Differential Equation with a Discontinuous Forcing Function

number of terms N

4

Consider the equation

2

d

u(t)

d

2

t

+

2

π

u(t)=g(t)

, where

g(t)

is a square-wave step function and

u(t)

is the oscillation of a spring-mass system in resonance with the square-wave forcing function. The graph of

g(t)

is drawn in purple and that of

u(t)

in blue. Using Laplace transforms, this solution is more compact than using a Fourier series expansion of the forcing function.

The first three terms of the Laplace transform of the homogeneous solution for

u(t)

are:

-2s



s(

2

s

+

2

π

)

-

-s



s(

2

s

+

2

π

)

+

1

s(

2

s

+

2

π

)

. The Laplace transform of the forcing function is

-2s



s

-

-s



s

+

1

s

. The phase synchronization between input

g(t)

and output

u(t)

gives rise to resonance.