# Deriving a unicycle crank length formula?

Re: Deriving a unicycle crank length formula?

On Tue, 27 Sep 2005 01:17:09 -0500, “Mikefule” wrote:

>However, if the contact patch were exactly beneath the hub, you could
>never accelerate - including from a standstill.

Ah no, the contact patch is still “exactly” beneath the hub (on a
horizontal riding surface). At least seen from the side (in 2D
projection). What you mean is that the contact patch is slightly
behind the centre of gravity. The CoG (of a uni plus rider) is not
located at the hub but rather more near the seat, probably somewhat
higher.

## Klaas Bil - Newsgroup Addict

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

You are exactly right and I was terribly wrong.

I am now lying face down on the floor, throwing dust over myself as an act of repentance.

The contact patch is under the hub on the flat. It is behind the centre of mass.

On a slope, the contact patch is no longer directly under the hub.

Somewhere, my feeble brain lost the subtle connection between reality and what I was writing.

:o :o :o :o :o

What I mean is:
Think about a motorbike or a car. If you have a low gear (eg long cranks) then you will have a lot of torque and be able to reach your top speed quite quickly- lots of power, but when you reach top speed you will start revving out and the motor will be suffering from extra wear and tear from the high revs- using more petrol than lower revs. Then if you use a different gear (eg shorter cranks) the motor will take slightly longer to reach max speed, but the max speed at reasonable revs will be higher than that reached with the lower gear- so using less energy to maintain a higher top speed. I think the same applies to unicycles to some extent. With the short cranks- top speed is not far away from no speed, so the time taken to reach top speed is not an issue. The power thing- Klaas mentioned it- is also not often an issue for me. Most often when I fall off climbing a hill, it is not because of lack of power, it’s been because the power is too much and the contact area does not get enough grip and I slip off. Either that or I hit a bump when my pedals are vertical- the extra momentum from the higher speed can help get past that, but those bumps can happen regardless of crank length- just gotta be more skilled in approaching potential UPD causes.

That’s not a bad analogy, but it’s imperfect.

A closer analogy is not to look at gears, but at the engine itself.

(1) A petrol or diesel engine has a cyclinder.
(2) Fuel and air in the cyclinder mix and burn.
(3) The explosion pushes a piston along the cyclinder.
(4) The piston is connected by connecting rods to a
(5) Crank, which spins a
(6) crankshaft and flywheel.

On the uni
(1) You have leg muscles
(2) These burn carbohydrates and oxygen. This powers a contraction of the muscles which…
(3) Pushes the lower leg downwards
(4) Which is connected by the foot and pedal to
(5) A crank, which spins…
(6) The axle and therefore the whole wheel.

This is slightly distorted tomake the analogy more plain than it would be if i kept it literally true.

In an engine, there are two general rules:
(1) Short cranks mean high revs.
(2) Long cranks mean high torque.

In an engine, the high revs are possible because the weight of the piston, conrod and crank have to move less distance with short cranks. However, at high revs, it’s difficult to get enough air and fuel into the cyclinder, which is why you need multiple valves.

In an engine, long cranks give more torque because the explosion has more time and distance to push that piston and exert force on the crank.

Again, I simplify, and engineers will wince.

The analogy with the unicycle:

Short cranks: the moving parts of the leg, foot, pedal and crank have less distance to travel. Every time a moving object changes direction, that takes energy. However, with really short cranks, you may only be using a small part of your muscle, and struggle to keep the carbohydrate/oxygen supply adequate.

With long cranks, all the moving bits have to move further, but you are using more of your leg muscle, so you are using your muscles more efficiently.

Now some of us have petrol legs, and some have diesel legs. Some have four valves per leg, and some have only two. So the type of legs you have will have some effect on what is the ideal crank length for you.

If you are young and fit, with lots of “sprint” muscle fibres, you may find it easy to maintain high revs with short cranks.

If you are older, or more of a stamina person, you may find it easier to plod away at low revs for long periods.

As the amount of force that tired muscles can exert is fairly limited, it’s easier to pedal smoothly at a steady speed than it is to keep accelerating. So if you reach the stage where you “stall” after each pedal stroke, and have to start the wheel going again (e.g. on a very steep or rough hill) then you will need long cranks so that what little force you can apply acts through a long lever to create plenty of torque.

If you are able to keep those cranks spinning smoothly, and you have “fast burn” legs, then you may find that the tight, focussed movement allowed by small cranks will let you power up the hill where you might find yourself flagging with long ungainly cranks.

If you are Aspenmike, you will just ride up the hill.

I’m not striving to be perfect- just tried to explain what I was thinking when I wrote it. My “short cranks are so and so length in this discussion” was not meant to determine other peoples generalisations, but so people wouldnt misinterpret my comment to mean rediculously short cranks like 2mm, but I chose my words poorly in that case.

Your way of simplifying seems to have become more complicated . I reckon I am young, not especially fit, and sprinting isn’t my strongest point, but my legs hardly ever get tired from unicycling unless I go for hours, cos unicycling has been my main form of transport for 9 years and the muscles must be getting used to it by now. Usually I suffer from saddle soreness before my legs tire out. I don’t think it’s a problem to get enough air and feul into my cylinders (legs) cos unicycling uses such a small amount of energy- it’s a pretty efficient way to travel if you don’t mind waiting a bit to get there!

Edit I forgot to mention that I find long cranks best on steep downhills, where you need the torque to brake and maintain control, where sometimes the short cranks get out of hand. The energy from the flailing of the legs is subsidised by the potential gravitational energy being converted into momentum means that you can spin fast without spending too much energy to maintain high rpm. On the flat or uphills the reality of the big circles kicks in and slows you down.

