Slip ratio and slip angle

A tire only produces force while it slips a little. Longitudinal force comes from slip ratio, the difference between how fast the wheel turns and how fast the ground goes by:

κ = (ω·R − Vx) / |Vx|

Positive when driving, negative when braking. Lateral force comes from slip angle, the angle between where the wheel points and where it actually travels:

α = atan(Vy / |Vx|)

Both curves share a shape: a linear region where force is proportional to slip, a peak, and a falloff to a sliding value. The slope at the origin is the stiffness, the peak height is peak friction × load, and the ratio of the sliding value to the peak is the falloff.

Sign convention is ISO 8855: positive slip angle means the contact patch moves toward the left, and the tire pushes back to the right, so the lateral force is negative. The explorer plots −Fy so the curve reads upward.

Explore

Drag the load and watch the peak move but the curve flatten relative to load: that is load sensitivity, and it is why load transfer changes a car's balance. Add camber to see the lateral curve shift. Add braking slip to see the lateral curve collapse: that is combined slip.

The explorer runs the real core, so what you see is what the car feels.

Sources

Pacejka, Tire and Vehicle Dynamics, ch. 1–4. Milliken & Milliken, Race Car Vehicle Dynamics, ch. 2.