Using the PAC-TIME Tire Model
The PAC-TIME Magic-Formula tire model has been developed by MSC Software according to a publication, A New Tyre Model for TIME Measurement Data, by J.J.M. van Oosten e.a. [5]. PAC-TIME has improved equations for side force and aligning moment under pure slip conditions. For longitudinal pure slip and combined slip, the tire model is similar to PAC-TIME.
Learn about:
When to Use PAC-TIME
Magic-Formula (MF) tire models are considered the state-of-the-art for modeling tire-road interaction forces in vehicle dynamics applications. Since 1987, Pacejka and others have published several versions of this type of tire model. The PAC-TIME model is similar to PAC2002, but has improved equations for side force (Fy) and aligning moment (Mz) under pure side slip conditions.
The following is background information about the PAC-TIME tire model, as stated in the paper, A New Tyre Model for TIME Measurement Data, J.J.M. van Oosten, E. Kuiper, G. Leister, D. Bode, H. Schindler, J. Tischleder, S. Köhne [5]:
In 1999 a new method for tyre Force and Moment (F&M) testing has been developed by a consortium of European tyre and vehicle manufacturers: the TIME procedure. For Vehicle Dynamics studies often a Magic Formula (MF) tyre model is used based upon such F&M data. However when calculating MF parameters for a standard MF model out of the TIME F&M data, several difficulties are observed. These are mainly due to the non-uniform distribution of the data points over the slip angle, camber and load area and the mutual dependency in between the slip angle, camber and load. A new MF model for pure cornering slip conditions has been developed that allows the calculation of the MF parameters despite of the dependency of the three input variables in the F&M data and shows better agreement with the measured F&M data points. From mathematical point of view the optimisation process for deriving MF parameters is better conditioned with the new MF-TIME, resulting in less sensitivity to starting values and better convergence to a global minimum. In addition the MF-TIME has improved extrapolation performance compared to the standard MF models for areas where no F&M data points are available. Next to the use for TIME F&M data, the new model is expected to have interesting prospects for converting ‘on-vehicle’ measured tyre data into a robust set of MF parameters.
In general, an MF tire model describes the tire behavior for rather smooth roads (road obstacle wavelengths longer than the tire radius) up to frequencies of 8 Hz. This makes the tire model applicable for all generic vehicle handling and stability simulations, including:
■Steady-state cornering
■Single- or double-lane change
■Braking or power-off in a turn
■Split-mu braking tests
■J-turn or other turning maneuvers
■ABS braking, when stopping distance is important (not for tuning ABS control strategies)
■Other common vehicle dynamics maneuvers on rather smooth roads (wavelength of road obstacles must be longer than the tire radius)
For modeling roll-over of a vehicle, you must pay special attention to the overturning moment characteristics of the tire (Mx), and the loaded radius modeling. The last item may not be sufficiently addressed in this model.
The PAC-TIME model has been developed for car tires with camber (inclination) angles to the road not exceeding 15 degrees.
Modeling of Tire-Road Interaction Forces
For vehicle dynamics applications, an accurate knowledge of tire-road interaction forces is inevitable because the movements of a vehicle primarily depend on the road forces on the tires. These interaction forces depend on both road and tire properties, and the motion of the tire with respect to the road.
In the radial direction, the MF tire models consider the tire to behave as a parallel linear spring and linear damper with one point of contact with the road surface. The contact point is determined by considering the tire and wheel as a rigid disc. In the contact point between the tire and the road, the contact forces in longitudinal and lateral direction strongly depend on the slip between the tire patch elements and the road.
The figure, Input and Output Variables of the Magic Formula Tire Model, presents the input and output vectors of the PAC2002 tire model. The tire model subroutine is linked to the Adams Solver through the Standard Tire Interface (STI) [3]. The input through the STI consists of:
■Position and velocities of the wheel center
■Orientation of the wheel
■Tire model (MF) parameters
■Road parameters
The tire model routine calculates the vertical load and slip quantities based on the position and speed of the wheel with respect to the road. The input for the Magic Formula consists of the wheel load (Fz), the longitudinal and lateral slip (
, 
), and inclination angle (
) with the road. The output is the forces (Fx, Fy) and moments (Mx, My, Mz) in the contact point between the tire and the road. For calculating these forces, the MF equations use a set of MF parameters, which are derived from tire testing data.
, ), and inclination angle () with the road. The output is the forces (Fx, Fy) and moments (Mx, My, Mz) in the contact point between the tire and the road. For calculating these forces, the MF equations use a set of MF parameters, which are derived from tire testing data.The forces and moments out of the Magic Formula are transferred to the wheel center and returned to Adams Solver through STI.
Input and Output Variables of the Magic Formula Tire Model
Axis Systems and Slip Definitions
Axis Systems
The PAC-TIME model is linked to Adams Solver using the TYDEX STI conventions, as described in the TYDEX-Format [2] and the STI [3].
The STI interface between the MF-TIME model and Adams Solver mainly passes information to the tire model in the C-axis coordinate system. In the tire model itself, a conversion is made to the W-axis system because all the modeling of the tire behavior, as described in this help, assumes to deal with the slip quantities, orientation, forces, and moments in the contact point with the TYDEX W-axis system. Both axis systems have the ISO orientation but have different origin as can be seen in the figure below.
TYDEX C- and W-Axis Systems Used in PAC-TIME , Source [2]
The C-axis system is fixed to the wheel carrier with the longitudinal xc-axis parallel to the road and in the wheel plane (xc-zc-plane). The origin of the C-axis system is the wheel center.
The origin of the W-axis system is the road contact-point defined by the intersection of the wheel plane, the plane through the wheel carrier, and the road tangent plane.
The forces and moments calculated by PAC-TIME using the MF equations in this guide are in the W-axis system. A transformation is made in the source code to return the forces and moments through the STI to Adams Solver.
The inclination angle is defined as the angle between the wheel plane and the normal to the road tangent plane (xw-yw-plane).
Units
The units of information transferred through the STI between Adams Solver and PAC-TIME are according to the SI unit system. Also, the equations for PAC-TIME described in this guide have been developed for use with SI units, although you can easily switch to another unit system in your tire property file. Because of the non-dimensional parameters, only a few parameters have to be changed.
However, the parameters in the tire property file must always be valid for the TYDEX W-axis system (ISO oriented). The basic SI units are listed in the table below.
