From: haverl@cats.ucsc.edu (Carl A Haverl)
Newsgroups: rec.bicycles.tech
Subject: Mathematics of gears and why 24-36-46 is bad
Date: 26 Jan 93 00:01:36 GMT
Organization: University of California, Santa Cruz
NNTP-Posting-Host: am.ucsc.edu


When I replaced the cranks on my mountain bike, the gearing changed
from 26-36-46 to 24-36-46.  With the new chainwheels and chain, shifting
from the smallest chainring was very difficult.  I generally had to
overshift into (if not onto) the largest chainwheel and then back the
chain off onto the middle ring.  Shifting performance didn't not improve
or "wear in" after several months of daily commuting (800ft vertical).

I investigated.  When shifting off the smallest ring, the chain never
lined up with the teeth of the middle ring.  Here is the problem.
The greatest common divisor, gcd(24,36), is 12.  This means that
there are 24/12 = 2 qualitatively different ways to shift the chain
off the smallest ring, depending on which set of 12 teeth the chain
leaves from.

I resolved to rotate the teeth on my smallest gear so that one set of
twelve teeth would be perfectly aligned.  The other set would be perfectly
misaligned of course.  The 24 teeth are spaced every 15 degrees.
Rotating the smallest gear 1/5 of a turn clockwise has the effect
of rotating the tooth pattern 3 degrees counterclockwise (72 degrees
is equivalent to -3 degrees mod 15).

Does it work?  No!  The problem is: there are two types of links in
the chain.  I call these "wide links" and "narrow links" because that's
what they look like when one views the chain from the seat.  Need
I mention that there are equal numbers of wide and narrow links?  Since
24 is and the total number of all links are both even, the position
of the wide links relative to the two alternating sets of twelve teeth
on the small chainring does not change (unless you shift).  The fact
is you need a WIDE link to catch a tooth of the middle chainring.
Thus is if the exact shift position has a narrow link trying to grab
the middle chainring, the shift will fail -- remember, the other
twelve teeth are misaligned for any link.

This explained why my new Shimano cranks came with the small chainring
misaligned with respect to the middle by 1/8 and 3/8 inches with respect
to the two alternating groups of twelve teeth on the 24 tooth gear.
(I measure this by how the chain seated on the 24 tooth gear contacts
the 36 tooth gear.)  This also explains why my changing the misalignment
to 0 and 1/4 inches resp. did not produce satisfactory results either.
I got good shifts and terrible shifts.  The Shimano alignment gives
consistently mediocre shifts.  Maybe they want to sell their middle
chainring with ramps ;-) .

The solution?  Going back to 26-36-46 solved my problems.  Since
gcd(26,36)=2, I have 26/2=13 different shift positions off the small
ring so one of them is bound to line up well.  Also, gcd(36,46)=2
so there are 36/2=18 different shift positions off the middle ring to
the larger ring.

Other good possibilities would be 24-38-48 or 24-34-46, etc.

Note that gcd problems are not important for downshifting as it is
not really necessary for the chain to catch a tooth to start a
downshift.  Also note that gcd problems do not plague the rear gears
because the steps between gears are small, and the gcd must divide
this step size.  Small gcd is good.

I think that gear-ratio programs should issue a warning when people
choose chainring combinations that will cause problems with upshifts.
I don't know how many shift positions are necessary.  2 is definitely
too few.


-- 
Carl Haverl
haverl@cats.ucsc.edu
Newsgroups: rec.bicycles.tech