I believe the following eight-planet system to be exact: ring = 53, sun = 27, offset = 20.00, planets = 3, 5, 5, 13, 13, 21, 21, 23 Here is an animation w/ the 23-tooth gear omitted (as it overlaps): Seven Planets 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...