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33.9 Matrix Perturbation Theory for Complex Modes

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In Section 33.2 to Section 33.8, the matrix perturbation for real modes of systems with real symmetric

mass and stiffness matrices, M and K; was given. However, in many engineering problems such as systems

with nonproportional damping (see Chapter 3 and Chapter 19), dynamic systems under nonconservative

forces, analysis of aero-elastic flutter, and structural vibration control systems, the system matrices are

not symmetric and may not be diagonalizable. In this case, the matrix perturbation for real modes cannot

be used, and we must use the matrix perturbation for complex modes (Murthy and Haftka, 1988; Chen,

1993; Liu, 1999; Adhikari and Friswell, 2001). In the following, we assume that the system is not defective;

that is, the system has a complete eigenvector set to span the eigenspace. The discussion in this chapter is

limited to the nondefective systems.

33.9.1 Basic Equations

The vibration equation of a linear system with n-DoFs is given by

Mq€ þ Cq_ þ Kq ¼ QðtÞ ð33:121Þ

where the matrices M; C; and K; are assumed to be real and unsymmetric. The free vibration equation of

the system is

Mq€ þ Cq_ þ Kq ¼ 0 ð33:122Þ

The corresponding right eigenvalues problem is

ðMs2 þ Cs þ KÞx ¼ 0 ð33:123Þ

and its adjoint eigenvalue problem is

ðMs2 þ Cs þ KÞTy ¼ 0

yTðMs2 þ Cs þ KÞ ¼ 0 ð33:124Þ

It is common in literature to call y the left eigenvector, while x in the original system, a column vector, is

called the right eigenvector.

Let us introduce a state vector

u ¼

( )

¼ Tx ð33:125Þ

where T is the state transformation matrix

T ¼

( )

ð33:126Þ

33-22 Vibration and Shock Handbook

Similarly, we introduce the state vector

v ¼

( )

¼ Ty ð33:127Þ

Hence, Equations 33.123 and Equation 33.124 become

ðAs þ BÞu ¼ 0 ð33:128Þ

ðAs þ BÞTv ¼ 0 ð33:129Þ

vTðAs þ BÞ ¼ 0

where

A ¼

2C 2K

I 0

" #

B ¼

M 0

0 I

" #

It is well known that the eigenvalues of the adjoint eigenproblem (Equation 33.129) are identical to

that of the original eigenproblem (Equation 33.128). The characteristic equation is

detðA þ sBÞ ¼ 0

This characteristic determinant is a polynomial of 2n order in s; and 2n eigenvalues si ði ¼ 1; 2; …; 2nÞ

can be found in the complex domain. The left and right modal vectors, vi and ui; corresponding to

si satisfy

Aui ¼ siBui ð33:130Þ

and

ATvi ¼ siBTvi ð33:131Þ

The orthogonality conditions are

j Bui ¼ 0 ð33:132Þ

j Aui ¼ 0 ð33:133Þ

The normalization conditions are

i Bui ¼ 1

i Bui ¼ 1 ð33:134Þ

Therefore, the orthogonality conditions can be written as

j Bui ¼ dij

j Aui ¼ sidij

ð33:135Þ

33.9.2 Matrix Perturbation Method for Distinct Modes

If small changes are made on the structural parameters, the mass, damping, and stiffness matrices of the

system also have small changes given by

M ¼ M0 þ 1M1

C ¼ C0 þ 1C1

K ¼ K0 þ 1K1

ð33:136Þ

Structural Dynamic Modification and Sensitivity Analysis 33-23

and we have

A ¼ A0 þ 1A1 ð33:137Þ

B ¼ B0 þ 1B1 ð33:138Þ

where 1 is a small parameter.

In the following, we first consider the case of distinct eigenvalues, s0i; of the original system. According

to the matrix perturbation theory, the eigenvalues and eigenvectors can be expressed as a power series in

1; that is

S ¼ S0 þ 1S1 þ 12S2 þ · · · ð33:139Þ

U ¼ U0 þ 1U1 þ 12U2 þ · · · ð33:140Þ

V ¼ V0 þ 1V1 þ 12V2 þ · · · ð33:141Þ

where S0; U0; and V0 are the eigensolutions of the original system; S1; U1; and V1 are the first-order

perturbations of eigensolutions; and S2; U2; and V2 the second-order perturbations.

