Gear pitch circle
= Gear pitch circle
A <gear pitch circle> is an imaginary <circle> used to represent the motion or reference geometry of a <gear>. A useful distinction is between the <gear reference circle>, specified for an individual <gear>, and the <operating pitch circle>, determined for a meshing pair. The terminology entry in <KHK Gear Nomenclature> calls the latter the intermeshing pitch circle and also uses the name <working pitch circle>.<KHK Gear Nomenclature>[“Intermeshing Pitch Circle”, definition and alternative names; The ABCs of Gears, A, §7, printed p. 18: reference-circle diagram]{ref}
For an external pair with fixed axes and constant <gear ratio>, imagine two <operating pitch circles> in contact and <rolling without slipping>. If their <radii> are $r_1,r_2$ and their axis separation is $a$, their geometry and signed <angular velocities> obey
$$
r_1+r_2=a,\qquad r_1\omega_1+r_2\omega_2=0,
\qquad \frac{r_2}{r_1}=\frac{N_2}{N_1}.
$$
The last equality follows by matching the <gear ratio> to tooth counts $N_1,N_2$. It is an ideal kinematic construction: the imaginary <circles> themselves are not physical rims carrying the contact force.
For standard unshifted straight-tooth <gears> at their standard axis separation, the <gear reference circles> coincide with the <operating pitch circles>. The <gear module> $m$ specifies reference diameter $d=mN$. With $m=2\,\mathrm{mm}$ and tooth counts 20 and 40, the reference diameters are $40\,\mathrm{mm}$ and $80\,\mathrm{mm}$, giving reference <radii> $20\,\mathrm{mm}$ and $40\,\mathrm{mm}$ and standard separation $60\,\mathrm{mm}$. The reference diameters differ from the larger tooth-tip diameters. If a compatible pair is operated at another axis separation, its <gear reference circles> remain properties of the individual <gears>, while its <operating pitch circles> must fit the new separation and the same tooth-count ratio. This is why an unspecified “pitch diameter” should not silently mix reference geometry with operating geometry.<KHK Gear Nomenclature>[“Calculation of Gear Dimensions”, §4.1, operating pitch circles of meshing profile-shifted spur gears; “Basic Gear Terminology and Calculation”, reference diameter]{ref}