| name | disc-brake |
| description | Disc brake design — contact pressure distribution, braking torque, thermal analysis (rotor temperature rise), fade, FMVSS 135, pad material selection, caliper design, NVH (squeal), duty cycle, ventilated rotor. |
| metadata | {"priority":7,"promptSignals":{"phrases":["disc brake","brake design","brake rotor","braking torque","brake fade","brake thermal"],"minScore":3}} |
Disc Brake Design — Complete Skill
Braking Torque
Braking torque (per caliper):
T_brake = μ_f × N × r_eff [N·m; μ_f = friction coefficient; N = normal clamp force; r_eff = effective radius]
For disc with uniform pressure distribution:
r_eff = (2/3) × (r_o³ - r_i³) / (r_o² - r_i²) [m; r_o, r_i = outer/inner pad radius]
For uniform wear distribution:
r_eff = (r_o + r_i) / 2 [m; uniform wear is more accurate for most pads]
Clamp force from hydraulic caliper:
N = P_brake × A_piston [N; P_brake = brake line pressure [Pa]; A_piston = piston area [m²]]
Typical: P_brake = 5–15 MPa; A_piston = 2000–5000 mm² → N = 10,000–75,000 N
Total vehicle deceleration:
a = Σ T_brake_all_wheels / (m × r_tire) [m/s²]
Or: a = μ_f × N_all × r_eff_all / (m × r_tire)
Vehicle deceleration required (FMVSS 135):
Maximum deceleration: 0.9g (high pedal force); normal stop: 0.3–0.5g
At 0.9g: F_brake = m × 0.9g per axle (front takes 65–75% load transfer)
Contact Pressure Distribution
Pressure distribution (uniform pressure assumption):
p = N / A_pad [Pa; A_pad = pad area]
Pressure distribution (uniform wear p × r = const):
p(r) = p_max × r_i / r [maximum pressure at inner pad edge]
p_max = N / (r_i × ∫_{r_i}^{r_o} ∫_0^θ r dr dθ) = N / (r_i × θ × (r_o - r_i)) [θ = pad angular span]
Contact stress limit:
p_max ≤ 3–5 MPa (metallic pads); 1–2 MPa (organic/non-asbestos pads)
Thermal Analysis
Temperature Rise During Single Stop
Energy converted:
E_stop = 0.5 × m × v₀² [J; v₀ = initial speed; 100% to 0]
E_per_rotor = E_stop × (front_brake_share) / 2 [front/rear split; 2 rotors front]
Average temperature rise (lumped thermal mass):
ΔT_avg = E_per_rotor / (m_rotor × c_p_rotor)
m_rotor = ρ_iron × V_rotor; c_p = 450 J/(kg·K) for cast iron; ρ = 7200 kg/m³
Surface temperature at end of stop (Newcomb formula):
ΔT_surface = 0.71 × q̇ × √(t_stop) / √(π × ρ × k × c_p) [J/m²·s = heat flux × √t/√(πρkc)]
q̇ = P_dissipated / A_rotor_swept [heat flux; W/m²]
A_rotor = π(r_o² - r_i²) [swept area]
Maximum surface temperature (simpler estimate):
T_max = T_ambient + E_stop / (m_rotor × c_p × 2 × 0.5) [rough; 50% front, 50% absorbed by rotor in stop]
Cast iron rotor properties at high T:
k ≈ 44 W/(m·K); c_p ≈ 500 J/(kg·K); ρ = 7200 kg/m³
Fade temperature threshold: pads begin to fade at surface T > 300°C (organic); > 500°C (low-metallic)
Duty Cycle Analysis (Mountain Pass)
Repeated braking:
E_total = N_stops × 0.5 × m × v² × duty_fraction
Heat soak: T_final = T_initial + E_total / (m_total_thermal_mass) [includes hub, bearing, tire rim]
Ventilated rotor: k_eff ≈ 2–3× solid rotor (due to internal vanes); convection cooling
Rotor Design
Solid vs. ventilated:
Solid: < 20 kW heat input; passenger cars (front); trucks (rear)
Ventilated (internal vanes): 20–100 kW; all front discs of performance cars; racing
Drilled/slotted: improves gas escape from pad; slightly lower mass; cracks more likely
Ventilated rotor vane analysis:
Mass flow: ṁ_air = ρ_air × A_vane × V_vane × N_vanes
