The combined effect of unicycle wheel diameter, crank length and hub gear ratio

by Klaas Bil

On a typical bike, the gearing choice is made 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 both). Crank length in bikes usually doesn’t vary that much: most cranks for adult bikes are around 170-175 mm. Because of this, bicyclists usually disregard the effect of crank length on gearing, and express their gearing as “gear inches”. This is to be understood as a virtual wheel diameter, in fact it is the physical diameter of the wheel times the gear ratio(s) in the hub and/or chain drive.

On a conventional unicycle there is no such thing as gearing in the hub or chain drive. Still, unicyclists have always had some sort of “poor man’s gearing” by appropriately choosing wheel diameter and crank length. Imagine a hypothetical 24″ unicycle (24″ refers to the wheel diameter) with 12″ long cranks. In this strange unicycle the crank length is equal to the wheel radius, i.e. the pedals are at the outside of the tyre. Riding this uni would be somewhat like walking with your feet going in a big circle. There would be no gearing at all, just as there is no gearing in a walking person. In other words, the gearing ratio in this situation is 1.00. Now if you would replace the 12″ cranks with more practical 6″ cranks (a fairly common setup), the gearing ratio would become 2.00. This simply means that for every one inch you move your foot, the unicycle moves 2.00 inches. (Want proof? When the pedal moves a full circle with a radius of 6″, the unicycle has progressed one circumference of a circle with a 12″ radius which obviously is twice as far. )

The gear ratio of this unicycle can be further increased, e.g. to 3.00, in two ways:
·    replacing the 6″ cranks with 4″ cranks (keeping the wheel at 24″), OR
·    replacing the 24″ wheel with a 36″ wheel (keeping the cranks at 6″).

All of the above is in fact old hat, as it refers to so-called fixed(*) unicycles. Over the last few years, geared(*) unicycles have become more common. Therefore, it is time to expand the gear ratio concept to include the effect of a geared hub(*). For this we simply multiply the wheel/crank gear ratio with the gear ratio of the hub, and we call the result the Total Gear Ratio (TGR). The TGR is the number of inches the unicycle travels when the pedaling foot has moved one inch in the pedal circle.

With this, we have a third way of increasing the gear ratio of the above-mentioned 24″ unicycle with 6″ cranks: we fit a geared hub with a 1.5 : 1 ratio (the commercially available Schlumpf unicycle hub is approximately this ratio). So now we have three different unicycles:
·    24″ wheel, 4″ cranks, fixed hub
·    36″ wheel, 6″ cranks, fixed hub
·    24″ wheel, 6″ cranks, 1.5 : 1 hub
These three unicycles have an objective criterion in common: their TGR is 3.00. Now the amazing thing is that because of their equal TGR, the “feel” of these unicycles is quite similar. This similarity applies to aspects such as accelerating, hill-climbing, balance corrections, or even the difficulty of freemounting or just riding. That is the amazing power of the TGR concept. Obviously, these three unicycles are quite different, and will feel different in some respects. Nevertheless, TGR is useful to compare widely different unicycles.

TGR and speed
Some people might be tempted to think of the TGR as a factor in an exact speed equation. This would imply that if you switch to an x times higher TGR, you would ride x times faster. While this idea has some merit, especially if one stays within “reasonable” values for TGR and crank length, there are many non-linearities involved having to do with the physical ability to push a certain gear, fear of falling holding you back at higher speeds, maximum cadence, non-optimality of crank length with respect to leg length, etc. In addition, all of these are rider-specific.

Having said that, if you are equally “at home” on two unicycles, and their TGR is about the same, then the speed you can achieve is probably similar. If one of the two has a higher TGR, then you can probably get a higher speed on that unicycle, but by how much remains to be seen.

More safely, therefore, one could think of the TGR as a factor in an equation for relative speed potential. In other words, if the TGR of one unicycle is 10% higher than that of another, it is potentially 10% faster. To what degree the potential will be realised depends on the rider.

How to calculate TGR
TGR is a dimensionless number, defined as 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. Note that wheel radius is used, which is half of the wheel diameter.

Finally, by way of example, Total Gear Ratios calculated for some popular unicycle setups:
·    Fixed 36″ Coker, 114 mm cranks: TGR = 4.01
·    Schlumpf-geared(*) 36″ 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 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 diameter with the components that you have on there. In these 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″.

—————————————————————————————————-
(*) Footnotes /definitions
·    “Fixed” means that the cranks are directly connected to the hub and wheel, and rotate at the same angular velocity.
·    “Geared” means that when the cranks rotate one revolution, the wheel rotates more than one revolution. This is actually “geared up”. Theoretically, a unicycle could also be “geared down”, but to my knowledge this has not been built.
·    Note that there are other gear systems than internally geared hubs (also for unicycles), but for the principle it doesn’t matter.
·    Hub gear ratio is the number of revolutions that the wheel turns, if the crankset goes through one full revolution. In the calculated examples, I used the exact Schlumpf gear ratio of 17/11 (equal to about 1.545), as opposed to the rounded value of 1.5.