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Linear

Linear tire formulation

Description

The Linear tire component provides a linear relationship between the slip angle and longitudinal slip of the tire, and the lateral and longitudinal forces developed at the contact patch. The aligning moment, rolling resistance and overturning moment are assumed to be zero.

The tire geometry is assumed to be a thin circular disk, which is common in automotive applications. A single point contact is considered for the tire-ground interaction.

The tire kinematics used in this component are described in detail in Tire Kinematics.

Several options are available for defining the surface on which the tire is operating. These options are explained in Surface.

Details

Normal Force

The normal force exerted by the surface to the tire is calculated using the given compliance parameters and surface geometry.

The tire loaded radius is calculated using the distance of the tire center from the surface, $rz$ (see Surface), and the inclination angle, $γ$ (see Tire Kinematics).

$r_{L} = \frac{rz}{\cos (γ)}$

Using a linear spring and saturated damping forces based on the tire compliance, the normal force, $F_{z}$ , is calculated as follows

$F_{z}^{C} = {\begin{array}{c} C (R_{0} - r_{L}) & r_{L} < R_{0} \\ 0 & otherwise \end{array}$

$F_{z}^{K} = {\begin{array}{c} K V_{z} & r_{L} < R_{0} \\ 0 & otherwise \end{array}$

$F_{z} = {\begin{array}{c} F_{z}^{C} + \min (F_{z}^{C}, F_{z}^{K}) & 0 < F_{z}^{C} + F_{z}^{K} \\ 0 & otherwise \end{array}$

where $V_{z}$ is the tire center speed with respect to ISO Z, $C$ is tire stiffness, $K$ is tire damping, and $R_{0}$ is tire unloaded radius. The use of the min function is to ensure that $F_{z}$ is continuous at $r_{L} = R_{0}$ .

Slip Calculations

The following equations for longitudinal slip, $κ$ , and slip angle, $α$ , hold true on a flat surface with no inclination angle:

$κ = {\begin{array}{c} \frac{Ω r_{e} - V_{x}}{|V_{x}|} & | V_{x} | > V_{x (\min)} \\ (Ω r_{e} - V_{x}) \frac{2 V_{x (\min)}}{V_{x}^{2} + V_{x (\min)}^{2}} & otherwise \end{array}$

$\tan (α) = {\begin{array}{c} \frac{V_{y}}{|V_{x}|} & | V_{x} | > V_{x (\min)} \\ V_{y} \frac{2 V_{x (\min)}}{V_{x}^{2} + V_{x (\min)}^{2}} & otherwise \end{array}$

Above, $r_{e}$ is the tire effective radius and is considered to be equal to the loaded radius, $r_{L}$ for the tire component. $Ω$ is the tire speed of revolution, and $V_{x}$ and $V_{y}$ are the speeds of the tire center with respect to ISO X and ISO Y axes, respectively. The component code implementation is such that the longitudinal slip and slip angle are continuous and differentiable in the neighborhood of $V_{x} = 0$ .

	Using Time Lags
	A first-order dynamics to the longitudinal slip and slip angle calculation can be introduced in the Time Lags section of the component properties. When active, the following slip formulation will be used: $T_{long} \frac{d κ}{d t} = r_{e} Ω - V_{x} - κ \|V_{x}\|$ $T_{lat} \frac{d}{d t} (\tan (α)) = V_{y} - \tan (α) \|V_{x}\|$

Equations

The formulation for resultant forces/moments of tire-surface interaction at the tire contact patch are summarized below for the linear tire component.

$F_{x} = {\begin{array}{c} C_{long} κ (\frac{F_{z}}{F_{z0}}) & {use F}_{z} = true \\ C_{long} κ \tanh ({kF}_{z} F_{z}) & otherwise \end{array}$

$F_{y} = {\begin{array}{c} - C_{lat} α (\frac{F_{z}}{F_{z0}}) & {use F}_{z} = true \\ - C_{lat} α \tanh ({kF}_{z} F_{z}) & otherwise \end{array}$

$M_{x} = M_{y} = M_{z} = 0$

When ${use F}_{z}$ is true, the longitudinal and lateral forces are linearly dependent on $F_{z}$ . When ${use F}_{z}$ is false, the longitudinal and lateral forces are not dependent on $F_{z}$ , however, a tanh function is used so that the tire does not generate longitudinal or lateral forces when not in contact with the ground, i.e., $F_{z} = 0$ .

