# Tire (Magic Formula)

Tire with longitudinal behavior given by Magic Formula coefficients

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• Simscape / Driveline / Tires & Vehicles

## Description

The Tire (Magic Formula) block models a tire with longitudinal behavior given by the Magic Formula [1], an empirical equation based on four fitting coefficients. The block can model tire dynamics under constant or variable pavement conditions.

The longitudinal direction of the tire is the same as its direction of motion as it rolls on pavement. This block is a structural component based on the Tire-Road Interaction (Magic Formula) block.

To increase the fidelity of the tire model, you can specify properties such as tire compliance, inertia, and rolling resistance. However, these properties increase the complexity of the tire model and can slow down simulation. Consider ignoring tire compliance and inertia if simulating the model in real time or if preparing the model for hardware-in-the-loop (HIL) simulation.

### Tire Model

The Tire (Magic Formula) 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
VxWheel hub longitudinal velocity
uTire longitudinal deformation
ΩWheel angular velocity
Ω′Contact point angular velocity. If there is no tire longitudinal deformation, that is, if $u=0$, ${\Omega }^{\prime }=\Omega$.
${r}_{w}{\Omega }^{\prime }$Tire tread longitudinal velocity
${V}_{sx}={r}_{w}\Omega -{V}_{x}$Wheel slip velocity
${{V}^{\prime }}_{sx}={r}_{w}{\Omega }^{\prime }-{V}_{x}$Contact slip velocity. If there is no tire longitudinal deformation, that is, if $u=0$, ${{V}^{\prime }}_{sx}={V}_{sx}$.
$k=\frac{{V}_{sx}}{|{V}_{x}|}$Wheel slip
${k}^{\prime }=\frac{{{V}^{\prime }}_{sx}}{|{V}_{x}|}$Contact slip. If there is no tire longitudinal deformation, that is, if $u=0$, ${k}^{\prime }=k$.
VthWheel hub threshold velocity
FxLongitudinal force exerted on the tire at the contact point
${C}_{{F}_{x}}={\left(\frac{\partial {F}_{x}}{\partial u}\right)}_{0}$Tire longitudinal stiffness under deformation
${b}_{{F}_{x}}={\left(\frac{\partial {F}_{x}}{\partial \stackrel{˙}{u}}\right)}_{0}$Tire longitudinal damping under deformation
IwWheel-tire inertia, such that the effective mass is equal to $\frac{{I}_{w}}{{r}_{w}^{2}}$
τdriveTorque applied by the axle to the wheel

### Tire Kinematics and Response

Roll and Slip

A nonslipping tire would roll and translate as ${V}_{x}={r}_{w}\Omega$. However, as tires do slip, they respond by developing a longitudinal force, Fx.

The wheel slip velocity is ${V}_{sx}={r}_{w}\Omega -{V}_{x}$. The wheel slip is $k=\frac{{V}_{sx}}{|{V}_{x}|}$. For a locked, sliding wheel, $k=-1$. For perfect rolling, $k=0$.

Slip at Low Speed

For low speeds, as defined by $|{V}_{x}|\le |{V}_{th}|$, the wheel slip becomes:

$k=\frac{2{V}_{sx}}{\left({V}_{th}+\frac{{V}_{x}^{2}}{{V}_{th}}\right)}$

This modification allows for a nonsingular, nonzero slip at zero wheel velocity. For example, for perfect slipping, that is, in the case of a nontranslating spinning tire, ${V}_{x}=0$ while $k=\frac{2{r}_{w}\Omega }{{V}_{th}}$ is finite.

Deformation

If the tire is modeled with compliance, it is also flexible. In this case, because 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 you model a tire without compliance, that is, if $u=0$, then there is no tire longitudinal deformation at any time in the simulation and:

• ${k}^{\prime }=k$

• ${{V}^{\prime }}_{sx}={V}_{sx}$

• ${\Omega }^{\prime }=\Omega$

### Tire and Wheel Dynamics

The full tire model is equivalent to this Simscape™/Simscape Driveline™ 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. The Tire-Road Interaction (Magic Formula) block models the longitudinal force Fx on the tire as a function of Fz and k′ using the Magic Formula, with k′ as the independent slip variable.

