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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
6 Single Degree-of-Freedom Vibration
Section 6.2: Deterministic SDOF Vibration
6.1. A single degree-of-freedom system with ω
n
= 1 rad/s and ζ = 0.24 is excited by an external force
per unit mass, F (t) /m = 2 sin ω
f
t. Derive the response if the forcing frequency coincides with the
damped natural frequency.
Solution: The damped natural frequency is ω
n
1 ζ
2
= 0.97 rad/s. The particular solution is given by
x
p
(t) = D sin (ω
f
t φ) ,
where
D =
2
ω
2
n
1
*
(1 ω
2
f
2
n
)
2
+ (2ζω
f
n
)
2
,
θ = tan
1
2ζω
f
n
1 ω
2
f
2
n
.
When ω
f
= ω
d
, we have
ω
f
ω
n
=
*
1 ζ
2
.
Then, the amplitude and phase of the particular solution become
D =
2
ω
2
n
1
ζ
*
ζ
2
+ 4
1 ζ
2
= 4.26
θ = 1.45 rad/s.
The particu
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Publisher Resources

ISBN: 9781439849897