Helical gear calculation: module, helix angle and thrust
A helical gear is a spur gear with the teeth cut on a slant. That one angle makes it quieter and stronger, but it also adds a force pushing along the shaft and a second module to keep track of. Here is every formula you need, with a worked example you can follow.
Why helical gears
On a spur gear, a whole tooth meets its partner at once, which is why a fast spur train whines. On a helical gear the contact starts at one end of the tooth and rolls across, so two or three teeth are always sharing the load. The result is smoother, quieter running and higher load capacity at the same size. The price you pay is an axial thrust force and a little extra friction. The whole calculation is about quantifying both.
Normal vs transverse module
The single biggest source of confusion in helical gears is that there are two modules. The teeth are cut perpendicular to the helix, so the cutting tool works in the normal plane. The gear runs in the transverse plane, perpendicular to the shaft.
- Normal module mn — set by the hob; this is what you order.
- Transverse module mt = mn / cosβ — governs the running geometry.
Because cosβ is always less than 1, the transverse module is always larger than the normal module. Mixing the two is the classic helical-gear error.
Geometry formulas
| Quantity | Formula |
|---|---|
| Transverse module | mt = mn / cosβ |
| Pitch diameter | d = z × mn / cosβ |
| Centre distance | a = mn(z1 + z2) / (2 cosβ) |
| Transverse pressure angle | αt = arctan(tanαn / cosβ) |
| Virtual tooth number | zv = z / cos³β |
| Gear ratio | i = z2 / z1 |
| Axial pitch | px = π mn / sinβ |
The virtual tooth number zv matters because it is the spur-equivalent tooth count. You use it to choose the form cutter and to run a bending-strength check, since a helical tooth behaves like a spur tooth with zv teeth.
Tooth forces and axial thrust
Three force components act at the tooth. Start from the tangential force, which carries the torque:
- Tangential Ft = 2T / d
- Radial Fr = Ft tanαt
- Axial (thrust) Fa = Ft tanβ
Worked example
Design a helical pair: pinion z1 = 20, gear z2 = 40, normal module mn = 2 mm, helix angle β = 20°, normal pressure angle αn = 20°, transmitting T = 32 N·m at the pinion.
| Step | Calculation | Result |
|---|---|---|
| Transverse module | 2 / cos20° = 2 / 0.9397 | 2.128 mm |
| Pinion pitch dia d1 | 20 × 2 / cos20° | 42.57 mm |
| Gear pitch dia d2 | 40 × 2 / cos20° | 85.14 mm |
| Centre distance a | 2(20 + 40) / (2 × 0.9397) | 63.85 mm |
| Gear ratio i | 40 / 20 | 2.0 |
| Transverse pressure angle αt | arctan(tan20° / cos20°) | 21.17° |
| Virtual teeth zv1 | 20 / cos³20° | 24.1 |
Now the forces, using d1 = 42.57 mm = 0.04257 m:
| Force | Calculation | Result |
|---|---|---|
| Tangential Ft | 2 × 32 / 0.04257 | 1503 N |
| Radial Fr | 1503 × tan21.17° | 582 N |
| Axial Fa | 1503 × tan20° | 547 N |
So a modest 32 N·m at the pinion still throws 547 N straight down the shaft. That thrust drives the bearing choice. Run your own numbers through the gear calculator to check the geometry instantly.
Common mistakes
- Using mn where mt belongs. Centre distance and pitch diameter both run on cosβ; forget it and your gears will not mesh.
- Ignoring axial thrust. A plain deep-groove bearing will not survive a steady 500 N of thrust for long.
- Picking the cutter on z, not zv. Tooth form follows the virtual tooth number.
- Mismatched helix hands. An external pair must have opposite hands, one left and one right, to mesh.
- Over-steep helix. Above about 30° the thrust grows faster than the smoothness gain; most power gears sit at 15–25°.
Frequently asked questions
What is the difference between normal and transverse module?
Normal module mn is measured across the teeth and sets the cutter; transverse module mt is measured in the plane of rotation and equals mn/cosβ, so it is always larger.
How do you find pitch diameter?
d = z × mn / cosβ. For the same module and tooth count it is larger than a spur gear.
Why do helical gears create thrust?
The angled teeth give the contact force a component along the axis: Fa = Ft tanβ. Herringbone gears cancel it.
Are helical gears stronger than spur gears?
Yes, because more teeth share the load at once, which also makes them quieter, at the cost of axial thrust and a little efficiency.