# the math to make 29" = 36"

I know someone has provided some sort of link at one time…I’d like to see it, and discuss it here too.

How can you make this a true equation: 29=36

Crank size matters:

When is 29" equal to 36"?

Would, say, 4" cranks on a 29" go as fast/easy as, say, 6" cranks on a 36" Coker? Could the two Unis take the same trip with the same speed and effort?

I’m pretty good in math, but I’ll need the initial explanation in laymans terms (then I’ll try NASA Engineering terms)

What’s the best formula for these two Unicyclists to ride together?

One way to think of it is that the classical formula is Rate = Distance/Time. For the two unicyclists to travel at the same rate, they have to travel the same distance in one minute. The Distance travelled is one Circumference per Revolution, and the Cadence is Revolutions per Minute, so we get

So a unicyclist with a cadence of 90 revolutions per minute on a 36" wheel travels 848 feet per minute, or 9.6 miles per hour.

For two uniists to match rates (ground speeds), we have

Or:

So if uni1 is a 29 inch, and uni2 is a 36 inch,

So the unicyclist on the smaller wheel has to pedal 36/29 times the cadence of the one on the larger wheel, or 24% faster.

Trying to calculate effort as a function of wheel size AND crank length, though is much more complicated and involves terrain, technique, and even a person’s natural muscle construction and training style. During the 24 hours of Adrenaline in California, Kris Holm did the fastest lap on his 24", faster than others on their 36". http://www.unicyclist.com/forums/showthread.php?s=&threadid=18257&highlight=24+hours+of+adrenaline

Re: the math to make 29" = 36"

Put the faster one on the 29".

Did you want a mathematical solution? So often the theoretical solutions to the math problems just don’t work out where the actual riding takes place. Whatever you do to equalize the cranks between the two cycles, for it to be useful you’ll have to find two equal riders.

Otherwise, I’m sure you’ll get more useful results by experimenting and seeing what works. Anything you come up with in numbers is likely to only be disproven in the field.

Remember to switch riders if one guy is getting too far ahead of the other…

But with smaller cranks it should be easier to maintain a higher cadence rate for a longer time. This phenom would speed up the little guy and slow down the bigger guy.

I just thought there was a formula that would tell us, for example, 170mm cranks for the 36 (with a slower cadence) should ride alongside a 29 with 140mm crank arms (faster cadence).

from your explanations, I see that the two riders will appear to be acting differently. But are they expending similar energy?

Experimentation is the key. (and enjoyable in this case)

Thanks for the brainwork. Mud.

If you can figure out crank lengths for the 29er and the Coker so that when both riders are riding the same speed they have the same footspeed then the two unicycles are as equal as you are going to get them.

The spreadsheet will help you in figuring out crank lengths for the 29er and Coker that will make the Coker and 29er more or less equal.

In order for the 29er to keep up with the Coker the 29er will have a higher cadence. For riders like me who are not good at maintaining (or even achieving) super high cadences, it is difficult to consider the effort required on a 29er to be equal to the effort required on a Coker no matter what the math and the spreadsheet say.

All these relationships are linear so to obtain the same footspeed the 29 incher would use cranks that are 29/36 times the length of the 36 incher. So if the Coker rider used 125mm cranks the 29er rider would use the closest to 100mm that he/she could find, say 102.

Like those above and elsewhere have said, that is not, however, the whole picture.

Re: the math to make 29" = 36"

“U-Turn” <U-Turn.drs5n@timelimit.unicyclist.com> wrote in message
news:U-Turn.drs5n@timelimit.unicyclist.com
>
[snip]

> During the 24 hours of Adrenaline in California, Kris Holm did the
> fastest lap on his 24", faster than others on their 36".
> http://tinyurl.com/2isl

Sorry if you got that idea from my write-up. Kris rode Bronson’s Coker on
both of his laps. Here’s the whole list - I don’t think I ever posted this:

Lap # Rider Time Cycle
1 Scot 1:13:13 Coker
2 Kris 1:09:12 Coker
3 Nathan 1:15:46 Coker
4 Gary 1:26:24 Coker
5 Bruce 1:33:15 Coker
6 Rob 1:28:31 Coker
7 Carl 1:36:42 26" Pashley
8 Geoff 1:56:15 24" Telford
9 Bronson 1:59:23 KH24
10 Scot 1:43:45 24" Wilder
11 Kris 1:17:30 Coker
12 Nathan 1:28:40 Coker
13 Gary 1:22:47 Coker
14 Bruce 1:37:10 Coker
15 Rob 1:32:54 Coker
16 Carl 1:30:30 26" Pashley

Laps 8 through 11 were completely dark and laps 7 and 12 were partly dark.

