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 topsThe 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 balanceThe 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 primesThe 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 locatorNote 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.
