Probably only a few people saw the earlier version of this post, but in any case I take it back. I thank Alex Wetmore for correcting my error. The 17-tooth planet gear of the SRAM S7 hub, not the 12-tooth, engages the ring gear. The ratios are as SRAM describes.
Also see the corrected chart for the earlier SRAM Torpedo hub, at the bottom of the page.
I will happily install a SRAM S7 hub on my Raleigh Twenty. Here is a table giving the actual ratios:
Drive ratios of SRAM Spectro S7 hubs
| Number of gear teeth | Ratios | |||||||
| 1 to 7 | Sun gear | Planet meshing with sun | Planet meshing with ring | Ring gear | Power path | Turns, Out/In | YES! SRAM says: | |
| 1 | 33 | 12 | 17 | 63 | R>P | 252/439 | 0.574 | |
| 2 | 30 | 17 | 63 | R>P | 21/31 | 0.677 | ||
| 3 | 21 | 24 | 17 | 63 | R>P | 72/89 | 0.809 | |
| 4 | Gears idling | 1/1 | 1/1 | 1.000 | ||||
| 5 | 21 | 24 | 17 | 63 | P>R | 89/72 | 1.236 | |
| 6 | 30 | 17 | 63 | P>R | 31/21 | 1.476 | ||
| 7 | 33 | 12 | 17 | 63 | P>R | 439/252 | 1.742 | |
Below is what I originally wrote, so you can see how even a purported bicycle expert can make a fool of himself!
Being a stickler for accuracy, I decided yesterday to count the gear teeth inside a SRAM Spectro S7 7-speed internally-geared hub to get exact (fractional) gear ratios. SRAM only provides decimal approximations. The results were much more interesting than I expected. I checked two different hubs, one with a drum brake and another with a coaster brake; plugged the tooth counts into my handy-dandy planetary gear ratio spreadsheet — which gives spot-on results — and with increasing incredulity checked against several versions of SRAM’s online technical manuals. The ratios and overall range are considerably narrower than what the company says.
[This is incorrect, please see correction at top of page!]
The claimed total range is 3.03. The actual range is 2.32, narrower than claimed for SRAM’s 5-speed hub (I’ll check it too, soon) and hardly wider than for the Sturmey-Archer 5-speed I have been using. This is significant to me as I was thinking of building up a wheel with a drum-brake S7 for my Raleigh Twenty folding bicycle. I am not getting any younger, I live on top of a hill, I want to pull a cargo trailer… Here’s a table showing what I found:
[This is incorrect, please see correction at top of page!]
| Number of gear teeth | Ratios | |||||||
| 1 to 7 | Sun gear | Plan- et mesh- ing with sun | Plan- et mesh- ing with ring | Ring gear | Pow- er path | Frac- tion | Deci- mal | Not! Mfr. says: |
| 1 | 33 | 12 | 63 | R>P | 21/32 | 0.656 | 0.574 | |
| 2 | 30 | 17 | 12 | 63 | R>P | 119/159 | 0.748 | 0.677 |
| 3 | 21 | 24 | 12 | 63 | R>P | 6/7 | 0.857 | 0.809 |
| 4 | Gears idling | 1/1 | 1/1 | 1.000 | 1.000 | |||
| 5 | 21 | 24 | 12 | 63 | P>R | 7/6 | 1.167 | 1.236 |
| 6 | 30 | 17 | 12 | 63 | P>R | 159/119 | 1.336 | 1.476 |
| 7 | 33 | 12 | 63 | P>R | 32/21 | 1.524 | 1.742 | |
Now let’s try to figure out whether the ratios that SRAM claims might work with different numbers of teeth.
[The paragraphs below are incorrect, please see correction at top of page!]
The top ratio of the actual hub is achieved through a simple planetary system where the sun gear is fixed to the axle, the ring gear is connected to the hub shell, the axles of the planet gears are connected to the driver, and the same 12-tooth planet gears engage both the ring gear and sun gear. In this type of planetary gearing, the drive ratio is 1 + the ratio of the number of teeth of the sun gear to that of the ring gear. The number of teeth of the planet gears does not figure into the calculation.
The claimed ratios have the same symmetry around the unity ratio as those of the actual hub, and so it is most logical to expect that the hypothetical hub is of a similar design — reversing the power path through the gear train for the higher and lower drive ratios. SRAM says that the top gear ratio of the S7 is 1.742 — rounded to three decimal places, so the actual ratio may be anywhere from 1.7415 to 1.7425 and the tooth count ratio of the sun gear to the ring gear may be anywhere from 0.7415 to 0.7425.
