33.3 Perturbation Method of Vibration Modes

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The perturbation methods of vibration modes are well developed (Fox and Kapoor, 1968; Rogers, 1977;

Chen and Wada, 1979; Hu, 1987; Chen, 1993). In this section, it is assumed that all eigenvalues of the

original structure are distinct.

33.3.1 First-Order Perturbation of Distinct Modes

According to the expansion theorem, the first-order perturbation, u1i; can be expanded by the modal

vectors, u0s; of the original structure as

u1i ¼

Xn

s¼1

c1su0s ð33:17Þ

where

c1s ¼

1

l0i 2 l0s ðuT

0sK1u0i 2 l0iuT

0sM1u0iÞ; s – i ð33:18Þ

c1i ¼ 2 1

2 uT

0iM1u0i ð33:19Þ

The first-order perturbation of the eigenvalues is

l1i ¼ uT

0iK1u0i 2 l0iuT

0iM1u0i ð33:20Þ

33.3.2 Second-Order Perturbation of Distinct Modes

If the parameter modification is fairly large, in order to obtain high computing accuracy, the secondorder

perturbation must be used. According to the expansion theorem, the second-order perturbation,

u2i; can be expanded by the modal vectors, u0s; of the original structure as

u2i ¼

Xn

s¼1

c2ju0s ð33:21Þ

where

c2s ¼

1

l0i 2 l0s ðuT

0sK1u1i 2 l0iuT

0sM1u1i 2 l1iuT

0sM0u1i 2 l1iuT

0sM1u0iÞ; s – i ð33:22Þ

c2s ¼ 2

1

2 ðuT

1iM0u1i þ uT

0iM1u1i þ uT

1iM1u0iÞ ð33:23Þ

The second perturbation of the eigenvalues is

l2i ¼ uT

0iK1u1i 2 l0iuT

0iM1u1i 2 l1iuT

0iM0u1i 2 l1iuT

0iM1u0i ð33:24Þ

33.3.3 Numerical Examples

As illustrations of the matrix perturbation method, several numerical examples are given now.

Example 33.1

Consider the five-degree-of-freedom (five-DoF) system shown in Figure 33.1. The physical parameters

are given as

m1 ¼ m2 ¼ m3 ¼ m4 ¼ 1:0; m5 ¼ 0:5; k1 ¼ k2 ¼ k3 ¼ k4 ¼ k5 ¼ 1:0

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© 2005 by Taylor & Francis Group, LLC

In order to study the computing accuracy of first- and second-order perturbations, let us assume that the

fifth mass undergoes a decrement of 5 to 30%, and the stiffness of the first spring undergoes a decrement

of 5 to 30%.

The computed results for the natural frequencies are presented in Table 33.1, in which the initial

solutions mean the eigensolutions of the original structure.

K1 K2 K3 K4 K5

m1 m2 m3 m4 m5

FIGURE 33.1 Mass – spring system for Example 33.1.

TABLE 33.1 Comparison of Natural Frequencies

Mode

Number

Changes of Structural Parameter (%)

