382
9 Forced Vibration of Continuous Systems
L
0
ρ AY
2
r dx =
L
0
ρ A sin
2 r π x
L
dx = ρ A
L
2
L
0
p(x)Y r dx = P 0 sin
r π
4
Therefore, the equation of motion for the rth mode is
¨
ξ r + p
2
r ξ r =
2P 0
ρ AL
sin
r π
4
f (t)
which on solution yields
ξ r =
2P 0
ρ AL
sin
r π
4
1
p r
t
0
f (τ ) sin p r (t − τ ) dτ
=
2P 0
ρ AL
sin
r π
4
1
p 2
r
(DLF) r
Therefore,
y(x, t) =
2P 0
ρ AL
∞
r =1,2,3
1
p 2
r
sin
r π
4
sin
r π x
L
(DL F) r
9.5 Forced Transverse Vibration of Uniform Damped Beam
The equation of motion of a beam with viscous damping and external loading is
given by
E I
∂
4 y
∂ x 4 + c
∂ y
∂t
+ m
∂
2 y
∂t 2 = P (x, t)
(9.56)
The applied force P (x, t) is expanded in terms of modes,
P (x, t) =
∞
j=1
P j (t) Y j (x)
(9.57)
Both sides of Eq. (9.57) be multiplied by m Y k (x) and then integration is carried
out over the span of the beam, yields