Adams Car Package > Adams Tire > Tire Models > Using Pacejka '89 and '94 Models > Using Pacejka '94 Handling Force Model

Using Pacejka '94 Handling Force Model

Learn about the Pacejka '94 handling force model:

Coordinate System and Units in Pacejka '94

The equations for the force and moment calculation in the Pacejka ‘94 tire model follow the SAE coordinate system, the units follow the conventions as specified in the Table ‘Conventions for Naming Variables’ below. Note that the ‘Magic Formula’ parameters in the tire property file will also follow these conventions. However, the results from Adams Solver as presented in the Adams Postprocessor can adhere a different coordinate system or use different units. This depends on the type of requests that are being used, see the Adams Tire request definitions in section Outputting Results.
 
Note:  
The section [UNITS] in the tire property file does not apply to the Magic Formula coefficients.
Figure 2 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.
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
(degree)
Positive when the top of the tire tilts to the right (when viewing the tire from the rear).
Sideslip angle
(degree)
Positive in a left turn.
Longitudinal slip
(%)
Negative in braking (-100%: wheel lock).
Positive in traction.

Force and Moment Formulation for Pacejka '94

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 - Stiffness Factor
B=BCD/(C*D)
Horizontal Shift
Sh=B9*Fz+B10
Vertical Shift
Sv=B11*FZ+B12
Composite
X1=(κ+Sh)
E Curvature Factor
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
Parameters:
Description:
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

Lateral Force for Pacejka '94

C - Shape Factor
C=A0
D - Peak Factor
D=((A1*FZ+A2) *(1-A15*)*FZ) * DLAT
BCD
BCD=(A3*SIN(ATAN(FZ/A4)*2.0)*(1-A5*ABS()))* BCDLAT
B - Stiffness Factor
B=BCD/(C*D)
Horizontal Shift
Sh=A8*FZ+A9+A10*
Vertical Shift
Sv=A11*FZ+A12+(A13*FZ2+A14*FZ)*
Composite
X1=(α+Sh)
E - Curvature Factor
E=(A6*FZ+A7)*(1-(((A16*γ)+A17)*SIGN(1,X1))))
FY Equation
FY=(D*SIN(C*ATAN(B*X1-E*(B*X1-ATAN(B*X1)))))+Sv
Parameters for Lateral Force
Parameters:
Description:
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

Self-Aligning Torque for Pacejka '94

C - Shape Factor
C=C0
D - Peak Factor
D=(C1*FZ2+C2*FZ)*(1-C18*)
BCD
BCD=(C3*FZ2+C4*FZ)*(1-(C6*ABS(γ)))*EXP(-C5*FZ)
B - Stiffness Factor
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 - Curvature Factor
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
Parameters:
Description:
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

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.

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.

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.)
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

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 = 'radian'
MASS = 'pound_mass'
TIME = 'second'
$--------------------------------------------------------------model
[MODEL]
! use mode   1   2   3   4   11   12   13   14
! ---------------------------------------------------------------
! smoothing          X   X             X    X   
! combined       X       X         X        X
! transient                  X     X   X    X
!
PROPERTY_FILE_FORMAT = 'PAC94'
USE_MODE = 12.0
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
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
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