Vehicle Dynamics Notes

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VEHIC

LEDYNAMIC

SFACHHOCHSCHULE REGENSBURG

UNIVERSITY OF APPLIED SCIENCES

HOCHSCHULE FUR

TECHNIK

WIRTSCHAFT

SOZIALES

LECTURE NOTES

Prof. Dr. Georg Rill

NOVEMBER, 2002

download: http://homepages.fh-regensburg.de/%7Erig39165/


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Contents

Contents I

1 Introduction 1

1.1 Terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1.1 Vehicle Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1.2 Driver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.1.3 Vehicle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.1.4 Load . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.1.5 Environment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2 Wheel/Axle Suspension Systems . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.2.1 General Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.2.2 Multi Purpose Suspension Systems . . . . . . . . . . . . . . . . . . . 4

1.2.3 Specific Suspension Systems . . . . . . . . . . . . . . . . . . . . . . . 5

1.3 Steering Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.3.1 Requirements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.3.2 Rack and Pinion Steering . . . . . . . . . . . . . . . . . . . . . . . . 6

1.3.3 Lever Arm Steering System . . . . . . . . . . . . . . . . . . . . . . . 6

1.3.4 Drag Link Steering System . . . . . . . . . . . . . . . . . . . . . . . 71.3.5 Bus Steer System . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.4 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.4.1 Coordinate Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.4.2 Forces and Torques in the Tire Contact Area . . . . . . . . . . . . . . 9

1.4.3 Dynamic Rolling Radius . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.4.4 Toe and Camber Angle . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.4.4.1 Definitions according to DIN 70 000 . . . . . . . . . . . . . 11

1.4.4.2 Calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

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1.4.5 Steering Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

1.4.5.1 Kingpin . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

1.4.5.2 Caster and Kingpin Angle . . . . . . . . . . . . . . . . . . . 13

1.4.5.3 Caster and Kingpin Offset . . . . . . . . . . . . . . . . . . . 14

2 Tire 15

2.1 Contact Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.1.1 Contact Point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.1.2 Local Track Plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.1.3 Contact Point Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.2 Tire Forces and Torques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.2.1 Wheel Load . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.2.2 Longitudinal Force and Longitudinal Slip . . . . . . . . . . . . . . . . 19

2.2.3 Lateral Slip, Lateral Force and Self Aligning Torque . . . . . . . . . . 22

2.2.4 Generalized Tire Characteristics . . . . . . . . . . . . . . . . . . . . . 24

2.2.5 Wheel Load Influence . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.2.6 Self Aligning Torque . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.2.7 Camber Influence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292.2.8 Bore Torque . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

2.2.9 Typical Tire Characteristics . . . . . . . . . . . . . . . . . . . . . . . 31

3 Longitudinal Dynamics 33

3.1 Accelerating and Braking . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.1.1 Simple Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.1.2 Maximum Acceleration . . . . . . . . . . . . . . . . . . . . . . . . . 34

3.1.3 Drive Torque at Single Axle . . . . . . . . . . . . . . . . . . . . . . . 343.1.4 Braking at Single Axle . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.1.5 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

3.1.6 Optimal Distribution of Drive and Brake Forces . . . . . . . . . . . . . 37

3.1.7 Different Distributions of Brake Forces . . . . . . . . . . . . . . . . . 38

3.1.8 Anti-Lock-Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

3.2 Drive and Brake Pitch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

3.2.1 Plane Vehicle Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

3.2.2 Position . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

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3.2.3 Linearization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

3.2.4 Equations of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

3.2.5 Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

3.2.6 Driving and Braking . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

3.2.7 Brake Pitch Pole . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

4 Lateral Dynamics 47

4.1 Steady State Cornering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

4.1.1 Overturning Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

4.1.2 Roll Support and Camber Compensation . . . . . . . . . . . . . . . . 50

4.1.3 Roll Center and Roll Axis . . . . . . . . . . . . . . . . . . . . . . . . 51

4.1.4 Roll Angle and Wheel Loads . . . . . . . . . . . . . . . . . . . . . . . 51

4.2 Kinematic Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

4.2.1 Kinematic Tire Model . . . . . . . . . . . . . . . . . . . . . . . . . . 53

4.2.2 Ackermann Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . 53

4.2.3 Vehicle Model with Trailer . . . . . . . . . . . . . . . . . . . . . . . . 54

4.2.3.1 Position . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

4.2.3.2 Vehicle Movements . . . . . . . . . . . . . . . . . . . . . . 564.2.3.3 Entering a Curve . . . . . . . . . . . . . . . . . . . . . . . 57

4.2.3.4 Trailer Movements . . . . . . . . . . . . . . . . . . . . . . . 58

4.2.3.5 Course Calculations . . . . . . . . . . . . . . . . . . . . . . 59

4.3 Simple Handling Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

4.3.1 Forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

4.3.2 Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

4.3.3 Lateral Slips . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

4.3.4 Equations of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . 624.3.5 Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

4.3.5.1 Eigenvalues . . . . . . . . . . . . . . . . . . . . . . . . . . 63

4.3.5.2 Low Speed Approximation . . . . . . . . . . . . . . . . . . . 64

4.3.5.3 High Speed Approximation . . . . . . . . . . . . . . . . . . 64

4.3.6 Steady State Solution . . . . . . . . . . . . . . . . . . . . . . . . . . 65

4.3.6.1 Side Slip Angle and Yaw Velocity . . . . . . . . . . . . . . . 65

4.3.6.2 Steering Tendency . . . . . . . . . . . . . . . . . . . . . . . 67

4.3.6.3 Slip Angles . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

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4.3.7 Influence of Wheel Load on Cornering Stiffness . . . . . . . . . . . . . 68

4.3.7.1 Linear Wheel Load Influence . . . . . . . . . . . . . . . . . 68

4.3.7.2 Digressive Wheel Load Influence . . . . . . . . . . . . . . . 69

4.3.7.3 Steering Tendency depending on Lateral Acceleration . . . . 70

5 Vertical Dynamics 71

5.1 Goals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

5.2 Basic Tuning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

5.2.1 Simple Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

5.2.2 Track . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

5.2.3 Spring Preload . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

5.2.4 Eigenvalues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

5.2.5 Free Vibrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

5.3 Nonlinear Force Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

5.3.1 Random Road Profile . . . . . . . . . . . . . . . . . . . . . . . . . . 77

5.3.2 Vehicle Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

5.3.3 Quality Criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

5.3.4 Optimal Parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . 795.3.4.1 Linear Characteristics . . . . . . . . . . . . . . . . . . . . . 79

5.3.4.2 Nonlinear Characteristics . . . . . . . . . . . . . . . . . . . 80

5.3.4.3 Limited Spring Travel . . . . . . . . . . . . . . . . . . . . . 81

5.4 Dynamic Force Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

5.4.1 System Response in the Frequency Domain . . . . . . . . . . . . . . . 83

5.4.1.1 First Harmonic Oscillation . . . . . . . . . . . . . . . . . . . 83

5.4.1.2 Sweep-Sine Excitation . . . . . . . . . . . . . . . . . . . . . 84

5.4.2 Hydro-Mount . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

5.4.2.1 Principle and Model . . . . . . . . . . . . . . . . . . . . . . 85

5.4.2.2 Dynamic Force Characteristics . . . . . . . . . . . . . . . . 87

5.5 Different Influences on Comfort and Safety . . . . . . . . . . . . . . . . . . . 88

5.5.1 Vehicle Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

5.5.2 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

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6 Driving Behavior of Single Vehicles 91

6.1 Standard Driving Maneuvers . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

6.1.1 Steady State Cornering . . . . . . . . . . . . . . . . . . . . . . . . . 91

6.1.2 Step Steer Input . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

6.1.3 Driving Straight Ahead . . . . . . . . . . . . . . . . . . . . . . . . . 93

6.1.3.1 Random Road Profile . . . . . . . . . . . . . . . . . . . . . 93

6.1.3.2 Steering Activity . . . . . . . . . . . . . . . . . . . . . . . . 95

6.2 Coach with different Loading Conditions . . . . . . . . . . . . . . . . . . . . 96

6.2.1 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

6.2.2 Roll Steer Behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . 966.2.3 Steady State Cornering . . . . . . . . . . . . . . . . . . . . . . . . . 97

6.2.4 Step Steer Input . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

6.3 Different Rear Axle Concepts for a Passenger Car . . . . . . . . . . . . . . . . 98

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1 Introduction

1.1 Terminology

1.1.1 Vehicle Dynamics

The Expression Vehicle Dynamics encompasses the interaction of

driver,

vehicle

load and

environment

Vehicle dynamics mainly deals with

the improvement of active safety and driving comfort as well as

the reduction of road destruction.

In vehicle dynamics

computer calculations

test rig measurements and

field tests

are employed.

The interactions between the single systems and the problems with computer calculationsand/or measurements shall be discussed in the following.

