clutch-brake

name	clutch-brake
description	Clutch and brake design — disc clutch/brake, drum brake, band brake, energy dissipation, thermal limits, actuation force, friction material selection, stopping time.
metadata	{"priority":5,"promptSignals":{"phrases":["clutch","brake","disc brake","drum brake","band brake","friction clutch","stopping time"],"minScore":4}}

Clutch & Brake Design — Complete Skill

Basic Equations

Torque Transmitted (Single Friction Pair)

T = μ × F_n × r_eff

Where:

• μ = coefficient of friction [0.3-0.5 dry, 0.1-0.2 wet/oil]
• F_n = normal force [N]
• r_eff = effective (mean) friction radius [m]

Stopping/Engagement Time

Δω = α × t → t = (ω_i - ω_f) / α α = T_net / I_system [rad/s²] T_net = T_brake - T_load (for braking)

Energy Dissipated

ΔKE = ½ I (ω_i² - ω_f²) [J] For stop: ΔKE = ½ I ω_i² Entirely converted to heat → temperature rise in friction material

Disc Clutch / Disc Brake

Single-Plate (Two Friction Surfaces)

Uniform pressure (new condition): p_max = F_a / (π(r_o² - r_i²)) T = (2/3) μ F_a (r_o³ - r_i³)/(r_o² - r_i²)

Uniform wear (worn-in condition): p × r = p_max × r_i = constant F_a = 2π p_max r_i (r_o - r_i) T = μ π p_max r_i (r_o² - r_i²) = μ F_a (r_o + r_i)/2

Uniform wear gives lower T for same F_a → use for design (conservative) Optimal r_i/r_o = 0.577 (maximum T per unit friction area)

Multiple-plate clutch: T = n_f × μ F_a (r_o + r_i)/2 [n_f = number of friction interfaces] For n discs: n_f = 2n - 1 (both sides of each intermediate plate)

Axial Actuation Force

F_a = T / (n_f × μ × r_eff) where r_eff = (r_o + r_i)/2 [mean radius, uniform wear]

Drum Brake (External Expanding)

Simplex (Leading/Trailing) Drum Brake

For rotating drum of radius r, shoe arc angle (2θ): Moment of friction force about pivot: M_f = μ r × (∫p dA × moment arm) = μ r b p_max/(sin θ_a) × [r/2 × (θ₂ - θ₁) - r/4 × (sin 2θ₂ - sin 2θ₁)] - μ × moment of normal forces

Where:

• b = shoe width
• θ_a = angle of max pressure (usually = 90°, so p_max at 90°)
• For p = p_max × sin(θ): self-energizing if geometry favorable

Leading shoe: friction assists activation → self-locking possible Trailing shoe: friction resists activation → more stable

Actuating force F_a (for leading shoe): F_a = (M_N - M_F) / c [c = distance from pivot to actuating force] M_N = p_max × b × r / sin(θ_a) × integral_N M_F = p_max × μ × b × r / sin(θ_a) × integral_F

If M_F > M_N → self-locking (F_a ≤ 0, irreversible) Self-locking condition: μ ≥ some geometric ratio (depends on brake geometry)

Band Brake

Band wraps around drum; capstan effect: F_tight / F_slack = e^(μθ) [belt friction equation] θ = wrap angle [rad]

T = (F_tight - F_slack) × r = F_slack × (e^(μθ) - 1) × r

Simple band brake actuation force: F_a × L = F_tight × a or F_slack × b [depending on hinge geometry]

Self-locking: if moment arm geometry makes F_a ≤ 0 (irreversible, e.g., hand of God brake) Used in: hoists, backstop mechanisms, Prony brake dynamometers

Thermal Analysis

Temperature Rise (Single Engagement)

ΔT = ΔKE / (m_c × c_p)

Where:

• m_c = mass of clutch/brake element that absorbs heat
• c_p = specific heat (steel: 477 J/kg·K, cast iron: 460 J/kg·K)

Allowable ΔT: friction material limits: Organic (paper/resin): max 200-260°C Sintered metal: max 350-500°C Carbon-carbon: max 600°C+ (aircraft brakes)

Continuous Operation (Power Dissipation)

q̇ = μ × p_avg × V_sliding × A_friction [W/m²] q̇_allowable: 0.3-1.2 MW/m² (organic, dry); up to 3 MW/m² (sintered metal)

Friction Materials

Material	μ (dry)	μ (wet)	Max p [MPa]	Max T [°C]
Organic (paper/resin)	0.35-0.45	0.10-0.15	0.5-1.4	200-260
Asbestos-based	0.35-0.50	—	1.0-2.0	260 (phased out)
Molded asbestos-free	0.35-0.45	—	1.0-1.7	220-250
Sintered metal (bronze)	0.3-0.35	0.08-0.12	1.5-3.0	350-500
Carbon-carbon	0.2-0.3	—	2.0-5.0	600+

Brake Factor C_BF

C_BF = T / (F_a × r) — dimensionless torque amplification

C_BF for disc (uniform wear): C_BF = n_f × μ × (r_o+r_i)/2 / r_o ≈ n_f × μ × 0.75 (typical) Higher C_BF → greater braking for same actuation force

Self-energizing design: C_BF → ∞ (requires zero actuation force) but sensitive to μ variation (danger)

Stopping Distance (Vehicle)

D = v²/(2a) where a = μ_tire × g (maximum deceleration) t_stop = v/a For μ_tire = 0.8, v = 100 km/h = 27.8 m/s: D = 27.8²/(2×0.8×9.81) = 49.2 m

Output

Provide: T_friction [N·m], actuation force F_a [N], stopping time t [s], energy dissipated [kJ], ΔT [°C], friction material recommendation, n_f (number of friction interfaces).