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Pacejka 2012 Tire

Tire component with Pacejka 2012 formulation and visualization

Description

The Pacejka 2012 Tire component employs the 2012 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 2012 tire model has about 180 parameters. Unlike the Linear and the Fiala tire components, where the required parameters are defined in the MapleSim GUI, to facilitate 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 > Pacejka Parameters in MapleSim. The generated parameter block will be located in the Local Components panel on the left side of the MapleSim GUI.

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

Override Parameters

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

Enabling one of these checkboxes 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 checkbox also exposes the associated parameters to MapleSim 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 in MapleSim should be used with $Side = 0$ , and those on the left side should have $Side = 1$ .

Normal Force and Effective Radius

The normal force exerted by the surface to the tire is calculated using the given compliance parameters and surface geometry. There are two implemented formulation in the Pacejka tire component for calculating the normal force: Pacejka formulation and Linear spring-damper.

Pacejka Formulation

The Pacejka formulation option uses the following equation for the normal force [1]

$F_{z} = \{1 + q_{V2} |Ω| \frac{R_{0}}{V_{0}} - {(q_{Fcx} \frac{F_{x}}{F_{z0}})}^{2} - {(q_{Fcy} \frac{F_{y}}{F_{z0}})}^{2}\} \{(γ^{2} q_{Fz3} + q_{Fz1}) \frac{ρ_{z}}{R_{0}} + q_Fz2 \frac{ρ_{z}^{2}}{R_{0}^{2}}\} ({dp}_{i} p_{pFz1} + 1) F_{z0}$

Note that with this option selected for the normal force, the Pacejka effective radius formulation will also be used internally. This formulation is as follows [1]

$r_{eff} = r_Ω - F_{z0} \frac{F_{reff} \frac{F_{z}}{F_{z0}} + D_{reff} \arctan (B_{reff} \frac{F_{z}}{F_{z0}})}{C}$

$r_{Ω} = R_{0} (q_{reo} + q_{V1} {(R_{0} \frac{Ω}{V_{0}})}^{2})$

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

Linear Spring-Damper

The Linear spring-damper option is the same formulation used for normal force calculation in the Linear and the Fiala tire components as explained below.

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

With this option selected for $F_{z}$ , the user can choose between the tire unloaded radius, $R_{0}$ , and the loaded radius, $r_{L}$ , to assign to the effective radius, $r_{eff}$ .

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

$κ = \frac{r_{e} Ω - V_{x}}{|V_{x}|}$

$\tan (α) = \frac{V_{y}}{|V_{x}|}$

where $r_{e}$ is the tire effective radius, $Ω$ 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 $V_{x} = 0$ .

	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 MapleSim 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 summarized below for the Pacejka 2012 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 and Effective Radius section.

The overturning couple is

$\begin{array}{c} M_{x} = R_{0} F_{z} ( & q_{sx1} λ_{VMx} - q_{sx2} γ ({dp}_{i} p_{pM} x1 + 1) + q_{sx3} \frac{F_{y}}{F_{z0}} \\ + q_{sx4} \cos (q_{sx5} {\arctan (q_{sx6} \frac{F_{z}}{F_{z0}})}^{2}) \sin (q_{sx7} γ + q_{sx8} \arctan (q_{sx9} \frac{F_{y}}{F_{z0}})) \\ + q_{sx10} \arctan (q_{sx11} \frac{F_{z}}{F_{z0}}) γ) λ_{Mx} \end{array}$

The rolling resistance moment is

$M_{y} = F_{z} R_{0} (q_{sy1} + q_{sy2} \frac{F_{x}}{F_{z0}} + q_{sy3} | \frac{V_{x}}{V_{0}} | + q_{sy4} {(\frac{V_{x}}{V_{0}})}^{4} + (q_{sy5} + q_{sy6} \frac{F_{z}}{F_{z0}}) γ^{2}) ({(\frac{F_{z}}{F_{z0}})}^{q_{sy7}} {(\frac{p_{i}}{p_{io}})}^{q_{sy8}}) λ_{My}$

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.

All the employed parameters in the equations above need to be defined in the tire parameters block and be accessible to the Pacejka tire components.

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
${en}_{in}$	[1] Vector input for surface normal vector	en_in
$r_{c}$	[1] Vector output for tire center position w.r.t. the inertial frame	r_c
${rz}_{in}$	[1] Signal input for tire center distance from the surface	rz_in
$p_{in}$	[2] Signal input for tire inflations pressure	p_in

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

[2] Available if Pressure Override parameter is true and Constant Pressure is false.