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Vehicle Dynamics FH Regensburg, University of Applied Sciences

1.1.2 Driver

By various means of interference the driver can interfere with the vehicle:

driver

steering wheel lateral dynamicsgas pedalbrake pedalclutchgear shift

longitudinal dynamics

vehicle

The vehicle provides the driver with some information:

vehiclevibrations: longitudinal, lateral, vertical

sound: motor, aerodynamics, tyresinstruments: velocity, external temperature, ...

driverThe environment also influences the driver:

environment

climatetraffic densitytrack

driver

A drivers reaction is very complex. To achieve objective results, an ideal driver is used incomputer simulations and in driving experiments automated drivers (e.g. steering machines)are employed.

Transferring results to normal drivers is often difficult, if field tests are made with test drivers.Field tests with normal drivers have to be evaluated statistically. In all tests, the drivers securitymust have absolute priority.

Driving simulators provide an excellent means of analyzing the behavior of drivers even in limitsituations without danger.

For some years it has been tried to analyze the interaction between driver and vehicle withcomplex driver models.

1.1.3 Vehicle

The following vehicles are listed in the ISO 3833 directive:

Motorcycles,

Passenger Cars,

Busses,

Trucks

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FH Regensburg, University of Applied Sciences Prof. Dr.-Ing. G. Rill

Agricultural Tractors,

Passenger Cars with Trailer Truck Trailer / Semitrailer,

Road Trains.

For computer calculations these vehicles have to be depicted in mathematically describablesubstitute systems. The generation of the equations of motions and the numeric solution aswell as the acquisition of data require great expenses.

In times of PCs and workstations computing costs hardly matter anymore.

At an early stage of development often only prototypes are available for field and/or laboratory

tests.Results can be falsified by safety devices, e.g. jockey wheels on trucks.

1.1.4 Load

Trucks are conceived for taking up load. Thus their driving behavior changes.

Load

mass, inertia, center of gravitydynamic behaviour (liquid load)

In computer calculations problems occur with the determination of the inertias and the mod-elling of liquid loads.

Even the loading and unloading process of experimental vehicles takes some effort. Whenmaking experiments with tank trucks, flammable liquids have to be substituted with water.The results thus achieved cannot be simply transferred to real loads.

1.1.5 Environment

The Environment influences primarily the vehicle:

Environment Road: bumps, coefficient of frictionAir: resistance, cross wind vehiclebut also influences the driver

Environment

climatevisibility

driver

Through the interactions between vehicle and road, roads can quickly be destroyed.

The greatest problem in field test and laboratory experiments is the virtual impossibility ofreproducing environmental influences.

The main problems in computer simulation are the description of random road bumps and theinteraction of tires and road as well as the calculation of aerodynamic forces and torques.

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1.2 Wheel/Axle Suspension Systems

1.2.1 General Remarks

The Automotive Industry uses different kinds of wheel/axle suspension systems. Importantcriteria are costs, space requirements, kinematic properties and compliance attributes.

1.2.2 Multi Purpose Suspension Systems

The Double Wishbone Suspension, the McPherson Suspension and the Multi-Link Suspensionare multi purpose wheel suspension systems, Fig. 1.1.

O

Q

D

R

B

E

N z

xy BB

Rz

x

y

R

R

M

PG

F1

S

S

U1

N3

1

O2

U2

1

2

U

O

R

G

B

F

Q

S

D

C

BA

z

xy

BB

Rz

x

y

R

R

M

P

R

G

Y

S

D

Rz

x

yR

R

V

ZW

E

UB

A

F

XP

Q

Figure 1.1: Double Wishbone, McPherson and Multi-Link Suspension

They are used as steered front or non steered rear axle suspension systems. These suspensionsystems are also suitable for driven axles.

In a McPherson suspension the spring is mounted with an inclination to the strut axis. Thusbending torques at the strut which cause high friction forces can be reduced.

At pickups, trucks and busses often rigid axles are used. The rigid axles are guided either by

X1

X2

Y1

Y2

Z2

Z1

xA

zA

yA

xA

zA

yA

Figure 1.2: Rigid Axles

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leaf springs or by rigid links, Fig. 1.2. Rigid axles tend to tramp on rough road.

Leaf spring guided rigid axle suspension systems are very robust. Dry friction between the leafsleads to locking effects in the suspension. Although the leaf springs provide axle guidance onsome rigid axle suspension systems additional links in longitudinal and lateral direction areused.

Rigid axles suspended by air springs need at least four links for guidance. In addition to a gooddrive comfort air springs allow level control.

1.2.3 Specific Suspension Systems

The Semi-Trailing Arm, the SLA and the Twist Beam axle suspension are suitable only for

non steered axles, Fig. 1.3.

xR

zR

yR

xA

yA

zA

Figure 1.3: Specific Wheel/Axles Suspension Systems

The semi-trailing arm is a simple and cheap design which requires only few space. It is mostlyused for driven rear axles.

The SLA axle design allows a nearly independent layout of longitudinal and lateral axle motions.It is similar to the Central Control Arm axle suspension, where the trailing arm is completelyrigid and hence only two lateral links are needed.

The twist beam axle suspension exhibits either a trailing arm or a semi-trailing arm character-istic. It is used for non driven rear axles only. The twist beam axle provides enough space for

spare tire and fuel tank.

1.3 Steering Systems

1.3.1 Requirements

The steering system must guarantee easy and safe steering of the vehicle. The entirety of themechanical transmission devices must be able to cope with all loads and stresses occurring inoperation.

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In order to achieve a good manuvrability a maximum steer angle of approx. 30 must beprovided at the front wheels of passenger cars. Depending on the wheel base busses and trucks

need maximum steer angles up to 55 at the front wheels.Recently some companies have started investigations on steer by wire techniques.

1.3.2 Rack and Pinion Steering

Rack and pinion is the most common steering system on passenger cars, Fig. 1.4. The rackmay be located either in front of or behind the axle. The rotations of the steering wheel L are

steerbox

rackdragli

nk

wheelandwheelbody

PQ

L

uZ

1 2

pinionL

Figure 1.4: Rack and Pinion Steering

firstly transformed by the steering box to the rack travel uZ = uZ(L) and then via the draglinks transmitted to the wheel rotations 1 = 1(uZ), 2 = 2(uZ). Hence the overall steeringratio depends on the ratio of the steer box and on the kinematics of the steer linkage.

1.3.3 Lever Arm Steering System

Using a lever arm steering system Fig. 1.5, large steer angles at the wheels are possible. This

steer box

draglink1

Q1

L2

1

G

P1 P2

Q2

draglink2

steerl

ever2steerlever1

wheel andwheel body

Figure 1.5: Lever Arm Steering System

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steering system is used on trucks with large wheel bases and independent wheel suspension atthe front axle. Here the steering box can be placed outside of the axle center.

The rotations of the steering wheel L are firstly transformed by the steering box to therotation of the steer levers G = G(L). The drag links transmit this rotation to the wheel1 = 1(G), 2 = 2(G). Hence, again the overall steering ratio depends on the ratio of thesteer box and on the kinematics of the steer linkage.

1.3.4 Drag Link Steering System

At rigid axles the drag link steering system is used, Fig. 1.6.

steer box(90o rotated)

drag link

steer linkage

steerlever

K

L

I

HO

H

1 2

wheelandwheelbody

Figure 1.6: Drag Link Steering System

The rotations of the steering wheel L are transformed by the steering box to the rotation ofthe steer lever arm H = H(L) and further on to the rotation of the left wheel, 1 = 1(H).The drag link transmits the rotation of the left wheel to the right wheel, 2 = 2(1).

1.3.5 Bus Steer System

In busses the driver sits more than 2 m in front of the front axle. Here, sophisticated steersystems are needed, Fig. 1.7.

The rotations of the steering wheel L are transformed by the steering box to the rotation ofthe steer lever arm H = H(L). Via the steer link the left lever arm is moved, H = H(G).This motion is transferred by a coupling link to the right lever arm. Via the drag links the leftand right wheel are rotated, 1 = 1(H) and 2 = 2(H).

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steer box

steer link

Q

L

2

1

G

draglink coupl.link

leftleverarm

steerlever

IJ

H

K

P

H

wheel andwheel body

Figure 1.7: Bus Steer System

1.4 Definitions

1.4.1 Coordinate Systems

In vehicle dynamics several different coordinate systems are used, Fig 1.8.

xy

z

F

F

F

xy

z

00

0

eSeU

eNeyR

Figure 1.8: Coordinate Systems

The inertial system with the axes x0, y0, z0 is fixed to the track. Within the vehicle fixedsystem the xF-axis is pointed forward, the yF-axis left and the zF-axis upward. The positionof the wheel is given by the unit vector eyR in direction of the wheel rotation axis.

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The unit vectors in the directions of circumferential and lateral forces eU and eS as well as thetrack normal eN follow from the contact geometry.

1.4.2 Forces and Torques in the Tire Contact Area

In any point of contact between tire and track normal and friction forces are delivered.

According to the tires profile design the contact area forms a not necessarily coherent area.

