Engee documentation

Tire-Road Interaction (Magic Formula)

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Dynamics of tyre-road interaction determined by the coefficients of H.Paceika’s empirical formula.

blockType: Engee1DMechanical.Vehicles.Tires.MagicFormulaRoadInteraction

tire road interaction magic formula

Path in the library:

/Physical Modeling/1D Mechanical/Tires & Vehicles/Tire Subcomponents/Tire-Road Interaction (Magic Formula)

Description

The block Tire-Road Interaction (Magic Formula) is a model of the interaction between the tyre tread and the road surface. The H.Paceika formula determines the longitudinal force resulting from this interaction using an empirical equation based on empirical coefficients. The block ignores tyre properties such as pliability and inertia.

Tyre-road interaction model

Block determines the longitudinal forces at the tyre-road contact patch using the empirical formula of H.Paceika [2].

The figure shows the forces acting on the tyre.

tire magic formula 1

Variables that define the model:

  • - angular velocity of the wheel;

  • - wheel radius;

  • - longitudinal velocity of the wheel hub;

  • - longitudinal velocity of the tyre tread;

  • - longitudinal deformation;

  • - longitudinal velocity of the tyre tread. Typically, tyre tread longitudinal velocity includes a component related to tyre rotation , and an additional component related to tyre deformation . Since is specified via the T port, the calculations for tyre rotation and deformation occur outside the block;

  • - the sliding velocity of the contact patch. If there is no longitudinal elastic deformation of the tyre, then ;

  • - is the wheel slip coefficient for a tyre with no elastic deformation;

  • - vertical load on the tyre;

  • - nominal vertical load on the tyre;

  • - longitudinal force acting on the tyre at the point of contact, - tyre characteristic function.

Tyre reaction

You can simulate tyre rolling and sliding.

Force and characteristic function

The block uses the characteristic function of the tyre in steady state , where

  • - is the longitudinal force acting on the tyre;

  • - vertical load;

  • - is the slip coefficient of the wheel.

Rolling and sliding

The equation of translational motion of a tyre without slip is . Tyre slip results in a change in longitudinal force .

The sliding velocity of the contact patch is . For a tyre without deformation .

The unsmoothed value of the contact patch slip rate is equal to:

The block defines the denominator of the slip coefficient as:

where is the value of the parameter Lower boundary of slip denominator, VXLOW.

The value of changes smoothly to in transient regions:

,

.

The block determines the sliding coefficient according to:

where

  • - is the value of the parameters Minimum valid wheel slip, KPUMIN;

  • - the value of the parameters Maximum valid wheel slip, KPUMAX.

The value of changes smoothly in transient regions:

,

.

The block defines the smoothing threshold of the slip coefficient as:

For this equation, a locked, sliding wheel has . For perfect rolling .

The coefficients of H. Paceika’s empirical formula for typical road conditions

This block uses numerical values based on empirical tyre data. These values represent typical sets of constant coefficients of the H.Paceika empirical formula for typical road conditions.

Surface

Dry asphalt

10

1.9

1

0.97

Wet Asphalt

12

2.3

0.82

1

Snow

5

2

0.3

1

Ice

4

2

0.1

1

Parametrization

Peak longitudinal force and corresponding slip

If the parameter Parameterize by is set to the value of Peak longitudinal force and corresponding slip, then the block uses a typical set of coefficients of the empirical formula of H.Paceika. The block scales the coefficients to obtain the peak longitudinal force at the corresponding slip coefficient , which is set, for the nominal vertical load .

Empirical formula with constant coefficients

If the parameters Parameterize by are set to the value of Constant Magic Formula coefficients, then the block uses dimensionless coefficients , , and or stiffness, shape, peak and curvature such that:

The slope of to is .

Empirical formula with load-dependent coefficients

If the parameters Parameterize by are set to the value of Load-dependent Magic Formula coefficients, the block uses dimensionless coefficients that are functions of the bus load. A set of parameters defines these functions:

where

  • - is the value of parameters Magic Formula C-coefficient parameter, p_Cx1;

  • - the value of the parameters Magic Formula D-coefficient parameters, [p_Dx1 p_Dx2];

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

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

  • - parameter value Magic Formula H-coefficient parameters, [p_Hx1 p_Hx2]

  • - parameter value Magic Formula V-coefficient parameters, [p_Vx1 p_Vx2]

  • and - displacements towards slip and longitudinal force in the function of force dependence on slip or horizontal and vertical displacements if the function is plotted as a curve;

  • - is the friction coefficient depending on the longitudinal load;

  • - is a small number that prevents division by zero as approaches zero.

