The Heaviside Step Function

Date: 09/03/2003 at 04:40:42
From: Trilochan
Subject: Dirac Delta Function

I have to calculate the strain energy of a beam, incorporating a term
that is zero when the input is below a certain limit, and starts
contributing when it reaches that limit.  Can I use the Dirac delta
function for this?  If so, how?  If not, is there something else I can
use? 

Date: 09/03/2003 at 15:22:12
From: Doctor Douglas
Subject: Re: Dirac Delta Function

Hi Trilochan,

You might wish to use not the Dirac delta function but rather
the Heaviside step function T(x), which is sometimes written with
a Greek capital Theta: 

  T(x) = 0           x <  0
       = 1           x >= 0

In certain contexts, T(x=0) may be defined to be zero, or perhaps
1/2.  The salient feature of this function, however, is that it is
zero for negative x, but "turns on" for positive x.  You can show
that the derivative of T(x) with respect to x is the Dirac delta 
function.

Your equation for the dependence of the strain y upon the stress x
might look something like this:

  y(x) = f(x) + T(x-X)g(x-X),

where f and g are functions that you must specify.  Note that the
second term "turns on" for x >= X.

- Doctor Douglas, The Math Forum
  http://mathforum.org/dr.math/