3018740E-04 0.2308740E-03 0.4099469E-03 0.3389347E-03 0.2174331E-03 -0.2280237E-03 -0.3254875E-03

Errors due to thepresent method O.l837137E-17 O.l833396E-17 O.l001193E-18 O.i054883E-18 O.l087873E-18 O.l064364E-18 O.l057914E-18 O.l054519E-18 O.l059390E-18 O.l061455E-18

0.5178245E-08 0.4712397E-08 O.l473439E-07 0.1120184E-08 0.7501532E-08 O.l543513E-08 0.5237630E-09 O.l695827E-08 O.I487116E-09 0.5567579E-09

-0.21158278-17 -O.l392102E-17 -0.3386300E-18 O.l169290E-18 0.2583183E-18 O.l159350E-18 0.9702022E-19 0.8987491E-19 O.l057487E-18 O.l111125E-18

O.l223021E-03 0.7137463E-04 -O.l823584E-05 -0.3205944E-04 -0.3981145E -04 -0.5695008E-05 -O.l606158E-05 O.l063890E-05 0.5150206E-05 0.5826460E-05

0.246725OE-07 O.l622238E-07 0.5930446E-07 0.4178865E-08 0.3054579E-07 06467985E-08 0.2118009E-08 0.7027185E-08 0.6548267E-09 0.2284955E-08

0.5102979E-06 0.2774590E-06 O.l122WE-05 0.7677866E-07 0.5792694E-06 0.1241339E -06 0.4050572E-07 O.l348012E-06 O.l220269E-07 0.4459849E-07

Modal trunc :ation method -O.l195315E-02 -0.6557699E-03 O.l040538E-03 0.3975780E-03 0.4519924E-03 -O.l611320E-03 -O.l82544OE-03 -0,1499662E-03 0.6706679E-04 O.l185148E-03

-0.65516418-03 -0.4535844E-03 -O.l371594E-03 0.28193888-04 O.l060984E-03 0.3998975E-03 0.3463861E-03 0.2321098E-03 -0.2276296E-03 -0.3301756E-03

method Errors due to them odaltruncation 0.2115827E-17 O.l392102E-17 0.3386300E-18 0.1169290E-18 0.2583183E-18 0.1159350E-18 0.9702022E-19 0.8987491E-19 O.l057487E-18 O.l111125E-18

O.l760504E-04 0.2827334E-04 0.2839385E-04 0.6253766E-06 0.98052OOE-05 0,6589834E-06 0.5774226E-06 O.l050081E-05 O.l650659E-07 0.3349917E-06

0.7580914E-04 O.l264384E-03 O.I302437E-03 0.1419518E-05 0.4654820E-04 0.3542467E-05 0.2766564E-05 0.531036OE-05 O.l237985E-06 0.16951618-05

0.1925138E-03 0.3291801E-03 0.34468788-03 O.l916745E-05 O.l253548E-03 O.l017352E-04 0.7491980E-05 O.l481146E-04 0.4062834E-06 0.4732707E-05

1141

Computation of eigenvector derivatives Table I. (conr.) dx, lb,

dx, ib,

h/b,

dxalb,

Exact solution 0.12258198 - 02 0.1225819E-02 0.1225818E - 02 O.l225818E-02 O.I225818E-02 0.1225818E - 02 O.I225818E-02 O.l225818E-02 O.l225818E-02 0.1225818E - 02

-0.6290168E -0.6080919E -0.56627578 -05305839E -0.49879478 -0.1920434E -0.39079858 -0.5344938E -0.8532206E -0.9 153’798E-

03 03 03 03 03 03 03 03 03 03

- 0.24660008 -0.13594328 0.2405 172E 0.90058528 0.10559308 0.3108265E 0.4146557E 0.5589069E 0.3695420E 0.30223 l5E -

0I 01 02 02 01 04 02 02 02 02

-O.i224680E-01 -0.8142346E - 02 -0.1579348E -02 0.1907257E - 02 0.3515009E - 02 0.85221688 - 02 O.l~257lE-01 0.83116lOE - 02 -0.407792 1E - 02 -0.70341 l4E - 02

Present method The first term retained

The first two terms retained

0.1225818E - 02 O.l225818E-02 0.1225818E - 02 0.1225818E - 02 0.1225818E - 02 0.1225818E - 02 0.1225818E - 02 O.i225818E-02 0.1225818E -02 0.12258188 - 02

