Schlumpf 29" (guni) calculations

I went for my loop tonight, hitting 22.5 mph again despite my exhaustion from yestereen.

Anyway, on my way home, I used my GPS watch to figure out some cadence info.

I found that in low gear (29"), it took me about 77 revs to hit a tenth of a mile, or about 770 revs per mile.

It only took about 47.5 revs to reach .1m. That means about 475 revs per mile.

I reached 22.5 mph (tho only for a nanosecond).

22.5 mph means 22.5/60 miles per minute, or .375 (or 3/8) miles per minute. If a mile is 475 revs, then 3/8 of 475 revs per minute is a cadence of 178.

Are my calcs correct? Did I really hit nearly 180 revs per minute on a unicycle with a 29" wheel. Jeez. That’s a high cadence.

77/47.5 = 1.62

If your figures are correct, then the hub gears up by 62%

That seems a lot. A Sturmey Archer hub is + 33%. I think Blue Shift is + 50%

Ar you sure you’ve measured correctly.

The arithmetic will be fairly simple if you can be sure of the gear ratio.

Mike’s SA and BlueShift numbers are correct. SA three speeds go up and down 33% as well as 1:1. On Schlumpf’s unicycle website under “technique,” “technical data” the gear ratio is listed as 1:1.5. I believe Florian’s hub is actually somewhat higher than that but not 1:1.62.

I don’t know what your true rollout is. I get

29"xPIx77/12"/foot=584’ and 584’/5280’/mile=0.11 mile

So your 77 revs really gets you 10% farther than your cyclometer reads if your tire is truly 29". I think my Big Apple rolls out at 29.25" diameter which would make 589’ in 77 revs or 0.111 mile which is pretty much the same thing. Three digit accuracy with rollouts and wheel wobble and loaded tire compression is fishy to begin with.

Could you have miscounted (different from dismounted)? Did you ride on an accurately measured 0.1 mile stretch or did you rely on your cyclometer for the distance?

I would question first my (own) ability to count correctly. Then I would question the accuracy of the cyclometer and its ability to count accurately which may be speed dependent. If you rode on a measured tenth mile course I would then question that accuracy.

Doing this right involves some thought.

I was fairly careful about the 77 and 47.5. I certainly can’t have been off by 10% - that would be like miscounting 7 revs! I’ll double check both of the #s.

Let’s suppose that the 77 was really a 76 and that the 47.5 is really 48. Then 76/48 is 1.58, and that’s pretty close to the 1.55:1 ratio that many of us have noted earlier about the Schlumpf hubs.

What I’ll do tonite is to check using the cyclo rather than my GPS watch, which is less accurate. I will also check using a longer distance, like maybe a half-mile.

It looks like you are mixing apples and oranges (basically what Greg said). If you are using the GPS to measure speed, that is ground speed. The actual distance travelled to achieve that ground speed is what a traditional cyclometer would measure. Your cadence would have to be higher than that calculated from the GPS to actually achieve the GPS speed because of higher-frequency changes in path direction (wobble of various types).

Another factor is how the GPS actually arrived at the top speed you mentioned. Something probably not knowable except by the engineers that designed that particular GPS. This includes number of averages and/or curve fitting of the speed data, noisy data rejection, ghost handling, predictive elements to the algorithm, and the like.

Finally, what’s the GPS accuracy anyway? I think it’s about 1m these days in absolutely clear conditions with 5 satellites and no tree cover or surrounding buildings, with a stationary measurement over time. Or at least something like that. Can someone confirm or shoot that down (it’s been a few years)? Anyhow, under typical recreational conditions, there’s going to be plenty of measurement error to make this kind of calculation, which is dependent on one single measurement, essentially meaningless. For recreational civilian use with no controls on environmental or experimental conditions, treating a max speed as anything near truth is probably a misuse of the instrument. An average speed, though, taken over a reasonable time (minutes?) should be good under many conditions, though not all.

I’ve seen people try to map single-track under less than dense leaf cover, and have a heck of a time trying to get a GPS to work properly.

My measurement of the BA diameter has been just about 30", though I did not do a rollout on it. This was the fatter BA at about 60 psi, unloaded.

Whatever the calculations, given what cadences track racers reach, and world-class cyclists train at, and your much shorter cranks, 178 cps is not an overwhelmingly high cadence to reach for a top speed.

Hey, Dave! It’s always great to communicate with you. I hope all’s well. As to your points:
I understand that there is a discrepancy bw the GPS watch and the cyclo. However, they both have the same readout for my relatively long ride around the loop: 3.35 miles. If there were even a 1% discrepancy, it would show up over such a long distance. Since it doesn’t, I believe that the two are fairly accurate. I also don’t wobble very much on these rides – the large wheel and my balance have kept the wobble down to a minimum, I think.
178 isn’t overwhelmingly high cadence…till you try it. Dang, it’s a bit scary! I’m not sure I’d like to go much faster, at least not without more padding!