Connections

Name	Description	Modelica ID
${frame}_{a}$	Multibody frame for tire center	frame_a
$Fz$	Signal output for the normal force	Fz
$IncAng$	Signal output for tire inclination angle or camber	IncAng
$LongSlip$	Signal output for longitudinal slip	LongSlip
$r_{eff}$	Signal output for tire effective radius	r_eff
$SlipAng$	Signal output for slip angle	SlipAng
$SpinRate$	Signal output for tire speed of revolution or spin rate	SpinRate
${en}_{in}$	[1] Vector signal input for surface normal vector	en_in
${rz}_{in}$	[1] Signal input for tire center distance from the surface	rz_in
$r_{c}$	[1] Vector signal output for tire center position w.r.t. the inertial frame	r_c

[1] Available if Surface parameters Flat surface is false and Defined externally is true.

Parameters

Coefficients

Name	Default	Units	Description	Modelica ID
Use Fz dependency	false		True (checked) means that longitudinal/lateral forces will also linearly depend on the normal force	useFz
$F_{z0}$	$4000$	$N$	Nominal load	Fz0
$C_{long}$	$1.15 \cdot 10^{5}$	$N$	Longitudinal force coefficient	Clong
$C_{lat}$	$1.17 \cdot 10^{5}$	$N$	Lateral force coefficient	Clat

Inertia

Name	Default	Units	Description	Modelica ID
Use inertia	$false$		True (checked) means use mass and inertia parameters for tire and enable the following two parameters	useInertia
$m$	$28$	$kg$	Tire mass	Mass
[I]	[1]	$kg m^{2}$	Rotational inertia, expressed in frame_a (center of tire)	Inertia

[1] $(\begin{array}{c} 0.78 & 0 & 0 \\ 0 & 1.56 & 0 \\ 0 & 0 & 0.78 \end{array})$

Initial Conditions

Name	Default	Units	Description	Modelica ID
Use Initial Conditions	$false$		True (checked) enables the following parameters	useICs
${IC}_{r, v}$	$Ignore$		Indicates whether to ignore, try to enforce, or strictly enforce the translational initial conditions	MechTranTree
${\overset{&conjugate0;}{r}}_{0}$	$[0, 0, 0]$	$m$	Initial displacement of frame_a (tire center) at the start of the simulation expressed in the inertial frame	InitPos
Velocity Frame	$Inertial$		Indicates whether the initial velocity is expressed in frame_a or inertial frame	VelType
${\overset{&conjugate0;}{v}}_{0}$	$[0, 0, 0]$	$\frac{m}{s}$	Initial velocity of frame_a (tire center) at the start of the simulation expressed in the frame selected in Velocity Frame	InitVel
${IC}_{θ, ω}$	$Ignore$		Indicates whether to ignore, try to enforce, or strictly enforce the rotational initial conditions	MechRotTree
$Quaternions$	$false$		Indicates whether the 3D rotations will be represented as a 4 parameter quaternion or 3 Euler angles. Regardless of setting, the initial orientation is specified with Euler angles.	useQuats
Euler Sequence	$[1, 2, 3]$		Indicates the sequence of body-fixed rotations used to describe the initial orientation of frame_a (center of mass). For example, [1, 2, 3] refers to sequential rotations about the x, then y, then z axis (123 - Euler angles)	RotType
${\overset{&conjugate0;}{θ}}_{0}$	$[0, 0, 0]$	$rad$	Initial rotation of frame_a (center of tire) at the start of the simulation (based on Euler Sequence selection)	InitAng
Angular Velocity Frame	$Euler$		Indicates whether the initial angular velocity is expressed in frame_a (body) or the inertial frame. If Euler is chosen, the initial angular velocities are assumed to be the direct derivatives of the Euler angles.	AngVelType
${\overset{&conjugate0;}{ω}}_{0}$	$[0, 0, 0]$	$\frac{rad}{s}$	Initial angular velocity of frame_a (center of tire) at the start of the simulation expressed in the frame selected in Angular Velocity Frame	InitAngVel

Radial Compliance

These parameters define the radial compliance of the tire.