U1 can be expressed as a linear combination of the right eigenvectors of the original system as

U1 ¼ U0C1 ð33:142Þ

where C1 is to be the determined matrix given by

C 1

ij ¼

S0j 2 S0i

ij; j – i; i; j ¼ 1; 2; … ð33:143Þ

Also

S1 ¼ diagðP1

11; P1

22; …Þ ð33:144Þ

where P1

ij are the elements of P1 given by

P1 ¼ VT

0 ð2A1U0 þ B1U0S0Þ ð33:145Þ

The V1 can be expressed as the expansion of V0

V1 ¼ V0D1 ð33:146Þ

where D1 is to be the determined coefficient matrix given by

ij ¼

S0j 2 S0i

ij; j – i; i; j ¼ 1; 2; … ð33:147Þ

and R1

ij are the nondiagonal elements of R1

R1 ¼ UT

0 ðBT

1 V0S0 2 AT

1 V0Þ ð33:148Þ

If the modification of the parameter is fairly large, the second-order perturbation must be used to

obtain high accuracy. According to the expansion theorem, the second-order perturbation of

eigenvectors, U2; can be expressed as

U2 ¼ U0C2 ð33:149Þ

In a similar manner, S2 and the elements C 2

ij can be obtained as

S2 ¼ diagðP 2

11; P 2

22; …Þ ð33:150Þ

C 2

ij ¼

S0j 2 S0i

P 2

ij; j – i; i; j ¼ 1; 2; … ð33:151Þ

where P 2

ij are the nondiagonal elements of P2

P2 ¼ VT

0 B0U1S1 þ VT

0 B1U0S1 þ VT

0 B1U1S0 2 VT

0 A1U1 ð33:152Þ

33-24 Vibration and Shock Handbook

V2 can be expressed as

V2 ¼ V0D2 ð33:153Þ

where D2 is to be the determined coefficient matrix given by

ij ¼

S0j 2 S0i

ij; j – i; i; j ¼ 1; 2; … ð33:154Þ

and R2

ij are the nondiagonal elements of R2

R2 ¼ UT

0 ðBT

0 V1S1 þ BT

1 V0S1 þ BT

1 V1S0 2 AT

1 V1Þ ð33:155Þ

If j ¼ i; the coefficients C 1

ii; D1

ii; C 2

ii ; and D2

ii can be computed as

C 1

ii ¼

0iðB0 þ BT

0 Þu0i

0iB1u0i þ

j¼1

j–i

CijuT

0jðB0 þ BT

0 Þu0i

BBB@

CCCA

ð33:156Þ

ii ¼ Q 1

ii 2 C 1

ii ð33:157Þ

where Q 1

ii is the diagonal element of Q1

Q1 ¼ 2VT

0 B1U0 ð33:158Þ

C 2

ii ¼

2uT

1iB0u1i þ uT

0iB1u1i þ uT

1iB1u0i 2

j¼1

j–i

CijuT

0jðB0 þ BT

0 Þu0i

0iðB0 þ BT

0 Þu0i ð33:159Þ

ii ¼ Q 2

ii 2 C 2

ii ð33:160Þ

and Q 2

ii is the diagonal element of Q2

Q2 ¼ 2VT

0 B0U1 2 VT

1 B0U1 2 VT

1 B1U0 ð33:161Þ

33.9.3 High-Accuracy Modal Superposition for Eigenvector Derivatives

For a large-scale structure, only a small number of the first lower L modes are extracted, and the higher

modes are truncated in order to reduce the computational cost. The modal superposition method may

not only give inaccurate result, but also may be misleading if the truncation is considerable. In this

section, we give a high-accuracy modal superposition method for derivatives of the complex mode of

nonsymmetric matrices.

