Tire (Magic Formula)

Tire with longitudinal dynamics and motion approximated by Magic Formula


Tires & Vehicles


The Tire (Magic Formula) block models the longitudinal dynamics of a vehicle axle-wheel-tire combination, with road contact represented by the Magic Formula and optional deformation compliance. 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 Model.

    Note:   Tire (Magic Formula) is based on the Tire-Road Interaction (Magic Formula) block and the Simscape™ Wheel and Axle, Translational Spring, and Translational Damper blocks. For more information, see the Tire-Road Interaction (Magic Formula) block reference page about how Tire (Magic Formula) behaves in contact with the road.


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

The wheel axle rotation is represented by the rotational conserving port A. The wheel axle transfer of horizontal thrust reaction to the vehicle is represented by the translational conserving port H.

Dialog Box and Parameters

Tire Force

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—Parameterize the Magic Formula directly with its 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


Rolling radius

Unloaded tire-wheel radius rw. The default is 0.3.

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



Select how to model the dynamical compliance of the tire. The default is No compliance — Suitable for HIL simulation.

  • No compliance — Suitable for HIL simulation — Tire is modeled with no dynamical compliance.

  • Specify stiffness and damping—Tire is modeled as a stiff, dampened spring and deforms under load. If you select this option, the panel changes from its default.

     Tire Stiffness and Damping


Select how to model the rotational inertia of the tire. The default is No inertia.

  • No inertia—Tire is modeled with no inertia.

  • Specify inertia and initial velocity—Tire is modeled with rotational inertia. If you select this option, the panel changes from its default.

     Tire Rotational Inertia and Initial Velocity

Rolling Resistance

Rolling resistance

Method used to specify the rolling resistance acting on a rotating wheel hub. The default value is No rolling resistance.

No rolling resistance

Select this option to ignore the effect of rolling resistance on a model.

Specify rolling resistance

Select between two rolling resistance models: Constant coefficient and Pressure and velocity dependent.

The default value is Constant coefficient.

 Constant coefficient

 Pressure and velocity dependent

Slip Calculation

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 Model

The Tire block models the tire as a rigid wheel-tire combination in contact with the road and subject to slip. When torque is applied to the wheel axle, the tire pushes on the ground (while subject to contact friction) and transfers the resulting reaction as a force back on the wheel. This action pushes the wheel forward or backward. If you include the optional tire compliance, the tire also flexibly deforms under load.

The figure shows the forces acting on the tire. The table defines the tire model variables.

Tire Model Variables

SymbolDescription and Unit
rwWheel radius
VxWheel hub longitudinal velocity
uTire longitudinal deformation
ΩWheel angular velocity
Ω′Contact point angular velocity = Ω if u = 0
rwΩ'Tire tread longitudinal velocity
Vsx = rWΩ – VxWheel slip velocity
V'sx = rWΩ' – VxContact slip velocity = Vsx if u = 0
κ = Vsx/|Vx|Wheel slip
κ'= Vsx/|Vx|Contact slip = κ if u = 0
VthWheel hub threshold velocity
FzVertical load on tire
FxLongitudinal force exerted on the tire at the contact point.
CFx = (∂Fx/∂u)0Tire longitudinal stiffness under deformation
bFx = (∂Fx/∂ů)0Tire longitudinal damping under deformation
IwWheel-tire inertia; effective mass = Iw/rw2
τdriveTorque applied by the axle to the wheel

Tire Kinematics and Response

Roll and Slip

If the tire did not slip, it would roll and translate as Vx = rwΩ. But the tire actually does slip and develops a longitudinal force Fx only in response to slip.

The wheel slip velocity is Vsx = rWΩ – Vx. The wheel slip is κ = Vsx/|Vx|. For a locked, sliding wheel, κ = –1. For perfect rolling, κ = 0.

Slip at Low Speed

For low speeds, |Vx| ≤ |Vth|, the wheel slip is modified to:

κ = 2Vsx/(Vth + Vx2/Vth) .

This modification allows for a nonsingular, nonzero slip at zero wheel velocity. For example, for perfect slipping (nontranslating spinning tire), Vx = 0 while κ = 2rwΩ/Vth is finite.


If the tire is modeled with compliance, it is also flexible. Because in this case, the tire deforms, the tire-road contact point turns at a slightly different angular velocity Ω′ from the wheel Ω and requires, instead of the wheel slip, the contact point or contact patch slip κ'. The block models the deforming tire as a translational spring-damper of stiffness CFx and damping bFx.

If the tire is not modeled with compliance, then Ω′ = Ω, V'sx = Vsx, and κ' = κ. In this case, the tire starts simulation undeformed and remains undeformed.

Tire and Wheel Dynamics

The full tire model is equivalent to this Simscape-SimDriveline™ component diagram. It simulates both transient and steady-state behavior and correctly represents starting from, and coming to, a stop. The Translational Spring and Translational Damper are equivalent to the tire stiffness CFx and damping bFx. Tire-Road Interaction (Magic Formula) models the longitudinal force Fx on the tire as a function of Fz and κ′ using the Magic Formula, with κ′ as the independent slip variable [1].

The Wheel and Axle radius is the wheel radius rw. The Mass value is the effective mass, Iw/rw2. The tire characteristic function f(κ′, Fz) determines the longitudinal force Fx. Together with the driveshaft torque applied to the wheel axis, Fx determines the wheel angular motion and longitudinal motion.

Without tire compliance, the Translational Spring and Translational Damper are omitted, and contact variables revert to wheel variables. In this case, the tire effectively has infinite stiffness, and port P of Wheel and Axle connects directly to port T of Tire-Road Interaction (Magic Formula).

Without tire inertia, the Mass is omitted.


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

Tire Compliance

Tire compliance implies a time lag in the tire response to the forces on it. Time lag simulation increases model fidelity but reduces simulation performance. See Adjust Model Fidelity.


[1] Pacejka, H. B. Tire and Vehicle Dynamics, Society of Automotive Engineers and Butterworth-Heinemann, Oxford, 2002, chapters 1,4,7, and 8

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