= Gear base-roll speed
{title2=$u_b=|\omega|r_b$}
<Gear base-roll speed> $u_b$ is the rate at which base-circle rolling advances an ideal <involute> contact along the <gear line of action>. For a fixed-axis external pair at constant speed,
$$
u_b=|\omega_1|r_{b1}=|\omega_2|r_{b2}.
$$
The equality follows from the tooth-count speed ratio and compatible base pitches. It describes the rate of change of contact position along a geometric line, rather than the <gear sliding velocity> between material points on the flanks. Base circles and their common tangent define this geometry.<KHK Gear Nomenclature>[Basic Guide B §3, printed pp.20–21, involute/base-circle construction; “Length of Path of Contact”, active line-of-action segment]{ref}
For standard module2 gears with20 and40 teeth at20degree reference angle, base radii are $r_{b1}=20\cos20\,{}^{\circ}\,\mathrm{mm}$ and $r_{b2}=40\cos20\,{}^{\circ}\,\mathrm{mm}$. Signed <angular velocities> $\omega_1=50\,\mathrm{rad/s}$ and $\omega_2=-25\,\mathrm{rad/s}$ give the same $u_b\approx0.93969\,\mathrm{m/s}$. The first gear's reference-circle tangential speed is1m/s, so base-roll and pitch-circle speeds are distinct.
In one tooth-entry interval, the base circle advances one <gear base pitch>:
$$
\frac{u_b}{f_m}=\frac{2\pi r_{b1}}{N_1}=p_b.
$$
With $f_m\approx159.155\,\mathrm{Hz}$, the example advances5.90426mm per entry. An active contact length9.65457mm therefore gives <gear contact duration> $L/u_b\approx10.2742\,\mathrm{ms}$.
For compatible involute teeth, changing center distance changes the <operating pressure angle> and operating pitch circles, while leaving the manufactured base circles unchanged. At the same shaft speeds, base-roll speed is therefore unchanged even though the usable contact length may change. This separates a kinematic rate from the active interval needed to calculate contact time. If speed varies, contact displacement follows $\int u_b(t)\,dt$; a constant-speed quotient is then only an approximation.
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