An accurate modal truncation method for eigenvector derivatives

Computers and Structures 73 (1999) 609±614

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An accurate modal truncation method for eigenvector derivatives You-qun Zhao a,*, Zhong-sheng Liu b, Su-huan Chen b, Gui-yu Zhang a

Department of Automobile Engineering, Shandong Institute of Technology, Zibo, 255012, People's Republic of China b Department of Mechanics, Jilin University of Technology, Changchun, 130025, People's Republic of China Received 22 January 1997; accepted 3 October 1998

Abstract This paper deals with obtaining accurate eigenvector derivatives in damped vibration systems with modal truncation method only by system matrices and a few available lower modes. A numerical example shows that the method of this paper is correct and ecient. # 1999 Elsevier Science Ltd. All rights reserved.

1. Introduction The usefulness of modal sensitivities for analysis and design of engineering systems is well known. Some speci®c applications include redesign of vibratory systems, identi®cation of dynamic systems, design of control systems by pole placement, and assessment of design changes on system performance. Therefore, methods for computing eigenvector (vibration mode shape) derivatives have been an active area of research since the work of Fox and Kapoor [1]. The development in this area was surveyed by Haftka and Adelmen [2]. Now the methods of computing eigenvector derivatives for real symmetric systems in structural dynamics have progressed signi®cantly [3±8]. The ®rst general expression for eigenvalue and eigenvector derivatives for non-self-adjoint systems (general eigenvalue problem) via modal expansion appears to be given by Plaut and Huseyin [9], but Rogers was the ®rst to recognize the need for two sets of normalization conditions for nonsymmetric eigensystems. Basically, there are two methods for computing eigenvector derivatives: algebraic methods and modal

* Corresponding author.

superposition methods. Algebraic methods using only the eigenvector of concern are available in Refs. [1,6,7,10]. Since algebraic methods require special manipulation of the system matrix for each eigenvector of concern, they cannot be implemented easily and are not ecient if a large number of eigenvector derivatives are required. To obtain exact solutions, Nelson's method [10] is the most ecient of the algebraic methods if few eigenvector derivatives are required. However, the algebraic methods require the solution of auxiliary sets of linear equations that may be ill conditioned, thus requiring careful attention from the user. When many eigenvector derivatives are required, modal methods are more ecient. Because of the cost of generating computer solutions for a dynamic analysis, it is often impractical to obtain all modes. Therefore, only the ®rst low-frequency modes are computed and are used as basis vectors of eigenvector derivatives. However, modal truncation includes errors that can be signi®cant if more high-frequency modes are truncated. An explicit method to improve the truncated modal superposition representation of eigenvector derivatives is presented by Wang [11] for real symmetric systems, in which a residual static mode is used to approximate the contribution due to unavailable high-frequency modes. The numerical perform-

0045-7949/99/$ - see front matter # 1999 Elsevier Science Ltd. All rights reserved. PII: S 0 0 4 5 - 7 9 4 9 ( 9 8 ) 0 0 2 2 7 - 2

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Y. Zhao et al. / Computers and Structures 73 (1999) 609±614

ance of the cited methods is compared in Refs. [3,12]. Liu has extended Wang's method in two senses: the ®rst 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 system matrices [13]. However, Liu's method can only be used in non-damped vibration systems. This paper tries to extend Liu's method to include the damped systems, and to extend Lim's modal truncation method by giving an accurate expression of contribution due to the higher unavailable modes in terms of the available modes and system matrix [14]. Therefore, the method of this paper signi®cantly improves the accuracy of the previously published method [14] and broadly extends the application of the previous one [13].

2. Lim's modal truncation method The generalized eigenvalue problem for damped vibration systems in structural dynamics is Ax i ˆ li Bx i ,

i ˆ 1, . . . n

AT yi ˆ li B T yi ,

…1†

i ˆ 1, . . . n

…7†

iˆk

gik ˆ

 yTi @ A=@ b ÿ …@ li =@ b†B ÿ li …@ B=@ b† x k , li ÿ lk

gik ˆ ÿyTi …@ B=@ b†x i ÿ aii ,

iˆk

i 6ˆ k

…8†

For a large-scale structure, there may be thousands of degrees of freedom. In this case only a few of the ®rst lower modes are calculated, while the other higher modes are truncated. Assume the ®rst L (L <
…9†

