adams tire models

Tire Models

Adams/Tire

12

Using the Fiala Handling Force ModelThis section of the help provides detailed technical reference material for defining tires on a mechanical

system model using Adams/Tire. It assumes that you know how to run Adams/Car, Adams/Solver, or

Adams/Chassis. It also assumes that you have a moderate level of tire-modeling proficiency.

The Fiala tire model is the standard tire model that comes with all Adams/Tire modules. This chapter

contains information for using the Fiala tire model:

• Assumptions

• Inputs

• Tire Slip Quantities and Transient Tire Behavior

• Force Evaluation

• Tire Carcass Shape

• Property File Format Example

• Contact Methods

Fiala Tire AssumptionsThe background of the Fiala tire model is a physical tire model, where the carcass is modeled as a beam

on an elastic foundation in the lateral direction. Elastic brush elements provide the contact between

carcass and road. Under these assumptions, analytical expressions for the steady-state slip characteristics

can be derived, which are the basis for the calculation of the longitudinal and lateral forces in Adams.

• Rectangular contact patch or footprint.

• Pressure distribution uniform across contact patch.

• No tire relaxation effects are considered.

• Camber angle has no effect on tire forces.

Fiala Tire InputsThe inputs to the Fiala tire model come from two sources:

• Input parameters from the tire property file (.tir), such as tire undeflected radius, that the tire

references.

• Tire kinematic states, such as slip angle ( ), which Adams/Tire calculates.

The following table summarizes the input that the Fiala tire model uses to calculate force.

α

13Tire Models

Input for Calculating Tire Forces

Quantity: Description: Use by Fiala: Source:

Mt Mass of tire • Damping

• Vertical force (Fz)

Alpha Slip angle Lateral force (Fy) Tire kinematic state from

Adams/Solver

Ss Longitudinal slip ratio Longitudinal force (Fx) Tire kinematic state from

Adams/Solver

pen Penetration (tire deflection) Vertical force (Fz) Tire kinematic state from

Adams/Solver

Vpen d/dt (penetration) Vertical force (Fz) Tire kinematic state from

Adams/Solver

Vertical_damping Vertical damping coefficient • Damping

• Vertical force (Fz)

Tire property file (.tir)

Vertical_stiffness Vertical tire stiffness Vertical force (Fz) Tire property file (.tir)

CSLIP Partial derivative of longitudinal

force (Fx) with respect to

longitudinal slip ratio (S) at zero

longitudinal slip

Longitudinal force (Fx) Tire property file (.tir)

CALPHA Partial derivative of lateral force

(Fy) with respect to slip angle

( ) at zero slip angle

Lateral force (Fy) Tire property file (.tir)

UMIN Coefficient of friction at zero

slip

Fx, Fy, Tz Tire property file (.tir)

UMAX Coefficient of friction when tire

is sliding

Fx, Fy, Tz Tire property file (.tir)

Rolling_resistance Rolling resistance coefficient Rolling resistance moment (Ty) Tire property file (.tir)

α

Adams/Tire

14

Tire Slip Quantities and Transient Tire Behavior

Definition of Tire Slip Quantities

Slip Quantities at combined cornering and braking/traction

The longitudinal slip velocity Vsx in the SAE-axis system is defined using the longitudinal speed Vx, the

wheel rotational velocity , and the loaded rolling radius Rl:

The lateral slip velocity is equal to the lateral speed in the contact point with respect to the road

plane:

The practical slip quantities (longitudinal slip) and (slip angle) are calculated with these slip

velocities in the contact point:

and

Note that for realistic tire forces the slip angle is limited to 900 and the longitudinal slip in

between -1 (locked wheel) and 1.

Lagged longitudinal and lateral slip quantities (transient tire behavior)

In general, the tire rotational speed and lateral slip will change continuously because of the changing

interaction forces in between the tire and the road. Often the tire dynamic response will have an important

role on the overall vehicle response. For modeling this so-called transient tire behavior, a first-order

system is used both for the longitudinal slip ? as the side slip angle, ?. Considering the tire belt as a

stretched string, which is supported to the rim with lateral springs, the lateral deflection of the belt can

be estimated (see also reference [1]). The figure below shows a top-view of the string model.

Ω Vsx Vx ΩRl–=

Vsy Vy=

κ α

κVsx

Vx

--------–= αtanVsy

Vx

---------=

α κ

15Tire Models

Stretched String Model for Transient Tire behavior

When rolling, the first point having contact with the road adheres to the road (no sliding assumed).

Therefore, a lateral deflection of the string will arise that depends on the slip angle size and the history

of the lateral deflection of previous points having contact with the road.

For calculating the lateral deflection v1 of the string in the first point of contact with the road, the

following differential equation is valid during braking slip:

with the relaxation length in the lateral direction. The turnslip can be neglected at radii larger than

10 m. This differential equation cannot be used at zero speed, but when multiplying with Vx, the equation

can be transformed to:

When the tire is rolling, the lateral deflection depends on the lateral slip speed; at standstill, the deflection

depends on the relaxation length, which is a measure for the lateral stiffness of the tire. Therefore, with

this approach, the tire is responding to a slip speed when rolling and behaving like a spring at standstill.

A similar approach yields the following for the deflection of the string in longitudinal

direction:

1

Vx

------dv1

dt--------

v1

σ α------+ αtan aφ+=

σ α φ

σ α

dv1dt-------- Vx v1+ σ αVsy=

σ κ

du1dt-------- Vx u1+ σ– κVsx=

Adams/Tire

16

Now the practical slip quantities, and , are defined based on the tire

deformation:

These practical slip quantities and are used instead of the usual and definitions for steady-

state tire behavior.

The longitudinal and lateral relaxation length are read from the tire property file, see Fiala Tire Property

File Format Example

Fiala Tire Force EvaluationTypes of force evaluation:

• Normal Force of Road on Tire

• Longitudinal Force

• Lateral Force

• Rolling Resistance Moment

• Aligning Moment

• Smoothing

Normal Force of Road on Tire

The normal force of a road on a tire at the contact patch in the SAE coordinates (+Z downward) is always

negative (directed upward). The normal force is:

Fz = min (0.0, Fzk + Fzc)

where:

• Fzk is the normal force due to tire vertical stiffness

• Fzc is the normal force due to tire vertical damping

• Fzk = - vertical_stiffness × pen

• Fzc = - vertical_damping × Vpen

Instead of the linear vertical tire stiffness, also an arbitrary tire deflection - load curve can be defined in

the tire property file in the section [DEFLECTION_LOAD_CURVE] (see the Property File Format

Example). If a section called [DEFLECTION_LOAD_CURVE] exists, the load deflection datap oints

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 file, but it does not play any

role.

κ' α '

κ'u1σ κ------ Vx( )sgn=

α 'v1

σ α------

atan=

κ' α ' κ α

17Tire Models

Longitudinal Force

The longitudinal force depends on the vertical force (Fz), the current coefficient of friction (U), the

longitudinal slip ratio (Ss), and the slip angle (Alpha). The current coefficient of friction depends on the

static (UMAX) and dynamic (UMIN) friction coefficients and the comprehensive slip ratio (SsAlpha).

UMAX specifies the tire/road coefficient of friction at zero slip and represents the static friction

coefficient. This is the y-intercept on the friction coefficient versus slip graph. Note that this value is an

unobtainable maximum friction value, because there is always slip within a footprint. This value is used

in conjunction with UMIN to define a linear friction versus slip relation. UMAX will normally be larger

than UMIN.

UMIN specifies the tire/road coefficient of friction for the full slip case and represents the sliding friction

coefficient. This is the friction coefficient at 100% slip, or pure sliding. This value is used in conjunction

with UMAZ to define a linear friction versus slip relationship.

The comprehensive slip ( ):

The current value coefficient of friction (U):

Fiala defines a critical longitudinal slip ( ):

This is the value of longitudinal slip beyond which the tire is sliding.

Case 1. Elastic Deformation State: |Ss| S_critical

Fx = -CSLIP × Ss

Case 2. Complete Sliding State: |Ss| S_critical

Fx = -sign(Ss)(Fx1- Fx2)

where:

Ssα

Ssα Ss2

tan2 α( )+=

U Umax Umax Umin–( ) Ssα⋅( )–=

Scritical

ScriticalU Fz⋅

2 CSLIP⋅-------------------------=

<

>

Fx1 U Fz⋅=

Fx2U Fz⋅( )2

4 Ss CSLIP⋅ ⋅--------------------------------------=

Adams/Tire

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Lateral Force

Like the longitudinal force, the lateral force depends on the vertical force (Fz) and the current coefficient

of friction (U). And similar to the longitudinal force calculation, Fiala defines a critical lateral slip

( ):

The lateral force peaks at a value equal to U × |Fz| when the slip angle (Alpha) equals the critical slip

angle ( ).

Case 1. Elastic Deformation State: |Alpha|

Fy = - U × |Fz|× (1-H3) × sign(Alpha)

where:

Case 2. Sliding State: |Alpha| Alpha_critical

Fy = -U|Fz|sign(Alpha)

Rolling Resistance Moment

When the tire is rolling forward: Ty = -rolling_resistance * Fz

When the tire is rolling backward: Ty = rolling_resistance * Fz

Aligning Moment

Case 1. Elastic Deformation State: |Alpha|

Mz = U × |Fz|× WIDTH × (1-H) × H3 × sign(Alpha)

where:

Case 2. Complete Sliding State: |Alpha|

Mz= 0.0

α critical

α critical

3 U Fz⋅ ⋅

CALPHA-------------------------

atan=

α critical

α critical≤

H 1CALPHA α( )tan⋅

3 U Fz⋅ ⋅--------------------------------------------------–=

α critical≤

H 1CALPHA α( )tan⋅

3 U Fz⋅ ⋅--------------------------------------------------–=

α critical>

19Tire Models

Smoothing

Adams/Tire can smooth 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

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, smoothing is off

• USE_MODE = 2, smoothing is on

Fiala Tire Carcass ShapeUsing Fiala tire, you can optionally supply a tire carcass cross-sectional shape in the tire property file in

the [SHAPE] block. The 3D-durability, tire-to-road contact algorithm uses this information when

calculating the tire-to-road volume of interference. To learn more about this topic, see Applying the Tire

Carcass Shape. If you omit the [SHAPE] block from a tire property file, the tire carcass cross-section

defaults to the rectangle that the tire radius and width define.

You specify the tire carcass shape by entering points in fractions of the tire radius and width. Because

Adams/Tire assumes that the tire cross-section is symmetrical about the wheel plane, you only specify

points for half the width of the tire. The following apply:

• For width, a value of zero (0) lies in the wheel center plane.

• For width, a value of one (1) lies in the plane of the side wall.

• For radius, a value of one (1) lies on the tread.

For example, suppose your tire has a radius of 300 mm and a width of 185 mm and that the tread is joined

to the side wall with a fillet of 12.5 mm radius. The tread then begins to curve to meet the side wall at

+/- 80 mm from the wheel center plane. If you define the shape table using six points with four points

along the fillet, the resulting table might look like the shape block that is at the end of the following

property format example.

Fiala Tire Property File Format Example$------------------------------------------------MDI_HEADER[MDI_HEADER]FILE_TYPE = 'tir'FILE_VERSION = 2.0FILE_FORMAT = 'ASCII'(COMMENTS)comment_string'Tire - XXXXXX''Pressure - XXXXXX'

Adams/Tire

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'Test Date - XXXXXX''Test tire''New File Format v2.1'$----------------------------------------------------units[UNITS]LENGTH = 'mm'FORCE = 'newton'ANGLE = 'degree'MASS = 'kg'TIME = 'sec'$-----------------------------------------------------model[MODEL]! use mode 12 11 12 ! -------------------------------------------- ! smoothingX X! transient X X!PROPERTY_FILE_FORMAT = 'FIALA'USE_MODE = 11.0$-------------------------------------------------dimension[DIMENSION]UNLOADED_RADIUS = 309.9WIDTH = 235.0ASPECT_RATIO = 0.45$-------------------------------------------------parameter[PARAMETER]VERTICAL_STIFFNESS = 310.0VERTICAL_DAMPING = 3.1ROLLING_RESISTANCE = 0.0CSLIP = 1000.0CALPHA = 800.0CGAMMA = 0.0UMIN = 0.9UMAX = 1.0RELAX_LENGTH_X = 0.05RELAX_LENGTH_Y = 0.15$---------------------------------------------carcass shape[SHAPE]radius width1.0000 0.00001.0000 0.50001.0000 0.86490.9944 0.92350.9792 0.98190.9583 1.0000$------------------------------------------------load_curve$ Maximum of 100 points (optional)[DEFLECTION_LOAD_CURVE]pen fz0 0.01 212.02 428.03 648.05 1100.0

21Tire Models

10 2300.020 5000.030 8100.0

Fiala Tire Contact MethodsThe Fiala tire model supports the following roads:

• 2D Roads, see Using the 2D Road Model.

• 3D Roads, see Adams/3D Road Model

Adams/Tire

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Using the PAC2002Tire ModelThe PAC2002 Magic-Formula tire model has been developed by MSC.Software according to Tyre and

Vehicle Dynamics by Pacejka [1]. PAC2002 is latest version of a Magic-Formula model available in

Adams/Tire.

Learn about:

• When to Use PAC2002

• Modeling of Tire-Road Interaction Forces

• Axis Systems and Slip Definitions

• Contact Point and Normal Load Calculation

• Basics of Magic Formula

• Steady-State: Magic Formula

• Transient Behavior

• Gyroscopic Couple

• Left and Right Side Tires

• USE_MODES OF PAC2002: from Simple to Complex

• Quality Checks for Tire Model Parameters

• Contact Methods

• Standard Tire Interface (STI)

• Definitions

• References

• Example of PAC2002 Tire Property File

When to Use PAC2002Magic-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 PAC2002 contains the latest developments that have been published in

Tyre and Vehicle Dynamics by Pacejka [1].

In general, a 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)

23Tire Models

• 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

accurate in this model.

The PAC2002 model has proven to be applicable for car, truck, and aircraft tires with camber

(inclination) angles to the road not exceeding 15 degrees.

PAC2002 and Previous Magic Formula Models

Compared to previous versions, PAC2002 is backward compatible with all previous versions of

PAC2002, MF-Tyre 5.x tire models, and related tire property files.

New Features

The enhancements for PAC2002 in Adams/Tire 2005 r2 are:

• More advanced tire-transient modeling using a contact mass in the contact point with the road.

This results in more realistic dynamic tire model response at large slip, low speed, and standstill

(usemode > 20).

• Parking torque and turn-slip have been introduced: the torque around the vertical axis due to

turning at standstill or at low speed (no need for extra parameters).

• Extended loaded radius modeling (see Contact Point and Normal Load Calculation) are suitable

for driving under extreme conditions like roll-over events and racing applications.

• The option to use a nonlinear spline for the vertical tire load-deflection instead of a linear tire

stiffness. See Contact Point and Normal Load Calculation.

• Modeling of bottoming of the tire to the road by using another spline for defining the bottoming

forces. Learn more about wheel bottoming.

• Online scaling of the tire properties during a simulation; the scaling factors of the PAC2002 can

now be changed as a function of time, position, or any other variable in your model dataset. See

Online Scaling of Tire Properties.

Modeling of Tire-Road Interaction ForcesFor 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.

Adams/Tire

24

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.

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

• Units

• Definition of Tire Slip Quantities

κ α γ

25Tire Models

Axis Systems

The PAC2002 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 PAC2002 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 PAC2002, 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 PAC2002 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 PAC2002 are according

to the SI unit system. Also, the equations for PAC2002 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.

Adams/Tire

26

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 PAC2002


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:

(1)

Slip Quantities at Combined Cornering and Braking/Traction

Variable type: Name: Abbreviation: Unit:

Angle Slip angle

Inclination angle

Radians

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 Radians per second

α

γ

ω

ω

Vsx Vx ΩRe–=

27Tire Models

The lateral slip velocity is equal to the lateral speed in the contact point with respect to the road plane:

(2)


velocities in the contact point with:

(3)

(4)

The rolling speed Vr is determined using the effective rolling radius Re:

(5)

Turn-slip is one of the two components that form the spin of the tire. Turn-slip is calculated using the

tire yaw velocity :

(6)

The total tire spin is calculated using:

(7)

The total tire spin has contributions of turn-slip and camber. denotes the camber reduction factor for

the camber to become comparable with turn-slip.

Vsy Vy=

κ α

κVsx

Vx

--------–=

αtanVsy

Vx

---------=

Vr ReΩ=

φ

ψ·

Wtψ·

Vx

------=

ϕ

ψ 11

Vx

------ ψ· 1 εγ–( )Ω γsin– =

ε

Adams/Tire

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Contact Point and Normal Load Calculation• Contact Point

• Loaded and Effective Tire Rolling Radius

• Wheel Bottoming

Contact Point

In the vertical direction, the tire is modeled as a parallel linear spring and damper having one point of

contact (C) with the road. This is valid for road obstacles with a wavelength larger than the tire radius

(for example, for car tires 1m).

For calculating the kinematics of the tire relative to the road, the road is approximated by its tangent plane

at the road point right below the wheel center (see the figure below).

Contact Point C: Intersection between Road Tangent Plane, Spin Axis Plane, and Wheel Plane

The contact point is determined by the line of intersection of the wheel center-plane with the road tangent

(ground) plane and the line of intersection of the wheel center-plane with the plane through the wheel

spin axis. The normal load Fz of the tire is calculated with the tire deflection as follows:ρ

29Tire Models

(8)

Using this formula, the vertical tire stiffness increases due to increasing rotational speed and

decreases by longitudinal and lateral tire forces. If qFz1 is zero, qFz1 will be CzR0/Fz0.

When you do not provide the coefficients qV2, qFcx, qFcy, qFz1, qFz2 and qFc in the tire property

file, the normal load calculation is compatible with previous versions of PAC2002, because, in that case,

the normal load is calculated using the linear vertical tire stiffness Cz and tire damping Kz according to:

(9)

Instead of the linear vertical tire stiffness Cz (= qFz1Fz0/R0), you can define an arbitrary tire deflection

- load curve in the tire property file in the section [DEFLECTION_LOAD_CURVE] (see the Example

of PAC2002 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 tire radius Rl defined as the distance of the wheel center to the contact point of the tire

with the road, the tire deflection can be calculated using the free tire radius R0 and a correction for the

tire radius growth due to the rotational tire speed :

(10)

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:

(11)

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.

Fz 1 qV2 ΩRo

Vo

------ qFcxlFxFz0--------

2

– qFcylFyFz0--------

2

– qFcylγ2

+ +

qFzlρR0

------ qFz2ρR0

------ 2+ Fz0 Kz ρ·•+

=

ω

γ

Fz Czρ λ Cz Kzρ·

+=

ω

ρ R0 R1– qV1R0 ΩR0

V0

------

2

+=

Re

Vx

Ω------=

Adams/Tire

30

Effective Rolling Radius and Longitudinal Slip

To represent the effective rolling radius Re, a MF-type of equation is used:

(12)

in which Fz0 is the nominal tire deflection:

Rf R0 qV1R0

ΩR0

V0

-----------

2

RFz0 DPeffarc BReffρd

( ) FReffρd

+tan[ ]–+=

ρ

31Tire Models

(13)

and is called the dimensionless radial tire deflection, defined by:

(14)

Example of Loaded and Effective Tire Rolling Radius as Function of Vertical Load

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 (if qFz1=0)

ρ Fz0Fz0

Czλ Cz----------------=

ρ

ρd ρ

ρ Fz0---------=

Adams/Tire

32

Wheel Bottoming

You can optionally supply a wheel-bottoming deflection, that is, a load curve in the tire property file in

the [BOTTOMING_CURVE] block. If the deflection of the wheel is so large that the rim will be hit

(defined by the BOTTOMING_RADIUS parameter in the [DIMENSION] section of the tire property

file), the tire vertical load will be increased according to the load curve defined in this section.

Note that the rim-to-road contact algorithm is a simple penetration method (such as the 2D contact) based

on the tire-to-road contact calculation, which is strictly valid for only rather smooth road surfaces (the

length of obstacles should have a wavelength longer than the tire circumference). The rim-to-road contact

algorithm is not based on the 3D-volume penetration method, but can be used in combination with the

3D Contact, which takes into account the volume penetration of the tire itself. If you omit the

[BOTTOMING_CURVE] block from a tire property file, no force due to rim road contact is added to the

tire vertical force.

You can choose a BOTTOMING_RADIUS larger than the rim radius to account for the tire's material

remaining in between the rim and the road, while you can adjust the bottoming load-deflection curve for

the change in stiffness.

Kz VERTICAL_DAMPING Tire vertical damping

qFz1 QFZ1 Tire vertical stiffness coefficient (linear)

qFz2 QFZ2 Tire vertical stiffness coefficient

(quadratic)

qFcx1 QFCX1 Tire stiffness interaction with Fx

qFcy1 QFCY1 Tire stiffness interaction with Fy

qFc 1 QFCG1 Tire stiffness interaction with camber

qV1 QV1 Tire radius growth coefficient

qV2 QV2 Tire stiffness variation coefficient with

speed

Name:Name Used in Tire Property

File: Explanation:

γ

33Tire Models

If (Pentire - (Rtire - Rbottom) - ½·width ·| tan(g) |) < 0, the left or right side of the rim has contact with

the road. Then, the rim deflection Penrim can be calculated using:

= max(0 , ½·width ·| tan( ) | ) + Pentire- (Rtire - Rbottom)

Penrim= 2/(2 · width ·| tan( ) |)

δ γ

δ γ

Adams/Tire

34

Srim= ½·width - max(width , /| tan( ) |)/3

with Srim, the lateral offset of the force with respect to the wheel plane.

If the full rim has contact with the road, the rim deflection is:

Penrim = Pentire - (Rtire - Rbottom)

Srim = width2 · | tan( ) | · /(12 · Penrim)

Using the load - deflection curve defined in the [BOTTOMING_CURVE] section of the tire property file,

the additional vertical force due to the bottoming is calculated, while Srim multiplied by the sign of the

inclination is used to calculate the contribution of the bottoming force to the overturning moment.

Further, the increase of the total wheel load Fz due to the bottoming (Fzrim) will not be taken into account

in the calculation for Fx, Fy, My, and Mz. Fzrim will only contribute to the overturning moment Mx using

the Fzrim·Srim.

Basics of the Magic Formula in PAC2002The 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 (see the figure,

Characteristic Curves for Fx and Fy Under Pure Slip Conditions). Because of the sine - arctangent

combination, the basic Magic Formula equation is capable of describing this shape:

(15)

where Y(x) is either Fx with x the longitudinal slip , or Fy and x the lateral slip .

δ γ

γ

γ

Note: Rtire is equal to the unloaded tire radius R0; Pentire is similar to effpen (= ).ρ

α

κ

Y x( ) D Carc Bx E Bx arc Bx( )tan–( )– tan[ ]cos=

κ α

35Tire Models

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 a has a cosine shape, a cosine version the Magic Formula is used:

(16)

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).


α

Adams/Tire

36

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 weighing functions.

Again, these weighting functions have a cosine-shaped MF equation.

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).

37Tire Models

Input Variables

The input variables to the Magic Formula are:

Output Variables

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:

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)

Longitudinal slip [-]

Slip angle [rad]

Inclination angle [rad]

Normal wheel load Fz [N]

Longitudinal force Fx [N]

Lateral force Fy [N]

Overturning couple Mx [Nm]

Rolling resistance

moment

My [Nm]

Aligning moment Mz [Nm]

Nominal (rated) load Fz0 [N]

Unloaded tire radius R0 [m]

Tire belt mass mbelt [kg]

κ

α

γ

fzdFz F'z0–

F'z0--------------------=

γΦ z0

F'z0 Fz0 λ Fz0•=

Adams/Tire

38

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

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, l and ,

are also used for the position-dependent friction in 3D Road Contact and 3D Road. An overview of all

scaling factors is shown in the following tables.

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


t Transient tire behavior

j = x Along the longitudinal axis

y Along the lateral axis

z About the vertical axis

k = 1, 2, ...

µξ γµψ ψ

39Tire Models

Scaling Factor Coefficients for Pure Slip

Name:Name used in tire property file: Explanation:

Fzo LFZO Scale factor of nominal (rated) load

Cz LCZ Scale factor of vertical tire stiffness

Cx LCX Scale factor of Fx shape factor

x LMUX Scale factor of Fx peak friction coefficient

Ex LEX Scale factor of Fx curvature factor

Kx LKX Scale factor of Fx slip stiffness

Hx LHX Scale factor of Fx horizontal shift

Vx LVX Scale factor of Fx vertical shift

x LGAX Scale factor of inclination for Fx

Cy LCY Scale factor of Fy shape factor

y LMUY Scale factor of Fy peak friction coefficient

Ey LEY Scale factor of Fy curvature factor

Ky LKY Scale factor of Fy cornering stiffness

Hy LHY Scale factor of Fy horizontal shift

Vy LVY Scale factor of Fy vertical shift

gy LGAY Scale factor of inclination for Fy

t LTR Scale factor of peak of pneumatic trail

Mr LRES Scale factor for offset of residual moment

LGAZ Scale factor of inclination for Mz

Mx LMX Scale factor of overturning couple

VMxMx LVMX Scale factor of Mx vertical shift

My LMY Scale factor of rolling resistance moment

λ

λ

λ

λ µ

λ

λ

λ

λ

λ γ

λ

λ µ

λ

λ

λ

λ

λ

λ

λ

λ γzγz

λ

λ

λ

Adams/Tire

40

Scaling Factor Coefficients for Combined Slip

Scaling Factor Coefficients for Transient Response

Note that the scaling factors change during the simulation according to any user-introduced function. See

the next section, Online Scaling of Tire Properties.

Online Scaling of Tire Properties

PAC2002 can provide online scaling of tire properties. For each scaling factor, a variable should be

introduced in the Adams .adm dataset. For example:

!lfz0 scaling! adams_view_name='TR_Front_Tires until wheel_lfz0_var'VARIABLE/53, IC = 1, FUNCTION = 1.0

This lets you change the scaling factor during a simulation as a function of time or any other variable in

your model. Therefore, tire properties can change because of inflation pressure, road friction, road

temperature, and so on.

You can also use the scaling factors in co-simulations in MATLAB/Simulink.

For more detailed information, see Knowledge Base Article 12732.