Name	Default	Units	Description	Modelica ID
Stiffness	$3.04 \cdot 10^{5}$	$\frac{N}{m}$	Tire radial stiffness	C
Damping	$500$	$\frac{N s}{m}$	Tire radial damping	K

Settings

Name	Default	Units	Description	Modelica ID
$e_{Spin}$	[0,1,0]		Tire's spin axis (local)	SymAxis

Size

Name	Default	Units	Description	Modelica ID
$R_{0}$	$0.355$	$m$	Unloaded tire radius	R_0

Surface

Name	Default	Units	Description	Modelica ID
Flat surface	$true$		True (checked) means the road surface is assumed flat. It is defined by a plane passing through (0,0,0) and the normal vector given by ${\overset{&Hat;}{e}}_{g}$	flatSurface
Defined externally	$false$		True (checked) means the road surface is defined external to the tire component. Additional input and output signal ports are activated.	externallyDefined
$δ_{L}$	$0.01$	$m$	Base distance for local surface patch approximation	deltaL
Data source	$inline$		Data source for the uneven surface. See following table.	datasourcemode
Surface data			Surface data; matrix or attached data set	table or data
Smoothness	$linear$		Smoothness of table interpolation	smoothness
$n_{Iter}$	$2$		Number of iterations to find the contact point candidate, recommended value between 1 and 5	nIter

Content of Data source matrix.

Surface normal	First Column	First Row
Global Z	x values	y values
Global Y	z values	x values
Global X	y values	z values

Time Lags

Name	Default	Units	Description	Modelica ID
Use time lags	$false$		True (checked) means use time lags in slip calculation and enable the following two parameters	useTimeLag
$T_{long}$	$0.3$	$s$	Time lag for longitudinal slip	Tlong
$T_{lat}$	$0.3$	$s$	Time lag for slip angle	Tlat

Visualization

Name	Default	Units	Description	Modelica ID
Show tire	$false$		True (checked) creates a tire visualization and enables following three parameters	showTire
$D_{w}$	$0.1$	$m$	Tire width (for visualization)	D_w
Tire color	$black$		Tire color	color00
Band color	$yellow$		Tire band color	color01
Tire transparency	$false$		True (checked) means the tire is transparent	transparent0
Show force arrow	$false$		True (checked) display a force vector and enables the following three parameters	showForceArrow
Show components	$false$		True (checked) means three arrows for force components in ISO axes will be shown instead of a single total force arrow	showForceComponents
Force arrow color	$red$		Specifies the color of the force arrow	color1
Force arrow transparency	$false$		True (checked means the force arrow is transparent	transparent1
Force arrow scale	$1$	$\frac{N}{m}$	Scales the length of the force arrow	scale1
Show torque arrow	$false$		True (checked) displays a torque vector and enables the following three parameters	showMomentArrow
Show components	$false$		True (checked) means three arrows for torque components in ISO axes will be shown instead of a single total torque arrow	showMomentComponents
Torque arrow color	$blue$		Specifies the color of the torque arrow	color2
Torque arrow transparency	$false$		True (checked) means the torque arrow is transparent	transparent2
Torque arrow scale	$1$	$\frac{N m}{m}$	Scales the length of the torque arrow	scale2
Show tangent plane	$false$		True (checked) displays the tangent plane of the contact patch and enables the following four parameters	ShowTanSurface
${th}_{0}$	$0.01$	$m$	Patch visualization thickness	th0
$r_{p}$	$0.2$	$m$	Patch visualization radius	r_patch
Patch color	Green		Color of the contact patch	color3
Patch transparency	$false$		True (checked) means contact patch is transparent	transparent3

Advanced Parameters

Name	Default	Units	Description	Modelica ID
${kF}_{z}$	$0.01$	$\frac{1}{N}$	Dependency factor of longitudinal/lateral forces on the normal force	kFz
$V_{x (\min)}$	$0.1$	$\frac{m}{s}$	Velocity threshold used for singularity avoidance in the slip calculations	V_x_min
$ε_{sgn}$	$0.001$		Used to smooth $sign (x)$ as $\tanh (\frac{x}{ε_{sgn}})$	epsilon_sign
$ε_{norm}$	$1 \cdot 10^{−8}$		Used to prevent singularity in vector normalization	epsilon_norm

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