The camber angle is the angle between the road normal and the z-axis of the carrier frame, indicated with γ in the figure above.
WHAT IS MAGIC FORMULA?
Magic formula (also named Pacejka's model) models the forces that are generated by the tire as a result of the load applied on it, the slip angle, the camber angle and the slip ratio. Magic formula is not a physics model in the sense that the equations governing the dynamics of the tire were not obtained from constitutive laws but from a curve-fitting approach. Sets of mathematical formulas were developed to allow an accurate description of the steady-state behavior of tires. Some of the parameters of the equations do have a physical meaning, but most of them only act as tuning coefficients to adjust the mathematical formulation to the force and moment measurements.
In practice, measurements are carried out with specific instrumented vehicles equipped with dynamometric hubs or more commonly with lab facilities such as drum test benches or test tracks.
Many versions of MF models and MF-like models exists in the scientific literature.
The MF model proposed here aims at calculating the forces Fx (longitudinal), Fy (lateral) and moment Mz (self-aligning moment) respectively in function of the sets of 3 parameters [slip ratio, camber angle, vertical load], [slip angle, camber angle, vertical load] and [slip angle, camber angle, vertical load].
Limitations : as mentioned above, the MF model proposed here is a steady-state model, i.e. transient behavior of the tire with respect to handling at frequencies up to 8 Hz is not correctly modeled by the sets of MF equations.
Another limitation is that only pure slip conditions are modeled. This means that combined slip conditions (combined braking/traction and cornering) require more sophisticated MF equations and moreover a larger number of measurements.
CAMBER ANGLE
The
camber angle is the angle between the road normal and the z-axis of
the carrier frame, indicated with γ in the figure above.
SLIP RATIO
The slip ratio is named k
and is defined as the ratio of vehicle longitudinal speed and wheel
longitudinal slip speed :
Re is called the effective rolling radius. There are at least 3 different tire radii : undeflected radius, R, effective rolling radius, Re and loaded radius, Rl. The loaded radius is the wheel centre’s height over ground, while the effective rolling radius is a measure of how far a free rolling tire travels per revolution. Generally, Rl < Re < R. This means that the rolling tire travels farther per revolution than determined by using the wheel’s centre height as rolling radius.
The spin velocity of the wheel is named W.
Vx is the longitudinal slip speed.
SLIP ANGLE
The
slip angle, named a, is the angle
between the instant slip speed and the wheel plane.
Fx FORMULATION
Fz0= rated load (N)
Fy FORMULATION
Mz FORMULATION
R0 : unloaded tire radius (m)
REFERENCES
Tyre and Vehicle Dynamics
Author: Hans B. Pacejka
Format: Hardcover, 256 pages
Publication Date: September 2002
Publisher: Butterworth-Heinemann