[math-fun] Somsky gears, aka unsymmetric off-center planetary gears. Don't really work.
https://www.youtube.com/watch?v=M_BUn4TDns8 demonstrates. And there is a proof, which appears momentarily in the video at 3:35, if you freeze the video and view at full screen magnification you can actually try to read the proof. The first line in the proof is the necessary condition about tooth count sum which makes it work. This is quite a beautiful thing, which is highly non-obvious, but has a very simple pre-college ancient-Greek sort of proof. However, if the radii b and b' are unequal then Somsky's proof indicates the two gears with centers B and B' will rotate at constant angular velocity hence necessarily eventually collide. Therefore, this kind of planetary gear is IMPOSSIBLE if all gears coplanar and you want to be able to rotate the central gear (center C) an infinite number of times without being killed by a collision. By making the gears have suitable nonzero thicknesses and with the gears with centers B' and B being thinner and in sufficiently offset parallel planes, then they can "pass through" each other rather than colliding... but that will not save us because gear with center C will be unable to mesh with them both. So, I conclude this kind of planetary gear is not possible unless the travel distance is restricted; you cannot do an infinite number of rotations. So it seems practically useless, unfortunately, except in the well known symmetric case where all planets same size. -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)
Oskar Deventer in his video remarks he can make Somsky gears with 1 sun and 4 planets, and also says Somsky showed one can make 1 sun and 6 planets. I point out that it is possible to make Somsky gears with 1 sun and 2N planets for ANY value of N>0, and in a countably infinite number of ways for each N. Begin with a symmetric centered sun and all planets of equal size. This is trivial to do for any value of N. Now perform a "geometric inversion" transformation http://mathworld.wolfram.com/Inversion.html https://en.wikipedia.org/wiki/Inversive_geometry to the whole configuration, causing the gear radii to become all unequal and making it unsymmetric. However if all the radii originally were rational and the coordinates of the gear centers ditto, and ditto for all the parameters of the inversive transformation (which we of course can easily assure; there is a dense infinite set of rational-coordinate points on the standard unit circle corresponding to pythagorean triples) then all that will still be true post-transformation. And note inversions map circles to circles. And then by scale-up by the LCM of the denominators we can make all radii be integer, hence result will work as Somsky gears for a limited but nonzero travel distance. QED. -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)
Excellent question, which had not occurred to me. Can the argument be refined to establish upper or lower bounds, or better still exact values, for the (maximum) distance travelled? [Memo to self: if in a hurry, do not accept lifts in vehicles designed by anyone called Deventer, Somsky, etc.] WFL On 7/9/15, Warren D Smith <warren.wds@gmail.com> wrote:
Oskar Deventer in his video remarks he can make Somsky gears with 1 sun and 4 planets, and also says Somsky showed one can make 1 sun and 6 planets.
I point out that it is possible to make Somsky gears with 1 sun and 2N planets for ANY value of N>0, and in a countably infinite number of ways for each N. Begin with a symmetric centered sun and all planets of equal size. This is trivial to do for any value of N. Now perform a "geometric inversion" transformation http://mathworld.wolfram.com/Inversion.html https://en.wikipedia.org/wiki/Inversive_geometry to the whole configuration, causing the gear radii to become all unequal and making it unsymmetric. However if all the radii originally were rational and the coordinates of the gear centers ditto, and ditto for all the parameters of the inversive transformation (which we of course can easily assure; there is a dense infinite set of rational-coordinate points on the standard unit circle corresponding to pythagorean triples) then all that will still be true post-transformation. And note inversions map circles to circles. And then by scale-up by the LCM of the denominators we can make all radii be integer, hence result will work as Somsky gears for a limited but nonzero travel distance. QED.
-- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)
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On Thu, Jul 9, 2015 at 12:14 PM, Warren D Smith <warren.wds@gmail.com> wrote:
However, if the radii b and b' are unequal then Somsky's proof indicates the two gears with centers B and B' will rotate at constant angular velocity hence necessarily eventually collide.
I just don't see this at all; why must they rotate at constant angular velocity? For any flat 2D system of gears not slipping, the speed of movement of the contact points on each gear is fixed across all gears, and the angular velocity of each gear is inversely proportional to the gear radius (big gears rotate more slowly). This should be the same for this system, and thus, I see the gears continuing to rotate forever without collision. Put another way, if I put an axle through each gear and lock it to the plane, the gears will rotate (clearly the axle through the outer gear has to be imagined). But you state it so confidently so I must be missing something important. Can you clue me in? Thanks! -- -- http://cube20.org/ -- [Golly link suppressed; ask me why] --
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Warren D Smith