Gear ratio calculator

How do I calculate a gear ratio?

Divide the teeth on the driven gear by the teeth on the driving gear: ratio = driven teeth ÷ driver teeth. A 20-tooth gear driving a 60-tooth gear gives 60 ÷ 20 = 3, written 3 : 1. The output turns three times slower than the input and, apart from losses, delivers three times the torque.

Why does the tooth count give the speed ratio?

At the point where two gears mesh, teeth pass by at the same rate on both. If the driver has 20 teeth and turns at 1,800 rpm, 36,000 teeth pass the mesh per minute, and a gear with 60 teeth must therefore turn at 600 rpm. The same holds for a belt or chain: its teeth or its pitch diameters take the place of tooth counts.

How do compound gear trains work?

In a compound train two gears sit on one shaft, so the ratios of the stages multiply. A 20:60 stage followed by a 15:45 stage gives 3 × 3 = 9 : 1. At 1,800 rpm in, the output turns at 200 rpm. Add one row per stage in the calculator; a stage is any driver and driven pair.

Does an idler gear change the ratio?

No. A gear that sits between the driver and the driven gear and is not on a shared shaft with any other changes only the direction of rotation and the distance between the shafts, because its own teeth are matched on both sides. Leave idlers out of the list and count them in the direction instead.

Which way does the output turn?

Each external mesh (two outside-toothed gears) reverses the direction, so a train with an odd number of external meshes turns the output backwards relative to the input, and an even number turns it the same way. An internal ring gear and a belt or chain keep the direction. The calculator counts these for you from the type you choose for each stage.

What efficiency should I use?

NASA researchers who measured spur-gear systems report efficiencies generally above 98 percent, except at very low torque (Anderson and Loewenthal, NASA TM-81426). That is why the default is 98 percent per stage. For two stages the overall figure is 0.98 × 0.98 = 96.04%. Other gear types, such as worm gears, and belts and chains, have different losses; use the maker’s figure.

What is the difference between this and a bike gear calculator?

This page handles any gear train and gives output speed and torque. The bike gear calculator is built for a bicycle: chainring and cog, wheel size, cadence and gear inches.

Speed and torque trade places

A gear train cannot create power; it exchanges speed for torque. Power is torque times angular speed, so if the output turns three times slower, it can deliver at most three times the torque. With losses the output torque is the input torque × the ratio × the product of the stage efficiencies. As an example, a motor delivering 10 lb-ft (13.6 N·m) at 1,800 rpm drives a 20:60 stage at 98 percent: the output is 600 rpm and 29.4 lb-ft, against 30 lb-ft with no losses.

Worked example: two stages

Take that motor through the 20:60 pair and then a 15:45 pair. The ratio is (60 ÷ 20) × (45 ÷ 15) = 9 : 1. The output speed is 1,800 ÷ 9 = 200 rpm. The torque is 10 × 9 × 0.98² = 86.4 lb-ft, which is 96% of the lossless 90 lb-ft. Both meshes are external, so the direction reverses twice and the output turns the same way as the motor. Reverse the roles (a 60-tooth driver and a 20-tooth driven gear at 300 rpm and 30 lb-ft) and the ratio is 0.33 : 1: the output turns at 900 rpm with 9.8 lb-ft.

Limits of this method

The ratio of a gear train is exact; the efficiency is a typical figure and not a promise. NASA’s measurements were for spur gearsets under stated test conditions, and the report itself notes lower efficiency at very low torque. Other types of mesh have different losses, and a stack of stages multiplies them. The calculator also assumes steady operation and ignores the torque needed to accelerate the gears, bearing and seal friction and the strength of the teeth: a ratio that works on paper can still break a tooth. For a design that matters, use the figures of the gear maker and a proper strength check.

Common mistakes

Swapping driver and driven turns a reduction into a speed-up. Counting an idler as a stage is the next error; it changes only direction. In a compound train the ratio is driven over driver for each stage, then multiplied; adding the stage ratios instead is wrong. Belts and chains take pitch diameters or tooth counts, not the outside diameter of the pulley. And the torque at the output is not free: it needs the matching drop in speed.

Sources: N. E. Anderson and S. H. Loewenthal, “Effect of geometry and operating conditions on spur gear system power loss,” NASA Technical Memorandum 81426 (1980), efficiency results for spur gearsets; the ratio rules (driven over driver, multiplication of stages, direction of external meshes) follow from equal tooth rates at the mesh, as in any machine-design text. Related: bike gear calculator, torque calculator, RPM to MPH calculator, pulley system calculator. Tooth strength and gear life are not checked here.