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Probabilistic Models for Dynamical Systems, 2nd Edition
book

Probabilistic Models for Dynamical Systems, 2nd Edition

by Haym Benaroya, Seon Mi Han, Mark Nagurka
May 2013
Intermediate to advanced content levelIntermediate to advanced
764 pages
6h 43m
English
CRC Press
Content preview from Probabilistic Models for Dynamical Systems, 2nd Edition
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}
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Publisher Resources

ISBN: 9781439849897