# Gear ratios

Let's start with a definition: gear ratio is a number which, multiplied by a traveled angle of input shaft, gives us a traveled angle of output shaft. The angle can be replaced by other things: number of revolutions, rotation speed etc.. The ratio is a dimensionless number. Values greater than 1 are overdrives, those less than 1 are reductions. Negative value reverses the direction (usually not used on bikes), zero usually means chain failure.

Torque works the other way: it grows when speed decreases, so we must divide it by gear ratio instead of multiplying.

There are several gears on a bike. First are cranks which transform circular movement of feet to rotation of a shaft: longer ones create more torque, shorter ones more speed. Second part is usually the chain transmission from chainring to sprocket, third may be a hub gearbox. Last gear is the transmission from the drive wheel to the road: the larger the wheel, the further it rolls per one revolution. The complete gear is usually expressed in metres and says how far we go per one revolution of the cranks. English speaking countries also use gear inches, which is a diameter of a hypothetical wheel driven directly by the cranks. If you want to convert gear inches to metres, multiply them by 0.0798 (that's *π and /0.0254).

## How to calculate gear ratio

(just to make things clear: "*" is a multiplication operator, "/" is division and "-" is a minus, I don't use these characters in any other meanings here)

### Chain

Chain gear ratio = number of teeth on the chainring / number of teeth on the sprocket. For example, with a fifty-tooth ring and twenty-tooth sprocket we get a ratio of z1/z2 = 50/20 = 2.5 (that's more than 1, so it's an overdrive gear).

The same equation applies for any other transmission where toothy cogs on stationary axles are involved: chain, toothed belt, pinions and even worms. And also friction gears, because number of teeth is nothing more than diameter multiplied by a constant. But it's not for epicyclic and harmonic gears.

### Epicyclic gearing

Epicyclic gear consists of three parts: central sun gear, planet cage with several planet pinions (the more of them, the stronger the gearbox is, but theoretically you only need one) and outer gear ring. One part is stationary (marked gray on the following picture) and the remaining two parts move:

If I'm not mistaken, the ratios are:

1. gear ring / planet cage = (z1+z3)/z3
2. planet cage / sun gear = z1/(z1+z3)
3. gear ring / sun gear = -z1/z3 (this is actually not an epicyclic drive)

In most cases hub gear manufacturers tell us the ratios directly, so we don't need to calculate anything.

### Gear composition

Gears connected in series, i.e. output of first one being input of second one, simply multiply. For example a 46T chainring, 22T sprocket and a 0.75 hub gear (reducer), we get a total gear ratio of (46/22)*0.75 = 2.091*0.75 = 1.568.

### Through wheels to the road

Distance traveled per one revolution of a wheel is equal to its circumference. We can either measure it directly (which is most accurate), or calculate as π*diameter. Express the circumference in metres, multiply by the gear ratio from cranks to the driven wheel and you get the sought distance per one crank turn (if you don't see the angle or speed of the input shaft, it's because it equals 1 (one crank revolution) and number one is very stealthy in multiplications).

### Crank length effect

If we divide distance per crank revolution by crank circumference (2π times crank length, it's something slightly over 1 m for most cranks), we get a ratio between foot speed and bike speed (which means how many inches the bike travels when the foot travels one inch). If all your cranks are of the same length, you don't have to worry about this. But if not, you'd better calculate this total gear ratio and check if it doesn't reach into some unpedalable area.

### Gear range

Gear range is usually expressed in percents. It says how many times the fastest gear is bigger than the slowest one. For example a range of 300 % means gmax = 3*gmin. It doesn't matter whether you divide dimensionless ratios or complete gears in metres or inches, all units cancel each other and you get the same result.

## Real world examples

Out of curiosity, I made a comparison of several transmission systems, both existing and hypothetic. Source data here (XLS), the most useful part are these plots (best viewed with browser window wide enough to fit them side by side):

First chart shows gears as distance traveled per one crank revolution. Second one shows them divided by circumference of that revolution to compensate for varying crank length. Light blue area in the middle is a range which works for level ground for me, faster gears are above and slower ones below. Little boxes labeled 40, 50 and 60 are roughly the top speeds in km/h which I can spin at the given gear value (only downhill in most cases). And now what the coloured lines mean:

• Favorit 2x5 - 175 mm cranks, double chainring 42..52 teeth, five-sprocket thread-on cluster 14..28 teeth, 28" wheels. My longest-used bike and primary reference sample. It climbs fairly well, but mostly due to its low weight and long cranks. When loaded, it has limits. It also needs one more gear for downhills. Overall ranking: not enough for my taste.
• Shimano 7 - seven speed Nexus hub. I only plotted the end points to find the range even narrower than the ten-speed above. Not interesting for me.
• Shimano 8 - eight speed Nexus, first of the possible candidates for my folder. Calculated for 170 mm cranks, chain transmission 52/16 and 20" wheels.
• Sturmey 8 - X-RK8 hub which won for the folder because of its more convenient chain ratio. 170 mm cranks, 42/25 chain, 20" wheels. Gear range suits me perfectly for city use. Fast end is just about right, I have never spun out even on the steepest of downhills (aero and rolling drag of the bike is fairly high), but I use it daily. Slow end nears its limits on some hills, but has never made me get off and push.
• Python 3x6 - cranks 170, triple chainring 28..48, cassette 11..28 (originally 7 sprockets, but the derailleur could only reach 6 of them, so I swapped the two biggest cogs to keep full range), 26" wheels. A bit untypical bike, so it's hard to judge. Fast end was just about right; the bike could go even faster downhill, but I was too afraid to pedal anyway. Slow end was too short pretty often, but partly because of power losses in the steering. It might be good enough for a classic geometry and lighter weight.
• Sturmey 3 - three speed S-RC3 hub we use on a city bike. Cranks 170, chain 46/22, wheels 28". Good for short errands and commutes, inadequate for touring over the hills and far away. It is possible to spin 40 km/h on level ground for several seconds with maximum effort, higher speeds can only be reached by coasting downhill. Uphills are a weak spot.
• Eska 3 - an old wreck which used to take up space in our basement. Cranks about 170, chainring 46, three-sprocket cluster 16..20, 26" wheels. Gear range is mostly symbolic; at the days when the bike was made, having any derailleur at all was a boon.
• Dualdrive 8x3 - CS-RK3 hybrid hub propelling my cargo 'bent. 155 mm cranks, 38-tooth chainring, 8-speed cassette 11..34, 26" driven wheel. The range is wide, but the shortened cranks move it further to the fast end than expected. I can comfortably spin more than 60 km/h which is not really necessary, but there are many steep hills I can't climb with this heavyweight monster (it climbs better than the folder mentioned above, but it's supposed to be a tourer which must be able to cope with steeper grades). Switching to a 36-tooth chainring looks like a perfect solution, but I haven't tried it yet.
• Sturmey 2 - S2C two speed hub. Tested on a folder: cranks 175, chain 46/16, wheels 20". That's a good choice: fast gear for the flats, slow one for small uphills. The range is no jackpot, the point is in easy replacement of a singlespeed hub with no added cables.
• Rohloff 2x14 - legendary fourteen speed hub which I combined with a 42..52 double chainring to extend the range (that's the big jump at the bottom of the plot). 155 mm cranks, 16-tooth sprocket, 20" driven wheel. Fast end is just right, I can spin 50 km/h and keep up with cars downhill in a city. Slow end is enough for steep hills with cargo, minimum speed at reasonable cadence is 4.5 km/h - probably too slow to balance a bike, but no problem for a trike.

## Calculator

If you are not fond of manual counting, you can use this automatic calculator: