= Gear path of approach
The <gear path of approach> is the part of the active <gear path of contact> followed before the contact reaches the <gear pitch point>. For a conventional external <involute> pair in the stated driving direction, contact starts where the driven gear's tip circle intersects the usable <gear line of action>, and then moves toward the pitch point. This is a straight-line distance along the common normal, not an arc around a pitch circle.<KHK Gear Nomenclature>[“Length of Path of Contact”, approach segment from first contact to the pitch point]{ref}
Let gear1 drive gear2. Write $r_{a2}$ for gear2's tip radius, $r_{b2}$ for its base radius, $r_{w2}$ for its operating radius and $\alpha_w$ for <operating pressure angle>. A right triangle from the base-circle tangency gives tip-to-tangency distance $\sqrt{r_{a2}^2-r_{b2}^2}$. Subtracting the pitch-point-to-tangency distance gives
$$
L_a=\sqrt{r_{a2}^2-r_{b2}^2}-r_{w2}\sin\alpha_w.
$$
This formula assumes the full tip-limited interval is usable without interference or profile truncation.
For an original standard module2,20/40-tooth pair at60mm center distance and20degree reference angle, gear2 has tip radius42mm, operating radius40mm and base radius $40\cos20\,{}^{\circ}\,\mathrm{mm}$. Consequently $L_a\approx5.0586\,\mathrm{mm}$. If gear1 rotates at $50\,\mathrm{rad/s}$, contact advances along the line at $u_b=|\omega_1|r_{b1}=0.93969\,\mathrm{m/s}$, giving an approach contribution to <gear contact duration> of about $5.383\,\mathrm{ms}$. This contact-location speed differs from <gear sliding velocity>, which tends to zero on arriving at the pitch point.
The total active length adds <gear path of recess>. Reversing the followed engagement direction exchanges which segment is called approach and which is recess. If altered usable flanks exclude the nominal pitch point, an assumed positive tip-limited formula must not replace an inspection of the actual active interval.
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