A bouncing basketball can not have harmonic motion – either damped or undamped. In order to create harmonic motion, a force has to depend on something’s position. For example, if you tied a weight to the end of a spring and bounced it up and down, the force of the string pulling on the weight would depend on how far the weight had moved (how stretched or compressed the spring was). Gravity doesn’t work like
this. The force of gravity is equal to an objects mass times the
acceleration of gravity (9.81 m/s^2 at the earths surface). This is the
same no matter where the object is so it can’t be harmonic motion.
Here’s a more intuitive way to look at this: think of the way that a graph of harmonic motion looks – it looks like a cosine or sine curve going smoothly from up to down and down to up. If you were to graph the location of a bouncing basketball versus time, it would look different. It starts at the top and slowly starts to fall. As it falls, gravity pulls on it, so it falls faster and faster, in a parabola-shape. But when it reaches the ground, it stops and suddenly reverses direction (there is a sharp turn in the graph). Now the graph looks like a mirror of the first part. It starts off moving upwards very quickly then slows down as it rounds the top of the parabola and starts to fall again.
The damping part of your question is a bit more reasonable. Let's pretend the basketball is hanging from a perfect spring rather than bouncing on the floor so that its motion is truly harmonic. The damping force in such a case would be due to air resistance, which depends on the basketball’s speed, and would cause it to gradually come to a stop.
(published on 10/22/2007)