33.9.3.1 Improved Modal Superposition

An improved modal superposition (IMS) to reduce the computation errors by modal truncation was

proposed (Lim et al., 1989). The derivatives of modes can be expressed as

›ui

›b ¼ a􀀊iiui þ z􀀊i ð33:162Þ

z􀀊i ¼

j¼1

j–i

j Fi

Si 2 Sj

uj þ A21Fi þ

j¼1

j Fi

uj ð33:163Þ

Fi ¼

›A

›b

›Si

›b

B 2 Si

›B

›b

􀀏 􀀐

ui ð33:164Þ

Structural Dynamic Modification and Sensitivity Analysis 33-25

where A21Fi is the contribution of the truncated higher modes to the derivatives of modes, as given by

a􀀊ii ¼ 2

›B

›b

ui þ uT

i Bz􀀊i þ z􀀊T

i Bui

􀀏 􀀐

ð33:165Þ

33.9.3.2 High-Accuracy Modal Superposition

Assume that the eigenvalues are ordered according to their modular magnitude, and satisfy the following

condition:

lSil , lSjl; j . L ð33:166Þ

The derivatives of modes can be expressed as

›ui

›b ¼ 􀀊 aiiui þ z􀀊i ¼ a􀀊iiui þ ziL þ ziH ð33:167Þ

where

z􀀊i ¼ ziL þ ziH ð33:168Þ

ziL ¼

j¼1

j–i

j Fi

Si 2 Sj

uj ð33:169Þ

ziH ¼ 2

j¼1

􀀍

A21ðBA21Þj21 2 ULððSLÞ21ÞjVTL

􀀎

Fi ð33:170Þ

Also, a􀀊ii can be obtained from Equation 33.165, and K denotes the number of terms used in series

(Equation 33.170).

It can be shown that for K ¼ 1; Equation 33.167 is equivalent to Equation 33.162.

33.9.3.3 Numerical Example

Example 33.6

Consider a 20-DoF system, as shown in Figure 33.6, with the parameters given by

m1 ¼ m2 ¼ · · · ¼ m19 ¼ 2m; m20 ¼ m ¼ 1:0 kg

k1 ¼ k2 ¼ · · · ¼ k21 ¼ 1:0 £ 103 N=m

c1 ¼ c2 ¼ · · · ¼ c7 ¼ 3c; c8 ¼ c9 ¼ · · · ¼ c14 ¼ 2c

c15 ¼ c16 ¼ · · · ¼ c21 ¼ c ¼ 0:1 N sec=m

K1 K2 Ki Ki+1

Ci Ci+1 C21

K21

m1 m2 mi m5

FIGURE 33.6 The 20-DoF system for Example 33.6.

33-26 Vibration and Shock Handbook

For the purpose of comparison, the errors of the truncation modal superposition (TMS), the IMS,

and the high-accuracy superposition (HAMS) in computing the derivatives of eigenvectors are listed

in Table 33.7.

For the sake of simplicity, the errors of eigenvector derivatives are represented by

›ui

›bj

􀁻 !

›ui

›bj

􀁻 !

􀀈 􀀈 􀀈 􀀈 􀀈

where ð›ui=›bjÞ1 denotes the exact solution and ð›ui=›bjÞa denotes those obtained by the three methods

presented above. In the computation of the eigenvector derivatives, the parameters m1 and m10 are

functions of the design variable b1; the parameters c8 and c15 are functions of design variable b2; and K

denotes the number of terms used in series (Equation 33.170).

The results in Table 33.7 confirm that the solution accuracy of the high-accuracy modal superposition

is much higher than that of the TMS and the IMS. For example, if only the first four modes are used,

the errors of ›u2=›b1 are 101.00 and 50.94% for the truncated modal superposition and the IMS, and the

errors are reduced to about 27.27, 7.64, and 1.51% for the high-accuracy modal superposition. If the first

12 modes are used, the error of ›u1=›b1 is 16.76% for the truncated modal superposition, and the errors

are reduced to 0.01, 0.00 and 0.00% for the case of K ¼ 2; 3, 4 in the series (Equation 33.170), where the

first four modes are used, respectively.

