# Deriving a unicycle crank length formula?

Yeah, I know when I’m beaten.

If I had to guess, I’d think cadence is the wiggle from side to side.

Re: Deriving a unicycle crank length formula?

rupert wrote:

> Rowan wrote:
>
>>How long does it take to reach max speed on short muni cranks (short
>>in this discussion means 140-150mm)? Only a few pedal revolutions at
>>most.

[snip]

> Seeing as this is this is your first foray into this discussion you
> have no business telling me what short in this discussion means!

And you have no business being so condescending. Relax a little and
remember that it is perfectly possible to read the entire discussion
many times without actually saying anything and so understanding what is
being discussed.

I think if you go back and have a look at the previous posts, you’ll
find that with the exception of tholub mentioning 125mm cranks,
everybody else has mentioned 150mm as “short”.

Either way, there is no call for rudeness.

Roger

Re: Re: Deriving a unicycle crank length formula?

Those posts are on a different thread, and hence a different discussion! In my post, as I have tried to explain, it didn’t matter how short it actually was, just that it was very short, not in absolute terms but relative to its previous value.

“Cadence” in the context of cycling is just pedalling speed in rpm. Interestingly, inexperienced cyclists will in general use a lower pedalling speed (i.e. ride a higher gear at a given speed). Most more experienced cyclists find a higher rpm to be more efficient. There are exeptions of course, but as a general rule it is often true. This can’t really be transferred to unicycling where shorter cranks (analogous to a “higher gear”) are used to allow a higher pedalling speed. On a bike, the rider picks the crank length to suit, then varies the gearing of the bike to give the most efficient/comfortable cadence at a given speed - to apply this to unicycles we shouldn’t be talking about crank length in isolation from wheel size (gearing).

Some interesting input even if there was a short side step, but hey it’s a discussion

John mentioned BMI which is something I hadn’t thought of but ought to have as I’ve lost quite a bit just lately :D. Whether it’d affect crank choice I don’t know but I’d just be interested what BMI’s unicyclist have

Q-factor I’d not thought of and could make quite a difference though you’d probably have to take hip width into consideration.

Short? Long? Whatever size you consider as short or long doesn’t matter with regards to a formula.

Speed? Cadence? These may be determining factors for crank length choice but I believe one size does not fit all people in a particular situation, which is what the formula idea is about. Rider 1 may be as fast as rider 2 with the same wheel size and the same situation but have different crank sizes and different body sizes. So have their body sizes contributed to their preferred crank size? Thus does it work the other way, know the body size choose the cranks?

Preferred crank size after experimentation is the key for the data. As has been pointed out it wouldn’t be very informative if someone had ridden with nothing but what came with their unicycle (even if it turned out to be the ideal length). Which leads to - wouldn’t it be better if you had an idea what crank length would suit you when you first bought a particular unicycle? One seat tube length doesn’t fit all, and although more complex, why should crank length?

Muni terrain could well be very difficult to quantify and categorise, any ideas?

Oh and am I thinking too much

Work and Feel of Wheel and Crank Geometry

Having been bred an engineer, I spent some time thinking about this and I offer the following. What we want to know is how will a given crank on a given wheel feel in terms of power and control. Where feel means many things as mentioned in the strings referenced–control, speed, torque, rotational diameter, power, knee pain…etc.

Work equals force times distance. For different wheel sizes on different rims to feel the same, will essentially require that the rider be accomplishing the same amount of work.

Each wheel size has a different diameter, when circumference is calculated…Pi times diameter, you can obtain a number for each wheel size. Then by multiplying by 2.54 you can determine the circumference in centimeter.

20 in diameter times Pi equals 62.80 in or 159.51 cm
24 in diameter times Pi equals 75.36 in or 191.41 cm
26 in diameter times Pi equals 81.64 in or 207.37 cm
29 in diameter times Pi equals 91.06 in or 231.29 cm
36 in diameter times Pi equals 113.04 in or 287.12 cm

By accomplishing a ratio of circumference divided by crank length in centimeters, you can determine a ratio for comparison. Using this approach each crank length and wheel diameter can be compared in a grid. I hope the chart works on the internet…

