Tire-Road Interaction (Magic Formula)

The dynamics of the interaction of the tire with the road, determined by the coefficients of the empirical formula of H. Paceyka.

blockType: Engee1DMechanical.Vehicles.Tires.MagicFormulaRoadInteraction

Path in the library:

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

Description

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

The model of bus-road interaction

The block determines the longitudinal forces in the contact spot of the tire with the road using the empirical formula of H. Pacejka [2].

The figure shows the forces acting on the tire.

tire magic formula 1

Variables that define the model:

— angular velocity of the wheel;
— wheel radius;
— the longitudinal speed of the wheel hub;
— the longitudinal speed of the tire tread;
— longitudinal deformation;
— the longitudinal speed of the tire tread. Typically, the longitudinal tread velocity of a tire includes a component related to tire rotation , and an additional component related to tire deformation . Because it is set via the T port, calculations for tire rotation and deformation take place outside the unit;
— the sliding speed of the contact spot. If there is no longitudinal elastic deformation of the tire, then ;
— wheel sliding coefficient for tires without elastic deformation;
— vertical tire load;
— nominal vertical tire load;
— the longitudinal force acting on the tire at the point of contact, — the characteristic function of the tire.

Tire reaction

You can simulate the rolling and sliding of a tire.

power and characteristic function

The unit uses the characteristic function of the bus in steady state mode , where

— the longitudinal force acting on the tire;
— vertical load;
— the coefficient of wheel slip.

Pumping and sliding_

The equation of translational motion of a tire without slipping has the form . The sliding of the tires leads to a change in the longitudinal force .

The sliding speed of the contact spot is . For a tire without deformation .

The non-smoothed value of the slip coefficient of the contact spot is:

The block defines the denominator of the slip coefficient as:

where — parameter value Lower boundary of slip denominator, VXLOW.

Meaning smoothly changes to in transition areas:

The unit determines the sliding coefficient according to:

where

— parameter value Minimum valid wheel slip, KPUMIN;
— parameter value Maximum valid wheel slip, KPUMAX.

Meaning it changes smoothly in transition areas:

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

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

coeffects of H. Paceyka’s empirical formula for typical road conditions

This block uses numerical values based on empirical tire data. These values are typical sets of constant coefficients of the empirical H. Paceyka formula for normal road conditions.

Surface

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`

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

Parameterization

_ Peak longitudinal force and corresponding slip_

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

Empirical formula with constant coefficients_

If for the parameter Parameterize by the value is set Constant Magic Formula coefficients, then the block uses dimensionless coefficients , , and or stiffness, shape, peak, and curvature such that:

Slant to equal to .

Empirical formula with load-dependent coefficients_

If for the parameter Parameterize by the value is set Load-dependent Magic Formula coefficients then the block uses dimensionless coefficients, which are functions of the tire load. A set of parameters defines these functions:

where

— parameter value Magic Formula C-coefficient parameter, p_Cx1;
— parameter value 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 sliding and longitudinal force as a function of the force dependence on sliding, or horizontal and vertical displacements if the function is constructed as a curve;
— this is the coefficient of friction, depending on the longitudinal load;
— this is a small number that prevents division by zero as it approaches to zero.

Assumptions and limitations

The block assumes only longitudinal movement and does not take into account camber, turn or lateral movement.

Ports

Input

# N — normal force, N
scalar

Details

The input port associated with the normal force acting on the tire is in H. The normal force is positive if it acts on the tire in the downward direction, pressing it to the road surface.

Data types	`Float64`
Complex numbers support	I don’t

# M — vector of coefficients of the empirical formula
vector

Details

The input port that sets the coefficients of the empirical formula of H. Paceyka.

Provide the coefficients as a vector .

Dependencies

To use this port, set the parameter Parameterize by value Physical signal Magic Formula coefficients.

Data types	`Float64`
Complex numbers support	I don’t

Conserving

# T — tire tread
rotational mechanics

Details

A mechanical rotary port connected to the tire tread.

Program usage name

tread_flange

# H — hub
translational mechanics

Details

Mechanical translational port connected to the wheel hub.

