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VE
HIC
LE D
YN
AM
ICS FACHHOCHSCHULE REGENSBURG
UNIVERSITY OF APPLIED SCIENCESHOCHSCHULE FÜR
TECHNIKWIRTSCHAFT
SOZIALES
LECTURE NOTESProf. Dr. Georg Rill© October 2005
download: http://homepages.fh-regensburg.de/%7Erig39165/
Contents
Contents I
1 Introduction 11.1 Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2.1 Vehicle Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2.2 Driver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2.3 Vehicle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2.4 Load . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.2.5 Environment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3 Denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.3.1 Reference frames . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.3.2 Toe and camber angle . . . . . . . . . . . . . . . . . . . . . . . . . 51.3.3 Design Position of Wheel Rotation Axis . . . . . . . . . . . . . . . 61.3.4 Steering Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.3.4.1 Kingpin . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.3.4.2 Caster and Kingpin Angle . . . . . . . . . . . . . . . . . . 81.3.4.3 Caster, Steering Oset and Disturbing Force Lever . . . . 9
2 Road 102.1 Modeling Aspects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.2 Deterministic Proles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.2.1 Bumps and Potholes . . . . . . . . . . . . . . . . . . . . . . . . . . 112.2.2 Sine Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.3 Random Proles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.3.1 Statistical Properties . . . . . . . . . . . . . . . . . . . . . . . . . . 122.3.2 Classication of Random Road Proles . . . . . . . . . . . . . . . . 152.3.3 Realizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.3.3.1 Sinusoidal Approximation . . . . . . . . . . . . . . . . . . 162.3.3.2 Shaping Filter . . . . . . . . . . . . . . . . . . . . . . . . 172.3.3.3 Two-Dimensional Model . . . . . . . . . . . . . . . . . . . 18
I
3 Tire 193.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.1.1 Tire Development . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.1.2 Tire Composites . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.1.3 Tire Forces and Torques . . . . . . . . . . . . . . . . . . . . . . . . 203.1.4 Measuring Tire Forces and Torques . . . . . . . . . . . . . . . . . . 21
3.2 Contact Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233.2.1 Basic Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233.2.2 Local Track Plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253.2.3 Tire Deection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263.2.4 Length of Contact Patch . . . . . . . . . . . . . . . . . . . . . . . . 283.2.5 Static Contact Point . . . . . . . . . . . . . . . . . . . . . . . . . . 293.2.6 Contact Point Velocity . . . . . . . . . . . . . . . . . . . . . . . . . 313.2.7 Dynamic Rolling Radius . . . . . . . . . . . . . . . . . . . . . . . . 32
3.3 Forces and Torques caused by Pressure Distribution . . . . . . . . . . . . . 343.3.1 Wheel Load . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343.3.2 Tipping Torque . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353.3.3 Rolling Resistance . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.4 Friction Forces and Torques . . . . . . . . . . . . . . . . . . . . . . . . . . 373.4.1 Longitudinal Force and Longitudinal Slip . . . . . . . . . . . . . . . 373.4.2 Lateral Slip, Lateral Force and Self Aligning Torque . . . . . . . . . 403.4.3 Wheel Load Inuence . . . . . . . . . . . . . . . . . . . . . . . . . . 413.4.4 Two-Dimensional Tire Characteristics . . . . . . . . . . . . . . . . . 433.4.5 Dierent Friction Coecients . . . . . . . . . . . . . . . . . . . . . 453.4.6 Self Aligning Torque . . . . . . . . . . . . . . . . . . . . . . . . . . 463.4.7 Camber Inuence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 483.4.8 Bore Torque . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 503.4.9 Typical Tire Characteristics . . . . . . . . . . . . . . . . . . . . . . 52
4 Suspension System 544.1 Purpose and Components . . . . . . . . . . . . . . . . . . . . . . . . . . . 544.2 Some Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
4.2.1 Multi Purpose Systems . . . . . . . . . . . . . . . . . . . . . . . . . 554.2.2 Specic Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
4.3 Steering Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 564.3.1 Requirements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 564.3.2 Rack and Pinion Steering . . . . . . . . . . . . . . . . . . . . . . . 574.3.3 Lever Arm Steering System . . . . . . . . . . . . . . . . . . . . . . 574.3.4 Drag Link Steering System . . . . . . . . . . . . . . . . . . . . . . . 584.3.5 Bus Steer System . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
4.4 Standard Force Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . 594.4.1 Springs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 594.4.2 Damper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 604.4.3 Rubber Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
II
4.5 Dynamic Force Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . 634.5.1 Testing and Evaluating Procedures . . . . . . . . . . . . . . . . . . 634.5.2 Simple Spring Damper Combination . . . . . . . . . . . . . . . . . 664.5.3 General Dynamic Force Model . . . . . . . . . . . . . . . . . . . . . 68
4.5.3.1 Hydro-Mount . . . . . . . . . . . . . . . . . . . . . . . . . 69
5 Vertical Dynamics 725.1 Goals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 725.2 Modelling Aspects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
5.2.1 Full Vehicle Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 725.2.2 Twodimensional Models . . . . . . . . . . . . . . . . . . . . . . . . 735.2.3 Simple Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
5.3 Basic Tuning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 765.3.1 Natural Frequency and Damping Rate . . . . . . . . . . . . . . . . 765.3.2 Spring Rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
5.3.2.1 Minimum Spring Rates . . . . . . . . . . . . . . . . . . . . 785.3.2.2 Nonlinear Springs . . . . . . . . . . . . . . . . . . . . . . . 79
5.3.3 Inuence of Damping . . . . . . . . . . . . . . . . . . . . . . . . . . 815.3.4 Optimal Damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
5.3.4.1 Avoiding Overshoots . . . . . . . . . . . . . . . . . . . . . 815.3.4.2 Fast Approach to Steady State . . . . . . . . . . . . . . . 82
5.4 Sky Hook Damper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 875.4.1 Modelling Aspects . . . . . . . . . . . . . . . . . . . . . . . . . . . 875.4.2 Eigenfrequencies and Damping Ratios . . . . . . . . . . . . . . . . . 885.4.3 Technical Realization . . . . . . . . . . . . . . . . . . . . . . . . . . 90
5.5 Nonlinear Force Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . 905.5.1 Quarter Car Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 905.5.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
6 Longitudinal Dynamics 946.1 Dynamic Wheel Loads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
6.1.1 Simple Vehicle Model . . . . . . . . . . . . . . . . . . . . . . . . . . 946.1.2 Inuence of Grade . . . . . . . . . . . . . . . . . . . . . . . . . . . 956.1.3 Aerodynamic Forces . . . . . . . . . . . . . . . . . . . . . . . . . . 96
6.2 Maximum Acceleration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 976.2.1 Tilting Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 976.2.2 Friction Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
6.3 Driving and Braking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 986.3.1 Single Axle Drive . . . . . . . . . . . . . . . . . . . . . . . . . . . . 986.3.2 Braking at Single Axle . . . . . . . . . . . . . . . . . . . . . . . . . 996.3.3 Optimal Distribution of Drive and Brake Forces . . . . . . . . . . . 1006.3.4 Dierent Distributions of Brake Forces . . . . . . . . . . . . . . . . 1026.3.5 Anti-Lock-Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
6.4 Drive and Brake Pitch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
III
6.4.1 Vehicle Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1036.4.2 Equations of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . 1056.4.3 Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1066.4.4 Driving and Braking . . . . . . . . . . . . . . . . . . . . . . . . . . 1076.4.5 Brake Pitch Pole . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
7 Lateral Dynamics 1097.1 Kinematic Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
7.1.1 Kinematic Tire Model . . . . . . . . . . . . . . . . . . . . . . . . . 1097.1.2 Ackermann Geometry . . . . . . . . . . . . . . . . . . . . . . . . . 1097.1.3 Space Requirement . . . . . . . . . . . . . . . . . . . . . . . . . . . 1107.1.4 Vehicle Model with Trailer . . . . . . . . . . . . . . . . . . . . . . . 112
7.1.4.1 Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . 1127.1.4.2 Vehicle Motion . . . . . . . . . . . . . . . . . . . . . . . . 1137.1.4.3 Entering a Curve . . . . . . . . . . . . . . . . . . . . . . . 1147.1.4.4 Trailer Motions . . . . . . . . . . . . . . . . . . . . . . . . 1157.1.4.5 Course Calculations . . . . . . . . . . . . . . . . . . . . . 116
7.2 Steady State Cornering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1177.2.1 Cornering Resistance . . . . . . . . . . . . . . . . . . . . . . . . . . 1177.2.2 Overturning Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . 1187.2.3 Roll Support and Camber Compensation . . . . . . . . . . . . . . . 1217.2.4 Roll Center and Roll Axis . . . . . . . . . . . . . . . . . . . . . . . 1247.2.5 Wheel Loads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
7.3 Simple Handling Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1257.3.1 Modeling Concept . . . . . . . . . . . . . . . . . . . . . . . . . . . 1257.3.2 Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1257.3.3 Tire Forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1267.3.4 Lateral Slips . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1267.3.5 Equations of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . 1277.3.6 Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
7.3.6.1 Eigenvalues . . . . . . . . . . . . . . . . . . . . . . . . . . 1287.3.6.2 Low Speed Approximation . . . . . . . . . . . . . . . . . . 1297.3.6.3 High Speed Approximation . . . . . . . . . . . . . . . . . 129
7.3.7 Steady State Solution . . . . . . . . . . . . . . . . . . . . . . . . . . 1307.3.7.1 Side Slip Angle and Yaw Velocity . . . . . . . . . . . . . . 1307.3.7.2 Steering Tendency . . . . . . . . . . . . . . . . . . . . . . 1327.3.7.3 Slip Angles . . . . . . . . . . . . . . . . . . . . . . . . . . 133
7.3.8 Inuence of Wheel Load on Cornering Stiness . . . . . . . . . . . 133
8 Driving Behavior of Single Vehicles 1368.1 Standard Driving Maneuvers . . . . . . . . . . . . . . . . . . . . . . . . . . 136
8.1.1 Steady State Cornering . . . . . . . . . . . . . . . . . . . . . . . . . 1368.1.2 Step Steer Input . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1378.1.3 Driving Straight Ahead . . . . . . . . . . . . . . . . . . . . . . . . . 138
IV
8.1.3.1 Random Road Prole . . . . . . . . . . . . . . . . . . . . 1388.1.3.2 Steering Activity . . . . . . . . . . . . . . . . . . . . . . . 140
8.2 Coach with dierent Loading Conditions . . . . . . . . . . . . . . . . . . . 1408.2.1 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1408.2.2 Roll Steering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1418.2.3 Steady State Cornering . . . . . . . . . . . . . . . . . . . . . . . . . 1418.2.4 Step Steer Input . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
8.3 Dierent Rear Axle Concepts for a Passenger Car . . . . . . . . . . . . . . 143
V
1 Introduction
1.1 Literature
ATZ: Automobiltechnische Zeitschrift
Fachbuchgruppe Fahrwerktechnik:Jörnsen Reimpell, Hrsg. Vogel Buchverlag Würzburg.
Dynamik der Kraftfahrzeuge: M. Mitschke, Bde. A,B,C; Springer-Verlag.
Mitschke, M.; Wallentowitz, H.: Dynamik der Kraftfahrzeuge. 4. Auage.Springer-Verlag Berlin Heidelberg 2004.
Popp, K.; Schiehlen, W.: Fahrzeugdynamik. Teubner Stuttgart 1993.
Simulation von Kraftfahrzeugen: G. Rill, Vieweg-Verlag 1994.
Radführungen der Straÿenfahrzeuge: W. Matschinsky, Springer-Verlag.
Blundell, M.; Harty, D.: The Multibody System Approach to Vehicle Dynamics.Elsevier Butterworth-Heinemann Publications, 2004.
Fundamentals of Vehicle Dynamics: Th., D. Gillespie, SAE, Inc.
ISO-Standards: (International Organisation for Standardization.)
z.B.: ISO 4138 Steady State Circular Test Procedure.
Kraftfahrtechnisches Handbuch: Robert Bosch GmbH (Hrsg.), 23. Au.,Vieweg-Verlag.
Proceedings:
VDI-Tagungen: z.B.: Berechnung im Automobilbau.
SAE-Congress: (Society of Automotive Engineers).
FISITA: ( Féd. Internat. des Sociétés d'Ingénieurs de Techniques de l'Automobile).
IAVSD: (International Assosiation for Vehicle System Dynamics).
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Vehicle Dynamics FH Regensburg, University of Applied Sciences
1.2 Terminology
1.2.1 Vehicle Dynamics
The expression `Vehicle Dynamics' encompasses the interaction of:
driver
vehicle
load
environment
Vehicle dynamics mainly deals with:
the improvement of active safety and driving comfort
the reduction of road destruction
In vehicle dynamics are employed:
computer calculations
test rig measurements
eld tests
In the following the interactions between the single systems and the problems with com-puter calculations and/or measurements shall be discussed.
1.2.2 Driver
By various means the driver can interfere with the vehicle:
driver
steering wheel lateral dynamicsaccelerator pedalbrake pedalclutchgear shift
longitudinal dynamics
−→ vehicle
The vehicle provides the driver with these information:
vehicle
vibrations: longitudinal, lateral, verticalsounds: motor, aerodynamics, tiresinstruments: velocity, external temperature, ...
−→ driver
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FH Regensburg, University of Applied Sciences © Prof. Dr.-Ing. G. Rill
The environment also inuences the driver:
environment
climatetrac densitytrack
−→ driver
The driver's reaction is very complex. To achieve objective results, an `ideal' driver isused in computer simulations, and in driving experiments automated drivers (e.g. steeringmachines) are employed.
Transferring results to normal drivers is often dicult, if eld tests are made with testdrivers. Field tests with normal drivers have to be evaluated statistically. Of course, thedriver's security must have absolute priority in all tests.
Driving simulators provide an excellent means of analyzing the behavior of drivers evenin limit situations without danger.
It has been tried to analyze the interaction between driver and vehicle with complex drivermodels for some years.
1.2.3 Vehicle
The following vehicles are listed in the ISO 3833 directive:
motorcycles
passenger cars
busses
trucks
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 motion, the numeric solution, aswell as the acquisition of data require great expenses. In times of PCs and workstationscomputing costs hardly matter anymore.
At an early stage of development, often only prototypes are available for eld and/orlaboratory tests. Results can be falsied by safety devices, e.g. jockey wheels on trucks.
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Vehicle Dynamics FH Regensburg, University of Applied Sciences
1.2.4 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 at the determination of the inertias and themodeling of liquid loads.
Even the loading and unloading process of experimental vehicles takes some eort. Whencarrying out experiments with tank trucks, ammable liquids have to be substituted withwater. Thus, the results achieved cannot be simply transferred to real loads.
1.2.5 Environment
The environment inuences primarily the vehicle:
Environment
road: irregularities, coecient of frictionair: resistance, cross wind
−→ vehicle
but also aects the driver:
environment
climatevisibility
−→ driver
Through the interactions between vehicle and road, roads can quickly be destroyed.
The greatest diculty with eld tests and laboratory experiments is the virtual impossi-bility of reproducing environmental inuences.
The main problems with computer simulation are the description of random road irreg-ularities and the interaction of tires and road as well as the calculation of aerodynamicforces and torques.
1.3 Definitions
1.3.1 Reference frames
A reference frame xed to the vehicle and a ground-xed reference frame are used todescribe the overall motions of the vehicle, Figure 1.1. The ground-xed reference framewith the axis x0, y0, z0 serves as an inertial reference frame. Within the vehicle-xedreference frame the xF -axis points forward, the yF -axis to the left, and the zF -axis upward.
The wheel rotates around an axis which is xed to the wheel carrier. The reference frameC is xed to the wheel carrier. In design position its axes xC , yC and zC are parallel tothe corresponding axis of vehicle-xed reference frame F .
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FH Regensburg, University of Applied Sciences © Prof. Dr.-Ing. G. Rill
x0
en eyR
z0
xF
y0
yF
zF
xC
zC
yC
Figure 1.1: Frames used in vehicle dynamics
The momentary position of the wheel is xed by the wheel center and the orientation ofthe wheel rim center plane which is dened by the unit vector eyR into the direction ofthe wheel rotation axis.
Finally, the normal vector en describes the inclination of the local track plane.
1.3.2 Toe and camber angle
front
rear
yF
xF
δ
leftwheel
rightwheel
δ
vehiclecenterplane
Figure 1.2: Positive toe-in angle
According to the DIN 70 000 directive the angle δ between the vehicle center plane inlongitudinal direction and the intersection line of the tire center plane with the trackplane is named toe or toe-in angle. It will be positive, if the front part of the wheel isoriented towards the vehicle center plane, Figure 1.2. Toe-in reduces the tendency of thewheels to shimmy.
The camber angle γ is the angle between the wheel center plane and the local track normalen. It will be positive, if the upper part of the wheel is inclined outwards, Figure 1.3. Acambered wheel causes a non symmetric tire wear.
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Vehicle Dynamics FH Regensburg, University of Applied Sciences
yF
zF
γγ
leftwheel
rightwheel
road
en en
Figure 1.3: Positive camber angle
1.3.3 Design Position of Wheel Rotation Axis
The unit vector eyR describes the wheel rotation axis. Its orientation with respect to thewheel carrier xed reference frame can be dened by the angles δ0 and γ0 or δ0 and γ∗0 ,Fig. 1.4. In design position the corresponding axes of the frames C and F are parallel.
γ0
eyR
zC = zF
δ0
xC = xF
yC = yFγ0*
Figure 1.4: Design position of wheel rotation axis
Then, for the left wheel we get
eyR,F = eyR,C =1√
tan2 δ0 + 1 + tan2 γ∗0
tan δ0
1− tan γ∗0
(1.1)
or
eyR,F = eyR,C =
sin δ0 cos γ0
cos δ0 cos γ0
− sin γ0
, (1.2)
where δ0 is the angle between the yF -axis and the projection line of the wheel rotationaxis into the xF - yF -plane, the angle γ∗0 describes the angle between the yF -axis and theprojection line of the wheel rotation axis into the yF - zF -plane, whereas γ0
0 is the anglebetween the wheel rotation axis eyR and its projection into the xF - yF -plane. Kinematics
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FH Regensburg, University of Applied Sciences © Prof. Dr.-Ing. G. Rill
and compliance test machines usually measure the angle γ∗0 . That is why, the automotiveindustry mostly uses this angle instead of γ0.
On a at and horizontal road where the track normal en points into the direction ofthe vertical axes zC = zF the angles δ0 and γ0 correspond with the toe angle δ and thecamber angle γ0. To specify the dierence between γ0 and γ∗0 the ratio between the thirdand second component of the unit vector eyR is considered. The Equations 1.1 and 1.2deliver
− tan γ∗01
=− sin γ0
cos δ0 cos γ0
or tan γ∗0 =tan γ0
cos δ0
. (1.3)
Hence, for small angles δ0 1 the dierence between the angles γ0 and γ∗0 is hardlynoticeable.
1.3.4 Steering Geometry
1.3.4.1 Kingpin
At the steered front axle, the McPherson-damper strut axis, the double wishbone axis,and the multi-link wheel suspension or the enhanced double wishbone axis are mostlyused in passenger cars, Fig. 1.5 and Fig. 1.6.
M
A
Rz
x
y
R
R
B
kingpin axis A-B
Figure 1.5: Double wishbone wheel suspension
The wheel body rotates around the kingpin at steering motions. At the double wishboneaxis the ball joints A and B, which determine the kingpin, are both xed to the wheelbody. Whereas the ball joint A is still xed to the wheel body at the standard McPhersonwheel suspension, the ball joint B is now xed to the vehicle body. At a multi-link axle thekingpin is no longer dened by real joints. Here, as well as with the enhanced McPhersonwheel suspension, the kingpin changes its position relative to the wheel body at wheeltravel and steering motions.
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Vehicle Dynamics FH Regensburg, University of Applied Sciences
B
MA
Rz
x
y
R
R
kingpin axis A-B
M
Rz
x
y
R
R
rotation axis
Figure 1.6: McPherson and multi-link wheel suspensions
1.3.4.2 Caster and Kingpin Angle
The unit vector eS describes the direction of the kingpin axis. Within the vehicle xedreference frame F it can be xed by two angles. The caster angle ν denotes the anglebetween the zF -axis and the projection line of eS into the xF -, zF -plane. In a similarway the projection of eS into the yF -, zF -plane delivers the kingpin inclination angle σ,Fig. 1.7.
zFFz
xF
ν
yF
σeS
Figure 1.7: Kingpin and caster angle
At many axles the kingpin and caster angle can no longer be determined directly. Here, thecurrent rotation axis at steering motions, which can be taken from kinematic calculationswill deliver a virtual kingpin. The current values of the caster angle ν and the kingpininclination angle σ can be calculated from the components of the unit vector eS in thedirection of the kingpin, described in the vehicle xed reference frame
tan ν =−e
(1)S,F
e(3)S,F
and tan σ =−e
(2)S,F
e(3)S,F
, (1.4)
where e(1)S,F , e
(2)S,F , e
(3)S,F are the components of the unit vector eS,F expressed in the vehicle
xed reference frame F .
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FH Regensburg, University of Applied Sciences © Prof. Dr.-Ing. G. Rill
1.3.4.3 Caster, Steering Offset and Disturbing Force Lever
The contact point P , the local track normal en and the unit vectors ex and ey whichpoint into the direction of the longitudinal and lateral tire force result from the contactgeometry. The axle kinematics denes the kingpin line. In general, the point S where anextension oft the kingpin line meets the road surface does not coincide with the contactpoint P , Fig. 1.8. As both points are located on the local track plane, for the left wheelthe vector from S to P can be written as
rSP = −c ex + s ey , (1.5)
where c names the caster and s is the steering oset. Caster and steering oset will bepositive, if S is located in front of and inwards of P .
SP
C d
exey
s c
en
kingpin line
Figure 1.8: Caster and steering oset
The distance d between the wheel center C and the king pin line represents the disturbingforce lever. It is an important quantity in evaluating the overall steering behavior.
9
2 Road
2.1 Modeling Aspects
Sophisticated road models provide the road height zR and the local friction coecient µL
at each point x, y, Fig. 2.1.
z(x,y)
x0y0
z0
µ(x,y)
Center Line L(s)
Friction
Segments
Road profile
Obstacle
Figure 2.1: Sophisticated road model
The tire model is then responsible to calculate the local road inclination. By separatingthe horizontal course description from the vertical layout and the surface properties ofthe roadway almost arbitrary road layouts are possible.
Besides single obstacles or track grooves the irregularities of a road are of stochasticnature. A vehicle driving over a random road prole mainly performs hub, pitch and rollmotions. The local inclination of the road prole also induces longitudinal and lateralmotions as well as yaw motions. On normal roads the latter motions have less inuenceon ride comfort and ride safety. To limit the eort of the stochastic description usuallysimpler road models are used.
If the vehicle drives along a given path its momentary position can be described by thepath variable s = s(t). Hence, a fully two-dimensional road model can be reduced to aparallel track model, Fig. 2.2.
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FH Regensburg, University of Applied Sciences © Prof. Dr.-Ing. G. Rill
z1(s)
s
xy
z
zR(x,y)
z1z2
Figure 2.2: Parallel track road model
Now, the road heights on the left and right track are provided by two one-dimensionalfunctions z1 = z1(s) and z2 = z2(s). Within the parallel track model no informationabout the local lateral road inclination is available. If this information is not provided byadditional functions the impact of a local lateral road inclination to vehicle motions is nottaken into account.
For basic studies the irregularities at the left and the right track can considered to beapproximately the same, z1(s) ≈ z2(s). Then, a single track road model with zR(s) =z1(x) = z2(x) can be used. Now, the roll excitation of the vehicle is neglected too.
2.2 Deterministic Profiles
2.2.1 Bumps and Potholes
Bumps and Potholes on the road are single obstacles of nearly arbitrary shape. Alreadywith simple rectangular cleats the dynamic reaction of a vehicle or a single tire to asudden impact can be investigated. If the shape of the obstacle is approximated by asmooth function, like a cosine wave, then, discontinuities will be avoided. Usually theobstacles are described in local reference frames, Fig. 2.3.
L
H
B B
x y
z
H
Lx
y
z
Figure 2.3: Rectangular cleat and cosine-shaped bump
Then, the rectangular cleat is simply dened by
z(x, y) =
H if 0 < x < L and − 1
2B < y < 1
2B
0 else(2.1)
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Vehicle Dynamics FH Regensburg, University of Applied Sciences
and the cosine-shaped bump is given by
z(x, y) =
12H(1− cos
(2π
x
L
))if 0 < x < L and − 1
2B < y < 1
2B
0 else(2.2)
where H, B and L denote height, width and length of the obstacle. Potholes are obtainedif negative values for the height (H < 0) are used.
2.2.2 Sine Waves
Using the parallel track road model, a periodic excitation can be realized by
z1(s) = A sin (Ω s) , z2(s) = A sin (Ω s−Ψ) , (2.3)
where s is the path variable, A denotes the amplitude, Ω the wave number, and the angleΨ describes a phase lag between the left and the right track. The special cases Ψ = 0 andΨ = π represent the in-phase excitation with z1 = z2 and the out of phase excitation withz1 = −z2.
If the vehicle runs with constant velocity ds/dt = v0, the momentary position of thevehicle is given by s = v0 t, where the initial position s = 0 at t = 0 was assumed. Byintroducing the wavelength
L =2π
Ω(2.4)
the term Ω s can be written as
Ω s =2π
Ls =
2π
Lv0 t = 2π
v0
Lt = ω t . (2.5)
Hence, in the time domain the excitation frequency is given by f = ω/(2π) = v0/L.
For most of the vehicles the rigid body vibrations are in between 0.5 Hz to 15 Hz. Thisrange is covered by waves which satisfy the conditions v0/L ≥ 0.5 Hz and v0/L ≤ 15 Hz.
For a given wavelength, lets say L = 4m, the rigid body vibration of a vehicle are excitedif the velocity of the vehicle will be varied from vmin
0 = 0.5Hz ∗ 4m = 2m/s = 7.2 km/hto vmax
0 = 15 Hz ∗ 4 m = 60 m/s = 216 km/h. Hence, to achieve an excitation in thewhole frequency range with moderate vehicle velocities proles with dierent varyingwavelengths are needed.
2.3 Random Profiles
2.3.1 Statistical Properties
Road proles t the category of stationary Gaussian random processes. Hence, the irregu-larities of a road can be described either by the prole itself zR = zR(s) or by its statisticalproperties, Fig. 2.4.
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FH Regensburg, University of Applied Sciences © Prof. Dr.-Ing. G. Rill
Histogram
Realization
0.15
0.10
0.05
0
-0.05
-0.10
-0.15-200 -150 -100 -50 0 50 100 150 200
Gaussiandensityfunction
m
+σ
−σ
[m]
[m]
zR
s
Figure 2.4: Road prole and statistical properties
By choosing an appropriate reference frame, a vanishing mean value
m = E zR(s) = limX→∞
1
X
X/2∫−X/2
zR(s) ds = 0 (2.6)
can be achieved, where E denotes the expectation operator. Then, the Gaussian densityfunction which corresponds with the histogram is given by
p(zR) =1
σ√
2πe− z2
R
2σ2 , (2.7)
where the deviation or the eective value σ is obtained from the variance of the processzR = zR(s)
σ2 = Ez2
R(s)
= limX→∞
1
X
X/2∫−X/2
zR(s)2 ds . (2.8)
Alteration of σ eects the shape of the density function. In particular, the points ofinexion occur at ±σ. The probability of a value |z| < ζ is given by
P (±ζ) =1
σ√
2π
+ζ∫−ζ
e− z2
2σ2 dz . (2.9)
In particular, one gets the values: P (±σ)= 0.683, P (±2σ)= 0.955, and P (±3σ)=0.997.Hence, the probability of a value |z| ≥ 3σ is 0.3%.
In extension to the variance of a random process the auto-correlation function is denedby
R(ξ) = E zR(s) zR(s+ξ) = limX→∞
1
X
X/2∫−X/2
zR(s) zR(s+ξ) ds . (2.10)
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Vehicle Dynamics FH Regensburg, University of Applied Sciences
The auto-correlation function is symmetric, R(ξ) = R(−ξ), and it plays an importantpart in the stochastic analysis. In any normal random process, as ξ increases the linkbetween zR(s) and zR(s+ξ) diminishes. For large values of ξ the two values are practicallyunrelated. Hence, R(ξ → ∞) will tend to 0. In fact, R(ξ) is always less R(0), whichcoincides with the variance σ2 of the process. If a periodic term is present in the processit will show up in R(ξ).
Usually, road proles are characterized in the frequency domain. Here, the auto-correlationfunction R(ξ) is replaced by the power spectral density (psd) S(Ω). In general, R(ξ) andS(Ω) are related to each other by the Fourier transformation
S(Ω) =1
2π
∞∫−∞
R(ξ) e−iΩξ dξ and R(ξ) =1
2π
∞∫−∞
S(Ω) eiΩξ dΩ , (2.11)
where i is the imaginary unit, and Ω in rad/m denotes the wave number. To avoid negativewave numbers, usually a one-sided psd is dened. With
Φ(Ω) = 2 S(Ω) , if Ω ≥ 0 and Φ(Ω) = 0 , if Ω < 0 , (2.12)
the relationship e±iΩξ = cos(Ωξ) ± i sin(Ωξ), and the symmetry property R(ξ) = R(−ξ)Eq. (2.11) results in
Φ(Ω) =2
π
∞∫0
R(ξ) cos (Ωξ) dξ and R(ξ) =
∞∫0
Φ(Ω) cos (Ωξ) dΩ . (2.13)
Now, the variance is obtained from
σ2 = R(ξ=0) =
∞∫0
Φ(Ω) dΩ . (2.14)
In reality the psd Φ(Ω) will be given in a nite interval Ω1 ≤ Ω ≤ ΩN , Fig. 2.5. Then,
ΩNΩ1
N ∆Ω∆Ω
Ωi
Φ(Ωi)
Φ
Ω
Figure 2.5: Power spectral density in a nite interval
Eq. (2.14) can be approximated by a sum, which for N equal intervals will result in
σ2 ≈N∑
i=1
Φ(Ωi)4Ω with 4Ω =ΩN − Ω1
N. (2.15)
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FH Regensburg, University of Applied Sciences © Prof. Dr.-Ing. G. Rill
2.3.2 Classification of Random Road Profiles
Road elevation proles can be measured point by point or by high-speed prolometers.The power spectral densities of roads show a characteristic drop in magnitude with thewave number, Fig. 2.6a. This simply reects the fact that the irregularities of the road mayamount to several meters over the length of hundreds of meters, whereas those measuredover the length of one meter are normally only some centimeter in amplitude.
