Vehicle Dynamics 2004

S C I M A N Y D E L C I H E V

FACHHOCHSCHULE REGENSBURG UNIVERSITY OF APPLIED SCIENCES

HOCHSCHULE FÜR TECHNIK WIRTSCHAFT SOZIALES

LECTURE NOTES Prof. Dr. Georg Rill © October 2004

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

Contents

Contents

I

1 Intro Introdu ducti ction on

1

1.1

1.2

1.3

1.4 1.4

Termino erminolog logy y. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.1 1.1.1 Vehicl ehiclee Dynam Dynamics ics . . . . . . . . . . . . . . . . . . . . . . . 1.1. 1.1.2 2 Dri Driver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1. 1.1.3 3 Vehic ehicle le . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.4 1.4 Loa Load . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.5 1.1.5 Envir Environm onment ent . . . . . . . . . . . . . . . . . . . . . . . . . . Wheel/Axl Wheel/Axlee Suspensi Suspension on Systems Systems . . . . . . . . . . . . . . . . . . . . 1.2.1 1.2.1 Genera Generall Remar Remarks ks . . . . . . . . . . . . . . . . . . . . . . . . 1.2.2 1.2.2 Multi Purpose Purpose Suspensi Suspension on Systems Systems . . . . . . . . . . . . . . 1.2.3 1.2.3 Specific Specific Suspensi Suspension on Systems Systems . . . . . . . . . . . . . . . . . . Steeri Steering ng System Systemss . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.1 1.3.1 Requir Requireme ements nts . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.2 1.3.2 Rack Rack and Pinion Pinion Steeri Steering ng . . . . . . . . . . . . . . . . . . . . 1.3.3 1.3.3 Leve Leverr Arm Steeri Steering ng Syste System m . . . . . . . . . . . . . . . . . . 1.3.4 1.3.4 Drag Drag Link Link Steeri Steering ng System System . . . . . . . . . . . . . . . . . . . 1.3.5 1.3.5 Bus Steer Steer Syste System m . . . . . . . . . . . . . . . . . . . . . . . . Defin Definit itio ions ns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.1 1.4.1 Coordi Coordinat natee Syste Systems ms . . . . . . . . . . . . . . . . . . . . . . 1.4.2 1.4.2 Toe and and Camber Camber Angle Angle . . . . . . . . . . . . . . . . . . . . . 1.4.2. 1.4.2.1 1 Defini Definitio tions ns accor accordin ding g to DIN 70 000 . . . . . . . . 1.4. 1.4.2. 2.2 2 Calc Calcul ulat atio ion n. . . . . . . . . . . . . . . . . . . . . . 1.4.3 1.4.3 Steeri Steering ng Geomet Geometry ry . . . . . . . . . . . . . . . . . . . . . . . 1.4. 1.4.3. 3.1 1 King Kingpi pin n . . . . . . . . . . . . . . . . . . . . . . . 1.4.3. 1.4.3.2 2 Caste Casterr and and Kingpi Kingpin n Angle Angle . . . . . . . . . . . . . . 1.4.3.3 1.4.3.3 Disturbing Disturbing Force Force Lever Lever,, Caster Caster and Kingpin Kingpin Offset Offset .

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2 The The Tire Tire 2.1

Introd Introduc uctio tion n . . . . . . . . . . . . . . . . . . . . . . 2.1.1 2.1.1 Tire Tire Deve Develop lopmen mentt . . . . . . . . . . . . . . . 2.1.2 2.1.2 Tire Tire Compo Composit sites es . . . . . . . . . . . . . . . 2.1.3 2.1.3 Forces Forces and and Torqu Torques es in in the the Tire Tire Conta Contact ct Area Area .

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2.2

2.3 2.3

2.4 2.5 2.6 2.7 2.8

Conta Contact ct Geomet Geometry ry . . . . . . . . . . . . . . . . . . . 2.2. 2.2.1 1 Cont Contac actt Poin Pointt . . . . . . . . . . . . . . . . . 2.2.2 2.2.2 Local Local Track rack Plane Plane . . . . . . . . . . . . . . Whee Wheell Load Load . . . . . . . . . . . . . . . . . . . . . . 2.3.1 2.3.1 Dynami Dynamicc Rollin Rolling g Radiu Radiuss . . . . . . . . . . . 2.3.2 2.3.2 Contact Contact Point Point Velocity elocity . . . . . . . . . . . . Longitud Longitudinal inal Force Force and and Longitudi Longitudinal nal Slip . . . . . . Lateral Lateral Slip, Slip, Latera Laterall Force Force and Self Aligni Aligning ng Torqu Torquee Cambe Camberr Influe Influence nce . . . . . . . . . . . . . . . . . . . Bore Bore Torque orque . . . . . . . . . . . . . . . . . . . . . . Typical ypical Tire Tire Characte Characteristic risticss . . . . . . . . . . . . . .

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3 Vertical ertical Dynami Dynamics cs 3.1 3.1 3.2

3.3

3.4

3.5

Goals . . . . . . . . . . . . . . . . . . . . . . . . Basic Basic Tuning uning . . . . . . . . . . . . . . . . . . . . 3.2. 3.2.1 1 Simp Simple le Mode Models ls . . . . . . . . . . . . . . . 3.2. 3.2.2 2 Track rack . . . . . . . . . . . . . . . . . . . . 3.2.3 3.2.3 Spring Spring Preloa Preload d . . . . . . . . . . . . . . . 3.2.4 3.2.4 Eigen Eigenva value luess . . . . . . . . . . . . . . . . . 3.2.5 3.2.5 Free Free Vibrati ibrations ons . . . . . . . . . . . . . . . Sky Sky Hook Hook Dampe Damperr . . . . . . . . . . . . . . . . . 3.3.1 3.3.1 Modell Modelling ing Aspect Aspectss . . . . . . . . . . . . . 3.3.2 3.3.2 System System Perfo Performa rmanc ncee . . . . . . . . . . . . Nonlinea Nonlinearr Force Force Elements Elements . . . . . . . . . . . . . 3.4.1 3.4.1 Quarte Quarterr Car Car Model Model . . . . . . . . . . . . . 3.4.2 3.4.2 Random Random Road Road Profile Profile . . . . . . . . . . . . 3.4.3 3.4.3 Vehicl ehiclee Data Data . . . . . . . . . . . . . . . . 3.4. 3.4.4 4 Meri Meritt Func Functio tion n . . . . . . . . . . . . . . . 3.4.5 3.4.5 Optima Optimall Param Paramete eterr . . . . . . . . . . . . . 3.4.5. 3.4.5.1 1 Linea Linearr Charac Character terist istics ics . . . . . . 3.4.5.2 3.4.5.2 Nonlinea Nonlinearr Characte Characteristi ristics cs . . . . 3.4.5. 3.4.5.3 3 Limite Limited d Spring Spring Trav Travel el . . . . . . Dynam Dynamic ic Force Force Eleme Elements nts . . . . . . . . . . . . . . 3.5.1 3.5.1 System System Response Response in the Frequenc Frequency y Domai Domain n 3.5.1. 3.5.1.1 1 First First Harmo Harmonic nic Oscilla Oscillatio tion n . . . 3.5.1. 3.5.1.2 2 Sweep Sweep-Si -Sine ne Excita Excitatio tion n. . . . . . 3.5. 3.5.2 2 Hydr Hydroo-Mo Moun untt . . . . . . . . . . . . . . . . 3.5.2. 3.5.2.1 1 Princi Principle ple and Model Model . . . . . . . 3.5.2.2 3.5.2.2 Dynamic Dynamic Force Force Characte Characteristic risticss .

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4 Longit Longitudi udinal nal Dynami Dynamics cs 4.1

II

15 15 17 17 18 20 21 24 25 27 29

Dynam Dynamic ic Wheel Wheel Loads Loads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.1 4.1.1 Simple Simple Vehicle ehicle Model Model . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.2 4.1.2 Influen Influence ce of Grade Grade . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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51 51 51 52

4.2

4.3

4.4

4.1.3 4.1.3 Aerody Aerodynam namic ic Force Forcess . . . . . . . . . . . . . . . Maxim Maximum um Accel Accelera eratio tion n . . . . . . . . . . . . . . . . . . 4.2.1 4.2.1 Tilti Tilting ng Limits Limits . . . . . . . . . . . . . . . . . . . 4.2.2 4.2.2 Fricti Friction on Limits Limits . . . . . . . . . . . . . . . . . . Drivin Driving g and and Brakin Braking g . . . . . . . . . . . . . . . . . . . 4.3.1 4.3.1 Single Single Axle Axle Drive Drive . . . . . . . . . . . . . . . . . 4.3.2 4.3.2 Brakin Braking g at Single Single Axle Axle . . . . . . . . . . . . . . 4.3.3 4.3.3 Optimal Optimal Distrib Distribution ution of Driv Drivee and and Brake Brake Forces Forces . 4.3.4 4.3.4 Differen Differentt Distrib Distributio utions ns of Brake Brake Forces Forces . . . . . 4.3.5 4.3.5 Anti-L Anti-Loc ock-S k-Syst ystems ems . . . . . . . . . . . . . . . . Drive Drive and and Brake Brake Pitch Pitch . . . . . . . . . . . . . . . . . . . 4.4.1 4.4.1 Vehicl ehiclee Model Model . . . . . . . . . . . . . . . . . . 4.4.2 4.4.2 Equati Equation onss of Motion Motion . . . . . . . . . . . . . . . 4.4. 4.4.3 3 Equi Equili libr briu ium m . . . . . . . . . . . . . . . . . . . . 4.4.4 4.4.4 Drivin Driving g and and Brakin Braking g . . . . . . . . . . . . . . . 4.4.5 4.4.5 Brake Brake Pitch Pitch Pole Pole . . . . . . . . . . . . . . . . .

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5 Latera Laterall Dynam Dynamics ics 5.1

5.2

5.3

Kinema Kinematic tic Appro Approac ach h . . . . . . . . . . . . . . . 5.1.1 5.1.1 Kinema Kinematic tic Tire Tire Model Model . . . . . . . . . . 5.1.2 5.1.2 Ackerm Ackermann ann Geome Geometry try . . . . . . . . . . 5.1.3 5.1.3 Space Space Requir Requireme ement nt . . . . . . . . . . . . 5.1.4 5.1.4 Vehicle ehicle Model Model with Trailer Trailer . . . . . . . . 5.1. 5.1.4. 4.1 1 Posi Positi tion on . . . . . . . . . . . . 5.1. 5.1.4. 4.2 2 Vehic ehicle le . . . . . . . . . . . . . 5.1.4. 5.1.4.3 3 Enter Entering ing a Curve Curve . . . . . . . . 5.1. 5.1.4. 4.4 4 Traile railerr . . . . . . . . . . . . . 5.1.4. 5.1.4.5 5 Cours Coursee Calcu Calculat lation ionss . . . . . . Steady Steady State State Corne Cornerin ring g . . . . . . . . . . . . . . 5.2.1 5.2.1 Corner Cornering ing Resist Resistan ance ce . . . . . . . . . . . 5.2.2 5.2.2 Overt Overturn urning ing Limit Limit . . . . . . . . . . . . 5.2.3 5.2.3 Roll Support Support and Camber Camber Compensa Compensation tion 5.2.4 5.2.4 Roll Roll Center Center and Roll Roll Axis Axis . . . . . . . . 5.2. 5.2.5 5 Whee Wheell Load Loadss . . . . . . . . . . . . . . . Simple Simple Handli Handling ng Model Model . . . . . . . . . . . . . . 5.3.1 5.3.1 Modell Modelling ing Concep Conceptt . . . . . . . . . . . . 5.3. 5.3.2 2 Kine Kinema matic ticss . . . . . . . . . . . . . . . . 5.3. 5.3.3 3 Tire ire Forc Forces es . . . . . . . . . . . . . . . . 5.3. 5.3.4 4 Late Latera rall Slip Slipss . . . . . . . . . . . . . . . 5.3.5 5.3.5 Equati Equation onss of Motion Motion . . . . . . . . . . . 5.3. 5.3.6 6 Stab Stabili ility ty . . . . . . . . . . . . . . . . . . 5.3.6. 5.3.6.1 1 Eigen Eigenva value luess . . . . . . . . . . 5.3.6. 5.3.6.2 2 Low Low Speed Speed Approx Approxima imatio tion n . . 5.3.6. 5.3.6.3 3 High High Speed Speed Appro Approxim ximati ation on . .

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III

5.3.7 5.3.7

5.3.8 5.3.8

Steady Steady State State Soluti Solution on . . . . . . . . . . . . . . . 5.3.7.1 5.3.7.1 Side Slip Angle Angle and Yaw Velocity elocity . . . 5.3.7. 5.3.7.2 2 Steeri Steering ng Tendenc endency y . . . . . . . . . . . 5.3. 5.3.7. 7.3 3 Slip Slip Angl Angles es . . . . . . . . . . . . . . Influence Influence of Whee Wheell Load Load on Cornering Cornering Stiffnes Stiffnesss .