Program usage name

hub_flange

Output

# S — sliding
scalar

Details

the output port associated with the slip factor between the tire and the road.

Data types	`Float64`
Complex numbers support	I don’t

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 unit parameterizes the tire using an empirical formula:

Peak longitudinal force and corresponding slip — parameterization of the empirical formula using the physical characteristics of the tire;
Constant Magic Formula coefficients — set the parameters that determine the constant coefficients , , and in the form of scalars;
Load-dependent Magic Formula coefficients — set the parameters in the form of vectors that determine the coefficients , , , , and load-dependent, one parameter for each coefficient;
Physical signal Magic Formula coefficients — set the coefficients of the empirical formula via 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	No

# Rated vertical load — rated load force
N | nN | uN | mN | kN | MN | GN | dyn | lbf | kgf

Details

Rated vertical load force .

Dependencies

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

Units	`N` \| `nN` \| `uN` \| `mN` \| `kN` \| `MN` \| `GN` \| `dyn` \| `lbf` \| `kgf`
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 | nN | uN | mN | kN | MN | GN | dyn | lbf | kgf

Details

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

Dependencies

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

Units	`N` \| `nN` \| `uN` \| `mN` \| `kN` \| `MN` \| `GN` \| `dyn` \| `lbf` \| `kgf`
Default value	`3500.0 N`
Program usage name	`F_longitudinal_load`
Evaluatable	Yes

# Slip at peak force at rated load (percent) — sliding coefficient as a percentage at maximum longitudinal force and rated load

Details

Sliding 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 for the parameter Parameterize by meaning Peak longitudinal force and corresponding slip.

Default value	`10.0`
Program usage name	`percent_slip`
Evaluatable	Yes

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

Details

Ratio in the empirical formula, it is independent of the load.

Dependencies

To use this parameter, set for the parameter Parameterize by meaning Constant Magic Formula coefficients.

Default value	`10.0`
Program usage name	`coefficient_B`
Evaluatable	Yes

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

Details

Ratio in the empirical formula, it is independent of the load.

Dependencies

To use this parameter, set for the parameter Parameterize by meaning Constant Magic Formula coefficients.

Default value	`1.9`
Program usage name	`coefficient_C`
Evaluatable	Yes

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

Details

Ratio in the empirical formula, it is independent of the load.

Dependencies

To use this parameter, set for the parameter Parameterize by meaning Constant Magic Formula coefficients.

Default value	`1.0`
Program usage name	`coefficient_D`
Evaluatable	Yes

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

Details

Ratio in the empirical formula, it is independent of the load.

Dependencies

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

Default value	`0.97`
Program usage name	`coefficient_E`
Evaluatable	Yes

# Tire nominal vertical load, FNOMIN — rated normal power
N | nN | uN | mN | kN | MN | GN | dyn | lbf | kgf

Details

Rated normal power on the bus.

Dependencies

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

Units	`N` \| `nN` \| `uN` \| `mN` \| `kN` \| `MN` \| `GN` \| `dyn` \| `lbf` \| `kgf`
Default value	`4000.0 N`
Program usage name	`F_vertical_nominal`
Evaluatable	Yes

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

Details

Ratio in the empirical formula, it depends on the load.

Dependencies

To use this parameter, set for the parameter Parameterize by meaning 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] — coefficients in the empirical formula

Details

Coefficients in the empirical formula, load-dependent.

Dependencies

To use this parameter, set for the parameter Parameterize by meaning 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] — coefficients in the empirical formula

Details

Coefficients in the empirical formula, load-dependent.

Dependencies

To use this parameter, set for the parameter Parameterize by meaning 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] — coefficients in the empirical formula

Details

Coefficients in the empirical formula, load-dependent.

Dependencies

To use this parameter, set for the parameter Parameterize by meaning 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] — coefficients in the empirical formula

Details

Coefficients in the empirical formula, load-dependent.

Dependencies

To use this parameter, set for the parameter Parameterize by meaning 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] — coefficients in the empirical formula

Details

Coefficients in the empirical formula, load-dependent.