33.9.4 Matrix Perturbation for Repeated Eigenvalues of Nondefective

Systems

33.9.4.1 Basic Equation

Consider a system having repeated eigenvalues, S0 ¼ S1 ¼ · · · ¼ S0m; with multiplicity m, and the

corresponding right and left modal matrices

U0m ¼ ½ u01 u02 · · · u0m􀀉 ð33:171Þ

TABLE 33.7 Errors of Eigenvector Derivatives (%)

Modes used TMS IMS HAMS

2 3 4

›u1 =›b1 4 60.18 13.14 1.41 0.01 0.00

8 26.66 1.01 0.03 0.00 0.00

12 16.76 0.66 0.01 0.00 0.00

›u1 =›b1 4 101.00 50.94 27.72 7.64 1.51

8 48.65 10.32 1.29 0.38 0.03

12 23.98 3.37 0.35 0.06 0.01

›u1 =›b1 4 69.74 14.82 3.07 0.02 0.00

8 26.60 1.39 0.23 0.00 0.00

12 25.92 0.85 0.12 0.00 0.00

›u1 =›b1 4 90.63 43.79 21.33 9.01 2.00

8 44.86 6.65 1.56 0.35 0.04

12 23.76 3.23 0.37 0.11 0.02

›u1 =›b1 4 72.52 15.07 8.46 1.11 0.03

8 37.95 1.22 0.57 0.03 0.00

12 28.54 0.52 0.03 0.00 0.00

›u1 =›b1 4 63.33 9.19 1.92 0.42 0.00

8 27.56 0.75 0.49 0.21 0.00

12 25.00 0.47 0.15 0.01 0.00

Structural Dynamic Modification and Sensitivity Analysis 33-27

V0m ¼ ½ v01 v02 · · · v0m􀀉 ð33:172Þ

the remaining eigenvalues being distinct.

The repeated eigenvalues satisfy the following equations:

A0U0m ¼ B0U0mS0 ð33:173Þ

0 V0m ¼ BT

0 V0mS0 ð33:174Þ

0mB0U0m ¼ I ð33:175Þ

0iB0u0i ¼ 1 ð33:176Þ

If small changes are made to the parameters, we have

A ¼ A0 þ 1A1 ð33:177Þ

B ¼ B0 þ 1B1 ð33:178Þ

The eigenvalues and eigenvectors of the perturbed system can be expressed as power series expansions

in 1:

Sm ¼ S0 þ 1S1 þ 12S2 þ · · · ð33:179Þ

Um ¼ U0 þ 1U1 þ 12U2 þ · · · ð33:180Þ

Vm ¼ V0 þ 1V1 þ 12V2 þ · · · ð33:181Þ

where

U0 ¼ U0ma ð33:182Þ

V0 ¼ V0mb ð33:183Þ

and am£m and bm£m are to be determined coefficient matrices.

33.9.4.2 The First-Order Perturbation of Eigenvalues

The first-order perturbation diagonal matrix, S1; of the repeated eigenvalues and the coefficient matrix,

a; can be obtained from the equations:

Wa ¼ aS1 ð33:184Þ

WTb ¼ bS1 ð33:185Þ

W ¼ VT

0mðA1 2 S0B1ÞU0m ð33:186Þ

and the normalization conditions

aTa ¼ I ð33:187Þ

bTa ¼ I ð33:188Þ

If matrix W has no repeated eigenvalues, a and b can be uniquely determined. If W has

repeated eigenvalues, we must consider the higher order perturbation equations for determining a

and b: Here, we assume that S1i are distinct eigenvalues, that is, S1i – S1j ði – j Þ; where S1i are the

diagonal elements of S1:

33.9.4.3 The First-Order Perturbation of Eigenvectors

According to the modal expansion theorem, the first-order perturbation of the right and left

eigenvectors, U1 and V1; can be expressed as

U1 ¼ U0maC1m

þ UAC1

A ð33:189Þ

V1 ¼ V0mbD1m

þ VAD1

A ð33:190Þ

33-28 Vibration and Shock Handbook

where C1m

and C1

A are coefficient matrices which are to be determined, and UA and VA are the right and

left modal matrices corresponding to the distinct eigenvalues:

A ¼ ðSA 2 S0Þ21VT

AðS0B1 2 A1ÞU0ma ð33:191Þ

A ¼ ðSA 2 S0Þ21UT

AðS0BT

1 2 AT

1 ÞV0mb ð33:192Þ

The elements of matrix C1m

can be computed by

mij ¼

l1j 2 l1i

; i – j; i; j ¼ 1; 2; …; m ð33:193Þ

where R1

ij are the nondiagonal elements of R1:

