using pacejka '89 and '94 models

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Pacejka '89/'94

Using Pacejka '89 and '94 Models

ADAMS/Tire provides you with the handling force models, Pacejka '89 and Pacejka '94. Learn about these models:

• About Pacejka '89 and '94 • Using Pacejka '89 Handling Force Model • Using Pacejka '94 Handling Force Model • Combined Slip • Left and Right Side Tires • Contact Methods

ADAMS/Tire 2005 r2 Help

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About Pacejka '89 and '94

The Pacejka '89 and '94 handling models are special versions of the Magic-Formula Tyre model as cited in the following publications:

• Pacejka '89 - H.B Pacejka, E. Bakker, and L. Lidner. A New Tire Model with an Application in Vehicle Dynamics Studies, SAE paper 890087, 1989.

• Pacejka '94 - H.B Pacejka and E. Bakker. The Magic Formula Tyre Model. Proceedings of the 1st International Colloquium on Tyre Models for Vehicle Dynamics Analysis, Swets & Zeitlinger B.V., Amsterdam/Lisse, 1993.

PAC2002 is technically superior, continuously kept up to date with latest Magic Formula developments, and MSC’s recommended handling model. However, because many ADAMS/Tire users have pre-existing tire data or new data from tire suppliers and testing organizations in a format that is compatible with the Pacejka '89 and '94 models, the ADAMS/Tire Handling module includes these models in addition to the PAC2002.

The material in this help is intended to illustrate only the formulas used in the Pacejka '89 and '94 tire models. For general information on the PAC2002 and the Magic Formula method, see the papers cited above or the PAC2002 help.

Learn more about:

• History of the Pacejka Name in ADAMS/Tire • About Coordinate Systems • Normal Force

History of the Pacejka Name in ADAMS/Tire

The formulas used in the Pacejka '89 and '94 tire models are derived from publications by Dr. H.B. Pacejka, and are commonly referred to as the Pacejka method in the automotive industry. Dr. Pacejka himself is not personally associated with the development of these tire models, nor does he endorse them in any way.

About Coordinate Systems

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The coordinate systems used in tire modeling and measurement are sometimes confusing. The coordinate systems employed in the Pacejka ’89 and ’94 tire models are no exception. They are derived from the tire-measurement systems that the majority of ADAMS/Tire customers were using at the time when the models were originally developed.

The Pacejka '89 and '94 tire models were developed before the implementation of the TYDEX STI. As a result, Pacejka ’89 conforms to a modified SAE-based tire coordinate system and sign conventions, and Pacejka ’94 conforms to the standard SAE tire coordinate system and sign conventions. MSC maintains these conventions to ensure file compatibility for ADAMS/Tire customers.

Future tire models will adhere to one single coordinate system standard, the TYDEX C-axis and W-axis system. For more information on the TYDEX standard, see Standard Tire Interface (STI).

Normal Force

The normal force Fz is calculated assuming a linear spring (stiffness: kz) and damper (damping constant cz), so the next equation holds:

If the tire loses contact with the road, the tire deflection and

deflection velocity become zero, so the resulting normal force Fzwill also be zero. For very small positive tire deflections, the value of the damping constant is reduced and care is taken to ensure that the normal force Fz will not become negative.

In stead of the linear vertical tire stiffness cz , also an arbitrary tire deflection - load curve can be defined in the tire property file in the section [DEFLECTION_LOAD_CURVE], see also the example tire property files, Example of Pacejka ’89 Property File and Example of Pacejka ’94 Property File. If a section called [DEFLECTION_LOAD_CURVE] exists, the load deflection datapoints with a cubic spline for inter- and extrapolation are used for the calculation of the vertical force of the tire. Note that you must specify VERTICAL_STIFFNESS in the tire property, but it does not play any role.


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Using Pacejka '89 Handling Force Model


Learn about the Pacejka '89 handling force model:

• Using Correct Coordinate System and Units • Force and Moment Formulation for Pacejka ’89 • Example of Pacejka ’89 Property File

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Using Correct Coordinate System and Units in Pacejka '89

The test data and resulting coefficients that come from the Pacejka '89 tire model conform to a modified SAE tire coordinate system. The standard SAE tire coordinate system is shown next and the modified sign conventions for Pacejka '89 are described in the table below.