Wheel – 20 24 26 29 36

Crank

85 – 188 225 244 272 338

110 – 145 174 189 210 261

114 – 140 167 182 203 252

125 – 127 153 166 185 230

140 – 114 137 148 165 205

150 – 106 127 138 154 191

165 – 97 116 126 140 174

170 – 94 113 122 136 169

175 – 91 109 118 132 164

These numbers provide a referenced baseline for the work of a given crank/wheel combination for the same distance output. Similar numbers between columns will represent similar feel in terms of unicycle performance–given that each of us has a different comfort level and range of comfortable skill. Note that a 20 inch with a 125 crank gives a rating of 127…which compares in terms of work and feel to a 24 inch with a 150 (similarly a 127 work rating). Likewise a 26 inch with a 165 crank should feel about the same in terms of work but recognizing that the larger rotation of the longer cranks has a subjective impact.

As I look at it, the ratio of around 125 to about 175 covers the range of most riders comfort zones. The higher number being harder to pedal with a smaller pedal arc and the opportunity for greater speed. The lower number makes it easier to pedal a give distance due to increases lever arm resulting in probably lower speeds due to larger pedal arcs. As the crank to wheel size ratio goes down to about 100, the torque skyrockets and traction probably becomes the limiter…although large pedal arcs may be less comfortable. In the opposite direction, as the ratio goes up…the opportunity for speed due to shorter pedal arcs increases with a penalty in power and hill climbing ability.

Note that AspenMike, who as I recall, rides 175s on his 36 inch Coker, carries a tougher pedal stroke than someone who rides a 20 inch unicycle with 110s–making his performance on the Iron Horse all the more remarkable. This chart should give riders the chance to decide how cranks compare on different wheel sizes–recognizing that every one of us will have a comfort zone on this chart of varying width based on skill and strength.

I bought a 24 inch Torker LX with 150 cranks (work ratio 127) and quickly went to 114s (work ratio 140) which are more comfortable in my riding situation. On the KH29XC I bought with 29 inch wheel and 150 cranks (work ratio 154), I also want to optimize for speed so I bought a set of 125 cranks (work ratio 185). By comparison, the 29 inch with 150s starts me at a higher work ratio than my previous riding situation (harder cranking, less control, less power) and my decision to learn to ride the same unicycle with only 125s is a commitment train to even less control and power but this is offset by the ability for a better spin and higher speed.

Hope this helps. Any subjective opinions on the field application of this rough science is welcome.

Carey

Work and Feel of Wheel and Crank Geometry

Having been bred an engineer, I spent some time thinking about this and I offer the following. What we want to know is how will a given crank on a given wheel feel in terms of power and control. Where feel means many things as mentioned in the strings referenced–control, speed, torque, rotational diameter, power, knee pain…etc.

Work equals force times distance. For different wheel sizes on different rims to feel the same, will essentially require that the rider be accomplishing the same amount of work.

Each wheel size has a different diameter, when circumference is calculated…Pi times diameter, you can obtain a number for each wheel size. Then by multiplying by 2.54 you can determine the circumference in centimeter.

20 in diameter times Pi equals 62.80 in or 159.51 cm
24 in diameter times Pi equals 75.36 in or 191.41 cm
26 in diameter times Pi equals 81.64 in or 207.37 cm
29 in diameter times Pi equals 91.06 in or 231.29 cm
36 in diameter times Pi equals 113.04 in or 287.12 cm

By accomplishing a ratio of circumference divided by crank length in centimeters, you can determine a ratio for comparison. Using this approach each crank length and wheel diameter can be compared in a grid. I hope the chart works on the internet…

Wheel – 20 24 26 29 36

Crank

85 – 188 225 244 272 338

110 – 145 174 189 210 261

114 – 140 167 182 203 252

125 – 127 153 166 185 230

140 – 114 137 148 165 205

150 – 106 127 138 154 191

165 – 97 116 126 140 174

170 – 94 113 122 136 169

175 – 91 109 118 132 164

These numbers provide a referenced baseline for the work of a given crank/wheel combination for the same distance output. Similar numbers between columns will represent similar feel in terms of unicycle performance–given that each of us has a different comfort level and range of comfortable skill. Note that a 20 inch with a 125 crank gives a rating of 127…which compares in terms of work and feel to a 24 inch with a 150 (similarly a 127 work rating). Likewise a 26 inch with a 165 crank should feel about the same in terms of work but recognizing that the larger rotation of the longer cranks has a subjective impact.

