This is an update – simpler than the original – of an adjacent post.
Bikes ‘gear up’ our feet’s movement so we can travel more quickly than with feet alone. A numerical representation of the gearing enables us to make comparisons.
Gearing up, the Gear gain, happens in two stages.
In the first penny-farthing (or high wheel) bicycles, gear gain was provided by one single factor – the fact that our feet travel in a small radius circle and the wheel itself has a larger diameter.
This gain, the radius ratio is the ratio of the bike wheel radius to the radius of the pedal cranks.
Radius ratio = (bike wheel radius)/(crank radius)
Typical large wheels (700c) on a modern bike have a wheel diameter of 350 mm (half the diameter). Crank lengths are typically 170 mm.
So radius ratio = (bike wheel radius)/(crank radius) ratio = 350/170 = 2 (near enough).
Since the wheel and crank radii for all normally large wheeled bikes are very similar, you can assume that the radius ratio for all bikes with large wheels is 2.
Modern bicycles have gears which give gain in addition to the radius ratio. The gear ratio tells you how many times the rear wheel goes round for each turn of the pedals. The cranks turn a chainwheel which drives a rear cog. The cog is mostly smaller than the chainwheel gearing up the motion. For very low gears the sprocket can be larger than the chainwheel, gearing it down.
Gear ratio = (chainwheel teeth)/(cog teeth)
For a chainwheel of 42 teeth driving a cog of 14 teeth,
Gear ratio = (chainwheel teeth)/(cog teeth) = 42/14 = 3
Gain ratio is the overall effect of the gearing caused by both the radius ratio and the gear ratio. You find the overall gain ratio by multiplying the two together. For the bike above,
Gain ratio = radius ratio x gear ratio = 2 x 3 = 6
With this gain ratio, your bike is going 6 times as fast as your feet are moving. When your feet are moving at 4 mph, your bike is going at 24 mph.
The concept of gain ratio was suggested by Sheldon Brown. Like many of his suggestions it made complete sense…but has largely been ignored.
Internal gear hub gain
Gearing gain can also be provided by an internal gear hub. Here is part of a Brompton folding bicycles with a compact 2-gear derailleur coupled with an internal gear hub.
The Brompton has 16″ x 1-3/8″ tyres with a radius of 204 mm. The crank length is 170 mm.
Radius ratio = 204/170 = 1.2
The Brompton chainwheel has 50 teeth and the 2-speed derailleur has 13 teeth on its smallest cog.
Derailleur gain = 50/13 = 3.85
The wide-range Sturmey-Archer S-RF5 hub has a gain of 1.6 in its top gear. When fitted to the Brompton with that derailleur,
overall top gear ratio = derailleur gain x hub gear gain
overall gear ratio = 3.85 x 1.6 = 6.16
Gain ratio = radius ratio x gear ratio = 1.2 x 6.16 = 7.38
The combination of the derailleur and internal hub gears gives this small-wheeled bike significantly more gain than the large-wheeled bike above.
Other ways of measuring gear gain
There are a couple of other common ways of measuring gear gain which are in common use. I include them for completeness.
Gear inches is the ‘equivalent diameter’ of a wheel that would be needed for a penny-farthing bike with the same gear gain. Think about the bike’s gear ratio ‘magnifying’ the diameter of the rear wheel.
Gear inches = gear ratio x diameter of the rear wheel.
The Brompton bike above has 16” diameter wheels.
So gear inches = 6.16 x 16” = 98.6”
Making the reasonable approximation that all cranks are 170 mm long, here is the conversion between gear inches and gain ratio.
Gear inches = Gain ratio x 13.4
Development in metres
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.
Development = circumference of wheel x gear ratio
= 2π x radius of wheel x gear ratio, where π is roughly 3.14
Any units may be used for this formula. You can calculate development in inches by using inches for measurements of the wheel.
For the Brompton above, where the radius of the wheel is 204 mm (0.204 metres) and the gear ratio 6.18,
Development = 2 x 3.14 x 0.204 metres x 6.18 = 7.9 metres.
If (again) we assume that cranks are 170 mm long, here is the interconversion:
Development = 1.07 x gain ratio.
Remembering that the absolute value of development is not very important, near enough the gain ratio (very easy to calculate) is equal to the the development in metres.
The simplicity of gain ratio
Now we’ve calculated development in metres and compared it with Sheldon Brown’s gear ratio, we see how much simpler gain ratio is to calculate and use as a method of comparison. One calculates the radius ratio once for whatever bike one is considering and multiplies it by the gear ratio.
Since for all large wheel bikes, the radius ratio is very close to 2, double the gear ratio and you have the gain ratio, which is near enough the development in metres.
Gain ratio, cadence and speed
To a close approximation, again assuming that the crank length is 170 mm,
Speed (in km/h) = (Gain ratio x Cadence)/15.6
Speed (in mph) = (Gain ratio x Cadence)/25
Given that a cadence of 75 per minute is pretty brisk and that
Speed (in mph) = (Gain ratio x cadence)/25 = Gain ratio x 75/25 = 3 x gain ratio.
At a cadence of 75 per minute, speed (in mph) = 3 x gain ratio (or 3 x development if you prefer).
For those living in fully metric countries,
At a cadence of 75 per minute, speed (in km/h) = 5 x gain ratio (or 5 x development)
With our Brompton bike, having a gain ratio of 7.38, being pedalled with a cadence of 75 rotations per minute,
Speed = 3 x 7.38 = 22 mph.
Ellse approximations for bikes with normally large wheels
- Development in metres = gain ratio = 2 x gear ratio
- Speed at cadence of 75 =
3 x development or gain ratio (mph)
5 x development or gain ratio (km/h).
Chainset 48-35T, cassette 10-33T
So top gear ratio = 48/10 = 4.8
Using Ellse approximations for top gear.
1. Development in metres = gain ratio = 2 x 4.8 = 9.6
2. Speed in top gear at cadence of 75
= 3 x 9.6 = 30 mph
= 5 x 9.6 = 50 km/h
Comparing the different units
For bikes with full-sized wheels, where radius ratio is about 2, this table provides a useful comparison.
|Gain ratio||Gear ratio||Gear inches||Development||Speed at cadence of 75 per minute|
|Highest gear on road racer||9.6||48/10 = 4.8||129”||10.3 m||28.8 mph|
|Middle gear on commuting bike||4.2||38/18 = 2.1||56”||4.5 m||15 mph|
|Low gear on mountain bike||1.5||34/46 = 0.74||20”||1.6 m||5.4 mph|