Machine Design

Optimizing Gear Teeth For Maximum Loads

Modifying gear geometry can reduce contact stresses and minimize pitting.

By Prokop Sroda
University of Mining and Metallurgy
Kraków, Poland


Pitting commonly appears on operating surfaces of gear teeth. A fundamental cause is excessive loading that raised contact stresses beyond critical levels.

Pitting chips first appear on the highest pressure surfaces of mating gear teeth. As the condition progresses, teeth profiles tend to deform, resulting in ever increasing dynamic forces as well as vibration and noise. Left unchecked, pitting will eventually destroy the teeth.

Fortunately, gear profiles can be modified to increase their durability and load capacity. The key involves determining the optimum involute profile that minimizes contact stresses. The process essentially changes kinetics and dynamics of the gear pair, compared with uncorrected gears, by increasing the correction coefficient. This moves the active profile of a tooth into an area of the involute with a greater radius of curvature. The result increases contact pressure strength of the teeth and reduces the prospect of pitting failure. Here's a look at the underlying theory.

Contact stress (Hertzian pressure) at the operating pitch circle is defined by ISO standards as:

where the basic contact stress is:

The zone factor ZH is defined as

For external and internal gears with normal pressure angle n = 20°, the Zone factor diagram provides ZH curves in terms of correction coefficients x1 and x2, the number of pinion and gear teeth z1 and z2, and reference helix angle .

The elasticity factor ZE is determined from:

The contact-ratio factor Z for spur gears is

For helical gears the contact ratio factor is

for < 1; and

for εβ >1.

Helix-angle factor Z is

The maximum contact stress recommended by ISO standards should be evaluated separately for pinion and gear. It is defined as

Surface load capacity is calculated based on Hertzian pressure on the operating pitch circle. The actual contact stress H at the pitch circle must be less than or equal to the allowable contact stress,

Substituting the equations for contact stress and allowable contact stress into the above relation-ship results in:

This equation lets designers determine the maximum permissible load QH before the onset of pitting.

Correcting the radius of the contacting teeth changes curvature profiles and, with it, the coefficient ZH in the previous equation. The result becomes an iterative optimization problem with the goal of determining the set of permissible correction coefficients that maximize teeth load capacity for both pinion and gear teeth.

An example illustrates the effect of correction on permissible unit loads. a set of gears with z1 = 17, z2 = 42, mn = 3.0, = 0°, and toothed rims width bw1 = 32 and bw2 = 30. Both wheels are of 18H2N8 grade steel, carburized and hardened to 52 HRC, manufactured to Grade 7 accuracy. The pinion gear operates at 1,500 rpm.

The Permissible unit loads graph shows the load-limit results for various gear parameters with a factor of safety of 1.1. Line "a" in the figure represents the contact ratio = 1.2, "b" are measures of tip relief, "c" are ures of tooth cut, and "e" and "f" are measures of tooth interference.

A computer program that analyzes the correction effect on permissible unit load lets designers calculate permissible loads for the entire range of possible rection coefficients. In the example, the optimum values of correction coefficients are at point A where x1 = 0.28 and x2 = 1.5 on line a. This permits a 19.5% increase in permissible specific load in relation to uncorrected gears.

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