The Wheel and Axle radius is the wheel radius rw. The Mass value is the effective mass, $\frac{{I}_{w}}{{r}_{w}^{2}}$. The tire characteristic function f(k′, 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.

## Assumptions and Limitations

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

• 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.

## Ports

### Input

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Physical signal input port associated with the normal force acting on the tire. The normal force is positive if it acts downward on the tire, pressing it against the pavement.

Physical signal input port associated with the Magic Formula coefficients. Provide the Magic Formula coefficients as a four-element vector, specified in the order [B, C, D, E].

#### Dependencies

Port M is exposed only if the Main > Parameterize by parameter is set to Physical signal Magic Formula coefficients. For more information, see Main Parameter Dependencies.

### Output

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

### Conserving

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Mechanical rotational port associated with the axle that the tire sits on.

Mechanical translational port associated with the wheel hub that transmits the thrust generated by the tire to the remainder of the vehicle.

## Parameters

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### Main

The table shows how the visibility of some Main parameters depends on the options that you choose for other parameters. To learn how to read the table, see Parameter Dependencies.

Main Parameter Dependencies

Main

Parameterize by — Choose Peak longitudinal force and corresponding slip, Constant Magic Formula coefficients, Load-dependent Magic Formula coefficients, or Physical signal Magic Formula coefficients

Peak longitudinal force and corresponding slipConstant Magic Formula coefficientsLoad-dependent Magic Formula coefficientsPhysical signal Magic Formula coefficients — Exposes physical signal input port M for providing the Magic Formula coefficients to the block as an array of elements in this order [B, C, D, E].

Magic Formula B coefficient

Magic Formula C-coefficient parameter, p_Cx1

Peak longitudinal force at rated load

Magic Formula C coefficient

Magic Formula D-coefficient parameters, [p_Dx1 p_Dx2]

Slip at peak force at rated load (percent)

Magic Formula D coefficient

Magic Formula E-coefficient parameters, [p_Ex1 p_Ex2 p_Ex3 p_Ex4]

Magic Formula E coefficient

Magic Formula BCD-coefficient parameters, [p_Kx1 p_Kx2 p_Kx3]

Magic Formula H-coefficient parameters, [p_Hx1 p_Hx2]

Magic Formula V-coefficient parameters, [p_Vx1 p_Vx2]

To model tire dynamics under constant pavement conditions, select one of these models:

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

• Constant Magic Formula coefficients — Specify the parameters that define the constant B, C, D, and E coefficients as scalars, with these default values.

CoefficientDefault Value
B10
C1.9
D1
E0.97

• Load-dependent Magic Formula coefficients — Specify the parameters that define the load-dependent C, D, E, K, H, and V coefficients as vectors, one for each coefficient, with these default values.

CoefficientParametersDefault Values
Cp_Cx11.685
D[ p_Dx1 p_Dx2 ][ 1.21 –0.037 ]
E[ p_Ex1 p_Ex2 p_Ex3 p_Ex4 ][ 0.344 0.095 –0.02 0 ]
K[ p_Kx1 p_Kx2 p_Kx3 ][ 21.51 –0.163 0.245 ]
H[ p_Hx1 p_Hx2 ][ –0.002 0.002 ]
V[ p_Vx1 p_Vx2 ][ 0 0 ]

To model tire dynamics under variable pavement conditions, select Physical signal Magic Formula coefficients. Selecting this model exposes a physical signal port M. Use the port to input the Magic Formula coefficients as a four-element vector, specified in the order [B, C, D,E].

#### Dependencies

Each parameterization method option exposes related parameters and hides unrelated parameters. Selecting Physical signal Magic Formula coefficients exposes a physical signal input port. For more information, see Main Parameter Dependencies.

