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Pacejka 2002

Tire component with Pacejka 2002 formulation and visualization

Description

The Pacejka 2002 tire component employs the 2002 formulation of the Pacejka tire model presented in [1].

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

Tire Parameters Block

The Pacejka 2002 tire model has about 130 parameters. Unlike the Linear and the Fiala tire components, where the required parameters are defined in the GUI, to facilitate the parameter handling process the Pacejka Parameters App should be used to generate a parameter block which contains the necessary tire parameters. To open this app, browse to Add Apps or Templates > Tires > Pacejka Parameters. The generated parameter block will be located in the Local Components panel on the left side of the GUI.

The user should place the generated parameter block into the workspace at the same or higher level as the Pacejka tire components that it defines.

Override Parameters

There is an Override check box in the Inertia, Radial Compliance, and Scaling Factors sections of the Pacejka tire component properties.

Enabling one of these check boxes allows the user to override the associated parameters otherwise defined in the tire parameters block. For example, the user can override the inertia properties as shown below.

Checking an Override check box also exposes the associated parameters to apps such as the Parameter Sweep app and the FMU Generation app.

ISO Axis

Unlike the Linear and Fiala tire models, the Pacejka tire model is typically asymmetric, that is $F_{x} (- κ) \neq - F_{x} (κ)$ or $F_{y} (- α) \neq - F_{y} (α)$ . To ensure the correct formulation, the ISO X axis of a tire should point towards the heading of the vehicle. The Show ISO axis option in the visualization section of the tire parameters can be helpful to visually confirm that the ISO axes have been assigned correctly.

If not assigned correctly, the user can change the integer parameter of ISO from 0 to 1 to rotate the ISO axis 180 degrees around ISO Z.

Sideness

The Pacejka tire parameters apply to a specific tire side. This denotes the side of the vehicle where the tire should be mounted. The Side parameter in the properties can be used to mirror the tire. For example, if the parameters of the generated parameter block are for a right side tire, then the tire components mounted on the right side of the vehicle model should be used with $Side = 0$ , and those on the left side should have $Side = 1$ .

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

Effective Radius

There are three options for defining the effective radius: Loaded Radius, Unloaded Radius, and Pacejka.

Upon selection of either of the first two options, the loaded or unloaded radius will be assigned to the effective radius.

With the Pacejka option selected, the effective radius is

$r_{eff} = R_{0} - F_{z0} \frac{F_{reff} C \frac{R_{0} - r_{L}}{F_{z0}} + D_{reff} \arctan (B_{reff} C \frac{R_{0} - r_{L}}{F_{z0}})}{C}$

where the nominal load, $F_{z0}$ , and the rest of the parameters used in this equation are defined in the tire parameters block.

Slip Calculations

Three options are available for tire slip calculation, Quasi-static, Constant time lags, and Stretched string.

Quasi-static

With the choice of Quasi-static, the following equations for longitudinal slip, $κ$ , and slip angle, $α$ , hold true on a flat surface with no inclination angle

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

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

where $r_{e}$ is the tire effective radius and considered equal to the loaded radius ( $r_{L}$ ), $Ω$ is the tire speed of revolution, $V_{x}$ and $V_{y}$ are the speeds of the tire center with respect to ISO X and ISO Y axes, respectively, and $V_{x (\min)}$ is the velocity threshold used for singularity avoidance in the slip calculations.

The longitudinal slip and slip angle are continuous and differentiable in the neighborhood of $V_{x} = 0$ .

	Constant Time Lags
	A first-order dynamics to the longitudinal slip and slip angle calculation can be introduced using the Constant time lags option. When active, the following slip formulation is used: $T_{long} \frac{d κ}{d t} = r_{e} Ω - V_{x} - κ \|V_{x}\|$ $T_{lat} (d \frac{\tan (α)}{d t}) = V_{y} - \tan (α) \|V_{x}\|$

Stretched String

With this option active, the relaxation lengths will be used in slip calculation as follows

$σ_{long} \frac{d κ}{d t} = r_{e} Ω - V_{x} - κ |V_{x}|$

$σ_{lat} (d \frac{\tan (α)}{d t}) = V_{y} - \tan (α) |V_{x}|$

where

$σ_{long} = \max (F_{z} (p_{Tx1} + p_{Tx2} {df}_{z}) \exp (p_{Tx3} {df}_{z}) (\frac{R_{0}}{F'_{z0}}) {LS}_{κ}, σ_{long}^{\min})$

$σ_{lat} = \max (p_{Ty1} \sin (2 \arctan (\frac{F_{z}}{p_{Ty2} F'_{z0}})) (1 - p_{Ty3} | γ |) R_{0} {LS}_{α}, σ_{lat}^{\min})$

Parameters in the above equations should be inserted using the GUI.

The load ratio, ${df}_{z}$ , is defined as

${df}_{z} = \frac{F_{z} - F_{z0}}{F_{z0}}$

Equations

The formulation for resultant forces/moments of tire-surface interaction at the tire contact patch are briefly summarized below for the Pacejka 2002 tire component.