SI Units Used in PAC-TIME
Variable type: | Name: | Abbreviation: | Unit: |
|---|---|---|---|
Angle | Slip angle Inclination angle |   | Radian |
Force | Longitudinal force Lateral force Vertical load | Fx Fy Fz | Newton |
Moment | Overturning moment Rolling resistance moment Self-aligning moment | Mx My Mz | Newton.meter |
Speed | Longitudinal speed Lateral speed Longitudinal slip speed Lateral slip speed | Vx Vy Vsx Vsy | Meters per second |
Rotational speed | Tire rolling speed |  | Radian per second |
Definition of Tire Slip Quantities
Slip Quantities at Combined Cornering and Braking/Traction
The longitudinal slip velocity Vsx in the contact point (W-axis system, see Slip Quantities at Combined Cornering and Braking/Traction) is defined using the longitudinal speed Vx, the wheel rotational velocity 
, and the effective rolling radius Re:
, and the effective rolling radius Re: | (1) |
The lateral slip velocity is equal to the lateral speed in the contact point with respect to the road plane:
 | (2) |
The practical slip quantities 
(longitudinal slip) and 
(slip angle) are calculated with these slip velocities in the contact point with:
(longitudinal slip) and (slip angle) are calculated with these slip velocities in the contact point with: | (3) |
 | (4) |
The rolling speed Vr is determined using the effective rolling radius Re:
 | (5) |
Contact Methods and Normal Load Calculation
Contact Methods
The PAC-TIME tire model supports all Adams Tire contact methods.
■One Point Follower Contact, used by default for 2D Road, 3D Spline Road, OpenCRG Road and RGR Road.
■3D Equivalent Volume Contact, used by default for 3D Shell Road.
■3D Enveloping Contact, can be used with all road types when the keyword CONTACT_MODEL = '3D_ENVELOPING' is specified in the [MODEL] section of the tire property file.
In vertical direction, the PAC-TIME tire is modeled as a parallel spring and damper. The spring deflection and damper velocity are derived with the (effective) road height and plane information supplied by the contact method.
The normal load Fz of the tire is calculated with:
 | (6) |
where 
is the tire deflection and 
is the deflection rate of the tire.
is the tire deflection and is the deflection rate of the tire.Instead of the linear vertical tire stiffness Cz, you can also define an arbitrary tire deflection - load curve in the tire property file in the section [DEFLECTION_LOAD_CURVE] (see the Example of PAC-TIME Tire Property File). If a section called [DEFLECTION_LOAD_CURVE] exists, the load deflection data points 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 Cz in the tire property file, but it does not play any role.
Loaded and Effective Tire Rolling Radius
With the loaded rolling tire radius R defined as the distance of the wheel center to the contact point of the tire with the road, where 
is the deflection of the tire, and R0 is the free (unloaded) tire radius, then the loaded tire radius Re is:
is the deflection of the tire, and R0 is the free (unloaded) tire radius, then the loaded tire radius Re is: | (7) |
In this tire model, a constant (linear) vertical tire stiffness Cz is assumed; therefore, the tire deflection 
can be calculated using:
can be calculated using: | (8) |
The effective rolling radius Re (at free rolling of the tire), which is used to calculate the rotational speed of the tire, is defined by:
 | (9) |
For radial tires, the effective rolling radius is rather independent of load in its load range of operation because of the high stiffness of the tire belt circumference. Only at low loads does the effective tire radius decrease with increasing vertical load due to the tire tread thickness. See the figure below.
Effective Rolling Radius and Longitudinal Slip
To represent the effective rolling radius Re, an MF type of equation is used:
 | (10) |
in which 
is the nominal tire deflection:
is the nominal tire deflection: | (11) |
and 
is called the dimensionless radial tire deflection, defined by:
is called the dimensionless radial tire deflection, defined by: | (12) |
Effective Rolling Radius and Longitudinal Slip
Normal Load and Rolling Radius Parameters
Name: | Name Used in Tire Property File: | Explanation: |
|---|---|---|
Fz0 | FNOMIN | Nominal wheel load |
Ro | UNLOADED_RADIUS | Free tire radius |
B | BREFF | Low load stiffness effective rolling radius |
D | DREFF | Peak value of effective rolling radius |
F | FREFF | High load stiffness effective rolling radius |
Cz | VERTICAL_STIFFNESS | Tire vertical stiffness |
Kz | VERTICAL_DAMPING | Tire vertical damping |
Basics of the Magic Formula in PAC-TIME
The Magic Formula is a mathematical formula that is capable of describing the basic tire characteristics for the interaction forces between the tire and the road under several steady-state operating conditions. We distinguish:
■Pure cornering slip conditions: cornering with a free rolling tire
■Pure longitudinal slip conditions: braking or driving the tire without cornering
■Combined slip conditions: cornering and longitudinal slip simultaneously
For pure slip conditions, the lateral force Fy as a function of the lateral slip 
, respectively, and the longitudinal force Fx as a function of longitudinal slip 
, have a similar shape. Because of the sine - arctangent combination, the basic Magic Formula example is capable of describing this shape:
, respectively, and the longitudinal force Fx as a function of longitudinal slip , have a similar shape. Because of the sine - arctangent combination, the basic Magic Formula example is capable of describing this shape: | (13) |
where Y(x) is either Fx with x the longitudinal slip 
, or Fy and x the lateral slip 
.
, or Fy and x the lateral slip .Characteristic Curves for Fx and Fy Under Pure Slip Conditions
The self-aligning moment Mz is calculated as a product of the lateral force Fy and the pneumatic trail t added with the residual moment Mzr. In fact, the aligning moment is due to the offset of lateral force Fy, called pneumatic trail t, from the contact point. Because the pneumatic trail t as a function of the lateral slip 
has a cosine shape, a cosine version the Magic Formula is used:
has a cosine shape, a cosine version the Magic Formula is used: | (14) |
in which Y(x) is the pneumatic trail t as function of slip angle 
.
.The figure, The Magic Formula and the Meaning of Its Parameters, illustrates the functionality of the B, C, D, and E factor in the Magic Formula:
■D-factor determines the peak of the characteristic, and is called the peak factor.
■C-factor determines the part used of the sine and, therefore, mainly influences the shape of the curve (shape factor).
■B-factor stretches the curve and is called the stiffness factor.
■E-factor can modify the characteristic around the peak of the curve (curvature factor).
The Magic Formula and the Meaning of Its Parameters
In combined slip conditions, the lateral force Fy will decrease due to longitudinal slip or the opposite, the longitudinal force Fx will decrease due to lateral slip. The forces and moments in combined slip conditions are based on the pure slip characteristics multiplied by the so-called weighting functions. Again, these weighting functions have a cosine-shaped MF examples.
The Magic Formula itself only describes steady-state tire behavior. For transient tire behavior (up to 8 Hz), the MF output is used in a stretched string model that considers tire belt deflections instead of slip velocities to cope with standstill situations (zero speed).
Inclination Effects in the Lateral Force
From a historical point of view, the camber stiffness always has been modeled implicit in the Magic Formulas. For deriving coefficients of a Pacejka tire model usually so-called tire tests with slip angle sweeps at various values of constant load and inclination are performed. In the resulting Force & Moment measurement data, the effects of camber on the side force Fy are relatively small compared to side force effects by slip angle, which can easily result in non-realistic camber stiffness properties. Because there is no explicit definition of the camber stiffness, the effects on camber stiffness cannot be controlled in the coefficient optimization process.