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6 Driving Driving Behav Behavior ior of of Single Single Vehic Vehicles les 6.1

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6.3 6.4

IV

Standard Standard Driving Driving Maneuve Maneuvers rs . . . . . . . . . . . 6.1.1 6.1.1 Steady Steady State State Corner Cornering ing . . . . . . . . . . 6.1.2 6.1.2 Step Step Steer Steer Input Input . . . . . . . . . . . . . . 6.1.3 6.1.3 Drivin Driving g Straig Straight ht Ahead Ahead . . . . . . . . . . 6.1.3. 6.1.3.1 1 Rando Random m Road Road Profile Profile . . . . . 6.1.3. 6.1.3.2 2 Steeri Steering ng Activ Activity ity . . . . . . . . Coach Coach with differen differentt Loadi Loading ng Condition Conditionss . . . . 6.2.1 2.1 Data Data . . . . . . . . . . . . . . . . . . . . 6.2.2 6.2.2 Roll Roll Steer Steer Behav Behavior ior . . . . . . . . . . . . 6.2.3 6.2.3 Steady Steady State State Corner Cornering ing . . . . . . . . . . 6.2.4 6.2.4 Step Step Steer Steer Input Input . . . . . . . . . . . . . . Differen Differentt Rear Rear Axle Axle Concept Conceptss for a Passen Passenger ger Car Car Differen Differentt Influen Influences ces on Comfort Comfort and and Safety Safety . . . 6.4.1 6.4.1 Vehicl ehiclee Model Model . . . . . . . . . . . . . . 6.4.2 6.4.2 Simula Simulatio tion n Resul Results ts . . . . . . . . . . . .

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1 Intr Introducti oduction on 1.1 Terminology erminology 1.1.1 Vehicle ehicle Dynamics Dynamics The Expression ’Vehicle Dynamics’ encompasses the interaction of • driver driver,, • vehicle vehicle • load load and and • environment environment Vehicle dynamics mainly deals with • the improvement improvement of active safety and and driving comfort as well as • the reduction reduction of road destruc destruction. tion. In vehicle dynamics • computer calculations • test rig measu measureme rements nts and • field field tests tests are employed. employed. The intera interacti ction onss betwe between en the single single syste systems ms and the proble problems ms with with comput computer er calcu calculat lation ionss and/or and/or measurements shall be discussed in the following.

1

Vehicle Dynamics

FH Regensburg, University of Applied Sciences

1.1.2 1.1.2 Dri Driver ver By various means of interference the driver can interfere with the vehicle:

    

driver

steering wheel gas pedal brake pedal clutch gear shift

 

lateral dynamics longitudinal dynamics

The vehicle provides the driver with some information:

vehicle

  −→    −→

vibrations vibrations:: longitudi longitudinal, nal, lateral, lateral, vertical vertical soun sound: d: moto motorr, aero aerody dyna nami mics cs,, tire tiress instruments: velocity, velocity, external temperature, ...

The environment also influences the driver: environment

 

climate traffic density track

  −→

vehicle

driver

driver

A driver’s reaction is very complex. To achieve objective results, an ”ideal” driver is used in computer simulations and in driving experiments automated drivers (e.g. steering machines) are employed. employed. Transferring results to normal drivers is often difficult, if field tests are made with test drivers. Field tests with normal drivers have to be evaluated evaluated statistically. In all tests, the driver’s security must have absolute priority. Driving simulators provide an excellent means of analyzing the behavior of drivers even in limit situations without danger. danger. For some years it has been tried to analyze the interaction between driver and vehicle with complex driver models.

1.1.3 Vehicle ehicle The following vehicles are listed in the ISO 3833 directive: • Motorcyc Motorcycles, les, • Passen Passenger ger Cars, Cars, • Busses, Busses, • Trucks Trucks

2


© Prof. Dr.-Ing. G. Rill

• Agricultural Tractors, Tractors, • Passenger Passenger Cars with Trailer Trailer • Truck Trailer / Semitrailer, Semitrailer, • Road Trains Trains.. For computer calculations these vehicles have to be depicted in mathematically describable substitute systems. The generation of the equations of motions and the numeric solution as well as the acquisition of data require great expenses. In times of PCs and workstations computing costs hardly matter anymore. At an early stage of development often only prototypes are available for field and/or laboratory tests. Results can be falsified by safety devices, e.g. jockey wheels on trucks.

1.1.4 1.1.4 Load Load Trucks are conceived for taking up load. Thus their driving behavior changes. Load

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mass, inertia, center of gravity dynamic behaviour (liquid load)

In computer calculations problems occur with the determination of the inertias and the modelling of liquid loads. Even the loading and unloading process of experimental vehicles takes some effort. When making experiments with tank trucks, flammable liquids have to be substituted with water. The results thus achieved cannot be simply transferred to real loads.

1.1.5 Enviro Environment nment The Environment influences primarily the vehicle: Environment

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Road: Road: irregulari irregularities, ties, coefficie coefficient nt of frictio friction n Air: Air: resi resist stan ance ce,, cros crosss wind wind

 −→

vehicle

but also influences the driver Environment



climate visibility

 −→

driver

Through the interactions between vehicle and road, roads can quickly be destroyed. The greatest problem in field test and laboratory experiments is the virtual impossibility of reproducing environmental environmental influences. influences. The main problems in computer simulation are the description of random road r oad irregularities and the interaction of tires and road as well as the calculation of aerodynamic forces and torques.

3

Vehicle Dynamics


1.2 Wheel/Axle Wheel/Axle Suspensi Suspension on Systems Systems 1.2.1 General Remarks The Automotive Industry uses different kinds of wheel/axle suspension systems. Important criteria are costs, space requirements, kinematic properties and compliance attributes.

1.2.2 Multi Purpose Purpose Suspension Suspension Systems Systems The Double Wishbone Suspension, the McPherson Suspension and the Multi-Link Suspension are multi purpose wheel suspension systems, Fig. 1.1. 1.1. E

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yR

P

Q

S

δS

O1

G

zR

F

O2

N3

O

xB

yB M

A

Figure 1.1: Double Wishbone, McPherson and Multi-Link Suspension They are used as steered front or non steered rear axle suspension systems. These suspension 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. zA zA

Z2

Y2 Z1 Y1

X2

xA

xA X1

yA

yA

Figure 1.2: Solid Axles At pickups, trucks and busses often solid axles are used. Solid axles are guided either by leaf springs or by rigid links, Fig. 1.2. Solid axles tend to tramp on rough road.

4



Leaf spring guided solid axle suspension systems are very robust. Dry friction between the leafs leads to locking effects in the suspension. Although the leaf springs provide axle guidance on some solid axle suspension systems additional links in longitudinal and lateral direction are used. Thus the typical wind up effect on braking can be avoided. Solid axles suspended by air springs need at least four links for guidance. In addition to a good driving comfort air springs allow level control too.

1.2.3 Specific Specific Suspension Suspension Systems Systems The Semi-Trailing Arm, the SLA and the Twist Beam axle suspension are suitable only for non steered axles, Fig. 1.3. zR

zA

yR

yA

xR

xA ϕ

Figure 1.3: Specific Wheel/Axles Suspension Systems The semi-trailing arm is a simple and cheap design which requires only few space. It is mostly used for driven rear axles. The SLA axle design allows a nearly independent layout of longitudinal and lateral axle motions. It is similar to the Central Control Arm axle suspension, where the trailing arm is completely rigid and hence only two lateral links are needed. The twist beam axle suspension exhibits either a trailing arm or a semi-trailing arm characteristic. It is used for non driven rear axles only. The twist beam axle provides enough space for spare tire and fuel tank.

1.3 Steering Steering Systems Systems 1.3.1 Requirements Requirements The steering system must guarantee easy and safe steering of the vehicle. The entirety of the mechanical transmission devices must be able to cope with all loads and stresses occurring in operation. In order to achieve a good maneuverability a maximum steer angle of approx. 30◦ must be provided at the front wheels of passenger cars. Depending on the wheel base busses and trucks need maximum steer angles up to 55◦ at the front wheels.

5

Vehicle Dynamics


Recently some companies have started investigations on ’steer by wire’ techniques.

1.3.2 Rack and and Pinion Pinion Steering Steering Rack and pinion is the most common steering system on passenger cars, Fig. 1.4. 1.4. The rack may be located either in front of or behind the axle. The rotations of the steering wheel δL are firstly

wheel and wheel body

Q

in k a g l i n d r a

uZ

P

δL

pinion

rack

steer box

δ1

δ2

L

Figure 1.4: Rack and Pinion Steering transformed by the steering box to the rack travel uZ = uZ (δL ) and then via the drag links transmitted to the wheel rotations δ1 = δ1 (uZ ), δ2 = δ2 (uZ ). Hence the overall steering ratio depends on the ratio of the steer box and on the kinematics of the steer linkage.

1.3.3 Lever Arm Arm Steering Steering System System δG

Q1

s t e e e r l e ev e r 1

drag link 1 P1 δ1

steer box 2 v e r e e l e e r s t e

P2

Q2

dr ag link 2

L

δ2

wheel and wheel body Figure 1.5: Lever Arm Steering System Using a lever arm steering system Fig. 1.5, 1.5, large steer angles at the wheels are possible. This steering system is used on trucks with large wheel bases and independent wheel suspension at the front axle. Here the steering box can be placed outside of the axle center.

6



The rotations of the steering wheel δL are firstly transformed by the steering box to the rotation of the steer levers δG = δG (δL ). The drag links transmit this rotation to the wheel δ1 = δ1 (δG ), δ2 = δ2 (δG ). Hence, again the overall steering ratio depends on the ratio of the steer box and on the kinematics of the steer linkage.

1.3.4 Drag Link Link Steering Steering System System At solid axles the drag link steering system is used, Fig. 1.6. e v e r e e r l e s t e δH

O

H


steer box

(90o rotated)

steer link

I L δ1

δ2

K

drag link Figure 1.6: Drag Link Steering System

The rotations of the steering wheel δL are transformed by the steering box to the rotation of the steer lever arm δH = δH (δL ) and further on to the rotation of the left wheel, δ1 = δ1 (δH ). The drag link transmits the rotation of the left wheel to the right wheel, δ2 = δ2 (δ1 ). The steering ratio is defined by the ratio of the steer box and the kinematics of the steer link. Here the ratio δ2 = δ2 (δ1 ) given by the kinematics of the drag link can be changed separately.

1.3.5 1.3.5 Bus Steer Steer System System In busse ussess the the driv driver er sits sits more more than than 2 m in front front of the the fron frontt axle axle.. Here Here,, soph sophis isti tica cate ted d stee steerr syst system emss are needed, Fig. 1.7. The rotations of the steering wheel δL are transformed by the steering box to the rotation of the steer lever arm δH = δH (δL ). Via the steer link the left lever arm is moved, δH = δH (δG ). This motion is transferred by a coupling link to the right lever arm. Via the drag links the left and right wheel are rotated, δ1 = δ1 (δH ) and δ2 = δ2 (δH ).

7

Vehicle Dynamics

FH Regensburg, University of Applied Sciences s t t e ee r e r l e ev e v er r δG

H steer box steer link l e ef f t t l e ev ve er r a r rm m

I

J K

Q

d r ra g l i in k

δH

P δ1

coupl. link

L

δ2


Figure 1.7: Bus Steer System

1.4 Definit Definition ions s 1.4.1 Coordinate Coordinate Systems In vehicle dynamics several different coordinate systems are used, Fig 1.8. The inertial system z0 x0 zF xF

ex

y0

yF

eyR

en ey

Figure 1.8: Coordinate Systems with the axes x0 , y0 , z0 is fixed to the track. Within the vehicle fixed system the xF -axis is

8



pointing forward, the yF -axis left and the zF -axis upward. The orientation of the wheel is given by the unit vector eyR in direction of the wheel rotation axis. The unit vectors in the directions of circumferential and lateral forces ex and ey as well as the track normal en follow from the contact geometry.

1.4.2 Toe and Camber Camber Angle 1.4.2.1 1.4.2.1 Definition Definitions s according according to DIN 70 000 The angle between the vehicle center plane in longitudinal direction and the intersection line of the tire center plane with the track plane is named toe angle. It is positive, if the front part of the δ

δ

front xF

yF

left

right rear

Figure 1.9: Positive Toe Angle wheel is oriented towards the vehicle center plane, Fig. 1.9. The The camb camber er angl anglee is the the angl anglee betw betwee een n the the whee wheell cent center er plan planee and and the the trac track k norm normal al.. It is posi positi tive ve,, γ

γ

top zF

yF

left

right bottom

Figure 1.10: Positive Camber Angle if the upper part of the wheel is inclined outwards, Fig. 1.10. 1.10.

1.4.2.2 Calculation The calculation of the toe angle is done for the left wheel. The unit vector eyR in direction of the wheel rotation axis is described in the vehicle fixed coordinate system F , Fig. 1.11

eyR,F =



(1) eyR,F

(2) eyR,F

(3) eyR,F



T

,

(1.1)

9

Vehicle Dynamics


eyR

zF yF (2)

e yR,F

xF

δV

(3) e yR,F (1) e yR,F

Figure 1.11: Toe Angle where the axis xF and zF span the vehicle center plane. The xF -axis points forward and the zF -axis points upward. The toe angle δ can then be calculated from (1)

tan δ =

eyR,F (2)

eyR,F

.

(1.2)

The The real real camb camber er angl anglee γ follo follows ws from from the the scal scalar ar prod produc uctt betw betwee een n the the unit unit vect vector orss in the the dire direct ctio ion n of the wheel rotation axis eyR and in the direction of the track normal en ,

sin γ =

T n eyR

.

(1.3)

(3) yR,F

.

(1.4)

−e

The wheel camber angle can be calculated by

sin γ =

−e

On a flat horizontal road both definitions are equal.

1.4.3 Steering Steering Geometry Geometry 1.4.3.1 1.4.3.1 Kingpin Kingpin At the steered front axle the McPherson-damper strut axis, the double wishbone axis and multilink wheel suspension or dissolved double wishbone axis are frequently employed in passenger cars, Fig. 1.12 and Fig. 1.13. The wheel body rotates around the kingpin at steering movements. At the double wishbone axis, the ball joints A and B , which determine the kingpin, are fixed to the wheel body. The ball joint point A is also fixed to the wheel body at the classic McPherson wheel wheel suspension, but the point B is fixed to the vehicle body. At a multi-link axle, the kingpin is no longer defined by real link points. Here, as well as with the McPherson wheel suspension, the kingpin changes its position against the wheel body at wheel travel and steer motions.

