ADAMS Tire Models

Tire Models

12 Adams/Tire

Using the Fiala Handling Force Model This 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 Assumptions The 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 Inputs The 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 ( α ), which Adams/Tire calculates.

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

13 Tire Models

Input for Calculating Tire Forces Quantity: Mt

Description: Mass of tire

Use by Fiala:

Source:

• Damping • Vertical force (Fz)

Alpha

Slip angle

Lateral force (Fy)

Tire kinematic state from Adams/Solver

Ss

Longitudinal slip ratio

Longitudinal force (Fx)


pen

Penetration (tire deflection)

Vertical force (Fz)


Vpen

d/dt (penetration)

Vertical force (Fz)


Vertical_damping

Vertical damping coefficient

• Damping

Tire property file (.tir)

• Vertical force (Fz)

Vertical_stiffness

Vertical tire stiffness

CSLIP

Partial derivative of longitudinal Longitudinal force (Fx) force (Fx) with respect to longitudinal slip ratio (S) at zero longitudinal slip


CALPHA

Partial derivative of lateral force Lateral force (Fy) (Fy) with respect to slip angle ( α ) at zero slip angle


UMIN

Coefficient of friction at zero slip

Fx, Fy, Tz


UMAX

Coefficient of friction when tire is sliding

Fx, Fy, Tz


Rolling_resistance Rolling resistance coefficient

Vertical force (Fz)


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

14 Adams/Tire

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

Ω, and the loaded rolling radius Rl: V sx = V x – ΩR l

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

= Vy

The practical slip quantities κ (longitudinal slip) and velocities in the contact point:

α (slip angle) are calculated with these slip

V sx V sy κ = – -------- and tan α = --------Vx Vx Note that for realistic tire forces the slip angle between -1 (locked wheel) and 1.

α is limited to 900 and the longitudinal slip κ 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 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.

15 Tire 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

v1 1 dv 1 ----+ - = tan α + aφ V x dt σ α

following differential equation is valid during braking slip: ------ --------

with the relaxation length σ α in the lateral direction. The turnslip φ 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

dv 1 + V x v 1 = σ α V sy dt

can be transformed to: σ α --------

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

du 1 + V x u 1 = – σ κ V sx dt

direction: σ κ --------

16 Adams/Tire

Now the practical slip quantities, deformation: κ'

κ' and α' , are defined based on the tire

u1 = ------ sgn ( V x ) σκ

v -----1- α' = atan  σ α  These practical slip quantities state tire behavior.

κ' and α' are used instead of the usual κ and α definitions for steady-

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

Fiala Tire Force Evaluation Types 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.

17 Tire 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 ( S s α ):

Ss α =

2

S s + tan 2 ( α )

The current value coefficient of friction (U):

U = U max – ( ( U max – U min ) ⋅S s α ) Fiala defines a critical longitudinal slip ( S critical ):

U ⋅F z S critical = -----------------------2 ⋅CSLIP 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|

>

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

F x1 = U ⋅F z ( U ⋅F z ) 2 F x2 = ------------------------------------4 ⋅ S s ⋅CSLIP

S_critical

18 Adams/Tire

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 ( α critical ):

3 ⋅U ⋅F z α critical = atan ------------------------- CALPHA The lateral force peaks at a value equal to U × |Fz| when the slip angle (Alpha) equals the critical slip angle ( α critical ).

≤ α critical

Case 1. Elastic Deformation State: |Alpha|

Fy = - U × |Fz|× (1-H3) × sign(Alpha) where:

CALPHA ⋅ tan ( α ) H = 1 – -------------------------------------------------3 ⋅U ⋅ F z 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|

≤ α critical

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

CALPHA ⋅ tan ( α ) H = 1 – -------------------------------------------------3 ⋅U ⋅ F z Case 2. Complete Sliding State: |Alpha| Mz= 0.0

> α critical

19 Tire 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 Shape Using 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 = 2.0 FILE_FORMAT = 'ASCII' (COMMENTS) {comment_string} 'Tire - XXXXXX' 'Pressure - XXXXXX'

20 Adams/Tire

'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 = 309.9 WIDTH = 235.0 ASPECT_RATIO = 0.45 $-------------------------------------------------parameter [PARAMETER] VERTICAL_STIFFNESS = 310.0 VERTICAL_DAMPING = 3.1 ROLLING_RESISTANCE = 0.0 CSLIP = 1000.0 CALPHA = 800.0 CGAMMA = 0.0 UMIN = 0.9 UMAX = 1.0 RELAX_LENGTH_X = 0.05 RELAX_LENGTH_Y = 0.15 $---------------------------------------------carcass shape [SHAPE] {radius width} 1.0000 0.0000 1.0000 0.5000 1.0000 0.8649 0.9944 0.9235 0.9792 0.9819 0.9583 1.0000 $------------------------------------------------load_curve $ Maximum of 100 points (optional) [DEFLECTION_LOAD_CURVE] {pen fz} 0 0.0 1 212.0 2 428.0 3 648.0 5 1100.0

21 Tire Models

10 20 30

2300.0 5000.0 8100.0

Fiala Tire Contact Methods The Fiala tire model supports the following roads: • 2D Roads, see Using the 2D Road Model. • 3D Roads, see Adams/3D Road Model

22 Adams/Tire

Using the PAC2002Tire Model The 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 PAC2002 Magic-Formula (MF) tire models are considered the state-of-the-art for modeling tire-road interaction forces in vehicle dynamics applications. Since 1987, Pacejka and others have published several versions of this type of tire model. The 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)

23 Tire 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 Forces For vehicle dynamics applications, an accurate knowledge of tire-road interaction forces is inevitable because the movements of a vehicle primarily depend on the road forces on the tires. These interaction forces depend on both road and tire properties, and the motion of the tire with respect to the road. In the radial direction, the MF tire models consider the tire to behave as a parallel linear spring and linear damper with one point of contact with the road surface. The contact point is determined by considering the tire and wheel as a rigid disc. In the contact point between the tire and the road, the contact forces in longitudinal and lateral direction strongly depend on the slip between the tire patch elements and the road.

24 Adams/Tire

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 ( κ , α ), and inclination angle ( γ ) 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

25 Tire 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.

26 Adams/Tire

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 Variable type: Angle

Force

Moment

Name:

Abbreviation:

Slip angle

α

Inclination angle

γ

Longitudinal force

Fx

Lateral force

Fy

Vertical load

Fz

Overturning moment

Mx

Unit: Radians

Newton

Newton.meter

Rolling resistance moment My

Speed

Self-aligning moment

Mz

Longitudinal speed

Vx

Lateral speed

Vy

Longitudinal slip speed

Vsx

Lateral slip speed

Vsy

Rotational speed Tire rolling speed

ω

Meters per second

Radians per second

Definition of Tire Slip Quantities 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

ω , and the effective rolling radius Re: V sx = V x – ΩR e Slip Quantities at Combined Cornering and Braking/Traction

(1)

27 Tire Models

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

V sy = V y

(2)

The practical slip quantities κ (longitudinal slip) and velocities in the contact point with:


V sx κ = – -------Vx

(3)

V sy tan α = --------Vx

(4)

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

Vr = Re Ω

(5)

Turn-slip is one of the two components that form the spin of the tire. Turn-slip tire yaw velocity

φ is calculated using the

· ψ:

· ψ W t = -----Vx The total tire spin

(6)

ϕ is calculated using:

1 · ψ 1 = ------ { ψ – ( 1 – εγ)Ωsin γ} Vx The total tire spin has contributions of turn-slip and camber. the camber to become comparable with turn-slip.

(7)

ε denotes the camber reduction factor for

28 Adams/Tire

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

ρ

as follows:

29 Tire Models

Ro F x 2  F y 2   ------------- + q Fcyl γ2  F z = 1 + q V2 Ω ------ –  q – q Fcxl Fcyl Vo  F z0  F z0  

(8)

ρ ρ 2 · q Fzl ------ + q Fz2 ------ F z0 + K z • ρ R 0 R0 Using this formula, the vertical tire stiffness increases due to increasing rotational speed decreases by longitudinal and lateral tire forces. If qFz1 is zero, qFz1 will be CzR0/Fz0.

ω and

When you do not provide the coefficients qV2, qFcx, qFcy, qFz1, qFz2 and qFc γ 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)

· F z = C z ρ λ Cz + K z ρ 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

R0 2 ρ = R 0 – R 1 + q V1 R 0 Ω------  V 0

ω: (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:

Vx R e = -----Ω

(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.

30 Adams/Tire

Effective Rolling Radius and Longitudinal Slip

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

ΩR 02 d d ----------R f = R 0 + q V1 R 0   V 0  – R Fz0 [ D Peff arc tan ( B Reff ρ ) + F Reff ρ ] in which

ρ Fz0 is the nominal tire deflection:

(12)

31 Tire Models

F z0 ρ Fz0 = ---------------C z λ Cz and

ρ

d

(13)

ρ is called the dimensionless radial tire deflection, defined by: ρ = --------ρ Fz0

(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)

32 Adams/Tire

Name:


Explanation:

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 γ1

QFCG1

Tire stiffness interaction with camber

qV1

QV1

Tire radius growth coefficient

qV2

QV2

Tire stiffness variation coefficient with speed

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.

33 Tire 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:

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

δ 2/(2 · width ·| tan( γ) |)

34 Adams/Tire

Srim= ½·width - max(width ,

δ /| tan( γ) |)/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( γ) | · /(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 γ 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.

Note:

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

Basics of the Magic Formula in PAC2002 The Magic Formula is a mathematical formula that is capable of describing the basic tire characteristics for the interaction forces between the tire and the road under several steady-state operating conditions. We distinguish: • Pure cornering slip conditions: cornering with a free rolling tire • Pure longitudinal slip conditions: braking or driving the tire without cornering • Combined slip conditions: cornering and longitudinal slip simultaneously

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

α , respectively, and the

longitudinal force Fx as a function of longitudinal slip κ , 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:

Y ( x ) = D cos [ Carc tan { Bx – E ( Bx – arc tan ( Bx ) ) } ] where Y(x) is either Fx with x the longitudinal slip

κ , or Fy and x the lateral slip α .

(15)

35 Tire 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:

Y ( x ) = D cos [ Carc tan { Bx – E ( Bx – arc tan ( Bx ) ) } ] in which Y(x) is the pneumatic trail t as function of slip angle

(16)

α.

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).

36 Adams/Tire

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).

37 Tire Models

Input Variables The input variables to the Magic Formula are: Longitudinal slip

κ

[-]

Slip angle

α

[rad]

Inclination angle

γ

[rad]

Normal wheel load

Fz

[N]

Output Variables Longitudinal force

Fx

[N]

Lateral force

Fy

[N]

Overturning couple

Mx

[Nm]

Rolling resistance moment

My

[Nm]

Aligning moment

Mz

[Nm]

The output variables are defined in the W-axis system of TYDEX. Basic Tire Parameters All tire model parameters of the model are without dimension. The reference parameters for the model are: Nominal (rated) load

Fz0

[N]

Unloaded tire radius

R0

[m]

Tire belt mass

mbelt

[kg]

As a measure for the vertical load, the normalized vertical load increment dfz is used:

F z – F' z0 df z = -------------------F' z0

(17)

with the possibly adapted nominal load (using the user-scaling factor, γΦ z0 ):

F' z0 = F z0 • λ

F z0

(18)

38 Adams/Tire

Nomenclature of the Tire Model Parameters In the subsequent sections, formulas are given with non-dimensional parameters aijk with the following logic: Tire Model Parameters Parameter: a=

i=

j=

k=

Definition:

p

Force at pure slip

q

Moment at pure slip

r

Force at combined slip

s

Moment at combined slip

B

Stiffness factor

C

Shape factor

D

Peak value

E

Curvature factor

K

Slip stiffness = BCD

H

Horizontal shift

V

Vertical shift

s


t

Transient tire behavior

x

Along the longitudinal axis

y

Along the lateral axis

z

About the vertical axis

1, 2, ...

User Scaling Factors A set of scaling factors is available to easily examine the influence of changing tire properties without the need to change one of the real Magic Formula coefficients. The default value of these factors is 1. You can change the factors in the tire property file. The peak friction scaling factors, l µξ and γµψ ψ , 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.

39 Tire Models

Scaling Factor Coefficients for Pure Slip

Name:

Name used in tire property file:

Explanation:

λ Fzo

LFZO

Scale factor of nominal (rated) load

λ Cz

LCZ

Scale factor of vertical tire stiffness

λ Cx

LCX

Scale factor of Fx shape factor

λ µx

LMUX

Scale factor of Fx peak friction coefficient

λ Ex

LEX

Scale factor of Fx curvature factor

λ Kx

LKX

Scale factor of Fx slip stiffness

λ Hx

LHX

Scale factor of Fx horizontal shift

λ Vx

LVX

Scale factor of Fx vertical shift

λ γx

LGAX

Scale factor of inclination for Fx

λ Cy

LCY

Scale factor of Fy shape factor

λ µy

LMUY

Scale factor of Fy peak friction coefficient

λ Ey

LEY

Scale factor of Fy curvature factor

λ Ky

LKY

Scale factor of Fy cornering stiffness

λ Hy

LHY

Scale factor of Fy horizontal shift

λ Vy

LVY

Scale factor of Fy vertical shift

λ gy

LGAY

Scale factor of inclination for Fy

λ t

LTR

Scale factor of peak of pneumatic trail

λ Mr

LRES

Scale factor for offset of residual moment

λ γz γz

LGAZ

Scale factor of inclination for Mz

λ Mx

LMX

Scale factor of overturning couple

λ VMxMx

LVMX

Scale factor of Mx vertical shift

λ My

LMY

Scale factor of rolling resistance moment

40 Adams/Tire

Scaling Factor Coefficients for Combined Slip

Name:


Explanation:

λ xα α

LXAL

Scale factor of alpha influence on Fx

λ yκ κ

LYKA


λ Vy κ κ

LVYKA

Scale factor of kappa-induced Fy

λ

LS

Scale factor of moment arm of Fx

s

Scaling Factor Coefficients for Transient Response

Name:


Explanation:

λ sk

LSGKP

Scale factor of relaxation length of Fx

λ sa

LSGAL

Scale factor of relaxation length of Fy

λ gyr

LGYR

Scale factor of gyroscopic moment

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.

41 Tire 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:

F x = F x0 ( κ , F z, γ)

(19)

F x0 = D x sin ( [ C x arc tan { B x κ x – E x ( B x κ x – arc tan ( B x κ x ) ) } ] + S Vx )

(20)

κ x = κ + S Hx

(21)

γx = γ ⋅λ

(22)

γx

with following coefficients:

C x = p Cx1 ⋅λ

(23)

Cx

D x = µx ⋅F z ⋅ζ1

(24) 2

µx = ( p Dx1 + p Dx2 df z ) ⋅( 1 – p Dx3 ⋅γ )λ

(25)

µx

2

E x = ( p Ex1 + p Ex2 df z ) ⋅{ 1 – p Ex4 sgn ( κ x ) } ⋅λ

Ex

with E x ≤ 1

(26)

the longitudinal slip stiffness:

K x = F z ⋅( p Kx1 + p Kx2 df z ) ⋅ exp ( p Kx3 df z ) ⋅λ Kx = Bx Cx Dx = B x = K x ⁄( C x D x )

Kx

(27)

∂ F x0 at κ = 0 ∂ κx (28)

42 Adams/Tire

S Hx = ( p Hx1 + p Hx2 ⋅df z )λ

(29)

Hx

S Vx = F z ⋅( p Hx1 + p Hx2 ⋅df z )λ

Vx

⋅λ

Hµx

⋅ζ 1

(30)

Longitudinal Force Coefficients at Pure Slip

Name:


Explanation:

pCx1

PCX1

Shape factor Cfx for longitudinal force

pDx1

PDX1

Longitudinal friction Mux at Fznom

pDx2

PDX2

Variation of friction Mux with load

pDx3

PDX3

Variation of friction Mux with inclination

pEx1

PEX1

Longitudinal curvature Efx at Fznom

pEx2

PEX2

Variation of curvature Efx with load

pEx3

PEX3

Variation of curvature Efx with load squared

pEx4

PEX4

Factor in curvature Efx while driving

pKx1

PKX1

Longitudinal slip stiffness Kfx/Fz at Fznom

pKx2

PKX2

Variation of slip stiffness Kfx/Fz with load

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

Formulas for the Lateral Force at Pure Slip

F y = F y0 ( α , γ, F z )

(31)

F y0 = D y sin [ C y arc tan { B y α y – E y ( B y α y – arc tan ( B y α y ) ) } ] + S Vy

(32)

α y = α + S Hy

(33)

The scaled inclination angle:

γy = γ ⋅λ

(34)

γy

with coefficients:

C y = p Cy1 ⋅λ

Cy

(35)

43 Tire Models

D y = µy ⋅F z ⋅ζ2

(36) 2

µy = ( p Dy1 + p Dy2 df z ) ⋅( 1 – p Dy3 γy ) ⋅λ

(37)

µy

E y = ( p Ey1 + p Ey2 df z ) ⋅{ 1 – ( p Ey3 + p Ey4 γy ) sgn ( α y ) } ⋅γEy with E y ≤ 1

(38)

The cornering stiffness:

Fz    K y0 = P Ky1 ⋅F z0 ⋅ sin  2ac tan ---------------------------  ⋅λ  P Ky2 F 0 λ Fz0 

F z0

⋅λ



Ky

(39)



K y = K y0 ⋅( 1 – p Ky3 γy ) ⋅ζ 3

(40)

B y = K y ⁄( C y D y )

(41)

S Hy = ( p Hy1 + p Hy2 df z ) ⋅λ

Hy

+ p Hy3 γy ⋅ζ0 + ζ4 – 1

S Vy = F z ⋅{ ( p Vy1 + p Vy2 df z ) ⋅λ

Vy

(42)

+ ( p Vy3 + p Vy4 df z ) ⋅γy } ⋅λ

µy

⋅ζ 4

(43)

The camber stiffness is given by:

K yγ0 = P Hy3 K y0 + F z ( p νy3 + p νy4 df z )

(44)

Lateral Force Coefficients at Pure Slip

Name:


Explanation:

pCy1

PCY1

Shape factor Cfy for lateral forces

pDy1

PDY1

Lateral friction Muy

pDy2

PDY2

Variation of friction Muy with load

pDy3

PDY3

Variation of friction Muy with squared inclination

pEy1

PEY1

Lateral curvature Efy at Fznom

pEy2

PEY2

Variation of curvature Efy with load

pEy3

PEY3

Inclination dependency of curvature Efy

pEy4

PEY4

Variation of curvature Efy with inclination

pKy1

PKY1

Maximum value of stiffness Kfy/Fznom

pKy2

PKY2

Load at which Kfy reaches maximum value

pKy3

PKY3

Variation of Kfy/Fznom with inclination

pHy1

PHY1

Horizontal shift Shy at Fznom

44 Adams/Tire

Name:


Explanation:

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

Formulas for the Aligning Moment at Pure Slip

M' z = M z0 ( α , γ, F z )

(45)

M z0 = – t ⋅F y0 + M zr with the pneumatic trail t:

t ( α t ) = D t cos [ C t arc tan { B t α t – E t ( B t α t – arc tan ( B t α t ) ) } ] cos ( α )

(46)

α t = α + S Ht

(47)

and the residual moment Mzr:

M zr ( α r ) = D r cos [ C r arc tan ( B r α r ) ] ⋅ cos ( α )

(48)

α r = α + S Hf

(49)

S Hf = S Hy + S Vy ⁄K y

(50)


γz = γ ⋅λ

(51)

γz

with coefficients: 2

B t = ( q Bz1 + q Bz2 df z + q Bz3 df z ) ⋅( 1 + q Bz4 γz + q Bz5 γz ) ⋅λ

Ky

⁄λ

µy

(52)

C t = q Cz1

(53)

R0 2 D t = F z ⋅( q Dz1 + q Dz2 df z ) ⋅( 1 + q Dz3 γz + q Dz4 γz ) ⋅-------- ⋅λ t ⋅ζ5 F z0

(54)

45 Tire Models

2

E t = ( q Ez1 + q Ez2 df z + q Ez3 df z )

(55)

 2- ⋅arc tan ( B ⋅C ⋅α )  w ith ( E ≤ 1 ) 1 + ( q Ez1 + q Ez2 γz ) -t t t  t π   S Ht = q Hz1 + q Hz2 df z + ( q Hz3 + q Hz4 ⋅df z )γz

(56)

λ Ky --------- + q Bz10 ⋅B y ⋅C y ⋅ζ6 Br =  q ⋅ Bz9   λ µy

(57)

Cr = ζ7 D r = F z ⋅[ ( q Dz6 + q Dz7 df z ) ⋅γr + ( q Dz8 + q Dz9 df z ) ⋅γz ] ⋅R o ⋅λ

µγ +

ζ8 – 1

(58)

An approximation for the aligning moment stiffness reads:

K z = – t ⋅K y

 ≈ – ∂ M at α  = 0 )   ∂α z

(59)

Aligning Moment Coefficients at Pure Slip

Name:


Explanation:

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

46 Adams/Tire

Name:


Explanation:

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

Turn-slip and Parking For situations where turn-slip may be neglected and camber remains small, the reduction factors ζ i i that appear in the equations for steady-state pure slip, are to be set to 1:

ζi = 1

i = 0.

