Gear Module Calculator

Module is the scaling unit of a metric gear tooth, defined as pitch diameter divided by tooth count, m = d / z, in millimetres. Every other proportion follows: addendum equals m, dedendum 1.25 m, whole depth 2.25 m, circular pitch pi x m, and centre distance for a mating pair m x (z1 + z2) / 2. Two gears mesh only if they share both module and pressure angle, which makes module the primary compatibility parameter rather than a size descriptor. Preferred values are standardized in ISO 54 - 1, 1.25, 1.5, 2, 2.5, 3, 4, 5, 6, 8, 10 and upward - because hobs are cut to those sizes, so a non-standard module means non-standard tooling. The imperial equivalent, diametral pitch, is the reciprocal in the other unit system, DP = 25.4 / m. Pressure angle sets the profile shape. Twenty degrees is the modern default; larger angles thicken the tooth root and improve bending strength at the cost of higher radial bearing loads and noise. It also fixes the undercutting threshold, since the minimum tooth count a rack-type cutter can generate without undercutting the root is about 2 / sin^2 of the pressure angle: roughly 17 teeth at 20 degrees, 12 at 25, and 32 at 14.5. Contact ratio, the average number of tooth pairs engaged, must exceed 1.0 and normally lands between 1.4 and 1.8 for spur gears. These formulas assume standard unshifted involute proportions per ISO 53, and say nothing about load capacity, which requires AGMA 2001 or ISO 6336.

The involute curve, traced by a point on a taut string unwinding from a base circle, was suggested for gear teeth by the French mathematician Philippe de La Hire in the 1690s, and Leonhard Euler developed the theory during the eighteenth century, showing that involute flanks transmit a constant angular velocity ratio and keep doing so when centre distance shifts slightly. That tolerance to mounting error is why the involute displaced the cycloidal profiles favoured by earlier millwrights and clockmakers everywhere except in horology. Robert Willis, professor at Cambridge, helped establish it in engineering practice with his 1838 work On the Teeth of Wheels, and the odd 14.5 degree pressure angle of early standards derives from its sine being close to one quarter, convenient for hand layout. Interchangeability required manufacturing to catch up with geometry. The generating principle, in which a straight-sided rack-form cutter produces correct involute flanks for any tooth count of the same module, let one tool serve a whole family of gears; Hermann Pfauter built the first universal gear hobbing machine in Germany in 1897. Standards bodies then fixed the numbers: the American Gear Manufacturers Association was founded in 1916, and ISO issued its cylindrical gear standards including ISO 54 for preferred modules and ISO 53 for the basic rack profile. The twenty degree pressure angle displaced 14.5 degrees through the mid-twentieth century as higher power densities made root bending strength the limiting factor.