New concept: Total Gear Ratio

General
1

Before the days of geared unicycles, our poor man’s gearing was in choosing other crank lengths and wheel sizes. Mikefule did some good research on this a few years ago and came up with the Constant Food Speed Hypothesis (CFSH). Maybe somewhat jokingly but he was onto something: if wheel size (and hub gear) are constant, then making the crank smaller by a factor x results in a speed increase by a factor x. In addition, making the wheel larger by a factor of y (and leaving crank length and hub gear ratio constant), results in a speed increase by a factor of y. Surely the CFSH model breaks down at extreme crank lengths or wheel sizes (both too large and too small), but it’s still a useful model.

In bicycles, crank length usually doesn’t vary that much. Most cranks for adult bikes are around 170 mm. Wheel size is somewhat of a variable, but indeed most of the gearing choice of bikes is done either through front and rear derailleurs putting the drive chain on other chain wheels or sprockets, or alternatively through internally geared hubs (some bikes have hybrid systems). Because of the semi-constant crank lengths used in bikes, bicyclists express their gear ratio as “gear inches”, i.e. how far the bicycle travels with one full revolution of the cranks.

Now that unicycles with internally geared hubs are becoming more common, I’ve also seen unicyclists referring to their gear ratios in terms of “gear inches”. E.g. Chuck did this in his excellent post Future of Unicycle Road Racing (here’s what I think; what do you think?). However, in my opinion, “gear inches” is an incomplete measure for gear ratio on a unicycle, as it doesn’t take crank length into account.

I would like to propose a new concept to express gear ratios for a unicycle, and call it Total Gear Ratio (TGR). It is a dimensionless number, being the number of length units that a unicycle travels per one length unit of travel of the pedal along the pedal circle (with respect to the moving unicycle). Algebraically, TGR = hub gear ratio * wheel radius / crank length. Wheel radius and crank length should be in the same length unit, e.g. inches or millimeters.

For example, if you would have a hypothetical unicycle with a fixed hub, a 24" wheel, and 12" long cranks, then the TGR would be 1.00, because the cranks are equal to the wheel radius. If you would replace the cranks with more practical 6" cranks, then the TGR would be 2.00. (When the pedal moves a full circle with a diameter of 6 inches, the wheel has progressed one circumference of a circle with a radius of 12 inches.) Can you still follow me?

The usefulness of the TGR would be in comparing the speed potential of various unicycle setups (or bikes for that matter), since the speed potential would be determined within reasonable limits by the TGR.

By way of example, Total Gear Ratios for some popular unicycle setups:

Fixed Coker, 114 mm cranks: TGR = 4.01
Schlumpf geared Coker, 137 mm cranks: TGR = 5.16
Fixed 29", 102 mm cranks: TGR = 3.61
Schlumpf geared 29", 127 mm cranks: TGR = 4.15
Standard unicycle (IUF definition = fixed 24.333", 125 mm cranks): TGR = 2.47

(Note that I used the exact Schlumpf gear ratio of 17/11 (equal to about 1.545), rather than the rounded value of 1.5.)

Comments welcome.

4 Likes
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Wow, this is pretty cool. How long did it take you to think of this?

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That has almost nothing to do with what Klaas Bil posted. It doesn’t matter what the components are that make up your uni, the number is found using effective diameter. That’s wrong anyways it’s a 15" rim with a 2.5" tire.

I like this system Klaas. Is there an easy to use equation for finding speed at certain cadences?

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As far as I’m aware the term gear inches refers to the apparent diameter of the wheel and was originally adopted for penny farthings.The late Sheldon Brown has some interesting stuff on gears and crank length.I think you might be on to something with the TGR.

http://sheldonbrown.com/gain.html

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Roger did a spread sheet you could have a look at;

http://www.unicycle.uk.com/Cranklength.xls

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You guys are both right. While the nominal wheel diameter is usually a good start if you have no measured wheel diameter, what the formula really needs is the actual radius with the components that you have on there. In my examples I used 36" and 29" as the exact diameter even though they are (also) nominal diameters, because I know (from measuring my own unis) that the actual diameter is quite close. My “IUF standard unicycle” on the other hand has an actual wheel diameter of almost 24.333", while nominally the tyre is 26" x 1".

