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I'm making a toy, partly to use a couple of ferrite ring magnets that I've liberated from a magnetron, and partly to see whether I'm capable of stabilising the control loops that will be required.

The idea is to use one magnet vertically above the other, oriented so they repel. This equilibrium is stable in height y, but is unstable in the x and z position, and also in the tilt of the floating magnet. I plan to sense the position and tilt of the magnet with capacitive sensing, and control them with air-core electromagnets.

Here's a very abbreviated sketch of the system, showing only the x axis control coils, cyan in the yz plane for torquing the magnet about the z axis, dark blue in the xz plane for displacing it along the x axis. There will be another set along the z axis. Not shown are the position sensors or the caging/support arrangement.

enter image description here

The symmetry of the picture invited the thought of what happens if the top magnet is spun, perhaps by a tangential jet of air. At rest, z axis torque will turn the magnet about the z axis. However at a high enough angular speed, gyroscopic action will mean that the magnet precesses about x rather than z. This would then require each tilt loop to control the other axis.

Finally the question(s).

What happens at low spin speed?
Do I need to switch from static behaviour to gyroscope behaviour at some critical spin speed?
Or is the behaviour always mixed, with the quadrature torque needed proportional to the spin speed? (which ratio between static and gyroscopic gives me the 'critical' speed?)

I've gone cross-eyed trying to do the thought experiment, and the physical one is quite a while off yet.

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About behavior at low speed: the correct view, I believe, is to see it in terms of proportion.

The usual mathematical expression for spin rate and the response to torque is an approximation, applicable only when the spin rate is far, far larger than the ensuing precession rate.

So in the case of a slow spinning wheel: the level of torque that would be fine for a fast spinning wheel would do a complete wipe-out to a slow spinning wheel.

I do expect, as you do, that with a proportionally smaller torque you get precession behavior, only everything much slower.

This is not a case where some idea of 'critical speed' is applicable. For a given torque the faster the spin rate the closer you are to the idealized case. (Which of course conversely means that for a given torque the slower the spin rate the further away the actual behavior will be from the idealized case.)

For additional information: a tabletop experiment has been done (Svilen Kostov and Daniel Hammer). To verify a property of the onset of gyroscopic precession they set up the onset of precession for a range of spin rates. The title of their paper is 'It has to go down a little, in order to go around'.

The following may be relevant to you: Kostov and Hammer point out that in the idealized case of perfectly frictionless suspension the nutation that accompanies the onset of precession will not decay. In real world circumstances friction in the bearings of the suspension drains the energy from the nutation. In the setup that you want to build you may have to take nutation into account, if only in the form of providing a way to drain the kinetic energy of nutation.

To understand the way that a spinning wheel responds to a torque being applied see my 2012 answer here on physics.stackexchange to a questiuon titled 'What determines the direction of precession of a gyroscope?'

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