10



zR B yR M

xR

A

kingpin axis A-B

Figure 1.12: Double Wishbone Wheel Suspension B zR

zR

yR

yR xR

M

xR

M A

kingpin axis A-B

rotation axis

Figure 1.13: McPherson and Multi-Link Wheel Suspensions

1.4.3.2 1.4.3.2 Caster Caster and Kingpin Angle Angle The current direction of the kingpin can be defined by two angles within the vehicle fixed coordinate system, Fig. 1.14. 1.14. If the kingpin is projected into the yF -, zF -plane, the kingpin inclination angle σ can be read as the angle between the zF -axis and the projection of the kingpin. The projection of the kingpin into the xF -, zF -plane delivers the caster angle ν with the angle between the zF -axis and the projection of the kingpin. With many axles the kingpin and caster angle can no longer be determined directly. The current rotation axis at steering movements, that can be taken from kinematic calculations here delivers a virtual kingpin. The current values of the caster angle ν and the kingpin inclination angle σ can be calculated from the components of the unit vector in

11

Vehicle Dynamics

FH Regensburg, University of Applied Sciences zF

eS

zF

σ

ν

yF xF

Figure 1.14: Kingpin and Caster Angle the direction of the kingpin, described in the vehicle fixed coordinate system

tan ν =

(1) S,F (3) eS,F

−e

and

tan σ =

(2) S,F (3) eS,F

−e

with

eS,F =



(1) eS,F

(2) eS,F

(3) eS,F



T

.

(1.5)

1.4.3.3 Disturbing Force Force Lever, Lever, Caster and Kingpin Offset The distance d between the wheel center and the king pin axis is called disturbing force lever. It is an important quantity in evaluating the overall steer behavior. In general, the point S where

C ey

P

d ex S

rS

nK

Figure 1.15: Caster and Kingpin Offset the kingpin runs through the track plane does not coincide with the contact point P , Fig. 1.15. 1.15. If the kingpin penetrates the track plane before the contact point, the kinematic kingpin offset is positive, positive, nK > 0. The caster offset is positive, rS > 0, if the contact point P lies outwards of S .

12

2 The Tire Tire 2.1 Introduct Introduction ion 2.1.1 Tire Developmen Developmentt The following table shows some important mile stones in the development of tires. 1839

Charles Charles Goodyear Goodyear:: vulcaniz vulcanization ation

1845

Robert Robert William William Thompson Thompson:: first pneumati pneumaticc tire (several thin inflated tubes inside a leather cover)

1888

John John Boyd Boyd Dunlop: Dunlop: patent patent for bicycle bicycle (pneu (pneumatic matic)) tires

1893

The Dunlop Dunlop Pneuma Pneumatic tic and Tyre Tyre Co. GmbH, GmbH, Hanau, Hanau, German Germany y

1895

André André and Edoua Edouard rd Michelin: Michelin: pneum pneumatic atic tires tires for Peugeo Peugeott Paris-Bordeaux-Paris Paris-Bordeaux-Paris (720 Miles): 50 tire deflations, 22 complete inner tube changes

1899

Continen Continental: tal: longe longerr life tires tires (approx (approx.. 500 Kilome Kilometer) ter)

1904

Carbon Carbon added: added: black black tires. tires.

1908

Frank Frank Seiberling Seiberling:: grooved grooved tires tires with improv improved ed road tractio traction n

1922

Dunlop: Dunlop: steel steel cord cord thread thread in the the tire tire bead bead

1943

Continen Continental: tal: patent patent for tubeless tubeless tires

1946 1946 .. .

Radia Radiall Tire

Table 2.1: Mile Stones in the Development of Tires

2.1.2 Tire Composites Composites A modern tire is a mixture of steel, fabric, and rubber.

13

Vehicle Dynamics


Reinfo Reinforce rcemen ments: ts: steel, steel, rayon rayon,, nylon nylon

16% 16%

Rubber: natural/synthetic

38%

Compou Compounds nds:: carbo carbon, n, silica silica,, chalk, chalk, ...

30% 30%

Softener: oil, resin

10%

Vulca ulcani niza zati tion on:: sulf sulfur ur,, zinc zinc oxid oxide, e, ... ...

4%

Miscellaneous

2%

Tire Mass

8.5 kg

Table 2.2: Tire Composites: 195/65 R 15 ContiEcoContact, Data from www.felge.de

2.1.3 Forces Forces and Torq Torques ues in the Tire Contact Area In any point of contact between tire and track normal and friction forces are delivered. According to the tire’s profile design the contact area forms a not necessarily coherent area. The effect of the contact forces can be fully described by a vector of force and a torque in reference to a point in the contact patch. The vectors are described in a track-fixed coordinate coordinate system. The z -axis is normal to the track, the x-axis is perpendicular to the z -axis and perpendicular to the wheel rotation axis eyR . The demand for a right-handed coordinate system then also fixes the y-axis.

F x F y F z

longitudinal or circumferential force lateral force vertical force or wheel load

M x M y M z

tilting torque rolling resistance torque self aligning and bore torque

Fy Mx Fx

Fz

Mz

My

Figure 2.1: Contact Forces and Torques The components of the contact force are named according to the direction of the axes, Fig. 2.1. 2.1. Non symmetric distributions of force in the contact patch cause torques around the x and y axes. The tilting torque M x occurs when the tire is cambered. M y also contains the rolling resistance of the tire. In particular the torque around the z -axis is relevant in vehicle dynamics. It consists of two parts, M z = M B + M S (2.1) S . Rotation of the tire around the z -axis causes the bore torque M B . The self aligning torque M S S respects the fact that in general the resulting lateral force is not applied in the center of the contact patch.

14



2.2 Contact Contact Geometry Geometry 2.2.1 Contact Contact Point Point The current position of a wheel in relation to the fixed x0 -, y0 - z0 -system is given by the wheel center M and the unit vector eyR in the direction of the wheel rotation axis, Fig. 2.2. 2.2. γ

rim centre plane

tire

ezR M

M

e yR

e yR

en

rS ex

en

P0

P0 x0

P*

y0

road: z = z ( x , y )

z0

ey

b

a

P

local road plane

0

Figure 2.2: Contact Geometry The irregularities of the track can be described by an arbitrary function of two spatial coordinates z = z (x, y ). (2.2) At an uneven track the contact point P can not be calculated directly. One can firstly get an estimated value with the vector rM P = r0 ezB , (2.3) ∗

−

where r0 is the undeformed tire radius and ezB is the unit vector in the z -direction of the body fixed reference frame. The position of P ∗ with respect to the fixed system x0 , y0 , z0 is determined by

r0P = r0M + rM P , ∗

(2.4)

∗

where the vector r0M states states the position position of the rim center M . Usually the point P ∗ lies not on the track. The corresponding track point P 0 follows from

r0P ,0 = 0

   z

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

∗

∗

∗

  

.

(2.5)

15

Vehicle Dynamics


In the point P 0 now the track normal en is calculated. Then the unit vectors in the tire’s circumferential direction and lateral direction can be calculated

ex =

eyR en , eyR en

× | × |

ey = en ex .

×

and

(2.6)

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

 

T γ = arcsi arcsin n eyR en

(2.7)

describes the inclination of the wheel rotation axis against the track normal. The vector from the rim center M to the track point P 0 is now split into three parts

rM P = 0

−r

S ezR

+ a ex + b ey ,

(2.8)

where rS names names the loaded loaded or static tire radius radius and a, b are displacements in circumferential and lateral direction. The unit vector

ex eyR . (2.9) ex eyR is perpendicular to ex and eyR . Because the unit vectors ex and ey are perpendicular to en , the scalar multiplication of (2.8 (2.8)) with en results in ezR =

enT rM P 0

=

−

× | × |

rS enT ezR

rS =

or

−

enT rM P . enT ezR 0

(2.10)

Now also the tire deflection can be calculated

r

= r0

−r

S ,

(2.11)

S ezR

(2.12)

with r0 marking the undeformed tire radius. The point P given by the vector

rM P =

−r

lies within the rim center plane. The transition from P 0 to P takes place according to (2.8 ( 2.8)) by terms a ex and b ey , standing perpendicular to the track normal. The track normal however was calculated in the point P 0 . Therefore with an uneven track P no longer lies on the track. With the newly estimated value P ∗ = P now the equations (2.5 ( 2.5)) to (2.12 (2.12)) can be recurred until the difference between P and P 0 is sufficiently small. Tire models which can be simulated within acceptable time assume that the contact patch is even. At an ordinary passenger-car tire, the contact patch has at normal load about the size of approximately 20 20 cm. There is obviously little sense in calculating a fictitious contact point to fractions of millimeters, when later the real track is approximated in the range of centimeters by a plane.

×

If the track in the contact patch is replaced by a plane, no further iterative improvement is necessary at the hereby used initial value.

16



2.2.2 Local Track Track Plane Plane A plane is given by three points. With the tire width b, the undeformed tire radius r0 and the length of the contact area LN at given wheel load, estimated values for three track points can be given in analogy to (2.4 ( 2.4))

rM L

=

rM R

= =

∗

∗

rM F

∗

LN 2

exB

−

b 2 b 2

eyR eyR

−r −r −r

0 ezB

,

0 ezB

,

0 ezB

.

(2.13)

The points lie left, resp. right and to the front of a point below the rim center. The unit vectors exB and ezB point in the longitudinal and vertical direction of the vehicle. The wheel rotation axis is given by eyR . According to (2.5 ( 2.5)) the corresponding points on the track L, R and F can be calculated. The vectors

rRF = r0F

−r

and

0R

rRL = r0L

−r

(2.14)

0R

lie within the track plane. The unit vector calculated by

en =

|

rRF rRL . rRF rRL

× ×

(2.15)

|

is perpendicular to the plane defined by the points L, R, and F and gives an average track normal over the contact area. Discontinuities which occur at step- or ramp-sized obstacles are smoothed that way. Of course it would be obvious to replace LN in (2.13) 2.13) by the actual length L of the contact area and the unit vector ezB by the unit vector ezR which points upwards in the wheel center plane. The values however, can only be calculated from the current track normal. Here also an iterative solution would be possible. Despite higher computing effort the model quality cannot be improved by this, because approximations in the contact calculation and in the tire model limit the exactness of the tire model.

2.3 Wheel Wheel Load Load The vertical tire force F z can be calculated as a function of the normal tire deflection enT r and the deflection velocity z˙ = enT r˙

z =

F z = F z ( z,

(2.16)







 z˙ ) .

Because the tire can only deliver pressure forces to the road, the restriction F z In a first approximation F z is separated into a static and a dynamic part

F z = F zS + F zD .

≥ 0 holds. (2.17)

17

Vehicle Dynamics


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

F zS = c0

z + κ (z)

2

.

(2.18)

The constants c0 and κ may be calculated from the radial stiffness at nominal payload and at double the payload. Results for a passenger car and a truck tire are shown in Fig. 2.3. 2.3. The parabolic approximation Eq. (2.18 (2.18)) fits very well to the measurements. Passenger Car Tire: 205/50 R15

10

Truck Tire: X31580 R22.5

100 80

8 ] N6 k [

] N k [

60

F 4

F

40

z

z

20

2 0

0

10

20 30 ∆z [mm]

40

0

50

0

20

40 ∆z

Figure 2.3: Tire Radial Stiffness:

60

80

[mm]

◦ Measurements, Measurements, — Approximation

200 N can be specified The radial tire stiffness of the passenger car tire at the payload of F z = 3 200 190 000 000N/m. The Payload F z = 35000 N and the stiffness c0 = 1250000N/m 1250000N/m of a with c0 = 190 truck tire are significantly larger. The dynamic part is roughly approximated by

F zD = dR

z˙ ,

(2.19)

where dR is a constant describing the radial tire damping.

2.3.1 Dynamic Dynamic Rolling Rolling Radius Radius At an angular rotation of ϕ, assuming assuming the tread particles particles stick to the track, the deflected deflected tire moves on a distance of x, Fig. 2.4. 2.4.



With r0 as unloaded and rS = r0

− r as loaded or static tire radius r sin ϕ = x 0

(2.20)

and

r0 cos hold.

ϕ = r

S .

(2.21)

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

r0

18

 sin ϕ = x = r ϕ . D

(2.22)



deflected tire

rigid wheel

Ω

Ω

r0 r S

rD

∆ϕ

vt

∆ϕ

x

x Figure 2.4: Dynamic Rolling Radius

Hence, the dynamic tire radius is given by

r0 sin ϕ . ϕ

rD = For

ϕ → 0 one gets the trivial solution r

D





(2.23)

= r0 .

At small, yet finite angular rotations the sine-function can be approximated by the first terms of its Taylor-Expansion. aylor-Expansion. Then, (2.23) 2.23) reads as

rD = r0

1 6

ϕ − ϕ ϕ

3

= r0

−  1 ϕ2 6

1

.

(2.24)

With the according approximation for the cosine-function

rS = cos r0

1 ϕ2 2

ϕ = 1 − 

one finally gets

rD = r0

or

 −  −  1

1 1 3

rS r0

2

ϕ =

= 2

−  1

rS r0

2 1 r0 + rS 3 3

(2.25)

(2.26)

remains. The radius rD depends on the wheel load F z because of rS = rS (F z ) and thus is named dynamic tire radius. With this first approximation it can be calculated from the undeformed radius r0 and the steady state radius rS . By

vt = rD Ω

(2.27)

the average velocity is given with which tread particles are transported through the contact area.

19

Vehicle Dynamics


2.3.2 Contact Contact Point Point Velocity Velocity The absolute velocity of the contact point one gets from the derivation of the position vector

v0P,0 P,0 = r˙0P,0 P,0 = r˙0M,0 M,0 + r˙M P,0 P,0 .

