Tire-Road Interaction (Magic Formula)

Tire tread in contact with road surface approximated by Magic Formula


Tires & Vehicles/Tire Subcomponents


The Tire-Road Interaction (Magic Formula) block models the longitudinal tire-road contact force using the Magic Formula. Use this block as a component in tire models.

The convention for the vertical load is positive downward. If the vertical load is zero or negative, the horizontal tire force vanishes. In that case, the tire is just touching the ground or has left the ground. The longitudinal direction lies along the forward-backward axis of the tire. For model details, see Tire-Road Interaction Model.


You specify the downward vertical load Fz (in newtons) through a physical signal input at port N. The block reports the developed tire slip κ, as a decimal fraction, through a physical signal output at port S.

The tire tread in contact with the road and the transfer of horizontal thrust reaction to the vehicle at the wheel hub are represented by the translational conserving ports T and H, respectively.

Dialog Box and Parameters

Parametrize by

Select how to use the Magic Formula to model the tire-road interaction. The default is Peak longitudinal force and corresponding slip.

  • Peak longitudinal force and corresponding slip — Parametrize the Magic Formula with physical characteristics of the tire.

     Peak Force and Corresponding Slip

  • Constant Magic Formula coefficients — Parametrize the Magic Formula directly with constant coefficients. If you select this option, the panel changes from its default.

     Constant Magic Formula Coefficients

  • Load-dependent Magic Formula coefficients — Parametrize the Magic Formula directly with load-dependent coefficients. If you select this option, the panel changes from its default.

     Load-Dependent Magic Formula Coefficients

Velocity threshold

The wheel hub velocity Vth below which the slip calculation is modified to avoid singular evolution at zero velocity. Must be positive. The default is 0.1.

From the drop-down list, choose units. The default is meters per second (m/s).

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 [7].

The figure displays the forces on the tire. The table defines the tire model variables.

Tire-Road Contact Variables

SymbolDescription and Unit
ΩWheel angular velocity
rwWheel radius
VxWheel hub longitudinal velocity
rwΩTire tread longitudinal velocity
Vsx = rwΩ – VxWheel slip velocity = tread velocity T – hub velocity H
κ = Vsx/|Vx|Wheel slip
Fz, Fz0Vertical load and nominal vertical load on tire
Fx = f(κ, Fz)Longitudinal force exerted on the tire at the contact point.
Also a characteristic function f of the tire.

Tire Response

Forces and Characteristic Function

A tire model provides a steady-state tire characteristic function Fx = f(κ, Fz), the longitudinal force Fx on the tire, based on:

  • Vertical load Fz

  • Wheel slip κ

Magic Formula with Constant Coefficients

The Magic Formula is a specific form for the tire characteristic function, characterized by four dimensionless coefficients (B, C, D, E), or stiffness, shape, peak, and curvature:

Fx = f(κ, Fz) = Fz·D·sin( C·arctan[ { BκE·[ Bκ – arctan(Bκ) ] } ] ) .

The slope of f at κ = 0 is BCD·Fz.

Magic Formula with Load-Dependent Coefficients

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 dialog box, specifies these functions:

Fx0 = Dx·sin( Cx·arctan[ { BxκxEx·[ Bxκx – arctan⁡(Bxκx) ] } ] ) + SVx ,


dfz = (FzFz0)/Fz ,

κx = κ + SHx ,

Cx = p_Cx1 ,

Dx = μx·Fz ,

μx = p_Dx1 + p_Dx2·dfz ,

Ex = (p_Ex1 + p_Ex2·dfz + p_Ex3·dfz2)[1 – p_Ex4·sgn(κx)] ,

K = Fz·(p_Kx1 + p_Kx2·dfz)·exp(p_Kx3·dfz) ,

Bx = K/(CxDx + εx) ,

SHx = p_Hx1 + p_Hx2·dfz ,

SVx = Fz·(p_Vx1 + p_Vx2·dfz) .

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.

Peak Longitudinal Force and Corresponding Slip

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.

Magic Formula Coefficients for Typical Road Conditions

Numerical values are based on empirical tire data. These values are typical sets of constant Magic Formula coefficients for common road conditions.

Dry tarmac101.910.97
Wet tarmac122.30.821


The Tire-Road Interaction (Magic Formula) block assumes longitudinal motion only and includes no camber, turning, or lateral motion.

Related Examples

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