§1The helix and why it helps
On a spur gear a tooth meets its mate all along its face at one instant — a small impact repeated every tooth, which is the source of gear whine. A helical tooth is angled, so contact begins at one end and sweeps across.
Because the load is picked up and released gradually, and because more than one tooth is always partway through its sweep, helical gears run markedly quieter and can carry more load at higher speed than an equivalent spur pair. The extra overlap that the helix provides — the face contact ratio, added on top of the profile contact ratio the spur page describes — is what smooths the mesh. The single complication is that the angled tooth pushes sideways as well as round, and that axial thrust (§4) has to go somewhere.
Contents§2Normal and transverse planes
A helical tooth is described in two planes, and keeping them straight is the whole of helical arithmetic: the normal plane (perpendicular to the tooth) and the transverse plane (perpendicular to the axis, the plane you see end-on).
The cutter works in the normal plane, so the normal module m_n is the standard tool size; but the pitch diameter is set in the transverse plane, so it uses the transverse module m_t, which is always the larger of the two. Everything the spur page said about pitch circles and ratio still holds — provided the transverse values are used. For the worked pair (m_n = 3 mm, β = 20°) the transverse module is 3/cos 20° = 3.193 mm.
Contents§3Geometry of a helical pair
With the transverse module in hand, the pair is dimensioned exactly like a spur pair.
Pinion z₁ = 20, gear z₂ = 40, normal module 3 mm, helix angle 20°. Transverse module 3.193 mm, so pitch diameters are 20 × 3.193 = 63.85 mm and 40 × 3.193 = 127.70 mm. Centre distance C = m_n(z₁+z₂)/(2 cos β) = 180/(2 × 0.9397) = 95.78 mm. Note the helix angle gives the designer a free adjustment: nudging β changes the centre distance continuously, so a helical pair can be tuned to an existing housing that no spur tooth count would fit.
§4Axial thrust
The angled tooth resolves the mesh force into three: the tangential load that does the work, the radial load that separates the shafts, and the axial thrust along the shaft — unique to the helix.
With a tangential load F_t = 2000 N and β = 20°, the axial thrust is F_a = 2000 × tan 20° = 728 N — over a third of the working load, pushing straight along the shaft. That thrust must be caught by a thrust bearing or a shoulder, and it is the reason helix angles are usually kept between 15° and 30°: steeper is quieter and smoother but throws ever more load at the bearings. The cure for eliminating it entirely is the double-helical gear (§6).
§5Equivalent number of teeth
For tooth strength and cutter selection, a helical tooth behaves like a spur tooth on a larger imaginary gear — the equivalent, or virtual, gear formed by the normal section.
For the pinion above, z_v = 20/cos³ 20° = 24.1 — so a 20-tooth helical pinion is as strong at the root, and is cut as if it were, a 24-tooth spur pinion. This matters twice: it lets the spur strength methods and Lewis form factors be reused directly, and it pushes the undercut limit down, so a helical pinion can carry fewer actual teeth than a spur one before undercutting — another reason the helix earns its keep in compact drives.
Contents§6Single, double and crossed
Three arrangements cover most helical work.
Single helical
One helix; simplest and cheapest, but carries the full axial thrust to the bearings.
Double (herringbone)
Two opposite helices on one gear; the thrusts cancel internally, leaving none for the bearings. Used for heavy, high-speed drives.
Crossed helical
Two helical gears on non-parallel, non-intersecting shafts; point contact only, so light duty — a motion drive, not a power one.
The herringbone is the standard answer wherever the thrust of a single helix would be a burden — it keeps the smooth, quiet, high-capacity mesh while returning the thrust to zero, at the cost of a more expensive gear to cut. Crossed helicals, by contrast, trade nearly all load capacity for the freedom to connect skew shafts, a role that otherwise falls to the worm (its own page).
Contents§7Quick reference
The working core of the page on one card rack.
Planes
m_t = m_n/cos β
d = m_n z/cos β
Forces
F_a = F_t tan β
F_r = F_t tan φ_n/cos β
Strength
z_v = z/cos³ β
Helix angle
15°–30° typical
steeper = smoother, more thrust
Thrust cure
double helical (herringbone)
→ net thrust zero