Random road proles can be approximated by a psd in the form of
Φ (Ω) = Φ (Ω0)
(Ω
Ω0
)−w
, (2.16)
where, Ω = 2π/L in rad/m denotes the wave number and Φ0 = Φ (Ω0) in m2/(rad/m)describes the value of the psd at a the reference wave number Ω0 = 1 rad/m. The dropin magnitude is modeled by the waviness w.
10-2 10-1 102101100
Wave number Ω [rad/m]10-2 10-1 102101100
10-4
10-3
10-5
10-6
10-7
10-8
10-9Pow
er s
pect
ral d
ensi
ty Φ
[m2 /
(rad
/m)]
Wave number Ω [rad/m]
a) Measurements (country road) b) Range of road classes (ISO 8608)
Class A
Class E
Φ0=256∗10−6
Φ0=1∗10−6
Figure 2.6: Road power spectral densities: a) Measurements, b) Classication
According to the international directive ISO 8608 typical road proles can be groupedinto classes from A to E. By setting the waviness to w = 2 each class is simply dened byits reference value Φ0. Class A with Φ0 = 1 ∗ 10−6 m2/(rad/m) characterizes very smoothhighways, whereas Class E with Φ0 = 256 ∗ 10−6 m2/(rad/m) represents rather roughroads, Fig. 2.6b.
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Vehicle Dynamics FH Regensburg, University of Applied Sciences
2.3.3 Realizations
2.3.3.1 Sinusoidal Approximation
A random prole of a single track can be approximated by a superposition of N → ∞sine waves
zR(s) =N∑
i=1
Ai sin (Ωi s−Ψi) , (2.17)
where each sine wave is determined by its amplitude Ai and its wave number Ωi. Bydierent sets of uniformly distributed phase angles Ψi, i = 1(1)N in the range between0 and 2π dierent proles can be generated which are similar in the general appearancebut dierent in details.
The variance of the sinusoidal representation is then given by
σ2 = limX→∞
1
X
X/2∫−X/2
(N∑
i=1
Ai sin (Ωi s−Ψi)
)(N∑
j=1
Aj sin (Ωj s−Ψj)
)ds . (2.18)
For i = j and for i 6= j dierent types of integrals are obtained. The ones for i = j canbe solved immediately
Jii =
∫A2
i sin2 (Ωis−Ψi) ds =A2
i
2Ωi
[Ωis−Ψi −
1
2sin(2 (Ωis−Ψi)
)]. (2.19)
Using the trigonometric relationship
sin x sin y =1
2cos(x−y) − 1
2cos(x+y) (2.20)
the integrals for i 6= j can be solved too
Jij =
∫Ai sin (Ωis−Ψi) Aj sin (Ωjs−Ψj) ds
=1
2AiAj
∫cos (Ωi−j s−Ψi−j) ds − 1
2AiAj
∫cos (Ωi+j s−Ψi+j) ds
= −1
2
AiAj
Ωi−j
sin (Ωi−j s−Ψi−j) +1
2
AiAj
Ωi+j
sin (Ωi+j s−Ψi+j)
(2.21)
where the abbreviations Ωi±j = Ωi±Ωj and Ψi±j = Ψi±Ψj were used. The sine and cosineterms in Eqs. (2.19) and (2.21) are limited to values of ±1. Hence, Eq. (2.18) simplyresults in
σ2 = limX→∞
1
X
N∑i=1
[Jii
] X/2
−X/2︸ ︷︷ ︸N∑
i=1
A2i
2Ωi
Ωi
+ limX→∞
1
X
N∑i,j=1
[Jij
] X/2
−X/2︸ ︷︷ ︸0
=1
2
N∑i=1
A2i . (2.22)
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FH Regensburg, University of Applied Sciences © Prof. Dr.-Ing. G. Rill
On the other hand, the variance of a sinusoidal approximation to a random road proleis given by Eq. (2.15). So, a road prole zR = zR(s) described by Eq. (2.17) will have agiven psd Φ(Ω) if the amplitudes are generated according to
Ai =√
2 Φ(Ωi)4Ω , i = 1(1)N , (2.23)
and the wave numbers Ωi are chosen to lie at N equal intervals 4Ω.
0.10
0.05
-0.10
-0.05
0
0 10 20 30 40 50 60 70 80 90 100[m]
[m]
Road profile z=z(s)
Figure 2.7: Realization of a country road
A realization of the country road with a psd of Φ0 = 10 ∗ 10−6 m2/(rad/m) is shown inFig. 2.7. According to Eq. (2.17) the prole z = z(s) was generated by N = 200 sine wavesin the frequency range from Ω1 = 0.0628 rad/m to ΩN = 62.83 rad/m. The amplitudesAi, i = 1(1)N were calculated by Eq. (2.23) and the MATLABr function rand was usedto produce uniformly distributed random phase angles in the range between 0 and 2π.
2.3.3.2 Shaping Filter
The white noise process produced by random number generators has a uniform spectraldensity, and is therefore not suitable to describe real road proles. But, if the white noiseprocess is used as input to a shaping lter more appropriate spectral densities will beobtained. A simple rst order shaping lter for the road prole zR reads as
d
dszR(s) = −γ zR(s) + w(s) , (2.24)
where γ is a constant, and w(s) is a white noise process with the spectral density Φw.Then, the spectral density of the road prole is obtained from
ΦR = H(Ω) ΦW HT (−Ω) =1
γ + i ΩΦW
1
γ − i Ω=
ΦW
γ2 + Ω2, (2.25)
where Ω is the wave number, and H(Ω) is the frequency response function of the shapinglter.
By setting ΦW = 10∗10−6 m2/(rad/m) and γ = 0.01rad/m the measured psd of a typicalcountry road can be approximated very well, Fig. 2.8.
The shape lter approach is also suitable for modeling parallel tracks. Here, the cross-correlation between the irregularities of the left and right track have to be taken intoaccount too.
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Vehicle Dynamics FH Regensburg, University of Applied Sciences
10-2 10-1 102101100
10-4
10-3
10-5
10-6
10-7
10-8
10-9Pow
er s
pect
ral d
ensi
ty Φ
[m2 /
(rad
/m)]
Wave number Ω [rad/m]
MeasurementsShaping filter
Figure 2.8: Shaping lter as approximation to measured psd
2.3.3.3 Two-Dimensional Model
The generation of fully two-dimensional road proles zR = zR(x, y) via a sinusoidal ap-proximation is very laborious. Because a shaping lter is a dynamic system, the result-ing road prole realizations are not reproducible. By adding band-limited white noiseprocesses and taking the momentary position x, y as seed for the random number gener-ator a reproducible road prole can be generated.
-4-20
24
05
1015
2025
3035
4045
50
-101
m
zx
y
Figure 2.9: Two-dimensional road prole
By assuming the same statistical properties in longitudinal and lateral direction two-dimensional proles, like the one in Fig. 2.9, can be obtained.
18
3 Tire
3.1 Introduction
3.1.1 Tire Development
Some important mile stones in the development of tires are shown in Table 3.1.
1839 Charles Goodyear: vulcanization1845 Robert William Thompson: rst pneumatic tire
(several thin inated tubes inside a leather cover)1888 John Boyd Dunlop: patent for bicycle (pneumatic) tires1893 The Dunlop Pneumatic and Tyre Co. GmbH, Hanau, Germany1895 André and Edouard Michelin: pneumatic tires for Peugeot
Paris-Bordeaux-Paris (720 Miles): 50 tire deations,22 complete inner tube changes
1899 Continental: long-lived tires (approx. 500 Kilometer)1904 Carbon added: black tires.1908 Frank Seiberling: grooved tires with improved road traction1922 Dunlop: steel cord thread in the tire bead1943 Continental: patent for tubeless tires1946 Radial Tire
Table 3.1: Milestones in tire development
Of course the tire development did not stop in 1946, but modern tires are still based onthis achievements.
3.1.2 Tire Composites
Tires are very complex. They combine dozens of components that must be formed, as-sembled and cured together. And their ultimate success depends on their ability to blendall of the separate components into a cohesive product that satises the driver's needs. Amodern tire is a mixture of steel, fabric, and rubber. The main composites of a passengercar tire with an overall mass of 8.5 kg are listed in Table 3.2.
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Vehicle Dynamics FH Regensburg, University of Applied Sciences
Reinforcements: steel, rayon, nylon 16%Rubber: natural/synthetic 38%Compounds: carbon, silica, chalk, ... 30%Softener: oil, resin 10%Vulcanization: sulfur, zinc oxide, ... 4%Miscellaneous 2%
Table 3.2: Tire composites: 195/65 R 15 ContiEcoContact, data from www.felge.de
3.1.3 Tire Forces and Torques
In any point of contact between the tire and the road surface normal and friction forcesare transmitted. According to the tire's prole design the contact patch forms a notnecessarily coherent area, Figure 3.1.
180 mm
140
mm
Figure 3.1: Tire footprint of a passenger car at normal loading condition:Continental 205/55 R16 90 H, 2.5 bar, Fz = 4700 N
The eect of the contact forces can be fully described by a resulting force vector appliedat a specic point of the contact patch and a torque vector. The vectors are described in atrack-xed reference frame. The z-axis is normal to the track, the x-axis is perpendicularto the z-axis and perpendicular to the wheel rotation axis eyR. Then, the demand for aright-handed reference frame also xes the y-axis.
The components of the contact force vector are named according to the direction of theaxes, Figure 3.2.
A non symmetric distribution of the forces in the contact patch causes torques aroundthe x and y axes. A cambered tire generates a tilting torque Tx. The torque Ty includesthe rolling resistance of the tire. In particular, the torque around the z-axis is importantin vehicle dynamics. It consists of two parts,
Tz = TB + TS . (3.1)
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FH Regensburg, University of Applied Sciences © Prof. Dr.-Ing. G. Rill
Fx longitudinal forceFy lateral forceFz vertical force or wheel load
Tx tilting torqueTy rolling resistance torqueTz self aligning and bore torque Fx
Fy
Fz
TxTy
Tz
eyR
Figure 3.2: Contact forces and torques
The rotation of the tire around the z-axis causes the bore torque MB. The self aligningtorque MS takes into account that ,in general, the resulting lateral force is not acting inthe center of the contact patch.
3.1.4 Measuring Tire Forces and Torques
To measure tire forces and torques on the road a special test trailer is needed, Figure3.4. Here, the measurements are performed under real operating conditions. Arbitrary
tire
test wheel
compensation wheel
real road
exact contact
Test trailer
Figure 3.3: Layout of a tire test trailer
surfaces like asphalt or concrete and dierent environmental conditions like dry, wet oricy are possible. Measurements with test trailers are quite cumbersome and in generalthey are restricted to passenger car tires.
Indoor measurements of tire forces and torques can be performed on drums or on a atbed, Figure 3.4.
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Vehicle Dynamics FH Regensburg, University of Applied Sciences
tire
tire
safety walkcoating
rotationdrum
too smallcontact area
too large contact area
tire
safety walk coating perfect contact
Figure 3.4: Drum and at bed tire test rig
On drum test rigs the tire is placed either inside or outside of the drum. In both casesthe shape of the contact area between tire and drum is not correct. That is why, onecan not rely on the measured self aligning torque. Due its simple and robust design, wideapplications including measurements of truck tires are possible.
The at bed tire test rig is more sophisticated. Here, the contact patch is as at as on theroad. But, the safety walk coating which is attached to the steel bed does not generatethe same friction conditions as on a real road surface.
-40 -30 -20 -10 0 10 20 30 40
Longitudinal slip [%]
-4000
-3000
-2000
-1000
0
1000
2000
3000
4000
Long
itud
forc
e F
x [N
]
Radial 205/50 R15, FN= 3500 N, dry asphalt
Driving
Braking
Figure 3.5: Typical results of tire measurements
Tire forces and torques are measured in quasi-static operating conditions. Hence, the mea-surements for increasing and decreasing the sliding conditions usually result in dierentgraphs, Figure 3.5. In general, the mean values are taken as steady state results.
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FH Regensburg, University of Applied Sciences © Prof. Dr.-Ing. G. Rill
3.2 Contact Geometry
3.2.1 Basic Approach
The current position of a wheel in relation to the xed x0-, y0- z0-system is given by thewheel center M and the unit vector eyR in the direction of the wheel rotation axis, Figure3.6.
road: z = z ( x , y )
eyR
M
en
0P
tire
x0
0y0
z0*P
P
x0
0y0
z0
eyR
M
en
ex
γ
ey
rimcentreplane
local road plane
ezR
rMP
wheelcarrier
0P ab
Figure 3.6: Contact geometry
The irregularities of the track can be described by an arbitrary function of two spatialcoordinates
z = z(x, y). (3.2)
At an uneven track the contact point P can not be calculated directly. At rst, one canget an estimated value with the vector
rMP ∗ = −r0 ezB , (3.3)
where r0 is the undeformed tire radius, and ezB is the unit vector in the z-direction of thebody xed reference frame.
The position of this rst guess P ∗ with respect to the earth xed reference frame x0, y0,z0 is determined by
r0P ∗,0 = r0M,0 + rMP ∗,0 =
x∗
y∗
z∗
, (3.4)
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Vehicle Dynamics FH Regensburg, University of Applied Sciences
where the vector r0M describes the position of the rim center M . Usually, the point P ∗
does not lie on the track. The corresponding track point P0 follows from
r0P0,0 =
x∗
y∗
z (x∗, y∗)
, (3.5)
where Eq. (3.2) was used to calculate the appropriate road height. In the point P0 thetrack normal en is calculated, now. Then the unit vectors in the tire's circumferentialdirection and lateral direction can be determined. One gets
ex =eyR×en
| eyR×en |and ey = en×ex , (3.6)
where eyR denotes the unit vector into the direction of the wheel rotation axis. Calculatingex demands a normalization, as eyR not always being perpendicular to the track. The tirecamber angle
γ = arcsin(eT
yR en
)(3.7)
describes the inclination of the wheel rotation axis against the track normal.
The vector from the rim center M to the track point P0 is split into three parts now
rMP0 = −rS ezR + a ex + b ey , (3.8)
where rS denotes the loaded or static tire radius, a, b are distances measured in circum-ferential and lateral direction, and the radial direction is given by the unit vector
ezR = ex×eyR (3.9)
which is perpendicular to ex and eyR. A scalar multiplication of Eq. (3.8) with en resultsin
eTn rMP0 = −rS eT
n ezR + a eTn ex + b eT
n ey . (3.10)
As the unit vectors ex and ey are perpendicular to en Eq. (3.10) simplies to
eTn rMP0 = −rS eT
n ezR . (3.11)
Hence, the static tire radius is given by
rS = − eTn rMP0
eTn ezR
. (3.12)
The contact point P given by the vector
rMP = −rS ezR (3.13)
lies within the rim center plane. The transition from the point P0 to the contact point Ptakes place according to Eq. (3.8) by the terms a ex and b ey perpendicular to the track
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FH Regensburg, University of Applied Sciences © Prof. Dr.-Ing. G. Rill
normal en. The track normal, however, was calculated in the point P0. With an uneventrack the point P no longer lies on the track and can therefor no longer considered ascontact point.
With the newly estimated value P ∗ = P now the Eqs. (3.5) to (3.13) can be repeateduntil the dierence between P and P0 is suciently small.
Tire models which can be simulated within acceptable time assume that the contact patchis suciently at. At an ordinary passenger car tire, the contact area has at normal loadapproximately the size of 15×20cm. It makes no sense to calculate a ctitious contact pointto fractions of millimeters, when later on the real track will be approximated by a planein the range of centimeters. If the track in the contact area is replaced by a local plane,no further iterative improvements will be necessary for the contact point calculation.
3.2.2 Local Track Plane
A plane is given by three points. In order to get a good approximation to the local trackunevenness four point will be used to determine the local track normal. Using the initialguess in Eq. (3.3) and the wheel rotation axis eyr the circumferential direction can beestimated by
e∗x =eyR×ezB
| eyR×ezB |. (3.14)
Similar to Eq. (3.4) four points are generated now
r0Q∗i ,0 = r0M,0 + rMQ∗i ,0 =
x∗iy∗iz∗i
, i = 1(1)4 . (3.15)
In order to sample the contact patch as good as possible the tire width b and the unloadedtire radius r0 are used to place the points via
rMQ∗1= λxr0 ex∗ − r0 ezB ,
rMQ∗2= −λxr0 ex∗ − r0 ezB ,
rMQ∗3= λyb eyR − r0 ezB ,
rMQ∗4= −λyb eyR − r0 ezB
(3.16)
in the front, in the rear, in the left, and in the right of the contact patch.
According to Eq. (3.5) the corresponding points on the track are given by
r0Qi,0 =
x∗iy∗i
z (x∗i , y∗i )
, i = 1(1)4 . (3.17)
The calculation of the track normal is straight forward now, Figure 3.7. The vectors
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Vehicle Dynamics FH Regensburg, University of Applied Sciences
rMP*
eyRM
P*
en
Q1*
Q1
Q2*
Q2
Q3*
Q3
Q4*
Q4
rQ2Q1
rQ3Q4
P
Figure 3.7: Local track plane
rQ2Q1 = r0Q1−r0Q2 and rQ4Q3 = r0Q3−r0Q4 dene the local track inclination in longitudinaland lateral direction. Hence, the local track normal is dened by
en =rQ2Q1×rQ4Q3
| rQ2Q1×rQ4Q3 |. (3.18)
The unit vectors ex, ey in longitudinal and lateral direction are calculated from Eq. (3.6).The mean value of the track points
r0P0,0 =1
4(r0Q1,0 + r0Q2,0 + r0Q3,0 + r0Q4,0) (3.19)
serves as rst improvement of the contact point, P ∗ → P0. Finally, the correspondingpoint P in the rim center plane is obtained by Eqs. (3.12) and (3.13). On rough roadsthe point P not always is located on the track. But, together with the local track normalit represents the local road plane very well. As in reality, sharp bends and discontinuitieswhich will occur at step- or ramp-sized obstacles are smoothed by this approach.
3.2.3 Tire Deflection
For a vanishing camber angle γ = 0 the deected zone has a rectangular shape, Figure3.8. Its area is given by
A0 = 4z b , (3.20)
where b is the width of the tire, and the tire deection is obtained by
4z = r0 − rS . (3.21)
Here, the width of the tire simply equals the width of the contact zone, wC = b.
On a cambered tire the deected zone of the tire cross section depends on the contactsituation. The magnitude of the tire ank radii
rSL = rs +b
2tan γ and rSR = rs −
b
2tan γ (3.22)
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FH Regensburg, University of Applied Sciences © Prof. Dr.-Ing. G. Rill
rS
r0
eyR
en
P∆z
wC = b
rSL
r0
eyR
en
P
b
rSR
γ
r0
eyR
en
P
b*
rSR
γ
full contact partial contact
γ = 0γ = 0
wC wC
/
rSrS
Figure 3.8: Tire deection
determines the shape of the deected zone.
The tire will be in full contact to the road if rSL ≤ r0 and rSR ≤ r0 hold. Then, thedeected zone has a trapezoidal shape with an area of
Aγ =1
2(r0−rSR + r0−rSL) b = (r0 − rS) b . (3.23)
Equalizing the cross sections A0 = Aγ results in
4z = r0 − rS . (3.24)
Hence, at full contact the tire camber angle γ has no inuence to the vertical tire force.But, due to
wC =b
cos γ(3.25)
the width of the contact area increases with the tire camber angle.
The deected zone will change to a triangular shape if one of the ank radii exceeds theundeected tire radius. Assuming rSL > r0 and rSR < r0 the area of the deected zone isobtained by
Aγ =1
2(r0−rSR) b∗ , (3.26)
where the width of the deected zone follows from
b∗ =r0−rSR
tan γ. (3.27)
Now, Eq. (3.26) reads as
Aγ =1
2
(r0−rSR)2
tan γ. (3.28)
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Vehicle Dynamics FH Regensburg, University of Applied Sciences
Equalizing the cross sections A0 = Aγ results in
4z =1
2
(r0 − rS + b
2tan γ
)2b tan γ
. (3.29)
where Eq. (3.22) was used to express the ank radius rSR by the static tire radius rS, thetire width b and the camber angle γ. Now, the width of the contact area is given by
wC =b∗
cos γ=
r0 − rSR
tan γ cos γ=
r0 − rS + b2
tan γ
sin γ, (3.30)
where the Eqs. (3.27) and (3.22) where used to simplify the expression. If tan γ and sin γare replaced by | tan γ | and | sin γ | then, the Eqs. (3.29) and (3.30) will hold for positiveand negative camber angles.
3.2.4 Length of Contact Patch
To approximate the length of the contact patch the tire deformation is split into twoparts, Figure 3.9. By 4zF and 4zB the average tire ank and the belt deformation aremeasured. Hence, for a tire with full contact to the road
4z = 4zF +4zB = r0 − rS (3.31)
will hold.
Fz
L
r0rS
Belt
Rim
L/2
r0
∆zF
∆zB ∆zB
undeformed belt
Figure 3.9: Length of contact patch
Assuming both deections being equal will lead to
4zF ≈ 4zB ≈ 1
24z . (3.32)
Approximating the belt deection by truncating a circle with the radius of the undeformedtire results in (
L
2
)2
+ (r0 −4zB)2 = r20 . (3.33)
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FH Regensburg, University of Applied Sciences © Prof. Dr.-Ing. G. Rill
In normal driving situations the belt deections are small, 4zB r0. Hence, Eq. (3.33)can be simplied and nally results in
L2
4= 2 r04zB or L =
√8 r04zB . (3.34)
Inspecting the passenger car tire footprint in Figure 3.1 leads to a contact patch length ofL ≈ 140mm. For this tire the radial stiness and the inated radius are specied with cR =265 000 N/m and r0 = 316.9 mm. The overall tire deection can be estimated by 4z =Fz/cR. At the load of Fz = 4700N the deection amounts to4z = 4700N / 265 000N/m =0.0177 m. Then, by approximating the belt deformation by the half of the tire deection,the length of the contact patch will become L =
√8 0.3169 m 0.0177/2 m = 0.1498 m =
150 mm which corresponds quite well with the length of the tire footprint.
3.2.5 Static Contact Point
Assuming that the pressure distribution on a cambered tire with full road contact cor-responds with the trapezoidal shape of the deected tire area, the acting point of theresulting vertical tire force FZ will be shifted from the geometric contact point P to thestatic contact point Q, Figure 3.10.
en
P
γ
wC
rS
Q
Fzr0-rSL r0-rSR
y
ey
A
A
Figure 3.10: Lateral deviation of contact point at full contact
The center of the trapezoidal area determines the lateral deviation yQ. By splitting thearea into a rectangular and a triangular section we will obtain
yQ = − y A + y4A4
A. (3.35)
The minus sign takes into account that for positive camber angles the acting point willmove to the right whereas the unit vector ey dening the lateral direction points to theleft. The area of the whole cross section results from
A =1
2(r0−rSL + r0−rSR) wC , (3.36)
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Vehicle Dynamics FH Regensburg, University of Applied Sciences
where the width of the contact area wC is given by Eq. (3.25). Using the Eqs. (3.22) and(3.24) the expression can be simplied to
A = 4z wC . (3.37)
As the center of the rectangular section is located on the center line which runs through thegeometric contact point, y = 0 will hold. The distance from the center of the triangularsection to the center line is given by
y4 =1
2wC −
1
3wC =
1
6wC . (3.38)
Finally, the area of the triangular section is dened by
A4 =1
2(r0−rSR − (r0−rSL)) wC =
1
2(rSL − rSR)) wC =
1
2(b tan γ) wC , (3.39)
where Eq. (3.22) was used to simplify the expression. Now, Eq. (3.35) can be written as
yQ = −16wC
12b tan γ wC
4z wC
= − b tan γ
124zwC = − b2
124z
tan γ
cos γ. (3.40)
If the cambered tire has only a partial contact to the road then, according to the deectionarea a triangular pressure distribution will be assumed, Figure 3.11.
en
γ
P
wC
Q
Fzy
ey
b/2
Figure 3.11: Lateral deviation of contact point at partial contact
Now, the location of the static contact point Q is given by
yQ = ±(
1
3wC −
b
2 cos γ
), (3.41)
where the width of the contact area wC is determined by Eq. (3.30) and the term b/(2 cos γ)describes the distance from the geometric contact point P to the outer corner of thecontact patch. The plus sign holds for positive and the minus sign for negative camberangles.
The static contact point Q described by the vector
r0Q = r0P + yQ ey (3.42)
represents the contact patch much better than the geometric contact point P .
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FH Regensburg, University of Applied Sciences © Prof. Dr.-Ing. G. Rill
3.2.6 Contact Point Velocity
To calculate the tire forces and torques which are generated by friction the contact pointvelocity will be needed. The static contact point Q given by Eq. (3.42) can be expressedas follows
r0Q = r0M + rMQ , (3.43)
where M denotes the wheel center and hence, the vector rMQ describes the position ofstatic contact point Q relative to the wheel center M . The absolute velocity of the contactpoint will be obtained from
v0Q,0 = r0Q,0 = r0M,0 + rMQ,0 , (3.44)
where r0M,0 = v0M,0 denotes the absolute velocity of the wheel center. The vector rMQ
takes part on all those motions of the wheel carrier which do not contain elements of thewheel rotation and it In addition, it contains the tire deection 4z normal to the road.Hence, its time derivative can be calculated from
rMQ,0 = ω∗0R,0×rMQ,0 + 4z en,0 , (3.45)
where ω∗0R is the angular velocity of the wheel rim without any component in the directionof the wheel rotation axis, 4z denotes the change of the tire deection, and en describesthe road normal. Now, Eq. (3.44) reads as
v0Q,0 = v0M,0 + ω∗0R,0×rMQ,0 + 4z en,0 . (3.46)
As the point Q lies on the track, v0Q,0 must not contain any component normal to thetrack
eTn,0 v0P,0 = 0 or eT
n,0
(v0M,0 + ω∗0R,0×rMQ,0
)+ 4z eT
n,0 en,0 = 0 . (3.47)
As en,0 is a unit vector, eTn,0 en,0 = 1 will hold, and then, the time derivative of the tire
deformation is simply given by
4z = − eTn,0
(v0M,0 + ω∗0R,0×rMQ,0
). (3.48)
Finally, the components of the contact point velocity in longitudinal and lateral directionare obtained from
vx = eTx,0 v0Q,0 = eT
x,0
(v0M,0 + ω∗0R,0×rMQ,0
)(3.49)
andvy = eT
y,0 v0P,0 = eTy,0
(v0M,0 + ω∗0R,0×rMQ,0
), (3.50)
where the relationships eTx,0 en,0 = 0 and eT
y,0 en,0 = 0 were used to simplify the expressions.
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Vehicle Dynamics FH Regensburg, University of Applied Sciences
x
r0 rS
ϕ∆
r
x
ϕ∆
D
deflected tire rigid wheel
Ω Ω
vt
Figure 3.12: Dynamic rolling radius
3.2.7 Dynamic Rolling Radius
At an angular rotation of4ϕ, assuming the tread particles stick to the track, the deectedtire moves on a distance of x, Figure 3.12.
With r0 as unloaded and rS = r0 −4r as loaded or static tire radius
r0 sin4ϕ = x (3.51)
andr0 cos4ϕ = rS (3.52)
hold.
If the motion of a tire is compared to the rolling of a rigid wheel, then, its radius rD willhave to be chosen so that at an angular rotation of 4ϕ the tire moves the distance
r0 sin4ϕ = x = rD4ϕ . (3.53)
Hence, the dynamic tire radius is given by
rD =r0 sin4ϕ
4ϕ. (3.54)
For 4ϕ → 0 one obtains the trivial solution rD = r0.
At small, yet nite angular rotations the sine-function can be approximated by the rstterms of its Taylor-Expansion. Then, Eq. (3.54) reads as
rD = r0
4ϕ− 164ϕ3
4ϕ= r0
(1− 1
64ϕ2
). (3.55)
With the according approximation for the cosine-function
rS
r0
= cos4ϕ = 1− 1
24ϕ2 or 4ϕ2 = 2
(1− rS
r0
)(3.56)
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FH Regensburg, University of Applied Sciences © Prof. Dr.-Ing. G. Rill
one nally gets
rD = r0
(1− 1
3
(1− rS
r0
))=
2
3r0 +
1
3rS . (3.57)
Due to rS = rS(Fz) the ctive radius rD depends on the wheel load Fz. Therefore, it iscalled dynamic tire radius. If the tire rotates with the angular velocity Ω, then
vt = rD Ω (3.58)
will denote the average velocity at which the tread particles are transported through thecontact patch.
0 2 4 6 8-20
-10
0
10
[mm
]
rD
- r0
Fz [kN]
Measurements− TMeasy tire model
Figure 3.13: Dynamic tire radius
In extension to Eq. (3.57), the dynamic tire radius is approximated in the tire modelTMeasy by
rD = λ r0 + (1− λ)
(r0 −
F Sz
c0
)︸ ︷︷ ︸
≈ rS
(3.59)
where the static tire radius rS = r0 −4r has been approximated by using the linearizedtire deformation 4r = F S
z /c0. The parameter λ is modeled as a function of the wheelload Fz
λ = λN + ( λ2N − λN )
(Fz
FNz
− 1
), (3.60)
where λN and λ2N denote the values for the pay load Fz = FNz and the doubled pay load
Fz = 2FNz .
With the TMeasy parameters for a passenger car tire
vertical tire stiffness at fz=fz0 [N/m], 190000.
vertical tire stiffness at fz=2*fz0 [N/m], 206000.
coefficient for dynamic tire radius fz=fz0 [-], 0.375
coefficient for dynamic tire radius fz=2*fz0 [-], 0.750
the approximation of measured tire data can be done very well, Figure 3.13.
