= Gear separating force
{title2=$F_r=F_t\tan\alpha_w$}
<Gear separating force> is the component of an external <gear> mesh load along the line joining the gear centers, tending to push those centers apart. For a frictionless external <spur gear> pair,
$$
F_r=F_n\sin\alpha_w=F_t\tan\alpha_w,
$$
where $F_n$ is tooth-normal force, $F_t$ is <gear tangential force> and $\alpha_w$ is <operating pressure angle>. This component acts on the shaft supports even when the mesh is ideal and lossless.<Elements of Metric Gear Technology by SDP-SI>[§3.1, pressure-angle and common-normal geometry]{ref}
For example, take operating radius $r_w=20\,\mathrm{mm}$ and transmitted <torque> $4\,\mathrm{N\,m}$. Then $F_t=200\,\mathrm N$. At $\alpha_w=20\,{}^{\circ}$, $F_r\approx72.79\,\mathrm N$; at $25\,{}^{\circ}$ and the same tangential load, $F_r\approx93.26\,\mathrm N$. The larger separating load is a change in load direction, not by itself an efficiency loss.
If the gear centers are ideally fixed, this radial component produces no work through a center-separation displacement, since that displacement is zero. Flexible supports can nevertheless deflect under it and store <elastic potential energy>. The resulting change in axis separation can change <operating pressure angle>, <backlash> and tooth contact, so the assumed geometry and support stiffness interact.
A support-load experiment can apply a known <torque> slowly to a mounted external pair and measure the reaction along the line of centers while maintaining the stated axis separation. Agreement with $F_r/F_t=\tan\alpha_w$ tests the force resolution under low-friction assumptions. Additional reactions, tooth <friction> and mounting preload must be accounted for before identifying a measured support force with this mesh component. Internal meshes and nonparallel-axis gearing require their own direction conventions; the external separating-force picture must not be transferred unchanged.
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