Before the days of geared unicycles, our poor man’s gearing was in choosing other crank lengths and wheel sizes. Mikefule did some good research on this a few years ago and came up with the Constant Food Speed Hypothesis (CFSH). Maybe somewhat jokingly but he was onto something: if wheel size (and hub gear) are constant, then making the crank smaller by a factor x results in a speed increase by a factor x. In addition, making the wheel larger by a factor of y (and leaving crank length and hub gear ratio constant), results in a speed increase by a factor of y. Surely the CFSH model breaks down at extreme crank lengths or wheel sizes (both too large and too small), but it’s still a useful model.
In bicycles, crank length usually doesn’t vary that much. Most cranks for adult bikes are around 170 mm. Wheel size is somewhat of a variable, but indeed most of the gearing choice of bikes is done 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 hybrid systems). Because of the semi-constant crank lengths used in bikes, bicyclists express their gear ratio as “gear inches”, i.e. how far the bicycle travels with one full revolution of the cranks.
Now that unicycles with internally geared hubs are becoming more common, I’ve also seen unicyclists referring to their gear ratios in terms of “gear inches”. E.g. Chuck did this in his excellent post Future of Unicycle Road Racing (here’s what I think; what do you think?). However, in my opinion, “gear inches” is an incomplete measure for gear ratio on a unicycle, as it doesn’t take crank length into account.
I would like to propose a new concept to express gear ratios for a unicycle, and call it Total Gear Ratio (TGR). It is a dimensionless number, being 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.
For example, if you would have a hypothetical unicycle with a fixed hub, a 24" wheel, and 12" long cranks, then the TGR would be 1.00, because the cranks are equal to the wheel radius. If you would replace the cranks with more practical 6" cranks, then the TGR would be 2.00. (When the pedal moves a full circle with a diameter of 6 inches, the wheel has progressed one circumference of a circle with a radius of 12 inches.) Can you still follow me?
The usefulness of the TGR would be in comparing the speed potential of various unicycle setups (or bikes for that matter), since the speed potential would be determined within reasonable limits by the TGR.
By way of example, Total Gear Ratios for some popular unicycle setups:
Fixed Coker, 114 mm cranks: TGR = 4.01
Schlumpf geared 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 I used the exact Schlumpf gear ratio of 17/11 (equal to about 1.545), rather than the rounded value of 1.5.)