Just fell over an interesting gear simulation maybe of interest to others.

[Stojan]

The reason for my interest in this was Orrery's.,.. Couldn't help wondering if one could design an orrery using gears in that particular way. It doesn't seem so as all the gears take the same amount of time to move in a circle. And why does it seem to be such a challenge to build?

http://www.mekanizmalar.com/looney_gears.html  This guy has spent a lot of time making animations...

Which was built because of an article/challenge at http://mathpuzzle.com/17Dec06.html

M. Oskar van Deventer: A few months ago I challenged people to design four gears fitting inside a fifth gear. Andreas R?ver took up the challenge. The sketch below and this gear animation shows some of Andreas' results. Andreas realised quickly that it would be hard or even impossible to find a solution that would result in an exact fit. In an exact solutions, the gears would fit perfectly and mesh perfectly. Instead, Andreas developped an algorithm that searches for approximate solution. Andreas found several criteria which a solution would satisfy, like a maximum mis-mesh of x%, gears not having common denominators and gears fitting "nicely" together. Finally, Andreas performed a quasi-random search for good-enough solutions. The resulting gear arrangements look quite nice. The Looney Gear animation demonstrates how they can rotate as a set of planetary gears, albeit quite asymmetrical. Two questions remain. 1) Do exact solutions exist, or only appromiate ones? 2) Do these Looney Gears have any useful applications? [Ed - The animation is fascinating. Other gears can be found at the Kinematic Models for Design library]

[ArtF]

Hi Stojan:

  >>1) Do exact solutions exist, or only appromiat e ones?

  I expect both exist. They do take some computing to find though. Even something like
the planetary calculator takes a lot of searching through tables to find the appropriate
mixes of tooth count and distances. Its a similar problem, in which I used tables of
ratio's to scroll through as it searches for pairs that will work with integer tooth counts.

Art

[JustinO]

I'm going to bet $5 that there are no exact, non-trivial solutions for "asymmetric planetary gear trains." The intuition is that both the circumference and the radius appear in the equations I see. The circumference and the radius are related by the factor Pi, which is irrational. Gears don't like irrational. Heck, the system is probably transcendental too!

By exact, I mean the pitch circles of the four gears are tangent and not slipping.
By non-trivial, I mean the four gears have four different radii, and the number of teeth is not zero or infinity.

[Stojan]

Hey G'day,

Thank you for the replies, Art I'll take an each way betm, if it is or isn't possible. I think that you are eminently better equipped to argue the point with Justino :P

The more I look at those gears the more I like the way it works, I think it would make a fantastic steampunk art type of project.

I admire the skills required to make some of these projects work and the challenge behind them. Attaching small planets to the pinions seeing as the gears themselves are what is holding the project together.

Cheers and avagreatday.

[ArtF]

>>'m going to bet $5 that there are no exact, non-trivial solutions for "asymmetric planetary gear trains."
>> The circumference and the radius are related by the factor Pi

  Hmm, all gears are related to PI, as are all things round... but thats not to say your wrong.  I think it would
take a fair amount of study for me to make a declaration of belief on either side. Id like to see it in equation
form..or try to develop one to see. Ill have to give it some thought .. its seems intuitive that at some combination
an exact match should be allowed..
  But then , intuition is a poor formula for anything.. :)

Art
 

[Nate]

...

By exact, I mean the pitch circles of the four gears are tangent and not slipping.
By non-trivial, I mean the four gears have four different radii, and the number of teeth is not zero or infinity.

https://www.youtube.com/watch?v=3iyO-OvIS_k

It's pretty easy to find exact solutions with 4 different sized planets.

It's not hard to work out that the centres of the planet gears are on an ellipse whose foci are the centres of the sun gear and the ring gear.  Working out the gear sizes can then be thought of as equivalent to finding a particular trig equation with enough 'co-rational' solutions, but I wasn't able to crack the 8 gear problem that way.    I'm not sure whether Oskar's friends have a different approach, or are just more sophisticated than me in their knowledge of trig functions.

[JustinO]

Intuition engine down for servicing.

[JustinO]

Whew, my bucket of brass gears has what I need  -- I can only think in metal.

Sun gear:    40
Ring gear:    96
Planet A1:    36
Planet A2:    20
Planet B1:    32
Planet B2:    24

They run well, but the backlash allows arrangements that work nicely, yet are clearly not "correct". I can tighten or loosen the train by jumping teeth -- convenient, but not very rigorous. These are off the shelf 48dp Boston Gear gears.

It is creepy that the three 'diagonal' gears are able to wiggle.

[Nate]

This video has more 'practical sense' to it.

https://www.youtube.com/watch?v=M_BUn4TDns8

[Stojan]

Crazy stuff I was compelled to watch more of his videos lol...

[ArtF]

Nice to know my intuition was right.. it so often isnt.. lol

Art

[Stojan]

So who won the $5  ::)

[Nate]

So who won the $5  ...

Dunno.  You probably got your money's worth if you hadn't seen Oskar's channel before.

I'm still chewing on the math problem.

[JustinO]

No one took the bet -- you have to risk loosing to win.

[kit]

My first thoughts on this are to calculate based on pitch diameters (equivalent to using un-toothed, smooth wheels) then look for a common factor for all the circumferences to decide on a tooth size. Iterate until the size/number of teeth is within machineable limits.

Whether or not this approach requires more or less computational power than a brute-force attack on 128-bit encryption I am not qualified to say.

Kit