The TIME measurement procedure guarantees more realistic tire test data, because they are performed under realistic tire operating conditions and specific parts of the test program concentrate on getting accurate cornering and camber stiffness. Because the inputs to the test program (side and longitudinal slip, inclination, and load) are not independent, for the parameter optimization process, a Pacejka tire model was required that has a better definition of cornering and camber stiffness from mathematical point of view (for a more detailed explanation, see [5]).
Therefore, the PAC-TIME tire model has an explicit definition of camber effects, similar to the tire model for motorcycle tires (PAC_MC). The basic Magic Formula sine function for the lateral force Fy has been extended with an argument for the inclination 
as follows:
as follows: | (15) |
In the PAC-TIME tire model, 
has been set to ½, and 
is not used (zero value). This approach results in an explicit definition of the camber stiffness, because:
has been set to ½, and is not used (zero value). This approach results in an explicit definition of the camber stiffness, because: | (16) |
Input Variables
The input variables to the Magic Formula are:
Input Variables
Longitudinal slip |  | [-] |
Slip angle |  | [rad] |
Inclination angle |  | [rad] |
Normal wheel load | Fz | [N] |
Output Variables
Its output variables are:
Output Variables.
Longitudinal force | Fx | [N] |
Lateral force | Fy | [N] |
Overturning couple | Mx | [Nm] |
Rolling resistance moment | My | [Nm] |
Aligning moment | Mz | [Nm] |
The output variables are defined in the W-axis system of TYDEX.
Basic Tire Parameters
All tire model parameters of the model are without dimension. The reference parameters for the model are:
Basic Tire Parameters
Nominal (rated) load | Fz0 | [N] |
Unloaded tire radius | R0 | [m] |
Tire belt mass | mbelt | [kg] |
As a measure for the vertical load, the normalized vertical load increment dfz is used:
 | (17) |
with the possibly adapted nominal load (using the user-scaling factor, 
):
): | (18) |
Nomenclature of the Tire Model Parameters
In the subsequent sections, formulas are given with non-dimensional parameters aijk with the following logic:
Tire Model Parameters
Parameter: | Definition: | |
|---|---|---|
a = | p | Force at pure slip |
q | Moment at pure slip | |
r | Force at combined slip | |
s | Moment at combined slip | |
i = | B | Stiffness factor |
C | Shape factor | |
D | Peak value | |
E | Curvature factor | |
K | Slip stiffness = BCD | |
H | Horizontal shift | |
V | Vertical shift | |
s | Moment at combined slip | |
t | Transient tire behavior | |
j = | x | Along the longitudinal axis |
y | Along the lateral axis | |
z | About the vertical axis | |
k = | 1, 2, ... | |
User Scaling Factors
A set of scaling factors is available to easily examine the influence of changing tire properties without the need to change one of the real Magic Formula coefficients. The default value of these factors is 1. You can change the factors in the tire property file. The peak friction scaling factors, factors, 
and 
', are also used for the position-dependent friction in 3D Road Contact and Adams 3D Road. An overview of all scaling factors is shown in the next tables.
and ', are also used for the position-dependent friction in 3D Road Contact and Adams 3D Road. An overview of all scaling factors is shown in the next tables.Scaling Factor Coefficients for Pure Slip
Name: | Name used in tire property file: | Explanation: |
|---|---|---|
 | LFZO | Scale factor of nominal (rated) load |
 | LCX | Scale factor of Fx shape factor |
 | LMUX | Scale factor of Fx peak friction coefficient |
 | LEX | Scale factor of Fx curvature factor |
 | LKX | Scale factor of Fx slip stiffness |
 | LVX | Scale factor of Fx vertical shift |
 | LHX | Scale factor of Fx horizontal shift |
 | LGAX | Scale factor of camber for Fx |
 | LCY | Scale factor of Fy shape factor for side slip |
 | LMUY | Scale factor of Fy peak friction coefficient |
 | LEY | Scale factor of Fy curvature factor |
 | LKY | Scale factor of Fy cornering stiffness |
 | LVY | Scale factor of Fy vertical shift |
 | LHY | Scale factor of Fy horizontal shift |
 | LKC | Scale factor of camber stiffness (K-factor) |
 | LGAY | Scale factor of camber force stiffness |
 | LTR | Scale factor of peak of pneumatic trail |
 | LRES | Scale factor for offset of residual torque |
 | LGAZ | Scale factor of camber torque stiffness |
 | LMX | Scale factor of overturning couple |
 | LVMX | Scale factor of Mx vertical shift |
 | LMY | Scale factor of rolling resistance torque |
Scaling Factor Coefficients for Combined Slip
Name: | Name used in tire property file: | Explanation: |
|---|---|---|
 | LXAL | Scale factor of alpha influence on Fx |
 | LYKA | Scale factor of alpha influence on Fx |
 | LVYKA | Scale factor of kappa induced Fy |
 | LS | Scale factor of moment arm of Fx |
Scaling Factor Coefficients for Transient Response
Name: | Name used in tire property file: | Explanation: |
|---|---|---|
 | LSGKP | Scale factor of relaxation length of Fx |
 | LSGAL | Scale factor of relaxation length of Fy |
 | LGYR | Scale factor of gyroscopic moment |
Steady-State: Magic Formula in PAC-TIME
Steady-State Pure Slip
Formulas for the Longitudinal Force at Pure Slip
For the tire rolling on a straight line with no slip angle, the formulas are:
 | (19) |
 | (20) |
 | (21) |
 | (22) |
with following coefficients:
 | (23) |
 | (24) |
 | (25) |
 | (26) |
the longitudinal slip stiffness:
 | (27) |
 | (28) |
 | (29) |
Longitudinal Force Coefficients at Pure Slip
Name: | Name used in tire property file: | Explanation: |
|---|---|---|
pCx1 | PCX1 | Shape factor Cfx for longitudinal force |
pDx1 | PDX1 | Longitudinal friction Mux at Fznom |
pDx2 | PDX2 | Variation of friction Mux with load |
pDx3 | PDX3 | Variation of friction Mux with inclination |
pEx1 | PEX1 | Longitudinal curvature Efx at Fznom |
pEx2 | PEX2 | Variation of curvature Efx with load |
pEx3 | PEX3 | Variation of curvature Efx with load squared |
pEx4 | PEX4 | Factor in curvature Efx while driving |
pKx1 | PKX1 | Longitudinal slip stiffness Kfx/Fz at Fznom |
pKx2 | PKX2 | Variation of slip stiffness Kfx/Fz with load |
Formulas for the Lateral Force at Pure Slip
 | (30) |
 | (31) |
 | (32) |
The scaled inclination angle:
 | (33) |
with coefficients:
 | (34) |
 | (35) |
 | (36) |
 | (37) |
The cornering stiffness:
 | (38) |
 | (39) |
 | (40) |
 | (41) |
 | (42) |
 | (43) |
Lateral Force Coefficients at Pure Slip
Name: | Name used in tire property file: | Explanation: |
|---|---|---|
pCy1 | PCY1 | Shape factor Cfy for lateral forces |
pDy1 | PDY1 | Lateral friction Muy |
pDy2 | PDY2 | Variation of friction Muy with load |
