Page 8 - Vibrations 5
P. 8
When there is no damping, this can be written as,
̈ + = 0
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
{ } are known as the eigenvalues.
{u i} is referred to as the principal mode shape.
Work through a simple example with no damping,
m 1 = 5 kg, m 2 = 10 kg, k 1 = k 2 = 2 N/m and k 3 = 4 N/m
0 ̈ + − 1 0
1
( 1 ) ( ) + ( 1 2 2 ) ( ) = ( )
0 2 ̈ 2 − 2 + 3 2 0
2
With values
5 0 ̈ 1 4 −2 1 0
( ) ( ) + ( ) ( ) = ( )
0 10 ̈ 2 −2 6 2 0
7