where zi ˆ

n X aij x j

…10†

jˆ1 j6ˆi

…2†

Using Eq. (7), Eq. (10) can be written as

with biorthogonality and normalization yTk Bx i ˆ dik ,

8 9 n X > 1> < T = T T aij x j …B ‡ B †x i , x i …@ B=@ b†x i ‡ aik ˆ ÿ 2> jˆ1 > : ; j6ˆi

i,k ˆ 1, . . . n

…3†

zi ˆ

n X yTj Fi xj l ÿ lj jˆ1 i

…11†

L yT F n X X yTj Fi j i xj ‡ xj l ÿ lj l ÿ lj jˆ1 i jˆL‡1 i

…12†

j6ˆi

x Ti Bx i

ˆ 1,

i ˆ 1, . . . n

…4†

where A and B are n  n real matrices, li a generally complex eigenvalue of A and B, and xi and yi the ith n  1 complex-valued associated eigenvectors. After some algebra analogous by using the exact modal method, it can be shown that the eigenvector derivatives take the form n X @ x i =@ b ˆ aik x k

…5†

kˆ1

@ yi =@ b ˆ

n X gik yk

…6†

or zi ˆ

j6ˆi

where  Fi ˆ …@ A=@ b† ÿ …@ li =@ b†B ÿ li …@ B=@ b† x i

…13†

and we have assumed that i R L. If the eigenvalues are numbered according to their magnitude in ascending order, then for the class of problems with a large frequency gap

kˆ1

lj ÿ li 1lj ,

where  yTk @ A=@ b ÿ …@ gi =@ b†B ÿ li …@ B=@ b† x i aik ˆ , li ÿ lk

i 6ˆ k

for j > L

…14†

It is clear that the above approximation is very accurate for j>>L. Thus, Eq. (13) can be written, letting zi be approximated by zi, as

Y. Zhao et al. / Computers and Structures 73 (1999) 609±614

zi 1z i ˆ

L yT F n yT F X X j i j i xj ÿ xj l ÿ l lj j jˆ1 i jˆL‡1

…15†

j6ˆi

Eq. (15) can be rewritten as z i ˆ

L yT F n yT F L yT F X X X j i j i j i xj ÿ xj ‡ xj l ÿ l l lj j j jˆ1 i jˆ1 jˆ1

…16†

j6ˆi

The biorthogonality conditions can be written in matrix form as T

Y AX ˆ L

…17†

where L ˆ diag …l1 , . . . ,ln † X ˆ ‰x 1 , . . . ,x n Š

a ii ˆ ÿ

1 T x i …@ B=@ b†x i ‡ x Ti Bz i ‡ z Ti Bx i 2

For the class of problems that satisfy the previously mentioned frequency conditions, the modal approximation of Eq. (21) may represent an improvement over Eq. (5), vis-aÁ-vis computation, and often provides acceptable precision. The derivatives of the left eigenvectors can be computed similarly. 3. The present method In order to obtain a more accurate solution we now focus on the contribution of the unavailable eigenvectors to the eigenvector derivatives. As assumed above: i R L, and the eigenvalues are numbered according to their magnitude in ascending order, requiring li < lj , for j > L …23†

zi ˆ ziL ‡ ziH

or

Thus

Aÿ1 ˆ

jˆ1

lj

…19†

Using the spectral decomposition of Eq. (19), the second summation on the right-hand side of Eq. (16) becomes ÿA ÿ1Fi, and Eq. (16) can be written as L yT F L yT F X X j i j i z i ˆ x j ÿ Aÿ1 Fi ‡ xj l ÿ lj lj jˆ1 i jˆ1

…20†

j6ˆi

We note that the ÿA ÿ1Fi term in Eq. (20) approximates the e€ect of lower-order modes. Presumably, for many cases, this approximation is more accurate than an initial expansion involving only the ®rst L modes. Now the modal representation of eigenvector derivative can be approximated as @ x i =@ b ˆ a ii x i ‡ z i

ziL ˆ

L X yTj Fi xj l ÿ lj jˆ1,j6ˆi i

…25†

ziH ˆ

n X yTj Fi xj l ÿ lj jˆL‡1 i

…26†

…18†

or T j yj

…24†

where

X ÿ1 Aÿ1 …Y T †ÿ1 ˆ Lÿ1

n x X

…22†

Rewrite Eq. (12) as

Y ˆ ‰ y1 , . . . ,yn Š

Aÿ1 ˆ XLÿ1 Y T

611

…21†

where ziL is the contribution of the available modes, and ziH is the contribution of the unavailable modes. Since vliv < vljv, for j>L, the diagonal matrix (LHÿliI )ÿ1 can be expanded into a convergent series [15], …LH ÿ li I †ÿ1 ˆ