@ dangerdog: As to what gear inches really are: yes, it works out to be apparent roll-out (apparent diameter times pi), which is the same as actual wheel rollout times the gear ratio in the chain drive. Which in turn equates to what I wrote “how far the bicycle travels with one full revolution of the cranks”.

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I’m both happy and proud to have ridden with someone smart enough to think of something like this. I usually just hop on and pedal and space. Once in a while–especially if I’m riding with Harper–I might kind of wonder if my shoelace is untied or something like that.

I bet I’m going to start falling more now on my 36, because every time I shift my 4-way adjustable cranks, I’m going to trip up on the math and crash. I’ll have to double down on the espresso and chocolate to help my brain keep up.

Perhaps–just to make things easy for myself–what I really need to be thinking about is building up a geared 36-er.

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Hey, nice! You explained this really, really well… way, way better than I ever have. Last year I made a post in a thread about short cranks on cokers where I did the same thing, only I was using the reciprocals of all the numbers you gave here… didn’t make much sense. Your greater-than-one numbers are a lot easier to understand. Also, TGR is a great way to talk about this. Sometimes you just need a cool acronym to represent an idea that would otherwise take many words to reference in a sentence… kind of like making a pointer to an object you use a lot in your code, if you write programs.

A quick suggestion, if I can take the liberty: it may help to keep in mind that (assuming the ability to choose any gear ratio) there is an optimal crank length for each rider, and that a departure from that crank length will make the rider slower. A Schlumpfed up Coker with approximately 154.545454mm (hehehe) cranks may have the same Total Gear Ratio as a Coker with 100s, but it is clearly the faster unicycle for those who’ve spent some time riding both setups. Changing the crank length from X to Y | Y<X will increase a rider’s speed by a factor of Y/X, and decrease the speed by some as yet unknown function of the loss in pedaling efficiency due to the departure from the optimal length, which is currently long enough (for average-height folks) that nobody has had a cause to actually use it for road unicycling, due to the lack of big enough gears to make it useful. TGR is definitely more complete a descriptor of the ride than simple gear inch counts, but we still need the crank length to know whether a uni will seriously be rock-on awesome for speed or not.

For simplicity, from here on out, when I talk about a wheel/gear-ratio combination of a certain number of gear inches, I’ll just refer to it as a gear. To push a gear… to push such a setup.

I think that each value (between TGR, gear-inches, and crank length) gives a distinct sense of what kind of ride you’ll have. TGR is how hard you’ll have to push to adjust your balance and accelerate, and gear-inches (assuming a known crank length) is that as well, though is more descriptive of cadence than exertion, in my opinion. Crank length… maybe just a matter of how it’ll feel? I think they’re all valuable when comparing unicycle setups, because the ability of a given rider to ride a unicycle fast is a function of (at least) those two variables (gear size and crank length). TGR is a really great way to feel out how fast a uni will be.

Also, I’d like to add that we’re all thinking about this in one direction, which is to ever-shorter cranks. This is because we’re all stuck on gears that are too small. Even our geared 36ers are still not yet big enough, at least, for anyone who opts to ride with cranks shorter than what they’d pick on an adequately geared bike. As far as I know, that is all of us right now. (Note that I’m assuming flat road for that statement!) Our gears have been too small pretty much as long as modern unicycling has existed. If, for some strange reason in some bizarro universe, we were all stuck on gears that were too big, we’d be talking about how lengthening the cranks would make you faster by a ratio of newlength/oldlength, with a loss of efficiency because we’re departing away from the optimal crank length to the longer direction, sacrificing a well-fit-to-your-body crank length for some leverage to push a suboptimally huge gear. Once we get unicycles with these huge gears, I think our understanding of what it means to have a given crank length will be opened up to the other side of things… the there-can-be-a-too-big side :), and we’ll all be thinking about things as a give-and-take situation in both directions instead of having an initial “shorter cranks are always faster” starting point.
edit: I’ve thought about this paragraph some more, and now realize that, though longer cranks (than are a good fit) would make you faster with a gear that is way way too big, there’d be no logical reason to think that this would be by a factor of longerlength/shorterlength, whereas there clearly is reason to start out thinking that way when talking about shorter cranks. Some of the logic in there was kind of dumb… but I think the point still exists in a less-strong fashion than I thought it did when I wrote it…

Also, for future reference, any time I ramble on about how great it’d be to have an n gear-inch setup (70 <= n <= 100), I’m talking about that optimal crank length, which, for me, would be 170-175mm. Why use short cranks if you can pick your dream gear? :-D… (my road bike cranks are that long… as much as I love unicycling, there is a soft spot in my heart for my 175 cranks and 20 gears ranging from TGR of 3.44 to 9.79!) Actually, I think it’s better to pick your crank length first by knowing what fits your body, then pick a gear you know you can kick some butt with for hours on end using those cranks.

