= Gear contact-set consistency check
A \b[gear contact-set consistency check] verifies that a proposed solution satisfies force balance and the unilateral contact conditions used to define a <gear mesh active-contact set>. Solving an equation for selected contacts is insufficient if an omitted tooth should be loaded or an included tooth would require tension.
For the model $F_j=k_j\max(\delta-e_j,0)$, choose a trial active set $\mathcal A$ and solve
$$
\delta=\frac{F_{\rm applied}+\sum_{j\in\mathcal A}k_je_j}{\sum_{j\in\mathcal A}k_j}.
$$
The check then requires positive $k_j(\delta-e_j)$ for included pairs and $\delta\le e_j$ for excluded pairs. Recomputing every contact with the unilateral law gives the force residual
$$
r_F=\sum_j k_j\max(\delta-e_j,0)-F_{\rm applied}.
$$
A consistent exact solution has $r_F=0$ together with the inclusion inequalities.
An original example has stiffnesses $100,50,150\,\mathrm{N/\mu m}$ and gaps $0,2,5\,\mathrm{\mu m}$. At $F_{\rm applied}=900\,\mathrm N$, a trial set containing only the first two pairs gives $\delta=20/3\,\mathrm{\mu m}$. It balances those selected forces but exceeds the third gap by $5/3\,\mathrm{\mu m}$. The omitted contact would carry $250\,\mathrm N$, giving $r_F=250\,\mathrm N$. Including all three gives the consistent approach $35/6\,\mathrm{\mu m}$.
Conversely, including all three at applied load $500\,\mathrm N$ gives the un-clipped solution $\delta=4.5\,\mathrm{\mu m}$ and a third force $-75\,\mathrm N$. Rejecting that tensile contact and solving with the first two gives $\delta=4\,\mathrm{\mu m}$ and forces $400,100,0\,\mathrm N$.
Numerical tolerances should keep their units and roles clear. A displacement tolerance near a <gear tooth engagement threshold> is not the same quantity as a force-residual tolerance. At an exact zero-force touch, either a geometric-touch flag or the positive-force active convention may be useful, but the reported convention must be stated. The one-sided <gear mesh tangent stiffness> can differ across that boundary.
Primary dynamic gear analysis couples tooth <stiffness> and calculated loads as contact position changes.<Predicted Effect of Dynamic Load on Pitting Fatigue Life for Low-Contact-Ratio Spur Gears by David G. Lewicki>[Summary and Introduction, printed p. 1]{ref} The scalar check above validates internal consistency of its quasistatic model; it does not establish that the model includes every deformation or dynamic effect in a real mesh.
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