Design help -- concentric rings that spin within themselves

[lynxz3]

I'm trying to design mechanism similar to this video https://www.youtube.com/watch?v=t1JNWnTpmkA where i have concentric rings that spin within them selves around x, y and z axis.

I intend to use track roller cam follower bearings to support each ring, with a bearing width of 10mm and a diameter of 12mm. Now what I'm trying to calculate is the depth of the track that would allow a smooth rotation for the bearings. To simplify things i am looking at each nested ring as an solid axle spining at its centre.

Is there an equation to calculate the track depth?

Thanks

 

[Simon Bridge]

Welcome to PF;

Is there an equation to calculate the track depth?

No. There isn't.

The device is a tad more complex than you described too - the inner ring spins about an axle that is attached to the middle ring. In turn, the axle position can slide around the middle ring. The middle ring has the same arrangement re the outer ring. In the animation, the outer ring is unsupported.

Is this just for art or do you have a purpose in mind?

A more usual arrangement as x-y-z axles that don't slide.

 

[lynxz3]

I actually want to make this wind a mechanical watch, through artificial simulation of being worn on the wrist, a regular areotrim was my initial idea, but the additional randomness that the slide adds, would result in a better winding mechanism.

I intend to you cam follower bearing with a integral axle something like this

on opposite sides of the nested rings, basically creating an axle, with ring acting as shaft between two pivots.

These bearings i believe would allow for both the sliding and rotation, provided a track with sufficient clearance is in place that would allow for the sliding action to take place.

As for the 'outer' most ring it would be supported to a fixed bracket that would allow for rotation along the z axis.