(2.28)

Here r˙0M,0 M,0 = v0M,0 M,0 is the absolute velocity of the wheel center and rM P,0 P,0 the vector from the wheel center M to the contact point P , expressed in the inertial frame 0. With (2.12 (2.12)) one gets

r˙M P,0 P,0 =

d ( rS ezR,0 zR, 0 ) = dt

−

−r˙

S ezR,0 zR, 0

−r

S e˙ zR,0 zR, 0

.

(2.29)

Due to r0 = const.

− r˙

S

=

r˙

(2.30)

follows from (2.11 (2.11). ). The unit vector ezR moves moves with the rim but does not perform rotations around the wheel rotation axis. Its time derivative is then given by ∗ e˙ zR,0 zR, 0 = ω0R,0 zR, 0 R,0 ezR,0

×

(2.31)

where ω0∗R is the angular velocity of the wheel rim without components in the direction of the wheel rotation axis. Now (2.29 ( 2.29)) reads as

r˙M P,0 P,0 =

r˙ e

zR,0 zR, 0

∗ S ω0R,0 R,0

−r

×e

(2.32)

ZR,0 ZR, 0

and the contact point velocity can be written as

v0P,0 P,0 = v0M,0 M,0 +

∗ S ω0R,0 R,0

r˙ e − r zR,0 zR, 0

×e

ZR,0 ZR, 0

.

(2.33)

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

enT v0P = 0 .

(2.34)

The tire deformation velocity is defined by this demand

r˙

=

T ∗ n (v0M + rS ω0R enT ezR

−e

×e

ZR )

.

(2.35)

Now, the contact point velocity v0P and its components in longitudinal and lateral direction

vx = exT v0P

(2.36)

vy = eyT v0P

(2.37)

and can be calculated.

20



2.4 Longitudin Longitudinal al Force Force and Longitudinal Longitudinal Slip To get some insight into the mechanism generating tire forces in longitudinal direction we consider a tire on a flat test rig. The rim is rotating with the angular speed Ω and the flat track runs with speed vx . The distance between the rim center an the flat track is controlled to the loaded tire radius corresponding to the wheel load F z , Fig. 2.5. 2.5. A tread particle enters at time t = 0 the contact area. If we assume adhesion between the particle and the track then the top of the particle runs with the track speed vx and the bottom with the average transport velocity vt = rD Ω. Depending on the speed difference v = rD Ω vx the tread particle is deflected in longitudinal direction



u = (rD Ω

−

−v )t.

(2.38)

x

rD Ω

vx Ω

rD u

vx L

u max

Figure 2.5: Tire on Flat Track Test Rig The time a particle spends in the contact area can be calculated by

T =

L , rD Ω

(2.39)

| |

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

| |

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

umax = (rD Ω

− v ) T = (r x

D

Ω

− v ) r L|Ω| . x

(2.40)

D

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

F xt = ctx u ,

(2.41)

21

Vehicle Dynamics


where ctx is the stiffness of one tread particle in longitudinal direction. On normal wheel loads more than one tread particle is in contact with the track, Fig. 2.6a. 2.6a. The number p of the tread particles can be estimated by

p =

L . s+a

(2.42)

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

b)

L

c)

L

r0 s

cxt *

a

∇

r

u

cut *

u max

L/2

Figure 2.6: a) Particles, b) Force Force Distribution, Distribution, c) Tire Deformation Deformation Particles entering the contact area are undeformed on exit the have the maximum deflection. According to (2.41 (2.41)) this results in a linear force distribution versus the contact length, Fig. 2.6b. 2.6b. For p particles the resulting force in longitudinal direction is given by

F x =

1 t p c umax . 2 x

(2.43)

With (2.42 (2.42)) and (2.40 ( 2.40)) this results in

F x =

1 L t c (rD Ω 2 s+a x

− v ) r L|Ω| . x

(2.44)

D

A first approximation of the contact length L is given by

(L/ L/2) 2)2 = r02

2

− (r − r) , (2.45) where r is the undeformed tire radius, and r denotes the tire deflection, Fig. 2.6c. 2.6c. With r  r one gets L ≈ 8 r r . (2.46) 0

0

0

2

0

The tire deflection can be approximated by

r

= F z /cR .

(2.47)

where F z is the wheel load, and cR denotes the radial tire stiffness. Now, (2.43 ( 2.43)) can be written as

r0 ctx rD Ω vx F x = 4 F z . s + a cR rD Ω

− | |

22

(2.48)



The non-dimensional relation between the sliding velocity of the tread particles in longitudinal direction vxS = vx rD Ω and the average transport velocity rD Ω forms the longitudinal slip

−

| |

sx =

−(v − r Ω) . r |Ω| x

D

(2.49)

D

In this first approximation the longitudinal force F x is proportional to the wheel load F z and the longitudinal slip sx F x = k F z sx , (2.50) where the constant k collects the tire properties r0 , s, a, ctx and cR . The relation (2.50 (2.50)) holds only as long as all particles stick to the track. At average slip values the particles at the end of the contact area start sliding, and at high slip values only the parts at the beginning of the contact area still stick to the road, Fig. . 2.7. 2.7. small slip values Fx = k * Fz* s x

moderate slip values Fx = Fz * f ( s x )

L

large slip values Fx = FG

L t

t

Fx <= FH adhesion

L t

Fx = FG

t t Fx = FH

adhesion

sliding

sliding

Figure 2.7: Longitudinal Force Distribution for different Slip Values The resulting nonlinear function of the longitudinal force F x versus the longitudinal slip sx can be defined by the parameters initial inclination (driving stiffness) dF x0 , location sM x and magnitude of the maximum F xM , start of full sliding sliding sxG and the sliding force F xG , Fig. 2.8. 2.8.

Fx M Fx G Fx

adhesion

sliding

dFx0

sM x

sGx

sx

Figure 2.8: Typical Longitudinal Force Characteristics

23

Vehicle Dynamics


2.5 Lateral Lateral Slip, Lateral Force Force and Self Aligning Torq Torque ue Similar to the longitudinal slip sx , given by (2.49 (2.49), ), the lateral slip can be defined by

sy =

S y

−v , r |Ω|

(2.51)

D

where the sliding velocity in lateral direction is given by

vyS = vy

(2.52)

and the lateral component of the contact point velocity vy follows from (2.37 (2.37). ). As long as the tread particles stick to the road (small amounts of slip), an almost linear distribution of the forces along the length L of the contact area appears. appears. At moderate moderate slip values values the particles at the end of the contact area start sliding, and at high slip values only the parts at the beginning of the contact area stick to the road, Fig. 2.9. 2.9. The nonlinea nonlinearr characteris characteristics tics

small slip values Fy = k * Fz * s y n o i s e h d a

n y

F

L

moderate slip values Fy = Fz * f ( s y ) n o i s e h d a

y

F

g n i d i l s

L

large slip values Fy = FG

g n i d i l s

y

F

L

Figure 2.9: Lateral Force Distribution over Contact Area of the lateral force versus the lateral slip can be described by the initial inclination (cornering M G stiffness) dF y0 , location sM y and magnitude F y of the maximum and start of full sliding sy and magnitude F yG of the sliding force. The distribution of the lateral forces over the contact area length also defines the acting point of the resulting lateral force. At small slip values the working point lies behind the center of the contact area (contact point P). With rising slip values, it moves forward, sometimes even before the center of the contact area. At extreme slip values, when practically all particles are sliding, the resulting force is applied at the center of the contact area. The resulting lateral force F y with the dynamic tire offset or pneumatic trail n as a lever generates the self aligning torque M S n F y . (2.53) S =

−

The lateral force F y as well as the dynamic tire offset are functions of the lateral slip sy . Typical plots of these quantities are shown in Fig. 2.10. Characteristic parameters for the lateral

24



n/L (n/L)0

adhesion adhesion/sliding

Fy

full sliding

adhesion/ adhesion sliding M Fy G Fy dFy0

sy0

full sliding

MS

sGy

sy

adhesion adhesion/sliding full sliding

sG y

sM y

sy

sy0

sGy

sy

Figure 2.10: Typical Plot of Lateral Force, Tire Offset and Self Aligning Torque force graph are initial inclination (cornering stiffness) dF y0, location sM y and magnitude of the M G G maximum F y , begin of full sliding sy , and the sliding force F y . The dynamic tire offset has been normalized by the length of the contact area L. The initial n/L)0 as well as the slip values sy0 and syG characterize the graph sufficiently. value (n/L) sufficiently.

2.6 Camber Camber Influence Influence At a cambered tire, Fig. 2.11, the angular velocity of the wheel Ω has a component normal to the road Ωn = Ω sin γ . (2.54) Now, the tread particles in the contact area possess a lateral velocity which depends on their position ξ and is given by

vγ (ξ ) =

−Ω

n

L ξ ,= 2 L/ L/22

L/22 ≤ ξ ≤ L/ L/22 . −Ω sin γ ξ , −L/

(2.55)

At the center of the contact area (contact point) it vanishes and at the end of the contact area it is of the same value but opposite to the value at the beginning of the contact area. Assuming that the tread particles stick to the track, the deflection profile is defined by

y˙γ (ξ ) = vγ (ξ ) .

(2.56)

The time derivative can be transformed to a space derivative

y˙ γ (ξ ) =

d yγ (ξ ) d ξ d yγ (ξ ) = rD Ω dξ dt dξ

| |

(2.57)

25

Vehicle Dynamics

en


γ

rim centre plane

eyR

Fy = Fy (s y ): Parameter γ

4000

γ

3000 2000

Ωn

Ω

1000 0 -1000

ex

rD |Ω|

vγ (ξ)

ey

yγ (ξ)

-2000 -3000

ξ

-4000 -0.5

0.5

0

Figure 2.11: Cambered Tire F y (γ ) at F z = 3.2 kN and γ = 0◦ , 2◦ , 4◦ , 6◦ , 8◦ where rD Ω denotes the average transport velocity. Now (2.56 ( 2.56)) reads as

| |

d yγ (ξ ) rD Ω = dξ

−Ω sin γ ξ ,

| |

(2.58)

which results in the parabolic deflection profile

1 Ω sin γ yγ (ξ ) = 2 rD Ω

| |

   −   2

L 2

ξ L/22 L/

1

2

.

(2.59)

Similar to the lateral slip sy which is by (2.51 (2.51)) we now can define a camber slip

sγ =

−Ω sin γ L . r |Ω| 2

(2.60)

D

The lateral deflection of the tread particles generates a lateral force

F yγ yγ =

−c

y

y¯γ ,

(2.61)

where cy denotes the lateral stiffness of the tread particles and L/2 L/2

1 L 1 y¯γ = ( sγ ) 2 2 L

−

  −    1

x L/22 L/

−L/2 L/2

is the average value of the parabolic deflection profile.

26

2

dξ =

− 16 s

γ L

(2.62)



A purely lateral tire movement without camber results in a linear deflexion profile with the average deflexion

y¯y =

− 12 s

y

L.

(2.63)

A comparison of (2.62 ( 2.62)) to (2.63 (2.63)) shows, that with

sγ y =

1 sγ 3

(2.64)

the lateral camber slip sγ can be converted to an equivalent lateral slip sγ y. In normal driving operation, the camber angle and thus the lateral camber slip are limited to small values. So the lateral camber force can be approximated by

F yγ

0 γ y y .

≈ dF s

(2.65)

If the “global” inclination dF y = F y /sy is used instead of the initial inclination dF y0 , one gets the camber influence on the lateral force as shown in Fig. 2.11. 2.11. The camber angle influences the distribution of pressure in the lateral direction of the contact area, and changes the shape of the contact area from rectangular to trapezoidal. It is thus extremely difficult if not impossible to quantify the camber influence with the aid of such simple models. But this approach turns out to be a quit good approximation.

2.7 Bore Torque orque If the angular velocity of the wheel

ω0W = ω0∗R + Ω eyR

(2.66)

has a component in direction of the track normal en

ωn = enT ω0W = 0 .



(2.67)

a very very comp compli lica cate ted d defle deflect ctio ion n profi profile le of the the trea tread d parti particl cles es in the the cont contac actt area area occu occurs rs.. By a simp simple le approach the resulting bore torque can be approximated by the parameter of the longitudinal force characteristics. characteristics.

0 , sy 0. Fig. 2.12 shows the contact area at zero camber, γ = 0 and small slip values, sx The contact area is separated into small stripes of width dy . The longitudinal slip in a stripe at position y is then given by

≈

sx ( y ) =

− (−ω y) . r |Ω| n

≈

(2.68)

D

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

d F x F x (y ) = d sx



sx =0

d sx y. dy

(2.69)

27

Vehicle Dynamics


dy

U(y) y

dy

x

x contact area

UG P

Q ω

P

L

n

L

y

-UG

ωn

contact area

B

B

Figure 2.12: Bore Torque generated by Longitudinal Forces With (2.68 (2.68)) one gets

d F x F x (y) = d sx



sx =0

ωn y. rD Ω

(2.70)

| |

The forces F x (y ) generate a bore torque in the contact point P +B 2

M B =

− B1



+B 2

y F x (y ) dy

=

− B1

−B 2

1 2 d F x = B 12 d sx where

   y

−B 2



−ω r |Ω| n

1 d F x B 12 d sx

=

D

sx =0

−ω |Ω|

n

sB =

d F x d sx

sx =0

sx =0

B rD

ωn y dy rD Ω

| |

−ω |Ω|

n

(2.71)

,

(2.72)

can be considered as bore slip. Via dF x /dsx the bore torque takes into account the actual friction and slip conditions. The bore torque calculated by (2.71 ( 2.71)) is only a first approximation. At large bore slips the longitudinal forces in the stripes are limited by the sliding values. Hence, the bore torque is limited by +B 2

max B

| M | ≤ M B

= 2

1 B

 0

where F xG denotes the longitudinal sliding force.

