Page 257 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

244 Computational Methods Chap. 8

an arbitrary test column:
0.5'
( - 0.2
1.0 .
The first iteration then becomes
0 - 4 . 3 2 - 3 . 0 ‘ i - 2 . 1 3 6 ] 1 - 0 . 8 0 1
'’ -^'1
0 1 . 6 7 0 i - 0 . 2 = j' - 0 . 3 3 4 = 2 . 6 6 6 - 0 . 1 2 5
0 1 . 6 7 3 . 0 1.0 [ 2 . 6 6 6 j ( 1 .0 0
With this normalized column, the second iteration becomes
0 - 4 . 3 2 - 3 . 0 ( - 0 . 8 0 1 \ 1 - 2 . 4 6 ) 1 - 0 . 8 8 1
0 1 . 6 7 0 - 0 . 1 2 5 = < - 0 . 2 1 = 2 . 7 9 - 0 . 0 7 5
0 1 . 6 7 3 . 0 i 1 .0 0 j i 2 . 7 9 j i 1 .0 0
The third iteration gives
0 - 4 . 3 2 - 3 . 0 ‘ 1 - 0 . 8 8 1 ] ( - 2 . 6 8 ] 1 - 0 . 9 3 3
0 1 . 6 7 0 - 0 . 0 7 5 = ^ - 0 . 1 2 5 = 2 . 8 7 - 0 . 0 4 4
0 1 . 6 7 3 . 0 i 1 .0 0 j i 2 . 8 7 j i 1 .0 0
After a few more iterations, the convergence is to
1.0
3 . 0 < 0
1.0
Thus, the eigenvalue and eigenvector for the second mode are
A 3/c r r
At = —^— = 3.0 ■

(x)^m
- 1.0
4>2- { 0
1.0

For the determination of the third mode, we impose the condition Cj = C2 = 0
from the orthogonality equation:
3
c , = Y . = 4 ( 0 . 2 5 ) X | + 2 ( 0 . 7 9 ) a-2 + 1 ( 1 . 0 ) a:3 = 0
/ - 1

c, = = 4 ( - I . 0 ) a:| + 2 (0 )t 2 + 1(1.0)a:3 = 0
/ = 1
From these two equations, we obtain
X , = 0 . 2 5 x , X t = - 0 . 7 9 x 3