-0.6289895E -0.6080954E -0.5662975E -0.5305904E -0.4987853E -0.19197148 -0.390802lE -0.5345533E -0.85323008 -0.9 153592E -

03 03 03 03 03 03 03 03 03 03

-0.2465919E - 01 -0.1359446E - 01 0.2404504E - 02 0.900565lE - 02 0.1055956E - 01 0.3304575E - 04 0.4146443E - 02 0.5587416E - 02 0.369515lE -02 0.3022877E - 02

-0.1224319E -01 -0.81429218-02 - 0.1582204E - 02 0.1906439E - 02 0.35161638 - 02 0.8530264E - 02 0.1002522E - 01 0.8304747E - 02 -0.40790378 - 02 -0.7031772E - 02

Errors due to the present method 0.72792298 - 09 0.58820943%- 09 0.3726380E - 09 0.3794782E - 09 0.39716078 - 09 O.l983125E-09 0.25622028 - 09 0.20~6E - 09 0.83146248 - 10 06466295E - IO

0.2732853E 0.34842lOE 0.2180916E 0.65384568 0.9347647E 0.72017378 0.36l~lE 0.5950976E 0.9416321E 0.20653768 -

07 08 07 08 08 07 08 07 08 07

0.81505638 - 06 0.1339367E - 06 0.66718468 - 06 0.2004937E - 06 0.26269448 - 06 0.19631038 - 05 0.1136045E - 06 0.16531238-05 0.2682587E - 06 0.5614073E - 06

0.3607282E - 05 0.5750590E - 06 0.2855917E - 05 0.8183558E - 06 0.1154093E - 05 0.80955238 - 05 0.4~9276E - 06 0.6862963E - 05 O.l116607E-05 0.2342071E - 05

Modal truncation method O.l225818E-02 O.l225818E-02 O.l225818E-02 0.1225818E-02 0.3225818E-02 0.3225818E - 02 0.1225818E - 02 0.1225818E-02 0.1225818E - 02 0.12258188 - 02

-0.6347762E -0.61117868 -0.56146088 -0.52178778 -0.4935871E -0.311239lE -0.3600733E -0.4581219E -0.8492184E -0.93644078

- 03 - 03 - 03 - 03 - 03 -03 - 03 - 03 - 03 - 03

-0.2442446E -0.1350197E 0.2202582E 0.87128278 0.1~2407E 0.3842426E 0.3245466E 0.3061396E 0.3558900E 0.371485OE-

01 01 02 02 01 02 02 02 02 02

-0.1209046E - 01 -0.8091028E - 02 -0.17146358 -02 0.1732010E -02 0.3445938E - 02 0.1075955E - 01 0.9517519E - 02 0.6807041E - 02 -0.4167107E -02 -0.6617564E - 02

Errors due to the modal truncation method 0.7279229E 0.5882~4E 0.37263808 0.3794782E 0.3971607E 0.1983125E 0.25622028 0.2084486E 0.83 14624E 0.64662958 -

OY 09 09 09 09 09 09 09 10 IO

0.57593888 - 05 0.3086~6E - 0.5 0.4814916E-05 0.87962128 - 05 0.5207560E - 05 0.1191957E-03 0.3072516E - 04 0.7637188E - 04 04002184E - 05 0:2106085E - 04

0.23554548 0.9235586E 0.20258998 0.293025I E 0.1352227E 0.38 113438 0.9010908E 0.2527673E 0.1365197E 0.6925348E -

03 04 03 03 03 02 03 02 03 03

0.156338lE -03 0.5131785E -04 0.1352870E - 03 0.1752468E - 03 0.6907154E - 04 0.2237378E - 02 0.5081901E - 03 0.1504569E - 02 0.8918646E - 04 0.4165502E - 03

1142

Zhong-sheng Table

Liu et al. 1. (conr.)

dsz lb4

d.u, lb,

% lb,

h/b,

Exact solution 0.245 1637E 0.245 1637E 0.2451637E 0.245 1637E 0.2451637E 0.2451635E 0.2451635E 0.2451635E-02 0.245 l636E 0.245 1636E -

02 02 02 02 02 02 02 02 02

O.l26003lE-01 0.1070830E 0.7 183879E 0.8880725E 0.8971486E 0.3282597E 0.25883468 0.1711216E-02 -0.3988900E -0.8255580E -

03 03

Present

The first term retained 0.245 l636E 0.245 l636E 0.2451636E 0.2451636E 0.2451636E 0.2451636E 0.245 l636E 0.245 l636E 0.245 l636E 0.245 l636E

