Rolling and Sliding of Spinning Things

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Last updated: March 27, 2007

Euler's Disk

Next, Euler's disk first, Jellett's egg further below. This commentary was stimulated by two papers in Nature on axi-symmetric objects in close-to steady precession while supported by a flat surface. Both papers make minor technical points. Both papers are also misleading and wrong in various ways, they don't acknowledge the state of human knowledge before the papers, and seem to try to get credit for interesting aspects of the phenomena which the papers don't address.

 

Joe Bendik invented and sells a toy he calls "Euler's Disk". It is a coin shaped object (a squat cylinder) but about 7.5 cm across and 1 cm thick. It has very finely machined circular rims. It comes with a roughly parabolic glass bowl held somewhat flexibly in a plastic base. When you let the disk fall from an edge with a little twist, it shudders around, like a coin does but for much longer. It does that for up to two minutes. It took Joe some work to get a manufacturable disk and base that would have low enough dissipation to yield such a long shuddering time. The exact relevant features of the toy are Bendik's trade secrets.
In April 2000 Keith Moffatt wrote an article about this disk in Nature:

Euler's disk and its finite time singularity
Keith Moffatt, Nature 404, 20 April 2000, Pages 833-834

This article got a lot of press (e.g., NY Times article by K. Chang on April 25, 2000). One common misconception in the press is that Moffatt explains for the first time why a disk shudders faster and faster as it gets more and more flat. This much was known well in the nineteenth century and the formula describing the relation between shudder (precession) speed and tip angle was printed on the side of the Euler's Disk toy box, but not in recent versions for some reason. The press misunderstanding that the Moffatt paper explains this increase in shuddering speed is undoubtedly due in part to the dramatic increase in shudder speed being the obviously appealing feature of the toy, and in part due to Moffatt's re-derivation of this old formula in his paper.

When it first came out, several people immediately pointed me to Moffatt's paper and asked my opinion about it. I read the paper, did some simple calculations, some with Anindya Chatterjee, did some simple experiments with disks with K R Y Simha and Chatterjee, and had a long frustrating email exchange with Keith Moffatt. This culminated in a letter I wrote to Nature critiquing the paper. That letter was not published, perhaps because it had a complex array of criticisms rather than a single simple one, perhaps because it was too dense, perhaps because the comments were not backed by experiments or formulas, or perhaps because the editors doubted my credibility. I think Moffatt's paper is slightly cute. What it does actually explain is hard to say since it is mostly some combination of wrong and misleading.

Here is a somewhat detailed review of Moffatt's disk paper (PDF). I wrote this up in a hurry. Please comment if you find it unclear or inaccurate in any way.

Nature did publish a critique of Moffatt's paper by people who did some good critical experiments but whose mechanical explanations reveal their training to be in disciplines other than mechanics.

Numismatic gyrations
Ger van den Engh, Peter Nelson, Jared Roach, Nature 408, 30 November 2000, Page 540

Moffatt replied, including responses to other unpublished critics including me.

Moffatt replies
Nature 408, 30 November 2000, Page 540

Moffatt's response includes:

"I chose to focus on viscous dissipation because that is the only mechanism for which a fundamental (rather than empirical) description is available, namely that based on the Navier-Stokes equations of fluid dynamics...[for rolling friction] determination of the associated rate of dissipation of energy (in terms of the physical properties of the disk and the surface) involves solution of the equations of (possibly plastic) deformation in both solids at the moving point of contact, a difficult problem, which so far as I am aware, still awaits definitive analysis."

I think Moffatt believes "it is more important to have beauty in one's equations than to have them fit experiment" as Dirac supposedly said, so Moffatt is in good company.

I still don't know what Euler knew about Euler's disk. Euler's paper on rigid body dynamics has never been translated to English and neither I nor anyone I have talked to has deciphered the Latin version. A web search on "Rolling Disk" or "Euler's Disk" will show various other people who have been interested in rolling disks both before and since Moffatt's paper. Much of the basic mechanics, leaving out all forms of dissipation, is in most books on analytical mechanics from the past 140 years.