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


sk LSGKP Scale factor of relaxation length of Fx

sa LSGAL Scale factor of relaxation length of Fy

gyr LGYR Scale factor of gyroscopic moment

λ xαα

λ yκκ

λ Vyκκ

λ s

λ

λ

λ

http://support.adams.com/kb/faq.asp?ID=kb12732.html

41Tire Models

Steady-State: Magic Formula in PAC2002• Steady-State Pure Slip

• Steady-State Combined Slip

Steady-State Pure Slip

• Longitudinal Force at Pure Slip

• Lateral Force at Pure Slip

• Aligning Moment at Pure Slip

• Turn-slip and Parking

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)

Fx Fx0 κ Fz γ, ,( )=

Fx0 Dx Cxarc Bxκx Ex Bxκx arc Bxκx( )tan–( )– tan[ ] SVx+( )sin=

κx κ SHx+=

γx γ λ γx⋅=

Cx pCx1 λ Cx⋅=

Dx µx Fz ζ1⋅ ⋅=

µx pDx1 pDx2 fzd+( ) 1 pDx3 γ2

⋅–( )λ µx⋅=

Ex pEx1 pEx2 fz2

d+( ) 1 pEx4 κx( )sgn– λ Ex with Ex 1≤⋅ ⋅=

Kx Fz pKx1 pKx2 fzd+( ) pKx3 fzd( ) λ Kx⋅exp⋅ ⋅=

Kx BxCxDx κx∂

∂ Fx0at κ 0= = =

Bx Kx CxDx( )⁄=

Adams/Tire

42

(29)

(30)

Longitudinal Force Coefficients at Pure Slip

Formulas for the Lateral Force at Pure Slip

(31)

(32)

(33)

The scaled inclination angle:

(34)

with coefficients:

(35)


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

pKx3 PKX3 Exponent in slip stiffness Kfx/Fz with load

pHx1 PHX1 Horizontal shift Shx at Fznom

pHx2 PHX2 Variation of shift Shx with load

pVx1 PVX1 Vertical shift Svx/Fz at Fznom

pVx2 PVX2 Variation of shift Svx/Fz with load

SHx pHx1 pHx2 dfz⋅+( )λ Hx=

SVx Fz pHx1 pHx2 dfz⋅+( )λ Vx λ Hµx ζ1⋅ ⋅⋅=

Fy Fy0 α γ Fz, ,( )=

Fy0 Dy Cyarc Byα y Ey Byαy arc Byα y( )tan–( )– tan[ ] SVy+sin=

α y α SHy+=

γy γ λ γy⋅=

Cy pCy1 λ Cy⋅=

43Tire Models

(36)

(37)

(38)

The cornering stiffness:

(39)

(40)

(41)

(42)

(43)

The camber stiffness is given by:

(44)

Lateral Force Coefficients at Pure Slip


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

pHy1 PHY1 Horizontal shift Shy at Fznom

Dy µy Fz ζ2⋅ ⋅=

µy pDy1 pDy2dfz+( ) 1 pDy3γy2

–( ) λ µy⋅ ⋅=

Ey pEy1 pEy2dfz+( ) 1 pEy3 pEy4γy+( ) α y( )sgn– γEy with Ey 1≤⋅ ⋅=

Ky0 PKy1 Fz0 2acFz

PKy2F0λ Fz0---------------------------

tan λ Fz0 λ Ky⋅ ⋅

sin⋅ ⋅=

Ky Ky0 1 pKy3 γy–( ) ζ3⋅ ⋅=

By Ky CyDy( )⁄=

SHy pHy1 pHy2dfz+( ) λ Hy pHy3γy ζ0 ζ4 1–+⋅+⋅=

SVy Fz pVy1 pVy2dfz+( ) λ Vy pVy3 pVy4dfz+( ) γy⋅+⋅ λ µy ζ4⋅ ⋅ ⋅=

Kyγ0 PHy3Ky0 Fz pνy3 pνy4dfz+( )+=

Adams/Tire

44

Formulas for the Aligning Moment at Pure Slip

(45)

with the pneumatic trail t:

(46)

(47)

and the residual moment Mzr:

(48)

(49)

(50)


(51)

with coefficients:

(52)

(53)

(54)

pHy2 PHY2 Variation of shift Shy with load

pHy3 PHY3 Variation of shift Shy with inclination

pVy1 PVY1 Vertical shift in Svy/Fz at Fznom

pVy2 PVY2 Variation of shift Svy/Fz with load

pVy3 PVY3 Variation of shift Svy/Fz with inclination

pVy4 PVY4 Variation of shift Svy/Fz with inclination and load


M'z Mz0 α γ Fz, ,( )=

Mz0 t Fy0 Mzr+⋅–=

t α t( ) Dt Ctarc Btα t Et Btα t arc Btα t( )tan–( )– tan[ ] α( )coscos=

α t α SHt+=

Mzr α r( ) Dr Crarc Brα r( )tan[ ] α( )cos⋅cos=

α r α SHf+=

SHf SHy SVy Ky⁄+=

γz γ λ γz⋅=

Bt qBz1 qBz2dfz qBz3dfz2

+ +( ) 1 qBz4γz qBz5 γz+ +( ) λ Ky λ µy⁄⋅ ⋅=

Ct qCz1=

Dt Fz qDz1 qDz2dfz+( ) 1 qDz3γz qDz4γz2

+ +( )R0

Fz0-------- λ t ζ5⋅ ⋅ ⋅ ⋅ ⋅=

45Tire Models

(55)

(56)

(57)

(58)

An approximation for the aligning moment stiffness reads:

(59)

Aligning Moment Coefficients at Pure Slip


qBz1 QBZ1 Trail slope factor for trail Bpt at Fznom

qBz2 QBZ2 Variation of slope Bpt with load

qBz3 QBZ3 Variation of slope Bpt with load squared

qBz4 QBZ4 Variation of slope Bpt with inclination

qBz5 QBZ5 Variation of slope Bpt with absolute inclination

qBz9 QBZ9 Slope factor Br of residual moment Mzr

qBz10 QBZ10 Slope factor Br of residual moment Mzr

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

qDz3 QDZ3 Variation of peak Dpt with inclination

qDz4 QDZ4 Variation of peak Dpt with inclination squared.

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

Et qEz1 qEz2dfz qEz3dfz2

+ +( )=

1 qEz1 qEz2γz+( )2

π---

arc Bt Ct α t⋅ ⋅( )tan⋅

+

with Et 1≤( )

SHt qHz1 qHz2dfz qHz3 qHz4 dfz⋅+( )γz+ +=

Br qBz9λ Kyλ µy--------- qBz10 By Cy⋅ ⋅+⋅

ζ6⋅=

Cr ζ7=

Dr Fz qDz6 qDz7dfz+( ) γr⋅ qDz8 qDz9dfz+( ) γz⋅+[ ] Ro λ µγ ζ8 1–+⋅ ⋅ ⋅=

Kz t Ky α∂∂

– Mz at α≈ ⋅– 0 )= =

Adams/Tire

46

Turn-slip and Parking

For situations where turn-slip may be neglected and camber remains small, the reduction factors i that

appear in the equations for steady-state pure slip, are to be set to 1:

For larger values of spin, the reduction factors are given below.

The weighting function is used to let the longitudinal force diminish with increasing spin, according

to:

with:

The peak side force reduction factor reads:

with:

The cornering stiffness reduction factor is given by:

qDz9 QDZ9 Variation of Dmr with inclination and load

qEz1 QEZ1 Trail curvature Ept at Fznom

qEz2 QEZ2 Variation of curvature Ept with load

qEz3 QEZ3 Variation of curvature Ept with load squared

qEz4 QEZ4 Variation of curvature Ept with sign of Alpha-t

qEz5 QEZ5 Variation of Ept with inclination and sign 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


ζi

ζi 1= i 0.= 1.… 8

ζ1

ζi arc Bxϕ R0ϕ( )tan[ ]cos=

Bxϕ pDxϕ 1 1 pDxϕ 2dfz+( ) arc pDx ϕ 3κ( )tan[ ]cos=

ζ2

ζ2 arc Byϕ R0 ϕ pDyϕ 4 R0 ϕ+( ) tan[ ]cos=

Byϕ pDxϕ 1 1 pDxϕ 2dfz+( ) arc pDx ϕ 3 αtan( )tan[ ]cos=

ζ3

47Tire Models

The horizontal shift of the lateral force due to spin is given by:

The factors are defined by:

The spin force stiffness KyR0 is related to the camber stiffness Ky0:

in which the camber reduction factor is given by:

The reduction factors and for the vertical shift of the lateral force are given by:

The reduction factor for the residual moment reads:

The peak spin torque Dr is given by:

The maximum value is given by:

ζ3 arc pKyϕ 1R02ϕ2

( )tan[ ]cos=

SHyϕ DHyϕ CHyϕ arc BHyϕ Roϕ EHyϕ BHyϕ arc BHyϕ R0ϕ( )tan–( )– tan[ ]sin=

CHyϕ pHyϕ 1

DHyϕ pHyϕ 2 pHyϕ 3dfz+( ) Vx( )

EHyϕ

sin⋅

PHyϕ 4

BHyϕKyRϕ 0

CyDyKy0

-----------------------

=

=

=

=

KyRϕ 0

Kyγ0

1 εγ–-------------=

εγ pεγϕ 1 1 pεγϕ 2dfz+( )=

ζ0 ζ4

ζ0 0

ζ4 1 SHyϕ SVyγ Ky⁄–+

=

=

ζ8 1 Drϕ+=

ϕ

Drϕ DDrϕ e CDrϕ arc BDrϕ R0ϕ EDrϕ BDrϕ R0ϕ arc BDrϕ R0ϕ( )tan–( )– tan[ ]sin=

Adams/Tire

48

The moment at vanishing wheel speed at constant turning is given by:

The shape factors are given by:

in which:

The reduction factor reads:

The spin moment at 90º slip angle is given by:

The spin moment at 90º slip angle is multiplied by the weighing function Gy to account for the action of

the longitudinal slip (see steady-state combined slip equations).

The reduction factor is given by:

DDrϕ

Mzϕ ∞

π2---CDrϕ

sin

-----------------------------=

Mzϕ ∞ qCrϕ 1µyR0Fz Fz Fz0⁄=

CDrϕ qDrϕ 1

EDrϕ qDrϕ 2

BDrϕ

Kzγr0

CDrϕ DDr ϕ 1 εy–( )--------------------------------------------

=

=

=

Kzγr0 FzR0 qDz8 qDz9dfz+( )=

ζ6

ζ6 arc qBrϕ 1R0ϕ( )tan[ ]cos=

Mzϕ 90 Mzϕ ∞2

π--- arc qCr ϕ 2R0ϕ( ) Gyx κ( )⋅tan⋅ ⋅=

ζ7

ζ72

π--- arc Mzϕ 90 DDrϕ⁄[ ]cos⋅=

49Tire Models

Turn-Slip and Parking Parameters

The tire model parameters for turn-slip and parking are estimated automatically. In addition, you can

specify each parameter individually in the tire property file (see example).

Steady-State Combined Slip

PAC2002 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

Name:Name used in

tire property file: Explanation:

p 1 PECP1 Camber spin reduction factor parameter in camber stiffness

p 2 PECP2 Camber spin reduction factor varying with load parameter in

camber stiffness

pDx 1 PDXP1 Peak Fx reduction due to spin parameter

pDx 2 PDXP2 Peak Fx reduction due to spin with varying load parameter

pDx 3 PDXP3 Peak Fx reduction due to spin with kappa parameter

pDy 1 PDYP1 Peak Fy reduction due to spin parameter

pDy 2 PDYP2 Peak Fy reduction due to spin with varying load parameter

pDy 3 PDYP3 Peak Fy reduction due to spin with alpha parameter

pDy 4 PDYP4 Peak Fy reduction due to square root of spin parameter

pKy 1 PKYP1 Cornering stiffness reduction due to spin

pHy 1 PHYP1 Fy-alpha curve lateral shift limitation

pHy 2 PHYP2 Fy-alpha curve maximum lateral shift parameter

pHy 3 PHYP3 Fy-alpha curve maximum lateral shift varying with load

parameter

pHy 4 PHYP4 Fy-alpha curve maximum lateral shift parameter

qDt 1 QDTP1 Pneumatic trail reduction factor due to turn slip parameter

qBr 1 QBRP1 Residual (spin) torque reduction factor parameter due to side

slip

qCr 1 QCRP1 Turning moment at constant turning and zero forward speed

parameter

qCr 2 QCRP2 Turn slip moment (at alpha=90deg) parameter for increase

with spin

qDr 1 QDRP1 Turn slip moment peak magnitude parameter

qDr 2 QDRP2 Turn slip moment peak position parameter

εϕεϕ

ϕϕϕϕϕϕϕϕϕϕϕ

ϕϕϕ

ϕ

ϕ

ϕϕ

Adams/Tire

50

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

• Longitudinal Force at Combined Slip

• Lateral Force at Combined Slip

• Aligning Moment at Combined Slip

• Overturning Moment at Pure and Combined Slip

• Rolling Resistance Moment at Pure and Combined Slip

Formulas for the Longitudinal Force at Combined Slip

(60)

with Gx the weighting function of the longitudinal force for pure slip.

We write:

(61)

(62)

with coefficients:

(63)

(64)

(65)

(66)

(67)

The weighting function follows as:

(68)

Fx Fx0 Gxα α κ Fz, ,( )⋅=

α

Fx Dxα Cxαarc Bxαα s Exα Bxαα s arc Bxαα s( )tan–( )– tan[ ]cos=

α s α SHxα+=

Bxα rBx1 arc rBx2κ tan[ ] λ xα⋅cos=

Cxα

Dxα

Fxo

Cxαarc BxαSHxα Exα BxαSHxα arc BxαSHxα( )tan–( )– tan[ ]cos------------------------------------------------------------------------------------------------------------------------------------------------------------------=

Exα rEx1 rEx2dfz with Exα 1≤+=

SHxα rHx1=

Gxα

Cxαarc Bxαα s Exα Bxαα s arc Bxαα s( )tan–( )– tan[ ]cos

Cxαarc BxαSHxα Exα BxαSHxα arc BxαSHxα( )tan–( )–[ ]tan[ ]cos----------------------------------------------------------------------------------------------------------------------------------------------------------------=

51Tire Models

Longitudinal Force Coefficients at Combined Slip

Formulas for Lateral Force at Combined Slip

(69)

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:

(70)

(71)

with the coefficients:

(72)

(73)

(74)

(75)

(76)

(77)


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

Fy Fy0 Gyκ α κ γ Fz, , ,( )⋅=

Fy Dyκ Cyκarc Byκκ s Eyκ Byκκs arc Byκκ s( )tan–( )– tan[ ] SVyκ+cos=

κ s κ SHyκ+=

Byκ rBy1 arc rBy2 α rBy3–( ) tan[ ] λ yκ⋅cos=

Cyκ rCy1=

Dyκ

Fyo

Cyκarc ByκSHyκ Eyκ ByκSHyκ arc ByκSHyκ( )tan–( )– tan[ ]cos---------------------------------------------------------------------------------------------------------------------------------------------------------------=

Eyκ rEy1 rEy2dfz with Eyκ 1≤+=

SHyκ rHy1 rHy2dfz+=

SVyκ DVyκ rVy5arc rVy6κ( )tan[ ]sin=

Adams/Tire

52

(78)

The weighting function appears is defined as:

(79)

Lateral Force Coefficients at Combined Slip

Formulas for Aligning Moment at Combined Slip

(80)

with:

(81)

Name:Name used in


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/Muy*Fz at Fznom

rVy2 RVY2 Variation of Svyk/Muy*Fz with load

rVy3 RVY3 Variation of Svyk/Muy*Fz with inclination

rVy4 RVY4 Variation of Svyk/Muy*Fz with alpha

rVy5 RVY5 Variation of Svyk/Muy*Fz with kappa

rVy6 RVY6 Variation of Svyk/Muy*Fz with atan (kappa)

DVyκ µyFz rVy1 rVy2dfz rVy3γ+ +( ) arc rVy4α( )tan[ ]cos⋅ ⋅=

Gyκ

Cyκarc Byκκ s Eyκ Byκκsarc Byκκs( )tan( )– tan[ ]cos

Cyκarc ByκSHyκ Eyκ ByκSHyκarc ByκSHyκ( )tan( )– tan[ ]cos---------------------------------------------------------------------------------------------------------------------------------------------------------=

M'z t Fy' Mzr s Fx⋅+ +⋅–=

t t α t eq,( )=

53Tire Models

(82)

(83)

(84)

(85)

with the arguments:

(86)

(87)

Aligning Moment Coefficients at Combined Slip

Formulas for Overturning Moment at Pure and Combined Slip

For the overturning moment, the formula reads both for pure and combined slip situations:

(88)

Name:Name used in


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

Dt Ctarc Btα t eq, Et Btα t eq, arc Btα t eq,( )tan–( )– tan[ ] α( )coscos=

F'y γ, 0= Fy SVyκ–=

Mzr Mzr α r eq,( ) Dr arc Brα r eq,( )tan[ ] α( )coscos= =

t t α t eq,( )=

α t eq, arc α2

t

Kx

Ky

------

2

κ2

α t( )sgn⋅+tantan=

α r eq, arc α2

r

Kx

Ky

------

2

κ2

α r( )sgn⋅+tantan=

Mx Ro Fz qSx1λ VMx qSx2 λ qSx3FyFz0--------⋅+⋅–

λ Mx⋅ ⋅=

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54

Overturning Moment Coefficients

Formulas for Rolling Resistance Moment at Pure and Combined Slip

The rolling resistance moment is defined by:

(89)

If qsy1 and qsy2 are both zero and FITTYP is equal to 5 (MF-Tyre 5.0), then the rolling resistance is

calculated according to an old equation:

(90)

Rolling Resistance Coefficients

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.


qsx1 QSX1 Lateral force induced overturning couple

qsx2 QSX2 Inclination induced overturning couple

qsx3 QSX3 Fy induced overturning couple


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

My Ro Fz qSy1 qSy3Fx Fz0⁄ qSy3 Vx Vref( )⁄ qSy4Vx Vref( )⁄4

+ + + ⋅ ⋅=

My R0 SVx Kx SHx⋅+( )=

κc κ SHxSVxKx

---------+ +=

55Tire Models

The following friction coefficients are defined:

The forces corrected for the combined slip conditions are:

α c α SHySVy

Ky

---------+ +=

α∗ α c( )sin=

βκc

κc2

α∗ 2+

-------------------------

acos=

µx act,

Fx 0, SVx–

Fz-------------------------= µy act,

Fy 0, SVy–

Fz-------------------------=

µx max,

Dx

Fz------= µy max,

Dy

Fz------=

µx1

1

µx act,-------------

2 βtan

µy max,----------------

2+

---------------------------------------------------------=

µyβtan

1

µx max,----------------

2 βtan

µy act,-------------

2+

---------------------------------------------------------=

Fxµx

µx act,-------------Fx 0,= Fy

µy

µy act,-------------Fy 0,=

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56

For aligning moment Mx, rolling resistance My and aligning moment Mz the formulae (76) until and

including (85) are used with =0.

Transient Behavior in PAC2002The 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 15 Hz) is called transient behavior. PAC2002 provides two methods to model

transient tire behavior:

• Stretched String

• Contact Mass

Stretched String Model

For accurate transient tire behavior, you can use the stretched string tire model (see reference [1]). The

tire belt is modeled as stretched string, which is supported to the rim with lateral (and longitudinal)

springs. Stretched String Model for Transient Tire Behavior shows a top-view of the string model. When

rolling, the first point having contact with the road adheres to the road (no sliding assumed). Therefore,

a lateral deflection of the string arises that depends on the slip angle size and the history of the lateral

deflection of previous points having contact with the road.

Stretched String Model for Transient Tire Behavior

SVyκ

57Tire Models


following differential equation is valid:

(91)

with the relaxation length in the lateral direction. The turnslip can be neglected at radii larger

than 10 m. This differential equation cannot be used at zero speed, but when multiplying with Vx, the

equation can be transformed to:

(92)




A similar approach yields the following for the deflection of the string in longitudinal direction:

(93)

Both the longitudinal and lateral relaxation length are defined as of the vertical load:

(94)

(95)

Now the practical slip quantities, and , are defined based on the tire deformation:

(96)

(97)

Using these practical slip quantities, and , the Magic Formula equations can be used to calculate

the tire-road interaction forces and moments:

(98)

1

Vx

------td

dv1 v1σ α------+ α( ) aφ+tan=

σ α φ

σ α td

dv1Vx v1+ σ αVsy=

σ x td

du1Vx u1+ σ xVsx=

σ x Fz pTx1 pTx2dfz+( ) pTx3dfz( ) R0 Fz0⁄( )λ σ x⋅exp⋅ ⋅=

σ α pTy1Fz0 2arcFz

pTy2Fz0λ Fz0( )---------------------------------

tan 1 pKy3 γy–( ) R0λ Fz0 λ σ α⋅⋅ ⋅sin=

κ α

κ'u1

σ x

------ Vx( )sin⋅=

α 'v1σ α------

atan=

κ α

Fx Fx α ' κ' Fz, ,( )=

Adams/Tire

58

(99)

(100)

Coefficients and Transient Response

Contact Mass Model

The contact mass model is based on the separation of the contact patch slip properties and the tire carcass

compliance (see reference [1]). Instead of using relaxation lengths to describe compliance effects, the

carcass springs are explicitly incorporated in the model. The contact patch is given some inertia to ensure

computational causality. This modeling approach automatically accounts for the lagged response to slip

and load changes that diminish at higher levels of slip. The contact patch itself uses relaxation lengths to

handle simulations at low speed.

The contact patch can deflect in longitudinal, lateral, and yaw directions with respect to the lower part of

the wheel rim. A mass is attached to the contact patch to enable straightforward computations.

The differential equations that govern the dynamics of the contact patch body are:

The contact patch body with mass mc and inertia Jc is connected to the wheel through springs cx, cy, and

c and dampers kx, ky, and k in longitudinal, lateral, and yaw direction, respectively.

The additional equations for the longitudinal u, lateral v, and yaw deflections are:


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

Fy Fy α ' κ' γ Fz, , ,( )=

M'z M'z α ' κ' γ Fz, , ,( )=

mc V·cx Vcyψ

·c–( ) kxu

· cxu+ + Fx=

mc V·cy Vcxψ

·c–( ) kyu

· cyu+ + Fy=

Jcψ··c kψ β· cψ β+ + Mz=

ψ ψ

β

u· Vcx Vsx–=

59Tire Models

in which Vcx, Vcy and are the sliding velocity of the contact body in longitudinal, lateral, and yaw

directions, respectively. Vsx, Vsy, and are the corresponding velocities of the lower part of the wheel.

The transient slip equations for side slip, turn-slip, and camber are:

where the calculated deflection angle has been used:

The tire total spin velocity is:

With the transient slip equations, the composite transient turn-slip quantities are calculated:

v· Vcy Vsy–=

β· ψ· c ψ–=

ψ· c

ψ·

σ c td

d α ' Vx α '+ Vcy Vxβ– Vx βst+=

σ c td

dα 'tVx α 't+ Vx α '=

σ c td

dϕ 'cVx ϕ 'c+ ψ· γ=

σ F2 td

dϕ 'F2Vx ϕ 'cF2+ ψ· γ=

σ ϕ 1 td

dϕ '1Vx ϕ '1+ ψ· γ=

σ ϕ 2 td

dϕ '2Vx ϕ '2+ ψ· γ=

βstMz

cφ-------=

ψ· γ ψ c 1 εγ–( )Ω γsin–=

ϕ 'F 2ϕ 'c ϕ 'F2–=

ϕ 'M εϕ ϕ 'c εϕ 12 ϕ '1 ϕ '2–( )+=

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60

The tire forces are calculated with and the tire moments with .

The relaxation lengths are reduced with slip:

Here a is half the contact length according to:

The composite tire parameter reads:

and the equivalent slip:


Name:Name used in


mc MC Contact body mass

Ic IC Contact body moment of inertia

kx KX Longitudinal damping

ky KY Lateral damping

k KP Yaw damping

cx CX Longitudinal stiffness

ϕ 'F ϕ 'M

σ c a 1 θζ–( )⋅=

σ 2

t0

a----σ c=

σ F2 bF2σ c=

σ ϕ 1 bϕ 1σ c=

σ ϕ 2 bϕ 2σ c=

a pA1R0 ρz

R0

------ pA2ρ zR0

------+

=

θKy0

2µyFx---------------=

ζ1

1 κ'+------------- α ' aεϕ 12 ϕ '1 ϕ '2–+

2 Kx0

Ky0

---------

2

κ'2

3---b ϕ 'c+

2

+=

ϕ

61Tire Models

The remaining contact mass model parameters are estimated automatically based on longitudinal and

lateral stiffness specified in the tire property file.

Gyroscopic Couple in PAC2002When 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:

(101)

with the parameter (in addition to the basic tire parameter mbelt):

(102)

and:

(103)

The total aligning moment now becomes:

(104)

cy CY Lateral stiffness

c CP Yaw stiffness

pA1 PA1 Half contact length with vertical tire deflection

pA2 PA2 Half contact length with square root of vertical tire

deflection

EP Composite turn-slip (moment)

EP12 Composite turn-slip (moment) increment

bF2 BF2 Second relaxation length factor

b 1 BP1 First moment relaxation length factor

b 2 BP2 Second moment relaxation length factor

Name:Name used in


ϕ

εϕεϕ 12

ϕϕ

Mz gyr, cgyrmbeltVrl td

dvarc Brα r eq,( )tan[ ]cos=

cgyr qTz1 λ gyr⋅=

arc Brα r eq,( )tan[ ]cos 1=

Mz M'z Mz gyr,+=

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62


Left and Right Side TiresIn 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 and 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.

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.

Create Wheel and Tire Dialog Box in Adams/View

Name:Name used in




pTx3 PTX3 Variation of longitudinal relaxation length with exponent of

load





63Tire Models

USE_MODES of PAC2002: from Simple to ComplexThe parameter USE_MODE in the tire property file allows you to switch the output of the PAC2002 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.

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64

USE_MODE Values of PAC2002 and Related Tire Model Output

Quality Checks for the Tire Model ParametersBecause PAC2002 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:

• Rolling Resistance

• Camber (Inclination) Effects

• Validity Range of the Tire Model Input

USE_MODE: State: Slip conditions:

PAC2002 output(forces and moments):

0 Steady state Acts as a vertical spring &

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)


14 Transient Combined slip Fx, Fy, Fz, Mx, My, Mz

15 Transient Combined slip and Fx, Fy, Fz, Mx, My, Mz

21 Advanced transient Pure longitudinal slip Fx, 0, Fz, My, 0

22 Advanced transient Pure lateral (cornering slip) 0, Fy, Fz, Mx, 0, Mz

23 Advanced transient Longitudinal and lateral (not

combined)


24 Advanced transient Combined slip Fx, Fy, Fz, Mx, My, Mz

25 Advanced transient Combined slip and Fx, Fy, Fz, Mx, My, Mz

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.

65Tire Models

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%); for heavy commercial truck tires, it can be around 0.006 (0.6%).

Tire property files with the keyword FITTYP=5 determine the rolling resistance in a different way (see

equation (85)). To avoid the ‘old’ rolling resistance calculation, remove the keyword FITTYP and add a

section like the following:

$---------------------------------------------------rolling resistance[ROLLING_COEFFICIENTS]QSY1 = 0.01QSY2 = 0QSY3 = 0QSY4 = 0

Camber (Inclination) Effects

Camber stiffness has not been explicitly defined in PAC2002; however, for car tires, positive inclination

should result in a negative lateral force at zero slip angle. 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 PAC2002 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 table below lists further checks on the PAC2002 parameters.

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66

Checklist for PAC2002 Parameters and Properties

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

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 (22))

pDx1/pDx2 < 0 Peak friction Fx must decrease with increasing load

Kx > 0 Long slip stiffness (see equation (25))

Dy > 0 Peak friction (see equation (34))

pDy1/pDy2 < 0 Peak friction Fx must decrease with increasing load

Ky < 0 Cornering stiffness (see equation (37))

qsy1 > 0 Rolling resistance, in the range of 0.005 - 0.015

67Tire Models

[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 PAC2002Because 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 also 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 used to develop 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 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

• Tire Kinematics

• Slip Quantities

• Force and Moments

Adams/Tire

68

General

General Definitions

Tire Kinematics

Tire Kinematics Definitions

Slip Quantities

Slip Quantities 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.

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 [rads-1]


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]

ρρρ

ω

69Tire Models

Forces and Moments

Force and Moment Definitions

References1. 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.

Example of PAC2002 Tire Property File[MDI_HEADER]FILE_TYPE

='tir'FILE_VERSION

=3.0FILE_FORMAT

Vy Lateral speed of tire contact center [ms-1]

Vr Linear speed of rolling [ms-1]


Slip angle [rad]


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]


καγ

Adams/Tire

70

='ASCII'! : TIRE_VERSION :

PAC2002! : COMMENT :

Tire 235/60R16

! : COMMENT : Manufacturer

! : COMMENT : Nom. section with

(m) 0.235

! : COMMENT : Nom. aspect ratio

(-) 60

! : COMMENT : Infl. pressure

(Pa) 200000

! : COMMENT : Rim radius

(m) 0.19

! : COMMENT : Measurement ID

! : COMMENT : Test speed

(m/s) 16.6! : COMMENT :

Road surface! : COMMENT :

Road condition Dry

! : FILE_FORMAT : ASCII

! : Copyright MSC.Software, Fri Jan 23 14:30:06 2004!! 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]

71Tire Models

LENGTH ='meter'

FORCE ='newton'

ANGLE ='radians'

MASS ='kg'

TIME ='second'

$----------------------------------------------------------------model[MODEL]PROPERTY_FILE_FORMAT='PAC2002'USE_MODE

= 14 $Tyre use switch (IUSED)

VXLOW = 1

LONGVL = 16.6

$Measurement speedTYRESIDE

= 'LEFT' $Mounted side of tyre at vehicle/test bench

$-----------------------------------------------------------dimensions[DIMENSION]UNLOADED_RADIUS

= 0.344 $Free tyre radius

WIDTH = 0.235

$Nominal section width of the tyreASPECT_RATIO

= 0.6 $Nominal aspect ratio

RIM_RADIUS = 0.19

$Nominal rim radiusRIM_WIDTH

= 0.16 $Rim width

$----------------------------------------------------------------shape[SHAPE]radial width 1.0

0.0 1.0

0.4 1.0

0.9 0.9

Adams/Tire

72

1.0$------------------------------------------------------------parameter[VERTICAL]VERTICAL_STIFFNESS = 2.1e+005 $Tyre vertical stiffness

VERTICAL_DAMPING = 50

$Tyre vertical dampingBREFF

= 8.4 $Low load stiffness e.r.r.

DREFF = 0.27

$Peak value of e.r.r.FREFF

= 0.07 $High load stiffness e.r.r.