1.… 8

For larger values of spin, the reduction factors are given below. The weighting function ζ 1 is used to let the longitudinal force diminish with increasing spin, according to:

ζ i = cos [ arc tan ( B xϕ R 0 ϕ ) ] with:

B xϕ = p Dxϕ 1 ( 1 + p Dxϕ 2 df z ) cos [ arc tan ( p Dxϕ 3 κ ) ] The peak side force reduction factor ζ 2 reads:

ζ 2 = cos [ arc tan { B yϕ ( R 0 ϕ + p Dyϕ

4

R0 ϕ ) } ]

with:

B yϕ = p Dxϕ 1 ( 1 + p Dxϕ 2 df z ) cos [ arc tan ( p Dxϕ 3 tan α ) ] The cornering stiffness reduction factor ζ 3 is given by:

47 Tire Models

2 2

ζ 3 = cos [ arc tan ( p Kyϕ 1 R 0 ϕ ) ] The horizontal shift of the lateral force due to spin is given by:

S Hyϕ = D Hyϕ sin [ C Hyϕ arc tan { B Hyϕ R o ϕ – E Hyϕ ( B Hyϕ – arc tan ( B Hyϕ R 0 ϕ ) ) } ] The factors are defined by:

C Hyϕ

= p Hyϕ

1

D Hyϕ = ( p Hyϕ 2 + p Hyϕ 3 df z ) ⋅ sin ( V x ) E Hyϕ = P Hyϕ

4

K yRϕ 0 B Hyϕ = ----------------------C y D y K y0 The spin force stiffness KyR0 is related to the camber stiffness Ky0:

K yRϕ

0

K yγ0 = ------------1 – εγ

in which the camber reduction factor is given by:

εγ = p εγϕ 1 ( 1 + p εγϕ 2 df z ) The reduction factors ζ 0 and ζ 4 for the vertical shift of the lateral force are given by:

ζ0 = 0 ζ 4 = 1 + S Hyϕ – S Vyγ ⁄K y The reduction factor for the residual moment reads:

ζ 8 = 1 + D rϕ The peak spin torque Dr ϕ is given by:

D rϕ = D Drϕ sin e [ C Drϕ arc tan { B Drϕ R 0 ϕ – E Drϕ ( B Drϕ R 0 ϕ – arc tan ( B Drϕ R 0 ϕ ) ) } ] The maximum value is given by:

48 Adams/Tire

M zϕ ∞ D Drϕ = ----------------------------π  --sin  2 C Drϕ  The moment at vanishing wheel speed at constant turning is given by:

M zϕ

∞

= q Crϕ 1 µy R 0 F z F z ⁄F z0

The shape factors are given by:

C Drϕ = q Drϕ

1

E Drϕ = q Drϕ

2

K zγr0 B Drϕ = -------------------------------------------C Drϕ D Drϕ ( 1 – εy ) in which:

K zγr0 = F z R 0 ( q Dz8 + q Dz9 df z ) The reduction factor ζ 6 reads:

ζ 6 = cos [ arc tan ( q Brϕ 1 R 0 ϕ ) ] The spin moment at 90º slip angle is given by:

M zϕ

90

= M zϕ

∞

2 ⋅--- ⋅arc tan ( q Crϕ 2 R 0 ϕ ) ⋅G yx ( κ ) π

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 ζ 7 is given by:

2 ζ 7 = --- ⋅arc cos [ M zϕ π

90

⁄D Drϕ ]

49 Tire Models

Turn-Slip and Parking Parameters

Name:


Explanation:

p εϕ 1

PECP1

Camber spin reduction factor parameter in camber stiffness

p εϕ 2

PECP2

Camber spin reduction factor varying with load parameter in camber stiffness

pDx ϕ 1

PDXP1

Peak Fx reduction due to spin parameter

pDx ϕ 2

PDXP2

Peak Fx reduction due to spin with varying load parameter

pDx ϕ 3

PDXP3

Peak Fx reduction due to spin with kappa parameter

pDy ϕ 1

PDYP1

Peak Fy reduction due to spin parameter

pDy ϕ 2

PDYP2

Peak Fy reduction due to spin with varying load parameter

pDy ϕ 3

PDYP3

Peak Fy reduction due to spin with alpha parameter

pDy ϕ 4

PDYP4

Peak Fy reduction due to square root of spin parameter

pKy ϕ 1

PKYP1

Cornering stiffness reduction due to spin

pHy ϕ 1

PHYP1

Fy-alpha curve lateral shift limitation

pHy ϕ 2

PHYP2

Fy-alpha curve maximum lateral shift parameter

pHy ϕ 3

PHYP3

Fy-alpha curve maximum lateral shift varying with load parameter

pHy ϕ 4

PHYP4

Fy-alpha curve maximum lateral shift parameter

qDt ϕ 1

QDTP1

Pneumatic trail reduction factor due to turn slip parameter

qBr ϕ 1

QBRP1

Residual (spin) torque reduction factor parameter due to side slip

qCr ϕ 1

QCRP1

Turning moment at constant turning and zero forward speed parameter

qCr ϕ 2

QCRP2

Turn slip moment (at alpha=90deg) parameter for increase with spin

qDr ϕ 1

QDRP1

Turn slip moment peak magnitude parameter

qDr ϕ 2

QDRP2

Turn slip moment peak position parameter

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

50 Adams/Tire

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

F x = F x0 ⋅G xα ( α , κ , F z )

(60)

with Gx α the weighting function of the longitudinal force for pure slip. We write:

F x = D xα cos [ C xα arc tan { B xα α s – E xα ( B xα α s – arc tan ( B xα α s ) ) } ]

(61)

α s = α + S Hxα

(62)

with coefficients:

B xα = r Bx1 cos [ arc tan { r Bx2 κ } ] ⋅λ

xα

(63)

C xα

(64)

F xo D xα = -----------------------------------------------------------------------------------------------------------------------------------------------------------------cos [ C xα arc tan { B xα S Hxα – E xα ( B xα S Hxα – arc tan ( B xα S Hxα ) ) } ]

(65)

E xα = r Ex1 + r Ex2 df z with E xα ≤ 1

(66)

S Hxα = r Hx1

(67)

The weighting function follows as:

cos [ C xα arc tan { B xα α s – E xα ( B xα α s – arc tan ( B xα α s ) ) } ] G xα = ---------------------------------------------------------------------------------------------------------------------------------------------------------------cos [ C xα arc tan [ B xα S Hxα – E xα ( B xα S Hxα – arc tan ( B xα S Hxα ) ) ] ]

(68)

51 Tire Models

Longitudinal Force Coefficients at Combined Slip

Name:


Explanation:

rBx1

RBX1

Slope factor for combined slip Fx reduction

rBx2

RBX2

Variation of slope Fx reduction with kappa

rCx1

RCX1

Shape factor for combined slip Fx reduction

rEx1

REX1

Curvature factor of combined Fx

rEx2

REX2

Curvature factor of combined Fx with load

rHx1

RHX1

Shift factor for combined slip Fx reduction

Formulas for Lateral Force at Combined Slip

F y = F y0 ⋅G yκ ( α , κ , γ, F z )

(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:

F y = D yκ cos [ C yκ arc tan { B yκ κ s – E yκ ( B yκ κ s – arc tan ( B yκ κ s ) ) } ] + S Vyκ

(70)

κ s = κ + S Hyκ

(71)

with the coefficients:

B yκ = r By1 cos [ arc tan { r By2 ( α – r By3 ) } ] ⋅λ

yκ

C yκ = r Cy1

(72) (73)

F yo D yκ = --------------------------------------------------------------------------------------------------------------------------------------------------------------cos [ C yκ arc tan { B yκ S Hyκ – E yκ ( B yκ S Hyκ – arc tan ( B yκ S Hyκ ) ) } ]

(74)

E yκ = r Ey1 + r Ey2 df z with E yκ ≤ 1

(75)

S Hyκ = r Hy1 + r Hy2 df z

(76)

S Vyκ = D Vyκ sin [ r Vy5 arc tan ( r Vy6 κ ) ]

(77)

52 Adams/Tire

D Vyκ = µy F z ⋅( r Vy1 + r Vy2 df z + r Vy3 γ) ⋅ cos [ arc tan ( r Vy4 α ) ]

(78)

The weighting function appears is defined as:

cos [ C yκ arc tan { B yκ κ s – E yκ ( B yκ κ s arc tan ( B yκ κ s ) ) } ] G yκ = --------------------------------------------------------------------------------------------------------------------------------------------------------cos [ C yκ arc tan { B yκ S Hyκ – E yκ ( B yκ S Hyκ arc tan ( B yκ S Hyκ ) ) } ]

(79)

Lateral Force Coefficients at Combined Slip

Name:


Explanation:

rBy1

RBY1

Slope factor for combined Fy reduction

rBy2

RBY2

Variation of slope Fy reduction with alpha

rBy3

RBY3

Shift term for alpha in slope Fy reduction

rCy1

RCY1

Shape factor for combined Fy reduction

rEy1

REY1

Curvature factor of combined Fy

rEy2

REY2

Curvature factor of combined Fy with load

rHy1

RHY1

Shift factor for combined Fy reduction

rHy2

RHY2

Shift factor for combined Fy reduction with load

rVy1

RVY1

Kappa induced side force Svyk/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)

Formulas for Aligning Moment at Combined Slip

M' z = – t ⋅F y' + M zr + s ⋅F x

(80)

with:

t = t ( α t, eq )

(81)

53 Tire Models

= D t cos [ C t arc tan { B t α t, eq – E t ( B t α t, eq – arc tan ( B t α t, eq ) ) } ] cos ( α ) F' y, γ = 0 = F y – S Vyκ

(82)

(83)

M zr = M zr ( α r, eq ) = D r cos [ arc tan ( B r α r, eq ) ] cos ( α ) t = t ( α t, eq )

(84)

(85)

with the arguments:

Kx 2 2 2 α t, eq = arc tan tan α t + ------ κ ⋅ sgn ( α t ) K y

(86)

Kx 2 2 2 α r, eq = arc tan tan α r + ------ κ ⋅ sgn ( α r ) K y

(87)

Aligning Moment Coefficients at Combined Slip

Name:


Explanation:

ssz1

SSZ1

Nominal value of s/R0 effect of Fx on Mz

ssz2

SSZ2

Variation of distance s/R0 with Fy/Fznom

ssz3

SSZ3

Variation of distance s/R0 with inclination

ssz4

SSZ4

Variation of distance s/R0 with load and inclination

Formulas for Overturning Moment at Pure and Combined Slip For the overturning moment, the formula reads both for pure and combined slip situations:

 M x = R o ⋅F z ⋅q Sx1 λ 

Fy  ------– q ⋅ λ + q ⋅ VMx Sx2 Sx3 F -  λ z0 

Mx

(88)

54 Adams/Tire

Overturning Moment Coefficients

Name:


Explanation:

qsx1

QSX1

Lateral force induced overturning couple

qsx2

QSX2

Inclination induced overturning couple

qsx3

QSX3

Fy induced overturning couple

Formulas for Rolling Resistance Moment at Pure and Combined Slip The rolling resistance moment is defined by: 4

M y = R o ⋅F z ⋅{ q Sy1 + q Sy3 F x ⁄F z0 + q Sy3 V x ⁄( V ref ) + q Sy4 V x ⁄( V ref ) }

(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:

M y = R 0 ( S Vx + K x ⋅S Hx )

(90)

Rolling Resistance Coefficients

Name:


Explanation:

qsy1

QSY1

Rolling resistance moment coefficient

qsy2

QSY2

Rolling resistance moment depending on Fx

qsy3

QSY3

Rolling resistance moment depending on speed

qsy4

QSY4

Rolling resistance moment depending on speed^4

Vref

LONGVL

Measurement speed

Combined Slip with friction ellipse In case the tire property file does not contain the coefficients for the 'standard' combined slip method (cosine 'weighing functions), the friction ellipse method is used, as described in this section. Note that the method employed here is not part of one of the Magic Formula publications by Pacejka, but is an inhouse development of MSC.Software.

S Vx κ c = κ + S Hx + --------Kx

55 Tire Models

S Vy α c = α + S Hy + --------Ky α ∗ = sin ( α c ) κc   β = acos -------------------------  κ 2 + α ∗ 2 c The following friction coefficients are defined:

F x, 0 – S Vx µx, act = ------------------------Fz Dx µx, max = -----Fz

F y, 0 – S Vy µy, act = ------------------------Fz

Dy µy, max = -----Fz

1 µx = --------------------------------------------------------1 2 --------------tan β -2 ------------+ µx, act µy, max tan β µy = --------------------------------------------------------1 -2 ------------tan β 2 --------------+ µx, max µy, act The forces corrected for the combined slip conditions are:

µx F x = ------------- F x, 0 µx, act

µy F y = ------------- F y, 0 µy, act

56 Adams/Tire

For aligning moment Mx, rolling resistance My and aligning moment Mz the formulae (76) until and including (85) are used with

S Vyκ =0.

Transient Behavior in PAC2002 The previous Magic Formula equations are valid for steady-state tire behavior. When driving, however, the tire requires some response time on changes of the inputs. In tire modeling terminology, the lowfrequency 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

57 Tire Models

For calculating the lateral deflection v1 of the string in the first point of contact with the road, the following differential equation is valid:

v1 1 dv 1 ---------+ - = tan ( α ) + aφ Vx d t σα

(91)

with the relaxation length σ α in the lateral direction. The turnslip φ 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:

σα


(92)

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:

σx

du 1 + V x u 1 = σ x V sx dt

(93)

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

σ x = F z ⋅( p Tx1 + p Tx2 df z ) ⋅ exp ( p Tx3 df z ) ⋅( R 0 ⁄F z0 )λ

Fz   σ α = p Ty1 F z0 sin 2arc tan ---------------------------------  ⋅( 1 – p Ky3 γy ) ⋅R 0 λ ( p Ty2 F z0 λ Fz0 )  Now the practical slip quantities,

(94)

σx

F z0

⋅λ

σα

(95)

κ and α , are defined based on the tire deformation:

u1 κ' = ------ ⋅ sin ( V x ) σx

(96)

v1 α' = atan ------ σ α 

(97)

Using these practical slip quantities, κ and α , the Magic Formula equations can be used to calculate the tire-road interaction forces and moments:

F x = F x ( α', κ', F z )

(98)

58 Adams/Tire

F y = F y ( α', κ', γ, F z )

(99)

M' z = M' z ( α', κ', γ, F z )

(100)

Coefficients and Transient Response

Name:


Explanation:

pTx1

PTX1

Longitudinal relaxation length at Fznom

pTx2

PTX2

Variation of longitudinal relaxation length with load

pTx3

PTX3

Variation of longitudinal relaxation length with exponent of load

pTy1

PTY1

Peak value of relaxation length for lateral direction

pTy2

PTY2

Shape factor for lateral relaxation length

qTz1

QTZ1

Gyroscopic moment constant

Mbelt

MBELT

Belt mass of the wheel

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:

· m c ( V· cx – V cy ψ c ) + k x u· + c x u = F x · m c ( V· cy – V cx ψ c ) + k y u· + c y u = F y ·· · Jc ψ c + kψ β + cψ β = Mz The contact patch body with mass mc and inertia Jc is connected to the wheel through springs cx, cy, and c ψ and dampers kx, ky, and k ψ in longitudinal, lateral, and yaw direction, respectively. The additional equations for the longitudinal u, lateral v, and yaw

u· = V cx – V sx

β deflections are:

59 Tire Models

v· = V cy – V sy · · β = ψc–ψ · ψ c are the sliding velocity of the contact body in longitudinal, lateral, and yaw · directions, respectively. Vsx, Vsy, and ψ are the corresponding velocities of the lower part of the wheel. in which Vcx, Vcy and

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

σ c d α' + V x α' = V cy – V x β + V x βst dt σc

dα' t + V x α' t = V x α' dt

σc

dϕ ' c · + Vx ϕ 'c = ψ γ dt

σ F2 σϕ σϕ

dϕ ' F2 · + V x ϕ ' cF2 = ψ γ dt dϕ ' 1

1dt

dϕ ' 2 2dt

· + Vx ϕ '1 = ψ γ · + Vx ϕ '2 = ψ γ

where the calculated deflection angle has been used:

Mz β st = ------cφ The tire total spin velocity is:

· ψ γ = ψ c – ( 1 – εγ)Ωsin γ With the transient slip equations, the composite transient turn-slip quantities are calculated:

ϕ ' F = 2ϕ ' c – ϕ ' F2 ϕ ' M = εϕ ϕ ' c + εϕ

12 ( ϕ ' 1

– ϕ '2 )

60 Adams/Tire

The tire forces are calculated with

ϕ ' F and the tire moments with ϕ ' M .

The relaxation lengths are reduced with slip:

σ c = a ⋅( 1 – θζ ) t0 σ 2 = ---- σ c a σ F2 = b F2 σ c σϕ

1

= bϕ 1 σ c

σϕ

2

= bϕ 2 σ c

Here a is half the contact length according to:

 z ρ  a = p A1 R 0 ρ ------ + p A2 -----z- R 0  R0 The composite tire parameter reads:

K y0 θ = --------------2µy F x and the equivalent slip:

1 ζ = ------------- { α' + aεϕ 1 + κ'

2

K x02   ---------  κ' + 2 --- b ϕ ' c  ϕ '1 – ϕ '2 } +  K y0 3   2

12


Name:


Explanation:

mc

MC

Contact body mass

Ic

IC

Contact body moment of inertia

kx

KX

Longitudinal damping

ky

KY

Lateral damping

kϕ

KP

Yaw damping

cx

CX

Longitudinal stiffness

61 Tire Models

Name:


Explanation:

cy

CY

Lateral stiffness

cϕ

CP

Yaw stiffness

pA1

PA1

Half contact length with vertical tire deflection

pA2

PA2

Half contact length with square root of vertical tire deflection

εϕ εϕ

EP

Composite turn-slip (moment)

EP12

Composite turn-slip (moment) increment

bF2

BF2

Second relaxation length factor

bϕ 1

BP1

First moment relaxation length factor

bϕ 2

BP2

Second moment relaxation length factor

12

The remaining contact mass model parameters are estimated automatically based on longitudinal and lateral stiffness specified in the tire property file.