Let’s check on what numbers of teeth might achieve this ratio by using the table of logarithms of gear ratios in Machinery’s Handbook. This table includes all ratios of gears from 16 to 120 teeth.
The table in the Handbook starts with a 1/1 ratio and works upwards, so we’ll use reciprocals. (But I started with the top ratio to minimize rounding errors).
The reciprocal of 0.7415 is 1.3486 and its logarithm is 0.12989.
The reciprocal of 0.7425 is 1.3468 and its logarithm is 0.12930.
The logarithm of a possible gear ratio must be between these numbers, or very close to them. Also, If there are three planet gears spaced 120 degrees apart, as in the S7 hub, then the numbers of teeth of both the sun gear and ring gear must be divisible by 3. A ratio that meets these requirements is 93/69, with a logarithm of .12963, but 93 teeth are unlikely for the ring gear — the teeth would be tiny. 93 and 69 are both divisible by 3. The resulting numbers, 31 and 23, are not, so they are out of the question. The number of planet-gear teeth would be very small in this case as well, probably only 5. Would 4 planet gears be a possibility? No, there are no ratios within the prescribed limits for which the number of teeth for the planet gear and sun gear are both divisible by 4.
[The paragraph below is incorrect, please see correction at top of page!]
So, the ratio which SRAM states for the top gear of the hub is not only incorrect, but impossible except, perhaps, with a hub of a rather different design or a larger ring gear. The ratio which SRAM gives for the bottom gear is the reciprocal of that for the top gear and is therefore equally unlikely. Other ratios can’t be calculated, because they would depend on the numbers of tewth of the planet gears. SRAM’s parts list for the hub gives the correct numbers of teeth for the sun gears — 21, 30 and 33 teeth — and these would in any case not be compatible with a 93-tooth ring gear. Strange, isn’t it!
[The paragraph below is incorrect, please see correction at top of page!]
And the mystery deepens. The ratios cited for the older Torpedo 7-speed hub don’t compute, either. The claimed top ratio of 1.685/1 is achievable with a 39-tooth sun gear and 57-tooth ring gear. But the parts list in the Sutherland’s Handbook for Bicycle Mechanics, 6th Edition indicates that the sun gears have 21, 30 and 33 teeth, same as with the Spectro S7 hub, strongly suggesting that all the tooth counts and drive ratios of both hubs are the same. Unfortunately, I don’t have a Torpedo hub on hand to check. I myself compiled the information on the Torpedo hub for Sutherland’s but I may not have actually counted the teeth and calculated the ratios. Big mistake!
[Now, an attempt at a correction. Here is a table of probable ratios for the Torpedo 7-speed. Some check out exactly, others are only off by 1/1000, possibly due to rounding error.]
I calculated the ratios for the Torpedo hub in the following way:
Knowing that the middle sun gear has 30 teeth and the ratio that uses this gear is 1.476/1, the ring gear must have 63 teeth. The resulting ratios for 5th gear checks out with the decimal numbers on Sheldon Brown’s gear calculator, which have three places to the right of the decimal point.
Next: the middle planet gear must have 17 teeth, or close to that, to fill the space between the sun gear and ring gear.
That planet gear and the 30-tooth sun gear it engages have a total of 47 teeth, give or take one or two. Each of the other planet gears and the sun gear it engages must add up to about the same number of teeth.
Now, trying middle planet gears of 15 through 19 teeth, and for each, combinations of the 21 and 33-tooth sun gears with different planet gears, the results below are the closest to Sheldon’s numbers.
The discrepancies of one thousandth are probably due to rounding error, as no other ring gear would give a result that is even close.
Probable drive ratios of Sachs Torpedo 7-speed hub (old model, discontinued)
| Number of gear teeth | Ratios | |||||||
| 1 to 7 | Sun gear | Plan- et mesh- ing with sun | Plan- et mesh- ing with ring | Ring gear | Pow- er path | Frac- tion | Deci- mal | Shel- don: |
| 1 | 33 | 13 | 17 | 63 | R>P | 273/460 | 0.593 | 0.594 |
| 2 | 30 | 17 | 63 | R>P | 21/31 | 0.677 | 0.677 | |
| 3 | 21 | 25 | 17 | 63 | R>P | 75/92 | 0.815 | 0.814 |
| 4 | Gears idling | 1/1 | 1/1 | 1.000 | 1.000 | |||
| 5 | 21 | 25 | 17 | 63 | P>R | 92/75 | 1.227 | 1.227 |
| 6 | 30 | 17 | 63 | P>R | 31/21 | 1.476 | 1.476 | |
| 7 | 33 | 13 | 17 | 63 | P>R | 460/273 | 1.685 | 1.684 |
John,
Thanks a lot for the nice tables! I was looking for quite a long time for *exact* ratios of the planet gears.