Note: The section [UNITS] in the tire property file does not apply to the Magic Formula coefficients

SAE Tire Coordinate System

Conventions for Naming Variables

Variable name and abbreviation:

Description:

Normal load Fz (kN) Positive when the tire is penetrating the road.*


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Lateral force Fy (N) Positive in a right turn. Negative in a left turn.

Longitudinal force

Fx (N) Positive during traction. Negative during braking.

Self-aligning torque

Mz (Nm)

Positive in a left turn. Negative in a right turn.

Inclination angle (deg) Positive when the top of the tire tilts to the right (when viewing the tire from the rear).*

Sideslip angle (deg)

Positive in a right turn.*

Longitudinal slip (%) Negative in braking (-100%: wheel lock). Positive in traction.

* Opposite convention to standard SAE coordinate system shown in SAE Tire Coordinate System.

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Force and Moment Formulation for Pacejka '89

Learn about:

• Longitudinal Force for Pacejka '89 • Overturning Moment for Pacejka '89 • Self-Aligning Torque • Lateral Stiffness • Rolling Resistance • Smoothing

Longitudinal Force for Pacejka '89

C - Shape Factor

C=B0

D - Peak Factor

D=(B1*FZ2+B2*FZ)

BCD

BCD=(B3*FZ2+B4*FZ)*EXP(-B5*FZ)

B - Stiffness Factor

B=BCD/(C*D)

Horizontal Shift

Sh=B9*FZ+B10

Vertical Shift

Sv=0.0

Composite

X1=(k+Sh)

E Curvature Factor


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E=(B6*FZ2+B7*FZ+B8)

FX Equation

FX=(D*SIN(C*ATAN(B*X1-E*(B*X1-ATAN(B*X1)))))+Sv

Longitudinal Force

Parameters: Description:

B0 Shape factor

B1, B2 Peak factor

B3, B4, B5 BCD calculation

B6, B,7 B8 Curvature factor

B9, B10 Horizontal shift

Example Longitudinal Force Plot for Pacejka ’89

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Lateral Force for Pacejka '89

C - Shape Factor

C=A0

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D - Peak Factor

D=(A1*FZ2+A2*FZ)

BCD

BCD=A3*SIN(ATAN(FZ/A4)*2.0)*(1.0-A5*ABS(g))


B=BCD/(C*D)

Horizontal Shift

Sh=A9*FZ+A10+A8*g

Vertical Shift

Sv=A11*FZ*g+A12*FZ+A13

Composite

X1=(a+Sh)

E - Curvature Factor

E=(A6*FZ+A7)

FY Equation

FY=(D*SIN(C*ATAN(B*X1-E*(B*X1-ATAN(B*X1)))))+Sv

Parameters for Lateral Force


A0 Shape factor

A1, A2 Peak factor

A3, A4, A5 BCD calculation

A6, A7 Curvature factor

A8, A9, A10 Horizontal shift

A11, A12, A13 Vertical shift

Example Lateral Force Plot for Pacejka ’89


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Self-Aligning Torque

C - Shape Factor

C=C0

D - Peak Factor

D=(C1*FZ2+C2*FZ)

BCD

BCD=(C3*FZ2+C4*FZ)*(1-C6*ABS(g))*EXP(-C5*FZ)


B=BCD/(C*D)

Horizontal Shift

Sh=C11*g+C12*FZ+C13

Vertical Shift

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Sv= (C14*FZ2+C15*FZ)*g+C16*FZ+C17

Composite

X1=(a+Sh)


E=(C7*FZ2)+C8*FZ+C9)*(1.0-C10*ABS(g))

MZ Equation

MZ=(D*SIN(C*ATAN(B*X1-E*(B*X1-ATAN(B*X1)))))+Sv

Parameters for Self-Aligning Torque


C0 Shape factor

C1, C2 Peak factor

C3, C4, C5, C6 BCD calculation

C7, C8, C9, C10 Curvature factor

C11, C12, C13 Horizontal shift

C14, C15, C16, C17 Vertical shift

Example Self-Aligning Torque Plot for Pacejka ’89


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Overturning Moment

The lateral stiffness is used to calculate an approximate lateral deflection of the contact patch when there is a lateral force present:

deflection = Fy / lateral_stiffness

This deflection, in turn, is used to calculate an overturning moment due to the vertical force:

Mx (overturning moment) = -Fz * deflection

And an incremental aligning torque due to longtiudinal force (Fx)

Mz = Mz,Magic Formula + Fx * deflection

Here Mz,Magic Formula is the magic formula for aligning torque and Fx * deflection is the contribution due to the longitudinal force.