I really like using the acronym TGR to describe this. I never really knew how to describe it in a precise way… I’ve tried to use a whole lot of different ways of trying to refer to the concept, and could never seem to explain myself. TGR just gets the job done. :slight_smile: Hopefully, when we compare setups in the future, we’ll talk about this kind of thing primarily, using TGR as our “pointer” to the idea. Much easier to talk that way. Kudos!

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The concept itself is compelling. Nevertheless, I don’t think there is a linear correlation of crank length and speed. I would rather expect a formula like
TGR = hub gear ratio * wheel radius / sqrt(crank length)
Of course that formula depends on the chosen units, so maybe you could get it more precise by defining
TGR = hub gear ratio * wheel radius/cm / sqrt(crank length/cm)
Additionally I am not sure, whether the power of 1/2 needs some adaption.

BTW, where did you get the information that the exact ratio of the Schlumpf hub is 17/11? I couldn’t find that anywhere, especially not on Florians internet page.

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Gear ratio of a schlumpf is quite easy to verify - turn the cranks 11 times and see how many times the wheel turns. Or ask Florian. Because of the design, it has to be an exact integer ratio - just line up the valve with the frame, and do several full turns of the pedals, counting how many turns you do, and how many the wheel does before the valve stops level with the frame at the same point as you finish a full turn.

We all know that total gear ratio isn’t exactly relative to speed, but it is a simple physically well defined, unitless quantity, based on the amount of leverage you have at the pedal. Whereas your calculation is unless you justify it better, just a magic equation made up from nowhere.

Obviously at the extremes either way, none of these equations work, but in the mid range where current unicycles are setup, the total gear ratio thing seems pretty consistent with what you experience riding - the schlumpf 29er with 125 cranks is roughly similar in speed, difficulty etc. to a coker with 102 cranks and way faster than a 29er with 102s.

Joe

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Important correction: I did not come up with the constant foot speed hypothesis. I came across the expression in this forum liked it and used it in a series of related posts. I take no credit for the expression.

Bicyclists talk about gears in terms of inches. The comparison is with the distance travelled per revolution by a directly driven wheel, like on a penny farthing.

However, the comparison is only useful if you assume a standard crank length. Bicycles virtually all have 165 or 170 mm cranks, which is more or less an industry standard.

However, unicles can have cranks of almost any practical length.

Therefore the expression of gear simply in terms of wheel size or inches falls apart as a useful method of comparison or prediction of performance.

The CFSH postulates that at any given level of effort and ability, your feet will move at a more or less constant speed. With short cranks, the feet move in smaller circles and therefore the wheel does more revolutions per minute. With longer cranks, the feet move in larger circles so they do fewer rpm.

If this hypothesis were literally true, then useful predictions about speed and performance could be made by simply comparing the length of the cranks with the radius (or diameter) of the wheel.

For example (in constant units) 5 inch (125 mm)cranks on a 20 inch wheel would perform identically to 6 inch (150mm) cranks on a 24 inch wheel.

In each case, the length of the cranks is identical to half the radius of the wheel, or a quarter of the diameter.

It would follow from this that with 6 inch cranks and a 24 inch wheel, your foot would move 37.7 inches per revolution, and the unicycle would cover 75 inches per revolution.

On 5 inch cranks, your feet would move 31.4 inches per revolution, and the unicycle would move 62.8 inches per revolution.

A simple comparison shows that in each case, if the crank length is half the wheel radius, the wheel moves twice as far as the foot.

The hypothesis is easy to disprove with a simple thought experiment:

consider two unicycles: the 120 inch wheel with the 30 inch cranks, and the 1 inch wheel with the 1/4 inch cranks.

These would each have the same crank:wheel ratio as the two examples above, but of course would not perform the same.

Where the CFSH did work was in making an approximate prediction of what would happen if you made a small change in the wheel diameter or the crank length. For example, you could use the hypothesis to make reasonable predictions about:

  • Changing from 125s to 150s on a given unicycle.
  • The relative performance of a 20 and a 24 with identical cranks.