pDy3 | PDY3 | Variation of friction Muy with squared inclination |
pEy1 | PEY1 | Lateral curvature Efy at Fznom |
pEy2 | PEY2 | Variation of curvature Efy with load |
pEy3 | PEY3 | Inclination dependency of curvature Efy |
pEy4 | PEY4 | Variation of curvature Efy with inclination |
pKy1 | PKY1 | Maximum value of stiffness Kfy/Fznom |
pKy2 | PKY2 | Load at which Kfy reaches maximum value |
pKy3 | PKY3 | Variation of Kfy/Fznom with inclination |
pKy4 | PKY4 | Shape factor of Kfy |
pKy5 | PKY5 | Linear variation of Kγ with load |
pKy6 | PKY6 | Quadratic variation of Kγ with load |
pHy1 | PHY1 | Horizontal shift Shy at Fznom |
pHy2 | PHY2 | Variation of shift Shy with load |
pVy1 | PVY1 | Vertical shift in Svy/Fz at Fznom |
pVy2 | PVY2 | Variation of shift Svy/Fz with load |
Formulas for the Aligning Moment at Pure Slip
 | (44) |
with the pneumatic trail t:
 | (45) |
and the residual moment Mzr:
 | (46) |
 | (47) |
 | (48) |
The scaled inclination angle:
 | (49) |
with coefficients:
 | (50) |
 | (51) |
 | (52) |
 | (53) |
 | (54) |
 | (55) |
 | (56) |
An approximation for the aligning moment stiffness reads:
 | (57) |
and the aligning stiffness for inclination is:
 | (58) |
Aligning Moment Coefficients at Pure Slip
Name: | Name used in tire property file: | Explanation: |
|---|---|---|
qBz1 | QBZ1 | Trail slope factor for trail Bpt at Fznom |
qBz2 | QBZ2 | Variation of slope Bpt with load |
qBz4 | QBZ4 | Variation of slope Bpt with inclination |
qBz5 | QBZ5 | Variation of slope Bpt with absolute inclination |
qCz1 | QCZ1 | Shape factor Cpt for pneumatic trail |
qDz1 | QDZ1 | Peak trail Dpt = Dpt*(Fz/Fznom*R0) |
qDz2 | QDZ2 | Variation of peak Dpt with load |
qDz6 | QDZ6 | Peak residual moment Dmr = Dmr/ (Fz*R0) |
qDz7 | QDZ7 | Variation of peak factor Dmr with load |
qDz8 | QDZ8 | Variation of peak factor Dmr with inclination |
qDz9 | QDZ9 | Variation of Dmr with inclination and load |
qEz1 | QEZ1 | Trail curvature Ept at Fznom |
qEz2 | QEZ2 | Variation of curvature Ept with load |
qEz4 | QEZ4 | Variation of curvature Ept with sign of Alpha-t |
qHz1 | QHZ1 | Trail horizontal shift Sht at Fznom |
qHz2 | QHZ2 | Variation of shift Sht with load |
qHz3 | QHZ3 | Variation of shift Sht with inclination |
qHz4 | QHZ4 | Variation of shift Sht with inclination and load |
Steady-State Combined Slip
PAC-TIME has two methods for calculating the combined slip forces and moments. If the user supplies the coefficients for the combined slip cosine 'weighing' functions, the combined slip is calculated according to Combined slip with cosine 'weighing' functions (standard method). If no coefficients are supplied, the so-called friction ellipse is used to estimate the combined slip forces and moments, see section Combined Slip with friction ellipse
Combined slip with cosine 'weighing' functions
Formulas for the Longitudinal Force at Combined Slip
 | (59) |
with 
the weighting function of the longitudinal force for pure slip.
the weighting function of the longitudinal force for pure slip.We write:
 | (60) |
 | (61) |
with coefficients:
 | (62) |
 | (63) |
 | (64) |
 | (65) |
 | (66) |
The weighting function follows as:
 | (67) |
Longitudinal Force Coefficients at Combined Slip
Name: | Name used in tire property file: | Explanation: |
|---|---|---|
rBx1 | RBX1 | Slope factor for combined slip Fx reduction |
rBx2 | RBX2 | Variation of slope Fx reduction with kappa |
rCx1 | RCX1 | Shape factor for combined slip Fx reduction |
rEx1 | REX1 | Curvature factor of combined Fx |
rEx2 | REX2 | Curvature factor of combined Fx with load |
rHx1 | RHX1 | Shift factor for combined slip Fx reduction |
Formulas for Lateral Force at Combined Slip
 | (68) |
with Gyk the weighting function for the lateral force at pure slip and SVyk the '
-induced' side force; therefore, the lateral force can be written as:
-induced' side force; therefore, the lateral force can be written as: | (69) |
 | (70) |
with the coefficients:
 | (71) |
 | (72) |
 | (73) |
 | (74) |
 | (75) |
 | (76) |
The weighting function appears is defined as:
 | (77) |
Lateral Force Coefficients at Combined Slip
Name: | Name used in tire property file: | Explanation: |
|---|---|---|
rBy1 | RBY1 | Slope factor for combined Fy reduction |
rBy2 | RBY2 | Variation of slope Fy reduction with alpha |
rBy3 | RBY3 | Shift term for alpha in slope Fy reduction |
rCy1 | RCY1 | Shape factor for combined Fy reduction |
rEy1 | REY1 | Curvature factor of combined Fy |
rEy2 | REY2 | Curvature factor of combined Fy with load |
rHy1 | RHY1 | Shift factor for combined Fy reduction |
rHy2 | RHY2 | Shift factor for combined Fy reduction with load |
rVy1 | RVY1 | Kappa induced side force SVyk/μy·Fz at Fznom |
rVy2 | RVY2 | Variation of SVyk/μy·Fz with load |
rVy3 | RVY3 | Variation of SVyk/μy·Fz with inclination |
rVy4 | RVY4 | Variation of SVyk/μy·Fz with α |
rVy5 | RVY5 | Variation of SVyk/μy·Fz with κ |
rVy6 | RVY6 | Variation of SVyk/μy·Fz with atan(κ) |
Formulas for Aligning Moment at Combined Slip
 | (78) |
with:
 | (79) |
 | (80) |
 | (81) |
 | (82) |
with the arguments:
 | (83) |
 | (84) |
Aligning Moment Coefficients at Combined Slip
Name: | Name used in tire property file: | Explanation: |
|---|---|---|
ssz1 | SSZ1 | Nominal value of s/R0 effect of Fx on Mz |
ssz2 | SSZ2 | Variation of distance s/R0 with Fy/Fznom |
ssz3 | SSZ3 | Variation of distance s/R0 with inclination |
ssz4 | SSZ4 | Variation of distance s/R0 with load and inclination |
Formulas for Overturning Moment at Pure and Combined Slip
For the overturning moment, the formula reads both for pure and combined slip situations:
 | (85) |
Overturning Moment Coefficients
Name: | Name used in tire property file: | Explanation: |
|---|---|---|
qsx1 | QSX1 | Vertical offset overturning couple |
qsx2 | QSX2 | Inclination induced overturning couple |
qsx3 | QSX3 | Fy induced overturning couple |
Formulas for Rolling Resistance Moment at Pure and Combined Slip
The rolling resistance moment is defined by:
 | (86) |
Rolling Resistance Coefficients
Name: | Name used in tire property file: | Explanation: |
|---|---|---|
qsy1 | QSY1 | Rolling resistance moment coefficient |
qsy2 | QSY2 | Rolling resistance moment depending on Fx |
qsy3 | QSY3 | Rolling resistance moment depending on speed |
qsy4 | QSY4 | Rolling resistance moment depending on speed^4 |
Vref | LONGVL | Measurement speed |
Combined Slip with friction ellipse
In case the tire property file does not contain the coefficients for the 'standard' combined slip method (cosine 'weighing functions), the friction ellipse method is used, as described in this section. Note that the method employed here is not part of one of the Magic Formula publications by Pacejka, but is an in-house development of MSC Software.