The effect of the contact forces can be fully described by a vector of force and torque inreference to a point in the contact patch. The vectors are described in a track-fixed coordinatesystem. The z-axis is normal to the track, the x-axis is perpendicular to the z-axis and per-pendicular to the wheel turning axis eyR. The demand for a right-handed coordinate system

then also fixes the y-axis.

Fx longitudinal or circumferential forceFy lateral forceFz vertical force or wheel load

Mx tilting torqueMy rolling resistance torqueMz self aligning and bore torque F

x

Mx

FzM

z

F

y

M

y

Figure 1.9: Contact Forces and Torques

The components of the contact force are named according to the direction of the axes, Fig. 1.9.

Non symmetric distributions of force in the contact patch cause torques around the x and yaxes. The tilting torque Mx occurs when the tire is cambered. My also contains the rollingresistance of the tire. In particular the torque around the z-axis is relevant in vehicle dynamics.It consists of two parts,

Mz = MB + MS . (1.1)

Rotation of the tire around the z-axis causes the bore torque MB. The self aligning torqueMS respects the fact that in general the resulting lateral force is not applied in the contactpoint.

1.4.3 Dynamic Rolling Radius

At an angular rotation of , assuming the tread particles stick to the track, the deflectedtire moves on a distance of x, Fig. 1.10.

With r0 as unloaded and rS = r0 r as loaded or static tire radius

r0 sin = x (1.2)

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x

r0 rS

r

x

D

deflected tire rigid wheel

vt

Figure 1.10: Dynamic Rolling Radius

andr0 cos = rS . (1.3)

hold.

If the movement of a tire is compared to the rolling of a rigid wheel, its radius rD then has tobe chosen so, that at an angular rotation of the tire moves the distance

x = rD . (1.4)

From (1.2) and (1.4) one gets

rD =r0 sin

, (1.5)

where the trivial solution rD = r0 follows from at 0.

At small, yet finite angular rotations the sine-function can be approximated by the first termsof its Taylor-Expansion. Then, (1.5) reads as

rD = r0 1

63

= r0 1 162 . (1.6)

With the according approximation for the cosine-function

rSr0

= cos = 1 1

22 or 2 = 2

1

rSr0

. (1.7)

follows from (1.3). Inserted into (1.6)

rD = r0

1

1

3

1

rSr0

=

2

3r0 +

1

3rS (1.8)

remains.

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The radius rD depends on the wheel load Fz because of rS = rS(Fz) and thus is named dy-namic tire radius. With the first approximation (1.8) it can be calculated from the undeformed

radius r0 and the steady state radius rS.At a rotation with the angular velocity , the tread particles are transported through thecontact area with the average velocity

vt = rD (1.9)

1.4.4 Toe and Camber Angle

1.4.4.1 Definitions according to DIN 70 000

The angle between the vehicle center plane in longitudinal direction and the intersection lineof the tire center plane with the track plane is named toe angle. It is positive, if the front partof the wheel is oriented towards the vehicle center plane.

The camber angle is the angle between the wheel center plane and the track normal. It ispositive, if the upper part of the wheel is inclined outwards.

1.4.4.2 Calculation

The calculation can be done via the unit vector eyR in the direction of the wheel turning axis.

For the calculation of the toe angle the unit vector eyR is described in the vehicle fixedcoordinate system F, Fig. 1.11

eyR,F =

e(1)yR,F e

(2)yR,F e

(3)yR,F

T, (1.10)

where the axes xF and zF span the vehicle center plane. The xF-axis points forward and thezF-axis points upward.

The toe angle V can then be calculated from

tan V =e(1)yR,F

e(2)yR,F

. (1.11)

The camber angle follows from the scalar product between the unit vectors in the direction ofthe wheel turning axis and in the direction of the track normal

sin = eTyR en . (1.12)

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eyR

yF

zF

xFV

eyR,F(1)

eyR,F(2) e

yR,F(3)

Figure 1.11: Toe Angle

1.4.5 Steering Geometry

1.4.5.1 Kingpin

At the steered front axle the McPherson-damper strut axis, the double wishbone axis and multi-link wheel suspension or dissolved double wishbone axis are frequently employed in passengercars, Fig. 1.12 and Fig. 1.13.

M

A

Rz

x

y

R

R

B

kingpin axis A-B

Figure 1.12: Double Wishbone Wheel Suspension

The wheel body rotates around the kingpin at steering movements.

At the double wishbone axis, the ball joints A and B, which determine the kingpin, are fixedto the wheel body.

The ball joint point A is also fixed to the wheel body at the classic McPherson wheel suspen-sion, but the point B is fixed to the vehicle body.

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B

M

A

Rz

x

y

R

R

kingpin axis A-B

M

R

z

x

y

R

R

rotation axis

Figure 1.13: McPherson and Multi-Link Wheel Suspensions

At a multi-link axle, the kingpin is no longer defined by real link points. Here, as well as withthe McPherson wheel suspension, the kingpin changes its position against the wheel body atwheel travel.

1.4.5.2 Caster and Kingpin Angle

The current direction of the kingpin can be defined by two angles within the vehicle fixedcoordinate system, Fig. 1.14.

If the kingpin is projected into the yF-, zF-plane, the kingpin inclination angle can be readas the angle between the zF-axis and the projection of the kingpin.

The projection of the kingpin into the xF-, zF-plane delivers the caster angle with the anglebetween the zF-axis and the projection of the kingpin.

With many axles the kingpin and caster angle can no longer be determined directly.

The current rotation axis at steering movements, that can be taken from kinematic calculations

here delivers the kingpin. The current values of the caster angle and the kingpin inclinationangle can be calculated from the components of the unit vector in the direction of thekingpin, described in the vehicle fixed coordinate system

tan =e

(1)S,F

e(3)S,Fand tan =

e(2)S,F

e(3)S,F(1.13)

with

eS,F =

e(1)S,F e

(2)S,F e

(3)S,F

T. (1.14)

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zF

Fz

xF

yF

eS

Figure 1.14: Kingpin and Caster Angle

1.4.5.3 Caster and Kingpin Offset

In general, the point S where the kingpin runs through the track plane does not coincide withthe contact point P, Fig. 1.15.

S

P exey

rS nK

Figure 1.15: Caster and Kingpin Offset

If the kingpin penetrates the track plane before the contact point, the kinematic kingpin offsetis positive, nK > 0.

The caster offset is positive, rS > 0, if the contact point P lies outwards of S.

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2 Tire

2.1 Contact Geometry

2.1.1 Contact Point

The current position of a wheel in relation to the fixed x0-, y0- z0-system is given by the wheelcenter M and the unit vector eyR in the direction of the wheel turning axis, Fig. 2.1.

P

eyRM

en

ex

ey

rim centre plane

local road plane

ezR

rS

P0 ab

road: z = z ( x , y )

eyR

M

en

0P

tire

0

y0

x

0

z0

*P

Figure 2.1: Contact Geometry

The irregularities of the track are described by an arbitrary function of two spatial coordinates

z = z(x, y). (2.1)

At an uneven track the contact point P can not be calculated directly. One can firstly get anestimated value with the vector

rM P = r0 ezB , (2.2)

where r0 is the undeformed tire radius and ezB is the unity vector in the z direction of thebody fixed reference system.

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The position of P with respect to the fixed system x0, y0, z0 is determined by

r0P

= r0M + rM P

, (2.3)

where the vector r0M states the position of the rim center M. Usually the point P lies not

on the track. The corresponding track point P0 follows from

r0P0,0 =

r0P,0(1)r0P,0(2)

z(r0P,0(1), r0P,0(2))

. (2.4)

In the point P0 now the track normal is calculated. Then the unit vectors in the tires circum-ferential direction and lateral direction can be calculated

ex =eyR en

| eyR en |, and ey = en ex . (2.5)

Calculating ex demands a normalization, for the unit vector in the direction of the wheelturning axis eyR is not always perpendicular to the track. The tire camber angle

= arcsin

eTyR en

(2.6)

describes the inclination of the wheel turning axis against the track normal.

The vector from the rim center M to the track point P0 is now split into three parts

rM P0 = rS ezR + a ex + b ey , (2.7)

where rS names the loaded or static tire radius and a, b are displacements in circumferentialand lateral direction.

The unit vector

ezR =ex eyR

| ex eyR |. (2.8)

is perpendicular to ex and eyR. Because the unit vectors ex and ey are perpendicular to en,the scalar multiplication of (2.7) with en results in

eTn rM P0 = rS eTn ezR or rS =

eTn rM P0eTn ezR

. (2.9)

Now also the tire deflection can be calculated

r = r0 rS , (2.10)

with r0 marking the undeflected tire radius.

The point P given by the vector

rM P = rS ezR (2.11)

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lies within the rim center plane. The transition from P0 to P takes place according to (2.7)by terms a ex and b ey, standing perpendicular to the track normal. The track normal however

was calculated in the point P0. Therefore with an uneven track P no longer lies on the track.With the newly estimated value P = P now the equations (2.4) to (2.11) can be recurreduntil the difference between P and P0 is sufficiently small.

