228
6 Free Vibration of Multiple Degrees of Freedom System
Without modifying the derived vectors, we proceed for two more iterations and
let us see what happens.
m
6k
⎡
⎣
8 5 2
5 5 2
2 2 2
⎤
⎦
⎧
⎨
⎩
− 1.756
1.000
3.078
⎫
⎬
⎭
=
2.376m
6k
⎧
⎨
⎩
− 1.217
1.000
1.955
⎫
⎬
⎭
m
6k
⎡
⎣
8 5 2
5 5 2
2 2 2
⎤
⎦
⎧
⎨
⎩
− 1.217
1.000
1.955
⎫
⎬
⎭
=
2.825m
6k
⎧
⎨
⎩
− 0.293
1.000
1.323
⎫
⎬
⎭
(a)
It is quite clear that convergence does not appear to be good in the third iteration
step. It may be due to creeping of round-off errors. It has already been stated that the
modification as given by Eq. (6.94) or Eq. (6.96) should be made at each iteration
step. Let us modify the vector from the second to the third iteration step. Therefore,
α 1 =
1.000 0.759 0.336
m
⎡
⎣
1.0 0 0
0 1.0 0
0 0 1.0
⎤
⎦
⎧
⎨
⎩
− 1.217
1.000
1.955
⎫
⎬
⎭
1.000 0.759 0.336
m
⎡
⎣
1 0 0
0 1.0 0
0 0 1
⎤
⎦
⎧
⎨
⎩
1.000
0.759
0.336
⎫
⎬
⎭
or α 1 = 0.117.
Therefore,
{φ} =
⎧
⎨
⎩
− 1.217
1.000
1.955
⎫
⎬
⎭
− 0.117
⎧
⎨
⎩
1.000
0.759
0.336
⎫
⎬
⎭
=
⎧
⎨
⎩
− 0.240
0.059
0.583
⎫
⎬
⎭
We can normalise {φ} such that
{φ} =
⎧
⎨
⎩
− 1.464
1.000
2.103
⎫
⎬
⎭
Therefore,
m
6k
⎡
⎣
8 5 2
5 5 2
2 2 2
⎤
⎦
⎧
⎨
⎩
− 1.464
1.000
2.103
⎫
⎬
⎭
=
1.886m
6k
⎧
⎨
⎩
− 1.319
1.000
1.982
⎫
⎬
⎭