Pleasure to hear from you, William! Ahh, so full circle back to Oskar's original question: maybe the problem with the offset problem (after making the sun larger and the outer and planet gears smaller) is that we can't get the symmetrical planets to mesh for the new size. (That is, placing the *third* small planet fails to mesh.) So, William, do you have a symbolic solution to the 34-18-10-8-6 case that shows perfect mesh? Or is it just numerically close? -tom On Mon, Jul 13, 2015 at 3:28 PM, William R Somsky <wrsomsky@gmail.com> wrote:
Yes, wrap a belt around any two planetary gears, and push the sun in until it meets the planets, letting them roll along the annulus. You can then add the complementary gears to each planet as per the geometric proof. This gives a 4 planet "somsky" system. In fact, by using different integral lengths for the belt, you can get multiple, geometrically distinct configurations for the same gears.
I used to have (maybe I can find it) a program to tabulate these pairs, listing them and (a numerical calculation of) the displacement of the sun from the center. To find systems w/ more planets, I looked for sets w/ the same annulus, sun and sun displacement. As the displacement is a computed numerical value, I can only say that they are "close", but cannot prove that they are exact matches.
You have to be careful about invoking symmetry, however. In placing the central sun w/ two different sized planets, the sun may end up in a phase that is not symmetric under reflection...
On 07/13/15 13:46, Tom Rokicki wrote:
Well, as posted on math fun, if every continuous belt that you can loop around a particular gear set has an integer length (in tooth count), the gears should mesh. (This is what the Somsky proof is saying, as I understand it, for a particular set of teeth and this is what lets it work.)
I claim that any two pairs of Somsky planets can mesh properly. Here's my explanation.
For any single pair of planets, the location of the sun gear is fairly arbitrary. Indeed, you can take the larger of the planets out, move the sun gear arbitrarily as long as it still touches the remaining planet, and always reinsert the remaining planet with everything meshing perfectly. Somsky's proof shows this clearly.
So let's take the sun gear, and the smaller planet from *one* Somsky pair of planets and the smaller planet from *another* Somsky pair of planets. Can we insert the two smaller planets in some way such that the sun meshes properly with both of these, and the two smaller planets mesh properly with the outer gear? I claim yes, and it's pretty easy to visualize; just put in one smaller planet, then the sun gear, then put the other planet against the outer gear and roll it up to the sun gear. When it touches it won't necessarily mesh, but you can move the newly inserted planet closer to the other planet and the sun brushing along the new planet until they mesh (the surfaces move in different directions as you do this so eventually you'll get a mesh).
At this point you just insert the larger planets of the two Somsky gears; they are guaranteed to mesh with the sun and the outer planet, so you are done.
Your case of *three* pairs of Somsky planets is just a situation where one of the pairs has equal-sized planets, which introduces symmetry, which can then be exploited to introduce a mirror of the second pair of planets.
Does this make sense?
On Mon, Jul 13, 2015 at 1:10 PM, M. Oskar van Deventer <m.o.vandeventer@planet.nl> wrote:
Hi Tom,
Thank you for accepting the challenge, and for responding so quickly.
do you mean, they won't mesh with the same centers as in the original (36-18-10-8-6) ...
That is indeed what I mean. I want to 3D print a Somsky-style contraption, where the sun gear is coaxial with another sun gear, and the same for each other planetary gears and the annulus.
I can follow your reasoning for the individual (35-17-9-9) meshes as does (35-17-11-7). However, I doubt whether your reasoning is correct for both together. How can we check that?
Maybe what happens is the two 11's clash?
I am not worrying about clashes. Mathematically, gears can overlap. Moreover, if your reasoning is correct, then there should exist plenty “Rokicki Gears” that don’t overlap. Or am I misunderstanding you?
Best regards,
Oskar
From: Tom Rokicki Sent: Monday, July 13, 2015 9:18 PM To: M. Oskar van Deventer Cc: Warren Smith ; William Somsky ; Julian Ziegler Hunts ; Bill Gosper ; math-fun Subject: Re: New challenge: Offset Sonsky Gears
Oskar,
Thanks for the challenge!
When you say the (35-17-11-9-7) gears don't mesh---do you mean, they won't mesh with the same centers as in the original (36-18-10-8-6), or they don't mesh in *general* (that is, permitting the centers of the gears to move)?
Clearly by Somsky (35-17-9-9) meshes as does (35-17-11-7), and I would think therefore so would (35-17-11-9-7) with only one 11 and one 7, and then by symmetry so would (35-17-11-9-7) with two 11s and two 7s. That is, I would *expect* that the (35-17-9-9) has enough freedom of the sun gear to permit an 11-7 pair to also mesh. Maybe what happens is the two 11's clash?
I'm not doing this mathematically, but strictly intuitively, which of course is always dangerous . . . but I'm curious where my intuition is breaking down.
-tom
On Mon, Jul 13, 2015 at 12:00 PM, M. Oskar van Deventer <m.o.vandeventer@planet.nl> wrote:
Gentlemen,
While you are still discussing new theorems about the Somsky Gears (which I am unable to parse as non-mathematician), I would like to take the liberty and pose a new challenge: Offset Somsky Gears.
What Bill Somksy has proven, is that there are plenty of exact solutions for planetary gears where the sun is offset from the annulus gear, with exactly meshing gears. Bill sent me the below 34-18-10-8-6 example mid 2012.
So how about offsetting the generating circle of each gear as shown in the image below? In this example, I offset Bill’s 34-18-10-8-6 geometry into a 35-17-11-9-7 geometry. So the circles fit in this geometry. However, when drawing the corresponding gears, you will discover that they won’t mesh. So offsetting these Somsky Gears does not yield more Somsky Gears.
For regular planetary gears, the classic threefold symmetrical (120-degrees) concentric geometry has many solutions with different gearing ratios that all mesh.
Now, the challenge is to find asymmetric concentric planetary-gear geometry and/or a Somsky geometry that meshes, AND where the above-described offset yields another exactly meshing configuration.
I hope that the challenge is a bit clear. Probably, a professional mathematician can provide a proper definition of the challenge.
Enjoy!
Oskar
-----Original Message----- From: Warren D Smith Sent: Monday, July 13, 2015 3:31 PM To: M. Oskar van Deventer ; William Somsky ; math-fun Subject: Gear topologies main theorem, revised & corrected
I got tired of posting wrong theorem+proofs that get refuted by return mail, so I wrote it more carefully, put it in a file and stuck it on the web via dropbox. You can read the theorem & proof here:
https://dl.dropboxusercontent.com/u/3507527/GearTopologies.txt
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