
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
rπ
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