M
Matt Cahill
Guest
This was posted by 'bostontrevor' over on the fixed gear/single speed
bike forum. I was so impressed that I thought it deserved posting
here:
To determine the number of potential "skid patches", you start by
factoring each gear. Find the least common multiple, which means
multiplying all of the unique terms in each gear. So if I had a 48/16
ratio, my terms are 2*2*2*2*3 = 48, 16 = 2*2*2*2--The set for 48
actually contains all of the unique terms, so multiplying them = 48 =
the LCM(48,16). On the other hand for 46/18, the terms are 46 = 2*23,
16 = 2*2*2*2, so the LCM(46,16) = 2*23*2*2*2 = 368. So whenever the
number of teeth the chain passes over counts a multiple of 368, that
means both the cranks and wheel have returned to the same position
that they started in.
That comes out to 8 rotations of the crank, so on any given skid
(assuming the same leg position), you have a 1-in-8 chance of hitting
any given skid patch on the wheel.
If your ring and cog have relatively prime numbers of teeth (that is,
they share no terms in common), then you have a best case scenario
where the number of skid patches is equavalent to the number of teeth
in the cog. My case, with a 47/16 ratio: LCM(47,16) = 47*2*2*2*2 =
752. 752/47 = 16 (big surprise) => 16 unique skid patches.
17 is a prime number, so it's always going to be relatively prime to a
given chainring (except those that are even multiples of 17: 17, 34,
51, 68, etc).
MC: So prime number cogs will generally give the same number of skid
points as the number of teeth on the cog.
bike forum. I was so impressed that I thought it deserved posting
here:
To determine the number of potential "skid patches", you start by
factoring each gear. Find the least common multiple, which means
multiplying all of the unique terms in each gear. So if I had a 48/16
ratio, my terms are 2*2*2*2*3 = 48, 16 = 2*2*2*2--The set for 48
actually contains all of the unique terms, so multiplying them = 48 =
the LCM(48,16). On the other hand for 46/18, the terms are 46 = 2*23,
16 = 2*2*2*2, so the LCM(46,16) = 2*23*2*2*2 = 368. So whenever the
number of teeth the chain passes over counts a multiple of 368, that
means both the cranks and wheel have returned to the same position
that they started in.
That comes out to 8 rotations of the crank, so on any given skid
(assuming the same leg position), you have a 1-in-8 chance of hitting
any given skid patch on the wheel.
If your ring and cog have relatively prime numbers of teeth (that is,
they share no terms in common), then you have a best case scenario
where the number of skid patches is equavalent to the number of teeth
in the cog. My case, with a 47/16 ratio: LCM(47,16) = 47*2*2*2*2 =
752. 752/47 = 16 (big surprise) => 16 unique skid patches.
17 is a prime number, so it's always going to be relatively prime to a
given chainring (except those that are even multiples of 17: 17, 34,
51, 68, etc).
MC: So prime number cogs will generally give the same number of skid
points as the number of teeth on the cog.
