Spur Gears

§1Why the involute

Gears exist to transmit rotation at an exact, constant ratio. The tooth profile that guarantees this is the involute — the curve a taut string traces as it unwinds from a circle.

The requirement is the fundamental law of gearing: for the velocity ratio to stay constant, the common normal at the point of tooth contact must always pass through a fixed point on the line of centres — the pitch point. The involute satisfies this automatically, and it brings two gifts besides: the ratio is unaffected by small errors in centre distance (the gears simply mesh a little differently but still constant-ratio), and every tooth can be cut with a straight-sided rack cutter or hob, which is why involute gears are cheap to make to precision. Almost every power-transmission gear in service is an involute.

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§2Module and pitch

Tooth size is set by one number. In metric practice it is the module; in inch practice, the diametral pitch — reciprocals of each other.

module m = dz (mm) circular pitch p = π m diametral pitch P = zd(in) = 25.4m

Here d is the pitch diameter and z the number of teeth. Two gears only mesh if they share a module (and pressure angle) — the module is the compatibility key. Standard proportions follow from it: addendum (tooth above the pitch circle) a = m, dedendum b = 1.25 m, so the whole depth is 2.25 m. For the worked pair below (m = 4 mm), the circular pitch is π × 4 = 12.566 mm, the addendum 4 mm and the whole depth 9 mm.

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§3The circles of a gear

A gear is described by a family of concentric circles; the pitch circle is the one that matters most, being the imaginary rolling circle on which the ratio is defined.

The defining circles (worked for the pinion: z = 20, m = 4 mm, φ = 20°)
Circle	Definition	Pinion value
Pitch	d = m z	80 mm
Base (involute origin)	d_b = d cos φ	75.18 mm
Addendum (outside)	d_a = d + 2m	88 mm
Dedendum (root)	d_f = d − 2.5m	70 mm
The involute exists only outside the base circle — below it the flank is a non-working fillet. This is why the base circle, not the pitch circle, is the true parent of the tooth, and why the hero flank is drawn unwinding from it.

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§4Pressure angle and the line of action

Teeth do not push along the tangent; they push along the line of action, inclined to it by the pressure angle φ — the standard value being 20°.

The line of action is tangent to both base circles and passes through the pitch point; all contact travels along it, and the force between teeth acts along it too. That force therefore has a component driving the gear round (the useful tangential load) and a component pushing the shafts apart (the separating, or radial, load = tangential × tan φ). A larger pressure angle gives stronger, stubbier teeth but higher separating force and bearing load; 20° is the near-universal compromise, with 14.5° surviving in older work and 25° used where tooth strength must be maximised.

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§5Ratio and centre distance

Two numbers fall straight out of the tooth counts — the ratio the gears provide and the distance their shafts must sit apart.

i = z₂z₁ = d₂d₁ = n₁n₂ centre distance C = d₁ + d₂2 = m(z₁ + z₂)2

Example 1 — the worked pair

Pinion z₁ = 20, gear z₂ = 60, module 4 mm. Pitch diameters 80 mm and 240 mm; ratio i = 60/20 = 3 : 1 (a 1500 rev/min input leaves at 500 rev/min); centre distance C = (80 + 240)/2 = 160 mm. Because C = m(z₁+z₂)/2, the only centre distances available for a given module are fixed by the tooth totals — a real constraint when a gearbox housing is already cast.

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§6Contact ratio

Smooth running needs more than one tooth pair engaged on average, so that load is always being handed over, never dropped. That average is the contact ratio.

m_c = √(r_a1² − r_b1²) + √(r_a2² − r_b2²) − C sin φp cos φ

Example 2 — contact ratio of the pair

Feeding the worked pair’s radii into the formula (addendum radii 44 and 124 mm, base radii 37.59 and 112.76 mm, C = 160, φ = 20°) gives a contact ratio of 1.67. It means that on average between one and two tooth pairs carry the load — for two-thirds of the mesh cycle two pairs share it, for the remaining third a single pair carries alone. A contact ratio below about 1.4 runs rough and noisy; below 1.0 the drive would momentarily disengage. More teeth, finer module or a smaller pressure angle all raise it.

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§7Undercut and minimum teeth

Cut too few teeth on a pinion and the generating tool gouges the flank below the base circle — undercut — weakening the root and removing working profile.

z_min = 2sin² φ — at φ = 20°, z_min = 17.1 → 18 teeth

Below the minimum, the standard cures are a larger pressure angle (25° drops the minimum to about 12), or profile shift — feeding the cutter outward to move the tooth off the undercut zone, at the cost of a modified tooth thickness and a corrected centre distance. Profile shift is also the standard trick for making a chosen tooth count land on a required centre distance. For a first design, keeping every pinion at 18 teeth or more sidesteps the problem entirely.

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§8Quick reference

The working core of the page on one card rack.

Size

m = d/z · p = πm

P = 25.4/m

Circles

d_b = d cos φ

d_a = d + 2m

Ratio

i = z₂/z₁ · C = m(z₁+z₂)/2

Meshing

contact ratio > 1.4

force ⟂ = F_t tan φ

Undercut

z_min = 2/sin²φ

20° → 18 teeth

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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 Spur Gears. 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 Spur Gears by beginning with the duty, not the component or software command. Convert the key ideas—ratio, gear, pitch, spur, involute—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.