Free damped motion of a mass *m*
is modeled by the 2nd order ODE:

where *x*(*t*) is the distance
from equilibrium, *c* is the damping constant and *k* is the
spring constant. We divide by *m* and write:

where 2λ = *c/m* and
ω^{2} = *k/m*. The qualitative nature of the
solution depends on whether λ is greater than, equal to or less
than ω.

**Overdamped (λ > ω):** The roots
of the auxiliary equation are −λ ±
(λ^{2} − ω^{2})^{½}
and so the general solution is:

**Critically
Damped (λ = ω):** The auxiliary equation has
−λ as a repeated root and so the general solution
is:

**Underdamped (λ < ω):**
The roots of the auxiliary equation are the complex numbers
−λ ± *i*ω_{0} where
ω_{0} = (ω^{2} −
λ^{2})^{½} and so the general solution
is:

Solution curves in the three cases and
also the undamped case (*c* =
0) are shown below. These functions all satisfy the initial
conditions *x*(0) = 6 and *x'*(0) = 0

Notice that the frequency in the
underdamped case (ω_{0}) is slightly less than in the
undamped case (ω); we see
this in the slightly longer wavelength in the underdamped case. The
dotted black line is the "exponential envelop" ±(6^{2} +
(ω_{0}/3)^{2})^{½}*e*^{−t/2}.

Images generated by sage.