= Gear mesh stiffness
{title2=$k_{\rm mesh}=\sum_jk_j$}
\b[Gear mesh stiffness] describes the force response of the engaged tooth contacts to a specified relative approach along the mesh direction. In the linear parallel-contact model underlying <gear tooth load sharing>, all active pairs have the same approach $\delta$ and
$$
F=\sum_jk_j\delta=k_{\rm mesh}\delta,\qquad k_{\rm mesh}=\sum_jk_j.
$$
The sum assumes compatible displacement coordinates. Adding a bending stiffness, a torsional stiffness and a normal contact stiffness without transforming their coordinates would be dimensionally and mechanically inconsistent.
For an original example, two zero-gap pairs with $k_1=80$ and $k_2=120\,\mathrm{N/\mu m}$ give $k_{\rm mesh}=200\,\mathrm{N/\mu m}$. A $1000\,\mathrm N$ load produces $5\,\mathrm{\mu m}$ normal approach. If only the first pair is engaged, the same ideal stiffness model instead gives $12.5\,\mathrm{\mu m}$.
With gaps or nonlinear response, specify whether the reported value is <secant stiffness> $F/\delta$ from a reference or the local <gear mesh tangent stiffness> $dF/d\delta$. Suppose
$$
F(\delta)=80\delta+120\max(\delta-3,0),
$$
where $\delta$ is measured in micrometres and the result in newtons. The second pair starts carrying load at $\delta=3\,\mathrm{\mu m}$, corresponding to $F=240\,\mathrm N$. At $F=720\,\mathrm N$, approach is $5.4\,\mathrm{\mu m}$, so the origin-based <secant stiffness> is about $133.3\,\mathrm{N/\mu m}$ while <tangent stiffness> is $200\,\mathrm{N/\mu m}$. At the ideal corner itself, a unique two-sided <derivative> does not exist.
A small-perturbation experiment can hold contact position and mean load approximately fixed, apply a controlled force increment and measure the corresponding relative displacement. Shaft and support <mechanical compliance> must be separated or included explicitly in the reported system response. Rotating the mesh changes contact position, so one scalar does not necessarily describe a whole cycle.
This matters dynamically: changes in <gear mesh stiffness> can alter oscillatory response and <gear tooth load fractions>. The primary NASA analysis used an improved tooth-stiffness model coupled to dynamic-load calculations, rather than prescribing stiffness independently of the load solution.<Predicted Effect of Dynamic Load on Pitting Fatigue Life for Low-Contact-Ratio Spur Gears by David G. Lewicki>[Summary, printed p. 1, modified TELSGE and stiffness–dynamic-load iteration]{ref} The elementary parallel-spring sum explains the linear limit, while real contacts require a stated engagement and deformation model.
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