Adding a damping force proportional to 
to the equation of simple
harmonic motion, the first derivative of 
with respect to time, the equation of motion for damped
simple harmonic motion is
 |
(1)
|
where 
is the damping constant. This equation arises, for example, in the analysis of the
flow of current in an electronic CLR circuit, (which contains a capacitor, an inductor,
and a resistor). The curve produced by two damped harmonic oscillators at right angles
to each other is called a harmonograph, and simplifies
to a Lissajous curve if 
.
The damped harmonic oscillator can be solved by looking for trial solutions of the form 
. Plugging this into (1) gives
 |
(2)
|
 |
(3)
|
This is a quadratic equation with solutions
 |
(4)
|
There are therefore three solution regimes depending on the sign of the quantity inside the square root,
 |
(5)
|
The three regimes are summarized in the following table.
If a periodic (sinusoidal) forcing term is added at angular frequency 
, the same three solution regimes are again obtained. Surprisingly,
the resulting motion is still periodic (after an initial transient response, corresponding
to the solution to the unforced case, has died out), but it has an amplitude
different from the forcing amplitude.
The particular solution 
to the forced second-order
nonhomogeneous ordinary differential
equation
 |
(6)
|
due to forcing can be found using variation of parameters to be given by the equation
 |
(7)
|
where 
and 
are the homogeneous solutions to the unforced equation
 |
(8)
|
and 
is the Wronskian of these two functions. Once the sinusoidal
case of forcing is solved, it can then be generalized to any periodic function
by expressing the periodic function in a Fourier series.
