Worm Gearing

§1A screw driving a wheel

The worm is a screw — usually one, two or four thread starts wound round a cylinder. The wheel is a gear whose teeth are curved to wrap the worm. They mesh on shafts that cross at a right angle without intersecting.

What makes the worm distinctive is the ratio available in a single mesh. Each full turn of a single-start worm advances the wheel by exactly one tooth, so a 40-tooth wheel needs 40 worm turns per wheel turn — a 40 : 1 reduction from one pair, where spur or bevel gearing would need two or three stages to match it. The trade is efficiency: the worm works by sliding, not rolling, and sliding means friction, heat and loss (§5). Worms are chosen where high reduction, compactness, quiet running or self-locking outweigh that cost.

§2Lead, lead angle and ratio

Two screw properties govern the worm: the lead (axial advance per turn) and the lead angle (the helix angle of the thread at the pitch cylinder).

Here z_w is the number of thread starts, m_x the axial module and d_w the worm’s pitch diameter. The ratio depends only on wheel teeth divided by worm starts — so a single-start worm on a 40-tooth wheel gives 40 : 1, a double-start worm 20 : 1, and so on. The lead angle is the hinge on which efficiency and self-locking both turn (§4–5): a small lead angle (few starts) tends toward self-locking and low efficiency; a large one toward free running and higher efficiency.

§3Geometry of a worm set

The wheel is sized like any gear from its axial module; the centre distance then follows from the two pitch diameters.

The worm’s own pitch diameter is chosen fairly freely — it is not tied to the module the way a gear’s is — because it sets the lead angle for a given lead. A smaller worm diameter raises the lead angle (better efficiency) but weakens the worm shaft; a larger one does the reverse. That single free choice is the worm designer’s main lever.

§4Self-locking

A worm set can be irreversible: the worm drives the wheel, but no torque on the wheel can turn the worm back. This is self-locking, and it is often the whole reason a worm is chosen.

Self-locking occurs, broadly, when the lead angle is smaller than the friction angle ρ = arctan μ — that is, when the thread is shallow enough that friction alone resists back-driving. In practice a lead angle below about 5–6° is usually self-locking, though the margin depends on μ, on lubrication, and on vibration (which can shake a marginal set loose). The property is prized in hoists, jacks and any drive that must hold its load with the motor off — a worm-driven gate or lift stays put by geometry, needing no brake. The same shallow angle that locks it, however, is exactly the one that makes it inefficient.

§5Sliding, efficiency and heat

Because the worm thread slides across the wheel teeth, friction dominates worm performance in a way it never does for rolling gears.

Efficiency rises with lead angle and falls with friction, and for a single-start worm it is often only 50–70 %. All the lost power becomes heat, so worm boxes need generous casings, cooling fins and the right lubricant — a mild extreme-pressure oil that survives the sliding — and continuous-duty worms are frequently thermally limited rather than strength limited. Multi-start worms with larger lead angles reach 85–95 % but give up both the high ratio and the self-locking. The efficiency equation is the same relation that governs a screw thread on the Mechanics page, which is exactly what a worm is.

§6Worked single-start drive

One set shows ratio, geometry and efficiency together.

§7Quick reference

The working core of the page on one card rack.