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[ta0]

When I visited Philippe Dyon years ago, he had finger tops with 2 and 3 rings and he showed me with a strobe light how they arranged symmetrically, balancing the top.

[Iacopo]

When I visited Philippe Dyon years ago, he had finger tops with 2 and 3 rings and he showed me with a strobe light how they arranged symmetrically, balancing the top.

This is what I expected. Also, I suppose that this self balancing dynamics can work only in tops where the TMI_tip is larger than the AMI, for this reason I asked it to Jeremy. 

[Jeremy McCreary]

When I visited Philippe Dyon years ago, he had finger tops with 2 and 3 rings and he showed me with a strobe light how they arranged symmetrically, balancing the top.

This is what I expected. Also, I suppose that this self balancing dynamics can work only in tops where the TMI_tip is larger than the AMI, for this reason I asked it to Jeremy.

Working on a way to bracket the moment ratio without actually measuring the moments. Meanwhile, why do you think that?

[Jeremy McCreary]

Also I would like to know, if possible, if the AMI is littler or larger than the TMI_tip, in that top. I would expect two different behaviours, in the two cases.

Using the largely geometric argument below, my axisymmetric top must have AMI < TMI_tip, with or without the rings. Good bet anyway, as this is true of most real tops with non-recessed tips.

For my top, max radius R = 43 mm, and CM-contact distance H = 33 mm. Its mass M = 91 g, but the mass turns out to be irrelevant in this problem.

For any axisymmetric rigid body, solid or hollow, the AMI is of the form

I3 = G3 M R²,

where G3 is a positive dimensionless geometric factor that can't exceed 1. And TMI_tip must have the form

I1 = ½ I3 + X + M H²,

where X = 0 for an axially thin disk, and X > 0 otherwise. Then

I1 / I3 = ½ + (H / R)² / G3 + X / I3

When (H / R)² / G3 > ½, this moment ratio exceeds 1 for any X. Given my values for R and H, this condition is met whenever G3 < 1.18. And that must be the case, since as noted earlier, G3 can never exceed 1.

Hence I1 / I3 > 1 for my top, which means that I1 > I3.

[Iacopo]

Using the largely geometric argument below, my axisymmetric top must have AMI < TMI_tip, with or without the rings. Good bet anyway, as this is true of most real tops with non-recessed tips.

My reasoning is that, with one ring, a balanced top becomes unbalanced;  the direction of tilting of the unbalanced top depends on the AMI/TMI_tip ratio; if the ratio is >1, the stem will stay tilted towards the heavy side, the side where the ring is protruding outwards. A second ring added, would be pushed by the centrifugal force to the position farest from the spin axis, which is the same position where there is already the first ring; so the two rings would tend to stay together and make the unbalance the worst.

But if the ratio is at least a bit larger than 1, the unbalanced top spins staying tilted towards the other side, the light side.
In this case too the second ring would be pushed by the centrifugal force to the farest position from the spin axis, but this position now is opposite to that with the first ring. So the two rings will tend to hover about 180° apart, each one looking for the lighter side of the top, and making it heavier with its own weight, so the top becomes self-balancing.

I should try with one of my tops and two rings, to see what it happens.
 

[Jeremy McCreary]

My reasoning is that, with one ring, a balanced top becomes unbalanced;  the direction of tilting of the unbalanced top depends on the AMI/TMI_tip ratio; if the ratio is >1, the stem will stay tilted towards the heavy side, the side where the ring is protruding outwards....
I should try with one of my tops and two rings, to see what it happens.

I think you'll find the 2-ring system quite surprising.

Made a series of small LEGO test tops designed to help me understand why some tops with static unbalance lean toward the heavy side (phase 0°) and others toward the light side (phase 180°). Never completed enough of the huge test matrix to come to any firm conclusions, but CM-contact distance seemed to play an important role.

The jury was still out on moment ratio, but sounds like you have data on that.

Will have to revist this investigation. Each test top had an independent adjustment for each likely suspect: AMI, central TMI, CM-contact distance, tip curvature, and degree of static or couple unbalance. Each stem had a simple whirl detector — a color wheel combining the principle behind the paint brush blancing method with the unmixing of wheel colors in the presence of whirl.

[Iacopo]

The jury was still out on moment ratio, but sounds like you have data on that.

Will have to revist this investigation. Each test top had an independent adjustment for each likely suspect: AMI, central TMI, CM-contact distance, tip curvature, and degree of static or couple unbalance. Each stem had a simple whirl detector — a color wheel combining the principle behind the paint brush blancing method with the unmixing of wheel colors in the presence of whirl.

The data are in the thread about resonance and phase shift.
The resonance, between spin speed and nutation, (nutation in the physical sense of torque free wobble and not in the kinematical sense), marks the phase shift, before which the unbalanced top stays tilted towards the heavy side, and after which it stays tilted towards the light side.
Since the speeds of nutation and spin are linked by the formula:
AMI : TMI_tip = nutation speed : spin speed
it means that if the AMI is larger than the TMI_tip, the unbalanced top stays tilted towards the heavy side, otherwise it stays tilted towards the light side.
The formula is not accurate at low speed but a good starting point for observations.

When you make your tests, the tip of the top must not slip, otherwise the observations are not reliable.



I tried my tops with two rings but the behaviour seems chaotical, I would need more time to afford the issue more calmly.