Can we make a rigid ball rotate steadily by shaking a curved surface on which it stands?

I would like to know how to analyze the dynamics of a ball on a curved surface (lower part of a sphere), which can move in the xy plane. To be clear, the surface does not rotate, only translates. The interaction between the ball and the surface is due to gravity and normal force, and friction. My main goal it to find a way to make the ball rotate steadily, with the goals:

1. Ball spins around a fixed axis, preferably vertical (z axis).
2. Ball’s center of mass Does not move (or its displacement is much smaller than its own radius).
3. Ball spin frequency is constant in time – ball reaches a stable rotational steady state.
4. Ball spins as fast as it can.

I should add that I conducted an experiment already. The experiment was done with a perfect spherical surface (a finely polished spherical mirror) with 5 mm radius and a 1 mm diameter glass bead (pretty spherical, but not perfect. I don’t have any specific details about how spherical was the bead). The mirror was epoxied to the surface of an xy shear piezo stack, which was driven in both x and y with same frequency and with a phase difference. The drive amplitude was about 0.15 microns peak to peak in both directions.

I got the bead to spin around a constant axis (to the precision of my measurement), and with roughly constant spin frequency of about 30 Hz (constant for about 3 seconds, after that varied with plus minus 10 Hz on average).
The spin was drive-frequency and drive-phase dependent. The best results (with a scan resolution of 100 Hz for the drive frequency) was at 7.6 kHz drive, and with phases of 0 and 180 between the x and y drives (I don’t have an explanation for the phases).

Just for context: The reason for this question is my need to rotate a sample inside a cryostat at 4 K, without magnetic field.

My general question: How can I model my system (what Lagrangian should I write and how to model the friction?), such that maybe I can predict the relevant parameters for the spin – the drive frequency, maybe phase, the maximal rotation frequency possible for the ball and the stability, and how do these relate to the setup parameters: ball mass, ratio of ball radius to surface radius and so forth.

Thanks for your input!