Gyroscopic Couple in PAC2002 When having fast rotations about the vertical axis in the wheel plane, the inertia of the tire belt may lead to gyroscopic effects. To cope with this additional moment, the following contribution is added to the total aligning moment:

M z, gyr = c gyr m belt V rl

dv cos [ arc tan ( B r α r, eq ) ] dt

(101)

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

c gyr = q Tz1 ⋅λ

gyr

(102)

and:

cos [ arc tan ( B r α r, eq ) ] = 1

(103)

The total aligning moment now becomes:

M z = M' z + M z, gyr

(104)

62 Adams/Tire


Name:


Explanation:

pTx1

PTX1


pTx2

PTX2


pTx3

PTX3


pTy1

PTY1


pTy2

PTY2


qTz1

QTZ1


Mbelt

MBELT


Left and Right Side Tires In general, a tire produces a lateral force and aligning moment at zero slip angle due to the tire construction, known as conicity and plysteer. In addition, the tire characteristics cannot be symmetric for positive and negative slip angles. A tire property file with the parameters for the model results from testing with a tire that is mounted in a tire test bench comparable either to the left or the right side of a vehicle. If these coefficients are used for both the left and the right side of the vehicle model, the vehicle does not drive straight at zero steering wheel angle. The latest versions of tire property files contain a keyword TYRESIDE in the [MODEL] section that indicates for which side of the vehicle the tire parameters in that file are valid (TIRESIDE = 'LEFT' or TIRESIDE = 'RIGHT'). If this keyword is available, Adams/Car corrects for the conicity 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

63 Tire Models

USE_MODES of PAC2002: from Simple to Complex The 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.

64 Adams/Tire

USE_MODE Values of PAC2002 and Related Tire Model Output

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


11

Transient


Fx, 0, Fz, 0, My, 0

12

Transient



13

Transient



14

Transient

Combined slip


15

Transient

Combined slip and


21

Advanced transient


Fx, 0, Fz, My, 0

22

Advanced transient

Pure lateral (cornering slip)


23

Advanced transient



24

Advanced transient

Combined slip


25

Advanced transient

Combined slip and


Quality Checks for the Tire Model Parameters Because 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

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.

65 Tire 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 = 0.01 QSY2 = 0 QSY3 = 0 QSY4 = 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.

66 Adams/Tire

Checklist for PAC2002 Parameters and Properties Parameter/property:

Requirement:

Explanation:

LONGVL

1 m/s

Reference velocity at which parameters are measured

VXLOW

Approximately 1 m/s

Threshold for scaling down forces and moments

Dx

>0

Peak friction (see equation (22))

pDx1/pDx2

<0

Peak friction Fx must decrease with increasing load

Kx

>0

Long slip stiffness (see equation (25))

Dy

>0


pDy1/pDy2

<0


Ky

<0

Cornering stiffness (see equation (37))

qsy1

>0

Rolling resistance, in the range of 0.005 - 0.015

Validity Range of the Tire Model Input In the tire property file, a range of the input variables has been given in which the tire properties are supposed to be valid. These validity range parameters are (the listed values can be different): $-----------------------------------------------------long_slip_range [LONG_SLIP_RANGE] KPUMIN = -1.5 $Minimum valid wheel slip KPUMAX = 1.5 $Maximum valid wheel slip $----------------------------------------------------slip_angle_range [SLIP_ANGLE_RANGE] ALPMIN = -1.5708 $Minimum valid slip angle ALPMAX = 1.5708 $Maximum valid slip angle $----------------------------------------------inclination_slip_range [INCLINATION_ANGLE_RANGE] CAMMIN = -0.26181 $Minimum valid camber angle CAMMAX = 0.26181 $Maximum valid camber angle $------------------------------------------------vertical_force_range

67 Tire 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 PAC2002 Because all Adams products use the Standard Tire Interface (STI) for linking the tire models to Adams/Solver, below is a brief background of the STI history (see 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

68 Adams/Tire

General General Definitions Term:

Definition:

Road tangent plane

Plane with the normal unit vector (tangent to the road) in the tire-road contact point C.

C-axis system

Coordinate system mounted on the wheel carrier at the wheel center according to TYDEX, ISO orientation.

Wheel plane

The plane in the wheel center that is formed by the wheel when considered a rigid disc with zero width.

Contact point C

Contact point between tire and road, defined as the intersection of the wheel plane and the projection of the wheel axis onto the road plane.

W-axis system

Coordinate system at the tire contact point C, according to TYDEX, ISO orientation.

Tire Kinematics Tire Kinematics Definitions Parameter:

Definition:

Units:

R0


[m]

R

Loaded tire radius

[m]

Re

Effective tire radius

[m]

ρ ρd ρ Fz0

Radial tire deflection

[m]

Dimensionless radial tire deflection

[-]

Radial tire deflection at nominal load

[m]

mbelt

Tire belt mass

[kg]

ω

Rotational velocity of the wheel

[rads-1]

Slip Quantities Slip Quantities Definitions Parameter:

Definition:

Units:

V

Vehicle speed

[ms-1]

Vsx

Slip speed in x direction

[ms-1]

Vsy

Slip speed in y direction

[ms-1]

Vs

Resulting slip speed

[ms-1]

Vx

Rolling speed in x direction

[ms-1]

69 Tire Models

Parameter:

Definition:

Units:

Vy

Lateral speed of tire contact center

[ms-1]

Vr

Linear speed of rolling

[ms-1]

κ α γ

Longitudinal slip

[-]

Slip angle

[rad]

Inclination angle

[rad]

Forces and Moments Force and Moment Definitions Abbreviation:

Definition:

Units:

Fz

Vertical wheel load

[N]

Fz0

Nominal load

[N]

dfz

Dimensionless vertical load

[-]

Fx

Longitudinal force

[N]

Fy

Lateral force

[N]

Mx

Overturning moment

[Nm]

My

Braking/driving moment

[Nm]

Mz

Aligning moment

[Nm]

References 1. H.B. Pacejka, Tyre and Vehicle Dynamics, 2002, Butterworth-Heinemann, ISBN 0 7506 5141 5. 2. H.-J. Unrau, J. Zamow, TYDEX-Format, Description and Reference Manual, Release 1.1, Initiated by the International Tire Working Group, July 1995. 3. A. Riedel, Standard Tire Interface, Release 1.2, Initiated by the Tire Workgroup, June 1995. 4. J.J.M. van Oosten, H.-J. Unrau, G. Riedel, E. Bakker, TYDEX Workshop: Standardisation of Data Exchange in Tyre Testing and Tyre Modelling, Proceedings of the 2nd International Colloquium on Tyre Models for Vehicle Dynamics Analysis, Vehicle System Dynamics, Volume 27, Swets & Zeitlinger, Amsterdam/Lisse, 1996.

Example of PAC2002 Tire Property File [MDI_HEADER] FILE_TYPE ='tir' FILE_VERSION =3.0 FILE_FORMAT

70 Adams/Tire

='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]

71 Tire 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 speed TYRESIDE = '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 tyre ASPECT_RATIO = 0.6 $Nominal aspect ratio RIM_RADIUS = 0.19 $Nominal rim radius RIM_WIDTH = 0.16 $Rim width $---------------------------------------------------------------shape [SHAPE] {radial width} 1.0 0.0 1.0 0.4 1.0 0.9 0.9

72 Adams/Tire

1.0 $-----------------------------------------------------------parameter [VERTICAL] VERTICAL_STIFFNESS = 2.1e+005 $Tyre vertical stiffness VERTICAL_DAMPING = 50 $Tyre vertical damping BREFF = 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 fz} 0.000 0.0 0.001 212.0 0.002 428.0 0.003 648.0 0.005 1100.0 0.010 2300.0 0.020 5000.0 0.030 8100.0 $-----------------------------------------------------long_slip_range [LONG_SLIP_RANGE] KPUMIN = -1.5 $Minimum valid wheel slip KPUMAX = 1.5 $Maximum valid wheel slip

73 Tire Models

$----------------------------------------------------slip_angle_range [SLIP_ANGLE_RANGE] ALPMIN = -1.5708 $Minimum valid slip angle ALPMAX = 1.5708 $Maximum valid slip angle $----------------------------------------------inclination_slip_range [INCLINATION_ANGLE_RANGE] CAMMIN = -0.26181 $Minimum valid camber angle CAMMAX = 0.26181 $Maximum valid camber angle $------------------------------------------------vertical_force_range [VERTICAL_FORCE_RANGE] FZMIN = 225 $Minimum allowed wheel load FZMAX = 10125 $Maximum allowed wheel load $-------------------------------------------------------------scaling [SCALING_COEFFICIENTS] LFZO = 1 $Scale factor of nominal (rated) 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

74 Adams/Tire

LCY = 1 $Scale factor of Fy shape factor LMUY = 1 $Scale factor of Fy peak friction coefficient LEY = 1 $Scale factor of Fy curvature factor LKY = 1 $Scale factor of Fy cornering stiffness LHY = 1 $Scale factor of Fy horizontal shift LVY = 1 $Scale factor of Fy vertical shift LGAY = 1 $Scale factor of camber for Fy LTR = 1 $Scale factor of Peak of pneumatic trail LRES = 1 $Scale factor for offset of residual torque LGAZ = 1 $Scale factor of camber for Mz LXAL = 1 $Scale factor of alpha influence on Fx LYKA = 1 $Scale factor of alpha influence on Fx LVYKA = 1 $Scale factor of kappa induced Fy LS = 1 $Scale factor of Moment arm of FxL SGKP = 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

75 Tire Models

LVMX = 1 $Scale factor of Mx vertical shiftL MY = 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 load PDX3 = 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 squared PEX4 = -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 Fznom PHX2 = 0.0004318 $Variation of shift Shx with load PVX1 = -8.8098e-006 $Vertical shift Svx/Fz at Fznom PVX2 = 1.862e-005 $Variation of shift Svx/Fz with load RBX1

76 Adams/Tire

= 13.276 $Slope factor for combined slip Fx reduction RBX2 = -13.778 $Variation of slope Fx reduction with kappa RCX1 = 1.2568 $Shape factor for combined slip Fx reduction REX1 = 0.65225 $Curvature factor of combined Fx REX2 = -0.24948 $Curvature factor of combined Fx with load RHX1 = 0.0050722 $Shift factor for combined slip Fx reduction PTX1 = 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 couple QSX3 = 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 load PDY3 = -2.8821 $Variation of friction Muy with squared camber PEY1

77 Tire Models

= -0.0074722 $Lateral curvature Efy at Fznom PEY2 = -0.0063208 $Variation of curvature Efy with load PEY3 = -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 camber PHY1 = 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 camber PVY1 = 0.037318 $Vertical shift in Svy/Fz at Fznom PVY2 = -0.010049 $Variation of shift Svy/Fz with load PVY3 = -0.32931 $Variation of shift Svy/Fz with camber PVY4 = -0.69553 $Variation of shift Svy/Fz with camber and load RBY1 = 7.1433 $Slope factor for combined Fy reduction RBY2 = 9.1916 $Variation of slope Fy reduction with alpha RBY3 = -0.027856 $Shift term for alpha in slope Fy reduction RCY1 = 1.0719 $Shape factor for combined Fy reduction REY1 = -0.27572 $Curvature factor of combined Fy REY2

78 Adams/Tire

= 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 load RVY3 = -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 Fx QSY3 = 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

79 Tire Models

QBZ3 = -0.52041 $Variation of slope Bpt with load squared QBZ4 = 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 Mzr QCZ1 = 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 load QDZ3 = -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 load QDZ8 = -0.18175 $Variation of peak factor Dmr" with camber QDZ9 = 0.029952 $Var. of peak factor Dmr" with camber and load QEZ1 = -1.5697 $Trail curvature Ept at Fznom QEZ2 = 0.33394 $Variation of curvature Ept with load QEZ3 = 0 $Variation of curvature Ept with load squared QEZ4 = 0.26711 $Variation of curvature Ept with sign of Alpha-t QEZ5 = -3.594

80 Adams/Tire

$Variation of Ept with camber and sign Alpha-t QHZ1 = 0.0047326 $Trail horizontal shift Sht at Fznom QHZ2 = 0.0026687 $Variation of shift Sht with load QHZ3 = 0.11998 $Variation of shift Sht with camber QHZ4 = 0.059083 $Variation of shift Sht with camber and load SSZ1 = 0.033372 $Nominal value of s/R0: effect of Fx on Mz SSZ2 = 0.0043624 $Variation of distance s/R0 with Fy/Fznom SSZ3 = 0.56742 $Variation of distance s/R0 with camber SSZ4 = -0.24116 $Variation of distance s/R0 with load and camber QTZ1 = 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

81 Tire 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

82 Adams/Tire

= 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

83 Tire 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 Fx QFCY1 = 0.3 $Tire stiffness interaction with Fy QFCG1 = 0.0 $Tire stiffness interaction with camber QFZ1 = 0.0 $Linear stiffness coefficient, if zero, VERTICAL_STIFFNESS is taken QFZ2 = 14.35 $Tire vertical stiffness coefficient (quadratic)

Contact Methods The PAC2002 model supports the following roads: • 2D Roads, see Using the 2D Road Model • 3D Roads, see Adams/3D 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.

84 Adams/Tire

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

85 Tire Models

Using the PAC-TIME Tire Model The PAC-TIME Magic-Formula tire model has been developed by MSC.Software according to a publication, A New Tyre Model for TIME Measurement Data, by J.J.M. van Oosten e.a. [5]. PAC-TIME has improved equations for side force and aligning moment under pure slip conditions. For longitudinal pure slip and combined slip, the tire model is similar to PAC-TIME. Learn about: • When to Use PAC-TIME • 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 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-TIME Magic-Formula (MF) tire models are considered the state-of-the-art for modeling tire-road interaction forces in vehicle dynamics applications. Since 1987, Pacejka and others have published several versions of this type of tire model. The PAC-TIME model is similar to PAC2002, but has improved equations for side force (Fy) and aligning moment (Mz) under pure side slip conditions. The following is background information about the PAC-TIME tire model, as stated in the paper, A New Tyre Model for TIME Measurement Data, J.J.M. van Oosten, E. Kuiper, G. Leister, D. Bode, H. Schindler, J. Tischleder, S. Köhne [5]: In 1999 a new method for tyre Force and Moment (F&M) testing has been developed by a consortium of European tyre and vehicle manufacturers: the TIME procedure. For Vehicle Dynamics studies often a Magic Formula (MF) tyre model is used based upon such F&M data. However when calculating MF parameters for a standard MF model out of the TIME F&M data, several difficulties are observed. These are mainly due to the non-uniform distribution of the data points over the slip angle, camber and load area and the mutual dependency in between the slip angle, camber and load. A new MF model for pure cornering slip conditions has been developed that allows the calculation of the MF parameters despite of the dependency of the three input variables in the F&M data and shows better agreement with the

86 Adams/Tire

measured F&M data points. From mathematical point of view the optimisation process for deriving MF parameters is better conditioned with the new MF-TIME, resulting in less sensitivity to starting values and better convergence to a global minimum. In addition the MF-TIME has improved extrapolation performance compared to the standard MF models for areas where no F&M data points are available. Next to the use for TIME F&M data, the new model is expected to have interesting prospects for converting ‘on-vehicle’ measured tyre data into a robust set of MF parameters. In general, an MF tire model describes the tire behavior for rather smooth roads (road obstacle wavelengths longer than the tire radius) up to frequencies of 8 Hz. This makes the tire model applicable for all generic vehicle handling and stability simulations, including: • Steady-state cornering • Single- or double-lane change • Braking or power-off in a turn • Split-mu braking tests • J-turn or other turning maneuvers • ABS braking, when stopping distance is important (not for tuning ABS control strategies) • Other common vehicle dynamics maneuvers on rather smooth roads (wavelength of road

obstacles must be longer than the tire radius) For modeling roll-over of a vehicle, you must pay special attention to the overturning moment characteristics of the tire (Mx), and the loaded radius modeling. The last item may not be sufficiently addressed in this model. The PAC-TIME model has been developed for car tires with camber (inclination) angles to the road not exceeding 15 degrees.

Modeling of Tire-Road Interaction Forces For vehicle dynamics applications, an accurate knowledge of tire-road interaction forces is inevitable because the movements of a vehicle primarily depend on the road forces on the tires. These interaction forces depend on both road and tire properties, and the motion of the tire with respect to the road. In the radial direction, the MF tire models consider the tire to behave as a parallel linear spring and linear damper with one point of contact with the road surface. The contact point is determined by considering the tire and wheel as a rigid disc. In the contact point between the tire and the road, the contact forces in longitudinal and lateral direction strongly depend on the slip between the tire patch elements and the road. The figure, Input and Output Variables of the Magic Formula Tire Model, presents the input and output vectors of the PAC2002 tire model. The tire model subroutine is linked to the Adams/Solver through the Standard Tire Interface (STI) [3]. The input through the STI consists of: • Position and velocities of the wheel center • Orientation of the wheel • Tire model (MF) parameters

87 Tire Models

• 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 ( κ , α ), and inclination angle ( γ) 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

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].

88 Adams/Tire

The STI interface between the MF-TIME model and Adams/Solver mainly passes information to the tire model in the C-axis coordinate system. In the tire model itself, a conversion is made to the W-axis system because all the modeling of the tire behavior, as described in this help, assumes to deal with the slip quantities, orientation, forces, and moments in the contact point with the TYDEX W-axis system. Both axis systems have the ISO orientation but have different origin as can be seen in the figure below. TYDEX C- and W-Axis Systems Used in PAC-TIME , Source [2]

The C-axis system is fixed to the wheel carrier with the longitudinal xc-axis parallel to the road and in the wheel plane (xc-zc-plane). The origin of the C-axis system is the wheel center. The origin of the W-axis system is the road contact-point defined by the intersection of the wheel plane, the plane through the wheel carrier, and the road tangent plane. The forces and moments calculated by PAC-TIME using the MF equations in this guide are in the W-axis system. A transformation is made in the source code to return the forces and moments through the STI to Adams/Solver. The inclination angle is defined as the angle between the wheel plane and the normal to the road tangent plane (xw-yw-plane). Units The units of information transferred through the STI between Adams/Solver and PAC-TIME are according to the SI unit system. Also, the equations for PAC-TIME described in this guide have been developed for use with SI units, although you can easily switch to another unit system in your tire property file. Because of the non-dimensional parameters, only a few parameters have to be changed. However, the parameters in the tire property file must always be valid for the TYDEX W-axis system (ISO oriented). The basic SI units are listed in the table below.

89 Tire Models

SI Units Used in PAC-TIME Variable type: Angle

Force

Moment

Name:

Abbreviation:

Slip angle

α

Inclination angle

γ

Longitudinal force

Fx

Lateral force

Fy

Vertical load

Fz

Overturning moment

Mx

Unit: Radians

Newton

Newton.meter

Rolling resistance moment My

Speed


Mz

Longitudinal speed

Vx

Lateral speed

Vy


Vsx

Lateral slip speed

Vsy

Rotational speed Tire rolling speed

ω

Meters per second

Radians per second

Definition of Tire Slip Quantities Slip Quantities at Combined Cornering and Braking/Traction

The longitudinal slip velocity Vsx in the contact point (W-axis system, see Slip Quantities at Combined Cornering and Braking/Traction) is defined using the longitudinal speed Vx, the wheel rotational velocity

ω , and the effective rolling radius Re:

90 Adams/Tire

V sx = V x – ΩR e

(105)


V sy = V y The practical slip quantities κ (longitudinal slip) and velocities in the contact point with:

(106)


V sx κ = – -------Vx

(107)

V sy tan α = --------Vx

(108)


Vr = Re Ω

(109)

Contact Point and Normal Load Calculation • Contact Point • Loaded and Effective Tire Rolling Radius

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

91 Tire Models

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 though the wheel spin axis. The normal load Fz of the tire is calculated with:

· F z = C z ρ + K z ⋅ρ where

(110)

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

Instead of the linear vertical tire stiffness Cz, you can also define an arbitrary tire deflection - load curve in the tire property file in the section [DEFLECTION_LOAD_CURVE] (see the Example of PAC-TIME Tire Property File). If a section called [DEFLECTION_LOAD_CURVE] exists, the load deflection data points with a cubic spline for inter- and extrapolation are used for the calculation of the vertical force of the tire. Note that you must specify Cz in the tire property file, but it does not play any role. Loaded and Effective Tire Rolling Radius With the loaded rolling tire radius R defined as the distance of the wheel center to the contact point of the tire with the road, where the loaded tire radius Rl is:

R1 = R0 – ρ

ρ is the deflection of the tire, and R0 is the free (unloaded) tire radius, then (111)

In this tire model, a constant (linear) vertical tire stiffness Cz is assumed; therefore, the tire deflection can be calculated using:

ρ

92 Adams/Tire

Fz ρ = -----Cz

(112)


Vx R e = -----Ω

(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 decrease with increasing vertical load due to the tire tread thickness. See the figure below. Effective Rolling Radius and Longitudinal Slip

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

d

R e = R 0 – ρ Fz0 ( Darc tan ( Bρ ) + Fρ )

(114)

93 Tire Models

in which


F z0 ρ Fz0 = -------Cz and

(115)

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

ρ ρ d = --------ρ Fz0

(116)

Effective Rolling Radius and Longitudinal Slip

Normal Load and Rolling Radius Parameters

Name:


Explanation:

Fz0

FNOMIN

Nominal wheel load

Ro

UNLOADED_RADIUS

Free tire radius

B

BREFF


D

DREFF


94 Adams/Tire

Name:


Explanation:

F

FREFF


Cz

VERTICAL_STIFFNESS Tire vertical stiffness

Kz

VERTICAL_DAMPING


Basics of the Magic Formula in PAC-TIME The Magic Formula is a mathematical formula that is capable of describing the basic tire characteristics for the interaction forces between the tire and the road under several steady-state operating conditions. We distinguish: • Pure cornering slip conditions: cornering with a free rolling tire • Pure longitudinal slip conditions: braking or driving the tire without cornering • Combined slip conditions: cornering and longitudinal slip simultaneously

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


longitudinal force Fx as a function of longitudinal slip κ , have a similar shape. Because of the sine arctangent combination, the basic Magic Formula example is capable of describing this shape:

Y ( x ) = D cos [ Carc tan { Bx – E ( Bx – arc tan ( Bx ) ) } ] where Y(x) is either Fx with x the longitudinal slip

κ , or Fy and x the lateral slip α .

