How does a gearbox work? The trade between speed and torque

You reach a steep hill on your bike. The pedals get heavy, so you shift to an “easy” gear: you pedal faster, the bike slows down, but you make it up the hill. A car does the same: you pull away in first gear and move up through the gears as you speed up. In both cases the device doing the work is the same: the gearbox, or transmission. In this post we look at what a gearbox can and cannot change, with numbers and with a drawing you can open up.

Gear ratio: which gear turns how many times?

When two gears mesh, each time a tooth of one passes the contact point, a tooth of the other passes too. If a small 14-tooth gear turns a large 28-tooth gear, then one full turn of the small gear turns the large gear only half a turn. The large gear turns twice as slowly.

This relationship is called the gear ratio: the number of teeth on the driven gear divided by the number of teeth on the driving gear.

We call the number of turns per minute the speed in revolutions per minute (rpm). If the driving gear turns at 1500 rpm, the driven gear turns at $1500 / 2 = 750$ rpm. The number of teeth and the speed are inversely proportional: their product stays the same, $14 \cdot 1500 = 28 \cdot 750$. You can try this from the maths side with two gears in the Inverse proportion lesson, and our post on direct and inverse proportion explains the difference with everyday examples. This post looks at the engineering side: what do we gain when the speed goes down?

Two meshing gears also change the direction: they turn in opposite directions. Put a third gear between them and the direction is back to the original.

Torque: the turning effect

When you tighten a nut, a long spanner turns it more easily than a short one for the same push. The turning effect of a force is called torque. Torque is the force times the length of the lever arm, and its unit is the newton metre (N·m).

Gears work the same way. The small gear turns the large gear by pushing on its teeth. The force where the two gears touch is the same for both, but on the large gear it acts farther from the centre. So the torque on the large gear’s shaft grows by the ratio. In the 28/14 pair the output torque is twice the input torque.

The trade: torque × speed stays almost the same

The power a turning shaft carries is its torque times its angular speed. Angular speed $\omega$ is measured in radians per second and comes from the rpm like this: $\omega = 2\pi n / 60$.

A gearbox does not make energy. If we leave friction aside, the power that goes in at the input comes out at the output:

With a ratio $i$, the output turns $i$ times more slowly and gives $i$ times more torque. Speed is traded for torque, and the product does not change. In a real gearbox the friction between the teeth and the drag of the oil turn part of the power into heat. In well-lubricated spur gear pairs this loss is usually a few per cent, so “almost the same” is the more accurate way to say it.

Worked example. The drawing below uses these numbers: the motor gives at most 2 N·m of torque and turns at 1500 rpm. A drum of radius 5 cm sits on the output shaft, and a rope wound on the drum tries to lift a 6 kg load. The torque the drum needs to lift the load is:

• Gear 3 (1 : 1): The output gives at most 2 N·m. The load needs 2.94 N·m, so it does not lift; the motor strains and stalls.
• Gear 2 (2 : 1): The output gives at most $2 \cdot 2 = 4$ N·m, so the load lifts. The output turns at 750 rpm. The motor gives only the torque that is needed: $2.94 / 2 = 1.47$ N·m. Power: $1.47 \cdot 157 \approx 231$ W goes in and $2.94 \cdot 78.5 \approx 231$ W comes out.
• Gear 1 (4 : 1): The output gives up to 8 N·m; the load lifts easily, but twice as slowly, at 375 rpm. The power halves too (about 116 W), because the load is the same and the speed is half.

Same motor, same load: a lower gear does not add power; it adds the torque available to lift the load, and the speed goes down.

A look inside the gearbox

The device in the drawing is a simple example of a layout used in older cars and still used in many machines today: a three-speed gearbox with its lid off. Move the gear lever, change the motor speed and the load; you can also type in your own numbers of teeth.