Tire models which can be simulated within acceptable time assume that the contact patch iseven. At an ordinary passenger-car tire, the contact patch has at normal load about the sizeof approximately 20 20 cm. There is obviously little sense in calculating a fictitious contactpoint to fractions of millimeters, when later the real track is approximated in the range ofcentimeters by a plane.

If the track in the contact patch is replaced by a plane, no further iterative improvement is

necessary at the hereby used initial value.

2.1.2 Local Track Plane

A plane is given by three points. With the tire width b, the undeformed tire radius r0 and thelength of the contact area LN at given wheel load, estimated values for three track points canbe given in analogy to (2.3)

rM L =b2

eyR r0 ezB ,

rM R = b2

eyR r0 ezB ,

rM F = LN2 exB r0 ezB .

(2.12)

The points lie left, resp. right and to the front of a point below the rim center. The unitvectors exB and ezB point in the longitudinal and vertical direction of the vehicle. The wheelturning axis is given by eyR. According to (2.4) the corresponding points on the track L, Rand F can be calculated.

The vectorsrRF = r0F r0R and rRL = r0L r0R (2.13)

lie within the track plane. The unit vector calculated by

en = rRF rRL| rRF rRL |

. (2.14)

is perpendicular to the plane defined by the points L, R, and F and gives an average tracknormal over the contact area. Discontinuities which occur at step- or ramp-sized obstacles aresmoothed that way.

Of course it would be obvious to replace LN in (2.12) by the actual length L of the contactarea and the unit vector ezB by the unit vector ezR which points upwards in the wheel centerplane. The values however, can only be calculated from the current track normal. Here also aniterative solution would be possible. Despite higher computing effort the model quality cannotbe improved by this, because approximations in the contact calculation and in the tire modellimit the exactness of the tire model.

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2.1.3 Contact Point Velocity

The absolute velocity of the contact point one gets from the derivation of the position vector

v0P,0 = r0P,0 = r0M,0 + rM P,0 . (2.15)

Here r0M,0 = v0M,0 is the absolute velocity of the wheel center and rM P,0 the vector from thewheel center M to the contact point P, expressed in the inertial frame 0. With (2.11) onegets

rM P,0 =d

dt(rS ezR,0) = rS ezR,0 rS ezR,0 . (2.16)

Due to r0 = const. rS = r (2.17)

follows from (2.10).

The unit vector ezR is fixed to the wheel body. Its time derivative is then given by

ezR,0 = 0RK,0 ezR,0 (2.18)

where 0RK is the angular velocity of the wheel body RK relative to the inertial frame 0.With rM P,0 = rS ezR,0 and the relations (2.17) and (2.18), (2.16) reads as

rM P,0 = r ezR,0 + 0RK,0 rM P,0 . (2.19)

The contact point velocity is then given by

v0P,0 = v0M,0 + r ezR,0 + 0RK,0 rM P,0 (2.20)

where the velocity components from the wheel rotation have not yet been taken into account.

Because the point P lies on the track, v0P,0 must not contain a component normal to thetrack

eTn v0P = 0 . (2.21)

The tire deformation velocity is defined by this demand

r =eTn (v0M + 0RK rM P)

eTn ezR. (2.22)

Then, one gets for the velocity components in longitudinal and lateral direction

vx = eTx v0P = e

Tx (v0M + 0RK rM P) (2.23)

andvy = e

Ty v0P = e

Ty (v0M + r ezR + 0RK rM P) , (2.24)

where the term which can be cancelled in v0P by the orthogonality relation ezRex has alreadybeen omitted in (2.23).

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2.2 Tire Forces and Torques

2.2.1 Wheel Load

The vertical tire force Fz can be calculated as a function of the normal tire deflection z =eTn r and the deflection velocity z = e

Tn r

Fz = Fz(z, z) . (2.25)

Because the tire can only deliver pressure forces to the road, the restriction Fz 0 holds.

In a first approximation Fz is separated into a static and a dynamic part

Fz = F

S

z + F

D

z . (2.26)

The static part is described as a nonlinear function of the normal tire deflection

FSz = a1 z + a2 (z)2 . (2.27)

The constants a1 and a2 are calculated from the radial stiffness at nominal payload

cNz =d FSzd z

FSz =F

Nz

(2.28)

and the radial stiffness at double payload

c2Nz =d FSzd z

FSz =2F

Nz

. (2.29)

The dynamic part is approximated by

FDz = dR z , (2.30)

where dR is a constant describing the radial tire damping.

2.2.2 Longitudinal Force and Longitudinal SlipTo get some insight into the mechanism generating tire forces in longitudinal direction weconsider a tire on a flat test rig. The rim is rotating with the angular speed and the flattrack runs with speed vx. The distance between the rim center an the flat track is controlledto the loaded tire radius corresponding to the wheel load Fz, Fig. 2.2.

A tread particle enters at time t = 0 the contact area. If we assume adhesion between theparticle and the track then the top of the particle runs with the track speed vx and thebottom with the average transport velocity vt = rD . Depending on the speed differencev = rD v the tread particle is deflected in longitudinal direction

u = (rD vx) t . (2.31)

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vx

L

rD

u

umax

rD

vx

Figure 2.2: Tire on Flat Track Test Rig

The time a particle spends in the contact area can be calculated by

T =L

rD ||, (2.32)

where L denotes the contact length, and T > 0 is assured by ||.

The maximum deflection occurs when the tread particle leaves at t = T the contact area

umax = (rD vx) T = (rD vx)L

rD || . (2.33)

The deflected tread particle applies a force to the tire. In a first approximation we get

Ftx = ctx u , (2.34)

where ctx is the stiffness of one tread particle in longitudinal direction.

On normal wheel loads more than one tread particle is in contact with the track, Fig. 2.3a.The number p of the tread particles follows from

p =L

s + a. (2.35)

where s is the length of one particle and a denotes the distance between the particles.

Particles entering the contact area are undeflected on exit the have the maximum deflection.According to (2.34) this results in a linear force distribution versus the contact length, Fig. 2.3b.For p particles the resulting force in longitudinal direction is given by

Fx =1

2p ctx umax . (2.36)

With (2.35) and (2.33) this results in

Fx =1

2

L

s + a ctx (rD vx)

L

rD || . (2.37)

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c u

b)L

max

tx*

c utu*

a) c)

L/2

0r

r

L

s a

Figure 2.3: a) Particles, b) Force Distribution, c) Tire Deformation

A first approximation of the contact length L is given by

(L/2)2 = r20 (r0 r)2 , (2.38)

where r0 is the undeflected tire radius, and r denotes the tire deflection, Fig. 2.3c. Withr r0 one gets

L2 8 r0 r . (2.39)

The tire deflection can be approximated by

r = Fz/cR . (2.40)

where Fz is the wheel load, and cR denotes the radial tire stiffness. Now, (2.36) can be writtenas

Fx = 4 r0s + a

ct

x

cRFz rD vx

rD ||. (2.41)

The non-dimensional relation between the sliding velocity of the tread particles in longitudinaldirection vSx = vx rD and the average transport velocity rD || is the longitudinal slip

sx =(vx rD )

rD ||. (2.42)

In this first approximation the longitudinal force Fx is proportional to the wheel load Fz andthe longitudinal slip sx

Fx = k Fz sx , (2.43)

where the constant k collects the tire properties r0, s, a, ctx and cR.

The relation (2.43) holds only as long as all particles stick to the track. At average slip valuesthe particles at the end of the contact area start sliding, and at high slip values only the partsat the beginning of the contact area stick to the road, Fig. . 2.4.

The resulting nonlinear function of the longitudinal force Fx versus the longitudinal slip sx canbe defined by the parameters initial inclination (longitudinal stiffness) dF0x , location s

Mx and

magnitude of the maximum FMx , start of full sliding sGx and the sliding force F

Gx , Fig. 2.5.

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L

adhesion

Fxt


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L

adhesion

Fy

small slip values

Ladhesion

Fy

sliding

moderate slip values

L

sliding

Fy

large slip values

n

F = k F sy ** y F = F f ( s )y * y F = Fy Gz z

Figure 2.6: Lateral Force Distribution over Contact Area

The resulting lateral force Fy with the dynamic tire offset or pneumatic trail n as a levergenerates the self aligning torque

MS = n Fy . (2.46)

The lateral force Fy as well as the dynamic tire offset are functions of the lateral slip sy.Typical plots of these quantities are shown in Fig. 2.7. Characteristic parameters for the lateral

Fy

yM

yG

dFy0

sysysyM G

F

Fadhesion

adhesion/sliding

full sliding

adhesionadhesion/sliding

n/L

0

sysyGsy

0

(n/L)

adhesion

adhesion/sliding

M

sysyGsy0

S

full sliding

full sliding

Figure 2.7: Typical Plot of Lateral Force, Tire Offset and Self Aligning Torque

force graph are initial inclination (cornering stiffness) dF0y , location sMy and magnitude of the

maximum FMy , begin of full sliding sGy , and the sliding force F

Gy .

