Tire-Road Interaction (Magic Formula)

Tire-road dynamics given by magic formula coefficients

• Library:

• 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.

Tire-Road Interaction Model

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.

Tire-Road Contact Variables

Symbol	Description and Unit
Ω	Wheel angular velocity
rw	Wheel radius
Vx	Wheel 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
$$k=\frac{V_{sx}}{\left|V_x\right|}$$	Wheel slip
Fz	Vertical load on tire
Fz0	Nominal vertical load on tire
$$F_x=f(\kappa ,F_z)$$	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, $$F_x=f(\kappa ,F_z)$$, the longitudinal force F_x on the tire, based on:

• Vertical load F_z

• 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:

$$F_x=f(\kappa ,F_z)=F_z\cdot D\cdot \sin \left(C\cdot \arctan \left\{B\kappa -E\left[B\kappa -\arctan \left(B\kappa \right)\right]\right\}\right)$$

The slope of f at $$k=0$$ is $$BCD\cdot F_z$$.

Magic Formula with Load-Dependent Coefficients

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:

$$F_{x0}=D_x\sin \left(C_x\arctan \left\{B_x{\kappa }_x-E_x\left[B_x{\kappa }_x-\arctan \left(B_x{\kappa }_x\right)\right]\right\}\right)+S_{Vx}$$

Where:

$$d{f}_{\text{z}}=\frac{\left(F_{\text{z}}–F_{\text{z}0}\right)}{F_{\text{z0}}}$$

$${\kappa }_{{}_x}=\kappa +S_{{}_{Hx}}$$

$$C_{\text{x}}=\text{ p}\_\text{Cx1}$$

$$D_{\text{x}}={\mu }_{\text{x}}·F_{\text{z}}$$

$${\mu }_{\text{x}}=\text{ p}\_\text{Dx1 }+\text{ p}\_\text{Dx2}·d{f}_{\text{z}}$$

$$E_{\text{x}}=\left(\text{p}\_\text{Ex1 }+\text{ p}\_\text{Ex2}·d{f}_{\text{z}}+\text{ p}\_\text{Ex3}·d{f}_{\text{z}}{}^{\text{2}}\right)\left[\text{1 }–\text{ p}\_\text{Ex4}·\text{sgn}\left({\kappa }_{\text{x}}\right)\right]$$

$$K_{\text{x}\kappa }=F_{\text{z}}\cdot \left(\text{p}\_\text{Kx1 }+\text{ p}\_\text{Kx2}·d{f}_{\text{z}}\right)\cdot e^{\left(\text{p}\_\text{Kx3}·d{f}_{\text{z}}\right)}$$

$$B_{\text{x}}=\frac{K_{\text{x}\kappa }}{C_{\text{x}}{D}_{\text{x}}+{\epsilon }_{\text{x}}}$$

$$S_{\text{Hx}}=\text{ p}\_\text{Hx1 }+\text{ p}\_\text{Hx2}·d{f}_{\text{z}}$$

$$S_{\text{Vx}}=F_{\text{z}}·\left(\text{p}\_\text{Vx1 }+\text{ p}\_\text{Vx2}·d{f}_{\text{z}}\right)$$

S_Hx and S_Vx 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. μ_x is the longitudinal load-dependent friction coefficient. ε_x is a small number inserted to prevent division by zero as F_z 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 F_x0 at the corresponding slip κ₀ that you specify, for rated vertical load F_z0.

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.

Surface	B	C	D	E
Dry tarmac	10	1.9	1	0.97
Wet tarmac	12	2.3	0.82	1
Snow	5	2	0.3	1
Ice	4	2	0.1	1

Assumptions and Limitations

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

Ports

Input

N — Normal force

physical signal

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.

M — Magic formula coefficients

physical signal | vector | [B, C, D, E]

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

S — Slip

physical signal

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

Conserving

A — Axle

mechanical rotational

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

H — Hub

mechanical translational

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

Parameters

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 slip	Constant Magic Formula coefficients	Load-dependent Magic Formula coefficients	Physical 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].
Rated vertical load	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

Parameterize by — Parameterization method

Peak longitudinal force and corresponding slip (default) | Constant Magic Formula coefficients | Load-dependent Magic Formula coefficients | Physical signal Magic Formula coefficients

Magic Formula tire-road interaction model.

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.

  Coefficient	Default 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.

  Coefficient	Parameters	Default Values
  C	p_Cx1	1.685
  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.

Rated vertical load — Rated load force

3000 N (default) | positive scalar

Rated vertical load force F_z0.

Dependencies

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

Peak longitudinal force at rated load — Maximum longitudinal force at rated load

3500 N (default) | positive scalar

Maximum longitudinal force F_x0 that the tire exerts on the wheel when the vertical load equals its rated value F_z0.

Dependencies

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

Slip at peak force at rated load (percent) — Percent slip at maximum longitudinal force and rated load

10 (default) | positive scalar

Contact slip κ'₀, expressed as a percentage (%), when the longitudinal force equals its maximum value F_x0 and the vertical load equals its rated value F_z0.

Dependencies

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

Magic Formula B coefficient — Constant Magic Formula B coefficient

10 (default) | positive scalar

Load-independent Magic Formula B coefficient.

Dependencies

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

Magic Formula C coefficient — Constant Magic Formula C coefficient

1.9 (default) | positive scalar

Load-independent Magic Formula C coefficient.

Dependencies

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

Magic Formula D coefficient — Constant Magic Formula D coefficient

1 (default) | positive scalar

Load-independent Magic Formula D coefficient.

Dependencies

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

Magic Formula E coefficient — Constant Magic Formula E coefficient

0.97 (default) | positive scalar

Load-independent Magic Formula E coefficient.

Dependencies

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

Tire nominal vertical load — Normal force

4000 N (default) | positive scalar

Nominal normal force F_z0 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.

Magic Formula C-coefficient parameter, p_Cx1 — Variable Magic Formula C coefficient

1.685 (default) | scalar

Load-dependent Magic Formula C coefficient.

Dependencies

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

Magic Formula D-coefficient parameters, [p_Dx1 p_Dx2] — Variable Magic Formula D coefficient

[1.21, -.037] (default) | vector

Load-dependent Magic Formula D coefficient.

Dependencies

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

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

[.344, .095, -.02, 0] (default) | vector

Load-dependent Magic Formula E coefficient.

Dependencies

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

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

[21.51, -.163, .245] (default) | vector

Load-dependent Magic Formula K coefficient.

Dependencies

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

Magic Formula H-coefficient parameters, [p_Hx1 p_Hx2] — Variable Magic Formula H coefficient

[-.002, .002] (default) | vector

Load-dependent Magic Formula H coefficient.

Dependencies

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

Magic Formula V-coefficient parameters, [p_Vx1 p_Vx2] — Variable Magic Formula V coefficient

[0, 0] (default) | vector

Load-dependent Magic Formula V coefficient.

Dependencies

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

Velocity threshold — Wheel hub velocity threshold for mathematical slip model

0.1 m/s (default) | positive scalar

Wheel hub velocity V_th 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.

Extended Capabilities

C/C++ Code Generation
Generate C and C++ code using Simulink® Coder™.

Version History

Introduced in R2011a