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Rolling Resistance

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The rolling resistance moment My is opposite to the wheel angular velocity. The magnitude is given by:

My = Fz * Lrad * rolling_resistance

Where Fz equals the vertical force and Lrad is the tyre loaded radius. The rolling resistance coefficient can be entered in the tire property file:

[PARAMETER]

ROLLING_RESISTANCE = 0.01

A value of 0.01 introduces a rolling resistance force that is 1% of the vertical load.

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Smoothing

When you indicate smoothing by setting the value of use mode in the tire property file, ADAMS/Tire smooths initial transients in the tire force over the first 0.1 seconds of simulation. The longitudinal force, lateral force, and aligning torque are multiplied by a cubic step function of time. (See STEP in the ADAMS/Solver online help.)

Longitudinal Force

FLon = S*FLon

Lateral Force

FLat = S*FLat

Overturning Moment

Mx = S*Mx

Rolling resistance moment

My = S*My

Aligning Torque

Mz = S*Mz


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The USE_MODE parameter in the tire property file allows you to switch smoothing on or off:

• USE_MODE = 1 or 2, smoothing is off • USE_MODE = 3 or 4, smoothing is on

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Example of Pacejka '89 Property File $-------------------------------------------------------------MDI_HEADER [MDI_HEADER] FILE_TYPE = 'tir' FILE_VERSION = 2.0 FILE_FORMAT = 'ASCII' (COMMENTS) {comment_string} 'Tire - XXXXXX' 'Pressure - XXXXXX' 'Test Date - XXXXXX' 'Test tire' $-------------------------------------------------------------UNITS [UNITS] LENGTH = 'mm' FORCE = 'newton' ANGLE = 'radians' MASS = 'kg' TIME = 'sec' $-------------------------------------------------------------MODEL [MODEL] ! use mode 1 2 3 4 ! ------------------------------------------- ! smoothing X X ! combined X X ! PROPERTY_FILE_FORMAT = PAC89' USE_MODE = 4.0 TYRESIDE = 'LEFT' $-------------------------------------------------------------DIMENSION [DIMENSION] UNLOADED_RADIUS = 326.0 WIDTH = 245.0 ASPECT_RATIO = 0.35 $-------------------------------------------------------------PARAMETER [PARAMETER] VERTICAL_STIFFNESS = 310.0 VERTICAL_DAMPING = 3.1 LATERAL_STIFFNESS = 190.0 ROLLING_RESISTANCE = 0.0 $---------------------------------------------------------------------LOAD_CURVE $ For a non-linear tire vertical stiffness (optional) $ Maximum of 100 points [DEFLECTION_LOAD_CURVE] {pen fz} 0 0.0 1 212.0 2 428.0 3 648.0 5 1100.0 10 2300.0


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20 5000.0 30 8100.0 $-------------------------------------------------------------LATERAL_COEFFICIENTS [LATERAL_COEFFICIENTS] a0 = 1.65000 a1 = -34.0 a2 = 1250.00 a3 = 3036.00 a4 = 12.80 a5 = 0.00501 a6 = -0.02103 a7 = 0.77394 a8 = 0.0022890 a9 = 0.013442 a10 = 0.003709 a11 = 19.1656 a12 = 1.21356 a13 = 6.26206 $-------------------------------------------------------------longitudinal [LONGITUDINAL_COEFFICIENTS] b0 = 2.37272 b1 = -9.46000 b2 = 1490.00 b3 = 130.000 b4 = 276.000 b5 = 0.08860 b6 = 0.00402 b7 = -0.06150 b8 = 1.20000 b9 = 0.02990 b10 = -0.17600 $-------------------------------------------------------------aligning [ALIGNING_COEFFICIENTS] c0 = 2.34000 c1 = 1.4950 c2 = 6.416654 c3 = -3.57403 c4 = -0.087737 c5 = 0.098410 c6 = 0.0027699 c7 = -0.0001151 c8 = 0.1000 c9 = -1.33329 c10 = 0.025501 c11 = -0.02357 c12 = 0.03027 c13 = -0.0647 c14 = 0.0211329 c15 = 0.89469 c16 = -0.099443 c17 = -3.336941

