Tire-Road Interaction (Magic Formula)
Tire-road dynamics given by magic formula coefficients
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Simscape / Driveline / Tires & Vehicles / Tire Subcomponents
Description
The Tire-Road Interaction (Magic Formula) block models the interaction between the tire tread and road pavement. The longitudinal force arising from this interaction is given by the magic formula, an empirical equation based on four fitting coefficients. Tire properties such as compliance and inertia are ignored.
Tire-Road Interaction Model
The Tire-Road Interaction (Magic Formula) block models the longitudinal forces at the tire-road contact patch using the Magic Formula of Pacejka [1].
The figure displays the forces on the tire. The table defines the tire model variables.
Tire-Road Contact Variables
Symbol | Description and Unit |
---|---|
Ω | Wheel angular velocity |
rw | Wheel radius |
Vx | Wheel hub longitudinal velocity |
Tire tread longitudinal velocity | |
Wheel slip velocity, defined as the difference between the longitudinal velocities of the wheel hub and the tire tread | |
Wheel slip | |
Fz | Vertical load on tire |
Fz0 | Nominal vertical load on tire |
Longitudinal force exerted on the tire at the contact
point. Also a characteristic function f of the tire. |
Tire Response
A tire model provides a steady-state tire characteristic function, , the longitudinal force Fx on the tire, based on:
Vertical load Fz
Wheel slip κ
The Magic Formula is a specific form for the tire characteristic function, characterized by four dimensionless coefficients, B, C, D, and E, or stiffness, shape, peak, and curvature:
The slope of f at is .
A more general Magic Formula uses dimensionless coefficients that are functions of the
tire load. A more complex set of parameters p_i
, entered in
the property inspector, specifies these functions:
Where:
SHx and SVx represent offsets to the slip and longitudinal force in the force-slip function, or horizontal and vertical offsets if the function is plotted as a curve. μx is the longitudinal load-dependent friction coefficient. εx is a small number inserted to prevent division by zero as Fz approaches zero.
The block uses a representative set of Magic Formula coefficients. The block scales the coefficients to yield the peak longitudinal force Fx0 at the corresponding slip κ0 that you specify, for rated vertical load Fz0.
Numerical values are based on empirical tire data. These values are typical sets of constant Magic Formula coefficients for common road conditions.
Surface | B | C | D | E |
---|---|---|---|---|
Dry tarmac | 10 | 1.9 | 1 | 0.97 |
Wet tarmac | 12 | 2.3 | 0.82 | 1 |
Snow | 5 | 2 | 0.3 | 1 |
Ice | 4 | 2 | 0.1 | 1 |
Assumptions and Limitations
The Tire-Road Interaction (Magic Formula) block assumes longitudinal motion only and includes no camber, turning, or lateral motion.
Ports
Input
Output
Conserving
Parameters
References
[1] Pacejka, H. B. Tire and Vehicle Dynamics. Elsevier Science, 2005.
Extended Capabilities
Version History
Introduced in R2011a