Tyre Grip

Vehicle Dynamics · Demo

Slip angle, temperature and one shared budget — the whole physics of grip, solved live.

The tyre

Slip angle3.0°

Vertical load4.0 kN

Surface temperature60 °C

Braking demand0%

Find the peak of the curve, then try to keep it while the temperature and the braking sliders fight you for it.

Readout

Sideways force3.89 kN

Grip budget4.20 kN

Budget demanded93%

Cornering stiffness1.83 kN/°

Peak at4.7°

Temperature factor100%

The budget is μ·load — everything the tyre can transmit, in any direction, right now.

In plain terms

A cornering tyre points a few degrees away from where the car is actually going — the slip angle — and the rubber in the contact patch stretches sideways like a rank of tiny springs. Stretch further and the force grows, up to a peak where the rubber starts sliding instead of gripping; past it, more steering gives less force. Where that peak sits, and how tall it is, depends on the compound, on how hard the tyre is pressed down, and — brutally — on temperature: rubber only grips properly inside its window, and a racing slick’s window is tall and narrow where a road tyre’s is low and forgiving.

The circle on the right is the honest summary: one budget of grip, spent on cornering, braking, or a blend — and a demand that pokes outside the circle is not delivered, it is dropped.

How this is computed

The slip curve is Pacejka’s Magic Formula — the industry-standard fit for measured tyre data:

F_y = D \,\sin\!\Big(C \arctan\!\big[\,B\alpha - E\,(B\alpha - \arctan B\alpha)\,\big]\Big)

with the peak $D = \mu F_z$ and cornering stiffness $C_\alpha = BCD$ . On top of it sit two physical effects. First, load sensitivity:

\mu = \mu_0\!\left(1 - s\,\frac{F_z - F_{z0}}{F_{z0}}\right)

— doubling the load gives less than double the force, which is why weight transfer always costs total grip. Second, a temperature window: a flat-topped Gaussian around the compound’s optimum (road ≈ 60 °C and forgiving; slick ≈ 100 °C and narrow), floored at the compound’s cold grip. Braking and cornering share one budget via the friction circle — the lateral force still available under braking is $\sqrt{F_{max}^2 - F_x^2}$ .

The model is steady-state: no transient relaxation length, no camber or pressure effects, no wear, and temperature is a slider rather than simulated from slip energy. Real cars are simulated with exactly this formula shape fitted to rig measurements, and the coefficients here land in published ranges — a road tyre peaks near 6° at $\mu \approx 1.0$ , a warm slick nearer 4° at $\mu \approx 1.8$ .