Characteristic Curves for Fx and Fy Under Pure Slip Conditions

(117)

95 Tire 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, called pneumatic trail t, from the contact point. Because the pneumatic trail t as a function of the lateral slip

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

Y ( x ) = D cos [ Carc tan { Bx – E ( Bx – arc tan ( Bx ) ) } ] in which Y(x) is the pneumatic trail t as function of slip angle

(118)

α.

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



96 Adams/Tire

In combined slip conditions, the lateral force Fy will decrease due to longitudinal slip or the opposite, the longitudinal force Fx will decrease due to lateral slip. The forces and moments in combined slip conditions are based on the pure slip characteristics multiplied by the so-called weighting functions. Again, these weighting functions have a cosine-shaped MF examples. The Magic Formula itself only describes steady-state tire behavior. For transient tire behavior (up to 8 Hz), the MF output is used in a stretched string model that considers tire belt deflections instead of slip velocities to cope with standstill situations (zero speed). Inclination Effects in the Lateral Force From a historical point of view, the camber stiffness always has been modeled implicit in the Magic Formulas. For deriving coefficients of a Pacejka tire model usually so-called tire tests with slip angle sweeps at various values of constant load and inclination are performed. In the resulting Force & Moment measurement data, the effects of camber on the side force Fy are relatively small compared to side force

97 Tire 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

γ as follows:

F y0 = D y cos [ C y arc tan { B y α y – E y ( B y α y – arc tan ( B y α y ) ) } + C γarc tan { B y α γ – E γ( B γα γ – arc tan ( B γα γ) ) } ]

(119)

In the PAC-TIME tire model, C γ has been set to ½, and E γ is not used (zero value). This approach results in an explicit definition of the camber stiffness, because:

F yo K γ = B γC γD γ = ∂ -------- at α γ = 0 δγ

(120)

Input Variables The input variables to the Magic Formula are: Input Variables

[rad]

Inclination angle

κ α γ

Normal wheel load

Fz

[N]

Longitudinal slip Slip angle

[-] [rad]

Output Variables Its output variables are: Output Variables. Longitudinal force

Fx

[N]

Lateral force

Fy

[N]

98 Adams/Tire

Overturning couple

Mx

[Nm]


My

[Nm]

Aligning moment

Mz

[Nm]

The output variables are defined in the W-axis system of TYDEX. Basic Tire Parameters All tire model parameters of the model are without dimension. The reference parameters for the model are: Basic Tire Parameters Nominal (rated) load

Fz0

[N]


R0

[m]

Tire belt mass

mbelt

[kg]


F z – F' z0 df z = -------------------F' z0

(121)

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

F' z0 = F z0 ⋅λ

λ Fz0): (122)

F z0


Definition:

p

Force at pure slip

q

Moment at pure slip

r


s


99 Tire Models

Parameter: i=

j=

k=

Definition:

B

Stiffness factor

C

Shape factor

D

Peak value

E

Curvature factor

K


H

Horizontal shift

V

Vertical shift

s


t


x


y


z


1, 2, ...

User Scaling Factors A set of scaling factors is available to easily examine the influence of changing tire properties without the need to change one of the real Magic Formula coefficients. The default value of these factors is 1. You can change the factors in the tire property file. The peak friction scaling factors, factors,

λ µξ and

λ γψ ', are also used for the position-dependent friction in 3D Road Contact and Adams/3D Road. An overview of all scaling factors is shown in the next tables. Scaling Factor Coefficients for Pure Slip

Name:

λ λ λ λ λ λ λ λ λ λ


Explanation:

Fzo

LFZO


Cx

LCX


µξ

LMUX


Ex

LEX


Kx

LKX


Vx

LVX


Hx

LHX

Scale factor of Fx horizontal shift

γxx

LGAX

Scale factor of camber for Fx

Cy

LCY

Scale factor of Fy shape factor for side slip

µy

LMUY


100 Adams/Tire

Name:

λ λ λ λ λ λ λ λ λ λ λ λ


Explanation:

EyEy

LEY


KyKy

LKY


VVy

LVY

Scale factor of Fy vertical shift

HyHyy

LHY


K γK γ

LKC

Scale factor of camber stiffness (K-factor)

γψ

LGAY

Scale factor of camber force stiffness

t

LTR


Mr

LRES

Scale factor for offset of residual torque

γz

LGAZ

Scale factor of camber torque stiffness

Mx

LMX


VMx

LVMX


My

LMY

Scale factor of rolling resistance torque


Name:

λ λ λ λ


Explanation:

xα

LXAL


yκ

LYKA


Vy κ

LVYKA

Scale factor of kappa induced Fy

s

LS

Scale factor of moment arm of Fx

Scaling Factor Coefficients for Transient Response

Name:

λ σ κκ λ σαα λ gyr


Explanation:

LSGKP


LSGAL


LGYR


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

101 Tire Models

Steady-State Pure Slip • Longitudinal Force at Pure Slip • Lateral Force at Pure Slip • Aligning Moment at Pure Slip


F x = F x0 ( κ , F z, γ)

(123)

F x0 = D x cos [ C x arc tan { B x κ x – E x ( B x κ x – arc tan ( B x κ s ) ) } ] + S Vx

(124)

κ x = κ + S Hx

(125)

γx = γ ⋅λ

(126)

γx


C x = p Cx1 ⋅λ

(127)

cx

D x = µx ⋅F z

(128)

µx = ( p Dx1 + p Dx2 df z ) ⋅1 – ( p Dx3 ⋅γx2 )γµx

(129)

E x = ( p Ex1 + p Ex2 df z + p Ex2 df z2 ) ⋅{ 1 – p Ex4 sgn ( κ x ) } ⋅λ

Ex

with E x ≤ 1

(130)

the longitudinal slip stiffness:

K x = F z ( p Kx1 + p Kx2 df z ) ⋅ exp ( p Kx3 df z )λ Kx = Bx Cx Dx =

K

(131)

∂ F x0 at κ x = 0 ∂ κx

B x = K x ⁄( C x D x ) S Hx = ( p Hx1 + p Hx2 df z )λ

(132) (133)

Hx

S Vx = F z ⋅( p Vx1 + p Vx2 df z ) ⋅λ

Vx

⋅λ

µx

102 Adams/Tire

Longitudinal Force Coefficients at Pure Slip Name used in tire property file:

Name:

Explanation:

pCx1

PCX1


pDx1

PDX1


pDx2

PDX2


pDx3

PDX3

Variation of friction Mux with inclination

pEx1

PEX1


pEx2

PEX2


pEx3

PEX3


pEx4

PEX4


pKx1

PKX1


pKx2

PKX2



F y = F y0 ( α , γ, F z )

(134)

1 F y0 = D y sin C y arc tan { B y α y – E y ( B y α y – arc tan ( B y α y ) ) } + --- arc tan ( B γγy ) (135) + S Vy 2 α y = α + S Hy

(136)


γy = γ ⋅λ

(137)

γy

with coefficients:

C y = p Cy1 ⋅λ

(138)

Cy

D y = µy ⋅F z µy = ( p Dy1 + p Dy2 df z ) ⋅( 1 – p Dy3 γy2 ) ⋅λ

(139)

E y = { p Ey1 + p Ey2 df z + ( p Ey3 + p Ey4 γy ) sgn ( α y ) } ⋅λ The cornering stiffness:

(140)

µy Ey

(141)

103 Tire Models

K () y =

∂ Fy Fz   = p Ky1 F z0 sin p Ky4 arc tan -------------------------------  ⋅( 1 – p Ky3 γy2 ) ⋅λ ∂α p Ky2 F z0 λ Fz0 

Ky λ Fz0 (142)

with p Ky4 ≤ 2 Kγ =

∂ Fy = F z ( p Ky5 + p Ky5 df z ) ⋅λ ∂γ

(143)

Ky

Ky B y = ------------Cy Dy

(144)

2K γ B γ = --------Dy

(145)

S Hy = ( p Hy1 + p Hy2 df z ) ⋅λ

(146)

Hy

S Hy = F z ( p Vy1 + p Vy2 df z ) ⋅λ

Vy λ µy

(147)


Name:


Explanation:

pCy1

PCY1


pDy1

PDY1


pDy2

PDY2

Variation of friction Muy with load

pDy3

PDY3

Variation of friction Muy with squared inclination

pEy1

PEY1


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


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

104 Adams/Tire


Name:

Explanation:

pHy1

PHY1


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 = M z0 ( α , γ, F z )

(148)

M z0 = – t ⋅( F y0 – S Vy ) γ = 0 + Mzr with the pneumatic trail t:


(149)

α t = α + S Ht and the residual moment Mzr:

M zr ( α r ) = D r cos [ arc tan ( B r α r ) ] cos ( α )

(150)

α r = α + S Ht

(151)

S Hr = 0

(152)


γz = γ ⋅λ

(153)

γz

with coefficients:

B t = ( q Bz1 + q Bz2 df z ) ( ( 1 + q Bz4 γz2 + q Bz5 γz ) ⋅λ k ⁄λ C t = q Cz1 D t = F z ( q Dz1 + q Dz2 df z ) ( R 0 ⁄F z0 ) ⋅λ

µy )

(154) (155)

t

(156)

105 Tire Models

  2 E t = ( q Ez1 + q Ez2 df z ) 1 + q Ez4 --- arc tan ( B t C t α t )  π  

(157)

with E t ≤ 1 S Ht = q Hz1 + q Hz2 df z + ( q Hz3 + q Hz4 df z )γz

(158)

Br = λ

(159)

Kz

⁄λ

µy

D r = F z { ( q Dz6 + q Dz7 df z )λ r + ( q Dz8 + q Dz9 df z )γz } R 0 λ

(160)

µy


∂ Mz = – tK y = F z ( q Dz1 + q Dz2 df z ) ( R 0 ⁄F z0 ) ( p Ky5 + p Ky6 df z )F z ⋅λ t λ ∂α

Ky

(161)

and the aligning stiffness for inclination is:

dM z = ( q Dz8 + q Dz9 df z )R 0 F z λ dγ

(162)

µy


Name:


Explanation:

qBz1

QBZ1


qBz2

QBZ2


qBz4

QBZ4

Variation of slope Bpt with inclination

qBz5

QBZ5

Variation of slope Bpt with absolute inclination

qCz1

QCZ1


qDz1

QDZ1


qDz2

QDZ2


qDz6

QDZ6


qDz7

QDZ7


qDz8

QDZ8

Variation of peak factor Dmr with inclination

qDz9

QDZ9

Variation of Dmr with inclination and load

qEz1

QEZ1


106 Adams/Tire

Name:


Explanation:

qEz2

QEZ2


qEz4

QEZ4


qHz1

QHZ1

Trail horizontal shift Sht at Fznom

qHz2

QHZ2

Variation of shift Sht with load

qHz3

QHZ3

Variation of shift Sht with inclination

qHz4

QHZ4

Variation of shift Sht with inclination and load

Steady-State Combined Slip PAC-TIME has two methods for calculating the combined slip forces and moments. If the user supplies the coefficients for the combined slip cosine 'weighing' functions, the combined slip is calculated according to Combined slip with cosine 'weighing' functions (standard method). If no coefficients are supplied, the so-called friction ellipse is used to estimate the combined slip forces and moments, see section Combined Slip with friction ellipse Combined slip with cosine 'weighing' functions • 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


F x = F x0 ⋅G xα ( α , κ , F z )

(163)

with Gx α the weighting function of the longitudinal force for pure slip. We write:


(164)

α s = α + S Hxα

(165)

with coefficients:

B xα = r Bx1 cos [ arc tan { r Bx2 κ } ] ⋅λ C xα = r Cx1

xα

(166) (167)

107 Tire Models


(168)


(169)

S Hxα = r Hx1

(170)


cos [ C xα arc tan { B xα α s – E xα ( B xα α s – arc tan ( B xα α s ) ) } ] G xα = -----------------------------------------------------------------------------------------------------------------------------------------------------------------cos [ C xα arc tan { B xα S Hxα – E xα ( B xα S Hxα – arc tan ( B xα S Hxα ) ) } ]

(171)

Longitudinal Force Coefficients at Combined Slip Name used in tire property file:

Name:

Explanation:

rBx1

RBX1


rBx2

RBX2


rCx1

RCX1


rEx1

REX1


rEx2

REX2


rHx1

RHX1



F y = F y0 ⋅G yκ ( α , κ , γ, F z ) + S Vyκ

(172)

with Gyk the weighting function for the lateral force at pure slip and SVyk the ' κ -induced' side force; therefore, the lateral force can be written as:

F y = D yκ cos [ C yκ arc tan { B yκ κ s – E yκ ( B xα κ s – arc tan ( B yκ κ s ) ) } ] + S Vyκ

(173)

κ s = κ + S Hyκ

(174)


B yκ = r By1 cos [ arc tan { r By2 ( α – r By3 ) } ] ⋅λ C yκ = r Cy1

yκ

(175) (176)

108 Adams/Tire

F yo D yκ = --------------------------------------------------------------------------------------------------------------------------------------------------------------cos [ C yκ arc tan { B yκ S Hyκ – E yκ ( B xα S Hyκ – arc tan ( B yκ S Hyκ ) ) } ]

(177)


(178)


(179)


(180)


cos [ C yκ arc tan { B yκ κ s – E yκ ( B xα κ s – arc tan ( B yκ κ s ) ) } ] G yκ = --------------------------------------------------------------------------------------------------------------------------------------------------------------cos [ C yκ arc tan { B yκ S Hyκ – E yκ ( B xα S Hyκ – arc tan ( B yκ S Hyκ ) ) } ]

(181)


Name:


Explanation:

rBy1

RBY1


rBy2

RBY2


rBy3

RBY3


rCy1

RCY1


rEy1

REY1


rEy2

REY2


rHy1

RHY1


rHy2

RHY2


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)


M' z = – t ⋅F' y + M zr + s ⋅F x

(182)

with:

t = t ( α t, eq )

(183)

109 Tire Models

= D t cos [ C t arc tan { B t α t, eq – E t ( B t α t, eq – arc tan ( B t α t, eq ) ) } ] cos ( α )

(184)

F' y, γ = 0 = F y – S Vyκ

(185)

M zr = M zr ( α r, eq ) = D r cos [ arc tan ( B r α r, eq ) ] cos ( α )

(186)

t = t ( α t, eq ) with the arguments:

K x2 2 2 -----α t, eq = arc tan tan α t +  K y κ ⋅ sgn ( α t )

(187)

Kx 2 2 α r, eq = arc tan tan α r + ------ κ 2 ⋅ sgn ( α r ) K y

(188)

Aligning Moment Coefficients at Combined Slip

Name:


Explanation:

ssz1

SSZ1


ssz2

SSZ2


ssz3

SSZ3


ssz4

SSZ4



 M x = R o ⋅F z ⋅q sx1 λ 

VMx

Fy  – q sx2 + q sx3 ⋅--------  λ F z0 

(189)

Mx


Name:


Explanation:

qsx1

QSX1

Lateral force induced overturning couple

qsx2

QSX2

Inclination induced overturning couple

qsx3

QSX3

Fy induced overturning couple

110 Adams/Tire

Formulas for Rolling Resistance Moment at Pure and Combined Slip The rolling resistance moment is defined by:

M y = R o ⋅F z ⋅{ q Sy1 + q Sy2 F x ⁄F z0 + q Sy3 V x ⁄( V ref ) q Sy4 ( V x ⁄V ref ) 4 }

(190)


Name:


Explanation:

qsy1

QSY1


qsy2

QSY2


qsy3

QSY3


qsy4

QSY4


Vref

LONGVL

Measurement speed


S Vx κ c = κ + S Hx + --------Kx S Vy α c = α + S Hy + --------Ky α ∗ = sin ( α c ) κc   β = acos -------------------------  κ 2 + α ∗ 2 c The following friction coefficients are defined:

F x, 0 – S Vx µx, act = ------------------------Fz Dx µx, max = -----Fz

F y, 0 – S Vy µy, act = ------------------------Fz


111 Tire Models

1 µx = --------------------------------------------------------1 2 --------------tan β -2 ------------+ µx, act µy, max tan β µy = --------------------------------------------------------1 -2 ------------tan β 2 --------------+ µx, max µy, act The forces corrected for the combined slip conditions are:

µx F x = ------------- F x, 0 µx, act

µy F y = ------------- F y, 0 µy, act


S Vyκ =0.

Transient Behavior in PAC-TIME The previous Magic Formula equations are valid for steady-state tire behavior. When driving, however, the tire requires some response time on changes of the inputs. In tire modeling terminology, the lowfrequency 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. Stretched String Model for Transient Tire Behavior

112 Adams/Tire

For calculating the lateral deflection v1 of the string in the first point of contact with the road, the following differential equation is valid:


(191)

with the relaxation length σ α in the lateral direction. The turnslip φ 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:

σα


(192)


113 Tire Models

σx

du 1 + V x u 1 = – σ x V sx dt

(193)



Fz   σ α = p Ty1 F z0 sin p Ky4 arc tan -----------------------------  ⋅( 1 – p Ky3 γ2 ) ⋅R 0 λ p Ty2 F z0 λ Fz0  Now the practical slip quantities,

(194)

σκ

F z0 λ σ α

(195)

κ' and α' , are defined based on the tire deformation:

u1 κ' = ------ sin ( V x ) σx

(196)

v -----1- α' = atan  σ α 

(197)

Using these practical slip quantities, κ' and α' , the Magic Formula equations can be used to calculate the tire-road interaction forces and moments:

F x = F x ( α', κ', F z )

(198)

F y = F y ( α', κ', γ, F z )

(199)

M z = M z ( α', κ', γ, F z )

(200)

Gyroscopic Couple in PAC-TIME When having fast rotations about the vertical axis in the wheel plane, the inertia of the tire belt may lead to gyroscopic effects. To cope with this additional moment, the following contribution is added to the total aligning moment:

M z, gyr = c gyr m belt V r1


(201)

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

c gyr = q Tz1 ⋅λ and:

gyr

(202)

114 Adams/Tire

cos [ arc tan ( B r α r, eq ) ] = 1

(203)


M z = M z' + M z, gyr Coefficients and Transient Response

Name:


Explanation:

pTx1

PTX1


pTx2

PTX2


pTx3

PTX3


pTy1

PTY1


pTy2

PTY2


qTz1

QTZ1


Mbelt

MBELT


Left and Right Side Tires In general, a tire produces a lateral force and aligning moment at zero slip angle due to the tire construction, known as conicity and plysteer. In addition, the tire characteristics cannot be symmetric for positive and negative slip angles. A tire property file with the parameters for the model results from testing with a tire that is mounted in a tire test bench comparable either to the left or the right side of a vehicle. If these coefficients are used for both the left and the right side of the vehicle model, the vehicle does not drive straight at zero steering wheel angle. The latest versions of tire property files contain a keyword TYRESIDE in the [MODEL] section that indicates for which side of the vehicle the tire parameters in that file are valid (TIRESIDE = 'LEFT' or TIRESIDE = 'RIGHT'). If this keyword is available, Adams/Car corrects for the conicity, plysteer, and asymmetry when using a tire property file on the opposite side of the vehicle. In fact, the tire characteristics are mirrored with respect to slip angle zero. In Adams/View, this option can only be used when the tire is generated by the graphical user interface: select Build -> Forces -> Special Force: Tire, as shown in the figure below. Create Wheel and Tire Dialog Box in Adams/View

115 Tire 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 SYMMETRIC, the tire characteristics are changed to symmetric behavior.