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Vehicle Dynamics FH Regensburg, University of Applied Sciences
3.3 Forces and Torques caused by Pressure Distribution
3.3.1 Wheel Load
The vertical tire force Fz can be calculated as a function of the normal tire deection 4zand the deection velocity 4z
Fz = Fz(4z, 4z) . (3.61)
Because the tire can only apply pressure forces to the road the normal force is restrictedto Fz ≥ 0. In a rst approximation Fz is separated into a static and a dynamic part
Fz = F Sz + FD
z . (3.62)
The static part is described as a nonlinear function of the normal tire deection
F Sz = a14z + a2 (4z)2 . (3.63)
The constants a1 and a2 may be calculated from the radial stiness at nominal and doublepayload
cN =dF S
z
d4z
∣∣∣∣F S
z =F Nz
and c2N =dF S
z
d4z
∣∣∣∣F S
z =2F Nz
. (3.64)
The derivative of Eq. (3.63) results in
dF Sz
d4z= a1 + 2 a24z . (3.65)
From Eq. (3.63) one gets
4z =−a1 ±
√a2
1 + 4a2FSz
2a2
. (3.66)
Because the tire deection is always positive, the minus sign in front of the square roothas no physical meaning, and can be omitted therefore. Hence, Eq. (3.65) can be writtenas
dF Sz
d4z= a1 + 2 a2
(−a1 +
√a2
1 + 4a2FSz
2a2
)=√
a21 + 4a2F
Sz . (3.67)
Combining Eqs. (3.64) and (3.67 results in
cN =√
a21 + 4a2F
Nz or c2
N = a21 + 4a2F
Nz ,
c2N =√
a21 + 4a22F
Nz or c2
2N = a21 + 8a2F
Nz
(3.68)
nally leading to
a1 =√
2 c2N − c2
2N and a2 =c22N − c2
N
4 FNz
. (3.69)
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FH Regensburg, University of Applied Sciences © Prof. Dr.-Ing. G. Rill
Results for a passenger car and a truck tire are shown in Figure 3.14. The parabolicapproximation in Eq. (3.63) ts very well to the measurements. The radial tire stinessof the passenger car tire at the payload of Fz = 3 200 N can be specied with c0 =190 000N/m. The Payload Fz = 35 000 N and the stiness c0 = 1 250 000N/m of a trucktire are signicantly larger.
0 10 20 30 40 500
2
4
6
8
10Passenger Car Tire: 205/50 R15
Fz
[kN
]
0 20 40 60 800
20
40
60
80
100Truck Tire: X31580 R22.5
Fz
[kN
]
∆z [mm] ∆z [mm]
Figure 3.14: Tire radial stiness: Measurements, Approximation
The dynamic part is roughly approximated by
FDz = dR4z , (3.70)
where dR is a constant describing the radial tire damping, and the derivative of the tiredeformation 4z is given by Eq. (3.48).
3.3.2 Tipping Torque
The lateral shift of the vertical tire force Fz from the geometric contact point P to thestatic contact point Q is equivalent to a force applied in P and the tipping torque
Mx = Fz y (3.71)
acting around a longitudinal axis in P , Figure 3.15.
Note: Figure 3.15 shows a negative tipping torque. Because a positive camber angle movesthe contact point into the negative y-direction and hence, will generate a negative tippingtorque.
As long as the cambered tire has full contact to the road the lateral displacement y isgiven by Eq. (3.40). Then, Eq. (3.71) reads as
Mx = − Fzb2
124z
tan γ
cos γ. (3.72)
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Vehicle Dynamics FH Regensburg, University of Applied Sciences
en
γ
P Q
Fzy
ey
en
γ
P
Fz
ey
Tx
∼
Figure 3.15: Tipping torque at full contact
If the wheel load is approximated by its linearized static part Fz ≈ cN 4z and smallcamber angles |γ| 1 are assumed, then, Eq. (3.72) simplies to
Mx = − cN 4zb2
124zγ = − 1
12cN b2 γ , (3.73)
where the term 112
cNb2 can be regarded as the tipping stiness of the tire.
en
γ
P
Q
Fzy
ey
Figure 3.16: Cambered tire with partial contact
The use of the tipping torque instead of shifting the contact point is limited to those caseswhere the tire has full or nearly full contact to the road. If the cambered tire has onlypartly contact to the road, the geometric contact point P may even be located outsidethe contact area whereas the static contact point Q is still a real contact point, Figure3.16. In the following the static contact Q will be used as the contact point, because itrepresents the contact area more precisely than the geometric contact point P .
3.3.3 Rolling Resistance
If a non-rotating tire has contact to a at ground the pressure distribution in the contactpatch will be symmetric from the front to the rear, Figure 3.17. The resulting verticalforce Fz is applied in the center C of the contact patch and hence, will not generate atorque around the y-axis.
If the tire rotates tread particles will be stued into the front of the contact area whichcauses a slight pressure increase, Figure 3.17. Now, the resulting vertical force is appliedin front of the contact point and generates the rolling resistance torque
ty = −Fz xR sign(Ω) , (3.74)
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FH Regensburg, University of Applied Sciences © Prof. Dr.-Ing. G. Rill
Fz
C
Fz
Cex
enrotating
ex
en
non-rotatingxR
Figure 3.17: Pressure distribution at a non-rotation and rotation tire
where sign(Ω) assures that ty always acts against the wheel angular velocity Ω. Thedistance xR from C to the working point of Fz usually is related to the unloaded tireradius r0
fR =xR
r0
. (3.75)
The dimensionless rolling resistance coecient slightly increases with the traveling velocityv of the vehicle
fR = fR(v) . (3.76)
Under normal operating conditions, 20 km/h < v < 200 km/h, the rolling resistancecoecient for typical passenger car tires is in the range of 0.01 < fR < 0.02.
The rolling resistance hardly inuences the handling properties of a vehicle. But it playsa major part in fuel consumption.
3.4 Friction Forces and Torques
3.4.1 Longitudinal Force and Longitudinal Slip
To get a certain insight into the mechanism generating tire forces in longitudinal direction,we consider a tire on a at bed test rig. The rim rotates with the angular velocity Ω andthe at bed runs with the velocity vx. The distance between the rim center and the atbed is controlled to the loaded tire radius corresponding to the wheel load Fz, Figure 3.18.
A tread particle enters at the time t = 0 the contact patch. If we assume adhesion betweenthe particle and the track, then the top of the particle will run with the bed velocity vx
and the bottom with the average transport velocity vt = rD Ω. Depending on the velocitydierence 4v = rD Ω− vx the tread particle is deected in longitudinal direction
u = (rD Ω− vx) t . (3.77)
The time a particle spends in the contact patch can be calculated by
T =L
rD |Ω|, (3.78)
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Vehicle Dynamics FH Regensburg, University of Applied Sciences
vx
Ω
L
rD
u
umax
ΩrD
vx
Figure 3.18: Tire on at bed test rig
where L denotes the contact length, and T > 0 is assured by |Ω|.The maximum deection occurs when the tread particle leaves the contact patch at thetime t = T
umax = (rD Ω− vx) T = (rD Ω− vx)L
rD |Ω|. (3.79)
The deected tread particle applies a force to the tire. In a rst approximation we get
F tx = ct
x u , (3.80)
where ctx represents the stiness of one tread particle in longitudinal direction.
On normal wheel loads more than one tread particle is in contact with the track, Figure3.19a. The number p of the tread particles can be estimated by
p =L
s + a, (3.81)
where s is the length of one particle and a denotes the distance between the particles.
c u
b) L
max
tx *
c utu*
a) L
s a
Figure 3.19: a) Particles, b) Force distribution,
Particles entering the contact patch are undeformed, whereas the ones leaving have themaximum deection. According to Eq. (3.80), this results in a linear force distribution
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FH Regensburg, University of Applied Sciences © Prof. Dr.-Ing. G. Rill
versus the contact length, Figure 3.19b. The resulting force in longitudinal direction forp particles is given by
Fx =1
2p ct
x umax . (3.82)
Using the Eqs. (3.81) and (3.79) this results in
Fx =1
2
L
s + actx (rD Ω− vx)
L
rD |Ω|. (3.83)
A rst approximation of the contact length L was calculated in Eq. (3.34). Approximatingthe belt deformation by 4zB ≈ 1
2Fz/cR results in
L2 ≈ 4 r0Fz
cR
, (3.84)
where cR denotes the radial tire stiness, and nonlinearities and dynamic parts in the tiredeformation were neglected. Now, Eq. (3.82) can be written as
Fx = 2r0
s + a
ctx
cR
FzrD Ω− vx
rD |Ω|. (3.85)
The nondimensional relation between the sliding velocity of the tread particles in lon-gitudinal direction vS
x = vx − rD Ω and the average transport velocity rD |Ω| form thelongitudinal slip
sx =−(vx − rD Ω)
rD |Ω|. (3.86)
The longitudinal force Fx is proportional to the wheel load Fz and the longitudinal slipsx in this rst approximation
Fx = k Fz sx , (3.87)
where the constant k summarizes the tire properties r0, s, a, ctx and cR.
Eq. (3.87) holds only as long as all particles stick to the track. At moderate slip valuesthe particles at the end of the contact patch start sliding, and at high slip values only theparts at the beginning of the contact patch still stick to the road, Figure 3.20.
L
adhesion
Fxt <= FH
t
small slip valuesF = k F sx ** x F = F f ( s )x * x F = Fx Gz z
L
adhesion
Fxt
FHt
moderate slip values
L
sliding
Fxt FG
high slip values
=
sliding
=
Figure 3.20: Longitudinal force distribution for dierent slip values
The resulting nonlinear function of the longitudinal force Fx versus the longitudinal slipsx can be dened by the parameters initial inclination (driving stiness) dF 0
x , locationsM
x and magnitude of the maximum FMx , start of full sliding sS
x and the sliding force F Sx ,
Figure 3.21.
39
Vehicle Dynamics FH Regensburg, University of Applied Sciences
Fx
xM
xS
dFx0
sxsxsxM S
FF
adhesion sliding
Figure 3.21: Typical longitudinal force characteristics
3.4.2 Lateral Slip, Lateral Force and Self Aligning Torque
Similar to the longitudinal slip sx, given by Eq. (3.86), the lateral slip can be dened by
sy =−vS
y
rD |Ω|, (3.88)
where the sliding velocity in lateral direction is given by
vSy = vy (3.89)
and the lateral component of the contact point velocity vy follows from Eq. (3.50).
As long as the tread particles stick to the road (small amounts of slip), an almost lineardistribution of the forces along the length L of the contact patch appears. At moderateslip values the particles at the end of the contact patch start sliding, and at high slipvalues only the parts at the beginning of the contact patch stick to the road, Figure 3.22.
L
adhe
sion
F y
small slip values
Ladhe
sion
F y
slid
ing
moderate slip values
L
slid
ing F y
large slip values
n
F = k F sy ** y F = F f ( s )y * y F = Fy Gz z
Figure 3.22: Lateral force distribution over contact patch
The nonlinear characteristics of the lateral force versus the lateral slip can be describedby the initial inclination (cornering stiness) dF 0
y , the location sMy and the magnitude FM
y
40
FH Regensburg, University of Applied Sciences © Prof. Dr.-Ing. G. Rill
of the maximum, the beginning of full sliding sSy , and the magnitude F S
y of the slidingforce.
The distribution of the lateral forces over the contact patch length also denes the pointof application of the resulting lateral force. At small slip values this point lies behindthe center of the contact patch (contact point P). With increasing slip values it movesforward, sometimes even before the center of the contact patch. At extreme slip values,when practically all particles are sliding, the resulting force is applied at the center of thecontact patch.
The resulting lateral force Fy with the dynamic tire oset or pneumatic trail n as a levergenerates the self aligning torque
TS = −n Fy . (3.90)
The lateral force Fy as well as the dynamic tire oset are functions of the lateral slipsy. Typical plots of these quantities are shown in Figure 3.23. Characteristic parameters
Fy
yM
yS
dFy0
sysysyM S
F
Fadhesion adhesion/
slidingfull sliding
adhesion
adhesion/sliding
n/L
0
sysySsy
0
(n/L)
adhesion
adhesion/sliding
M
sysySsy
0
S
full sliding
full sliding
Figure 3.23: Typical plot of lateral force, tire oset and self aligning torque
of the lateral force graph are initial inclination (cornering stiness) dF 0y , location sM
y andmagnitude of the maximum FM
y , begin of full sliding sSy , and the sliding force F S
y .
The dynamic tire oset has been normalized by the length of the contact patch L. Theinitial value (n/L)0 as well as the slip values s0
y and sSy suciently characterize the graph.
3.4.3 Wheel Load Influence
The resistance of a real tire against deformations has the eect that with increasingwheel load the distribution of pressure over the contact area becomes more and moreuneven. The tread particles are deected just as they are transported through the contactarea. The pressure peak in the front of the contact area cannot be used, for these treadparticles are far away from the adhesion limit because of their small deection. In the rearof the contact area the pressure drop leads to a reduction of the maximally transmittable
41
Vehicle Dynamics FH Regensburg, University of Applied Sciences
Longitudinal force Fx Lateral force Fy
Fz = 3.0 kN Fz = 6.0 kN Fz = 3.0 kN Fz = 6.0 kN
dF 0x = 70 kN dF 0
x = 220 kN dF 0y = 72 kN dF 0
y = 130 kN
sMx = 0.160 sM
x = 0.120 sMy = 0.180 sM
y = 0.200
FMx = 2.90 kN FM
x = 5.60 kN FMy = 2.85 kN FM
y = 5.40 kN
sSx = 0.500 sS
x = 0.500 sSy = 0.500 sS
y = 0.700
F Sx = 2.65 kN F S
x = 5.10 kN F Sy = 2.80 kN F S
y = 5.30 kN
Table 3.3: Characteristic tire data with degressive wheel load inuence
friction force. With rising imperfection of the pressure distribution over the contact area,the ability to transmit forces of friction between tire and road lessens.
In practice, this leads to a degressive inuence of the wheel load on the characteristiccurves of longitudinal and lateral forces. In order to respect this fact in a tire model, thecharacteristic data for two nominal wheel loads FN
z and 2 FNz are given in Table 3.3.
From this data the initial inclinations dF 0x , dF 0
y , the maximal forces FMx , FM
x and thesliding forces F S
x , FMy for arbitrary wheel loads Fz are calculated by quadratic functions.
For the maximum longitudinal force it reads as
FMx (Fz) =
Fz
FNz
[2 FM
x (FNz )− 1
2FM
x (2FNz )−
(FM
x (FNz )− 1
2FM
x (2FNz ))Fz
FNz
]. (3.91)
-0.4 -0.2 0 0.2 0.4-6
-4
-2
0
2
4
6
-20 -10 0 10 20-6
-4
-2
0
2
4
6
Fx
[kN
]
sx [-]
Fy
[kN
]
α [deg]
Figure 3.24: Longitudinal and lateral force characteristics: Fz = 1.8, 3.2, 4.6, 5.4, 6.0 kN
The location of the maxima sMx , sM
y , and the slip values, sSx , sS
y , at which full slidingappears, are dened as linear functions of the wheel load Fz. For the location of the
42
FH Regensburg, University of Applied Sciences © Prof. Dr.-Ing. G. Rill
maximum longitudinal force this results in
sMx (Fz) = sM
x (FNz ) +
(sM
x (2FNz )− sM
x (FNz ))( Fz
FNz
− 1
). (3.92)
With the numeric values from Tab. 3.3 a slight shift of the maxima with an increasingwheel load is also modeled, Figure 3.24.
3.4.4 Two-Dimensional Tire Characteristics
The longitudinal force as a function of the longitudinal slip Fx = Fx(sx) and the lateralforce depending on the lateral slip Fy = Fy(sy) can be dened by their characteristicparameters initial inclination dF 0
x , dF 0y , location sM
x , sMy and magnitude of the maximum
FMx , FM
y as well as sliding limit sSx , sS
y and sliding force F Sx , F S
y , Figure 3.25. Duringgeneral driving situations, e.g. acceleration or deceleration in curves, longitudinal sx andlateral slip sy appear simultaneously.
Fy
sx
ssy
S
ϕ
FS
M
FM
dF0
F(s)
dF
S
y
FyFy
M
SsyMsy
0
Fy
sy
dFx0
FxM Fx
SFx
sxM
sxS
sx
Fx
s
s
Figure 3.25: Generalized tire characteristics
The longitudinal slip sx and the lateral slip sy can vectorally be added to a generalizedslip
s =
√(sx
sx
)2
+
(sy
sy
)2
, (3.93)
where the slips sx and sy were normalized by appropriate weighting factors sx and sy.
43
Vehicle Dynamics FH Regensburg, University of Applied Sciences
Similar to the graphs of the longitudinal and lateral forces the graph of the generalized tireforce is dened by the characteristic parameters dF 0, sM , FM , sS and F S. The parametersare calculated from the corresponding values of the longitudinal and lateral force
dF 0 =
√(dF 0
x sx cos ϕ)2 +(dF 0
y sy sin ϕ)2
,
sM =
√(sM
x
sx
cos ϕ
)2
+
(sM
y
sy
sin ϕ
)2
,
FM =
√(FM
x cos ϕ)2 +(FM
y sin ϕ)2
,
sS =
√(sS
x
sx
cos ϕ
)2
+
(sS
y
sy
sin ϕ
)2
,
F S =
√(F S
x cos ϕ)2 +(F S
y sin ϕ)2
,
(3.94)
where the slip normalization have also to be considered at the initial inclination. Theangular functions
cos ϕ =sx/sx
sand sin ϕ =
sy/sy
s(3.95)
grant a smooth transition from the characteristic curve of longitudinal to the curve oflateral forces in the range of ϕ = 0 to ϕ = 90.
The function F = F (s) is now described in intervals by a broken rational function, a cubicpolynomial, and by the sliding force F S
F (s) =
sM dF 0 σ
1 + σ
(σ + dF 0 sM
FM− 2
) , σ =s
sM, 0 ≤ s ≤ sM ;
FM − (FM − F S) σ2 (3− 2 σ) , σ =s− sM
sS − sM, sM < s ≤ sS ;
F S , s > sS .
(3.96)
When dening the curve parameters, one just has to make sure that the condition dF 0 ≥2 F M
sM is fullled, because otherwise the function has a turning point in the interval 0 <s ≤ sM .
Now, the longitudinal and the lateral force follow from the according projections in lon-gitudinal and lateral direction
Fx = F cos ϕ and Fy = F sin ϕ . (3.97)
Hence, within TMeasy the one-dimensional characteristics are automatically converted toa two-dimensional combination characteristics, Figure 3.26.
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FH Regensburg, University of Applied Sciences © Prof. Dr.-Ing. G. Rill
-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 %; |α| = 1, 2, 4, 6, 10, 14
Figure 3.26: Two-dimensional tire characteristics at Fz = 3.2 kN / Fz = 35 kN
3.4.5 Different Friction Coefficients
The tire characteristics are valid for one specic tire road combination only. Hence, dif-ferent tire road combinations will demand for dierent model parameter.
-0.5 0 0.5-4000
-3000
-2000
-1000
0
1000
2000
3000
4000 µF
sy [-]
Fy
[ N]
µL/µ0
0.20.40.60.81.0
Fz = 3.2 kN
Figure 3.27: Lateral force characteristics for dierent friction coecients
If only the coecient of friction is changed a simple but eective adaption of given modeldata is possible. A reduced or changed friction coecient mainly inuences the maximumforce and the sliding force, whereas the initial inclination will remain unchanged. So, bysetting
sM → µL
µ0
sM , FM → µL
µ0
FM , sS → µL
µ0
sS , F S → µL
µ0
F S , (3.98)
45
Vehicle Dynamics FH Regensburg, University of Applied Sciences
the essential tire model parameter which are valid for the friction coecient µ0 are ad-justed to the new friction coecient µL. The result of this simple approach is shown inFigure 3.27.
If the road model provides not only the roughness information z = fR(x, y) but also thelocal friction coecient [z, µL] = fR(x, y) then, braking on µ-split maneuvers can easilybe simulated.
3.4.6 Self Aligning Torque
According to Eq. (3.90) the self aligning torque can be calculated via the dynamic tireoset. The dynamic tire oset n can be normalized by the length L of the contact area,nN = n/L. It mainly depends on the lateral slip sy. The normalized tire oset startsat sy = 0 with an initial value (n/L)0. It tends to zero, n/L → 0 at large slip values,sy ≥ sS
y . Sometimes the dynamic tire oset overshoots to negative values before it reacheszero again. This behavior can be modeled by introducing the parameter s0
y < sSy , Figure
3.28.
n/L
0
sysySsy
0
(n/L)
n/L
0
sysy0
(n/L)
Figure 3.28: Normalized tire oset with and without overshoot
In order to achieve a simple and smooth approximation of the normalized tire oset versusthe lateral slip, a linear and a cubic function are overlayed in the rst section sy ≤ s0
y
n
L=(n
L
)0
[(1−w) (1−s) + w
(1− (3−2s) s2
)]|sy| ≤ s0
y and s =|sy|s0
y
− (1−w)|sy| − s0
y
s0y
(sS
y − |sy|sS
y − s0y
)2
s0y < |sy| ≤ sS
y
0 |sy| > sSy
(3.99)
where the factor
w =s0
y
sSy
(3.100)
weights the linear and the cubic function according to the values of the parameter s0y
and sSy . No overshoot occurs for s0
y = sSy . Here, w = 1 and (1 − w) = 0 will produce a
cubic transition from n/L = (n/L)0 to n/L = 0 with vanishing inclinations at sy = 0 andsy = s0
y.
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FH Regensburg, University of Applied Sciences © Prof. Dr.-Ing. G. Rill
The characteristic curve parameters, which are used for the description of the dynamictire oset, are at rst approximation not wheel load dependent. Similar to the descriptionof the characteristic curves of longitudinal and lateral force, here also the parameters forsingle and double pay load are given. The calculation of the parameters of arbitrary wheelloads is done similar to Eq. (3.92) by linear inter- or extrapolation.
Tire oset parameter
Fz = 3.0 kN Fz = 6.0 kN
(n/L)0 = 0.170 (n/L)0 = 0.190
s0y = 0.200 s0
y = 0.220
sEy = 0.420 sE
y = 0.400
Fz
0 10 20 30-30 -20 -10
0
50
100
150
200
-200
-150
-100
-50
Tz
[ Nm
]
α [deg]
Figure 3.29: Self aligning torque: Fz = 1.5, 3.0, 4.5, 6.0, 7.5 kN
The value of (n/L)0 can be estimated very well. At small values of lateral slip sy ≈ 0 onegets at rst approximation a triangular distribution of lateral forces over the contact arealength cf. Figure 3.22. The working point of the resulting force (dynamic tire oset) isthen given by
n(Fz→0, sy =0) =1
6L . (3.101)
The value n = 16L can only serve as reference point, for the uneven distribution of pressure
in longitudinal direction of the contact area results in a change of the deexion proleand the dynamic tire oset.
The self aligning torque in Figure 3.29 has been calculated with the tire parameter fromTable 3.3. The degressive inuence of the wheel load on the self aligning torque can beseen here as well.
With the parameters for the description of the tire oset it has been assumed that atthe payload Fz = FN
z the related tire oset reaches the value of (n/L)0 = 0.17 ≈ 1/6 atsy = 0. The slip value s0
y, at which the tire oset passes the x-axis, has been estimated.Usually the value is somewhat higher than the position of the lateral force maximum.With rising wheel load it moves to higher values. The values for sS
y are estimated too.
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Vehicle Dynamics FH Regensburg, University of Applied Sciences
3.4.7 Camber Influence
At a cambered tire, Figure 3.30, the angular velocity of the wheel Ω has a componentnormal to the road
Ωn = Ω sin γ . (3.102)
eyR
vγ(ξ)
rimcentreplane
Ω
γ
yγ(ξ)
Ωn
ξ
rD |Ω|ex
ey
en
F = Fy y (sy): Parameter γγ
-0.5 0 0.5-4000
-3000
-2000
-1000
0
1000
2000
3000
4000
Figure 3.30: Cambered tire Fy(γ) at Fz = 3.2 kN and γ = 0, 2 , 4 , 6 , 8
Now, the tread particles in the contact patch possess a lateral velocity which depends ontheir position ξ and is provided by
vγ(ξ) = −ΩnL
2
ξ
L/2, = −Ω sin γ ξ , −L/2 ≤ ξ ≤ L/2 . (3.103)
At the contact point it vanishes whereas at the end of the contact patch it takes on thesame value as at the beginning, however, pointing into the opposite direction. Assumingthat the tread particles stick to the track, the deection prole is dened by
yγ(ξ) = vγ(ξ) . (3.104)
The time derivative can be transformed to a space derivative
yγ(ξ) =d yγ(ξ)
d ξ
d ξ
d t=
d yγ(ξ)
d ξrD |Ω| (3.105)
where rD |Ω| denotes the average transport velocity. Now, Eq. (3.104) can be written as
d yγ(ξ)
d ξrD |Ω| = −Ω sin γ ξ or
d yγ(ξ)
d ξ= −Ω sin γ
rD |Ω|L
2
ξ
L/2, (3.106)
48
FH Regensburg, University of Applied Sciences © Prof. Dr.-Ing. G. Rill
where L/2 was used to achieve dimensionless terms. Similar to the lateral slip sy whichis dened by Eq. (3.88) we can introduce a camber slip now
sγ =−Ω sin γ
rD |Ω|L
2. (3.107)
Then, Eq. (3.106) simplies to
d yγ(ξ)
d ξ= sγ
ξ
L/2. (3.108)
The shape of the lateral displacement prole is obtained by integration
yγ = sγ1
2
L
2
(ξ
L/2
)2
+ C . (3.109)
The boundary condition y(ξ = 1
2L)
= 0 can be used to determine the integration constantC. One gets
C = −sγ1
2
L
2. (3.110)
Then, Eq. (3.109) reads as
yγ(ξ) = −sγ1
2
L
2
[1−
(ξ
L/2
)2]
. (3.111)
The lateral displacements of the tread particles caused by a camber slip are comparednow with the ones caused by pure lateral slip, Figure 3.31. At a tire with pure lateral
yγ(ξ)
ξ
y
-L/2 0 L/2
yy(ξ)
ξ
y
-L/2 0 L/2
a) camber slip b) lateral slip
yy
_
yγ_
Figure 3.31: Displacement proles of tread particles
slip each tread particle in the contact patch possesses the same lateral velocity whichresults in dyy/dξ rD |Ω| = vy, where according to Eq. (3.105) the time derivative yy wastransformed to the space derivative dyy/dξ . Hence, the deection prole is linear, andreads as yy = vy/(rD |Ω|) ξ = −sy ξ , where the denition in Eq. (3.88) was used tointroduce the lateral slip sy . Then, the average deection of the tread particles underpure lateral slip is given by
yy = −syL
2. (3.112)
49
Vehicle Dynamics FH Regensburg, University of Applied Sciences
The average deection of the tread particles under pure camber slip is obtained from
yγ = −sγ1
2
L
2
1
L
L/2∫−L/2
[1−
(x
L/2
)2]
dξ = −1
3sγ
L
2. (3.113)
A comparison of Eq. (3.112) with Eq. (3.113) shows, that by using
sγy =
1
3sγ (3.114)
the lateral camber slip sγ can be converted to an equivalent lateral slip sγy .
In normal driving conditions, the camber angle and thus, the lateral camber slip arelimited to small values. So, the lateral camber force can be approximated by
F γy ≈ dF 0
y sγy . (3.115)
If the global inclination dFy = Fy/sy is used instead of the initial inclination dF 0y , one
gets the camber inuence on the lateral force as shown in Figure 3.30.
The camber angle inuences the distribution of pressure in the lateral direction of thecontact patch, and changes the shape of the contact patch from rectangular to trape-zoidal. Thus, it is extremely dicult, if not impossible, to quantify the camber inuencewith the aid of such simple models. But, it turns out that this approach is quit a goodapproximation.
3.4.8 Bore Torque
In particular during steering motions the angular velocity of the wheel
ω0W = ω∗0R + Ω eyR (3.116)
has a component in direction of the track normal en
ωn = eTn ω0W 6= 0 . (3.117)
Then, a very complicated deection prole of the tread particles in the contact patchoccurs. However, by a simple approach the resulting bore torque can be approximatedquite good by the parameter of the generalized tire force characteristics.
At rst, the complex shape of a tire's contact patch is approximated by a circle, Figure3.32. By setting
R =1
2
(L
2+
B
2
)=
1
4(L + B) (3.118)
the radius of the circle can be adjusted to the length L and the width B of the actualcontact patch. The integration over the whole circle area results in the bore torque
TB =1
A
∫A
F r dA , (3.119)
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FH Regensburg, University of Applied Sciences © Prof. Dr.-Ing. G. Rill
normal shape of contact patchcircularapproximation
ϕdϕ
F
R
r
dr
ωn
B
L
ex
ey
Figure 3.32: Bore torque approximation
where F denotes the force transmitted by the patch element dA, and A is the area of thecircle. With dA = r dϕ dr and A = R2 π Eq. (3.119) reads as
TB =1
R2 π
R∫0
2π∫0
F r rdϕ dr (3.120)
which immediately results in
TB =1
R2 π
R∫0
F r r dr
2π
0
=2
R2
R∫0
F r2 dr . (3.121)
For small slip values the force transmitted in the patch element can be approximated by
F = F (s) ≈ dF 0 s (3.122)
where s denotes the slip of the patch element, and dF 0 is the initial inclination of thegeneralized tire force characteristics. Similar to Eqs. (3.86 and (3.88) we dene
s =−r ωn
rD |Ω|(3.123)
where r ωn describes the sliding velocity in the patch element, and rD and Ω denote thedynamic tire radius and the angular velocity of the wheel.
Now, Eq. (3.121) reads as
TB =2
R2
R∫0
dF 0 −r ωn
rD |Ω|r2 dr (3.124)
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Vehicle Dynamics FH Regensburg, University of Applied Sciences
which nally results in
TB = − 2
R2dF 0 ωn
rD |Ω|
R∫0
r3 dr = − 2
R2dF 0 ωn
rD |Ω|R4
4=
1
2R dF 0 −R ωn
rD |Ω|(3.125)
wheresB =
−R ωn
rD |Ω|(3.126)
can be considered as bore slip. Via the initial inclination dF 0 the bore torque TB takesthe actual tire properties into account.
The bore torque calculated by Eq. (3.125) is only a rst approximation. At large bore slipsthe generalized tire force F is limited to the sliding force F S. Then, Eq. (3.121) changesto
TmaxB =
2
R2
R∫0
F S r2 dr =2
R2F S R3
3=
2
3F S R . (3.127)
Due to the generalized sliding force F S the maximum bore torque TmaxB depends on the
tire properties and the actual friction value. Now, the bore torque is given by
TB =1
2R dF 0 −R ωn
rD |Ω|with |TB| ≤
2
3F S R (3.128)
where according to Eq. (3.118) the circle radius R can be replaced by the length L andthe width B of the contact patch.
