Problem 620

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albert
Posts: 52
Joined: Sat Aug 02, 2008 11:36 am

Problem 620

"The circles can overlap."

One could leave this out, as in the sequel the circles are represented by gears, which obviously can't overlap.
Also the drawing suggest that ghost gears that are somehow occupying the same space are not allowed.
So it started out as circles out that roll off each other without sliding, but the remainder to makes it into a Diophantine problem is murky.

Or have I misunderstood the problem totally?

Groetjes Albert

RobertStanforth
Posts: 307
Joined: Mon Dec 30, 2013 11:25 pm

Re: Problem 620

Hi Albert,

The 'planet' gears can overlap if they occupy different locations in the $z$ axis (i.e. the direction coming out of the page).

Steppenwolf99
Posts: 2
Joined: Sun Feb 11, 2018 2:46 pm

Re: Problem 620

What arrangements are counted as distinct? It doesn't look like two arrangements of gears that differ only in orientation are counted as distinct.

RobertStanforth
Posts: 307
Joined: Mon Dec 30, 2013 11:25 pm

Re: Problem 620

Rotations and reflections do not give rise to distinct arrangements.

Posts: 1
Joined: Sun Feb 11, 2018 6:37 pm

Re: Problem 620

Can two planets (with this same circumference) have exactly the same position?

RobertStanforth
Posts: 307
Joined: Mon Dec 30, 2013 11:25 pm

Re: Problem 620

Sun Feb 11, 2018 6:42 pm
Can two planets (with this same circumference) have exactly the same position?
No, the four planets must all be distinct: no two may have the same size and position.

abcwuhang
Posts: 2
Joined: Mon Nov 28, 2016 5:14 pm

Re: Problem 620

As an epicyclic gear train allows the planets to rotate, what is the meaning of perfectly meshing of gears? Could you show how to drive the whole system? Thanks.

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RobertStanforth
Posts: 307
Joined: Mon Dec 30, 2013 11:25 pm

Re: Problem 620

abcwuhang wrote:
Mon Feb 12, 2018 1:18 pm
As an epicyclic gear train allows the planets to rotate, what is the meaning of perfectly meshing of gears? Could you show how to drive the whole system? Thanks.
Each of the six gears rotates about its own axis. The axes themselves don't move.

Steppenwolf99
Posts: 2
Joined: Sun Feb 11, 2018 2:46 pm

Re: Problem 620

I left this alone for 2 days and finally solved it on my second attempt today. A comment: some of the clarifications on this section should be included in the problem description. The trouble I had was more about parsing semantics and I made a couple of wrong assumptions which could have been avoided if I had come back to this page earlier. Thanks for an elegant problem.

Posts: 2
Joined: Tue Jan 16, 2018 9:27 pm

Re: Problem 620

A correct configuration of gears will ALWAYS have their underlying circles as tangent to each other as described in the problem, correct?

How important is the shape of the teeth? If they were more square or more rounded, would it affect the answer? What if the teeth stuck out further or lesser? (Is this what is meant by "pitch" in the problem?)

Animus