How gears change speed

A car engine delivers the most force within a certain range of speeds. The wheels, though, must turn at everything from a standstill to highway speed. A transmission sits in between to bridge the two. The five works here are “simplified teaching models” based on real machines. Friction, heat loss and small parts are left out so that only the flow of speed and force stays visible.

First: trading speed for torque

In a pair of gears, the ratio of the tooth counts swaps rotation speed for torque (turning force). In this article the gear ratio means the output teeth divided by the input teeth. The bigger the ratio, the slower the output and the stronger the force. With no losses, speed times torque (the power) stays the same. When gears are chained, the overall ratio is the product of the individual ratios.

Manual gearbox (the site’s “Manual Gearbox”)

Many manual gearboxes in passenger cars have three shafts: input, countershaft and output. The gear pairs are always meshed, and a dog clutch chooses which pair is locked to the output shaft. To choose, the friction clutch between the engine and the gearbox is opened first. A friction clutch can be slipped while it engages; a dog clutch is poorly suited to slipping. The jobs are split. If the speeds are mismatched the teeth clash, so a cone-shaped friction ring brings the speeds together before the dog clutch locks. This is called synchromesh.

The work uses five forward ratios and a reverse as examples: 3.6 / 2.1 / 1.4 / 1.0 / 0.8, reverse −3.6. These are invented for the model, not figures from any real car. The program checked that in every gear the output speed is engine speed divided by the ratio, that speed times torque stays constant, and that reverse flips only the direction (manual-gearbox/engine.test.ts).

Reverse puts one extra gear (an idler) between two gears to flip the direction. The idler’s tooth count does not change the ratio. As for history, five-speed boxes are said to have become common in the 1980s. Claims about the “first” gearbox with this or that many speeds are easy to contradict, so we leave them out.

Planetary gears (the site’s “Planetary Gear”)

A sun gear in the middle, a carrier holding planet gears that circle around it, and an outer ring gear: these three parts make a planetary gear set. Let Zs and Zr be the tooth counts of the sun and ring, and ωs, ωr and ωc the rotation speeds of the sun, ring and carrier. They are linked by one relation.

(Zs + Zr) × ωc = Zs × ωs + Zr × ωr

Hold one of the three still, use another as input and the third as output, and the ratio depends on which one you hold. Lock any two together and the whole set turns as one block (direct drive, ratio 1). The work uses a sun of 30 teeth and a ring of 70 teeth.

• Ring held, input at the sun, output at the carrier: a reduction of 1 + 70/30 ≈ 3.33, the largest reduction.
• Carrier held: the sun and ring turn in opposite directions. The ratio is −70/30 ≈ −2.33, which can serve as reverse.
• Sun held, input at the ring, output at the carrier: a reduction of 1 + 30/70 ≈ 1.43, a small one.
• Two parts locked together: direct drive.

The program checked that the relation holds in every mode and in both directions of rotation, and that the values 3.33, direct drive and reverse come out as stated (planetary-gear/engine.test.ts). An automatic transmission builds its gears by switching, with clutches and brakes, which part of a planetary set is held and which parts are joined. The switching is done by oil pressure or by electronics. Around 1940, a mass-produced automatic transmission that combined a fluid coupling (a part that passes rotation through the flow of oil) with planetary gears is said to have appeared.

Stepless drive (the site’s “CVT”)

Run a belt over two pulleys (wheels with a V-shaped groove). Each pulley is made of two cone-shaped discs whose gap can change. Narrow the gap and the belt rides higher; widen it and the belt sinks. The belt length and the distance between the shafts do not change, so when one pulley’s belt rises the other’s sinks. The ratio therefore changes continuously instead of in steps. If the belt does not slip, the belt speed is the same at both pulleys, so the output speed is the input speed times the input radius divided by the output radius.

The program checked that wherever the slider is, the two pulley radii work out so that the belt length stays exactly fixed (cvt/engine.test.ts). As for history, a passenger car from 1958 is said to be the first mass-produced car with a stepless drive. But a “first” like this is easy to contradict. A story that Leonardo da Vinci devised one also circulates, but we could not confirm which manuscript or drawing it refers to, so we leave it out.

Differential (the site’s “Differential”)

In a curve the outer wheel travels farther than the inner one. Join the two wheels with one rigid axle and one of them must skid on the road. A differential divides the speeds. The speed of the case that receives power from the engine is the average of the two wheel speeds. case speed = (left + right) ÷ 2

For example, if the middle of the car follows a circle of radius 10 m and the wheels are 1.5 m apart, the outer and inner wheel speeds are in the ratio 10.75 : 9.25, about 1.16 : 1. The program checked that this ratio matches the ratio of the distances travelled and that the case speed is the mean of the two wheels (differential-gear/engine.test.ts).

It has a weak point too. In the simplest, open kind, if one wheel spins on ice the force goes to that side, little reaches the other wheel, and the car may not move. The work includes that case with one wheel on ice: the car stays still while one wheel spins (this scene is also a simplified model checked by the program).

As for history, in the 1820s the Frenchman Pecqueur is said to have included a differential in a patent for a steam-driven cart. Sources disagree on the exact year. There are also claims that the ancient Chinese south-pointing chariot or the Antikythera mechanism used differential gearing, but those are hypotheses without firm evidence.

Bicycle gears (the site’s “Bicycle Gears”)

The gear ratio of a bicycle is the front (pedal side) tooth count divided by the rear (wheel side) tooth count. With 50 teeth in front and 25 behind, the ratio is 2.0: one turn of the pedals turns the rear wheel twice. With a tyre diameter of 0.67 m, one turn of the pedals moves you about 4.2 m. The program checked this value (bicycle-gears/engine.test.ts). The bigger the ratio, the farther you go per turn, and the harder the pedals are to push.

With front gears of 30, 40 and 50 teeth and rear gears of 15, 20, 25 and 30, some pairs give the same ratio (2.0): 30/15, 40/20 and 50/25. The program also checked that such pairs can be found exactly by cross-multiplying. Pairs with the same ratio are wasteful, so a bike can look as if it has many gears while offering fewer distinct ratios. As for history, in 1905 the Frenchman de Vivie is said to have devised a two-speed rear changer and used it in the Alps, and gear changers were allowed in the Tour de France in 1937.

What they share

• Speed for force: in gears, belts or chains alike, speed times torque stays the same when nothing is lost.
• Ratios multiply: chained stages multiply their ratios.
• Different ways to switch: a manual box picks a gear pair, planetary gears pick which part is held, and a stepless drive changes the radius continuously.

All five are simplified teaching models. Losses, slipping, and figures from real cars or parts are not covered here.