= Gear tooth load sharing
<Gear tooth load sharing> is the division of transmitted mesh load among simultaneously contacting tooth pairs. <Gear contact ratio> determines how engagement intervals overlap, but it does not by itself specify each pair's <force>. Tooth and support deformation, profile geometry and operating speed affect that distribution. A primary dynamic analysis explicitly coupled tooth stiffness with calculated dynamic loads rather than assigning a fixed full load to every active pair.<Predicted Effect of Dynamic Load on Pitting Fatigue Life for Low-Contact-Ratio Spur Gears by David G. Lewicki>[Summary, printed p.1, tooth-stiffness model, stiffness/load iteration and varying contact-position loads]{ref}
An illuminating quasistatic model treats active pairs as parallel <springs> with a common normal approach $\delta$. With pair stiffnesses $k_j$ and no gaps, $F_j=k_j\delta$, so
$$
F=\sum_jF_j=\left(\sum_jk_j\right)\delta,\qquad
\eta_j=\frac{F_j}{F}=\frac{k_j}{\sum_\ell k_\ell}.
$$
The stiffness sum is <gear mesh stiffness>, and $\eta_j$ is a <gear tooth load fraction>. If two active pairs have stiffness100 and150N/µm under total normal load1000N, their common approach is4µm and loads are400 and600N. Equal sharing requires equal effective stiffness in this particular model.
Include a geometric gap $e_j$ by writing $F_j=k_j\max(\delta-e_j,0)$. If the second pair has a2µm gap, the same1000N total load gives $100\delta+150(\delta-2)=1000$ with $\delta$ measured in micrometres. The result is $\delta=5.2\,\mathrm{\mu m}$, $F_1=520\,\mathrm N$ and $F_2=480\,\mathrm N$. At only100N, the first pair alone compresses1µm, leaving the second pair out of contact.
For an ideal <spur gear> mesh with transverse contact ratio between1 and2, the fraction of a tooth-entry cycle having two pairs in contact is $\epsilon_\alpha-1$. A ratio1.635 therefore gives about63.5% two-pair contact and36.5% single-pair contact. This is geometric timing, not a guarantee of equal forces in the overlapping part. A <gear friction work per engagement> calculation must use each pair's own load history; dynamic motion can require more than the quasistatic spring approximation.
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