|Solving for numbers of gear teeth based on scanty information is
very much like solving a Sudoku puzzle -- but for a real-world problem. The
example here is for the Sachs Torpedo 7-speed hub. Illustrations in the
manufacturer's literature show that this hub has single-stage compound
planetary gears, and that the middle planet gear engages the ring gear.
Increase ratios and decrease ratios are the inverse of each other, and so,
use the same gears. Starting with that information, I turned the common formulas
for planetary gearing around to solve for tooth counts when output ratios are
known. I also made the results for the table cells with colored backgrounds
(above) propagate to other cells that describe the same gears -- the other cells which, as you can see, hold
the same numbers. I then looked for plausible results by trying different
numbers in the cells in yellow, green and blue, in that order. The hub's
having 3 compound planet gears reduced the number of possible solutions, as
the number of teeth on the ring gear and sun gears must be divisible by 3.
The ratio of the middle sun gear to the ring gear is the key to the problem,
as the middle sun gear must then have 30 teeth (or a multiple, which would
make for an exceedingly large ring gear). Read the notes to individual cells
in the order, pink, yellow, green, blue for further explanations. With this
hub, I was able to check my results against sun-gear tooth counts in the
manufacturer's literature. Having confirmed that I could solve the problem, I
also reverse-engineered the Sturmey-Archer 7-speed and SRAM i-Motion 9-speed,
for which I did not have any tooth counts.