Pacejka

Pacejka refers to a family of tire models collectively known as the Pacejka Magic Formula, used to describe the relationship between tire forces and slip. The model is empirical and designed to capture how a tire’s longitudinal and lateral forces vary with slip angle or slip ratio, normal load, and other conditions. It is widely used in vehicle dynamics simulations and tire design studies.

The most widely cited version was developed in the 1980s by Hans B. Pacejka and colleagues. Over time it has been extended and refined, resulting in several variants used in automotive engineering. The core idea is to provide a compact, parameterized description of tire behavior that can be calibrated to different tires and operating conditions.

The general governing equation for a tire force F as a function of slip x is F = D * sin(C * arctan(B * x)) + E * [cos(C * arctan(B * x)) - 1], where x is slip (slip angle for lateral forces or slip ratio for longitudinal forces). The coefficients B, C, D, and E are dimensionless factors that depend on factors such as normal load and tire construction. D controls the peak force, B and C shape the curve, and E adds curvature near the peak. Different coefficient sets are used for longitudinal and lateral forces.

Calibration requires tire test data across varying normal loads, slip angles, and speeds. The resulting coefficients are stored in tables or fit with regression techniques, enabling simulations and control systems to predict tire behavior under diverse conditions.

Applications include vehicle dynamics modeling, ESC and ABS development, traction analysis, and performance tuning. Limitations include its empirical basis, dependence on representative test data, and restricted accuracy outside the calibrated operating range.