USE_MODES of PAC-TIME: from Simple to Complex The parameter USE_MODE in the tire property file allows you to switch the output of the PAC-TIME tire model from very simple (that is, steady-state cornering) to complex (transient combined cornering and braking).

116 Adams/Tire

The options for the USE_MODE and the output of the model have been listed in the table below. USE_MODE Values of PAC-TIME and Related Tire Model Output

USE_MODE:

State:

Slip conditions:

PAC-TIME output (forces and moments):

0

Steady state

Acts as a vertical spring and damper

0, 0, Fz, 0, 0, 0

1

Steady state


Fx, 0, Fz, 0, My, 0

2

Steady state



3

Steady state



4

Steady state

Combined slip


11

Transient


Fx, 0, Fz, 0, My, 0

12

Transient



13

Transient



14

Transient

Combined slip


Quality Checks for the Tire Model Parameters Because PAC-TIME uses an empirical approach to describe tire - road interaction forces, incorrect parameters can easily result in non-realistic tire behavior. Below is a list of the most important items to ensure the quality of the parameters in a tire property file: • Rolling Resistance • Camber (Inclination) Effects • Validity Range of the Tire Model Input

Note:


Rolling Resistance For a realistic rolling resistance, the parameter qsy1 must be positive. For car tires, it can be in the order of 0.006 - 0.01 (0.6% - 1.0%). $---------------------------------------------------rolling resistance [ROLLING_COEFFICIENTS] QSY1 = 0.01 QSY2 = 0 QSY3 = 0

117 Tire Models

QSY4 = 0 Camber (Inclination) Effects Camber stiffness has been explicitly defined in PAC-TIME, so camber stiffness can be easily checked by the tire model parameters itself, see the table, Checklist for PAC-TIME Parameters and Properties, below. For car tires, positive inclination should result in a negative lateral force at zero slip angle (see Effect of Positive Camber on the Lateral Force in TYDEX W-axis (ISO) System below). If positive inclination results in an increase of the lateral force, the coefficient may not be valid for the ISO, but for the SAE coordinate system. Note that PAC-TIME only uses coefficients for the TYDEX W-axis (ISO) system. Effect of Positive Camber on the Lateral Force in TYDEX W-axis (ISO) System

The following table lists further checks on the PAC-TIME parameters. Checklist for PAC-TIME Parameters and Properties Parameter/property:

Requirement:

Explanation:

LONGVL

1 m/s


VXLOW

Approximately 1 m/s Threshold for scaling down forces and moments

Dx

>0


118 Adams/Tire

Parameter/property:

Requirement:

Explanation:

pDx1/pDx2

<0


Kx

>0


Dy

>0


pDy1/pDy2

<0


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

Validity Range of the Tire Model Input In the tire property file, a range of the input variables has been given in which the tire properties are supposed to be valid. These validity range parameters are (the listed values can be different): $-----------------------------------------------------long_slip_range [LONG_SLIP_RANGE] KPUMIN = -1.5 $Minimum valid wheel slip KPUMAX = 1.5 $Maximum valid wheel slip $----------------------------------------------------slip_angle_range [SLIP_ANGLE_RANGE] ALPMIN = -1.5708 $Minimum valid slip angle ALPMAX = 1.5708 $Maximum valid slip angle $----------------------------------------------inclination_slip_range [INCLINATION_ANGLE_RANGE] CAMMIN = -0.26181 $Minimum valid camber angle CAMMAX = 0.26181 $Maximum valid camber angle $------------------------------------------------vertical_force_range [VERTICAL_FORCE_RANGE] FZMIN = 225 $Minimum allowed wheel load FZMAX = 10125

119 Tire Models

$Maximum allowed wheel load If one of the input parameters exceeds a minimum or maximum validity value, the calculation in the tire model is performed with the minimum or maximum value of this range to avoid non-realistic tire behavior. In that case, a message appears warning you that one of the inputs exceeds a validity value.

Standard Tire Interface (STI) for PAC-TIME Because all Adams products use the Standard Tire Interface (STI) for linking the tire models to Adams/Solver, below is a brief background of the STI history (see reference [4]). At the First International Colloquium on Tire Models for Vehicle Dynamics Analysis on October 21-22, 1991, the International Tire Workshop working group was established (TYDEX). The working group concentrated on tire measurements and tire models used for vehicle simulation purposes. For most vehicle dynamics studies, people previously developed their own tire models. Because all car manufacturers and their tire suppliers have the same goal (that is, development of tires to improve dynamic safety of the vehicle), it aimed for standardization in the tire behavior description. In TYDEX, two expert groups, consisting of participants of vehicle industry (passenger cars and trucks), tire manufacturers, other suppliers and research laboratories, had been defined with following goals: • The first expert group's (Tire Measurements - Tire Modeling) main goal was to specify an




120 Adams/Tire


Definition:

Road tangent plane Plane with the normal unit vector (tangent to the road) in the tire-road contact point C. C-axis system


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


W-axis system



Definition:

Units:

R0


[m]

R

Loaded tire radius

[m]

Re


[m]

ρ ρd ρ Fz0


[m]


[-]


[m]

mbelt

Tire belt mass

[kg]

ω


[rads-1]


Definition:

Units:

V

Vehicle speed

[ms-1]

Vsx


[ms-1]

Vsy


[ms-1]

Vs


[ms-1]

Vx

Rolling speed in x direction

[ms-1]

121 Tire Models

Parameter:

Definition:

Units:

Vy


[ms-1]

Vr


[ms-1]

κ α γ

Longitudinal slip

[-]

Slip angle

[rad]

Inclination angle

[rad]


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


[Nm]

Mz

Aligning moment

[Nm]

References 1. H.B. Pacejka, Tyre and Vehicle Dynamics, 2002, Butterworth-Heinemann, ISBN 0 7506 5141 5. 2. H.-J. Unrau, J. Zamow, TYDEX-Format, Description and Reference Manual, Release 1.1, Initiated by the International Tire Working Group, July 1995. 3. A. Riedel, Standard Tire Interface, Release 1.2, Initiated by the Tire Workgroup, June 1995. 4. J.J.M. van Oosten, H.-J. Unrau, G. Riedel, E. Bakker, TYDEX Workshop: Standardisation of Data Exchange in Tyre Testing and Tyre Modelling, Proceedings of the 2nd International Colloquium on Tyre Models for Vehicle Dynamics Analysis, Vehicle System Dynamics, Volume 27, Swets & Zeitlinger, Amsterdam/Lisse, 1996. 5. J.J.M. van Oosten, E. Kuiper, G. Leister, D. Bode, H. Schindler, J. Tischleder, S. Köhne,A new tyre model for TIME measurement data,Tire Technology Expo 2003, Hannover.

Example of PAC-TIME Tire Property File [MDI_HEADER] FILE_TYPE ='tir'

122 Adams/Tire

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

123 Tire 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 = 14 $Tyre use switch (IUSED) VXLOW = 2 LONGVL = 30 $Measurement speed TYRESIDE = 'LEFT' $Mounted side of tyre at vehicle/test bench $----------------------------------------------------------dimensions [DIMENSION] UNLOADED_RADIUS = 0.317 $Free tyre radius WIDTH = 0.205 $Nominal section width of the tyre ASPECT_RATIO = 0.55 $Nominal aspect ratio RIM_RADIUS = 0.203

124 Adams/Tire

$Nominal rim radius RIM_WIDTH = 0.165 $Rim width $---------------------------------------------------------------shape [SHAPE] {radial width} 1.0 0.0 1.0 0.4 1.0 0.9 0.9 1.0 $----------------------------------------------------------load_curve $ For a non-linear tire vertical stiffness (optional) $ Maximum of 100 points [DEFLECTION_LOAD_CURVE] {pen fz} 0.000 0.0 0.001 212.0 0.002 428.0 0.003 648.0 0.005 1100.0 0.010 2300.0 0.020 5000.0 0.030 8100.0 $-----------------------------------------------------------parameter [VERTICAL] VERTICAL_STIFFNESS = 2.648e+005 $Tyre vertical stiffness VERTICAL_DAMPING = 500 $Tyre vertical damping BREFF = 4.90

125 Tire 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 = -1.5 $Minimum valid wheel slip KPUMAX = 1.5 $Maximum valid wheel slip $----------------------------------------------------slip_angle_range [SLIP_ANGLE_RANGE] ALPMIN = -1.5708 $Minimum valid slip angle ALPMAX = 1.5708 $Maximum valid slip angle $----------------------------------------------inclination_slip_range [INCLINATION_ANGLE_RANGE] CAMMIN = -0.26181 $Minimum valid camber angle CAMMAX = 0.26181 $Maximum valid camber angle $------------------------------------------------vertical_force_range [VERTICAL_FORCE_RANGE] FZMIN = 140 $Minimum allowed wheel load

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FZMAX = 10800 $Maximum allowed wheel load $-------------------------------------------------------------scaling [SCALING_COEFFICIENTS] LFZO = 1 $Scale factor of nominal load LCX = 1 $Scale factor of Fx shape factor LMUX = 1 $Scale factor of Fx peak friction coefficient LEX = 1 $Scale factor of Fx curvature factor LKX = 1 $Scale factor of Fx slip stiffness LHX = 1 $Scale factor of Fx horizontal shift LVX = 1 $Scale factor of Fx vertical shift LGAX = 1 $Scale factor of camber for Fx LCY = 1 $Scale factor of Fy shape factor LMUY = 1 $Scale factor of Fy peak friction coefficient LEY = 1 $Scale factor of Fy curvature factor LKY = 1 $Scale factor of Fy cornering stiffness

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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

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= 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

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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

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$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

131 Tire 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

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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

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[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

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$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

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QTZ1 = 0 $Gyroscopic torque constant MBELT = 0 $Belt mass of the wheel -kg-

Contact Methods The PAC-TIME model supports the following roads: • 2D Roads, see Using the 2D Road Model • 3D Roads, see Adams/3D Road Model

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.

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Using Pacejka '89 and '94 Models Adams/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 • Left and Right Side Tires • 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.

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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: F z

· = kz ρ + cz ρ ·

If the tire loses contact with the road, the tire deflection ρ and deflection velocity ρ 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. Definition of Tire Slip Quantities Slip Quantities at combined cornering and braking/traction


Ω, and the loaded rolling radius Rl:

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V sx = V x – ΩR l The lateral slip velocity is equal to the lateral speed in the contact point with respect to the road plane:

V sy = V y The practical slip quantities κ (longitudinal slip) and velocities in the contact point:


V sx V sy κ = – -------- and tan α = --------Vx Vx Note that for realistic tire forces the slip angle between -1 (locked wheel) and 1.

α is limited to 900 and the longitudinal slip κ 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 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. 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.

139 Tire Models

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:

v 1 dv ------ --------1 + -----1- = tan α + aφ V x dt σ α with the relaxation length σ α in the lateral direction. The turnslip φ 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:

dv 1 σ α -------- + V x v 1 = σ α V sy dt 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:

du 1 σ κ -------- + V x u 1 = – σ κ V sx dt Now the practical slip quantities,


u1 κ' = ------ sgn ( V x ) σκ v1 α' = atan ------ σ α  These practical slip quantities state tire behavior.

κ' and α' are used instead of the usual κ and α definitions for steady-

The longitudinal and lateral relaxation length are estimated with the longitudinal and lateral stiffness of the non-rolling tire:

BCD x σ κ = -------------------------------------------------------longitudinal_stiffness

and

BCD y σ α = ------------------------------------------lateral_stiffness

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:

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longitudinal_stiffness = 4 × lateral_stiffness 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 '89 The test data and resulting coefficients that come from the Pacejka '89 tire model conform to a modified SAE tire coordinate system. The standard SAE tire coordinate system is shown next and the modified sign conventions for Pacejka '89 are described in the table below. Note:

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

SAE Tire Coordinate System

141 Tire Models

Conventions for Naming Variables Variable name and abbreviation:

Description:

Normal load

Fz (kN)

Positive when the tire is penetrating the road.*

Lateral force

Fy (N)

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

Longitudinal force

Fx (N)

Positive during traction. Negative during braking.

Self-aligning torque

Mz (Nm)

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

Inclination angle

γ (deg)

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

Sideslip angle

α (deg) κ (%)

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.

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

Longitudinal Force for Pacejka '89 C - Shape Factor C=B0 D - Peak Factor D=(B1*FZ2+B2*FZ) BCD BCD=(B3*FZ2+B4*FZ)*EXP(-B5*FZ) B - Stiffness Factor B=BCD/(C*D)

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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 Parameters:

Description:

B0

Shape factor

B1, B2

Peak factor

B3, B4, B5

BCD calculation

B6, B,7 B8

Curvature factor

B9, B10

Horizontal shift

Example Longitudinal Force Plot for Pacejka ’89

143 Tire 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 - Stiffness Factor 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)

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FY Equation FY=(D*SIN(C*ATAN(B*X1-E*(B*X1-ATAN(B*X1)))))+Sv Parameters for Lateral Force Parameters:

Description:

A0

Shape factor

A1, A2

Peak factor

A3, A4, A5

BCD calculation

A6, A7

Curvature factor

A8, A9, A10

Horizontal shift

A11, A12, A13

Vertical shift

Example Lateral Force Plot for Pacejka ’89

Self-Aligning Torque C - Shape Factor C=C0 D - Peak Factor

145 Tire Models

D=(C1*FZ2+C2*FZ) BCD BCD=(C3*FZ2+C4*FZ)*(1-C6*ABS(g))*EXP(-C5*FZ) B - Stiffness Factor 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 - Curvature Factor 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 Parameters:

Description:

C0

Shape factor

C1, C2

Peak factor

C3, C4, C5, C6

BCD calculation

C7, C8, C9, C10

Curvature factor

C11, C12, C13

Horizontal shift

C14, C15, C16, C17

Vertical shift

Example Self-Aligning Torque Plot for Pacejka ’89

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Overturning Moment The lateral stiffness is used to calculate an approximate lateral deflection of the contact patch when there is a lateral force present: deflection = Fy / lateral_stiffness This deflection, in turn, is used to calculate an overturning moment due to the vertical force: Mx (overturning moment) = -Fz * deflection And an incremental aligning torque due to longtiudinal force (Fx) Mz = Mz,Magic Formula + Fx * deflection Here Mz,Magic Formula is the magic formula for aligning torque and Fx * deflection is the contribution due to the longitudinal force. 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.

147 Tire 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 Rolling resistance moment My = S*My Aligning Torque Mz = S*Mz The USE_MODE parameter in the tire property file allows you to switch smoothing on or off: • USE_MODE = 1 or 2, smoothing is off • USE_MODE = 3 or 4, smoothing is on

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

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[MODEL] ! use mode 123411121314 ! ----------------------------------------------------------------------! smoothingXXXX ! combinedXXXX ! transient X X X X ! PROPERTY_FILE_FORMAT = 'PAC89' USE_MODE = 12.0 TYRESIDE = 'LEFT' $------------------------------------------------------------DIMENSION [DIMENSION] UNLOADED_RADIUS = 326.0 WIDTH = 245.0 ASPECT_RATIO = 0.35 $------------------------------------------------------------PARAMETER [PARAMETER] VERTICAL_STIFFNESS = 310.0 VERTICAL_DAMPING = 3.1 LATERAL_STIFFNESS = 190.0 ROLLING_RESISTANCE = 0.0 $--------------------------------------------------------------------LOAD_CURVE $ For a non-linear tire vertical stiffness (optional) $ Maximum of 100 points [DEFLECTION_LOAD_CURVE] {pen fz} 0 0.0 1 212.0 2 428.0 3 648.0 5 1100.0 10 2300.0 20 5000.0 30 8100.0 $------------------------------------------------------------LATERAL_COEFFICIENTS [LATERAL_COEFFICIENTS] a0 = 1.65000 a1 = -34.0 a2 = 1250.00 a3 = 3036.00 a4 = 12.80 a5 = 0.00501 a6 = -0.02103 a7 = 0.77394 a8 = 0.0022890 a9 = 0.013442 a10 = 0.003709 a11 = 19.1656 a12 = 1.21356

149 Tire Models

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

150 Adams/Tire

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

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

Conventions for Naming Variables Variable name and abbreviation:

Description:

Normal load

Fz (kN)

Positive when the tire is penetrating the road.

Lateral force

Fy (N)

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

Longitudinal force

Fx (N)

Positive during traction. Negative during braking.

Self-aligning torque

Mz (Nm)

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

Inclination angle

γ(deg)

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

Sideslip angle

α (deg) κ (%)

Positive in a left turn.

Longitudinal slip

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

Force and Moment Formulation for Pacejka '94 • Longitudinal Force for Pacejka '94 • Lateral Force for Pacejka '94 • Self-Aligning Torque • Overturning Moment • Rolling Resistance • Smoothing

151 Tire 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 - Stiffness Factor 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 Parameters:

Description:

B0

Shape factor

B1, B2

Peak factor

B3, B4, B5

BCD calculation

B6, B7, B8, B13

Curvature factor

B9, B10

Horizontal shift

B11, B12

Vertical shift

DLON, BCDLON

Scale factor

Lateral Force for Pacejka '94 C - Shape Factor C=A0

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D - Peak Factor D=((A1*FZ+A2) *(1-A15* γ2)*FZ) * DLAT BCD BCD=(A3*SIN(ATAN(FZ/A4)*2.0)*(1-A5*ABS( γ)))* BCDLAT B - Stiffness Factor B=BCD/(C*D) Horizontal Shift Sh=A8*FZ+A9+A10* γ Vertical Shift Sv=A11*FZ+A12+(A13*FZ2+A14*FZ)* γ Composite X1=(a+Sh) E - Curvature Factor 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 Parameters:

Description:

A0

Shape factor

A1, A2, A15

Peak factor

A3, A4, A5

BCD calculation

A6, A7, A16, A17

Curvature factor

A8, A9, A10

Horizontal shift

A11, A12, A13, A14

Vertical shift

DLAT, BCDLAT

Scale factor

Self-Aligning Torque for Pacejka '94 C - Shape Factor C=C0 D - Peak Factor

153 Tire Models

D=(C1*FZ2+C2*FZ)*(1-C18* γ2) BCD BCD=(C3*FZ2+C4*FZ)*(1-(C6*ABS(g)))*EXP(-C5*FZ) B - Stiffness Factor B=BCD/(C*D) Horizontal Shift Sh=C11*FZ+C12+C13* γ Vertical Shift Sv=C14*FZ+C15+(C16*FZ2+C17*FZ)* γ Composite X1=( α +Sh) E - Curvature Factor E=(((C7*FZ2)+(C8*FZ)+C9)*(1-(((C19* γ)+C20)*SIGN(1,X1))))/(1-(C10*ABS( γ))) MZ Equation MZ=(D*SIN(C*ATAN(B*X1-E*(B*X1-ATAN(B*X1)))))+Sv Parameters for Self-Aligning Torque Parameters:

Description:

C0

Shape factor

C1, C2, C18

Peak factor

C3, C4, C5, C6

BCD calculation

C7, C8, C9, C19, C20

Curvature factor

C11, C12, C13

Horizontal shift

C14, C15, C16, C17

Vertical shift

Overturning Moment The lateral stiffness is used to calculate an approximate lateral deflection of the contact patch when there is a lateral force present: deflection = Fy / lateral_stiffness This deflection, in turn, is used to calculate an overturning moment due to the vertical force: Mx (overturning moment) = -Fz * deflection

154 Adams/Tire

And an incremental aligning torque due to longtiudinal force (Fx): Mz = Mz,Magic Formula + Fx * deflection Here Mz,Magic Formula is the magic formula for aligning torque and Fx * deflection is the contribution due to the longitudinal force. Rolling Resistance The rolling resistance moment My is opposite to the wheel angular velocity. The magnitude is given by: My = Fz * Lrad * rolling_resistance Where Fz equals the vertical force and Lrad is the tyre loaded radius. The rolling resistance coefficient can be entered in the tire property file: [PARAMETER] ROLLING_RESISTANCE = 0.01 A value of 0.01 will introduce a rolling resistance force, which is 1% of the vertical load. Smoothing Adams/Tire smooths initial transients in the tire force over the first 0.1 seconds of simulation. The longitudinal force, lateral force, and aligning torque are multiplied by a cubic step function of time. (See STEP in the Adams/Solver online help.) Longitudinal Force FLon = S*FLon Lateral Force FLat = S*FLat Overturning Moment Mx = S*Mx Rolling resistance moment My = S*My Aligning Torque Mz = S*Mz The USE_MODE parameter in the tire property file allows you to switch smoothing on or off: • USE_MODE = 1 or 2, smoothing is off • USE_MODE = 3 or 4, smoothing is on

Example of Pacejka '94 Property File !:FILE_TYPE: !:FILE_VERSION:

tir 2

155 Tire 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 fz} 0.000 0 0.039 943 0.079 1904 0.118 2882 0.197 4893 0.394 10231 0.787 22241 1.181 36031 $----------------------------------------------------------------------scaling

156 Adams/Tire

[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

157 Tire 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 '94 The combined slip calculation of the Pacejka '89 and '94 tire models is identical. Note that the method employed here is not part of the Magic Formula as developed by Professor Pacejka, but is an in-house development of MSC. Inputs: • Dimensionless longitudinal slip 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 + S hx α * = α + S hy SAG = sin ( α * )

158 Adams/Tire

  κ* β = arc cos ---------------------------------------  ( κ * ) 2 + SAG 2 Friction coefficients:

F x – S vx µx, act = -------------------Fz

F y – S vy µy, act = -------------------Fz

Dx µx, max = -----Fz


1 µx = --------------------------------------------------------1 2 --------------tan β -2 ------------+ µx, act µy, max

tan β µy = -----------------------------------------------------------1 -2 --------------tan β -2 --------------+ µx, max µy, max

Forces corrected for combined slip conditions:

µx F x, comb = ------------- ( F x + S vx ) µx, act

µy F y, comb = ------------- ( F y + S vy ) µy, act

Contact Methods The 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.