On this course, Cokers win (assuming someone who has ridden a Coker as much
as a Muni).

—Nathan

Very, very interesting, Nathan. Thanks!

Were y’all using light sources, I presume? If so, do you mind telling us which?

Re: the math to make 29" = 36"

Each person riding starting from dusk to dawn had to carry a flashing red
rear light, a headlight (bike and/or helmet mounted), plus a spare light. We
all had different lighting systems, but of course all were helmet mounted. I
have a NiteRider Digital 12 volt system with 5 different power settings from
6 to 32 watts. 6 to 12 is all you need at unicycle speeds though. Others had
other brands - I don’t remember them all. You really do need a real light
though, a simple headlamp just doesn’t cut it. The tough part was that all
the suspended dust made depth perception virtually nil, so it was really
hard to see bumps coming.

Looks like we may do it again next May, and I’m considering entering in a
2-man team - am I crazy or what? Coker by day, Muni by night.

—Nathan

“U-Turn” <U-Turn.dsyr0@timelimit.unicyclist.com> wrote in message
news:U-Turn.dsyr0@timelimit.unicyclist.com
>
> Very, very interesting, Nathan. Thanks!
>
> Were y’all using light sources, I presume? If so, do you mind telling
> us which?

I’ve been thinking about the relationship between cranks length, wheel size and speed since this thread was started. I’m lucky to have 5 wheel sizes and 5 crank sizes to play with and too much time on my hands, so I’ve tried several combinations over the last few months.

The speed of the unicycle is governed by the cadence (rpm) and the wheel diameter. More diameter and/or more rpm = more speed.

The cadence is influenced by the length of the cranks. Broadly speaking, shorter cranks allow a faster cadence, so for a given wheel size, shorter cranks = more speed.

One hypothesis suggests that the foot speed is constant, and that this is why shorter cranks allow a faster cadence. By ‘constant’ I assume we are saying that the comfortable cruising footspeed remains constant across a range of crank sizes and/or the maximum possible footspeed remains constant across a range of crank sizes. Obviously, the footspeed varies depending on a variety of other factors such as terrain and technique. I do not think the constant footspeed hypothesis is completely reliable, but it is a reasonable guide.

The foot goes round in a circle, and the distance the foot has to travel to make one complete revolution is directly related to the crank length. (The circumference of the circle is Pi x 2 x the crank length.)

To aid imagination, think of the tyre on a treadmill: the tread of the tyre goes round in a circle. (When riding, the tread describes a more complex shape.) The distance travelled by the tread of the tyre is directly related to the radius of the wheel. (Pi x 2 x radius.)

As these are two measurements of distance, rather than area or volume, the ratio between them is a simple matter, with no squares or cubes or exponential growth to worry about.

Thus the ratio of crank length to wheel radius is exactly the same as the ratio of the distance travelled by the foot to the distance travelled by the tread of the tyre.

Now introduce time to the calculation. Distance travelled measured over a period of time = speed. The ratio of the speed of the foot to the speed of the tread of the tyre is exactly the same as the ratio of crank length to wheel radius.

This would suggest at first sight that for any given ratio, the speeds of two unicycles would be identical. By this I mean the comfortable cruising speed and/or the maximum possible speed.

For example:
A 20 inch uni with 5 inch cranks has a crank:wheel radius ratio of 1:2 or 50%.
A 24 inch uni with 6 inch cranks has a crank:wheel radius ratio of 1:2 or 50%.

Therefore, on the ‘constant footspeed hypothesis’ the two unis should cruise at a similar speed, and rev. out at a similar top speed. This seems plausible.

