Gear tooth failure and surface durability

Modes of failure of gear teeth

Gears are usually designed for beam strength (bending) and then checked for wear strength (surface durability) and dynamic load.

Surface fatigue (pitting)

The contact between two teeth is like two cylinders pressed together (Hertz contact). The maximum contact stress is high and repeats with every mesh, so subsurface shear stresses start a crack that grows to the surface, and a small pit forms. Pitting begins near the pitch line, where there is pure rolling and the oil film is thin.

Buckingham's equation for limiting wear load

$$F_w = d_1\, b\, Q\, K$$

where

• $F_w$ is the limiting (maximum) wear load, N;
• $d_1$ is the pitch diameter of the pinion, mm;
• $b$ is the face width, mm;
• $Q$ is the ratio factor: $Q = \dfrac{2 z_2}{z_1 + z_2}$ for external gears and $Q = \dfrac{2z_2}{z_2 - z_1}$ for internal gears ($z_1$ = pinion teeth, $z_2$ = gear teeth);
• $K$ is the load-stress factor (N/mm$^2$), obtained from the Hertz stress:

$$K = \frac{\sigma_{es}^2 \sin\phi}{1.4}\left(\frac{1}{E_1} + \frac{1}{E_2}\right)$$

with $\sigma_{es}$ the surface endurance limit (often taken as $2.75\,\text{BHN} - 70$ MPa for steel; the value depends on the textbook), $\phi$ the pressure angle, and $E_1$, $E_2$ the moduli of elasticity.

To be safe against pitting, the wear load must be at least equal to the dynamic load: $F_w \ge F_d$. For this reason the surface hardness is raised by through hardening, case hardening, or flame or induction hardening.