234
6 Free Vibration of Multiple Degrees of Freedom System
for q eigenvalues [] r + 1 (a diagonal matrix) and eigenvectors [Q] r + 1 .
(d) An improved approximation to the original system can be calculated as
follows
{δ} r + 1 = [δ] r + 1 [Q] r + 1
(6.117)
The eigenvalues [] r + 1 and eigenvectors [δ] r + 1 converge to the lowest
eigenvalues and eigenvectors as r → ∞.
The first step in the subspace iteration method is the selection of starting vector
[δ] 1 . For details, the reader may refer to References [11–13].
6.17 Simultaneous Iteration Method and Algorithm
Simultaneous iteration method is similar in technique to subspace iteration method
[12].
In this method, [K ] is decomposed as
[K ] = [L] [L]
T
(6.118)
where [L] is a lower triangular matrix.
Using Eq. (6.115a, 6.115b), the free vibration Eq. (6.118) can be written as
[L]
− 1
[M]
[L]
− 1
T [L]
T
{δ}
=
1
2
[L]
T
{δ}
(6.119)
or
[A] {x} = λ {x}
(6.120)
where
[A] =
[L]
− 1
[M]
[L]
− 1
T
{x} = [L]
T
{δ}
λ =
1
p 2
The procedure for solving Eq. (6.120) by the simultaneous iteration method is
summarised as follows:
1. Set a trial vector [U ] and orthonormalise it,