As I look at it, the ratio of around 125 to about 175 covers the range of most riders comfort zones. The higher number being harder to pedal with a smaller pedal arc and the opportunity for greater speed. The lower number makes it easier to pedal a give distance due to increases lever arm resulting in probably lower speeds due to larger pedal arcs. As the crank to wheel size ratio goes down to about 100, the torque skyrockets and traction probably becomes the limiter…although large pedal arcs may be less comfortable. In the opposite direction, as the ratio goes up…the opportunity for speed due to shorter pedal arcs increases with a penalty in power and hill climbing ability.

Note that AspenMike, who as I recall, rides 175s on his 36 inch Coker, carries a tougher pedal stroke than someone who rides a 20 inch unicycle with 110s–making his performance on the Iron Horse all the more remarkable. This chart should give riders the chance to decide how cranks compare on different wheel sizes–recognizing that every one of us will have a comfort zone on this chart of varying width based on skill and strength.

I bought a 24 inch Torker LX with 150 cranks (work ratio 127) and quickly went to 114s (work ratio 140) which are more comfortable in my riding situation. On the KH29XC I bought with 29 inch wheel and 150 cranks (work ratio 154), I also want to optimize for speed so I bought a set of 125 cranks (work ratio 185). By comparison, the 29 inch with 150s starts me at a higher work ratio than my previous riding situation (harder cranking, less control, less power) and my decision to learn to ride the same unicycle with only 125s is a commitment train to even less control and power but this is offset by the ability for a better spin and higher speed.

Hope this helps. Any subjective opinions on the field application of this rough science is welcome.

Carey

Sorry

Didn’t mean to put that in twice.

Carey

As you said, there have been many, many threads discussing crank length throughout the years here on this forum. My personal belief is that there is no way to quantify all of the variables involved and derive a useful formula for this.

I looked at the site you listed about bicycle crank length, and to be honest, I disagree with the writer’s basic premise even for a bicycle. If I were to follow his formula, I would have 188mm cranks on my bike. I can’t imagine how uncomfortable that would be even if I could find a crankset that long! I use a 165mm crankset on my fixed-gear, and 170mm on my commuter. I often wish I could find a crankset with much shorter cranks to experiment with on my bicycles. On a bicycle, leverage isn’t nearly as big a factor as it is on a unicycle, since gearing is easily manipulated.

A standard unicycle always has a 1:1 gear ratio. Every time your feet make a circle, the wheel makes a circle. The only way to manipulate gear/inches is to change the size of the wheel.

Optimum crank length is based more on individual strengths and preferences. I prefer fairly short cranks on my unicycles, but many stronger riders prefer even shorter cranks. I use 125mm on my Coker, and 110’s on my 29er. I don’t have a 24x3 muni anymore, but when I did I discovered that I didn’t like the 170mm cranks that came on it, and that I could ride it much better using 150’s. With 170’s I felt like my knees were trying to hit me in the chin!

This is just one old mans humble opinion, but I don’t think that there is any formula that could determine an ideal crank length for a unicycle or a bicycle. Individual preference, strength, skill level and the terrain that you ride determine what crank length is “right”.

Chuck

Rough Science

Chuck,

Your preferences chart out at 210 and 230, the top right portion of the chart–in my book a power rider! A work ratio of over two hundred looks to me like it would take strength as well as skill.

Carey

Roger from unicycle.com UK worked up a spreadsheet a few years ago somewhat along those same lines, Carey. I don’t know if it is still available on his website, but it is fun to play with. It calculates the relationship between wheel size, crank length, speed, cadence and footspeed.

I’ve got it somewhere, I’ll try to find it if you are interested.

Chuck

Re: Deriving a unicycle crank length formula?

On Sat, 24 Sep 2005 18:08:19 -0500, “Carey” wrote:

>For different wheel sizes on
>different rims to feel the same, will essentially require that the rider
>be accomplishing the same amount of work.

Using such a table is a fruitful approach, and it is useful, in that
if I look up to different set-ups with the same work ratio and imagine
riding them, they would indeed feel about the same. However, I don’t
buy the logic on which the table is based (i.e. your copied statement
above). In the first place, the actual work done is larger than you
assume, because both legs exert force on the pedals, and moreover it
happens in a sort of erratic way (in the time domain), as required to
keep balance. But more importantly, I think that a skilled rider
cruising along a horizontal path will not be bothered about power, as
the required power output is generally lowish. He’ll be more concerned
with control and log movement.

