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Finding the Impulse Function Using Laplace Transform
.

The impulsive response is

( )
1 1 1
−1 −2t
y(t) = L = e sin 2t.
2
2 (s + 1) + 2 2 4
. . . .

.

Example 3.2, A 16 g weight is attached to a spring with a spring
constant equal to 2 g/m. Neglect damping. The weight is released
from rest at 3 m below the equilibrium position. At t = 2π sec,

it is struck with a hammer, providing an impulse of 4 g-sec.
Determine the displacement function y(t) of the weight.
. . . . .

.

Solution. This situation is modelled by the initial value problem

16
y + 2y = 4δ(t − 2π), y(0) = 3, y (0) = 0.
′
′′
32
Apply Laplace transform to both sides to obtain

2
s Y (s) − 3s + 4Y (s) = 8e −2πs .

Solving for Y (s) we find

3s e −2πs
Y (s) = + .
2
2
s + 4 s + 4
Now take the inverse Laplace transform to get

−1
y(t) = L [Y (s)] = 3 cos 2t + 8u (t − 2π)f(t − 2π)
0
where
1 1
−1
f(t) = L { } = sin 2t/
2
s + 4 2
Hence,

y(t) = 3 cos 2t + 4u (t − 2π) sin 2(t − 2π) = 3 cos 2t + 4u (t − 2π) sin 2t
0
0
or more explicitly

{
3 cos 2t, t < 2π,
y(t) =
3 cos 2t + 4 sin 2t, t ≥ 2π.
. . . .

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