159 Tire Models

Using the UA-Tire Model Learn 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-Tire The 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

γ can be found in About Tire Kinematic and Force Outputs. The tire · deflection ρ and deflection velocity ρ are determined using either a point follower or durability contact slip a and camber angle

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

160 Adams/Tire


161 Tire Models


ω , and the effective rolling radius Re:

V sx = V x – ΩR e The lateral slip velocity is equal to the lateral speed in the contact point with respect to the road plane:

V sy = V y The practical slip quantities κ (longitudinal slip) and velocities in the contact point:

V sx κ = – -------Vx


V sy and tan α = --------Vx

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

V sx κ = – -------Vr

V sy and tan α = --------Vr


Vr = Re Ω Note that for realistic tire forces the slip angle (=

α is limited to 45 degrees and the longitudinal slip Ss

κ ) 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 firstorder system is used both for the longitudinal slip κ 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 be estimated (see also reference [1]). The figure below shows a top-view of the string model. Stretched String Model for Transient Tire Behavior

162 Adams/Tire

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:

v1 1 dv 1 ---------+ - = tan ( α ) + aφ Vx d t σα with the relaxation length σ α in the lateral direction. The turnslip φ 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:

σα


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

163 Tire Models

σx

dv 1 + V x v 1 = – σ α V sx dt

Now the practical slip quantities,

κ ’ and α ’, are defined based on the tire deformation:

u1 κ' = ------ – k Vlowx V sx sin ( V x ) σ κ  v1 α' = atan ------ – k Vlowy v sy σ α   These practical slip quantities κ' and α' are used instead of the usual κ and α definitions for steadystate 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 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 κ and α .

Tire Model Parameters Definition of Tire Parameters

Symbol:

Name in tire property file:

Units*: L

Description:

r1

UNLOADED_R ADIUS

Tire unloaded radius

kz

VERTICAL_STI F/L FFNESS

Vertical stiffness

cz

VERTICAL_DA MPING

FT/L

Vertical damping

Cr

ROLLING_RES ISTANCE

L

Rolling resistance parameter

Cs

CSLIP

F

Longitudinal slip stiffness,

Cα

CALPHA

F/A

Cγ

CGAMMA

F/A

UMIN

UMIN

-

∂ Fx ∂κ

∂ Fy ∂ F∂y α α =0 Camber stiffness, ∂ γ γ= 0 Cornering stiffness,

κ=0

Minimum friction coefficient (Sg=1)

164 Adams/Tire

Symbol:

Name in tire property file:

Units*: -

Description:

UMAX

UMAX

σx σy

REL_LEN_LON L

Relaxation length in longitudinal direction

REL_LEN_LAT

Relaxation length in lateral direction

L

Maximum friction coefficient (Ssg=0)

* 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:

· Fz = kz ρ + cz ρ ·

If the tire loses contact with the road, the tire deflection ρ and deflection velocity ρ 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:

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

165 Tire Models

Lateral Slip Ratios: Sa , Sg , Sag The lateral slip ratio due to slip angle, S α , is defined as:

S α* =

tan α during braking ( 1 – S s ) tan α during traction

S α = min ( 1.0, S α* ) The lateral slip ratio due to inclination angle, S γ, is defined as:

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

S α*

l sin γ tan α – -----------2r l

=

where

during braking

l sin γ ( 1 – S s ) tan α ------------ during traction 2r l

l =

8r 1 ρ is the length of the contact patch.

S α γ = min ( 1.0, S * α γ) Comprehensive Slip Ratio: Ssag A comprehensive slip ratio due to longitudinal slip, slip angle, and inclination angle may be defined as:

S * sα γ =

S s2 + S 2 α γ

S α γ = min ( 1.0, S * α γ)

166 Adams/Tire

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 α γ) 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 α γ and µ, 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 ( µx and

µy) but limits the maximum value for both coefficients to µ. See the figure below.

Friction Circle Concept

167 Tire Models

The relationship that defines the friction circle follows: 2 µ 2 µ -----x + -----y = 1  µ  µ

or

µx = µcos β and µy = µsin β

where:

Ss Sα γ cos β = ---------- and sin β = ---------S sα γ S sα γ 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:

168 Adams/Tire

• Case 1:

αγ< 0

• Case 2:

α γ ≥ 0 and C α S α γ < C γ S γ

• Case 3:

α γ < 0 and C α S α γ < C γ S γ

• 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

α γ < 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 γc, then it can be evaluated as:

Fz S γc = µ-----Cγ If Ssc represents a slip ratio due to the critical (longitudinal) slip ratio, then it can be evaluated as:

Fz S sc = 3µ----Cs If a slip ratio due to the critical slip angle is denoted by S α c, then it can be determined as:

Cs Sγ 2 – S 2 – 3C -----S α c = ------- S sc s γ Cα Cα 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:

B 2 + B 22 – B 1 B 3 S n = ------------------------------------------B1 where:

169 Tire Models

B 1 = ( 3µF z ) 2 – ( 3C γS γ) 2 B 2 = 2C α S α C γS γ B3 = –[ ( Cs Ss ) 2 + ( Cα Sα ) 2 ] A nondimensional contact patch length is determined as:

ln = 1 – Sn A modified lateral friction coefficient

µy( m ) is evaluated as:

C γS γ ----------µy( m ) = µy –   Fz  where

µy = µsin β 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 γ <

S γc, Ss

<

Ssc, and S α

<

Sα c

In the elastic deformation state, the longitudinal force F ξ , the lateral force F τ , 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:

170 Adams/Tire

F ξ = C s S s l n2 + µx F z ( 1 – 3l n2 + 2l n3 ) F η = C α S s l n2 + µy( m ) F z ( 1 – 3l n2 + 2l n3 ) + C γS γ 1 2 3 M zα = C α S α – --- + --- l n + --- µy( m ) F z S n2 ll n2  2 3  2 M zsα

3µx µy F z2 2 3 = --- C s S s S α l n + ---------------------- ( 1 – 10l n3 + 15l n4 – 6l n5 ) 3 5C α

M zsγ = η F ξ where: •

η = S γ r l2 – l' 2 ⁄4 is the offset between the wheel plane center and the tire tread base.

•

r l2 – l 2 ⁄4 is set to zero if it is negative.

•

l =

8r l ρ the length of the contact patch.

Mz α is the portion of the self-aligning torque generated by the slip angle α . Mzs α and Mzs γ are other components of the self-aligning torque produced by the longitudinal force, which has an offset between

α and the inclination angle γ, respectively. The self-aligning torque Mz is determined as combinations of Mz α , Mzs α and Mzs γ. the wheel center plane and the tire tread base, due to the slip angle

(ii) Complete Sliding State: S γ

≥

S γc, Ss

≥

Ssc, and S α

≥

Sα c

In the complete sliding state, the longitudinal force, the lateral force, and three components of the selfaligning torque are determined as functions of the normal force and the friction coefficients without any elastic stiffness and slip ratio as:

171 Tire Models

F ξ = µx F z F η = µy F z M zα = 0 3µx µy F z2 l M zα s = -----------------------5C α M zsγ = η F ξ Case 2:

αγ

≥

0 and C α S α

≥

Cγ Sγ

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

Fz S γc = 3µ-----Cγ A slip ratio due to the critical value of the slip angle can be found as:

Cs Sγ 2 – S 2 + 3C -----S α c = ------- S sc s γC Cα α when Ss

≤ Ssc.

The nondimensional slip ratio Sn, is determined as:

B 2 + B 22 – B 1 B 3 S n = ------------------------------------------B1 where:

172 Adams/Tire

B 1 = ( 3µF z ) 2 – ( 3C γS γ) 2 B 2 = 2C α S α C γS γ B3 = –[ ( Cs Ss ) 2 + ( Cα Sα ) 2 ] The nondimensional contact patch length ln is found from the equation ln = 1 - Sn, and the modified lateral friction coefficient

µy( m ) is expressed as:

C γS γ µy( m ) = µy + -----------  Fz  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 α S α angle.

<

(i) Elastic Deformation State: Ss

≥ C γ S γ, replaces the critical condition of the inclination Ssc and S

α

<

Sac

In the elastic deformation state:

F ξ = C s S s l n2 + µx F z ( 1 – 3l n2 + 2l n3 ) F η = C α S s l n2 + µy( m ) F z ( 1 – 3l n2 + 2l n3 ) + C γS γ 1 2 3 M zα = C α S α – --- + --- l n + --- µy( m ) F z S n2 ll n2  2 3  2 3µx µy F z2 2 M zsα = --- C s S s S α l n3 + ---------------------- ( 1 – 10l n3 + 15l n4 – 6l n5 ) 5C α 3 M zsγ = η F ξ (ii) Complete Sliding State: Ss

≥

Ssc and S α

≥

Sac

173 Tire Models

F ξ = µx F z F η = µy F z M zα = 0 3µx µy F z2 l M zα s = -----------------------5C α M zsγ = η F ξ Case 3:

αγ

≥

0 and C α S α

<

Cγ Sγ

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

3µF z + C α S α S γc = -------------------------------3C γ 1 S sc = ----- ( 3µF z ) 2 – ( C α S α – 3C γS γ) Cs The nondimensional slip ratio Sn, is expressed as:

B 2 + B 22 – B 1 B 3 S n = ------------------------------------------B1 where:

B 1 = ( 3µF z ) 2 – ( 3C γS γ) 2 B 2 = 2C α S α C γS γ B3 = –[ ( Cs Ss ) 2 + ( Cα Sα ) 2 ]

174 Adams/Tire

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 α S α the slip angle. (i) Elastic Deformation State: S γ

<

<

S γc and Ss

C γ S γ , replaces the critical condition of

<

Ssc

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

F ξ = C s S s l n2 + µx F z ( 1 – 3l n2 + 2l n3 ) F η = C α S s l n2 + µy( m ) F z ( 1 – 3l n2 + 2l n3 ) + C γS γ 1 2  3 (m) 2 2 --- ----M zα = C α S α  – 2 + 3 l n + 2 µy F z S n ll n 3µx µy F z2 2 M zsα = --- C s S s S α l n3 + ---------------------- ( 1 – 10l n3 + 15l n4 – 6l n5 ) 3 5C α M zsγ = η F ξ (ii) Complete Sliding State: S γ

≥

S γc and Ss

≥

Ssc

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

F ξ = µx F z F η = µy F z M zα = 0 3µx µy F z2 l M zα s = -----------------------5C α M zsγ = η F ξ

175 Tire Models

respectively. The longitudinal force F ξ , the lateral force F η , and three components of the self-aligning torques, Mz α , Mzs α , and Mzs γ, always have positive values, but they can be transformed to have positive or negative values depending on the slip ratio s, the slip angle the SAE coordinate system.

α , and the inclination angle γ in

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 ξ , for all cases Lateral Force: Fy = -sin( α ) F η , for cases 1 and 2 Fy = sin( γ) F η , for case 3 Self-aligning Torque: Mz = sin( α ) Mz α - sin( κ ) [-sin( α ) Mzs α + sin( γ)Mzs γ] Rolling Resistance Moment: My = -Cr Fz, for a forward rolling tire. My = Cr Fz, for a backward rolling tire.

Operating Mode: USE_MODE You 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

176 Adams/Tire

• 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 Shape 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 tireto-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 = 2.0 FILE_FORMAT = 'ASCII' (COMMENTS) {comment_string} 'Tire - XXXXXX' 'Pressure - XXXXXX' 'TestDate - XXXXXX' 'Test tire' 'New File Format v2.1' $-------------------------------------------------------------units [UNITS] LENGTH = 'meter' FORCE = 'newton'

177 Tire 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 = 0.295 WIDTH = 0.195 ASPECT_RATIO = 0.55 $---------------------------------------------------------parameter [PARAMETER] VERTICAL_STIFFNESS = 190000 VERTICAL_DAMPING = 50 ROLLING_RESISTANCE = 0.003 CSLIP = 80000 CALPHA = 60000 CGAMMA = 3000 UMIN = 0.8 UMAX = 1.1 REL_LEN_LON = 0.6 REL_LEN_LAT = 0.5

178 Adams/Tire

$-------------------------------------------------------------shape [SHAPE] {radial width} 1.0 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 fz} 0.000 0.0 0.001 212.0 0.002 428.0 0.003 648.0 0.005 1100.0 0.010 2300.0 0.020 5000.0 0.030 8100.0

Contact Methods The 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.

179 Tire Models

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

When to Use PAC Motorcycle Magic-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 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 • Lane-change maneuvers • 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) • 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

obstacles must be longer than the tire radius) 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 Forces For vehicle dynamics applications, 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.

180 Adams/Tire

In the radial direction, the MF tire models consider the tire to behave as a parallel linear spring and linear damper with one point of contact with the road surface. The contact point is determined by considering the tire and wheel as a rigid disc. In the contact point between the tire and the road the contact forces in longitudinal and lateral direction strongly depend on the slip between the tire patch elements and the road. The figure, Input and Output Variables of the Magic Formula Tire Model, presents the input and output vectors of the 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: • 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 ( F z ), the longitudinal and lateral slip ( κ ,

α ), and inclination angle ( γ) with the road. The output is the forces ( F x , F y ) and moments ( M x , M y , M z ) 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 the STI. Input and Output Variables of the Magic Formula Tire Model

181 Tire Models

Axis Systems and Slip Definitions • Axis System • Units • Definition of Tire Slip Quantities

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 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 a different origin as can be seen in the figure below. TYDEX C- and W-Axis Systems Used in PAC MC, Source[2]

The C-axis system is fixed to the wheel carrier with the longitudinal xc-axis parallel to the road and in the wheel plane (xc-zc-plane). The origin of the C-axis system is the wheel center. The origin of the W-axis system is the road contact-point defined by the intersection of the wheel plane, the plane through the wheel carrier, and the road tangent plane. The forces and moments calculated by PAC MC 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).

182 Adams/Tire

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. However, the parameters in the tire property file must always be valid for the TYDEX W-axis system (ISO oriented). The basic SI units are listed in the table below. SI Units Used in PAC MC Variable Type: Angle

Force

Moment

Name:

α

Radians

Inclination angle Longitudinal force

γ Fx

Newton

Lateral force

Fy

Vertical load

Fz Mx

Overturning moment


Rotational speed

Unit:

Slip angle


Speed

Abbreviation:

My Mz

Longitudinal speed

Vx

Lateral speed

Vy


V sx

Lateral slip speed

V sy ω

Tire rolling speed

Newton.meter


Meters per second

Radians per second

183 Tire Models

The longitudinal slip velocity

V sx in the contact point (W-axis system, see the figure, Slip Quantities

at Combined Cornering) is defined using the longitudinal speed V x , the wheel rotational velocity ω , and the effective rolling radius R e :

V sx = V x – ΩR e

(204)


V sy = V y The practical slip quantities κ (longitudinal slip) and velocities in the contact point with:

(205)


V sx κ = – -------Vx

(206)

V sy tan ( α ) = --------Vx

(207)


Vr = Re Ω

(208)

Contact-Point and Normal Load Calculation • Contact Point • Loaded and Effective Tire Rolling Radius

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).

184 Adams/Tire

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 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 though the wheel spin axis. The normal load F z of the tire is calculated with:

· F z = C z ρ + K z ⋅ρ where

(209)

· ρ is the tire deflection and ρ 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 C z in the tire property file, but it does not play any role.

185 Tire Models

Loaded and Effective Tire Rolling Radius With the loaded rolling tire radius R defined as the distance of the wheel center to the contact point of the tire with the road (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:

R1 = R0 – ρ In this tire model, a constant (linear) vertical tire stiffness

(210)

C z is assumed; therefore, the tire deflection

ρ can be calculated using: Fz ρ = -----Cz

(211)


Vx R e = -----Ω

(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 decrease with increasing vertical load due to the tire tread thickness. See the figure below. Effective Rolling Radius and Longitudinal Slip

186 Adams/Tire

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

R e = R 0 – ρ Fz0 ( D ⋅ atan ( ( Bρ d ) + Fρ d ) ) in which


F z0 ρ Fz0 = -------Cz and

(213)

(214)

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

ρ ρ d = --------ρ Fz0

(215)

187 Tire Models

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

Normal Load and Rolling Radius Parameters Nam e:

F z0 R0 B D F Cz KZ


Explanation:

FNOMIN

Nominal wheel load

UNLOADED_RADIUS

Free tire radius

BREFF


DREFF


FREFF


VERTICAL_STIFFNESS

Tire vertical stiffness

VERTICAL_DAMPING


Basics of the Magic Formula in PAC MC The Magic Formula is a mathematical formula that is capable of describing the basic tire characteristics for the interaction forces between the tire and the road under several steady-state operating conditions. We distinguish: • Pure cornering slip conditions: cornering with a free rolling tire

188 Adams/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 F y as a function of the lateral slip


longitudinal force F x as a function of longitudinal slip κ , 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:

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

(216)

where Y(x) is either F x with x the longitudinal slip κ , or F y and x the lateral slip α . Characteristic Curves for Fx and Fy Under Pure Slip Conditions

The self-aligning moment

M z is calculated as a product of the lateral force F y and the pneumatic trail t

added with the residual moment M zr . In fact, the aligning moment is due to the offset of lateral force F y , called pneumatic trail t, from the contact point. Because the pneumatic trail t as a function of the lateral slip

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

Y ( x ) = D ⋅ cos [ C ⋅ atan ( B x – E ( B x – atan ( B x ) ) ) ] in which Y(x) is the pneumatic trail t as function of slip angle

(217)

α.

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.

