Gear separating force ($F_r=F_t\tan {\alpha }_w$)

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,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.^[ref]

For example, take operating radius $r_w=20\mathrm{mm}$ and transmitted torque $4\mathrm{N}\mathrm{m}$. Then $F_t=200\mathrm{N}$. At ${\alpha }_w=20{}^{\circ }$, $F_r\approx 72.79\mathrm{N}$; at $25{}^{\circ }$ and the same tangential load, $F_r\approx 93.26\mathrm{N}$. 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.