The following friction coefficients are defined:
The forces corrected for the combined slip conditions are:
For aligning moment Mx, rolling resistance My and aligning moment Mz the formulae (76) until and including (84) are used with 
.
.Transient Behavior in PAC-TIME
The previous Magic Formula equations are valid for steady-state tire behavior. When driving, however, the tire requires some response time on changes of the inputs. In tire modeling terminology, the low-frequency behavior (up to 8 Hz) is called transient behavior.
For estimating this transient tire behavior, a linear transient model is used as described in [1].
In this linear transient model the tire contact point S' is suspended to the wheel-rim plane with a longitudinal and lateral spring, with respectively stiffness's CFx and CFy., see Reference. In the figure below a top view of the tire with the single contact point S' and the longitudinal (u) and lateral (v) carcass deflections is shown.
The contact point may move with respect to the wheel-rim plane and road. Movements relative to the road will result in tire-road interaction forces. Differences in slip velocities at point S and point S' will result in the tire carcass to deflect. The change of the longitudinal deflection u can be defined as:
 | (87) |
and the lateral deflection v as:
 | (88) |
For small values of slip the side force Fy can be calculated using the cornering stiffness CFα as follows:
 | (89) |
While the lateral force on the carcass reads:
 | (90) |
When introducing the lateral relaxation length σα as:
 | (91) |
the differential equation for the lateral deflection can be written as follows:
 | (92) |
For linear small slip we can define the practical slip quantity α' as:
 | (93) |
With α' the equation for the lateral deflection becomes:
 | (94) |
Similar the differential equation for longitudinal direction with the longitudinal relaxation length σκ can be derived:
 | (95) |
with the practical slip quantity 
 | (96) |
Both the longitudinal and lateral relaxation length are defined as of the vertical load:
 | (97) |
 | (98) |
Using these practical slip quantities, 
and 
, the Magic Formula equations can be used to calculate the tire-road interaction forces and moments:
and , the Magic Formula equations can be used to calculate the tire-road interaction forces and moments: | (99) |
 | (100) |
 | (101) |
 | (102) |
 | (103) |
With this linear transient model the effective lateral compliance of the tire at stand-still is
 | (104) |
And similar in longitudinal direction the compliance is:
 | (105) |
Reference
1. H.B. Pacejka, Tyre and Vehicle Dynamics, 2002, Butterworth-Heinemann, ISBN 0 7506 5141.
Gyroscopic Couple in PAC-TIME
When having fast rotations about the vertical axis in the wheel plane, the inertia of the tire belt may lead to gyroscopic effects. To cope with this additional moment, the following contribution is added to the total aligning moment:
 | (106) |
with the parameters (in addition to the basic tire parameter mbelt):
 | (107) |
and:
 | (108) |
The total aligning moment now becomes:
Coefficients and Transient Response
Name: | Name used in tire property file: | Explanation: |
|---|---|---|
pTx1 | PTX1 | Longitudinal relaxation length at Fznom |
pTx2 | PTX2 | Variation of longitudinal relaxation length with load |
pTx3 | PTX3 | Variation of longitudinal relaxation length with exponent of load |
pTy1 | PTY1 | Peak value of relaxation length for lateral direction |
pTy2 | PTY2 | Shape factor for lateral relaxation length |
qTz1 | QTZ1 | Gyroscopic moment constant |
Mbelt | MBELT | Belt mass of the wheel |
Left and Right Side Tires
In general, a tire produces a lateral force and aligning moment at zero slip angle due to the tire construction, known as conicity and plysteer. In addition, the tire characteristics cannot be symmetric for positive and negative slip angles.
A tire property file with the parameters for the model results from testing with a tire that is mounted in a tire test bench comparable either to the left or the right side of a vehicle. If these coefficients are used for both the left and the right side of the vehicle model, the vehicle does not drive straight at zero steering wheel angle.
The latest versions of tire property files contain a keyword TYRESIDE in the [MODEL] section that indicates for which side of the vehicle the tire parameters in that file are valid (TIRESIDE = 'LEFT' or TIRESIDE = 'RIGHT').
If this keyword is available, Adams Car corrects for the conicity, plysteer, and asymmetry when using a tire property file on the opposite side of the vehicle. In fact, the tire characteristics are mirrored with respect to slip angle zero. In Adams View, this option can only be used when the tire is generated by the graphical user interface: select Build Forces Special Force: Tire, as shown in the figure below.
Create Wheel and Tire Dialog Box in Adams View
Next to the LEFT and RIGHT side option of TYRESIDE, you can also set SYMMETRIC: then the tire characteristics are modified during initialization to show symmetric performance for left and right side corners and zero conicity and plysteer (no offsets). Also, when you set the tire property file to SYMMETRIC, the tire characteristics are changed to symmetric behavior.
USE_MODES of PAC-TIME: from Simple to Complex
The parameter USE_MODE in the tire property file allows you to switch the output of the PAC-TIME tire model from very simple (that is, steady-state cornering) to complex (transient combined cornering and braking).
The options for the USE_MODE and the output of the model have been listed in the table below.
USE_MODE Values of PAC-TIME and Related Tire Model Output
USE_MODE: | State: | Slip conditions: | PAC-TIME output (forces and moments): |
|---|---|---|---|
0 | Steady state | Acts as a vertical spring and damper | 0, 0, Fz, 0, 0, 0 |
1 | Steady state | Pure longitudinal slip | Fx, 0, Fz, 0, My, 0 |
2 | Steady state | Pure lateral (cornering) slip | 0, Fy, Fz, Mx, 0, Mz |
3 | Steady state | Longitudinal and lateral (not combined) | Fx, Fy, Fz, Mx, My, Mz |
4 | Steady state | Combined slip | Fx, Fy, Fz, Mx, My, Mz |
11 | Transient | Pure longitudinal slip | Fx, 0, Fz, 0, My, 0 |
12 | Transient | Pure lateral (cornering) slip | 0, Fy, Fz, Mx, 0, Mz |
13 | Transient | Longitudinal and lateral (not combined) | Fx, Fy, Fz, Mx, My, Mz |
14 | Transient | Combined slip | Fx, Fy, Fz, Mx, My, Mz |
Quality Checks for the Tire Model Parameters
Because PAC-TIME uses an empirical approach to describe tire - road interaction forces, incorrect parameters can easily result in non-realistic tire behavior. Below is a list of the most important items to ensure the quality of the parameters in a tire property file:
Note: | Do not change Fz0 (FNOMIN) and R0 (UNLOADED_RADIUS) in your tire property file. It will change the complete tire characteristics because these two parameters are used to make all parameters without dimension. |
Rolling Resistance
For a realistic rolling resistance, the parameter qsy1 must be positive. For car tires, it can be in the order of 0.006 - 0.01 (0.6% - 1.0%).
