Gears

Gears transmit motion and torque with a fixed ratio. Learn spur, helical, bevel and worm gears, module and pitch geometry, gear ratios and trains, tooth forces, Lewis bending and AGMA/ISO contact stress, and gear materials.

From wristwatches to wind turbines, gears change speed and torque precisely and efficiently (up to 99 % per mesh). A small pinion driving a large gear reduces speed and multiplies torque.

Designers choose tooth size (module), tooth counts, face width and material so the teeth survive bending fatigue and surface pitting for the required life.

An analogy

Bicycle chainrings and sprockets: a small front ring with a big rear sprocket makes climbing easy (more torque, less speed) — gears do the same without a chain.

Gear types

Spur: straight teeth, parallel shafts, simple, noisy at speed. Helical: angled teeth, smoother and quieter, create axial thrust. Bevel: intersecting shafts (differentials). Worm: high ratios (10–100:1) at right angles, can be self-locking but less efficient. Planetary: compact high ratios with torque split among planets.

Involute teeth and module

Involute tooth profiles keep the speed ratio constant even if centre distance varies slightly. Mating gears must share module and pressure angle. Fewer than ~17 teeth (20°) causes undercutting unless profile-shifted.

Gear trains

Compound train ratio = product of individual ratios. Planetary sets: with sun input and ring fixed, i = 1 + z_ring/z_sun. Each mesh loses ~1–2 % (spur/helical); worms lose much more.

Rating: bending and pitting

Teeth fail by root bending fatigue (Lewis/AGMA J-factor) and surface pitting from Hertzian contact stress (σ_H ∝ √(F_t/(b d))). ISO 6336/AGMA 2001 apply factors for dynamic load, load distribution, size, reliability and life. Case-hardened (carburised) steel gears, ground teeth, profile/lead modifications and good lubrication maximise capacity and minimise noise — critical in EV gearboxes.

Formulas

• Geometry: d = m·z, centre distance a = m(z₁ + z₂)/2
• Ratio: i = z₂/z₁ = n₁/n₂ = T₂/T₁ (ideal)
• Tooth forces: F_t = 2T/d, F_r = F_t tan φ
• Lewis bending stress: σ = F_t / (b · m · Y)
• Pitch-line velocity: v = π d n / 60 000 [m/s]

Worked example: Gear geometry and forces

Given: Pinion z = 20, gear z = 60, m = 3 mm, φ = 20°; 10 kW at 1440 rpm on the pinion

Find: Diameters, centre distance, forces.

• d₁ = 60 mm, d₂ = 180 mm, a = 120 mm
• T = 10 000/(2π × 1440/60) = 66.3 N·m
• F_t = 2 × 66 300/60 = 2210 N; F_r = 2210 × tan 20° = 804 N

Answer: a = 120 mm; F_t ≈ 2.21 kN, F_r ≈ 0.80 kN

Worked example: Lewis bending check

Given: F_t = 2210 N, b = 30 mm, m = 3; Y = 0.32 (20 teeth)

Find: Root bending stress.

• σ = 2210/(30 × 3 × 0.32) = 76.7 MPa
• v = π × 0.06 × 1440/60 = 4.5 m/s (moderate dynamic factor)

Answer: ≈ 77 MPa — acceptable for through-hardened steel

Common mistakes

• Wrong: Mating gears of different modules or pressure angles. — Right: Both gears must share m and φ.
• Wrong: Using too few pinion teeth. — Right: Keep z ≥ 17 (20°) or apply profile shift to avoid undercut.

Applications

• Automotive & EV: Transmissions, differentials, single-speed EV reducers (quiet, ground helical gears).
• Industrial gearboxes: Conveyors, mixers, cranes, wind-turbine planetary gearboxes.
• Robotics: Harmonic and cycloidal drives for zero-backlash high ratios.

Key takeaways

• d = mz; mating gears share module and pressure angle.
• i = z₂/z₁; torque rises as speed falls.
• F_t = 2T/d, F_r = F_t tan φ.
• Check bending (Lewis/AGMA) and pitting (contact stress).

Going further

• Transmission error: Tooth deflection and manufacturing errors cause mesh-frequency whine; micro-geometry modifications minimise it.
• Planetary load sharing: Floating suns and flexible pins equalise load among planets.
• Gear materials: Carburised 18CrNiMo7-6 for heavy duty; nitrided, induction-hardened steels; plastic gears for quiet, light-duty mechanisms.