189 Tire Models

• C-factor determines the part used of the sine and, therefore, mainly influences the shape of the



190 Adams/Tire

In combined slip conditions, the lateral force F y decreases due to longitudinal slip or the opposite, the longitudinal force F x decreases due to lateral slip. The forces and moments in combined slip conditions

191 Tire 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. The Magic Formula itself only describes steady-state tire behavior. For transient tire behavior (up to 8 Hz), the MF output is used in a stretched string model that considers tire belt deflections instead of slip velocities to cope with standstill situations (zero speed). Inclination Effects in the Lateral Force From a historical point of view, the 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

γ has also been added within the MF sine function:

F y0 = D y sin [ C y arc tan { B y α y – E y ( B y α y – arc tan ( B y α y ) ) } + C γarc tan { B γγy – E γ( B γγy – arc tan ( B γγy ) ) } ]

(218)

This elegant formulation has the advantage of an explicit definition of the camber stiffness, because this results now in:

K γ = B γC γD γ =

∂ F yo at ∂γ

αy = 0 (219)

Input Variables The input variables to the Magic Formula are: Input Variables [-]

Inclination angle

κ α γ

Normal wheel load

Fz

[N]

Longitudinal force

Fx

[N]

Lateral force

Fy

[N]

Longitudinal slip Slip angle

[rad] [rad]

Output Variables Its output variables are: Output Variables

192 Adams/Tire

Overturning couple

Mx

[Nm]


My

[Nm]

Aligning moment

Mz

[Nm]

The output variables are defined in the W-axis system of TYDEX. Basic Tire Parameters All tire model parameters of the model are without dimension. The reference parameters for the model are: Basic Tire Parameters Nominal (rated) load

Fz0

[N]


R0

[m]

Tire belt mass

mbelt

[kg]


F z – F' z0 df z = -------------------F' z0

(220)

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

F' z0 = F z0 ⋅λ

λ

F z0 ):

(221)

F z0


Definition:

p

Force at pure slip

q

Moment at pure slip

r


s


193 Tire Models

Parameter: i=

j=

k=

Definition:

B

Stiffness factor

C

Shape factor

D

Peak value

E

Curvature factor

K


H

Horizontal shift

V

Vertical shift

s


t


x


y


z


1, 2, ...

User Scaling Factors A set of scaling factors is available to easily examine the influence of changing tire properties without the need to change one of the real Magic Formula coefficients. The default value of these factors is 1. You can change the factors in the tire property file. The peak friction scaling factors, λ µξ and λ γψ , are also used for the position-dependent friction in 3D Road Contact and Adams/3D Road. An overview of all scaling factors is shown in the next tables. Scaling Factor Coefficients for Pure Slip

Name:

λ λ λ λ λ λ λ λ


Explanation:

F z0

LFZO


Cx

LCX


µξ

LMUX


Ex

LEX


Kx

LKX


Vx

LVX


γx

LGAX

Scale factor of camber for Fx

Cy

LCY

Scale factor of Fy shape factor

194 Adams/Tire

Name:

λ λ λ λ λ λ λ λ λ λ λ λ λ λ


Explanation:

µy

LMUY


Ey

LEY


Ky

LKY


Cγ

LCC

Scale factor of camber shape factor

Kγ

LKC

Scale factor of camber stiffness (K-factor)

Eγ

LEC

Scale factor of camber curvature factor

Hyy

LHY


γψ

LGAY

Scale factor of camber force stiffness

t

LTR


Mr

LRES

Scale factor for offset of residual torque

gz

LGAZ

Scale factor of camber torque stiffness

Mx

LMX


VMxMx

LVMX


My

LMY

Scale factor of rolling resistance torque

Scaling Factor Coefficients for Combined Slip Name used in tire property file:

Name:

λ λ λ λ

Explanation:

xα

LXAL


yκ

LYKA


Vy κ

LVYKA


s

LS

Scale factor of Moment arm of Fx


Name:

λ λ λ λ


Explanation:

xα

LXAL


yκ

LYKA


Vy κ

LVYKA


s

LS

Scale factor of Moment arm of Fx

195 Tire Models

Scaling Factor Coefficients for Transient Response Name used in tire property file:

Name:

σκ σα λ gyr

Explanation:

LSGKP


LSGAL


LGYR


Steady-State: Magic Formula for PAC MC • 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


F x = F x0 ( κ , F z, γ)

(222)

F x0 = D x sin [ C x arc tan { B x κ x – E x ( B x κ x – arc tan ( B x κ x ) ) } ] + S Vx

(223)

κ x = κ + S Hx

(224)

γx = γ ⋅λ

(225)

γx


C x = p Cx1 ⋅λ

(226)

Cx

D x = µx ⋅F z ⋅ζ1 µx = ( p Dx1 + p Dx2 df z ) ⋅( 1 – p Dx3 ⋅γx2 )λ

(227)

E x = ( p Ex1 + p Ex2 df z + p Ex3 df z2 ) ⋅{ 1 – p Ex4 sgn ( κ x ) } ⋅λ the longitudinal slip stiffness:

(228)

µx Ex

with E x ≤ 1

(229)

196 Adams/Tire

K x = F z ⋅( p Kx1 + p Kx2 df z ) ⋅ exp ( p Kx3 df z ) ⋅λ

(K x = B x C x D x =

(230)

K

∂ F x0 at κ x = 0 ) ∂ κx

B x = K x ⁄( C x D x ) S Hx = – ( p sy1 F z λ

My

(231)

+ S Vx ) ⁄K x

S Vx = F z ⋅( p Vx1 + p Vx2 df z ) ⋅λ

(232)

Vx

⋅λ

µx

⋅ζ 1

(233)

Longitudinal Force Coefficients at Pure Slip

Name:


Explanation:

pCx1

PCX1


pDx1

PDX1


pDx2

PDX2


pDx3

PDX3

Variation of friction Mux with camber

pEx1

PEX1


pEx2

PEX2


pEx3

PEX3


pEx4

PEX4


pKx1

PKX1


pKx2

PKX2


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


F y = F y0 ( α , γ, F z )

(234)

F y0 = D y sin [ C y arc tan { B y α y – E y ( B y α y – arc tan ( B y α y ) ) }

(235)

+ C γarc tan { B γγy – E γ( B γγy – arc tan ( B γγy ) ) } ]

197 Tire Models

α y = α + S Hy

( Cy + Cγ < 2 )

(236)


γy = γ ⋅λ

(237)

γy

with coefficients:

C y = p Cy1 ⋅λ

(238)

Cy

D y = µy ⋅F z ⋅ζ2

(239)

µy = p Dy1 ⋅ exp ( p Dy2 df z ) ⋅( 1 – p Dy3 γy2 ) ⋅λ

(240)

µy

E y = { p Ey1 + p Ey2 γy2 – ( p Ey3 + p Ey4 γy ) ⋅ sin ( α y ) } ⋅λ

Ey

with E y ≤ 1

(241)

The cornering stiffness:

Fz   - K y = p Ky1 F zo sin p Ky2 arc tan ---------------------------------------------------------2 ( p Ky3 + p Ky4 γy )F zo λ Fzo  ( 1 – p Ky5 γy2 ) ⋅λ

Fzo

(K y = B y C y D y =

⋅λ

(242)

Ky

∂ F yo at α y = 0 ) ∂ αy

B y = K y ⁄( C y D y )

(243)

S Hy = p Hy1 ⋅λ

(244)

C γ = p Cy2 ⋅λ

Hy

(245)

Cγ

and the explicit camber stiffness:

K γ = ( p Ky6 + p Ky7 df z ) ⋅F z ⋅λ E γ = p Ey5 ⋅λ

Eγ

with E γ ≤ 1

B γ = K γ ⁄( C γD γ)

Kγ

(=B γC γD γ =

∂ F yo at α y = 0 ∂γ

(246) (247) (248)

198 Adams/Tire


Name:


Explanation:

pCy1

PCY1


pCy2

PCY2

Shape factor Cfc for camber forces

pDy1

PDY1


pDy2

PDY2

Exponent lateral friction Muy

pDy3

PDY3

Variation of friction Muy with squared camber

pEy1

PEY1


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

pKy1

PKY1


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

pHy1

PHY1



M z' = M z0 ( α , γ, F z ) M z0 = – t ⋅F y0 + M zr

(249)

with the pneumatic trail t:


(250)

α t = α + S Ht

(251)

and the residual moment Mzr:

M zr ( α r ) = D r cos [ C r arc tan ( B r α r ) ] ⋅ cos ( α )

(252)

199 Tire Models

α r = α + S Hr

(253)


γz = γ ⋅λ

(254)

γz

with coefficients:

B t = ( q Bx1 + q Bx2 df z + q Bx3 df z2 ) ⋅{ 1 + q Bx4 γz + q Bz5 γz } ⋅λ

Ky

⁄λ

µy

(255)

C t = q Cz1

(256)

D t = F z ⋅( q Dz1 + q Dz2 df z ) ⋅( 1 + q Dz3 γz + q Dz4 γz2 ) ⋅( R 0 ⁄F z0 )

(257)

E t = ( q Ex1 + q Ex2 df z + q Ex3 df z2 )

(258)

 2- ⋅arc tan ( B ⋅C ⋅α )  with E ≤ 1 1 + ( q Ez4 + q Ez5 γz ) ⋅-t t t  t π  

(259)

S Ht = 0

(260)

B r = q Bz9 ⋅λ

Ky

⁄λ

(261)

µy

D r = F z [ ( q Dz6 + q Dz7 df z )λ r + ( q Dz8 + q Dz9 df z )γz + ( q Dz10 + q Dz11 df z ) ( γz ⋅ γz (262) ) ] R0 λ S Hr = q Hz1 + q Hz2 df z + ( q Hz3 + q Hz4 df z )γz

(263)


 ≈ – dM z at α  = 0 )   dα

K z = – t ⋅K y

(264)


Name:


Explanation:

qBz1

QBZ1


qBz2

QBZ2


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

µy

200 Adams/Tire

Name:


Explanation:

qBz9

QBZ9

Slope factor Br of residual torque Mzr

qCz1

QCZ1


qDz1

QDZ1


qDz2

QDZ2


qDz3

QDZ3

Variation of peak Dpt with camber

qDz4

QDZ4

Variation of peak Dpt with camber squared.

qDz6

QDZ6


qDz7

QDZ7


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

qEz1

QEZ1


qEz2

QEZ2


qEz3

QEZ3

Variation of curvature Ept with load squared

qEz4

QEZ4


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

Steady-State Combined Slip PAC-TIME has two methods for calculating the combined slip forces and moments. If the user supplies the coefficients for the combined slip cosine 'weighing' functions, the combined slip is calculated according to Combined slip with cosine 'weighing' functions (standard method). If no coefficients are supplied, the so-called friction ellipse is used to estimate the combined slip forces and moments, see section Combined Slip with friction ellipse Combined slip with cosine 'weighing' functions • Longitudinal Force at Combined Slip • Lateral Force at Combined Slip • Aligning Moment at Combined Slip • Overturning Moment at Pure and Combined Slip

201 Tire Models

• Rolling Resistance Moment at Pure and Combined Slip


F x = F x0 ⋅G xα ( α , κ , F z )

(265)

with Gx α o the weighting function of the longitudinal force for pure slip. We write:


(266)

α s = α + S Hxα

(267)

with coefficients:

B xα = ( r Bx1 + r Bx3 γ2 ) cos [ arc tan { r Bx2 κ } ] ⋅λ

(268)

xα

C xα = r Cx1

(269)


(270)


(271)

S Hxα = r Hx1

(272)


cos [ C xα arc tan { B xα α s – E xα ( B xα α s – arc tan ( B xα α s ) ) } ] G xα = -----------------------------------------------------------------------------------------------------------------------------------------------------------------cos [ C xα arc tan { B xα S Hxα – E xα ( B xα S Hxα – arc tan ( B xα S Hxα ) ) } ] Longitudinal Force Coefficients at Combined Slip

Name:


Explanation:

rBx1

RBX1


rBx2

RBX2


rBx3

RBX3

Influence of camber on stiffness for Fx reduction

rCx1

RCX1


(273)

202 Adams/Tire

Name:


Explanation:

rEx1

REX1


rEx2

REX2


rHx1

RHX1



F x = F x0 ⋅G xα ( α , κ , F z )

(274)

with Gyk the weighting function for the lateral force at pure slip and SVyk the ' κ -induced' side force; therefore, the lateral force can be written as:

F y = D yκ cos [ C yκ arc tan { B yκ κ s – E yκ ( B yκ κ s – arc tan ( B yκ κ s ) ) } ] + S Vyκ

(275)

κ s = κ + S Hyk

(276)


B yκ = ( r By1 + r By4 γ2 ) cos [ arc tan { r By2 ( α – r By3 ) } ] ⋅λ

yκ

(277)

C yκ = r Cy1

(278)

F yo D yκ = --------------------------------------------------------------------------------------------------------------------------------------------------------------cos [ C yκ arc tan { B yκ S Hyκ – E yκ ( B yκ S Hyκ – arc tan ( B yκ S Hyκ ) ) } ]

(279)


(280)


(281)

S Vyk = D Vyκ sin [ r Vy5 arc tan ( r vy6 κ ) ] ⋅λ

Vyκ


(282) (283)


cos [ C yκ arc tan { B yκ κ s – E yκ ( B yκ κ s – arc tan ( B yκ κ s ) ) } ] G yκ = --------------------------------------------------------------------------------------------------------------------------------------------------------------cos [ C yκ arc tan { B yκ S Hyκ – E yκ ( B yκ S Hyκ – arc tan ( B yκ S Hyκ ) ) } ]

(284)

203 Tire Models


Name:


Explanation:

rBy1

RBY1


rBy2

RBY2


rBy3

RBY3


rCy1

RCY1


rEy1

REY1


rEy2

REY2


rHy1

RHY1


rHy2

RHY2


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)


M z' = – t ⋅F y' + M zr + s ⋅F x

(285)

with:

t = t ( α t, eq )

(286)

= D t cos [ C t arc tan { B t α t, eq – E t ( B t α t, eq – ac tan ( B t α t, eq ) ) } ] cos ( α )

(287)

F' y, γ = 0 = F y – S Vyκ

(288)

M zr = M zr ( α r, eq ) = D r cos [ arc tan ( B r α r, eq ) ] cos ( α )

(289)

s = { s sz1 + s sz2 ( F y ⁄F z0 ) + ( s sz3 + s sz4 df z )γ} ⋅R 0 ⋅λ

(290)

with the arguments:

s

204 Adams/Tire

Kx 2 2 α t, eq = arc tan tan α t + ------ κ 2 ⋅ sgn ( α t ) K y

(291)

Kx 2 2 α r, eq = arc tan tan α r + ------ κ 2 ⋅ sgn ( α r ) K y

(292)

Aligning Moment Coefficients at Combined Slip Name used in tire property file:

Name:

Explanation:

ssz1

SSZ1


ssz2

SSZ2


ssz3

SSZ3


ssz4

SSZ4



 M x = R 0 ⋅F z ⋅q sx1 λ 

VMx

Fy  – q sx2 ⋅γ + q sx3 ⋅--------  λ F z0 

(293)

Mx


Name:


Explanation:

qsx1

QSX1

Lateral force-induced overturning couple

qsx2

QSX2

Inclination-induced overturning couple

qsx3

QSX3

Fy-induced overturning couple

Formulas for Rolling Resistance Moment at Pure and Combined Slip The rolling resistance moment is defined by:

205 Tire Models

M y = R 0 ⋅F z ⋅{ q Sy1 + q Sy2 F x ⁄F z0 + q Sy3 V x ⁄V ref + q Sy4 ( V x ⁄V ref ) 4 }

(294)


Name:


Explanation:

qsy1

QSY1


qsy2

QSY2


qsy3

QSY3


qsy4

QSY4


Vref

LONGVL

Measurement speed


S Vx κ c = κ + S Hx + --------Kx S Vy α c = α + S Hy + --------Ky α ∗ = sin ( α c ) κc   β = acos -------------------------  κ 2 + α ∗ 2 c The following friction coefficients are defined:

F x, 0 – S Vx µx, act = ------------------------Fz

F y, 0 – S Vy µy, act = ------------------------Fz

206 Adams/Tire

Dx µx, max = -----Fz


1 µx = --------------------------------------------------------1 2  tan β 2 ------------+ ---------------µx, act µy, max tan β µy = --------------------------------------------------------1 2  tan β 2 --------------- + ------------µx, max µy, act The forces corrected for the combined slip conditions are:

µx F x = ------------- F x, 0 µx, act

µy F y = ------------- F y, 0 µy, act


S Vyκ =0.

Transient Behavior in PAC MC The previous Magic Formula examples 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 lowfrequency behavior (up to 8 Hz) is called transient behavior. Stretched String Model for Transient Tire Behavior

207 Tire 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. For calculating the lateral deflection v1 of the string in the first point of contact with the road, the following differential equation is valid:


(295)

with the relaxation length σ α in the lateral direction. The turnslip φ can be neglected at radii larger than 10 m. This differential example cannot be used at zero speed, but when multiplying with Vx, the example can be transformed to:

208 Adams/Tire

σα


(296)


σx

du 1 + V x u 1 = – σ x V sx dt

(297)



σκ

Fz   -  ( 1 – p Ky5 γ2 ) ⋅R 0 λ σ α = p Ty1 sin p Ty2 arc tan -------------------------------------------------------2 ( p Ty3 + p Ky4 γ )F z0 λ Fz0  Now the practical slip quantities,

(298)

F z0 λ σ α (299)


u1 κ' = ------ ⋅ sin ( V x ) σx

(300)

v1 α' = atan ------ σ α 

(301)

Using these practical slip quantities, κ' and α' , the Magic Formula examples can be used to calculate the tire-road interaction forces and moments:

F x' = F x ( α', κ', F z )

(302)

F y' = F y ( α', κ', γ, F z )

(303)

M z' = M z' ( α', κ', γ, F z )

(304)

Gyroscopic Couple in PAC MC When having fast rotations about the vertical axis in the wheel plane, the inertia of the tire belt may lead to gyroscopic effects. To cope with this additional moment, the following contribution is added to the total aligning moment:

209 Tire Models

M z, gyr = c gyr m belt V r1


(305)

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

c gyr = q Tz1 ⋅λ

(306)

gyr

and:

cos { arc tan ( B r α r, eq ) } = 1

(307)


M z = M z' + M z, gyr

(308)


Name:


Explanation:

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


pTy3

PTY3

Load where lateral relaxation is at maximum

qTz1

QTZ1

Gyroscopic torque constant

Mbelt

MBELT


Left and Right Side Tires In general, a tire produces a lateral force and aligning moment at zero slip angle due to the tire construction, known as conicity and plysteer. In addition, the tire characteristics cannot be symmetric for positive and negative slip angles. A tire property file with the parameters for the model results from testing with a tire that is mounted in a tire test bench comparable either to the left or the right side of a vehicle. If these coefficients are used

210 Adams/Tire

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 (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 SYMMETRIC, the tire characteristics are changed to symmetric behavior. Create Wheel and Tire Dialog Box in Adams/View

211 Tire Models

USE_MODES of PAC MC: from Simple to Complex The 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.

212 Adams/Tire

USE_MODE Values of PAC MC and Related Tire Model Output

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


Fx, 0, Fz, 0, My, 0

2

Steady state



3

Steady state



4

Steady state

Combined slip


11

Transient


Fx, 0, Fz, 0, My, 0

12

Transient



13

Transient



14

Transient

Combined slip


Contact Methods The PAC MC model supports the following roads: • 2D Roads, see Using the 2D Road Model • 3D Roads, see Adams/3D Road Model

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.

