= Gear contact duration
{title2=$T=L/u_b$}
<Gear contact duration> is the time for a specified tooth pair to traverse its active <gear path of contact>. For a fixed-axis external <spur gear> pair at constant speed, the contact location advances along the <gear line of action> at the common <gear base-roll speed>
$$
u_b=|\omega_1|r_{b1}=|\omega_2|r_{b2}.
$$
This is the speed of the contact location along a geometric line, not the relative tooth-surface <gear sliding velocity>. The latter vanishes at the <gear pitch point>, while the contact location continues moving. For active path length $L$, duration is $T=L/u_b$. The geometry relates <gear contact ratio> to active path length divided by <gear base pitch>.<KHK Gear Nomenclature>[“Length of Path of Contact”, active segment and base-pitch contact ratio]{ref}
The time between successive tooth-pair entries is $1/f_m$, with <gear mesh frequency> $f_m=N_1|\omega_1|/(2\pi)$. Since $r_{b1}=r_{w1}\cos\alpha_w$ for compatible operating geometry, the base-circle distance advanced per entry is
$$
\frac{u_b}{f_m}=\frac{2\pi r_{b1}}{N_1}=p_b.
$$
Consequently
$$
T f_m=\frac L{p_b}=\epsilon_\alpha.
$$
A contact ratio above1 means one pair remains engaged after the next pair enters; it does not mean one pair contacts for an entire shaft revolution.
For the standard module2,20/40-tooth example with20degree reference angle, active length is9.65457mm and base pitch is5.90426mm. At driving <angular velocity> $50\,\mathrm{rad/s}$, $u_b=0.93969\,\mathrm{m/s}$, $f_m\approx159.155\,\mathrm{Hz}$ and $T\approx10.2742\,\mathrm{ms}$. The entry interval is6.28319ms, giving $Tf_m\approx1.63519$. Doubling both shaft speeds halves duration while preserving geometric <gear contact ratio>. Variable speed requires integrating the changing base-roll speed; a <helical gear> also needs a stated transverse slice or a whole-face contact convention.
Back to article page
