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Author Topic: Balancing math  (Read 420 times)

ta0

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Balancing math
« on: November 29, 2019, 10:02:03 PM »

Almost spintop related: the math of balancing a (test tube) centrifuge:

https://www.youtube.com/watch?v=7DHE8RnsCQ8

If somebody posts a top made this way (preferably not obviously balanced), I'll move the topic  ;)
« Last Edit: December 01, 2019, 12:27:47 PM by ta0 »
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ta0

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Re: Balancing math
« Reply #1 on: December 01, 2019, 01:37:47 AM »

I was inspired by the Lucky Penny Top (I think an invention of Don Olney):



and developed the Centrifuge PennyTop that I 3D printed:



In addition to using a penny as a tip/stem, it has slots for up to 12 pennies.
According to the centrifuge theorem, if you have N slots you can put a number K of tubes if (and only if) both K and N-K can be written as sums that use the prime factors of N.
As the prime factors of 12 are 2 and 3, you can balance it with any number of coins, except 1 and 11. On the photo is the only solution for 7 pennies (not counting rotations). There is only one for 5, but I count four distinctive solutions for 6 pennies.

When you spin it, the top becomes obviously unbalanced if you shift any coin by just one position. As a mathematical problem, it gets more interesting with more pennies, but experimentally the unbalance produced by 1 coin becomes less obvious as the number of coins becomes larger.
« Last Edit: December 01, 2019, 11:54:07 AM by ta0 »
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jim in paris

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Re: Balancing math
« Reply #2 on: December 01, 2019, 02:16:26 AM »

very very cool , Jorge!
you should get it patented :D :D

jim
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ta0

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Re: Balancing math
« Reply #3 on: December 01, 2019, 11:49:31 AM »

very very cool , Jorge!
you should get it patented :D :D

jim

I thought about it. But I'm working on other spinning top ideas even more interesting.  :-X  ;)
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ta0

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Re: Balancing math
« Reply #4 on: December 01, 2019, 03:17:25 PM »

I google searched the centrifuge problem and found this blog post: https://mattbaker.blog/2018/06/25/the-balanced-centrifuge-problem/
The proof of the "only if" part of the theorem is not easy (proved in 2010). The blog has this picture of a 24 tube centrifuge:





You can check that the 13 tubes are well balanced (hint: pairs are 12 spaces apart, triplets 8 spaces apart).
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cecil

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Re: Balancing math
« Reply #5 on: December 01, 2019, 04:19:29 PM »

When I took a Machine shop at City College my teacher let me make spin tops. He said the true way to balance a top - ls with two weights - the Y method. And when I just can’t get a top to balance perfect - I use this method. It works. He was the Best manual Machinist I have ever seen in 50 years.
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mailman

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Re: Balancing math
« Reply #6 on: December 01, 2019, 04:55:34 PM »

When I took a Machine shop at City College my teacher let me make spin tops. He said the true way to balance a top - ls with two weights - the Y method. And when I just can’t get a top to balance perfect - I use this method. It works. He was the Best manual Machinist I have ever seen in 50 years.

Could you please describe this Y method, Cecil?
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the Earl of Whirl

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Re: Balancing math
« Reply #7 on: December 01, 2019, 09:41:53 PM »

The penny penny top looks like fun.  I am impressed!!!
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cecil

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Re: Balancing math
« Reply #8 on: December 02, 2019, 02:09:34 AM »

Just look at the letter Y - the bottom of the Y is heavy - so you put the two weights at the top of the Y. This will really give you a truer spin. I only do this when I have to. And it works.
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Jeremy McCreary

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Re: Balancing math
« Reply #9 on: December 02, 2019, 11:16:51 PM »

Just look at the letter Y - the bottom of the Y is heavy - so you put the two weights at the top of the Y.

