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.

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.

SymbolDescription and Unit
ΩWheel angular velocity
VxWheel hub longitudinal velocity
${r}_{w}\Omega$Tire tread longitudinal velocity
${V}_{sx}={r}_{w}\Omega -{V}_{x}$Wheel slip velocity, defined as the difference between the longitudinal velocities of the wheel hub and the tire tread
$k=\frac{{V}_{sx}}{|{V}_{x}|}$Wheel slip
${F}_{x}=f\left(\kappa ,{F}_{z}\right)$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, ${F}_{x}=f\left(\kappa ,{F}_{z}\right)$, the longitudinal force Fx on the tire, based on:

• 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, and E, or stiffness, shape, peak, and curvature:

`${F}_{x}=f\left(\kappa ,{F}_{z}\right)={F}_{z}\cdot D\cdot \mathrm{sin}\left(C\cdot \mathrm{arctan}\left\{B\kappa -E\left[B\kappa -\mathrm{arctan}\left(B\kappa \right)\right]\right\}\right)$`

The slope of f at $k=0$ is $BCD\cdot {F}_{z}$.

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:

`${F}_{x0}={D}_{x}\mathrm{sin}\left({C}_{x}\mathrm{arctan}\left\{{B}_{x}{\kappa }_{x}-{E}_{x}\left[{B}_{x}{\kappa }_{x}-\mathrm{arctan}\left({B}_{x}{\kappa }_{x}\right)\right]\right\}\right)+{S}_{Vx}$`

Where:

`${\kappa }_{{}_{x}}=\kappa +{S}_{{}_{Hx}}$`

`${D}_{\text{x}}={\mu }_{\text{x}}·{F}_{\text{z}}$`

`${B}_{\text{x}}=\frac{{K}_{\text{x}\kappa }}{{C}_{\text{x}}{D}_{\text{x}}+{\epsilon }_{\text{x}}}$`

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.

SurfaceBCDE
Dry tarmac101.910.97
Wet tarmac122.30.821
Snow520.31
Ice420.11

## Assumptions and Limitations

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

## 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 Parameterize by parameter is set to ```Physical signal Magic Formula coefficients```. For more information, see 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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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.

Parameter Dependencies

Parameters

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]

Velocity threshold

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

• ```Peak longitudinal force and corresponding slip``` — Parametrize 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
B`10`
C`1.9`
D`1`
E`0.97`

• ```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 Parameter Dependencies.

#### Dependencies

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

#### Dependencies

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

#### Dependencies

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

#### Dependencies

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

#### Dependencies

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

#### Dependencies

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

#### Dependencies

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

#### Dependencies

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

#### Dependencies

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

#### Dependencies

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

#### Dependencies

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

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

## References

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

## Version History

Introduced in R2011a