 Gear contact-stress load factor

The gear contact-stress load factor $\sqrt{K_{H\beta }}$ is the multiplier associated with the face-load term in a contact-stress calculation where Hertzian contact pressure scales with the square root of local line load. It distinguishes a stress ratio from the underlying gear face load factor. The historical definition of $K_{H\beta }$ compares maximum line-load intensity with the mean over the evaluation width.^[ref]

For ideal elastic line contact, hold the effective geometry and material fixed. Write the peak pressure aswhere $w$ is load per contact length, $E^{\ast }$ the effective elastic modulus and $R^{\ast }$ the effective radius. If the largest line load is $w_{\max }=K_{H\beta }\overline{w}$, division by the pressure calculated from the mean line load givesThe square root follows from the pressure model; it is not a second independent multiplier on the load.

For an original example, a face-load ratio $K_{H\beta }=2.25$ gives a stress multiplier $1.5$. A mean-line-load reference pressure $600\mathrm{MPa}$ would become $900\mathrm{MPa}$ under these assumptions. Multiplying pressure directly by $2.25$ would instead give $1350\mathrm{MPa}$ and count the load ratio as though it were a stress ratio. Increasing effective line load fourfold doubles this model pressure.

Small relative uncertainties also scale differently:Thus a small $8\%$ relative change in $K$ produces about a $4\%$ relative change in its square root to first order. This sensitivity concerns that term alone; uncertainties in material, geometry and the reference load remain.

A local stress scaling does not predict gear surface pitting life on its own. Contact position, curvature, load history and material response matter. A primary NASA analysis explicitly combined varying tooth loads and contact geometry with a fatigue-life model.^[ref] A real rating must keep all factors and load conventions consistent with the chosen method.