Vehicle Cornering Lab (2D)

Slip angle and the tyre force

A tyre does not roll exactly where it points. The angle between the wheel plane and the direction the contact patch actually moves is the slip angle alpha. The rubber twists in the contact patch and produces a sideways force that rises roughly linearly at small angles, Fy = C_alpha alpha, then bends over and peaks near the friction limit mu Fz. This page uses the Pacejka magic formula Fy = D sin(C atan(B alpha - E (B alpha - atan(B alpha)))) with D = mu Fz, shape factors C = 1.45 and E = -0.6, and the stiffness factor B fixed by the cornering stiffness C_alpha = k Fz (1 - 0.2 (Fz/Fz0 - 1)). The stiffness grows less than proportionally with load (load sensitivity), which is why weight transfer costs grip.

Single-track (bicycle) model

Two axles, two states (lateral speed vy and yaw rate r): m (dvy/dt + u r) = Fyf cos(delta) + Fyr and Iz dr/dt = a Fyf cos(delta) - b Fyr, with front slip alpha_f = delta - atan((vy + a r)/u) and rear slip alpha_r = -atan((vy - b r)/u). Speed u is held by the driver (or reduced by the lift-off button). Each axle is two tyres; the lateral acceleration moves load dFz = m ay h / track to the outer wheels, shared front/rear by the roll-stiffness split. Braking moves load forward by m ax h / L and uses part of the friction circle, Fy_max -> Fy_max sqrt(1 - (ax/(mu g))^2).

Understeer gradient and characteristic speed

In the linear range the steady turn needs delta = L/R + K ay with K = (m/L)(b/C_f - a/C_r) (a, b: CG to front and rear axle; C_f, C_r: axle cornering stiffnesses). K > 0 is understeer: the front slips more, more steering is needed as speed rises, and the yaw gain r/delta = u / (L + K u^2) peaks at the characteristic speed sqrt(L/K). K < 0 is oversteer: the denominator vanishes at the critical speed sqrt(-L/K), above which the linear model has a positive real eigenvalue, i.e. the car spins unless the driver corrects it. At the limit, whichever axle saturates first decides the final behaviour: the handling curve on graph 1 shows where the front or rear runs out.

Numerical checks on this page

(1) A 0.3 degree steering step is integrated for 14 s and the steady yaw rate is compared with u delta / (L + K u^2). (2) The integrated path is fitted with a circle through three points and the radius compared with sqrt(u^2 + vy^2) / r. (3) Understeer cars: the maximum of the closed-form yaw gain is located by a scan and compared with sqrt(L/K); oversteer cars: the sign change of the largest eigenvalue across the critical speed.

Honest simplifications

No tyre relaxation length, no combined-slip ellipse apart from the braking scale factor, no steering compliance, no camber thrust, no aerodynamic downforce, no suspension kinematics or roll dynamics (roll angle is a fixed 4.5 deg/g scaled by CG height and mass), constant speed except in the lift-off manoeuvre, flat road, equal front and rear track, and the same Pacejka shape for all tyres. Real vehicles show a more gradual limit and a driver always corrects. The numbers illustrate the physics; they are not a tuning tool for a real car.

• Load Family car: understeer and watch graph 1 rise: the front axle needs a larger slip angle than the rear. Press Skidpad ramp to see the car push wide when the front tyres saturate.
• Load Rear-biased sports car: oversteer, choose Skidpad ramp and then Lift off / brake in the corner: load moves forward, the rear loses grip and the rear slip angle grows past the front.
• Compare Tall SUV with the family car in the Slalom: a high centre of gravity gives more load transfer, so the tyre curves on graph 3 flatten and the roll angle grows.
• Drop the road friction to 0.5 (wet) and watch the grip limit readout fall and the car run wide at a much lower lateral acceleration.
• Move front weight rearward until K turns negative: the readout changes to oversteer with a critical speed, and above it the yaw rate keeps growing.