Up: Table of Contents Final results for this hub
Sachs Torpedo 7-speeds - example of reverse engineering[1]
  Number of gear teeth Drive path Ratios  
Shifter Position Sun Gear 1 Pinion meshing with sun  Pinion meshing with ring Ring gear 1=R to P, 2=Direct, 3=P to R Drive ratio as fraction[2] Drive ratio from Sheldon Brown[3] Ratio incre- ment
1 33[4] 13.0187[5] 17[6] 63[7] 1  250/421 0.594 0.594  
2 29.988[8] 17 17 63 1  250/369 0.678[9] 0.677 1.141
3 21[10] 24.9633[11] 17 63 1  163/200 0.815[12] 0.814 1.203
4 21 24.9633 17 63 2 1        1.000 1.000 1.227
5 21 24.9633 17 63 3 1 190/837 1.227 1.227 1.227
6 29.988 17 17 63 3 1 119/250 1.476 1.476 1.203
7 33 13.0187 17 63 3 1 171/250 1.684 1.684 1.141
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.
[1]
John Allen:
Ratios are calculated but are necessarily correct, as the sun-gear tooth counts are given in the parts list, and the sum of tooth counts of each sun gear and the planet gear meshing with it must be about the same. Discrepancies with Sheldon Brown's numbers reflect rounding errors. Other notes on this page tell how tooth counts are derived from ratios.
[2]
John Allen:
These drive ratios are incorrect due to the non-integer tooth counts. These fractions may not even represent them accurately, as Excel only finds the nearest fraction with three places or fewer in the denominator. That's usually good enough if the tooth counts are all integers, but not here.
[3]
John Allen:
These are the ratios given in Sheldon Brown's Internal Hub Gear Calculator, http://www.sheldonbrown.com/gears/internal.html at the time this workbook was prepared. As the final results on the previous page show, there are some minor rounding errors.
[4]
John Allen:
Values in the blue cells must be divisible by 3. I try different values until I get integer or near-integer results in the pink cells in the same rows. If nothing works, I go back and try a different number in the green cell.
[5]
John Allen:
The formula for this planet gear is entered before trying different solutions. The tooth count output by the formula generates the specified drive ratio in combination with the sun gear, ring gear and other planet gear in the same row of the spreadsheet. The tooth count must be an integer, give or take a small error due to rounding of the specified ratio. One other cell is linked to this one, as the same gear combination is used for both a decrease ratio and an increase ratio.
[6]
John Allen:
This cell gives the number of teeth for the middle planet gear, which engages the ring gear at all times and which engages the middle sun gear in the 2nd and 6th ratios. All of the other cells with the same tooth count are linked to this cell. Note that two of those cells are in the previous column. That is because the same planet gear engages both the ring gear and sun gear in 2nd and 6th. We already know the tooth counts of the ring gear and sun gear for these combinations. To fit between the ring gear and sun gear, this planet gear must have a tooth count of approximately half the difference between their tooth counts. Here, 63 - 30 is 33, and 17 is half of 34, meeting that requirement. But we don't know for sure whether this gear has 17, or maybe 16, teeth until we try values in the cells with blue backgrounds.
[7]
John Allen:
As there are 3 planet gears -- and they must be equally-spaced due to their 3 sets of teeth -- the tooth count for the ring gear must be an integer divisible by 3, and must produce an integer also divisible by 3 in the pink cell in the column "Sun Gear 1".
[8]
John Allen:
The formula for this sun gear is calculated before trying different solutions. The formulas for a simple planetary system require that the tooth count here be 0.476 times the ring gear tooth count, give or take a small rounding error, and in this hub with 3 compound planet gears, the number of teeth of all sun gears must be divisible by 3. The same planet gear engages both this sun gear and the ring gear, and so the tooth count of the planet gear does not figure into the calculation. One other cell is linked to this cell, as the same gear is used for both a decrease ratio and an increase ratio.
[9]
John Allen:
The calculation in the pink cell in this row uses the 0.476 from the increase ratio, and so this number differs from the one in the next column due to rounding.
[10]
John Allen:
Values in the blue cells must be divisible by 3. I try different values until I get integer or near-integer results in the pink cells in the same rows. If nothing works, I go back and try a different number in the green cell.
[11]
John Allen:
The formula for this planet gear is entered before trying different solutions. The tooth count output by the formula generates the specified drive ratio in combination with the sun gear, ring gear and other planet gear in the same row of the spreadsheet. The tooth count must be an integer, give or take a small error due to rounding of the specified ratio. One other cell is linked to this one, as the same gear combination is used for both a decrease ratio and an increase ratio.
[12]
John Allen:
The calculation in the pink cell in this row uses the 0.227 from the increase ratio, and so this number differs from the one in the next column due to rounding.