28

y F xG dy =

1 B F xG , 4

(2.73)



2.8 Typical ypical Tire Characteristic Characteristics s The tire model TMeasy 1 which is based on this simple approach can be used for passenger car tires as well as for truck tires. It approximates the characteristic curves F x = F x (sx ), F y = F y (α) and M z = M z (α) quite well even for different wheel loads F z , Fig. 2.13. 6

40

4 ] N k [ x

20

2

] N k [

0

F

x

-4

10 kN 20 kN 30 kN 40 kN 50 kN

F

1.8 kN 3.2 kN 4.6 kN 5.4 kN

-2

0

-20 -40

-6 -40

-20

0 s [%]

20

-40

40

-20

0 s [%]

20

40

x

x

6

40

4 ] N k [

y

20

2

] N k [

0

F

1.8 kN 3.2 kN 4.6 kN 6.0 kN

-2 -4

y

0

F

-20 -40

-6 150

1500

100

1000

50

] m N [

0

z

z

M

-50

1.8 kN 3.2 kN 4.6 kN 6.0 kN

-100 -150

500

] m N [

0

10 kN 20 kN 30 kN 40 kN

-20

-10

0 α

[o]

10

20

M

-500

18.4 kN 36.8 kN 55.2 kN

-1000 -1500

-20

-10

0 o α [ ]

Figure 2.13: Longitudinal Force, Lateral Force and Self Aligning Torque:

1

10

20

◦ Meas., − TMeasy

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

29

Vehicle Dynamics


Within Within TMeasy TMeasy the one-dime one-dimensio nsional nal characte characteristic risticss are automatic automatically ally conver converted ted to a twodimensional combination characteristics, Fig. 2.14. 2.14.

] N k [

y

3

30

2

20

1

] N k [

0

y

F

10 0

F

-1

-10

-2

-20

-3 -4

-2

0 F [kN]

2

4

x

10, 15%; |s | = 1, 2, 4, 6, 10, x

-30

-20

0 F [kN]

20

x

◦

10, 14 |α| = 1, 2, 4, 6, 10,

Figure 2.14: Two-dimensional wo-dimensional Tire Characteristics at F z = 3.2 kN / F z = 35 kN

30

3 Vertical ertical Dynami Dynamics cs 3.1 3. 1 Goal Goals s The aim of vertical dynamics is the tuning of body suspension and damping to guarantee good driving comfort, resp. a minimal stress of the load at sufficient safety. The stress of the load can be judged fairly well by maximal or integral values of the body accelerations. The wheel load F z is linked to the longitudinal F x and lateral force F y by the coefficient of friction. The digressive influence of F z on F x and F y as well as instationary processes at the increase of F x and F y in the average lead to lower longitudinal and lateral forces at wheel load variations. Maximal driving safety can therefore be achieved with minimal variations of wheel load. Small variations of wheel load also reduce the stress on the track. The comfort of a vehicle is subjectively judged by the driver. In literature, different approaches of describing the human sense of vibrations by different metrics can be found. Transferred to vehicle vertical dynamics, the driver primarily registers the amplitudes and accelerations of the body vibrations. These values are thus used as objective criteria in practice.

3.2 Basic Tuning 3.2.1 Simple Simple Models Models Fig. 3.1 shows simple quarter car models, that are suitable for basic investigations of body and axle vibrations. At normal vehicles the wheel mass m is in relation r elation to the respecti r espective ve body mass M much smaller m M . The coupling of wheel and body movement can thus be neglected for basic investigations.



In describing the vertical movements of the body, the wheel movements remain unrespected. If the wheel movements are in the foreground, then body movements can be neglected. The equations of motion for the models read as

M z¨B + dS z˙B + cS zB = dS z˙R + cS zR

(3.1)

31

Vehicle Dynamics


M  `  ` ` 

` cS  `  ` 

  ` ` `  cS   dS  ` ` ` 

6 zB

m

dS

c

6 zW

 ` `  `  ` cT  `  `  6 zR

6 zR

c

Figure 3.1: Simple Vehicle and Suspension Model and

m z¨W + dS z˙W + (cS + cT ) zW = cT zR ,

(3.2)

where zB and zW label the vertical movements of the body and the wheel mass out of the equilibrium position. The constants cS , dS describe the body suspension and damping, and cT the vertical stiffness of the tire. The tire damping is hereby neglected against the body damping.

3.2.2 Track The track is given as function in the space domain

zR = zR (x) .

(3.3)

In (3.1 ( 3.1)) also the time gradient of the track irregularities is necessary. From ( 3.3) 3.3) firstly follows

z˙R =

d zR dx . dx dt

(3.4)

At the simple model the speed, with which the track irregularities are probed equals the vehicle speed dx/dt = v . If the vehicle speed is given as time function v = v (t), the covered distance x can be calculated by simple integration.

3.2.3 Spring Spring Preload Preload The suspension spring is loaded with the t he respective respective vehicle load. At linear spring characteristics the steady state spring deflection is calculated from

f 0 =

Mg . cS

(3.5)

M + M leads At a conventional suspension without niveau regulation a load variation M f 0 + f . In analogy to (3.5 to changed spring deflections f 0 ( 3.5)) the additional deflection follows from

→

→



f = cM g . S

32



(3.6)



If for the maximum load variation M max the additional spring deflection is limited to the suspension spring rate can be estimated by a lower bound



max

cS

≥ M f

g

max

max

f

.

(3.7)

3.2.4 Eigenvalu Eigenvalues es At an ideally even track the right side of the equations of motion ( 3.1), 3.1), (3.2) 3.2) vanishes because of zR = 0 and z˙R = 0. The remaining homogeneous second order differential equations can be written as z¨ + 2 δ z˙ + ω02 z = 0 . (3.8) The respective attenuation constants δ and the undamped natural circular frequency ω0 for the models in Fig. 3.1 can be determined from a comparison of ( 3.8) 3.8) with (3.1 (3.1)) and (3.2 (3.2). ). The results are arranged in table 3.1.

attenuation constant

Motions

Differential Equation

Body

M z¨B + dS z˙B + cS zB = 0

Wheel

m z¨W + dS z˙W + (cS + cT ) zW = 0

dS 2 M dS δR = 2m

δB =

undamped Eigenfrequency

cS M cS + cT = m

2 ωB = 0

2 ωW

0

Table 3.1: Attenuation Constants and undamped natural Frequencies

With

z = z0 eλt

(3.9)

(λ2 + 2 δ λ + ω02) z0 eλt = 0 .

(3.10)

λ2 + 2 δ λ + ω02 = 0

(3.11)

the equation follows from (3.8 (3.8). ). For also non-trivial solutions are possible. The characteristical equation (3.11 ( 3.11)) has got the solutions

λ1,2 =

 − ± − δ

δ2

ω02

(3.12)

ω02 the eigenvalues λ1,2 are real and, because of δ 0 not positive, λ1,2 For δ 2 bances z (t =0) = z0 with z˙ (t =0) = 0 then subside exponentially.

≥

≥

≤ 0. Distur-

33

Vehicle Dynamics


With δ 2 < ω02 the eigenvalues eigenvalues become complex

λ1,2 =

 − ± − δ

ω02

i

δ2 .

(3.13)

The system now executes damped oscillations. The case

δ2 = ω02 ,

bzw.

δ = ω0

(3.14)

describes, in the sense of stability, an optimal system behavior. Wheel and body mass, as well as tire stiffness are fixed. The body spring rate can be calculated via load variations, cf. section 3.2.3. With the abbreviations from table 3.1 now damping parameters can be calculated from ( 3.14) 3.14) which provide with

(dS )opt = 2 M 1

optimal body vibrations and with

(dS )opt = 2 m 2

 



cS = 2 M

cS + cT = 2 m

optimal wheel vibrations.

cS M



(cS + cT ) m

(3.15)

(3.16)

3.2.5 Free Vibrations Vibrations Fig. 3.2 shows the time response of a damped single-mass oscillator to an initial disturbance as results from the solution of the differential equation (3.8 ( 3.8). ). The system here has been started withou withoutt initia initiall speed speed z˙ (t =0) = 0 but but with with the initia initiall distur disturba bance nce z (t =0) = z0 . If the the atte attenu nuat atio ion n constant δ is increased at first the system approaches the steady state position zG = 0 faster and faster, but then, a slow asymptotic behavior occurs. z0

z(t) t

Figure 3.2: Damped Vibration

34



Counting differences from the steady state positions as errors (t) = z (t) the quality of the vibration. The overall error is calculated by

−z

G, allows judging

t=tE

2G



=

z (t)2 dt ,

(3.17)

t=0

where the time tE have to be chosen appropriately. If the overall error becomes a Minimum

2G

→ Minimum

(3.18)

the system approaches the steady state position as fast as possible. To judge driving comfort and safety the deflections zB and accelerations z¨B of the body and the dynamic wheel load variations are used. The system behavior is optimal if the parameters M , m, cS , dS , cT result from the demands for comfort t=tE

2GC =

          → 2

g1 z¨B

g2 zB

+

2

dt

t=0

and safety

→ Minimum

(3.19)

t=tE

2GS =

2

cT zW

dt

Minimum .

(3.20)

t=0

With the factors g1 and g2 accelerations and deflections can be weighted differently. In the equations of motion for the body (3.1 ( 3.1)) the terms M z¨B and cS zB are added. With g1 = M and g2 = cS or g1 = 1 and g2 = cS /M one gets system-fitted weighting factors. At the damped single-mass oscillator, the integrals in ( 3.19) 3.19) can, for tE analytically. analytically. One gets

2GC = zB2 and 2 2GS = zW

0

0



cS 1 M 2

2 1 cT 2



dS cS + 2 M dS

 

dS m + cS + cT dS

→ ∞, still be solved (3.21)

.

(3.22)

0 or large body masses M Small body suspension stiffnesses cS 2 0 and so guarantee a high driving comfort. criteria (3.21 (3.21)) small GC

→

→

→ ∞ make the comfort

A great body mass however is uneconomic. The body suspension stiffness cannot be reduced arbitrary low values, because then load variations would lead to too great changes in static deflection. At fixed values for cS and M the damper can be designed in a way that minimizes the comfort criteria (3.21 (3.21). ). From the necessary condition for a minimum

∂ 2GC cS 1 = zB2 ∂d S M 2 0



1 M

−

cS 2 2 dS



= 0

(3.23)

35

Vehicle Dynamics


the optimal damper parameter

(dS )opt = 3

that guarantees optimal comfort follows.



2 cS M ,

(3.24)

0 make the safety criteria (3.22 0 and thus Small tire spring stiffnesses cT ( 3.22)) small 2GS reduce dynamic wheel load variations. The tire spring stiffness can however not be reduced to arbitrary low values, because this would cause too great tire deformation. Small wheel masses m 0 and/or a hard body suspension cS also reduce the safety criteria (3.22 ( 3.22). ). The use of light metal rims increases, because of wheel weight reduction, the driving safety of a car.

→

→

→

→∞

Hard body suspensions contradict driving comfort. With fixed values for cS , cT and m here the damper can also be designed to minimize the safety criteria (3.22 (3.22). ). From the necessary condition of a minimum

∂ 2GS 2 2 1 = zW cT ∂d S 2 0



1 cS + cT

−

m 2 dS



= 0

(3.25)

the optimal damper parameter

(dS )opt = 4

follows, which guarantees optimal safety. safety.



(cS + cT ) m ,

(3.26)

3.3 Sky Hook Hook Damp Damper er 3.3.1 Modelling Modelling Aspects Aspects In standard vehicle suspension systems the damper is mounted between the wheel and the body. body. Hence, the damper affects body and wheel/axle motions simultaneously. To take this situation into account the simple quarter car models of section 3.2.1 must be combined to a more enhanced model, Fig. 3.3a. 3.3a. Assuming a linear characteristics the suspension damper force is given by

F D =

−d

S (z˙B

− z˙

W )

,

(3.27)

where dS denotes the damping constant, and z˙B , z˙W are the time derivatives of the absolute vertical body and wheel displacements. The sky hook damping concept starts with two independent dampers for the body and the wheel/axle mass, Fig. 3.3b. 3.3b. A practical realization in form of a controllable damper will then provide the damping force F D = dB z˙B + dW z˙W , (3.28)

−

where instead of the single damping constant dS now two design parameter dB and dW are available.

36


© Prof. Dr.-Ing. G. Rill sky zB

zB M

dB

M

dW cS

cS

dS

zW

zW m zR

FD

cT

m cT

zR

a) Standard Damper

b) Sky Hook Damper

Figure 3.3: Quarter Car Model with Standard and Sky Hook Damper The equations of motion for the quarter car model are given by

M z¨B = F S S + F D m z¨W = F T F S T S

−M g, − − F − m g ,

(3.29)

D

where M , m are the sprung and unsprung mass, zB , zW denote their vertical displacements, displacements, and g is the constant of gravity. The suspension spring force is modelled by 0 F S S = F S

−c

S (zB

−z

W )

,

(3.30)

where F S 0 = mB g is the spring preload, and cS is the spring stiffness. Finally, the vertical tire force is given by 0 F T T = F T

−c

S (zW

−z

R)

,

(3.31)

where F T 0 = (M + m) g is the tire preload, cS the vertical tire stiffness, and zR describes the 0 takes the tire lift off into account. road roughness. The condition F T T

≥

3.3.2 System Perfo Performance rmance To perform an optimization the merit functions (3.19 ( 3.19)) and (3.20 (3.20)) were combined to one merit function t=tE

2GC

=

       

t=0

z¨B g

2

+

cS zB Mg

2

+

cT zW F T 0

2

dt

→ Minimum ,

(3.32)

37

Vehicle Dynamics


where the constant of gravity g and the tire preload F T 0 were used to weight the comfort and safety parts.

