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.

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§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).

L = z_w · p_x = z_w · π m_x tan λ = Lπ d_w = z_w m_xd_w i = z_wheelz_w

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.

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§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.

d_wheel = m_x z_wheel C = d_w + d_wheel2

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.

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§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.

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§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.

η = tan λtan(λ + ρ) ρ = arctan μ

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.

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§6Worked single-start drive

One set shows ratio, geometry and efficiency together.

Example 1 — a 40 : 1 self-locking worm

Single-start worm (z_w = 1) on a 40-tooth wheel, axial module 5 mm, worm pitch diameter 50 mm, μ = 0.05. Ratio i = 40/1 = 40 : 1 in one stage. Lead L = π × 5 = 15.708 mm, so tan λ = 15.708/(π × 50) = 0.1 and the lead angle λ = 5.71° — shallow enough to be self-locking under most conditions. Wheel pitch diameter = 5 × 40 = 200 mm; centre distance C = (50 + 200)/2 = 125 mm. Efficiency η = tan 5.71°/tan(5.71° + 2.86°) = 66 % — the price of the high ratio and the self-locking, one third of the input power leaving as heat.

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§7Quick reference

The working core of the page on one card rack.

Ratio

i = z_wheel / z_w

one stage, very high

Lead angle

tan λ = z_w m_x / d_w

Self-locking

λ ≲ arctan μ

holds load, no brake

Efficiency

η = tan λ / tan(λ+ρ)

single-start 50–70 %

Heat

sliding drive → cool the box

EP lubricant

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Handbook application: from concept to controlled practice

Purpose. This expanded section turns the original page into a practical handbook. It preserves the supplied material and adds a repeatable way to apply, check and review Worm Gearing. It does not replace a contract, legislation, a controlled standard, competent engineering judgement or specialist advice.

The operating aim is to carry the subject from function and assumptions through design evidence, verification and controlled release. Read the original explanation first, then use the workflow and checks below to convert knowledge into evidence.

Apply Worm Gearing by beginning with the duty, not the component or software command. Convert the key ideas—worm, lead, gearing, wheel, angle—into measurable requirements and interfaces. Record operating and non-operating environments, duty cycle, expected life, loads, energy sources, human interaction and reasonably foreseeable abnormal conditions. When a value is not a project requirement or verified supplier datum, identify it as an assumption or illustrative value.

Create a calculation and evidence trail that another competent person can audit. Every input should carry a source, unit, revision and uncertainty or tolerance where relevant. Every model should state its boundary conditions and limitations. Keep nominal capacity separate from design capacity, and keep verification margin separate from an arbitrary safety factor. If a code or standard governs the work, confirm the applicable edition and contractual status rather than copying a number from a secondary summary.

Design for manufacture, assembly, inspection, operation and maintenance at the same time. A technically valid geometry can still fail because it cannot be fixtured, measured, cleaned, guarded, reached or replaced. Review process capability, datum or reference strategy, tolerance accumulation, access, error-proofing and changeover. Where people interact with plant, apply the hierarchy of controls and consult those who will operate, clean, maintain and recover the equipment.

Plan verification before release. Define the characteristic, method, equipment, sample or test condition, acceptance criterion, record and responsible person. Validation then asks a different question: whether the resulting system is effective and suitable in the intended use context. A passed drawing check or analysis does not by itself validate usability, maintainability or production performance.

Step-by-step operating method

Define the duty. Capture the required function, interfaces, operating environment, life, loads and unacceptable outcomes.
Establish the model. Identify governing principles, units, material or process data, assumptions and uncertainty.
Develop alternatives. Compare feasible concepts against performance, manufacturability, safety, maintainability and cost.
Verify the design. Use analysis, test, inspection or demonstration with acceptance criteria defined before execution.
Release and learn. Baseline the design, control changes, retain evidence and feed operating results into the next revision.

Illustrative design review record

Illustrative values only. Build a one-page record with the required function, input sources, assumptions, governing load or process condition, failure consequences, selected concept, verification method and acceptance criterion. Mark every numerical input as project requirement, verified supplier data, measured value, calculation output or assumption. Review the weakest evidence first. If an assumption can change safety, compliance, interchangeability or capacity, it must be resolved before release rather than buried in a calculation note.

Evidence class	Question	Release expectation
Requirement	What must the design do and under which conditions?	Approved and traceable
Input	Where did the load, property, tolerance or process limit come from?	Source, unit and revision recorded
Analysis	Which model and assumptions connect input to result?	Checkable calculation or simulation
Verification	How will conformity be demonstrated?	Method and acceptance criterion agreed
Validation	Will the solution work for intended users and conditions?	Representative use evidence

Common failure modes and recovery actions

1. Watch for

Starting detailed design before interfaces and operating limits are agreed.

Recovery: Return to the governing definition or requirement and restate the decision in one sentence.

2. Watch for

Using catalogue or typical values as though they were certified project inputs.

Recovery: Separate evidence from assumption, assign an owner and set a date for validation.

3. Watch for

Checking nominal performance while ignoring tolerances, degradation and foreseeable misuse.

Recovery: Run a small counterexample, boundary test, pilot or independent check before proceeding.

4. Watch for

Confusing verification of requirements with validation of user need.

Recovery: Record the consequence, decision and rationale, then update the controlled baseline.

5. Watch for

Releasing drawings or procedures without configuration, inspection and change controls.

Recovery: Escalate when the issue affects safety, compliance, acceptance, material value or an agreed tolerance.