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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
Section 8.3: Deterministic Vibration
8.7. Determine the response of a fixed-fixed string with uniform cross-section to the step initial displace-
ment,
y (x, 0) =
A, 0.45L < x < 0.55L
0, elsewhere.
Assume that the initial velocity is zero.
Solution: The natural frequencies and mode shapes were previously obtained, and they are
Y
r
(x) =
3
2
mL
sin
rπx
L
ω
r
=
3
T
m
L
, r = 1, 2, 3, ··· .
The free response is given by
y(x, t) =
r=1
Y
r
(x) (A
r
sin ω
r
t + B
r
cos ω
r
t) ,
where
A
r
=
1
ω
r
L
0
m ˙y(x, 0)Y
r
(x) dx
B
r
=
L
0
my(x, 0)Y
r
(x) dx, r = 1, 2, ··· .
Since the initial velocities equal zero, A
r
= 0. The coefficients of cosine functions are given by
B
r
= Am
0.55L
0.45L
Y
r
(x) dx
= Am
3
2
mL
0.55L
0
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Publisher Resources

ISBN: 9781439849897