FNOMIN = 4850

$Nominal wheel load$-----------------------------------------------------------load_curve$ For a non-linear tire vertical stiffness (optional)$ Maximum of 100 points[DEFLECTION_LOAD_CURVE]pen

fz0.000

0.00.001

212.00.002

428.00.003

648.00.005

1100.00.010

2300.00.020

5000.00.030

8100.0$------------------------------------------------------long_slip_range[LONG_SLIP_RANGE]KPUMIN


KPUMAX = 1.5

$Maximum valid wheel slip

73Tire Models

$-----------------------------------------------------slip_angle_range[SLIP_ANGLE_RANGE]ALPMIN


ALPMAX = 1.5708

$Maximum valid slip angle$-----------------------------------------------inclination_slip_range[INCLINATION_ANGLE_RANGE]CAMMIN


CAMMAX = 0.26181

$Maximum valid camber angle$-------------------------------------------------vertical_force_range[VERTICAL_FORCE_RANGE]FZMIN


FZMAX = 10125

$Maximum allowed wheel load$--------------------------------------------------------------scaling[SCALING_COEFFICIENTS]LFZO

= 1 $Scale factor of nominal (rated) load

LCX = 1

$Scale factor of Fx shape factorLMUX

= 1 $Scale factor of Fx peak friction coefficient

LEX = 1

$Scale factor of Fx curvature factorLKX

= 1 $Scale factor of Fx slip stiffness

LHX = 1

$Scale factor of Fx horizontal shiftLVX

= 1 $Scale factor of Fx vertical shift

LGAX = 1

$Scale factor of camber for Fx

Adams/Tire

74

LCY = 1

$Scale factor of Fy shape factorLMUY

= 1 $Scale factor of Fy peak friction coefficient

LEY = 1

$Scale factor of Fy curvature factorLKY

= 1 $Scale factor of Fy cornering stiffness

LHY = 1

$Scale factor of Fy horizontal shiftLVY

= 1 $Scale factor of Fy vertical shift

LGAY = 1

$Scale factor of camber for FyLTR

= 1 $Scale factor of Peak of pneumatic trail

LRES = 1

$Scale factor for offset of residual torqueLGAZ

= 1 $Scale factor of camber for Mz

LXAL = 1

$Scale factor of alpha influence on FxLYKA

= 1 $Scale factor of alpha influence on Fx

LVYKA = 1

$Scale factor of kappa induced FyLS

= 1 $Scale factor of Moment arm of FxL

SGKP = 1

$Scale factor of Relaxation length of FxLSGAL

= 1 $Scale factor of Relaxation length of Fy

LGYR = 1

$Scale factor of gyroscopic torqueLMX

= 1 $Scale factor of overturning couple

75Tire Models

LVMX = 1

$Scale factor of Mx vertical shiftLMY

= 1 $Scale factor of rolling resistance torque

$---------------------------------------------------------longitudinal[LONGITUDINAL_COEFFICIENTS]PCX1

= 1.6411 $Shape factor Cfx for longitudinal force

PDX1 = 1.1739 $Longitudinal friction Mux at Fznom

PDX2 = -0.16395

$Variation of friction Mux with loadPDX3

= 0 $Variation of friction Mux with camber

PEX1 = 0.46403 $Longitudinal curvature Efx at Fznom

PEX2 = 0.25022 $Variation of curvature Efx with load

PEX3 = 0.067842

$Variation of curvature Efx with load squaredPEX4

= -3.7604e-005 $Factor in curvature Efx while driving

PKX1 = 22.303 $Longitudinal slip stiffness Kfx/Fz at Fznom

PKX2 = 0.48896 $Variation of slip stiffness Kfx/Fz with load

PKX3 = 0.21253 $Exponent in slip stiffness Kfx/Fz with load

PHX1 = 0.0012297

$Horizontal shift Shx at FznomPHX2

= 0.0004318 $Variation of shift Shx with load

PVX1 = -8.8098e-006

$Vertical shift Svx/Fz at FznomPVX2

= 1.862e-005 $Variation of shift Svx/Fz with loadRBX1

Adams/Tire

76

= 13.276 $Slope factor for combined slip Fx reduction

RBX2 = -13.778

$Variation of slope Fx reduction with kappaRCX1

= 1.2568 $Shape factor for combined slip Fx reduction

REX1 = 0.65225

$Curvature factor of combined FxREX2

= -0.24948 $Curvature factor of combined Fx with load

RHX1 = 0.0050722

$Shift factor for combined slip Fx reductionPTX1

= 2.3657 $Relaxation length SigKap0/Fz at Fznom

PTX2 = 1.4112 $Variation of SigKap0/Fz with load

PTX3 = 0.56626

$Variation of SigKap0/Fz with exponent of load$----------------------------------------------------------overturning[OVERTURNING_COEFFICIENTS]QSX1

= 0 $Lateral force induced overturning moment

QSX2 = 0

$Camber induced overturning coupleQSX3

= 0 $Fy induced overturning couple

$--------------------------------------------------------------lateral[LATERAL_COEFFICIENTS]PCY1

= 1.3507 $Shape factor Cfy for lateral forces

PDY1 = 1.0489 $Lateral friction Muy

PDY2 = -0.18033

$Variation of friction Muy with loadPDY3

= -2.8821 $Variation of friction Muy with squared camber

PEY1

77Tire Models

= -0.0074722 $Lateral curvature Efy at FznomPEY2

= -0.0063208 $Variation of curvature Efy with loadPEY3

= -9.9935 $Zero order camber dependency of curvature Efy

PEY4 = -760.14 $Variation of curvature Efy with camber

PKY1 = -21.92 $Maximum value of stiffness Kfy/Fznom

PKY2 = 2.0012 $Load at which Kfy reaches maximum value

PKY3 = -0.024778

$Variation of Kfy/Fznom with camberPHY1

= 0.0026747 $Horizontal shift Shy at Fznom

PHY2 = 8.9094e-005$Variation of shift Shy with load

PHY3 = 0.031415

$Variation of shift Shy with camberPVY1

= 0.037318 $Vertical shift in Svy/Fz at Fznom

PVY2 = -0.010049

$Variation of shift Svy/Fz with loadPVY3

= -0.32931 $Variation of shift Svy/Fz with camber

PVY4 = -0.69553

$Variation of shift Svy/Fz with camber and loadRBY1

= 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 reductionRCY1

= 1.0719 $Shape factor for combined Fy reduction

REY1 = -0.27572

$Curvature factor of combined FyREY2

Adams/Tire

78

= 0.32802 $Curvature factor of combined Fy with load

RHY1 = 5.7448e-006$Shift factor for combined Fy reduction

RHY2 = -3.1368e-005

$Shift factor for combined Fy reduction RVY1

= -0.027825 $Kappa induced side force Svyk/Muy*Fz at Fznom

RVY2 = 0.053604

$Variation of Svyk/Muy*Fz with loadRVY3

= -0.27568 $Variation of Svyk/Muy*Fz with camber

RVY4 = 12.12 $Variation of Svyk/Muy*Fz with alpha

RVY5 = 1.9$Variation of Svyk/Muy*Fz with kappa

RVY6 = -10.704

$Variation of Svyk/Muy*Fz with atan(kappa)PTY1

= 2.1439 $Peak value of relaxation length SigAlp0/R0

PTY2 = 1.9829 $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 FxQSY3

= 0 $Rolling resistance torque depending on speed

QSY4 = 0

$Rolling resistance torque depending on speed ^4$-------------------------------------------------------------aligning[ALIGNING_COEFFICIENTS]QBZ1

= 10.904 $Trail slope factor for trail Bpt at Fznom

QBZ2 = -1.8412

$Variation of slope Bpt with load

79Tire Models

QBZ3 = -0.52041

$Variation of slope Bpt with load squaredQBZ4

= 0.039211 $Variation of slope Bpt with camber

QBZ5 = 0.41511 $Variation of slope Bpt with absolute camber

QBZ9 = 8.9846 $Slope factor Br of residual torque Mzr

QBZ10 = 0

$Slope factor Br of residual torque MzrQCZ1

= 1.2136 $Shape factor Cpt for pneumatic trail

QDZ1 = 0.093509

$Peak trail Dpt" = Dpt*(Fz/Fznom*R0)QDZ2

= -0.0092183 $Variation of peak Dpt" with loadQDZ3

= -0.057061 $Variation of peak Dpt" with camber

QDZ4 = 0.73954 $Variation of peak Dpt" with camber squared

QDZ6 = -0.0067783 $Peak residual torque Dmr" = Dmr/(Fz*R0)

QDZ7 = 0.0052254

$Variation of peak factor Dmr" with loadQDZ8

= -0.18175 $Variation of peak factor Dmr" with camber

QDZ9 = 0.029952

$Var. of peak factor Dmr" with camber and loadQEZ1

= -1.5697 $Trail curvature Ept at Fznom

QEZ2 = 0.33394 $Variation of curvature Ept with load

QEZ3 = 0

$Variation of curvature Ept with load squaredQEZ4

= 0.26711 $Variation of curvature Ept with sign of Alpha-t

QEZ5 = -3.594

Adams/Tire

80

$Variation of Ept with camber and sign Alpha-tQHZ1

= 0.0047326 $Trail horizontal shift Sht at Fznom

QHZ2 = 0.0026687

$Variation of shift Sht with loadQHZ3

= 0.11998 $Variation of shift Sht with camber

QHZ4 = 0.059083

$Variation of shift Sht with camber and loadSSZ1

= 0.033372 $Nominal value of s/R0: effect of Fx on Mz

SSZ2 = 0.0043624

$Variation of distance s/R0 with Fy/FznomSSZ3

= 0.56742 $Variation of distance s/R0 with camber

SSZ4 = -0.24116

$Variation of distance s/R0 with load and camberQTZ1

= 0.2 $Gyration torque constant

MBELT = 5.4

$Belt mass of the wheel$-----------------------------------------------turn-slip parameters

[TURNSLIP_COEFFICIENTS]

PECP1 = 0.7

$Camber stiffness reduction factor

PECP2 = 0.0

$Camber stiffness reduction factor with load

PDXP1 = 0.4

$Peak Fx reduction due to spin

PDXP2 = 0.0

$Peak Fx reduction due to spin with load

PDXP3 = 0.0

$Peak Fx reduction due to spin with longitudinal slip

81Tire Models

PDYP1 = 0.4

$Peak Fy reduction due to spin

PDYP2 = 0.0

$Peak Fy reduction due to spin with load

PDYP3 = 0.0

$Peak Fy reduction due to spin with lateral slip

PDYP4 = 0.0

$Peak Fy reduction with square root of spin

PKYP1 = 1.0

$Cornering stiffness reduction due to spin

PHYP1 = 1.0

$Fy lateral shift shape factor

PHYP2 = 0.15

$Maximum Fy lateral shift

PHYP3 = 0.0

$Maximum Fy lateral shift with load

PHYP4 = -4.0

$Fy lateral shift curvature factor

QDTP1 = 10.0

$Pneumatic trail reduction factor

QBRP1 = 0.1

$Residual torque reduction factor with lateral slip

QCRP1 = 0.2

$Turning moment at constant turning with zero speed

QCRP2 = 0.1

$Turning moment at 90 deg lateral slip

QDRP1

Adams/Tire

82

= 1.0 $Maximum turning moment

QDRP2 = -1.5

$Location of maximum turning moment

$----------------------------------------------contact patch parameters

[CONTACT_COEFFICIENTS]

PA1 = 0.4147

$Half contact length dependency on Fz)

PA2 = 1.9129

$Half contact length dependency on sqrt(Fz/R0)

$-----------------------------------------------contact patch slip model

[DYNAMIC_COEFFICIENTS]

MC = 1.0 $Contact mass

IC = 0.05 $Contact moment of inertia

KX = 409.0

$Contact longitudinal damping

KY = 320.8

$Contact lateral damping

KP = 11.9 $Contact yaw damping

CX = 4.350e+005 $Contact longitudinal stiffness

CY = 1.665e+005 $Contact lateral stiffness

CP = 20319

$Contact yaw stiffness

83Tire Models

EP = 1.0

EP12 = 4.0

BF2 = 0.5

BP1 = 0.5

BP2 = 0.67

$--------------------------------------------------------loaded radius[LOADED_RADIUS_COEFFICIENTS]QV1

= 0.000071 $Tire radius growth coefficient

QV2 = 2.489 $Tire stiffness variation coefficient with speed

QFCX1 = 0.1

$Tire stiffness interaction with FxQFCY1

= 0.3 $Tire stiffness interaction with Fy

QFCG1 = 0.0

$Tire stiffness interaction with camberQFZ1

= 0.0 $Linear stiffness coefficient, if zero, VERTICAL_STIFFNESS is

takenQFZ2

= 14.35 $Tire vertical stiffness coefficient (quadratic)

Contact MethodsThe PAC2002 model supports the following roads:

• 2D Roads, see Using the 2D Road Model


Note that the PAC2002 model has only one point of contact with the road; therefore, the wavelength of

road obstacles must be longer than the tire radius for realistic output of the model. In addition, the contact

force computed by this tire model is normal to the road plane. Therefore, the contact point does not

generate a longitudinal force when rolling over a short obstacle, such as a cleat or pothole.

Adams/Tire

84

For ride and comfort analyses, we recommend more sophisticated tire models, such as Ftire.

85Tire Models

Using the PAC-TIME Tire ModelThe 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

• Modeling of Tire-Road Interaction Forces

• Axis Systems and Slip Definitions

• Contact Point and Normal Load Calculation

• Basics of Magic Formula

• Steady-State: Magic Formula

• Transient Behavior

• Gyroscopic Couple


• USE_MODES OF PAC-TIME: from Simple to Complex

• Quality Checks for Tire Model Parameters

• Standard Tire Interface (STI)

• Definitions

• References

• Example of PAC-TIME Tire Property File

• Contact Methods

When to Use PAC-TIMEMagic-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

Adams/Tire

86

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




• Single- or double-lane change





• Other common vehicle dynamics maneuvers on rather smooth roads (wavelength of road


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 ForcesFor vehicle dynamics applications, an accurate knowledge of tire-road interaction forces is inevitable


forces depend on both road and tire properties, and the motion of the tire with respect to the road.



the tire and wheel as a rigid disc. In the contact point between the tire and the road, the contact forces in


road.


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:




87Tire Models

• Road parameters


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.


Adams/Solver through STI.


Axis Systems and Slip Definitions• Axis Systems

• Units


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].

κ α γ

Adams/Tire

88

The STI interface between the MF-TIME model and Adams/Solver mainly passes information to the tire


because all the modeling of the tire behavior, as described in this help, assumes to deal with the slip


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 forces and moments calculated by PAC-TIME using the MF equations in this guide are in the W-axis


to Adams/Solver.



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.



89Tire Models

SI Units Used in PAC-TIME



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:

Variable type: Name: Abbreviation: Unit:

Angle Slip angle

Inclination angle

Radians


Lateral force

Vertical load

Fx

Fy

Fz

Newton




Mx

My

Mz

Newton.meter


Lateral speed


Lateral slip speed

Vx

Vy

Vsx

Vsy

Meters per second


α

γ

ω

ω

Adams/Tire

90

(105)


(106)



(107)

(108)


(109)

Contact Point and Normal Load Calculation• Contact Point


Contact Point





at the road point right below the wheel center (see the figure below).


Vsx Vx ΩRe–=

Vsy Vy=

κ α

κVsx

Vx

--------–=

αtanVsy

Vx

---------=

Vr ReΩ=

91Tire Models


(ground) plane and the line of intersection of the wheel center-plane with the plane though the wheel spin

axis.

The normal load Fz of the tire is calculated with:

(110)

where 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.


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 Rl is:

(111)

In this tire model, a constant (linear) vertical tire stiffness Cz is assumed; therefore, the tire deflection

can be calculated using:

Fz Czρ Kz ρ·⋅+=

ρ ρ

ρ

R1 R0 ρ–=

ρ

Adams/Tire

92

(112)



(113)

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



To represent the effective rolling radius Re, an MF type of equation is used:

(114)

ρFzCz

------=

Re

Vx

Ω------=

Re R0 ρ Fz0 Darc Bρd

( ) Fρd

+tan( )–=

93Tire Models

in which Fz0 is the nominal tire deflection:

(115)

and d is called the dimensionless radial tire deflection, defined by:

(116)



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

ρ

ρ Fz0Fz0Cz

--------=

ρ

ρ d ρρ Fz0---------=

Adams/Tire

94

Basics of the Magic Formula in PAC-TIMEThe Magic Formula is a mathematical formula that is capable of describing the basic tire characteristics


We distinguish:




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:

(117)

where Y(x) is either Fx with x the longitudinal slip , or Fy and x the lateral slip .


F FREFF High load stiffness effective rolling radius

Cz VERTICAL_STIFFNESS Tire vertical stiffness

Kz VERTICAL_DAMPING Tire vertical damping

Name:Name Used in Tire Property File: Explanation:

α

κ


κ α

95Tire Models

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,


slip has a cosine shape, a cosine version the Magic Formula is used:

(118)










α


α

Adams/Tire

96

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.




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

97Tire Models

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:

(119)

In the PAC-TIME tire model, C has been set to ½, and E is not used (zero value). This approach

results in an explicit definition of the camber stiffness, because:

(120)

Input Variables


Input Variables

Output Variables

Its output variables are:

Output Variables.


Slip angle [rad]





γ

Fy0 Dy Cyarc Byα y Ey Byα y arc Byαy( )tan–( )– Cγarc Byαγ Eγ Bγαγ arc Bγαγ( )tan–( )– tan+

tan[]

cos=

γ γ

Kγ BγCγDγ

Fyo

γδ-------- at αγ∂ 0= = =

καγ

Adams/Tire

98




are:



(121)

with the possibly adapted nominal load (using the user-scaling factor, Fz0):

(122)



logic:



Rolling resistance moment My [Nm]










dfzFz F'z0–

F'z0--------------------=

λ

F'z0 Fz0 λ Fz0⋅=

99Tire Models




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.



C Shape factor

D Peak value

E Curvature factor


H Horizontal shift

V Vertical shift






k = 1, 2, ...

Name:Name used in


Fzo LFZO Scale factor of nominal (rated) load


LMUX Scale factor of Fx peak friction coefficient




Hx LHX Scale factor of Fx horizontal shift

xx LGAX Scale factor of camber for Fx

Cy LCY Scale factor of Fy shape factor for side slip



λ µξ

λ γψ

λλλ µξλλλλλ γλλ µ

Adams/Tire

100



Steady-State: Magic Formula in PAC-TIME• Steady-State Pure Slip


EyEy LEY Scale factor of Fy curvature factor

KyKy LKY Scale factor of Fy cornering stiffness

VVy LVY Scale factor of Fy vertical shift

HyHyy LHY Scale factor of Fy horizontal shift

K K LKC Scale factor of camber stiffness (K-factor)

LGAY Scale factor of camber force stiffness


Mr LRES Scale factor for offset of residual torque

z LGAZ Scale factor of camber torque stiffness


VMx LVMX Scale factor of Mx vertical shift

My LMY Scale factor of rolling resistance torque


x LXAL Scale factor of alpha influence on Fx

y LYKA Scale factor of alpha influence on Fx

Vy LVYKA Scale factor of kappa induced Fy

s LS Scale factor of moment arm of Fx


LSGKP Scale factor of relaxation length of Fx

LSGAL Scale factor of relaxation length of Fy


Name:Name used in


λλλλλ γ γλ γψλλλ γλλλ

λ αλ κλ κλ

λ σ κκλ σ ααλ

101Tire Models







(123)

(124)

(125)

(126)


(127)

(128)

(129)

(130)


(131)

(132)

(133)


Fx0 Dx Cxarc Bxκx Ex Bxκx arc Bxκs( )tan–( )– tan[ ] SVx+cos=

κx κ SHx+=

γx γ λ γx⋅=

Cx pCx1 λ cx⋅=

Dx µx Fz⋅=

µx pDx1 pDx2dfz+( ) 1 pDx3 γx2⋅( )γµx–⋅=

Ex pEx1 pEx2dfz pEx2dfz2+ +( ) 1 pEx4 κx( )sgn– λ Ex with Ex 1≤⋅ ⋅=

Kx Fz pKx1 pKx2dfz+( ) pKx3dfz( )λ K

Kx

exp⋅

BxCxDx κx∂

∂ Fx0at κx 0

=

= = =

Bx Kx CxDx( )⁄=

SHx pHx1 pHx2dfz+( )λ Hx=

SVx Fz pVx1 pVx2dfz+( ) λ Vx λ µx⋅ ⋅ ⋅=

Adams/Tire

102



(134)

(135)

(136)


(137)

with coefficients:

(138)

(139)

(140)

(141)






pDx3 PDX3 Variation of friction Mux with inclination








Fy0 Dy Cyarc Byαy Ey Byα y arc Byα y( )tan–( )– 1

2---arc Bγγy( )tan+tan SVy+sin=

α y α SHy+=

γy γ λ γy⋅=

Cy pCy1 λ Cy⋅=

Dy µy Fz⋅=

µy pDy1 pDy2dfz+( ) 1 pDy3γy2–( ) λ µy⋅ ⋅=

Ey pEy1 pEy2dfz pEy3 pEy4γy+( ) α y( )sgn+ + λ Ey⋅=

103Tire Models

() (142)

(143)

(144)

(145)

(146)

(147)





pDy2 PDY2 Variation of friction Muy with load

pDy3 PDY3 Variation of friction Muy with squared inclination


pEy2 PEY2 Variation of curvature Efy with load

pEy3 PEY3 Inclination dependency of curvature Efy

pEy4 PEY4 Variation of curvature Efy with inclination


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 Kg with load

pKy6 PKY6 Quadratic variation of Kg with load

Ky α∂

∂ FypKy1Fz0 pKy4arc

FzpKy2Fz0λ Fz0-------------------------------

tan 1 pKy3γy2–( ) λ Kyλ Fz0⋅ ⋅sin= =

with pKy4 2≤

Kγ γ∂

∂ FyFz pKy5 pKy5dfz+( ) λ Ky⋅= =

By

Ky

CyDy

-------------=

Bγ

2Kγ

Dy

---------=

SHy pHy1 pHy2dfz+( ) λ Hy⋅=

SHy Fz pVy1 pVy2dfz+( ) λ Vyλ µy⋅=

Adams/Tire

104


(148)


(149)


(150)

(151)

(152)


(153)

with coefficients:

(154)

(155)

(156)


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


M'z Mz0 α γ Fz, ,( )=

Mz0 t Fy0 SVy–( )γ 0=Mzr+⋅–=


α t α SHt+=

Mzr α r( ) Dr arc Brα r( )tan[ ] α( )coscos=

α r α SHt+=

SHr 0=

γz γ λ γz⋅=

Bt qBz1 qBz2dfz+( ) 1 qBz4γz2 qBz5 γz+ +( ) λ k λ µy⁄⋅( )=

Ct qCz1=

Dt Fz qDz1 qDz2dfz+( ) R0 Fz0⁄( ) λ t⋅=

105Tire Models

(157)

(158)

(159)

(160)


(161)

and the aligning stiffness for inclination is:

(162)





qBz4 QBZ4 Variation of slope Bpt with inclination

qBz5 QBZ5 Variation of slope Bpt with absolute inclination






qDz8 QDZ8 Variation of peak factor Dmr with inclination

qDz9 QDZ9 Variation of Dmr with inclination and load


Et qEz1 qEz2dfz+( ) 1 qEz42

π---arc BtCtα t( )tan+

with Et 1≤

=

SHt qHz1 qHz2dfz qHz3 qHz4dfz+( )γz+ +=

Br λ Kz λ µy⁄=

Dr Fz qDz6 qDz7dfz+( )λ r qDz8 qDz9dfz+( )γz+ R0λ µy=

α∂

∂ MztKy– Fz qDz1 qDz2dfz+( ) R0 Fz0⁄( ) pKy5 pKy6dfz+( )Fz λ tλ Ky⋅= =

γd

dMzqDz8 qDz9dfz+( )R0Fzλ µy=

Adams/Tire

106


PAC-TIME has two methods for calculating the combined slip forces and moments. If the user supplies












(163)

with Gx the weighting function of the longitudinal force for pure slip.

We write:

(164)

(165)

with coefficients:

(166)

(167)



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



α


α s α SHxα+=

Bxα rBx1 arc rBx2κ tan[ ] λ xα⋅cos=

Cxα rCx1=

107Tire Models

(168)

(169)

(170)


(171)



(172)



(173)

(174)


(175)

(176)








Dxα

FxoCxαarc BxαSHxα Exα BxαSHxα arc BxαSHxα( )tan–( )– tan[ ]cos

------------------------------------------------------------------------------------------------------------------------------------------------------------------=


SHxα rHx1=

Gxα



Fy Fy0 Gyκ α κ γ Fz, , ,( ) SVyκ+⋅=

κ

Fy Dyκ Cyκarc Byκκ s Eyκ Bxακs arc Byκκs( )tan–( )– tan[ ] SVyκ+cos=

κ s κ SHyκ+=

Byκ rBy1 arc rBy2 α rBy3–( ) tan[ ] λ yκ⋅cos=

Cyκ rCy1=

Adams/Tire

108

(177)

(178)

(179)

(180)


(181)



(182)

with:

(183)










rVy1 RVY1 Kappa induced side force SVyk/my·Fz at Fznom

rVy2 RVY2 Variation of SVyk/my·Fz with load

rVy3 RVY3 Variation of SVyk/my·Fz with inclination

rVy4 RVY4 Variation of SVyk/my·Fz with a

rVy5 RVY5 Variation of SVyk/my·Fz with k

rVy6 RVY6 Variation of SVyk/my·Fz with atan(k)

Dyκ

FyoCyκarc ByκSHyκ Eyκ BxαSHyκ arc ByκSHyκ( )tan–( )– tan[ ]cos

---------------------------------------------------------------------------------------------------------------------------------------------------------------=




Gyκ

Cyκarc Byκκs Eyκ Bxακs arc Byκκs( )tan–( )– tan[ ]cos

Cyκarc ByκSHyκ Eyκ BxαSHyκ arc ByκSHyκ( )tan–( )– tan[ ]cos---------------------------------------------------------------------------------------------------------------------------------------------------------------=

M'z t F'y Mzr s Fx⋅+ +⋅–=

t t α t eq,( )=

109Tire Models

(184)

(185)

(186)

with the arguments:

(187)

(188)




(189)



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


qsx1 QSX1 Lateral force induced overturning couple

qsx2 QSX2 Inclination induced overturning couple

qsx3 QSX3 Fy induced overturning couple

Dt Ctarc Btα t eq, Et Btα t eq, arc Btα t eq,( )tan–( )– tan[ ] α( )coscos=



t t α t eq,( )=

α t eq, arc α2

t

Kx

Ky

------

2

κ2 α t( )sgn⋅+tantan=

α r eq, arc α2

r

Kx

Ky

------

2

κ2 α r( )sgn⋅+tantan=

Mx Ro Fz qsx1λ VMx qsx2 qsx3FyFz0--------⋅+–

λ Mx⋅ ⋅=

Adams/Tire

110



(190)









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


My Ro Fz qSy1 qSy2Fx Fz0⁄ qSy3 Vx Vref( )⁄ qSy4 Vx Vref⁄( )4+ + ⋅ ⋅=

κc κ SHxSVx

Kx

---------+ +=

α c α SHySVyKy

---------+ +=

α∗ α c( )sin=

βκc

κc2

α∗ 2+

-------------------------

acos=

µx act,

Fx 0, SVx–

Fz-------------------------= µy act,

Fy 0, SVy–

Fz-------------------------=

µx max,

Dx

Fz------= µy max,

Dy

Fz------=

111Tire Models




Transient Behavior in PAC-TIMEThe previous Magic Formula equations are valid for steady-state tire behavior. When driving, however,


frequency behavior (up to 8 Hz) is called transient behavior.

For accurate transient tire behavior, you can use the stretched string tire model (see reference [1]). The

tire belt is modeled as stretched string, which is supported to the rim with lateral (and longitudinal)

springs. Stretched String Model for Transient Tire Behavior shows a top-view of the string model. When

rolling, the first point having contact with the road adheres to the road (no sliding assumed). Therefore,

a lateral deflection of the string arises that depends on the slip angle size and the history of the lateral

deflection of previous points having contact with the road.


µx1

1

µx act,-------------

2 βtan

µy max,----------------

2+

---------------------------------------------------------=

µyβtan

1

µx max,----------------

2 βtan

µy act,-------------

2+

---------------------------------------------------------=

Fxµx

µx act,-------------Fx 0,= Fy

µy

µy act,-------------Fy 0,=

SVyκ

Adams/Tire

112



(191)




(192)





1

Vx

------td

dv1 v1σ α------+ α( ) aφ+tan=

σ α φ

σ α td

dv1Vx v1+ σ αVsy=

113Tire Models

(193)


(194)

(195)


(196)

(197)

Using these practical slip quantities, and , the Magic Formula equations can be used to calculate


(198)

(199)

(200)

Gyroscopic Couple in PAC-TIMEWhen having fast rotations about the vertical axis in the wheel plane, the inertia of the tire belt may lead



(201)

with the parameters (in addition to the basic tire parameter mbelt):

(202)

and:

σ x td

du1Vx u1+ σ xVsx–=

σ x Fz pTx1 pTx2dfz+( ) pTx3dfz( ) R0 Fz0⁄( )λ σ κ⋅exp⋅ ⋅=

σ α pTy1Fz0 pKy4arcFz

pTy2Fz0λ Fz0-----------------------------

tan 1 pKy3γ2–( ) R0λ Fz0λ σ α⋅ ⋅sin=

κ' α '

κ'u1σ x

------ Vx( )sin=

α 'v1

σ α------

atan=

κ' α '

Fx Fx α ' κ' Fz, ,( )=

Fy Fy α ' κ' γ Fz, , ,( )=

Mz Mz α ' κ' γ Fz, , ,( )=

Mz gyr, cgyrmbeltVr1 td



Adams/Tire

114

(203)






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.




