380
9 Forced Vibration of Continuous Systems
Fig. 9.3 Example 9.1
Therefore, Eq. (9.52) can be written as
y (x, t) =
Y r ξ rst (DLF) r
(9.55)
Example 9.1 A simply supported uniform beam having mass density ρ and crosssectional area A has a distributed load, whose variation with time is shown in Fig. 9.3.
Derive the expression for the dynamic deflection of the beam.
The normal modes of the beam are
Y r (x) = sin
r π x
L
The natural frequencies of the beam are
p r = (r π)
2
E I
ρ AL 3
Now,
L
0
ρ AY
2
r dx =
L
0
A sin
2 r π x
L
dx =
ρ AL
2
.
and
L
0
p(x)Y r dx =
L
0
ρ 0 sin
r π x
L
dx =
2P 0
r π
pt,
9 Forced Vibration of Continuous Systems
Fig. 9.3 Example 9.1
Therefore, Eq. (9.52) can be written as
y (x, t) =
Y r ξ rst (DLF) r
(9.55)
Example 9.1 A simply supported uniform beam having mass density ρ and crosssectional area A has a distributed load, whose variation with time is shown in Fig. 9.3.
Derive the expression for the dynamic deflection of the beam.
The normal modes of the beam are
Y r (x) = sin
r π x
L
The natural frequencies of the beam are
p r = (r π)
2
E I
ρ AL 3
Now,
L
0
ρ AY
2
r dx =
L
0
A sin
2 r π x
L
dx =
ρ AL
2
.
and
L
0
p(x)Y r dx =
L
0
ρ 0 sin
r π x
L
dx =
2P 0
r π
pt,