3.4.9 Typical Tire Characteristics
-40 -20 0 20 40-6
-4
-2
0
2
4
6
Fx [k
N]
1.8 kN3.2 kN4.6 kN5.4 kN
-40 -20 0 20 40
-40
-20
0
20
40
Fx [k
N]
10 kN20 kN30 kN40 kN50 kN
passenger car tire truck tire
sx [%] sx [%]
Figure 3.33: Longitudinal force: Meas., − TMeasy
The tire model TMeasy which is based on this approach, can be used for passenger cartires as well as for truck tires. It approximates the characteristic curves Fx = Fx(sx),
52
FH Regensburg, University of Applied Sciences © Prof. Dr.-Ing. G. Rill
-6
-4
-2
0
2
4
6F y
[kN
]
1.8 kN3.2 kN4.6 kN6.0 kN
-20 -10 0 10 20-150
-100
-50
0
50
100
150
α [o]
Mz
[Nm
]
1.8 kN3.2 kN4.6 kN6.0 kN
-40
-20
0
20
40
Fy [k
N]
10 kN20 kN30 kN40 kN
-20 -10 0 10 20-1500
-1000
-500
0
500
1000
1500
α
Mz
[Nm
]
18.4 kN36.8 kN55.2 kN
[o]
passenger car truck
Figure 3.34: Lateral force and self aligning torque: Meas., − TMeasy
Fy = Fy(α) and Mz = Mz(α) quite well even for dierent wheel loads Fz, Figures 3.33and 3.34.
When experimental tire values are missing, the model parameters can be pragmaticallyestimated by adjustment of the data of similar tire types. Furthermore, due to their phys-ical signicance, the parameters can subsequently be improved by means of comparisonsbetween the simulation and vehicle testing results as far as they are available.
53
4 Suspension System
4.1 Purpose and Components
The automotive industry uses dierent kinds of wheel/axle suspension systems. Importantcriteria are costs, space requirements, kinematic properties, and compliance attributes.
The main purposes of a vehicle suspension system are
carry the car and its weight,
maintain correct wheel alignment,
control the vehicles direction of travel,
keep the tires in contact with the road,
reduce the eect of shock forces.
Vehicle suspension systems consist of
guiding elements:
control arms, links,
struts,
leaf springs,
force elements:
coil spring, torsion bar, air spring, leaf spring,
anti-roll bar,
damper,
bushings, hydro-mounts,
tires.
Tires are air springs that support the total weight of the vehicle. The air spring action ofthe tire is very important to the ride quality and safe handling of the vehicle.
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4.2 Some Examples
4.2.1 Multi Purpose Systems
The double wishbone suspension, the McPherson suspension and the multi-link suspensionare multi purpose wheel suspension systems, Fig. 4.1.
Figure 4.1: Double wishbone, McPherson and multi-link suspension
They are used as steered front or non steered rear axle suspension systems. These sus-pension systems are also suitable for driven axles.
In a McPherson suspension the spring is mounted with an inclination to the strut axis.Thus, bending torques at the strut, which cause high friction forces, can be reduced.
leaf springs
links
Figure 4.2: Solid axles guided by leaf springs and links
At pickups, trucks, and busses solid axles are used often. They are guided either by leafsprings or by rigid links, Fig. 4.2. Solid axles tend to tramp on rough roads.
Leaf-spring-guided solid axle suspension systems are very robust. Dry friction betweenthe leafs leads to locking eects in the suspension. Although the leaf springs provideaxle guidance on some solid axle suspension systems, additional links in longitudinal andlateral direction are used. Thus, the typical wind-up eect on braking can be avoided.
Solid axles suspended by air springs need at least four links for guidance. In addition toa good driving comfort air springs allow level control too.
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4.2.2 Specific Systems
The semi-trailing arm, the short-long-arm axle (SLA), and the twist beam axle suspensionare suitable only for non-steered axles, Fig. 4.3.
Figure 4.3: Specic wheel/axles suspension systems
The semi-trailing arm is a simple and cheap design which requires only few space. It ismostly used for driven rear axles.
The short-long-arm axle design allows a nearly independent layout of longitudinal andlateral axle motions. It is similar to the central control arm axle suspension, where thetrailing arm is completely rigid and hence, only two lateral links are needed.
The twist beam axle suspension exhibits either a trailing arm or a semi-trailing armcharacteristic. It is used for non driven rear axles only. The twist beam axle providesenough space for spare tire and fuel tank.
4.3 Steering Systems
4.3.1 Requirements
The steering system must guarantee easy and safe steering of the vehicle. The entiretyof the mechanical transmission devices must be able to cope with all loads and stressesoccurring in operation.
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In order to achieve a good maneuverability a maximum steering angle of approx. 30 mustbe provided at the front wheels of passenger cars. Depending on the wheel base, bussesand trucks need maximum steering angles up to 55 at the front wheels.
Recently some companies have started investigations on `steer by wire' techniques.
4.3.2 Rack and Pinion Steering
Rack-and-pinion is the most common steering system of passenger cars, Fig. 4.4. The rackmay be located either in front of or behind the axle. Firstly, the rotations of the steering
steeringbox
rackdrag link
wheelandwheelbody
uR
δ1 δ2
pinionδS
Figure 4.4: Rack and pinion steering
wheel δS are transformed by the steering box to the rack travel uR = uR(δS) and then viathe drag links transmitted to the wheel rotations δ1 = δ1(uR), δ2 = δ2(uR). Hence, theoverall steering ratio depends on the ratio of the steering box and on the kinematics ofthe steering linkage.
4.3.3 Lever Arm Steering System
steering box
drag link 1
δ2δ1
δL drag link 2
steering lever 2steering lever 1
wheel andwheel body
Figure 4.5: Lever arm steering system
Using a lever arm steering system Fig. 4.5, large steering angles at the wheels are possible.This steering system is used on trucks with large wheel bases and independent wheel
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Vehicle Dynamics FH Regensburg, University of Applied Sciences
suspension at the front axle. Here, the steering box can be placed outside of the axlecenter.Firstly, the rotations of the steering wheel δS are transformed by the steering box to therotation of the steer levers δL = δL(δS). The drag links transmit this rotation to the wheelδ1 = δ1(δL), δ2 = δ2(δL). Hence, the overall steering ratio again depends on the ratio ofthe steering box and on the kinematics of the steering linkage.
4.3.4 Drag Link Steering System
At solid axles the drag link steering system is used, Fig. 4.6. The rotations of the steering
steer box(90o rotated)
drag link
steering link
steering
lever
OδL
δ1 δ2
wheelandwheelbody
Figure 4.6: Drag link steering system
wheel δS are transformed by the steering box to the rotation of the steering lever armδL = δL(δS) and further on to the rotation of the left wheel, δ1 = δ1(δL). The drag linktransmits the rotation of the left wheel to the right wheel, δ2 = δ2(δ1). The steering ratiois dened by the ratio of the steering box and the kinematics of the steering link. Here,the ratio δ2 = δ2(δ1) given by the kinematics of the drag link can be changed separately.
4.3.5 Bus Steer System
In busses the driver sits more than 2 m in front of the front axle. In addition, large steeringangles at the front wheels are needed to achieve a good manoeuvrability. That is why,more sophisticated steering systems are needed, Fig. 4.7. The rotations of the steeringwheel δS are transformed by the steering box to the rotation of the steering lever armδL = δL(δS). The left lever arm is moved via the steering link δA = δA(δL). This motionis transferred by a coupling link to the right lever arm. Finally, the left and right wheelsare rotated via the drag links, δ1 = δ1(δA) and δ2 = δ2(δA).
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steering box
steering link
δ2δ1
drag link coupl.link
leftlever arm
steering lever
δA
wheel andwheel body
δL
Figure 4.7: Typical bus steering system
4.4 Standard Force Elements
4.4.1 Springs
Springs support the weight of the vehicle. In vehicle suspensions coil springs, air springs,torsion bars, and leaf springs are used, Fig. 4.8.
Coil spring
Leaf springTorsion baru
u
u Air springu
FS
FS
FS
FS
Figure 4.8: Vehicle suspension springs
Coil springs, torsion bars, and leaf springs absorb additional load by compressing. Thus,the ride height depends on the loading condition. Air springs are rubber cylinders lledwith compressed air. They are becoming more popular on passenger cars, light trucks, andheavy trucks because here the correct vehicle ride height can be maintained regardless ofthe loading condition by adjusting the air pressure.
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Vehicle Dynamics FH Regensburg, University of Applied Sciences
c
L
FS
LF
∆L
u
FSc
L0
u
FS
FS
0
Figure 4.9: Linear coil spring and general spring characteristics
A linear coil spring may be characterized by its free length LF and the spring stiness c,Fig. 4.9. The force acting on the spring is then given by
FS = c(LF − L
), (4.1)
where L denotes the actual length of the spring. Mounted in a vehicle suspension the springhas to support the corresponding chassis weight. Hence, the spring will be compressed tothe conguration length L0 < LF . Now, Eq. (4.1) can be written as
FS = c(LF − (L0 − u)
)= c
(LF − L0
)+ c u = F 0
S + c u , (4.2)
where F 0S is the spring preload and u describes the spring displacement measured from
the spring's conguration length.
In general the spring force FS can be dened by a nonlinear function of the spring dis-placement u
FS = FS(u) . (4.3)
Now, arbitrary spring characteristics can be approximated by elementary functions, likepolynomials, or by tables which are then inter- and extrapolated by linear functions orcubic splines.
The complex behavior of leaf springs and air springs can only be approximated by simplenonlinear spring characteristics, FS = FS(u). For detailed investigations sophisticated oreven dynamic spring models have to be used.
4.4.2 Damper
Dampers are basically oil pumps, Fig. 4.10. As the suspension travels up and down, thehydraulic uid is forced by a piston through tiny holes, called orices. This slows downthe suspension movement.
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Remote Oil Ch.
Rebound Ch.
Remote Gas Chamber
CompressionChamber
PistonFD
v
FD
Piston orifice
Remote orifice
Figure 4.10: Principle of a mono-tube damper
Today twin-tube and mono-tube dampers are used in vehicle suspension systems. Dynamicdamper models compute the damper force via the uid pressure applied to each side ofthe piston. The change in uid pressures in the compression and rebound chambers arecalculated by applying the conservation of mass.
In standard vehicle dynamics applications simple characteristics
FD = FD(v) (4.4)
are used to describe the damper force FD as a function of the damper velocity v. Toobtain this characteristics the damper is excited with a sinusoidal displacement signalu = u0 sin 2πft. By varying the frequency in several steps from f = f0 to f = fE dierentforce displacement curves FD = FD(u) are obtained, Fig. 4.11. By taking the peak valuesof the damper force at the displacement u = u0 which corresponds with the velocityv = ±2πfu0 the characteristics FD = FD(v) is generated now. Here, the rebound cycle isassociated with negative damper velocities.
1000
0-0.04 0
-1000
-2000
-3000
-0.4-0.8-1.2-1.6-0.02 0.02 0.04
0
-4000-0.06 0.06 0.8 1.61.20.4
FD = FD(u) FD = FD(v)
Rebound
Compression
FD
[N
]
u [m] v [m/s]
fE
f0
Figure 4.11: Damper characteristics generated from measurements
Typical passenger car or truck dampers will have more resistance during its rebound cyclethen its compression cycle.
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4.4.3 Rubber Elements
Force elements made of natural rubber or urethane compounds are used in many locationson the vehicle suspension system, Fig. 4.12. Those elements require no lubrication, isolateminor vibration, reduce transmitted road shock, operate noise free, oer high load carryingcapabilities, and are very durable.
Control armbushings
Subframe mounts
Topmount
Stop
Figure 4.12: Rubber elements in vehicle suspension
During suspension travel, the control arm bushings provide a pivot point for the controlarm. They also maintain the exact wheel alignment by xing the lateral and verticallocation of the control arm pivot points. During suspension travel the rubber portion ofthe bushing must twist to allow control arm movement. Thus, an additional resistance tosuspension movement is generated.
Bump and rebound stops limit the suspension travel. The compliance of the topmountavoids the transfer of large shock forces to the chassis. The subframe mounts isolate thesuspension system from the chassis and allow elasto-kinematic steering eects of the wholeaxle.
It turns out, that those elastic elements can hardly be described by simple spring anddamper characteristics, FS = FS(u) and FD = FD(v), because their stiness and dampingproperties change with the frequency of the motion. Here, more sophisticated dynamicmodels are needed.
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4.5 Dynamic Force Elements
4.5.1 Testing and Evaluating Procedures
The eect of dynamic force elements is usually evaluated in the frequency domain. Forthis, on test rigs or in simulation the force element is excited by sine waves
xe(t) = A sin(2π f t) , (4.5)
with dierent frequencies f0 ≤ f ≤ fE and amplitudes Amin ≤ A ≤ Amax. Starting att = 0, the system will usually be in a steady state condition after several periods t ≥ nT ,where T = 1/f and n = 2, 3, . . . have to be chosen appropriately. Due to the nonlinearsystem behavior the system response is periodic, F (t + T ) = F (T ), where T = 1/f , yetnot harmonic. That is why, the measured or calculated force F will be approximatedwithin one period n T ≤ t ≤ (n + 1)T , by harmonic functions as good as possible
F (t)︸︷︷︸measured/calculated
≈ α sin(2π f t) + β cos(2π f t)︸ ︷︷ ︸rst harmonic approximation
. (4.6)
The coecients α and β can be calculated from the demand for a minimal overall error
1
2
(n+1)T∫nT
(α sin(2π f t)+β cos(2π f t)− F (t)
)2
dt −→ Minimum . (4.7)
The dierentiation of Eq. (4.7) with respect to α and β yields two linear equations asnecessary conditions
(n+1)T∫nT
(α sin(2π f t)+β cos(2π f t)− F (t)
)sin(2π f t) dt = 0
(n+1)T∫nT
(α sin(2π f t)+β cos(2π f t)− F (t)
)cos(2π f t) dt = 0
(4.8)
with the solutions
α =
∫F sin dt
∫cos2 dt−
∫F cos dt
∫sin cos dt∫
sin2 dt∫
cos2 dt− 2∫
sin cos dt
β =
∫F cos dt
∫sin2 dt−
∫F sin dt
∫sin cos dt∫
sin2 dt∫
cos2 dt− 2∫
sin cos dt
, (4.9)
where the integral limits and arguments of sine and cosine no longer have been written.Because it is integrated exactly over one period nT ≤ t ≤ (n + 1)T , for the integrals inEq. (4.9) ∫
sin cos dt = 0 ;∫
sin2 dt =T
2;∫
cos2 dt =T
2(4.10)
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Vehicle Dynamics FH Regensburg, University of Applied Sciences
hold, and as solution
α =2
T
∫F sin dt , β =
2
T
∫F cos dt (4.11)
remains. However, these are exactly the rst two coecients of a Fourier-"-Approximation.
The rst order harmonic approximation in Eq. (4.6) can now be written as
F (t) = F sin (2π f t + Ψ) (4.12)
where amplitude F and phase angle Ψ are given by
F =√
α2 + β2 and tan Ψ =β
α. (4.13)
A simple force element consisting of a linear spring with the stiness c and a linear damperwith the constant d in parallel would respond with
F (t) = c xe + d xe = c A sin 2πft + d 2πf A cos 2πft . (4.14)
Here, amplitude and phase angle are given by
F = A
√c2 + (2πfd)2 and tan Ψ =
d 2πf A
c A= 2πf
d
c. (4.15)
Hence, the response of a pure spring, c 6= 0 and d = 0 is characterized by F = A c andtan Ψ = 0 or Ψ = 0, whereas a pure damper response with c = 0 and d 6= 0 results inF = 2πfdA and tan Ψ → ∞ or Ψ = 90. Hence, the phase angle Ψ which is also calledthe dissipation angle can be used to evaluate the damping properties of the force element.The dynamic stiness, dened by
cdyn =F
A(4.16)
is used to evaluate the stiness of the element.
In practice the frequency response of a system is not determined punctually, but continu-ously. For this, the system is excited by a sweep-sine. In analogy to the simple sine-function
xe(t) = A sin(2π f t) , (4.17)
where the period T = 1/f appears as pre-factor at dierentiation
xe(t) = A 2π f cos(2π f t) =2π
TA cos(2π f t) . (4.18)
A generalized sine-function can be constructed, now. Starting with
xe(t) = A sin(2π h(t)) , (4.19)
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the time derivative results in
xe(t) = A 2π h(t) cos(2π h(t)) . (4.20)
In the following we demand that the function h(t) generates periods fading linearly intime, i.e:
h(t) =1
T (t)=
1
p− q t, (4.21)
where p > 0 and q > 0 are constants yet to determine. Eq. (4.21) yields
h(t) = − 1
qln(p− q t) + C . (4.22)
The initial condition h(t = 0) = 0 xes the integration constant
C =1
qln p . (4.23)
With Eqs. (4.23) and (4.22) Eq. (4.19) results in a sine-like function
xe(t) = A sin(2π
qln
p
p− q t
), (4.24)
which is characterized by linear fading periods.
The important zero values for determining the period duration lie at
1
qln
p
p− q tn= 0, 1, 2, or
p
p− q tn= en q , mit n = 0, 1, 2, (4.25)
andtn =
p
q(1− e−n q) , n = 0, 1, 2, . (4.26)
The time dierence between two zero values yields the period
Tn = tn+1 − tn =p
q(1−e−(n+1) q − 1+e−n q)
Tn =p
qe−n q (1− e−q)
, n = 0, 1, 2, . (4.27)
For the rst (n = 0) and last (n = N) period one nds
T0 =p
q(1− e−q)
TN =p
q(1− e−q) e−N q = T0 e−N q
. (4.28)
With the frequency range to investigate, given by the initial f0 and nal frequency fE,the parameters q and the ratio q/p can be calculated from Eq. (4.28)
q =1
Nln
fE
f0
,q
p= f0
1−
[fE
f0
] 1N
, (4.29)
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with N xing the number of frequency intervals. The passing of the whole frequency rangethen takes the time
tN+1 =1− e−(N+1) q
q/p. (4.30)
Hence, to test or simulate a force element in the frequency range from 0.1Hz to f = 100Hzwith N = 500 intervals will only take 728 s or 12min.
4.5.2 Simple Spring Damper Combination
Fig. 4.13 shows a simple dynamic force element where a linear spring with the stiness cand a linear damper with the damping constant d are arranged in series.
s u
c d
Figure 4.13: Spring and damper in series
The displacements of the force element and the spring itself are described by u and s.Then, the the forces acting in the spring and damper are given by
FS = c s and FD = d (u− s) . (4.31)
The force balance FD = FS results in a linear rst order dierential equation for thespring displacement s
d (u− s) = c s ord
cs = −s +
d
cu , (4.32)
where the ratio between the damping coecient d and the spring stiness c acts as timeconstant, T = d/c. Hence, this force element will responds dynamically to any excitation.
The steady state response to a harmonic excitation
u(t) = u0 sin Ωt respectively u = u0Ω cos Ωt (4.33)
can be calculated easily. The steady state response will be of the same type as the exci-tation. Inserting
s∞(t) = u0 (a sin Ωt + b cos Ωt) (4.34)
into Eq. (4.32) results in
d
cu0 (aΩ cos Ωt− bΩ sin Ωt)︸ ︷︷ ︸
s∞
= − u0 (a sin Ωt + b cos Ωt)︸ ︷︷ ︸s∞
+d
cu0Ω cos Ωt︸ ︷︷ ︸
u
. (4.35)
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Collecting all sine and cosine terms we obtain two equations
− d
cu0 bΩ = −u0 a and
d
cu0 a Ω = −u0 b +
d
cu0Ω (4.36)
which can be solved for the two unknown parameter
a =Ω2
Ω2 + (c/d)2 and b =c
d
Ω
Ω2 + (c/d)2 . (4.37)
Hence, the steady state force response reads as
FS = c s∞ = c u0Ω
Ω2 + (c/d)2
(Ω sin Ωt +
c
dcos Ωt
)(4.38)
which can be transformed to
FS = FS sin (Ωt + Ψ) (4.39)
where the force magnitude FS and the phase angle Ψ are given by
FS =c u0 Ω
Ω2 + (c/d)2
√Ω2 + (c/d)2 =
c u0 Ω√Ω2 + (c/d)2
and Ψ = arctanc/d
Ω. (4.40)
The dynamic stiness cdyn = FS/u0 and the phase angle Ψ are plotted in Fig. 4.14 fordierent damping values.
0
100
200
300
400
0 20 40 60 80 1000
50
100
cdyn
[N/mm]
[o]Ψ
f [Hz]
c = 400 N/mm
d1 = 1000 N/(m/s)d2 = 2000 N/(m/s)
d1 = 3000 N/(m/s)d2 = 4000 N/(m/s)
1
1
23
4
234
c
d
Figure 4.14: Frequency response of a spring damper combination
With increasing frequency the spring damper combination changes from a pure damperperformance, cdyn → 0 and Ψ ≈ 90 to a pure spring behavior, cdyn ≈ c and Ψ → 0. Thefrequency range, where the element provides stiness and damping is controlled by thevalue for the damping constant d.
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4.5.3 General Dynamic Force Model
To approximate the complex dynamic behavior of bushings and elastic mounts dierentspring damper models can be combined. A general dynamic force model is constructedby N parallel force elements, Fig. 4.15. The static load is carried by a single spring withthe stiness c0 or an arbitrary nonlinear force characteristics F0 = F0(u).
c1 c2 cN
d1
s1
d2
s2
dN
sN
u
FF1FM FF2FM FFNFM
c0
Figure 4.15: Dynamic force model
Within each force element the spring acts in serial to parallel combination of a damperand a dry friction element. Now, even hysteresis eects and the stress history of the forceelement can be taken into account.
The forces acting in the spring and damper of force element i are given by
FSi = −ci si and FDi = di (si − u) , (4.41)
were u and si describe the overall element and the spring displacement.
As long as the absolute value of the spring force FSi is lower than the maximum frictionforce FM
F the damper friction combination will not move at all
u− si = 0 for |FSi| ≤ FMF . (4.42)
In all other cases the force balance
FSi = FDi ± FMF (4.43)
holds. Using Eq. 4.41 the force balance results in
di (si − u) = FSi ∓ FMF (4.44)
which can be combined with Eq. 4.42 to
di si =
FSi + FM
F FSi <−FMF
di u for −FMF ≤FSi≤+FM
F
FSi − FMF +FM
F <FSi
(4.45)
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where according to Eq. 4.41 the spring force is given by FSi = −ci si.
In extension to this linear approach nonlinear springs and dampers may be used. To deriveall the parameters an extensive set of static and dynamic measurements is needed.
4.5.3.1 Hydro-Mount
For the elastic suspension of engines in vehicles very often specially developed hydro-mounts are used. The dynamic nonlinear behavior of these components guarantees agood acoustic decoupling but simultaneously provides sucient damping.
main spring
chamber 1
membrane
ring channel
xe
c2T
cF
MF
uF
__ c2T__
d2F__d
2F__
chamber 2
Figure 4.16: Hydro-mount
Fig. 4.16 shows the principle and mathematical model of a hydro-mount. At small de-formations the change of volume in chamber 1 is compensated by displacements of themembrane. When the membrane reaches the stop, the liquid in chamber 1 is pressedthrough a ring channel into chamber 2. The ratio of the chamber cross section to the ringchannel cross section is very large. Thus the uid is moved through the ring channel atvery high speed. This results in remarkable inertia and resistance forces (damping forces).
The force eect of a hydro-mount is combined from the elasticity of the main spring andthe volume change in chamber 1.
With uF labeling the displacement of the generalized uid mass MF ,
FH = cT xe + FF (xe − uF ) (4.46)
holds, where the force eect of the main spring has been approximated by a linear springwith the constant cT .
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With MFR as the actual mass in the ring channel and the cross sections AK , AR ofchamber and ring channel the generalized uid mass is given by
MF =(AK
AR
)2
MFR . (4.47)
The uid in chamber 1 is not being compressed, unless the membrane can evade no longer.With the uid stiness cF and the membrane clearance sF , one gets
FF (xe − uF ) =
cF
((xe − uF ) + sF
)(xe − uF ) < −sF
0 for |xe − uf | ≤ sF
cF
((xe − uF ) − sF
)(xe − uf ) > +sF
(4.48)
The hard transition from clearance FF = 0 and uid compression resp. chamber deforma-tion with FF 6= 0 is not realistic and leads to problems, even with the numeric solution.Therefore, the function (4.48) is smoothed by a parabola in the range |xe − uf | ≤ 2 ∗ sF .
The motions of the uid mass cause friction losses in the ring channel, which are as a rstapproximation proportional to the speed,
FD = dF uF . (4.49)
Then, the equation of motion for the uid mass reads as
MF uF = − FF − FD . (4.50)
The membrane clearance makes Eq. (4.50) nonlinear and only solvable by numerical in-tegration. The nonlinearity also aects the overall force in the hydro-mount, Eq. (4.46).
The dynamic stiness and the dissipation angle of a hydro-mount are displayed in Fig. 4.17versus the frequency.
The simulation is based on the following system parameters
mF = 25 kg generalized uid mass
cT = 125 000 N/m stiness of main spring
dF = 750 N/(m/s) damping constant
cF = 100 000 N/m uid stiness
sF = 0.0002 mm clearance in membrane bearing
By the nonlinear and dynamic behavior a very good compromise can be achieved betweennoise isolation and vibration damping.
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0
100
200
300
400
100
101
0
10
20
30
40
50
60Dissipation Angle [deg] at Excitation Amplitudes A = 2.5/0.5/0.1 mm
Excitation Frequency [Hz]
Dynamic Stiffness [N/m] at Excitation Amplitudes A = 2.5/0.5/0.1 mm
Figure 4.17: Dynamic stiness [N/mm] and dissipation angle [deg] for a hydro-mount
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5 Vertical Dynamics
5.1 Goals
The aim of vertical dynamics is the tuning of body suspension and damping to guaranteegood ride comfort, resp. a minimal stress of the load at sucient safety.
The stress of the load can be judged fairly well by maximal or integral values of the bodyaccelerations.
The wheel load Fz is linked to the longitudinal Fx and lateral force Fy by the coecientof friction. The digressive inuence of Fz on Fx and Fy as well as non-stationary processesat the increase of Fx and Fy in the average lead to lower longitudinal and lateral forcesat wheel load variations.
Maximal driving safety can therefore be achieved with minimal variations of the wheelload. Small variations of the wheel load also reduce the stress on the track.
The comfort of a vehicle is subjectively judged by the driver. In literature dierent ap-proaches of describing the human sense of vibrations by dierent metrics can be found.
Transferred to vehicle vertical dynamics, the driver primarily registers the amplitudes andaccelerations of the body vibrations. These values are thus used as objective criteria inpractice.
5.2 Modelling Aspects
5.2.1 Full Vehicle Model
For detailed investigations of ride comfort and ride safety sophisticated road and vehiclemodels are needed. The three-dimensional vehicle model, shown in Fig. 5.1, includesan elastically suspended engine, and dynamic seat models. The elasto-kinematics of thewheel suspension was described fully nonlinear. In addition, dynamic force elements forthe damper elements and the hydro-mounts are used. Such sophisticated models not onlyprovide simulation results which are in good conformity to measurements but also makeit possible to investigate the vehicle dynamic attitude in an early design stage.
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XXXXXXXXX YYYYYYYYY
ZZZZZZZZZ
Time = 0.000000
Thilo Seibert Ext. 37598Vehicle Dynamics, Ford Research Center Aachen
/export/ford/dffa089/u/tseiber1/vedyna/work/results/mview.mvw 07/02/98 AA/FFA
Ford
Figure 5.1: Full vehicle model
5.2.2 Twodimensional Models
Much simpler models can be used, however, for basic studies on ride comfort and ridesafety. A two-dimensional vehicle model, for instance, suits perfectly with a single trackroad model, Fig. 5.2. Neglecting longitudinal accelerations, the vehicle chassis only per-
Ca1
a2
M*M1
M2
m1
m2
zA1
zC1
zR(s-a2)
zC2
zR(s+a1)
zB
xB
yB
C
hubM, Θ
zA2
pitch
zR(s) s
Figure 5.2: Vehicle model for basic comfort and safety analysis
forms hub and pitch motions. Here, the chassis is considered as one rigid body. Then,mass and inertia properties can be represented by three point masses which are located inthe chassis center of gravity and on top of the front and the rear axle. The lumped massmodel has 4 degrees of freedom. The hub and pitch motion of the chassis are representedby the vertical motions of the chassis in the front zC1 and in the rear zC2. The coordinateszA1 and zA2 describe the vertical motions of the front and rear axle. The function zR(s)provides road irregularities in the space domain, where s denotes the distance covered bythe vehicle and measured at the chassis center of gravity. Then, the irregularities at thefront and rear axle are given by zR(s + a1) and zR(s − a2) respectively, where a1 and a2
locate the position of the chassis center of gravity C.
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Vehicle Dynamics FH Regensburg, University of Applied Sciences
The point masses must add up to the chassis mass
M1 + M∗ + M2 = M (5.1)
and they have to provide the same inertia around an axis located in the chassis center Cand pointing into the lateral direction
a21M1 + a2
2M2 = Θ . (5.2)
The correct location of the center of gravity is assured by
a1M1 = a2M2 . (5.3)
Now, Eqs. (5.2) and (5.3) yield the main masses
M1 =Θ
a1(a1+a2)and M2 =
Θ
a2(a1+a2), (5.4)
and the coupling mass
M∗ = M
(1− Θ
Ma1a2
)(5.5)
follows from Eq. (5.1).
If the mass and the inertia properties of a real vehicle happen to result in Θ = Ma1a2 then,the coupling mass vanishes M∗ = 0, and the vehicle can be represented by two uncoupledtwo mass systems describing the vertical motion of the axle and the hub motion of thechassis mass on top of each axle.
vehicles
properties
midsizecar
fullsizecar
sportsutilityvehicle
commercialvehicle
heavytruck
front axlemass m1 [kg] 80 100 125 120 600
rear axlemass m2 [kg] 80 100 125 180 1100
centerof
gravity
a1 [m]
a2 [m]
1.10
1.40
1.40
1.40
1.45
1.38
1.90
1.40
2.90
1.90
chassismass M [kg] 1100 1400 1950 3200 14300
chassisinertia Θ [kg m2] 1500 2350 3750 5800 50000
lumpedmassmodel
M1
M∗M2
[kg]
545
126
429
600
200
600
914
76
960
925
1020
1255
3592
5225
5483
Table 5.1: Mass and inertia properties of dierent vehicles
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FH Regensburg, University of Applied Sciences © Prof. Dr.-Ing. G. Rill
Depending on the actual mass and inertia properties the vertical dynamics of a vehiclecan be investigated by two simple decoupled mass models describing the vibrations of thefront and rear axle and the corresponding chassis masses. By using half of the chassis andhalf of the axle mass we nally end up in quarter car models.
