Gears, gears, gears!

October 16, 2026

Gears are toothed wheels that pass rotation from one shaft to another. They are among the most common parts in any machine: car gearboxes, clocks, cordless drills, wind turbines, robot arms and printers all depend on them. Gears are useful because they change the speed and the torque (twisting force) of a shaft by an exact, fixed ratio while passing on nearly all of its power. They can also reverse the direction of rotation, carry it around a corner, or turn it into straight-line motion.

The ratio comes from the teeth. Each tooth on one gear pushes exactly one tooth on the other past the point where they touch, so the gears can't slip. When a 12-tooth gear drives a 36-tooth gear, the small gear turns three times to push all 36 teeth past, so the big gear turns at a third of the speed.

The torque changes by the same ratio in the other direction. The teeth push on each other with one force, and torque is that force multiplied by the radius it acts at. The 36-tooth gear's radius is three times larger, so it gets three times the torque. Power is torque multiplied by speed, so it comes through unchanged apart from the 1–2% lost to friction between the teeth.

Each section below has a picture, and the models turn on their own until you click one to pause it.

1. The parts of a gear

For teeth to pass one for one, both gears need teeth of the same size at the same spacing. So every gear's size comes from one number, the module m: the tooth size in millimetres. A gear with z teeth has a pitch diameter of m·z, so a module-2 gear with 30 teeth is 60 mm across.

The pitch circle is the circle the gear would be if it were a smooth wheel with no teeth. Two gears in mesh behave exactly like their two pitch circles rolling on each other without slipping, and every other dimension starts from them.

Each tooth sticks out one module beyond the pitch circle (the addendum) and is cut 1.25 modules below it (the dedendum). The extra quarter module is clearance, so the tip of one gear never hits the bottom of the other's space. Along the pitch circle, one tooth and one space take up one circular pitch, πm, and they split it equally. And since the two pitch circles touch, the centres sit one pitch radius plus the other apart. For a standard 20° tooth:

Two gears mesh only if their module and their pressure angle match (the 20°, explained in section 4). Standard modules run 0.5, 0.8, 1, 1.25, 1.5, 2, 2.5, 3, 4, 5, 6 and up. Inch-unit gears use diametral pitch instead, P = 25.4/m.

2. Ratio: counting teeth

Teeth can't skip, so counting them gives the ratio exactly.

When the driver turns once, all z₁ of its teeth pass the contact point, and each one pushes one tooth of the driven gear past it. That's z₁ teeth out of the driven gear's z₂, so the driven gear turns z₁/z₂ of a turn. The ratio i is the number of input turns for one output turn:

Because d = m·z and both gears share the same m, the ratio is also the ratio of the pitch diameters. That's the same no-slip rim, measured as distance instead of teeth.

Worked example. A motor turns a 12-tooth pinion at 1,500 rpm, and it drives a 48-tooth gear. The ratio is 48/12 = 4, so the gear turns at 1,500/4 = 375 rpm.

3. Torque: one force, two levers

Speed went down by the ratio. Power can't come from nowhere, and it doesn't vanish either, so something has to go up by the same factor, and that something is torque.

At the contact point the teeth push on each other with one force F, equal and opposite (Newton's third law). Each gear turns that force into torque with its own pitch radius as the lever arm:

Same force on a longer lever gives more torque, in exactly the ratio of the radii, which is the tooth ratio again. So torque goes up by the same factor that speed goes down, and power, torque × speed, comes through unchanged. Real teeth slide slightly as they mesh, and that friction costs 1–2% per mesh. The efficiency η accounts for it:

Worked example. The motor above makes 2 N·m. Through the 4:1 pair the output gets about 0.98 × 4 × 2 ≈ 7.8 N·m at 375 rpm. Check the power: 2 N·m × (1,500 × 2π/60) ≈ 314 W in, and 7.8 × (375 × 2π/60) ≈ 307 W out, so about 6 W became heat.

The same lever gives the force on the teeth: F = T/r = 2T/d. For a given torque, a smaller pinion has a shorter lever, so its teeth push harder. That's why the pinion is usually the part that fails first.

4. Why teeth have that shape

So far the teeth have acted like a perfect rolling rim. Real teeth only copy that rim, and their shape decides how well. If the teeth were square pegs, the point where each one pushes would slide in and out as it swung into mesh. The lever arm would keep changing, so the output speed would wobble once per tooth and the gears would buzz and hammer.

