Gear axial force
= Gear axial force
{title2=$F_a=F_t\tan\beta$}
<Gear axial force> is the component of tooth-contact load parallel to a gear's rotation axis. In a standard parallel-axis external <helical gear> pair, ideal force resolution at the reference mounting gives
$$
|F_a|=|F_t|\tan|\beta|,
$$
where $F_t$ is <gear tangential force> and $\beta$ is reference <gear helix angle>. Its direction depends jointly on helix hand and the transmitted-torque direction. Reversing the torque can reverse axial thrust even though the cut helix remains unchanged.<Elements of Metric Gear Technology by SDP-SI>[§7.3, equation(7–8), axial thrust and Figure7-1, direction versus gear hand and rotation]{ref}
For an original example, transmit $6\,\mathrm{N\,m}$ through reference radius $30\,\mathrm{mm}$ at standard mounting. Then $|F_t|=200\,\mathrm N$. A30degree helix gives axial load $200\tan30\,{}^{\circ}\approx115.47\,\mathrm N$; a15degree helix at the same tangential load gives about $53.59\,\mathrm N$. The support must sustain the appropriate axial reaction in addition to the mesh's radial and tangential loads.
The axial load does not itself imply energy loss. If an ideal gear has no axial velocity, that force does no work through an axial displacement. Real support friction and deflection can nevertheless affect efficiency and tooth alignment. A useful bench check applies a controlled transmitted torque and measures the axial reaction while recording helix hand and the actual power-flow direction; an unsigned force magnitude alone does not establish the direction rule.
For the ideal <spur gear> limit, $\beta=0$ and this geometric thrust term vanishes. Opposite-handed helical sections can be arranged to cancel net axial thrust under suitable symmetric loading, but unequal loading prevents assuming exact cancellation. Away from the stated reference mounting, force formulas must use a geometrically consistent operating convention rather than mixing operating pitch radii with an unrelated reference angle.