By imperfect, I simply meant incomplete. It wasn’t meant as an insult. Sometimes I use fairly sterile language like that and it can be misinterpreted. Sorry.

I certainly agree that long cranks really come into their own on steep descents. That is their biggest single advantage.

This is a mixture of truth and fiction. You are correct that the contact patch is slightly behind the center of mass when you are accelerating. On the other hand, when you are decelerating the contact patch is in front of you. It has to be or else you fall forwards. And, guess what? If you are maintaining a constant speed the contact patch is exactly beneath the center of mass.

In reality you are never maintaining a perfectly constant speed. You constantly waver between accelerating and decelerating, and otherwise wobble, but it averages out to beneath you, with expert riders keeping the contact patch consistently closer to under the center of mass.

I think you’ll find that at constant speed on a perfectly smooth level surface, the centre of mass is still very slightly forward of the contact patch.

For a simple demonstration, balance a broom on your finger and walk with it.

However, I agree that the better the rider, and the smoother the surface, etc., the more constant the position of the centre of mass relative to the contact point. A good rider is not constantly accelerating and decelerating, but exactly matches his speed to counteract the downwards deceleration of the centre of mass. It’s a bit like an object in orbit, constantly falling but never landing.

Re: Deriving a unicycle crank length formula?

On Thu, 6 Oct 2005 01:17:43 -0500, Mikefule wrote:

>I think you’ll find that at constant speed on a perfectly smooth level
>surface, the centre of mass is still very slightly forward of the
>contact patch.

I think you’re right, but you don’t state the reason.

>For a simple demonstration, balance a broom on your finger and walk
>with it.

Balancing a broom on your finger and then walk with it is relatively
easy indeed. Nevertheless, it’s not a simple demonstration. How can
you “simply” demonstrate an effect that is “very slight”?

The reason for the contact patch being forward (both in unicycling and
in the broom case) is air resistance. This effect is probably too
small to discern casually by the naked eye. If you think you see it,
then either your observation is distorted because you focus on the
combination of tasks, or you are still accellerating overall.

## Klaas Bil - Newsgroup Addict

“dit dit diddle diddle dit dit did-it, dit dit diddle diddle dit dit did-it, dit diddle dit dit dit diddle dit dit, diddle-diddle-diddle-diddle-dit dit diddle diddle dit dit did-it,… - Spudman”

Klaas Bil, I think I was wrong. I think you are right in that, in the absence of air resistance, it would be possible to ride at a steady speed without leaning forwards.

If you could breathe. ;0)

Re: Deriving a unicycle crank length formula?

In message
<8e0862740723911d209e5f1f07fd89e0.1wjjmm@NoEmail.Message.Poster.at.Unicyc
list.com>, Mikefule <Mikefule@NoEmail.Message.Poster.at.Unicyclist.com>
writes
>
>Klaas Bil, I think I was wrong. I think you are right in that, in the
>absence of air resistance, it would be possible to ride at a steady
>speed without leaning forwards.

Only in the absence of rolling resistance and friction in the bearings.
In that case, one could take ones feet off the pedals and continue in
uniform, rectilinear motion until acted on by a force…

Martin/

Martin E Phillips nb Boden, Splatt Bridge
http://www.g4cio.demon.co.uk martin/at/g4cio/dot/demon/dot/co/dot/uk
Homebrewing, black pudding, boats, morris dancing, ham radio and more!
The Gloucester-Sharpness canal page http://www.glos-sharpness.org.uk

I think that’s probably right. Yes, Klaas was incomplete in his answer.

If a force is tending to slow down the unicycle, and the unicyclist wants to maintain a steady speed, he will exert a force on the pedals to turn the wheel. Newton’s law of equal and opposite reaction will then make the centre of mass move in the opposite direction. Therefore there needs to be a tilt forwards to allow the effect of gravity to counterbalance this tendency for the entire unicycle frame and rider to rotate in the opposite direction to the wheel.

So, although I hadn’t got all the details right in my head,I think my original intuition was right after all.

Re: Deriving a unicycle crank length formula?

On Fri, 7 Oct 2005 18:55:18 +0100, Martin Phillips wrote:

>Only in the absence of rolling resistance and friction in the bearings.

I do agree that my analysis was incomplete. Let’s go through some
other effects that push you off-vertical and that you have to
compensate for in order not to fall…

Like Mikefule hints at, friction in the bearings would actually cause
the centre of mass to be BEHIND the contact patch, to compensate for
the fact that the turning wheel tends to push the frame forward. So
with the “right” amount of friction in the bearings to compensate for
air resistance at a certain speed, one would still be upright - apart
from other effects mentioned below.

Rolling resistance, however, acts in the tyre-wheel system and would
have no effect on the uprightness of the unicycle.

Celestial bodies, such as the moon, will attract you (gravitational
force), like they do attract ocean water and cause tides. In fact,
every body of mass will attract you: a mountain, a fellow unicyclist,
a mosquito. And density variations in the Earth, although this depends

If there is more light coming from one side, e.g. if you are riding
towards a car’s headlights, then the light falling upon you will tend
to push you off-vertical. If you have a headlight yourself, it will
exert a reaction force upon you.

If you are not riding exactly towards the North or the South, the
turning of the Earth will “push” you towards the equator.

If you breathe out, the reaction force will push you backwards. This
is different from air resistance, it also works in vacuum - though
breathing out would be a different experience there. A similar story
holds if you breathe in, except that this is not possible in vacuum.

I’d rather stop now

## Klaas Bil - Newsgroup Addict

“dit dit diddle diddle dit dit did-it, dit dit diddle diddle dit dit did-it, dit diddle dit dit dit diddle dit dit, diddle-diddle-diddle-diddle-dit dit diddle diddle dit dit did-it,… - Spudman”