-

02 02 02 02 02 02 02 02 02 02

O.l260334E-01 0.1070735E 0.7181691E 0.888076lE 0.8972685E 0.3282749E 0.2588265E 0.171 l013E -0.3989020E -0.8254873E -

01 01 03 02 02 02 03 02 02 02

-O.l879862E-01 -0.1248779E -0.24050478 -O.l78957lE-02 -0.5895447E 0.3207855E 0.9750238E 0.8195477E - 0.38023 l4E - 0.668458 I E

-

- 01 - 01 - 02 - 02 - 03 -01 - 02 - 02 - 02 - 02

-01 - 02 03 01 02 02 02 02

method

08 09 09 09 09 09 09 09 09

-

02 02 02 02 02 02 02 02 02 02

0.1455846E O.l176419E-08 0.7452760E 0.7589564E 0.7943215E 0.3966249E 0.5124403E 0.4168972E 0.1662925E O.l29326lE-09

08

0.3029020E 0.9548129E 0.2188605E 0.3649556E 0. Il98438E 0.15169OOE 0.809693 I E 0.2028586E 0.1204942E 0.7063995E

-

O.l119477E-01 0.1052974E 0.93934OOE 0.863255OE 0.8072809E 0.3299408E 0.2570434E 0.1831 l73E -0.3908372E -0.86429638 Errors

09 09 09 09 09 09 09

01 02 02 02 02 02 02 03 03

-0.3283569E -0.1971810E - 0.2354036E 0.4284893E 0.5706447E 0.2936793E 0.9546220E 0.2902089E 0.4628082E 0.4808 l29E

due to the present

Modal 0.2451636E 0.2451636E 0.2451636E 0.2451636E 0.2451636E 0.2451636E 0.245 l636E 0.245 l636E 0.245 l636E 0.245 l636E

-

The first two terms retained

Errors 0.1455846E O.l176419E-08 0.74527608 0.7589564E 0.7943215E 0.3966249E 0.5124403E 0.4168972E 0.1662925E 0.129326lE -

-0.328384OE -0.1971723E -0.23346068 0.4284848E 0.5705379E 0.2936722E 0.9546542E 0.2902237E 0.4628 l24E 0.4808058E

01 02 02 02 02 02

05 06 05 07 05 06 07 06 07 07

01 01 03 02 02 02 03 02 02 02

-0.1877320E -0.1249599E -0.2423008E -0.1789196E -0.57967098 0.1207949E 0.97497418 0.8193930E -0.3802557E -0.6683938E

-

05 06 05 07 05 07 07 06 07 07

0.2541354E 0.8195442E 0.1796077E 0.3752239E 0.9873859E 0.9360825E 0.4976849E 0.1547247E 0.2438936E 0.64310538

-

01 01 02 02 02 02 02 02 02 02

-0.1669442E -0.1239042E -0.5401754E -0.1451339E 0.6769342E O.Il57756E 0.10507668 0.773 l434E -0.39210228 -0.65355238

method

0.2710752E 0.8652249E 0.1942964E 0.4461562E 0.10681348 0.7140407E 0.3222688E 0.1478599E 0.414264lE 0.7127493E

-

04 05 04 06 05 06 06 05 06 06

trunc ation method

01 02 02 02 02 02 02 03 03

due to the modal

0.1405543E 0.1785603E 0.220952lE 0.248 l750E 0.8986769E 0.1681032E O.l791178E-04 O.ll99573E 0.8052745E 0.3873832E -

-

02 03 02 03 03 04 03 05 04

-0.3131645E -0.1956266E -0.2580047E 0.45994868 0.6665659E 0.2216214E 0.2039392E 0.234853OE 0.4471469E 0.4979870E

Iruncation

- 01 - 01 - 02 -02 - 03 - 01 - 0I - 02 -02 - 02

method

O.l521944E-02 O.l545718E-03 0.2346587E 0.3 146379E 0.9602801 E 0.7205080E 0.10847378 0.5537073E 0. I566543E 0.1718123E -

02 03 03 03 02 03 03 03

0.2104194E 0.9737205E 0.2996707E 0.3382324E 0. I266479E 0.5009917E 0.7574192E 0.4640433E 0. I 187089E 0.1490582E

-

02 04 02 03 02 03 03 03 03 03

1143

Computation of eigenvector derivatives

truncation method, of 10-2-10-4. It can also be seen that the errors of dx,/db,, and dx,/db, are smaller than the errors of dx4jdbi. This is because p is closer to 1, and RJ than i4, resulting in a faster convergent speed associated with dx,/db, and dx, ldb, .

