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The gear mesh active-contact set identifies the tooth pairs that carry positive compressive normal force in a specified contact model. It is not simply the set of geometrically overlapping teeth from the gear contact ratio. A pair touching at zero force is excluded under the positive-force convention used here.

For unilateral linear contacts,Here $e_j$ is the initial gap, $k_j>0$ the pair stiffness, and $\delta$ the common approach. For a proposed nonempty set $\mathcal{A}$, force balance givesA gear contact-set consistency check then verifies $\delta >e_j$ for included pairs and $\delta \le e_j$ for excluded ones. Solving the linear equation without this check can retain a negative force or omit a contact that should engage.

An original three-pair example uses stiffnesses $100,50,150N/\mu \mathrm{m}$ and gaps $0,2,5\mu \mathrm{m}$. At total load $500\mathrm{N}$, the set $\{1,2\}$ gives approach $(500+50\cdot 2)/150=4\mu \mathrm{m}$ and forces $400,100,0\mathrm{N}$. This set is consistent.

At $900\mathrm{N}$, keeping only those two contacts would give approach $1000/150\simeq 6.667\mu \mathrm{m}$, beyond the third gap. Including all three instead givesand forces approximately $583.33,191.67,125\mathrm{N}$, which sum to $900\mathrm{N}$. The third gear tooth engagement threshold occurs at total load $100\cdot 5+50(5-2)=650\mathrm{N}$.

Within a fixed active set, gear mesh tangent stiffness is the sum of its pair stiffnesses. At a transition, force can remain continuous while that derivative changes. For positive total load the scalar force law is strictly increasing after the first contact, giving a unique approach in this model; the unloaded state need not specify one unique separation.

Real deformation, local contact nonlinearity and dynamics can require more than one common-approach coordinate. A primary NASA analysis coupled tooth stiffness with dynamic-load calculations over contact position and speed.^[ref] The active-set calculation is an explicit quasistatic model, not a substitute for that fuller mechanical description.