Bevel Gears

§1Cones instead of cylinders

A spur or helical pair is two cylinders rolling together; a bevel pair is two cones. Their apexes meet at the point where the shaft axes cross, and the teeth are cut on the cone surfaces.

Everything tapers: the tooth is largest at the outer (heel) end and shrinks toward the apex (the toe), so its module, pitch and depth all vary along the face. By convention the gear is dimensioned at the outer end, where the numbers are largest and the drawing is clearest. Because the cones share an apex, a bevel pair transmits motion between intersecting shafts at any shaft angle — though the right angle is overwhelmingly the common case, and the one worked here.

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§2Pitch cone angles

The half-angle of each pitch cone is fixed by the tooth counts and the shaft angle. For the usual 90° shaft angle the two cone angles are complementary and follow directly from the ratio.

shaft angle 90°: tan γ₁ = z₁z₂ tan γ₂ = z₂z₁ γ₁ + γ₂ = 90°

The pinion, with fewer teeth, gets the slim cone; the gear, the wide one. A special case names itself: when the two gears are equal (z₁ = z₂), both cone angles are 45° and the pair is a mitre gear — a right-angle drive at 1 : 1. Where the shaft angle is not 90°, the general relation tan γ₁ = sin Σ /(z₂/z₁ + cos Σ) applies, but the right-angle forms above cover the great majority of designs.

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§3Cone distance and tooth size

The slant length from apex to the outer end — the cone distance — sets the scale of the gear and the length of its teeth.

d = m z A_o = d₁2 sin γ₁ = √( r₁² + r₂² ) (at 90°)

The face width is then taken as a fraction of the cone distance — commonly no more than a third — because a tooth cut too far toward the apex becomes impractically small and weak. The outer pitch diameters d₁ and d₂ come straight from the module and tooth counts exactly as for a spur gear, since the outer end is where the gear is dimensioned.

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§4The back cone and equivalent teeth

A bevel tooth’s profile is worked out by pretending, at the outer end, that it is a spur tooth — Tredgold’s back-cone approximation, the bevel analogue of the helical equivalent gear.

z_v = zcos γ — equivalent (virtual) spur teeth on the back cone

The back cone is a cone tangent to the pitch cone at the outer end, unrolled into a flat sector; the tooth drawn on it is a spur tooth of z_v teeth, and its involute is used as the bevel profile. This is what lets the entire spur toolkit — profile, strength, contact — carry across to bevels. The equivalent counts are always larger than the real ones (the wide gear’s especially so), which is why bevel pinions resist undercut better than their tooth count alone would suggest.

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§5Straight, spiral and hypoid

Three tooth forms trade cost against smoothness and capacity, mirroring the spur-to-helical step.

Straight bevel

Teeth radial to the apex; the simplest, but engages abruptly like a spur — noisier, lower speed.

Spiral bevel

Curved, angled teeth engage gradually; quieter and stronger at speed — the helical of the bevel world. Produces thrust that varies with rotation sense.

Hypoid

Like spiral bevel but the axes are offset, not intersecting; allows a lower drive line and heavy sliding contact — the classic car rear axle.

The progression is the same lesson as spur-to-helical: gradual engagement buys quietness and load capacity at the price of complexity and induced thrust. Hypoids step beyond true bevels — the offset means the pitch surfaces are hyperboloids, not cones — and demand extreme-pressure lubricants because of the sliding they introduce.

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§6Worked right-angle pair

A single pair pulls the whole page together.

Example 1 — a 2 : 1 right-angle bevel pair

Pinion z₁ = 20, gear z₂ = 40, module 5 mm (outer end), shaft angle 90°. Cone angles: tan γ₁ = 20/40 = 0.5, so γ₁ = 26.57° and γ₂ = 63.43° (they sum to 90°, as they must). Outer pitch diameters d₁ = 100 mm, d₂ = 200 mm. Cone distance A_o = √(50² + 100²) = 111.8 mm. Equivalent teeth: pinion z_v1 = 20/cos 26.57° = 22.4, gear z_v2 = 40/cos 63.43° = 89.4 — so the gear’s tooth is profiled as an 89-tooth spur tooth, nearly straight-flanked, while the pinion is profiled as a 22-tooth one.

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§7Quick reference

The working core of the page on one card rack.

Cone angles (90°)

tan γ₁ = z₁/z₂

γ₁ + γ₂ = 90°

Scale

d = m z · A_o = √(r₁²+r₂²)

face ≤ A_o/3

Profile

z_v = z/cos γ (back cone)

Mitre

z₁ = z₂ → 45°/45°

1 : 1 right angle

Forms

straight · spiral · hypoid

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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 Bevel 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 Bevel Gears by beginning with the duty, not the component or software command. Convert the key ideas—cone, bevel, gears, pitch, cones—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.