Helical, bevel and worm gear design

Why it matters

Car gearboxes use helical gears because they are quiet and strong; differentials and rear axles use bevel (usually spiral bevel or hypoid) gears to turn the drive through 90°; worm gears appear in steering boxes, wiper drives, window regulators and seat adjusters, where a large reduction and self-locking are wanted. Each type produces different bearing loads, and getting the axial and radial forces right is as important as the tooth strength.

Key ideas

Helical gears have teeth on a helix at angle ψ to the axis. Contact starts at one end of the tooth and spreads across the face, so several teeth share the load and engagement is gradual: they are quieter and stronger than spur gears of the same size. The price is an axial (thrust) force F_a = F_t·tan ψ that the bearings must carry; double-helical (herringbone) gears cancel it. Typical ψ = 15°–30° (up to about 45° for herringbone).

• Normal module m_n (the cutter module) and transverse module m_t are related by m_n = m_t·cos ψ, so the pitch diameter is d = m_n·z / cos ψ.
• For strength, a helical gear behaves like a spur gear with the virtual (formative) number of teeth z_v = z / cos³ψ, which is used to read the Lewis form factor. Beam strength S_b = σ_b·b·m_n·Y_v.
• Mating helical gears on parallel shafts have equal helix angles of opposite hand. (Crossed helical gears can connect non-parallel shafts but carry little load because they touch at a point.)

Bevel gears connect intersecting shafts, usually at 90°. Teeth are cut on cones meeting at the apex. For a 90° pair the pitch cone angles satisfy tan γ_p = z_p / z_g and γ_p + γ_g = 90°. Tooth forces are taken at the mean radius of the face. On the pinion the tooth force splits into tangential F_t, radial F_r = F_t·tan φ·cos γ_p and axial F_a = F_t·tan φ·sin γ_p (and the pinion's axial force equals the gear's radial force for a 90° pair). For strength, the virtual number of teeth is z_v = z / cos γ and the Lewis beam strength is reduced for the tapering tooth: S_b = σ_b·b·m·Y_v·(1 − b/A₀), where A₀ is the cone distance and b is usually limited to about A₀/3. Spiral bevel and hypoid gears (offset axes) are used in car rear axles for quietness and a lower propeller shaft.

Worm gears connect non-intersecting perpendicular shafts. The worm is a screw with z₁ starts; the wheel has z₂ teeth.

• Speed ratio i = z₂ / z₁ (a single-start worm with a 40-tooth wheel gives 40:1 in one stage).
• Lead l = p_x·z₁ = π·m·z₁; lead angle λ from tan λ = l / (π·d₁), where d₁ is the worm pitch diameter.
• Efficiency (worm driving), ignoring the pressure angle: η = tan λ / tan(λ + ϕ), where tan ϕ = μ. Efficiency rises with lead angle, so multi-start worms are more efficient.
• Self-locking: the wheel cannot drive the worm when λ ≤ ϕ (in the simple model); this needs a small lead angle and means the forward efficiency is below about 50 %. Self-locking cannot be fully relied on under vibration, so safety-critical lifts still use brakes.
• Sliding between worm and wheel generates heat, so worm drives need a thermal (heat-dissipation) check and dissimilar materials (case-hardened steel worm, phosphor bronze wheel).

Formulas

Torque: T = 60·P / (2π·N), F_t = 2T / d Helical: m_n = m_t·cos ψ, d = m_n·z / cos ψ, z_v = z / cos³ψ Helical forces: F_r = F_t·tan φ_n / cos ψ, F_a = F_t·tan ψ Bevel (90°): tan γ_p = z_p / z_g, z_v = z / cos γ Bevel pinion forces: F_r = F_t·tan φ·cos γ_p, F_a = F_t·tan φ·sin γ_p Bevel strength: S_b = σ_b·b·m·Y_v·(1 − b/A₀) Worm: i = z₂ / z₁, tan λ = π·m·z₁ / (π·d₁) = m·z₁ / d₁ Worm efficiency: η = tan λ / tan(λ + ϕ), tan ϕ = μ Self-locking: λ ≤ ϕ

