5. Tooth forces and flank pressure
The force between two involute teeth acts along the line of action, at the working pressure angle. Resolve it and you get the two components the shaft and its bearings actually feel:
The radial component does no work — it just pushes the wheels apart and loads the bearings. That is what the gear entry in ShaftLab wants from you.
Pitting: the flank rating
A gear tooth does not usually break. It pits: the Hertzian pressure at the contact line fatigues the surface until flakes come away. ISO 6336-2 writes that pressure as a nominal stress multiplied by a set of factors:
| Factor | What it carries | Spur, 20°, steel |
|---|---|---|
| ZH | the curvature of the flanks at the pitch point | 2.495 |
| ZE | the elasticity of the material pair | 189.8 √MPa |
| Zε | load sharing between teeth | 0.87 – 0.90 |
| Zβ | helix angle | 1 (spur) |
What the square root is telling you
σH goes with √Ft, not with Ft. Four times the torque is only twice the flank stress. It also goes with 1/√b, so doubling the face width buys 30 %, not 50 %. This is the signature of a contact problem and it is why gear sizing feels unlike beam sizing: the cheap lever is the diameter, which appears both in Ft and in d₁.
A case-hardened robot pinion at the first stage of a joint: 3 N·m at 3000 rpm into a 1:4 pair.