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[kindofdoon]

Hello all, I have an apparently simple question to which I have not been able to find a satisfying answer.

For a solid cylindrical spinning top of a certain uniform material density ρ, body radius r, body elevation e, and initial angular velocity w, what body height h maximizes the total spin duration? In other words, what is the optimal value of h/r for a solid top? See attached for a diagram. For simplicity, I am content to neglect the mass of the tip and handle.

Increasing h/r has two basic and opposing effects: increasing the axial moment of inertia (good), and raising the center of gravity (bad). Intuitively, it seems that there must exist a well-posed optimum between the suboptimal extremes of a "thin" top with h/r→0, and a "thick" top with h/r→1. Empirically, this optimal value of h/r seems to be about 1/2 to 2/3. I am seeking analysis, research, experiments, etc. that explores this ratio, and how it arises.

Thanks in advance. -Daniel

[ta0]

Hi Daniel

Welcome to the forum!

That's an interesting question. And the answer is not simple if you also have to consider air resistance.

There is an equation for critical spin, the minimum spin at which it will stay vertical. So that equation can be maximized for the variables. But once you consider air resistance, it throws everything off unless you spin it very slowly to start with. I'm sure Jeremy and others (Ortwin?) would like to contribute. We have discussed the subject before. I'm very busy with the raffle, etc. now, but give me a couple of days and I will dig the posts.

We have in the forum the world record holder for a finger top spin duration, Iacopo: over 1 hour (with recessed tip and multiple finger twirls but no tungsten). If you haven't yet, look for his posts.

[ortwin]

Hello Daniel!
A warm "Welcome to THE forum" also from me!


You can find some topics dealing with the question you have, by searching for example for "critical speed".
One topic is this one.
I'm sure Jeremy has a few words and formulas  for you, not only general advice and vaguely fitting links.
Looking forward to interesting discussions!

[Iacopo]

Hi Daniel, welcome to the Forum.

I believe that it's very difficult to find the best h/r for a top just with calculations, because there are many interconnected variables which make the matter very complex, we don't even have all the needed data about air drag and tip friction..

The simplest way seems to make some different tops and directly test them.
Certainly there isn't just one optimal h/r ratio, because this ratio itself is variable, depending on other data, like for example the starting speed, the density of the material, the size of the top...
The optimal ratios you found empirically, between 1/2 and 2/3, seem high to me, probably the tops you used were littler and/or lighter than the mine.

Once I made three cylindrical brass tops, with three different h/r ratios, and external tip. The ratio of the best one turned out to be about 1/3,(weight 107 grams, diameter 59.9 mm, longest spin 26 m 20 s, by one twirl of the fingers). The tallest top was more efficient at high speed, but the flatter one had a lowest toppling down speed, and this made it spin longer.

You can see the test with the three cylindrical tops in this video:


     

[kindofdoon]

Thank you all for the warm welcome! As a bit of background, I am a mechanical engineer and a novice woodturner. Below is a photo of the tops I've made in maple, oak, and pine, which are up to about 1.5 inches in diameter, and at best have nearly reached 2 minutes. My tooling setup currently only allows for spindle turning. I appreciate that this is a complex problem with many interrelated parameters, but still I suspect that with some simplifying assumptions, an insightful solution could be obtained. For example, perhaps we could produce a result showing the optimal h/r for an air drag-dominated top, versus the optimal h/r for a tip friction-dominated top. Or perhaps there is even a form that could be deduced in the simpler, idealized case of no losses, but I'm less certain about that.

Iacopo, I was especially impressed with your video studying the effect of body proportions on performance. The findings about the relative importance of air drag versus tip friction were illuminating! Still, I would not consider these "solid, uniform" tops, because the material density varies within the top, e.g. in the brass ring versus the wood core. I might characterize these bodies as having r1 (inner radius), r2 (outer radius), and h (height), so it's not quite the same problem as I posed. I understand the benefits of multi-material tops such as yours, but still I am interested in the "solid, uniform" case as arguably the simplest form of the problem that might yield analytical insight.

ortwin, I followed your link, and it appears to discuss an idealized case of h/r→0, or a "thin" disk, which I think would not give much guidance as to the ideal body proportion.

I will continue thinking on this. 🤔

[Iacopo]

I am interested in the "solid, uniform" case as arguably the simplest form of the problem that might yield analytical insight.

If you are mainly interested in formulas, some others here can help you much better than me.
   
If you want your tops to spin longer, a few tips I can offer are to make them bigger, to use the densest material you can, and make the core of the top lighter, even simply by making the core hollow, like in tippe tops, if you don't want to use different materials in the same top.

My first tops were made of wood, like the your, (at that time I didn't have a metal lathe yet). The shape of your ones remind me my Nr. 1, (photo below). I have always been interested in longest spins; in this case it seems like we made a similar reasoning about the shape of the bottom of the top, as an attempt for to lower the height of the center of mass on the tip, without reducing the scrape angle.

 

[ortwin]

@Daniel: the problem can't be reduced just to (h/r) I think. For example the e (body elevation)  factors in also. It can strongly effect the TMI/AMI ratio which is an important thing one has to watch in this context. Large e  leads to higher TMI (bad) but also to a larger scrape angle (good). And e can also be smaller than 0. Those would be tops with recessed tips. Iacopo made  quite some tops with that type of tip. Bringing the center of mass (CM) very close to the contact point in that way can help a lot for longer spin times, but it can lead to some other problems.
 If you search the forum for TMI/AMI, CM, scrape angle you'll  find a lot of stuff especially by Jeremy. In this thread there might also be something of interest to you.

... Or perhaps there is even a form that could be deduced in the simpler, idealized case of no losses, but I'm less certain about that.

...

I'm not sure what you mean. In the idealized case of no losses things become simple: every shape and h/r value is optimal. The thing will spin forever if started beyond its critical speed!