Gear Design, Brakes and Clutches
Lewis bending equation and wear strength of spur gears, gear forces, friction clutches under uniform pressure and uniform wear, and block, band and disc brakes.
This lesson covers the design side of gears and the friction devices that start and stop machines: clutches and brakes.
Forces on spur gears
For transmitted power P at pitch-line speed v (or torque T on a gear of pitch diameter d):
- Tangential force F_t = P/v = 2T/d.
- Radial force F_r = F_t tan φ (φ = pressure angle).
- Resultant force F = F_t / cos φ.
Gear tooth strength
- Lewis equation (bending): F_t = σ_b b m Y, with face width b, module m and Lewis form factor Y (depends on the number of teeth).
- Dynamic effects are added through a velocity factor; Buckingham's equation gives the dynamic load.
- Wear (surface) strength: F_w = d_p b Q K, with ratio factor Q = 2T_g/(T_g + T_p) and load-stress factor K.
Friction clutches
A single-plate clutch with inner radius r₁, outer radius r₂, axial force W and friction μ has two design assumptions:
| Assumption | Torque per friction surface | When used |
|---|---|---|
| Uniform pressure | T = μW × (2/3)(r₂³ − r₁³)/(r₂² − r₁²) | New clutches |
| Uniform wear (pw = constant) | T = μW (r₁ + r₂)/2 | Worn clutches; safer design |
Multiply by the number of friction surfaces (2 for a single plate gripped on both sides). Uniform wear gives a slightly lower torque, so it is used for design.
Cone clutch: T = μW R_m / sin α (α = semi-cone angle).
Brakes
- Block (shoe) brake: braking torque T = μN r. Take moments about the lever pivot to relate the applied force to N. A brake where friction helps apply the shoe is self-energising; if it needs no applied force at all, it is self-locking.
- Band brake: T₁/T₂ = e^(μθ), braking torque = (T₁ − T₂) r.
- Energy absorbed while stopping = change in kinetic energy (½mv² + ½Iω²); it becomes heat.
Worked example
A single-plate clutch (both sides effective) has r₁ = 80 mm, r₂ = 120 mm, μ = 0.3 and W = 4 kN. Find the torque by uniform wear.
- Mean radius = 100 mm. Torque per surface = 0.3 × 4000 × 0.1 = 120 N·m.
- Two surfaces: T = 240 N·m.
Practice
A spur gear with pitch diameter 200 mm transmits 400 N·m; the pressure angle is 20°. What is the radial force in N? (Round to the nearest integer.)
Show answer and explanation
Answer: 1456F_t = 2T/d = 4000 N; F_r = 4000 × tan 20° ≈ 1456 N.
A single-plate clutch, both sides effective, has r₁ = 50 mm and r₂ = 150 mm, μ = 0.25 and W = 2 kN. What is the torque by uniform wear in N·m?
Show answer and explanation
Answer: 100Mean radius = 0.1 m; T = 2 × 0.25 × 2000 × 0.1 = 100 N·m.
For a new clutch, which assumption is normally used?
Show answer and explanation
Answer: AA new clutch has flat, unworn surfaces, so the pressure is nearly uniform. After wearing in, uniform wear applies.
A band brake has tight-side tension 1000 N, slack-side tension 400 N and drum radius 0.25 m. What is the braking torque in N·m?
Show answer and explanation
Answer: 150T = (T₁ − T₂) r = 600 × 0.25 = 150 N·m.