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.





pTx3 PTX3 Variation of longitudinal relaxation length with exponent

of load





arc Brα r eq,( )tan[ ]cos 1=

Mz Mz' Mz gyr,+=

115Tire Models

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


USE_MODES of PAC-TIME: from Simple to ComplexThe 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).

Adams/Tire

116

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

Quality Checks for the Tire Model ParametersBecause PAC-TIME uses an empirical approach to describe tire - road interaction forces, incorrect






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 = 0.01QSY2 = 0QSY3 = 0

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)


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)


14 Transient Combined slip Fx, Fy, Fz, Mx, My, Mz




117Tire Models

QSY4 = 0


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.


The following table lists further checks on the PAC-TIME parameters.

Checklist for PAC-TIME Parameters and Properties


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 (22))

Adams/Tire

118




$------------------------------------------------------long_slip_range[LONG_SLIP_RANGE]KPUMIN


KPUMAX = 1.5







$-------------------------------------------------vertical_force_range[VERTICAL_FORCE_RANGE]FZMIN


FZMAX = 10125

pDx1/pDx2 < 0 Peak friction Fx must decrease with increasing load

Kx > 0 Long slip stiffness (see equation (25))

Dy > 0 Peak friction (see equation (34))

pDy1/pDy2 < 0 Peak friction Fx must decrease with increasing load

Ky < 0 Cornering stiffness (see equation (37a))

Kg < 0 Camber stiffness (see equation (37b))

qsy1 > 0 Rolling resistance, in the range of 0.005 - 0.015


119Tire Models

$Maximum allowed wheel load


model is performed with the minimum or maximum value of this range to avoid non-realistic tire


Standard Tire Interface (STI) for PAC-TIMEBecause 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]).




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.











• Tire Kinematics

• Slip Quantities


Adams/Tire

120

General

General Definitions

Tire Kinematics


Slip Quantities


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.






d Dimensionless radial tire deflection [-]

Fz0 Radial tire deflection at nominal load [m]


Rotational velocity of the wheel [rads-1]







ρρρ

ω

121Tire Models

Forces and Moments










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'

Vy Lateral speed of tire contact center [ms-1]



Slip angle [rad]












καγ

Adams/Tire

122

FILE_VERSION =3.0

FILE_FORMAT ='ASCII'

! : TIRE_VERSION : PAC-TIME

! : COMMENT : Tire

205/55 R16 90H ! : COMMENT :

Manufacturer Continental

! : 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

! : FILE_FORMAT : ASCII

! : Copyright MSC.Software, Thu Oct 14 13:52:26 2004!! 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

123Tire Models

! +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

='radians'MASS

='kg'TIME

='second'$----------------------------------------------------------------model[MODEL]PROPERTY_FILE_FORMAT

='PAC-TIME'USE_MODE


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

Adams/Tire

124

$Nominal rim radius

RIM_WIDTH = 0.165 $Rim width

$----------------------------------------------------------------shape[SHAPE]radial width1.0

0.0 1.0

0.4 1.0

0.9 0.9

1.0$-----------------------------------------------------------load_curve$ For a non-linear tire vertical stiffness (optional)$ Maximum of 100 points[DEFLECTION_LOAD_CURVE]pen

fz0.000

0.00.001

212.00.002

428.00.003

648.00.005

1100.00.010

2300.00.020

5000.00.030

8100.0$------------------------------------------------------------parameter[VERTICAL]VERTICAL_STIFFNESS

= 2.648e+005 $Tyre vertical stiffness

VERTICAL_DAMPING = 500

$Tyre vertical damping

BREFF = 4.90

125Tire Models

$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


KPUMAX = 1.5

$Maximum valid wheel slip

$-----------------------------------------------------slip_angle_range[SLIP_ANGLE_RANGE]ALPMIN


ALPMAX = 1.5708

$Maximum valid slip angle



CAMMAX = 0.26181

$Maximum valid camber angle



Adams/Tire

126

FZMAX = 10800


$--------------------------------------------------------------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

127Tire Models

LHY = 1

$Scale factor of Fy horizontal shift

LVY = 1

$Scale factor of Fy vertical shift

LGAY = 1

$Scale factor of camber for Fy

LKC = 1

$Scale factor of camber stiffness

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

Adams/Tire

128

= 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

129Tire Models

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

Adams/Tire

130

$Variation of SigKap0/Fz with load

PTX3 = -1.3131

$Variation of SigKap0/Fz with exponent of load

$----------------------------------------------------------overturning[OVERTURNING_COEFFICIENTS]QSX1

= 0 $Lateral force induced 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

131Tire Models

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

Adams/Tire

132

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

133Tire Models

[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

Adams/Tire

134

$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

135Tire Models

QTZ1 = 0

$Gyroscopic torque constant

MBELT = 0 $Belt mass of the wheel -kg-

Contact MethodsThe PAC-TIME model supports the following roads:



Note that the PAC-TIME model has only one point of contact with the road; therefore, the wavelength

of road obstacles must be longer than the tire radius for realistic output of the model. In addition, the

contact force computed by this tire model is normal to the road plane. Therefore, the contact point does

not generate a longitudinal force when rolling over a short obstacle, such as a cleat or pothole.

For ride and comfort analyses, we recommend more sophisticated tire models, such as Ftire.

Adams/Tire

136

Using Pacejka '89 and '94 ModelsAdams/Tire provides you with the handling force models, Pacejka '89 and Pacejka '94.

• About Pacejka '89 and '94

• Using Pacejka '89 Handling Force Model

• Using Pacejka '94 Handling Force Model

• Combined Slip


• Contact Methods

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.

• 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

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.

137Tire Models

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.


Slip Quantities at combined cornering and braking/traction



Fz kzρ czρ·

+=

ρ ρ·

Ω

Adams/Tire

138




and

Note that for realistic tire forces the slip angle is limited to 900 and the longitudinal slip in






system is used both for the longitudinal slip ? as the side slip angle, ?. Considering the tire belt as a



Stretched String Model for Transient Tire behavior




Vsx Vx ΩRl–=

Vsy Vy=

κ α

κVsx

Vx

--------–= αtanVsy

Vx

---------=

α κ

139Tire Models













The longitudinal and lateral relaxation length are estimated with the longitudinal and lateral stiffness of

the non-rolling tire:

For BCDx and BCDy see section Force and Moment Formulation for Pacejka '89 or '94.

In case the longitudinal stiffness is not available in the tire property file the longitudinal stiffness is

estimated with:

1

Vx

------dv1

dt--------

v1

σ α------+ αtan aφ+=

σ α φ

σ α

dv1

dt-------- Vx v1+ σ αVsy=

σ κ

du1dt-------- Vx u1+ σ– κVsx=

κ' α '

κ'u1

σ κ------ Vx( )sgn=

α 'v1

σ α------

atan=

κ' α ' κ α

σ κ

BCDx

longitudinal_stiffness--------------------------------------------------------= and σ α

BCDy

lateral_stiffness-------------------------------------------=

Adams/Tire

140

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

Using Correct Coordinate System and Units in Pacejka '89The 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.

SAE Tire Coordinate System

longitudinal_stiffness 4 lateral_stiffness×=

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

coefficients.

141Tire Models

Conventions for Naming Variables

Force and Moment Formulation for Pacejka '89• Longitudinal Force for Pacejka '89

• Lateral Force

• Self-Aligning Torque

• Overturning Moment


• 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)

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 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.

γ

ακ

Adams/Tire

142

Horizontal Shift

Sh=B9*FZ+B10

Vertical Shift

Sv=0.0

Composite

X1=(k+Sh)

E Curvature Factor

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

FX Equation

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

Longitudinal Force

Example Longitudinal Force Plot for Pacejka ’89

Parameters: Description:

B0 Shape factor

B1, B2 Peak factor

B3, B4, B5 BCD calculation

B6, B,7 B8 Curvature factor

B9, B10 Horizontal shift

143Tire Models

Lateral Force for Pacejka '89

C - Shape Factor

C=A0

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)

Adams/Tire

144

FY Equation

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

Parameters for Lateral Force

Example Lateral Force Plot for Pacejka ’89

Self-Aligning Torque

C - Shape Factor

C=C0

D - Peak Factor


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

145Tire Models

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

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

Example Self-Aligning Torque Plot for Pacejka ’89


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

Adams/Tire

146

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

147Tire Models

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


My = S*My

Aligning Torque

Mz = S*Mz


• USE_MODE = 1 or 2, smoothing is off

• USE_MODE = 3 or 4, smoothing is on

Example of Pacejka '89 Property File$-------------------------------------------------------------MDI_HEADER[MDI_HEADER]FILE_TYPE = 'tir'FILE_VERSION = 2.0FILE_FORMAT = 'ASCII'(COMMENTS)comment_string'Tire - XXXXXX''Pressure - XXXXXX''Test Date - XXXXXX''Test tire'$-------------------------------------------------------------UNITS[UNITS]LENGTH = 'mm'FORCE = 'newton'ANGLE = 'radians'MASS = 'kg'TIME = 'sec'$--------------------------------------------------------------------------MODEL

Adams/Tire

148

[MODEL]! use mode 123411121314! ------------------------------------------------------------------------! smoothingXXXX! combinedXXXX! transient X X X X!PROPERTY_FILE_FORMAT = 'PAC89'USE_MODE = 12.0TYRESIDE = 'LEFT'$-------------------------------------------------------------DIMENSION[DIMENSION]UNLOADED_RADIUS = 326.0WIDTH = 245.0ASPECT_RATIO = 0.35$-------------------------------------------------------------PARAMETER[PARAMETER]VERTICAL_STIFFNESS = 310.0VERTICAL_DAMPING = 3.1LATERAL_STIFFNESS = 190.0ROLLING_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 1100.010 2300.020 5000.030 8100.0$-------------------------------------------------------------LATERAL_COEFFICIENTS[LATERAL_COEFFICIENTS]a0 = 1.65000a1 = -34.0a2 = 1250.00a3 = 3036.00a4 = 12.80a5 = 0.00501a6 = -0.02103a7 = 0.77394a8 = 0.0022890a9 = 0.013442a10 = 0.003709a11 = 19.1656a12 = 1.21356

149Tire Models

a13 = 6.26206$-------------------------------------------------------------longitudinal[LONGITUDINAL_COEFFICIENTS]b0 = 1.67272b1 = -9.46000b2 = 1490.00b3 = 30.000b4 = 176.000b5 = 0.08860b6 = 0.00402b7 = -0.06150b8 = 0.20000b9 = 0.02990b10 = -0.17600$-------------------------------------------------------------aligning[ALIGNING_COEFFICIENTS]c0 = 2.34000c1 = 1.4950c2 = 6.416654c3 = -3.57403c4 = -0.087737c5 = 0.098410c6 = 0.0027699c7 = -0.0001151c8 = 0.1000c9 = -1.33329c10 = 0.025501c11 = -0.02357c12 = 0.03027c13 = -0.0647c14 = 0.0211329c15 = 0.89469c16 = -0.099443c17 = -3.336941$--------------------------------------------------------------------------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

Using Pacejka '94 Handling Force Model

Learn about the Pacejka '94 handling force model:

Adams/Tire

150

• Using Correct Coordinate System and Units

• Force and Moment Formulation for Pacejka ’94

• Example of Pacejka ’94 Property File

Using Correct Coordinate System and Units in Pacejka '94The 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

Conventions for Naming Variables

Force and Moment Formulation for Pacejka '94• Longitudinal Force for Pacejka '94

• Lateral Force for Pacejka '94

• Self-Aligning Torque

• Overturning Moment


• Smoothing

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

coefficients.

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.

γ

ακ

151Tire Models

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

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

Lateral Force for Pacejka '94

C - Shape Factor

C=A0


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

Adams/Tire

152

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*

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

Self-Aligning Torque for Pacejka '94

C - Shape Factor

C=C0

D - Peak Factor


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

γ

γ

γ

γ

153Tire Models

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

BCD

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

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


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

γ

γ

γ

α

γ γ

Adams/Tire

154

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


My = S*My

Aligning Torque

Mz = S*Mz


• 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

155Tire Models

!: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 12341234! ------------------------------------------------------------------------! smoothingXXXX! combinedXXXX! 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 fz0.000 00.039 9430.079 19040.118 28820.197 48930.394 102310.787 222411.181 36031$-----------------------------------------------------------------------scaling

Adams/Tire

156

[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

157Tire Models

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

Combined Slip of Pacejka '89 and '94The 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 k (range –1 to 1) and side slip angle a 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:

κ* κ Shx+=

α * α Shy+=

SAG α *( )sin=

Adams/Tire

158

Friction coefficients:

Forces corrected for combined slip conditions:

Contact MethodsThe Pacejka '89 and '94 models support the following roads:

• 2D roads, see Using the 2D Road Model.

• 3D roads, see Adams/3D Road Model

These tire models use a one point of contact method; therefore, the wavelength of road obstacles must be

longer than the tire radius for realistic output of the model.

β arcκ*

κ*

( )2

SAG2+

---------------------------------------

cos=

µx act,

Fx Svx–

Fz-------------------- µy act,

Fy Svy–

Fz--------------------

µx max,

Dx

Fz------ µy max,

Dy

Fz------

= =

= =

µx1

1

µx act,-------------

2 βtan

µy max,----------------

2+

--------------------------------------------------------- µyβtan

1

µx max,----------------

2 βtan

µy max,----------------

2+

------------------------------------------------------------= =

Fx comb,

µx

µx act,------------- Fx Svx+( ) Fy comb,

µy

µy act,------------- Fy Svy+( )= =

159Tire Models

Using the UA-Tire ModelLearn about using the University of Arizona (UA) tire model:

• Background Information

• Tire Model Parameters

• Force Evaluation

• Operating Mode: USE_MODE

• Tire Carcass Shape

• Property File Format Example

Background Information for UA-TireThe University of Arizona tire model was originally developed by Drs. P.E. Nikravesh and G. Gim.

Reference documentation: G. Gim, Vehicle Dynamic Simulation with a Comprehensive Model for

Pneumatic Tires, PhD Thesis, University of Arizona, 1988. The UA-Tire model also includes relaxation

effects, both in the longitudinal and lateral direction.

The UA-Tire model calculates the forces at the ground contact point as a function of the tire kinematic

states, see Inputs and Output of the UA-Tire Model. A description of the inputs longitudinal slip k, side

slip a and camber angle can be found in About Tire Kinematic and Force Outputs. The tire

deflection and deflection velocity are determined using either a point follower or durability contact

model. For more information, see Road Models in Adams/Tire. A description of outputs, longitudinal

force Fx, lateral force Fy, normal force Fz, rolling resistance moment My and self aligning moment Mz

is given in About Tire Kinematic and Force Outputs. The required tire model parameters are described

in Tire Model Parameters.

Inputs and Output of the UA-Tire Model

γ

ρ ρ·

Adams/Tire

160



161Tire Models

The longitudinal slip velocity Vsx in the SAE-axis system is defined using the longitudinal speed Vx,

the wheel rotational velocity , and the effective rolling radius Re:




When the UA Tire is used for the force calculation the slip quantities during positive Vsx (driving) are

defined as:


Note that for realistic tire forces the slip angle is limited to 45 degrees and the longitudinal slip Ss

(= ) in between -1 (locked wheel) and 1.



interaction forces in between the tire and the road. Often the tire dynamic response will have an

important role on the overall vehicle response. For modeling this so-called transient tire behavior, a first-

order system is used both for the longitudinal slip as the side slip angle, . Considering the tire belt as

a stretched string, which is supported to the rim with lateral spring, the lateral deflection of the belt can



ω

Vsx Vx ΩRe–=

Vsy Vy=

κ α

κVsx

Vx

-------- and αtan–Vsy

Vx

---------= =

κVsx

Vr


Vr

---------= =

Vr ReΩ=

α

κ

κ

Adams/Tire

162












When the UA Tire is used for the force calculations, at positive Vsx (traction) the Vx should be replaced

by Vr in these differential equations.


1

Vx

------td

dv1 v1

σ α------+ α( ) aφ+tan=

σ α φ

σ α td

dv1Vx v1+ σ αVsy=

163Tire Models

Now the practical slip quantities, ’ and ’, are defined based on the tire deformation:


state tire behavior. kVlow_x and kVlow_y are the damping rates at low speed applied below the

LOW_SPEED_THRESHOLD speed. For the LOW_SPEED_DAMPING parameter in the tire property

file yields:

kVlow_x= 100 · kVlow_y= LOW_SPEED_DAMPING

Tire Model ParametersDefinition of Tire Parameters

σ x td

dv1Vx v1+ σ– αVsx=

κ α

κ'u1σ κ------ kVlowxVsx–

Vx( )sin=

α 'v1

σ α------

kVlowyvsy–

atan=

κ' α ' κ α

Note: If the tire property file's REL_LEN_LON or REL_LEN_LAT = 0, then steady-state tire

behavior is calculated as tire response on change of the slip and .κ α

Symbol:Name in tire property file: Units*: Description:

r1 UNLOADED_R

ADIUS

L Tire unloaded radius

kz VERTICAL_STI

FFNESS

F/L Vertical stiffness

cz VERTICAL_DA

MPING

FT/L Vertical damping

Cr ROLLING_RES

ISTANCE

L Rolling resistance parameter

Cs CSLIP F Longitudinal slip stiffness,

C CALPHA F/A Cornering stiffness,

C CGAMMA F/A Camber stiffness,

UMIN UMIN - Minimum friction coefficient (Sg=1)

κ∂

∂ Fx

κ 0=αα∂

∂ Fy

α 0=γγ∂

∂ Fy

γ 0=

Adams/Tire

164

* L=length, F=force, A=angle, T=time

Force Evaluation in UA-Tire• Normal Force

• Slip Ratios

• Friction Coefficient

Normal Force

The normal force F z is calculated assuming a linear spring (stiffness: k z ) and damper (damping constant

c z ), 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 F z will 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 Property File

Format Example. 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 file but it does not

play any role.

Slip Ratios

For the calculation of the slip forces and moments a number of slip ratios will be introduced:

Longitudinal Slip Ratio: Ss

The absolute value of longitudinal slip ratio, Ss, is defined as:

Where k is limited to be within the range -1 to 1.

UMAX UMAX - Maximum friction coefficient (Ssg=0)

x REL_LEN_LON L Relaxation length in longitudinal direction

y REL_LEN_LAT L Relaxation length in lateral direction

Symbol:Name in tire property file: Units*: Description:

σσ

Fz kzρ czρ·

+=

ρ ρ·

Ss κ=

165Tire Models

Lateral Slip Ratios: Sa , Sg , Sag

The lateral slip ratio due to slip angle, S , is defined as:

The lateral slip ratio due to inclination angle, S , is defined as:

A combined lateral slip ratio due to slip and inclination angles, S , is defined as:

where is the length of the contact patch.

Comprehensive Slip Ratio: Ssag

A comprehensive slip ratio due to longitudinal slip, slip angle, and inclination angle may be defined as:

α

Sα* αtan during braking

1 Ss–( ) αtan during traction

Sα min 1.0 Sα*,( )

=

=

γ

Sγ γsin=

αγ

Sα*

αl γsin

2rl------------–tan during braking

1 Ss–( ) αl γsin

2rl------------tan during traction

=

l 8r1ρ=

Sαγ min 1.0 S*αγ,( )=

S*sαγ Ss2 S2αγ+

Sαγ min 1.0 S*αγ,( )=

=

Adams/Tire

166

Friction Coefficient

The resultant friction coefficient between the tire tread base and the terrain surface is determined as a

function of the resultant slip ratio (Ss ) and friction parameters (UMAX and UMIN ). The friction

parameters are experimentally obtained data representing the kinematic property between the surfaces of

tire tread and the terrain.

A linear relationship between Ss and , the corresponding road-tire friction coefficient, is assumed.

The figure below depicts this relationship.

Linear Tire-Terrain Friction Model

This can be analytically described as:

m = UMAX - (UMAX - UMIN) * Ssag

The friction circle concept allows for different values of longitudinal and lateral friction coefficients ( x

and y) but limits the maximum value for both coefficients to . See the figure below.

Friction Circle Concept

αγ

αγ µ

µ

µ µ

167Tire Models

The relationship that defines the friction circle follows:

or and

where:

Slip Forces and Moments

To compute longitudinal force, lateral force, and self-aligning torque in the SAE coordinate system, you

must perform a test to determine the precise operating conditions. The conditions of interest are:

µx

µ-----

2 µy

µ-----

2

+ 1=

µx µ βcos= µy µ βsin=

βcosSs

Ssαγ

---------- and βsinSαγ

Ssαγ----------= =

Adams/Tire

168

• Case 1: 0

• Case 2: 0 and C S C S

• Case 3: 0 and C S C S

• Forces and moments at the contact point

The lateral force Fh can be decomposed into two components: Fha and Fhg. The two components are in

the same direction if a· g < 0 and in opposite direction if 0.

Case 1. ag < 0

Before computing the longitudinal force, the lateral force, and the self-aligning torque, some slip

parameters and a modified lateral friction coefficient should be determined. If a slip ratio due to the

critical inclination angle is denoted by S c, then it can be evaluated as:

If Ssc represents a slip ratio due to the critical (longitudinal) slip ratio, then it can be evaluated as:

If a slip ratio due to the critical slip angle is denoted by S c, then it can be determined as:

when Ss Ssc.

The term critical stands for the maximum value which allows an elastic deformation of a tire during pure

slip due to pure slip ratio, slip angle, or inclination angle. Whenever any slip ratio becomes greater than

its corresponding critical value, an elastic deformation no longer exists, but instead complete sliding state

represents the contact condition between the tire tread base and the terrain surface.

A nondimensional slip ratio Sn is determined as:

where:

αγ<

αγ≥ α αγ< γ γ

αγ< α αγ< γ γ

αγ<

γ

Sγc µFzCγ------=

Ssc 3µFzCs

-----=

α

Sαc

Cs

Cα------- Ssc

2 Ss2– 3Cγ

Sγ

Cα-------–=

≤

SnB2 B2

2 B1B3–+

B1

-------------------------------------------=

169Tire Models

A nondimensional contact patch length is determined as:

A modified lateral friction coefficient is evaluated as:

where is the available friction as determined by the friction circle.

To determine the longitudinal force, the lateral force, and the self-aligning torque, consider two subcases

separately. The first case is for the elastic deformation state, while the other is for the complete sliding

state without any elastic deformation of a tire. These two subcases are distinguished by slip ratios caused

by the critical values of the slip ratio, the slip angle, and the inclination angle. Specifically, if all of slip

ratios are smaller than those of their corresponding critical values, then there exists an elastic

deformation state, otherwise there exists only complete sliding state between the tire tread base and the

terrain surface.

(i) Elastic Deformation State: S S c, Ss Ssc, and S S c

In the elastic deformation state, the longitudinal force F , the lateral force F , and three components

of the self-aligning torque are written as functions of the elastic stiffness and the slip ratio as well as the

normal force and the friction coefficients, such as:

B1 3µFz( )2 3CγSγ( )2

B2

–

2CαSαCγSγ

B3 CsSs( )2 CαSα( )2+[ ]–

=

=

=

ln 1 Sn–=

µym( )

µym( ) µy

CγSγ

Fz-----------

–=

µy µ βsin=

γ< γ < α < α

ξ τ

Adams/Tire

170

where:

• is the offset between the wheel plane center and the tire tread base.

• is set to zero if it is negative.

• the length of the contact patch.

Mz is the portion of the self-aligning torque generated by the slip angle . Mzs and Mzs are other

components of the self-aligning torque produced by the longitudinal force, which has an offset between

the wheel center plane and the tire tread base, due to the slip angle and the inclination angle ,

respectively. The self-aligning torque Mz is determined as combinations of Mz , Mzs and Mzs .

(ii) Complete Sliding State: S S c, Ss Ssc, and S S c

In the complete sliding state, the longitudinal force, the lateral force, and three components of the self-

aligning torque are determined as functions of the normal force and the friction coefficients without any

elastic stiffness and slip ratio as:

Fξ CsSsln2 µxFz 1 3ln

2– 2ln3+( )

Fη

+

CαSsln2 µy

m( )Fz 1 3ln2– 2ln

3+( ) CγSγ

Mzα

+ +

CαSα1

2---–

2

3---ln+

3

2---µy

m( )FzSn2+ lln

2

Mzsα2

3---CsSsSα ln

33µxµyFz

2

5Cα---------------------- 1 10ln

3– 15ln4 6ln

5–+( )

Mzsγ

+

η Fξ

=

=

=

=

=

η Sγ rl2 l'2 4⁄–=

rl2 l2 4⁄–

l 8rlρ=

α α α γ

α γ

α α γ

γ ≥ γ ≥ α ≥ α

171Tire Models

Case 2: 0 and C S C S

As in Case 1, a slip ratio due to the critical value of the slip ratio can be obtained as:

A slip ratio due to the critical value of the slip angle can be found as:

when Ss Ssc.

The nondimensional slip ratio Sn, is determined as:

where:

Fξ µxFz

Fη µyFz

Mzα 0

Mzα s

3µxµyFz2l

5Cα------------------------

Mzsγ η Fξ

=

=

=

=

=

α γ ≥ α α ≥ γ γ

Sγc 3µFz

Cγ------=

Sαc

Cs

Cα------- Ssc

2 Ss2– 3Cγ

Sγ

Cα-------+=

≤

SnB2 B2

2 B1B3–+

B1

-------------------------------------------=

Adams/Tire

172

The nondimensional contact patch length ln is found from the equation ln = 1 - Sn, and the modified

lateral friction coefficient is expressed as:

For the longitudinal force, the lateral force and the self-aligning torque two subcases should also be

considered separately. A slip ratio due to the critical value of the inclination angle is not needed here since

the required condition for Case 2, C S C S , replaces the critical condition of the inclination

angle.

(i) Elastic Deformation State: Ss Ssc and S Sac

In the elastic deformation state:

(ii) Complete Sliding State: Ss Ssc and S Sac

B1 3µFz( )2 3CγSγ( )2

B2

–

2CαSαCγSγ

B3 CsSs( )2 CαSα( )2+[ ]–

=

=

=

µym( )

µym( ) µy

CγSγ

Fz-----------

+=

α α ≥ γ γ

< α <


2– 2ln3+( )

Fη

+

CαSsln2 µy

m( )Fz 1 3ln2– 2ln

3+( ) CγSγ

Mzα

+ +

CαSα1

2---–

2

3---ln+

3

2---µy

m( )FzSn2+ lln

2

Mzsα2

3---CsSsSα ln

33µxµyFz

2

5Cα---------------------- 1 10ln

3– 15ln4 6ln

5–+( )

Mzsγ

+

η Fξ

=

=

=

=

=

≥ α ≥

173Tire Models

Case 3: 0 and C S C S

Similar to Cases 1 and 2, slip ratios due to the critical values of the inclination angle and the slip ratio

are obtained as:

The nondimensional slip ratio Sn, is expressed as:

where:

Fξ µxFz

Fη µyFz

Mzα 0

Mzα s

3µxµyFz2l

5Cα------------------------

Mzsγ η Fξ

=

=

=

=

=

α γ ≥ α α < γ γ

Sγc

3µFz CαSα+

3Cγ--------------------------------

Ssc1

Cs

----- 3µFz( )2 CαSα 3CγSγ–( )–

=

=

SnB2 B2

2 B1B3–+

B1

-------------------------------------------=

B1 3µFz( )2 3CγSγ( )2

B2

–

2CαSαCγSγ

B3 CsSs( )2 CαSα( )2+[ ]–

=

=

=

Adams/Tire

174

For the longitudinal force, the lateral force, and the self-aligning torque, two subcases should also be

considered similar to Cases 1 and 2. A slip ratio due to the critical value of the slip angle is not needed

here since the required condition for Case 3, C S C S , replaces the critical condition of

the slip angle.

(i) Elastic Deformation State: S S c and Ss Ssc

In the elastic deformation state, F and Mz can be written:

(ii) Complete Sliding State: S S c and Ss Ssc

In the complete sliding state, F , F , Mz , Mzs , and Mzs can be determined by using:

α α < γ γ

γ < γ <

η α


2– 2ln3+( )

Fη

+

CαSsln2 µy

m( )Fz 1 3ln2– 2ln

3+( ) CγSγ

Mzα

+ +

CαSα1

2---–

2

3---ln+

3

2---µy

m( )FzSn2+ lln

2

Mzsα2

3---CsSsSα ln

33µxµyFz

2

5Cα---------------------- 1 10ln

3– 15ln4 6ln

5–+( )

Mzsγ

+

η Fξ

=

=

=

=

=

γ ≥ γ ≥

ξ η α α γ

Fξ µxFz

Fη µyFz

Mzα 0

Mzα s

3µxµyFz2l

5Cα------------------------

Mzsγ η Fξ

=

=

=

=

=

175Tire Models

respectively. The longitudinal force F , the lateral force F , and three components of the self-aligning

torques, Mz , Mzs , and Mzs , always have positive values, but they can be transformed to have

positive or negative values depending on the slip ratio s, the slip angle , and the inclination angle in

the SAE coordinate system.