7. W. C. Mills-Curran, Calculation of eigenvector derivatives for structures with repeated eigenvalues, AIAA J. 26, 86-F-871 (1988). 8. R. L. Dailey, Eigenvector derivatives with repeated eigenvalues. AIAA J. 27, 486-491 (1989). 9. E. J. Haug and B. Rousselet. Design sensitivity analysis

in structural J. Sfrucf.

CONCLUDING REMARKS

mechanics

II, eigenvalue

variations.

.Mech. 8, 161-186 (1980).

10. E. U. Ojatvo, Efficient computation

An accurate modal method for computing the eigenvector derivatives for free-free structures with rigid-body modes is proposed in this paper. First, the contribution due to the truncated modes to the eigenvector derivatives is exactly and explicitly formulated using the available modes and the system matrices with non-singular stiffness matrix. Second, a transformation is introduced in which the eigenproblem with singular stiffness matrix is transformed in to its equivalent eigenproblem with non-singular stiffness matrix, in the sense that both eigenproblems have the same eigenvalue derivatives and eigenvector derivatives. This transformation is realized by the well-known eigenvalue shift, i.e. R = K - PM. As a result, the contribution due to the truncated modes can also be evaluated for the free-free structures. The extra computational cost required for this method is to factorize the matrix R. However, if many eigenvector derivatives are to be computed, the extra computational cost per eigenvector derivative is much less. When [L][U] factorization of R has been obtained in the process of eigensolution analysis of the free-free structures, this [L][U] result can be used for computing S,. Thus, little extra computational cost for obtaining S, is needed.

of modal sensitivities for systems with repeated frequencies. AfAA

J. 26, 361-366 (1988). II.

B. P. Wang, Improved approximate methods for computing eigenvector derivatives in structural dynamics.

AfAA J. 29, 1018-1020 (1991). 12. R. B. Nelson, Simplified ~lculation of eigenvector derivatives, AIAA J. 24, 823-832 (1986). APPENDIX

Consider the ejgenproblem

K[X,X,,l= M[X,X,]dia&4,, 4,)

641)

where

From eqns (Al) and (A2), we have

[XLXhlTKtXLXhl = diag(&, 4,).

(A31

Inverting eqn (A3) yields Acknowledgement-This

research was supported by the

National Natural Science Foundation of China and Open Laboratory of CAD/CAM Technology Manufactu~ng, Academia Sinica.

K- ’ = [X,X,,]diag(n;‘,

n;‘)[X,X$.

(A4)

X L A-‘X’ L t.,

(AS)

for Advanced Thus we have X,A,‘X,r=

REFERENCES

1. Z. S. Liu and S. H. Chen, Derivatives of eigenvalue for torsional vibration of a geared-shaft system. J. Vibr. Acoast. (accepted). 2. R. T. Haftka and R. H. Adelman, Sensitivity analysis of discrete structural systems. AIAA J. 24. 823-832 (1986).

3. J. F. Baldwin and S. G. Hutton, Natural modified structures. AIAA J. 23. 1737-1743 4. R. L. Fox and M. P. Kapoor; Rates of eigenvalues and eigenvectors. AfAA J. 6,

modes of (19851.

change of 2426-2429

K-l-

Post-multiplying eqn (A4) with MK-’ K-‘MK-’

yields

= [X~X~]diag(~~‘, n;‘)[X,X,,]r,%fK-‘.

(A6) Substituting eqn (A4) into the right side of eqn (A6). we have K-‘,%fK-’ = [X,X,]diag(,4,‘,

.4;‘)[XLX,,]‘.

(A7)

(1968). 5. Zhong-sheng

Eigensolution

Liu, Su-huan Chen and You-qun Zhao, derivatives in vibration systems. J.

Chinese Sot. Asrronautics. 4, 96-102 (1992). 6. J. N. Juang, P. Ghaemmaghamj and K. B. Lim, Eigen-

value and eigenvector derivatives of a nondefective matrix. J. Guidance, Control Dynam. 12, 480-486 (1989).

Thus we have X,A,zX;=

K-‘/UK-‘-

XLA;2Xl.

In a similar way, it can be shown that

(.48)