Helical vs spur vs herringbone gears: contact ratio, axial thrust and module conversion compared

The three gear types at a glance

Spur gears have teeth parallel to the axis; helical gears have teeth inclined at a helix angle β; herringbone (double helical) gears join a right-hand and a left-hand helical gear back to back in a V. All three drive parallel shafts — they differ in how teeth engage and what forces they produce.

Contact ratio: why helical gears are quieter

Spur teeth enter and leave mesh along their full width at once. Each time the mesh drops from two tooth pairs to one, mesh stiffness jumps, causing vibration and noise. Helical teeth make contact gradually, so besides the transverse contact ratio εα they gain an overlap ratio εβ along the face:

εβ = b · sin β / (π · mn)　　εγ = εα + εβ

Values for 20 and 40 teeth, mn = 1, b = 10 mm. A spur pair typically has εα ≈ 1.4–1.8, so at times only one pair carries the load. At β = 20° the total ratio is about 2.6 — always two or more pairs sharing the load, so it is quieter and stronger.

TIPAim for εβ ≥ 1, i.e. face width b ≥ π·mn / sin β. For mn1 and β20° that is at least 9.2 mm; a narrow helical gear loses most of its smoothness advantage.

Axial thrust Fa = Ft·tanβ and bearing choice

Ft = 2T / d　　Fa = Ft · tan β　　Fr = Ft · tan αn / cos β

The price of a helix is axial thrust, growing with β. The table shows axial force Fa and radial force Fr for a tangential force Ft = 500 N (e.g. 10 N·m on a 40 mm pitch diameter).

• At β = 15°–20°, Fa is 27–36% of Ft — fine for deep-groove ball bearings. For more, use angular-contact or tapered roller bearings, or a thrust washer.
• Thrust direction depends on hand and rotation; reversing drives need axial location on both sides.
• Herringbone halves produce equal and opposite thrust, allowing 30°–45° helix angles without thrust bearings. Usually only one shaft is located axially; the other floats so the teeth self-centre.

Converting normal and transverse module

Measured perpendicular to the slanted teeth you get the normal module mn (the hob and cutter standard); measured perpendicular to the axis you get the transverse module mt. Pitch diameter and centre distance use mt:

mt = mn / cos β　　tan αt = tan αn / cos β　　d = mt · z　　a = mn · (z₁ + z₂) / (2 cos β)

For m2 with 20 and 40 teeth, a spur pair sits at 60 mm and a β = 20° helical pair at 63.85 mm. To hit exactly 62 mm, solve cos β = mn(z₁ + z₂) / (2a) = 120 / 124, giving β ≈ 14.59° — helical gears reach round centre distances without profile shift.

TIPThe virtual tooth count zv = z / cos³β is larger than z, so helical gears tolerate fewer teeth before undercut than the 17 of a spur gear — 12 teeth at β = 30°.

3D printing and selection advice

• Prototypes and teaching: spur gears — easiest to print, measure, debug and tune backlash.
• Quiet or fast stages (e.g. the first stage on a motor): helical, β 15°–25°, with a thrust washer or angular-contact bearing.
• High load without thrust handling: herringbone, β 30°–45°. The V keeps FDM overhangs symmetric, so no supports are needed.
• A herringbone planetary set can't be assembled axially — print it in place or split the ring gear into two bolted halves.
• Print helical and herringbone gears with the axis vertical, 0.12–0.2 mm layers, and start backlash tuning at 0.1 mm.

For production parts, SLS / MJF nylon and CNC machining handle both helical and herringbone gears. Hobbed herringbones need a central relief groove (double helical); the herringbone generator lets you set its width.