I may be wrong on the odd-sun case... On 07/13/15 17:24, William R Somsky wrote:
Yes, I do believe that symmetry allows us to state that this is exact.
In fact, I believe that it we can then say that (ignoring overlap -- mesh conditions only) that there are one or more exact configurations -- differing in the offset of the sun gear -- for any set of the form: sun = a, annulus = a+2n, 2x"shepherd" planets = n, 2x"minor" planets = b = one of {1,...,n-1} (yes, a one tooth gear is odd, but works in an odd sort of way) , 2x"major" planets = 2n-b.
Construction works approximately thus (loose description): set your ring gear w/ phase 0 straight down. Insert the two "shepherd" planets, meshed, at 3 o'clock and 9 o'clock w/in the ring. The sun can now be inserted, meshed, into the center w/ either phase 0 or 1/2 (depends on if a is odd or even) (symmetry grants this phase). Place the two minor planets, meshed, symmetrically into the lower gap between the sun and the ring, and wrap with a "timing belt" of integral length. By symmetry, the mark/phase on the timing belt on the centerline nearest the sun gear will be 0 or 1/2 (depending on if the belt is even or odd length). Select either odd or even to make this match the phase of the sun gear. The sun gear can then be lowered into contact w/ the belt and the minor planets, w/ the shepherd planets herding the sun down symmetrically. Once contact is made, the major planets complementary to the minor planets can be inserted w/ guaranteed mesh due to the geometric proof.
This might be what you were saying Tom, or close to.
This works by symmetry, but it is still unclear if any more general exact sets greater than 4 planets exist. As I've said, I can demonstrate ones that are numerically very close, but lack any analytic proof confirming or denying exact matches. They seem to come too close numerically to be purely coincidental, but that's far from a proof.
- WRSomsky
On 07/13/15 16:02, William R Somsky wrote:
In general, I can only get "numerically close" (something like 10^-6), but it's been a while so I don't remember exactly w/ this one. Symmetry may be able to make this one provable exact.
On 07/13/15 15:43, Tom Rokicki wrote:
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...