388
9 Forced Vibration of Continuous Systems
Fig. 9.6 A simply supported
beam
9.7.2 Response of the Beam when the Mass of the Vehicle is
Small
A harmonic force P sin (ωt + α) due to unbalanced parts of a vehicle moves with a
uniform velocity v from left to right. At time t = 0, the load is assumed to remain at
the left hand support.
Assuming the beam to possess uniform flexural rigidity EI, the equation of motion
for the beam is (Fig. 9.6)
E I
∂
4 y
dx 4 + ρ A
∂
2 y
∂t 2 = P sin (ωt + α)
(9.80)
Assuming a solution in terms of normal modes
y =
√
2 sin
r π x
L
ξ r (t)
(9.81)
The factor
√
2 is for normalising the mode shape and substituting it in Eq. (9.80),
and following steps similar to that presented in Art. 9.4, we get the rth equation
[Eq. (9.49)] as
¨
ξ r + p
2
r ξ r =
√
2 P sin(ωt + α)
ρ AL
sin
r πvt
L
(9.82)
The right-hand side of Eq. (9.54) is
RHS =
P
√
2 ρ AL
cos
ωt + α −
r πvt
L
− cos
ωt + α +
r πvt
L
(9.83)
The complete solution of Eq. (9.82) is
ξ r = A r sin ω r t + B r cos ω r t
+
P
√
2 ρ AL
cos
ωt + α −
r πvt
L
ω 2
r −
ω −
r πv
L
2 −
cos
ωt + α +
r πvt
L
ω 2
r −
ω +
r πv
L
2
(9.84)