Spur gears: why teeth break and surfaces pit

Teeth shaped to keep the speed steady

Almost every gear tooth is cut to an involute, the curve traced by the end of a taut string unwound from a circle. Two involute teeth touch along one straight line, the line of action, which crosses the line between the gear centres at the pressure angle, usually 20°. Because that line never moves, the speed ratio stays exactly constant as each tooth rolls through the mesh, even if the centres are not quite where the drawing puts them.

The module describes the size of the teeth: the pitch diameter divided by the number of teeth. Any two gears with the same module and pressure angle mesh, and the ratio is simply the ratio of their tooth counts. A 17-tooth pinion driving a 52-tooth gear turns 3.06 times for each turn of the gear and multiplies the torque by the same amount, less friction.

How many teeth share the load

The contact ratio is the average number of tooth pairs in contact. Above 1, a new pair picks up the load before the old one lets go; typical spur gears run between about 1.4 and 1.8, carrying the load on two pairs for part of each cycle and on one pair for the rest. Designers avoid ratios below about 1.2, where small errors leave moments with no tooth properly in contact.

Small pinions have a limit of their own. With too few teeth, the tips of the gear dig into the pinion's flanks below the base circle, a fault called interference that undercuts the teeth as they are cut and weakens them. A 20° full-depth pinion needs 18 teeth to mesh with a rack; a 13-tooth pinion can drive nothing larger than 16 teeth.

A tooth is a short cantilever

In 1892 Wilfred Lewis, an engineer at the machine-tool maker William Sellers & Co. in Philadelphia, treated a gear tooth as a cantilever beam loaded at its tip. The bending stress is highest at the root, where the tooth joins the rim, and he captured the shape of the tooth in a single form factor Y: σ = W_t/(b·m·Y), with W_t the force along the pitch circle, b the face width and m the module.

Two things follow. Stress falls in proportion to the module, so bigger teeth are stronger; and pinions with few teeth have narrow roots and a small Y, so the pinion is usually the weaker of the pair in bending. The modern AGMA method keeps Lewis's form but replaces Y with a geometry factor J that adds the stress concentration in the root fillet and the sharing of load between teeth.

Pitting: two cylinders pressed together

Where two teeth touch, their curved faces behave like two cylinders pressed together. The contact is a narrow band and the pressure in it is very high, often several hundred megapascals even in a modest gearbox. The theory of contact between elastic bodies gives that stress, and with it a surprise: it grows only with the square root of the load, because a heavier load also flattens the surfaces into a wider band.

Millions of passes through that band start small fatigue cracks just beneath the surface. They grow and break out as pits, usually on or just below the pitch line, where a single pair of teeth carries the whole load and the sliding between the faces reverses direction. Pitting spreads, the teeth grow noisy and rough, and eventually a tooth breaks from the damage. Hardness is the defence: the allowable contact stress of through-hardened steel rises steeply with Brinell hardness.

From a textbook stress to a rating

Real gears are not perfect, so the AGMA method multiplies the ideal load by factors. The dynamic factor K_v accounts for the small impacts of tooth spacing errors, which grow with speed and shrink with more accurately cut gears. The overload factor K_o covers shocks from the motor and the machine. The load-distribution factor K_m allows for misalignment that puts more of the load on one end of the face, and a size factor K_s for the lower strength of larger teeth.

On the strength side, allowable stress numbers come from tests at 10⁷ cycles and 99% reliability. Stress-cycle factors adjust them for the life required, the reliability factor for the fraction of gears that must survive, and a hardness-ratio factor rewards a pinion harder than its gear, which work-hardens the gear's surface as they run. Pinions are often made harder anyway: in a 3:1 reduction each pinion tooth meshes three times for every gear tooth.

Which failure comes first

The two safety factors cannot be compared directly. Double the load and the bending factor S_F halves, but the contact stress rises only by √2, so the pitting factor S_H falls by that much. To see which limit a growing load reaches first, AGMA compares S_F with S_H², or S_H³ for crowned teeth, whose contact is closer to a point.

A pair with S_F = 2.0 and S_H = 1.6 looks weakest in pitting, yet S_H² = 2.56: the load can grow 2.56 times before the surfaces reach their limit but only twice before the pinion's teeth do. In practice through-hardened gears are often limited by pitting, and the fix is harder surfaces rather than bigger teeth.

How engineers learned this

Lewis's equation of 1892 was the first rational method for tooth strength, and it held for decades. Contact stress came from elsewhere: in the early 1880s Heinrich Hertz, years before his experiments with radio waves, worked out the stresses where two curved elastic bodies touch, and his solution became the basis of pitting calculations.

As machines ran faster in the twentieth century, gears failed at loads Lewis's formula said were safe. Research led by Earle Buckingham at MIT measured the extra dynamic loads that tooth errors cause at speed, and the American Gear Manufacturers Association gathered this work, with decades of test results, into the rating standards used for spur and helical gears today.

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