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

· F z = C z ρ + K z ⋅ρ

ρ ) of the tire through the tangent (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

214 Adams/Tire

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 Parameters Because PAC MC 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: • Camber (Inclination) Effects • Validity Range of the Tire Model Input

Note:


Camber (Inclination) Effects 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. Effect of Positive Camber on the Lateral Force in TYDEX W-axis (ISO) System

215 Tire Models

The table below lists further checks on the PAC MC parameters. Checklist for PAC MC Parameters and Properties Parameter/property:

Requirement:

Explanation:

LONGVL

> 1 m/s


VXLOW

Approximately 1m/s

Threshold for scaling down forces and moments

Dx

>0


pDx1/pDx2

< 0


Kx

< 0


Dy

>0


pDy1/pDy2

< 0


216 Adams/Tire

Parameter/property:

Requirement:

Explanation:

Ky

< 0

Cornering stiffness (see equation (152))

qsy1

< 0

Rolling resistance, should in range of 0.005 - 0.015

Kγ

< 0

Camber stiffness (see equation (156))

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 = -1.0996 $Minimum valid camber angle CAMMAX = 1.0996 $Maximum valid camber angle $------------------------------------------------vertical_force_range [VERTICAL_FORCE_RANGE] FZMIN = 73.75 $Minimum allowed wheel load FZMAX = 3319.5 $Maximum allowed wheel load

217 Tire Models

If one of the input parameters exceeds a minimum or maximum validity value, the calculation in the tire model will be performed with the minimum or maximum value of this range to avoid non-realistic tire behavior. In that case, a message appears warning you that one of the inputs exceeds a validity value.

Standard Tire Interface (STI) for PAC MC Because all Adams products use the Standard Tire Interface (STI) for linking the tire models to Adams/Solver, below is a brief background of the STI history (see reference [4]). At the First International Colloquium on Tire Models for Vehicle Dynamics Analysis on October 21-22, 1991, the International Tire Workshop working group was established (TYDEX). The working group concentrated on tire measurements and tire models used for vehicle simulation purposes. For most vehicle dynamics studies, people previously developed their own tire models. Because all car manufacturers and their tire suppliers have the same goal (that is, development of tires to improve dynamic safety of the vehicle), it aimed for standardization in 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




218 Adams/Tire


Definition:

Road tangent plane

Plane with the normal unit vector (tangent to the road) in the tire-road contact point C.

C-axis system


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


W-axis system



Definition:

Units:

R0


[m]

R

Loaded tire radius

[m]

Re


[m]

ρ ρd


[m]


[-]

ρ Fz0


[m]

mbelt

Tire belt mass

[kg]

ω


[rads-1]

219 Tire Models


Definition:

Units:

V

Vehicle speed

[ms-1]

Vsx


[ms-1]

Vsy


[ms-1]

Vs


[ms-1]

Vx

Rolling speed in x direction [ms-1]

Vy


[ms-1]

Vr


[ms-1]

κ α γ

Longitudinal slip

[-]

Slip angle

[rad]

Inclination angle

[rad]


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


[Nm]

Mz

Aligning moment

[Nm]

References 1. H.B. Pacejka, Tyre and Vehicle Dynamics, 2002, Butterworth-Heinemann, ISBN 0 7506 5141 5. 2. H.-J. Unrau, J. Zamow, TYDEX-Format, Description and Reference Manual, Release 1.1, Initiated by the International Tire Working Group, July 1995. 3. A. Riedel, Standard Tire Interface, Release 1.2, Initiated by the Tire Workgroup, June 1995.

220 Adams/Tire

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 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]

221 Tire Models

PROPERTY_FILE_FORMAT ='PAC_MC' USE_MODE = 14 $Tyre use switch (IUSED) 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 width} 1.0 0.0 0.994 0.211 0.975 0.423 0.947 0.634 0.894 0.845 0.841 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 0.0000000 0.00300 0.0000468 0.00600 0.0001877 0.00900 0.0004235 0.01200 0.0007561 0.01500

222 Adams/Tire

0.0011877 0.01800 0.0017216 0.02100 0.0023613 0.02400 0.0031114 0.02700 0.0039770 0.03000 0.0049639 0.03300 0.0060785 0.03600 0.0073282 0.03900 0.0087207 0.04200 0.0102646 0.04500 0.0119694 0.04800 0.0138449 0.05100 0.0159018 0.05400 0.0181517 0.05700 0.0206065 0.06000 0.0232793 0.06300 0.0261836 0.06600 0.0293337 0.06900 0.0327447 0.07200 0.0364323 0.07500 0.0404132 0.07800 0.0447047 0.08100 0.0493248 0.08400 0.0542923 0.08700 0.0596270 $-----------------------------------------------------------parameter [VERTICAL] VERTICAL_STIFFNESS = 2e+005 $Tyre vertical stiffness $

223 Tire 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 fz} 0.000 0.0 0.001 212.0 0.002 428.0 0.003 648.0 0.005 1100.0 0.010 2300.0 0.020 5000.0 0.030 8100.0 $-----------------------------------------------------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 $

224 Adams/Tire

$------------------------------------------------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 (Kfactor) $ 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 $

225 Tire 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

226 Adams/Tire

$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 $

227 Tire 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 $

228 Adams/Tire

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 $

229 Tire 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

230 Adams/Tire

$Variation of distance s/R0 with load and camber $ QTZ1 = 0 $Gyroscopic torque constant $ MBELT = 0 $Belt mass of the wheel -kg- $

231 Tire Models

521-Tire Model Overview 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


232 Adams/Tire

The longitudinal slip velocity Vsx in the SAE-axis system is defined using the longitudinal speed Vx, the wheel rotational velocity Ω , and the loaded rolling radius Rl:

V sx = V z – ΩR 1 The lateral slip velocity is equal to the lateral speed in the contact point with respect to the road plane:

V sy = V y The practical slip quantities κ (longitudinal slip) and α (slip angle) are calculated with these slip velocities in the contact point:

V sx κ = – -------Vx

and

V sy tan α = --------Vx

Note that for realistic tire forces the slip angle α is limited to 90 degrees and the longitudinal slip κ 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 κ as the side slip angle, α . 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. Stretched String Model for Transient Tire Behavior

233 Tire Models

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: v 1 dv -------------1 + ------1 = tan ( α ) + aφ V x dt σ α

with the relaxation length σ α in the lateral direction. The turnslip φ 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: dv 1 σ α -------- + V x v 1 = – σ κ V sx dt

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: du 1 σ α -------- + V x u 1 = – σ κ V sx dt

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

234 Adams/Tire

u1 κ' = ------ sin ( V x ) σκ v1 α' = atan ------ σ α  These practical slip quantities κ ′ and α ′ are used instead of the usual κ and α definitions for steadystate tire behavior. 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.

235 Tire 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 Force The tire normal force Fz is calculated based on the tire deflection and radial velocity. A progressive spring and linear damping constant are employed:

F z = F stiff – F damp where Fstiff is tire stiffness force and Fdamp is tire damping force. The vertical stiffness force is calculated from: θ

F stiff = K z δ

where Kz is the tire vertical stiffness, δ is tire deflection, and damping force is calculated from:

θ is the stiffness exponent. The tire

F damp = C z × RadialVelocity 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

236 Adams/Tire

, 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 Force The 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:

F x = µF z where the friction coefficient µis 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

237 Tire Models

The µcurve 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:

M y = coef rr ⋅F z 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 twodimensional 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.

238 Adams/Tire

You must select the Equivalent Plane Method for generating these durability forces.

Lateral Force and Aligning Torque Two 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:

F y = – ( µstat F z ⋅( 1 – e

–Kα α

) ⋅sign ( α ) )

where Kα denotes the tire cornering stiffness coefficient. The aligning moment Mz is calculated using the pneumatic trail t according to:

M z = – t ⋅F y while the pneumatic trails are calculated with half the contact length a: –K α 1 t = --- ⋅a ⋅e α 3

a =

2

R0 – R1

2

with R0 and Rl are, respectively, the unloaded and loaded tire radius.

Overturning Moment In both methods, the overturning moment Mx calculation is based on the lateral tire force Fy, the lateral tire stiffness Ky, and the vertical load:

239 Tire Models

Fy M x = ------ F z Ky Tire Lateral Force as a Function of Slip Angle

• The contribution of the camber • The cornering stiffness equals

γ is disregarded in the Equation Method.

– µstat F z K a .

Combined Slip of 5.2.1 The 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

κ (range -1 to 1) and side slip angle α 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: •

  k β = acos ------------------------------  k 2 + sin α 2

Friction coefficients:

240 Adams/Tire

Fx µx, act = ----Fz

F y – F y, α = 0 µy, act = -----------------------------Fz

1 tan β µx = ---------------------------------------------------- µy = --------------------------------------------------1 2 tan β2 1 2  tan β 2 --------------------- + ------------+ ----------µx, act µstat  µstat µy, act Forces corrected for combined slip conditions:

µx µy F x, comb = ------------- F x F y, comb = ------------- ( F y + F y, α µx, act µy, act

= 0)

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:

F y, comb M z, comb = M z, pure + F x, comb ⋅-----------------Ky and the overturning moment uses the lateral force for combined slip:

F y, comb M x, comb = ------------------ F z Ky

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 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_MODE You 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.

241 Tire 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. USE_MODE:

Smoothing:

Combined slip correction:

1

off

off

2

off

on

3

on

off

4

on

on

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:

ωa – ωf κ = -----------------ωf where: •

κ = Slip ratio

•

ωf = Free rolling rotational velocity (no slip)

•

ωa = Actual rotational velocity

Kinematic relationships between translational and rotational velocities and the effective rolling radius give:

V x – V sx ωa = --------------------Re Vx ωf = -----Re where: •

V sx = Contact patch velocity reletive to road surface

•

V x = Actual longitudinal velocity

•

R e = Effective rolling radius

242 Adams/Tire

Substituting these relationships into the original slip ratio equation with some cancelling of variables gives:

V sx κ = – -------Vx Therefore:

V sx = – V x κ 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 Method The point-follower contact method assumes a single contact point between the tire and road. The contact point is the point nearest to the wheel center that lies on the line formed by the intersection of the tire (wheel) plane with the local road plane. The contact force computed by the point-follower contact method is normal to the road plane. Therefore, in a simulation of a tire hitting a pothole, the point-follower contact method does not generate the expected longitudinal force.

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

243 Tire 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

244 Adams/Tire

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

245 Tire 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. 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

µstatic

Mu_Dynamic

Figure 4

µdynamic

Mu_Static_velocity

Figure 4

velocity µstatic

Mu_Dynamic_Velocity

Figure 4

velocity µdynamic

rolling_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.

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'

246 Adams/Tire

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

247 Tire 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

248 Adams/Tire

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 Mu_Dynamic Mu_Static_Velocity Mu_Dynamic_Velocity

= = = =

0.95 0.75 3000 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 !

249 Tire Models

! !

pnt1

....

pnt225

ALIGNING_TORQUE_DATA_LIST 225 -1355.7504 5.31 0.11 0.47 0.04 0.80 1.75 2.54 -1.28 1.59

6.52 2.84 -12.44 -21.38 -3.70 17.43 11.08 0.02 -3.77

0.06 14.23 5.95 5.54 -1.29 -9.45 -5.05 -17.73 0.46 -2.48 4.71 26.10 4.26 16.60 2.41 4.28 -0.92 0.22

22.88 5.49 -37.99 -69.04 -27.94 52.20 40.53 14.82 -17.17

26.41 -3.92 -67.22 -111.44 -44.25 81.97 73.54 2.93 6.60

30.58 -14.04 -116.07 -168.11 -53.74 145.78 95.55 10.35 -11.91

22.93 11.45 15.74 13.72 -1.65 -15.64 -26.98 -57.25 -107.71 -62.62 -109.03 -161.88 -19.48 -33.54 -49.52 60.80 90.85 119.51 52.46 93.32 141.34 2.21 9.11 30.44 12.61 2.51 -18.77

0.43 6.70 -2.20 -1.39 2.87 6.99 7.10 1.63 -0.78

-4.62 15.92 -5.53 -17.28 -0.38 24.54 18.78 2.91 10.13

15.36 0.14 -13.28 -52.17 -14.27 66.06 58.20 8.33 -9.94

7.16 -4.20 -47.48 -102.80 -29.03 93.27 104.51 20.32 -13.02

11.70 -11.81 -92.88 -161.71 -42.42 126.38 156.39 42.09 -11.95

5.62 2.31 7.87 1.40 2.10 5.60 3.56 -0.08 1.38

4.36 6.41 1.33 -10.04 0.56 26.48 20.63 1.81 -2.13

23.16 14.10 -16.31 -50.94 -16.15 62.92 60.74 14.39 -2.42

38.03 6.03 -40.24 -93.06 -27.15 90.16 108.26 34.98 -4.08

8.73 -11.66 -82.58 -157.50 -40.13 122.03 162.97 59.72 -2.72

3.69 3.09 6.08 0.76 0.70 5.92 4.36 -2.03 0.39

1.71 7.15 0.38 -7.65 4.37 34.39 29.81 5.94 -5.52

29.06 -7.92 -2.69 -37.28 -7.59 72.55 76.70 26.18 -6.06

10.05 13.53 -32.10 -89.05 -23.71 92.88 118.91 53.59 10.16

11.38 -5.78 -62.17 -145.09 -28.49 129.34 180.59 89.76 7.81

250 Adams/Tire

!-----------------LATERAL FORCE VALUES-------------------------------------! No. of pnt Conversion ! factor ! pnt1 .... pnt225 ! LATERAL_FORCE_DATA_LIST 225 4.448 234.08 269.79 213.70 150.79 11.52 -116.75 -224.15 -242.08 -213.99

585.56 628.82 565.29 452.18 50.58 -367.42 -588.24 -612.70 -597.29

1000.29 1040.78 974.49 752.21 199.87 -618.68 -1001.01 -1059.55 -988.14

1307.77 1331.72 1198.82 885.23 199.50 -683.16 -1235.88 -1344.53 -1343.86

1603.78 1624.83 1387.74 960.13 208.75 -857.81 -1488.88 -1658.66 -1689.35

234.40 239.27 252.34 167.55 32.23 -122.59 -208.93 -261.05 -241.50

572.75 647.77 603.75 481.45 78.77 -423.13 -576.28 -634.90 -607.16

981.30 1007.37 1033.50 826.41 231.31 -552.58 -948.45 -1064.15 -1021.87

1352.37 1357.22 1288.76 962.64 250.14 -613.52 -1149.44 -1338.52 -1322.30

1698.90 1666.30 1483.64 1028.74 254.32 -607.61 -1314.69 -1581.84 -1598.25

210.20 237.91 226.60 154.74 34.37 -130.29 -226.48 -270.70 -254.64

578.56 600.60 629.48 496.21 74.19 -339.00 -557.52 -595.22 -602.76

968.72 1025.67 1084.97 878.72 240.00 -509.04 -884.91 -1059.76 -1032.71

1344.05 1377.57 1354.12 1028.03 284.42 -543.75 -1083.18 -1314.74 -1313.22

1730.40 1733.03 1575.22 1095.59 283.85 -555.05 -1175.12 -1564.43 -1609.96

238.28 227.13 221.76 195.50 28.51 -145.10 -230.33 -230.62 -218.36

531.25 594.51 633.49 505.90 68.59 -319.56 -548.99 -597.10 -570.13

945.70 1038.87 1135.31 899.88 241.99 -464.11 -815.88 -1009.76 -1049.72

1305.28 1365.33 1375.28 1059.92 311.15 -499.27 -991.78 -1261.43 -1344.94

1786.96 1733.29 1619.82 1135.28 331.84 -500.83 -1108.36 -1504.09 -1589.60

228.49 221.19 224.63 178.96 42.58

564.69 595.52 590.58 474.70 65.26

954.06 1019.74 1108.01 918.87 230.69

1332.84 1378.35 1408.87 1125.97 306.58

1687.50 1749.40 1707.09 1242.75 428.45

251 Tire 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 fz} 0.0 0.0 1.0 212.0 2.0 428.0 3.0 648.0 5.0 1100.0 10.0 2300.0 20.0 5000.0 30.0 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

252 Adams/Tire

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 41. INITIAL_HEIGHT

! or FLAT

= 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

253 Tire 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

254 Adams/Tire

Using FTire Tire Model Learn 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 FTire The 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.

255 Tire 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

256 Adams/Tire

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 Approach FTire 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).

257 Tire 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. 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.

• 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 largeand/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

258 Adams/Tire

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 runout. 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 Models FTire 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].

259 Tire Models

Using FTire with Adams FTire 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).

260 Adams/Tire

FTire Parameters • About FTire Parameters • Procedure for Parameterizing FTire • Listing of FTire Parameters

About FTire Parameters FTire 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 overdetermined, 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.

261 Tire 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-ofplane 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 FTire A 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

262 Adams/Tire

• 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.

263 Tire 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 Parameters The 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.

264 Adams/Tire

The parameter:

Means:

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 it is deflected by 20 mm, with zero camber 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.

265 Tire Models

The parameter: volume_gradient

Means: Relative decrease in volume, of a small tire segment, when that segment is deflected vertically. Note:

rel_long_belt_memb_tension

This parameter is optional and only marginally affects the model’s behavior.

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, rigidbody 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.

266 Adams/Tire

The parameter:

Means:

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 it is deflected by 10 mm, with zero camber 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:

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 overdetermined 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:

cleat_width

For truck tires, you can specify wheel_load_at_20_mm_defl_ cleat, as well.

The weights are optional. Default value is 1.

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.

267 Tire Models

The parameter: cleat_bevel_edge_width

Means: 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 leastsquares approach, the contribution of the respective parameter. If a weight is set to zero, the related parameter is completely ignored. Note:

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.

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The parameter: belt_twist_stiffn

Means: 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:

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:

belt_torsion_lat_displ_coupl

Optional, and only needed for full 3D variant. Unit is angle/length. Default value is 0.

Curvature radius of belt cross section perpendicular to mid-plane. Note:

belt_lat_bend_stiffn

Only needed for full 3D variant.Unit is force/angle.

If belt twist angle is 0, value is the kinematic belt torsion angle at 1 mm lateral belt displacement. Note:

belt_lat_curvature_radius

Only needed for full 3D variant. Unit is force*length2/angle.

Optional, and only needed for full 3D variant. Default value is (nearly) infinity.

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.

269 Tire Models

The parameter: wheel_load_at_10_mm_defl_lo_cl

Means: 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:

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 leastsquares approach, the contribution of the related parameter. If a weight is set to zero, the respective parameter is completely ignored. Note:

belt_lat_bend_damp

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.


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.

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The parameter: 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

Means: 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).

271 Tire Models

The parameter: damping_tread_rubber

Means: 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 that defines the pressure dependency of the friction coefficient. Default value is 2 bar.

high_ground_pressure

The third of three ground-pressure values that defines the pressure dependency of the 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.

272 Adams/Tire

The parameter:

Means:

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 velocity defined by parameter 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 velocity defined by parameter 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:

static_balance_ang_position

The angular position at the rim where the static balance weight would have been placed. Note:

dynamic_balance_weight

This parameter is optional.

Amplitude of the harmonic radial stiffness variation as percentage of the mean radial stiffness. Note:

radial_non_unif_ang_position

Parameter is optional.

The angular position at the rim where the left dynamic balance weight would have been placed. Note:

radial_non_uniformity


One of the two equal weights that would have been placed on the rim outer and inner horns for dynamic balancing. Note:

dynamic_balance_ang_position



Angular position where radial stiffness reaches its maximum. Note:

This parameter is optional.

273 Tire Models

The parameter: tang_non_uniformity

Means: Amplitude of the harmonic tangential stiffness variation as percentage of the mean tangential stiffness. Note:

tang_non_unif_ang_position

Angular position where tangential stiffness reaches its maximum. Note:

conicity


Small rotation angle of belt elements at zero moment, about circumferential axis, resulting in a conical shape of the unloaded belt. Note:

ply_steer_percentage


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.

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.

274 Adams/Tire

The parameter: number_tread_strips

Means: 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.

275 Tire Models

The parameter: BDF_parameter

Means: 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.

About the FTire Tire Data File As 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

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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 0.4 1.0 0.9 0.9 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 =

277 Tire 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] $

278 Adams/Tire

$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 =

279 Tire 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

280 Adams/Tire

$ [-] 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] $

281 Tire 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 Conditions You 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.

282 Adams/Tire

OPERATING_CONDITIONS Section Parameters The parameter:

Means:

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 instances that are parameterized by this 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 instances that are parameterized by this 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).

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.

283 Tire 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 = 2.2 INFLATION_PRESSURE_WHEEL_3 = 2.2 INFLATION_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

284 Adams/Tire

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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