300kg and m = 50kg 50kg , the suspe The optimi optimiza zatio tion n was was done done numeri numerica cally lly.. The The masse massess M = 300kg suspensi nsion on stiffness cS = 18000 N/m and the vertical vertical tire stiffnes stiffnesss cT = 220000 N/m correspond to a passenger car. This parameter were kept unchanged. Using the simple model approach the standard damper can be designed according to the t he comfort (3.24) 3.24) or to the safety criteria criteria (3.26 ( 3.26). ). One gets C (dS )opt =

(dS )S opt =

√2 c

S M

=

√218000300 = 3286. 3286.3 N/ N/((m/s) m/s) , (3.33)



(cS + cT ) m =



(18 (18 000 000 + 220 220 000) 000) 50 = 3449 3449..6 N/ N/((m/s) m/s) ,

An optimization with the quarter car model results in

(dS )qcm N/((m/s) m/s) , opt = 2927 N/

(3.34)

where, according to the merit function ( 3.32) 3.32) a weighted compromise between comfort and safety was demanded. This ”optimal” damper value is 10% smaller than the one calculated with the simple model approach. 10 ] 2 ^ s / m [

0.02 Standard Damper

8

0

Sky Hook Damper

0

] m [ l e -0.02 v a r t n o -0.04 i s n e p s -0.06 u s

-2

-0.08

5000

0.02

s n o i t a r e l e c c a y d o b

6 4 2

Standard Damper Sky Hook Damper

wheel

Standard Damper

] N 4000 [

] m [

Sky Hook Damper

d a o 3000 l l e e h 2000 w c i m1000 a n y d 0

0

s t n -0.02 e m e c -0.04 a l p s i d

body Standard Damper

-0.06

-1000 0

0 .2

0 .4 0 .6 time [s]

0.8

1

-0.08

Sky Hook Damper 0

0 .2

0 .4 0.6 time [s]

Figure 3.4: Standard and Sky Hook Damper Performance

38

0.8

1



The optimization of the sky hook damper results in results in

(dC )qcm N/((m/s) m/s) opt = 3580 N/

(dW )qcm N/((m/s) m/s) . opt = 1732 N/

(3.35)

In Fig. 3.4 the simulation results of a quarter car model with optimized standard and sky hook damper are plotted. The free vibration manoeuver was performed with the initial displacements zB (t = 0) = 0.08 m, zW (t = 0) = 0.02 m and vanishing initial velocities z˙B (t = 0) = 0.0 m/s, z˙W (t = 0) = 0. 0.0 m/s.

−

−

The sky hook damper provides an larger potential to optimize vehicle vibrations. The improvement in the merit function amounts to 7%. Here, especially the part evaluating the body acceleration changed significantly. significantly.

3.4 Nonlinear Nonlinear Force Force Elements Elements 3.4.1 3.4.1 Quarter Quarter Car Car Model Model The principal influence of nonlinear characteristics on driving comfort and safety can already be displayed on a quarter car model Fig. 3.5. 3.5. zB

progressive spring

M

FF

degressive damper FD

FR v

xR

zW

x

m zR

cT

Figure 3.5: Quarter Car Model with nonlinear Characteristics The equations of motion are given by

M z¨B = F m z¨W =

− Mg F − F − m g ,

(3.36)

z

81 m/s2 labels the constant of gravity and M , m are the masses of body and wheel. where g = 9.81m/s The coordinates zB and zW are measured from the equilibrium position. Thus, the wheel load F z is calculated from the tire deflection zW

F z = (M + m) g + cT (zR

−z

W )

−z

R

.

via the tire stiffness cT (3.37)

39

Vehicle Dynamics


The first term in (3.37 (3.37)) describes the static part. The condition F z into consideration. consideration.

≥ 0 takes the wheel lift off

Body suspension and damping are described with nonlinear functions of the spring travel

x = zW

−z

(3.38)

B

and the spring velocity

v = z˙W

− z˙

B

,

(3.39)

where x > 0 and v > 0 marks the spring and damper compression. The damper characteristics are modelled as digressive functions with the parameters pi

i = 1(1)4 F D (v ) =

  

p1 v p3 v

1 v 1 + p2 v 1 1

−p

4

≥0

≥ 0,

(Druck)

.

(3.40)

v < 0 (Zug)

v

A linear damper with the constant d is described by p1 = p3 = d and p2 = p4 = 0. For the spring characteristics the approach

F F F (x) = M g +

F R x xR

− p |x| 1 − p x 1

5

5

(3.41)

R

p5 < 1, is used, where M g marks the spring preload. With parameters within the range 0 one gets differently progressive characteristics. characteristics. The special case p5 = 0 describes a linear spring with the constant c = F R /xR . All spring characteristics run through the operating point xR , F R . Thus, at a real vehicle, one gets the same roll angle, independent from the chosen progression at a certain lateral acceleration.

≤

3.4.2 Random Random Road Profile Profile The vehicle moves with the constant speed vF = const. When starting at t = 0 at the point xF = 0, the current position of the car is given by

xF (t) = vF t .

∗

(3.42)

The irregularities of the track can thus be written as time function zR = zR (xF (t)) The calculation of optimal characteristics, i.e. the determination of the parameters p1 to p5 , is done for three different tracks. Each track consists of a number of single obstacles, which lengths and heights are distributed randomly. Fig. 3.6 shows the first track profile zS (x). Profiles number two and three are generated from the first by multiplication with the factors 3 and 5, zS (x) = 3 zS (x), zS (x) = 5 zS (x). 1

2

40

∗

1

3

∗

1



road profil [m] 0.1 0.05 0 -0.05 -0.1

0

20

40

60 [m] 80

100

Figure 3.6: Track profile 1

3.4.3 Vehicle ehicle Data Data The values, arranged in table 3.2, describe the respective body mass of a fully loaded and an empty bus over the rear axle, the mass of the rear axle and the sum of tire stiffnesses at the twin tire rear axle. vehicle data M [kg] m [kg] F R [N] xR [m] cT [N/m] fully loaded 11 000 800 40 000 0.100 3 200 000 unloaded 6 0 00 00 800 22 500 0. 0.100 3 2 00 00 0 00 00 Table 3.2: Vehicle Data

The vehicle possesses niveau-regulation. niveau-regulation. Therefore also the force F R at the reference deflection xR has been fitted to the load. The vehicle drives at the constant speed vF = 20 m/s.

1(1)5, which describe the nonlinear spring-damper characteristics, The five parameters, pi , i = 1(1)5 are calculated by minimizing merit functions.

3.4.4 Merit Function Function In a first merit function, driving comfort and safety are to be judged by body accelerations and wheel load variations

GK 1

=

tE

                 tE

1

−t

0

t0

z¨B g

2

comfort

+

F zD F zS

2

.

(3.43)

safety

The body acceleration z¨B has been normalized to the constant of gravity g . The dynamic share zW ) follows from (3.37 of the normal force F zD = cT (zR (3.37)) with the static normal force S F z = (M + m) g.

−

41

Vehicle Dynamics


At real cars the spring travel is limited. The merit function is therefore extended accordingly

GK 2

=

tE

                      tE

1

−t

0

z¨B g

t0

2

+

P D P S S

2

safety

comfort

x xR

+

2

,

(3.44)

spring trave tr avell

where the spring travel x, defined by (3.38 ( 3.38), ), has been related to the reference travel xr . According to the covered distance and chosen driving speed, the times used in ( 3.43) 3.43) and (3.44 (3.44)) have been set to t0 = 0 s and tE = 8 s

3.4.5 Optimal Optimal Paramete Parameterr 3.4.5.1 Linear Characteristics Judging the driving comfort and safety after the criteria GK 1 and restricting to linear characteristics, with p1 = p3 and p2 = p4 = p5 = 0, one gets the results arrayed in table 3.3. 3.3. The spring optima optimall parame parameter ter road road load load

p1

p2

p3

parts parts in merit merit functi function on p4 p5 comfort safety

1 2 3

+ 357 35766 0 35766 0 0 0.00 0.002 2886 886 0.00 .00266 2669 + 357 35763 0 35763 0 0 0.02 0.025 5972 972 0.02 .02401 4013 + 357 35762 0 35762 0 0 0.07 0.072 2143 143 0.06 .06670 6701

1 2 3

− − −

202 20298 0 20298 0 0 0.00 0.003 3321 321 203 20300 0 20300 0 0 0.02 0.029 9889 889 199 19974 0 19974 0 0 0.08 0.083 3040 040

0.00 .00396 3961 0.03 .03564 5641 0.09 .09838 8385

Table 3.3: Linear Spring and Damper Damper Parameter optimized via GK 1

constants c = F R /xr for the fully loaded and the empty vehicle are defined by the numerical 000N/m and cloaded = 400 000 000N/m. values in table 3.2. 3.2. One gets: cempty = 225 000 As expe expect cted ed the the resu result ltss are are almo almost st inde indepe pend nden entt from from the the trac track. k. The The opti optima mall value alue of the the damp dampin ing g parameter d = p1 = p3 however is strongly dependent on the load state. The optimizing quasi fits the damper constant to the changed spring rate. The loaded vehicle is more comfortable and safer.

3.4.5.2 Nonlinear Characteristics The results of the optimization with nonlinear characteristics are arrayed in the table 3.4. The optimizing has been started with the linear parameters from table 3.3. 3.3. Only at the extreme track irregularities of profile 3, linear spring characteristics, with p5 = 0, appear, Fig. 3.8. At moderate track irregularities, one gets strongly progressive springs.

42



optim ptimal al para parame mete terr road road load load

p1

p2

p3

p4

parts arts in merit erit fun functio ction n comfort safety

p5

1 2 3

+ 1618 16182 2 0.000 0.000 2002 20028 8 1.31 1.316 6 0.967 0.9671 1 0.00 0.0002 0265 65 0.00 0.0011 1104 04 + 5217 52170 0 2.689 2.689 5789 57892 2 1.17 1.175 5 0.698 0.6983 3 0.00 0.0090 9060 60 0.01 0.0127 2764 64 + 1875 1875 3.048 3.048 311 311773 773 4.29 4.295 5 0.000 0.0000 0 0.0408 0.040813 13 0.050 0.050069 069

1 2 3

− − −

1396 13961 1 0.000 0.000 1725 17255 5 0.33 0.337 7 0.920 0.9203 3 0.00 0.0008 0819 19 0.00 0.0034 3414 14 1608 16081 1 0.808 0.808 2770 27703 3 0.45 0.454 4 0.656 0.6567 7 0.01 0.0129 2947 47 0.03 0.0312 1285 85 9942 9942 0.22 0.227 7 6434 64345 5 0.71 0.714 4 0.00 0.0000 00 0.06 0.0609 0992 92 0.09 0.0902 0250 50

Table 3.4: Nonlinear Spring and Damper Characteristics optimized via GK 1

The dampers are digressive and differ in jounce and rebound. In comp compar aris ison on to the the line linear ar mode modell a sign signifi ifica cant nt impr improv ovem emen entt can can be note noted, d, espe especi cial ally ly in comf comfor ort. t. While driving over profile 2 with the loaded vehicle, the body accelerations are displayed in Fig. 3.7. body accelerations [m/s2 ] 10 5 0 -5 -10 0

2

4

[s]

6

Figure 3.7: Body Accelerations optimized via GK 1

8

· · · linear, — nonlinear)

(

spring force [kN] 40 20 0 -20 -40 -0.1

-0.05

0 0.05 spring travel [m]

0.1

Figure 3.8: Optimal Spring Characteristics for fully loaded Vehicle; Criteria: GK 1

43

Vehicle Dynamics


The extremely progressive progressive spring characteristics, optimal at smooth tracks (profile 1), cannot be realize practically in that way. Due to the small spring stiffness around the equilibrium position, small disturbances cause only small aligning forces. Therefore it would take long to reach the equilibrium position again. Additionally, friction forces in the body suspension would cause a large deviation of the equilibrium position.

3.4.5.3 Limited Spring Travel Practically relevant results can only be achieved, if additionally the spring travels are judged. Firstly, linear characteristics are assumed again, table 3.5. 3.5. opti optima mall para parame mete terr road road load load

p1

p2

p3

p4 p5

part partss in meri meritt func functi tion on comfort safety s. travel

1 2 3

+ 6872 68727 7 0 6872 68727 7 0 0 0.00 0.0038 3854 54 0.00 0.0036 3673 73 0.00 0.0063 6339 39 + 6866 68666 6 0 6866 68666 6 0 0 0.03 0.0346 4657 57 0.03 0.0330 3025 25 0.05 0.0570 7097 97 + 7288 72882 2 0 7288 72882 2 0 0 0.09 0.0989 8961 61 0.09 0.0944 4431 31 0.14 0.1487 8757 57

1 2 3

− − −

3533 35332 2 0 3533 35332 2 0 0 0.00 0.0044 4417 17 0.00 0.0047 4701 01 0.00 0.0066 6638 38 3565 35656 6 0 3565 35656 6 0 0 0.04 0.0400 0049 49 0.04 0.0425 2507 07 0.05 0.0591 9162 62 3748 37480 0 0 3748 37480 0 0 0 0.11 0.1121 2143 43 0.11 0.1167 6722 22 0.15 0.1552 5290 90

Table 3.5: Linear Spring and Damper Characteristics optimized via GK 2

The judging numbers for comfort and safety have worsened by limiting the spring travel in comparison to the values from table 3.3. In order to receive realistic spring characteristics, now the parameter p5 has been limited upward wardss to p5 0.6. Star Startin ting g with with the the line linear ar para parame mete ters rs from from tabl tablee 3.5, an optimizati optimization on via criteria criteria

≤

optimal parameter road road load load

p1

p2

p3

p4

p5

parts in merit function comfort safety s. trav ravel

1 2 3

+ 175530 175530 12.89 12.89 102997 102997 3.437 3.437 0.4722 0.4722 0.001747 0.001747 0.002044 0.002044 0.005769 0.005769 + 204674 204674 5.505 5.505 107498 107498 1.234 1.234 0.6000 0.6000 0.015877 0.015877 0.018500 0.018500 0.050073 0.050073 + 327864 327864 4.844 4.844 152732 152732 1.165 1.165 0.5140 0.5140 0.064980 0.064980 0.068329 0.068329 0.116555 0.116555

1 2 3

− − −

66391 66391 5.244 5.244 50353 50353 2.082 2.082 0.5841 0.5841 0.00238 0.002380 0 0.003 0.003943 943 0.005 0.005597 597 37246 37246 0.601 0.601 37392 37392 0.101 0.101 0.5459 0.5459 0.02452 0.024524 4 0.033 0.033156 156 0.059 0.059717 717 89007 89007 1.668 1.668 68917 68917 0.643 0.643 0.3614 0.3614 0.08500 0.085001 1 0.102 0.102876 876 0.125 0.125042 042

Table 3.6: Nonlinear Spring and Damper Characteristics optimized via GK 2

GK 2 delivers the results arranged in table 3.6. A vehicle with GK 2 -optimized characteristics manages the travel over uneven tracks with significantly less spring travel than a vehicle with GK 1 -optimized characteristics, Fig. 3.9.

