33.8 Matrix Perturbation Method for Closely Spaced Eigenvalues

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The vibration modes with close frequencies, that is, with clusters of frequencies, often occur in certain

structural systems including large space structures, multispan beams, and in some nearly periodic

structures and symmetric structures. Therefore, it is important here to present the perturbation method

for vibration modes with close eigenvalues (Liu, 2000).

The perturbation analysis of close eigenvalues can be transformed into a problem with a repeated

eigenvalue, which is equal to the average value of the close eigenvalues (Chen, 1993).

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33.8.1 Method One of Perturbation Analysis for Close Eigenvalues

Consider vibration eigenproblem

K0½U0

.. .

UA􀀉 ¼ M0½U0

.. .

UA􀀉diagðL0; LAÞ ð33:92Þ

½U0

.. .

UA􀀉TM0½U0

.. .

UA􀀉 ¼ I ð33:93Þ

where K0 and M0 are n £ n real symmetric matrices, and L0 and U0 are the m £ m diagonal matrix of

close eigenvalues and the corresponding n £ m modal matrix.

Using the spectral decomposition of K0; the problem can be expressed as

K0 ¼ K􀀊 0 þ 1 dK0 ð33:94Þ

where

K􀀊 0 ¼ M0ðl0U0UT

0 ÞM0 þ M0ðUALAUT

AÞM0 ð33:95Þ

1 dK0 ¼ M0ðU0ð1 dL0ÞUT

0 ÞM0 ð33:96Þ

1 dL0 ¼ L0 2 l0I ¼ L0 2

Xm

i¼1

l0i

m

0

BBBB@

1

CCCCA

I ð33:97Þ

It can be seen that K􀀊 0 given by Equation 33.95 satisfies

K􀀊 0½U0

.. .

UA􀀉 ¼ M0½U0

.. .

UA􀀉diagðl0I; LAÞ ð33:98Þ

½U0

.. .

UA􀀉TM0½U0

.. .

UA􀀉 ¼ I ð33:99Þ

This indicates that l0 and U0 are the repeated eigenvalues and the corresponding eigenvector subspace

with multiplicity m of the eigenproblem (Equation 33.92), and LA and UA are also the eigensolution of

eigenproblem (Equation 33.92).

If L0 !l0I; 1 dL0 !0; and K􀀊 0 !K0; and if the small parameter modifications 1K1 and 1M1 are

introduced to the matrices K0 and M0; the eigenproblem with close eigenvalues becomes

ðK0 þ 1K1ÞU ¼ ðM0 þ 1M1ÞUL ð33:100Þ

UðM0 þ 1M1ÞUT ¼ I ð33:101Þ

Substituting Equation 33.94 into Equation 33.100, we obtain

ð K􀀊 0 þ 1K􀀊 1ÞU ¼ ðM0 þ 1M1ÞUL ð33:102Þ

UðM0 þ 1M1ÞUT ¼ I ð33:103Þ

where

1K􀀊 1 ¼ 1 dK0 þ 1K1 ð33:104Þ

L ¼ l0I þ 1L1 þ 12L2 þ · · · ð33:105Þ

U ¼ U0a þ 1U1 þ 12U2 þ · · · ð33:106Þ

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Therefore, the eigenproblem of Equation 33.102 and Equation 33.103 can be considered to be a

perturbed eigenproblem with the perturbation matrices equal to ðdK0 þ K1Þ and M1; respectively.

The eigensolutions, L and U; can be obtained from Equation 33.102 and Equation 33.103 by using

the perturbation method for repeated eigenvalues as discussed in Section 33.7. Accordingly, the

perturbation problem of modes with close eigenvalues is transformed into one of the repeated

eigenvalues.

The complete algorithm for L and U is given below.

(1) Compute

l0 ¼

Xm

i¼1

l0i

m

(2) Compute

W ¼ UT

0 ðdK0 þ K1 2 l0M1ÞU0

(3) Solve the eigenvalue problem

Wa ¼ aL1

aTa ¼ I

(4) Compute the perturbed eigenvalues of the close eigenvalues

L ¼ l0I þ L1

(5) Compute the new eigenvectors U0a corresponding to l0.

