Tire (Magic Formula)

Tire with longitudinal behavior given by Magic Formula coefficients

Library:
Simscape / Driveline / Tires & Vehicles

Description

The Tire (Magic Formula) block represents 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

Symbol	Description and Unit
r_w	Tire rolling radius.
V_x	Wheel hub longitudinal velocity.
u	Tire longitudinal deformation.
Ω	Wheel angular velocity.
$Ω^{'} = Ω + \frac{\dot{u}}{r_{W}}$	Contact point angular velocity. If there is no tire longitudinal deformation, that is, if $u = 0$ , then $Ω^{'} = Ω$ .
$r_{w} Ω^{'}$	Tire tread longitudinal velocity.
$V_{s x} = r_{w} Ω - V_{x}$	Wheel slip velocity for a tire without compliance.
${V^{'}}_{s x} = r_{w} Ω + \dot{u} - V_{x}$	Contact slip velocity for a tire with compliance. If there is no tire longitudinal deformation, that is, if $u = 0$ , then ${V^{'}}_{s x} = V_{s x}$ .
$k = \frac{V_{s x}}{\sqrt{V_{t h}^{2} + V_{x}^{2}}}$	Wheel slip for a tire without compliance.
$k^{'} = \frac{{V^{'}}_{s x}}{\sqrt{V_{t h}^{2} + V_{x}^{2}}}$	Contact patch slip. If there is no tire longitudinal deformation, that is, if $u = 0$ , then $k^{'} = k$ .
V_th	Wheel hub threshold velocity.
F_z	Vertical load on tire.
F_x	Longitudinal force exerted on the tire at the contact point.
$C_{F_{x}} = {(\frac{\partial F_{x}}{\partial u})}_{0}$	Tire longitudinal stiffness under deformation.
$b_{F_{x}} = {(\frac{\partial F_{x}}{\partial \dot{u}})}_{0}$	Tire longitudinal damping under deformation.
I_w	Wheel-tire inertia, such that the effective mass is equal to $\frac{I_{w}}{r_{w}^{2}}$
τ_drive	Torque applied by the axle to the wheel.

Tire Kinematics and Response

Roll and Slip

The equation for translational motion of a non-slipping, non-compliant tire is $V_{x} = r_{w} Ω$ . When tires experience slip, they respond by developing a longitudinal force, F_x.

The contact patch slip velocity is ${V^{'}}_{s x} = r_{w} Ω + \dot{u} - V_{x}$ . For a tire without compliance, u = 0. The block defines the contact patch slip as

$k^{'} = \frac{{V^{'}}_{s x}}{\sqrt{V_{t h}^{2} + V_{x}^{2}}},$

where the square root expression provides numerical robustness when V_x = 0. For example, in the case of a non-translating, spinning tire, $V_{x} = 0$ while $k = \frac{r_{w} Ω}{V_{t h}}$ is finite.

When the tire deformation is negligible and the wheel hub longitudinal velocity is sufficiently large, the equation for wheel slip approaches

$k = \frac{V_{s x}}{| V_{x} |} .$

For this equation, a locked, sliding wheel has k = -1. For perfect rolling, k = 0.

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, C_{F_x}, and damping, b_{F_x}.

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:

$k^{'} = k$
${V^{'}}_{s x} = V_{s x}$
$Ω^{'} = Ω$

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 C_{F_x} and damping b_{F_x}. The Tire-Road Interaction (Magic Formula) block models the longitudinal force F_x on the tire as a function of F_z, and k′ using the Magic Formula, with k′ as the independent slip variable. Here, F_z is the input signal at port N.

The Wheel and Axle radius is the tire rolling radius r_w. The Mass value is the effective mass, $\frac{I_{w}}{r_{w}^{2}}$ . The tire characteristic function f(k′, F_z) determines the longitudinal force F_x. Together with the driveshaft torque applied to the wheel axis, F_x 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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`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 Main > Parameterize by parameter is set to Physical signal Magic Formula coefficients. For more information, see Main Parameter Dependencies.

Output

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`S` — Slip
physical signal

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

Conserving

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`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

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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 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]

`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 — 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.

Coefficient Default Value
B 10
C 1.9
D 1
E 0.97

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 Main 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 Main 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 Main 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 Main 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 Main 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 Main 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 Main 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 Main Parameter Dependencies.