1 X ÿkÿ1 lki LH

…27†

kˆ0

Thus, using Eqs. (23) and (27), Eq. (26) can be expressed as following: ziH ˆ

n X yTj Fi x j ˆ ÿXH …LH ÿ li I †ÿ1 Y TH Fi l ÿ lj jˆL‡1 i

  T ÿ1 2 T ˆ ÿ XH Lÿ1 Y F ‡ l X …L † Y F ‡


i i H i H H H H where

where aii can be computed by requiring

LH ˆ diag …lL‡1 , . . . ,ln †


@ ÿ T x Bx i ˆ 0 @b i

XH ˆ ‰x L‡1 , . . . ,x n Š

…28†

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Y. Zhao et al. / Computers and Structures 73 (1999) 609±614

Fig. 1. Mass±damp±spring system.

aii ˆ a ii

YH ˆ ‰ yL‡1 , . . . ,yn Š and h

. Aÿ1 ˆ XL ..XH

ih

. LL ..LH

iÿ1 h

. YL ..YH

iT

T ÿ1 T ˆ XL Lÿ1 L Y L ‡ XH LH Y H

ziL ˆ …29†

where

L X yTj Fi xj l ÿ lj jˆ1,j6ˆi i

ziH ˆ ÿ

 1  X k T Aÿ1 …BAÿ1 †kÿ1 ÿ XL …Lÿ1 L † Y L Fi kˆ1

De®ne SH(k ) as

LL ˆ diag …l1 , . . . ,lL †

1  X

 k T Aÿ1 …BAÿ1 †kÿ1 ÿ XL …Lÿ1 † Y L L Fi

XL ˆ ‰x 1 , . . . ,x L Š

SH …k† ˆ ÿ

YL ˆ ‰ y1 , . . . ,yL Š

Using the de®nition, the given iteration process can be terminated if the next inequality

kˆ1

Thus T ÿ1 T XH Lÿ1 ÿ XL Lÿ1 H YH ˆA L YL

…30†

Post-multiplying Eq. (30) with BA ÿ1 yields 2 T 2 T ÿ1 ÿ1 XH …Lÿ1 ÿ XL …Lÿ1 H † Y H ˆ A BA L † YL

…32†

T XH Lÿ1 H Y H Fi

…33†

j6ˆi

The formulas of computing @xi/@b are summarized follows: @ x i =@ b ˆ aii zi ‡ ziL ‡ ziH where

Consider a 20-degree-of-freedom mass-damp-spring system, shown in Fig. 1, with parameters given by m1 ˆ m2 ˆ


ˆ m19 ˆ 2m,

m20 ˆ m ˆ 1 kg

k1 ˆ k2 ˆ


ˆ k21 ˆ 1  103 N=m c1 ˆ c2 ˆ


ˆ c7 ˆ 3c,

c8 ˆ c9 ˆ


ˆ c14 ˆ 2c

c15 ˆ c16 ˆ


ˆ c21 ˆ c ˆ 0:1 N s=m

T ˆ ziL ÿ Aÿ1 Fi ‡ XL Lÿ1 L Y L Fi L yT F L yT F X X j i j i ˆ x j ÿ Aÿ1 Fi ‡ xj l ÿ l lj i j jˆ1 jˆ1

is satis®ed, where e is some speci®ed accuracy requirement. 4. Numerical example

Substituting Eq. (32) into Eq. (28), we can yield the accurate value of ziH using the available modes and the system matrices. The following formulation proves that Lim's method [14] is a special case of the present method, provided k = 1 (refer to Eq. (20)). zi 1ziL ÿ

…36†

…31†

In a similar way, it can be shown that k T k T ÿ1 ÿ1 kÿ1 XH …Lÿ1 ÿ XL …Lÿ1 H † Y H ˆ A …BA † L † YL

kSH … j† ÿ SH … j ÿ 1†k< e

…35†

…34†

The free vibration equation of the system in Fig. 1 can be expressed as Mfq g ‡ Cfq_ g ‡ Kfqg ˆ 0 The elements in matrices M, C and K can be obtained using Lagrange equation. So the matrices A and B in Eqs. (1) and (2) are     ÿM 0 0 M Aˆ and B ˆ 0 K M C

Y. Zhao et al. / Computers and Structures 73 (1999) 609±614

613

Table 1 Errors of eigenvector derivatives Present method (%); number of series terms