$---------------------------------------------------rolling resistance
[ROLLING_COEFFICIENTS]
QSY1 = 0.01
QSY2 = 0
QSY3 = 0
QSY4 = 0
Camber (Inclination) Effects
Camber stiffness has been explicitly defined in PAC-TIME, so camber stiffness can be easily checked by the tire model parameters itself, see the table, Checklist for PAC-TIME Parameters and Properties, below. For car tires, positive inclination should result in a negative lateral force at zero slip angle (see Effect of Positive Camber on the Lateral Force in TYDEX W-axis (ISO) System below). If positive inclination results in an increase of the lateral force, the coefficient may not be valid for the ISO, but for the SAE coordinate system. Note that PAC-TIME only uses coefficients for the TYDEX W-axis (ISO) system.
Effect of Positive Camber on the Lateral Force in TYDEX W-axis (ISO) System
The following table lists further checks on the PAC-TIME parameters.
Checklist for PAC-TIME Parameters and Properties
Parameter/property: | Requirement: | Explanation: |
|---|---|---|
LONGVL | 1 m/s | Reference velocity at which parameters are measured |
VXLOW | Approximately 1 m/s | Threshold for scaling down forces and moments |
Dx | > 0 | Peak friction (see equation (24)) |
pDx1/pDx2 | < 0 | Peak friction Fx must decrease with increasing load |
Kx | > 0 | Long slip stiffness (see equation (27)) |
Dy | > 0 | Peak friction (see equation (35)) |
pDy1/pDy2 | < 0 | Peak friction Fx must decrease with increasing load |
Ky | < 0 | Cornering stiffness (see equation (38)) |
 | < 0 | Camber stiffness (see equation (39)) |
qsy1 | > 0 | Rolling resistance, in the range of 0.005 - 0.015 |
Validity Range of the Tire Model Input
In the tire property file, a range of the input variables has been given in which the tire properties are supposed to be valid. These validity range parameters are (the listed values can be different):
$-----------------------------------------------------long_slip_range
[LONG_SLIP_RANGE]
KPUMIN = -1.5 $Minimum valid wheel slip
KPUMAX = 1.5 $Maximum valid wheel slip
$----------------------------------------------------slip_angle_range
[SLIP_ANGLE_RANGE]
ALPMIN = -1.5708 $Minimum valid slip angle
ALPMAX = 1.5708 $Maximum valid slip angle
$----------------------------------------------inclination_slip_range
[INCLINATION_ANGLE_RANGE]
CAMMIN = -0.26181 $Minimum valid camber angle
CAMMAX = 0.26181 $Maximum valid camber angle
$------------------------------------------------vertical_force_range
[VERTICAL_FORCE_RANGE]
FZMIN = 225 $Minimum allowed wheel load
FZMAX = 10125 $Maximum allowed wheel load
If one of the input parameters exceeds a minimum or maximum validity value, the calculation in the tire model is performed with the minimum or maximum value of this range to avoid non-realistic tire behavior. In that case, a message appears warning you that one of the inputs exceeds a validity value.
Standard Tire Interface (STI) for PAC-TIME
Because all Adams products use the Standard Tire Interface (STI) for linking the tire models to Adams Solver, below is a brief background of the STI history (see reference [4]).
At the First International Colloquium on Tire Models for Vehicle Dynamics Analysis on October 21-22, 1991, the International Tire Workshop working group was established (TYDEX).
The working group concentrated on tire measurements and tire models used for vehicle simulation purposes. For most vehicle dynamics studies, people previously developed their own tire models. Because all car manufacturers and their tire suppliers have the same goal (that is, development of tires to improve dynamic safety of the vehicle), it aimed for standardization in the tire behavior description.
In TYDEX, two expert groups, consisting of participants of vehicle industry (passenger cars and trucks), tire manufacturers, other suppliers and research laboratories, had been defined with following goals:
■The first expert group's (Tire Measurements - Tire Modeling) main goal was to specify an interface between tire measurements and tire models. The result was the TYDEX-Format [2] to describe tire measurement data.
■The second expert group's (Tire Modeling - Vehicle Modeling) main goal was to specify an interface between tire models and simulation tools, which resulted in the Standard Tire Interface (STI) [3]. The use of this interface should ensure that a wide range of simulation software can be linked to a wide range of tire modeling software.
Definitions
General
General Definitions
Term: | Definition: |
|---|---|
Road tangent plane | Plane with the normal unit vector (tangent to the road) in the tire-road contact point C. |
C-axis system | Coordinate system mounted on the wheel carrier at the wheel center according to TYDEX, ISO orientation. |
Wheel plane | The plane in the wheel center that is formed by the wheel when considered a rigid disc with zero width. |
Contact point C | Contact point between tire and road, defined as the intersection of the wheel plane and the projection of the wheel axis onto the road plane. |
W-axis system | Coordinate system at the tire contact point C, according to TYDEX, ISO orientation. |
Tire Kinematics
Tire Kinematics Definitions
Parameter: | Definition: | Units: |
|---|---|---|
R0 | Unloaded tire radius | [m] |
R | Loaded tire radius | [m] |
Re | Effective tire radius | [m] |
 | Radial tire deflection | [m] |
 d | Dimensionless radial tire deflection | [-] |
 Fz0 | Radial tire deflection at nominal load | [m] |
mbelt | Tire belt mass | [kg] |
 | Rotational velocity of the wheel | [radian-1] |
Slip Quantities
Slip Quantities Definitions
Parameter: | Definition: | Units: |
|---|---|---|
V | Vehicle speed | [ms-1] |
Vsx | Slip speed in x direction | [ms-1] |
Vsy | Slip speed in y direction | [ms-1] |
Vs | Resulting slip speed | [ms-1] |
Vx | Rolling speed in x direction | [ms-1] |
Vy | Lateral speed of tire contact center | [ms-1] |
Vr | Linear speed of rolling | [ms-1] |
 | Longitudinal slip | [-] |
 | Slip angle | [rad] |
 | Inclination angle | [rad] |
Forces and Moments
Force and Moment Definitions
Abbreviation: | Definition: | Units: |
|---|---|---|
Fz | Vertical wheel load | [N] |
Fz0 | Nominal load | [N] |
dfz | Dimensionless vertical load | [-] |
Fx | Longitudinal force | [N] |
Fy | Lateral force | [N] |
Mx | Overturning moment | [Nm] |
My | Braking/driving moment | [Nm] |
Mz | Aligning moment | [Nm] |
References
1. H.B. Pacejka, Tyre and Vehicle Dynamics, 2002, Butterworth-Heinemann, ISBN 0 7506 5141 5.
2. H.-J. Unrau, J. Zamow, TYDEX-Format, Description and Reference Manual, Release 1.1, Initiated by the International Tire Working Group, July 1995.