44



spring travel [m] 0.1 0.05 0 -0.05 -0.1 0

2

4

6

[s]

Figure Figure 3.9: 3.9: Sprin Spring g Tra Trave vels ls on on Profil Profilee 2

8

(- - - GK 1 , — GK 2 )

The reduced spring travel however reduces comfort and safety. Still, in most cases, the according part of the merit function in table 3.6 lie even below the values of the linear model from table 3.3, where the spring travels have not been evaluated. By the use of nonlinear characteristics, the comfort and safety of a vehicle can so be improved, despite limitation of the spring travel. The optimal damper characteristics strongly depend on the roughness of the track, Fig. 3.10. damper force [kN]

100 rebound

50 0 -50 compression

-100 -1

-0.5

0 [m/s] 0.5

1

Figure 3.10: Optimal Damper Characteristics according to Table 3.6 Optimal comfort and safety are only guaranteed if the dampers are fitted to the load as well as to the roughness of the track.

3.5 Dynamic Dynamic Force Force Elements Elements 3.5.1 System Response Response in the Frequency Frequency Domain Domain 3.5.1.1 First Harmonic Harmonic Oscillation Oscillation The effect of dynamic force elements is usually judged in the frequency domain. For this, on test rigs or in simulation, the force element is periodically excited with different frequencies

45

Vehicle Dynamics

f 0


≤ f ≤ f

E E and amplitudes

i

Amin

≤A ≤A j

max

xe (t) = A j sin(2π sin(2π f i t) .

(3.45)

Starting at t = 0 at t = T 0 with T 0 = 1/f 0 the system usually is in a steady state condition. Due to the nonlinear system behavior the system response is periodic, yet not harmonic. For the evaluation thus the answer, e.g. the measured or calculated force F , each within the intervals tS i t tS i + T i , is approximated by harmonic functions as good as possible

≤ ≤

F ( F (t)

αi sin(2π sin(2π f i t) + β i cos(2π cos(2π f i t) .

 ≈  measured or calculated



first harmonic approximation

(3.46)



The coefficients coefficients αi and β i can be calculated from the demand for a minimal overall error

1 2

tSi+T i

 

αi sin(2π sin(2π f i t) + β i cos(2π cos(2π f i t)

tSi



2

F (t) − F (

dt

−→

Minimum .

(3.47)

The differentiation of (3.47 (3.47)) with respect to αi and β i delivers delivers two linear equations as necessary conditions tSi+T i

   

αi sin(2π sin(2π f i t)+ β i cos(2π cos(2π f i t)

tSi tSi+T i

αi sin(2π sin(2π f i t)+ β i cos(2π cos(2π f i t)

tSi

 

F (t) − F ( F (t) − F (

2

sin(2π sin(2π f i t) dt = 0 (3.48)

2

cos(2π cos(2π f i t) dt = 0

with the solutions

αi = β i =

    −    −     −    −

F sin dt cos2 dt F cos dt sin cos dt sin2 dt cos2 dt 2 sin cos dt F cos dt sin2 dt F sin dt sin cos dt sin2 dt cos2 dt 2 sin cos dt

,

(3.49)

where the integral limits and arguments of sine and cosine have no longer been written. Because it is integrated exactly over one period tS i



sin cos dt = 0 ;

holds, and as solution

2 αi = T i





≤t≤t

sin2 dt =

F sin dt ,

S i + T i , for

T i ; 2

βi =

2 T i

 

the integrals integrals in (3.49 ( 3.49))

cos2 dt =

F cos dt .

T i 2

(3.50)

(3.51)

remains. These however are exactly the first two coefficients of a Fourier–Approximation. In practice, the frequency response of a system is not determined punctual, but continuous. For this, the system is excited by a sweep-sine.

46



3.5.1.2 Sweep-Sine Excitation In analogy to the simple sine-function

xe (t) = A sin(2π sin(2π f t) ,

(3.52)

where the period duration T = 1/f appears as pre-factor at differentiation

x˙ e (t) = A 2π f cos(2π cos(2π f t) =

2π A cos(2π cos(2π f t) , T

(3.53)

now a generalized sine-function can be constructed. Starting with

xe (t) = A sin(2π sin(2π h(t))

(3.54)

x˙ e (t) = A 2π h˙ (t) cos(2π cos(2π h(t)) .

(3.55)

the time derivative results in

Now we demand, that the function h(t) delivers a period, that fades linear in time, i.e:

h˙ (t) =

1 1 = , T ( T (t) p q t

(3.56)

−

where p > 0 and q > 0 are constants yet to determine. From (3.56 ( 3.56))

ln( p − q t) + C − 1q ln( p

h(t) =

(3.57)

follows. The initial condition h(t = 0) = 0 fixes the integration constant

C =

1 ln p ln p . q

(3.58)

Inserting (3.58 (3.58)) in (3.57 (3.57), ), a sine-like function follows from (3.54 ( 3.54))





2π p xe (t) = A sin ln , q p q t delivering linear fading period durations.

−

(3.59)

The important zero values for determining the period duration lie at

1 p ln = 0, 1, 2, q p q tn

or

−

and

tn =

p (1 q

−e

p = en q , mit n = 0, 1, 2, p q tn

−

−n q

) , n = 0, 1, 2, .

(3.60)

(3.61)

The time difference between two zero points determines the period

T n = tn+1 T n

−t

n

p −n q = e (1 q

p (1 e−(n+1) q q

=

−

−q

−e

−n q

− 1+ e

) , n = 0, 1, 2, .

(3.62)

)

47

Vehicle Dynamics


N ) period one finds For the first (n = 0) and last (n = N ) p (1 q p = (1 q

T 0 = T N N

−q

−e −e

−q

) )e

−N q

.

−N q

(3.63)

= T 0 e

With the frequency range to investigate given by the initial f 0 and final f E E frequency, the parameters q and the relation q/p can be calculated from (3.63 ( 3.63)) 1

 −  

1 f E E q = ln , N f 0

q = f 0 1 p

f E E f 0

N

,

(3.64)

with N fixing the number of frequency frequency intervals. The passing of the whole frequency range then takes

tN +1 N +1 =

1

−(N +1) +1) q

−e

(3.65)

q/p

seconds.

3.5.2 Hydro-Mo Hydro-Mount unt 3.5.2.1 3.5.2.1 Principle Principle and Model Model For elastic suspension of engines in vehicles very often specially developed hydro-mounts are used. The dynamic nonlinear behavior of these components guarantees a good acoustic decoupling, but simultaneously provides sufficient damping.

xe main spring chamber 1 membrane

c_ _ T 2

cF uF MF

ring channel chamber 2

dF __ 2

Figure 3.11: Hydro-Mount Fig. 3.11 shows the principle and mathematical model of a hydro-mount.

48

dF __ 2

cT __ 2



At small deformations the change of volume in chamber 1 is compensated by displacements of the membrane. When the membrane reaches the stop, the liquid in chamber 1 is pressed through a ring channel into chamber 2. The relation of the chamber cross section to ring channel cross section is very large. Thus the fluid is moved through the ring channel at very high speed. From this remarkable inertia and resistance forces (damping forces) result. The force effect of a hydro-mount is combined from the elasticity of the main spring and the volume change in chamber 1. With uF labelling the displacement of the generalized fluid mass M F F ,

F H H = cT xe + F F F (xe

−u

F )

(3.66)

holds, where the force effect of the main spring has been approximated by a linear spring with the constant cT . With M F R as actual mass in the ring channel and the cross sections AK , AR of chamber and ring channel the generalized fluid mass is given by

M F F =

  AK AR

2

M F R .

(3.67)

The fluid in chamber 1 is not being compressed, unless the membrane can evade no longer. With the fluid stiffness cF and the membrane clearance sF one gets

   

cF (xe

F F F (xe

−u

F )

=

0

cF (xe

−u

F )

 

+ sF

−u ) − s F

− u ) < −s |x − u | ≤ s ( x − u ) > +s

for

F

(xe

F

e

f

e

f

F

F

(3.68)

F

The hard transition from clearance F F F = 0 and fluid compression, resp. chamber deformation with F F F = 0 is not realistic and leads to problems, even with the numeric solution. The function 2 sF . (3.68) 3.68) is therefore smoothed by a parable in the range xe uf



| − |≤ ∗

The motions of the fluid mass cause friction losses in the ring channel, which are, at first approximation, proportional to the speed,

F D = dF u˙ F .

(3.69)

The equation of motion for the fluid mass then reads as

¨F = M F F u

− F − F F F

D

.

(3.70)

The membrane clearing makes (3.70 ( 3.70)) nonlinear, and only solvable by numerical integration. The nonlinearity also affects the overall force ( 3.66) 3.66) in the hydro-mount.

49

Vehicle Dynamics

FH Regensburg, University of Applied Sciences Dynamic Stiffness [N/m] at Excitation Amplitudes A = 2.5/0.5/0.1 mm

400

300

200

100

0 60

Dissipation Angle [deg] at Excitation Amplitudes A = 2.5/0.5/0.1 mm

50 40 30 20 10 0

0

1

10

Excitation Frequency [Hz]

10

Figure 3.12: Dynamic Stiffness [N/mm] and Dissipation Angle [deg] for a Hydro-Mount

3.5.2.2 Dynamic Force Characteristics The dynamic stiffness and the dissipation angle of a hydro bearing are displayed in Fig. 3.12 over the frequency. The dissipation angle is a measurement for the damping. The simulation is based on the following system parameters

mF = 25 kg cT

generalized fluid mass

=

125 000 N/m

stiffness of main spring

dF = cF =

750 N/ N/((m/s) m/s) 100 000 N/m

damping constant

sF =

0.0002 mm

clearance in membrane bearing

fluid stiffness

By the nonlinear and dynamic behavior a very good compromise between noise isolation and vibration damping can be achieved.

50

4 Longit Longitudi udinal nal Dynamic Dynamics s 4.1 Dynamic Dynamic Wheel Wheel Loads 4.1.1 Simple Simple Vehicle Vehicle Model Model The vehicle is considered as one rigid body which moves along an ideally even and horizontal road. At each axle the forces in the wheel contact points are combined into one normal and one longitudinal force.

v

S

h

Fz1

Fx1

mg a2

a1

Fx2 Fz2

Figure 4.1: Simple Vehicle Model If aerodynamic forces (drag, positive and negative lift) are neglected at first, then the equations of motions in the x-, z -plane read as

m v˙ = F x1 + F x2 , 0 = F z1 + F z2 0 = F z 1 a1

− F

z2

− mg,

a2 + (F (F x1 + F x2 ) h ,

(4.1) (4.2)

(4.3)

where v˙ indicates the vehicle’s acceleration, m is the mass of the vehicle, a1 + a2 is the wheel base, and h is the height of the center of gravity. This are only three equations for the four unknown forces F x1 , F x2 , F z1 , F z 2 . But, if we insert (4.1) 4.1) in (4.3 (4.3)) we can eliminate two unknowns by one stroke

0 = F z1 a1

− F

z2

a2 + m v˙ h .

(4.4)

51

Vehicle Dynamics


The equations (4.2 ( 4.2)) and (4.4 (4.4)) can now be resolved for the axle loads

F z1 = m g

a2 a1 + a2

h m v˙ , a1 + a2

(4.5)

F z 2 = m g

a1 h + m v˙ . a1 + a2 a1 + a2

(4.6)

The static parts

F zst1 = m g

−

a2 , a1 + a2

F zst2 = m g

a1 a1 + a2

(4.7)

describe the weight distribution according to the horizontal position of the center of gravity. gravity. The height of the center of gravity has influence only on the dynamic part of the axle loads,

F zdyn 1 =

h v˙ , 1 + a2 g

F zdyn 2 = +m g

−m g a

h v˙ . a1 + a2 g

(4.8)

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

4.1.2 Influence Influence of of Grade Grade

z

v x F x

1

mg

F z

1

h

a 1

F x a 2

2 2

α

F z

2 2

Figure 4.2: Vehicle on Grade For a vehicle on a grade, Fig.4.2 Fig. 4.2,, the equations of motions (4.1 ( 4.1)) to (4.3 (4.3)) can easily be extended to

m v˙ = F x1 + F x2 0 = F z 1 + F z2 0 = F z 1 a1

52

− m g sin α , − m g cos α ,

− F

z2

a2 + (F (F x1 + F x2 ) h ,

(4.9)



where α denotes the grade angle. Now, the axle loads are given by

F z 1 = m g cos α

F z2 = m g cos α

a2

− h tan α − a +a

h m v˙ , a1 + a2

(4.10)

a1 + h tan α h + m v˙ , a1 + a2 a1 + a2

(4.11)

1

2

where the dynamic parts remain unchanged, and the static parts also depend on the grade angle and the height of the center of gravity.

