= Gear mesh active-contact set
The \b[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,
$$
F_j=k_j\max(\delta-e_j,0),\qquad
\mathcal A(\delta)=\{j:\delta>e_j\},\qquad
F=\sum_jF_j.
$$
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 gives
$$
\delta=\frac{F+\sum_{j\in\mathcal A}k_je_j}{\sum_{j\in\mathcal A}k_j}.
$$
A <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 $100,50,150\,\mathrm{N/\mu m}$ and gaps $0,2,5\,\mathrm{\mu m}$. At total load $500\,\mathrm N$, the set $\{1,2\}$ gives approach $(500+50\cdot2)/150=4\,\mathrm{\mu m}$ and forces $400,100,0\,\mathrm N$. This set is consistent.
At $900\,\mathrm N$, keeping only those two contacts would give approach $1000/150\simeq6.667\,\mathrm{\mu m}$, beyond the third gap. Including all three instead gives
$$
\delta=\frac{900+50\cdot2+150\cdot5}{300}=\frac{35}{6}\,\mathrm{\mu m},
$$
and 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 $100\cdot5+50(5-2)=650\,\mathrm N$.
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.<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, stiffness–load iteration and contact-position dependence]{ref} The active-set calculation is an explicit quasistatic model, not a substitute for that fuller mechanical description.
Back to article page