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Learn about the Pacejka '94 handling force model:

• Using Correct Coordinate System and Units • Force and Moment Formulation for Pacejka ’94 • Example of Pacejka ’94 Property File


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Using Correct Coordinate System and Units in Pacejka '94

The test data and resulting coefficients that come from the Pacejka '94 tire model conform to the standard SAE tire coordinate system. The standard SAE coordinates are shown in SAE Tire Coordinate System. (See also About Coordinate Systems.) The corresponding sign conventions for Pacejka '94 are described next.

Note: The section [UNITS] in the tire property file does not apply to the Magic Formula coefficients.

Conventions for Naming Variables

Variable name and abbreviation:

Description:

Normal load Fz (kN)

Positive when the tire is penetrating the road.

Lateral force Fy (N)

Positive in a right turn. Negative in a left turn.

Longitudinal force

Fx (N) Positive during traction. Negative during braking.

Self-aligning torque

Mz (Nm)

Positive in a left turn. Negative in a right turn.

Inclination angle

(deg) Positive when the top of the tire tilts to the right (when viewing the tire from the rear).

Sideslip angle (deg)

Positive in a left turn.

Longitudinal slip (%) Negative in braking (-100%: wheel lock). Positive in traction.

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Force and Moment Formulation for Pacejka '94

Learn about:

• Longitudinal Force for Pacejka '94 • Lateral Force for Pacejka '94 • Self-Aligning Torque • Lateral Stiffness • Rolling Resistance • Smoothing

Longitudinal Force for Pacejka '94

C - Shape Factor

C=B0

D - Peak Factor

D=(B1*FZ2+B2*FZ) * DLON

BCD

BCD=((B3*FZ2+B4*FZ)*EXP(-B5*FZ)) * BCDLON


B=BCD/(C*D)

Horizontal Shift

Sh=B9*Fz+B10

Vertical Shift

Sv=B11*FZ+B12

Composite

X1=(k+Sh)

E Curvature Factor


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E=((B6*FZ+B7)*FZ+B8)*(1-(B13*SIGN(1,X1))))

FX Equation

FX=(D*SIN(C*ATAN(B*X1-E*(B*X1-ATAN(B*X1)))))+Sv

Parameters for Longitudinal Force


B0 Shape factor

B1, B2 Peak factor

B3, B4, B5 BCD calculation

B6, B7, B8, B13 Curvature factor

B9, B10 Horizontal shift

B11, B12 Vertical shift

DLON, BCDLON Scale factor

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Lateral Force for Pacejka '94

C - Shape Factor

C=A0

D - Peak Factor

D=((A1*FZ+A2) *(1-A15* 2)*FZ) * DLAT

BCD

BCD=(A3*SIN(ATAN(FZ/A4)*2.0)*(1-A5*ABS( )))* BCDLAT


B=BCD/(C*D)

Horizontal Shift

Sh=A8*FZ+A9+A10*

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Vertical Shift

Sv=A11*FZ+A12+(A13*FZ2+A14*FZ)*

Composite

X1=(a+Sh)


E=(A6*FZ+A7)*(1-(((A16*g)+A17)*SIGN(1,X1))))

FY Equation

FY=(D*SIN(C*ATAN(B*X1-E*(B*X1-ATAN(B*X1)))))+Sv

Parameters for Lateral Force


A0 Shape factor

A1, A2, A15 Peak factor

A3, A4, A5 BCD calculation

A6, A7, A16, A17 Curvature factor

A8, A9, A10 Horizontal shift

A11, A12, A13, A14 Vertical shift

DLAT, BCDLAT Scale factor

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Self-Aligning Torque for Pacejka '94

C - Shape Factor

C=C0

D - Peak Factor

D=(C1*FZ2+C2*FZ)*(1-C18* 2)

BCD


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BCD=(C3*FZ2+C4*FZ)*(1-(C6*ABS(g)))*EXP(-C5*FZ)


