33.5 High-Accuracy Modal Superposition for Sensitivity Analysis of Modes

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The modal superposition method is often used to compute the derivatives of modal vectors. Because of

the cost of generating computer solutions for a dynamic analysis, it is impractical to obtain all modes.

TABLE 33.5 Sensitivities of the First Four Chassis Frequencies

1 NE 15 11 19 24 6 37 34 36

le1

J 9.97 5.23 5.23 3.64 3.64 3.34 3.32 3.31

NE 24 6 23 7 22 8 33 21

le1

Iy 0.015 0.15 £ 1022 0.33 £ 1023 0.33 £ 1023 0.28 £ 1023 0.19 £ 1023 0.11 £ 1023 0.11 £ 1023

2 NE 16 14 11 19 18 12 13 17

le2

J 6.67 £ 1022 0.67 £ 1022 0.45 £ 1022 0.45 £ 1022 0.11 £ 1022 0.11 £ 1022 0.53 £ 1023 0.53 £ 1023

NE 38 10 21 9 22 8 24 6

le2

Iy 2.92 2.92 1.78 1.78 1.49 1.49 1.44 1.44

3 NE 15 19 11 37 31 30 32 1

le3

J 95.9 47.2 47.2 26.5 25.4 25.2 23.8 12.5

NE 10 38 9 21 8 22 6 24

le3

Iy 2.96 2.96 1.97 1.97 1.74 1.74 1.70 1.70

4 NE 14 16 19 11 12 18 17 13

le4

J 0.19 0.19 0.12 0.12 0.03 0.03 0.015 0.015

NE 10 38 39 20 5 25 6 24

le4

Iy 22.4 22.4 15.7 15.7 11.8 11.8 10.9 10.9

Structural Dynamic Modification and Sensitivity Analysis 33-11

© 2005 by Taylor & Francis Group, LLC

Therefore, only the first L low-frequency modes are computed and are used as basis vectors of eigenvector

derivatives. However, as noted above, modal truncation induces errors, which can be significant if more

high-frequency modes are truncated. An explicit method to improve the truncated modal superposition

representation of eigenvector derivatives is presented (Wang, 1991), in which a residual static mode is

used to approximate the contribution due to unavailable high-frequency modes (method one).

In this section a more accurate modal superposition method (method two; Chen, 1993a; Liu and

Chen, 1994a) than method one is given. In this method, the contribution of the truncated modes to

the eigenvector derivatives is expressed exactly, as a convergent series that can be evaluated by a simple

iterative procedure.

33.5.1 Method One

The modal sensitivity can be expressed as

ui;j ¼

XN

s¼1

csus ¼

XL

j¼1

cjuj þ SR ð33:45Þ

where

SR ¼

XN

j¼Lþ1

cjuj ð33:46Þ

Since li ,, lLþ1; Equation 33.46 can be approximated as

SR < SRA ¼ H􀀊 0 2W􀀊 0 ð33:47Þ

where

H􀀊 0 ¼ K21ð2K;j þ li;jM þ liM;jÞ ð33:48Þ

W􀀊 0 ¼

XL

j¼1

1

lj

uT

j ð2K;j þ li;jM þ liM;jÞuj ð33:49Þ

33.5.2 Method Two

The contribution of ui;j; SR due to truncated high-frequencies modes is as follows:

SR ¼

X1

j¼0

lj

iðHj 2 WjÞ ð33:50Þ

where

Wj ¼ ULL2j21

L UTL

ð2K;j þ li;jM þ liM;jÞ ð33:51Þ

UL ¼ ½u1; u2; …; uL􀀉 ð33:52Þ

Hj can be obtained with the following iterative procedure:

H0 ¼ K21ð2K;j þ Ki;jM þ liM;jÞ

F0j21 ¼ MHj21; j $ 1

Hj ¼ K21F0j21

9>>>=

>>>;

ð33:53Þ

33-12 Vibration and Shock Handbook

© 2005 by Taylor & Francis Group, LLC

Define SRðkÞ as

SRðkÞ ¼

Xk

j¼0

lj

iðHj 2 WjÞ ð33:54Þ

Using this definition, the given iterative process can be terminated if the following inequality

kSRðkÞ 2 SRðk 2 1Þk2 # 1 ð33:55Þ

is satisfied, where 1 is a specified accuracy requirement.

It should be noted that, if only the first term in the series (Equation 33.50) is retained with all the other

terms neglected, then method two is reduced to method one. In addition, the series (Equation 33.50) can

be used to estimate the errors induced by the modal truncation.

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