Parameters

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]		$kg m^{2}$	Rotational inertia, expressed in frame_a (center of tire)	Inertia

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 MapleSim will 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 MapleSim will 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
${\overset{&Hat;}{e}}_{spin}$	$[0, 0, 0]$		Tire's spin axis (local)	SymAxis
$Side$			0: default, 1: mirrored	Side
$ISO$	$0$		0: Keep ISO, 1: Rotate ISO pi radians around Z axis	intISO
$F_{z}$	$1$		Normal force equation	FzMode
$r_{eff}$	$0$		Effective radius	reffMode
$Slip$	$Quasi - static$		Choose type of slip calculation (Quasi-static, Constant time lags, or Stretched-string)	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

Pressure

Name	Default	Units	Description	Modelica ID
$Override$	$false$		True (checked) overrides override the pressure parameters and enables the following parameters	overridePressure
$p_{io}$	$2.2 \cdot 10^{5}$	$Pa$	Nominal tire pressure	over_p_io
$Constant pressure$	$true$		True (checked) uses constant pressure; false provides an input port for the tire pressure	isConstantPressure
$p_{i}$	$2.4 \cdot 10^{5}$	$Pa$	Tire pressure	over_p_cons

Scaling Factors

Name	Default	Units	Description	Modelica ID
$Override$	$false$		True (checked) override the scaling factors and enables the following parameter	overrideFactors
$λ_{Fz0}$	$1$		Nominal load scaling factor	over_lambda_Fz0
$λ_{mux}$	$1$		Peak friction coefficient (x) scaling factor	over_lambda_mux
$λ_{muy}$	$1$		Peak friction coefficient (y) scaling factor	over_lambda_muy
$λ_{muV}$	$0$		Slip speed decaying friction scaling factor	over_lambda_muV
$λ_{Kx κ}$	$1$		Brake slip stiffness scaling factor	over_lambda_KxKap
$λ_{Ky α}$	$1$		Cornering stiffness scaling factor	over_lambda_KyAlp
$λ_{Cx}$	$1$		Shape factor (x) scaling factor	over_lambda_Cx
$λ_{Cy}$	$1$		Shape factor (y) scaling factor	over_lambda_Cy
$λ_{Ex}$	$1$		Curvature factor (x) scaling factor	over_lambda_Ex
$λ_{Ey}$	$1$		Curvature factor (y) scaling factor	over_lambda_Ey
$λ_{Hx}$	$1$		Horizontal shift (x) scaling factor	over_lambda_Hx
$λ_{Hy}$	$1$		Horizontal shift (y) scaling factor	over_lambda_Hy
$λ_{Vx}$	$1$		Vertical shift (x) scaling factor	over_lambda_Vx
$λ_{Vy}$	$1$		Vertical shift (y) scaling factor	over_lambda_Vy
$λ_{Ky γ}$	$1$		Camber force stiffness scaling factor	over_lambda_KyGam
$λ_{Kz γ}$	$1$		Camber torque stiffness scaling factor	over_lambda_KzGam
$λ_{t}$	$1$		Pneumatic trail scaling factor	over_lambda_t
$λ_{Mr}$	$1$		Residual torque scaling factor	over_lambda_Mr
$λ_{x α}$	$1$		Alpha influence on $F_{x}$ (kappa) scaling factor	over_lambda_xAlp
$λ_{y κ}$	$1$		Kappa influence on $F_{y}$ (alpha) scaling factor	over_lambda_yKap
$λ_{Vy κ}$	$1$		Kappa induces ply-steer $F_{y}$ scaling factor	over_lambda_VyKap
$λ_{s}$	$1$		$M_{z}$ moment arm of $F_{x}$ scaling factor	over_lambda_s
$λ_{Cz}$	$1$		Radial tire stiffness scaling factor	over_lambda_Cz
$λ_{Mx}$	$1$		Overturning couple stiffness scaling factor	over_lambda_Mx
$λ_{My}$	$1$		Rolling resistance moment scaling factor	over_lambda_My
$λ_{VMx}$	$1$		Overturning couple vertical shift scaling factor	over_lambda_VMx

Surface

Name	Default	Units	Description	Modelica ID
Flat surface	$true$		True (checked) means theroad 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

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