= Gear contact ratio
{title2=$\epsilon_\alpha$}
<Gear contact ratio> measures how much successive tooth engagements overlap. For an external <spur gear> pair, its transverse value is
$$
\epsilon_\alpha=\frac{L}{p_b},
$$
where $L$ is usable <gear path of contact> length along the <gear line of action> and $p_b$ is <gear base pitch>. One tooth pair remains engaged while contact advances through length $L$; successive pairs begin after an advance of $p_b$. Thus $\epsilon_\alpha$ is also the ideal mean number of tooth pairs simultaneously engaged.<KHK Gear Nomenclature>[“Length of Path of Contact”, contact-length/base-pitch definition]{ref}
For a tip-limited, noninterfering external <involute> pair with tip radii $r_{a1},r_{a2}$, base radii $r_{b1},r_{b2}$, axis separation $a$ and <operating pressure angle> $\alpha_w$, right-triangle geometry gives
$$
L=\sqrt{r_{a1}^2-r_{b1}^2}+\sqrt{r_{a2}^2-r_{b2}^2}-a\sin\alpha_w.
$$
For an original example, use module $2\,\mathrm{mm}$, tooth counts 20 and 40, reference angle $20\,{}^{\circ}$ and standard separation $60\,\mathrm{mm}$. With tip radii $22$ and $42\,\mathrm{mm}$, the formula gives $L\approx9.6546\,\mathrm{mm}$ and $p_b\approx5.9043\,\mathrm{mm}$, hence $\epsilon_\alpha\approx1.6352$.
In the ideal periodic timing model, two tooth pairs are engaged for about $63.52\%$ of a tooth-passage period and one pair for about $36.48\%$. This does not mean that force is divided equally between engaged pairs: tooth flexibility and contact geometry affect load sharing. A value below one would leave gaps in ideal engagement, while a value exactly one leaves no overlap margin for tolerances and deformation.
\Image[/gear-contact-ratio-overlap.png]
{title=Ideal tooth-contact overlap for a spur-gear contact ratio of 1.635}
The plot shows contact timing, rather than physical tooth shapes or force sharing. For <helical gears>, the transverse measure must additionally be distinguished from <gear overlap ratio> caused by the inclination of teeth across the gear face. Finally, real usable contact can be shortened by modified or unavailable flank portions; the tip-circle formula is conditional on its stated geometric assumptions.
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