(6) Compute the matrix CA

CA ¼ ðLA 2 l0IÞ21UT

A ðl0M1 2 K1 2 dK0ÞU0a

(7) Compute

R ¼ ½Rij􀀉

R ¼ 2aTUT

0 M1U0aL1 2 l0aTUT

0 M1UACA 2 aTUT

0 ðdK0 þ K1ÞUACA 2 aUT

0 M0UACAL1

(8) Compute

Cm ¼ ½C m

ij 􀀉

C m

ij ¼

Rij

l1j 2 l1i

; i – j; i; j ¼ 1; 2; …; m

C m

ii ¼

1

2

Q ii

Q ¼ 2aTUT

0 M1U0a

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(9) Compute the perturbed eigenvectors U

U ¼ U0a þ U0aCm þ UACA

33.8.2 Method Two of Perturbation Analysis for Close Eigenvalues

Because of the importance of the problem in both theory and practice, we now present method two of

perturbation analysis for close eigenvalues, which is equivalent to method one given above.

Using the spectral decomposition of M0; the problem can be expressed as

M0 ¼ M􀀊 0 þ 1 dM0 ð33:107Þ

Then, the following equations hold:

K0½U0

.. .

UA􀀉 ¼ M􀀊 0½U0

.. .

UA􀀉diagðl0I; LAÞ ð33:108Þ

½U0

.. .

UA􀀉TM0½U0

.. .

UA􀀉 ¼ I ð33:109Þ

where

M􀀊 ¼ l22

0 K0U0UT

0 K0 þ K0½UAðL21

A Þ2UT

A􀀉K0 ð33:110Þ

1 dM0 ¼ K0½U01 dðL21

0 Þ2UT

0 􀀉K0 ð33:111Þ

1 dL22

0 ¼ L22

0 2 l22

0 I ð33:112Þ

and

l0 ¼

Xm

i¼1

l0i

m ð33:113Þ

It can be seen that M􀀊 0 and 1 dM0 given by Equation 33.110 and Equation 33.111 satisfy Equation 33.108

and Equation 33.109; that is, l0 and U0 are the repeated eigenvalues and corresponding modal matrix of

Equation 33.108 and Equation 33.109. LA and UA are the eigenvalue diagonal matrix and the

corresponding modal matrix excluding L0 and U0; respectively.

If K0 and M0 are modified to K0 þ 1K1 and M0 þ 1M1; the eigenvalue problem becomes

ðK0 þ 1K1ÞU ¼ ðM0 þ 1M1ÞUL ð33:114Þ

UðM0 þ 1M1ÞUT ¼ I ð33:115Þ

Substituting Equation 33.107 into Equation 33.114 yields

ðK0 þ 1K1ÞU ¼ ð M􀀊 0 þ 1M􀀊 1ÞUL ð33:116Þ

Uð M􀀊 0 þ 1M􀀊 1ÞUT ¼ I ð33:117Þ

where

1M􀀊 1 ¼ 1 dM0 þ 1M1 ð33:118Þ

Thus, Equation 33.116 and Equation 33.117 can be considered to be a perturbed eigenproblem with

repeated eigenvalues, and the perturbation method for repeated eigenvalues can be used to obtain the

perturbed eigensolutions of Equation 33.116 and Equation 33.117:

L ¼ l0I þ 1L1 þ · · · ð33:119Þ

U ¼ U0a þ 1U1 þ · · · ð33:120Þ

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Example 33.5

For the six-DoF mass – spring system shown in Figure 33.5, the stiffness and mass matrices K0 and M0

are given by

K0 ¼

1500 21000

21000 1200 2200

2200 15; 200 25000 25000 25000

25000 5000

25000 5000

25000 5000

2

6666666666664

3

7777777777775

ðN=mÞ

M0 ¼ diagð200; 300; 50; 20; 20; 20:004Þ (kg)

The perturbation eigensolutions are computed for the following three cases:

Case 1

1K1 ¼

0 0 0 0 0 0

0 0 0 0 0 0

0 0 5 0 25 0

0 0 0 0 0 0

0 0 25 0 5 0

0 0 0 0 0 0

2

6666666666664

3

7777777777775

ðN=mÞ

1M1 ¼ diagð0; …; 0Þ (kg)

Case 2

1K1 ¼

5:00 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

2

6666666666664

3

7777777777775

ðN=mÞ

1M1 ¼ diagð0; 0; 0; 0; 0; 0:5Þ (kg)