Assumptions and limitations

The block assumes longitudinal motion only and does not account for camber, rotation, or lateral motion.

Ports

Input

# N — normal force, N
scalar

Details

Input port related to the normal force acting on the tyre, in N. The normal force is positive if it acts on the tyre in a downward direction, pressing it against the road surface.

Data types

Float64.
Complex numbers support

No

# M — vector of coefficients of the empirical formula
vector

Details

Input port specifying the coefficients of H.Paceika’s empirical formula.

Provide the coefficients as a vector .

Dependencies

To use this port, set the parameters Parameterize by to Physical signal Magic Formula coefficients.

Data types

Float64
Complex numbers support

No

Conserving

# T — tyre tread
`rotational mechanics

Details

A mechanical rotational port associated with the tyre tread.

Program usage name

tread_flange

# H — hub
translational mechanics

Details

A mechanical progressive port connected to the hub of a wheel.

Program usage name

hub_flange

Output

# S — slip
scalar

Details

output port related to the slip coefficient, , between the tyre and the road.

Data types

Float64.
Complex numbers support

No

Parameters

Main

# Parameterize by — parameterization method
Peak longitudinal force and corresponding slip | Constant Magic Formula coefficients | Load-dependent Magic Formula coefficients | Physical signal Magic Formula coefficients

Details

Select how the block parameterises the bus using an empirical formula:

  • Peak longitudinal force and corresponding slip - parameterization of the empirical formula using the physical characteristics of the tyre.

  • Constant Magic Formula coefficients - set the parameters defining the constant coefficients , , and as scalars.

  • Load-dependent Magic Formula coefficients - set the parameters defining the load-dependent coefficients , , , , , and as vectors, one parameter for each coefficient.

  • Physical signal Magic Formula coefficients - set the coefficients of the empirical formula through the M port as a four-element vector .

Values

Peak longitudinal force and corresponding slip | Constant Magic Formula coefficients | Load-dependent Magic Formula coefficients | Physical signal Magic Formula coefficients
Default value

Peak longitudinal force and corresponding slip
Program usage name

friction_model
Evaluatable

Yes

# Rated vertical load — rated load force
N | kN | lb | mN | dyn | lbf

Details

Rated vertical load force .

Dependencies

To use this parameter, set the Parameterize by parameters to . Peak longitudinal force and corresponding slip.

Units

N | kN | lb | mN | dyn | lbf
Default value

3000.0 N
Program usage name

F_vertical_load
Evaluatable

Yes

# Peak longitudinal force at rated load — maximum longitudinal force at rated load
N | kN | lb | mN | dyn | lbf

Details

Maximum longitudinal force , which the tyre exerts on the wheel when the vertical load is equal to its nominal value .

Dependencies

To use this parameter, set the parameters Parameterize by to . Peak longitudinal force and corresponding slip.

Units

N | kN | lb | mN | dyn | lbf
Default value

3500.0 N
Program usage name

F_longitudinal_load
Evaluatable

Yes

# Slip at peak force at rated load (percent) — slip coefficient in per cent at maximum longitudinal force and rated load

Details

Slip coefficient , expressed as a percentage (%) when the longitudinal force is equal to the maximum value , and the vertical load is equal to the nominal value .

Dependencies

To use this parameter, set the parameters Parameterize by to Peak longitudinal force and corresponding slip.

Default value

10.0
Program usage name

percent_slip
Evaluatable

Yes

# Magic Formula B coefficient — constant factor B in the empirical formula

Details

Coefficient in the empirical formula, independent of load.

Dependencies

To use this parameter, set the parameter Parameterize by to the value of Constant Magic Formula coefficients.

Default value

10.0
Program usage name

coefficient_B
Evaluatable

Yes

# Magic Formula C coefficient — constant coefficient C in the empirical formula

Details

Coefficient in the empirical formula, independent of load.

Dependencies

To use this parameter, set the parameter Parameterize by to the value of Constant Magic Formula coefficients.