Now try this:
24 inch uni with 4 inch cranks = a ratio of 1:3 = 33%
36 inch uni with 6 inch cranks = a ratio of 1:3 = 33%
So these two unis should have a similar comfortable cruising speed and a similar top speed. My own experience is that this isn’t far off the truth, but in real life the larger wheel tends to have an advantage over any big distance - possibly because keeping up a high cadence requires more concentration.

So if we accept the ‘constant footspeed hypothesis’ then the speed of any unicycle (for a given rider) is directly related to the ratio of crank length to wheel radius. All you need to do is to convert the crank length into inches or the wheel radius into mm and divide the smaller number by the bigger number. The lower the result, the faster the uni.

However, there are many other factors to consider. I suspect many riders are less afraid of ‘revving out’ on a small wheel as there is less distance to fall. A big wheel has more momentum and takes more starting and stopping, so on a given ride, you will lose more time accelerating and decelerating on a big wheel. A big wheel will roll over small obstacles which would slow down a small wheel. On very rough ground a small wheel will keep plugging on when a big wheel might stall.

There is also ‘absolute’ crank length to consider, rather than relative crank length. When decelerating, you only put pressure on the rising crank for a small part of the circle - perhaps 90 degrees, maybe a bit more. The pedal may be moving at a ‘constant’ speed, but it is only in the ‘deceleration sector’ for a short distance, requiring more skill and muscle control. So ultra short cranks on a small wheel might put you at a disadvantage compared to a big wheel with cranks at the same ratio.

Re: the math to make 29" = 36"

On Fri, 8 Nov 2002 12:15:09 -0600, Mikefule
<Mikefule.dtajz@timelimit.unicyclist.com> wrote:

>I’ve been thinking about the relationship between cranks length, wheel
>size and speed since this thread was started. I’m lucky to have 5 wheel
>sizes and 5 crank sizes to play with and too much time on my hands, so
>I’ve tried several combinations over the last few months.
<BIG SNIP>
>So if we accept the ‘constant footspeed hypothesis’ then the speed of
>any unicycle (for a given rider) is directly related to the ratio of

That was a lot of words Mikefule but it was brilliantly clear. As to
the first quoted paragraph above: what were your findings? Did they
confirm the constant footspeed hypothesis?

Klaas Bil

All my posts are made with 100% recycled electrons.

Re: Re: the math to make 29" = 36"

:o Too kind. It’s difficult to balance length against clarity so there’s always something missed out.

I think the ‘constant footspeed hypothesis’ (CFSH) is a reasonable guide for reasonable circumstances. It is less accurate on broken or hilly ground where other factors (fear, absolute wheel diameter, skill, etc.) come into play.

So I think if you have a 24 inch uni and take off the 150s and put on 125s, the change in cruising speed and top speed will be consistent with the CFSH.

I think if you compare a 24 and a 28, the CFSH will give a good guide.

If you put 102s on a Coker, or 170s on a 24, you may find that the CFSH is less reliable. On the Coker, the fear factor would cut in; on the 24, the coordination factor would cut in.

Then you have to consider what you are really interested in measuring:
Absolute top speed.
Comfortable cruising speed on fast tarmac.
Average speed when riding on varied terrain.
Average speed over a journey, including rests, UPDs and time spent trying to remount.

Alan, Mark and I discussed some of this at the Unimeet. What is best for a ride like the Red Bull? A Coker (Alan said that remounting when tired was a major problem) or a 28/29 (Mark felt the Coker had better ‘rollover’ for small obstacles).

P.S. :o

Another factor which influences top speed:

A unicycle is a bit like an object in orbit. It is constantly falling towards the Earth. In the case of the satellite, the Earth curves away so the satellite never lands. In the case of a unicycle, the pressure on the pedals must counteract the downwards acceleration of the centre of mass caused by gravity.

In real life circumstances, the road is never completely flat and smooth. if the wheel hits an imperfection in the surface, the wheel is slowed down. This means that the rider falls further forward and MUST ACCELERATE to regain balance.

On a big wheel, the rider is higher off the ground. The leverage required to ‘push’ the rider back up to the balance position is therefore greater. If the rider is already going flat out, and using cranks which provide a very low leverage ratio (short cranks, big wheel) then the rider has less chance of regaining balance.

