
7 Multi Degree-of-Freedom Vibration
Section 7.1: Deterministic Vibration
7.1. Consider an undamped three degree-of-freedom system with property matrices:
[m] =
1 0.5 0
0.5 2 0.3
0 0.3 3
kg, [k] =
1 0 0
0 1 0
0 0 2
N/m.
The initial conditions are
{x (0)} =
0
0.5
0
m, {˙x (0)} =
0
0
0.1
m/s.
Obtain the response using modal analysis.
Solution: The associated eigenvalue problem is given by
[K] − ω
2
[M]
{u} = {0}.
The natural frequencies are obtained by letting the determinant of [K] − ω
2
[M] equal zero, and they are
found to be
ω
2
= 0.4422, 0.6856, 1.2766.
The mode shapes or eigenvectors are obtained by obtaining the vector {u}
i
that satisfies
[K] −ω
2
i
[M]
{u}