Default value

1.9
Program usage name

coefficient_C
Evaluatable

Yes

# Magic Formula D coefficient — constant factor D in the empirical formula

Details

Coefficient in the empirical formula, independent of load.

Dependencies

To use this parameter, set the parameter Parameterize by to the value of Constant Magic Formula coefficients.

Default value

1.0
Program usage name

coefficient_D
Evaluatable

Yes

# Magic Formula E coefficient — constant coefficient E in the empirical formula

Details

Coefficient in the empirical formula, independent of load.

Dependencies

To use this parameter, set the parameter Parameterize by to the value of Load-dependent Magic Formula coefficients.

Default value

0.97
Program usage name

coefficient_E
Evaluatable

Yes

# Tire nominal vertical load, FNOMIN — nominal normal force
N | kN | lb | mN | dyn | lbf

Details

Nominal normal force on the tyre.

Dependencies

To use this parameter, set the parameters Parameterize by to Load-dependent Magic Formula coefficients.

Units

N | kN | lb | mN | dyn | lbf
Default value

4000.0 N
Program usage name

F_vertical_nominal
Evaluatable

Yes

# Magic Formula C-coefficient parameter, p_Cx1 — C coefficient in the empirical formula

Details

Load-dependent coefficient in the empirical formula.

Dependencies

To use this parameter, set the parameter Parameterize by to the value of Load-dependent Magic Formula coefficients.

Default value

1.685
Program usage name

p_C_x
Evaluatable

Yes

# Magic Formula D-coefficient parameters, [p_Dx1 p_Dx2] — D coefficients in the empirical formula

Details

Load-dependent coefficients in the empirical formula.

Dependencies

To use this parameter, set the parameters Parameterize by to Load-dependent Magic Formula coefficients.

Default value

[1.21, -0.037]
Program usage name

p_D_x
Evaluatable

Yes

# Magic Formula E-coefficient parameters, [p_Ex1 p_Ex2 p_Ex3 p_Ex4] — E coefficients in the empirical formula

Details

Load-dependent coefficients in the empirical formula.

Dependencies

To use this parameter, set the parameters Parameterize by to Load-dependent Magic Formula coefficients.

Default value

[0.344, 0.095, -0.02, 0.0]
Program usage name

p_E_x
Evaluatable

Yes

# Magic Formula BCD-coefficient parameters, [p_Kx1 p_Kx2 p_Kx3] — K coefficients in the empirical formula

Details

Load-dependent coefficients in the empirical formula.

Dependencies

To use this parameter, set the parameters Parameterize by to Load-dependent Magic Formula coefficients.

Default value

[21.51, -0.163, 0.245]
Program usage name

p_K_x
Evaluatable

Yes

# Magic Formula H-coefficient parameters, [p_Hx1 p_Hx2] — H coefficients in the empirical formula

Details

Load-dependent coefficients in the empirical formula.

Dependencies

To use this parameter, set the parameters Parameterize by to Load-dependent Magic Formula coefficients.

Default value

[-0.002, 0.002]
Program usage name

p_H_x
Evaluatable

Yes

# Magic Formula V-coefficient parameters, [p_Vx1 p_Vx2] — V coefficients in the empirical formula

Details

Load-dependent coefficients in the empirical formula.

Dependencies

To use this parameter, set the parameters Parameterize by to Load-dependent Magic Formula coefficients.

Default value

[0.0, 0.0]
Program usage name

p_V_x
Evaluatable

Yes

Advanced

# Lower boundary of slip denominator, VXLOW — lower limit of the denominator of the slip coefficient
fpm | fps | kph | mph | m/s | cm/s | ft/s | in/s | km/s | mi/s | mm/s

Details

Lower boundary of the slip factor denominator .

VXLOW - TIR file identifier.

Units

fpm | fps | kph | mph | m/s | cm/s | ft/s | in/s | km/s | mi/s | mm/s
Default value

1.0 m/s
Program usage name

v_slip_threshold
Evaluatable

Yes

# Velocity threshold — tyre slip mode threshold
fpm | fps | kph | mph | m/s | cm/s | ft/s | in/s | km/s | mi/s | mm/s

Details

Threshold speed , which the unit uses to transition between slip modes.

For details, see [Rolling and Sliding]. Rolling and Sliding for details.