Tip the uni upside down. Start with the crank exactly in line with the fork leg. Mark the tyre with chalk or tape. Turn the crank exactly 1 revolution. How far does the wheel turn? It should be easy to tell whether it turns:

  • Exactly 1.5 times
  • More than 1.5 times
  • Less than 1.5 times

Distance covered when in direct drive is a simple matter of arithmetic. No need to do empirical testing. Simply measure the diameter of the wheel and multiply by Pi.

For comparison of distances covered in direct drive or in “overdrive”, minor errors in measurement of the tyre (e.g. not allowing for distortion/pressure) won’t matter, as the proportions will be the same whether the measurement is accurate or not.

If you can establish the diameter of the tyre, and the ratio of the gear, and specify whether you want a metric or imperial calculation, I can do it for you.

The effective decrease in a wheel’s diameter due to the interaction of rider’s weight, tire pressure, tire design/construction, and rim characteristics make this approach inadequate, Mike. There is significant change in diameter when the rider climbs on. Determing a wheel’s diameter in action requires some sort of on-unicycle rollout approach, hopefully with at least 10 revolutions, and with anti-wobble control (e.g., using an overhead pipe).

In truth, a wheel’s diameter in practice will also change with speed, as the tire/tube move in and out of the unloaded/loaded diameters with inertia, but for a unicycle this would be a tertiary effect at most.

The change in effective diameter from gear setting to gear setting will be a hard change, though, which is what I think you are saying, and could be determined from an unloaded experiment like one you are describing.

That is completely fascinating and surprising, Dave.

I agree completely. I doubt that I’ve been much over 150 on 150s. To make 178 a matter-of-fact cadence would require lots of time at high cadences. And 178 on a uni on a public path is hugely different than 178 on a track bike in a velodrome.

Of course, but assume that the radius is not going to change by more than 1 cm (seems reasonable on a tyre pumped up for speed riding):

Radius of a 28" (for example) is 35.56 cm.

Thus the error would be 1/35.56 = 2.8%.

For normal every day purposes, I can live with under 3% error on the speedo on my unicycle. It’s likely to be more accurate than my car or scooter.

In reality, I’d expect the deformation on my own 28 to be somewhat less than 1 cm, and the error to be proportionately reduced.

I’ve not had time to measure it properly, but by Mikes brief test, I think it’s more than 1.5:1 and significantly less than 1.75:1

The Schlumpf bike bottom bracket ‘Speed-drive’ is 1.65:1, maybe it’s the same as that, although I’d guess it’s lower than that.


David, did you manage the 22.5mp/h on a flat road? And was there any wind?
I’ve done 33.1km/h on flat with wind but I can’t get above 30km/h in the velodrome… More questions! What size cranks are you using and what’s your cruising speed? That’s enough for now…



P.S. I want a guni

My top speed (achieved on consecutive nights) was on the same stretch of DOWNHILL. The hill ends, and then there is a nice flat stretch where I maintained a speed ranging from 20-22 mph for a minute. I also don’t think I could reach a speed of 20 in a velodrome – the downhill helped considerably. My fastest speed on a flat was about 18.5. I’ve also noticed that it takes a while to hit top speed, tho maybe I haven’t tested that enough. My cranks are 127mm. I don’t have a ‘cruising speed’ because I so rarely get to ride on long, flat straightaways. I think that the top speed I’ve maintained for even a few minutes was about 16 or 17. I think it’s possible I could maintain 16 for an hour in a velodrome, but I’d have to train for that one a little.

Assuming a 29 inch wheel

29 inch diameter x Pi = circumference of 91.12 inches.

A mile = 1,760 yards.

X36 = 63,630 inches to the mile

63,636 / 91.12 = 695.34 revolutions to the mile.

1 mile an hour = 695.34 / 60 = 11.59 rpm.

10 miles per hour = 115.90 rpm.

Assume gearing up of 1.65 and in overdrive, 115.90 rpm = 16.5 mph.

Or, 115.90 / 16.5 shows that 7.02 rpm = 1 mph.

22 mph = 7.02 x 22 = 154.44 rpm, which is brisk but not unachievable.

Re: Schlumpf 29" (guni) calculations

On Fri, 28 Apr 2006 22:26:23 -0500, David_Stone wrote:

>Are my calcs correct? Did I really hit nearly 180 revs per minute on a
>unicycle with a 29" wheel. Jeez. That’s a high cadence.

I tweaked my roadride.xls calculation and on my setup I would need 173
revs/min to reach 22.5 mph (which I can’t, BTW). That’s with a 3%
wobble. If you say you have less wobble, you’ll need somewhat less
rpm, up to 5 rpm less, so probably 168 - 170 rpm. The discrepancy with
your calculation will be because of the indirect measurement, probably
mostly in the GPS inaccuracy.

