The Dirac Delta Function
Impulsive Inputs and 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
mx′′(t) + cx′(t) + kx(t) = f(t)
where t is time and x(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 x′(0) = x(0) = 0. At
t = 0 the mass is struck by an “instantaneous” hammer blow. This situation actually occurs
frequently in practice—a system sustains an 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
, 0 ≤ t ≤ ε
Here A is some constant and 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(t) dt = A,
a quantity which is called the total impulse delivered by the hammer blow, with units of
force times time. In what follows let’s normalize the impulse by taking A = 1. With this
normalization we can write f in terms of Heaviside functions, as
where I’ve now subscripted f with ε to explicitly denote that dependence. Our ultimate
interest is the behavior of the solution to equation (1) with forcing function fε in the limit
that ε → 0.
However, it’s easy to see that in the limit that ε → 0, fε(t) makes no sense as a function.
You can easily check that for any t
6= 0, limε→0 fε(t) = 0, while limε→0 fε(0) is undefined.
The appearance of fε(t) for various ε is shown below.
We’ll have to make sense of limε→0 fε in some other way.
Let xε(t) denote the solution to equation (1) with f(t) =