The data in Table 5.1 show that for a wide range of passenger cars the coupling mass issmaller than the corresponding chassis masses, M∗ < M1 and M∗ < M2. Here, the twomass model or the quarter car model represent a quite good approximation to the lumpedmass model. For commercial vehicles and trucks, where the coupling mass has the samemagnitude as the corresponding chassis masses, the quarter car model serves for basicstudies only.
5.2.3 Simple Models
At most vehicles, c.f. Table 5.1, the axle mass is much smaller than the correspondingchassis mass, mi Mi, i = 1, 2. Hence, for a rst basic study axle and chassis motionscan be investigated independently. The quarter car model is now further simplied to twosingle mass models, Fig. 5.3.
zR6c
cS
M
dS
zC6
zR6
cTc
zW6m
cS
dS
Figure 5.3: Simple vertical vehicle models
The chassis model neglects the tire deection and the inertia forces of the wheel. For thehigh frequent wheel motions the chassis can be considered as xed to the inertia frame.
The equations of motion for the models read as
M zC + dS zC + cS zC = dS zR + cS zR (5.6)
andm zW + dS zW + (cS + cT ) zW = cT zR , (5.7)
where zC and zW label the vertical motions of the corresponding chassis mass and thewheel mass with respect to the steady state position. The constants cS, dS describe thesuspension stiness and damping. The dynamic wheel load is calculated by
FDT = cT (zR − zW ) (5.8)
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Vehicle Dynamics FH Regensburg, University of Applied Sciences
where cT is the vertical or radial stiness of the tire and zR denotes the road irregularities.In this simple approach the damping eects in the tire are not taken into account.
5.3 Basic Tuning
5.3.1 Natural Frequency and Damping Rate
At an ideally even track the right side of the equations of motion (5.6), (5.7) vanishesbecause of zR = 0 and zR = 0. The remaining homogeneous second order dierentialequations can be written in a more general form as
z + 2 ζ ω0 z + ω20 z = 0 , (5.9)
where ω0 represents the undamped natural frequency, and ζ is a dimensionless parametercalled viscous damping ratio. For the chassis and the wheel model the new parameter aregiven by
Chassis: z → zC , ζ → ζC =dS
2√
cSM, ω2
0 → ω20C =
cS
M;
Wheel: z → zW , ζ → ζW =dS
2√
(cS+cT )m, ω2
0 → ω20W =
cS+cT
m.
(5.10)
The solution of Eq. (5.9) is of the type
z(t) = z0 eλt , (5.11)
where z0 and λ are constants. Inserting Eq. (5.11) into Eq. (5.9) results in
(λ2 + 2 ζ ω0 λ + ω20) z0 eλt = 0 . (5.12)
Non-trivial solutions z0 6= 0 are possible, if
λ2 + 2 ζ ω0 λ + ω20 = 0 (5.13)
will hold. The roots of the characteristic equation (5.13) depend on the value of ζ
ζ < 1 : λ1,2 = −ζ ω0 pm i ω0
√1−ζ2 ,
ζ ≥ 1 : λ1,2 = −ω0
(ζ ∓
√ζ2−1
).
(5.14)
Figure 5.4 shows the root locus of the eigenvalues for dierent values of the viscousdamping rate ζ.
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FH Regensburg, University of Applied Sciences © Prof. Dr.-Ing. G. Rill
-3 -2.5 -2 -1.5 -1 -0.5
-1.0
-0.5
0
0.5
1.0
Re(λ)/ω0
Im(λ)/ω0
ζ=1 ζ=1.25ζ=1.25 ζ=1.5ζ=1.5
ζ=0
ζ=0
ζ=0.2
ζ=0.2
ζ=0.5
ζ=0.5
ζ=0.7
ζ=0.7
ζ=0.9
ζ=0.9
Figure 5.4: Eigenvalues λ1 and λ2 for dierent values of ζ
For ζ ≥ 1 the eigenvalues λ1,2 are both real and negative. Hence, Eq. (5.11) will produce aexponentially decaying solution. If ζ < 1 holds, the eigenvalues λ1,2 will become complex,where λ2 is the complex conjugate of λ1. Now, the solution can be written as
z(t) = A e−ζω0t sin(ω0
√1−ζ2 t−Ψ
), (5.15)
where A and Ψ are constants which have to be adjusted to given initial conditions z(0) = z0
and z(0) = z0. The real part Re (λ1,2) = −ζω0 is negative and determines the decay ofthe solution. The imaginary Im (λ1,2) = ω0
√1−ζ2 part denes the actual frequency of
the vibration. The actual frequency
ω = ω0
√1−ζ2 (5.16)
tends to zero, ω → 0, if the viscous damping ratio will approach the value one, ζ → 1.In a more general way the relative damping may be judged by the ratio
Dλ =−Re(λ1,2)
|λ1,2 |. (5.17)
For complex eigenvalues which characterize vibrations
Dλ = ζ (5.18)
holds, because the absolute value of the complex eigenvalues is given by
|λ1,2 | =√
Re(λ1,2)2 + Im(λ1,2)2 =
√(−ζ ω0)
2 +(ω0
√1−ζ2
)2
= ω0 , (5.19)
and hence, Eq. (5.17) results in
Dλ =+ζ ω0
ω0
= ζ . (5.20)
For ζ ≥ 1 the eigenvalues become real and negative. Then, Eq. (5.17) will always producethe relative damping value Dλ = 1. In this case the viscous damping rate ζ is moresensitive.
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Vehicle Dynamics FH Regensburg, University of Applied Sciences
5.3.2 Spring Rates
5.3.2.1 Minimum Spring Rates
The suspension spring is loaded with the corresponding vehicle weight. At linear springcharacteristics the steady state spring deection is calculated from
u0 =M g
cS
. (5.21)
At a conventional suspension without niveau regulation a load variation M → M +4Mleads to changed spring deections u0 → u0 + 4u. In analogy to (5.21) the additionaldeection follows from
4u =4M g
cS
. (5.22)
If for the maximum load variation 4M the additional spring deection is limited to 4uthe suspension spring rate can be estimated by a lower bound
cS ≥ 4M g
4u. (5.23)
In the standard design of a passenger car the engine is located in the front and the trunkin the rear part of the vehicle. Hence, most of the load is supported by the rear axlesuspension.
For an example we assume that 150 kg of the permissible load of 500 kg are going to thefront axle. Then, each front wheel is loaded by 4MF = 150 kg/2 = 75 kg and each rearwheel by 4MR = (500− 150) kg/2 = 175 kg.
The maximum wheel travel is limited, u ≤ umax. At standard passenger cars it is in therange of umax ≈ 0.8 m to umax ≈ 0.10 m. By setting 4u = umax/2 we demand that thespring deection caused by the load should not exceed half of the maximum value. Then,according to Eq. (5.23) a lower bound of the spring rate at the front axle can be estimatedby
cminS = ( 75 kg ∗ 9.81 m/s2 )/(0.08/2) m = 18400 N/m . (5.24)
The maximum load over one rear wheel was estimated here by 4MR = 175 kg. Assumingthat the suspension travel at the rear axle is slightly larger, umax ≈ 0.10 m the minimumspring rate at the rear axle can be estimated by
cminS = ( 175 kg ∗ 9.81 m/s2 )/(0.10/2) m = 34300 N/m , (5.25)
which is nearly two times the minimum value of the spring rate at the front axle. In orderto reduce this dierence we will choose a spring rate of cS = 20 000N/m at the front axle.
In Tab. 5.1 the lumped mass chassis model of a full size passenger car is described byM1 = M2 = 600 kg and M∗ = 200. To approximate the lumped mass model by twodecoupled two mass models we have to neglect the coupling mass or, in order to achieve
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FH Regensburg, University of Applied Sciences © Prof. Dr.-Ing. G. Rill
the same chassis mass, to distribute M∗ equally to the front and the rear. Then, thecorresponding cassis mass of a quarter car model is given here by
M =(M1 + M∗/2
)/2 = (600 kg + 200/2 kg)/2 = 350 kg . (5.26)
According to Eq. 5.10 the undamped natural eigen frequency of the simple chassis modelis then given by ω2
0C = cS/M . Hence, for a spring rate of cS = 20000 N/m the undampednatural frequency of the unloaded car amounts to
f0C =√
20000 N/m ∗ 350 kg/(2 π) = 1.2 Hz , (5.27)
which is a typical value for most of all passenger cars. Due to the small amount of loadthe undamped natural frequency for the loaded car does not change very much,
f0C =√
20000 N/m ∗ (350 + 75) kg/(2 π) = 1.1 Hz . (5.28)
The corresponding cassis mass over the rear axle is given here by
M =(M2 + M∗/2
)/2 = (600 kg + 200/2 kg)/2 = 350 kg . (5.29)
Now the undamped natural frequencies for the unloaded
f 00C =
√34300 N/m/350 kg/(2 π) = 1.6 Hz (5.30)
and the loaded car
fL0C =
√34300 N/m/(350 + 175) kg/(2 π) = 1.3 Hz (5.31)
are larger and dier more.
5.3.2.2 Nonlinear Springs
In order to reduce the spring rate at the rear axle and to avoid too large spring deectionswhen loaded nonlinear spring characteristics are used, Fig. 5.5. Adding soft bump stopsthe overall spring force in the compression mode u ≥ 0 can be modeled by the nonlinearfunction
FS = F 0S + c0 u
(1 + k
(u
4u
)2)
, (5.32)
where F 0S is the spring preload, cS describes the spring rate at u = 0, and k > 0 charac-
terizes the intensity of the nonlinearity. The linear characteristic provides at u = 4u thevalue F lin
S (4u) = F 0S + cS 4u. To achieve the same value with the nonlinear spring
F 0S + c04u (1 + k) = F 0
S + cS 4u or c0 (1 + k) = cS (5.33)
must hold, where cS describes the spring rate of the corresponding linear characteristics.The local spring rate is determined by the derivative
dFS
du= c0
(1 + 3 k
(u
4u
)2)
. (5.34)
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Vehicle Dynamics FH Regensburg, University of Applied Sciences
FS
u
FS0
∆u
∆M g
dFSdu u=∆u
dFS
du u=0
cS
FS [N]
u [m]
63 kN/m44 kN/m
20 kN/m29 kN/m
0 0.05 0.12000
4000
6000
8000
Figure 5.5: Principle and realizations of nonlinear spring characteristics
Hence, the spring rate for the loaded car at u = 4u is given by
cL = c0 (1 + 3 k) . (5.35)
The intensity of the nonlinearity k can be xed, for instance, by choosing an appropriatespring rate for the unloaded vehicle. With c0 = 20000 N/m the spring rates on the frontand rear axle will be the same for the unloaded vehicle. With cS = 34300 N/m Eq. (5.33)yields
k =cS
c0
− 1 =34300
20000− 1 = 0.715 . (5.36)
The solid line in Fig. 5.5 shows the resulting nonlinear spring characteristics which ischaracterized by the spring rates c0 = 20 000 N/m and cL = c0 (1 + 3k) = 20 000 ∗ (1 +3 ∗ 0.715) = 62 900 N/m for the unloaded and the loaded vehicle. Again, the undampednatural frequencies
f 00C =
√20000 N/m
350 kg
1
2 π= 1.20 Hz or fL
0C =
√92000 N/m
(350+175) kg
1
2 π= 1.74 Hz (5.37)
for the unloaded and the loaded vehicle dier quite a lot.
The unloaded and the loaded vehicle have the same undamped natural frequencies if
c0
M=
cL
M +4Mor
cL
c0
=M +4M
M(5.38)
will hold. Combing this relationship with Eq. (5.35) one obtains
1 + 3 k =M
M +4Mor k =
1
3
(M +4M
M− 1
)=
1
3
4M
M. (5.39)
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FH Regensburg, University of Applied Sciences © Prof. Dr.-Ing. G. Rill
Hence, for the quarter car model with M = 350 kg and 4M = 175 the inten-sity of the nonlinear spring amounts to k = 1/3 ∗ 175/350 = 0.1667. This valueand cS = 34300 N/m will produce the dotted line in Fig. 5.5. The spring ratesc0 = cS/(1 + k) = 34 300 N/m / (1 + 0.1667) = 29 400 N/m for the unloaded andcL = c0 (1 + 3k) = 29 400 N/m ∗ (1 + 3 ∗ 0.1667) = 44 100 N/m for the loaded vehicle fol-low from Eqs. (5.34) and (5.35). Now, the undamped natural frequency for the unloadedf 0
0C =√
c0/M = 1.46 Hz and the loaded vehicle f 00C =
√cL/(M +4M) = 1.46 Hz are
in deed the same.
5.3.3 Influence of Damping
To investigate the inuence of the suspension damping to the chassis and wheel motion thesimple vehicle models are exposed to initial disturbances. Fig. 5.6 shows the time responseof the chassis zC(t) and wheel displacement zW (t) as well as the chassis acceleration zC
and the wheel load FT = F 0T + FD
T for dierent damping rates ζC and ζW . The dynamicwheel load follows from Eq. (5.8), and the static wheel load is given by F 0
T = (M + m) g,where g labels the constant of gravity.
To achieve the same damping rates for the chassis and the wheel model dierent valuesfor the damping parameter dS were needed.
With increased damping the overshoot eect in the time history of the chassis displace-ment and the wheel load becomes smaller and smaller till it vanishes completely at ζC = 1and ζW = 1. The viscous damping rate ζ = 1
5.3.4 Optimal Damping
5.3.4.1 Avoiding Overshoots
If avoiding overshoot eects is the design goal then, ζ = 1 will be the optimal dampingratio. For ζ = 1 the eigenvalues of the single mass oscillator change from complex to real.Thus, producing a non oscillating solution without any sine and cosine terms.
According to Eq. (5.10) ζC = 1 and ζW = 1 results in the optimal damping parameter
doptS
∣∣ζC=1
Comfort= 2
√cSM , and dopt
S
∣∣ζW =1
Safety= 2
√(cS+cT )m . (5.40)
So, the damping values
doptS
∣∣ζC=1
Comfort= 5292
N
m/sand dopt
S
∣∣ζW =1
Safety= 6928
N
m/s(5.41)
will avoid an overshoot eect in the time history of the chassis displacement zC(t) or in thein the time history of the wheel load FT (t). Usually, as it is here, the damping values foroptimal comfort and optimal ride safety will be dierent. Hence, a simple linear dampercan either avoid overshoots in the chassis motions or in the wheel loads.
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Vehicle Dynamics FH Regensburg, University of Applied Sciences
0 0.5 1 1.5-100
-50
0
50
100
150
200displacement [mm]
t [s]
ζC [ - ] dS [Ns/m]
0.250.500.751.001.25
13232646396952926614
0 0.05 0.1 0.150
1000
2000
3000
4000
5000
6000
50 kg
20000 N/m
220000 N/m
dS
wheel load [N]
t [s]
20000 N/m
350 kg
dS
ζW [ - ] dS [Ns/m]
0.250.500.751.001.25
17323464519669288660
-1
-0.5
0
0.5
1
0 0.5 1 1.5t [s]
acceleration [g]
0 0.05 0.1 0.15-10
-5
0
5
10
15
20
chassis model wheel model
displacement [mm]
t [s]
ζC ζW
ζC
ζW
Figure 5.6: Time response of simple vehicle models to initial disturbances
The overshot in the time history of the chassis accelerations zC(t) will only vanish forζC →∞ which surely is not a desirable conguration, because then, it takes a very longtime till the initial chassis displacement has fully disappeared.
5.3.4.2 Fast Approach to Steady State
Instead of avoiding overshoot eects we better demand that the time history of the systemresponse will approach the steady state value as fast as possible. Fig. 5.7 shows the typicaltime response of a damped single-mass oscillator to the initial disturbance z(t=0) = z0
and z(t=0) = 0.
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FH Regensburg, University of Applied Sciences © Prof. Dr.-Ing. G. Rill
z(t) t
z0
tE
zS
Figure 5.7: Evaluating a damped vibration
Counting the dierences of the system response z(t) from the steady state value zS = 0 aserrors allows to judge the attenuation. If the overall quadratic error becomes a minimum
ε2 =
t=tE∫t=0
z(t)2 dt → Min , (5.42)
the system approaches the steady state position as fast as possible. In theory tE → ∞holds, for practical applications a nite tE have to be chosen appropriately.
To judge ride comfort and ride safety the hub motion of the chassis zC , its accelerationzC and the variations of the dynamic wheel load FD
T can be used. In the absence of roadirregularities zR = 0 the dynamic wheel load from Eq. (5.8) simplies to FD
T = −cT zW .Hence, the demands
ε2C =
t=tE∫t=0
[ (g1 zC
)2+(
g2 zC
)2 ]dt → Min (5.43)
and
ε2S =
t=tE∫t=0
(−cT zW
)2dt → Min (5.44)
will guarantee optimal ride comfort and optimal ride safety. By the factors g1 and g2 theacceleration and the hub motion can be weighted dierently.
The equation of motion for the chassis model can be resolved for the acceleration
zC = −(ω2
0C zC + 2δC zC
), (5.45)
where, the system parameter M , dS and cS were substituted by the damping rate δC =ζC ω0C = dS/(2M) and by the undamped natural frequency ω0C = cS/M . Then, theproblem in Eq. (5.43) can be written as
ε2C =
t=tE∫t=0
[g21
(ω2
0CzC + 2δC zC
)2+ g2
2 z2C
]dt
=
t=tE∫t=0
[zC zC
] g21 (ω2
0C)2+ g2
2 g21 ω2
0C 2δC
g21 ω2
0C 2δC g21 (2δC)2
zC
zC
→ Min ,
(5.46)
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Vehicle Dynamics FH Regensburg, University of Applied Sciences
where xTC =
[zC zC
]is the state vector of the chassis model. In a similar way Eq. (5.44)
can be transformed to
ε2S =
t=tE∫t=0
c2T z2
W dt =
t=tE∫t=0
[zW zW
] [ c2T 0
0 0
][zW
zW
]→ Min , (5.47)
where xTW =
[zW zW
]denotes the state vector of the wheel model.
The problems given in Eqs. (5.46) and (5.47) are called disturbance-reaction problems.They can be written in a more general form
t=tE∫t=0
xT (t) Qx(t) dt → Min (5.48)
where x(t) denotes the state vector and Q = QT is a symmetric weighting matrix. Forsingle mass oscillators described by Eq. (5.9) the state equation reads as[
zz
]︸ ︷︷ ︸
x
=
[0 1−ω2
0 −2δ
]︸ ︷︷ ︸
A
[zz
]︸ ︷︷ ︸
x
. (5.49)
For tE →∞ the time response of the system exposed to the initial disturbance x(t=0) =x0 vanishes x(t→∞) = 0, and the integral in Eq.(5.48) can be solved by
t=tE∫t=0
xT (t) Qx(t) dt = xT0 R x0 , (5.50)
where the symmetric matrix R = RT is given by the Ljapunov equation
AT R + R A + Q = 0 . (5.51)
For the single mass oscillator the Ljapunov equation[0 −ω2
0
1 −2δ
] [R11 R12
R12 R22
]+
[R11 R12
R12 R22
] [0 1−ω2
0 −2δ
]+
[Q11 Q12
Q12 Q22
]. (5.52)
results in 3 linear equations
−ω20 R12 − ω2
0 R12 + Q11 = 0
−ω20 R22 + R11 − 2δ R12 + Q12 = 0
R12 − 2δ R22 + R12 − 2δ R22 + Q22 = 0
(5.53)
which easily can be solved for the elements of R
R11 =
(δ
ω20
+1
4δ
)Q11 −Q12 +
ω20
4δQ22 , R12 =
Q11
2ω20
, R22 =Q11
4δ ω20
+Q22
4δ. (5.54)
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FH Regensburg, University of Applied Sciences © Prof. Dr.-Ing. G. Rill
For the initial disturbance x0 = [ z0 0 ]T Eq. (5.50) nally results in
t=tE∫t=0
xT (t) Qx(t) dt = z20 R11 = z2
0
[(δ
ω20
+1
4δ
)Q11 −Q12 +
ω20
4δQ22
]. (5.55)
Now, the integral in Eq. (5.46) evaluating the ride comfort is solved by
ε2C = z2
0C
[(δC
ω20C
+1
4δC
)(g21
(ω2
0C
)2+ g2
2
)− g2
1 ω20C 2 δC +
ω20C
4δC
g21 (2δC)2
]= z2
0C ω20C
[ω0C
4ζC
(g21 +
(g2
ω20C
)2)
+
(g2
ω20C
)2
ζC ω0C
].
(5.56)
where the abbreviation δC was nally replaced by ζC ω0C .
By setting g1 = 1 and g2 = 0 the time history of the chassis acceleration zC is weightedonly. Eq. (5.56) then simplies to
ε2C
∣∣zC
= z20C ω2
0C
ω0C
4ζC
(5.57)
which will become a minimum for ω0C → 0 or ζC → ∞. As mentioned before, ζC → ∞surely is not a desirable conguration. A low undamped natural frequency ω0C → 0 isachieved by a soft suspension spring cS → 0 or a large chassis mass M →∞. However, alarge chassis mass is uneconomic and the suspension stiness is limited by the the loadingconditions. Hence, weighting the chassis accelerations only does not lead to a specicresult for the system parameter.
The combination of g1 = 0 and g2 = 1 weights the time history of the chassis displacementonly. Then, Eq. (5.56) results in
ε2C
∣∣zC
=z20C
ω0C
[1
4ζC
+ ζC
](5.58)
which will become a minimum for ω0C →∞ or
d ε2C |zC
d ζC
=z20C
ω0C
[−1
4ζ2C
+ 1
]= 0 . (5.59)
A high undamped natural frequency ω0C → ∞ contradicts the demand for rapidly van-ishing accelerations. The viscous damping ratio ζC = 1
2solves Eq. (5.59) and minimizes
the merit function in Eq. (5.58). But again, this value does not correspond with ζC →∞which minimizes the merit function in Eq. (5.57).
Hence, practical results can be achieved only if the chassis displacements and the chassisaccelerations will be evaluated simultaneously. To do so, appropriate weighting factorshave to be chosen. In the equation of motion for the chassis (5.6) the terms M zC andcS zC are added. Hence, g1 = M and g2 = cS or
g1 = 1 and g2 =cS
M= ω2
0C (5.60)
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Vehicle Dynamics FH Regensburg, University of Applied Sciences
provide system-tted weighting factors. Now, Eq. (5.56) reads as
ε2C = z2
0C ω20C
[ω0C
2ζC
+ ζC ω0C
]. (5.61)
Again, a good ride comfort will be achieved by ω0C → 0. For nite undamped naturalfrequencies Eq. (5.61) becomes a minimum, if the viscous damping rate ζC will satisfy
d ε2C |zC
d ζC
= z20C ω2
0C
[−ω0C
2ζ2C
+ ω0C
]= 0 . (5.62)
Hence, a viscous damping rate of
ζC =1
2
√2 (5.63)
or a damping parameter of
doptS
∣∣ζC= 12
√2
Comfort=√
2 cSM , (5.64)
will provide optimal comfort by minimizing the merit function in Eq. (5.61).
For the passenger car with M = 350 kg and cS = 20 000 N/m the optimal dampingparameter will amount to
doptS
∣∣ζC= 12
√2
Comfort= 3742
N
m/s(5.65)
which is 70% of the value needed to avoid overshot eects in the chassis displacements.
The integral in Eq. (5.47) evaluating the ride safety is solved by
ε2S =
z20W
ω0W
(ζW +
1
4ζW
)c2T (5.66)
where the model parameter m, cS, dS and cT where replaced by the undamped naturalfrequency ω2
0W = (cS + cT )/m and by the damping ratio δW = ζW ω0W = dS/(2m).
A soft tire cT → 0 make the safety criteria Eq. (5.66) small ε2S → 0 and thus, reduces the
dynamic wheel load variations. However, the tire spring stiness can not be reduced toarbitrary low values, because this would cause too large tire deformations. Small wheelmasses m → 0 and/or a hard body suspension cS → ∞ will increase ω0W and thus,reduce the safety criteria Eq. (5.66). The use of light metal rims improves, because ofwheel weight reduction, the ride safety of a car. Hard body suspensions contradict a gooddriving comfort.
With xed values for cT and ω0W the merit function in Eq. (5.66) will become a minimumif
∂ε2S
∂ζW
=z20W
ω0W
(1 +
−1
4ζ2W
)c2T = 0 (5.67)
will hold. Hence, a viscous damping rate of
ζW =1
2(5.68)
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or the damping parameter
doptS
∣∣Safety
=√
(cS + cT ) m (5.69)
will guarantee optimal ride safety by minimizing the merit function in Eq. (5.66).
For the passenger car with M = 350 kg and cS = 20 000 N/m the optimal dampingparameter will now amount to
doptS
∣∣ζW = 12
Safety= 3464
N
m/s(5.70)
which is 50% of the value needed to avoid overshot eects in the wheel loads.
5.4 Sky Hook Damper
5.4.1 Modelling Aspects
In standard vehicle suspension systems the damper is mounted between the wheel andthe body. Hence, the damper aects body and wheel/axle motions simultaneously.
dScS
cT
M
m
zC
zW
zR
sky
dW
dB
cS
cT
M
m
zC
zW
zR
FD
a) Standard Damper b) Sky Hook Damper
Figure 5.8: Quarter car model with standard and sky hook damper
To take this situation into account the simple quarter car models of section 5.2.3 must becombined to a more enhanced model, Fig. 5.8a.
Assuming a linear characteristics the suspension damper force is given by
FD = dS (zW − zC) , (5.71)
where dS denotes the damping constant, and zC , zW are the time derivatives of the absolutevertical body and wheel displacements.
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The sky hook damping concept starts with two independent dampers for the body andthe wheel/axle mass, Fig. 5.8b. A practical realization in form of a controllable damperwill then provide the damping force
FD = dW zW − dC zC , (5.72)
where instead of the single damping constant dS now two design parameter dW and dC
are available.
The equations of motion for the quarter car model are given by
M zC = FS + FD −M g ,
m zW = FT − FS − FD −m g ,(5.73)
where M , m are the sprung and unsprung mass, zC , zW denote their vertical displace-ments, and g is the constant of gravity.
The suspension spring force is modeled by
FS = F 0S + cS (zW − zC) , (5.74)
where F 0S = mC g is the spring preload, and cS the spring stiness.
Finally, the vertical tire force is given by
FT = F 0T + cT (zR − zW ) , (5.75)
where F 0T = (M + m) g is the tire preload, cS the vertical tire stiness, and zR describes
the road roughness. The condition FT ≥ 0 takes the tire lift o into account.
5.4.2 Eigenfrequencies and Damping Ratios
Using the force denitions in Eqs. (5.72), (5.74) and (5.75) the equations of motion inEq. (5.73) can be transformed to the state equation
zC
zW
zC
zW
︸ ︷︷ ︸
x
=
0 0 1 0
0 0 0 1
− cS
McS
M−dC
MdW
McS
m− cS+cT
mdC
m−dW
m
︸ ︷︷ ︸
A
zC
zW
zC
zW
︸ ︷︷ ︸
x
+
0
0
0cT
m
︸ ︷︷ ︸
B
[zR
]︸ ︷︷ ︸
u
, (5.76)
where the weight forces Mg, mg were compensated by the preloads F 0S , F 0
T , the term B udescribes the excitation, x denotes the state vector, and A is the state matrix. In thislinear approach the tire lift o is no longer taken into consideration.
The eigenvalues λ of the state matrix A will characterize the eigen dynamics of the quartercar model. In case of complex eigenvalues the damped natural eigenfrequencies are given
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0 1000 2000 3000 4000 50000
0.2
0.4
0.6
0.8
1
0 1000 2000 3000 4000 50000
2
4
6
8
10
12Damping ratio ζ = DλFrequencies [Hz]
dS [N/(m/s)] dS [N/(m/s)]
Chassis
Wheel0.7
0.5
32203880
350 kg
50 kg
dS
220000 N/m
20000 N/m
Figure 5.9: Quarter car model with standard damper
by the imaginary parts, ω = Im(λ), and according to Eq. (vdyn-eq: relative dampingratio lambda) ζ = Dλ = −Re(λ)/ |λ|. evaluates the damping ratio.
By setting dC = dS and dW = dS Eq. (5.76) represents a quarter car model with thestandard damper described by Eq. (5.71). Fig. 5.9 shows the eigenfrequencies f = ω/(2π)and the damping ratios ζ = Dλ for dierent values of the damping parameter dS.
Optimal ride comfort with a damping ratio of ζC = 12
√2 ≈ 0.7 for the chassis motion
could be achieved with the damping parameter dS = 3880 N/(m/s), and the dampingparameter dS = 3220 N/(m/s) would provide for the wheel motion a damping ratio ofζW = 0.5 which correspond to minimal wheel load variations. This damping parameterare very close to the values 3742 N/(m/s) and 3464 N/(m/s) which very calculated inEqs. (5.65) and (5.70) with the single mass models. Hence, the very simple single massmodels can be used for a rst damper layout. Usually, as it is here, optimal ride comfortand optimal ride safety cannot achieved both by a standard linear damper.
The sky-hook damper, modeled by Eq. (5.72), provides with dW and dS two design pa-rameter. Their inuence to the eigenfrequencies f and the damping ratios ζ is shown inFig. 5.10.
The the sky-hook damping parameter dC , dW have a nearly independent inuence onthe damping ratios. The chassis damping ratio ζC mainly depends on dC , and the wheeldamping ratio ζW mainly depends on dW . Hence, the damping of the chassis and thewheel motion can be adjusted to nearly each design goal. Here, a sky-hook damper withdC = 3900N/(m/s) and dW = 3200N/(m/s) would generate the damping ratios dC = 0.7and dW = 0.5 hence, combining ride comfort and ride safety within one damper layout.