The ratio stays exactly constant only if the push between the teeth always points through one fixed spot on the line between the centres. That spot is the pitch point, where the two pitch circles touch, and this rule is the law of gearing. The tooth curve that obeys it is the involute: the path traced by the end of a taut string as you unwind it from a circle, the base circle. Two involutes pressed together always touch somewhere on one straight line, tangent to both base circles, so the push always points along that line and through the pitch point.

That line is the line of action, and its angle to the common tangent at the pitch point is the pressure angle φ, standardised at 20°. Because the push never leaves that line:

• The force direction never changes, so the bearings feel a steady load.
• The ratio doesn't depend on the centre distance. The ratio comes from the base circles, and pulling the gears apart doesn't change them. The line of action just tilts slightly, and the teeth get a little more play. No gearbox is ever assembled at the perfect distance, so this matters a lot.
• The force always splits the same way: a tangential part Ft that does the work, and a radial part Fr = Ft·tan φ that pushes the shafts apart, about 0.36·Ft at 20°.

Contact ratio

Each tooth pair touches only while its contact point travels along part of the line of action. So before one pair lets go, the next has to have started, or the drive would stop for an instant. The contact ratio ε is the length of line of action the gears use, divided by the tooth spacing along it. It says how many pairs are in contact on average:

Spur gears usually land between 1.4 and 1.8, so two pairs share the load part of the time and one pair carries it the rest. Below about 1.2, small manufacturing errors leave gaps in the handover, so keep ε above that.

5. Gear trains

A single pair tops out around 5–7:1. Past that, either the pinion has so few teeth that the cutter undercuts them (section 8), or the gear gets huge. For more ratio, you chain pairs together.

Simple train. Each gear sits on its own shaft and meshes with the next. Every tooth that passes one contact also passes the next, so the middle gears cancel and only the first and last count:

A middle gear, called an idler, still matters for direction. Each mesh reverses the rotation, so one idler makes the output turn the same way as the input. Idlers also bridge a gap between shafts.

Compound train. Fix two gears of different sizes to one shaft and they turn together. The first stage's output becomes the second stage's input, so the stage ratios multiply:

Two 3:1 stages make 9:1, and three make 27:1, with every gear still a sensible size. For the smallest gearbox, split the total ratio roughly evenly between the stages. The losses multiply too: three stages at 98% pass about 94%.

Hunting teeth. If the two tooth counts share a factor, say 20 and 60, each pinion tooth only ever meets the same three teeth on the gear. A damaged tooth keeps wearing the same partners. With counts that share no factor, like 19 and 60, every tooth eventually meets every other, and the wear spreads evenly.

6. The kinds of gears

Everything so far assumed parallel shafts and straight teeth. Real machines need shafts that meet at an angle, cross, or share one axis, and many need to run quietly. Each type below is the same no-slip contact, rearranged for one of those needs.

Spur

Straight teeth on parallel shafts: the gears in every figure so far. They are cheap, easy to make and inspect, pass 98–99% of the power through each mesh, and push nothing along the shaft. But the whole width of a tooth comes into contact at once, so each tooth arrives with a small impact, and at speed those impacts add up to noise.

Helical

Twist the teeth into a helix, at a helix angle β, and each tooth enters the mesh at one end and rolls across the face. The load builds up gradually instead of all at once, so the drive runs smoother and quieter, and more teeth share the load at any moment. Two helical gears on parallel shafts need equal angles of opposite hand, like a left-hand and a right-hand screw.

The slant has two costs. Across the slanted tooth, the module (mₙ) is smaller than in the plane of rotation, so the same tooth count makes a bigger gear: d = mₙ·z / cos β. And the slanted contact pushes along the shaft, so the bearings have to take thrust.

Herringbone (double helical)

A helix of one hand pushes along the shaft one way, and a helix of the other hand pushes the other way. Join two helical gears of opposite hand face to face, and the two thrusts cancel inside the gear. You keep the smoothness, and the bearings feel no thrust at all. The V makes them harder to make: a hob can't run straight through it, so most have a groove down the middle. One of the two gears must also float sideways, so that both halves settle into sharing the load.

Internal

Cut the teeth on the inside of a ring and the pinion runs inside it. Now both turn the same way, because their edges at the contact point move in the same direction and both centres sit on the same side of it. The centres are close together, so the drive is compact. And a convex tooth pressing into a concave one touches over a wider patch than two convex teeth do, so the contact stress drops.