Dependencies

To use this parameter, set for the parameter Parameterize by meaning 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 — the lower bound of the denominator of the slip coefficient
m/s | mm/s | cm/s | km/s | m/hr | km/hr | in/s | ft/s | mi/s | ft/min | mi/hr | kn

Details

The lower bound of the denominator of the slip coefficient .

VXLOW — id of the TIR file.

Units	`m/s` \| `mm/s` \| `cm/s` \| `km/s` \| `m/hr` \| `km/hr` \| `in/s` \| `ft/s` \| `mi/s` \| `ft/min` \| `mi/hr` \| `kn`
Default value	`1.0 m/s`
Program usage name	`v_slip_threshold`
Evaluatable	Yes

# Velocity threshold — threshold of the tire slip mode
m/s | mm/s | cm/s | km/s | m/hr | km/hr | in/s | ft/s | mi/s | ft/min | mi/hr | kn

Details

Threshold speed , which the block uses to switch between sliding modes.

For more information, see Rolling and sliding.

Units	`m/s` \| `mm/s` \| `cm/s` \| `km/s` \| `m/hr` \| `km/hr` \| `in/s` \| `ft/s` \| `mi/s` \| `ft/min` \| `mi/hr` \| `kn`
Default value	`0.1 m/s`
Program usage name	`v_threshold`
Evaluatable	Yes

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

Details

The minimum allowable value of the wheel slip coefficient. A negative value means that the wheel is sliding in the opposite direction relative to rotation.

Default value	`-1.5`
Program usage name	`min_wheel_slip`
Evaluatable	Yes

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

Details

The maximum allowable value of the wheel sliding coefficient.

Default value	`1.5`
Program usage name	`max_wheel_slip`
Evaluatable	Yes

Scaling

# Enable scaling coefficients — scaling rolling resistance

Details

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

Dependencies

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

Default value	`false (switched off)`
Program usage name	`enable_scaling_factors`
Evaluatable	No

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

Details

Zoom level rated vertical load .

LFZO — id of the TIR file.

Dependencies

To use this parameter, set for the parameter Parameterize by meaning Load-dependent Magic Formula coefficients and check the box Enable scaling coefficients.

Default value	`1.0`
Program usage name	`lambda_F_z0`
Evaluatable	Yes

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

Details

Zoom level The shape factor .

LCX — this is the identifier of the TIR file.

Dependencies

To use this parameter, set for the parameter Parameterize by meaning Load-dependent Magic Formula coefficients and check the box 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

Details

Zoom level peak coefficient of friction .

LMUX — id of the TIR file.

Dependencies

To use this parameter, set for the parameter Parameterize by meaning Load-dependent Magic Formula coefficients and check the box Enable scaling coefficients.

Default value	`1.0`
Program usage name	`lambda_mu_x`
Evaluatable	Yes

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

Details

Zoom level the coefficient of curvature .

LEX — id of the TIR file.

Dependencies

To use this parameter, set for the parameter Parameterize by meaning Load-dependent Magic Formula coefficients and check the box Enable scaling coefficients.

Default value	`1.0`
Program usage name	`lambda_E_x`
Evaluatable	Yes

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

Details

Zoom level sliding stiffness .

LKX — id of the TIR file.

Dependencies

To use this parameter, set for the parameter Parameterize by meaning Load-dependent Magic Formula coefficients and check the box Enable scaling coefficients.

Default value	`1.0`
Program usage name	`lambda_K_x`
Evaluatable	Yes

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

Details

Zoom level horizontal shift .

LHX — id of the TIR file.

Dependencies

To use this parameter, set for the parameter Parameterize by meaning Load-dependent Magic Formula coefficients and check the box Enable scaling coefficients.

Default value	`1.0`
Program usage name	`lambda_H_x`
Evaluatable	Yes

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

Details

Zoom level vertical shift .

LVX — id of the TIR file.

Dependencies

To use this parameter, set for the parameter Parameterize by meaning Load-dependent Magic Formula coefficients and check the box Enable scaling coefficients.

Default value	`1.0`
Program usage name	`lambda_V_x`
Evaluatable	Yes

Literature

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): 337–352. https://doi.org/10.1080/00423111003748088.
H. B. Pacejka., Tire and Vehicle Dynamics, Elsevier Science, 2005.