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0 1000 2000 3000 4000 50000
2
4
6
8
10
12
0 1000 2000 3000 4000 50000
0.2
0.4
0.6
0.8
1
350 kg
50 kg
dC
220000 N/m
20000 N/m dWdC
4500400035003000250020001500
dC [N/(m/s)]
dW [N/(m/s)]dW [N/(m/s)]
Damping ratios ζC, ζWFrequencies [Hz]
0.7
0.5
ζW
ζC
Figure 5.10: Quarter car model with sky-hook damper
5.4.3 Technical Realization
By modifying the damper law in Eq. (5.72) to
FD = dW zW − dC zC+ =dW zW − dC zC
zW − zC︸ ︷︷ ︸d∗S
(zW − zC) = d∗S (zW − zC) (5.77)
the sky-hook damper can be realized by a standard damper in the form of Eq. (5.71). Thenew damping parameter d∗S now nonlinearly depends on the absolute vertical velocities ofthe chassis and the wheel d∗S = d∗S(zC , zW ). As, a standard damper operates in a dissipativemode only the damping parameter will be restricted to positive values, d∗S > 0. Hence, thepassive realization of a sky-hook damper will only match with some properties of the idealdamper law in Eq. (5.72). But, compared with the standard damper it still can provide abetter ride comfort combined with an increased ride safety.
5.5 Nonlinear Force Elements
5.5.1 Quarter Car Model
The principal inuence of nonlinear characteristics on driving comfort and safety canalready be studied on a quarter car model Fig. 5.11.
The equations of motion read as
M zC = FS + FD − M g
m zW = FT − FS − FD − m g ,(5.78)
where g = 9.81 m/s2 labels the constant of gravity, M , m are the masses of the chassisand the wheel, FS, FD, FT describe the spring, the damper, and the vertical tire force,
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cT
m
Mnonlinear dampernonlinear spring
zC
zW
zR
FD
v
u
FS
FS FD
Figure 5.11: Quarter car model with nonlinear spring and damper characteristics
and the vertical displacements of the chassis zC and the wheel zW are measured from theequilibrium position.
In extension to Eq. (5.32) the spring characteristics is modeled by
FS = F 0S +
c0 u
(1 + kr
(u
4ur
)2)
u < 0
c0 u
(1 + kc
(u
4uc
)2)
u ≥ 0
(5.79)
where F 0S = M g is the spring preload, and
u = zW − zC (5.80)
describes the spring travel. Here, u < 0 marks tension (rebound), and u ≥ 0 compres-sion. Two sets of kr, ur and kc, uc dene the spring nonlinearity during rebound andcompression. For kr = 0 and kc = 0 a linear spring characteristics is obtained.
A degressive damper characteristics can be modeled by
FD(v) =
d0 v
1 − pr vv < 0 ,
d0 v
1 + pc vv ≥ 0 ,
(5.81)
where d0 denotes the damping constant at v = 0, and the damper velocity is dened by
v = zW − zC . (5.82)
The sign convention of the damper velocity was chosen consistent to the spring travel.Hence, rebound is characterized by v < 0 and compression by v ≥ 0. The parameter pr
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and pc make it possible to model the damper nonlinearity dierently in the rebound andcompression mode. A linear damper characteristics is obtained with pr = 0 and pc = 0.
The nonlinear spring design in Section 5.3.2 holds for the compression mode. Hence,using the same data we obtain: c0 = 29 400N/m, uc = 4u = umax/2 = 0.10/2 = 0.05 andkc = k = 0.1667. By setting ur = uc and kr = 0 a simple linear is used in the reboundmode, Fig. 5.12a.
0
1000
2000
3000
4000
5000
6000
7000
-5000
-2500
0
2500
5000
-0.05-0.1 0.050 0.1 -0.5-1 0.50 1
cS = 34300 N/mc0 = 29400 N/mur = 0.05 mkr = 0uc = 0.05 mkc = 0.1667
u [m]
FS [
N/m
]
compressionu > 0
reboundu < 0
compressionv > 0
reboundv < 0
d0 = 4200 N/(m/s)pr = 0.4 1/(m/s)pc = 1.2 1/(m/s)
v [m/s]
FD
[N
/m]
a) Spring b) Damper
Figure 5.12: Spring and damper characteristics: - - - linear, nonlinear
According to Section 5.3.4 damping coecients optimizing the ride comfort and the ridesafety can be calculated from Eqs. (5.65) and (5.69). For cS = 34 300 N/m which is theequivalent linear spring rate, M = 350 kg, m = 50 kg and cT = 220 000 N/m we obtain
(dS)Copt =
√2 cS M =
√2 34 300 350 = 4900 N/(m/s) ,
(dS)Sopt =
√(cS + cT ) m =
√(18 000 + 220 000) 50 = 3570 N/(m/s) .
(5.83)
The mean value d0 = 4200N/(m/s) may serve as compromise. With pr = 0.4(m/s)−1 andpc = 1.2 (m/s)−1 the nonlinearity becomes more intensive in compression than rebound,Fig. 5.12b.
5.5.2 Results
The quarter car model is driven with constant velocity over a single obstacle. Here, acosine shaped bump with a height of H = 0.08 m and a length of L = 2.0 m was used.The results are plotted in Fig. 5.13.
Compared to the linear model the nonlinear spring and damper characteristics result insignicantly reduced peak values for the chassis acceleration (6.0m/s2 instead of 7.1m/s2)
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0 0.5 1 1.5-15
-10
-5
0
5
10
0 0.5 1 1.50
1000
2000
3000
4000
5000
6000
7000
0 0.5 1 1.5-0.06
-0.04
-0.02
0
0.02
0.04Chassis acceleration [m/s2] Wheel load [N] Suspension travel [m]
time [s] time [s] time [s]
linearnonlinear
666061607.1
6.0
Figure 5.13: Quarter car model driving with v = 20 km h over a single obstacle
and for the wheel load (6160 N instead of 6660 N). Even the tire lift o at t ≈ 0.25 scan be avoided. While crossing the bump large damper velocities occur. Here, the degres-sive damper characteristics provides less damping compared to the linear damper whichincreases the suspension travel.
0 0.5 1 1.5 0 0.5 1 1.5 0 0.5 1 1.5
Chassis acceleration [m/s2] Wheel load [N] Suspension travel [m]
time [s] time [s] time [s]
-15
-10
-5
0
5
10
0
1000
2000
3000
4000
5000
6000
7000
-0.08
-0.06
-0.04
-0.02
0
0.02
0.04
linear, low damping
nonlinear
Figure 5.14: Results for low damping compared to nonlinear model
A linear damper with a lower damping coecient, d0 = 3000N/m for instance, also reducesthe peaks in the chassis acceleration and in the wheel load, but then the attenuation ofthe disturbances will take more time. Fig. 5.14. Which surely is not optimal.
93
6 Longitudinal Dynamics
6.1 Dynamic Wheel Loads
6.1.1 Simple Vehicle Model
The vehicle is considered as one rigid body which moves along an ideally even and hor-izontal road. At each axle the forces in the wheel contact points are combined in onenormal and one longitudinal force.
S
h
1a 2a
mg
v
Fx1Fx2
Fz2Fz1
Figure 6.1: Simple vehicle model
If aerodynamic forces (drag, positive and negative lift) are neglected at rst, the equationsof motions in the x-, z-plane will read as
m v = Fx1 + Fx2 , (6.1)
0 = Fz1 + Fz2 −m g , (6.2)
0 = Fz1 a1 − Fz2 a2 + (Fx1 + Fx2) h , (6.3)
where v indicates the vehicle's acceleration, m is the mass of the vehicle, a1 +a2 is thewheel base, and h is the height of the center of gravity.
These are only three equations for the four unknown forces Fx1, Fx2, Fz1, Fz2. But, if weinsert Eq. (6.1) in Eq. (6.3), we can eliminate two unknowns at a stroke
0 = Fz1 a1 − Fz2 a2 + m v h . (6.4)
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The equations Eqs. (6.2) and (6.4) can be resolved for the axle loads now
Fz1 = m ga2
a1 + a2
− h
a1 + a2
m v , (6.5)
Fz2 = m ga1
a1 + a2
+h
a1 + a2
m v . (6.6)
The static partsF st
z1 = m ga2
a1 + a2
, F stz2 = m g
a1
a1 + a2
(6.7)
describe the weight distribution according to the horizontal position of the center ofgravity. The height of the center of gravity only inuences the dynamic part of the axleloads,
F dynz1 = −m g
h
a1 + a2
v
g, F dyn
z2 = +m gh
a1 + a2
v
g. (6.8)
When accelerating v > 0, the front axle is relieved as the rear axle is when deceleratingv<0.
6.1.2 Influence of Grade
mg
a1
a2
Fx1
Fz1 Fx2
Fz2
h
α
v z
x
Figure 6.2: Vehicle on grade
For a vehicle on a grade, Fig.6.2, the equations of motion Eq. (6.1) to Eq. (6.3) can easilybe extended to
m v = Fx1 + Fx2 −m g sin α ,
0 = Fz1 + Fz2 −m g cos α ,
0 = Fz1 a1 − Fz2 a2 + (Fx1 + Fx2) h ,
(6.9)
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where α denotes the grade angle. Now, the axle loads are given by
Fz1 = m g cos αa2 − h tan α
a1 + a2
− h
a1 + a2
m v , (6.10)
Fz2 = m g cos αa1 + h tan α
a1 + a2
+h
a1 + a2
m v , (6.11)
where the dynamic parts remain unchanged, whereas now the static parts also depend onthe grade angle and the height of the center of gravity.
6.1.3 Aerodynamic Forces
The shape of most vehicles or specic wings mounted at the vehicle produce aerodynamicforces and torques. The eect of these aerodynamic forces and torques can be representedby a resistant force applied at the center of gravity and down forces acting at the frontand rear axle, Fig. 6.3.
mg
a1
h
a2
FD1
FAR
FD2
Fx1 Fx2
Fz1 Fz2
Figure 6.3: Vehicle with aerodynamic forces
If we assume a positive driving speed, v > 0, the equations of motion will read as
m v = Fx1 + Fx2 − FAR ,
0 = Fz1−FD1 + Fz2−FD2 −m g ,
0 = (Fz1−FD1) a1 − (Fz2−FD2) a2 + (Fx1 + Fx2) h ,
(6.12)
where FAR and FD1, FD2 describe the air resistance and the down forces. For the dynamicaxle loads we get
Fz1 = FD1 + m ga2
a1 + a2
− h
a1 + a2
(m v + FAR) , (6.13)
Fz2 = FD2 + m ga1
a1 + a2
+h
a1 + a2
(m v + FAR) . (6.14)
The down forces FD1, FD2 increase the static axle loads, and the air resistance FAR
generates an additional dynamic term.
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6.2 Maximum Acceleration
6.2.1 Tilting Limits
Ordinary automotive vehicles can only apply pressure forces to the road. If we take thedemands Fz1 ≥ 0 and Fz2 ≥ 0 into account, Eqs. (6.10) and (6.11) will result in
v
g≤ a2
hcos α− sin α and
v
g≥ −a1
hcos α− sin α . (6.15)
These two conditions can be combined in one
− a1
hcos α ≤ v
g+ sin α ≤ a2
hcos α . (6.16)
Hence, the maximum achievable accelerations (v > 0) and decelerations (v < 0) arelimited by the grade angle α and the position a1, a2, h of the center of gravity. For v → 0the tilting condition Eq. (6.16) results in
− a1
h≤ tan α ≤ a2
h(6.17)
which describes the climbing and downhill capacity of a vehicle.
The presence of aerodynamic forces complicates the tilting condition. Aerodynamic forcesbecome important only at high speeds. Here, the vehicle acceleration is normally limitedby the engine power.
6.2.2 Friction Limits
The maximum acceleration is also restricted by the friction conditions
|Fx1| ≤ µ Fz1 and |Fx2| ≤ µ Fz2 (6.18)
where the same friction coecient µ has been assumed at front and rear axle. In the limitcase
Fx1 = ±µ Fz1 and Fx2 = ±µ Fz2 (6.19)
the linear momentum in Eq. (6.9) can be written as
m vmax = ±µ (Fz1 + Fz2)−m g sin α . (6.20)
Using Eqs. (6.10) and (6.11) one obtains(v
g
)max
= ±µ cos α − sin α . (6.21)
That means climbing (v > 0, α > 0) or downhill stopping (v < 0, α < 0) requires at leasta friction coecient µ ≥ tan |α|.
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According to the vehicle dimensions and the friction values the maximal acceleration ordeceleration is restricted either by Eq. (6.16) or by Eq. (6.21).
If we take aerodynamic forces into account, the maximum acceleration and decelerationon a horizontal road will be limited by
− µ
(1 +
FD1
mg+
FD2
mg
)− FAR
mg≤ v
g≤ µ
(1 +
FD1
mg+
FD2
mg
)− FAR
mg. (6.22)
In particular the aerodynamic forces enhance the braking performance of the vehicle.
6.3 Driving and Braking
6.3.1 Single Axle Drive
With the rear axle driven in limit situations, Fx1 = 0 and Fx2 = µ Fz2 hold. Then, usingEq. (6.6) the linear momentum Eq. (6.1) results in
m vR WD = µ m g
[a1
a1 + a2
+h
a1 + a2
vR WD
g
], (6.23)
where the subscript R WD indicates the rear wheel drive. Hence, the maximum accelerationfor a rear wheel driven vehicle is given by
vR WD
g=
µ
1− µh
a1 + a2
a1
a1 + a2
. (6.24)
By setting Fx1 = µ Fz1 and Fx2 = 0, the maximum acceleration for a front wheel drivenvehicle can be calculated in a similar way. One gets
vF WD
g=
µ
1 + µh
a1 + a2
a2
a1 + a2
, (6.25)
where the subscript F WD denotes front wheel drive. Depending on the parameter µ, a1,a2 and h the accelerations may be limited by the tilting condition v
g≤ a2
h.
The maximum accelerations of a single axle driven vehicle are plotted in Fig. 6.4. For rearwheel driven passenger cars, the parameter a2/(a1+a2) which describes the static axle loaddistribution is in the range of 0.4 ≤ a2/(a1+a2) ≤ 0.5. For µ = 1 and h = 0.55 this resultsin maximum accelerations in between 0.77 ≥ v/g ≥ 0.64. Front wheel driven passengercars usually cover the range 0.55 ≤ a2/(a1+a2) ≤ 0.60 which produces accelerations inthe range of 0.45 ≤ v/g ≥ 0.49. Hence, rear wheel driven vehicles can accelerate muchfaster than front wheel driven vehicles.
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0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1
a2 / (a1+a2)
RWD
FWD
range of load distribution
v / g
.
FWD
RW
DFigure 6.4: Single axle driven passenger car: µ = 1, h = 0.55 m, a1+a2 = 2.5 m
6.3.2 Braking at Single Axle
If only the front axle is braked, in the limit case Fx1 =−µ Fz1 and Fx2 =0 will hold. WithEq. (6.5) one gets from Eq. (6.1)
m vF WB = −µ m g
[a2
a1 + a2
− h
a1 + a2
vF WB
g
], (6.26)
where the subscript F WB indicates front wheel braking. Then, the maximum decelerationis given by
vF WB
g= − µ
1− µh
a1 + a2
a2
a1 + a2
. (6.27)
If only the rear axle is braked (Fx1 = 0, Fx2 = −µ Fz2), one will obtain the maximumdeceleration
vR WB
g= − µ
1 + µh
a1 + a2
a1
a1 + a2
, (6.28)
where the subscript R WB denotes a braked rear axle. Depending on the parameters µ, a1,a2, and h, the decelerations may be limited by the tilting condition v
g≥ −a1
h.
The maximum decelerations of a single axle braked vehicle are plotted in Fig. 6.5. Forpassenger cars the load distribution parameter a2/(a1+a2) usually covers the range of 0.4to 0.6. If only the front axle is braked, decelerations from v/g = −0.51 to v/g = −0.77will be achieved. This is a quite large value compared to the deceleration range of a brakedrear axle which is in the range of v/g = −0.49 to v/g = −0.33. Therefore, the brakingsystem at the front axle has a redundant design.
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0 0.2 0.4 0.6 0.8 1-1
-0.8
-0.6
-0.4
-0.2
0
a2 / (a1+a2)
range ofloaddistribution
v / g
.FWB
RWB
Figure 6.5: Single axle braked passenger car: µ = 1, h = 0.55 m, a1+a2 = 2.5 m
6.3.3 Optimal Distribution of Drive and Brake Forces
The sum of the longitudinal forces accelerates or decelerates the vehicle. In dimensionlessstyle Eq. (6.1) reads as
v
g=
Fx1
m g+
Fx2
m g. (6.29)
A certain acceleration or deceleration can only be achieved by dierent combinations ofthe longitudinal forces Fx1 and Fx2. According to Eq. (6.19) the longitudinal forces arelimited by wheel load and friction.
The optimal combination of Fx1 and Fx2 will be achieved, when front and rear axle havethe same skid resistance.
Fx1 = ± ν µFz1 and Fx2 = ± ν µFz2 . (6.30)
With Eq. (6.5) and Eq. (6.6) one obtains
Fx1
m g= ± ν µ
(a2
h− v
g
)h
a1 + a2
(6.31)
andFx2
m g= ± ν µ
(a1
h+
v
g
)h
a1 + a2
. (6.32)
With Eq. (6.31) and Eq. (6.32) one gets from Eq. (6.29)
v
g= ± ν µ , (6.33)
where it has been assumed that Fx1 and Fx2 have the same sign. Finally, if Eq. (6.33 isinserted in Eqs. (6.31) and (6.32) one will obtain
Fx1
m g=
v
g
(a2
h− v
g
)h
a1 + a2
(6.34)
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FH Regensburg, University of Applied Sciences © Prof. Dr.-Ing. G. Rill
andFx2
m g=
v
g
(a1
h+
v
g
)h
a1 + a2
. (6.35)
Depending on the desired acceleration v > 0 or deceleration v < 0, the longitudinal forcesthat grant the same skid resistance at both axles can be calculated now.
h=0.551
2
-2-10dFx2
0
a1=1.15
a2=1.35
µ=1.20
a 2/h
-a1/hFx1/mg
braking
tilting limits
driv
ing
dFx1
F x2/
mg
B1/mg
B2/
mg
Figure 6.6: Optimal distribution of driving and braking forces
Fig. 6.6 shows the curve of optimal drive and brake forces for typical passenger car values.At the tilting limits v/g = −a1/h and v/g = +a2/h, no longitudinal forces can be appliedat the lifting axle. The initial gradient only depends on the steady state distribution ofthe wheel loads. From Eqs. (6.34) and (6.35) it follows
dFx1
m g
dv
g
=
(a2
h− 2
v
g
)h
a1 + a2
(6.36)
andd
Fx2
m g
dv
g
=
(a1
h+ 2
v
g
)h
a1 + a2
. (6.37)
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Vehicle Dynamics FH Regensburg, University of Applied Sciences
For v/g = 0 the initial gradient remains as
dFx2
dFx1
∣∣∣∣0
=a1
a2
. (6.38)
6.3.4 Different Distributions of Brake Forces
Practical applications aim at approximating the optimal distribution of brake forces byconstant distribution, limitation, or reduction of brake forces as good as possible. Fig. 6.7.
Fx1/mg
F x2/
mg constant
distribution
Fx1/mgF x
2/m
g limitation reduction
Fx1/mg
F x2/
mg
Figure 6.7: Dierent distributions of brake forces
When braking, the stability of a vehicle depends on the potential of generating a lateralforce at the rear axle. Thus, a greater skid (locking) resistance is realized at the rear axlethan at the front axle. Therefore, the brake force distribution are all below the optimalcurve in the physically relevant area. This restricts the achievable deceleration, speciallyat low friction values.
Because the optimal curve depends on the center of gravity of the vehicle an additionalsafety margin have to be installed when designing real brake force distributions. Thedistribution of brake forces is often tted to the axle loads. There, the inuence of theheight of the center of gravity, which may also vary much on trucks, is not taken intoaccount and has to be compensated by a safety margin from the optimal curve. Only thecontrol of brake pressure in anti-lock-systems provides an optimal distribution of brakeforces independently from loading conditions.
6.3.5 Anti-Lock-Systems
On hard braking maneuvers large longitudinal slip values occur. Then, the stability and/orsteerability is no longer given because nearly no lateral forces can be generated. By control-ling the brake torque or brake pressure respectively, the longitudinal slip can be restrictedto values that allow considerable lateral forces.
Here, the angular wheel acceleration Ω is used as a control variable. Angular accelerationsof the wheel are derived from the measured angular speeds of the wheel by dierentiation.The rolling condition is fullled with a longitudinal slip of sL = 0. Then
rD Ω = x (6.39)
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holds, where rD labels the dynamic tire radius and x names the longitudinal accelerationof the vehicle. According to Eq. (6.21), the maximum acceleration/deceleration of a vehicledepends on the friction coecient, |x| = µ g. For a given friction coecient µ a simplecontrol law can be realized for each wheel
|Ω| ≤ 1
rD
|x| . (6.40)
Because no reliable possibility to determine the local friction coecient between tire androad has been found until today, useful information can only be gained from Eq. (6.40)at optimal conditions on dry road. Therefore, the longitudinal slip is used as a secondcontrol variable.
In order to calculate longitudinal slips, a reference speed is estimated from all measuredwheel speeds which is used for the calculation of slip at all wheels, then. This methodis too imprecise at low speeds. Therefore, no control is applied below a limit velocity.Problems also arise when all wheels lock simultaneously for example which may happenon icy roads.
The control of the brake torque is done via the brake pressure which can be increased,held, or decreased by a three-way valve. To prevent vibrations, the decrement is usuallymade slower than the increment.
To prevent a strong yaw reaction, the select low principle is often used with µ-split brakingat the rear axle. Here, the break pressure at both wheels is controlled by the wheel runningon lower friction. Thus, at least the brake forces at the rear axle cause no yaw torque.However, the maximum achievable deceleration is reduced by this.
6.4 Drive and Brake Pitch
6.4.1 Vehicle Model
The vehicle model in Fig. 6.8 consists of ve rigid bodies. The body has three degreesof freedom: Longitudinal motion xA, vertical motion zA and pitch βA. The coordinatesz1 and z2 describe the vertical motions of wheel and axle bodies relative to the body.The longitudinal and rotational motions of the wheel bodies relative to the body can bedescribed via suspension kinematics as functions of the vertical wheel motion:
x1 = x1(z1) , β1 = β1(z1) ;
x2 = x2(z2) , β2 = β2(z2) .(6.41)
The rotation angles ϕR1 and ϕR2 describe the wheel rotations relative to the wheel bodies.
The forces between wheel body and vehicle body are labeled FF1 and FF2. At the wheelsdrive torques MA1, MA2 and brake torques MB1, MB2, longitudinal forces Fx1, Fx2 andthe wheel loads Fz1, Fz2 apply. The brake torques are directly supported by the wheelbodies, whereas the drive torques are transmitted by the drive shafts to the vehicle body.
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ϕR2
ϕR1 MB1
MA1
MB2
MA2
βA
xA
zA
MB1
MB2
MA1
MA2
z2
z1
FF2
FF1
Fz1 Fx1
Fz2 Fx2
a1R
a2
hR
Figure 6.8: Simple vehicle model
The forces and torques that apply to the single bodies are listed in the last column of thetables 6.1 and 6.2.
The velocity of the vehicle body and its angular velocity are given by
v0A,0 =
xA
00
+
00zA
; ω0A,0 =
0
βA
0
. (6.42)
At small rotational motions of the body one gets for the velocities of the wheel bodiesand wheels
v0RK1,0 = v0R1,0 =
xA
00
+
00zA
+
−hR βA
0
−a1 βA
+
∂x1
∂z1z1
0z1
; (6.43)
v0RK2,0 = v0R2,0 =
xA
00
+
00zA
+
−hR βA
0
+a2 βA
+
∂x2
∂z2z2
0z2
. (6.44)
The angular velocities of the wheel bodies and wheels are obtained from
ω0RK1,0 =
0
βA
0
+
0
β1
0
and ω0R1,0 =
0
βA
0
+
0
β1
0
+
0ϕR1
0
(6.45)
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FH Regensburg, University of Applied Sciences © Prof. Dr.-Ing. G. Rill
as well as
ω0RK2,0 =
0
βA
0
+
0
β2
0
and ω0R2,0 =
0
βA
0
+
0
β2
0
+
0ϕR2
0
(6.46)
Introducing a vector of generalized velocities
z =[
xA zA βA β1 ϕR1 β2 ϕR2
]T, (6.47)
the velocities and angular velocities given by Eqs. (6.42), (6.43), (6.44), (6.45), and (6.46)can be written as
v0i =7∑
j=1
∂v0i
∂zj
zj and ω0i =7∑
j=1
∂ω0i
∂zj
zj (6.48)
6.4.2 Equations of Motion
The partial velocities ∂v0i
∂zjand partial angular velocities ∂ω0i
∂zjfor the ve bodies i=1(1)5
and for the seven generalized speeds j =1(1)7 are arranged in the tables 6.1 and 6.2.
partial velocities ∂v0i/∂zj applied forcesbodies xA zA βA z1 ϕR1 z2 ϕR2 F e
i
chassismA
100
001
000
000
000
000
000
00
FF1+FF2−mAg
wheel bodyfrontmRK1
100
001
−hR
0−a1
∂x1
∂z1
01
000
000
000
00
−FF1−mRK1g
wheelfrontmR1
100
001
−hR
0−a1
∂x1
∂z1
01
000
000
000
Fx1
0Fz1−mR1g
wheel bodyrear
mRK2
100
001
−hR
0a2
000
000
∂x2
∂z2
01
000
00
−FF2−mRK2g
wheelrearmR2
100
001
−hR
0a2
000
000
∂x2
∂z2
01
000
Fx2
0Fz2−mR2g
Table 6.1: Partial velocities and applied forces
With the aid of the partial velocities and partial angular velocities the elements of themass matrix M and the components of the vector of generalized forces and torques Q canbe calculated.
M(i, j) =5∑
k=1
(∂v0k
∂zi
)T
mk∂v0k
∂zj
+5∑
k=1
(∂ω0k
∂zi
)T
Θk∂ω0k
∂zj
; i, j = 1(1)7 ; (6.49)
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Vehicle Dynamics FH Regensburg, University of Applied Sciences
partial angular velocities ∂ω0i/∂zj applied torquesbodies xA zA βA z1 ϕR1 z2 ϕR2 M e
i
chassisΘA
000
000
010
000
000
000
000
0−MA1−MA2−a1 FF1+a2 FF2
0
wheel bodyfrontΘRK1
000
000
010
0∂β1
∂z1
0
000
000
000
0MB1
0
wheelfrontΘR1
000
000
010
0∂β1
∂z1
0
010
000
000
0MA1−MB1−R Fx1
0
wheel bodyrear
ΘRK2
000
000
010
000
000
0∂β2
∂z2
0
000
0MB2
0
wheelrearΘR2
000
000
010
000
000
0∂β2
∂z2
0
010
0MA2−MB2−R Fx2
0
Table 6.2: Partial angular velocities and applied torques
Q(i) =5∑
k=1
(∂v0k
∂zi
)T
F ek +
5∑k=1
(∂ω0k
∂zi
)T
M ek ; i = 1(1)7 . (6.50)
Then, the equations of motion for the plane vehicle model are given by
M z = Q . (6.51)
6.4.3 Equilibrium
With the abbreviations
m1 = mRK1 + mR1 ; m2 = mRK2 + mR2 ; mG = mA + m1 + m2 (6.52)
andh = hR + R (6.53)
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 ;
(6.54)
Q(4) = Fz1 − FF1 + ∂x1
∂z1Fx1 −m1 g + ∂β1
∂z1(MA1 −R Fx1) ;
Q(5) = MA1 −MB1 −R Fx1 ;(6.55)
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FH Regensburg, University of Applied Sciences © Prof. Dr.-Ing. G. Rill
Q(6) = Fz2 − FF2 + ∂x2
∂z2Fx2 −m2 g + ∂β2
∂z2(MA2 −R Fx2) ;
Q(7) = MA2 −MB2 −R Fx2 .(6.56)
Without drive and brake forces
MA1 = 0 ; MA2 = 0 ; MB1 = 0 ; MB2 = 0 (6.57)
from Eqs. (6.54), (6.55) and (6.56) one gets the steady state longitudinal forces, the springpreloads, and the wheel loads
F 0x1 = 0 ; F 0
x2 = 0 ;
F 0F1 = b
a+bmA g ; F 0
F2 = aa+b
mA g ;
F 0z1 = m1g + a2
a1+a2mA g ; F 0
z2 = m2g + a1
a1+a2mA g .
(6.58)
6.4.4 Driving and Braking
Assuming that on accelerating or decelerating the vehicle the wheels neither slip nor lock,
R ϕR1 = xA − hR βA + ∂x1
∂z1z1 ;
R ϕR2 = xA − hR βA + ∂x2
∂z2z2
(6.59)
hold. In steady state the pitch motion of the body and the vertical motion of the wheelsreach constant values
βA = βstA = const. , z1 = zst
1 = const. , z2 = zst2 = const. (6.60)
and Eq. (6.59) simplies to
R ϕR1 = xA ; R ϕR2 = xA . (6.61)
With Eqs. (6.60), (6.61) and (6.53) the equation of motion (6.51) results in
mG xA = F ax1 + F a
x2 ;
0 = F az1 + F a
z2 ;
−hR(m1+m2) xA + ΘR1xA
R+ ΘR2
xA
R= −a F a
z1 + b F az2 − (hR + R)(F a
x1 + F ax2) ;
(6.62)∂x1
∂z1m1 xA + ∂β1
∂z1ΘR1
xA
R= F a
z1 − F aF1 + ∂x1
∂z1F a
x1 + ∂β1
∂z1(MA1 −R F a
x1) ;
ΘR1xA
R= MA1 −MB1 −R F a
x1 ;(6.63)
∂x2
∂z2m2 xA + ∂β2
∂z2ΘR2
xA
R= F a
z2 − F aF2 + ∂x2
∂z2F a
x2 + ∂β2
∂z2(MA2 −R F a
x2) ;
ΘR2xA
R= MA2 −MB2 −R F a
x2 ;(6.64)
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Vehicle Dynamics FH Regensburg, University of Applied Sciences
where the steady state spring forces, longitudinal forces, and wheel loads have been sep-arated into initial and acceleration-dependent terms
F stxi = F 0
xi + F axi ; F st
zi = F 0zi + F a
zi ; F stF i = F 0
Fi + F aF i ; i=1, 2 . (6.65)
With given torques of drive and brake the vehicle acceleration xA, the wheel forces F ax1,
F ax2, F a
z1, F az2 and the spring forces F a
F1, F aF2 can be calculated from Eq. (6.62), Eq. (6.63)
and Eq. (6.64)
Via the spring characteristics which have been assumed as linear the acceleration-dependent forces also cause a vertical displacement and pitch motion of the body besidesthe vertical motions of the wheels,
F aF1 = cA1 za
1 ,
F aF2 = cA2 za
2 ,
F az1 = −cR1 (za
A − a βaA + za
1) ,
F az2 = −cR2 (za
A + b βaA + za
2) .