If you want to know cadence, why not measure cadence? Have a stopwatch
in your hand, rev up, count every second right foot down (so I mean
only the even ones, not the odd ones), like this:

and that is then 1
and that is then 2

and that is then 10.”

Start the stopwatch at 0, stop it at 10 (that’s 100 revolutions), and
do the calculation.

For the record:

The high gear ratio of the Schlumpf uni is exactly 1 : 17/11, or about
1 : 1.5455.

The gear ratio of Frank Bonsch’s unicycle is 1 : 1.5833.

Hey, I have to say that at my top speed, flying down a hill where a misstep could mean a trip to hospital, I can hardly imagine counting strokes. However, I did count cadence on my Coker at about 21 mph on that same hill, where I hit a cadence of about 200. The difference is that the Coker rides more securely – a fall isn’t quite as dangerous – so I felt safer in paying att’n to cadence. And anyway, I can’t rely on counting because my cadence is going to vary over the time it takes me to count out revs. In addition, unless I count for a long time, I may add or subtract a rev, which in a short distance will alter my overall cadence.

Where’d you get that 17/11? That’s a weird fraction, but it certainly is about how much I’ve calculated the shift to be.

I have one of these radar speed signs near my house. I’m not sure how accurate it is but it’s a great way of knowing how fast you’re going.

I’m okay to get to 19mph either on the Schlumpf or the Coker, but I’m still not sure if I’ve passed it going faster than 20mph on either (I’ve had higher figures on the coker but I think a car interfered). I think the Schlumpf will be first if I do it, just because I’m commuting on that at the moment. It’s a bit hard to use the sign, as sometimes it doesn’t catch you, and often there are cars in the way, and there’s only about 100m of slightly downhill to accelerate up to it.

I did some RPM tests too, on the flat, with a smooth surface, doing 120rpm is fine, and 100rpm is easy-tastic normal cruising speed. 120rpm is about 16mph I believe.


I have counted rpm’s easily at 160 rpm using my method (on a 20"). I actually find that I can go faster if I count. Like yours, my cadence varies over time, but my counting can keep up.

For the 17/11, I first counted like a 100 revs with the unicycling hanging from the ceiling to get a decimal number, then I did some math to find an almost matching fraction with not too high numbers, because I figured that there must be some exact fraction there. Then I checked and it turned out to be exact. Must have to do with the number of teeth on the sun and planet cogs.

A simple planetary gear system with fixed sun and driven planets has a gear ratio of 1:1+S/R where S is the number of teeth in the sun gear and R is the number of teeth in the ring gear. The fraction 17/11, also written 1+6/11, suggests that the ratio of the number of teeth in the sun gear to the ring gear is 6/11.

Florian’s hub has 6 planet gears which means that both the number of teeth in the sun and ring gears must be evenly divisible by 6 in order to mesh. My guess is that the sun gear has 36 teeth and the ring gear has 66 teeth. This could be made small and rugged.

The gear diameters must add such that two planet diameters plus one sun diameter fit into one ring diameter. The difference between the ring and sun gears is 30 teeth so that each planet gear would have 15 teeth. This is a workable combination for any number of gear pitches which could be determined if you knew the diameter of any one of the gears.

That’s my guess. His ring gear has 66 teeth, his sun gear has 36 teeth, and his planet gears have 15 teeth each. This is where the 1:17/11 ratio comes from.

Crank Force Calculations for Schlumpf Hub

Based on Greg Harper’s insights as to the number of teeth on respective Sun, Planet and Ring gears in Florian Schlumpf’s wonderful hub, I have calculated the respective required crank arm force during the time when we are accelerating the wheel.

It turns out that during accelerated motion, the required perpendicular force that you must apply to the crank arm to achieve the same angular acceleration of the wheel in 1.54:1 mode compared to 1:1 mode is:

Fcrank(1.54:1) = 1.54 * Fcrank(1:1)

So a few conclusions are made:

  1. You speed up 1.54 times faster in “high gear” than in “1:1 gear” (i.e., you achieve the higher speed sooner).
  2. You can achieve 1.54 times the ground speed in “high gear” than in 1:1 mode with the same cadence used in 1:1 mode.
  3. In order to keep the crank force manageable though, you must already have some angular speed built up in the wheel. So don’t slow down too much!
  4. Finally, since the force that must be applied to the crank when climbing hills increases as the incline becomes steeper, it becomes increasingly more difficult to maintain speed in “high gear” mode. Build up as much speed as you can before beginning the climb because the required force to maintain your uphill speed will grow at 1.54 times that experienced in “1:1” mode.