Tire Forces and Moments in the SAE Coordinate System

For the general formulations of the longitudinal force Fx, lateral force Fy, and self-aligning torque Mz,

in the SAE coordinate system, the three possible combinations of the slip ratio, the slip angle, and the

inclination angle are also considered.

Longitudinal Force:

Fx = sin(k) F , for all cases

Lateral Force:

Fy = -sin( ) F , for cases 1 and 2

Fy = sin( ) F , for case 3

Self-aligning Torque:

Mz = sin( ) Mz - sin( ) [-sin( ) Mzs + sin( )Mzs ]

Rolling Resistance Moment:

My = -Cr Fz, for a forward rolling tire.

My = Cr Fz, for a backward rolling tire.

Operating Mode: USE_MODEYou can change the behavior of the tire model through the switch USE_MODE in the [MODEL] section

of the tire property file.

• USE_MODE = 0: Steady-state forces and moments

• The tire forces and moments react instantaneously to changes in the tire kinematic states.

• USE_MODE = 1: Transient tire behavior

• The tire will have a lagged response because of the so-called relaxation length in both

longitudinal and lateral direction. See Lagged Longitudinal and Lateral Slip Quantities

(transient tire behavior).

• The effect of the relaxation lengths will be most pronounced at low forward velocity and/or high

excitation frequencies.

• USE_MODE = 2: Smoothing of forces and moments on startup of the simulation

ξ η

α α γ

α γ

ξ

α η

γ η

α α κ α α γ γ

Adams/Tire

176

• 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

Aligning Torque Mz = S*Mz

Tire Carcass ShapeYou can optionally supply a tire carcass cross-sectional shape in the tire property file in the [SHAPE]

block. The 3D-durability, tire-to-road contact algorithm uses this information when calculating the tire-

to-road volume of interference. If you omit the [SHAPE] block from a tire property file, the tire carcass

cross-section defaults to the rectangle that the tire radius and width define.

You specify the tire carcass shape by entering points in fractions of the tire radius and width. Because

Adams/Tire assumes that the tire cross-section is symmetrical about the wheel plane, you only specify

points for half the width of the tire. The following apply:

• For width, a value of zero (0) lies in the wheel center plane.

• For width, a value of one (1) lies in the plane of the side wall.

• For radius, a value of one (1) lies on the tread.

For example, suppose your tire has a radius of 300 mm and a width of 185 mm and that the tread is joined

to the side wall with a fillet of 12.5 mm radius. The tread then begins to curve to meet the side wall at

>+/- 80 mm from the wheel center plane. If you define the shape table using six points with four points

along the fillet, the resulting table might look like the shape block that is at the end of the property format

example (see SHAPE).

Property File Format Example$--------------------------------------------------------MDI_HEADER[MDI_HEADER]FILE_TYPE = 'tir' FILE_VERSION = 2.0FILE_FORMAT = 'ASCII'(COMMENTS) comment_string'Tire - XXXXXX''Pressure - XXXXXX''TestDate - XXXXXX''Test tire''New File Format v2.1'$-------------------------------------------------------------units[UNITS]LENGTH

= 'meter'FORCE

= 'newton'

177Tire Models

ANGLE = 'rad'

MASS = 'kg'

TIME = 'sec'

$-------------------------------------------------------------model[MODEL]! use mode

1 2 3

! ------------------------------------------! relaxation lengths

X

! smoothing X

!PROPERTY_FILE_FORMAT

= 'UATIRE'USE_MODE

= 2$---------------------------------------------------------dimension[DIMENSION]UNLOADED_RADIUS

= 0.295WIDTH

= 0.195ASPECT_RATIO

= 0.55$---------------------------------------------------------parameter[PARAMETER]VERTICAL_STIFFNESS

= 190000VERTICAL_DAMPING

= 50ROLLING_RESISTANCE

= 0.003CSLIP

= 80000CALPHA

= 60000CGAMMA

= 3000UMIN

= 0.8UMAX

= 1.1REL_LEN_LON

= 0.6REL_LEN_LAT

= 0.5

Adams/Tire

178

$-------------------------------------------------------------shape[SHAPE]radial width1.0 0.0 1.0 0.2 1.0 0.4 1.0 0.6 1.0 0.8 0.9 1.0$---------------------------------------------------------------------load_curve$ For a non-linear tire vertical stiffness (optional)$ Maximum of 100 points[DEFLECTION_LOAD_CURVE]pen

fz0.000

0.00.001

212.00.002

428.00.003

648.00.005

1100.00.010

2300.00.020

5000.00.030

8100.0

Contact MethodsThe UA-Tire Model supports the following roads

• 2D roads, see Using the 2D Road Model.

• 3D roads, see Adams/3D Road Model

The UA-Tire Model uses a one point of contact method; therefore, the wavelength of road obstacles must

be longer than the tire radius for realistic output of the model.

179Tire Models

PAC MCLearn about using the University of Arizona (UA) tire model:

When to Use PAC MotorcycleMagic-Formula (MF) tire models are considered the state-of-the-art for modeling of the tire-road

interaction forces in Vehicle Dynamics applications. First versions of the mode that were published by

Pacejka considered tire models for car and truck tires. In his book, Tyre and Vehicle Dynamics [1], he

also described a model for motorcycle tires that is backwards compatible with the MF-MCTyre,

previously resold by MSC.Software, and contains the latest developments in this field.

In general, a MF tire model describes the tire behavior for rather smooth roads (road obstacle




• Lane-change maneuvers





• Shimmy and weave phenomena, which can be analyzed when the tire model is used in transient

mode (see USE_MODES of PAC MC: from Simple to Complex)

• All other common vehicle dynamics maneuvers on rather smooth road (wavelength of road


The PAC MC model has proven to be applicable to motorcycle tires with inclination angles to the road

up to 60 degrees. In some cases, it can be used for car tires when exposed to large camber.

PAC MC and Previous Magic Formula Models

Compared to previous versions, PAC MC is backward compatible with all MF-MCTyre 1.x tire models,

generates the same output, and deals with all previous versions of MF-MCTyre property files.

In addition to PAC MC in Adams, the PAC MC in v2 contins a more advanced tire-road contact modeling

method that takes the tire's cross-section shape into account, which plays an important role at large

inclination angles of the wheel with the road. Learn more about the tire cross-section profile contact

method.

Modeling Tire-Road Interaction ForcesFor vehicle dynamics applications, accurate knowledge of tire-road interaction forces is inevitable


forces depend on both road and tire properties and the motion of the tire with respect to the road.

Adams/Tire

180



the tire and wheel as a rigid disc. In the contact point between the tire and the road the contact forces in


road.


vectors of the PAC MC 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 the:




• Road parameters


the wheel with respect to the road. The input for the Magic Formula consists of the wheel load ( ), the

longitudinal and lateral slip ( , ), and inclination angle ( ) with the road. The output is the forces

( , ) and moments ( , , ) 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.


Adams/Solver through the STI.


Fz

κ α γ

Fx Fy Mx My Mz

181Tire Models

Axis Systems and Slip Definitions• Axis System

• Units


Axis System

The PAC MC 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 PAC MC model and Adams/Solver mainly passes information to the tire


because all the modeling of the tire behavior, as described in this help, assumes to deal with the slip


axis systems have the ISO orientation but have a different origin as can be seen in the figure below.

TYDEX C- and W-Axis Systems Used in PAC MC, Source[2]





The forces and moments calculated by PAC MC using the MF equations in this guide are in the W-axis


to Adams/Solver.



Adams/Tire

182

Units

The units of information transferred through the STI between Adams/Solver and PAC MC are according

to the SI unit system. Also, the equations for PAC MC 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 units to be changed.



SI Units Used in PAC MC



Variable Type: Name: Abbreviation: Unit:

Angle Slip angle

Inclination angle

Radians


Lateral force

Vertical load

Newton


Rolling resistance

moment


Newton.meter


Lateral speed


Lateral slip speed

Meters per second


α

γFx

Fy

FzMx

My

Mz

Vx

Vy

Vsx

Vsy

ω

183Tire Models

The longitudinal slip velocity in the contact point (W-axis system, see the figure, Slip Quantities

at Combined Cornering) is defined using the longitudinal speed , the wheel rotational velocity ,

and the effective rolling radius :

(204)


(205)



(206)

(207)


(208)

Contact-Point and Normal Load Calculation• Contact Point


Contact Point




Vsx

Vx ω

Re

Vsx Vx ΩRe–=

Vsy Vy=

κ α

κVsx

Vx

--------–=

α( )tanVsy

Vx

---------=

Vr ReΩ=

Adams/Tire

184


at the road point right below the wheel center (see figure below).



(ground) plane and the line of intersection of the wheel center-plane with the plane though the wheel spin

axis.

The normal load of the tire is calculated with:

(209)

where is the tire deflection and is the deflection rate of the tire.

To take into account the effect of the tire cross-section profile, you can choose a more advanced method

(see the Tire Cross Section Profile Contact Method).

Instead 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]. 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

in the tire property file, but it does not play any role.

Fz


ρ ρ·

Cz

185Tire Models


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 (see Effective Rolling Radius and Longitudinal Slip), where r is the deflection of

the tire, and R0 is the free (unloaded) tire radius, then the loaded tire radius Rl reads:

(210)

In this tire model, a constant (linear) vertical tire stiffness is assumed; therefore, the tire deflection

can be calculated using:

(211)



(212)

For radial tires, the effective rolling radius is rather independent of load in its load range of operation due

to the high stiffness of the tire belt circumference. Only at low loads does the effective tire radius



R1 R0 ρ–=

Cz

ρ

ρFzCz

------=

Re

Vx

Ω------=

Adams/Tire

186

To represent the effective rolling radius Re, a MF-type of equation is used:

(213)

in which is the nominal tire deflection:

(214)

and is called the dimensionless radial tire deflection, defined by:

(215)

Re R0 ρ Fz0 D Bρ d( ) Fρ d+( )atan⋅( )–=

ρ Fz0

ρ Fz0Fz0

Cz

--------=

ρ d

ρ d ρρ Fz0---------=

187Tire Models

Example of the Loaded and Effective Tire Rolling Radius as a Function of the Vertical Load


Basics of the Magic Formula in PAC MCThe Magic Formula is a mathematical formula that is capable of describing the basic tire characteristics


We distinguish:


Name:

Name Used in Tire Property File: Explanation:

FNOMIN Nominal wheel load

UNLOADED_RADIUS Free tire radius

BREFF Low load stiffness effective rolling radius

DREFF Peak value of effective rolling radius

FREFF High load stiffness effective rolling radius

VERTICAL_STIFFNESS Tire vertical stiffness

VERTICAL_DAMPING Tire vertical damping

Fz0R0

B

D

F

Cz

KZ

Adams/Tire

188



For pure slip conditions, the lateral force as a function of the lateral slip , respectively, and the

longitudinal force as a function of longitudinal slip , have a similar shape (see the figure,

Characteristic Curves for Fx and Fy Under Pure Slip Conditions). Because of the sine - arctangent

combination, the basic Magic Formula example is capable of describing this shape:

(216)

where Y(x) is either with x the longitudinal slip , or and x the lateral slip .


The self-aligning moment is calculated as a product of the lateral force and the pneumatic trail t

added with the residual moment . In fact, the aligning moment is due to the offset of lateral force ,


slip has a cosine shape, a cosine version the Magic Formula is used:

(217)





Fy α

Fx κ

Y x( ) D c bx E bx bx( )atan–( )–( )atan⋅[ ]cos⋅=

Fx κ Fy α

Mz Fy

Mzr Fy

α

Y x( ) D C Bx E Bx Bx( )atan–( )–( )atan⋅[ ]cos⋅=

α

189Tire Models






Adams/Tire

190

In combined slip conditions, the lateral force decreases due to longitudinal slip or the opposite, the

longitudinal force decreases due to lateral slip. The forces and moments in combined slip conditions

Fy

Fx

191Tire Models

are based on the pure slip characteristics multiplied by the so-called weighting functions. Again, these

weighting functions have a cosine-shaped MF examples.




Inclination Effects in the Lateral Force

From a historical point of view, the basic Magic Formulas have always been developed for car and truck

tires, which cope with inclinations angles of not more than 10 degrees. To be able to describe the effects

at large inclinations, an extension of the basic Magic Formula for the lateral force Fy has been developed.

A contribution of the inclination has also been added within the MF sine function:

(218)

This elegant formulation has the advantage of an explicit definition of the camber stiffness, because this

results now in:

(219)

Input Variables


Input Variables

Output Variables

Its output variables are:

Output Variables


Slip angle [rad]





γ

Fy0 Dy Cyarc Byαy Ey Byα y arc Byα y( )tan–( )– Cγarc Bγγy Eγ Bγγy arc Bγγy( )tan–( )– tan+

tan[]

sin=

Kγ BγCγDγ γ∂

∂ Fyo= = at α y 0=

καγ

Adams/Tire

192




are:



(220)

with the possibly adapted nominal load (using the user-scaling factor, ):

(221)



logic:



Rolling resistance moment My [Nm]










dfzFz F'z0–

F'z0--------------------=

λ Fz0

F'z0 Fz0 λ Fz0⋅=

193Tire Models




You can change the factors in the tire property file. The peak friction scaling 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.



C Shape factor

D Peak value

E Curvature factor


H Horizontal shift

V Vertical shift






k = 1, 2, ...

Name:

Name used in tire

property file: Explanation:

LFZO Scale factor of nominal (rated) load


LMUX Scale factor of Fx peak friction coefficient




x LGAX Scale factor of camber for Fx

Cy LCY Scale factor of Fy shape factor


λ µξ λ γψ

λ Fz0λλ µξλλλλ γλ

Adams/Tire

194




Ey LEY Scale factor of Fy curvature factor

Ky LKY Scale factor of Fy cornering stiffness

C LCC Scale factor of camber shape factor

K LKC Scale factor of camber stiffness (K-factor)

E LEC Scale factor of camber curvature factor

Hyy LHY Scale factor of Fy horizontal shift

LGAY Scale factor of camber force stiffness


Mr LRES Scale factor for offset of residual torque

gz LGAZ Scale factor of camber torque stiffness


VMxMx LVMX Scale factor of Mx vertical shift

My LMY Scale factor of rolling resistance torque

Name:

Name used in tire property

file: Explanation:




s LS Scale factor of Moment arm of Fx





s LS Scale factor of Moment arm of Fx

Name:

Name used in tire

property file: Explanation:

λ µλλλ γλ γλ γλλ γψλλλλλλ

λ αλ κλ κλ

λ αλ κλ κλ

195Tire Models


Steady-State: Magic Formula for PAC MC• Steady-State Pure Slip








(222)

(223)

(224)

(225)


(226)

(227)

(228)

(229)



LSGKP Scale factor of relaxation length of Fx

LSGAL Scale factor of relaxation length of Fy


σ κσ αλ


Fx0 Dx Cxarc Bxκx Ex Bxκx arc Bxκx( )tan–( )– tan[ ] SVx+sin=

κx κ SHx+=

γx γ λ γx⋅=

Cx pCx1 λ Cx⋅=

Dx µx Fz ζ1⋅ ⋅=

µx pDx1 pDx2dfz+( ) 1 pDx3 γx2⋅–( )λ µx⋅=

Ex pEx1 pEx2dfz pEx3dfz2+ +( ) 1 pEx4 κx( )sgn– λ Ex with Ex 1≤⋅ ⋅=

Adams/Tire

196

(230)

(231)

(232)

(233)



(234)

(235)





pDx3 PDX3 Variation of friction Mux with camber







pKx3 PKX3 Exponent in slip stiffness Kfx/Fz with load

pVx1 PVX1 Vertical shift Svx/Fz at Fznom

pVx2 PVX2 Variation of shift Svx/Fz with load

Kx Fz pKx1 pKx2dfz+( ) pKx3dfz( ) λ K

(Kx

⋅exp⋅ ⋅

BxCxDx κx∂

∂ Fx0at κx 0 )

=

= = =

Bx Kx CxDx( )⁄=

SHx psy1Fzλ My SVx+( ) Kx⁄–=

SVx Fz pVx1 pVx2dfz+( ) λ Vx λ µx ζ1⋅ ⋅ ⋅ ⋅=


Fy0 Dy Cyarc Byα y Ey Byαy arc Byα y( )tan–( )–

Cγarc Bγγy Eγ Bγγy arc Bγγy( )tan–( )– tan+

tan[

]

sin=

197Tire Models

(236)


(237)

with coefficients:

(238)

(239)

(240)

(241)


(242)

(243)

(244)

(245)

and the explicit camber stiffness:

(246)

(247)

(248)

α y α SHy Cy Cγ 2<+( )+=

γy γ λ γy⋅=

Cy pCy1 λ Cy⋅=

Dy µy Fz ζ2⋅ ⋅=

µy pDy1 pDy2dfz( ) 1 pDy3γy2–( ) λ µy⋅ ⋅exp⋅=

Ey pEy1 pEy2γy2 pEy3 pEy4γy+( ) α y( )sin⋅–+ λ Ey with Ey 1≤⋅=

Ky pKy1Fzo pKy2arcFz

pKy3 pKy4γy2+( )Fzoλ Fzo

-----------------------------------------------------------

tan

1 pKy5γy2–( ) λ Fzo λ Ky

(Ky

⋅ ⋅

sin

ByCyDy α y∂

∂ Fyoat α y 0 )

=

= = =

By Ky CyDy( )⁄=

SHy pHy1 λ Hy⋅=

Cγ pCy2 λ Cγ⋅=

Kγ pKy6 pKy7dfz+( ) Fz λ Kγ (=BγCγDγ⋅ ⋅γ∂

∂ Fyoat α y 0= = =

Eγ pEy5 λ Eγ with Eγ 1≤⋅=

Bγ Kγ CγDγ( )⁄=

Adams/Tire

198



(249)


(250)

(251)


(252)

Name:Name used in



pCy2 PCY2 Shape factor Cfc for camber forces


pDy2 PDY2 Exponent lateral friction Muy

pDy3 PDY3 Variation of friction Muy with squared camber


pEy2 PEY2 Variation of curvature Efy with camber squared

pEy3 PEY3 Asymmetric curvature Efy at Fznom

pEy4 PEY4 Asymmetric curvature Efy with camber

pEy5 PEY5 Camber curvature Efc


pKy2 PKY2 Curvature of stiffness Kfy

pKy3 PKY3 Peak stiffness factor

pKy4 PKY4 Peak stiffness variation with camber squared

pKy5 PKY5 Lateral stiffness dependency with camber squared

pKy6 PKY6 Camber stiffness factor Kfc

pKy7 PKY7 Vertical load dependency of camber stiffness Kfc


Mz' Mz0 α γ Fz, ,( )=

Mz0 t Fy0 Mzr+⋅–=


α t α SHt+=

Mzr α r( ) Dr Crarc Brα r( )tan[ ] α( )cos⋅cos=

199Tire Models

(253)


(254)

with coefficients:

(255)

(256)

(257)

(258)

(259)

(260)

(261)

(262)

(263)


(264)





qBz3 qBz3 Variation of slope Bpt with load squared

qBz4 QBZ4 Variation of slope Bpt with camber

qBz5 QBZ5 Variation of slope Bpt with absolute camber

α r α SHr+=

γz γ λ γz⋅=

Bt qBx1 qBx2dfz qBx3dfz2+ +( ) 1 qBx4γz qBz5 γz+ + λ Ky λ µy⁄⋅ ⋅=

Ct qCz1=

Dt Fz qDz1 qDz2dfz+( ) 1 qDz3 γz qDz4γz2+ +( ) R0 Fz0⁄( )⋅ ⋅ ⋅=

Et qEx1 qEx2dfz qEx3dfz2+ +( )=

1 qEz4 qEz5γz+( )2

π---

arc Bt Ct α t⋅ ⋅( )tan⋅ ⋅+

with Et 1≤

SHt 0=

Br qBz9 λ Ky λ µy⁄⋅=

Dr Fz qDz6 qDz7dfz+( )λ r qDz8 qDz9dfz+( )γz qDz10 qDz11dfz+( ) γz γz⋅( )+ +[ ]R0λ µy=

SHr qHz1 qHz2dfz qHz3 qHz4dfz+( )γz+ +=

Kz t Ky αd

dMzat α–≈

⋅– 0 )= =

Adams/Tire

200


PAC-TIME has two methods for calculating the combined slip forces and moments. If the user supplies










qBz9 QBZ9 Slope factor Br of residual torque Mzr




qDz3 QDZ3 Variation of peak Dpt with camber

qDz4 QDZ4 Variation of peak Dpt with camber squared.



qDz8 QDZ8 Variation of peak factor Dmr with camber

qDz9 QDZ9 Variation of Dmr with camber and load

qDz10 QDZ10 Variation of peak factor Dmr with camber squared

qDz11 QDZ11 Variation of Dmr with camber squared and load



qEz3 QEZ3 Variation of curvature Ept with load squared


qEz5 QEZ5 Variation of Ept with camber and sign Alpha-t

qHz1 QHZ1 Trail horizontal shift Shr at Fznom

qHz2 QHZ2 Variation of shift Shr with load

qHz3 QHZ3 Variation of shift Shr with camber

qHz4 QHZ4 Variation of shift Sht with camber and load


201Tire Models



(265)

with Gx o the weighting function of the longitudinal force for pure slip.

We write:

(266)

(267)

with coefficients:

(268)

(269)

(270)

(271)

(272)


(273)


Name:Name used in tire property

file: Explanation:



rBx3 RBX3 Influence of camber on stiffness for Fx reduction



α


α s α SHxα+=

Bxα rBx1 rBx3γ2+( ) arc rBx2κ tan[ ] λ xα⋅cos=

Cxα rCx1=

Dxα

FxoCxαarc BxαSHxα Exα BxαSHxα arc BxαSHxα( )tan–( )– tan[ ]cos

------------------------------------------------------------------------------------------------------------------------------------------------------------------=


SHxα rHx1=

Gxα



Adams/Tire

202


(274)



(275)

(276)


(277)

(278)

(279)

(280)

(281)

(282)

(283)


(284)




Name:Name used in tire property

file: Explanation:


κ

Fy Dyκ Cyκarc Byκκs Eyκ Byκκs arc Byκκs( )tan–( )– tan[ ] SVyκ+cos=

κs κ SHyk+=

Byκ rBy1 rBy4γ2+( ) arc rBy2 α rBy3–( ) tan[ ] λ yκ⋅cos=

Cyκ rCy1=

Dyκ

FyoCyκarc ByκSHyκ Eyκ ByκSHyκ arc ByκSHyκ( )tan–( )– tan[ ]cos

---------------------------------------------------------------------------------------------------------------------------------------------------------------=



SVyk DVyκ rVy5arc rvy6κ( )tan[ ] λ Vyκ⋅sin=


Gyκ

Cyκarc Byκκs Eyκ Byκκ s arc Byκκs( )tan–( )– tan[ ]cos

Cyκarc ByκSHyκ Eyκ ByκSHyκ arc ByκSHyκ( )tan–( )– tan[ ]cos---------------------------------------------------------------------------------------------------------------------------------------------------------------=

203Tire Models



(285)

with:

(286)

(287)

(288)

(289)

(290)

with the arguments:










rVy1 RVY1 Kappa-induced side force Svyk/Muy*Fz at

Fznom

rVy2 RVY2 Variation of Svyk/Muy*Fz with load

rVy3 RVY3 Variation of Svyk/Muy*Fz with inclination

rVy4 RVY4 Variation of Svyk/Muy*Fz with alpha

rVy5 RVY5 Variation of Svyk/Muy*Fz with kappa

rVy6 RVY6 Variation of Svyk/Muy*Fz with atan (kappa)

Mz' t Fy' Mzr s Fx⋅+ +⋅–=

t t α t eq,( )=

Dt Ctarc Btα t eq, Et Btα t eq, ac Btα t eq,( )tan–( )– tan[ ] α( )coscos=



s ssz1 ssz2 Fy Fz0⁄( ) ssz3 ssz4dfz+( )γ+ + R0 λ s⋅ ⋅=

Adams/Tire

204

(291)

(292)




(293)





ssz1 SSZ1 Nominal value of s/R0

effect of Fx on Mz

ssz2 SSZ2 Variation of distance s/R0

with Fy/Fznom


with inclination


with load and inclination


qsx1 QSX1 Lateral force-induced

overturning couple

qsx2 QSX2 Inclination-induced

overturning couple

qsx3 QSX3 Fy-induced overturning

couple

α t eq, arc α2

t

Kx

Ky

------

2

κ2+tan α t( )sgn⋅tan=

α r eq, arc α2

r

Kx

Ky

------

2

κ2+tan α r( )sgn⋅tan=

Mx R0 Fz qsx1λ VMx qsx2 γ qsx3Fy

Fz0--------⋅+⋅–

λ Mx⋅ ⋅=

205Tire Models

(294)









qsy1 QSY1 Rolling resistance moment

coefficient


depending on Fx


depending on speed


depending on speed^4


My R0 Fz qSy1 qSy2Fx Fz0⁄ qSy3 Vx Vref⁄ qSy4 Vx Vref⁄( )4+ + + ⋅ ⋅=

κc κ SHxSVxKx

---------+ +=

α c α SHySVyKy

---------+ +=

α∗ α c( )sin=

βκc

κc2

α∗ 2+

-------------------------

acos=

µx act,

Fx 0, SVx–

Fz-------------------------= µy act,

Fy 0, SVy–

Fz-------------------------=

Adams/Tire

206




Transient Behavior in PAC MCThe previous Magic Formula examples are valid for steady-state tire behavior. When driving, however,


frequency behavior (up to 8 Hz) is called transient behavior.


µx max,

Dx

Fz------= µy max,

Dy

Fz------=

µx1

1

µx act,-------------

2 βtan

µy max,----------------

2+

---------------------------------------------------------=

µyβtan

1

µx max,----------------

2 βtan

µy act,-------------

2+

---------------------------------------------------------=

Fxµx

µx act,-------------Fx 0,= Fy

µy

µy act,-------------Fy 0,=

SVyκ

207Tire Models

For accurate transient tire behavior, you can use the "stretched string" tire model (see also reference [1]).

The tire belt is modeled as stretched string, which is supported to the rim with lateral (and longitudinal)

springs. The figure, Stretched String Model for Transient Tire Behavior, shows a top-view of the string

model. When rolling, the first point having contact with the road adheres to the road (no sliding

assumed). Therefore, a lateral deflection of the string arises that depends on the slip angle size and the

history of the lateral deflection of previous points having contact with the road.



(295)


than 10 m. This differential example cannot be used at zero speed, but when multiplying with Vx, the

example can be transformed to:

1

Vx

------td

dv1 v1σ α------+ α( ) aφ+tan=

σ α φ

Adams/Tire

208

(296)





(297)


(298)

(299)


(300)

(301)

Using these practical slip quantities, and , the Magic Formula examples can be used to calculate


(302)

(303)

(304)

Gyroscopic Couple in PAC MCWhen having fast rotations about the vertical axis in the wheel plane, the inertia of the tire belt may lead



σ α td

dv1Vx v1+ σ αVsy=

σ x td

du1Vx u1+ σ xVsx–=

σ x Fz pTx1 pTx2dfz+( ) pTx3dfz( ) R0 Fz0⁄( )λ σ κ⋅exp⋅ ⋅=

σ α pTy1 pTy2arcFz

pTy3 pKy4γ2+( )Fz0λ Fz0

---------------------------------------------------------

tan 1 pKy5γ2–( ) R0λ Fz0λ σ α⋅sin=

κ' α '

κ'u1

σ x

------ Vx( )sin⋅=

α 'v1σ α------

atan=

κ' α '

Fx' Fx α ' κ' Fz, ,( )=

Fy' Fy α ' κ' γ Fz, , ,( )=

Mz' Mz' α ' κ' γ Fz, , ,( )=

209Tire Models

(305)

with the parameters (in addition to the basic tire parameter mbelt):

(306)

and:

(307)


(308)





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


pTx1 PTX1 Relaxation length

sigKap0/Fz at Fznom

pTx2 PTX2 Variation of sigKap0/Fz

with load

pTx3 PTX3 Variation of sigKap0/Fz

with exponent of load

pTy1 PTY1 Peak value of relaxation

length Sig_alpha

pTy2 PTY2 Shape factor for lateral

relaxation length

pTy3 PTY3 Load where lateral

relaxation is at maximum

qTz1 QTZ1 Gyroscopic torque constant


Mz gyr, cgyrmbeltVr1 td



arc Brα r eq,( )tan cos 1=

Mz Mz' Mz gyr,+=

Adams/Tire

210

for both the left and the right side of the vehicle model, the vehicle does not drive straight at zero steering

wheel angle.




If this keyword is available, Adams/Car corrects for the conicity and 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 (see figure of dialog box below).

Next to the LEFT and RIGHT side option of TYRESIDE, you can also select 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



211Tire Models

USE_MODES of PAC MC: from Simple to ComplexThe parameter USE_MODE in the tire property file allows you to switch the output of the PAC MC tire

model from very simple (that is, steady-state cornering) to complex (transient combined cornering and

braking).

