Tire with longitudinal behavior given by magic formula coefficients
Simscape / Driveline / Tires & Vehicles
This block models a tire with longitudinal behavior given by the magic formula, an empirical equation based on four fitting coefficients. 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.
Port A represents the axle on which the tire sits. Port H represents the wheel hub that transmits the thrust generated by the tire to the remainder of the vehicle. Port N accepts a physical signal input of the normal force acting on the tire. The normal force is positive if it acts downward on the tire, pressing it against the pavement. Port S outputs a physical signal with the tire slip measured during simulation.
The block provides two friction variants. The default variant,
friction coefficients, accepts the magic formula coefficients
as block parameters. This variant treats the coefficients as constants
or load-dependent parameters. Use this variant to model tire dynamics
under constant pavement conditions.
The alternative variant,
Variable friction coefficients,
accepts the magic formula coefficients as a physical signal input.
Use this variant to model tire dynamics under variable pavement conditions.
Selecting this variant exposes physical signal port M. Use this port
to provide the magic formula coefficients as a four-element vector,
specified in the order [b, c, d, e].
To change block variants:
Right-click the tire block.
In the context-sensitive menu, select Simscape > Block choice.
Select the desired block variant.
To increase the fidelity of the tire model, the block enables you to 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-Loop (HIL) simulation.
The Tire 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|
|Vx||Wheel hub longitudinal velocity|
|u||Tire longitudinal deformation|
|Ω||Wheel angular velocity|
|Ω′||Contact point angular velocity = Ω if u = 0|
|rwΩ'||Tire tread longitudinal velocity|
|Vsx = rWΩ – Vx||Wheel slip velocity|
|V'sx = rWΩ' – Vx||Contact slip velocity = Vsx if u = 0|
|κ = Vsx/|Vx|||Wheel slip|
|κ'= V′sx/|Vx|||Contact slip = κ if u = 0|
|Vth||Wheel hub threshold velocity|
|Fz||Vertical load on tire|
|Fx||Longitudinal force exerted on the tire at the contact point.|
|CFx = (∂Fx/∂u)0||Tire longitudinal stiffness under deformation|
|bFx = (∂Fx/∂ů)0||Tire longitudinal damping under deformation|
|Iw||Wheel-tire inertia; effective mass = Iw/rw2|
|τdrive||Torque applied by the axle to the wheel|
If the tire did not slip, it would roll and translate as Vx = rwΩ. But the tire actually does slip and develops a longitudinal force Fx only in response to slip.
The wheel slip velocity is Vsx = rWΩ – Vx. The wheel slip is κ = Vsx/|Vx|. For a locked, sliding wheel, κ = –1. For perfect rolling, κ = 0.
For low speeds, |Vx| ≤ |Vth|, the wheel slip is modified to:
κ = 2Vsx/(Vth + Vx2/Vth) .
This modification allows for a nonsingular, nonzero slip at zero wheel velocity. For example, for perfect slipping (nontranslating spinning tire), Vx = 0 while κ = 2rwΩ/Vth is finite.
If the tire is modeled with compliance, it is also flexible. Because in this case, 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 CFx and damping bFx.
If the tire is not modeled with compliance, then Ω′ = Ω, V'sx = Vsx, and κ' = κ. In this case, the tire starts simulation undeformed and remains undeformed.
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 CFx and damping bFx. Tire-Road Interaction (Magic Formula) models the longitudinal force Fx on the tire as a function of Fz and κ′ using the Magic Formula, with κ′ as the independent slip variable .
The Wheel and Axle radius is the wheel radius rw. The Mass value is the effective mass, Iw/rw2. The tire characteristic function f(κ′, Fz) determines the longitudinal force Fx. Together with the driveshaft torque applied to the wheel axis, Fx 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.
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.
Conserving mechanical rotational port associated with the axle of the tire
Conserving mechanical translational port associated with the hub of the tire
Physical signal input port associated with the normal force on the tire
Physical signal input associated with the static and kinetic friction coefficients of the tire
Physical signal output port associated with the relative slip between the tire and road
Select how to use the Magic Formula to model the tire-road
interaction. The default is
Peak longitudinal force and
corresponding slip. This tab appears only when the block
variant is set to
coefficients. For more information about the block
variants, see the block description.
Peak longitudinal force and corresponding
slip — Parametrize the Magic Formula with
physical characteristics of the tire.
Constant Magic Formula coefficients
— Parameterize the Magic Formula directly with its
coefficients. If you select this option, the panel changes from
Load-dependent Magic Formula
coefficients—Parametrize the Magic Formula
directly with load-dependent coefficients. If you select this
option, the panel changes from its default.
Unloaded tire-wheel radius
rw. The default is
From the drop-down list, choose units. The default is meters
Select how to model the dynamical compliance of the tire. The default
No compliance — Suitable for HIL
No compliance — Suitable for HIL
simulation — Tire is modeled with no
Specify stiffness and damping—Tire
is modeled as a stiff, dampened spring and deforms under load.
If you select this option, the panel changes from its
Select how to model the rotational inertia of the tire. The default is
No inertia—Tire is modeled with no
Specify inertia and initial
velocity—Tire is modeled with rotational
inertia. If you select this option, the panel changes from its
Method used to specify the rolling resistance acting on a rotating
wheel hub. The default value is
resistance. Options include:
No rolling resistance
Select this option to ignore the effect of rolling resistance on a model.
Select between two rolling resistance models:
Constant coefficient and
Pressure and velocity
The default value is
Wheel hub velocity Vth below
which the slip calculation is modified to avoid singular evolution at
zero velocity. Must be positive. The default is
From the drop-down list, choose units. The default is meters per
For optimal simulation performance, set the Dynamics > Compliance parameter to
No compliance - Suitable for HIL
 Pacejka, H. B. Tire and Vehicle Dynamics, Society of Automotive Engineers and Butterworth-Heinemann, Oxford, 2002, chapters 1,4,7, and 8