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This is a bit of a weird question because the problem is more about how you could even go about formalizing a hypothesis more than how to prove it — but it seemed like a fun idea and I figured someone has to have some kind of results like this about rigid body mechanics.

Anyway, a CVT is a coupling that allows transfer of angular motion from one rod to another with $\omega_o = \omega_i*c$ for all $c$ in some non-empty interval $(0, m)$ where these are the output angular velocity and input angular velocity respectively.

Now there are a whole bunch of CVT designs built on various frictional mechanisms. For instance, you can have cones facing opposite directions on the end of the input and output shafts and use freely spinning ring/ball to bridge the gap. By adjusting where the ring makes contact you can change the value of c. However, every design seems to use a frictional coupling.

And intuitively it seems obvious something like that has to be the case. Any rigid connection like gear teeth would seem to force a fixed relationship but I started thinking about how one could actually prove something like that I had no idea. I assume there must be results like this but that's so far from my area I don't even have a clue.

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Before I get immediately counterexampled let me specify that I want to restrict the question to the interaction of rigid bodies and I want to exclude things like chains or other situations where the connection between input/output has any kind of free play like paddles intermittently batting something or non-uniqueness like one way ratchets. I'd like to say something like maybe forward motion of input shaft must always result in a unique non-zero movement of output shaft and there must be a unique configuration at all times -- but I fear this actually rules out too much even classical gears.

So I have no real idea on how you would even formalize the idea but I feel like there must be math about this sort of thing. Thoughts?

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EDIT: I foolishly assumed but didn't specify that it should take input from only a single rotating shaft. For the more general case of multiple input shafts the condition should have been formulated as saying the output should be able to have unboundedly large torque given bounded input torques. Hence why I wouldn't regard the planetary setup mention below as qualifying as the output torque is limited to a finite multiple (given by gear ratio) of the torque on input driving the exterior ring.

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  • $\begingroup$ If you search CVT transmission you can get an idea of what I mean and see they are all frictional rather than based on rigid geometric interactions -- sorry I'm sure that's not even the right term. $\endgroup$
    – Peter Gerdes
    Commented Jun 28 at 9:10
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    $\begingroup$ Where you wrote $\omega_i*c,$ I wonder whether you may have meant $\omega_i\cdot c.$ $\endgroup$
    – Michael Hardy
    Commented Jun 28 at 14:52
  • $\begingroup$ @MichaelHardy Are those not just stylistic variants in your area of math? They are in mine? I meant real valued multiplication. $\endgroup$
    – Peter Gerdes
    Commented Jul 3 at 16:44
  • $\begingroup$ The use of an asterisk for ordinary multiplication was introduced for contexts in which one is limited to characters on the keyboard. Sometimes people use it because they don't know how to type $5\cdot3$ or $5\times3.$ Sometimes they use it because they have adopted an affected style. Does any publisher of books or scholarly papers use it? In the eyes of some readers, unless it is obvious that you're using a deliberately affected style, it will appear somewhat less than literate, somewhat like the say in which (as often happens here) someone writes that they want to$\,\ldots\qquad$ $\endgroup$
    – Michael Hardy
    Commented Jul 3 at 16:50
  • $\begingroup$ $\ldots\,$solve the equation $ax^2+bx+c=0$ for x, rather than that they want to solve it for $x. \qquad$ $\endgroup$
    – Michael Hardy
    Commented Jul 3 at 16:50

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I believe an epicyclic or planetary cvt is a counterexample.

https://www.youtube.com/watch?v=xOiAfOH-fU8

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  • $\begingroup$ That's very interesting. But I don't think it quite fits the original conception which was a mechanism that translated the input from a single rotating input shaft to a single rotating output shaft. I admit I didn't define that quite correctly but I don't believe this allows arbitrary torque multiplcation However, it seems like a very good start on such a counterexample if you could somehow allow the single input to drive both inputs but not sure you could without regress. $\endgroup$
    – Peter Gerdes
    Commented Jul 3 at 16:42
  • $\begingroup$ Yes, I foolishly presumed a single input shaft in my statement of the problem but when I meant that the output speed could be an arbitrarily small multiple of the input I intended that to mean an arbitrarily large multiplication in force. A planetary gear setup like this is limited in maximal torque by that supplied by the element driving the ring (adjusted by the fixed gear ratio with said ring). I'll clarify. $\endgroup$
    – Peter Gerdes
    Commented Jul 3 at 16:49

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