The options for USE_MODE and the output of the model are listed in the table below.

Adams/Tire

212

USE_MODE Values of PAC MC and Related Tire Model Output

Contact MethodsThe PAC MC model supports the following roads:



By default the PAC-MC uses a one point of contact model similar to all the other Adams/Tire Handling

models. However the PAC-MC has an option to take the tire cross section shape into account:

Tire Cross-Section Profile Contact Method

In combination with the 2D Road Model and the 3D Road Model, you can improve the tire-road contact

calculation method by providing the tire's cross-section profile, which has an important influence on the

wheel center height at large inclination angles with the road.

USE MODE: State: Slip conditions:

PAC MC 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

213Tire Models

If the tire model reads a section called [SECTION_PROFILE_TABLE] in the tire property file, the cross

section profile will be taken into account for the vertical load calculation of the tire. The method assumes

that the tire deformation will not influence the position of the point with largest penetration (P), which

is valid for motor cycle tires.

The vertical tire load Fz is calculated using the penetration (effpen = ) of the tire through the tangent

road plane in the point C, see Figure above, according to:

(309)

Because in this method the tangent to the cross section profile determines the point P, a high accuracy of

the cross section profile is required. The section height y as function of the tire width x must be a

continous and monotone increasing function. To avoid singularities and instability, it is highly

ρ


Adams/Tire

214

recommended to fit measured cross section data with a polynom (for example y = a·x2 + b·x4 + c·x6 +

..) and provide the y cross section height data (y) from the polynom in the tire property file up to the

maximum width of the tire. The profile is assumed to be symmetric with respect to the wheel plane.

Note that the PAC MC model has only one point of contact with the road; therefore, the wavelength of

road obstacles must be longer than the tire radius for realistic output of the model. In addition, the contact

force computed by this tire model is normal to the road plane. Therefore, the contact point does not

generate a longitudinal force when rolling over a short obstacle, such as a cleat or pothole.

For ride and comfort analysis, we recommend more sophisticated tire models, such as Ftire.

Quality Checks for the Tire Model ParametersBecause PAC MC uses an empirical approach to describe tire - road interaction forces, incorrect






Camber stiffness has been explicitly defined in PAC MC (see equation (246). For realistic tire behavior,

the sign of the camber stiffness must be negative (TYDEX W-axis (ISO) system). If the sign is positive,

the coefficients may not be valid for the ISO but for the SAE coordinate system. Note that PAC MC only

uses coefficients for the TYDEX W-axis (ISO) system.





215Tire Models

The table below lists further checks on the PAC MC parameters.

Checklist for PAC MC Parameters and Properties


LONGVL 1 m/s Reference velocity at which

parameters are measured

VXLOW Approximately 1m/s Threshold for scaling down

forces and moments

Dx 0 Peak friction (see equation

(137))

pDx1/pDx2 0 Peak friction Fx must

decrease with increasing

load

Kx 0 Long slip stiffness (see

equation (140))

Dy 0 Peak friction (see equation

(149))

pDy1/pDy2 0 Peak friction Fx must

decrease with increasing

load

>

>

<

<

>

<

Adams/Tire

216




$------------------------------------------------------long_slip_range [LONG_SLIP_RANGE]KPUMIN


KPUMAX = 1.5








= 73.75 $Minimum allowed wheel load

FZMAX = 3319.5


Ky 0 Cornering stiffness (see

equation (152))

qsy1 0 Rolling resistance, should in

range of 0.005 - 0.015

K 0 Camber stiffness (see

equation (156))


<

<

γ <

217Tire Models


model will be performed with the minimum or maximum value of this range to avoid non-realistic tire


Standard Tire Interface (STI) for PAC MCBecause 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]).




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 tire behavior description.











• Tire Kinematics

• Slip Quantities


Adams/Tire

218

General

General Definitions

Tire Kinematics


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 centre 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.






d Dimensionless radial tire

deflection

[-]

Fz0 Radial tire deflection at

nominal load

[m]


Rotational velocity of the

wheel

[rads-1]

ρρ

ρ

ω

219Tire Models

Slip Quantities


Forces and Moments












Vy Lateral speed of tire contact

center

[ms-1]



Slip angle [rad]











καγ

Adams/Tire

220





Example of PAC MC Tire Property File[MDI_HEADER]FILE_TYPE ='tir'FILE_VERSION =3.0FILE_FORMAT ='ASCII'! : TIRE_VERSION : PAC Motorcycle! : COMMENT : Tire 180/55R17! : COMMENT : Manufacturer $! : COMMENT : Nom. section with (m) 0.18$! : COMMENT : Nom. aspect ratio (-) 55! : COMMENT : Infl. pressure (Pa) 200000! : COMMENT : Rim radius (m) 0.216$! : COMMENT : Measurement ID$! : COMMENT : Test speed (m/s) 16.7$! : COMMENT : Road surface $! : COMMENT : Road condition Dry! : FILE_FORMAT : ASCII! : Copyright MSC.Software, Mon Oct 20 10:46:57 2003!! USE_MODE specifies the type of calculation performed:! 0: Fz only, no Magic Formula evaluation! 1: Fx,My only! 2: Fy,Mx,Mz only! 3: Fx,Fy,Mx,My,Mz uncombined force/moment calculation! 4: Fx,Fy,Mx,My,Mz combined force/moment calculation! +10: including relaxation behaviour! *-1: mirroring of tyre characteristics!! example: USE_MODE = -12 implies:! -calculation of Fy,Mx,Mz only! -including relaxation effects! -mirrored tyre characteristics!$----------------------------------------------------------------units[UNITS]LENGTH ='meter'FORCE ='newton'

ANGLE ='radians'

MASS ='kg'

TIME ='second'

$----------------------------------------------------------------model[MODEL]

221Tire Models

PROPERTY_FILE_FORMAT ='PAC_MC'USE_MODE


VXLOW = 1

LONGVL = 16.7

$Longitudinal speed during measurements $TYRESIDE

= 'SYMMETRIC' $Mounted side of tyre at vehicle/test bench

$-----------------------------------------------------------dimensions[DIMENSION]UNLOADED_RADIUS

= 0.322 $Free tyre radius $

WIDTH = 0.18

$Nominal section width of the tyre $RIM_RADIUS

= 0.216 $Nominal rim radius $

RIM_WIDTH = 0.135

$Rim width $$----------------------------------------------------------------shape[SHAPE]radial width1.0 0.00.994 0.2110.975 0.4230.947 0.6340.894 0.8450.841 1.0$------------------------------------------------section_profile_table $ For taking the tire's cross shape into account (optional), max 100 pnts[SECTION_PROFILE_TABLE] x y 0.00000

0.00000000.00300

0.00004680.00600

0.00018770.00900

0.00042350.01200

0.00075610.01500

Adams/Tire

222

0.00118770.01800

0.00172160.02100

0.00236130.02400

0.00311140.02700

0.00397700.03000

0.00496390.03300

0.00607850.03600

0.00732820.03900

0.00872070.04200

0.01026460.04500

0.01196940.04800

0.01384490.05100

0.01590180.05400

0.01815170.05700

0.02060650.06000

0.02327930.06300

0.02618360.06600

0.02933370.06900

0.03274470.07200

0.03643230.07500

0.04041320.07800

0.04470470.08100

0.04932480.08400

0.05429230.08700

0.0596270$------------------------------------------------------------parameter[VERTICAL]VERTICAL_STIFFNESS = 2e+005

$Tyre vertical stiffness $

223Tire Models

VERTICAL_DAMPING = 50 $Tyre vertical damping $

BREFF = 8.4 $Low load stiffness eff. rolling radius $

DREFF = 0.27 $Peak value of eff. rolling radius $

FREFF = 0.07 $High load stiffness eff. rolling radius $

FNOMIN = 1475 $Nominal wheel load

$-----------------------------------------------------------load_curve$ For a non-linear tire vertical stiffness (optional)$ Maximum of 100 points[DEFLECTION_LOAD_CURVE]pen

fz0.000

0.00.001

212.00.002

428.00.003

648.00.005

1100.00.010

2300.00.020

5000.00.030

8100.0$------------------------------------------------------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 = -1.0996

$Minimum valid camber angle $CAMMAX = 1.0996

$Maximum valid camber angle $

Adams/Tire

224

$-------------------------------------------------vertical_force_range[VERTICAL_FORCE_RANGE]FZMIN = 73.75

$Minimum allowed wheel load $FZMAX = 3319.5

$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 $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 $LCC = 1

$Scale factor of camber shape factor $LKC = 1

$Scale factor of camber stiffness (K-factor) $LEC = 1

$Scale factor of camber curvature factor $LHY = 1

$Scale factor of Fy horizontal shift $LGAY = 1

$Scale factor of camber force stiffness $LTR = 1

$Scale factor of Peak of pneumatic trail $LRES = 1

$Scale factor of Peak of residual torque $

225Tire Models

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 = 1.7655

$Shape factor Cfx for longitudinal force $PDX1 = 1.2839

$Longitudinal friction Mux at Fznom $PDX2 = -0.0078226

$Variation of friction Mux with load $PDX3 = 0

$Variation of friction Mux with camber $PEX1 = 0.4743

$Longitudinal curvature Efx at Fznom $PEX2 = 9.3873e-005

$Variation of curvature Efx with load $PEX3 = 0.066154

$Variation of curvature Efx with load squared $PEX4 = 0.00011999

$Factor in curvature Efx while driving $PKX1 = 25.383

$Longitudinal slip stiffness Kfx/Fz at Fznom $PKX2 = 1.0978

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226

$Variation of slip stiffness Kfx/Fz with load $PKX3 = 0.19775

$Exponent in slip stiffness Kfx/Fz with load $PVX1 = 2.1675e-005

$Vertical shift Svx/Fz at Fznom $PVX2 = 4.7461e-005

$Variation of shift Svx/Fz with load $RBX1 = 12.084

$Slope factor for combined slip Fx reduction $RBX2 = -8.3959

$Variation of slope Fx reduction with kappa $RBX3 = 2.1971e-009

$Influence of camber on stiffness for Fx combined $RCX1 = 1.0648

$Shape factor for combined slip Fx reduction $REX1 = 0.0028793

$Curvature factor of combined Fx $REX2 = -0.00037777

$Curvature factor of combined Fx with load $RHX1 = 0

$Shift factor for combined slip Fx reduction $PTX1 = 0.83

$Relaxation length SigKap0/Fz at Fznom $PTX2 = 0.42

$Variation of SigKap0/Fz with load $PTX3 = 0.21

$Variation of SigKap0/Fz with exponent of load $$----------------------------------------------------------overturning[OVERTURNING_COEFFICIENTS]QSX1 = 0

$Lateral force induced overturning moment $QSX2 = 0.16056

$Camber induced overturning moment $QSX3 = 0.095298

$Fy induced overturning moment $$--------------------------------------------------------------lateral[LATERAL_COEFFICIENTS]PCY1 = 1.1086

$Shape factor Cfy for lateral forces $PCY2 = 0.66464

$Shape factor Cfc for camber forces $PDY1 = 1.3898

$Lateral friction Muy $PDY2 = -0.0044718

$Exponent lateral friction Muy $

227Tire Models

PDY3 = 0.21428 $Variation of friction Muy with squared

camber $PEY1 = -0.80276

$Lateral curvature Efy at Fznom $PEY2 = 0.89416

$Variation of curvature Efy with camber squared $PEY3 = 0

$Asymmetric curvature Efy at Fznom $PEY4 = 0

$Asymmetric curvature Efy with camber $PEY5 = -2.8159

$Camber curvature Efc $PKY1 = -19.747

$Maximum value of stiffness Kfy/Fznom $PKY2 = 1.3756

$Curvature of stiffness Kfy $PKY3 = 1.3528

$Peak stiffness factor $PKY4 = -1.2481

$Peak stiffness variation with camber squared $PKY5 = 0.3743

$Lateral stiffness depedency with camber squared $PKY6 = -0.91343

$Camber stiffness factor Kfc $PKY7 = 0.2907

$Vertical load dependency of camber stiffn. Kfc $PHY1 = 0

$Horizontal shift Shy at Fznom $RBY1 = 10.694

$Slope factor for combined Fy reduction $RBY2 = 8.9413

$Variation of slope Fy reduction with alpha $RBY3 = 0

$Shift term for alpha in slope Fy reduction $RBY4 = -1.8256e-010

$Influence of camber on stiffness of Fy combined $RCY1 = 1.0521

$Shape factor for combined Fy reduction $REY1 = -0.0027402

$Curvature factor of combined Fy $REY2 = -0.0094269

$Curvature factor of combined Fy with load $RHY1 = -7.864e-005

$Shift factor for combined Fy reduction $RHY2 = -6.9003e-006

$Shift factor for combined Fy reduction with load $

Adams/Tire

228

RVY1 = 0 $Kappa induced side force Svyk/Muy*Fz

at Fznom $RVY2 = 0

$Variation of Svyk/Muy*Fz with load $RVY3 = -0.00033208

$Variation of Svyk/Muy*Fz with camber $RVY4 = -4.7907e+015

$Variation of Svyk/Muy*Fz with alpha $RVY5 = 1.9

$Variation of Svyk/Muy*Fz with kappa $RVY6 = -30.082

$Variation of Svyk/Muy*Fz with atan(kappa) $PTY1 = 0.75

$Peak value of relaxation length Sig_alpha $PTY2 = 1

$Shape factor for Sig_alpha $PTY3 = 0.6

$Value of Fz/Fznom where Sig_alpha is maximum $$---------------------------------------------------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 = 9.246

$Trail slope factor for trail Bpt at Fznom $QBZ2 = -1.4442

$Variation of slope Bpt with load $QBZ3 = -1.8323

$Variation of slope Bpt with load squared $QBZ4 = 0

$Variation of slope Bpt with camber $QBZ5 = 0.15703

$Variation of slope Bpt with absolute camber $QBZ9 = 8.3146

$Slope factor Br of residual torque Mzr $QCZ1 = 1.2813

$Shape factor Cpt for pneumatic trail $

229Tire Models

QDZ1 = 0.063288 $Peak trail Dpt = Dpt*(Fz/Fznom*R0) $

QDZ2 = -0.015642 $Variation of peak Dpt with load $

QDZ3 = -0.060347 $Variation of peak Dpt with camber $

QDZ4 = -0.45022 $Variation of peak Dpt with camber squared $

QDZ6 = 0 $Peak residual torque Dmr = Dmr/(Fz*R0)

$QDZ7 = 0 $Variation of peak factor Dmr with load

$QDZ8 = -0.08525

$Variation of peak factor Dmr with camber $QDZ9 = -0.081035

$Variation of peak factor Dmr with camber and load $QDZ10 = 0.030766

$Variation of peak factor Dmr with camber squared $QDZ11 = 0.074309

$Variation of Dmr with camber squared and load $QEZ1 = -3.261

$Trail curvature Ept at Fznom $QEZ2 = 0.63036

$Variation of curvature Ept with load $QEZ3 = 0

$Variation of curvature Ept with load squared $QEZ4 = 0

$Variation of curvature Ept with sign of Alpha-t $QEZ5 = 0

$Variation of Ept with camber and sign Alpha-t $QHZ1 = 0

$Trail horizontal shift Sht at Fznom $QHZ2 = 0

$Variation of shift Sht with load $QHZ3 = 0

$Variation of shift Sht with camber $QHZ4 = 0

$Variation of shift Sht with camber and load $SSZ1 = 0

$Nominal value of s/R0: effect of Fx on Mz $SSZ2 = 0.0033657

$Variation of distance s/R0 with Fy/Fznom $SSZ3 = 0.16833

$Variation of distance s/R0 with camber $SSZ4 = 0.017856

Adams/Tire

230

$Variation of distance s/R0 with load and camber $QTZ1 = 0

$Gyroscopic torque constant $MBELT = 0

$Belt mass of the wheel -kg- $

231Tire Models

521-Tire ModelOverview

The 521-Tire model is a simple model that requires a small set of parameters or experimental data to

simulate the behavior of tires.

This chapter includes the following sections:

• About 521-Tire

• Tire Slip Quantities and Transient Tire Behaviour

• Force Calculations

• Converting Slip Ratio Data to Velocity Data

• Contact Methods

• 521-Tire Tire and Road Property Files

About 521-Tire

The 521-Tire is the first tire model incorporated in Adams. The name “521” (actually “5.2.1”) refers to

the version number of Adams/Tire when it was first released.

The slip forces and moments can be calculated in two ways:

• Using the Equation method

• Using the Interpolation method

Two dedicated contact methods exist for the 521-Tire:

• Point Follower, used for Handling analysis models

• Equivalent Plane Method, used for 3D Contact analysis models

Any combination of force and contact method is allowed.

The road data files used for the 521-Tire are unique and cannot be used in combination with any other

Handling tire model. The 521 road file format is described in Road Data File 521_pnt_follow.rdf.

Note that the capability and generality of the 521-Tire have been superseded by other, newer tire models,

described throughout this guide. We’ve retained the 521-Tire model primarily for backward

compatibility. We recommend that you use other tire models for new work.

Tire Slip Quantities and Transient Tire Behaviour

Definition of Tire Slip QuantitiesSlip Quantities at Combined Cornering and Braking/Traction

Adams/Tire

232






Note that for realistic tire forces the slip angle is limited to 90 degrees and the longitudinal slip in


Lagged longitudinal and lateral slip quantities (transient tire behavior)In general, the tire rotational speed and lateral slip will change continuously because of the changing



system is used both for the longitudinal slip as the side slip angle, . Considering the tire belt as a




Ω

Vsx Vz ΩR1–=

Vsy Vy=

κ α

κVsx

Vx


Vx

---------= =

α κ

κ α

233Tire Models













Now the practical slip quantities, and are defined based on the tire deformation:

1

Vx

------dv1

dt--------

v1

σ α------+ α( )tan aφ+=

σ α φ

σ α

dv1

dt-------- Vx v1+ σ– κVsx=

σ α

du1

dt-------- Vx u1+ σ– κVsx=

κ′ α′

Adams/Tire

234



The longitudinal and lateral relaxation length are read from the tire property file, see Tire Property File

521_equation.tir and 521_interpol.tir

Force Calculations

You can use the 521-Tire model for handling and durability analyses.

Directional Vectors for the Application of Tire Forces and Torques at the Center of the Tire-Road

Surface Contact Patch

The forces act along the directional vectors. From the tire spin vector and various information you supply

in the tire property and the road profile data files, Adams/Tire determines the positions and orientations

of the tire vertical, lateral, and longitudinal directional vectors. Figure 3 shows these directional vectors.

The tire vertical force acts along the vertical directional vector, the tire aligning torque acts about the

same vector, the tire lateral force acts along the lateral directional vector, and the tire longitudinal force

acts along the longitudinal directional vector. At this point, Adams/Tire determines the force directions

as if it were going to apply the tire aligning torque and all of the tire forces at the center of the tire-road

surface contact patch.

κ'u1σ κ------ Vx( )sin=

α 'v1σ α------

atan=

κ′ α′ κ α

235Tire Models

The tire-road surface contact patch may deflect laterally. Adams/Tire calculates the lateral deflection in

the direction (and with the sign) of the lateral force. The magnitude of the deflection is equal to the lateral

force divided by the tire lateral stiffness you provide in the tire property data file.

The tire vertical, lateral, and longitudinal forces are forces in the tire vertical, lateral, and longitudinal

directions (as determined at the tire-road surface contact patch). The tire aligning torque is a torque about

the tire vertical vector. The vehicle durability force has components in both the tire vertical and the tire

longitudinal directions.

Normal ForceThe tire normal force Fz is calculated based on the tire deflection and radial velocity. A progressive

spring and linear damping constant are employed:

where Fstiff is tire stiffness force and Fdamp is tire damping force. The vertical stiffness force is calculated

from:

where Kz is the tire vertical stiffness, δ is tire deflection, and is the stiffness exponent. The tire

damping force is calculated from:

where Cz is the tire damping constant.

The damping constant is reduced for small tire deflections, which are below 5% of the unloaded tire

radius.

The tire vertical stiffness can also be described using a spline function (force versus deflection) in the

Adams dataset. The user array is used to switch between tire property file stiffness and spline stiffness.

If the first value in the user array is equal to '5215', the spline vertical stiffness is used. The second value

of the user array refers to the ID of the spline. The message, 'Using spline data for the vertical spring', is

shown in the message file. If the first value in the user array is not equal to '5215', the tire property file

stiffness is used.

The following is an example of using the spline vertical stiffness:

! adams_view_name='spline_vertical_stiffness'SPLINE/10, X = -1,0,10,30, Y = 0,0,2000,6000!! adams_view_name='wheel_user_array'ARRAY/102

Fz Fstiff Fdamp–=

Fstiff Kzδθ

=

θ

Fdamp Cz RadialVelocity×=

Adams/Tire

236

, NUM=5215,10

Another option for having a non-linear tire stiffness is to introduce a deflection-load table in the tire

property file in a section called [DEFLECTION_LOAD_CURVE]. See 521-Tire Tire and Road Property

Files on page 20. 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.

Longitudinal ForceThe tire longitudinal force Fx can have up to three contributions:

• Traction/braking force

• Rolling resistance force

• Durability force (in case of durability contact)

Traction/Braking Force

Traction force is developed if the vehicle is starting to move and a braking force if the vehicle is

beginning to stop. In either case, the absolute magnitude of the force is calculated from:

where the friction coefficient µ is a function of the longitudinal slip velocity Vsx in the contact patch.

Note that this is somewhat unusual, since all the other Handling tire models in Adams/Tire assume that

the longitudinal force Fx is a function of the slip ratio.

Schematic of Friction Coefficient Versus Local Slip Velocity

Fx µFz=

237Tire Models

The µ curve as a function of longitudinal slip velocity is created using standard Adams STEP functions (see body 4 on page 10). You have to specify two points on the curve to define this characteristic:

• The coordinates of the curve at µstatic: (velocity µstatic, µstatic)

• The coordinates of the curve at µdynamic: (velocity µdynamic, µdynamic)

The friction values may be available to you as function of slip ratio instead of slip velocity. Converting

Slip Ratio Data to Velocity Data on page 16 explains how the slip ratios can be converted to slip

velocities.

Rolling Resistance Force

Rolling resistance Moment My is calculated from:

where coefrr is the rolling resistance coefficient that should be supplied in the tire property data file.

Durability Force

Durability force, sometimes known as radial planar force, is a special kind of tire vertical force. It is the

durability force that resists the action of road bumps. This force acts along the instantaneous vertical

directional vector calculated by Adams/Tire. The Adams/Tire durability tire forces are limited to two-

dimensional forces that lie in the plane of the tire and are directed toward the wheel-center marker.

Adams/Tire superimposes these forces upon any traction or lateral forces developed in the tire-road

surface interaction.

My coefrr Fz⋅=

Adams/Tire

238

You must select the Equivalent Plane Method for generating these durability forces.

Lateral Force and Aligning TorqueTwo methods exist for calculating the lateral force Fy and self-aligning moment Mz:

• Interpolation Method

• Equation Method

Interpolation Method

The AKIMA spline is employed to calculate Fy and Mz as a function of the slip angle α , camber angle γ,

and vertical load Fz. You should provide the data in the SAE axis system.

Note that the slip angle α and vertical load Fz input for the force and moment calculation of Fx, Fy, Mx,

My, and Mz are limited to minimum and maximum values in the input to avoid unrealistic extrapolated

values.

Equation Method

The Equation Method uses the following equation to generate the lateral force Fy:

where Kα denotes the tire cornering stiffness coefficient.

The aligning moment Mz is calculated using the pneumatic trail t according to:

while the pneumatic trails are calculated with half the contact length a:

with R0 and Rl are, respectively, the unloaded and loaded tire radius.

Overturning MomentIn both methods, the overturning moment Mx calculation is based on the lateral tire force Fy, the lateral

tire stiffness Ky, and the vertical load:

Fy µstatFz 1 eKα α–

–( )⋅ sign α( )⋅( )–=

Mz t– Fy⋅=

t1

3--- a e

Kα α–⋅⋅=

a R02

R12

–=

239Tire Models

Tire Lateral Force as a Function of Slip Angle

• The contribution of the camber is disregarded in the Equation Method.

• The cornering stiffness equals .

Combined Slip of 5.2.1The combined slip calculation of the 5.2.1. is using the friction ellipse and is similar to the combined slip

calculation of the Pacejka '89 and '94 tire models.

Inputs:

• Dimensionless longitudinal slip (range -1 to 1) and side slip angle in radians

• Longitudinal force Fx and lateral force Fy calculated using the equations of 521-Tire

• The vertical shift of Fy,a=0 is Fy calculated at zero slip angle

Output:

• Adjusted longitudinal force Fx and lateral force Fy incorporates the reduction due to combined

slip:

•

Friction coefficients:

Mx

FyKy

------= Fz

γ

µstatFzKa–

κ α

βk

k2

α2

sin+

------------------------------

acos=

Adams/Tire

240

Forces corrected for combined slip conditions:

Due to the lateral deflection of the tire patch, the aligning moment under combined slip conditions

increases by the effect of the longitudinal force Fx and the lateral tire stiffness Ky:

and the overturning moment uses the lateral force for combined slip:

SmoothingWhen 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 the 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 Fx = SFx.

• Lateral Force Fy = SFy

• Overturning moment torque Mx = SMz

• Aligning torque Mz = SMz

Changing the Operating Mode: USE_MODEYou can change the behavior of the tire model by changing the value of USE_MODE in the [MODEL]

section of the tire property file. If USE_MODE equals zero, or when it is absent, the smoothing time

equals 0.001 seconds and the

521-Tire model is compatible with the previous Adams/Solver implementation.

µx act,

Fx

Fz-----= µy act,

Fy Fy α 0=,–

Fz------------------------------=

µx1

1

µx act,-------------

2 βtan

µstat

----------- 2+

----------------------------------------------------= µyβtan

1

µstat

---------- 2 βtan

µy act,-------------

2+

---------------------------------------------------=

Fx comb,

µx

µx act,-------------Fx= Fy comb,

µy

µy act,------------- Fy Fy α 0=,+( )=

Mz comb, Mz pure, Fx comb,+=Fy comb,

Ky

------------------⋅

Mx comb,

Fy comb,

Ky

------------------Fz=

241Tire Models

By selecting a value of USE_MODE between 1 and 4, smoothing and combined slip correction can be

switched on and off, as shown in Table 1. The smoothing time equals 0.1 seconds for these values of

USE-MODE.

Converting Slip Ratio Data to Velocity Data

Adams/Tire requires that you enter the velocities that correspond to µstatic and µdynamic. You will often

obtain this information as the coefficient of friction versus slip ratio. You can calculate the velocities

required by Adams/Tire from the coefficient of friction versus slip ratio curve in the following way:

where:

• = Slip ratio

• = Free rolling rotational velocity (no slip)

• = Actual rotational velocity

Kinematic relationships between translational and rotational velocities and the effective rolling radius

give:

where:

• = Contact patch velocity reletive to road surface

• = Actual longitudinal velocity

• = Effective rolling radius

USE_MODE: Smoothing: Combined slip correction:

1 off off

2 off on

3 on off

4 on on

κωa ωf–

ωf

------------------=

κ

ωf

ωa

ωa

Vx Vsx–

Re

---------------------=

ωf

Vx

Re

------=

Vsx

Vx

Re

Adams/Tire

242

Substituting these relationships into the original slip ratio equation with some cancelling of variables

gives:

Therefore:

During testing for the coefficient of friction as a function of slip ratio, the longitudinal velocity Vx is held

constant. Therefore, you can obtain Vsx, the relative velocity of the contact patch with respect to the road

surface, from the test data curves for the static and dynamic values of friction.

Contact Methods

For handling analyses (which use a flat road surface profile), the 521-Tire model uses the point-follower

contact method. For durability analyses (which use uneven road surface profiles), the Equivalent Plane

Method yields the instantaneous tire radius directly, while finding the new road surface orientation

vector.

About the Point-Follower MethodThe 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.

About the Equivalent Plane Method 521-Tire uses the Equivalent Plane method to reorient the vertical road surface vector, which gives the

direction of the vertical force, and to calculate the new tire radius. To do this, a new smooth road surface

is generated at an angle calculated such that only the shape of the tire is different (see body 6 on page 18).

Equivalent Plane Method

κVsx

Vx

--------–=

Vsx Vxκ–=

243Tire Models

Both the deflected tire area and its centroid remain unchanged. The vector between the deflected area

centroid and the wheel-center marker then determines the orientation of the. vertical vector

perpendicular to the road surface.

The Equivalent Plane method is best suited for relatively large obstacles because it assumes the tire

encompasses the obstacle uniformly. In reality, the pneumatics and the bending stiffness of the tire

carcass prevent this. The result is an uneven pressure distribution and possibly gaps between the tire and

the road. If the obstacle is larger than the tire contact patch (such as a pothole or curb), the uniform

assumption is good. If the obstacle is much smaller than the tire patch, however (such as a tar strip or

expansion joint), the assumption is poor, and the Equivalent Plane method may greatly underestimate the

durability force.