Error of @x1/@b1 Error of @x2/@b1 Error of @x3/@b1 Error of @x1/@b2 Error of @x2/@b2 Error of @x3/@b2

Number of modes used

Modal truncation method (%)

Lim's method (%)

k=2

k=3

4 8 12 4 8 12 4 8 12 4 8 12 4 8 12 4 8 12

60.18 26.66 16.76 101.00 48.65 23.98 69.74 26.60 25.92 90.63 44.86 23.76 72.52 37.95 28.54 63.33 27.56 25.00

13.14 1.01 0.66 50.94 10.32 3.37 14.82 1.39 0.85 43.79 6.65 3.23 15.07 1.22 0.52 9.19 0.75 0.47

1.41 0.03 0.01 27.72 1.29 0.35 3.07 0.23 0.12 21.33 1.56 0.37 8.46 0.57 0.03 1.92 0.49 0.15

0.01 0.00 0.00 7.64 0.38 0.06 0.02 0.00 0.00 9.01 0.35 0.11 1.11 0.03 0.00 0.42 0.21 0.01

This numerical example is used to demonstrate and compare the solution accuracy for three di€erent methods including the modal truncation method that neglects ziH, Lim's method proposed in Ref. [14] and the present method. For the sake of simplicity, the error of the eigenvector derivative is de®ned as v(@xi/@bj )eÿ(@xi/@bj )av, where (@xi/@bj )e represents the exact solution for the ith eigenvector derivative with respect to the jth design parameter, and (@xi/@bj )a represents the approximate solution corresponding to (@xi/@bj )e. Here, the Nelson's method is used to compute (@xi/@bj )e [10]. Now select m1 and m10, c8 and c15 to be design parameters, i.e. m1 and m10 are functions of scalar b1, c8 and c15 are functions of scalar b2. The numerical values in Table 1 con®rm the following. 1. The error of modal truncation method is sometimes signi®cant if only a few lower frequency modes are retained while a large number of higher frequency modes are truncated. It can be seen from the third column in Table 1 that the errors associated with the modal truncation method are relatively large ranging from 16.76% to 101.00%. The reason is that eigenvector derivatives are usually more local than the eigenvector of concern. Therefore, the eigenvector derivatives may not be dominated by the ®rst modes, which are generally global.

k=4 0.00 0.00 0.00 1.51 0.03 0.01 0.00 0.00 0.00 2.00 0.04 0.02 0.03 0.00 0.00 0.00 0.00 0.00

2. When the same number of the ®rst modes is used in superposition, the present method is the most accurate of all the three methods, and Lim's method is more accurate than the modal truncation method. It can be seen from each row in Table 1 that the error of the present method is the smallest, whereas the error of the modal truncation method is the largest. For instance, when using four modes, the error of @x2/@b1 by the modal truncation method is 101.00%, that by Lim's method is reduced to 50.94%, and that by the present method is further reduced to 27.72%, 7.64% and 1.51%, respectively. 3. The present method is more accurate than the modal truncation method even if the total sum of modes used and series terms retained in the present method is equal to the number of modes used in the modal truncation method. Taking @x1/@b1 as an example, when 12 modes are used in the modal truncation method, its error is 16.76%, whereas four modes and two, three, four series terms (k = 2, 3, 4) are used in the present method, the error is 1.41%, 0.01% and 0.00%. 4. From Table 1 it can be shown that both adding modes and adding additional iterations can improve accuracy of the eigenvector derivatives, but we advise the users to improve the accuracy by adding additional iterations when a large number of modes are truncated.

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Y. Zhao et al. / Computers and Structures 73 (1999) 609±614

5. Concluding remarks An accurate modal method for computing the eigenvector derivatives for damped vibration systems 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. Second, we have demonstrated that the present method is the development of the method in Ref. [13] and Lim's modal truncation method is a special case of the present method [14]. The present method is that the contribution due to the unavailable high-frequency modes is given by a convergent series that can be computed without using the truncated modes. This series not only improves the solution accuracy but also provides the error estimates of the computed results. When the LU factorization of matrix A has been obtained when extracting the eigendata, the computational e€ort needed to compute this series is trivial. Therefore the present method can enhance the solution accuracy with little extra computational cost. Acknowledgements This research is supported by National Science Foundation of China and Open Laboratory of CAD/ CAM Technology for Advanced Manufacturing, Academia Sinica. References [1] Fox RL, Kapoor MP. Rates of change of eigenvalues and eigenvectors. AIAA J 1968;24:2426±9.

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