torqueless
Senior Retro Guru
Re:
For devotees of the "Every chainring tooth must be visited by any given link" school of thought, the Fibonacci numbers 21x55 look good. If my calculations are correct, (no guarantee of that) it will take 28 revolutions of the cranks for a link to visit all 55 teeth, which (I think) it does in it's own systematic way, like some crazy postman: "Let's see, I've just been to number one, I'd better go to number twenty-two next, then forty-three, then I'll go back to number nine.."
That's a 69 inch gear on road wheels, just like doctor-bond's 13x34... or about 65 inches on 26" wheels.
Addendum:
Well... I don't mind admitting that the behaviour of this system is baffling me now. I've just been doing some empirical tests on a 12-speed with 42 & 52 rings, a 13-20 block, and 106 (edit: 10
links in the chain. I tied a bit of thread around one chain link, and counted how many revolutions of the chain it took for that link to return to the same
tooth on the chainring.
No matter which sprocket was engaged, in the 42 ring it took only six chain revolutions for the link to return to the same chainring tooth it started from, evidence that any given link only ever visits six teeth on the chainring. Whether the link visited the same six teeth in every gear I don't know, but I think it must have.
On the 52 ring, no matter which sprocket was engaged, it took twelve chain revolutions for the link to return to the same chainring tooth, evidence that any given link only ever visits twelve teeth on the chainring. 52 is not even divisible by twelve. What's going on?
Over forty odd years of cycling, I have accumulated a small collection of chainrings, but none of them have an odd-number tooth count, let alone a prime number. So that's about as far as my empirical tests can go... and all that stuff about prime number tooth count remains, at least for me, theoretical. Phew!!
For devotees of the "Every chainring tooth must be visited by any given link" school of thought, the Fibonacci numbers 21x55 look good. If my calculations are correct, (no guarantee of that) it will take 28 revolutions of the cranks for a link to visit all 55 teeth, which (I think) it does in it's own systematic way, like some crazy postman: "Let's see, I've just been to number one, I'd better go to number twenty-two next, then forty-three, then I'll go back to number nine.."
That's a 69 inch gear on road wheels, just like doctor-bond's 13x34... or about 65 inches on 26" wheels.
Addendum:
Well... I don't mind admitting that the behaviour of this system is baffling me now. I've just been doing some empirical tests on a 12-speed with 42 & 52 rings, a 13-20 block, and 106 (edit: 10
links in the chain. I tied a bit of thread around one chain link, and counted how many revolutions of the chain it took for that link to return to the same tooth on the chainring.
No matter which sprocket was engaged, in the 42 ring it took only six chain revolutions for the link to return to the same chainring tooth it started from, evidence that any given link only ever visits six teeth on the chainring. Whether the link visited the same six teeth in every gear I don't know, but I think it must have.
On the 52 ring, no matter which sprocket was engaged, it took twelve chain revolutions for the link to return to the same chainring tooth, evidence that any given link only ever visits twelve teeth on the chainring. 52 is not even divisible by twelve. What's going on?
Over forty odd years of cycling, I have accumulated a small collection of chainrings, but none of them have an odd-number tooth count, let alone a prime number. So that's about as far as my empirical tests can go... and all that stuff about prime number tooth count remains, at least for me, theoretical. Phew!!