(6.66)
Especially the pitch of the vehicle βaA 6=0, caused by drive or brake will be felt as annoying,
if too distinct.
By an axle kinematics with 'anti dive' and/or 'anti squat' properties, the drive and/orbrake pitch angle can be reduced by rotating the wheel body and moving the wheel centerin longitudinal direction during the suspension travel.
6.4.5 Brake Pitch Pole
For real suspension systems the brake pitch pole can be calculated from the motions ofthe wheel contact points in the x-, z-plane, Fig. 6.9.
x-, z- motion of the contact pointsduring compression and rebound
pitch pole
Figure 6.9: Brake pitch pole
Increasing the pitch pole height above the track level means a decrease in the brake pitchangle. However, the pitch pole is not set above the height of the center of gravity inpractice, because the front of the vehicle would rise at braking then.
108
7 Lateral Dynamics
7.1 Kinematic Approach
7.1.1 Kinematic Tire Model
When a vehicle drives through a curve at low lateral acceleration, small lateral forceswill be needed for course holding. Then, hardly lateral slip occurs at the wheels. In theideal case at vanishing lateral slip the wheels only move in circumferential direction. Thevelocity component of the contact point in the lateral direction of the tire vanishes then
vy = eTy v0P = 0 . (7.1)
This constraint equation can be used as 'kinematic tire model' for course calculation ofvehicles moving in the low lateral acceleration range.
7.1.2 Ackermann Geometry
Within the validity limits of the kinematic tire model the necessary steering angle of thefront wheels can be constructed via the given momentary pivot pole M , Fig. 7.1.
At slowly moving vehicles the lay out of the steering linkage is usually done according tothe Ackermann geometry. Then, the following relations apply
tan δ1 =a
Rand tan δ2 =
a
R + s, (7.2)
where s labels the track width and a denotes the wheel base. Eliminating the curve radiusR, we get
tan δ2 =a
a
tan δ1
+ sor tan δ2 =
a tan δ1
a + s tan δ1
. (7.3)
The deviations 4δ2 = δa2−δA
2 of the actual steering angle δa2 from the Ackermann steering
angle δA2 , which follows from Eq. (7.3), are used, especially on commercial vehicles, to
judge the quality of a steering system.
At a rotation around the momentary pole M , the direction of the velocity is xed forevery point of the vehicle. The angle β between the velocity vector v and the longitudinalaxis of the vehicle is called side slip angle. The side slip angle at point P is given by
tan βP =x
Ror tan βP =
x
atan δ1 , (7.4)
where x denes the distance of P to the inner rear wheel.
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Vehicle Dynamics FH Regensburg, University of Applied Sciences
M
v
βP
δ1δ2
R
a
s
δ1δ2
x
P
βP
Figure 7.1: Ackermann steering geometry at a two-axled vehicle
7.1.3 Space Requirement
The Ackermann approach can also be used to calculate the space requirement of a vehicleduring cornering, Fig. 7.2. If the front wheels of a two-axled vehicle are steered accordingto the Ackermann geometry, the outer point of the vehicle front will run on the maximumradius Rmax, whereas a point on the inner side of the vehicle at the location of the rearaxle will run on the minimum radius Rmin. Hence, it holds
R2max = (Rmin + b)2 + (a + f)2 , (7.5)
where a, b are the wheel base and the width of the vehicle, and f species the distance fromthe front of the vehicle to the front axle. Then, the space requirement 4R = Rmax−Rmin
can be specied as a function of the cornering radius Rmin for a given vehicle dimension
4R = Rmax −Rmin =
√(Rmin + b)2 + (a + f)2 − Rmin . (7.6)
The space requirement 4R of a typical passenger car and a bus is plotted in Fig. 7.3versus the minimum cornering radius. In narrow curves Rmin = 5.0 m, a bus requires aspace of 2.5 times the width, whereas a passenger car needs only 1.5 times the width.
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FH Regensburg, University of Applied Sciences © Prof. Dr.-Ing. G. Rill
M
Rmin
a
b
Rmax
f
Figure 7.2: Space requirement
0 10 20 30 40 500
1
2
3
4
5
6
7
Rmin [m]
∆ R
[m
]
car: a=2.50 m, b=1.60 m, f=1.00 mbus: a=6.25 m, b=2.50 m, f=2.25 m
Figure 7.3: Space requirement of a typical passenger car and bus
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7.1.4 Vehicle Model with Trailer
7.1.4.1 Kinematics
Fig. 7.4 shows a simple lateral dynamics model for a two-axled vehicle with a single-axledtrailer. Vehicle and trailer move on a horizontal track. The position and the orientationof the vehicle relative to the track xed frame x0, y0, z0 is dened by the position vectorto the rear axle center
r02,0 =
x
y
R
(7.7)
and the rotation matrix
A02 =
cos γ − sin γ 0sin γ cos γ 0
0 0 1
. (7.8)
Here, the tire radius R is considered to be constant, and x, y as well as the yaw angle γare generalized coordinates.
K
A1
A2
A3
x 1y 1
x 2
x 3
y 2
y 3
c
a
b γ
δ
κ
x0
y0
Figure 7.4: Kinematic model with trailer
The position vector
r01,0 = r02,0 + A02 r21,2 with r21,2 =
a00
(7.9)
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FH Regensburg, University of Applied Sciences © Prof. Dr.-Ing. G. Rill
and the rotation matrix
A01 = A02 A21 with A21 =
cos δ − sin δ 0sin δ cos δ 0
0 0 1
(7.10)
describe the position and the orientation of the front axle, where a = const labels thewheel base and δ the steering angle.
The position vectorr03,0 = r02,0 + A02
(r2K,2 + A23 rK3,3
)(7.11)
with
r2K,2 =
−b00
and rK3,2 =
−c00
(7.12)
and the rotation matrix
A03 = A02 A23 with A23 =
cos κ − sin κ 0sin κ cos κ 0
0 0 1
(7.13)
dene the position and the orientation of the trailer axis, with κ labeling the bend anglebetween vehicle and trailer, and b, c marking the distances from the rear axle 2 to thecoupling point K and from the coupling point K to the trailer axis 3.
7.1.4.2 Vehicle Motion
According to the kinematic tire model, cf. section 7.1.1, the velocity at the rear axle canonly have a component in the longitudinal direction of the tire which here correspondswith the longitudinal direction of the vehicle
v02,2 =
vx2
00
. (7.14)
The time derivative of Eq. (7.7) results in
v02,0 = r02,0 =
xy0
. (7.15)
The transformation of Eq. (7.14) into the system 0
v02,0 = A02 v02,2 = A02
vx2
00
=
cos γ vx2
sin γ vx2
0
(7.16)
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Vehicle Dynamics FH Regensburg, University of Applied Sciences
compared to Eq. (7.15) results in two rst order dierential equations for the positioncoordinates x and y
x = vx2 cos γ , y = vx2 sin γ . (7.17)
The velocity at the front axle follows from Eq. (7.9)
v01,0 = r01,0 = r02,0 + ω02,0 × A02 r21,2 . (7.18)
Transformed into the vehicle xed system x2, y2, z2 we obtain
v01,2 =
vx2
00
︸ ︷︷ ︸
v02,2
+
00γ
︸ ︷︷ ︸ω02,2
×
a00
︸ ︷︷ ︸r21,2
=
vx2
a γ0
. (7.19)
The unit vectors
ex1,2 =
cos δsin δ
0
and ey1,2 =
− sin δcos δ0
(7.20)
dene the longitudinal and lateral direction at the front axle. According to Eq. (7.1) thevelocity component lateral to the wheel must vanish,
eTy1,2 v01,2 = − sin δ vx2 + cos δ a γ = 0 . (7.21)
Whereas in longitudinal direction the velocity
eTx1,2 v01,2 = cos δ vx2 + sin δ a γ = vx1 (7.22)
remains. From Eq. (7.21) a rst order dierential equation follows for the yaw angle
γ =vx2
atan δ . (7.23)
7.1.4.3 Entering a Curve
In analogy to Eq. (7.2) the steering angle δ can be related to the current track radius Ror with k = 1/R to the current track curvature
tan δ =a
R= a k . (7.24)
Then, the dierential equation for the yaw angle reads as
γ = vx2 k . (7.25)
With the curvature gradient
k = k(t) = kCt
T, (7.26)
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FH Regensburg, University of Applied Sciences © Prof. Dr.-Ing. G. Rill
the entering of a curve is described as a continuous transition from a straight line withthe curvature k = 0 into a circle with the curvature k = kC .The yaw angle of the vehicle can be calculated by simple integration now
γ(t) =vx2 kC
T
t2
2, (7.27)
where at time t = 0 a vanishing yaw angle, γ(t = 0) = 0, has been assumed. Then, theposition of the vehicle follows with Eq. (7.27) from the dierential equations Eq. (7.17)
x = vx2
t=T∫t=0
cos
(vx2 kC
T
t2
2
)dt , y = vx2
t=T∫t=0
sin
(vx2 kC
T
t2
2
)dt . (7.28)
At constant vehicle speed, vx2 = const., Eq. (7.28) is the parameterized form of a clothoide.From Eq. (7.24) the necessary steering angle can be calculated, too. If only small steeringangles are necessary for driving through the curve, the tan-function can be approximatedby its argument, and
δ = δ(t) ≈ a k = a kCt
T(7.29)
holds, i.e. the driving through a clothoide is manageable by a continuous steer motion.
7.1.4.4 Trailer Motions
The velocity of the trailer axis can be obtained by dierentiation of the position vectorEq. (7.11)
v03,0 = r03,0 = r02,0 + ω02,0 × A02 r23,2 + A02 r23,2 . (7.30)
The velocity r02,0 = v02,0 and the angular velocity ω02,0 of the vehicle are dened inEqs. (7.16) and (7.19). The position vector from the rear axle to the axle of the trailer isgiven by
r23,2 = r2K,2 + A23 rK3,3 =
−b − c cos κ−c sin κ
0
, (7.31)
where r2K,2 and rK3,3 are dened in Eq. (7.12). The time derivative of Eq. (7.31) resultsin
r23,2 =
00κ
︸ ︷︷ ︸ω23,2
×
−c cos κ−c sin κ
0
︸ ︷︷ ︸
A23 rK3,3
=
c sin κ κ−c cos κ κ
0
. (7.32)
Eq. (7.30) is transformed into the vehicle xed frame x2, y2, z2 now
v03,2 =
vx2
00
︸ ︷︷ ︸v02,2
+
00γ
︸ ︷︷ ︸ω02,2
×
−b − c cos κ−c sin κ
0
︸ ︷︷ ︸
r23,2
+
c sin κ κ−c cos κ κ
0
︸ ︷︷ ︸
r23,2
=
vx2 + c sin κ (κ+γ)−b γ − c cos κ (κ+γ)
0
.
(7.33)
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The longitudinal and lateral direction at the trailer axle are dened by the unit vectors
ex3,2 =
cos κsin κ
0
and ey3,2 =
− sin κcos κ0
. (7.34)
At the trailer axis the lateral velocity must also vanish
eTy3,2 v03,2 = − sin κ
(vx2 + c sin κ (κ+γ)
)+ cos κ
(−b γ − c cos κ (κ+γ)
)= 0 , (7.35)
whereas in longitudinal direction the velocity
eTx3,2 v03,2 = cos κ
(vx2 + c sin κ (κ+γ)
)+ sin κ
(−b γ − c cos κ (κ+γ)
)= vx3 (7.36)
remains. If Eq. (7.23) is inserted into Eq. (7.35) now, one will get a rst order dierentialequation for the bend angle
κ = −vx2
a
(a
csin κ +
(b
ccos κ + 1
)tan δ
). (7.37)
The dierential equations Eq. (7.17) and Eq. (7.23) describe position and orientationwithin the x0, y0 plane. The position of the trailer relative to the vehicle follows fromEq. (7.37).
7.1.4.5 Course Calculations
0 5 10 15 20 25 300
10
20
30
[s]
front axle steering angle δ
-30 -20 -10 0 10 20 30 40 50 600
10
20
[m]
[m]
front axle rear axle trailer axle
[o]
Figure 7.5: Entering a curve
For a given set of vehicle parameters a, b, c, and predened time functions of the vehiclevelocity, vx2 = vx2(t) and the steering angle, δ = δ(t), the course of vehicle and trailercan be calculated by numerical integration of the dierential equations Eqs. (7.17), (7.23)and (7.37). If the steering angle is slowly increased at constant driving speed, the vehicledrives a gure which will be similar to a clothoide, Fig. 7.5.
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7.2 Steady State Cornering
7.2.1 Cornering Resistance
In a body xed reference frame B, Fig. 7.6, the velocity state of the vehicle can bedescribed by
v0C,B =
v cos βv sin β
0
and ω0F,B =
00ω
, (7.38)
where β denotes the side slip angle of the vehicle measured at the center of gravity. Theangular velocity of a vehicle cornering with constant velocity v on an at horizontal roadis given by
ω =v
R, (7.39)
where R denotes the radius of curvature.
C
v
yB
xB
βω
Fx1 Fy1
Fx2
Fy2
δ
a1
a2
R
Figure 7.6: Cornering resistance
In the body xed reference frame, linear and angular momentum result in
m
(−v2
Rsin β
)= Fx1 cos δ − Fy1 sin δ + Fx2 , (7.40)
m
(v2
Rcos β
)= Fx1 sin δ + Fy1 cos δ + Fy2 , (7.41)
0 = a1 (Fx1 sin δ + Fy1 cos δ)− a2 Fy2 , (7.42)
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where m denotes the mass of the vehicle, Fx1, Fx2, Fy1, Fy2 are the resulting forces inlongitudinal and vertical direction applied at the front and rear axle, and δ species theaverage steer angle at the front axle.
The engine torque is distributed by the center dierential to the front and rear axle. Then,in steady state condition we obtain
Fx1 = k FD and Fx2 = (1− k) FD , (7.43)
where FD is the driving force and by k dierent driving conditions can be modeled:
k = 0 rear wheel drive Fx1 = 0, Fx2 = FD
0 < k < 1 all wheel driveFx1
Fx2
=k
1− k
k = 1 front wheel drive Fx1 = FD, Fx2 = 0
If we insert Eq. (7.43) into Eq. (7.40) we will get(k cos δ + (1−k)
)FD − sin δ Fy1 = −mv2
Rsin β ,
k sin δ FD + cos δ Fy1 + Fy2 =mv2
Rcos β ,
a1k sin δ FD + a1 cos δ Fy1 − a2 Fy2 = 0 .
(7.44)
These equations can be resolved for the driving force
FD =
a2
a1 + a2
cosβ sin δ − sin β cosδ
k + (1− k) cos δ
mv2
R. (7.45)
The driving force will vanish, if
a2
a1 + a2
cosβ sin δ = sin β cosδ ora2
a1 + a2
tan δ = tan β (7.46)
holds. This fully corresponds with the Ackermann geometry. But, the Ackermann geom-etry applies only for small lateral accelerations. In real driving situations, the side slipangle of a vehicle at the center of gravity is always smaller than the Ackermann side slipangle. Then, due to tan β < a2
a1+a2tan δ a driving force FD > 0 is needed to overcome the
'cornering resistance' of the vehicle.
7.2.2 Overturning Limit
The overturning hazard of a vehicle is primarily determined by the track width and theheight of the center of gravity. With trucks however, also the tire deection and the bodyroll have to be respected., Fig. 7.7.
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m g
m ay
αα 12
h2
h1
s/2 s/2FzLFzR
FyL F yR
Figure 7.7: Overturning hazard on trucks
The balance of torques at the height of the track plane applied at the already inclinedvehicle results in
(FzL − FzR)s
2= m ay (h1 + h2) + m g [(h1 + h2)α1 + h2α2] , (7.47)
where ay describes the lateral acceleration, m is the sprung mass, and small roll angles ofthe axle and the body were assumed, α11, α21.
On a left-hand tilt, the right tire raises
F TzR = 0 , (7.48)
whereas the left tire carries the complete vehicle weight
F TzL = m g . (7.49)
Using Eqs. (7.48) and (7.49) one gets from Eq. (7.47)
aTy
g=
s
2h1 + h2
− αT1 − h2
h1 + h2
αT2 . (7.50)
The vehicle will turn over, when the lateral acceleration ay rises above the limit aTy . Roll
of axle and body reduce the overturning limit. The angles αT1 and αT
2 can be calculatedfrom the tire stiness cR and the roll stiness of the axle suspension.
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If the vehicle drives straight ahead, the weight of the vehicle will be equally distributedto both sides
F statzR = F stat
zL =1
2m g . (7.51)
WithF T
zL = F statzL + 4Fz (7.52)
and Eqs. (7.49), (7.51), one obtains for the increase of the wheel load at the overturninglimit
4Fz =1
2m g . (7.53)
Then, the resulting tire deection follows from
4Fz = cR4r , (7.54)
where cR is the radial tire stiness.
Because the right tire simultaneously rebounds with the same amount, for the roll angleof the axle
24r = s αT1 or αT
1 =24r
s=
m g
s cR
(7.55)
holds. In analogy to Eq. (7.47) the balance of torques at the body applied at the rollcenter of the body yields
cW ∗ α2 = m ay h2 + m g h2 (α1 + α2) , (7.56)
where cW names the roll stiness of the body suspension. In particular, at the overturninglimit ay = aT
y
αT2 =
aTy
g
mgh2
cW −mgh2
+mgh2
cW −mgh2
αT1 (7.57)
applies. Not allowing the vehicle to overturn already at aTy = 0 demands a minimum of
roll stiness cW > cminW = mgh2. With Eqs. (7.55) and (7.57) the overturning condition
Eq. (7.50) reads as
(h1 + h2)aT
y
g=
s
2− (h1 + h2)
1
c∗R− h2
aTy
g
1
c∗W − 1− h2
1
c∗W − 1
1
cR∗, (7.58)
where, for abbreviation purposes, the dimensionless stinesses
c∗R =cR
m g
s
and c∗W =cW
m g h2
(7.59)
have been used. Resolved for the normalized lateral acceleration
aTy
g=
s
2
h1 + h2 +h2
c∗W − 1
− 1
c∗R(7.60)
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0 10 200
0.1
0.2
0.3
0.4
0.5
0.6
normalized roll stiffness cW*0 10 20
0
5
10
15
20
T T
normalized roll stiffness cW*
overturning limit ay roll angle α=α1+α2
Figure 7.8: Tilting limit for a typical truck at steady state cornering
remains.
At heavy trucks, a twin tire axle may be loaded with m = 13 000 kg. The radial stinessof one tire is cR = 800 000 N/m, and the track width can be set to s = 2 m. The valuesh1 = 0.8m and h2 = 1.0m hold at maximal load. These values produce the results shownin Fig. 7.8. Even with a rigid body suspension c∗W →∞, the vehicle turns over at a lateralacceleration of ay ≈ 0.5 g. Then, the roll angle of the vehicle solely results from the tiredeection.
At a normalized roll stiness of c∗W = 5, the overturning limit lies at ay ≈ 0.45 g andso reaches already 90% of the maximum. The vehicle will turn over at a roll angle ofα = α1 + α2 ≈ 10 then.
7.2.3 Roll Support and Camber Compensation
When a vehicle drives through a curve with the lateral acceleration ay, centrifugal forceswill be applied to the single masses. At the simple roll model in Fig. 7.9, these are theforces mA ay and mR ay, where mA names the body mass and mR the wheel mass.
Through the centrifugal force mA ay applied to the body at the center of gravity, a torqueis generated, which rolls the body with the angle αA and leads to an opposite deectionof the tires z1 = −z2.
At steady state cornering, the vehicle forces are balanced. With the principle of virtualwork
δW = 0 , (7.61)
the equilibrium position can be calculated.
At the simple vehicle model in Fig. 7.9 the suspension forces FF1, FF2 and tire forcesFy1, Fz1, Fy2, Fz2, are approximated by linear spring elements with the constants cA and
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FF1
z1 α1
y1
Fy1Fz1
S1
Q1
zA αA
yA
b/2 b/2
h0
r0
SA
FF2
z2 α2
y2
Fy2Fy2
S2
Q2
mA ay
mRay mR ay
Figure 7.9: Simple vehicle roll model
cQ, cR. The work W of these forces can be calculated directly or using W = −V via thepotential V . At small deections with linearized kinematics one gets
W = −mA ay yA
−mR ay (yA + hR αA + y1)2 − mR ay (yA + hR αA + y2)
2
−12cA z2
1 − 12cA z2
2
−12cS (z1 − z2)
2
−12cQ (yA + h0 αA + y1 + r0 α1)
2 − 12cQ (yA + h0 αA + y2 + r0 α2)
2
−12cR
(zA + b
2αA + z1
)2 − 12cR
(zA − b
2αA + z2
)2,
(7.62)
where the abbreviation hR = h0 − r0 has been used, and cS describes the spring constantof the anti roll bar, converted to the vertical displacement of the wheel centers.
The kinematics of the wheel suspension are symmetrical. With the linear approaches
y1 =∂y
∂zz1 , α1 =
∂α
∂zα1 and y2 = −∂y
∂zz2 , α2 = −∂α
∂zα2 (7.63)
the work W can be described as a function of the position vector
y = [ yA, zA, αA, z1, z2 ]T . (7.64)
Due toW = W (y) (7.65)
the principle of virtual work Eq. (7.61) leads to
δW =∂W
∂yδy = 0 . (7.66)
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Because of δy 6= 0, a system of linear equations in the form of
K y = b (7.67)
results from Eq. (7.66). The matrix K and the vector b are given by
K =
2 cQ 0 2 cQ h0∂yQ
∂zcQ −∂yQ
∂zcQ
0 2 cR 0 cR cR
2 cQ h0 0 cαb2cR+h0
∂yQ
∂zcQ − b
2cR−h0
∂yQ
∂zcQ
∂yQ
∂zcQ cR
b2cR+h0
∂yQ
∂zcQ c∗A + cS + cR −cS
−∂yQ
∂zcQ cR − b
2cR−h0
∂yQ
∂zcQ −cS c∗A + cS + cR
(7.68)
and
b = −
mA + 2 mR
0
(m1 + m2) hR
mR ∂y/∂z
−mR ∂y/∂z
ay . (7.69)
The following abbreviations have been used:
∂yQ
∂z=
∂y
∂z+ r0
∂α
∂z, c∗A = cA + cQ
(∂y
∂z
)2
, cα = 2 cQ h20 + 2 cR
(b
2
)2
. (7.70)
The system of linear equations Eq. (7.67) can be solved numerically, e.g. with MATLAB.Thus, the inuence of axle suspension and axle kinematics on the roll behavior of thevehicle can be investigated.
Aα
1γ 2γ
a)
roll centerroll center
Aα
1γ 2γ0
b)
0
Figure 7.10: Roll behavior at cornering: a) without and b) with camber compensation
If the wheels only move vertically to the body at jounce and rebound, at fast corneringthe wheels will be no longer perpendicular to the track Fig. 7.10 a. The camber anglesγ1 > 0 and γ2 > 0 result in an unfavorable pressure distribution in the contact area,which leads to a reduction of the maximally transmittable lateral forces. Thus, at more
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sportive vehicles axle kinematics are employed, where the wheels are rotated around thelongitudinal axis at jounce and rebound, α1 = α1(z1) and α2 = α2(z2). Hereby, a cambercompensation can be achieved with γ1 ≈ 0 and γ2 ≈ 0. Fig. 7.10 b. By the rotation ofthe wheels around the longitudinal axis on jounce and rebound, the wheel contact pointsare moved outwards, i.e against the lateral force. By this, a 'roll support' is achieved thatreduces the body roll.
7.2.4 Roll Center and Roll Axis
roll center rearroll axis
roll center front
Figure 7.11: Roll axis
The 'roll center' can be constructed from the lateral motion of the wheel contact pointsQ1 and Q2, Fig. 7.10. The line through the roll center at the front and rear axle is called'roll axis', Fig. 7.11.
7.2.5 Wheel Loads
PF0-∆PPF0+∆P
PR0-∆PPR0+∆P
PF0-∆PFPF0+∆PF
PR0-∆PR
PR0+∆PR
-TT+TT
Figure 7.12: Wheel loads for a exible and a rigid chassis
The roll angle of a vehicle during cornering depends on the roll stiness of the axle andon the position of the roll center. Dierent axle layouts at the front and rear axle mayresult in dierent roll angles of the front and rear part of the chassis, Fig. 7.12.
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On most passenger cars the chassis is rather sti. Hence, front and rear part of the chassisare forced by an internal torque to an overall chassis roll angle. This torque aects thewheel loads and generates dierent wheel load dierences at the front and rear axle. Dueto the degressive inuence of the wheel load to longitudinal and lateral tire forces thesteering tendency of a vehicle can be aected.
7.3 Simple Handling Model
7.3.1 Modeling Concept
x0
y0
a1
a2
xB
yB
C
δ
βγ
Fy1
Fy2
x2
y2
x1
y1
v
Figure 7.13: Simple handling model
The main vehicle motions take place in a horizontal plane dened by the earth-xedframe 0, Fig. 7.13. The tire forces at the wheels of one axle are combined to one resultingforce. Tire torques, rolling resistance, and aerodynamic forces and torques, applied at thevehicle, are not taken into consideration.
7.3.2 Kinematics
The vehicle velocity at the center of gravity can be expressed easily in the body xedframe xB, yB, zB
vC,B =
v cos βv sin β
0
, (7.71)
where β denotes the side slip angle, and v is the magnitude of the velocity.
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The velocity vectors and the unit vectors in longitudinal and lateral direction of the axlesare needed for the computation of the lateral slips. One gets
ex1,B =
cos δsin δ
0
, ey1,B =
− sin δcos δ0
, v01,B =
v cos βv sin β + a1 γ
0
(7.72)
and
ex2,B =
100
, ey2,B =
010
, v02,B =
v cos βv sin β − a2 γ
0
, (7.73)
where a1 and a2 are the distances from the center of gravity to the front and rear axle,and γ denotes the yaw angular velocity of the vehicle.
7.3.3 Tire Forces
Unlike with the kinematic tire model, now small lateral motions in the contact pointsare permitted. At small lateral slips, the lateral force can be approximated by a linearapproach
Fy = cS sy , (7.74)
where cS is a constant depending on the wheel load Fz, and the lateral slip sy is dened byEq. (3.88). Because the vehicle is neither accelerated nor decelerated, the rolling conditionis fullled at each wheel
rD Ω = eTx v0P . (7.75)
Here, rD is the dynamic tire radius, v0P the contact point velocity, and ex the unit vectorin longitudinal direction. With the lateral tire velocity
vy = eTy v0P (7.76)
and the rolling condition Eq. (7.75), the lateral slip can be calculated from
sy =−eT
y v0P
| eTx v0P |
, (7.77)
with ey labeling the unit vector in the lateral direction direction of the tire. So, the lateralforces are given by
Fy1 = cS1 sy1 ; Fy2 = cS2 sy2 . (7.78)
7.3.4 Lateral Slips
With Eq. (7.73), the lateral slip at the front axle follows from Eq. (7.77):
sy1 =+ sin δ (v cos β)− cos δ (v sin β + a1 γ)
| cos δ (v cos β) + sin δ (v sin β + a1 γ) |. (7.79)
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The lateral slip at the rear axle is given by
sy2 = −v sin β − a2 γ
| v cos β |. (7.80)
The yaw velocity of the vehicle γ, the side slip angle β and the steering angle δ areconsidered to be small
| a1 γ | |v| ; | a2 γ | |v| (7.81)
| β | 1 and | δ | 1 . (7.82)
Because the side slip angle always labels the smaller angle between the velocity vectorand the vehicle longitudinal axis, instead of v sin β ≈ v β the approximation
v sin β ≈ |v| β (7.83)
has to be used. Now, Eqs. (7.79) and (7.80) result in
sy1 = −β − a1
|v|γ +
v
|v|δ (7.84)
andsy2 = −β +
a2
|v|γ , (7.85)
where the consequences of Eqs. (7.81), (7.82), and (7.83) were already taken into consid-eration.
7.3.5 Equations of Motion
The velocities, angular velocities, and the accelerations are needed to derive the equationsof motion, For small side slip angles β 1, Eq. (7.71) can be approximated by
vC,B =
v|v| β
0
. (7.86)
The angular velocity is given by
ω0F,B =
00γ
. (7.87)
If the vehicle accelerations are also expressed in the vehicle xed frame xF , yF , zF , onewill nd at constant vehicle speed v = const and with neglecting small higher-order terms
aC,B = ω0F,B × vC,B + vC,B =
0
v γ + |v| β0
. (7.88)
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Vehicle Dynamics FH Regensburg, University of Applied Sciences
The angular acceleration is given by
ω0F,B =
00ω
, (7.89)
where the substitutionγ = ω (7.90)
was used. The linear momentum in the lateral direction of the vehicle reads as
m (v ω + |v| β) = Fy1 + Fy2 , (7.91)
where, due to the small steering angle, the term Fy1 cos δ has been approximated by Fy1,and m describes the vehicle mass. With Eq. (7.90) the angular momentum yields
Θ ω = a1 Fy1 − a2 Fy2 , (7.92)
where Θ names the inertia of vehicle around the vertical axis. With the linear descriptionof the lateral forces Eq. (7.78) and the lateral slips Eqs. (7.84), (7.85), one gets fromEqs. (7.91) and (7.92) two coupled, but linear rst order dierential equations
β =cS1
m |v|
(−β − a1
|v|ω +
v
|v|δ)
+cS2
m |v|
(−β +
a2
|v|ω)− v
|v|ω (7.93)
ω =a1 cS1
Θ
(−β − a1
|v|ω +
v
|v|δ)− a2 cS2
Θ
(−β +
a2
|v|ω)
, (7.94)
which can be written in the form of a state equation
[βω
]︸ ︷︷ ︸
x
=
− cS1 + cS2
m |v|a2 cS2 − a1 cS1
m |v||v|− v
|v|
a2 cS2 − a1 cS1
Θ− a2
1 cS1 + a22 cS2
Θ |v|
︸ ︷︷ ︸
A
[βω
]︸ ︷︷ ︸
x
+
v
|v|cS1
m |v|
v
|v|a1 cS1
Θ
︸ ︷︷ ︸
B
[δ]︸︷︷︸
u
. (7.95)
If a system can be at least approximatively described by a linear state equation, stability,steady state solutions, transient response, and optimal controlling can be calculated withclassic methods of system dynamics.
