Worm Gear Efficiency⚠ unverified

Understanding the Parameters

• Friction coefficient $\mu$ — dominates worm efficiency because the contact is almost pure sliding (unlike the rolling of spur/helical gears). $\mu \approx 0.03$–$0.05$ for a well-lubricated hardened-steel worm on a bronze wheel, higher when poorly lubricated. Lower friction raises efficiency but weakens self-locking.

• Lead angle $\lambda$ — the helix angle of the worm thread (small for a single-start worm, larger for multi-start). Efficiency rises steeply with $\lambda$: a shallow lead angle (large reduction, single start) is very inefficient but strongly self-locking; a steep lead angle (multi-start) is efficient but can back-drive. The self-locking boundary is $\lambda < \arctan\mu$ (the friction angle) — exactly the power-screw self-locking condition.

• The efficiency–self-locking trade-off — a worm is self-locking precisely when it is inefficient (small $\lambda$, so $\mu > \tan\lambda$). This is by design: a self-locking hoist worm accepts ~40 % efficiency to guarantee the load can't drive it backward.

Derivation (Approaching a Proof)

A worm drive is mechanically a screw and nut: the worm is the screw, the wheel tooth is the nut riding up the thread. So worm efficiency is the power-screw efficiency (Power Screw Efficiency) in gear clothing. Consider the wheel tooth as a load being driven up (and around) the worm thread's inclined plane of lead angle $\lambda$ against friction $\mu$.

The efficiency is (ideal, frictionless work)/(actual work). For motion up an inclined plane at angle $\lambda$ against friction, the ratio of the frictionless to the actual tangential force gives

$$\eta = \frac{\tan\lambda\,(1 - \mu\tan\lambda)}{\tan\lambda + \mu} = \frac{1 - \mu\tan\lambda}{1 + \mu\cot\lambda} = \frac{1 - \mu\tan\lambda}{1 + \mu/\tan\lambda},$$

which is exactly the registry form (the three expressions are algebraically identical, dividing through by $\tan\lambda$). Including the tooth pressure angle $\phi_n$ increases the normal force by $\sec\phi_n$ and modifies the friction terms, giving the fuller $\eta = (\cos\phi_n - \mu\tan\lambda)/(\cos\phi_n + \mu\cot\lambda)$; the registry uses $\phi_n = 0$. As $\mu \to 0$, $\eta \to 1$; as $\lambda \to 0$ (or $\mu \to \tan\lambda$ from above), $\eta \to 0$ and the drive self-locks.

Dimensional check. $\mu$ and $\tan\lambda$ are dimensionless, so $\eta$ is dimensionless (0–1). ✓

History and Development

The identity of worm gearing with the screw-and-nut, and its efficiency/self-locking behaviour, is classical machine design (Shigley). The self-locking property made worm drives the standard for hoists, winches, tuning pegs, and older automotive steering; the efficiency penalty is accepted for the holding capability and the large, quiet, compact reduction. Modern high-efficiency alternatives (hypoid, cycloidal) compete where back-driving is acceptable.

Related Concepts: Power Screw Efficiency, Screw Lead Angle, Worm Gear Center Distance, Torque Lower Load, Gear Power Capacity

Notes: Simplified zero-pressure-angle form (see note for the $\cos\phi_n$ version). Same physics as a power screw. Self-locking when $\lambda < \arctan\mu$; efficiency and self-locking trade off. $\lambda$ in degrees.