So on a tall unicycle, the tendency would be for a rider to hold back slightly from absolutely maximum speed. Likewise with very short cranks. The sensible rider leaves a margin for error. I think that the margin has to be wider on a taller wheel or with very short cranks, and particularly when the two factors are combined.

To this extent, I feel that the CFSH breaks down when considering maximum speed.

On the whole, I think bigger wheels make more difference to comfortable cruising speed, and smaller cranks make more difference to maximum speed. Others may have found different results.

True only if the perturbation (bump) is unexpected. In off-road riding, one partly expects unexpected bumps and a) holds back further in the balance envelope so that the perturbation leaves one in balance, and b) holds some weight off the seat so that the legs act more as shock absorbers, allowing the wheel to travel upwards easier and minimizing the slowing action. This is true too in road riding, but much more subtle. Since this behavior requires extra energy, as one gets tired or too-efficient, the accelerate-to-regain action becomes more prevalent.

This is what I have found too, especially with off-road. This is made even more so off-road because one lowers the seat for better legs-shock-absorption, which makes spinning at personally higher rates even harder. Longer cranks also have their own speed penalty, because the extra effort of throwing one’s legs mass around for a given speed increases.

I agree. These boundary conditions are messy.

I, too, have found that the 36" is harder to mount when I’m tired, although I must say that I’m not yet a very good 36" mounter. I did find that mounting the 36" was easier in all conditions (fatigue, slope, ground roughness) with 170mm cranks than with 150mm cranks.

Re: the math to make 29" = 36"

Mikefule,

Thanks for your further clarifications and expectations. However,
didn’t you say that you have tried several combinations over the last
few months? So do your /findings/ confirm the CSFH?

Klaas Bil

All my posts are made with 100% recycled electrons.

Re: the math to make 29" = 36"

On Sat, 9 Nov 2002 07:20:12 -0600, U-Turn
<U-Turn.durkb@timelimit.unicyclist.com> wrote:

>Longer cranks also have their own speed penalty, because
>the extra effort of throwing one’s legs mass around for a given speed
>increases.

IMHO this is not some /additional/ effect, but the very reason that
there is such thing as a constant foot speed hypothesis.

Klaas Bil

All my posts are made with 100% recycled electrons.

I’ve tried several combinations over the last few months. I’ve measured ‘results’ on an ad hoc basis with cycle computers. I have not set out to ride identical routes or anything like that.

My ‘findings’ are therefore not scientifically accurate, but where I have made measurements, they have tended to confirm the CFSH within certain broad limits.

A direct comparison of a Coker on 125s and a 24 on 102s, each ridden over an hour, suggested the CFSH was ‘accurate enough’ for top speed, but the bigger wheel gave the slight edge on average speed, with me riding.

There are many many variables, some of which are not susceptible to measurement (confidence, technique). I didn’t propose the CFSH - it is well established - but I did feel it was a reasonable ‘working hypothesis’ when considering the potential influence of changes in crank length and wheel diameter on certain aspects of speed.

My only answer to your very direct question, therefore, must be a slightly indirect one: “The CFSH is a useful guide to the influence of changes in wheel size and crank length.”

Derived from this, a direct comparison of crank length: wheel radius ratios for two unicycles will give a reasonably reliable indication of which will have the highest top speed for a given rider on a flat surface. However, given identical ratios, the larger wheel (and therefore more relaxed cadence) will tend to give a slightly higher average speed. On very difficult terrain, the medium sized wheel will probably be the best choice because of acceleration/deceleration time, UPDs and mounting.

This is somewhere between scientific and common sense and experience - what some scientists would call ‘a hand waving argument’.

I addressed the question in a fairly methodical manner because that is my style, because I was genuinely interested in the subject, and because I wanted to put forward a hypothesis (crank:wheel ratio is the clearest indicator of potential top and cruising speeds) to further the debate. If the hypothesis can be disproved or improved, I will be glad to learn something new from this - although I’ve been riding for donkey’s years, it is only a few months since I became aware of the possibilities for ‘tuning’ a uni in this way.

I’m not setting myself up as an expert here, and I’m well aware that many other people in and outside this forum have more experience and technical knowledge than I have.