R1 ¼ bTVT

0mB0UAC1

AS1 þ bTVT

0mB1U0maS1 þ bTVT

0mB1UAC1

AS0 2 bTVT

0mA1UAC1

A ð33:194Þ

mii ¼

0iðB0 þ BT

0 Þu0i

0iB1u0i þ

j¼1

j–i

mijuT

0jðB0 þ BT

0 Þu0i þ

j¼mþ1

AijuT

0jðB0 þ BT

0 Þu0i

BBB@

CCCA

ð33:195Þ

mij ¼

S1j 2 S1i

; i – j; i; j ¼ 1; 2; …; m ð33:196Þ

where R2

ij are the nondiagonal elements of R2:

R2 ¼ aTUT

0mB0VAD1

AS1 þ aTUT

0mBT

1 V0mbS1 þ aTUT

0mBT

1 VAD1

AS0 2 aTUT

0mAT

1 VAD1

A ð33:197Þ

mii ¼ Q 2

ii 2 C1

mii ð33:198Þ

where Q 2

ii are the diagonal elements of Q2:

Q2 ¼ 2bTVT

0mB1U0ma ð33:199Þ

33.9.5 Matrix Perturbation for Close Eigenvalues of Unsymmetric Matrices

Assume that S0 is a diagonal matrix with m close eigenvalues; U0n£m and V0n£m are the corresponding

right and left eigenvectors matrices; SA is the remaining distinct eigenvalue diagonal matrix; UAn£ðn2mÞ

and VAn£ðn2mÞ are the corresponding right and left eigenvector matrices. They satisfy the following

equations:

A0½U0

.. .

UA􀀉 ¼ B0½U0

.. .

UA􀀉diagðS0; SAÞ ð33:200Þ

0 ½V0

.. .

VA􀀉 ¼ BT

0 ½V0

.. .

VA􀀉diagðS0; SAÞ ð33:201Þ

½V0

.. .

VA􀀉TB0½U0

.. .

UA􀀉 ¼ I ð33:202Þ

0iB0u0i ¼ 1 ð33:203Þ

Construct the matrix A􀀊 0 as

A0 ¼ A􀀊 0 þ 1A􀀊 0 ð33:204Þ

where

A􀀊 0 ¼ B0½U0

.. .

UA􀀉diagðS0I; SAÞ½V0

.. .

VA􀀉TB0 ð33:205Þ

1A􀀊 0 ¼ B0U0ð1½dS0􀀉ÞVT

0 B0 ð33:206Þ

1½dS0􀀉 ¼ S0 2 S0I ð33:207Þ

Structural Dynamic Modification and Sensitivity Analysis 33-29

S0 ¼

k¼1

S0i

􀁻 !

ð33:208Þ

Here, S0i are the close eigenvalues, and S0 is the average of S0i ði ¼ 1; 2; …; mÞ:

It can be shown that the following equations hold:

A􀀊 0U0 ¼ B0U0S0I ð33:209Þ

A􀀊 T

0 V0 ¼ BT

0 V0S0I ð33:210Þ

These equations indicate that S0 is the repeated eigenvalue with multiplicity, m; for the eigenproblem

defined by Equation 33.209 and Equation 33.210, and U0 and V0 are the corresponding right and left

modal matrices, respectively.

If small modifications 1A1 and 1B1 are imposed on the matrices A0 and B0; then the eigenproblems of

the perturbed system become

ð A􀀊 0 þ 1A􀀊 1ÞU ¼ ðB0 þ 1B1ÞUS ð33:211Þ

ð A􀀊 0 þ 1A􀀊 1ÞTV ¼ ðB0 þ 1B1ÞTVS ð33:212Þ

where

1A􀀊 1 ¼ 1A􀀊 0 þ 1A1 ð33:213Þ

The eigensolutions of Equation 33.211 and Equation 33.212 are given by

U ¼ U0a þ 1U1 ð33:214Þ

V ¼ V0a þ 1V1 ð33:215Þ

S ¼ S0 þ 1S1 ð33:216Þ

It should be noted that Equation 33.211 and Equation 33.212 are the eigenproblem for repeated

eigenvalues. That is, the perturbation analysis for close eigenvalues has been transferred into that of

repeated eigenvalues. Hence, the methods given by Section 33.9.4 can be used to compute S1; a; b; U1;

and V1 in Equation 33.214 to Equation 33.216.

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