The dynamic tire offset has been normalized by the length of the contact area L. The initialvalue (n/L)0 as well as the slip values s

0y and s

Gy characterize the graph sufficiently.

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2.2.4 Generalized Tire Characteristics

The longitudinal force as a function of the longitudinal slip Fx = Fx(sx) and the lateral forcedepending on the lateral slip Fy = Fy(sy) can be defined by their characteristic parametersinitial inclination dF0x , dF

0y , location s

Mx , s

My and magnitude of the maximum F

Mx , F

My as

well as sliding limit sGx , sGy and sliding force F

Gx , F

Gy , Fig. 2.8.

Fy

sxssy

G

FG

M

FM

dF0

F(s)

dF

G

y

Fy FyM

GsyMsy

0

Fy

sy

dFx0

FxM Fx

GFx

sxM

sxG

sx

Fx

s

s

Figure 2.8: Generalized Tire Characteristics

When experimental tire values are missing, the model parameters can be pragmatically es-timated by adjustment of the data of similar tire types. Furthermore, due to their physicalsignificance, the parameters can subsequently be improved by means of comparisons betweenthe simulation and vehicle testing results as far as they are available.

During general driving situations, e.g. acceleration or deceleration in curves, longitudinal sx and

lateral slip sy appear simultaneously. The combination of the more or less differing longitudinaland lateral tire forces requires a normalization process. One way to perform the normalizationis described in the following.

The longitudinal slip sx and the lateral slip sy can vectorially be added to a generalized slip

s =

sxsx

2+

sysy

2, (2.47)

where the slips sx and sy were normalized in order to perform their similar weighting in s. Fornormalizing, the normation factors sx and sy are calculated from the location of the maxima

sMx , sMy the maximum values FMx , FMy and the initial inclinations dF0x , dF0x .

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Similar to the graphs of the longitudinal and lateral forces the graph of the generalized tireforce is defined by the characteristic parameters dF0, sM, FM, sG and FG. The parameters

are calculated from the corresponding values of the longitudinal and lateral force

dF0 =

(dF0x sx cos )

2 +

dF0y sy sin 2

,

sM =

sMxsx

cos

2+

sMysy

sin

2,

FM =

(FMx cos )

2 +

FMy sin

2

,

sG = sGx

sxcos

2+ sGy

sysin

2,

FG =

(FGx cos )

2 +

FGy sin 2

,

(2.48)

where the slip normalization have also to be considered at the initial inclination. The angularfunctions

cos =sx/sx

sand sin =

sy/sys

(2.49)

grant a smooth transition from the characteristic curve of longitudinal to the curve of lateralforces in the range of = 0 to = 90.

The function F = F(s) is now described in intervals by a broken rational function, a cubicpolynomial and a constant FG

F(s) =

sM dF0

1 +

+ F0

sM

FM 2

, = ssM

, 0 s sM ;

FM (FM FG) 2 (3 2 ) , =s sM

sG sM, sM < s sG ;

FG

, s > sG

.

(2.50)

When defining the curve parameters, one just has to make sure that the condition dF0 2 FM

sM

is fulfilled, because otherwise the function has a turning point in the interval 0 < s sM.

Longitudinal and lateral force now follow from the according projections in longitudinal andlateral direction

Fx = F cos and Fy = F sin . (2.51)

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2.2.5 Wheel Load Influence

The resistance of a real tire against deformations has the effect that with increasing wheel loadthe distribution of pressure over the contact area becomes more and more uneven. The treadparticles are deflected just as they are transported through the contact area. The pressurepeak in the front of the contact area cannot be used, for these tread particles are far awayfrom the adhesion limit because of their small deflection. In the rear of the contact area thepressure drop leads to a reduction of the maximally transmittable friction force. With risingimperfection of the pressure distribution over the contact area, the ability to transmit forcesof friction between tire and road lessens.

In practice, this leads to a digressive influence of the wheel load on the characteristic curvesof longitudinal and lateral forces.

Longitudinal Force Fx Lateral Force Fy

Fz = 3.2 kN Fz = 6.4 kN Fz = 3.2 kN Fz = 6.4 kN

dF0x = 90 kN dF 0

x = 160 kN dF 0

y = 70 kN dF 0

y = 100 kN

sMx = 0.090 sMx = 0.110 s

My = 0.180 s

My = 0.200

FMx = 3.30 kN FM

x = 6.50 kN FM

y = 3.10 kN FM

y = 5.40 kN

sGx = 0.400 sGx = 0.500 s

Gy = 0.600 s

Gy = 0.800

FGx = 3.20 kN FGx = 6.00 kN FGy = 3.10 kN FGy = 5.30 kN

Table 2.1: Characteristic Tire Data with Digressive Wheel Load Influence

In order to respect this fact in a tire model, the characteristic data for two nominal wheelloads FNz and 2 F

Nz are given in Tab. 2.1.

From this data the initial inclinations dF0x , dF0

y , the maximal forces FM

x , FM

x and the slidingforces FGx , F

My for arbitrary wheel loads Fz are calculated by quadratic functions. For the

maximum longitudinal force it reads as

FMx (Fz) =Fz

FNz

2 FMx (F

Nz )

12

FMx (2FN

z ) FMx (F

Nz )

12

FMx (2FN

z )Fz

FNz

. (2.52)

The location of the maxima sMx , sMy , and the slip values, s

Gx , s

Gy , at which full sliding ap-

pears, are defined as linear functions of the wheel load Fz. For the location of the maximumlongitudinal force this results in

sMx (Fz) = sMx (F

Nz ) +

sMx (2F

Nz ) s

Mx (F

Nz ) Fz

FNz 1

. (2.53)

With the numeric values from Tab. 2.1 a slight shift of the maxima towards higher slip valuesis also modelled, Fig. 2.9.

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-0.5 0 0.5

-8000

-6000

-4000

-2000

0

2000

4000

6000

8000

Fx

= Fx

(sx): Parameter F

z

Fz

-0.5 0 0.5

-8000

-6000

-4000

-2000

0

2000

4000

6000

8000

Fy = Fy (s y): Parameter Fz

Fz

Figure 2.9: Wheel Load Influence to Tire Forces

0.5 0 0.54000

3000

2000

1000

0

1000

2000

3000

4000

Fx

= Fx(sx): Parameter s

y

sy

sy = 0.0, 0.0375, 0.075, 0.1125, 0.15

0.5 0 0.54000

3000

2000

1000

0

1000

2000

3000

4000

Fy

= Fy(s

y): Parameter s

x

sx

sx = 0.0, 0.0375, 0.075, 0.1125, 0.15

Figure 2.10: Tire Forces vs. Longitudinal and Lateral Slip: Fz = 3.2 kN

The bilateral influence of longitudinal sx and lateral slip sy on the longitudinal Fx and lateralforce Fy is depicted in Fig. 2.10.

With the 20 parameters, which are, according to Tab. 2.1, necessary for the definition of thecharacteristic curves of longitudinal and lateral force, the tire model can be easily fitted tomeasured characteristics. Because for description of the characteristic curves of longitudinaland lateral force only characteristic curve parameters are used, a desired tire behavior can alsobe constructed in a convenient manner.

2.2.6 Self Aligning Torque

According to Eq. (2.46) the self aligning torque can be calculated via the dynamic tire offset.

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The approximation as a function of the lateral slip is done by a line and a cubic polynomial

n

L=

(n/L)0 (1 |sy|/s0Q) |sy| s0Q

(n/L)0|sy| s

0Q

s0Q

sEQ |sy|

sEQ s0Q

2s0Q < |sy| s

EQ

0 |sy| > sEQ

(2.54)

The cubic polynomial reaches the sliding limit sEQ with a horizontal tangent and is continuedwith the value zero.

The characteristic curve parameters, which are used for the description of the dynamic tireoffset, are at first approximation not wheel load dependent. Similar to the description of the

characteristic curves of longitudinal and lateral force, here also the parameters for single anddouble wheel load are given.

The calculation of the parameters of arbitrary wheel loads is done similar to Eq. (2.53) bylinear inter- or extrapolation.

Tire Offset Parameters

Fz = 3.2 kN Fz = 6.4 kN

(n/L)0 = 0.150 (n/L)0 = 0.130

s0y = 0.200 s0y = 0.230

sEy = 0.500 sEy = 0.450

-0.5 0 0.5-150

-100

-50

0

50

100

150M = Mz z(sy): Parameter Fz

Fz

Figure 2.11: Self Aligning Torque: Fz = 0, 2, 4, 6, 8 kN

The value of(n/L)0 can be estimated very well. At small values of lateral slip sy 0 one gets

at first approximation a triangular distribution of lateral forces over the contact area lengthcf. Fig. 2.6. The working point of the resulting force (dynamic tire offset) is then given by

n(Fz 0, sy =0) =1

6L . (2.55)

The value n = 16

L can only serve as reference point, for the uneven distribution of pressure inlongitudinal direction of the contact area results in a change of the deflexion profile and thedynamic tire offset.

