The quality factor is thus  Q = 2π/0.5 = 4π ≈ 12.5.

Simple Harmonic Oscillator

Damped Harmonic Oscillator

Underdamped:

Look at the exponential envolope of the decaying oscillation:

The oscillator appears to go through x=0 at exactly the same times
with and without damping, however that's not quite true. The frequency
of the damped oscillator is slightly lower, . Try increasing
the damping to gamma  = 4 to see the difference in the frequencies.

Overdamped:

Critically damped:

Forced Oscillator

Here you can see the transients as the oscillator is forced up to its steady state motion starting from rest. In the second case the transients exhibit beats before dying away. If you go back and turn off the damping, you should see that the beats continue forever.

Near resonance  (ω = = 2π):

Below resonance (ω = 2π - 1) :