So how does this play out in practice? For example, how do you find the top's heavy side?
« Last Edit: December 03, 2019, 03:01:27 PM by Jeremy McCreary »
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Jeremy McCreary

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Re: Balancing math
« Reply #10 on: December 04, 2019, 10:53:52 PM »

Almost spintop related: the math of balancing a (test tube) centrifuge:

Great find! I'd say totally related for tops of modular construction. An N = 12 LEGO example, here with K = 7...





The stiff black and red chassis has mostly 6-fold rotational symmetry, meaning that it mostly looks the same after 1/6 of a full revolution about the spin axis. The outer ends of the red arms bear small round black "mounts" for the gold hats -- here with 5 mounts unoccupied. Somewhat miraculously, the mounts came out exactly 12-fold.

Love both the math and the visuals here. Covering just 7 of the mounts with gold hats per the "centrifuge rule" sets up an interesting visual tension with the 6-fold symmetry of the chassis. And without mucking up dynamic balance!

Granted, air resistance limits spin time to ~10 s. But there's no visible wobble -- with or without gold hats!



That means 2 things: (1) The centrifuge rule balances at least the N = 12, K = 7 case to high precision, and (2) the top's structure quickly damps out any elastic vibrations excited during spin-up, as hoped. (Vibration-induced wobble is a non-issue in most one-piece tops but a common problem in LEGO tops with spokes.)

Centrifuge rule in LEGO tops

The video lecture tacitly assumes (a) a perfectly balanced empty centrifuge rotor, (b) K identical test tubes, and (c) N identical test tube mounts, all at the same level, with perfect N-fold symmetry about the spin axis.

In this LEGO top, the previously balanced black and red chassis corresponds to (a), and the identical gold hats to (b). LEGO parts of the same kind are identical in size, shape, and mass properties to very high precision, regardless of color.

As for (c), you can easily build a workable LEGO top chassis with N = 2, 3, 4, 6, or 16. Getting to N = 8, 10, or 12 is much harder, and you can forget about N = 14 or any odd N > 3. (A top with an N = 2 chassis will stay up only if structures with N > 2 carry most of its rotational inertia.)

Static, couple, and dynamic balance

The centrifuge rule's mainly about maintaining static balance as tubes or ornaments are added to a previously balanced rotor or top. The included no-stacking rule effectively maintains couple balance. The dynamic balance needed for a wobble-free centrifuge or top requires both static and couple balance.

As many of you know, a rotor or top is in static balance when its CM lies precisely on its spin axis. And it's in couple balance when its spin axis coincides with one of its principal axes of inertia.

Perhaps counterintuitively, couple unbalance tends to cause a lot more wobble than static. But it's easily avoided in LEGO tops: You just make sure that every "layer" of parts along the spin axis has static balance in its own right. Better yet, you can apply the centrifuge rule to as many of these layers as you like.

Sums of primes

The centrifuge rule states that to balance K tubes in N holes, both K and N - K must be sums of prime factors of N. Moreover, each prime subgroup of tubes must have its own dynamic balance.

I color-coded the dark blue N = 12 top below to show how this works in detail. The only prime factors for N = 12 are 2 and 3, K = 7 = 2 + 2 + 3, and N - K = 5 = 2 + 3. Hence, the 7 bright decorations come in 1 orange pair, 1 green pair, and 1 blue triad, each balanced separately...



A simple blacklight unbalance locator

Note the small orange, green, and blue fluorescent "emitters" around the dark blue top's stem. Under blacklight, these work together as a built-in unbalance locator -- and a very sensitive one at that.

When dynamic balance is perfect, the 3 emitters trace out perfectly overlapping orbits at speed. Under blacklight, their emitted colors then fuse into a bright, sharp-edged ring with a uniform greenish white glow.

Pure static unbalance separates the ring colors in a useful way. Say you were to balance the top and then add a small test mass outboard of the blue emitter. Under blacklight, this would add an inner blue border and an outer orange + green = yellow border to the previously uniform glowing ring. And the greater the static unbalance, the wider the borders.

