230
6 Free Vibration of Multiple Degrees of Freedom System
A reasonably good estimate for the fundamental mode of MDF system is the shape
obtained by applying a static load proportional to the mass, or the mass moment of
inertia of the system. Based on this concept, the following is the mode shape for the
given problem
{φ } =
⎧
⎨
⎩
1.00
0.80
0.40
⎫
⎬
⎭
{φ} can be obtained with the value of Example 6.7. Therefore,
{φ}
T
[K ] {φ} =
1.0 0.8 0.4
⎡
⎣
2k − 2k 0
− 2k 4k − 2k
0 − 2k 5k
⎤
⎦
⎧
⎨
⎩
1.0
0.8
0.4
⎫
⎬
⎭
= 0.88k
and
{φ}
T
[M] {φ} =
1.0 0.8 0.4
m
⎡
⎣
1
1
1
⎤
⎦
⎧
⎨
⎩
1.0
0.8
0.4
⎫
⎬
⎭
= 1.8m
Therefore, from Eq. (6.104),
p
2
=
0.88k
1.8m
= 0.49
k
m
or
p = 0.699
k
m
Comparing this value of p with those obtained in Example 6.9 where p =
0.695
k
m
, the estimate is quite close.