= Gear overlap ratio
{title2=$\epsilon_\beta$}
<Gear overlap ratio> measures the additional engagement overlap caused by a <helical gear>'s tooth inclination across its <gear face width>. It is distinct from the transverse <gear contact ratio>, which measures contact length within one transverse section. For face width $b$, reference <gear helix angle> $\beta$ and <gear normal module> $m_n$,
$$
\epsilon_\beta=\frac{b\sin|\beta|}{\pi m_n}
=\frac{b\tan|\beta|}{\pi m_t}
=\frac{b}{p_x},
$$
where $m_t$ is <gear transverse module> and $p_x$ is <gear axial pitch>.<Elements of Metric Gear Technology by SDP-SI>[§11, Table11-3, “Axial (Overlap) Contact Ratio”, helical-gear equation]{ref}
Unwrapping the reference cylinder explains the ratio: moving across axial width $b$ shifts the tooth pattern circumferentially by $b\tan|\beta|$. Dividing this shift by <gear transverse pitch> $\pi m_t$ counts how many tooth spacings the engagement progresses across the face.
For an original example, $b=20\,\mathrm{mm}$, $m_n=2\,\mathrm{mm}$ and $|\beta|=30\,{}^{\circ}$ give
$$
\epsilon_\beta=\frac5\pi\approx1.59155,
\qquad p_x=4\pi\,\mathrm{mm}\approx12.566\,\mathrm{mm}.
$$
Doubling the face width doubles this geometric overlap if the other parameters stay fixed. At the <spur gear> limit $\beta=0$, the overlap contribution tends to zero.
In the ideal cylindrical-helical model, total geometric contact ratio is the sum of transverse and axial contributions, $\epsilon_\gamma=\epsilon_\alpha+\epsilon_\beta$. This does not specify a uniform division of load: flexibility, alignment and the usable flank region affect actual contact. Increasing helix angle can improve overlap while also increasing <gear axial force>; increasing width also changes the alignment and support demands. These are related geometric tradeoffs, not a guarantee of a specified noise level or load capacity.
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