= Operating pressure angle
{title2=$\alpha_w$}
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.<Elements of Metric Gear Technology by SDP-SI>[§3.1, pressure-angle definition and line-of-action geometry]{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> gives
$$
\cos\alpha_w=\frac{r_{b1}+r_{b2}}a.
$$
If $a_0=r_1+r_2$ is the reference separation and both reference <gear pressure angles> equal $\alpha$, then $r_{bi}=r_i\cos\alpha$, hence
$$
\cos\alpha_w=\frac{a_0}a\cos\alpha.
$$
The <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 $a=61\,\mathrm{mm}$ gives
$$
\alpha_w=\arccos\left(\frac{60}{61}\cos20\,{}^{\circ}\right)\approx22.439\,{}^{\circ}.
$$
At 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.<KHK Gear Nomenclature>[The ABC's of Gears, Basic Guide B, §4, printed pp.22–23, axis separation, working pressure angle and contact-ratio limits]{ref}
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