Symbols: P = power (W); N = speed (rpm); T = torque (N·mm); d = pitch diameter (mm); F_t, F_r, F_a = tangential, radial and axial forces (N); ψ = helix angle; φ_n, φ = normal and transverse pressure angles; m_n, m_t, m = modules (mm); z = number of teeth; z_v = virtual number of teeth; γ_p, γ_g = pitch cone angles; A₀ = cone distance (mm); b = face width (mm); z₁ = worm starts; z₂ = wheel teeth; d₁ = worm pitch diameter (mm); λ = lead angle; ϕ = friction angle; μ = coefficient of friction; η = efficiency.

Worked examples

Example 1 (standard). A helical pinion with 24 teeth, normal module 3 mm, helix angle 25° and normal pressure angle 20° transmits 10 kW at 1500 rpm. Find the pitch diameter, the three force components and the virtual number of teeth.

1. d = 3 × 24 / cos 25° = 79.44 mm.
2. T = 60 × 10 000 / (2π × 1500) = 63.66 N·m = 63 662 N·mm.
3. F_t = 2 × 63 662 / 79.44 = 1603 N.
4. F_r = 1603 × tan 20° / cos 25° = 644 N.
5. F_a = 1603 × tan 25° = 747 N.
6. z_v = 24 / cos³25° = 32.2 (read Y for about 32 teeth).

Example 2 (GATE level). A two-start worm with module 8 mm and pitch diameter 64 mm drives a 40-tooth wheel. μ = 0.05. The worm runs at 1440 rpm with 5 kW input. Find the lead angle, efficiency, wheel speed and output torque, and say whether the drive is self-locking.

1. Lead l = π × 8 × 2 = 50.27 mm; tan λ = 50.27 / (π × 64) = 0.25, so λ = 14.04°.
2. Friction angle ϕ = tan⁻¹ 0.05 = 2.86°.
3. η = tan 14.04° / tan 16.90° = 0.25 / 0.3038 = 0.823 (82.3 %).
4. Wheel speed = 1440 × 2/40 = 72 rpm.
5. Output torque = η·P / ω₂ = 0.823 × 5000 × 60 / (2π × 72) = 546 N·m.
6. λ = 14.04° > ϕ = 2.86°, so the drive is not self-locking; the wheel could back-drive the worm.

Common mistakes

• Calculating torque as P·D/(2πN). Torque does not depend on gear diameter: T = 60P/(2πN); the diameter only converts torque into tooth force.
• Using the transverse module where the normal module is required (or vice versa).
• Forgetting that a helical gear produces axial thrust and that its direction depends on hand and rotation.
• Using z instead of z/cos³ψ when reading the form factor for a helical gear.
• Writing worm efficiency with the angles reversed, or claiming any worm drive is self-locking.
• Treating the worm speed ratio as d₂/d₁; it is z₂/z₁.

For GATE ME

Expect: force components on helical and bevel gears (often to find bearing reactions); virtual number of teeth; module conversions; worm gear ratio, lead angle, efficiency and self-locking; and qualitative comparisons of gear types. Practise resolving a helical or bevel tooth force into three components quickly.

Quick check

1. Helix angle 30°, F_t = 2 kN. Axial force?
2. Transverse module 4 mm, helix angle 20°. Normal module?
3. A three-start worm drives a 60-tooth wheel. Speed ratio?
4. For a 90° bevel pair with z_p = 20, z_g = 40, what is the pinion pitch cone angle?
5. What is the condition for a worm drive to be self-locking?

Answers: 1. 2000 × tan 30° = 1155 N. 2. 4 × cos 20° = 3.76 mm. 3. 20. 4. tan⁻¹(0.5) = 26.57°. 5. Lead angle less than or equal to the friction angle.