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6 Free Vibration of Multiple Degrees of Freedom System
Fig. 6.17 Example 6.17
rise to (n − 1) independent equations for a n degree of freedom system. This reduced
matrix is known as the sweeping matrix, as this sweeps out φ 1 component. Matrix
iteration as mentioned earlier can then be carried out with this matrix of size (n − 1)
× (n − 1), to obtain the second natural frequency and the second mode shape.
The same procedure is to be repeated for other natural frequencies and mode
shapes.
Example 6.9 Determine the natural frequencies and mode shapes for the framed
structure shown in Fig. 6.17. The floor is considered to be absolutely rigid.
The equivalent spring–mass system of the framed structure is shown in Fig. 6.17b.
The mass and the stiffness matrices of the problem based on the coordinate system
are
[M] =
⎡
⎣
m 0 0
0 m 0
0 0 m
⎤
⎦ = m
⎡
⎣
1 0 0
0 1 0
0 0 1
⎤
⎦
and
[K ] =
⎡
⎣
2k − 2k 0
− 2k 4k − 2k
0 − 2k 5k
⎤
⎦
By performing the matrix inversion of [K], we obtain
[K ]
− 1
=
1
6k
⎡
⎣
8 5 2
5 5 2
2 2 2
⎤
⎦
Therefore,