Ratio
-Efficiency
-Output torque
-Output speed
-All results
| Lead angle | - |
| Lead | - |
| Wheel pitch diameter | - |
| Center distance | - |
| Input torque at worm | - |
| Power lost as heat | - |
| Wheel tangential force (= worm axial) | - |
| Worm tangential force | - |
| Radial (separating) force | - |
| Self-locking tendency | - |
Method and formulas
Standard worm gear kinematics and the Coulomb-friction force analysis, so every figure can be checked by hand.
| Quantity | Formula |
|---|---|
| Ratio | i = z2 / z1 |
| Lead angle | tan γ = m z1 / d1 |
| Lead | L = π m z1 |
| Wheel pitch diameter, center distance | d2 = m z2, a = (d1 + d2) / 2 |
| Efficiency (worm driving) | η = (cos αn − μ tan γ) / (cos αn + μ / tan γ) |
| Output torque and speed | T2 = T1 i η, n2 = n1 / i |
| Wheel tangential force | Wgt = 2 T2 / d2 |
| Worm tangential force | Wwt = Wgt (cos αn sin γ + μ cos γ) / (cos αn cos γ − μ sin γ) |
| Radial force | Wr = Wgt sin αn / (cos αn sin γ + μ cos γ) |
| Self-locking tendency | μ ≥ cos αn tan γ |
US inputs are converted to metric internally: module m = 25.4 / P (P is the axial diametral pitch in 1/in), 1 in = 25.4 mm, 1 hp = 0.7457 kW, 1 lbf = 4.4482 N. The module is the axial module of the worm, which equals the transverse module of the wheel at 90 degree shaft angle.
Worked example: 2-start worm, 40-tooth wheel, 4 mm module
With d1 = 40 mm, 1450 rpm, 2.2 kW, μ = 0.05 and αn = 20°: i = 40 / 2 = 20, tan γ = 4 × 2 / 40 = 0.2 so γ = 11.310°, d2 = 160 mm and a = 100 mm. Efficiency is 78.1 %. T1 = 9549 × 2.2 / 1450 = 14.49 N·m, so T2 = 14.49 × 20 × 0.7815 = 226.4 N·m at 72.5 rpm. Wgt = 2 × 226.4 / 0.16 = 2831 N, and about 0.48 kW is lost as heat. cos αn tan γ = 0.1879 is above μ, so there is no self-locking.
Questions
How do you calculate the worm gear ratio?
The ratio is the number of wheel teeth divided by the number of worm starts, i = z2 / z1. A single-start worm with a 40-tooth wheel gives 40:1, a two-start worm with the same wheel gives 20:1. The output speed is the input speed divided by the ratio.
What is the lead angle and why does it matter?
The lead angle gamma satisfies tan(gamma) = lead / (pi d1) = m z1 / d1, where m is the axial module, z1 the number of starts and d1 the worm pitch diameter. A small lead angle (about 5 degrees or less) gives high ratio and a self-locking tendency but low efficiency; a large lead angle gives higher efficiency and no self-locking.
How do you calculate worm gear efficiency?
With friction coefficient mu and normal pressure angle alpha_n, the driving efficiency is eta = (cos(alpha_n) - mu tan(gamma)) / (cos(alpha_n) + mu / tan(gamma)). It is the same relation the output torque uses: T2 = T1 i eta. The friction coefficient depends on materials, surface finish, lubricant and sliding speed, so use a value from your gear supplier.
When is a worm gear self-locking?
Roughly when the friction coefficient is at least cos(alpha_n) tan(gamma), the wheel cannot drive the worm. This is a tendency, not a safety guarantee: vibration, shock loads and a lower running friction coefficient can let the load back-drive. Where a load must be held safely, use a brake or other positive device rather than relying on self-locking.
Where do the tooth forces come from?
The wheel tangential force equals the worm axial force, W = 2 T2 / d2. The worm tangential force is larger by the friction factor, and the radial force follows from the pressure angle. The worm axial force must be carried by thrust bearings, which is why worm shafts often need tapered or angular-contact bearings.
What does this calculator not cover?
Tooth strength and wear rating (AGMA 6034 or ISO/TR 14521 style checks), thermal rating and oil temperature, efficiency changes with sliding speed, bearing loads and lubrication. It is a first-pass geometry and torque estimate: verify with the manufacturer's catalog rating.
Engineering aid for guidance. It uses a constant friction coefficient and covers geometry and torque only, not tooth strength, wear or thermal rating. Verify critical designs against the manufacturer's catalog and applicable standards.