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COF

Coefficient of friction-based tire formulation

Description

The COF tire component provides isotropic friction force formulation for the tire forces at the contact path. Tire force calculation for this tire is not based on the slip values, but rather depends on the sliding velocity, the relative velocity of the contact path with respect to the ground. There are three friction formulation for this tire: Coulomb, Stribeck, and Custom.

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 $r_{L} = R_{0}$ .

Equations

There are three different options for the friction formulation as discussed below.

	Coulomb
	In this case, the Coulomb friction model is used for the tire-surface friction force $F_{f} = - \tanh (\frac{v_{r}}{v_{0}}) μ_{c} F_{z} e_{v}$ where $v_{r}$ and $e_{v}$ are the relative sliding velocity magnitude and unit vector, respectively. The tanh function is used as a smoothing function around $v_{r} = 0$ , with the scale of $v_{0}$ .

Stribeck

With this option chosen, additional terms for Stribeck effect and viscous friction are included in the friction equation as shown below.

$F_{f} = - (μ_{d} v_{r} + \tanh (\frac{v_{r}}{v_{0}}) \cdot (μ_{c} + μ_{c} (peak - 1) \exp (- {(\frac{v_{r}}{v_{s}})}^{n}))) F_{z} e_{v}$

where $μ_{d}$ is the viscous coefficient of friction, $peak$ is the ratio of static friction to sliding friction, and $v_{s}$ is the Stribeck sliding velocity coefficient.

	Custom
	In this case, an additional input of $μ_{in}$ appears on the component. The value of $μ_{in}$ needs to be externally calculated and then fed to the component. The friction force is $F_{f} = - \tanh (\frac{v_{r}}{v_{0}}) μ_{in} F_{z} e_{v}$

Connections

Name	Description	Modelica ID
${frame}_{a}$	Tire center frame	frame_a
$Fz$	Signal output for normal force	Fz
$IncAng$	Signal output for tire inclination angle or camber	IncAng
$SpinRate$	Signal output for tire spin rate	SpinRate
$F_{f}$	Vector output of friction force	Ff
$e_{v}$	Vector output of relative sliding velocity unit vector	ev
$v_{r}$	Signal output for magnitude of relative sliding velocity	v_r
$μ_{in}$	[1] Signal input for custom externally defined coefficient of friction	mu_in
${en}_{in}$	[2] Vector signal input for surface normal vector	en_in
${rz}_{in}$	[2] Signal input for tire center distance from the surface	rz_in
$r_{c}$	[2] Vector signal output for tire center position w.r.t. the inertial frame	r_c

[1] Available if friction formulation is chosen as Custom.

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

Parameters

Friction

Name	Default	Units	Description	Modelica ID
$Formulation$	Coulomb		Enumeration defining the friction formulation	Formulation
$μ_{c}$	$0.5$		Coulomb coefficient of friction	mu_c
$peak$	$1.2$		Peak of ratio of static friction to sliding friction	peak
$μ_{d}$	$0$		Viscous coefficient of friction	mu_d
$v_{s}$	$0.1$	$\frac{m}{s}$	Stribeck sliding velocity coefficient	vs
$n$	$1$		Decay exponent	n
$v_{0}$	$0.01$	$\frac{m}{s}$	Smoothing function scale, i.e. tanh(v_r/v0)	v0

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

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
${\overset{&Hat;}{e}}_{spin}$	$[0, 1, 0]$		Tire's spin axis (local)	SymAxis
ISO	$0$		0: keep ISO, 1: rotate $π$ radians around Z-axis	intISO

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

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
Show ISO axis	$false$		True (checked) displays the ISO axes and enables the following two parameters	showISO
Axis scale	$1$	$m$	Length of each XYZ ISO axis in the visualization	scaleISO
Axis transparency	$true$		True (checked) means axes are transparent	transparentISO

Advanced Parameters

Name	Default	Units	Description	Modelica ID
$ε_{norm}$	$1 \cdot 10^{−8}$		Used to prevent singularity in vector normalization	epsilon_norm

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