Entfaltung

Gain Ratio - Die Sheldon Brown Methode zur Berechnung der Entfaltung

Measuring Bicycle Gears

Cyclists often find it useful to have a numeric representation of the gearing provided by their bicycles. This allows them to make meaningful choices in customizing their gearing, and can be useful in comparing the performance of one bicycle with another.

There are several systems for doing this, none of them entirely satisfactory. I would like to propose a new, more accurate and more universal system.

Existing Systems

Front/Rear

Cyclists who are only associated with one narrow ghetto of the cycling world frequently make do by just naming the chainwheel and rear sprocket they are using. This is a bit cumbersome, using two numbers where only one is really needed, and can also be confusing. For example, a 39/14 is the same as a 53/19, but this is not obvious. Since there are only 4 chainwheel sizes in common use on road racing bikes (39,42,52 & 53) this is still usable for the cyclist who only deals with this type of machine.

More well-rounded cyclists, who are likely to deal with wider variations in chainwheel and tire size, need a more sophisticated system to realize that the 46/16 on their mountain bike, and the 52/12 on their Moulton and the 52/14 on their Bike Friday are all about the same as the 53/19 and 39/14 on their road racer.

This system is also pretty much useless when planetary gears are involved.

Gear Inches

The simplest system in common use is the "gear inch" system. This dates back to before the invention of the chain-drive bicycle. It originally was the diameter of the drive wheel of a high-wheel bicycle. When chain-drive "safety" bikes came in, the same system was used, multiplying the drive wheel diameter by the sprocket ratio. It is very easy to calculate: the diameter of the drive wheel, times the size of the front sprocket divided by the size of the rear sprocket. This gives a convenient two- or three-digit number. The examples listed above are all around 74-75 inches. The lowest gear on most mountain bikes is around 22-26 inches. The highest gear on road racing bikes is usually around 108-110 inches. Unfortunately, the handwriting is on the wall for all inch-based measurement systems.

Developement in Meters

In countries that use metric measurements, the usual system is "development" in meters. This is the distance that the bicycle moves with each revolution of the pedals. This system is a bit more cumbersome than the gear inch system, for two reasons. First, it is a little more difficult to calculate: wheel diameter in meters x front sprocket / rear sprocket x pi. Having to multiply by a constant (an irrational one, no less!) needlessly complicates things. Also, the resulting value is a less convenient number to work with, a single digit plus two decimals. For example a road bike's 52/13 would be exressed: 8.64. A mountain bike's 24/28 would be: 1.78.

What About Crank Length?

All of these systems share a common inadequacy: none of them takes crank length into account! The fact is that a mountain bike with a 46/16 has the same gear as a road bike with a 53/19 only if they have the same length cranks. If the mountain bike has 175's and the road bike 170's, the gear on the mountain bike is really about 3% lower!

A New Standard Proposed

I would like to propose a new system, which does take crank length into account. This system is independent of units, being expressed as a pure ratio.

This ratio would be calculated as follows: divide the wheel radius by the crank length; this will yield a single radius ratio applicable to all of the gears of a given bike. The individual gear ratios are calculated as with gear inches, using this radius ratio instead of the wheel size.

You can calculate gain ratios, gear inches or meters development with my Online Gear Calculator or with your slide rule Spoke Divider

An Example:

   A road bike with 170 mm cranks: (The usual generic diameter value for road wheels is 680 mm, so the radius would be 340 mm.)
   340 mm / 170 mm = 2.0. (The radius ratio)
   2.0 X 53 / 19 = 5.58
   This number is a pure ratio, the units cancel out. I call this a "gain ratio" (with thanks to Osman Isvan for suggesting this term.) What it means is that for every inch, or kilometer, or furlong the pedal travels in its orbit around the bottom bracket, the bicycle will travel 5.58 inches, or kilometers, or furlongs.

Another example:

   A mountain bike with 26 inch wheels (13 inch radius) and 6 3/4" cranks:
   13" / 6 3/4" = 1.93
   1.93 X 46 / 16 = 5.54
   Remember, the "radius ratio" only has to be figured out once for a given bike, because it is the same in all gears. Any individual gear is calculated as:
   Radius ratio X front(teeth) / rear(teeth)
   Any measurement units may be used, as long as the same units are used for both the wheel diameter and crank length.

You can calculate gain ratios, gear inches or meters development with my Online Gear Chart or with your slide rule Spoke Divider

Radius Ratios for Common Crank Sizes:

Tire Size	Tire Radius	165 mm	170 mm	172.5 mm	175 mm	180 mm
I.S.O. 630:
27 X 1 3/8	345	2.091	2.029	2.000	1.971	1.917
27 X 1 1/4	343	2.079	2.018	1.988	1.960	1.906
27 X 1 1/8	342	2.073	2.012	1.983	1.954	1.900
27 X 1	340	2.061	2.000	1.971	1.943	1.889
I.S.O. 622:
700 X 56	370	2.242	2.176	2.145	2.114	2.056
700 X 50	365	2.212	2.147	2.116	2.086	2.023
700 X 44	354	2.145	2.082	2.052	2.023	1.967
700 X 38	347	2.103	2.041	2.012	1.983	1.927
700 X 35	345	2.091	2.029	2.00	1.971	1.917
700 X 32	342	2.073	2.012	1.983	1.954	1.900
700 X 28	336	2.036	1.976	1.948	1.920	1.867
700 X 25	335	2.030	1.971	1.942	1.914	1.861
700 X 20	332	2.012	1.953	1.925	1.897	1.844
I.S.O. 559:
26 X 2.125	330	2.000	1.941	1.913	1.886	1.833
26 X 1.9	324	1.964	1.906	1.878	1.851	1.800
26 X 1.5	312	1.891	1.835	1.809	1.783	1.733
26 X 1.25	311	1.884	1.829	1.803	1.778	1.728
26 X 1.0 (559 mm)	305	1.848	1.794	1.768	1.743	1.694
I.S.O. 571:
26 x 1 (650C)	311	1.884	1.829	1.803	1.778	1.728
Other:
Wide Tubular	338	2.048	1.988	1.959	1.931	1.878
Narrow Tubular	335	2.030	1.971	1.942	1.914	1.861
26 X 1 3/8 (590 mm)	330	2.000	1.941	1.913	1.886	1.933
24"	305	1.848	1.794	1.768	1.743	1.694
24 x 1 (520)	279	1.691	1.641	1.617	1.594	1.550
20 X 1.75 (406 mm)	254	1.539	1.494	1.472	1.451	1.411
20 X 1 1/4 (451 mm)	257	1.558	1.512	1.490	1.469	1.428
17 x 1 1/4 (369 mm)	211	1.279	1.241	1.223	1.206	1.172
16 x 1 3/8 (349 mm)	204	1.236	1.200	1.183	1.166	1.133

Thanks to Galen Evans and Osman Isvan for their assistance.