33.7 Matrix Perturbation Theory for Repeated Modes

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33.7.1 Basic Equations

In this section, let us consider the case of repeated eigenvalues, namely, l0i ¼ l0iþ1 ¼ · · · ¼ l0iþm21: The

system is known as a degenerate system. In engineering, many complex and large structures, such as

airplanes, rockets, tall towers, bridges, and ocean platforms, often have multiple or cluster eigenvalues.

The matrix perturbation for the repeated modes is presented in Haug, et al. (1980), Chen and Pan (1986),

Hu (1987), Mills-Curran (1988), Ojalvo (1988), Dailey (1989), Lim et al. (1989) and Shaw and Jayasuriya

(1992).

Assume that l0 ¼ l01 ¼ l02 ¼ · · · ¼ l0m; that is, l0 is a repeated eigenvalue with multiplicity equal to

m; and u01; u02; · · ·; u0m are the eigenvectors associated with l0: Then, a linear combination of u0j

ðj ¼ 1; 2; …; mÞ; denoted as U0; will also be the eigenvector associated with l0:

U0 ¼ U0ma ð33:63Þ

where

U0m ¼ ½u01; u02; …; u0m􀀉 ð33:64Þ

aTa ¼ I ð33:65Þ

and

a ¼ ½a1; a2; …; am􀀉T ð33:66Þ

Note that a is a constant matrix to be determined.

According to the matrix perturbation method, the eigenvalues and eigenvectors of the structure with

repeated eigenvalues for the perturbed structure can be expressed as

Lm ¼ L0 þ 1L1 ð33:67Þ

Um ¼ U0ma þ 1ðU0Cm þ UACAÞ ¼ U0ma þ 1ðU0maCm þ UACAÞ ð33:68Þ

where UA is the n £ ðn 2 mÞ modal matrix containing all the eigenvectors except U0m; Lm is the m £ m

eigenvalue diagonal matrix of the perturbed structure, L1 is the m £ m diagonal matrix with its diagonal

elements equal to the first-order perturbations of eigenvalues, Cm is an m £ m matrix to be determined,

and CA is an ðn 2 mÞ £ ðn 2 mÞ matrix to be determined.

33.7.2 The First-Order Perturbation of Eigensolutions

L1 and a can be computed from the following ðm £ mÞ eigenproblem:

Wa ¼ aL1; aTa ¼ I ð33:69Þ

where

W ¼ UT

0mðK1 2 l0M1ÞU0m ð33:70Þ

Solving the m £ m eigenproblem of Equation 33.69 can produce L1 and a:

If matrix W has no repeated eigenvalues, a can be uniquely determined; if matrix W has repeated

eigenvalues, a can be determined using the higher order perturbation equations. Here, we assume that

matrix W has no repeated eigenvalues; that is, l1i – l1j; ði – jÞ; where l1k ð0 , k # mÞ are the elements

of the diagonal matrix L1:

The matrix CA is

CA ¼ ðLA 2 l0IÞ21UT

Aðl0M1 2 K1ÞU0ma ð33:71Þ

33-14 Vibration and Shock Handbook

© 2005 by Taylor & Francis Group, LLC

The elements of Cm are

C m

ij ¼

Rij

lð1Þ jm 2 lð1Þ im

; i – j i; j ¼ 1; 2; …; m ð33:72Þ

where Rij are the elements of R given by

R ¼ 2aTUT

0mM1U0maL1 þ aTU0m K1UACA 2 l0aTU0mM1UACA 2 aTUT

0mM0UACAL1 ð33:73Þ

and

C m

ii ¼

1

2

Q ii ð33:74Þ

where Q ii is the diagonal elements of Q; given by

Q ¼ 2aTUT

0mM1U0ma ð33:75Þ

33.7.3 High-Accuracy Modal Superposition for the First-Order Perturbation

of Repeated Modes

In Section 33.5, the high-accuracy modal superposition for the first-order perturbation of eigenvectors of

distinct eigenvalues is given. In this section, we extend these methods to the situation with repeated

modes.

33.7.3.1 Method One for Computing U1

Assuming UAL and LAL are the first L modes and eigenvalues excluding the repeated modes, the firstorder

perturbation of eigenvectors is

U1 ¼ U0maCm þ UALCAL þ SR ð33:76Þ

SR ¼ US 2 ½U0m

.. .

UAL􀀉diagðl21

0 ; L21

AL Þ½U0m

.. .

UAL􀀉TT ð33:77Þ

where US is the static displacement obtained by

KUS ¼ T ð33:78Þ

and

T ¼ M0U0maL1 þ l0M1U0ma 2 K1U0ma ð33:79Þ

In Equation 33.79, L1 and a can be obtained from Equation 33.69.

The matrix CAL is given by

CAL ¼ ðLAL 2 l0IÞ21UT

ALðl0M1 2 K1ÞU0ma ð33:80Þ

and the elements of matrix Cm are

C m

ij ¼

Rij

lð1Þ jm 2 lð1Þ im

; i – j; i; j ¼ 1; 2; …; m ð33:81Þ

where R is given by

R ¼ aTUT

0mM1U0mL1 2 aTUT

0mðl0M1 2 K1ÞðUALCAL þ SRÞ 2 aTUT

0mM0SRL1 ð33:82Þ

The diagonal elements of Cm are

C m

ii ¼

1

2

Q ii ð33:83Þ

Structural Dynamic Modification and Sensitivity Analysis 33-15

© 2005 by Taylor & Francis Group, LLC

where

Q ¼ 2aTUT

0mM1U0ma 2 aTUT

0mM0SR 2 STR

M0U0ma ð33:84Þ

33.7.3.2 Method Two for Computing U1

The first-order perturbation of eigenvectors can be expressed as

U1 ¼ U0maCm þ UALCAL þ SR ð33:85Þ

where CAL can also be calculated using Equation 33.80; that is

CAL ¼ ðLAL 2 l0IÞ21UT

ALðl0M1 2 K1ÞU0m a

and SR is given by

SR ¼

X1

j¼0

lj

0ðHj 2 WjÞ ð33:86Þ

where

Wj ¼ ½U0m

.. .

UAL􀀉L2j21

0 ½U0m

.. .

UAL􀀉TT; j $ 0 ð33:87Þ

T ¼ M0U0maL1 þ l0M1U0ma 2 K1U0ma ð33:88Þ

The iterative method for computing Hj is as follows:

H0 ¼ K21T;

F0j21 ¼ MHj21;

Hj ¼ K21F0j21;

j $ 1 ð33:89Þ

This iterative process can be terminated according to the accuracy requirement. If we define SRðkÞ as

SRðkÞ ¼

Xk

j¼0

lj

0ðHj 2 WjÞ ð33:90Þ

the termination condition can be stated as

kSRðkÞ 2 SRðk 2 1Þk2 # 1; j ¼ 1; 2; …; m ð33:91Þ

where 1 is a specified accuracy requirement.

The computation method for Cm in Equation 33.85 is similar to that of Equation 33.81 to Equation

33.84. The only difference is that SR in Equation 33.82 and Equation 33.84 can be replaced with SRðkÞ in

Equation 33.90.

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