Gear and gear train calculator

Build simple, compound or planetary trains with spur, helical, internal, bevel, worm-and-wheel or rack-and-pinion gears. Get the ratio, the speed, direction and torque of every shaft, the involute geometry (diameters, center distance, contact ratio), the tangential, radial and axial forces, and the bending (Lewis) and contact (Hertz) stresses with their safety factors.

What you can do

Animated mesh with a true involute profile.
Planetary set with Willis's equation and assembly conditions.
Worm efficiency and self-locking.
Undercut check and minimum number of teeth.
Metric module or diametral pitch.

Frequently asked questions

How is a gear train ratio calculated?

It is the product of the driven gear teeth divided by the product of the driver teeth: i = Π z_driven / Π z_driver.

What is the contact ratio?

The average number of tooth pairs in contact; it should exceed 1.2 for smooth meshing.

How does a planetary gear set work?

Sun, planets, ring and carrier satisfy (ωs − ωc)/(ωr − ωc) = −Zr/Zs; fixing one member gives the ratio between the other two.

Guide: spur gears, gear ratio and tooth strength

Gears transmit rotation and torque between shafts with a fixed speed ratio. Designing a gear pair means choosing the tooth counts for the ratio you need, computing the geometry and the tooth forces, and checking that the tooth will not break in bending or wear out at the contact. This guide covers those steps for spur gears and works an example.

Geometry with the module

In the metric system, tooth size is given by the module m (in mm). Two gears can only mesh if they share the same module and the same pressure angle φ (20° is standard). For full-depth teeth:

Pitch diameter: d = m · z, where z is the number of teeth.
Addendum: m; dedendum: 1.25 m. Outside diameter: d + 2m; root diameter: d − 2.5 m.
Base diameter: d · cos φ. Circular pitch: π · m.
Center distance: a = (d₁ + d₂)/2.

Ratio, speeds and torque

The gear ratio is i = z₂/z₁. The gear turns i times slower than the pinion and, without losses, with i times more torque. In an external pair, the shafts turn in opposite directions. In a train, the ratios of the stages multiply.

Tooth forces

Torque is transmitted by a force along the line of action, inclined at the pressure angle. It splits into

W_t = 2T / d (tangential, transmits the torque) W_r = W_t · tan φ (radial, pushes the shafts apart)

Strength: Lewis and Hertz

Bending (Lewis): the tooth is treated as a cantilever beam. σ = K_v · W_t / (b · m · Y), with b the face width, Y the Lewis form factor (it depends on the number of teeth) and K_v the dynamic factor. The tool uses Barth's, K_v = (6.1 + V)/6.1 with V in m/s, as in Shigley.
Contact (Hertz): the contact stress between the flanks governs surface wear (pitting). It is usually the most limiting check.

Worked example

A 3 kW motor at 1500 rpm drives a 20-tooth pinion meshing with a 50-tooth gear, module 2 mm, φ = 20°, face width 25 mm.

Ratio: i = 50/20 = 2.5; the gear turns at 1500/2.5 = 600 rpm, in the opposite direction.
Diameters: d₁ = 2·20 = 40 mm and d₂ = 2·50 = 100 mm; outside diameters 44 and 104 mm; center distance 70 mm.
Pinion torque: T = P/ω = 3000 / (2π·1500/60) ≈ 19.1 N·m.
Forces: W_t = 2·19.1/0.040 ≈ 955 N; W_r = 955·tan 20° ≈ 348 N.
Pitch-line velocity: V = π·0.040·1500/60 ≈ 3.14 m/s, so K_v = (6.1 + 3.14)/6.1 ≈ 1.515.
Pinion bending: with Y ≈ 0.322 for 20 teeth, σ = 1.515·955/(25·2·0.322) ≈ 89.9 MPa.
Output: with the typical efficiency of a spur pair (98 %), the gear delivers about 2.94 kW and 46.8 N·m.

All these values match the tool's, which also computes the contact stress, the safety factors for the chosen material and the contact ratio (1.66 here: there are always one or two pairs of teeth in mesh).

How to use the calculator

Type the input speed, the direction of rotation and the load (power or torque).
Add stages: spur, helical, internal, bevel, worm and wheel, rack and pinion or a planetary train. Each stage takes the output shaft of the previous one.
For each stage, give the teeth, module, pressure angle, face width, materials and, if you want, the efficiency.
Read the total ratio, the speed and torque on each shaft, and each stage's geometry, forces and strength, with the step by step.

Common mistakes

Meshing gears with different modules or pressure angles.
Using too few pinion teeth: with φ = 20° and full-depth teeth, below about 17 teeth interference appears when cutting with a rack.
Computing torque from rpm without converting to rad/s: ω = 2π·n/60.
Forgetting the dynamic factor: the higher the speed, the larger the effective load on the tooth.

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