What are your Scene Rigid Body World settings?
Have you changed the defaults to use a higher number of Steps Per Second?
I have tried to replicate your setup (with Cylinder collision shape) and got a stable simulation with 60 steps per second (the default) but got "bouncing" behaviour when I increased that to 600 steps per second. I think this could be an issue with how the active rigid body is interacting with the animated passive rigid body - since the animated one only changes position each frame but the active one is simulated many times per frame so that the 'jumps' in position of the surrounding mesh on each frame change 'kick' the cylinder on each frame.
I need to do more investigation into this but if I can get together a suitable test case and prove out the circumstances I'll raise it with the developers as a potential bug.
Simulated at 60 frames per second :
Simulated at 600 frames per second (should be more stable) :
Note that the simulation with more steps should be more stable but is producing more unexpected 'bounce'. This appears to be due to the interaction with the animated surface and may be a bug. 6000 steps per second (which should be more accurate) produces an even more unstable simulation!
Taking this a step further, I replaced the keyframe animation of the rotation with a Motor Rigid Body Constraint - to keep the animation all within the simulation. I ran this with 6000 steps per second with the following result :
As you can see this is running fine without any bouncing on the collisions. To me this seems fairly conclusive that it's the interaction with the keyframed animated mesh which is causing the 'bounce' in my case.