The CFSH would be no use for making predictions about the relative performances of a Coker on 80mm cranks and a 20 on 125mm cranks.

So the CFSH and all its derivatives are only useful for making rough predictions about small changes in crank length or wheel radius, all other things being equal.

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Mikefule don’t dash my spirits. Being a foodie, I was getting all excited that someone worked up a Constant FOOD Speed Hypothesis. It makes sense though even if it was “somewhat jokingly”. What is the rate of food that you need to take in in order to maintain that speed? Ultimately, that is what I’m most interested in.
This, of course, is dependent on wheel diameter, crank length and gear ratio and Klaas Bil is very insightful in his elegant solution to this portion of the problem.
But then we need to incorporate the differing caloric values of carbs and proteins (4 kCal per gram) and fats (9 kCal per gram).
Other factors would be the digestive efficiency of the rider, the efficiency/skill level of the rider (less skilled riders make bigger mistakes that require bigger corrections that require more energy), the rider’s weight and O2 Max, the inspiration and therefore speed gained from a really yummy meal…
I think we need to meld the geometry of TGR with the biochemistry of the Constant Food Speed Hypothesis, combining the potential speed of the machine with the energy requirements of the rider. What is the most efficient combination?
To me, this sounds like the Grand Unified Theory of long distance unicycling. The GUT. WOW! This must be more than a coincidence!
The Constant Food Speed Hypothesis needs to be fleshed out. Any takers?

1 Like
15

it’s a 15 inch rim… bmx wheels are 16"

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I’ve been thinking about this for a long time, and I really like the way that you’ve come up with. One change that I would like to suggest is that you keep the gear ratio separate. EX:

1 x 4.572 = 4.572 36" 100mm ungeared would have the same TGR as
17/11 x 2.958 = 4.572 36" 154.54mm geared but I would suggest that you say it like ‘1 by 4.5’ for the ungeared and ‘1.5 by 3’ for the geared one.

They both have the same TGR, but it shows that one is geared and the other isn’t so that you can tell the efficiency difference. Doing a little math in your head (multiplying 1.5 x 3) shows you that they have the same TGR but a different efficiency.

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Thanks all for supporting the TGR concept. I initially thought about this in order to compare my ungeared shortcranked 29" with my Schlumpf longcranked 29", and the latter with my ungeared mediumcranked 36". And then again, the latter two with my possibly-to-be-acquired Schlumpf 36" with yet undecided cranklength. I was looking for a way to compare them in my mind. I’m happy that the TGR struck a chord with more people.

Like many responses pointed out, there’s more to speed than just TGR. The physical ability to push a certain gear comes into play, as does fear of falling at higher speeds. Also, a certain crank length is optimal for a certain rider, which has to do with leg length and other things. Within limits, usually a shorter crank is faster even if it’s shorter than optimal, but not by the factor by which it was shortened.

However, the Total Gear Ratio is just that, a pure gear ratio. It is not a coefficient in an exact speed formula. Still, the TGR can give you a rough idea about the speed potential for a given setup.

@dangerdog:
Thanks for the Sheldon Brown link. His “gain ratio” is in essence the same as the Total Gear Ratio, although more “geared” towards bike use (pun intended). And oops, I apparently had the definition of “gear inches” wrong after all, I should have known though. There’s no pi (I mean 3.14159…) in there.

@tomblackwood:
What are 4-way adjustable cranks?

@wolfgang.schaper:
(1) You’re right that there isn’t a linear correlation of crank length and speed. But I wouldn’t want to put a square root (or another power) of crank length into the TGR, because then it’s not a gear ratio anymore. However, there might be some merit in developing a speed predictor formula that takes the non-linearities into account. One of the biggest difficulties with that approach is that it’s different for each rider. I leave that one to you.
(2) Indeed, Schlumpf’s website states the gear ratio as 1.5 to 1. But I determined by counting (much as Joe Marshall described) that it is not exactly true. If you do count enough revolutions (I think I did 200!) you get for the ratio (number of wheel revs) / (number of crank revs) a number that’s accurate to maybe three decimal places. Then try and find a fraction with lowish whole numbers that matches the ratio.

@mikefule:
Sorry for crediting you with the CFSH. Who did introduce it, then? And you’re right that the CFSH is not strictly true, especially in extreme regimes. The arguments are much along the same lines as why the TGR is not an exact predictor of speed.

