On 11/8/15, M. Oskar van Deventer <m.o.vandeventer@planet.nl> wrote:
Dear math-fans,
Here is a video of the Offset Somsky Gears: https://www.youtube.com/watch?v=ekZGCVFajqc
People are asking me about the mathematics of Somsky Gears and the Offset Somsky Gears. Unfortunately, the math-fun discussions are private and unstructured. Is there any public explanation available? So far, I know only
Tom Rokicki's Somsky Solver at http://tomas.rokicki.com/somsky.html. This is
great for providing insight, but it does not explain how or why it works.
Thank you.
Oskar
Note that the ring radius also increases when the planets increase, which I don't think was mentioned in the commentary. I'm still trying to put a coherent account of the mathematics together, but it keeps running away in different directions ... maybe WRS has had better luck? Along with the video are listed a number of other ingenious gear models, including "Flower Gears" at https://www.youtube.com/watch?v=AFsKvanfYbQ --- an epically disastrous attempt to construct an elaborate multi-level planetary gear system which disintegrates in a shower of cogs when any attempt is made to demonstrate its intended function. It drew my attention because of the analogy with my recent abortive "octocog" project, and I am fairly sure that it fails for essentially the same reason. Inspection reveals 3 distinct hexagonal circuits of gears, each of which must admit an endless belt of exactly integer length. In addition, the outer ring must have radius exactly an integer (given the radii of the planets, here all equal to 5). The total number of continuous constraints is 4, so there is almost certainly no nontrivial exact solution. By tinkering with the planet radii and floral pattern --- all discrete variables --- it will be possible to find approximate solutions, of which I am fairly sure "Flower Gears" is an example: and the larger the radii involved, the better the potential approximation. But I assume that Oskar found his via experimentation, without actually calculating either exact belt lengths or ring radius. The fudge duly came home to roost; although even if exact, such a mechanism would probably require subframes supporting intermediate levels. Fred Lunnon