3. A. Riedel, Standard Tire Interface, Release 1.2, Initiated by the Tire Workgroup, June 1995.
4. J.J.M. van Oosten, H.-J. Unrau, G. Riedel, E. Bakker, TYDEX Workshop: Standardisation of Data Exchange in Tyre Testing and Tyre Modelling, Proceedings of the 2nd International Colloquium on Tyre Models for Vehicle Dynamics Analysis, Vehicle System Dynamics, Volume 27, Swets & Zeitlinger, Amsterdam/Lisse, 1996.
5. J.J.M. van Oosten, E. Kuiper, G. Leister, D. Bode, H. Schindler, J. Tischleder, S. Köhne, A new tyre model for TIME measurement data,Tire Technology Expo 2003, Hannover.
Example of PAC-TIME Tire Property File
[MDI_HEADER]
FILE_TYPE ='tir'
FILE_VERSION =3.0
FILE_FORMAT ='ASCII'
! : TIRE_VERSION : MF-TIME
! : COMMENT : Tire 205/55 R16
! : COMMENT : Manufacturer
! : COMMENT : Nom. section with (m) 0.205
! : COMMENT : Nom. aspect ratio (-) 55
! : COMMENT : Infl. pressure (Pa) 250000
! : COMMENT : Rim radius (m) 0.2032
! : COMMENT : Measurement ID
! : COMMENT : Test speed (m/s) 11.11
! : COMMENT : Road surface
! : COMMENT : Road condition Dry
! : FILE_FORMAT : ASCII
! : Copyright (C) 2004-2011 MSC Software Corporation
!
! USE_MODE specifies the type of calculation performed:
! 0: Fz only, no Magic Formula evaluation
! 1: Fx,My only
! 2: Fy,Mx,Mz only
! 3: Fx,Fy,Mx,My,Mz uncombined force/moment calculation
! 4: Fx,Fy,Mx,My,Mz combined force/moment calculation
! +10: including relaxation behaviour
! *-1: mirroring of tyre characteristics
!
! example: USE_MODE = -12 implies:
! -calculation of Fy,Mx,Mz only
! -including relaxation effects
! -mirrored tyre characteristics
!
$---------------------------------------------------------------units
[UNITS]
LENGTH ='meter'
FORCE ='newton'
ANGLE ='radian'
MASS ='kg'
TIME ='second'
$---------------------------------------------------------------model
[MODEL]
PROPERTY_FILE_FORMAT ='PAC-TIME'
USE_MODE = 14 $Tyre use switch (IUSED)
VXLOW = 2
LONGVL = 30 $Measurement speed
TYRESIDE = 'LEFT' $Mounted side of tyre at vehicle/test bench
$----------------------------------------------------------dimensions
[DIMENSION]
UNLOADED_RADIUS = 0.317 $Free tyre radius
WIDTH = 0.205 $Nominal section width of the tyre
ASPECT_RATIO = 0.55 $Nominal aspect ratio
RIM_RADIUS = 0.203 $Nominal rim radius
RIM_WIDTH = 0.165 $Rim width
$---------------------------------------------------------------shape
[SHAPE]
{radial width}
1.0 0.0
1.0 0.4
1.0 0.9
0.9 1.0
$----------------------------------------------------------load_curve
$ For a non-linear tire vertical stiffness
$ Maximum of 100 points
[DEFLECTION_LOAD_CURVE]
{pen fz}
0.000 0.0
0.001 212.0
0.002 428.0
0.003 648.0
0.005 1100.0
0.010 2300.0
0.020 5000.0
0.030 8100.0
$-----------------------------------------------------------parameter
[VERTICAL]
VERTICAL_STIFFNESS = 2.648e+005 $Tyre vertical stiffness
VERTICAL_DAMPING = 500 $Tyre vertical damping
BREFF = 4.90 $Low load stiffness e.r.r.
DREFF = 0.41 $Peak value of e.r.r.
FREFF = 0.09 $High load stiffness e.r.r.
FNOMIN = 4704 $Nominal wheel load
$-----------------------------------------------------long_slip_range
[LONG_SLIP_RANGE]
KPUMIN = -1.5 $Minimum valid wheel slip
KPUMAX = 1.5 $Maximum valid wheel slip
$----------------------------------------------------slip_angle_range
[SLIP_ANGLE_RANGE]
ALPMIN = -1.5708 $Minimum valid slip angle
ALPMAX = 1.5708 $Maximum valid slip angle
$----------------------------------------------inclination_slip_range
[INCLINATION_ANGLE_RANGE]
CAMMIN = -0.26181 $Minimum valid camber angle
CAMMAX = 0.26181 $Maximum valid camber angle
$------------------------------------------------vertical_force_range
[VERTICAL_FORCE_RANGE]
FZMIN = 140 $Minimum allowed wheel load
FZMAX = 10800 $Maximum allowed wheel load
$-------------------------------------------------------------scaling
[SCALING_COEFFICIENTS]
LFZO = 1 $Scale factor of nominal load
LCX = 1 $Scale factor of Fx shape factor
LMUX = 1 $Scale factor of Fx peak friction coefficient
LEX = 1 $Scale factor of Fx curvature factor
LKX = 1 $Scale factor of Fx slip stiffness
LHX = 1 $Scale factor of Fx horizontal shift
LVX = 1 $Scale factor of Fx vertical shift
LGAX = 1 $Scale factor of camber for Fx
LCY = 1 $Scale factor of Fy shape factor
LMUY = 1 $Scale factor of Fy peak friction coefficient
LEY = 1 $Scale factor of Fy curvature factor
LKY = 1 $Scale factor of Fy cornering stiffness
LHY = 1 $Scale factor of Fy horizontal shift
LVY = 1 $Scale factor of Fy vertical shift
LKC = 1 $Scale factor of camber stiffness
LGAY = 1 $Scale factor of camber for Fy
LTR = 1 $Scale factor of Peak of pneumatic trail
LRES = 1 $Scale factor of Peak of residual torque
LGAZ = 1 $Scale factor of camber torque stiffness
LXAL = 1 $Scale factor of alpha influence on Fx
LYKA = 1 $Scale factor of kappa influence on Fy
LVYKA = 1 $Scale factor of kappa induced Fy
LS = 1 $Scale factor of Moment arm of Fx
LSGKP = 1 $Scale factor of Relaxation length of Fx
LSGAL = 1 $Scale factor of Relaxation length of Fy
LGYR = 1 $Scale factor of gyroscopic torque
LMX = 1 $Scale factor of overturning couple
LVMX = 1 $Scale factor of Mx vertical shift
LMY = 1 $Scale factor of rolling resistance torque
