| name | vehicle-dynamics |
| description | Vehicle dynamics — handling, tire forces, bicycle model, oversteer/understeer, roll stiffness, pitch/bounce, load transfer, stability derivatives, braking performance. |
| metadata | {"priority":7,"promptSignals":{"phrases":["vehicle dynamics","oversteer","understeer","handling","tire force","bicycle model","roll stiffness","cornering"],"minScore":3}} |
Vehicle Dynamics — Complete Skill
Tire Forces
Tire Coordinate System
F_x = longitudinal (traction/braking)
F_y = lateral (cornering)
F_z = normal (vertical load)
Slip angle: α = arctan(v_y / v_x) [angle between wheel plane and velocity vector]
Cornering stiffness: C_α = ∂F_y/∂α|_α=0 [N/rad]
F_y ≈ C_α × α (linear range, |α| < 5-8°)
Magic Formula (Pacejka):
F_y = D sin(C arctan(B α - E(B α - arctan(Bα))))
B = stiffness, C = shape (≈1.3-1.4 lateral), D = peak, E = curvature
Load sensitivity: C_α increases with F_z (sublinear)
C_α ≈ C_α0 × F_z/F_z0 (normalized for load normalization)
Bicycle Model (Linear, Steady-State)
Equations of Motion (Lateral + Yaw)
m(v̇_y + V ψ̇) = F_yf + F_yr
I_z ψ̈ = l_f F_yf - l_r F_yr
V = forward speed, ψ = yaw angle, l_f/l_r = CG to front/rear axle
Steady-state yaw rate gain:
ψ̇/δ = V/L × 1/(1 + K_us V²)
L = wheelbase = l_f + l_r
K_us = understeer gradient [s²/m²]
Understeer gradient:
K_us = m/L²(l_r/C_αf - l_f/C_αr)
C_αf, C_αr = front, rear cornering stiffness [N/rad]
Oversteer/Understeer/Neutral:
K_us > 0: understeer (self-correcting)
K_us = 0: neutral steer
K_us < 0: oversteer (potentially unstable)
Characteristic Speed and Critical Speed
Characteristic speed (understeer):
V_ch = √(g L / K_us) [speed at which yaw gain = (1/2) × low-speed gain]
Critical speed (oversteer — K_us < 0):
V_cr = √(-g L / K_us) [speed above which unstable; yaw diverges]
Design target: V_cr > 200 km/h (or no critical speed for understeer vehicles)
Load Transfer
Lateral Load Transfer
ΔF_z = m a_y h_cg / track_width
a_y = lateral acceleration [m/s²], h_cg = CG height [m]
Front/rear distribution:
ΔF_z,f = ΔF_z,total × k_φf/k_φ (proportional to roll stiffness)
k_φf = front roll stiffness [N·m/rad], k_φ = total roll stiffness
Roll angle:
φ = m a_y h_cg / k_φ [rad]
Braking Load Transfer
ΔF_z = m a_x h_cg / L (transferred to front axle under braking)
a_x = deceleration
Front normal force: N_f = m g l_r/L + m a_x h_cg/L
Rear normal force: N_r = m g l_f/L - m a_x h_cg/L
Optimal braking distribution (constant μ):
F_xf/F_xr = (l_r + h μ)/(l_f - h μ) × (1/brake_split)
Suspension Design Fundamentals
Natural Frequencies (Ride)
Bounce: f_b = (1/2π)√(k_s/m_sprung) [Hz]; target 1-2 Hz (passenger car)
Wheel (unsprung): f_w = (1/2π)√((k_s+k_t)/m_unsprung); target 10-15 Hz
k_s = spring rate, k_t = tire spring rate (~175 kN/m typical)
Roll stiffness:
K_φ = k_s × (t/2)² [for single spring at track t] + anti-roll bar contribution
Suspension Parameters
Static camber: -0.5 to -1.5° (slight negative) → improves lateral tire loading
Toe: slight toe-in (0.05°-0.1°) → straight-line stability
Caster angle: +3 to +7° (positive) → steering return moment
King pin inclination: 10-15° → scrub radius
Ackermann geometry: inner wheel steers more than outer for low-speed turns
Braking Performance
Maximum Deceleration
a_max = μ × g (limited by tire-road friction)
μ_dry_asphalt = 0.8-1.0; μ_wet = 0.5-0.7; μ_ice = 0.1-0.2
Braking distance (from V):
s = V² / (2 μ g) [ignores aero, grade]
At 100 km/h (27.8 m/s), μ=0.85: s = 27.8²/(2×0.85×9.81) = 46 m
ABS role: maintains slip ratio λ = (V-rω)/V ≈ 0.15-0.20 for max μ
Output
Provide: understeer gradient K_us [s²/m²], characteristic speed V_ch [km/h], steady-state yaw rate gain ψ̇/δ [rad/s per rad], lateral load transfer ΔF_z [N], roll angle φ [°] at a_y, natural frequencies f_b and f_w [Hz], braking distance [m].