#### Dependencies

This parameter is exposed only when you select the Peak longitudinal force and corresponding slip parameterization method. For more information, see Main Parameter Dependencies.

Maximum longitudinal force Fx0 that the tire exerts on the wheel when the vertical load equals its rated value Fz0.

#### Dependencies

This parameter is exposed only when you select the Peak longitudinal force and corresponding slip parameterization method. For more information, see Main Parameter Dependencies.

Contact slip κ'0, expressed as a percentage (%), when the longitudinal force equals its maximum value Fx0 and the vertical load equals its rated value Fz0.

#### Dependencies

This parameter is exposed only when you select the Peak longitudinal force and corresponding slip parameterization method. For more information, see Main Parameter Dependencies.

#### Dependencies

This parameter is exposed only when you select the Constant Magic Formula coefficients parameterization method. For more information, see Main Parameter Dependencies.

#### Dependencies

This parameter is exposed only when you select the Constant Magic Formula coefficients parameterization method. For more information, see Main Parameter Dependencies.

#### Dependencies

This parameter is exposed only when you select the Constant Magic Formula coefficients parameterization method. For more information, see Main Parameter Dependencies.

#### Dependencies

This parameter is exposed only when you select the Constant Magic Formula coefficients parameterization method. For more information, see Main Parameter Dependencies.

Nominal normal force Fz0 on tire.

#### Dependencies

This parameter is exposed only when you select the Load-dependent Magic Formula coefficients parameterization method. For more information, see Main Parameter Dependencies.

#### Dependencies

This parameter is exposed only when you select the Load-dependent Magic Formula coefficients parameterization method. For more information, see Main Parameter Dependencies.

#### Dependencies

This parameter is exposed only when you select the Load-dependent Magic Formula coefficients parameterization method. For more information, see Main Parameter Dependencies.

#### Dependencies

This parameter is exposed only when you select the Load-dependent Magic Formula coefficients parameterization method. For more information, see Main Parameter Dependencies.

#### Dependencies

This parameter is exposed only when you select the Load-dependent Magic Formula coefficients parameterization method. For more information, see Main Parameter Dependencies.

#### Dependencies

This parameter is exposed only when you select the Load-dependent Magic Formula coefficients parameterization method. For more information, see Main Parameter Dependencies.

#### Dependencies

This parameter is exposed only when you select the Load-dependent Magic Formula coefficients parameterization method. For more information, see Main Parameter Dependencies.

### Dynamics

The table shows how the visibility of some Dynamics parameters depends on the options that you choose for other parameters. To learn how to read the table, see Parameter Dependencies.

Dynamics Parameter Dependencies

Dynamics

Compliance — Choose No compliance - Suitable for HIL simulation or Specify stiffness and damping

No compliance - Suitable for HIL simulationSpecify stiffness and damping

Longitudinal stiffness

Longitudinal damping

Inertia — Choose No Inertia or Specify inertia and initial velocity

No InertiaSpecify inertia and initial velocity

Tire inertia

Initial velocity

Model for the dynamical compliance of the tire.

• 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.

#### Dependencies

Selecting the Specify stiffness and damping parameterization method, exposes stiffness and damping parameters. For more information, see Dynamics Parameter Dependencies.

Tire longitudinal stiffness CFx.

#### Dependencies

Selecting Specify stiffness and damping for the Compliance parameter, exposes this parameter. For more information, see Dynamics Parameter Dependencies.

Tire longitudinal damping bFx.

#### Dependencies

Selecting Specify stiffness and damping for the Compliance parameter, exposes this parameter. For more information, see Dynamics Parameter Dependencies.

Model for the rotational inertia of the tire.

• No inertia — Tire is modeled with no dynamical compliance.

• Specify inertia and initial velocity — Tire is modeled as a stiff, dampened spring and deforms under load.

#### Dependencies

Selecting the Specify inertia and initial velocity parameterization method, exposes inertia and velocity parameters. For more information, see Dynamics Parameter Dependencies.