Planetary (epicyclic)

Put several pinions inside a ring gear, all meshing with one gear in the middle, and you have a planetary set. The middle gear is the sun, the pinions are planets on a carrier, and the ring goes around everything. That makes three members you can connect to the outside: sun, carrier and ring. Hold one still, drive another, and the third is the output. The buttons in the figure try each one.

The ratios look strange until you ride along on the carrier. From there the planets just spin in place, so the sun, planets and ring form an ordinary gear train, with the planets as idlers between sun and ring. That train's ratio is −z_r/z_s, negative because the sun and ring turn opposite ways. Measure every speed relative to the carrier, and you get the Willis equation:

Set the held member's speed to zero and solve for the other two:

Two rules make the teeth fit together:

The first says the planets exactly fill the gap: the ring's pitch diameter is the sun's plus two planet diameters. The second lets equally spaced planets all mesh at once. Here (12 + 48)/3 = 20.

The same layout explains why planetaries are everywhere:

• Torque density. The load splits across three to five planet meshes instead of one. And the planets press into the ring's concave teeth, which carry more than two external gears could. So a planetary stage carries several times the torque of a single gear pair of the same diameter.
• Coaxial. Input and output share one axis, so stages stack like coins. Two stages at 5 : 1 make 25 : 1 in a short cylinder.
• Balanced. The planets push on the sun from equally spaced sides, so their radial forces cancel. The shafts carry torque with almost no side load.
• Several ratios from one set. Holding a different member gives a different ratio, so brakes and clutches can change gear without sliding any gears in or out. That's how a conventional automatic transmission works, and how a hybrid such as the Toyota Prius splits engine power between the wheels and a generator.

The costs come from the same layout: more parts, tolerances tight enough that every planet really takes its share, and some backlash, typically a few arcminutes in a precision unit.

How gear reductions work compares planetaries with harmonic and cycloidal reducers for robots.

Rack and pinion

Make a gear bigger and bigger, and its pitch circle flattens until it becomes a straight line, which is a rack. The no-slip rule still holds, so the rack moves exactly as far as the pinion's pitch circle rolls: one turn moves it π·d. The lever between torque and force is the pinion's pitch radius, so the push on the rack is F = T/r = 2T/d. A rack turns rotation into straight-line motion over any length you can make the rack.

Bevel

To turn a corner, the two rolling surfaces have to meet at an angle. Two cones that share a tip roll on each other without slipping, the way two discs do. Every point along the line where they touch is the same fraction of the way to the tip on both cones, so their surface speeds match all the way along. So bevel gears are cut on cones, with teeth that shrink toward the shared tip. The ratio is still the tooth ratio, and it fixes the cone angles: tan δ₁ = z₁/z₂.

Straight bevel teeth have the spur gear's problem, an impact as each tooth arrives. Spiral bevels curve the teeth to fix it, for the same reason helicals slant them. Hypoid gears go one step further and offset the pinion below the gear's centre. That makes the pinion larger and stronger and lets a car's driveshaft sit lower, at the cost of more sliding and hotter oil.

Worm

A worm is a screw, and the wheel's teeth act as its nut. Each turn of a single-start screw moves the nut by one thread, and here one thread is one tooth:

So a 30-tooth wheel gives 30:1 in one compact stage, at right angles. The price comes from the same screw action. The thread slides across the teeth instead of rolling, and sliding under load turns power into heat. Roughly η ≈ tan γ / tan(γ + ρ), where γ is the thread's lead angle and ρ = atan μ is the friction angle. A shallower thread means a smaller γ, so friction takes a bigger share. Once γ drops below ρ, the wheel can't drive the worm backwards at all. Don't rely on that as a brake, though: friction changes with temperature, vibration and wear. The usual pairing is a hardened steel worm against a bronze wheel, because bronze slides well on steel.

