= Gear tooth load fraction
{title2=$\eta_j=F_j/F$}
The \b[gear tooth load fraction] $\eta_j$ is the fraction of a stated total mesh <force> carried by tooth pair $j$ at a particular contact position or time:
$$
\eta_j=\frac{F_j}{F},\qquad F=\sum_jF_j>0.
$$
For nonnegative forces resolved in the same direction, $0\le\eta_j\le1$ and $\sum_j\eta_j=1$. A normal-force fraction must not mix one pair's normal load with another convention's tangential total. At zero total load the ratio is undefined.
An original quasistatic model treats engaged pairs as parallel <springs> with the same normal approach $\delta$. If $F_j=k_j\delta$ and every pair has zero initial gap, then
$$
\eta_j=\frac{k_j}{\sum_\ell k_\ell}.
$$
Two pairs with stiffnesses $60$ and $140\,\mathrm{N/\mu m}$ under total load $1000\,\mathrm N$ approach by $5\,\mathrm{\mu m}$ and carry $300$ and $700\,\mathrm N$. Their fractions are $0.3$ and $0.7$, rather than one half each. Counting two geometrically overlapping contacts does not establish equal <gear tooth load sharing>.
An initial gap $e_j$ changes the model to
$$
F_j=k_j\max(\delta-e_j,0).
$$
A pair below its <gear tooth engagement threshold> carries no load in this unilateral model. Changing total load can change both the active pairs and their fractions, even if all the individual <stiffnesses> remain constant. <Gear contact ratio> specifies geometric overlap; load fractions require deformation and engagement information.
The fractions also prevent double counting. If total load is $1000\,\mathrm N$, assigning that full load to both of two pairs gives an impossible sum of $2000\,\mathrm N$ within this model. A <gear friction work per engagement> calculation must use each pair's own $F_j(t)$ before summing its dissipation.
Dynamic motion can alter the quasistatic distribution. A 1986 NASA analysis calculated tooth stiffness and loads as contact position and speed varied, using a coupled stiffness/load iteration.<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 model, iteration and varying contact-position loads]{ref} Fractions inferred from the simple spring model should therefore be identified as model results, rather than universal operating values.
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