Gears

A gear is a wheel with teeth that mesh with the teeth of another wheel, so one shaft can drive another without slipping. By choosing the number of teeth, a builder trades speed for force, changes the direction of motion or holds an exact ratio.

What it is

A gear is a wheel with teeth around its edge. When those teeth mesh with the teeth of a second gear, turning one wheel forces the other to turn without slipping. Gears drive mills, clocks, bicycles and car gearboxes.

A small gear driving a large one turns it more slowly but with more turning force, or torque. Two meshing spur gears turn in opposite directions, and a face wheel or a bevel gear turns the motion through a right angle. Because teeth cannot slip, a gear train also holds its ratio for years, which is what a clock depends on.

How it works

Take two meshing gears with 60 and 15 teeth. Every time a tooth of the large gear passes the point of contact, one tooth of the small gear passes too, so one turn of the large gear makes four turns of the small one. The speed ratio is the ratio of the tooth counts. The power that goes in comes out again, less friction, so torque changes the other way: the small gear turns four times faster with a quarter of the torque.

The plate shows two families. On the left is the wooden gearing of a mill: a face wheel, with pegs called cogs set into one side of its rim, drives a lantern pinion made of round staves held between two discs. A worn cog can be knocked out and replaced on its own. On the right are metal spur gears with shaped teeth on parallel shafts.

Each turn of the worm moves the wheel on by a single tooth.

With plain pegs, the driven wheel speeds up and slows down slightly as each tooth comes into contact and leaves it, which brings noise, shaking and wear. For a steady ratio, the tooth flanks must follow curves chosen so that the push between two teeth always passes through the same point on the line between the shafts. The cycloid, long used in clocks, does this. So does the involute, the curve traced by the end of a taut string unwound from a circle. Involute gears keep their ratio even when the shafts sit slightly too far apart, and they are easier to cut, so nearly all modern gears use them.

A worm is a screw that meshes with a toothed wheel. It can have one or several thread starts. The worm shown here has a single start and advances the wheel by one tooth per turn, giving a 30 to 1 reduction with a 30-tooth wheel. Whether the wheel can drive the worm backwards depends on geometry and friction. Self-locking can fail under vibration, so hoists and lifting winches need an independent load-holding brake. A rack, a straight toothed bar, turns rotation into straight-line motion.

How to build

Accurate metal gears need a lathe with a dividing plate or a gear-cutting machine. A wooden face wheel and lantern pinion, outlined here, can be made with hand tools.

Choose the ratio. Pick tooth counts for the output speed you need, for example 45 cogs and 8 staves, about 5.6 to 1. Counts with no common factor spread the wear, because every cog meets every stave in turn.

Lay out the pitch. Cogs and staves must be spaced at exactly the same distance, the pitch, measured along their pitch circles. Scribe the circle and step it out with dividers; a millimetre of error per step adds up all the way round.

Make the wheel. Build the rim from seasoned hardwood such as oak, in overlapping segments so the grain follows the curve, and fix it to arms mortised through the axle. Cut a hole at each mark and drive in cogs of a dense, tough wood, with a tenon passing through the rim and wedged behind. Round off the working ends.

Make the lantern. Fix two discs on the shaft, drill them on the same pitch circle and drive round staves between them.

Set the mesh. Mount both shafts in solid bearings so that the cogs reach about halfway into the gaps between the staves, with a little play. Turn the gears slowly by hand, mark tight spots with chalk and pare them down. Grease the teeth with tallow.

Safety. Meshing gears pull in fingers, hair and loose clothing at the point where the teeth come together. Fit a guard, tie back hair and sleeves, and never reach into gearing while it can turn: stop and lock the drive first.

History and evolution

A rack and pinion turns rotation into straight-line motion.

The Mechanical Problems, written in Aristotle’s school around 280 BCE, describes wheels in contact turning in opposite directions, but says nothing of teeth; they may have been plain wheels relying on friction. The oldest gears that survive are small metal gears from China’s Warring States period, in the fourth century BCE, kept at the museum in Luoyang.

In Alexandria in the third century BCE, Ctesibius built a water clock in which a rising float moved a rod and drums with equal teeth, as Vitruvius later described it. Archimedes is credited with the worm around 250 BCE.

The largest fragment of the Antikythera mechanism, with its four-spoked main wheel.

In 1901 divers found the most complex ancient gearing known in a wreck off the Greek island of Antikythera. The mechanism, made in the second or first century BCE, survives in 82 fragments holding 30 bronze gearwheels, about a third of the original machine. Turned by hand, it showed the positions of the Sun, the Moon and the planets and predicted eclipses.

Around 25 BCE Vitruvius described a water mill in which a toothed drum on the wheel’s axle drove a larger horizontal wheel that turned the millstone. Face wheels and lantern pinions of this kind drove mills for the next fifteen centuries. Mechanical clocks spread in Europe from about 1300. Giovanni de’ Dondi’s astronomical clock is traditionally dated to 1348–1364, though the construction dates are disputed: Milan’s science and technology museum gives 1365–1380. In the 1490s Leonardo da Vinci drew tooth shapes in the notebooks now called the Madrid Codices.

Philippe de La Hire analysed tooth curves in about 1694, and around 1754 Leonhard Euler set out the rule that teeth must follow for a steady ratio. In the nineteenth century cast-iron and then machine-cut steel gears drove the factories, and the involute replaced the cycloid in most machines.