B=BCD/(C*D)

Horizontal Shift

Sh=C11*FZ+C12+C13*

Vertical Shift

Sv=C14*FZ+C15+(C16*FZ2+C17*FZ)*

Composite

X1=( +Sh)


E=(((C7*FZ2)+(C8*FZ)+C9)*(1-(((C19* )+C20)*SIGN(1,X1))))/(1-

(C10*ABS( )))

MZ Equation

MZ=(D*SIN(C*ATAN(B*X1-E*(B*X1-ATAN(B*X1)))))+Sv

Parameters for Self-Aligning Torque


C0 Shape factor

C1, C2, C18 Peak factor

C3, C4, C5, C6 BCD calculation

C7, C8, C9, C19, C20 Curvature factor

C11, C12, C13 Horizontal shift

C14, C15, C16, C17 Vertical shift

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Overturning Moment

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The lateral stiffness is used to calculate an approximate lateral deflection of the contact patch when there is a lateral force present:

deflection = Fy / lateral_stiffness

This deflection, in turn, is used to calculate an overturning moment due to the vertical force:

Mx (overturning moment) = -Fz * deflection

And an incremental aligning torque due to longtiudinal force (Fx):

Mz = Mz,Magic Formula + Fx * deflection

Here Mz,Magic Formula is the magic formula for aligning torque and Fx * deflection is the contribution due to the longitudinal force.

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Rolling Resistance

The rolling resistance moment My is opposite to the wheel angular velocity. The magnitude is given by:

My = Fz * Lrad * rolling_resistance

Where Fz equals the vertical force and Lrad is the tyre loaded radius. The rolling resistance coefficient can be entered in the tire property file:

[PARAMETER]

ROLLING_RESISTANCE = 0.01

A value of 0.01 will introduce a rolling resistance force, which is 1% of the vertical load.

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Smoothing

ADAMS/Tire smooths initial transients in the tire force over the first 0.1 seconds of simulation. The longitudinal force, lateral force, and aligning torque are multiplied by a cubic step function of time. (See STEP in the ADAMS/Solver online help.)


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Longitudinal Force

FLon = S*FLon

Lateral Force

FLat = S*FLat

Overturning Moment

Mx = S*Mx

Rolling resistance moment

My = S*My

Aligning Torque

Mz = S*Mz

The USE_MODE parameter in the tire property file allows you to switch smoothing on or off:

• USE_MODE = 1 or 2, smoothing is off • USE_MODE = 3 or 4, smoothing is on

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Example of Pacejka '94 Property File !:FILE_TYPE: tir !:FILE_VERSION: 2 !:TIRE_VERSION: PAC94 !:COMMENT: New File Format v2.1 !:FILE_FORMAT: ASCII !:TIMESTAMP: 1996/02/15,13:22:12 !:USER: ncos $--------------------------------------------------------------------------units [UNITS] LENGTH = 'inch' FORCE = 'pound_force' ANGLE = 'radians' MASS = 'pound_mass' TIME = 'second' $-------------------------------------------------------------------------model [MODEL] ! use mode 1 2 3 4 ! ------------------------------------------- ! smoothing X X ! combined X X ! ! USER_SUB_ID = 903 PROPERTY_FILE_FORMAT = 'PAC94' FUNCTION_NAME = 'TYR903' USE_MODE = 4 TYRESIDE = 'LEFT' $--------------------------------------------------------------------dimensions [DIMENSION] UNLOADED_RADIUS = 12.95 WIDTH = 10.0 ASPECT_RATIO = 0.30 $---------------------------------------------------------------------parameter [PARAMETER] VERTICAL_STIFFNESS = 2500 VERTICAL_DAMPING = 250.0 LATERAL_STIFFNESS = 1210.0 ROLLING_RESISTANCE = 0.01 $---------------------------------------------------------------------load_curve $ Maximum of 100 points (optional) [DEFLECTION_LOAD_CURVE] {pen fz} 0.000 0 0.039 943 0.079 1904


