Operating pitch circle ($r_{wi}=a{N}_i/(N_1+N_2)$)

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]

The operating pressure angle ${\alpha }_w$ is the gear pressure angle of an actual meshing pair at its mounted axis separation. For external spur gears with involute profiles, it is the angle between the gear line of action and the common tangent to the operating pitch circles. Changing the separation changes this angle even though the individual gear base circles remain fixed.^[ref]

Let the two base radii be $r_{b1},r_{b2}$ and the mounted axis separation be $a$. The right-triangle geometry of the gear line of action givesIf $a_0=r_1+r_2$ is the reference separation and both reference gear pressure angles equal $\alpha$, then $r_{bi}=r_i\cos \alpha$, henceThe gear reference circles and gear ratio remain unchanged by this mounting adjustment; the operating pitch circles expand or contract to fit $a$.

For example, reference radii $20\mathrm{mm}$ and $40\mathrm{mm}$ with $\alpha =20{}^{\circ }$ give $a_0=60\mathrm{mm}$. Mounting a compatible pair at $a=61\mathrm{mm}$ givesAt an imposed gear tangential force $100\mathrm{N}$, the associated gear separating force is about $41.30\mathrm{N}$, compared with $36.40\mathrm{N}$ at the reference angle. Holding torque instead of tangential load would additionally require using the changed operating radius.

The angle formula is only a geometric condition. Usable tooth flanks must still reach the contact region and maintain an adequate gear contact ratio; excessive separation can destroy useful contact. Likewise, the existence of a real inverse cosine does not establish that a proposed mounting has suitable backlash or mechanical strength.^[ref]