| 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).
