Clutches and brakes

Why it matters

Clutches connect and disconnect a driven machine from a running motor or engine; brakes absorb kinetic energy to slow or hold it. Both are friction devices sized by the same few ideas: how much torque a friction surface can transmit, how pressure is distributed, and where the heat goes. In mechatronic systems, spring-applied electromagnetic brakes hold vertical axes and robot joints safely when power fails.

Key ideas

Friction plate (disc) clutch. An axial force W presses annular friction surfaces (outer radius Ro, inner radius Ri) together. Torque capacity depends on the assumed pressure distribution:

• Uniform pressure - valid for new, accurately flat surfaces: friction radius Rf = (2/3)·(Ro³ − Ri³)/(Ro² − Ri²).
• Uniform wear - after initial running-in, wear is proportional to p·v, so p·r = constant; maximum pressure occurs at the inner radius. Friction radius Rf = (Ro + Ri)/2. This gives a slightly smaller torque, so it is the conservative and usual design assumption.
• Torque T = n·μ·W·Rf, where n is the number of pairs of friction surfaces: n = 2 for a single-plate clutch with both sides effective; n = n₁ + n₂ − 1 for a multi-plate clutch with n₁ driving and n₂ driven plates.
• Multi-plate clutches give high torque in a small diameter (motorcycles, automatic transmissions); wet clutches run in oil for cooling (lower μ).

Cone clutch. Friction surfaces on a cone of semi-cone angle α wedge together, so a small axial force gives a large normal force: T = μ·W·Rf/sin α. α is usually about 12.5° or more to avoid jamming.

Centrifugal clutch. Shoes are flung outwards against a drum as speed rises; engages automatically above a set speed (mopeds, chain saws).

Brakes.

• Block (shoe) brake: a block pressed on a drum with normal force N gives friction force μ·N and torque μ·N·r. Lever analysis decides the actuating force; depending on the pivot position relative to the friction force line, the friction moment either assists (self-energising) or opposes the applied force. If it assists so much that no actuating force is needed, the brake is self-locking (normally undesirable). For long shoes, the equivalent friction coefficient μ′ = 4μ·sin θ/(2θ + sin 2θ) (θ = half the contact angle) is used.
• Band brake: a flexible band wrapped through angle θ; tensions obey T₁/T₂ = e^(μθ) (T₁ tight side), and braking torque = (T₁ − T₂)·r. In a differential band brake both ends are attached to the lever, which can make it self-energising.
• Internal expanding shoe (drum) brake - vehicles; disc brake - calipers squeeze pads on a disc; open to air so it cools well and resists fade.

Energy and heat. All the kinetic energy (and any potential energy of a lowered load) is converted to heat in the friction surfaces: E = ½·m·(v₁² − v₂²) + ½·I·(ω₁² − ω₂²) + m·g·h. Temperature rise of the drum or disc ΔT = E/(m_d·c). Excess temperature reduces μ (fade) and damages linings, so heat capacity and cooling (and pV limits on lining pressure and speed) are design checks.

Formulas

Rf = (2/3)·(Ro³ − Ri³)/(Ro² − Ri²) (uniform pressure) · Rf = (Ro + Ri)/2 (uniform wear)

• Rf: friction radius (m); Ro, Ri: outer and inner radii of the friction surface (m).

T = n·μ·W·Rf (plate clutch) · n = n₁ + n₂ − 1 (multi-plate)

• T: torque capacity (N·m); n: number of pairs of friction surfaces; μ: coefficient of friction; W: axial force (N).

p_max = W / (2π·Ri·(Ro − Ri)) (uniform wear, at the inner radius)

• p_max: maximum pressure (Pa).

T = μ·W·Rf / sin α (cone clutch)

• α: semi-cone angle.

T = μ·N·r (short-shoe brake) · μ′ = 4μ·sin θ / (2θ + sin 2θ) (long shoe, θ = half contact angle)

• N: normal force on the shoe (N); r: drum radius (m).

T₁ / T₂ = e^(μθ) · T_b = (T₁ − T₂)·r (band brake)

• T₁, T₂: tight and slack side tensions (N); θ: angle of wrap (rad); T_b: braking torque (N·m).

E = ½·I·(ω₁² − ω₂²) + ½·m·(v₁² − v₂²) + m·g·h · ΔT = E / (m_d·c)

• E: energy absorbed (J); I: moment of inertia (kg·m²); ω: angular speed (rad/s); m_d: mass of drum/disc (kg); c: specific heat (J/(kg·K)).

Worked examples

Example 1 (standard). A single-plate clutch, both sides effective, has friction surfaces with Ro = 150 mm and Ri = 100 mm, μ = 0.3 and axial force 4 kN. Find the torque by uniform wear and by uniform pressure, and the maximum pressure (uniform wear).

1. Uniform wear: Rf = (150 + 100)/2 = 125 mm; T = n·μ·W·Rf = 2 × 0.3 × 4000 × 0.125 = 300 N·m.
2. Uniform pressure: Rf = (2/3)(150³ − 100³)/(150² − 100²) = (2/3)(2,375,000/12,500) = 126.7 mm; T = 2 × 0.3 × 4000 × 0.1267 = 304 N·m.
3. p_max = W/(2π·Ri·(Ro − Ri)) = 4000/(2π × 100 × 50) = 0.127 MPa at the inner radius.

Example 2 (GATE level). A simple band brake acts on a 400 mm diameter drum with a wrap angle of 270° and μ = 0.25. It must give a braking torque of 300 N·m. The tight side is anchored at the lever fulcrum and the slack side is attached to the lever 100 mm from the fulcrum; the operating force acts 600 mm from the fulcrum. Find T₁, T₂ and the operating force.

1. θ = 270° = 4.712 rad; e^(μθ) = e^(0.25 × 4.712) = e^1.178 = 3.248.
2. T₁ − T₂ = T_b/r = 300/0.2 = 1500 N.
3. T₂ = 1500/(3.248 − 1) = 667 N; T₁ = 3.248 × 667 = 2167 N.
4. Moments about the fulcrum (T₁ passes through it): P × 600 = T₂ × 100 → P = 667 × 100/600 = 111 N.

Energy check (quick). A rotor with I = 2 kg·m² is stopped from 600 rpm: ω = 62.83 rad/s, E = ½ × 2 × 62.83² = 3948 J; a 5 kg steel disc (c = 460 J/(kg·K)) rises by about 1.7 K per stop.

Common mistakes

• Forgetting that a single-plate clutch usually has two friction surfaces (n = 2).
• Using uniform pressure for worn clutches - uniform wear is the standard design assumption.
• Taking the wrap angle in degrees in e^(μθ).
• Mixing up tight and slack sides; the tight side is where the drum drags the band into tension.
• Ignoring direction of rotation in block and band brake lever analysis - it changes whether the brake is self-energising.
• Forgetting potential energy when braking a descending load.

For GATE ME

Expect torque capacity of plate and cone clutches under uniform wear or uniform pressure, number of friction surfaces in multi-plate clutches, band-brake tension ratio and braking torque, block-brake lever force, and energy absorbed or temperature rise. Practise free-body diagrams of brake levers with both directions of drum rotation.

Quick check

1. Multi-plate clutch with 4 driving and 3 driven plates. Number of friction pairs?
2. Which assumption gives the lower torque, uniform wear or uniform pressure?
3. μ = 0.3, wrap 180°. Tension ratio T₁/T₂?
4. Why is a spring-applied electromagnetic brake used on a vertical servo axis? Answers: 1. 6. 2. Uniform wear. 3. e^(0.3π) = 2.57. 4. It engages automatically on power loss, so the axis cannot fall (fail-safe).