The self aligning torque in Fig. 2.11 has been calculated with the tire parameters from Tab. 2.1,the tire stiffness cR = 180 kN/m and the undeflected tire radius r0 = 0.293 m. The digressiveinfluence of the wheel load on the lateral force can be seen here as well.

28


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With the parameters for the description of the tire offset it has been assumed that at doublepayload Fz = 2 F

Nz the related tire offset reaches the value of (n/L)0 = 0.13 at sy = 0.

Because for Fz = 0 the value 1/6 0.17 can be assumed, a linear interpolation provides thevalue (n/L)0 = 0.15 for Fz = F

Nz . The slip value s

0y, at which the tire offset passes the x-axis,

has been estimated. Usually the value is somewhat higher than the position of the lateral forcemaximum. With rising wheel load it moves to higher values. The values for sEy are estimatedtoo.

2.2.7 Camber Influence

If the wheel rotation axis is inclined against the road a lateral force appears, dependent onthe inclination angle. At a non-vanishing camber angle, = 0 the tread particles possess a

lateral velocity when entering the contact area, which is dependent on wheel rotation speed and the camber angle , Fig. 2.12. At the center of the contact area (contact point) this

eyR

en

ex

velocity

rimcentreplane

ey deflection profile

F = Fy y (sy): Parameter

-0.5 0 0.5-4000

-3000

-2000

-1000

0

1000

2000

3000

4000

Figure 2.12: Cambered Tire Fy() at Fz = 3.2 kN and = 0, 2 , 4 , 6 , 8

component vanishes and at the end of the contact area it is of the same value but opposingthe component at the beginning of the contact area. At normal friction and even distribution

of pressure in the longitudinal direction of the contact area one gets a parabolic deflectionprofile, which is equal to the average deflection

y =L

2

sin

R || s

1

6L (2.56)

s defines a camber-dependent lateral slip. A solely lateral tire movement without camberresults in a linear deflexion profile with the average deflexion

yvy =vy

R || sy

1

2L . (2.57)

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a comparison of Eq. (2.56) to Eq. (2.57) shows, that with sy =13

s the lateral camber slips can be converted to the equivalent lateral slip s

y .

In normal driving operation, the camber angle and thus the lateral camber slip are limited tosmall values. So the lateral camber force can be calculated over the initial inclination of thecharacteristic curve of lateral forces

Fy dF0

y sy . (2.58)

If the global inclination dFy Fy/sy is used instead of the initial inclination dF0

y , one getsthe camber influence on the lateral force as shown in Fig. 2.12.

The camber angle influences the distribution of pressure in the lateral direction of the contactarea, and changes the shape of the contact area from rectangular to trapezoidal. It is thusextremely difficult if not impossible to quantify the camber influence with the aid of simple

models. Therefore a plain approximation has been used, which still describes the camberinfluence rather exactly.

2.2.8 Bore Torque

If the wheel rotation 0R has a component in direction of the track normal en

n = eTn 0R = 0 . (2.59)

a very complicated deflection profile of the tread particles in the contact area occurs. By asimple approach the resulting bore torque can be calculated by the parameter of the longitudinal

force characteristics.Fig. 2.13 shows the contact area at zero camber ( = 0) and small slip values (sx 0,sy 0). The contact area is separated into small stripes of width dy. The longitudinal slip ina stripe at position y is then given by

sx(y) = (n y)

rD ||. (2.60)

For small slip values the nonlinear tire force characteristics can be linearized. The longitudinalforce in the stripe can then be approximated by

Fx(y) =d Fx

d sx

d sx

d y

y . (2.61)

With (2.60) one gets

Fx(y) =d Fxd sx

nrD ||

y . (2.62)

The forces Fx(y) generate a bore torque in the contact point P

MB = 1

B

+B2

B2

y Fx(y) dy = 1

B

+B2

B2

yd Fxd sx

nrD ||

y dy

=1

12 B2

d Fxd sx

nrD || =

1

12 Bd Fxd sx

B

rD

n| | ,

(2.63)

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y

B

L

U(y)

dy x

P

n

Q

contactarea

y

B

L

U

dy x

P

n

G

-UG

contactarea

Figure 2.13: Bore Torque generated by Longitudinal Forces

where

sB =n| |

(2.64)

can be considered as bore slip. Via dFx/dsx the bore torque takes into account the actualfriction and slip conditions.

The bore torque calculated by (2.63) is only a first approximation. At large bore slips thelongitudinal forces in the stripes are limited by the sliding values. Hence, the bore torque islimited by

MmaxB = 2 1B

+B2

0

y FGx dy = 14B FGx , (2.65)

where FGx denotes the longitudinal sliding force.

2.2.9 Typical Tire Characteristics

The tire model TMeasy1 approximates the characteristic curves Fx = Fx(sx), Fy = Fy()and Mz = Mz() quite well even for different wheel loads Fz, Fig. 2.14.

TMeasy is able to handle the different tire types in a suitable manner. The soft truck tireof type Radial 315/80 R22.5 at p=8.5 bar (right column) and the large differences betweenlongitudinal and lateral force characteristics at the passenger car tire of type Radial 205/50R15, 6J at p=2.0 bar (left column) are represented without any considerable fitting problems.

The one-dimensional characteristics are automatically converted to a two-dimensional combi-nation characteristics which are shown in Fig. 2.15.

1 Hirschberg, W; Rill, G. Weinfurter, H.: User-Appropriate Tyre-Modelling for Vehicle Dynamics inStandard and Limit Situations. Vehicle System Dynamics 2002, Vol. 38, No. 2, pp. 103-125. Lisse:Swets & Zeitlinger.

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-40 -20 0 20 40-6

-4

-2

0

2

4

6

sx[%]

Fx

[kN]

1.8 kN3.2 kN4.6 kN5.4 kN

-6

-4

-2

0

2

4

6

Fy

[kN]

1.8 kN3.2 kN4.6 kN6.0 kN

-20 -10 0 10 20-150

-100

-50

0

50

100

150

[o]

Mz

[N

m]

1.8 kN3.2 kN

4.6 kN6.0 kN

-40 -20 0 20 40

-40

-20

0

20

40

sx[%]

Fx

[kN]

10 kN20 kN30 kN40 kN50 kN

-40

-20

0

20

40

Fy

[kN]

10 kN20 kN30 kN40 kN

-20 -10 0 10 20-1500

-1000

-500

0

500

1000

1500

Mz

[N

m]

18.4 kN36.8 kN55.2 kN

[o]

Figure 2.14: Tire Characteristics at Different Wheel Loads: Meas., TMeasy

-4 -2 0 2 4

-3

-2

-1

0

1

2

3

Fx

[kN]

Fy

[kN]

-20 0 20-30

-20

-10

0

10

20

30

Fx

[kN]

Fy

[kN]

|sx| = 1, 2, 4, 6, 10, 15%; || = 1, 2, 4, 6, 10, 14

Figure 2.15: Two-dimensional Tire Characteristics at Fz = 3.2 kN / Fz = 35 kN

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3 Longitudinal Dynamics

3.1 Accelerating and Braking

3.1.1 Simple Model

S

h

1a2a

mg

v

Fx1Fx2

Fz2 Fz1

Figure 3.1: Simple Model

The forces in the wheel contact points are combined into one vertical and one circumferentialforce per axle. Aerodynamic forces (drag, positive and negative lift) are neglected.

The road runs horizontally and be ideally even. Then no vertical acceleration and no pitchacceleration around the lateral vehicle axle occur:

0 = Fz1 + Fz2 m g (3.1)

and0 = Fz1 a1 Fz2 a2 + (Fx1 + Fx2) h . (3.2)

The linear momentum in longitudinal direction results in

m v = Fx1 + Fx2 , (3.3)

where v indicates the vehicles acceleration. This are only three equations for the four unknownforces Fx1, Fx2, Fz1, Fz2.

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If we insert (3.3) in (3.2) we can eliminate two unknowns by one stroke

0 = Fz1 a1 Fz2 a2 + m v h . (3.4)

The equations (3.1) and (3.4) can now be resolved for the axle loads

Fz1 = m g

a2

a1 + a2

h

a1 + a2

v

g

(3.5)

and

Fz2 = m g

a1

a1 + a2+

h

a1 + a2

v

g

. (3.6)

The weight mg is distributed among the axles according to position of the center of gravity.

When accelerating v > 0, the front axle is relieved, as is the rear axle when decelerating v < 0.

3.1.2 Maximum Acceleration

Ordinary vehicles can only deliver pressure forces to the road. According to equation (3.6), theconditions Fz1 0 and Fz2 0 lead to the tilting conditions

a1h

v

g

a2h

. (3.7)

The maximum acceleration is also limited by the friction conditions

|Fx1| Fz1 and |Fx2| Fz2 (3.8)

where the same friction coefficient has been assumed at front and rear axle.

In the limit case|Fx1| = Fz1 and |Fx2| = Fz2 (3.9)

the maximally achievable acceleration resp. deceleration follows from (3.3) and (3.1)

|vmax| = g . (3.10)

According to the vehicle dimensions and the friction values the maximal acceleration or decel-eration is restricted either by (3.7) or by (3.10).

