tire-mechanics

name	tire-mechanics
description	Tire mechanics — Pacejka Magic Formula (lateral force, longitudinal force, aligning moment Mz), contact patch geometry, slip angle and slip ratio, cornering stiffness Cα, pneumatic trail, combined slip (ellipse model), tire-road friction (µ-slip curve, peak µ and µ-slide), camber thrust, tire structural mechanics (belt/carcass/sidewall), inflation pressure effects, rolling resistance coefficient Cr, and automotive handling simulation.
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Tire Mechanics — Complete Skill

Tire Structure and Contact Patch

Tire Construction

Radial tire (predominant in passenger/truck): Radial carcass plies: steel or polyester cords at 90° to circumference → high lateral stiffness; compliant sidewall Belt: steel belt at ±15–25° bias angle → controls contact patch geometry; stabilizes tread Tread: wear-resistant rubber compound; grooves for water drainage (hydroplaning prevention) Sidewall: rubber protects carcass; flex without cracking; lower modulus

Forces transmitted through contact patch: Longitudinal F_x: traction and braking Lateral F_y: cornering (slip angle), camber (camber angle) Vertical F_z: load (affects all other forces) Overturning moment M_x; rolling resistance moment M_y; aligning moment M_z

Contact Patch Geometry

Contact patch (footprint) approximation: Ellipse: semi-axes a (longitudinal) ≈ √(R_tire × δ) × correction; b (lateral) ≈ width-dependent For passenger tire: a ≈ 60–90 mm; b ≈ 80–110 mm; contact area A_c = π × a × b ≈ 0.013–0.03 m²

Contact patch length: a ≈ √(3 × F_z × R_tire / (2 × C_z)) [C_z = tire vertical stiffness; R_tire = loaded radius] C_z typically 150–300 kN/m for passenger tires; 800–2,000 kN/m for truck tires

Contact pressure distribution: non-uniform; highest at center; varies with inflation Average pressure ≈ inflation pressure p_i (rough approximation for stiff tires) p_avg = F_z / A_c → F_z = p_i × A_c [but actual contact pressure distribution is Hertz-like]

Longitudinal Tire Forces

Slip Ratio and Traction

Slip ratio κ (longitudinal): Braking: κ = (v_x − ω × r_eff) / v_x [v_x = vehicle speed; ω = wheel angular velocity; r_eff = effective rolling radius] Driving: κ = (ω × r_eff − v_x) / (ω × r_eff) κ = 0: free rolling; κ = 1: locked wheel (braking) or spinning wheel (drive)

µ-κ curve shape: Peak friction µ_peak at κ ≈ 0.10–0.15 (dry asphalt); then decreasing to µ_slide at κ = 1 µ_peak (dry asphalt) ≈ 0.8–1.1; µ_slide ≈ 0.6–0.8 µ_peak (wet) ≈ 0.6–0.8; µ_peak (ice) ≈ 0.1–0.2

Longitudinal force: F_x = µ(κ) × F_z [at low κ: linear region; F_x ≈ C_κ × κ; C_κ = longitudinal slip stiffness] C_κ (slip stiffness): 30,000–80,000 N/unit slip ratio for passenger tire at F_z = 5 kN

Lateral Tire Forces

Slip Angle and Cornering

Slip angle α: α = arctan(v_y / v_x) ≈ v_y / v_x [for small α; v_y = lateral velocity; v_x = longitudinal velocity] α > 0: tire pointing inward from travel direction; positive α → positive F_y in SAE convention

Cornering force F_y vs. slip angle: Linear region (|α| < 5°): F_y = −C_α × α [C_α = cornering stiffness] C_α (cornering stiffness): 50,000–120,000 N/rad for passenger tire at F_z = 5 kN; increases with F_z

Cornering stiffness C_α: C_α = ∂F_y/∂α|_{α=0} [slope of F_y-α curve at zero slip angle] C_α_normalized = C_α / F_z ≈ 15–25 N/(N·deg) = 860–1,430 N/(N·rad)

C_α vs. normal load: C_α = a × F_z − b × F_z² [parabolic; peaks at nominal load; decreases with overload] Tire load sensitivity: handling deteriorates with excessive load

Pacejka Magic Formula (MF)

Lateral Force

Magic Formula F_y: F_y = D × sin(C × arctan(B × α − E × (B × α − arctan(B × α)))) + S_v α_modified = α + S_h [horizontal shift S_h accounts for ply steer, conicity]

Coefficients: B = BCD / (C × D) [stiffness factor] C = shape factor [1.3 for F_y lateral; fixed by shape of F-α curve] D = peak value = µ × F_z [≈ 0.8–1.1 × F_z] E = curvature factor [controls shape between peak and slide; negative values flatten peak] BCD = C_α (cornering stiffness) [B × C × D = cornering stiffness slope at origin]

Product BCD = C_α: C_α = BCD ≈ a₁ × F_z² + a₂ × F_z [quadratic fit to load-dependent C_α; a₁, a₂ from curve fit]