• The input shaft comes from the motor. Its small gear keeps turning the large gear on the countershaft behind it. This is the fixed pair; with the default gears the countershaft always turns at half the input speed.
• The countershaft carries three more gears. They are always in mesh with gears on the output shaft in front. But the gears on the output shaft are not fixed to it; they spin freely on the shaft.
• Between these gears sit sliding collars that fit on the output shaft through splines. When a collar slides along the shaft onto the teeth on the side of a gear, it locks that gear to the shaft. The power goes through whichever gear is locked: that is the path shown in purple in the drawing.
• The collars are pushed by selector forks, and the forks are fixed to rails. The lower end of the gear lever sits in a slot on one rail; pushing the lever slides the rail and its fork.
• In gear 3 the collar locks the output shaft straight to the input shaft. The ratio is 1 : 1; this is called direct drive.

Every gear change passes through neutral: the lever pulls the rail back to the middle and then moves across to the other rail. In neutral the gears still turn, but none of them is locked to the output shaft. Real cars also have friction rings called synchronisers that match the speeds of the two parts before the teeth engage, so the change is smooth. The drawing leaves this detail out.

Bicycle gears: the same idea, with a chain

On a bike the gears do not touch each other; a chain runs between them. But the sums are the same: at the front is the chainring on the pedals, at the back the sprocket on the wheel. For every turn of the pedals the wheel turns $N_{\text{front}} / N_{\text{rear}}$ times.

• 50 teeth at the front, 11 at the back: the wheel turns about 4.5 times per pedal turn. A “hard” gear for going fast on the flat.
• 34 teeth at the front, 32 at the back: the wheel turns about 1.06 times per pedal turn. An “easy” gear for climbing.

Note that on a bike the large sprocket at the back is the easy gear. Your legs are the motor of this machine. At a comfortable pedalling rate you can only keep up so much force for long; the gears bring the torque the road asks for into the range your legs can give.

Car gearboxes and low gear on a hill

A car engine works efficiently and strongly only within a certain range of speeds: at very low rpm it labours, at very high rpm it strains. The wheels, though, must turn at every speed from standstill to motorway speed. The gearbox links these two worlds. In many cars first gear has a ratio between 3 and 4, and the top gear is close to 1 or below it. On top of that, a final drive pair (the gears in the differential) multiplies the ratio once more before the wheels.

Why a low gear on a hill? Going uphill, the part of the car’s weight that pulls it back asks for a bigger force at the wheels, that is, a bigger torque. In a low gear the ratio is large: the engine’s torque is multiplied more and the engine stays at a comfortable speed. The price is speed; at the same engine speed the car goes more slowly. A low gear also helps on a long downhill: the wheels try to spin the engine, the engine’s internal drag slows the car (engine braking) and the brakes do not overheat.

Common mix-ups

• “A gearbox increases power.” It does not. It raises the torque and lowers the speed by the same factor; at best the power stays the same, and in reality it drops a little.
• “The big gear is always the slow one.” It depends on which gear does the driving. A small gear turning a big one slows the motion down; a big gear turning a small one speeds it up.
• “I can raise both speed and torque at once.” Not with the same power. If you want both, you need a more powerful motor.
• “A low gear means a small gear.” A “low gear” means a large ratio: slow and strong. On a bike, it is the large sprocket at the back.

Gears are not a new invention. At the start of the 13th century, the water machines in al-Jazari’s book used toothed wheels to change the direction of motion; you can look inside that workshop in our post on the elephant clock.

Summary

• Gear ratio = teeth on the driven gear / teeth on the driving gear. The speed drops by this ratio and the torque rises by it.
• Power = torque × angular speed. A gearbox makes no power; leaving friction aside, the power in equals the power out.
• Changing gear means locking a different gear pair to the output shaft with a sliding collar.
• On a hill and when pulling away, use a low gear (large ratio): more torque, less speed.
• On a bike the large rear sprocket is the easy gear; in a car the final drive multiplies the ratio once more.