bostontrevor

Ok, this is rapidly spinning out of control. But here's an approximate answer... I think.

My drivetrain looks like this (because I'm a badass, sucka!):



So we're interested in the quantity z, the distance from the center of our chainring to center of our cog which is what changes when we add or remove cog teeth. Let's simplify and say that there is a quadrilateral whose edges are of length r1, r2, z, x as shown. It's not quite true because x is really tangent to the two circles BUT GIVE A GUY A DAMN BREAK!

That can be broken into a rectangle and a right triangle. The right triangle will have base z, hypotenuse x, and leg (r2-r1). Therefore Lord Pythagorus and some algebra tells us

z = sqrt(x**2 - (r2-r1)**2)

Add in two teeth such that r1' = r1 + pi/4. Since we have one more tooth engaged (the other will be on the opposite cog), x' = x - 0.5". More algebra sez

z' = sqrt((x-0.5)**2 - (r2-r1-pi/4)**2)

We're interested in the change from z to z'.

z - z' = sqrt(x**2 - (r2-r1)**2) - sqrt((x-0.5)**2 - (r2-r1-pi/4)**2)

Some more algebra gives us:

z**2 - z'**2 = (x**4 - 2x**2(r2-r1)**2 + (r2-r1)**4) - ((x-0.5)**4 - 2(x-0.5)**2(r2-r1-pi/4)**2 + (r2-r1-pi/4)**4)

Anyone who's really excited can try to simplify this bastard. Me, I'm getting tired.

BostonFixed

Chain length would cary *slightly* depending on the presence/absence of a straight chainline. This comes into play when there is a flop/flopper hub being used.

ryan_c

I started to work on it, and realized I wasn't nearly as excited about it as I would need to be to slog through that. Maybe if I get some time during class tomorrow I will work on it, although I promise nothing

Nice work, BT