Longitudinal Force

Magic Formula F_x: F_x = D_x × sin(C_x × arctan(B_x × κ − E_x × (B_x × κ − arctan(B_x × κ)))) + S_vx BCD_x = C_κ (longitudinal slip stiffness); C_x = 1.65; D_x = µ_x × F_z

Aligning Moment

Aligning moment M_z: M_z = −t × F_y + M_z,residual [t = pneumatic trail; F_y = lateral force] M_z,residual from ply steer and conicity

Pneumatic trail t: t = t_0 × (1 − |α|/α_peak) [decreases with increasing slip angle; t_0 at α = 0] t_0 ≈ a/3 to a/4 [a = contact patch half-length; t_0 ≈ 20–40 mm passenger tire]

Physical significance of M_z: Positive M_z (for positive α): restores wheel to straight-ahead → steering feel M_z reversal near slide: loss of restoring moment → reduced driver feedback near limit

MF Combined Slip (Ellipse Model)

Combined slip (simultaneous α and κ): F_x_combined = σ_x / σ × F_x0(σ) [σ = combined slip; F_x0 = pure longitudinal at κ_eq] F_y_combined = σ_y / σ × F_y0(σ)

σ_x = κ / (1 + κ) [normalized longitudinal slip] σ_y = tan(α) / (1 + κ) [normalized lateral slip] σ = √(σ_x² + σ_y²) [combined normalized slip]

Friction ellipse: at each F_z: (F_x / µ_x F_z)² + (F_y / µ_y F_z)² ≤ 1 [simplified; actual surface is not perfectly elliptic] Traction circle: F_x² + F_y² ≤ (µ × F_z)² [circular approximation; adequate for vehicle dynamics]

Camber Effects

Camber thrust (F_y from camber angle γ): F_y_camber = −C_γ × γ [C_γ = camber stiffness; typically C_γ ≈ 0.1–0.2 × C_α for radial tires] γ > 0: wheel tilts inward (top toward vehicle centerline)

Motorcycle tires: much larger camber angles (45–55°); C_γ ≈ C_α for two-wheelers

Inflation Pressure Effects

Pressure effect on C_α: C_α increases with inflation pressure (stiffer carcass → shorter contact patch → higher stiffness per unit angle) ΔC_α ≈ +3% per +0.1 bar pressure change (approximate)

Rolling resistance coefficient C_r: F_rolling = C_r × F_z [Cr = 0.010–0.015 for passenger tires; 0.005–0.008 for low-rolling-resistance EV tires] C_r decreases with higher inflation pressure; increases with lower pressure, higher speed EU tire labeling: C-class = 0.0077–0.0090; A-class < 0.0060

Hydroplaning: Critical speed: v_hydro ≈ 10.35 × √p_i [km/h; p_i in kPa; empirical Horne-Dreher] p_i = 220 kPa (32 psi): v_hydro ≈ 10.35 × √220 = 153 km/h Groove depth > 3 mm, drainage area > 30% of tread width required above 80 km/h

Vehicle Handling Relevance

Understeer gradient: K_US = m × (a/C_αf − b/C_αr) / (l × g) [a, b = CG distances to front/rear axle; l = wheelbase] K_US > 0: understeer (stable); K_US < 0: oversteer (unstable above characteristic speed)

Characteristic speed (neutral steer speed): v_ch = √(g × l / K_US) [vehicle at neutral steer above this speed if K_US > 0; always neutral if K_US < 0]

Front-rear cornering stiffness balance: C_αf = F_zf × (C_α/F_z)_f; C_αr = F_zr × (C_α/F_z)_r Typical design: C_αr > C_αf slightly → mild understeer → stability

Standards and References

Standard	Scope
ISO 8855	Vehicle dynamics — road vehicle — terms and definitions
SAE J670	Vehicle dynamics terminology
UNECE R117	Tire rolling resistance (EU labeling)
ISO 28580	Tire rolling resistance measurement
Pacejka "Tire and Vehicle Dynamics" (3rd ed.)	Definitive reference for Magic Formula
TNO MF-Tyre	Commercial MF tire model implementation

Output

Provide: tire specification (size: e.g., 225/45R17; load index; speed rating; construction: radial; inflation p_i [bar]; F_z_rated [N]), contact patch (a [mm]; b [mm]; A_c [cm²]; p_avg [MPa]; vertical stiffness C_z [kN/m]), pure lateral (cornering stiffness C_α [N/rad] at F_z_rated; MF coefficients B, C, D, E; F_y at α = 3°/5°/10° [N]; aligning moment M_z at α = 3° [N·m]; pneumatic trail t_0 [mm]), pure longitudinal (slip stiffness C_κ [N/unit κ]; F_x at κ = 0.1/0.5/1.0; µ_peak and µ_slide values), combined slip (friction ellipse limit at rated F_z; F_x when F_y/F_z = 0.5), camber (C_γ [N/rad]; F_y_camber at γ = 3° [N]), rolling resistance (C_r; F_r at v = 100 km/h; EU tire label class), hydroplaning (v_hydro at p_i [km/h]; groove depth requirement), vehicle handling (K_US [deg/g]; C_αf, C_αr if vehicle data given; understeer/oversteer assessment), and applicable standard (SAE J670; ISO 8855; Pacejka MF for tire model source).