Definition of Equivalent Plane Parameters

Adams/Tire

244

When using the Equivalent Plane method the following parameters need to be specified in the tire

property file:

Equivalent_plane_angle

Specifies the subtended angle (in degrees) bisected by the z-axis of the wheel-center marker, as shown

in Figure 7. This angle determines the extent of the road the tire can envelop. The value of the

equivalent_plane_angle must be between 0 and 180 degrees.

Equivalent_plane_increments

Specifies the number of increments into which the shadow of the tire subtended section is divided, as

shown in Figure 7.

521-Tire Tire and Road Property Files

This section contains four example input data files. For reference, the files are called:

• 521_equation.tir

• 521_interpol.tir

• 521_pnt_follow.rdf

• 521_equiv_plane.rdf

The first two files are tire property files, and the last two are road files. The file 521_equation.tir

illustrates the required format and parameters when you use the Equation method. The file

521_interpol.tir illustrates the Interpolation method. The two *.rdf files show how road data files must

be specified when either of the contact methods is used.

Tire Property File 521_equation.tir and 521_interpol.tir

You can select the method for calculating the normal force by setting the

VERTICAL_FORCE_METHOD parameter to either POINT_FOLLOWER (for the Point Follower

245Tire Models

method) or EQUIVALENT_PLANE (for the Equivalent Plane method). See Contact Methods on page

17 for details on these methods.

You can select the method for calculating the lateral force by setting the LATERAL_FORCE_METHOD

parameter to either INTERPOLATION or symbol. See Lateral Force and Aligning Torque on page 11

for details on these calculation methods.

The following table specifies how some of the parameter names used in the tire property file correspond

to parameters introduced in the equations that were presented in the previous sections.

521-equation.tir

The 521-equation.tir example tire property file starts here.

$---------------------------------------------------------------------MDI_HEADER[MDI_HEADER] FILE_TYPE = 'tir' FILE_VERSION = 3.0 FILE_FORMAT = 'ASCII'(COMMENTS)comment_string'Tire - XXXXXX''Pressure - XXXXXX''Test Date - XXXXXX''Test tire'$--------------------------------------------------------------------------units[UNITS] LENGTH = 'mm'

Parameter in file: Used in equation: As parameter:

vertical_stiffness [10] Kz

vertical_damping [11] Cz

lateral_stiffness [18] Ky

cornering_stiffness_coefficient [6] Kα

Mu_Static Figure 4 µstaticMu_Dynamic Figure 4 µdynamicMu_Static_velocity Figure 4 velocity µstaticMu_Dynamic_Velocity Figure 4 velocity µdynamicrolling_resistance_coefficient [13] coeffrr

vertical_stiffness_exponent [141] Note: If you do not specify vertical_stiffness_exponent in the tire property file, 521-Tire uses the default value of 1.1.

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FORCE = 'newton' ANGLE = 'rad' MASS = 'kg' TIME = 'second'$-------------------------------------------------------------------------model[MODEL]! use mode 12341234! ------------------------------------------------------------------------! smoothingXXXX! combinedXXXX! transient X X X X! PROPERTY_FILE_FORMAT = '5.2.1' FUNCTION_NAME = 'TYR913' USE_MODE = 12$---------------------------------------------------------------------dimension[DIMENSION] UNLOADED_RADIUS = 310.0 WIDTH = 195.0 ASPECT_RATIO = 0.70 RIM_RADIUS = 195,0 RIM_WIDTH = 139.7$---------------------------------------------------------------------parameters! VERTICAL_FORCE_METHOD = EQUIVALENT_PLANE LATERAL_FORCE_METHOD = EQUATION! vertical_stiffness = 206.0 vertical_stiffness_exponent = 1.1 vertical_damping = 2.06! lateral_stiffness = 50 cornering_stiffness_coefficient = 50! Mu_Static = 0.95 Mu_Dynamic = 0.75 Mu_Static_Velocity = 3000 Mu_Dynamic_Velocity = 6000! rolling_resistance_coefficient = 0.01! EQUIVALENT_PLANE_ANGLE= 100 EQUIVALENT_PLANE_INCREMENTS= 50! RELAX_LENGTH_X = 0.10 RELAX_LENGTH_Y = 0.30

247Tire Models

521_equation.tir

The 521-interpol.tir example tire property file starts here. In addition to the file for 521_equation.tir, it

contains data that is used for calculating the lateral force and aligning moment, instead of using formula

6 to 9. Note that the [DEFLECTION_LOAD_CURVE] can also be used in the tire property file for the

Equation method.

$---------------------------------------------------------------------MDI_HEADER[MDI_HEADER] FILE_TYPE = 'tir' FILE_VERSION = 3.0 FILE_FORMAT = 'ASCII'

(COMMENTS)comment_string'Tire - XXXXXX''Pressure - XXXXXX''Test Date - XXXXXX''Test tire'$--------------------------------------------------------------------------units[UNITS] LENGTH = 'mm' FORCE = 'newton' ANGLE = 'rad' MASS = 'kg' TIME = 'second'$-------------------------------------------------------------------------model[MODEL]! use mode 12341234! ------------------------------------------------------------------------! smoothingXXXX! combinedXXXX! transient X X X X! PROPERTY_FILE_FORMAT = '5.2.1' FUNCTION_NAME = 'TYR913' USE_MODE = 12$---------------------------------------------------------------------dimension[DIMENSION] UNLOADED_RADIUS = 310.0 WIDTH = 195.0 ASPECT_RATIO = 0.70 RIM_RADIUS = 195,0 RIM_WIDTH = 139.7$---------------------------------------------------------------------parameters! VERTICAL_FORCE_METHOD = POINT_FOLLOWER ! or EQUIVALENT_PLANE

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LATERAL_FORCE_METHOD = INTERPOLATION ! or EQUATION! vertical_stiffness = 206.0 vertical_stiffness_exponent = 1.1 vertical_damping = 2.06 lateral_stiffness = 50 cornering_stiffness_coefficient = 50! Mu_Static = 0.95 Mu_Dynamic = 0.75 Mu_Static_Velocity = 3000 Mu_Dynamic_Velocity = 6000! rolling_resistance_coefficient = 0.01! EQUIVALENT_PLANE_ANGLE= 100 EQUIVALENT_PLANE_INCREMENTS= 50!! RELAX_LENGTH_X = 0.10 RELAX_LENGTH_Y = 0.30!------------------CAMBER ANGLE VALUES------------------------------------------! Conversion! No. of pnts factor(D to R) pnt1 pnt2 pnt3 pnt4 pnt5! CAMBER_ANGLE_DATA_LIST 5 0.017453292 -3.0 0.0 3.0 6.0 10.0!!------------------SLIP ANGLE VALUES--------------------------------------------! Conversion! No. of pnts factor(D to R) pnt1 ...... pnt9! SLIP_ANGLE_DATA_LIST 9 0.017453292 -15.0 -10.0 -5.0 -2.5 0.0 2.5 5.0 10.0 15.0!!-----------------VERTICAL FORCE VALUES-----------------------------------------! Conversion ! No. of pnts factor! pnt1 pnt2 pnt3 pnt4 pnt5 ! VERTICAL_FORCE_DATA_LIST 5 4.448 200.0 600.0 1100.0 1500.0 1900.0!!-----------------ALLIGNING TORQUE VALUES---------------------------------------! No. of pnts Conversion! factor!

249Tire Models

! pnt1 .... pnt225! ALIGNING_TORQUE_DATA_LIST 225 -1355.7504

5.31 6.52 22.88 26.41 30.58 0.11 2.84 5.49 -3.92 -14.04 0.47 -12.44 -37.99 -67.22 -116.07 0.04 -21.38 -69.04 -111.44 -168.11 0.80 -3.70 -27.94 -44.25 -53.74 1.75 17.43 52.20 81.97 145.78 2.54 11.08 40.53 73.54 95.55 -1.28 0.02 14.82 2.93 10.35 1.59 -3.77 -17.17 6.60 -11.91

0.06 14.23 22.93 11.45 15.74 5.95 5.54 13.72 -1.65 -15.64 -1.29 -9.45 -26.98 -57.25 -107.71 -5.05 -17.73 -62.62 -109.03 -161.88 0.46 -2.48 -19.48 -33.54 -49.52 4.71 26.10 60.80 90.85 119.51 4.26 16.60 52.46 93.32 141.34 2.41 4.28 2.21 9.11 30.44 -0.92 0.22 12.61 2.51 -18.77

0.43 -4.62 15.36 7.16 11.70 6.70 15.92 0.14 -4.20 -11.81 -2.20 -5.53 -13.28 -47.48 -92.88 -1.39 -17.28 -52.17 -102.80 -161.71 2.87 -0.38 -14.27 -29.03 -42.42 6.99 24.54 66.06 93.27 126.38 7.10 18.78 58.20 104.51 156.39 1.63 2.91 8.33 20.32 42.09 -0.78 10.13 -9.94 -13.02 -11.95

5.62 4.36 23.16 38.03 8.73 2.31 6.41 14.10 6.03 -11.66 7.87 1.33 -16.31 -40.24 -82.58 1.40 -10.04 -50.94 -93.06 -157.50 2.10 0.56 -16.15 -27.15 -40.13 5.60 26.48 62.92 90.16 122.03 3.56 20.63 60.74 108.26 162.97 -0.08 1.81 14.39 34.98 59.72 1.38 -2.13 -2.42 -4.08 -2.72

3.69 1.71 29.06 10.05 11.38 3.09 7.15 -7.92 13.53 -5.78 6.08 0.38 -2.69 -32.10 -62.17 0.76 -7.65 -37.28 -89.05 -145.09 0.70 4.37 -7.59 -23.71 -28.49 5.92 34.39 72.55 92.88 129.34 4.36 29.81 76.70 118.91 180.59 -2.03 5.94 26.18 53.59 89.76 0.39 -5.52 -6.06 10.16 7.81

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!-----------------LATERAL FORCE VALUES--------------------------------------- ! No. of pnt Conversion ! factor ! pnt1 .... pnt225 ! LATERAL_FORCE_DATA_LIST 225 4.448 234.08 585.56 1000.29 1307.77 1603.78 269.79 628.82 1040.78 1331.72 1624.83 213.70 565.29 974.49 1198.82 1387.74 150.79 452.18 752.21 885.23 960.13 11.52 50.58 199.87 199.50 208.75 -116.75 -367.42 -618.68 -683.16 -857.81 -224.15 -588.24 -1001.01 -1235.88 -1488.88 -242.08 -612.70 -1059.55 -1344.53 -1658.66 -213.99 -597.29 -988.14 -1343.86 -1689.35

234.40 572.75 981.30 1352.37 1698.90 239.27 647.77 1007.37 1357.22 1666.30 252.34 603.75 1033.50 1288.76 1483.64 167.55 481.45 826.41 962.64 1028.74 32.23 78.77 231.31 250.14 254.32 -122.59 -423.13 -552.58 -613.52 -607.61 -208.93 -576.28 -948.45 -1149.44 -1314.69 -261.05 -634.90 -1064.15 -1338.52 -1581.84 -241.50 -607.16 -1021.87 -1322.30 -1598.25

210.20 578.56 968.72 1344.05 1730.40 237.91 600.60 1025.67 1377.57 1733.03 226.60 629.48 1084.97 1354.12 1575.22 154.74 496.21 878.72 1028.03 1095.59 34.37 74.19 240.00 284.42 283.85 -130.29 -339.00 -509.04 -543.75 -555.05 -226.48 -557.52 -884.91 -1083.18 -1175.12 -270.70 -595.22 -1059.76 -1314.74 -1564.43 -254.64 -602.76 -1032.71 -1313.22 -1609.96

238.28 531.25 945.70 1305.28 1786.96 227.13 594.51 1038.87 1365.33 1733.29 221.76 633.49 1135.31 1375.28 1619.82 195.50 505.90 899.88 1059.92 1135.28 28.51 68.59 241.99 311.15 331.84 -145.10 -319.56 -464.11 -499.27 -500.83 -230.33 -548.99 -815.88 -991.78 -1108.36 -230.62 -597.10 -1009.76 -1261.43 -1504.09 -218.36 -570.13 -1049.72 -1344.94 -1589.60

228.49 564.69 954.06 1332.84 1687.50 221.19 595.52 1019.74 1378.35 1749.40 224.63 590.58 1108.01 1408.87 1707.09 178.96 474.70 918.87 1125.97 1242.75 42.58 65.26 230.69 306.58 428.45

251Tire Models

-144.43 -290.91 -368.02 -398.98 -394.66 -224.99 -494.65 -761.78 -886.03 -941.20 -246.51 -563.13 -980.33 -1249.57 -1462.88 -239.34 -567.10 -1050.56 -1348.66 -1611.11$---------------------------------------------------------------------load_curve$ For a non-linear tire vertical stiffness (optional)$ Maximum of 100 points[DEFLECTION_LOAD_CURVE]pen fz0.0 0.01.0 212.02.0 428.03.0 648.05.0 1100.010.0 2300.020.0 5000.030.0 8100.0

521-Tire Road Data Files

The road data files used with the 521-Tire are unique and cannot be used with any other tire model. The

data files are fully described by the following two examples.

Road Data File 521_pnt_follow.rdf

This example file shows that, if you use the Point Follower method and indicate it in the associated tire

property file, the road_profile_type parameter must be set to FLAT.

1. $-----------------------------------------------------------MDI_HEADER

2. FILE_TYPE = ’rdf’

3. FILE_VERSION = 2.0

4. FILE_FORMAT = ’ASCII’

5. (COMMENTS)

6. comment_string

7. ’Example of 521-Tire, point follower flat road’

8. $----------------------------------------------------------------units

9. [UNITS]

10. LENGTH = ’mm’

11. FORCE = ’newton’

12. ANGLE = ’radians’

13. MASS = ’kg’

14. TIME = ’sec’

15. $-----------------------------------------------------------definition

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16. [MODEL]

17. METHOD = ’5.2.1’

18. $-----------------------------------------------------------parameters

19. road_profile_type = FLAT

20. initial_height = 0

21. Road Data File 521_equiv_plane.rdf

The following example shows which data the road data file must contain if the Equivalent Plane method

is used and specified in the associated tire property file. The main difference with the road data file used

in association with the Point Follower method is that here the ROAD_PROFILE_TYPE parameter is set

to INPUT and a ROAD_INPUT_DATA_LIST is specified.

22. $---------------------------------------------------------MDI_HEADER

23. FILE_TYPE = ’rdf’

24. FILE_VERSION = 2.0

25. FILE_FORMAT = ’ASCII’

26. (COMMENTS)

27. comment_string

28. ’Example of 521-Tire, equivalent plane method 2-D road’

29. $--------------------------------------------------------------units

30. [UNITS]

31. LENGTH = ’mm’

32. FORCE = ’newton’

33. ANGLE = ’radians’

34. MASS = ’kg’

35. TIME = ’sec’

36. $---------------------------------------------------------definition

37. [MODEL]

38. METHOD = ’5.2.1’

39. $---------------------------------------------------------parameters

40. ROAD_PROFILE_TYPE = INPUT ! or FLAT

41. INITIAL_HEIGHT = 0.000

42. !

43. ! if "ROAD_PROFILE_TYPE = INPUT" the road must be specified with a table

44. !

45. ROAD_INPUT_DATA_LIST

46. 23, 1

253Tire Models

47. -10000.00, 0.00

48. 1740.00, 0.00

49. 1740.94, 1.92

50. (.... a number of columns were removed here for clarity)

51. 1892.40, 0.00

52. 40000.00, 0.00

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Using FTire Tire ModelLearn about:

• About FTire

• Modeling Approach

• Using FTire with Road Models

• Using FTire with Adams

• Parameters

• About FTire Parameters

• Procedure for Parameterizing FTire

• List of FTire Parameters

• About the Tire Data File

• Choosing Operating Conditions

This help describes the Flexible Ring Tire Model (FTire)™, as it is invoked from Adams.

© Michael Gipser, Cosin Consulting

About FTireThe tire model, FTire (Flexible ring tire model), is a sophisticated tire force element. You can use it in

MBS-models for vehicle-ride comfort investigations and other vehicle dynamics simulations on even or

uneven roadways.

255Tire Models

The main benefits of FTire are:

• Fully nonlinear.

• Valid in frequency domain up to 120 Hz, and beyond.

• Valid for obstacle wave lengths up to half the length of the contact patch, and less.

• Parameters, among others, are the natural frequencies and damping factors of the linearized

model, and easy-to-obtain global static properties.

• Models both the in-plane and out-of-plane forces and moments.

• Computational effort no more than 5 to 20 times real time, depending on platform and model

level.

• High accuracy when passing single obstacles, such as cleats and potholes.

• Applicable in extreme situations like many kinds of tire misuse and sudden pressure loss.

• Sufficiently accurate in predicting steady-state tire characteristics.

In contrast to other tire models, FTire does not need any complicated road data preprocessing. Rather, it

takes and resolves road irregularities, and even extremely high and sharp-edged obstacles, just as they

are defined.

We recommend that you visit www.ftire.com, to learn more about FTire theory, validation, data supply,

and application. Also, at the FTire Web site, you will be kept informed about the latest FTire

improvements, and how to receive them. In the download section, you will find a set of auxiliary

programs, called FTire/tools for Windows™. These tools help to analyze and parameterize an FTire

http://www.ftire.com/

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Model. FTire/tools is free for FTire licensees. It comprises static, steady-date, and modal analysis,

linearization, data estimation, identification and validation tools, road data visualization, and more. In the

site's documentation section, you will find a more detailed and permanently updated FTire

documentation, together with as some additional literature.

Modeling ApproachFTire uses the following modeling approach:

• The tire belt is described as an extensible and flexible ring carrying bending stiffnesses,

elastically founded on the rim by distributed, partially dynamic stiffnesses in radial, tangential,

and lateral directions. The degrees of freedom of the ring are such that rim in-plane as well as

out-of-plane movements are possible. The ring is numerically approximated by a finite number

of discrete masses, the belt elements. These belt elements are coupled with their direct neighbors

by stiff springs and by bending stiffnesses both in-plane and out-of-plane.

Belt In-Plane and Out-Of-Plane Bending Stiffness outlines in-plane and out-of-plane bending

stiffness placing. In-plane bending stiffness is realized by means of torsional springs about the

lateral axis. The torsional deflection of these springs is determined by the angle between three

consecutive belt elements, projected onto the rim mid-plane. Similarly, the out-of-plane bending

stiffness is described by means of torsional springs about the radial axis. Here, the torsional

deflection is determined by the angle between three consecutive belt elements, projected onto the

belt tangential plane. Note that in the figure, the yellow plates do not represent the belt elements

themselves, but rather the connecting lines between the elements.

Belt In-Plane (left) and Out-Of-Plane (right) Bending Stiffness

• FTire calculates all stiffnesses, bending stiffnesses, and damping factors during preprocessing,

fitting the prescribed modal properties (see list of data below).

257Tire Models

• A number of massless tread blocks (5 to 50, for example) are associated with every belt element.

These blocks carry nonlinear stiffness and damping properties in the radial, tangential, and

lateral direction. The radial deflections of the blocks depend on the road profile, focus, and

orientation of the associated belt elements. FTire determines tangential and lateral deflections

using the sliding velocity on the ground and the local values of the sliding coefficient. The latter

depends on ground pressure and sliding velocity.

• FTire calculates all six components of tire forces and moments acting on the rim by integrating

the forces in the elastic foundation of the belt.

Because of this modeling approach, the resulting overall tire model is accurate up to relatively high

frequencies both in longitudinal and in lateral directions. There are few restrictions in its applicability

with respect to longitudinal, lateral, and vertical vehicle dynamics situations. FTire deals with large-

and/or short-wave-length obstacles. It works out of, and up to, a complete standstill, with no additional

computing effort nor any model switching. Finally, it is applicable with high accuracy in such delicate

simulations as ABS braking on extremely uneven roadways, and so on.

In a full 3D variant, FTire additionally takes into account belt element rotation and bending about the

circumferential axis. These new degrees of freedom enable FTire to use contact elements that are

distributed not only along a single line, but over the whole contact patch. You can choose the

arrangement of the contact elements to be either randomly distributed, or distributed along several

parallel lines.

In the full 3D variant, belt torsion about the circumferential axis is described by:

• Torsional stiffnesses between belt elements and rim, about circumferential axis (represented by

red torsion springs in the left side of the figure, Belt).

• Torsional stiffnesses between adjacent belt elements, about circumferential axis (represented by

blue torsion springs in the left side of figure, Belt).

The right side of the figure, Belt outlines the belt bending stiffness about the circumferential axis. This

is done in a somewhat simplified manner. Actually, lateral belt bending is taken into account by

introducing a parabolic shape function for each belt element. The curvature of this shape function is

treated as a belt elements’ additional degree of freedom.

Belt Torsional and Twisting Stiffness, and Belt Lateral Bending Stiffness

Note: Radial, tangential, and lateral are relative to the orientation of the belt element,

whereas sliding velocity is the block end-point velocity projected onto the road

profile tangent plane. By polynomial interpolation, certain precautions have been

taken not to let the ground pressure distribution mirror the polygonal shape of the

belt chain.

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258

You should chose the full 3D variant, which takes about 30% more computing time, in situations where

a considerable excitation of tire vibrations in lateral direction is expected. This, for example, will happen

when the tire runs over cleats that are placed in an oblique direction relative to the tire rolling direction.

Similarly, such an excitation will happen when the tire is running over obstacles with large camber angle.

Optionally, FTire can take into account tire non-uniformity, that is, a harmonic variation of vertical or

longitudinal stiffness, as well as static and dynamic imbalance, conicity, ply-steer, and geometrical run-

out.

All stiffness values may depend on the actual inflation pressure. To take full advantage of that option, it

is necessary to provide basic FTire input data, such as radial stiffness data and natural frequencies at two

different pressure values. Actual inflation pressure is one of the ‘operating conditions variables,’ which

can be made time-dependent, and therefore, can be changed even during a simulation.

There are two more operating conditions: tread depth and model level. The latter signal allows you to

switch between the reduced variant of FTire (all contact elements are arranged in one single line near the

rim mid-plane), and the full 3D variant (contact elements cover the whole contact patch).

The kernel of the FTire implementation is an implicit integration algorithm (BDF) that calculates the belt

shape. The integrator runs parallel but synchronized with the Adams main integrator. By using this

specialized implicit BDF integrator, you can choose the belt extensibility so it is extremely small. This

also allows the simulation of an inextensible belt without any numerical drawbacks.

Using FTire with Road ModelsFTire supports all MSC road definitions, including Motorsports and all 3D roads. It also supports several

customer-specific and third-party roads. For more information about available road descriptions, please

contact [email protected].

259Tire Models

Using FTire with AdamsFTire is a high-resolution tire model, with respect to road irregularities and tire vibration modes. To take

full advantage of that precision, we recommend that you choose a small step size for the Adams

integrator. There should be a minimum of 1,000 steps per one second simulation time (that is, an output

time step of 1 ms or less).

Controlling integrator step size in:

• Adams/Car

• Adams/Chassis

• Adams/View

• Adams/Solver

Controlling Integrator Step Size in Adams/Car

In Adams/Car, you can control the integrator step size by selecting:

Settings → Solver → Dynamics

and entering 1ms in the Hmax text box.

Alternatively, you can edit the driver control file (.dcf) that Adams/Car automatically generates when

performing a new dynamic maneuver. In that file, override the integrator step size, which is defined in

[EXPERIMENT] block, by entering the value 0.001 or less. After editing the file, you can launch

subsequent simulation experiments with the same driver's control (and, of course, the new integrator step

size) by selecting the following from Adams/Car:

Simulate → Full-Vehicle Analysis → DCF Driven → Driver Control Files → Browse

and selecting the .dcf you just edited.

Controlling Integrator Step Size in Adams/Chassis

In Adams/Chassis, you can control the integrator step size by setting the HMAX value to 0.001 or less.

HMAX is defined by selecting the following from Adams/Chassis:

System file → Properties → system_parameters → solver → hmax

Controlling Integrator Step Size in Adams/View

In Adams/View, you can control the integrator step size by checking:

Settings → Solver → Dynamics → Customized Settings

size, Min Step Size, and Max Step Size.

Controlling Integrator Step Size in Adams/Solver

In Adams/Solver, you can control the integrator step size by setting INTEGRATOR/HMAX to the

desired value in the Adams dataset (adm).

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260

FTire Parameters• About FTire Parameters

• Procedure for Parameterizing FTire

• Listing of FTire Parameters

About FTire ParametersFTire parameters can be divided into several groups. There are parameters that define:

• Tire size and geometry

• Stiffness, damping, and mass distribution of the belt/sidewall structure

• Tire imperfections (non-uniformity, imbalance, conicity, and so on)

• Stiffness and damping properties of the tread rubber

• Friction characteristics of the tread rubber

• Numerical properties of the model

For convenience, FTire tries to use data that can be measured as easy as possible. As a consequence, the

number of basic data might be larger than the number of internal parameters defined by these basic

parameters.

For example, the following four parameters together, after preprocessing, actually result in only two

values used in FTire: compression and shear stiffness of the idealized blocks that represent tread rubber:

• tread_depth

• tread_base_height

• stiffness_tread_rubber

• tread_positive

Also, sometimes different combinations of parameters are possible. This is true especially for data of the

second group, which determine the structural stiffness and damping properties of FTire. Your choice of

which combination of parameters to supply depends on the types of measurements that are available and

their accuracy.

Moreover, it is possible to prescribe over-determined subsets of parameters. For example, you may define

the belt in-plane bending stiffness by prescribing the frequency of the first bending mode, and at the same

time the radial stiffness on a transversal cleat. Both parameters are strongly influenced by the bending

stiffness, but might contradict each other.

In such a case, FTire automatically recognizes that the system of equations to be solved is over-

determined, and applies an appropriate solver (Householder QR factorization) to determine the solution

in the sense of least squares fit. That means, FTire is looking for a compromise to meet both conditions

as much as possible. Users can control the compromise by optionally defining weights for the

contradicting conditions.

261Tire Models

Note that, among others, FTire uses modal data to calculate internal structural stiffness and damping

coefficients. They are processed in such a way that the mathematical model, for small excitations, shows

exactly the measured behavior in the frequency domain. FTire is not a modal model, nor is it linear.

First Six Vibration Modes Of An Unloaded Tire With Fixed Rim

When parametrizing FTire, the bending mode frequencies rather sensitively influence the respective

bending stiffness. As an alternative, determining the radial stiffness both on a flat surface and on a short

obstacle (cleat) is an inexpensive and very accurate way to get both the vertical stiffness between belt

nodes and rim and the in-plane bending stiffness.

Other ways to determine the bending stiffness (and other data, as well) are to use the software tools

FTire/fit (time- and frequency-domain parameter identification) and FTire/estim (qualified parameter

estimation by comparison with a reference tire). For more information, see www.ftire.com.

Unfortunately, there is no direct analogy of the ‘radial stiffness on cleat’ measuring procedure to get the

out-of-plane bending stiffness. But this parameter does not seem to be as relevant as the in-plane bending

stiffness for ride comfort and durability. An indirect, but also very accurate, way to validate the out-of-

plane bending stiffness is to check resulting side-force and self-aligning characteristic. The cornering

stiffness, the pneumatic trail, as well as the difference between maximum side force and side force for

very large side-slip angles, are very sensitively determined both by the tread rubber friction characteristic

and by the out-of-plane bending stiffness. Similarly, the fourth mode (see figure, First Six Vibration

Modes Of An Unloaded Tire With Fixed Rim), being itself determined by the stiffness between belt nodes

and rim in lateral direction, very strongly influences the side-slip angle where maximum side force

occurs.

Procedure for Parameterizing FTireA typical procedure to parametrize FTire might be:

1. Either from tire data sheets, by some simple and inexpensive measurements, or directly from the

tire supplier, obtain:

• Tire size, load index, and speed symbol

• Rolling circumference

• Rim diameter

• Tread width

• Tire mass


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262

• Tread depth

• Rubber height over steel belt

• Shore-A stiffness or Young's modulus of tread rubber

• Tread pattern positive

2. Determine the natural frequencies and damping moduli of the first six modes, for an unloaded,

inflated tire, where the rim is fixed. Normally, you do this by exciting the tire structure with an

impulse hammer, measuring the time histories of at least four acceleration sensors in all three

directions, distributed along the tire circumference, and processing these using an FFT signal

analyzer. Optionally, repeat this step for a second inflation pressure value.

3. Determine the tire radial stiffness on a flat surface and on a short obstacle, for one or two inflation

pressure value(s).

4. Determine (or estimate) the lateral belt curvature radius from the unloaded tire's cross-section.

Determine the belt lateral bending stiffness to get a reasonable pressure distribution in the lateral

direction.

5. Determine (or estimate) tread rubber adhesion and sliding friction coefficients for ground

pressure values 0.5 bar, 2 bar, and 10 bar.