It now happens that if you take the reciprocal of each “work ratio” in
your chart, you end up with numbers which are proportional to what
others have called “gearing ratio”. One definition of gear ratio (by
Mikefule I think) is crank length divided by wheel radius, both in the
same length units. E.g. for a 20" unicycle with 5" cranks (a very
common combination), the gearing ratio is 0.5. For a 24" unicycle with
6" cranks, the gearing ratio is also 0.5. And indeed (I think) these
two combination feel about the same, or let’s say, you can’t make them
feel more similar by changing cranks. The reason for that, I think, is
that gear ratio determines how easy a unicycle can be controlled, and
not so much that the required power is the same. This breaks down at
the point where too high crank lengths make pedaling uncomfortable in
itself.

All in all, I think that gear ratio would be a good indicator of
unicycle “feel” with various wheel sizes and cranks. Also, the numbers
nicely vary between 0 (pedals at axle like a BC wheel) and 1 (pedals
at circumference of wheel), and are therefore easily understandable.

## Klaas Bil - Newsgroup Addict

“Unicycling is like glue: you have to stick with it, and it’s not to be sniffed at - Mikefule”

Work and Feel

Klaas Bil,

I think you are right. The work ratio I mention is an imprecisely defined approximate for comparison reasons only, with the ratio numerator and denominator randomly selected but consistently applied. The logic of a 0 to 1 scale is a good approach. In fact, it sounds like the gear ratio you mention is a similar ratio that works from only diameter and crank length…which makes sense since Pi is a constant in the equation. It would be a comparable estimate to my effort–and I notice, makes the same point about a 20 with 125s and a 24 with 150s.

All this data aside, what I know is that the 150s on my 24 are very powerful and precise but too much arc, the 114s made it smooth and fast, and I can still control it well. The 150s on the 29 feel good now but as I get better on it I will move to 125s. Kind of scary as I UPD’d off the front last night at pretty high speed and was barely able to keep my feet.

I am looking for speed that makes me immune to mosquito attacks here.

Chuck–If you can find that chart, I’d be interested in it. Sounds like you guys have thought about this in depth before. My approach was to give me a feel for where I would start with cranks on different wheel sizes, given that I knew what I liked on one.

Carey

Re: Deriving a unicycle crank length formula?

On Sun, 25 Sep 2005 09:49:50 -0500, “Carey” wrote:

>Chuck–If you can find that chart, I’d be interested in it. Sounds like
>give me a feel for where I would start with cranks on different wheel
>sizes, given that I knew what I liked on one.

I’ve seen Roger’s chart (it was a spreadsheet) but not saved it. I
have made one in Dutch that is roughly similar - you can have it if
you want. But you can make one yourself easily, the calculations
involved are very simple.

## Klaas Bil - Newsgroup Addict

“Unicycling is like glue: you have to stick with it, and it’s not to be sniffed at - Mikefule”

That spreadsheet of Roger’s is useful and it’s here;

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

Crank chart

Thanks for posting the chart. I will spend some time looking at it. His inclusion of foot speed for a given velocity is a signifianct additional piece of information. There is a lot one can play with in these calculations–but the output all depends on what you select as constants. Interesting!!

Carey

I think that’s interesting as an idea, but I’m not entirely convinced maximum speed is anywhere near that maximum torque limit, except maybe for incredibly short cranks like 40mm ones, or in very high winds. I think maximum speed is usually about ability to keep feet on the pedals, rather than the torque you can exert. In a strong headwind (20mph+) on a big wheel, this effect probably does come into play though, I know of coker riders who’ve changed to longer cranks in windy weather and found themselves riding faster into the wind.

Joe

It may be misleading to speak of “short” or “long” cranks because these words carry assumptions about what a “normal” length is. It is safer to speak of changing to “shorter” or “longer” cranks, relative to what you are using now.

My own experiments have suggested that there is some truth in what was half-jokingly called the “Constant Footspeed Hypothesis”. That is, at a given level of effort, your feet will move at a given speed. If the pedalling circle is smaller, the feet will do proportionately more revolutions in a given time.

However, the CFH is only a long winded name for a genral principle - and one that does not stand up to rigorous scrutiny.

For a small change in crank length (e.g. 10%) the change in unicycle top speed will be approximately 10%. It might be 9%, it might be 11%, but it will be about 10%.

