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Author Topic: Bracelet top  (Read 92 times)

Jeremy McCreary

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Bracelet top
« on: March 26, 2020, 06:28:34 PM »

Having lots of fun making centrifugally formed tops lately. This new gray/black "bracelet top" is one of my best -- partly for its long, wobble-free spins, and partly for a very nice feel during hand starts. The lion's share of its mass and rotational inertia resides in the floppy light and dark gray rotor, originally a gag bracelet for my wife.



The link design (left photo below) allows the bracelet to take on any in-plane shape with negligible resistance.



In this 82 g version, a bracelet of total length L = 384 mm mounts on 3 equally spaced spokes of length R = 64 mm. Best launch speeds and spin times are respectively ~1,000 RPM and ~90 s by hand and 1,520 RPM and 110 s with an electric starter. The structure could probably handle a lot more speed, but I have yet to push that envelope.

Centrifugal force is amazingly efficient at expanding, shaping, stiffening, and balancing the bracelet. At high speed (left), the bracelet blurs into a circular ring. But photos at lower speeds (right) hint at the true equilibrium shape, and it's definitely not circular. This shape forms at speeds well below critical and depends only on the bracelet / spoke length ratio B = L / R.



Here, I chose R = 64 mm first for structural reasons and then optimized L to minimize wobble. The result: L = 384 mm and B = 6.00. Compare that B to the B = 5.20 of a bracelet just long enough to form a straight-sided equilateral triangle, the B = 5.44 of a Rouleaux triangle, and the B = 2 pi = 6.28 of a true circle. Smooth spins at B > 6.28 are rare -- presumably due to bracelet waves between spokes.

Smaller blue/gray version with R = 56 mm and L = 336 mm -- here stopped gently enough to retain the bracelet's equilibrium shape. This is also the gray/black's equilibrium shape, as B = 6.00 in both cases.



This version shows some vibration wobble, as its core (combined stem+hub+spoke+tip assemblies) isn't as stiff.
« Last Edit: March 26, 2020, 08:13:13 PM by Jeremy McCreary »
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Playing with the physical world through LEGO

ta0

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Re: Bracelet top
« Reply #1 on: March 26, 2020, 08:38:31 PM »

Interesting investigation into chain tops, specially your study of the effect of the ratio of circumference to radius (you must be a follower of the tau movement  ;) ).

But the chain has a discrete number of links and the resolution of the shape is limited. In addition, air drag is significant. A string would give cleaner results.
Are the sections between the spokes arcs of a circle? Probably more like a catenary (the shape of a hanging string), but deformed (more steep) because of the dependence of the centrifugal force with the radius. I wonder if its a known shape.

PS: It looks like your wife returned the gag bracelet  >:D
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Jeremy McCreary

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Re: Bracelet top
« Reply #2 on: March 26, 2020, 10:21:31 PM »

Interesting investigation into chain tops, specially your study of the effect of the ratio of circumference to radius (you must be a follower of the tau movement  ;) ).

Wait, chain tops are a thing??

Yes, a partial follower: All for tau = 2 pi -- mainly because a single symbol would simplify many important formulas and equations. But unwilling to kick the pi habit, as that would complicate many others. I'd like to see both in common use.

PS: It looks like your wife returned the gag bracelet  >:D

Let's just say that her view of LEGO has been tainted by its effect on our budget. :D

But the chain has a discrete number of links and the resolution of the shape is limited. In addition, air drag is significant. A string would give cleaner results.

Not necessarily. Not sure that the bracelet's discrete links have a big impact on its overall shape at speed -- at least not with this many links. Also, drag shapes a high-speed jump rope, so no a priori reason to think that a string in place of the bracelet would be immune. Great free PDF with some bearing on this question: Aristoff & Stone, 2011,The aerodynamics of jumping rope.

Are the sections between the spokes arcs of a circle? Probably more like a catenary (the shape of a hanging string), but deformed (more steep) because of the dependence of the centrifugal force with the radius. I wonder if its a known shape.

Good questions. Maybe something akin to a catenary or the drag-free jump rope curve, aka troposkein. Problem is, the catenary comes from a force field of uniform magnitude and direction. Here, the bracelet sees a radial centrifugal force field varying in both magnitude and direction with some dependence on B. Someone may well have tackled that curve, but no name I know of.

One thing I can tell you: It's easy to spin the top fast enough for centrifugal force to erase all vestiges of gravitational sag, even by hand. Ditto for tank tops with partial granular fills, fast jump ropes (per Aristoff & Stone), and probably true for string in lieu of bracelet as well.
« Last Edit: March 26, 2020, 11:52:29 PM by Jeremy McCreary »
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ta0

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Re: Bracelet top
« Reply #3 on: March 27, 2020, 12:09:14 AM »

Yes, a partial follower: All for tau = 2 pi -- mainly because a single symbol would simplify many important formulas and equations.
Nah, extra factors of 2 in many equations is not the problem. The real problem is that we measure "natural" angles in radians where 1 turn is 2 pi, 1/2 turn is 1 pi, 1/4 of a turn is pi/2. But 1 tau = 1 turn, 1/2 tau = 1/2 turn, 1/4 tau = 1/4 turn. Much more logical and natural. A second but minor thing, is that the radius is generally more fundamental than the diameter (think physical forces).

Not necessarily. Not sure that the bracelet's discrete links have a big impact on its overall shape at speed -- at least not with this many links. 
I'm concerned about the visible shape. Slight differences between different possible curves would be lost with the big chain links, and might be more visible with a thin string.

Maybe something akin to a catenary or the drag-free jump rope curve, aka troposkein. Problem is, the catenary comes from a force field of uniform magnitude and direction.
I didn't know about the troposkein (aka jump rope curve). It takes into account the increase of centrifugal force with radius. But still doesn't take into account the change of direction of the force so it would be different. The radial direction would make it rounder. In fact, if the length is that of a circle (B=2 Pi) the shape should be circular!
« Last Edit: March 27, 2020, 12:57:48 AM by ta0 »
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Jeremy McCreary

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Re: Bracelet top
« Reply #4 on: March 27, 2020, 01:33:09 AM »

Good points about turns and radius vs. diameter. I've always preferred radius.

Alas, a LEGO bracelet top with R large enough to get a good approximation of B = tau with these links would probably have too much vibration wobble for my taste (due to core flexure).

If there are solutions with shorter links of comparable flexibility, they're eluding me at the moment. But never say never in the LEGO realm!
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