Okay, to reduce things down to something simple and testable: I believe he can crank his model forever. And I believe this is also true in a "perfect" no-slip case with no mechanical slop. On Thu, Jul 9, 2015 at 4:35 PM, Warren D Smith <warren.wds@gmail.com> wrote:
The "angular velocity" I was speaking of meant, more precisely, Omega'.
Here is a CC of an email I sent to Deventer when he asked for some clarification, maybe it'll help:
First of all, I should have made it clearer in proof of theorem 2 that you should start with N of the 2N planets (say the ones above the x-axis) before performing the inversive map. Then the remaining N "mate" planets can be filled in opposite in such a way as to fulfil the N Somsky sum-of-radii = outer-ring-radius conditions.
Re gears "passing through" this is only true in the 2D projection of the 3D reality. In 3D the planet gears would hit, or if in displaced parallel planes, pass by each other. But either way, it would not work for long. The central sun gear could not be rotated infinite number of times, it would only have a finite range of motion before either the planets collided, or passed "through" each other enough that the 4 contact points could no longer all be contacting, or the sun hit the outer ring. E.g. imagine when the two planets (of differing radii) both touch the outer ring at the same location (which would happen eventually if you kept going). Obviously at that time, the sun could not touch them both from outside, while remaining inside the outer ring.
Just try to keep turning the crank on the Somsky model you made and you won't be able to do it forever (right?).
I could indeed easily provide examples for 2N>6, if I tried. Again, even with offsets used, the range of motion would be finite. See bottom for a try.
Re mechanic vs mathematician, you have a considerable advantage over me in that your building it prevents you from making a mistake. I am more likely than you to make a mistake, especially since I have not tried hard to check myself. Caveat emptor.
An "inversion map" of the plane to itself is: Any point (x,y) is mapped to the point (X,Y) such that dist{ (X,Y), P } * dist{ (x,y), P } = 1 where P is a fixed point you choose, and both (x,y) and (X,Y) lie in the same direction as seen from P. ("Inversion with respect to P.") Inversion maps circles to circles. (Warning: the center of the circle, is mapped to a point which is NOT the center of the new circle!) Inversion map can be described using rational function formulas, so it maps rational numbers to rational numbers if P has rational coordinates.
The inversion of a POINT (x,y) with respect to (a,b) is (a, b) + (x-a, y-b) / ((x-a)^2+(y-b)^2) This inversion map applied to a CIRCLE with center=(x,y) and radius=r yields the circle with new radius = r/abs(S^2 - r^2) where S^2 =(x-a)^2+(y-b)^2 so that S=distance{ (x,y), (a,b) }; and the new center is (a,b) + (x-a, y-b) / (S^2 - r^2). The point formula is the same as the circle formula in the limit r=0.
Another mechanical application of inversion map is the so-called "Peaucellier Inversor" or "Peaucellier-Lipkin linkage." It converts circular motion to linear motion without any sliding. http://mathworld.wolfram.com/PeaucellierInversor.html https://www.youtube.com/watch?v=-Y90r8ykrl0 There are several other such mechanisms linked there, I think they would appeal to you. Mechanical engineers always use the sliding-based "crankshaft" mechanism and not these, far as I know.
OK, trying to produce explicit examples. Really, computer programs should be written to automate the search for examples to find especially nice ones with all integers small. But I will just do this by hand to get an example involving all integers, but some of the integers will be a lot larger than necessary. Not trying to be optimal. Several rational points on the unit circle x^2+y^2=1 are (1, 0) (3/5, 4/5) (5/13, 12/13) (15/17, 8/17) The last 3 arise from pythagorean triples. A trivial 16-planet configuration with everything rational and all planets same size then has the planet centers located at the 4 points (x,y) above, and (-x,y) and (-x,-y) and (x,-y). Let the planet radii all equal 1/24, the sun is centered at (0,0) with radius=23/24, and the outer ring is centered at (0,0) with radius=25/24. Now apply the inversive map arising from P=(53, 0), say, to the sun, outer ring, and the 8 planets with 999999*y+x>0. The result is a set of circles (the mapped versions of the sun, 8 planets, and outer ring) which all have rational coordinates for centers, and all rational radii, now all unequal. Now insert the 8 opposing-mate circles with the right radii to satisfy all the Somsky conditions. You then can also multiply by LCM(denominators) to get all radii integer. That will yield a 16-planet Somsky gear example with a nonzero finite range of motion.
Still more about inversive maps: http://xahlee.info/SpecialPlaneCurves_dir/Inversion_dir/inversion.html
Hope this reaches you, my internet connection breaking down.
-- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)
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