5 10 15 20 25 30

1 A 0.3022 0.2922 0.2821 0.2724 0.2629 0.2536

B 0.3128 0.3128 0.3128 0.3128 0.312 0.3128

C 3.52 7.14 10.9 14.38 18.97 23.3

D 0.3017 0.2903 0.2783 0.2658 0.2527 0.2388

C 0.14 0.58 1.34 2.43 3.9 5.81

E 0.3022 0.2922 0.2827 0.2740 0.2660 0.2588

C 0.0033 0.068 0.24 0.58 0.93 2.08

2 A 0.8788 0.8512 0.8249 0.7998 0.7756 0.7523

B 0.9079 0.98079 0.9079 0.98079 0.9079 0.98079

C 3.31 6.66 10.1 13.52 17.06 20.68

D 0.8775 0.8460 0.8133 0.7792 0.7435 0.7060

C 0.15 0.61 1.41 2.58 4.14 6.15

E 0.3789 0.8518 0.8267 0.8089 0.7835 0.7659

C 0.0076 0.062 0.21 0.52 0.91 1.80

3 A 1.3732 1.3348 1.2989 1.2650 1.2332 1.2031

B 1.1421 1.1421 1.1421 1.1421 1.1421 1.1421

C 2.99 5.94 8.88 11.79 14.7 17.5

D 1.371 1.3266 1.2806 1.2328 1.1832 1.1313

C 0.15 0.62 1.41 2.54 4.05 5.96

E 1.3733 1.3356 1.3015 1.2712 1.2449 1.2231

C 0.0074 0.060 0.20 0.49 0.74 1.66

4 A 1.7355 1.6923 1.6520 1.6143 1.5790 1.5457

B 1.7820 1.7820 1.7820 1.7820 1.7820

C 2.68 5.30 7.78 10.38 12.85 15.28

D 1.7331 1.6828 1.6319 1.5773 1.5209 1.5209

C 0.14 0.56 1.28 2.29 3.62 5.27

E 1.7356 1.6933 1.6552 1.6216 1.5830 1.5695

C 0.0070 0.057 0.19 0.45 0.68 1.53

5 A 1.9273 1.8825 1.8408 1.8017 1.7649 1.7304

B 1.9753 1.9753 1.9753 1.9753 1.9753 1.9753

C 2.49 4.93 7.31 9.63 11.92 14.16

D 1.9248 1.8929 1.896 1.7696 1.7079 1.6492

C 0.1300 0.51 1.15 2.06 3.23 4.69

E 1.9274 1.8835 1.8439 1.8090 1.7790 1.7541

C 0.0064 0.051 0.17 0.41 0.83 1.37

Structural Dynamic Modification and Sensitivity Analysis 33-5

© 2005 by Taylor & Francis Group, LLC

As can be seen from the results, if the change of the structural parameter is 15%, the average change of

the natural frequencies is 9.0%. Using the first-order perturbation, the average error of the frequencies is

reduced to 1.32%.

If the change of the structural parameter is 30%, the average error of the natural frequencies is 18%.

Using the first-order perturbation, the average error of natural frequencies is reduced to 1.6%.

The notation used in Table 33.1 and Table 33.2 is as follows:

* A: the exact solutions of the modified structure

* B: the initial solutions of the original structure

* C: percent error

* D: the first-order perturbation solutions

* E: the second-order perturbation solutions

* F: the percent errors of the initial solutions

* G: the percent errors of the first-order perturbation

* H: the percent errors of the second perturbation

Example 33.2

Consider a truss structure (as shown in

Figure 33.2) with 20 rods. The cross section area

of the second rod is changed from 1:0 to 2:0 cm2.

The results calculated are listed in Table 33.2.

Example 33.3

Consider a torsional vibration system with five

disks, as shown in Figure 33.3. The physical

parameters of the system are as follows:

* I1 ¼ 10:78 kg cm sec2

* I2 ¼ 82:82 kg cm sec2

* I3 ¼ 14:27 kg cm sec2

* I4 ¼ 29:56 kg cm sec2

* I5 ¼ 21:66 kg cm sec2

* K1 ¼ 10:48 £ 104 kg cm=rad

* K2 ¼ 34:30 £ 104 kg cm=rad

* K3 ¼ 24:40 £ 104 kg cm=rad

* K4 ¼ 40:60 £ 104 kg cm=rad

The corresponding constrained system is shown

in Figure 33.3b, in which the hung stiffness is

Ks ¼ 4060 kg cm=rad:

The exact eigensolutions of the constrained

system are taken as the initial results. Using matrix

TABLE 33.2 Comparison of Natural Frequencies

Mode No. A (Hz) B (Hz) F (%) D (Hz) G (%) E (Hz) H (%)

1 27.78 25.45 8.39 29.27 5.36 28.14 1.29

2 109.1 107.7 1.28 110.39 1.18 109.35 0.28

3 157.4 153.2 2.67 159.53 1.35 158.79 0.88

4 230.5 233.2 1.17 231.24 0.32 231.02 0.022

5 320.7 325.6 1.53 320.58 0.04 320.89 0.04

6 391.1 393.7 0.66 392.06 0.24 319.35 0.06

1

2

3

4

5

6

7

FIGURE 33.2 Truss structure for Example 33.2.