But don’t you think that S + Ps + Pr = R should always hold, where
S = number of teeth in the sun,
Ps = number of teeth in the planet, engaging with the sun,
Pr = number of teeth in the planet, engaging with the ring,
R = number of teeth in the ring.
In your numbers, it does not hold for gears 2 and 6, where 30 + 17 + 17 != 63.
Ratios, pretty much coinciding with the figures from the vendor, sound convincing, but I fail to understand how the rings, in which the above equation does not hold, can work together. The diameters do not add up to the ring. The only explanation I see is that teeth on the two tracks (S – Ps) and (Pr – R) are not of the same size.
Thanks. Have a look now at this page. I have now collected information or done tooth counts and calculations on most current hubs and many older ones. Still to do: Shimano 7-speed, Rohloff, Sturmey-Archer 8-speed, older Fichtel & Sachs. Done but not yet posted: Shimano 8-speed.
Through adjustments to tooth form, some minor variation around the theoretical number of teeth is possible. I’ve seen this in a number of hubs whose teeth I have counted myself. As the number of teeth of the sun gears is given in the parts list and it is known which gears engage which other ones, no other numbers are possible for the SRAM S7 or Sachs Torpedo 7 besides the ones I gave. I actually have had a SRAM S7 apart now and counted its teeth, and they are as I calculated!
@jsallen
Here you go for the Rohloff: http://www.minortriad.com/twist.html (bottom of the page).
I see that also there the numbers of teeth add up exactly only in one case out of three. In two other cases, there is one tooth difference. Hm, good that there is always something new to learn 🙂
On your sheet for SRAM-3, you write that S + R must be divisible by 3. Why so? I thought I can argue that S – R must be divisible by 3 🙂 (when the same planet ring engages with both sun and ring), but this does not hold true for your example of Shimano 333 hub. So far, this seems to be the only counterexample.
In the sheet for Torpedo you even claim that S and P both must be divisible by 3. But as I understand it’s not necessary for the three planet gears to be spaced uniformly at 120 deg from each other. Their tooth positions do not have to be central symmetric.
BTW, I just got my hands on an old Nexus-7. Hope to count the teeth soon.
And thanks again for some food for thought 🙂
The Shimano 333 hub has four planet gears, as do the Sturmey-Archer AW and similar Sturmey-Archer hubs. The sun gear and ring gear tooth counts of these hubs are multiples of four.
The Sachs/SRAM 3-speed hubs have the 3 planet gears spaced slightly unevenly. Teeth of the three planet gears engage in sequence. Tooth form must be more-carefully maintained to prevent the planet cage from moving off-center. The sequential tooth engagement probably makes the hub feel somewhat smoother — similar advantage as with helical gear teeth in other applications, but without loading the bearings longitudinally.
The Sturmey-Archer SW used 3 planet pinions, and a ring gear and sun gear with tooth counts not a multiple of 3 (but summing to a multiple of 3) to achieve sequential gear-tooth engagement.
A hub with compound planet gears (SRAM 5, 7 and 9-speed, Sturmey-Archer 5- and 7-speed, Shimano with 4 or more speeds…) must have sun gear and ring gear tooth counts divisible by the number of compound planet gears (3, for all of these hubs), because uneven spacing or sequential engagement would not be the same for planet gears with differing numbers of teeth.
Thanks for help with the Nexus 7 and the link to Rohloff information!