3.1.3 Drive Torque at Single Axle

With the rear axle driven in limit situations

Fx1 = 0 and Fx2 = Fz2 . (3.11)

holds. With this, one gets from (3.3)

m v0+ = Fz2 , (3.12)

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where the subscript 0+ indicates that the front axle is neither driven nor braked and the rearaxle is driven. Using 3.6 one gets

m v0+ = m g

a1

a1 + a2+

h

a1 + a2

v0+g

. (3.13)

Hence, the maximum acceleration for a rear wheel driven vehicle is given by

v0+g

=

1 h

a1 + a2

a1a1 + a2

. (3.14)

With front wheel drive one gets with

Fx1 = Fz1 and Fx2 = 0 (3.15)

the maximum acceleration

v+0g

=

1 + h

a1 + a2

a2a1 + a2

, (3.16)

where the subscript +0 indicates now a driven front axle and a rear axle which is neither drivennor braked.

3.1.4 Braking at Single Axle

With an unbraked rear axle in the limit case it holds

Fx1 = Fz1 and Fx2 = 0 . (3.17)

With that one gets from (3.3)m v0 = Fz1 , (3.18)

where the subscript 0 indicates a braked front axle and a rear axle which is neither driven nor

braked. With 3.5 one gets

m v0 = m g

a2

a1 + a2

h

a1 + a2

v0g

(3.19)

orv0

g=

1 h

a1 + a2

a2a1 + a2

. (3.20)

If only the rear axle is braked, using

Fx1 = 0 and Fx2 = Fz2 , (3.21)

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one gets now the maximal deceleration

v0g = 1 +

h

a1 + a2

a1a1 + a2, (3.22)

where the subscript 0 indicates a front axle which is neither driven nor braked and a brakedrear axle.

3.1.5 Example

Typical values for a passenger car are:

a1 = 1.25 m; a2 = 1.25 m; h = 0.55 m

With a friction coefficient of = 1 the maximal accelerations calculated from (3.14) and(3.16) result in

v0+g

=1

1 10.55

1.25 + 1.25

1.25

1.25 + 1.25= 0.64

andv+0

g=

1

1 + 10.55

1.25 + 1.25

1.25

1.25 + 1.25= 0.41

The maximal decelerations follow from (3.20) and (3.22)

v0g

= 1

1 10.55

1.25 + 1.25

1.25

1.25 + 1.25= 0.64

andv0

g=

1

1 + 10.55

1.25 + 1.25

1.25

1.25 + 1.25= 0.41

Because a load distribution of 50/50 between the axles was assumed, the maximal accelerationsan decelerations have the same absolute value.

If only the front axle is braked, the maximal deceleration still is about 2/3 of the maximallypossible of v/g = = 1. Braking only the rear axle is often not sufficient, because hereonly about 40% of the maximally possible deceleration can be achieved.

In vehicles with front drive the maximal acceleration is augmented by shifting the center ofgravity to the front. On the other hand the maximal acceleration can also be augmented byshifting the center of gravity to the rear.

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3.1.6 Optimal Distribution of Drive and Brake Forces

The sum of the circumferential forces accelerates or decelerates the vehicle. In dimensionlessstyle (3.3) reads

v

g=

Fx1m g

+Fx2m g

. (3.23)

A certain acceleration or deceleration can only be achieved by different combinations of thecircumferential forces Fx1 and Fx2. According to (3.9) the circumferential forces are limitedby wheel load and friction.

The optimal combination of Fx1 and Fx2 is achieved, when front and rear axle have the sameskid resistance.

Fx1 = Fz1 and Fx2 = Fz2 . (3.24)

With (3.5) and (3.6) one gets

Fx1m g

=

a2h

v

g

h

a1 + a2(3.25)

andFx2m g

=

a1h

+v

g

h

a1 + a2. (3.26)

With (3.25) and (3.26) one gets from (3.23)

v

g

= , (3.27)

where it has been assumed that Fx1 and Fx2 have the same sign.

With (3.27 inserted in (3.25) and (3.26) one gets

Fx1m g

=v

g

a2h

v

g

h

a1 + a2(3.28)

andFx2m g

=v

g

a1h

+v

g

h

a1 + a2. (3.29)

remain.

Depending on the desired acceleration v > 0 or deceleration v < 0 the circumferential forcesthat grant the same skid resistance at both axles can now be calculated.

Fig.3.2 shows the curve of optimal drive and brake forces for typical passenger car values. Atthe tilting limits v/g = a1/h and v/g = +a2/h no circumferential forces can be deliveredat the lifting axle.

The initial gradient only depends on the steady state distribution of wheel loads. From (3.28)and (3.29) it follows

dFx1m g

dv

g

= a2

h

2v

gh

a1 + a2

(3.30)

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h=0.551

2

-2-10dFx2

0

a1=1.15

a2=1.35

=1.20

a2/h

-a1/hFx1/mg

braking

tilting limits

driving

dFx1

Fx2/mg

Figure 3.2: Optimal Distribution of Drive and Brake Forces

and

dFx2m g

dv

g

=

a1h

+ 2v

g

h

a1 + a2. (3.31)

For v/g = 0 the initial gradient remains as

d Fx2d Fx1 0

=a1a2

. (3.32)

3.1.7 Different Distributions of Brake Forces

In practice it is tried to approximate the optimal distribution of brake forces by constantdistribution, limitation or reduction of brake forces as good as possible. Fig. 3.3.

When braking, the vehicles stability is dependent on the potential of lateral force (corneringstiffness) at the rear axle. In practice, a greater skid (locking) resistance is thus realized atthe rear axle than at the front axle. Because of this, the brake force balances in the physicallyrelevant area are all below the optimal curve. This restricts the achievable deceleration, specially

at low friction values.

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Fx1/mg

Fx2/mg constant

distribution

Fx1/mg

Fx2/mg

limitation reduction

Fx1/mg

Fx2/mg

Figure 3.3: Different Distributions of Brake Forces

Because the optimal curve is dependent on the vehicles center of gravity additional safetieshave to be installed when designing real distributions of brake forces.

Often the distribution of brake forces is fitted to the axle loads. There the influence of theheight of the center of gravity, which may also vary much on trucks, remains unrespected and

has to be compensated by a safety distance from the optimal curve.

Only the control of brake pressure in anti-lock-systems provides an optimal distribution ofbrake forces independent from loading conditions.

3.1.8 Anti-Lock-Systems

Lateral forces can only be scarcely transmitted, if high values of longitudinal slip occur whendecelerating a vehicle. Stability and/or steerability is then no longer given.

By controlling the brake torque, respectively brake pressure, the longitudinal slip can be re-stricted to values that allow considerable lateral forces.

The angular wheel acceleration is used here as control variable. Angular wheel accelerationsare derived from the measured angular wheel speeds by differentiation. With a longitudinal slipof sL = 0 the rolling condition is fulfilled. Then

rD = x (3.33)

holds, where rD labels the dynamic tyre radius and x is the vehicles acceleration. Accordingto (3.10), the maximum acceleration/deceleration of a vehicle is dependent on the frictioncoefficient, |x| = g. With a known friction coefficient a simple control law can be realized

for every wheel||

1

rD|x| (3.34)

. Because until today no reliable possibility to determine the local friction coefficient betweentyre and road has been found, useful information can only be gained from (3.34) at optimalconditions on dry road. Therefore the longitudinal slip is used as a second control variable.

In order to calculate longitudinal slips, a reference speed is estimated from all measured wheelspeeds which is then used for the calculation of slip at all wheels. This method is too impreciseat low speeds. Below a limit velocity no control occurs therefore. Problems also occur whenfor example all wheels lock simultaneously which may happen on icy roads.

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The control of the brake torque is done via the brake pressure which can be increased, held ordecreased by a three-way valve. To prevent vibrations, the decrement is usually made slower

than the increment.To prevent a strong yaw reaction, the select low principle is often used with -split brakingat the rear axle. The break pressure at both wheels is controlled the wheel running on lowerfriction. Thus the brake forces at the rear axle cause no yaw torque. The maximally achievabledeceleration however is reduced by this.

3.2 Drive and Brake Pitch

3.2.1 Plane Vehicle Model

R2

R1

MB1

MA1

MB2

MA2

A

xA

zA

MB1

MB2

MA1

MA2

z2

z1

FF2

FF1

Fz1 Fx1

Fz2 Fx2

a1

R

a2

hR

Figure 3.4: Plane Vehicle Model

The vehicle model drawn in Fig. 3.4 consists of five rigid bodies. The body has three degreesof freedom: Longitudinal motion xA, vertical motion zA and pitch A. The coordinates z1 andz2 describe the vertical motions of wheel and axle bodies relative to the body. The longitudinaland rotational motions of the wheel bodies relative to the body can be described via suspensionkinematics as functions of the vertical wheel motion:

x1 = x1(z1) , 1 = 1(z1) ;

x2 = x2(z2) , 2 = 2(z2) .(3.35)

The rotation angles R1 and R2 describe the wheel rotations relative to the wheel bodies.