I1 I2 I3 I4

K1 K2 K3 K4

K5

I5

(a)

(b)

FIGURE 33.3 Torsional vibration system for

Example 33.3.

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© 2005 by Taylor & Francis Group, LLC

perturbation, the eigensolutions of the free – free system can be obtained. The results are listed in

Table 33.3 and Table 33.4.

The notation used in Table 33.3 is as follows:

* v0: exact solution of the natural frequency of the free – free system (l/sec)

* v00: perturbation solutions of the natural frequency of the free – free system (l/sec)

* vx: natural frequency of the constrained system (l/sec)

1v1 ¼ v00 2 vx the perturbation of the natural frequency

d ¼

lv0 2 vx l

v0 ð%Þ

d0 ¼

lv0 2 v00l

v0 ð%Þ

TABLE 33.3 Comparison for Natural Frequencies

No.

1 2 3 4 5

v0 0.000000 62.554934 105.668169 177.680405 224.361444

vx 1.595950 62.604815 105.669360 177.704224 224.386235

d 0.079700 0.001070 0.013406 0.011000

1v1 2 1.525707 2 0.049895 2 0.001132 2 0.023834 2 0.002508

v 00 0.080243 62.554920 105.448168 177.680390 224.383727

d 0 2.24 £ 1025 9.46 £ 1027 8.22 £ 1028 9.93 £ 1025

TABLE 33.4 Comparison for Eigenvectors

No. ui0

ui

x 1ui1

u0i0

1 0.079283 0.097363 2 0.000080 0.079823

0.079283 0.079342 2 0.000059 0.079283

0.079283 0.079287 2 0.000004 0.079283

0.079283 0.079198 0.000085 0.079283

0.079283 0.079189 0.000153 0.079342

2 2 0.095647 2 0.095611 0.000036 2 0.095647

2 0.057148 2 0.057064 2 0.000084 2 0.057148

0.008612 0.008719 2 0.000107 0.008612

0.099081 0.099192 2 0.000111 0.099081

0.125223 0.125259 0.000036 0.125223

3 0.277894 0.277884 0.000010 0.277894

2 0.041278 2 0.041284 0.000005 2 0.41279

2 0.027509 2 0.027497 2 0.000013 2 0.027510

0.009810 0.009840 2 0.000030 0.009810

0.024263 0.024279 2 0.000016 0.024263

4 2 0.009038 2 0.009033 2 0.000005 2 0.009038

0.020312 2 0.020309 0.000002 0.020312

2 0.125560 2 0.125580 0.000024 2 0.125556

2 0.098791 0.098735 2 0.000053 2 0.098788

0.144375 0.144413 2 0.000038 0.144375

5 0.004823 0.004823 0.000001 0.004824

2 0.020156 2 0.020154 2 0.0000002 2 0.020156

0.217244 0.217227 0.000018 0.217245

2 0.088717 2 0.088725 0.000008 2 0.088717

0.052618 0.052652 2 0.000034 0.052618

Structural Dynamic Modification and Sensitivity Analysis 33-7

© 2005 by Taylor & Francis Group, LLC

The notation used in Table 33.4 is as follows:

* ui0

: exact solution of eigenvectors of the free – free system

* u0i0

: perturbation solution of eigenvectors of the free – free system

* ui

x: eigenvector of the constrained system

* 1ui1

: first-order perturbation of eigenvectors

As can be seen from Table 33.3, the natural frequencies of the free – free system are increased by the hung

elastic elements. For example, the frequency of the rigid mode is increased to 1.595950 (l/sec), and the

frequency of the first elastic mode is increased by 0.8124%. By modifying the eigensolutions with the

perturbation method, the frequency of the rigid mode is reduced to 0.079700 (l/sec), which is nearly

equal to zero, and all the frequencies of the elastic modes become almost exact solutions. The results in

Table 33.4 show that the mode shapes of the free – free system, u0i0

; are close to the exact solution, ui0

:

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