6. Take natural frequencies and damping moduli of modes 1, 2, and 4, together with the radial

stiffness on flat surface and on a cleat, for one or two inflation pressure value(s), as well as the

remaining basic data. These values result in a first, complete FTire input file for the basic variant

(belt circumferential rotation, twisting, and bending not taken in to account; all contact elements

are arranged in one line).

7. Let FTire preprocess these data. Compare the resulting additional modal properties of the model

with the modal data that are not used so far (modes 3, 5, and 6). If necessary, adjust the

preprocessed data to find a compromise with respect to accuracy.

8. If respective measurements are available, validate the data determined so far by means of side

force and aligning torque characteristics, and by measurements of vertical and longitudinal

force variations induced during rolling over cleats both with low and high speed. The validation

can be extended to a full parameter fitting procedure by using TIRE/fit, as mentioned earlier.

9. Estimate the following additional data that are only relevant for 'out-of-plane' excitation:

• Belt element torsional stiffness relative to rim (represented by red torsion springs between

yellow belt elements and gray rim in the figure, Belt)

• Belt twisting stiffness (represented by blue torsion springs between adjacent yellow belt

elements in the figure, Belt)

• Belt bending stiffness/damping about circumferential direction

• Belt lateral curvature radius

• Coupling coefficient between belt lateral displacement and belt rotation.

Start with the respective values of the sample data file. Then, adjust the values by fitting the model's

response to obliquely oriented cleats and handling characteristics for large camber angles at the same

time. This identification procedure can be made easier by using the the additional tool FTire/fit.

263Tire Models

Clearly, the performance of this procedure is not very easy in practice. On the other hand, every tire

model that is accurate enough for ride comfort and durability calculations will need as much or even

more data.

List of FTire ParametersThe following is a comprehensive list of all mandatory and optional FTire parameters. However, many

items are explained in greater detail in the extended documentation to be downloaded from the restricted

area in www.ftire.com. You will receive your pass-code from [email protected].

FTIRE_DATA Section Parameters

The parameter: Means:

tire_section_width Tire section width as specified in the tire

size designation (using length unit as

specified in the [UNITS] section).

tire_aspect_ratio Tire aspect ratio as specified in the tire size

designation. Unit is %.

rim_diameter Rim diameter as specified in the tire size

designation (using length unit as specified

in the [UNITS] section).

rim_width Inner distance between the two rim flanges.

load_index Load index of tire, as displayed in tire

service description.

tread_width Width of tread that comes into contact with

the road under normal running conditions

at LI load, without camber angle.

rolling_circumference Rolling circumference of tire under the

following running conditions:

Free rolling at v = 60 km/h and zero camber

angle

Vertically loaded by half of the maximum

load

The circumference is the distance traveled

with one complete wheel revolution.

tire_mass Overall tire mass.

inflation_pressure Inflation pressure, at which tire data

measurements have been taken.


Adams/Tire

264

inflation_pressure_2 Second inflation pressure, at which tire data

measurements have been taken (optional).

stat_wheel_load_at_10mm_defl Static wheel load of the inflated tire, when

it is deflected by 10 mm, with zero camber

angle, on a flat surface, during stand-still, at

very low friction value.

stat_wheel_load_at_20mm_defl Static wheel load of the inflated tire, when


angle, on a flat surface, during stand-still, at

very low friction value.

Note: Instead of using:

stat_wheel_load_at_10mm_defl and stat_wheel_load_at_20mm_defl

Note: You can equally define:

stat_wheel_load_at_20mm_defl and stat_wheel_load_at_40mm_defl.

Note: This will better fit typical operating conditions of truck tires. For extremely heavy vehicles, there are even more pairs of deflection values predefined. These can be found at the extended documentation at www.ftire.com.

dynamical_stiffening Increase of the overall radial stiffness at

high speed as compared to radial stiffness

during standstill. Unit is %.

speed_at_half_dyn_stiffening Running speed at which dynamic stiffening

reaches half of the final value.

belt_extension_at_200_kmh Percentage of rolling circumference growth

at a running speed of 200 km/h = 55.55 m/s

= 124.3 mph, compared to low speed.

interior_volume Interior tire volume when the tire is

mounted on the rim and inflated with

inflation_pressure.

Note: This parameter is only needed if you specify the next parameter (volume_gradient) and it is nonzero.



265Tire Models

volume_gradient Relative decrease in volume, of a small tire

segment, when that segment is deflected

vertically.

Note: This parameter is optional and only marginally affects the model’s behavior.

rel_long_belt_memb_tension The percentage by which inflation pressure

forces in the belt region are compensated

with membrane tension in longitudinal

direction, as compared to the total

compensation in lateral and longitudinal

direction.

Note: This parameter is optional, and can only be calculated using a finite-element (FE) model, or estimated by parameter identification. A value of 70 to 80% seems to be appropriate for many tires. The value will increase with increasing belt lateral curvature radius.

f1 First natural frequency: in-plane, rigid-

body rotation around wheel spin axis. Rim

is fixed. See the figure, First Six Vibration

Modes.

f2 Second natural frequency: rigid-body

movement in fore-aft direction. Rim is

fixed. See the figure, First Six Vibration

Modes.

f4 Fourth natural frequency: out-of-plane,

rigid-body rotation around road normal

axis. Rim is fixed. See the figure, First Six

Vibration Modes.

Note: f3 (out-of-plane, rigid-body movement) is not needed because it is closely related to f4.

At least one of:

f5 Fifth natural frequency: first in-plane

bending mode (quadrilateral-shaped). Rim

is fixed.


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266

belt_in_plane_bend_stiffn In-plane bending stiffness of the belt ring of

deflated and unloaded tire.

wheel_load_at_10_mm_defl_cleat Static wheel load of the inflated tire, when


angle, on a cleat as specified below, during

stand-still. Cleat must be high enough that

the tire does not touch the ground apart

from the cleat. The cleat is oriented in the

lateral direction, perpendicular to the tire’s

rolling direction.

Note: For truck tires, you can specify wheel_load_at_20_mm_defl_ cleat, as well.

weight_f5

weight_in_plane_bend_stiffn

weight_wheel_load_cleat

If you provide at least two of the data on the

previous page to define the in-plane

bending stiffness, they constitute an over-

determined system of equations for the

respective FTire's internal stiffness values.

FTire will try to find a compromise. You

can control this compromise by setting

these weight values. Their relative size

controls, in a least-squares approach, the

contribution of the respective parameter. If

a weight is set to zero, the related parameter

is completely ignored.

Note: The weights are optional. Default value is 1.

cleat_width Width of cleat that was used to determine

all parameters that require a cleat:

wheel_load_at_10_mm_defl_ cleat

wheel_load_at_10_mm_defl_ cl_lo

and so on.

Note: Parameter is optional. Default value is 20 mm.


267Tire Models

cleat_bevel_edge_width Bevel edge width (measured after

projection to x-y-plane) of cleat that was

used to determine all parameters that

require a cleat:

wheel_load_at_10_mm_defl_ cleat

wheel_load_at_10_mm_defl_ cl_lo

and so on.

Note: Parameter is optional. Default value is 0 mm.

At least one of:

f6 Sixth natural frequency: first out-of-plane

bending mode (banana-shaped).

belt_out_of_plane_bend_stiffn Out-of-plane bending stiffness of the belt

ring of inflated but unloaded tire.

weight_f6

weight_out_of_plane_bend_st

If you provide both data above (f6 and

belt_out_of_plane_bend_stiff) to define the

out-of-plane bending stiffness, they

constitute an over-determined system of

equations for the respective FTire's internal

stiffness values. FTire will try to find a

compromise. You can control the

compromise by setting these weight values.

Their relative size controls, in a least-

squares approach, the contribution of the

respective parameter. If a weight is set to

zero, the related parameter is completely

ignored.


D1 Damping of f1, between 0 and 1:

0 = undamped, ..., 1 = aperiodic limit case

D2 Damping of f2.

D4 Damping of f4.

Note: D5 and D6 cannot be prescribed, but result from D1, D2, and D4.


Adams/Tire

268

belt_twist_stiffn Belt-twisting stiffness: if the mean torsion

angle relative to the rim is 0, the value is the

moment in longitudinal direction per 1

degree twist angle for a unit length belt

segment. This value is independent on the

number of belt segments.

Note: Only needed for full 3D variant. Unit is force*length2/angle.

belt_torsion_stiffn Belt-torsional stiffness: if twist angle is 0,

the value is the moment in longitudinal

direction per 1 degree torsion angle relative

to rim, for a unit-length belt segment. This

value is independent on the number of belt

segments.

Note: Only needed for full 3D variant.Unit is force/angle.

belt_torsion_lat_displ_coupl If belt twist angle is 0, value is the

kinematic belt torsion angle at 1 mm lateral

belt displacement.

Note: Optional, and only needed for full 3D variant. Unit is angle/length. Default value is 0.

belt_lat_curvature_radius Curvature radius of belt cross section

perpendicular to mid-plane.

Note: Optional, and only needed for full 3D variant. Default value is (nearly) infinity.

belt_lat_bend_stiffn Bending stiffness of belt elements about

circumferential direction.

Note: Optional, and only needed for full 3D variant. Unit is force*length2. Default value is (nearly) infinity.


269Tire Models

wheel_load_at_10_mm_defl_lo_cl Wheel load at 10 mm deflection on

longitudinal cleat. Static wheel load of the

inflated tire, when it is deflected by 10 mm,

with zero camber angle, on a cleat as

specified above, during stand-still. Cleat

must be high enough that the tire does not

touch the ground apart from the cleat. The

cleat is oriented in longitudinal direction,

along foot-print centerline.

Note: This parameter is optional and you can specify it instead of, or in addition to, belt_lat_bend_stiffn. For truck tires, if you specify wheel_load_at_40_mm, FTire looks for wheel_load_at_20_mm_defl_ lo_cl instead.

weight_lat_bend_st

weight_wheel_load_lo_cl

If you provide both data on the previous

page (belt_lat_bend_stiffn and

wheel_load_at_10_mm_defl_lo_cl) to

define the lateral belt bending stiffness,

they constitute an over-determined system

of equations for the respective FTire's

internal stiffness values. FTire will try to

find a compromise. You can control the

compromise by setting these weight values.

Their relative size controls, in a least-

squares approach, the contribution of the

related parameter. If a weight is set to zero,

the respective parameter is completely

ignored.


belt_lat_bend_damp Quotient of bending damping and bending

stiffness of belt elements about

circumferential direction.

Note: Optional, and only needed for full 3D variant. Unit is time. Default value is 1 ms.


Adams/Tire

270

f1_p2

f2_p2

f4_p2

f5_p2

f6_p2

D1_p2

D2_p2

D4_p2

belt_in_plane_bend_st_p2

wheel_load_at_10_mm_defl_cl_p2

wheel_load_at_20_mm_defl_cl_p2

belt_out_of_plane_bend_st_p2

belt_lat_bend_stiffn_p2

belt_twist_st_p2

belt_torsion_st_p2

If measurements for a second inflation

pressure (inflation_pressure_2) are

available, these are the respective values of

the following taken at that pressure:

• f1

• f2

• f4

• f5

• f6

• D1

• D2

• D4

• belt_in_plane_bend_stiffn

• wheel_load_at_10_mm_defl_cleat

• wheel_load_at_20_mm_defl_cleat

• belt_out_of_plane_bend_stiffn

• belt_lat_bend_stiffn

• belt_twist_stiffn

• belt_torsion_stiffn

Note: These data are optional.

tread_depth Mean groove depth in tread.

tread_base_height Rubber height over steel belt for zero tread

depth, which is the distance between steel

belt and grooves.

stiffness_tread_rubber Stiffness of tread rubber in Shore-A units.

tread_positive Percentage of gross tread contact area with

respect to overall footprint area (tread

pattern positive).


271Tire Models

damping_tread_rubber Quotient of tread rubber damping modulus

and tread rubber elasticity modulus.

Note: Deflection/force phase-lag of elastomers is often assumed to be independent of excitation frequency. This behavior is not yet implemented in FTire; instead, viscous damping is used. The parameter damping_tread_rubber is nothing but the quotient of damper coefficient and spring stiffness of the coupling of blocks and belt. For that reason, the parameter carries the unit time.

sliding_velocity The sliding velocity of a tread rubber block,

when its friction coefficient reaches the

my_sliding values.

blocking_velocity The sliding velocity of a tread rubber block,

when its friction coefficient reaches the

my_blocking values.

low_ground_pressure The first of three ground-pressure values

that defines the pressure dependency of the

friction coefficient. Default value is 0.1 bar.

med_ground_pressure The second of three ground-pressure values


friction coefficient. Default value is 2 bar.

high_ground_pressure The third of three ground-pressure values


friction coefficient. Default value is 10 bar.

my_adhesion_at_low_p Coefficient of adhesion friction (which is

equal to static friction) between tread

rubber and road, at first ground pressure

value.

Note: For this parameter and the parameters in the following eight rows, you can still use the parameter names my_..._at_..._bar, used in the previous FTire version. To avoid confusion with the actual ground pressure values, however, we recommend you use the more general names.


Adams/Tire

272

my_sliding_at_low_p Coefficient of sliding friction, at a sliding

velocity defined by parameter

sliding_velocity, between tread rubber and

road, at first ground pressure value.

my_blocking_at_low_p Coefficient of sliding friction, at a sliding


blocking_velocity, between tread rubber

and road, at first ground pressure value.

my_adhesion_at_med_p Coefficient of adhesion friction (which is

equal to static friction) between tread

rubber and road, at second ground pressure

value.

my_sliding_at_med_p Coefficient of sliding friction, at a sliding


sliding_velocity, between tread rubber and

road, at second ground pressure value.

static_balance_weight Weight that would have put up on the rim

horn for static balancing.

Note: Parameter is optional.

static_balance_ang_position The angular position at the rim where the

static balance weight would have been

placed.


dynamic_balance_weight One of the two equal weights that would

have been placed on the rim outer and inner

horns for dynamic balancing.


dynamic_balance_ang_position The angular position at the rim where the

left dynamic balance weight would have

been placed.

Note: This parameter is optional.

radial_non_uniformity Amplitude of the harmonic radial stiffness

variation as percentage of the mean radial

stiffness.


radial_non_unif_ang_position Angular position where radial stiffness

reaches its maximum.

Note: This parameter is optional.


273Tire Models

tang_non_uniformity Amplitude of the harmonic tangential

stiffness variation as percentage of the

mean tangential stiffness.


tang_non_unif_ang_position Angular position where tangential stiffness

reaches its maximum.


conicity Small rotation angle of belt elements at

zero moment, about circumferential axis,

resulting in a conical shape of the unloaded

belt.

Note: Parameter is optional and can only be used with the full 3D variant. Nonzero conicity will cause a small side-force without side-slip angle. The sign of that force is independent of the tire’s rolling direction.

ply_steer_percentage Percentage of lateral belt displacement

relative to radial belt displacement, when a

radial force is applied.

Note: Ply-steer, besides conicity, is one of the reasons for nonzero side forces at zero side-slip angle. In contrast to the conicity side-force, this residual side force changes sign when the tire rolling direction is reversed.

run_out The maximum deviation of the local tire

radius from the mean tire radius. Run-out is

assumed to be a harmonic function of the

angular position.

run_out_ang_position The angular belt element position relative

to the rim, where maximum run-out occurs.

number_belt_segments Number of numerical belt segments.

Maximum value is 200, but can be changed

upon request.

number_blocks_per_belt_segm Number of numerical blocks (= contact

elements) per belt segment. Maximum

value is 50, but can be changed upon

request.


Adams/Tire

274

number_tread_strips Number of strips, into which the contact

points are arranged in the full 3D variant,

using an equal spacing.

Note: If value is greater than or equal to 1000, the contact points are scattered randomly over the tread. Alternatively, it is possible to place tread elements according to the actual tread pattern of the tire. This is done by specifying a bitmap file of the footprint. For more information, see the extended documentation at www.ftire.com.

If you specify neither number_tread_strips nor the bitmap file, FTire uses the basic FTire variant instead of the full 3D variant, regardless of the model-level specification in the operating_conditions section.

maximum_time_step Maximum integration time step allowed.

Note: You can call FTire with very large time steps (if this makes sense for your model). Internally, FTire uses multi-step integration with an internal time step that is chosen on basis of maximum_time_step. This internal time step is kept constant if the external time step does not change.

Changing the external time step can result in considerable longer computation time, because certain time-consuming preprocessing calculations have to be repeated. For that reason, you should avoid changing the external time step whenever possible.



275Tire Models

About the FTire Tire Data FileAs with all TeimOrbit files, entries in the [UNITS] block define the physical units of all parameters.

The basic parameters are preprocessed during initialization, resulting in the preprocessed parameters.

These parameters are saved in a separate TeimOrbit-style file, which can be used in further simulations

instead of the basic data file. By this, you can omit the preprocessing calculation phase, which may result

in a considerable saving of time.

This preprocessed data file is a copy of the original one; the preprocessed data are appended after the

bottom line, using a hexadecimal, space-saving coding. In contrast to earlier versions of FTire, it is

possible to use this file for parameter changes instead of the original one.

You should, of course, not change the hexadecimal data but only the readable part of the file. The

hexadecimal section does not only contain the preprocessed data, but a copy of the original one, as well.

Moreover, it carries coded information about the FTire version that was used for creation.

This information helps to automatically determine whether or not an update of the preprocessed data is

required. This means that whenever you change some basic data or you download a new FTire version,

preprocessing will be repeated automatically, and the preprocessed data file saved in your current

working directory. You can (and should) replace the FTire data file in your database with this one,

without any loss of information.

From www.ftire.com, you can download a tool (being a member of FTire/tools) to carry out

preprocessing outside of Adams.

The FTire interface routine automatically recognizes whether several wheels of the car share the same

basic data file. In that case, preprocessing is done only once for all these files. Also, FTire automatically

recognizes whether the data file contains basic parameters or pre-processed ones.

FTire does not use the data in the section [VERTICAL]. It is only included for compatibility with other

tire models. It is recommended that you set Vertical_Stiffness to the value of

BDF_parameter Numerical parameter to control the internal

FTire implicit (BDF) integration scheme,

which is independent of the Adams

integrator.

0 = Euler explicit

0.5 = Trapezoidal rule

1 = Euler implicit

Theoretically, every value between 0 and 1

are allowed. 0.505 or greater is

recommended.



Adams/Tire

276

stat_wheel_load_at_10_mm_defl, after dividing by 10 mm. For Vertical_Damping, choose 0 (or a small

nonzero value). The actual vertical damping of FTire is not just one single value, but will depend on

rolling speed, inflation pressure, load, camber, and so on.

The following is an examples of a basic FTire data file. Note that by far not all possible data are defined.

For examples, only data for one inflation pressure are provided.

$--------------------------------------------------------MDI_HEADER[MDI_HEADER] FILE_TYPE

= 'tir' FILE_VERSION = 4.0 FILE_FORMAT = 'ASCII'

(COMMENTS)comment_string 'Tire Manufacturer

- unknown' 'Tire Type

- unknown' 'Tire Dimension

- 195/65 R 15' 'Pressure

- 2.0 bar' 'File Generation Date - 03/03/11 10:32'$-------------------------------------------------------------SHAPE[SHAPE]radial width 1.0 0.0 1.0 0.41.0 0.90.9 1.0$-------------------------------------------------------------UNITS[UNITS] FORCE = 'NEWTON'MASS = 'GRAM'LENGTH = 'MM'TIME = 'MILLISECOND' ANGLE = 'DEGREE'$---------------------------------------------------------DIMENSION[DIMENSION] UNLOADED_RADIUS =

326.0 $ [mm]$----------------------------------------------------------VERTICAL[VERTICAL] VERTICAL_STIFFNESS =

170.0 $ [N/mm]VERTICAL_DAMPING =

0.0 $ [Nms/mm]$-------------------------------------------------------------MODEL[MODEL] PROPERTY_FILE_FORMAT =

'FTIRE' $

separate_animation =

277Tire Models

0 $ [0/1] additional_output_file =

0 $ [0/1]

verbose = 0

$ [0/1]$----------------------------------------------OPERATING_CONDITIONS[OPERATING_CONDITIONS] inflation_pressure =

2.0 $ [bar]

tread_depth = 8.0

$ [m] model_level =

7 $ [-]

$---------------------------------------------------------PARAMETER[FTIRE_DATA]$basic data and geometry ******************************************* tire_section_width =

195 $ [mm]

tire_aspect_ratio = 65

$ [%] rim_diameter =

381 $ [mm]

rim_width = 152.4

$ [mm]load_index =

91 $ [-]

rolling_circumference = 1975

$ [mm] tread_lat_curvature_radius = 800 $ [mm] tread_width =

160 $ [mm]

tire_mass = 9000

$ [g]interior_volume =

0.03e9 $ [mm^3] volume_gradient =

1.0 $ [%/mm] belt_torsion_lat_displ_coupl =

0.0 $ [deg/mm]

$

Adams/Tire

278

$static and modal data for 1st infl. pressure *********************** stat_wheel_load_at_10_mm_defl = 1690

$ [N] stat_wheel_load_at_20_mm_defl = 3600 $ [N]

dynamic_stiffening = 20

$ [%] speed_at_half_dyn_stiffening = 5.55

$ [mm/ms]=[m/s] radial_hysteretic_stiffening = 0

$ [%]radial_hysteresis_force =

0 $

[N] tang_hysteretic_stiffening = 0

$ [%]tang_hysteresis_force =

0 $ [N] belt_extension_at_200_kmh =

1.0 $ [%] rel_long_belt_memb_tension =

82.0 $ [%]

$f1 =

62.1 $ in-plane rotat. [Hz]

f2 = 81.4

$ in-plane transl. [Hz]f4 =

80.0 $ out-of-plane rotat. [Hz]

$D1 =

0.05 $ in-plane rotat. [-]

D2 = 0.08

$ in-plane transl. [-]D4 =

0.05 $ out-of-plane rotat. [-]

$belt_in_plane_bend_stiffn =

2.0e6 $ [Nmm^2] belt_out_of_plane_bend_stiffn = 200.0e6 $ [Nmm^2]

belt_lat_bend_stiffn = 20.0e6 $ [Nmm^2]

belt_twist_stiffn = 1.0e6

$ [Nmm^2/deg] belt_torsion_stiffn =

279Tire Models

100.0 $ [N/deg]

$rim_flange_contact_stiffness = 3000.0

$ [N/mm] rim_to_flat_tire_distance = 30.0

$ [mm]$$tread properties ************************************************** tread_depth =

8.0 $ [mm]

tread_base_height = 3.0

$ [mm] stiffness_tread_rubber = 64

$ [Shore A]tread_positive =

65 $ [%]

damping_tread_rubber = 0.025

$ [ms]

$sliding_velocity =

0.1 $ [mm/ms]

blocking_velocity = 50.0

$ [mm/ms]low_ground_pressure =

0.01 $ [bar]

med_ground_pressure = 2.0

$ [bar]high_ground_pressure =

10.0 $ [bar]

mu_adhesion_at_low_p = 1.3

$ [-]mu_sliding_at_low_p =

1.1 $ [-]

mu_blocking_at_low_p = 0.8

$ [-]mu_adhesion_at_med_p =

1.3 $ [-]

mu_sliding_at_med_p = 1.0

Adams/Tire

280

$ [-]mu_blocking_at_med_p =

0.8 $ [-]

mu_adhesion_at_high_p = 1.3

$ [-]mu_sliding_at_high_p =

1.0 $ [-]

mu_blocking_at_high_p = 0.8

$ [-]$$tire imperfections ************************************************ static_balance_weight =

0.0 $ [g] static_balance_ang_position = 0.0 $ [deg] dynamic_balance_weight =

0.0 $ [g] dynamic_balance_ang_position =

0.0 $ [deg]

radial_non_uniformity = 0.0

$ [%] radial_non_unif_ang_position = 0.0 $ [deg]

tang_non_uniformity = 0.0

$ [%] tang_non_unif_ang_position = 0.0

$ [deg] ply_steer_percentage =

0.0 $ [%] conicity =

0.0 $ [deg]

run_out = 0.0

$ [mm] run_out_angular_position = 0.0

$ [deg]$$measuring conditions ********************************************** inflation_pressure =

2.0 $ [bar]

rim_inertia = 0.25e9 $ [g*mm^2]

$

281Tire Models

$numerical data **************************************************** number_belt_segments =

80 $

number_blocks_per_belt_segm = 32 $

number_tread_strips = 8

$ maximum_time_step =

0.2 $ [ms]

BDF_parameter = 0.505

$ 0.5 .. 1.0 [-]

Choosing FTire Operating ConditionsYou can control certain tire data during a simulation, without rerunning preprocessing. These parameters,

listed below, are called operating condition parameters:

• Inflation pressure - The operating condition value of inflation_pressure defines the actual,

possibly time-dependent inflation pressure, whereas the [FTIRE_DATA] value describes the

inflation pressure at which the remainder of the data measurements had been taken.

• Tread depth -The operating condition value of tread_depth defines the actual, possibly time-

dependent tread depth, whereas the [FTIRE_DATA] value describes the tread depth at which the

remainder of the data measurements had been taken.

• Model level - The operating condition value of model_level defines what variant of FTire is to

be used: the basic version (=6) or the full 3D version (=7). The list of possible variants will be

extended in the next release.

Also in the next FTire release, ambient temperature, will be added to the list of operating conditions.

To determine the actual operating conditions, FTire looks for the section [OPERATING_CONDITIONS]

in the basic or preprocessed tire data file. If it does not find this section, or it does not contain the

respective definitions, FTire uses the data in the sections [FTIRE_DATA] or

[FTIRE_PREPROCESSED_DATA] as the measurement conditions.

In case the section [OPERATING_CONDITIONS] is defined, FTire first tries to read a constant value

for each operating condition. This value may either be the same for all tires using the data file, or it can

have individual values for each such tire instance.

You can enter constant operating conditions as shown the table below.

Adams/Tire

282

OPERATING_CONDITIONS Section Parameters

If no constant value is found, FTire looks for a table that is defining data points for operating condition

versus time. These data points then will be piecewise linearly interpolated with respect to simulation

time.


inflation_pressure Actual inflation pressure, used for all FTire

instances that are parameterized by this

data file.

inflation_pressure_wheel_i Actual inflation pressure, used only for

FTire instance with GFORCE ID i. This

value overrides the inflation_pressure

value (i is to be replaced by a numerical

GFORCE ID value of the tire instance).

tread_depth Actual tread depth, used for all FTire


data file.

tread_depth_wheel_i Actual tread depth, used only for FTire

instance with GFORCE ID i. This value

overrides the tread_depth value (i is to be

replaced by the numerical GFORCE ID

value of the tire instance).

model_level Actual model level, used for all FTire


data file. In the current release, the

following model levels are implemented:

• 6: FTire basic version (three

degrees of freedom for each belt

element, one line of contact

elements.

• 7: FTire full 3D version (five

generalized degrees of freedom for

each belt element, several lines of,

or irregularly scattered, contact

elements).

model_level_wheel_i Actual model level, used for FTire instance

with GFORCE ID i. This value will

override the model_level value (’i’ is to be

replaced by the numerical GFORCE ID

value of the tire instance).

283Tire Models

You enter such look-up tables as subsections of the section [OPERATING_CONDITIONS]. These

subsections can each contain up to 200 data pairs, one pair per line. Every data pair consists of a value

for time and a corresponding value for the operating condition. Units are the same as for constant values.

Similarly as for constant values, tables which are valid for all tires, or individual tables for each instance

are allowed.

The names of these table subsections, with obvious meanings, are:

• (TIME_TABLE_INFLATION_PRESSURE)

• (TIME_TABLE_INFLATION_PRESSURE_WHEEL_ i)

• (TIME_TABLE_TREAD_DEPTH)

• (TIME_TABLE_TREAD_DEPTH_WHEEL_ i)

• (TIME_TABLE_MODEL_LEVEL)

• (TIME_TABLE_MODEL_LEVEL_WHEEL_ i)

The following examples defines a sudden pressure loss (between 5 and 5.2 s of simulation time) in tire

with GFORCE ID 2. In addition, it specifies constant inflation pressure (2.2 bar) for the other tires, and

a certain, equal and constant extreme tread wear (0.1 mm every100 s) for all tires. Model level is chosen

to be the full 3D variant for all tires at any time:

$------------------------------------------------------OPERATING_CONDITIONS[OPERATING_CONDITIONS]

MODEL_LEVEL = 7

INFLATION_PRESSURE_WHEEL_1 = 2.2INFLATION_PRESSURE_WHEEL_3 = 2.2INFLATION_PRESSURE_WHEEL_4 = 2.2

(TIME_TABLE_TREAD_DEPTH)0 8.0

100 7.9

(TIME_TABLE_INFLATION_PRESSURE_WHEEL_2)0

2.2

5 2.2

5.2 1.2

Adams/Tire

284

Note: If you use the preprocessed data file in subsequent simulations, don't forget to copy the

[operating_conditions] section from the basic data file manually into the preprocessed data

file. This is not done automatically, because tire operating conditions are not considered to

be part of the tire data.

adams tire models

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