$--------------------------------------------------------longitudinal
[LONGITUDINAL_COEFFICIENTS]
PCX1 = 1.3178 $Shape factor Cfx for longitudinal force
PDX1 = 1.0455 $Longitudinal friction Mux at Fznom
PDX2 = 0.063954 $Variation of friction Mux with load
PDX3 = 0 $Variation of friction Mux with camber
PEX1 = 0.15798 $Longitudinal curvature Efx at Fznom
PEX2 = 0.41141 $Variation of curvature Efx with load
PEX3 = 0.1487 $Variation of curvature Efx with load squared
PEX4 = 3.0004 $Factor in curvature Efx while driving
PKX1 = 23.181 $Longitudinal slip stiffness Kfx/Fz at Fznom
PKX2 = -0.037391 $Variation of slip stiffness Kfx/Fz with load
PKX3 = 0.80348 $Exponent in slip stiffness Kfx/Fz with load
PHX1 = -0.00058264 $Horizontal shift Shx at Fznom
PHX2 = -0.0037992 $Variation of shift Shx with load
PVX1 = 0.045118 $Vertical shift Svx/Fz at Fznom
PVX2 = 0.058244 $Variation of shift Svx/Fz with load
RBX1 = 13.276 $Slope factor for combined slip Fx reduction
RBX2 = -13.778 $Variation of slope Fx reduction with kappa
RCX1 = 1.0 $Shape factor for combined slip Fx reduction
REX1 = 0 $Curvature factor of combined Fx
REX2 = 0 $Curvature factor of combined Fx with load
RHX1 = 0 $Shift factor for combined slip Fx reduction
PTX1 = 0.85683 $Relaxation length SigKap0/Fz at Fznom
PTX2 = 0.00011176 $Variation of SigKap0/Fz with load
PTX3 = -1.3131 $Variation of SigKap0/Fz with exponent of load
$---------------------------------------------------------overturning
[OVERTURNING_COEFFICIENTS]
QSX1 = 0 $Vertical offset overturning moment
QSX2 = 0 $Camber induced overturning moment
QSX3 = 0 $Fy induced overturning moment
$-------------------------------------------------------------lateral
[LATERAL_COEFFICIENTS]
PCY1 = 1.18 $Shape factor Cfy for lateral forces
PDY1 = 0.90312 $Lateral friction Muy
PDY2 = -0.17023 $Exponent lateral friction Muy
PDY3 = -0.76512 $Variation of friction Muy with squared camber
PEY1 = -0.57264 $Lateral curvature Efy at Fznom
PEY2 = -0.13945 $Variation of curvature Efy with load
PEY3 = 0.030873 $Zero order camber dependency of curvature Efy
PEY4 = 0 $Variation of curvature Efy with camber
PKY1 = -25.128 $Maximum value of stiffness Kfy/Fznom
PKY2 = 3.2018 $Load with peak of cornering stiffness
PKY3 = 0 $Variation with camber squared of cornering stiffness
PKY4 = 1.9998 $Shape factor for cornering stiffness with load
PKY5 = -0.50726 $Camber stiffness/Fznom
PKY6 = 0 $Camber stiffness depending on Fz squared
PHY1 = 0.0031414 $Horizontal shift Shy at Fznom
PHY2 = 0 $Variation of shift Shy with load
PVY1 = 0.0068801 $Vertical shift in Svy/Fz at Fznom
PVY2 = -0.0051 $Variation of shift Shv with load
RBY1 = 7.1433 $Slope factor for combined Fy reduction
RBY2 = 9.1916 $Variation of slope Fy reduction with alpha
RBY3 = -0.027856 $Shift term for alpha in slope Fy reduction
RCY1 = 1.0 $Shape factor for combined Fy reduction
REY1 = 0 $Curvature factor of combined Fy
REY2 = 0 $Curvature factor of combined Fy with load
RHY1 = 0 $Shift factor for combined Fy reduction
RHY2 = 0 $Shift factor for combined Fy reduction with load
RVY1 = 0 $Kappa induced side force Svyk/Muy*Fz at Fznom
RVY2 = 0 $Variation of Svyk/Muy*Fz with load
RVY3 = 0 $Variation of Svyk/Muy*Fz with camber
RVY4 = 0 $Variation of Svyk/Muy*Fz with alpha
RVY5 = 0 $Variation of Svyk/Muy*Fz with kappa
RVY6 = 0 $Variation of Svyk/Muy*Fz with atan(kappa)
PTY1 = 4.1114 $Peak value of relaxation length SigAlp0/R0
PTY2 = 6.1149 $Value of Fz/Fznom where SigAlp0 is extreme
$--------------------------------------------------rolling resistance
[ROLLING_COEFFICIENTS]
QSY1 = 0.01 $Rolling resistance torque coefficient
QSY2 = 0 $Rolling resistance torque depending on Fx
QSY3 = 0 $Rolling resistance torque depending on speed
QSY4 = 0 $Rolling resistance torque depending on speed^4
$------------------------------------------------------------aligning
[ALIGNING_COEFFICIENTS]
QBZ1 = 5.6241 $Trail slope factor for trail Bpt at Fznom
QBZ2 = -2.2687 $Variation of slope Bpt with load
QBZ4 = 6.891 $Variation of slope Bpt with camber
QBZ5 = -0.35587 $Variation of slope Bpt with absolute camber
QCZ1 = 1.0904 $Shape factor Cpt for pneumatic trail
QDZ1 = 0.082871 $Peak trail Dpt = Dpt*(Fz/Fznom*R0)
QDZ2 = -0.012677 $Variation of peak Dpt with load
QDZ6 = 0.00038069 $Peak residual torque Dmr = Dmr/(Fz*R0)
QDZ7 = 0.00075331 $Variation of peak factor Dmr with load
QDZ8 = -0.083385 $Variation of peak factor Dmr with camber
QDZ9 = 0 $Variation of peak factor Dmr with camber and load
QEZ1 = -34.759 $Trail curvature Ept at Fznom
QEZ2 = -37.828 $Variation of curvature Ept with load
QEZ4 = 0.59942 $Variation of curvature Ept with sign of Alpha-t
QHZ1 = 0.0025743 $Trail horizontal shift Sht at Fznom
QHZ2 = -0.0012175 $Variation of shift Sht with load
QHZ3 = 0.038299 $Variation of shift Sht with camber
QHZ4 = 0.044776 $Variation of shift Sht with camber and load
SSZ1 = 0.0097546 $Nominal value of s/R0: effect of Fx on Mz
SSZ2 = 0.0043624 $Variation of distance s/R0 with Fy/Fznom
SSZ3 = 0 $Variation of distance s/R0 with camber
SSZ4 = 0 $Variation of distance s/R0 with load and camber
QTZ1 = 0 $Gyroscopic torque constant
MBELT = 0 $Belt mass of the wheel -kg-