Page 50 - 4811

An Example of Impulse Response

electrical network) over a very short time frame. In this case, the output

corresponding to this sudden force is referred to as the ”impulse response”.
Mathematically, an impulse can be modelled by an initial value problem with a
special type of function known as the Dirac delta function as the external
force, i.e., the non-homogeneous term. To solve such IVP requires finding the

Laplace transform of the delta function which is the main topic of this section.

An Example of Impulse Response

Consider a spring-mass system with a time-dependent force f(t) applied to the
mass. The situation is modelled by the second-order differential equation

my + γy + ky = f(t) (3.1)

where t is time and y(t) is the displacement of the mass from equilibrium. Now
suppose that for t ≤ 0 the mass is at rest in its equilibrium position, so y(0) =
y (0) = 0. Hence, the situation is modelled by the initial value problem
′

my + γy + ky = f(t), y(0) = 0, y(0) = 0. (3.2)

Solving this equation by the method of variation of parameters one finds the
unique solution
t
∫
y(t) = ϕ(t − s)f(s)ds (3.3)
0
where
( )
√ 2
e (−γ/2m)t sin t k − γ
m 4m 2
ϕ(t) = √ .
m k − γ 2
m 4m 2
Next, we consider the problem of strucking the mass by an ”instantaneous”
hammer blow at t = 0. This situation actually occurs frequently in practice — a

system sustains a forceful, almost instantaneous input. Our goal is to model the
situation mathematically and determine how the system will respond.
In the above situation we might describe f(t) as a large constant force applied
on a very small time interval. Such a model leads to the forcing function

{
1 , 0 ≤ t ≤ ε,
f (t) = ε
ε
0, otherwise

where ε is a small positive real number. When ε is close to zero the applied force
is very large during the time interval 0 ≤ t ≤ ε and zero afterwards.
In this case it’s easy to see that for any choice of ε we have

∫
∞
f dt = 1
ε
−∞
49