= Gear path of recess
The <gear path of recess> is the part of the active <gear path of contact> followed after passing the <gear pitch point>, in the stated driving direction. For a standard external <involute> pair driven by gear1, contact ends at the driving gear's tip circle when that tip bounds the usable profile. The recess length lies on the <gear line of action>; it is not an arc of the <gear pitch circle>.<KHK Gear Nomenclature>[“Length of Path of Contact”, recess segment from pitch point to final contact]{ref}
Use driving-gear tip radius $r_{a1}$, base radius $r_{b1}$, operating radius $r_{w1}$ and <operating pressure angle> $\alpha_w$. The base-circle tangent triangle gives
$$
L_r=\sqrt{r_{a1}^2-r_{b1}^2}-r_{w1}\sin\alpha_w.
$$
For a standard module2,20/40-tooth pair at60mm separation and20degree reference angle, $r_{a1}=22\,\mathrm{mm}$, $r_{w1}=20\,\mathrm{mm}$ and $r_{b1}=20\cos20\,{}^{\circ}\,\mathrm{mm}$ yield $L_r\approx4.5960\,\mathrm{mm}$. The corresponding <gear path of approach> is about5.0586mm, so the pitch point does not bisect the active contact interval.
At driving <angular velocity> $50\,\mathrm{rad/s}$, the contact-location speed is $u_b=|\omega_1|r_{b1}\approx0.93969\,\mathrm{m/s}$. Recess therefore lasts about4.891ms. Adding approach gives <gear contact duration> $T\approx10.274\,\mathrm{ms}$. Tooth-passage frequency is $20(50)/(2\pi)\approx159.155\,\mathrm{Hz}$, so $Tf_m\approx1.6352$, agreeing with the transverse <gear contact ratio> calculated as active length divided by <gear base pitch>.
A shorter usable driving flank reduces recess and can change <gear sliding-friction loss>, because sliding grows with distance from the pitch point. The tip-circle formula alone cannot describe profile modifications or an interfering contact. The geometry and direction must be specified before applying the approach/recess labels.
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