Rotational inertia Iw of the wheel-tire assembly.

#### Dependencies

Selecting Specify inertia and initial velocity for the Inertia parameter, exposes this parameter. For more information, see Dynamics Parameter Dependencies.

Initial angular velocity, Ω(0), of the tire.

#### Dependencies

Selecting Specify inertia and initial velocity for the Inertia parameter, exposes this parameter. For more information, see Dynamics Parameter Dependencies.

### Rolling Resistance

The table shows how the visibility of some parameters depends on the options that you choose for other parameters. To learn how to read the table, see Parameter Dependencies.

Rolling Resistance Parameter Dependencies

Rolling Resistance

Rolling resistance — Choose Off or On

OffOn

Resistance model — Choose Constant coefficient or Pressure and velocity dependent

Constant coefficientPressure and velocity dependent

Constant coefficient

Tire pressure

Alpha

Beta

Coefficient A

Coefficient B

Coefficient C

Velocity threshold

Options for modeling rolling resistance are:

• Off — Neglect rolling resistance.

• On — Include rolling resistance.

#### Dependencies

Selecting On exposes rolling resistance parameters. For more information, see Rolling Resistance Parameter Dependencies.

Model for the rolling resistance of the tire.

• Constant coefficient — Neglect rolling resistance.

• Pressure and velocity dependent — Include rolling resistance.

#### Dependencies

Each Resistance model option exposes related parameters. For more information, see Rolling Resistance Parameter Dependencies.

Coefficient that sets the proportionality between the normal force and the rolling resistance force. The parameter must be greater than zero.

#### Dependencies

Selecting On for the Rolling resistance parameter and Constant coefficient for the Resistance model parameter exposes this parameter. For more information, see Rolling Resistance Parameter Dependencies.

Inflation pressure of the tire. The parameter must be greater than zero.

#### Dependencies

Selecting On for the Rolling resistance parameter and Pressure and velocity dependent for the Resistance model parameter exposes this parameter. For more information, see Rolling Resistance Parameter Dependencies.

Exponent of the tire pressure in the model equation. See Pressure and Velocity Dependent Model.

#### Dependencies

Selecting On for the Rolling resistance parameter and Pressure and velocity dependent for the Resistance model parameter exposes this parameter. For more information, see Rolling Resistance Parameter Dependencies.

Exponent of the normal force model equation. See Pressure and Velocity Dependent Model.

#### Dependencies

Selecting On for the Rolling resistance parameter and Pressure and velocity dependent for the Resistance model parameter exposes this parameter. For more information, see Rolling Resistance Parameter Dependencies.

Velocity-independent force component in the model equation. The parameter must be greater than zero. See Pressure and Velocity Dependent Model.

#### Dependencies

Selecting On for the Rolling resistance parameter and Pressure and velocity dependent for the Resistance model parameter exposes this parameter. For more information, see Rolling Resistance Parameter Dependencies.

Velocity-dependent force component in the model equation. The parameter must be greater than zero. See Pressure and Velocity Dependent Model.

#### Dependencies

Selecting On for the Rolling resistance parameter and Pressure and velocity dependent for the Resistance model parameter exposes this parameter. For more information, see Rolling Resistance Parameter Dependencies.

Force component that depends on the square of the velocity term in the model equation. The parameter must be greater than zero. See Pressure and Velocity Dependent Model.

#### Dependencies

Selecting On for the Rolling resistance parameter and Pressure and velocity dependent for the Resistance model parameter exposes this parameter. For more information, see Rolling Resistance Parameter Dependencies.

Velocity at which the full rolling resistance force is transmitted to the rolling hub. The parameter ensures that the force remains continuous during velocity direction changes, which increases the numerical stability of the simulation. The parameter must be greater than zero.

#### Dependencies

Selecting On for the Rolling resistance parameter exposes this parameter. For more information, see Rolling Resistance Parameter Dependencies.

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

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## References

[1] Pacejka, H. B. Tire and Vehicle Dynamics. Elsevier Science, 2005.