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The forces between wheel body and vehicle body are labelled FF1 and FF2. At the wheelstorques of drive MA1, MA2 and brake MB1, MB2, circumferential forces Fx1, Fx2 and the

wheel loads Fz1, Fz2 apply. The brake torques are supported directly by the wheel bodies, thedrive torques are supported directly by the vehicle via the drive train. The forces and torquesthat apply to the single bodies are listed in the last column of the tables 3.1 and 3.2.

3.2.2 Position

Position vector and rotation matrix

r0A,0 = xA0

R + hR + zA ; A0A = cos A 0 sin A

0 1 0

sin A 0 cos A (3.36)describe the position of the bodys center of gravity relative to the earth fixed coordinatesystem 0.

With R = const. and hR = const one can immediately get from this the velocity and angularvelocity of the body.

v0A,0 =

xA00

+

00

zA

; 0A,0 =

0

A0

. (3.37)

The position of the wheel bodies is given by

r0RK1,0 = r0A,0 + A0A rARK1,A with rARK1,A =

a1 + x10

hR + z1

(3.38)

and

A0RK1 = A0A AARK1 with AARK1 =

cos 1 0 sin 1

0 1 0 sin 1 0 cos 1

(3.39)

as well as

r0RK2,0 = r0A,0 + A0A rARK2,A with rARK2,A =

a2 + x20

hR + z2

(3.40)

and

A0RK2 = A0A AARK2 with AARK2 =

cos 2 0 sin 20 1 0

sin 2 0 cos 2

(3.41)

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With well balanced wheels the position vectors rARK1,A and rARK2,A also tell the position ofthe wheel pivots. If toe and camber angle are neglected, the following holds for the wheels

rotation matrices:

A0R1 = A0A AARK1 ARKR1 with ARRK1 =

cos R1 0 sin R10 1 0

sin R1 0 cos R1

, (3.42)

A0R2 = A0A AARK2 ARKR2 with ARRK2 =

cos R2 0 sin R20 1 0

sin R2 0 cos R2

. (3.43)

3.2.3 Linearization

At small rotational motions of the body and small spring motions one gets for the speed ofthe wheel bodies and wheels

v0RK1,0 = v0R1,0 =

xA0

0

+

00

zA

+

hR A0

a1 A

+

x1z1 z10

z1

; (3.44)

v0RK2,0 = v0R2,0 = xA0

0 + 00

zA + hR A

0

+a2 A + x2z2

z20

z2 . (3.45)The angular velocities of the wheel bodies and wheels are given by

0RK1,0 =

0A

0

+

01

0

and 0R1,0 =

0A

0

+

01

0

+

0R1

0

(3.46)

as well as

0RK2,0 = 0

A0 +

0

20 and 0R2,0 =

0

A0 +

0

20 +

0

R20

(3.47)

With the generalized velocities

z =

xA zA A 1 R1 2 R2T

(3.48)

the velocities and angular velocities (3.37), (3.44), (3.45), (3.46), (3.47) can be written as

v0i =7

j=1v0i

zjzj and 0i =

7

j=10i

zjzj (3.49)

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3.2.4 Equations of Motion

The partial velocitiesv0izj and partial angular velocities

0izj for the five bodies i =1(1)5 and

for the 7 generalized speeds j =1(1)7 are arranged in the tables 3.1 and 3.2. With the aid

partial velocities v0i/zj applied forces

bodies xA zA A z1 R1 z2 R2 Fe

i

chassismA

100

001

000

000

000

000

000

00

FF1+FF2mAg

wheel bodyfrontmRK1

100

001

hR0

a1

x1z1

01

000

000

000

00

FF1mRK1g

wheelfrontmR1

100

001

hR0

a1

x1z1

01

000

000

000

Fx10

Fz1mR1g

wheel bodyrear

mRK2

100

001

hR0

a2

000

000

x2z2

01

000

00

FF2mRK2g

wheelrearmR2

100

001

hR0

a2

000

000

x2z2

01

000

Fx20

Fz2mR2g

Table 3.1: Partial Velocities and Applied Forces

partial angular velocities 0i/zj applied torques

bodies xA zA A z1 R1 z2 R2 Mei

chassisA

000

000

010

000

000

000

000

0MA1MA2a1 FF1+a2 FF2

0

wheel bodyfrontRK1

000

000

010

01z1

0

000

000

000

0MB1

0

wheelfrontR1

000

000

010

01z1

0

010

000

000

0MA1MB1R Fx1

0

wheel bodyrear

RK2

000

000

010

000

000

02z2

0

000

0MB2

0

wheelrearR2

000

000

010

000

000

02z2

0

010

0MA2MB2R Fx2

0

Table 3.2: Partial Angular Velocities and Applied Torques

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of the partial velocities and partial angular velocities the elements of the mass matrix M andthe components of the vector of generalized forces and torques Q can be calculated.

M(i, j) =5

k=1

v0kzi

Tmk

v0kzj

+5

k=1

0kzi

Tk

0kzj

; i, j = 1(1)7 ; (3.50)

Q(i) =5

k=1

v0kzi

TFek +

5k=1

0kzi

TMek ; i = 1(1)7 . (3.51)

The equations of motion for the plane vehicle model are then given by

Mz = Q . (3.52)

3.2.5 Equilibrium

With the abbreviations

m1 = mRK1 + mR1 ; m2 = mRK2 + mR2 ; mG = mA + m1 + m2 (3.53)

andh = hR + R (3.54)

The components of the vector of generalized forces and torques read as

Q(1) = Fx1 + Fx2 ;

Q(2) = Fz1 + Fz2 mG g ;

Q(3) = a1Fz1 + a2Fz2 h(Fx1 + Fx2) + a1 m1 g a2 m2 g ;

(3.55)

Q(4) = Fz1 FF1 +x1z1

Fx1 m1 g +1z1

(MA1 R Fx1) ;

Q(5) = MA1 MB1 R Fx1 ;(3.56)

Q(6) = Fz2 FF2 +x2z2

Fx2 m2 g +2z2

(MA2 R Fx2) ;

Q(7) = MA2 MB2 R Fx2 .

(3.57)

Without drive and brake forces

MA1 = 0 ; MA2 = 0 ; MB1 = 0 ; MB2 = 0 (3.58)

from (3.55), (3.56) and (3.57) one gets the steady state circumferential forces, the springpreloads and wheel loads

F0x1 = 0 ; F0

x2 = 0 ;

F0F1 =b

a+bmA g ; F

0F2 =

aa+b

mA g ;

F0z1 = m1g + ba+b mA g ; F0z2 = m2g + aa+b mA g .

(3.59)

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3.2.6 Driving and Braking

Assuming that on accelerating or decelerating the vehicle xA = 0 the wheels neither slip norlock,

R R1 = xA hR A +x1z1

z1 ;

R R2 = xA hR A +x2z2

z2 .(3.60)

holds. In steady state the pitch motion of the body and the vertical motion of the wheels reachconstant values

A = stA = const. ; z1 = z

st1 = const. ; z2 = z

st2 = const. (3.61)

and (3.60) simplifies to

R R1 = xA ; R R2 = xA . (3.62)

With(3.61), (3.62) and (3.54) the equation of motion (3.52) results in

mG xA = Fa

x1 + Fa

x2 ;

0 = Faz1 + Fa

z2 ;

hR(m1+m2) xA + R1xAR

+ R2xAR

= a Faz1 + b Fa

z2 (hR + R)(Fa

x1 + Fa

x2) ;

(3.63)x1z1

m1 xA +1z1

R1xAR

= Faz1 Fa

F1 +x1z1

Fax1 +1z1

(MA1 R Fa

x1) ;

R1 xAR = MA1 MB1 R Fa

x1 ;(3.64)

x2z2

m2 xA +2z2

R2xAR

= Faz2 Fa

F2 +x2z2

Fax2 +2z2

(MA2 R Fa

x2) ;

R2xAR

= MA2 MB2 R Fa

x2 ;(3.65)

where the steady state spring forces, circumferential forces and wheel loads have been separatedinto initial and acceleration-dependent terms

Fstxi = F0

xi + Fa

xi ; Fst

zi = F0

zi + Fa

zi ; Fst

F i = F0

F i + Fa

F i ; i = 1, 2 . (3.66)

With given torques of drive and brake the vehicle acceleration xA, the wheel forces Fa

x1, Fa

x2,Faz1, Fa

z2 and the spring forces Fa

F1, Fa

F2 can be calculated from (3.63), (3.64) and (3.65)

Via the spring characteristics which have been assumed as linear the acceleration-dependentforces also cause a vertical displacement and pitch motion of the body

FaF1 = cA1 za1 ,

FaF2 = cA2 za2 ,

Faz1 = cR1 (zaA a

aA + z

a1) ,

Faz2 = cR2 (zaA + b

aA + z

a2) .

(3.67)

besides the vertical motions of the wheels.

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Vehicle Dynamics Notes