Gear ratio, pitch diameters, centre distance, output speed and torque for a spur gear pair — plus tangential load, pitch line velocity and AGMA-corrected Lewis bending stress with safety factor against the material allowable.
Geometry follows directly from module and tooth count. Strength is checked with the Lewis equation, which treats a tooth as a cantilever beam, then corrected by the AGMA velocity, size and load-distribution factors to reflect real running conditions.
m module (mm) ·
z tooth count ·
F face width (mm) ·
Y Lewis form factor ·
Kv dynamic factor ·
Ks size factor ·
Km load distribution factor ·
Sall allowable bending stress (MPa)
A module 4 spur pair, 20-tooth pinion driving a 60-tooth gear, 40 mm face width, transmitting 15 kW at 1450 rpm through hardened steel gears.
The AGMA correction factors nearly double the basic Lewis stress here — from 47.94 to 97.69 MPa. Most of that comes from the dynamic factor at 6 m/s pitch line velocity. Ignoring the correction factors would suggest a safety factor of 4.2 rather than the realistic 2.05.
| Quantity | Symbol | Unit | Accepted range |
|---|---|---|---|
| Module | m | mm | > 0 |
| Tooth count | z | — | ≥ 6 each gear |
| Face width | F | mm | ≥ 8 × m |
| Pinion speed | n1 | rpm | > 0 |
| Power | P | kW or hp | > 0 |
| Mesh efficiency | η | — | 0.5 – 1.0 |
| Allowable stress | Sall | MPa | > 0 |
| Pitch line velocity | Vp | m/s | output, flagged above 25 |
| Bending stress | σAGMA | MPa | output |
Power in horsepower is converted at 1 hp = 0.7457 kW. Face width below 8 × module is rejected as an AGMA violation; above 16 × module produces a warning because load cannot be distributed evenly across the face.
| Teeth z | Y | Teeth z | Y |
|---|---|---|---|
| 12 | 0.245 | 30 | 0.359 |
| 14 | 0.277 | 38 | 0.384 |
| 16 | 0.296 | 50 | 0.409 |
| 18 | 0.309 | 60 | 0.422 |
| 20 | 0.322 | 100 | 0.447 |
| 24 | 0.337 | 300 | 0.472 |
| Material | Sall (MPa) | Typical use |
|---|---|---|
| Steel, hardened | 200 | Industrial drives, gearboxes |
| Steel, soft | 83 | Light duty, low-cost machined gears |
| Cast iron | 50 | Slow-speed machinery, good damping |
| Bronze | 40 | Worm wheels, corrosion service |
Module is the pitch diameter divided by the number of teeth, in millimetres, and it sets the physical size of the teeth. A module 4 gear with 20 teeth has a pitch diameter of 80 mm. Two gears can only mesh if they share the same module and the same pressure angle. Module is the metric equivalent of diametral pitch, which is teeth per inch of pitch diameter.
Gear ratio is the number of teeth on the driven gear divided by the number on the driving gear. A 20-tooth pinion driving a 60-tooth gear gives a ratio of 3:1, so output speed is one third of input speed and output torque is three times input torque before losses. Because module is common to both gears, the ratio of pitch diameters equals the ratio of tooth counts.
The Lewis form factor Y accounts for tooth shape when treating a gear tooth as a cantilever beam in bending. It increases with tooth count because teeth on larger gears are less curved and therefore stronger in bending. Values run from about 0.245 at 12 teeth to 0.480 at 400 teeth. The weaker of the two gears governs, so the calculation uses the smaller of the two form factors.
These correct the basic Lewis stress for real operating conditions. Kv is the dynamic factor, which grows with pitch line velocity and accounts for tooth impact from manufacturing error. Ks is the size factor, which penalises large teeth for their lower material strength. Km is the load distribution factor, which accounts for load concentrating at one end of the face when the gear is wide relative to its diameter or the shafts deflect.
Common AGMA practice puts face width between 8 and 16 times the module, with 9 to 14 times module a typical target. Narrower than 8 × m makes the teeth unnecessarily highly stressed for the space used. Wider than 16 × m means load cannot be distributed evenly across the face because shaft and housing deflection concentrate it at one end, which shows up as a rising load distribution factor.