7.3.6 Stability
7.3.6.1 Eigenvalues
The homogeneous state equationx = A x (7.96)
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FH Regensburg, University of Applied Sciences © Prof. Dr.-Ing. G. Rill
describes the eigen-dynamics. If the approach
xh(t) = x0 eλt (7.97)
is inserted into Eq. (7.96), the homogeneous equation will remain
(λ E − A) x0 = 0 . (7.98)
One gets non-trivial solutions x0 6= 0 for
det |λ E − A| = 0 . (7.99)
The eigenvalues λ provide information concerning the stability of the system.
7.3.6.2 Low Speed Approximation
The state matrix
Av→0 =
− cS1 + cS2
m |v|a2 cS2 − a1 cS1
m |v||v|− v
|v|
0 − a21 cS1 + a2
2 cS2
Θ |v|
(7.100)
approximates the eigen-dynamics of vehicles at low speeds, v → 0. The matrix inEq. (7.100) has the eigenvalues
λ1v→0 = − cS1 + cS2
m |v|and λ2v→0 = − a2
1 cS1 + a22 cS2
Θ |v|. (7.101)
The eigenvalues are real and always negative independent from the driving direction.Thus, vehicles possess an asymptotically stable driving behavior at low speed!
7.3.6.3 High Speed Approximation
At high driving velocities, v →∞, the state matrix can be approximated by
Av→∞ =
0 − v
|v|a2 cS2 − a1 cS1
Θ0
. (7.102)
Using Eq. (7.102) one receives from Eq. (7.99) the relation
λ2v→∞ +
v
|v|a2 cS2 − a1 cS1
Θ= 0 (7.103)
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Vehicle Dynamics FH Regensburg, University of Applied Sciences
with the solutions
λ1,2v→∞ = ±√− v
|v|a2 cS2 − a1 cS1
Θ. (7.104)
When driving forward with v > 0, the root argument will be positive, if
a2 cS2 − a1 cS1 < 0 (7.105)
holds. Then however, one eigenvalue is positive, and the system is unstable. Two zero-eigenvalues λ1 = 0 and λ2 = 0 are obtained for
a1 cS1 = a2 cS2 . (7.106)
The driving behavior is indierent then. Slight parameter variations, however, can leadto an unstable behavior. With
a2 cS2 − a1 cS1 > 0 or a1 cS1 < a2 cS2 (7.107)
and v > 0 the root argument in Eq. (7.104) becomes negative. Then, the eigenvaluesare imaginary, and disturbances lead to undamped vibrations. To avoid instability, high-speed vehicles have to satisfy the condition Eq. (7.107). The root argument in Eq. (7.104)changes at backward driving its sign. Hence, a vehicle showing stable driving behavior atforward driving becomes unstable at fast backward driving!
7.3.7 Steady State Solution
7.3.7.1 Side Slip Angle and Yaw Velocity
At a given steering angle δ=δ0, a stable system reaches steady state after a certain time.With xst =const. or xst =0, the state equation Eq. (7.95) is reduced to a system of linearequations
A xst = −B u . (7.108)
With the elements from the state matrix A and the vector B, one gets from Eq. (7.108)two equations to determine the steady state side slip angle βst and the steady state angularvelocity ωst at a constant given steering angle δ=δ0
|v| (cS1 + cS2) βst + (m v |v|+ a1 cS1−a2 cS2) ωst = v cS1 δ0 , (7.109)
|v| (a1 cS1 − a2 cS2) βst + (a21 cS1 + a2
2 cS2) ωst = v a1 cS1 δ0 , (7.110)
where the rst equation has been multiplied by −m |v| |v| and the second with −Θ |v|.The solution can be derived from
βst =
v cS1 δ0 m v |v|+ a1 cS1−a2 cS2
v a1 cS1 δ0 a21 cS1 + a2
2 cS2
|v| (cS1 + cS2) m v |v|+ a1 cS1−a2 cS2
|v| (a1 cS1 − a2 cS2) a21 cS1 + a2
2 cS2
(7.111)
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and
ωst =
|v| (cS1 + cS2) v cS1 δ0
|v| (a1 cS1 − a2 cS2) v a1 cS1 δ0
|v| (cS1 + cS2) m v |v|+ a1 cS1−a2 cS2
|v| (a1 cS1 − a2 cS2) a21 cS1 + a2
2 cS2
(7.112)
The denominator results in
detD = |v|(
cS1 cS2 (a1 + a2)2 + m v |v| (a2 cS2 − a1 cS1)
). (7.113)
For a non vanishing denominator detD 6=0, steady state solutions exist
βst =v
|v|
a2 − m v |v| a1
cS2 (a1 + a2)
a1 + a2 + m v |v| a2 cS2 − a1 cS1
cS1 cS2 (a1 + a2)
δ0 , (7.114)
ωst =v
a1 + a2 + m v |v| a2 cS2 − a1 cS1
cS1 cS2 (a1 + a2)
δ0 . (7.115)
At forward driving vehicles v > 0, the steady state side slip angle starts with the kinematicvalue
βv→0st =
v
|v|a2
a1 + a2
δ0 and ωv→0st =
v
a1 + a2
δ0 (7.116)
and decreases with increasing speed. At speeds larger than
vβst=0 =
√a2 cS2 (a1 + a2)
a1 m(7.117)
the side slip angle changes the sign. Using the kinematic value of the yaw velocityEq. (7.115) can be written as
ωst =v
a1 + a2
1
1 +|v|v
(v
vch
)2
δ0 , (7.118)
where
vch =
√cS1 cS2 (a1 + a2)
2
m (a2 cS2 − a1 cS1)(7.119)
is called the 'characteristic' speed of the vehicle.
In Fig. 7.14 the side slip angle β, and the driven curve radius R are plotted versus thedriving speed v. The steering angle has been set to δ0 = 1.4321, in order to let the vehicle
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0 10 20 30 40-10
-8
-6
-4
-2
0
2
v [m/s]
β [d
eg]
steady state side slip angle
a1*cS1/a2*cS2 = 0.66667a1*cS1/a2*cS2 = 1 a1*cS1/a2*cS2 = 1.3333
0 10 20 30 400
50
100
150
200
v [m/s]
r [m
]
radius of curvrature
a1*cS1/a2*cS2 = 0.66667a1*cS1/a2*cS2 = 1 a1*cS1/a2*cS2 = 1.3333
m=700 kg;Θ=1000 kg m2;
a1=1.2 m;a2=1.3 m;
cS1 = 80 000 Nm; cS2 =110 770 Nm73 846 Nm55 385 Nm
Figure 7.14: Steady state cornering
drive a circle with the radius R0 = 100 m at v → 0. The actually driven circle radius Rhas been calculated via
ωst =v
R. (7.120)
Some concepts for an additional steering of the rear axle were trying to keep the sideslip angle of the vehicle, measured at the center of the vehicle to zero by an appropriatesteering or controlling. Due to numerous problems, production stage could not yet bereached.
7.3.7.2 Steering Tendency
After reaching the steady state solution, the vehicle moves on a circle. When insertingEq. (7.120) into Eq. (7.115) and resolving for the steering angle, one gets
δ0 =a1 + a2
R+ m
v2
R
v
|v|a2 cS2 − a1 cS1
cS1 cS2 (a1 + a2). (7.121)
The rst term is the Ackermann steering angle, which follows from Eq. (7.2) with thewheel base a = a1 + a2 and the approximation for small steering angles tan δ0 ≈ δ0.The Ackermann-steering angle provides a good approximation for slowly moving vehicles,because the second expression in Eq. (7.121) becomes very small at v → 0. Depending onthe value of a2 cS2 − a1 cS1 and the driving direction (forward: v > 0, backward: v < 0),the necessary steering angle diers from the Ackermann-steering angle at higher speeds.The dierence is proportional to the lateral acceleration
ay =v2
R. (7.122)
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At v > 0 the steering tendency of a vehicle is dened by the position of the center ofgravity a1, a2 and the cornering stinesses at the axles cS1, cS2. The various steeringtendencies are arranged in the table 7.1.
• understeering δ0 > δA0 or a1 cS1 < a2 cS2 or
a1 cS1
a2 cS2
< 1
• neutral δ0 = δA0 or a1 cS1 = a2 cS2 or
a1 cS1
a2 cS2
= 1
• oversteering δ0 < δA0 or a1 cS1 > a2 cS2 or
a1 cS1
a2 cS2
> 1
Table 7.1: Steering tendency of a vehicle at forward driving
7.3.7.3 Slip Angles
With the conditions for a steady state solution βst = 0, ωst = 0 and the relationEq. (7.120), the equations of motion Eq. (7.91) and Eq. (7.92) can be dissolved for thelateral forces
Fy1st =a2
a1 + a2
mv2
R,
Fy2st =a1
a1 + a2
mv2
R
ora1
a2
=Fy2st
Fy1st
. (7.123)
With the linear tire model in Eq. (7.74) one gets
F sty1 = cS1 sst
y1 and F sty2 = cS2 sst
y2 , (7.124)
where sstyA1
and sstyA2
label the steady state lateral slips at the axles. Now, from Eqs. (7.123)and (7.124) it follows
a1
a2
=F st
y2
F sty1
=cS2 sst
y2
cS1 ssty1
ora1 cS1
a2 cS2
=sst
y2
ssty1
. (7.125)
That means, at a vehicle with understeering tendency (a1 cS1 < a2 cS2) during steadystate cornering the slip angles at the front axle are larger than the slip angles at the rearaxle, sst
y1 > ssty2. So, the steering tendency can also be determined from the slip angle at
the axles.
7.3.8 Influence of Wheel Load on Cornering Stiffness
With identical tires at the front and rear axle, given a linear inuence of wheel load onthe raise of the lateral force over the lateral slip,
clinS1 = cS Fz1 and clin
S2 = cS Fz2 . (7.126)
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Vehicle Dynamics FH Regensburg, University of Applied Sciences
holds. The weight of the vehicle G = mg is distributed over the axles according to theposition of the center of gravity
Fz1 =a2
a1 + a2
G and .Fz2 =a1
a1 + a2
G (7.127)
With Eq. (7.126) and Eq. (7.127) one obtains
a1 clinS1 = a1 cS
a2
a1 + a2
G (7.128)
anda2 clin
S2 = a2 cSa1
a1 + a2
G . (7.129)
Thus, a vehicle with identical tires would be steering neutrally at a linear inuence of thewheel load on the cornering stiness, because of
a1 clinS1 = a2 clin
S2 (7.130)
The lateral force is applied behind the center of the contact patch at the caster osetdistance. Hence, the lever arms of the lateral forces change to a1 → a1 − v
|v| nL1 anda2 → a2+
v|v| nL1 , which will stabilize the vehicle, independently from the driving direction.
0 1 2 3 4 5 6 7 80
1
2
3
4
5
6
α
Fz [kN]
F y [
kN]
Fz [N ] Fy [N ]0 0
1000 7582000 14383000 20434000 25765000 30396000 34347000 37628000 4025
Figure 7.15: Lateral force Fy over wheel load Fz at dierent slip angles
At a real tire, a degressive inuence of the wheel load on the tire forces is observed,Fig. 7.15. According to Eq. (7.92) the rotation of the vehicle is stable, if the torque fromthe lateral forces Fy1 and Fy2 is aligning, i.e.
a1 Fy1 − a2 Fy2 < 0 (7.131)
holds. At a vehicle with the wheel base a = 2.45m the axle loads Fz1 = 4000N and Fz2 =3000 N yield the position of the center of gravity a1 = 1.05 m and a2 = 1.40 m. At equalslip on front and rear axle one gets from the table in 7.15 Fy1 = 2576N and Fy2 = 2043N .
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With this, the condition in Eq. (7.131) yields 1.05 ∗ 2576 − 1.45 ∗ 2043 = −257.55 . Thevalue is signicantly negative and thus stabilizing.
Vehicles with a1 < a2 have a stable, i.e. understeering driving behavior. If the axle load atthe rear axle is larger than at the front axle (a1 > a2), generally a stable driving behaviorcan only be achieved with dierent tires.
At increasing lateral acceleration the vehicle is more and more supported by the outerwheels. The wheel load dierences can dier at a suciently rigid vehicle body, because ofdierent kinematics (roll support) or dierent roll stiness. Due to the degressive inuenceof wheel load, the lateral force at an axle decreases with increasing wheel load dierence.If the wheel load is split more strongly at the front axle than at the rear axle, the lateralforce potential at the front axle will decrease more than at the rear axle and the vehiclewill become more stable with an increasing lateral force, i.e. more understeering.
135
8 Driving Behavior of Single Vehicles
8.1 Standard Driving Maneuvers
8.1.1 Steady State Cornering
The steering tendency of a real vehicle is determined by the driving maneuver calledsteady state cornering. The maneuver is performed quasi-static. The driver tries to keepthe vehicle on a circle with the given radius R. He slowly increases the driving speed vand, with this also the lateral acceleration due ay = v2
Runtil reaching the limit. Typical
results are displayed in Fig. 8.1.
0
20
40
60
80
lateral acceleration [g]
stee
r ang
le [d
eg]
-4
-2
0
2
4
side
slip
ang
le [d
eg]
0 0.2 0.4 0.6 0.80
1
2
3
4
roll
angl
e [d
eg]
0 0.2 0.4 0.6 0.80
1
2
3
4
5
6
whe
el lo
ads
[kN
]
lateral acceleration [g]
Figure 8.1: Steady state cornering: rear-wheel-driven car on R = 100 m
The vehicle is under-steering and thus stable according to Eq. (7.121) with Eq. (7.122).The inclination in the diagram steering angle versus lateral velocity decides about thesteering tendency and stability behavior.
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The nonlinear inuence of the wheel load on the tire performance is here used to designa vehicle that is weakly stable, but sensitive to steer input in the lower range of lateralacceleration, and is very stable but less sensitive to steer input in limit conditions.
With the increase of the lateral acceleration the roll angle becomes larger. The overturningtorque is intercepted by according wheel load dierences between the outer and innerwheels. With a suciently rigid frame the use of an anti roll bar at the front axle allowsto increase the wheel load dierence there and to decrease it at the rear axle accordingly.
Thus, the digressive inuence of the wheel load on the tire properties, cornering stinessand maximum possible lateral force, is stressed more strongly at the front axle, and thevehicle becomes more under-steering and stable at increasing lateral acceleration, until itdrifts out of the curve over the front axle in the limit situation.
Problems occur at front driven vehicles, because due to the demand for traction, the frontaxle cannot be relieved at will.
Having a suciently large test site, the steady state cornering maneuver can also becarried out at constant speed. There, the steering wheel is slowly turned until the vehiclereaches the limit range. That way also weakly motorized vehicles can be tested at highlateral accelerations.
8.1.2 Step Steer Input
The dynamic response of a vehicle is often tested with a step steer input. Methods forthe calculation and evaluation of an ideal response, as used in system theory or controltechnics, can not be used with a real car, for a step input at the steering wheel is notpossible in practice. A real steering angle gradient is displayed in Fig. 8.2.
0 0.2 0.4 0.6 0.8 10
10
20
30
40
time [s]
stee
ring
angl
e [d
eg]
Figure 8.2: Step steer input
Not the angle at the steering wheel is the decisive factor for the driving behavior, but thesteering angle at the wheels, which can dier from the steering wheel angle because ofelasticities, friction inuences, and a servo-support. At very fast steering movements, alsothe dynamics of the tire forces plays an important role.
In practice, a step steer input is usually only used to judge vehicles subjectively. Exceedsin yaw velocity, roll angle, and especially sideslip angle are felt as annoying.
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Vehicle Dynamics FH Regensburg, University of Applied Sciences
0
0.1
0.2
0.3
0.4
0.5
0.6
late
ral a
ccel
erat
ion
[g]
0
2
4
6
8
10
12
yaw
vel
ocity
[deg
/s]
0 2 40
0.5
1
1.5
2
2.5
3
roll
angl
e [d
eg]
0 2 4-2
-1.5
-1
-0.5
0
0.5
1
[t]
side
slip
ang
le [d
eg]
Figure 8.3: Step steer: passenger car at v = 100 km/h
The vehicle under consideration behaves dynamically very well, Fig. 8.3. Almost no over-shoots occur in the time history of the roll angle and the lateral acceleration. However,small overshoots can be noticed at yaw the velocity and the sideslip angle.
8.1.3 Driving Straight Ahead
8.1.3.1 Random Road Profile
The irregularities of a track are of stochastic nature. Fig. 8.4 shows a country road prole indierent scalings. To limit the eort of the stochastic description of a track, one usuallyemploys simplifying models. Instead of a fully two-dimensional description either twoparallel tracks are evaluated
z = z(x, y) → z1 = z1(s1) , and z2 = z2(s2) (8.1)
or one uses an isotropic track. The statistic properties are direction-independent at anisotropic track. Then, a two-dimensional track can be approximated by a single randomprocess
z = z(x, y) → z = z(s) ; (8.2)
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0 10 20 30 40 50 60 70 80 90 100 0 1 2 3 4 5-0.05-0.04-0.03-0.02-0.01
00.010.020.030.040.05
Figure 8.4: Track irregularities
A normally distributed, stationary and ergodic random process z = z(s) is completelycharacterized by the rst two expectation values, the mean value
mz = lims→∞
1
2s
s∫−s
z(s) ds (8.3)
and the correlation function
Rzz(δ) = lims→∞
1
2s
s∫−s
z(s) z(s− δ) ds . (8.4)
A vanishing mean value mz = 0 can always be achieved by an appropriate coordinatetransformation. The correlation function is symmetric,
Rzz(δ) = Rzz(−δ) , (8.5)
and
Rzz(0) = lims→∞
1
2s
s∫−s
(z(s)
)2ds (8.6)
describes the variance of zs.
Stochastic track irregularities are mostly described by power spectral densities (abbrevi-ated by psd). Correlating function and the one-sided power spectral density are linked bythe Fourier-transformation
Rzz(δ) =
∞∫0
Szz(Ω) cos(Ωδ) dΩ (8.7)
where Ω denotes the space circular frequency. With Eq. (8.7) follows from Eq. (8.6)
Rzz(0) =
∞∫0
Szz(Ω) dΩ . (8.8)
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Vehicle Dynamics FH Regensburg, University of Applied Sciences
Thus, the psd gives information, how the variance is compiled from the single frequencyshares.
The power spectral densities of real tracks can be approximated by the relation
Szz(Ω) = S0
[Ω
Ω0
]−w
, (8.9)
where the reference frequency is xed to Ω0 = 1 m−1. The reference psd S0 = Szz(Ω0)acts as a measurement for unevennes and the waviness w indicates, whether the track hasnotable irregularities in the short or long wave spectrum. At real tracks, the reference-psd S0 lies within the range from 1 ∗ 10−6 m3 to 100 ∗ 10−6 m3 and the waviness can beapproximated by w = 2.
8.1.3.2 Steering Activity
-2 0 20
500
1000
highway: S0=1*10-6 m3; w=2
-2 0 20
500
1000
country road: S0=2*10-5 m3; w=2
[deg] [deg]
Figure 8.5: Steering activity on dierent roads
A straightforward drive upon an uneven track makes continuous steering corrections nec-essary. The histograms of the steering angle at a driving speed of v = 90 km/h aredisplayed in Fig. 8.5. The track quality is reected in the amount of steering actions. Thesteering activity is often used to judge a vehicle in practice.
8.2 Coach with different Loading Conditions
8.2.1 Data
The dierence between empty and laden is sometimes very large at trucks and coaches.In the table 8.1 all relevant data of a travel coach in fully laden and empty condition arelisted.
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FH Regensburg, University of Applied Sciences © Prof. Dr.-Ing. G. Rill
vehicle mass [kg] center of gravity [m] inertias [kg m2]
empty 12 500 −3.800 | 0.000 | 1.50012 500 0 0
0 155 000 00 0 155 000
fully laden 18 000 −3.860 | 0.000 | 1.60015 400 0 250
0 200 550 0250 0 202 160
Table 8.1: Data for a laden and empty coach
The coach has a wheel base of a = 6.25m. The front axle with the track width sv = 2.046mhas a double wishbone single wheel suspension. The twin-tire rear axle with the trackwidths so
h = 2.152 m and sih = 1.492 m is guided by two longitudinal links and an a-arm.
The air-springs are tted to load variations via a niveau-control.
8.2.2 Roll Steering
-1 0 1-10
-5
0
5
10
susp
ensi
on tr
avel
[cm
]
steer angle [deg]
Figure 8.6: Roll steer: - - front, rear
While the kinematics at the front axle hardly cause steering movements at roll motions,the kinematics at the rear axle are tuned in a way to cause a notable roll steering eect,Fig. 8.6.
8.2.3 Steady State Cornering
Fig. 8.7 shows the results of a steady state cornering on a 100 m-Radius. The fully occu-pied vehicle is slightly more understeering than the empty one. The higher wheel loadscause greater tire aligning torques and increase the degressive wheel load inuence on theincrease of the lateral forces. Additionally roll steering at the rear axle occurs.
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Vehicle Dynamics FH Regensburg, University of Applied Sciences
0 0.1 0.2 0.3 0.450
100
150
200
250
lateral acceleration ay [g]
steer angle δLW [deg]
-100 0 1000
50
100
150
200
[m]
[m]
vehicle course
0 0.1 0.2 0.3 0.40
50
100wheel loads [kN]
0 0.1 0.2 0.3 0.40
50
100wheel loads [kN]
lateral acceleration ay [g] lateral acceleration ay [g]
Figure 8.7: Steady state cornering: coach - - empty, fully occupied
Both vehicles can not be kept on the given radius in the limit range. Due to the high posi-tion of the center of gravity the maximal lateral acceleration is limited by the overturninghazard. At the empty vehicle, the inner front wheel lift o at a lateral acceleration ofay ≈ 0.4 g . If the vehicle is fully occupied, this eect will occur already at ay ≈ 0.35 g.
8.2.4 Step Steer Input
The results of a step steer input at the driving speed of v = 80 km/h can be seen inFig. 8.8. To achieve comparable acceleration values in steady state condition, the stepsteer input was done at the empty vehicle with δ = 90 and at the fully occupied onewith δ = 135. The steady state roll angle is 50% larger at the fully occupied bus thanat the empty one. By the niveau-control, the air spring stiness increases with the load.Because the damper eect remains unchanged, the fully laden vehicle is not damped aswell as the empty one. This results in larger overshoots in the time histories of the lateralacceleration, the yaw angular velocity, and the sideslip angle.
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FH Regensburg, University of Applied Sciences © Prof. Dr.-Ing. G. Rill
0 2 4 6 80
0.1
0.2
0.3
0.4
lateral acceleration a y [g]
0 2 4 6 80
2
4
6
8
10
yaw velocity ωZ [deg/s]
0 2 4 6 80
2
4
6
8
[s]
roll angle α [deg]
0 2 4 6 8
-2
-1
0
1
2
[s]
side slip angle β [deg]
Figure 8.8: Step steer input: - - coach empty, coach fully occupied
8.3 Different Rear Axle Concepts for a Passenger Car
A medium-sized passenger car is equipped in standard design with a semi-trailing rearaxle. By accordingly changed data this axle can easily be transformed into a trailing armor a single wishbone axis. According to the roll support, the semi-trailing axle realizedin serial production represents a compromise between the trailing arm and the singlewishbone, Fig. 8.9, .
The inuences on the driving behavior at steady state cornering on a 100 m radius areshown in Fig. 8.10.
Substituting the semi-trailing arm at the standard car by a single wishbone, one gets,without adaption of the other system parameters a vehicle oversteering in the limit range.Compared to the semi-trailing arm the single wishbone causes a notably higher roll sup-port. This increases the wheel load dierence at the rear axle, Fig. 8.10. Because the wheelload dierence is simultaneously reduced at the front axle, the understeering tendency isreduced. In the limit range, this even leads to an oversteering behavior.
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Vehicle Dynamics FH Regensburg, University of Applied Sciences
-5 0 5-10
-5
0
5
10
lateral motion [cm]ve
rtica
l mot
ion
[cm
]
Figure 8.9: Rear axle: semi-trailing arm, - - single wishbone, · · · trailing arm
0 0.2 0.4 0.6 0.80
50
100
steer angle δLW [deg]
0 0.2 0.4 0.6 0.80
1
2
3
4
5roll angle α [Grad]
0 0.2 0.4 0.6 0.80
2
4
6wheel loads front [kN]
0 0.2 0.4 0.6 0.80
2
4
6
lateral acceleration ay [g]
wheel loads rear [kN]
lateral acceleration ay [g]
Figure 8.10: Steady state cornering, semi-trailing arm, - - single wishbone, · · · trailingarm
The vehicle with a trailing arm rear axle is, compared to the serial car, more understeering.The lack of roll support at the rear axle also causes a larger roll angle.
144
Index
Ackermann geometry, 109Ackermann steering angle, 109, 132Aerodynamic forces, 96Air resistance, 96All wheel drive, 118Anti dive, 108Anti roll bar, 122Anti squat, 108Anti-lock-system, 102Auto-correlation, 13Axle kinematics, 108
Double wishbone, 7McPherson, 7Multi-link, 7
Axle load, 95Axle suspension
Solid axle, 55Twist beam, 56
Bend angle, 113, 116Brake pitch angle, 103Brake pitch pole, 108Braking force distribution, 100
Camber angle, 6, 24Camber compensation, 121, 124Camber slip, 49Caster, 8, 9Characteristic speed, 131Climbing capacity, 97Comfort, 72Contact point, 24Cornering resistance, 117, 118Cornering stiness, 41, 133Curvature gradient, 114
Damping rate, 76Deviation, 13Disturbance-reaction problem, 84Disturbing force lever, 9Down forces, 96Downhill capacity, 97Drag link, 57, 58Drive pitch angle, 103Driver, 3Driving force distribution, 100Driving safety, 72Dynamic axle load, 95Dynamic force elements, 63Dynamic wheel loads, 94
Eective value, 13Eigenvalues, 128Environment, 4
First harmonic oscillation, 63Fourier-approximation, 64Frequency domain, 63Friction, 97Front wheel drive, 98, 118
Generalized uid mass, 70Grade, 95
Hydro-mount, 69
Kingpin, 7Kingpin Angle, 8
Lateral acceleration, 121, 132Lateral force, 126Lateral slip, 126
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Vehicle Dynamics FH Regensburg, University of Applied Sciences
Ljapunov equation, 84Load, 4
Maximum acceleration, 97, 98Maximum deceleration, 97, 99Mean value, 13
Natural frequency, 76
Optimal damping, 81, 87Oversteering, 133Overturning limit, 118
Parallel track model, 10Parallel tracks, 138Pinion, 57Pivot pole, 109Power spectral density, 139
Quarter car model, 87, 90
Rack, 57Random road prole, 138Rear wheel drive, 98, 118Reference frames
Ground xed, 5Inertial, 5Vehicle xed, 5
Relative damping rate, 77Ride comfort, 83Ride safety, 83Road, 10, 23Roll axis, 124Roll center, 124Roll steer, 141Roll stiness, 120Roll support, 121, 124Rolling condition, 126
Safety, 72Side slip angle, 109Sky hook damper, 87Space requirement, 110Spring rate, 78Stability, 128State equation, 128State matrix, 88
State vector, 88Steady state cornering, 117, 136, 141Steering activity, 140Steering angle, 114Steering box, 57, 58Steering lever, 58Steering oset, 9Steering system
Drag link steering system, 58Lever arm, 57Rack and pinion, 57
Steering tendency, 125, 132Step steer input, 137, 143Suspension model, 75Suspension spring rate, 78System response, 63
Tilting condition, 97Tire
Bore slip, 52Bore torque, 20, 50Camber angle, 24Camber inuence, 48Characteristics, 52Circumferential direction, 24Composites, 19Contact forces, 20Contact patch, 20Contact point, 23Contact point velocity, 31Contact torques, 20Cornering stiness, 41Deection, 26Deformation velocity, 31Development, 19Dynamic oset, 41Dynamic radius, 32, 33Friction coecient, 45Lateral direction, 24Lateral force, 20Lateral force characteristics, 41Lateral force distribution, 40Lateral slip, 40Lateral velocity, 31
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FH Regensburg, University of Applied Sciences © Prof. Dr.-Ing. G. Rill
Lift o, 88Linear model, 126Loaded radius, 24, 32Longitudinal force, 20, 38, 39Longitudinal force characteristics, 39Longitudinal force distribution, 39Longitudinal slip, 39Longitudinal velocity, 31Model, 52Normal force, 20Pneumatic trail, 41Radial damping, 35Radial direction, 24Radial stiness, 34, 121Rolling resistance, 20, 36, 37Self aligning torque, 20, 41Sliding velocity, 40Static radius, 24, 32, 33Tilting torque, 20Track normal, 24, 26Transport velocity, 33Tread deection, 38Tread particles, 37Unloaded radius, 32Vertical force, 34Wheel load inuence, 41
Tire ModelKinematic, 109Linear, 133
TMeasy, 52Toe angle, 6Toe-in angle, 6Track, 23Track curvature, 114Track normal, 5Track radius, 114Track width, 109, 121Trailer, 112, 115
Understeering, 133
Variance, 13Vehicle, 3Vehicle comfort, 72Vehicle dynamics, 2
Vehicle model, 75, 90, 94, 103, 112, 121,125
Vertical dynamics, 72Virtual work, 121
Waviness, 140Wheel
Angular velocity, 50Wheel base, 109Wheel load, 20Wheel loads, 94Wheel rotation axis, 5Wheel Suspension
Semi-trailing arm, 143Single wishbone, 143Trailing arm, 143
Wheel suspensionCentral control arm, 56Double wishbone, 55McPherson, 55Multi-Link, 55Semi-trailing arm, 56SLA, 56
Yaw angle, 112, 115Yaw velocity, 126
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