The same effect underlies Iacopo's "paint brush method" for locating static unbalances. Couple unbalance also separates the emitter colors, but in much more complicated 3D patterns.
« Last Edit: December 05, 2019, 11:49:21 AM by Jeremy McCreary »
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ta0

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Re: Balancing math
« Reply #11 on: December 05, 2019, 01:11:38 AM »

Nice!  8)

I was sure you were going to build a Lego version and I was wondering why it was taking you so long!  ;D

Quote
I'd say totally related for tops of modular construction
Yes, you are right. This balance problem comes naturally for a top built with Legos.



« Last Edit: December 05, 2019, 01:17:37 AM by ta0 »
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Jeremy McCreary

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Re: Balancing math
« Reply #12 on: December 05, 2019, 03:37:45 AM »

I was sure you were going to build a Lego version and I was wondering why it was taking you so long!  ;D

You know me too well. Had the tops built the same day. The pictures were the hold up.

I like your version, too. Pennies were a good choice, given that they're probably also identical to high precision.

Between your 3D printer and my LEGO, we're in a position to explore lots of ideas quickly.
« Last Edit: December 05, 2019, 11:51:57 AM by Jeremy McCreary »
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Jeremy McCreary

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Re: Balancing math
« Reply #13 on: December 08, 2019, 12:25:16 AM »

So how far can you take the centrifuge rule (CR) in modular top design?

Well, if you're after a well-balanced LEGO top with an unexpected appearance at rest, turns out that N = 12, K = 7 or 5 is about the best you can do under the CR as stated in the video...



That's because LEGO-doable values of N are limited to 3, 4, 6, 8, 10, 12, and 16 at top scales, and all other allowable N, K combos produce highly symmetric patterns seen in lots of tops.

But the design space really opens up when you apply the CR separately to multiple layers or concentric rings in the same LEGO top -- especially when you design for viewing under a variable-speed strobe (VSS).

This perfectly balanced example has an N = 6 chassis bearing 3 coaxial "layers" of decorations at different levels Z along the spin axis. Each layer has its own colors and CR-valid K value.







Here, all decorations orbit the spin axis at the same radius R. But each layer could just as well have had a different R, as overall dynamic balance is automatic when the chassis and each axial layer has dynamic balance on its own.

I find this top interesting for 2 reasons: (1) It looks far from balanced at first glance. (2) The layers obviously spin together, but a VSS can make them look like they're spinning at different angular speeds -- or even in different directions! (Pretty cool in person but impossible to capture in video.)

Here's another perfectly balanced example with a different N = 6 chassis bearing 3 concentric "rings" of decorations orbiting the spin axis at the same axial level Z. Each ring has its own colors, radius R, and CR-valid K value.





A viewer expecting a simple relationship between visual symmetry and balance would find this top equally improbable. A VSS can make the colored rings appear to rotate independently.

And that's just the beginning, as there are many CR-valid combos of N, K, Z, and R with perfect dynamic balance. You can even offset layers or rings by arbitrary angles about the spin axis without affecting balance. Stay tuned.
« Last Edit: December 08, 2019, 12:31:26 AM by Jeremy McCreary »
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ta0

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Re: Balancing math
« Reply #14 on: December 08, 2019, 01:45:42 AM »

Very nice. I love how unbalanced those tops look. I have also been thinking along the same lines (it's easy to stack coins). The N-k limitation is removed by using levels or rings.

By the way, I also searched for "centrifuge" tops that have only odd prime factors but can be balanced by even number of weights in a non-trivial way. You can balance a 9 with 6 weights but it's too symmetric. Unfortunately, 15, 21, 27 and 35 won't work except if you only use one prime and you have to go all the way to 45 for something interesting, and that seems too much.

You say you can have N=10 with Lego. Then you automatically have N=5 by skipping each other  ;)
« Last Edit: December 08, 2019, 01:36:40 PM by ta0 »
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