 Operating pitch circle

The operating pitch circles of two meshing gears are the imaginary circles that reproduce their motion by rolling without slipping. They are properties of the working pair and its mounting, whereas a gear reference circle specifies one gear's nominal geometry. Working pitch circle and intermeshing pitch circle are alternative names.^[ref]

Consider an external pair of spur gears on fixed parallel axes, with axis separation $a$ and tooth counts $N_1,N_2$. Their operating radii $r_{w1},r_{w2}$ must satisfy both contact geometry and the tooth-count gear ratio:Solving givesThus the operating radii scale with the actual separation, while the gear ratio stays fixed by tooth counts. These are kinematic equations, conditional on the teeth actually maintaining compatible contact.

For an original example, choose gear module $m=2\mathrm{mm}$ and tooth counts 20 and 40. Reference radii are $20\mathrm{mm}$ and $40\mathrm{mm}$; their standard separation is $60\mathrm{mm}$. If a compatible mounting uses $a=61\mathrm{mm}$ instead, operating radii become $20.333\dots \mathrm{mm}$ and $40.666\dots \mathrm{mm}$. At input angular velocity $30rad/s$ the output has $-15rad/s$, and both operating tangential speed magnitudes are $0.61m/s$. Using the reference radii in that speed calculation would incorrectly give $0.60m/s$.

For tooth flanks formed as involutes, the fixed gear base circles permit constant-ratio motion over a range of separations. If their radii are $r_{b1},r_{b2}$, the common-normal geometry giveswhere ${\alpha }_w$ is the operating pressure angle. Increasing separation therefore increases that angle, while leaving each gear reference circle and gear base circle unchanged. This geometric possibility does not guarantee adequate tooth contact, clearance or strength at an arbitrary separation: the physical tooth extents and mounting must still be checked.^[ref]