My original too-concise letter to Nature is on Kirk McDonald's web page, as is a good unpublished paper on the the rolling disk by Kirk and his son.

Jellett's Egg

Spinning eggs --- a paradox resolved, H.K. Moffatt and Y. Shimomura, (Nature 416, 28 March, 2002, Pages 385-386) claims to explain why when you spin a hard-boiled egg on its side you see that it rises on to one end. The paper does not mention the 1909 book by Crabtree that discusses this problem. But for a very minor technical point (Crabtree assumes a spherical egg tip, the Nature paper does not, see figure (PDF)), Crabtree seems to have the spinning-egg story written up as completely and more understandably than the new Nature paper.

Here are some scanned excerpts from Crabtree on eggs (PDF).

The egg theory is similar in many ways to the theory of the disk above. With the egg, however, the Nature authors (reasonably I think) neglect fluid effects (which would dominate if the geometry were as described in the paper with its smooth contact) and surrender to contact mechanisms (again, reasonably) of the type Moffatt found too distasteful to use for the disk (with no consideration for accuracy of the model).

Let's divide up the time of egg rising into four regimes:

First, there is the horizontal state. In this state the steady precession the paper assumes cannot hold. The instability away from this rolling state might be described by the (uncited) 1895 paper of Walker, however.

Second is the regime for which the paper might have a slightly more accurate approximation of reality for some eggs.

Third, from about 45 degrees up, the tip of an egg is incredibly close to a sphere. The case of a spherically tipped spinning egg was treated, a little more thoroughly than the treatment in this paper, in the 1872 book by Jellett which the authors do cite.

Fourth, the final rise, the last few degrees, is not described by Jellett or by this paper. Actually, the final rise has never been explained as far as Tom Kane or I know. But, as is never alluded to in the Moffatt-Shimomura paper, the theory they present is not valid as the egg rises all the way up. Near the vertical orientation, the friction is no longer a small effect and rolling takes over (as explained in Kane and Levinson, 1977, another uncited paper).

If there is a "paradox" to resolve, it is in this fourth regime. Why does a top or egg rise all the way when all the simple mechanisms say it should not? This regime the paper does not address. Rather their paper would lead you to think this regime did not exist.

The actual mechanism(s) for the final rise has to be a high-order effect. Candidates, none mentioned in the paper because the authors seem unaware of the issues, include:

1. Scrubbing torque (I think this has the wrong sign).
2. Creep due to rolling always having a slight slip when the sideways force is not zero.
3. Coupling of twist to rolling due to contact deformation (like I-think-its-called camber thrust in the context of tires).
4. Collisional contact at points that are not quite the nominal contact point.
5. Oscillatory effects because the inertia axes is not aligned with the outer surface axes.
6. Air friction (I think this mostly just slows the spin, and probably not even too much of that).
7. The spherical tip not being exactly isotropic so rattle-back (Walker 1895) effects kick in. This is a possible high-order effect, but it can't be primary because it means spinning the opposite direction would be unstable.

Lets return to the second regime, from somewhat above horizontal up to 45 degrees. In this regime the paper does perhaps reasonably describe the rising of a spinning egg with a technically new calculation. But the results are hardly a surprise. The egg acts basically like a spherically tipped top. Given any knowledge of tops, it's hard to see where the word "paradox" fits in here.

Finally, the authors claim that the generalization of the Jellett constant to ellipses (eggs) from circles (spheres) requires the neglect of a term that Jellett did not have to neglect (the ground reaction force). I do not believe that this is true. I have not carried out the calculation in detail, but I believe I understand the structure of the calculation and see no impediment to generalizing the Jellett constant while not neglecting the normal contact reaction (work in progress with Ishan Sharma and Manoj Srinivasen).