Page 14 - Vibrations 5

Assuming harmonic motion,

̈ = −

Therefore,

− + = 0

2
where = (for undamped).

-1
Multiply by M and let = to give,

−1
( − ) = 0

The characteristic equation is given by,

−1
| − | = 0

This time M and k are 3 by 3 matrices

Work through another example with no damping,

m 1 = 5 kg, m 2 = 10 kg, m 3 = 7 kg, k 1 = k 2 = 2 N/m, k 3 = 4

N/m and k 4 = 3 N/m

1 0 0 ̈ 1 + 2 − 2 0 1 0
1

( 0 2 0 ) (̈ ) + ( − 2 + 3 − 3 ) ( 2) = (0)
2
2
0 0 3 ̈ 3 0 − 3 + 4 3 0
3
This becomes

5 0 0 ̈ 1 2 + 2 −2 0 1 0

(0 10 0) (̈ ) + ( −2 2 + 4 −4 ) ( 2) = (0)
2
0 0 7 ̈ 3 0 −3 4 + 3 3 0

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