The rest of the family

These are rarer, and each one does a single job better than the main types:

• Crossed helical (screw gears). Two helical gears on shafts that cross without meeting. The teeth touch at a single point, so they carry only light loads. Used in distributor and oil-pump drives in older engines, and in speedometer take-offs.
• Spiral bevel. A bevel gear with curved teeth, which gives it the smoothness of a helical. Used in helicopter tail-rotor drives, motorcycle shaft drives, and high-speed right-angle gearboxes.
• Face gear. Teeth cut into the flat face of a disc, meshing with an ordinary spur pinion at 90°. The pinion can sit anywhere along its axis and still mesh, so it's cheap to assemble. Used in power tools and some helicopter transmissions.
• Strain wave (harmonic). An oval cam squeezes a flexible thin-walled gear into an oval inside a rigid ring with two more teeth. Each turn of the cam walks the flexible gear back by two teeth, which gives 50–160 : 1 in one stage with almost no backlash. Used in robot arm joints and spacecraft mechanisms.
• Cycloidal. A lobed disc wobbles inside a ring of pins, one lobe short, so each wobble advances it by one lobe. Many lobes share the load at once, so it's very stiff and tolerates shocks. Used in industrial robot bases and positioners.
• Non-circular gears. Elliptical or irregular pitch curves change the effective radius through each turn, so the speed changes too. Used in flow meters, some presses, and mechanisms whose return stroke must take less time than the working stroke.
• Geneva drive and sector gears. These move in steps rather than continuously: a Geneva wheel turns a quarter turn and then locks. Used in film projectors, indexing tables, and watch mechanisms.

How gear reductions work animates and compares the strain wave and cycloidal drives.

7. Forces on the teeth

Section 4 showed that the push between the teeth always runs along the line of action. A machine cares about that push in three directions: the tangential part turns the gear, and the radial part pushes the shafts apart. On slanted or cone-shaped teeth, a third, axial part also pushes along the shaft. The tangential part comes straight from the torque, Ft = 2T/d, and the tooth geometry sets the other two. The bearings and housing have to carry all three.

Worked example. Put 10 N·m on a 72 mm pitch diameter and the tangential force is about 278 newtons (Ft = 2 × 10 / 0.072). On a spur gear the radial force is about 101 newtons (278 × tan 20°), so the bearings see about 296 newtons in total. Make it helical at 20° and you add about 101 newtons of axial thrust, which needs a thrust bearing.

8. Making teeth

An involute is awkward to cut directly, but it has a useful property: an involute gear meshes perfectly with a rack that has straight-sided teeth. So instead of copying the curve, gear makers roll a rack-shaped cutter against the blank as if the two were meshing, and cut away everything the cutter passes through. What's left is exactly the shape that meshes with the rack, which is the involute. Gear makers call this generating, and they make almost all metal gears this way.

The machines differ in how they hold that rack and keep it rolling:

• Hobbing. The hob is a rack wrapped into a screw, spinning in step with the blank. It's the workhorse for external spur and helical gears. It cuts a small gear in minutes, holds good accuracy, and one hob cuts any tooth count of its module.
• Shaping. A hob can't reach inside a ring or right up to a shoulder, so a gear-shaped cutter strokes up and down there instead, rolling with the blank. Shaping cuts internal gears and gears next to a shoulder.
• Power skiving. A tilted gear-shaped cutter that cuts as it rolls. It's much faster than shaping for internal gears.
• Form milling. A cutter shaped like one tooth space, with the blank indexed round between cuts. It doesn't generate, so each tooth count needs its own cutter, which makes it a choice for one-offs.
• Broaching. A long multi-tooth tool pulled through in one pass, for internal gears and splines in volume.
• Grinding and honing finish hardened gears. Hardening distorts them, so precise gears are cut soft, hardened, then ground.
• Without cutting. Powder metal, injection-moulded plastic, stamping for thin clock gears, wire EDM for prototypes, and 3D printing for low loads.

Undercut. On a pinion with few teeth, the rack cutter's tips reach inside the base circle and carve away the bottom of the flank (try 10 teeth in the figure). That thins the tooth exactly where the bending stress peaks (section 9). For a 20° rack it starts below z = 2/sin²φ ≈ 17 teeth. The cure is to move the cutter out by x·m, called profile shift, with x ≈ 1 − z·sin²φ/2. The tooth grows thicker at the root and thinner at the tip, so keep the tip above about 0.25–0.4·m.

Backlash. Section 4 showed that moving the gears apart changes only the play, never the ratio. Gears need some play: room for oil, for heat growth, and for manufacturing error. You get it by thinning the teeth slightly or by adding centre distance. Each 0.1 mm of extra distance adds about 2 × 0.1 × tan 20° ≈ 0.07 mm of play. General gearing uses about 0.04–0.1·m.

Materials.