For a larger change, the “error” increases, and this seems to me because it takes the rider a while to adapt, and because levels of skill vary.

The CFH is easily refuted by reduction to the absurd: how fast would you go on 10 mm cranks?

As for longer cranks giving more torque, there is a certain amount of mythology about this. It is rare in normal riding to stop on a hill because it is literally impossible to propel the unicycle up the hill due to lack of torque. It is usually lack of stamina or lack of skill.

In MUni, it is common to come to sections which are literally too steep to ride. However, at these angles of gradient, there is another major factor to consider: the displacement of the contact patch of the tyre, relative to the hub. On the flat, the contact patch is almost exactly under the hub (slightly behind it, really). On a vertical cliff (for the sake of argument) the contact patch would be level with the hub, exactly in front of it. As a hill gets steeper, the contact patch gradually works forwards (backwards, if descending).

This contact patch is a fulcrum of sorts. If your cranks are so short that the centre of the pedal spindle is on the wrong side of the contact patch (i.e. away from the hill) then you will not be able to power the unicycle up the hill. (You may have sufficient momentum to continue to move up the hill, briefly, and it is theoretically possible to get a small amount of upwards acceleration by pulling on the seat, but the effect will be minimal._

And, depending on the size of the wheel, the length of crank needed to “reach past” the contact point wiill change.

For normal riding on flattish terrain, a simple comparison of crank length to wheel diameter (or radius) will suffice for approximate predictions of how two unciycle will compare. I find it easier to use wheel radius. It makes sense to use common units.

So, a 24 inch uni has a 12 inch radius. Put 6 inch (150 mm) cranks on it. The crank:radius ratio is 1:2, which can be expressed as 50%

Now take a 20 inch uni (10 inch radius) and put 5 inch (125 mm) cranks on it. Again, the ratio is 1:2 or 50%. The prediction is that the two unis will behave broadly the same interms of top speed and average speed. Experiment broadly bears this out.

But put 9 inch cranks on a Coker? It will behave very differently indeed. So already, our neat mathematical theory falls apart.

Go back to the 24 with 6 inch cranks. Now put 5 inch cranks on it. The maths says we should go about 20% faster. (Not 17% - think about it.) Experience shows we go roughly 20% faster. However, if we go straight from 150s to 89s, experience shows we wobble about a bit then fall off.

Then look at changes to the rest of the uni. By reducing rotating weight on my 700c (lighter tyre), I have made it possible for me to ride comfortably on 102s instead of 110s. If I put a heavy tyre and tube on, I’d go back to 110s, no doubt.

I have noticed a general increase in speed and acceleration, but I have laso noticed that the uni will “stall” a little bit more easily on short very steep hills. On the other hand, it will “rush” short hills better.

Too many variables for a useful formula. This is a case for fuzzy logic and experience. Any attempt at mathematical precision will be illusory.

Re: Deriving a unicycle crank length formula?

On Mon, 26 Sep 2005 11:00:24 -0500, “Mikefule”
<Mikefule@NoEmail.Message.Poster.at.Unicyclist.com> wrote:

>On the flat, the contact patch is almost exactly
>under the hub (slightly behind it, really).

Why is it “slightly behind”?? Mathematically, I think it should be
EXACTLY under the hub. Unless maybe you take tangential deformation of
the tyre into account, and/or spoke stretch, and/or indenting the
riding surface, but come on.

>This contact patch is a fulcrum of sorts. If your cranks are so short
>that the centre of the pedal spindle is on the wrong side of the contact
>patch (i.e. away from the hill) then you will not be able to power the
>unicycle up the hill.

That is correct only if you don’t pull on the seat. If you do though,
you could theoretically even climb your vertical cliff. You would
however have to have some supernatural friction between tyre and
cliff, and be very strong too.

## Klaas Bil - Newsgroup Addict

“Unicycling is like glue: you have to stick with it, and it’s not to be sniffed at - Mikefule”

Slightly behind. To ride a unicycle, you allow it to start to fall forwards, then you accelerate the wheel so that the uni never actually falls. When you slow down, you allow the wheel to overtake you, slightly.

The better the rider, the more control, the smoother the ride, the more vertical the relationship between the hub and the contact patch. However, if the contact patch were exactly beneath the hub, you could never accelerate - including from a standstill.