Q_conv = ṁ_air × c_p_air × ΔT_air = h_vane × A_vane_internal × (T_rotor - T_air)
h_vane_internal = 35–60 W/(m²·K) for radial vanes at 20 m/s
Rotor minimum thickness (post-wear):
Minimum hat-to-rotor: per vehicle manufacturer spec (typically 80% of new thickness)
Below minimum: discard
Pad Material Selection
| Material | μ_f | Fade T [°C] | Wear rate | Noise | Application |
|---|
| NAO (non-asbestos organic) | 0.35–0.45 | 300 | Medium | Low | Passenger car |
| Low-metallic | 0.40–0.50 | 450 | Medium-high | Medium | Performance street |
| Semi-metallic | 0.45–0.55 | 600 | High | Higher | High-performance |
| Carbon-ceramic | 0.35–0.50 | 800+ | Very low | Low | Racing, luxury sport |
| Sintered metallic | 0.40–0.55 | 700 | Low | Medium | Motorcycle, railway |
R90 regulation (Europe, ECE R90): replacement pads must achieve equivalent retardation to OEM within ±15%
NVH — Brake Squeal
Squeal mechanism: friction-induced limit cycle oscillation; complex eigenvalue problem
Mode coupling instability: two modes couple → become unstable (negative damping)
CEA (Complex Eigenvalue Analysis):
Eigenvalue: λ_i = α_i + iω_i [α_i = real part → instability if α_i > 0]
Instability index: I_i = α_i / |ω_i| × 100% [% damping loss]
Squeal frequency bands:
Low frequency (< 5 kHz): groan, judder, low-frequency squeal → pad geometry, suspension
Mid-frequency (5–10 kHz): high-frequency squeal (most common complaint)
High frequency (> 10 kHz): not audible (ultrasonic)
Countermeasures:
Shim on pad back (elastomeric isolator): decouples bending modes; absorbs energy
Chamfered or slotted pad: changes effective mode shapes → shifts instability
Damping paste (adhesive): add damping to rotor hub
Abutment clips/springs: change contact stiffness at pad ears
FMVSS 135 (US Braking Standard)
Requirements for passenger cars:
- First effective stop: 0.8g deceleration from 60 mph
- Service brake performance: 0.5g from 60 mph at low pedal force
- Parking brake: hold 30% grade
- Fade test: 15 consecutive stops from 60 mph → deceleration ≥ 0.67g on 15th stop
ECE R13/R13H (European):
Similar requirements; MFDD (Mean Fully Developed Deceleration); ≥ 0.8g for passenger vehicles
Caliper Design
Single-piston floating caliper: one piston; pad on piston side pushes; caliper reacts (pulls other pad)
Simpler; lower cost; uneven wear possible if guide pins stick
Multi-piston fixed caliper: pistons both sides; better performance; higher cost; racing preferred
Piston material:
Steel: standard; high stiffness
Phenolic resin (plastic): lower thermal conductivity → less fluid vapor lock
Titanium: high-performance; lower thermal mass
Standards
| Standard | Scope |
|---|
| FMVSS 135 | Passenger car braking requirements |
| ECE R13H | Passenger car brakes (European) |
| SAE J2430 | Brake noise test procedure |
| ISO 15484 | Friction lining test methods |
| ASTM D2714 | Block-on-ring friction test |
Output
Provide: braking torque T [N·m] per caliper, effective radius r_eff [mm], clamp force N [kN], brake line pressure [MPa], contact pressure p_max [MPa] vs. limit, energy per stop E [kJ], rotor temperature rise ΔT [°C] and peak T_surface [°C] vs. fade threshold, rotor type (solid/ventilated/drilled), ventilated rotor cooling rate [kW], pad material (type, μ_f, fade T [°C]), friction-induced squeal instability index [%] (CEA top unstable mode), FMVSS 135 deceleration [g] vs. requirement (0.8g), and applicable standard (FMVSS 135, SAE J2430, ECE R13H).