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0.118 2882 0.197 4893 0.394 10231 0.787 22241 1.181 36031 $-----------------------------------------------------------------------scaling [SCALING_COEFFICIENTS] DLAT = 0.10000E+01 DLON = 0.10000E+01 BCDLAT = 0.10000E+01 BCDLON = 0.10000E+01 $-----------------------------------------------------------------------lateral [LATERAL_COEFFICIENTS] A0 = 1.5535430E+00 A1 = -1.2854474E+01 A2 = -1.1133711E+03 A3 = -4.4104698E+03 A4 = -1.2518279E+01 A5 = -2.4000120E-03 A6 = 6.5642332E-02 A7 = 2.0865589E-01 A8 = -1.5717978E-02 A9 = 5.8287762E-02 A10 = -9.2761963E-02 A11 = 1.8649096E+01 A12 = -1.8642199E+02 A13 = 1.3462023E+00 A14 = -2.0845180E-01 A15 = 2.3183540E-03 A16 = 6.6483573E-01 A17 = 3.5017404E-01 $----------------------------------------------------------------longitudinal [LONGITUDINAL_COEFFICIENTS] B0 = 1.4900000E+00 B1 = -2.8808998E+01 B2 = -1.4016957E+03 B3 = 1.0133759E+02 B4 = -1.7259867E+02 B5 = -6.1757933E-02 B6 = 1.5667623E-02 B7 = 1.8554619E-01 B8 = 1.0000000E+00 B9 = 0.0000000E+00 B10 = 0.0000000E+00 B11 = 0.0000000E+00 B12 = 0.0000000E+00 B13 = 0.0000000E+00 $---------------------------------------------------------------------aligning [ALIGNING_COEFFICIENTS] C0 = 2.2300000E+00 C1 = 3.1552342E+00 C2 = -7.1338826E-01 C3 = 8.7134880E+00

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C4 = 1.3411892E+01 C5 = -1.0375348E-01 C6 = -5.0880786E-03 C7 = -1.3726071E-02 C8 = -1.0000000E-01 C9 = -6.1144302E-01 C10 = 3.6187314E-02 C11 = -2.3679781E-03 C12 = 1.7324400E-01 C13 = -1.7680388E-02 C14 = -3.4007351E-01 C15 = -1.6418691E+00 C16 = 4.1322424E-01 C17 = -2.3573702E-01 C18 = 6.0754417E-03 C19 = -4.2525059E-01 C20 = -2.1503067E-01 $--------------------------------------------------------------------------shape [SHAPE] {radial width} 1.0 0.0 1.0 0.2 1.0 0.4 1.0 0.5 1.0 0.6 1.0 0.7 1.0 0.8 1.0 0.85 1.0 0.9 0.9 1.0


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Combined Slip of Pacejka '89 and '94

The combined slip calculation of the Pacejka '89 and '94 tire models is identical. Note that the method employed here is not part of the Magic Formula as developed by Professor Pacejka, but is an in-house development of MSC.

Inputs:

• Dimensionless longitudinal slip κ (range –1 to 1) and side slip angle α in radians

• Longitudinal force Fx and lateral force Fy calculated using the Magic Formula

• Horizontal/vertical shifts and peak values of the Magic Formula (Sh, Sv, D)

Output:

• Adjusted longitudinal force Fx and lateral force Fy to incorporate the reduction due to combined slip:

Friction coefficients:

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Forces corrected for combined slip conditions:


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Contact Methods

The Pacejka '89 and '94 models use the following contact methods depending on the road model chosen:

Contact Methods Uses

For the road model: Uses the contact method:

2D Point-follower

3D 3D contact

About the Point-Follower Method

The point-follower contact method assumes a single contact point between the tire and road. The contact point is the point nearest to the wheel center that lies on the line formed by the intersection of the tire (wheel) plane with the local road plane.

The contact force computed by the point-follower contact method is normal to the road plane. Therefore, in a simulation of a tire hitting a pothole, the point-follower contact method does not generate the expected longitudinal force.

For more information about 2D roads, see Using the 2D Road Model.

About the 3D Contact Road Method

The 3D Contact road method determines an equivalent contact point and vertical deflection from the volume of intersection of the tire carcass with the road. The 3D Contact road method assumes the tire carcass is a cylinder, unless you input the tire carcass cross section in the [SHAPE] table of the tire property file. Triangular facets describe the road surface.

For more about 3D roads, see Using the 3D Contact Road Model.

using pacejka '89 and '94 models

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