= Gear mesh tangent stiffness
The \b[gear mesh tangent stiffness] is the local derivative of total mesh force with respect to a specified relative approach, with contact position and other state variables held fixed:
$$
k_{\rm tan}(\delta_0)=\left.\frac{dF}{d\delta}\right|_{\delta_0}.
$$
It predicts a sufficiently small incremental force $\Delta F\simeq k_{\rm tan}\Delta\delta$. Unlike a reference-based <secant stiffness> $F/\delta$, it describes the local response rather than the whole loading interval.
For $F(\delta)=\sum_j k_j\max(\delta-e_j,0)$ away from any <gear tooth engagement threshold>,
$$
k_{\rm tan}(\delta)=\sum_{j:\,\delta>e_j} k_j.
$$
Only members of the <gear mesh active-contact set> contribute. With $k_1=80$, $k_2=120\,\mathrm{N/\mu m}$ and gaps $0$ and $3\,\mathrm{\mu m}$, the <gear mesh tangent stiffness> is $80\,\mathrm{N/\mu m}$ before the second contact and $200\,\mathrm{N/\mu m}$ after it. At approach $4.8\,\mathrm{\mu m}$, an additional $0.1\,\mathrm{\mu m}$ gives $20\,\mathrm N$ more force within the same branch.
The ideal transition at $3\,\mathrm{\mu m}$ has distinct left and right derivatives. A centered <finite difference> with any sufficiently small positive $h$ gives
$$
\frac{F(3+h)-F(3-h)}{2h}=140\,\mathrm{N/\mu m},
$$
where numerical displacements are expressed in micrometres. The value $140$ is the mean of the two branch slopes; it does not establish a unique <derivative> at the corner, even as $h$ shrinks.
\Image[/gear-engagement-tangent-stiffness.png]
{title=Original unilateral-contact model: continuous force, a <gear tooth engagement threshold>, and distinct one-sided <gear mesh tangent stiffness> values}
The measured displacement channel also matters. If a contact stiffness $200\,\mathrm{N/\mu m}$ is in series with support stiffness $300\,\mathrm{N/\mu m}$, their <mechanical compliances> add, giving
$$
k_{\rm measured}=\left(\frac1{200}+\frac1{300}\right)^{-1}=120\,\mathrm{N/\mu m}.
$$
A measurement spanning the whole assembly would therefore not isolate the contact stiffness.
For a perturbation experiment, report the mean load, mesh position, displacement endpoints and whether the increment crosses a contact transition. A primary NASA analysis used a coupled tooth-stiffness/dynamic-load calculation; the derivative above explains a local quasistatic limit, rather than replacing that dynamical model.<Predicted Effect of Dynamic Load on Pitting Fatigue Life for Low-Contact-Ratio Spur Gears by David G. Lewicki>[Summary, printed p. 1, modified tooth-stiffness model and dynamic-load iteration]{ref}
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