• Through-hardened steel (about 300 HB) is cheap.
• Carburized steel has a 58–62 HRC case over a tough core and carries about twice the contact stress.
• Nitriding gives a thin, hard case with little distortion.
• Bronze is the usual partner for a steel worm.
• Acetal and nylon are quiet, self-lubricating and cheap in volume, but about ten times weaker than steel, and they soften with heat.

Accuracy. Standards (ISO 1328, AGMA 2015) grade gears by their errors, and a lower number means more accurate. Precision ground gears sit around grades 2–5, good industrial gears around 6–8, and moulded or powder-metal gears at 8 and coarser.

9. How teeth fail

Each tooth is a short, stubby beam, fixed at the root and pushed near the tip. So it can fail as a beam, by cracking at the root, or at its surface, by wearing out under the contact pressure. The two failures have different causes and different cures.

Bending. The tangential force bends the tooth like a cantilever, and a cantilever's stress peaks where it's held, at the root fillet. Lewis's 1892 formula is still the first check:

The force spreads over the face width b, and the module sets how thick the tooth is, so bigger teeth mean lower bending stress. Overloaded, the tooth grows a fatigue crack from the root until it snaps off.

Contact. Two curved flanks pressed together touch along a thin line, so even a moderate force makes a very high local pressure, called Hertzian stress. After millions of cycles, that pressure lifts tiny flakes out of the surface near the pitch line, leaving the pits that give the failure its name, pitting.

ZE ≈ 190 √MPa for steel on steel, and ZH ≈ 2.5 for standard spur teeth. The pressure depends on how sharply the flanks curve, and the pinion diameter sets that, not the module. And because the stress grows with the square root of the load per width, doubling the face width lowers it by only about √2.

Diameter controls contact and module controls bending, so size a gear in that order. Pick the pinion diameter and face width for contact first, then pick a module that passes bending. Try it:

Other ways teeth fail:

• Scuffing. The oil film breaks under high sliding and load, and the surfaces weld and tear.
• Micropitting. A grey, frosted band of tiny pits on hard flanks, from a thin oil film.
• Wear. From dirt, or slow running with no oil film.
• Plastic gears also fail by heat: softening, creep and melting.

Full ratings (ISO 6336, AGMA 2001) start from these two equations and multiply in factors for shock, speed, misalignment, size and life.

10. Heat, oil, and noise

The teeth roll at the pitch point but slide everywhere else along the line of action, and sliding under load is where the lost power goes. All of it ends up as heat:

A 10 kW gearbox at 97% efficiency has to get rid of 300 W. At 75%, a typical worm, it has to get rid of 2.5 kW. The oil's job is to keep the sliding surfaces apart, and the right oil depends on speed. Slow, heavily loaded gears need thick oil or grease to hold a film. Gears with pitch-line speeds above about 15 m/s need thinner oil sprayed straight at the mesh. Too much oil runs hotter than the right amount, because the gears waste power churning it.

Noise has a cause too. Each time one tooth pair hands the load to the next, the output jitters very slightly. So gears whine at the mesh frequency, the rate at which teeth meet:

Engineers call that jitter transmission error: under load the teeth bend, and their profiles aren't perfect. Each cure attacks it from one side. A higher contact ratio, or helical teeth, overlaps the handover. Tip relief eases each tooth into contact. Crowning, a slight barrel across the face, stops misalignment from loading one edge. And a better accuracy grade shrinks the profile errors themselves.

11. Where gears are used

The same few types turn up again and again, and what changes from machine to machine is which constraint wins.

12. Picking gears

Pick in this order. The shaft layout comes first, because it rules out most of the table, and the ratio comes second, because it sets how many stages you need. Tooth form comes last, once those two have narrowed the choice. Then:

1. Find the forces at the pitch circle, and give them somewhere stiff to go.
2. Size for contact first: pick the pinion diameter and face width (about 8–12 × the module). Then pick the module for bending, with 17 or more pinion teeth or a profile shift.
3. Choose hunting tooth counts, and keep the contact ratio above 1.2.
4. Account for every watt that doesn't reach the output. It's heat, and it has to go somewhere.

Sources

• NASA TM-105388: Contact stresses in meshing spur gear teeth
• NASA: A method for thermal analysis of spiral bevel gears
• NASA: Experimental and analytical spiral bevel gear-tooth stress
• ISO 53 (basic rack), ISO 1328 (accuracy), ISO 6336 (load capacity)
• Budynas and Nisbett, Shigley's Mechanical Engineering Design, the gear chapters
• Dudley, Handbook of Practical Gear Design