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I have recently come across Earnshaw's theorem in my research which states that it is not possible to stabilize an object using only static permanent magnets. I understand that and it makes sense to me.

My question is, is it possible to achieve stability with only one electromagnet and the help of gravity? The setup is illustrated below (sorry for the crude diagram):

enter image description here

(please imagine a heavy spinning load attached to the middle of the shaft between both sets of magnets)

The ferromagnetic shaft is pinned against the bottom passive magnet against gravity. The electromagnet at the top is controlled by a PID controller that measures the distance of the shaft from the electromagnet, and attempts to keep it at a constant distance.

The central magnets fix two rotational degrees of freedom and two translational degrees of freedom, the electromagnet fixes the last translational degree of freedom. The remaining DOF is the desired spinning motion.

Is this configuration stable, or are there any unstable paths the shaft can take that would lead to it eventually touching a magnet? Perhaps an unforeseen oscillatory motion of some kind? (since the shaft is fully encased in magnets and rapidly spinning, any failure mode is catastrophic)

Thanks

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  • $\begingroup$ Earnshaw's theorem doesn't get better with gravity, but it is easily bypassed with a rotating body. You may have heard of a toy called the "Levitron". If what you want is a functional magnetic bearing, you can buy them from several companies, no need to reinvent the wheel. I am sure you can find entire engineering textbooks with a complete analysis of technically useful designs in the library. $\endgroup$
    – CuriousOne
    Mar 22, 2016 at 7:16
  • $\begingroup$ @CuriousOne I didn't think Earnshaw's theorem applied to a dynamic electromagnet... I was not able to find affordable small magnetic bearings, all I could find was company websites requiring asking for quotes (meaning it's in the tens of thousands of dollars), could you share a link in case I am missing a keyword? $\endgroup$
    – Thomas
    Mar 22, 2016 at 7:19
  • $\begingroup$ That's the trouble with moving from an idea (physics) to a product (engineering)... it often requires decades long additional research and investment, hence the engineering section of the university library and the four, five or six figure price tags of such products. As the Levitron proves, one can build a functional magnetic bearing (that's what it is) for a few dollars. I doubt that active magnetic bearings make much sense for small loads, maybe you can find some in high reliability disk drives? $\endgroup$
    – CuriousOne
    Mar 22, 2016 at 7:30

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