Here is a brief treatment of damped oscillations with a friction force that is proportional to the velocity:

Without a proof, here is the general solution:

(for w_{0} > w_{d
}) Here are: w_{d} = b/2m,
w' = (w_{0}^{2}
- w_{d}^{2})^{1/2},
and w_{0 }= the frequency of the
oscillator without the friction force present =
(k/m)^{1/2}.

Instead of fighting with a derivation, we can experimentally explore the damped oscillator in this applet:

© W. Bauer, 1999