Case 3

1K1 ¼ 0; 1M1 ¼ diagð0; 0; 0; 0; 2:0; 0Þ (kg)

The unperturbed eigensolutions have a single pair of close eigenvalues given by

L0 ¼ diagð249:966642; 250:000000Þ

UT

0 ¼

0:00000 0:00000 20:00012 20:091283 20:091283 0:182560

0:00000 0:00000 0:00000 20:158114 0:158114 0:00000

" #

The other unperturbed eigensolutions are as follows:

LA ¼ diagð0:594885; 2:478725; 10:234656; 552:175102Þ

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UA

¼

20:058595 20:027542 0:028427 20:000001

0:032047 20:027659 0:039259 0:000127

20:006732 0:074593 0:058387 20:104793

20:007020 0:075340 0:058527 0:086699

20:007020 0:075340 0:058527 0:086699

20:007020 0:075340 0:058527 0:086667

2

6666666666664

3

7777777777775

The perturbed eigensolutions associated with

the single pair of close eigenvalues for the three

cases are summarized in Table 33.6. These

results show that the perturbation analysis of

distinct eigenvalues is not only inaccurate but

also misleading when applied to close eigenvalues,

and that the perturbed eigensolutions

given by the present method are in good

agreement with the exact solutions.

For example, in Case 3, the eigenvalue errors

induced by the present method are reduced to

1.047950 and 0.000000, while the errors induced by the perturbation of distinct eigenvalues are 4.174100

and 3.025595. The eigenvectors obtained by the perturbation method of distinct eigenvalues are not only

TABLE 33.6 Comparison of Eigensolutions with Close Eigenvalues

Exact Perturbation Method of Distinct

Eigenvalues

Perturbation Method of

Close Eigenvalues

Case 1

Eigenvalues 249.973872 250.159351 250.008324 250.125000 249.973872 250.159451

Eigenvectors 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000

0.000000 0.000000 0.000000 0.000000 0.000000 0.000000

0.000018 2 0.000066 2 0.000043 0.000079 0.000011 0.000005

0.150492 0.103411 2 0.433574 0.039471 0.150501 2 0.103354

0.014254 2 0.181965 0.251058 0.355699 0.014251 0.182015

2 0.164753 0.078701 0.182585 2 0.395285 2 0.164745 2 0.078658

Case 2

Eigenvalues 243.878599 247.932109 244.759777 246.875000 243.725997 247.908461

Eigenvectors 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000

0.000000 2 0.000004 0.000002 0.000005 0.000000 0.000000

2 0.000015 0.001506 2 0.000772 2 0.001976 0.000011 0.000006

2 0.000622 0.182090 8.467433 2 5.098718 0.001103 0.183060

2 0.155859 2 0.091163 2 8.648272 2 4.784467 2 0.157555 2 0.092069

0.156472 2 0.090082 0.180521 9.882118 0.158052 2 0.090988

Case 3

Eigenvalues 234.474405 249.975015 245.800915 237.500000 233.326455 249.975015

Eigenvectors 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000

0.000016 0.000000 2 0.000008 0.000021 0.000000 2 0.000000

2 0.005558 2 0.000015 0.003030 2 0.007905 0.000006 0.000010

2 0.089507 2 0.158189 34.141638 2 19.920531 2 0.091153 0.158189

0.175420 0.000158 2 34.321586 2 19.612210 0.182572 2 0.000158

2 0.089778 0.158022 0.181586 39.528471 2 0.091415 2 0.158025

m4 m5 m6

K4 K5 K6

m3

K3

m2

K2

m1

K1

FIGURE 33.5 Six-DoF mass – spring system for

Example 33.5.

Structural Dynamic Modification and Sensitivity Analysis 33-21

© 2005 by Taylor & Francis Group, LLC

inaccurate but also misleading, while good agreement with the exact eigenvectors has been obtained by

the present method.

33.8.3 Concluding Remarks

Perturbation analysis of vibration modes with close frequencies is presented in this section. It can be

regarded as a general treatment of perturbation analysis, because the perturbation analysis of both

distinct eigenvalues and repeated eigenvalues is contained in the present method. The results obtained by

this method allow one to analyze the influence of parameter changes in a system on the dynamic

characteristics of the system, which is very important for effective structural design.

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