3.12 Response to SDF Systems to a General Type of Forcing Function
95
x =
F (τ ) dτ
mp
sin pt
(3.89)
Having obtained the displacement under the effect of one impulse, we may
consider this impulse situated at a time instant τ . Then, the displacement after the
application of the impulse is given by Eq. (3.89), except that in place of t, we should
put (t − τ ), which is the time after impulse application. With this
x =
F (τ ) dτ
mp
sin p (t − τ )
(3.90)
Total displacement then is given by
x =
t
0
F (τ )
mp
sin p (t − τ ) dτ
(3.91)
The above integral gives the complete solution, after the constants of integrations
are evaluated from the initial conditions.
The expression is known as Duhamel integral or convolution integral.
If the initial conditions are not zero, then the solution becomes
x = x 0 cos pt +
˙
x 0
p
sin pt +
1
mp
t
0
F (τ ) sin p (t − τ ) dτ
(3.92)
If viscous damping is included in the system, then Eq. (3.92) becomes
x = e
− n t
(C 1 cos
p 2 − n 2 t + C 2 sin
p 2 − n 2 t )
+
1
m
p 2 − n 2
t
0
F (τ ) e
− n ( t−τ ) sin
p 2 − n 2 (t − τ ) dτ
(3.93)
where C 1 and C 2 are obtained from initial conditions.
If the expression of F(t) is known in analytical form, it can be substituted in
Eq. (3.93), and the necessary integration may be performed. If F(t) expression is
complicated, then the integration may have to be done numerically.
Example 3.12 A SDF system is subjected to a suddenly applied load with a limited
duration t d as shown in Fig. 3.24. Using Duhamel integral, determine the response
of the undamped system. The system starts at rest.