9.4 Forced Vibration of Flexural Member
381
r = 1, 3, 5, . . . .
f (t) = 1
Therefore, the equation of motion for the rth mode is
¨
ξ r + p
2
r ξ r =
2P 0 L · 2
r πρ AL
=
4 P 0
ρ A rπ
which has the solution
ξ r =
4P 0
ρ Aπ
1
r p 2
r
(1 − cos p r t)
Therefore, total dynamic displacement of the beam is expressed as
y(x, t) =
∞
r =1,3,5,
4P 0
ρ Aπ
1
r p 2
r
sin
r π x
L
(1 − cos p r t)
=
4P 0
ρ Aπ
∞
r −1,3,5
1
r p 2
r
sin
r π x
L
(1 − cos p r t)
Example 9.2 A simply supported uniform beam is subjected to a time dependent
concentrated load at quarter span. Express the dynamic displacement of the beam in
terms of dynamic load factor (Fig. 9.4).
The normal modes of the beam are
Y r (x) = sin
r π x
L
Now
Fig. 9.4 Example 9.2