33.6 Sensitivity of Eigenvectors for Free – Free Structures

Back

As can be seen from Equation 33.48 and Equation 33.53, both method one and method two fail to deal

with the free – free structures with rigid-body modes because they involve the inversion of the stiffness

matrix. However, we can transform the eigenproblem with a singular stiffness matrix into its equivalent

eigenproblem with a nonsingular stiffness matrix, in the sense that these two eigenproblems have the

same derivatives of eigenvalues and eigenvectors (Liu and Chen, 1994b).

Consider the eigenvalue problem

K􀀊 u􀀊 i ¼ l􀀊iMu􀀊 i ð33:56Þ

where

K􀀊 ; K 2mM ð33:57Þ

Here, m is a nonzero scalar parameter and K􀀊 is nonsingular if m – li ði ¼ 1; 2;…; nÞ

It can be shown that

l􀀊i ¼ li 2 m ð33:58Þ

u􀀊 i ¼ ui; i ¼ 1; 2;…;N ð33:59Þ

and

dl􀀊i

db ¼

dli

db ð33:60Þ

du􀀊 i

db ¼

dui

db ð33:61Þ

The derivatives dui=db can be obtained from the derivatives du􀀊 i=db of the eigenproblem of

Equation 33.56, in which K􀀊 is nonsingular. In this context, both method one and method two,

discussed in Section 33.5.1 and Section 33.5.2, can be applied to deal with the free – free structures with

rigid-body modes.

To achieve a faster average convergent speed for all the first m eigenvector derivatives, m can be

determined as

m ¼

Xm

j¼1

lj

0

@

1

A

m

;

m – lj;

8>>>><

>>>>:

j ¼ 1; 2; …; m ð33:62Þ

Structural Dynamic Modification and Sensitivity Analysis 33-13

© 2005 by Taylor & Francis Group, LLC

Навигация