If you take a hanging chain and oscillate the top back and forth, you can find that there are certain frequencies that give the chain a stable (though still oscillating) shape. These shapes are Bessel functions and the first few modes are shown here:
What I’ve been thinking about recently is that, even though it’s much easier to get those patterns by doing it, when you twirl the top in a circle instead of going back and forth, you get the same shapes but they’re possible at all kinds of frequencies, not just the ones that you can derive to get the above movie.
Here’s an example of the 2nd mode being done at lots of frequencies (thanks to my son Charlie for the cinematography):
(Note that we did this with a broken beaded (Mardi Gras) necklace. A chain of paper clips works just fine too.)
I find this really interesting. The discrete resonance (single frequency where it happens) is broken because of the need for the tension to increase to account for the centripetal force (which increases as I twirl it faster). I’m still trying to monkey around with modeling this, but I thought I’d get this out there in case others’ google-search-fu is better than mine. The closest paper I’ve found is this, which talks about the oscillating solutions, but also talks about the shape of a jump rope, which suffers from the same centripetal force problem.