The longitudinal force is

$F_{x} = G_{x α} F_{x0}$

where

$F_{x0} = D_{x} \sin (C_{x} \arctan (B_{x} κ_{x} - E_{x} (B_{x} κ_{x} - \arctan (B_{x} κ_{x})))) + S_{Vx}$

$G_{x α} = \frac{1}{G_{x α 0}} \cos (C_{x α} \arctan (B_{x α} α_{S} - E_{x α} (B_{x α} α_{S} - \arctan (B_{x α} α_{S}))))$

The lateral force is

$F_{y} = F_{y0} G_{y κ} + S_{V κ}$

where

$F_{y0} = D_{y} \sin (C_{y} \arctan (B_{y} α_{y} - E_{y} (B_{y} α_{y} - \arctan (B_{y} α_{y})))) + S_{Vy}$

$G_{y κ} = \frac{1}{G_{y κ 0}} \cos (C_{y κ} \arctan (B_{y κ} κ_{S} - E_{y κ} (B_{y κ} κ_{S} - \arctan (B_{y κ} κ_{S}))))$

The normal force, $F_{z}$ , has been discussed in the Normal Force section.

The overturning couple is

$M_{x} = R_{0} F_{z} (- q_{sx2} γ + q_{sx1} λ_{VMx} + q_{sx3} \frac{F_{y}}{F_{z0}}) λ_{Mx}$

The rolling resistance moment is

$M_{y} = F_{z} R_{0} signum (Ω) (q_{sy1} + q_{sy2} \frac{F_{x}}{F_{z0}} + q_{sy3} | \frac{V_{x}}{V_{0}} | + q_{sy4} {(\frac{V_{x}}{V_{0}})}^{4}) λ_{My}$

The self-aligning torque is

$M_{z} = M'_{z} + M_{zr} + s F_{x}$

where $M'_{z}$ is the torque due to pneumatic trail, $t$ , $M_{zr}$ is the residual torque, and $s F_{x}$ is the longitudinal force contribution to the self-aligning torque. Each of these terms has a specific expression, discussed in [1] in more detail.

Connections

Name	Description	Modelica ID
${frame}_{a}$	Multibody frame for tire center	frame_a
$Fz$	Signal output for the normal force	Fz
$LongSlip$	Signal output for longitudinal slip	LongSlip
$SlipAng$	Signal output for slip angle	SlipAng
$SpinRate$	Signal output for tire speed of revolution or spin rate	SpinRate
$r_{eff}$	Signal output for tire effective radius	r_eff
$IncAng$	Signal output for tire inclination angle or camber	IncAng
$r_{c}$	[1] Vector signal output for tire center position w.r.t. the inertial frame	r_c
${en}_{in}$	[1] Vector signal input for surface normal vector	en_in
${rz}_{in}$	[1] Signal input for tire center distance from the surface	rz_in
$F_{x}^{ISO}$	Signal input for the longitudinal force at the contact patch w.r.t. the ISO axis	FxISO
$F_{y}^{ISO}$	Signal input for the lateral force at the contact patch w.r.t. the ISO axis	FyISO
$F_{z}^{ISO}$	[2] Signal input for the normal force at the contact patch w.r.t. the ISO axis	FzISO
$M_{x}^{ISO}$	Signal input for the overturning couple at the contact patch w.r.t. the ISO axis	MxISO
$M_{y}^{ISO}$	Signal input for the rolling resistance moment at the contact patch w.r.t. the ISO axis	MyISO
$M_{z}^{ISO}$	Signal input for the self-aligning torque at the contact patch w.r.t. the ISO axis	MzISO

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

[2] Available if Settings parameter Calculate Fz internally is false.

Parameters

Inertia

Name	Default	Units	Description	Modelica ID
Override	$false$		True (checked) overrides the inertia parameters 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

Settings

Name	Default	Units	Description	Modelica ID
${\overset{&Hat;}{e}}_{spin}$	[0,1,0]		Tire's spin axis (local)	SymAxis
Calculate Fz internally	$false$		True (checked) means Fz is calculated internally using the compliance parameters	useFz
$Side$	$0$		0: default, 1: mirrored	Side
$ISO$	$0$		0: Keep ISO, 1: Rotate ISO pi radians around Z axis	intISO
$r_{eff}$	Loaded radius		Effective radius (Loaded radius, Unloaded radius, or Pacejka)	reffMode
$Slip$	Quasi-static		Choose type of slip calculation (Quasi-static, Constant time lags, Stretched string, or Damped transient)	slipMode
$T_{long}$	$0.3$	$s$	Time lag for longitudinal slip	TlongIn
$T_{lat}$	$0.3$	$s$	Time lag for slip angle	TlatIn
$Params$	[2]		Parameters for stretched-string formulation: [LSkappa, LSalpha, p_Tx1, p_Tx2, p_Tx3, p_Ty1, p_Ty2, p_Ty3]	ssParams
$σ_{{long}_{\min}}$	$0.1$		Minimum longitudinal relaxation length	TlongMin
$σ_{{lat}_{\min}}$	$0.1$		Minimum lateral relaxation length	TlatMin

[2] $[1, 1, 2.3657, 1.4112, 0.56626, 2.1439, 1.9829, −0.90729]$

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
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
$V_{x (\min)}$	$0.1$	$\frac{m}{s}$	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

References

[1] Pacejka, Hans. Tire and vehicle dynamics. SAE, 2002.

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