Units

fpm | fps | kph | mph | m/s | cm/s | ft/s | in/s | km/s | mi/s | mm/s
Default value

0.1 m/s
Program usage name

v_threshold
Evaluatable

Yes

# Minimum valid wheel slip, KPUMIN — minimum value of wheel slip coefficient

Details

The minimum permissible value of the wheel slip coefficient. A negative value means that the wheel slides in the opposite direction to the rotation.

Default value

-1.5
Program usage name

min_wheel_slip
Evaluatable

Yes

# Maximum valid wheel slip, KPUMAX — maximum value of wheel slip coefficient

Details

Maximum permissible value of the wheel slip coefficient.

Default value

1.5
Program usage name

max_wheel_slip
Evaluatable

Yes

Scaling

# Enable scaling coefficients — rolling resistance scaling

Details

Select this check box to include scaling factors in the parameterization of the empirical formula.

Dependencies

To use this parameter, set the parameter Parameterize by to Load-dependent Magic Formula coefficients.

Default value

false (switched off)
Program usage name

enable_scaling_factors
Evaluatable

Yes

# Scale factor of Fx nominal vertical load, LFZO — rated vertical load Fx

Details

Scaling factor nominal vertical load .

LFZO is the identifier of the TIR file.

Dependencies

To use this parameter, set the Parameterize by parameters to Load-dependent Magic Formula coefficients and tick the checkbox Enable scaling coefficients.

Default value

1.0
Program usage name

lambda_F_z0
Evaluatable

Yes

# Scale factor of Fx shape factor, LCX — shape factor Fx

Details

Scaling factor shape factor .

LCX is the identifier of the TIR file.

Dependencies

To use this parameter, set the Parameterize by parameters to Load-dependent Magic Formula coefficients and tick the checkbox Enable scaling coefficients.

Default value

1.0
Program usage name

lambda_C_x
Evaluatable

Yes

# Scale factor of Fx peak friction coefficient, LMUX — peak coefficient of friction Fx

Details

Scaling factor peak friction coefficient .

LMUX is the identifier of the TIR file.

Dependencies

To use this parameter, set the Parameterize by parameters to Load-dependent Magic Formula coefficients and tick the checkbox Enable scaling coefficients.

Default value

1.0
Program usage name

lambda_mu_x
Evaluatable

Yes

# Scale factor of Fx curvature factor, LEX — curvature coefficient Fx

Details

Scaling factor curvature coefficient .

LEX is the identifier of the TIR file.

Dependencies

To use this parameter, set the Parameterize by parameters to Load-dependent Magic Formula coefficients and tick the checkbox Enable scaling coefficients.

Default value

1.0
Program usage name

lambda_E_x
Evaluatable

Yes

# Scale factor of Fx slip stiffness, LKX — sliding stiffness Fx

Details

Scaling factor sliding stiffness .

LKX is the identifier of the TIR file.

Dependencies

To use this parameter, set the Parameterize by parameters to Load-dependent Magic Formula coefficients and tick the checkbox Enable scaling coefficients.

Default value

1.0
Program usage name

lambda_K_x
Evaluatable

Yes

# Scale factor of Fx horizontal shift, LHX — horizontal displacement Fx

Details

Scaling factor horizontal shift .

LHX is the identifier of the TIR file.

Dependencies

To use this parameter, set the Parameterize by parameters to Load-dependent Magic Formula coefficients and tick the checkbox Enable scaling coefficients.

Default value

1.0
Program usage name

lambda_H_x
Evaluatable

Yes

# Scale factor of Fx vertical shift, LVX — vertical displacement Fx

Details

Scaling factor vertical shift .

LVX is the identifier of the TIR file.

Dependencies

To use this parameter, set the Parameterize by parameters to Load-dependent Magic Formula coefficients and tick the checkbox Enable scaling coefficients.

Default value

1.0
Program usage name

lambda_V_x
Evaluatable

Yes

Literature

  1. I.J.M. Besselink, A.J.C. Schmeitz, H. B. Pacejka, An Improved Magic Formula/Swift Tyre Model That Can Handle Inflation Pressure Changes, Vehicle System Dynamics 48, no. sup1 (December 2010): 337-52. https://doi.org/10.1080/00423111003748088.

  2. H.B. Pacejka, Tire and Vehicle Dynamics, Elsevier Science, 2005.