How to calculate gear mesh forces

A meshing gear pushes its mate three ways at once, and the shaft and bearings have to carry all three. Every one of them comes from the transmitted torque and a bit of trigonometry — no gear-rating tables needed.

The three force components

Ft = 2T / d · Fr = Ft · tan(αt) · Fa = Ft · tan(β),  αt = atan(tan(αn) / cos β)

The tangential force Ft = 2T/d is the useful one — it transmits the torque (T) across the pitch circle (diameter d). The radial force Fr pushes the gears apart and does no work; it uses the transverse pressure angle αt. On a helical gear the axial (thrust) force Fa = Ft·tan β runs along the shaft — a spur gear (β = 0) has none. The gear mesh force calculator runs all three plus the resultant; size the gear geometry first with the involute gear calculator.

For the source-style query, the helical gear axial force formula is Fa = Ft tan beta, where beta is the helix angle at the pitch cylinder. Treat this as thrust magnitude; the helix hand and rotation direction set which bearing sees the axial load.

Worked example — 200 N·m through a helical gear

Pitch diameter d = 80 mm, normal pressure angle αn = 20°, helix β = 15°, torque T = 200 N·m:

Ft = 2·200/0.080 = 5000 N · αt = atan(tan20°/cos15°) = 20.6° · Fr = 5000·tan20.6° = 1884 N · Fa = 5000·tan15° = 1340 N

The same gear as a spur (β = 0) would separate with Fr = 5000·tan20° = 1820 N and zero thrust. Going helical for a quieter mesh costs a 1340 N axial load the bearings now have to react — the trade every gearbox designer makes. The total normal force on the tooth is Fn = √(Ft²+Fr²+Fa²) ≈ 5509 N.

Common mistakes

• Using the nominal torque. Apply the service factor first — shock loads and duty cycle drive the real tooth force, and the bearings are sized to that.
• Using αn for the radial force on a helical gear. The separating force resolves in the transverse plane: use αt = atan(tan αn / cos β), not the normal angle αn.
• Forgetting the axial thrust. A helical gear's Fa = Ft·tan β must go somewhere — a thrust bearing, or a back-to-back pair / double-helical gear that cancels it. Spur gears skip this; helical gears don't.
• Sizing the bearing on Ft alone. The radial bearing load is the vector sum of Ft and Fr (and the moment from Fa); the tangential force is only one leg of it. Feed that combined side load into the overhung load calculator to turn it into the actual bearing reactions and shaft bending moment.

Formula sources

The calculator page links the force-analysis references used for Ft = 2T/d, Fr = Ft·tan(αt) and Fa = Ft·tan(β). Use the gear mesh force calculator when you need the sourced formula and the numeric bearing-load inputs on one page.

Frequently asked questions

How do you calculate the forces on a gear tooth?

Start from the transmitted torque. The tangential (driving) force is Ft = 2T/d, where T is the torque and d the pitch diameter. The radial (separating) force is Fr = Ft·tan(αt), and on a helical gear the axial (thrust) force is Fa = Ft·tan(β), where αt is the transverse pressure angle and β the helix angle. For T = 200 N·m, d = 80 mm, αn = 20°, β = 15°: Ft = 5000 N, Fr ≈ 1884 N, Fa ≈ 1340 N.

What is the helical gear axial force formula?

Fa = Ft·tan(β), where Ft = 2T/d is the tangential force and β is the helix angle. A spur gear (β = 0) has no axial thrust; a 15° helix on a 5000 N tangential load throws 5000·tan(15°) ≈ 1340 N along the shaft, which the bearings must take. The thrust is the price of the helical gear's smoother, quieter mesh.

Why does the radial force use the transverse pressure angle?

On a helical gear the tooth is cut at the normal pressure angle αn, but the force resolves in the transverse plane, so the separating force uses αt = atan(tan(αn)/cos(β)). For αn = 20° and β = 15°, αt ≈ 20.6° — close to αn, but it grows as the helix steepens. On a spur gear β = 0, so αt = αn and Fr = Ft·tan(αn).

What do I do with these forces?

They size the shaft and bearings. The tangential and radial forces combine into the radial bearing load and the shaft bending moment; the axial force sets the thrust-bearing requirement and (with the helix hand) which bearing takes it. Apply a service factor to the torque first so the forces reflect the real, shock-loaded duty, not the nominal rating.