Pacejka 2012 Tire - MapleSim Help

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Pacejka 2012 Tire

Tire component with Pacejka 2012 formulation and visualization

 

Description

Details

Equations

Connections

Parameters

References

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 FxκFxκ or FyαFyα. 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]

Fz=1+qV2ΩR0V0qFcxFxFz02qFcyFyFz02γ2qFz3+qFz1ρzR0+q_Fz2 ρz2R02dpippFz1+1Fz0

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]

reff=r_ΩFz0FreffFzFz0+DreffarctanBreffFzFz0C

rΩ=R0qreo+qV1R0ΩV02

where the nominal load, Fz0 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).

rL=rzcosγ

Using a linear spring and saturated damping forces based on the tire compliance, the normal force, Fz, is calculated as follows

FzC&equals;{CR0rLrL<R00otherwise

FzK&equals;{KVzrL<R00otherwise

Fz&equals;{FzC&plus;minFzC&comma;FzK0<FzC&plus;FzK0otherwise

where Vz  is the tire center speed with respect to ISO Z, C is tire stiffness, K is tire damping, and R0  is tire unloaded radius. The use of the min function is to ensure that Fz is continuous at rL&equals;R0.

With this option selected for Fz, the user can choose between the tire unloaded radius, R0, and the loaded radius, rL, to assign to the effective radius, reff.

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&equals;reΩVxVx

tanα&equals;VyVx

where re is the tire effective radius, Ω is the tire speed of revolution, and Vx and Vy 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 Vx&equals;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:

Tlongdκdt&equals;reΩVxκVx

Tlatdtanαdt&equals;VytanαVx

Stretched string

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

σlongdκdt&equals;reΩVxκVx

σlatdtanαdt&equals;VytanαVx

where

σlong&equals;maxFzpTx1&plus;pTx2dfzexppTx3dfzR0Fz0LSκ&comma;σlongmin

σlat&equals;maxpTy1sin2arctanFzpTy2Fz01pTy3&verbar;γ&verbar;R0LSα&comma;σlatmin

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

The load ratio, dfz, is defined as

dfz&equals;FzFz0Fz0

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

Fx&equals;GxαFx0

where

Fx0&equals;DxsinCxarctanBxκxExBxκxarctanBxκx&plus;SVx

Gxα&equals;1Gxα0cosCxαarctanBxααSExαBxααSarctanBxααS

The lateral force is

Fy&equals;Fy0Gyκ&plus;SVκ

where

Fy0&equals;DysinCyarctanByαyEyByαyarctanByαy&plus;SVy

Gyκ&equals;1Gyκ0cosCyκarctanByκκSEyκByκκSarctanByκκS

The normal force, Fz, has been discussed in the Normal Force and Effective Radius section.

The overturning couple is

Mx&equals;R0Fz(qsx1λVMxqsx2γdpippMx1&plus;1&plus;qsx3FyFz0&plus;qsx4cosqsx5arctanqsx6FzFz02sinqsx7γ&plus;qsx8arctanqsx9FyFz0&plus;qsx10arctanqsx11FzFz0γ)λMx

The rolling resistance moment is

My&equals;FzR0qsy1&plus;qsy2FxFz0&plus;qsy3&verbar;VxV0&verbar;&plus;qsy4VxV04&plus;qsy5&plus;qsy6FzFz0γ2FzFz0qsy7pipioqsy8λMy

The self-aligning torque is

Mz&equals;Mz&plus;Mzr&plus;sFx

where Mz is the torque due to pneumatic trail, t, Mzr is the residual torque, and sFx 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

framea

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

reff

Signal output for tire effective radius

r_eff

IncAng

Signal output for tire inclination angle or camber

IncAng

enin

[1] Vector input for surface normal vector

en_in

rc

[1] Vector output for tire center position w.r.t. the inertial frame

r_c

rzin

[1] Signal input for tire center distance from the surface

rz_in

pin

[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]

 

kgm2

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

ICr&comma;v

Ignore

 

Indicates whether MapleSim will ignore, try to enforce, or strictly enforce the translational initial conditions

MechTranTree

r&conjugate0;0

0&comma;0&comma;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

v&conjugate0;0

0&comma;0&comma;0

ms

Initial velocity of frame_a (tire center) at the start of the simulation expressed in the frame selected in Velocity Frame

InitVel

ICθ&comma;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&comma;2&comma;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

θ&conjugate0;0

0&comma;0&comma;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

ω&conjugate0;0

0&comma;0&comma;0

rads

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

e&Hat;spin

0&comma;0&comma;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

Fz

1

 

Normal force equation

FzMode

reff

0

 

Effective radius

reffMode

Slip

Quasistatic

 

Choose type of slip calculation (Quasi-static, Constant time lags, or Stretched-string)

slipMode

Tlong

0.3

s

Time lag for longitudinal slip

TlongIn

Tlat

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

σlongmin

0.1

 

Minimum longitudinal relaxation length

TlongMin

σlatmin

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

pio

2.2·105

Pa

Nominal tire pressure

over_p_io

Constantpressure

true

 

True (checked) uses constant pressure; false provides an input port for the tire pressure

isConstantPressure

pi

2.4·105

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

λFz0

1

 

Nominal load scaling factor

over_lambda_Fz0

λmux

1

 

Peak friction coefficient (x) scaling factor

over_lambda_mux

λmuy

1

 

Peak friction coefficient (y) scaling factor

over_lambda_muy

λmuV

0

 

Slip speed decaying friction scaling factor

over_lambda_muV

λKxκ

1

 

Brake slip stiffness scaling factor

over_lambda_KxKap

λKyα

1

 

Cornering stiffness scaling factor

over_lambda_KyAlp

λCx

1

 

Shape factor (x) scaling factor

over_lambda_Cx

λCy

1

 

Shape factor (y) scaling factor

over_lambda_Cy

λEx

1

 

Curvature factor (x) scaling factor

over_lambda_Ex

λEy

1

 

Curvature factor (y) scaling factor

over_lambda_Ey

λHx

1

 

Horizontal shift (x) scaling factor

over_lambda_Hx

λHy

1

 

Horizontal shift (y) scaling factor

over_lambda_Hy

λVx

1

 

Vertical shift (x) scaling factor

over_lambda_Vx

λVy

1

 

Vertical shift (y) scaling factor

over_lambda_Vy

λKyγ

1

 

Camber force stiffness scaling factor

over_lambda_KyGam

λKzγ

1

 

Camber torque stiffness scaling factor

over_lambda_KzGam

λt

1

 

Pneumatic trail scaling factor

over_lambda_t

λMr

1

 

Residual torque scaling factor

over_lambda_Mr

λxα

1

 

Alpha influence on Fx (kappa) scaling factor

over_lambda_xAlp

λyκ

1

 

Kappa influence on Fy (alpha) scaling factor

over_lambda_yKap

λVyκ

1

 

Kappa induces ply-steer Fy scaling factor

over_lambda_VyKap

λs

1

 

Mz moment arm of Fx scaling factor

over_lambda_s

λCz

1

 

Radial tire stiffness scaling factor

over_lambda_Cz

λMx

1

 

Overturning couple stiffness scaling factor

over_lambda_Mx

λMy

1

 

Rolling resistance moment scaling factor

over_lambda_My

λVMx

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 e&Hat;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

δL

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

nIter

2

 

Number of iterations to find the contact point candidate, recommended value between 1 and 5

nIter

Content of Data source matrix.