Gear Tooth Tangential Force

Torque is a force times a distance, so a force is a torque divided by a distance — and for a gear the distance is the pitch radius. Write it with the diameter, which is what everyone actually knows, and the 2 appears:

\[ W_t = \frac{T}{r} = \frac{2T}{d} \]

This tangential load is where every gear stress calculation begins. AGMA's bending strength equation and its pitting-resistance equation both take $W_t$ as their first input, multiply it by a stack of factors for overload, dynamic effects, size, load distribution and rim thickness, and divide by the face width and a geometry factor. The rest of the method is refinement; this is the load.

The two forces this equation does not give you

A spur tooth does not push tangentially. It pushes along the line of action, which is inclined by the pressure angle $\varphi$ — 20° on almost everything made since the 1950s, 14.5° on older stock. Resolve that push and you get the tangential component above plus a radial or separating component:

\[ W_r = W_t \tan\varphi \]

which at 20° is about 36% of $W_t$. The separating force does no work at all — it never moves in its own direction — but it is carried in full by the bearings, and it is trying to push the two gears apart, which is what puts the load on the housing bores. A helical gear adds a third, axial component $W_a = W_t \tan\psi$ from the helix angle, which is why a helical box needs a bearing arrangement that can take thrust and a spur box does not.

Where the mistakes are

Three of them, all cheap to avoid. First: use the pitch diameter, not the tip diameter — see the pitch diameter page for why they differ. Second: use the torque on this gear. In a train every shaft carries its own torque, rising as the speed falls, and the pinion's torque is not the wheel's. Third: the two meshing gears see the same $W_t$ but different torques, precisely because their radii differ — that is the whole mechanism by which a gear pair trades speed for torque. The force at the mesh is common; the moment arms are not.

One practical extension. The dynamic load at the mesh is higher than this steady figure, sometimes considerably, because of tooth spacing errors, deflection under load and the shock of the driven machine. AGMA handles it with a dynamic factor $K_v$ that depends on pitch line velocity and the accuracy grade of the gear, and an overload factor $K_o$ for the character of the driven load. A reciprocating compressor is not a fan.