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Tire-Road Interaction (Magic Formula)

Tire-road dynamics given by magic formula coefficients


Tires & Vehicles/Tire Subcomponents


This 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.

Port T represents the tire tread. Port H represents the wheel hub that transmits the thrust generated by the tire to the remainder of the vehicle. Port N accepts a physical signal input of the normal force acting on the tire. The normal force is positive if it acts downward on the tire, pressing it against the pavement. Port S outputs a physical signal with the tire slip measured during simulation.

The block provides two friction variants. The default variant, Fixed friction coefficients, accepts the magic formula coefficients as block parameters. This variant treats the coefficients as constants or load-dependent parameters. Use this variant to model tire dynamics under constant pavement conditions.

The alternative variant, Variable friction coefficients, accepts the magic formula coefficients as a physical signal input. Use this variant to model tire dynamics under variable pavement conditions. Selecting this variant exposes physical signal port M. Use this port to provide the magic formula coefficients as a four-element vector, specified in the order [b, c, d, e].

To change block variants:

  1. Right-click the tire block.

  2. In the context-sensitive menu, select Simscape > Block choice.

  3. Select the desired block variant.


Parameterize by

Select how to use the Magic Formula to model the tire-road interaction. The default setting is Peak longitudinal force and corresponding slip. This and all dependent parameters are visible only when the block variant is set to Fixed friction coefficients. For more information about the block variants, see the block description.

  • 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.




Conserving mechanical rotational port associated with the axle of the tire


Conserving mechanical translational port associated with the hub of the tire


Physical signal input port associated with the normal force on the tire


Physical signal input associated with the static and kinetic friction coefficients of the tire


Physical signal output port associated with the relative slip between the tire and road

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