There has not been yet any 🙂 But I could not wait and before disassembling the hub, did my own “rectification” of tooth count for Nexus-7. Here is its scheme (in German, but page 11 of the pdf shows in the table, which rings are engaged in which gear). Instead of trying the values in a spreadsheet, I wrote a script which did the exhaustive search of the possibilities. Restricting the search space by some reasonable minimum and maximum values, at most 1 difference between R and S + Ps + Pr, and at most 1 decimal place difference from the decimal value of the transmission ratio from the vendor, gives already just a few candidate results. After throwing out these, where (R – S) mod 3 != 0, the only one result remains:
Step-down ring (ratio: Sun, Planet-to-Sun, Planet-to-Ring, Ring):
0.741: 24, 22, 22, 69
0.632: 31, 17, 22, 69
Step-up ring:
1.335: 31, 21, 15, 66
1.545: 36, 15, 15, 66
It was so nice to get a consistent result. Thanks a lot John for this idea! (Now I’m not sure any more that I want to disassemble my older Nexus-7… 🙂 )
Thanks — not sure about this with the 31T sun gear. I’m going to wait till I do counts (or you do, or someone else does) before I reach a definitive conclusion.
I already counted teeth of this hub once, over a decade ago, but I sent the results to Howard Sutherland and they are lost somewhere in his files!
Ouch!!!
I opened the hub today and counted teeth. Now I’m missing the possibility to erase my previous comments from your blog, and from the whole Internet 🙂 Here are the real counts:
Step-up ring:
30, 19, 14, 66
36, 14, 14, 66.
Step-down ring:
36, 20, 15, 72
42, 15, 15, 72
So, all suns and rings are divisible by 3, as you claimed/expected. 42 + 2*15 = 72. Surprisingly, 36 + 2*14 = 64, differs by 2 from 66 (that’s why my exhaustive search missed this combination, I assumed at most 1 tooth “inconsistency”). Leaving to do the homework and find out what made me miss also the correct combination on the step-down ring…
@KonstantinShemyak
Thank you. I will use your counts and give you credit. also remember, this entire conversation started with a big mistake of my own. I went on for several paragraphs!
Now what is suspicious is that one ratio out of four falls quite far apart from the value declared in the manuals. It’s gear 2. Set (36, 20, 15, 72) yields (72*20)/(72*20 + 36*15) = 8/11 ~ 0.727. Manual says 0.741. Other three one-step gears (1, 6 and 7) are within the last digit precision. Strange.
Yes, now I’ve done the calculations. You miscounted the smaller planet gear in the step-down stage. If I record it as 14 teeth, all the ratios check out with the published ones. I’ve made counting mistakes like that too. We have provided a simple example of how theory and experimental data check on each other.
Here’s one question which you may be able to answer if you still have the hub apart: why the non-unity middle ratio, when direct drive would be more efficient? I think that the key to this is the shifting mechanism, which engages and disengages pawls that bear on the axle.
Looking at the disassembly/reassembly instructions for the hub, I see that the step-down stage is first, nearest the sprocket at the right side of the hub. The planet cages of both stages are a single assembly, locked together with splines.
In the two lowest ratios, the pawls of the second- (step-up)- stage sun gears do not engage, and so the first stage operates alone. Drive is transmitted to the hub shell by pawls on the planet cage assembly, at the left side of the hub.
In the three middle gears, power is transmitted through both stages, and to the hub shell through the pawls on the outside of the second-stage ring gear. These pawls turn the hub shell faster than the the planet-cage pawls, overriding them.
In the top two gears, the planet cages are connected directly to the driver (through pawls shown most clearly inside the driver, un-numbered part between parts 9 and 10 in the exploded drawing near the end of the linked document). Only the step-up stage is active. Having the 20/27 ratio of the first stage slightly narrower than the 279/209 of the second stage makes the middle ratio slightly lower than unity and avoids having the planet cages potentially drive the hub when shifting between 4th and 5th gear (I think…).
Now, why not just retract the pawls of the second stage, engage the pawls inside the driver and let the planet cages drive the hub shell directly for the middle gear?
I speculate that shifting in and out of the middle gear would then become tricky. There would be the potential for “in between” positions shifting up or down by three steps. In the actual hub, “in between” upshifts are avoided by careful sequencing of engagement of pawls with the axle.
The pawls engaging the hub shell can not be disengaged by the shifting mechanism, and so can not be sequenced in this way.
Also, the mechanism needed to engage the driver pawls may not be capable of engagement and disengagement only one step apart in the sequence.
Smooth shifting is one advantage of this hub. Another is its very even sequence of drive ratios. Disadvantages are relatively-low efficiency (especially vexing in the non-unity middle gear), complexity, and that the diameter of the ratchets around the axle is small, resulting in high stress and rather short service life.
Pingback: Drive ratios of bicycle hub gears | John S. Allen's Bicycle Blog