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.
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.