Gear ratios: trading speed for torque

TL;DR:

• A gear ratio of 10:1 divides speed by 10 and multiplies torque by 10, before losses.

• Power does not increase. Output power is input power times efficiency, so you trade speed for torque, never get both.

• Each gear stage wastes some energy, and gears add backlash, so use only as much reduction as the job needs.

A small hobby motor spins at thousands of revolutions per minute, yet it cannot lift a book by a lever, never mind move a robot up a ramp. Fast and weak is the natural state of most small motors. A gearbox changes that: it gives up speed to buy force.

Why a gearbox helps

A lever gives the idea. A long handle lets a small force turn a bolt, because the force acts over a bigger distance from the pivot (see torque and levers). Meshing gears do the same for rotation. A small gear on the motor drives a larger gear on the output. The teeth of the two gears move at the same speed where they meet. The big gear has more teeth, so it turns fewer times, but it is pushed at a larger radius, so it turns with more torque.

The ratio math

Count the teeth. For a driver (input) gear with N_in teeth driving a driven (output) gear with N_out teeth, the gear ratio is N_out / N_in. A 12-tooth gear driving a 60-tooth gear gives 60 / 12 = 5, written 5:1.

For an ideal, lossless pair with ratio r:

1. Output speed = input speed / r.

2. Output torque = input torque × r.

Power is torque times angular speed, so with both changes the product is the same: (τ × r) × (ω / r) = τ × ω. The gearbox does not create energy. It converts a fast, light turn into a slow, strong one.

When gear stages are stacked (a compound train), the ratios multiply. A 2:1 stage followed by a 5:1 stage gives 2 × 5 = 10:1.

Worked example: a 10:1 gearbox

Say a motor delivers 0.05 N·m of torque at 3000 rpm, and it drives a 10:1 gearbox that is 0.8 efficient (so 0.8 of the input power gets through; that figure is an example, not a real part).

1. Output speed. 3000 / 10 = 300 rpm.

2. Output torque. Ideal 0.05 × 10 = 0.5 N·m. With losses, 0.5 × 0.8 = 0.4 N·m.

3. Power check. Input: 3000 rpm = 3000 × 2π / 60 = 314.2 rad/s, so P_in = 0.05 × 314.2 = 15.7 W. Output: 300 rpm = 31.4 rad/s, so P_out = 0.4 × 31.4 = 12.6 W.

4. Efficiency. 12.6 / 15.7 = 0.8. This matches the efficiency we started with, so the numbers are consistent.

Now put a wheel of radius 0.05 m on the output shaft.

1. Push force at the ground. F = τ / r = 0.4 / 0.05 = 8 N.

2. Speed. v = ω × r = 31.4 × 0.05 = 1.57 m/s.

Without the gearbox, the wheel would have 0.05 / 0.05 = 1 N of push and a speed ten times higher, 15.7 m/s, which no small robot could use. The gearbox turned a useless combination into a usable one.

Efficiency losses

Real gears waste energy as friction between the teeth and in the bearings, and as churned grease. Each stage has its own efficiency, and they multiply. Two stages that are each 0.9 efficient give 0.9 × 0.9 = 0.81 overall. So a big ratio built from many small stages loses more than a ratio built from fewer stages. Spur gears are usually more efficient than worm gears, which have a lot of sliding contact.

Efficiency has a useful side effect. A gearbox with a high ratio and plenty of friction, like a worm drive, can be non-back-drivable: pushing on the output does not turn the motor. That holds an arm up without power. But it also means the gears take the full shock if the arm is knocked, so they can break.

Where it goes wrong

• Backlash. Gears have a small gap between the teeth so they do not jam. Reverse direction and the output sits still until the gap is taken up. For a robot arm that wants to be accurate, backlash shows up as slop. Precision gearboxes minimize it, at a cost.

• Too much reduction. A ratio that gives lots of torque also gives little speed. A 100:1 gearbox on that wheel would move at 0.157 m/s, slow enough that the robot barely seems to move.

• Overload. Plastic gears strip when the output stalls. If a motor is stalled hard against a gearbox, the teeth carry the force that the motor could generate at its stall torque times the ratio.

• Heat and noise. Friction turns lost energy into heat, which is what the efficiency number is hiding.

Safety: Meshing gears trap fingers, hair, loose sleeves and cables. Keep covers on gearboxes, power off before you touch anything, and never try to stop a geared motor with your hand. A gearbox multiplies torque, so even a small motor can pull a finger in with force.

How real robots use gear ratios

Wheeled robots gear motors down so that a small motor can push the robot at walking pace. Robot arm joints use high-ratio reducers so that small, light motors can hold the arm up. Legged robots balance low ratios, which keep joints quick to respond and easy to back-drive, against high ratios, which give strong joints. A drone does the opposite: it usually connects the propeller directly to the motor, because it wants speed and a lighter airframe more than torque. The best ratio depends on the load, and a controller such as a PID controller will work with whatever the gearbox gives.

Try it yourself

Take a geared hobby motor and a wheel. Write down the datasheet gear ratio and the no-load speed of the motor, predict the output speed, and then measure it by counting revolutions in 10 seconds. Then predict the pushing force from the stall torque and wheel radius, and check by pulling the robot with a small spring scale from rest. Expect the real values to be lower than the ideal ones, by the efficiency.

Frequently asked questions

Does a gearbox give me more power?

No. Power out is power in times the efficiency, which is always less than one. A gearbox trades speed for torque, or the reverse.

How do I calculate the ratio from the gears?

Divide the number of teeth on the output gear by the number on the input gear. For several stages, multiply the stage ratios together.