@naturequack:
:o but also :slight_smile:

@scotthue:
I think the strong point of the TGR is that it encompasses all three influences on what you could call “gearing”. If you can do a little math in your head, you might as well just want to mention crank length, wheel diameter and hub (or other) gear all separately.

18

how about a quadratic regression?

You might be getting “2 pi r” (circumference) and “pi r squared” (area)confused, but you are right it is not linear.

If I’ve understood TGR correctly a 20” with 5” cranks would be the same as a 24” with 6” cranks.

10” radius / 5” radius = TGR 2.0
12” radius / 6” radius = TGR 2.0

However, they are not the same. I own both of these configurations and the 24 is faster on timed laps.

To help understand why the two unicycles described above are not the same, please consider exaggerated examples of unicycles with a 1.0 TGR. Imagine riding/walking on a little 2” radius wheel with 2” cranks. You would be taking tiny steps every time a crank hit the ground. However, on a 20” wheel with 10” inch cranks you could move at a much faster pace. The small wheel is the equivalent of walking with your shoelaces tied together. The larger wheel fit’s more with a natural human stride.

That’s the problem. Unicycles are easy to make mathematical models of, humans are not. How people “fit” with their unicycles is so hard to define, describe, calculate.

I believe an accurate model would be quadratic with the vertex being the ideal setup. If we had a way to collect the data, could we use a large number of riders timed riding a variety of unicycles and take a quadratic regression to find the best fitting equation.

It wouldn’t be as easy to remember or use as TGR, but it might be useful to some of us.

19

I like it, and just so I don’t have to do all those calculations I will just put 114s on my coker and call it a day.

20

I enjoy the TGR concept, but I want to add a concern about cadence:

Cadence

One obvious problem with the TGR equivalence of different unicycles is that most riders are more comfortable (and probably more efficient) in a given cadence range. (The TGR assumes that a constant foot velocity is the key, but as others have pointed out, this assumption is obviously wrong with very small or very large cranks.) Apparently bike riders often use 60-80 RPM cadences for most comfortable riding. Maybe we can push it to 100+ RPM on an appropriate unicycle. These numbers directly give us the speed of the unicycle for any configuration (and completely ignore the crank size.) Presumably different crank sizes make faster and slower cadences more effective and extend the range normally quoted for bicycle riders.

Here is a formula for MPH:

MPH = RPM X WD X GR X 0.00297

Where MPH = miles per hour
RPM = cadence in revolutions per minute
WD = Wheel diameter in inches
GR = hub gear ratio (1.00 unless you spent a lot of money on your unicycle)

(Notice that the crank length does not show up in this equation—it is implicit in the RPM term.)

Here is a bike cadence discussion:

http://www.kenkifer.com/bikepages/touring/gears.htm

The TGR theory suggests that the preferred cadence should be inversely proportional to the crank length. Has anyone measured comfortable cadences with different cranks on a unicycle? If not, I would suggest having someone with a large selection of different cranks try this test. Mount the cranks on a 24" unicycle and try a fast but comfortable pace with each set. Record the comfortable maximum RPM with each crank set. (One could measure MPH and back calculate RPM if that is an easier measurement.) That might give us a sense of how a comfortable cadence varies with crank size. I am suggesting a relatively small unicycle (24") so that the resistance is not very high (the TGR is small). One could then compare these measured cadence with the prediction of TGR theory that cadence should increase linearly with decreasing crank length.

21

As I mentioned in my initial post, Mikefule did some good (hands-on, or rather feet-on) research on this subject. He reported the results in this forum, but I was too lazy at the time, and I am still, to search for it and provide a link. Mikefule used not only various crank lengths but also various wheel sizes. He concluded that while cadence is not exactly inversely proportional to crank length, for a single step in crank length, and then within certain reasonable limits, the Constant Foot Speed Hypothesis is useful.

But let me say again that the Total Gear Ratio is a gear ratio. It may be interpreted as a predictor for speed, or somewhat more safely as a predictor for speed potential, but entirely at your own risk. YMMV.

1 Like
22

I’d just like to ping this thread to let everyone know that I’ve put an article about Total Gear Ratio on Adventure Unicyclist, kindly written by Klaas Bil.

I think it’s a great way to conceptualise unicycle gearing, and hopefully will be a good reference for all unicyclists in future.

Read about it here: Total Gear Ratio