4.1.3 Aerodyn Aerodynamic amic Forces Forces The shape of most vehicles or specific wings mounted at the vehicle produce aerodynamic forces and torques. The effect of this aerodynamic forces and torques can be represented by a resistant force applied at the center of gravity and ”down forces” acting at the front and rear axle, Fig. 4.3. 4.3. FD1

FD2 FAR h mg

Fx1

a2

a1

Fz1

Fx2 Fz2

Figure 4.3: Vehicle with Aerodynamic Aerodynamic Forces If we assume a positive driving speed, v >, then the equations of motion read as

m v˙ = F x1 + F x2 F AR AR , 0 = F z 1 F D1 + F z 2 F D2 m g , 0 = (F z1 F D1 ) a1 (F z2 F D2) a2 + (F (F x1 + F x2 ) h ,

− −

−

− − − −

(4.12)

where F AR AR and F D 1 , F D 2 describe the air resistance and the down forces. For the dynamic axle loads we get

F z 1 = F D1 + m g

a2 a1 + a2

h (m v˙ + F AR AR ) , a1 + a2

(4.13)

F z2 = F D2 + m g

a1 h + (m v˙ + F AR AR ) . a1 + a2 a1 + a2

(4.14)

−

The down forces F D1 , F D2 increase the static axle loads, and the air resistance F AR AR generates an additional dynamic term.

53

Vehicle Dynamics


4.2 Maximum Maximum Accelera Acceleration tion 4.2.1 Tilting Tilting Limits Limits Ordinary automotive vehicles can only deliver pressure forces to the road. If we apply the de0 and F z2 0 to (4.10 mands F z1 ( 4.10)) and (4.11 (4.11)) we get

≥

≥

v˙ g

≤

a2 cos α h

− sin α

v˙ g

and

≥ − ah cos α − sin α , 1

(4.15)

which can be combined to

− ah

1

cos α

v˙ + sin α g

≤

≤

a2 cos α . h

(4.16)

Hence, the maximum achievable accelerations ( v˙ > 0) and decelerations ( v˙ > 0) are limited by 0 the tilting condition (4.16 the grade angle and the position position of the center of gravity. gravity. For v˙ (4.16)) results in a1 a2 tan α (4.17)

→

−h ≤

≤

h

which describes the climbing and downhill capacity of a vehicle. The presence of aerodynamic forces complicates the tilting condition. Aerodynamic forces become important only at high speeds. Here the vehicle acceleration normally is limited by the engine power.

4.2.2 Friction Friction Limits The maximum acceleration is also limited by the friction conditions

|F | ≤ µ F x1

z1

and

|F | ≤ µ F x2

z2

(4.18)

where the same friction coefficient µ has been assumed at front and rear axle. In the limit case

F x1 =

± µ F

z1

and

F x2 =

± µ F

(4.19)

− m g sin α .

(4.20)

z2

the first equation in (4.9 (4.9)) can be written as

m v˙ max =

± µ (F

z1

+ F z 2 )

Using (4.10 (4.10)) and (4.11 (4.11)) one gets

 v˙ g

max

=

± µ cos α − sin α .

(4.21)

That means climbing ( v˙ > 0, α > 0) or downhill stopping (v˙ < 0, α < 0) requires at least a friction coefficient coefficient µ tan α.

≥

54



According to the vehicle dimensions and the friction values the maximal acceleration or deceleration is restricted either by (4.16 ( 4.16)) or by (4.21 (4.21). ). If we take aerodynamic forces into account the maximum acceleration on a horizontal road is limited by

−µ



F D1 F D2 1 + + mg mg

−

F AR AR mg

v˙ g

≤

µ

≤



F D1 F D2 1 + + mg mg

−

F AR AR . (4.22) mg

In particular the aerodynamic forces enhance the braking performance of the vehicle.

4.3 Dri Drivin ving g and Brakin Braking g 4.3.1 Single Single Axle Drive With the rear axle driven in limit situations F x1 = 0 and F x2 = µ F z2 holds. Then, using (4.6 ( 4.6)) the linear momentum (4.1 (4.1)) results in

m v˙ R WD = µ m g



a1 h v˙ R WD + a1 + a2 a1 + a2 g



,

(4.23)

where the subscript R WD indicates the rear wheel drive. Hence, the maximum acceleration for a rear wheel driven vehicle is given by

v˙ R WD = g

µ 1

−

h µ a1 + a2

a1 . a1 + a2

(4.24)

By setting F x1 = µ F z 1 and F x2 = 0 the maximum acceleration for a front wheel driven vehicle can be calculated in a similar way. One gets

v˙ F WD = g

µ h 1+µ a1 + a2

a2 , a1 + a2

(4.25)

where the subscript F WD denotes front wheel drive. Depending on the parameter µ, a1 , a2 and a h the accelerations may be limited by the tilting condition vg˙ h.

≤

2

The maximum accelerations of a single axle driven vehicle are plotted in Fig. 4.4. 4.4. For rear wheel driven passenger cars the parameter a2 /(a1 + a2 ) which describes the static axle load distribution is in the range of 0.4 a2 /(a1+a2 ) 0.5. For µ = 1 and h = 0.55 this results v/g ˙ 0.64. Front wheel driven passenger cars in maximum accelerations in between 0.77 a2 /(a1 + a2 ) 0.60 which produces accelerations in the range usually cover the range 0.55 ˙ 0.49. Hence, rear wheel driven vehicles can accelerate much faster than front of 0.45 v/g wheel driven vehicles. vehicles.

≤

≤

≥

≤

≥ ≤

≤ ≥

55

Vehicle Dynamics

FH Regensburg, University of Applied Sciences range of load distribution

1

g / v

D W R

D W F

FWD

0.8

.

0.6

0.4

RWD

0.2

0 0

0.2

0.4

0.6

0.8

1

a2 / (a1+a2)

Figure 4.4: Single Axle Driven Passenger Car: µ = 1, h = 0.55 m, a1 + a2 = 2.5 m

4.3.2 Braking at Single Single Axle Axle If only the front axle is braked then in the limit case F x1 = µ F z1 and F x2 = 0 holds. With (4.5 (4.5)) one gets from (4.1 (4.1))

−

m v˙ F WB = where the subscript given by

F WB

−µ m g



a2 a1 + a2

−

h v˙ F WB a1 + a2 g



(4.26)

indicates front wheel braking. The maximum deceleration is then

v˙ F WB = g

−

v˙ R WB = g

−

µ

a2 . a1 + a2

(4.27)

a1 , a1 + a2

(4.28)

h 1 µ a1 + a2 If only the rear axle is braked ( F x1 = 0, F x2 = µ F z2 ) one gets the maximal deceleration

− −

µ h 1+µ a1 + a2

where the subscript R WB indicates a braked rear axle. Depending on the parameter µ, a1 , a2 and a h the decelerations may be limited by the tilting condition gv˙ h.

≥−

1

The maximum decelerations of a single axle braked vehicle are plotted in Fig. 4.5. 4.5. For passenger cars the load distribution parameter a2 /(a1 + a2 ) usually covers the range from 0.4 to 0.6. If only the front axle is braked then decelerations from v/g ˙ = 0.51 to v/g ˙ = 0.77 can be achieved. This is pretty much compared to the deceleration range of a braked rear axle ˙ = 0.49 to v/g ˙ = 0.33. which is in the range from v/g

−

−

−

That is why the braking system at the front axle has a redundant design.

56

−

FH Regensburg, University of Applied Sciences 0

range of load distribution

-0.2

g / .


FWB

v

-0.4

-0.6

-0.8

RWB -1

0

0.2

0.4

0.6

0.8

1

a2 / (a1+a2)

Figure 4.5: Single Axle Braked Passenger Car: µ = 1, h = 0.55 m, a1 + a2 = 2.5 m

4.3.3 Optimal Optimal Distribution Distribution of Drive Drive and Brake Forces The sum of the longitudinal forces accelerates or decelerates the vehicle. In dimensionless style (4.1) 4.1) reads

v˙ F x1 F x2 = + . g mg mg

(4.29)

A certain acceleration or deceleration can only be achieved by different combinations of the longitudinal forces F x1 and F x2 . According to (4.19 (4.19)) the longitudinal forces are limited by wheel load and friction. The optimal combination of F x1 and F x2 is achieved, when front and rear axle have the same skid resistance. F x1 = ν µ F z1 and F x2 = ν µ F z2 . (4.30) With (4.5 (4.5)) and (4.6 (4.6)) one gets

and

±

±

F x1 = mg

±νµ

F x2 = mg

±νµ

With (4.31 (4.31)) and (4.32 ( 4.32)) one gets from (4.29 ( 4.29))

 −   a2 h

v˙ g

a1 v˙ + h g

v˙ = g

h a1 + a2

(4.31)

h . a1 + a2

(4.32)

±νµ ,

(4.33)

where it has been assumed that F x1 and F x2 have the same sign. With (4.33 (4.33 inserted in (4.31 (4.31)) and (4.32 (4.32)) one gets

F x1 v˙ = mg g



a2 h

−

v˙ g



h a1 + a2

(4.34)

57

Vehicle Dynamics


and

F x2 v˙ = mg g



a1 v˙ + h g



h . a1 + a2

(4.35)

remain. Depending on the desired acceleration v˙ > 0 or deceleration v˙ < 0 the longitudinal forces that grant the same skid resistance at both axles can now be calculated. Fig.4.6 Fig.4.6 shows the curve of optimal drive and brake forces for typical passenger car values. At g m / 2 B

braking

Fx1 /mg

-a1 /h

dFx2

0

dFx1 0

-1

-2

B1 /mg

a =1.15 1

a =1.35

g n i v i r d

2

h=0.55

1

µ=1.20

tilting limits 2

h / 2 a

g m / 2 x

F

Figure 4.6: Optimal Distribution of Drive and Brake Forces

˙ = the tilting limits v/g lifting axle.

˙ = +a /h no longitudinal forces can be delivered at the −a /h and v/g 1

2

The initial gradient only depends on the steady state distribution of wheel loads. From ( 4.34) 4.34) and (4.35 (4.35)) it follows follows

d

58

F x1 mg = v˙ d g



a2 h

−

v˙ 2 g



h a1 + a2

(4.36)


and

F x2 mg = v˙ d g

d




a1 v˙ +2 h g



h . a1 + a2

(4.37)

˙ = 0 the initial gradient remains as For v/g d F x2 d F x1



= 0

a1 . a2

(4.38)

4.3.4 Different Different Distribution Distributions s of Brake Forces Forces In practice it is tried to approximate the optimal distribution of brake forces by constant distribution, limitation or reduction of brake forces as good as possible. Fig. 4.7.

Fx1 /mg

Fx1 /mg g m / 2 x

F

constant distribution

Fx1 /mg g m / 2 x

F

limitation

g m / 2 x

reduction

F

Figure 4.7: Different Distributions of Brake Forces When braking, the vehicle’s stability is dependent on the potential of lateral force (cornering stiffness) at the rear axle. In practice, a greater skid (locking) resistance is thus realized at the rear axle than at the front axle. Because of this, the brake force balances in the physically relevant relevant area are all below the optimal curve. This restricts the achievable achievable deceleration, specially at low friction values. Because the optimal curve is dependent on the vehicle’s center of gravity additional safeties have to be installed when designing real distributions of brake forces. Often the distribution of brake forces is fitted to the axle loads. There the influence of the height of the center of gravity, which may also vary much on trucks, remains unrespected and has to be compensated by a safety distance from the optimal curve. Only the control of brake brake pressure in anti-lock-systems anti-lock-systems provides provides an optimal optimal distribution of brake forces independent from loading conditions.

4.3.5 Anti-Lock-S Anti-Lock-Systems ystems Lateral forces can only be scarcely transmitted, if high values of longitudinal slip occur when decelerating a vehicle. Stability and/or steerability is then no longer given. By controlling the brake torque, respectively brake pressure, the longitudinal slip can be restricted to values that allow considerable lateral forces.

59

Vehicle Dynamics


˙ is used here as control variable. Angular wheel accelerations The angular wheel acceleration Ω are derived from the measured angular wheel speeds by differentiation. With a longitudinal slip of sL = 0 the rolling condition is fulfilled. Then rD Ω˙ = x¨

(4.39)

¨ is the vehicle’s acceleration. According holds, where rD labels the dynamic tyre radius and x to (4.21 ( 4.21), ), the maximum acceleration/deceleration of a vehicle is dependent on the friction coef¨ = µ g . With a known friction coefficient µ a simple control law can be realized for ficient, x every wheel

||

|Ω˙ | ≤

1 x¨ . rD

||

(4.40)

Because until today no reliable possibility to determine the local friction coefficient between tyre and road has been found, useful information can only be gained from ( 4.40) 4.40) at optimal conditions on dry road. Therefore the longitudinal slip is used as a second control variable. In order to calculate longitudinal slips, a reference speed is estimated from all measured wheel speeds which is then used for the calculation of slip at all wheels. This method is too imprecise at low speeds. Below a limit velocity no control occurs therefore. Problems also occur when for example all wheels lock simultaneously which may happen on 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 usually made slower than the increment. To prevent a strong yaw reaction, the select low principle is often used with µ-split braking at the rear axle. The break pressure at both wheels is controlled the wheel running on lower friction. Thus the brake forces at the rear axle cause no yaw torque. The maximally achievable deceleration however is reduced by this.

4.4 Dri Drive ve and Brake Brake Pitc Pitch h 4.4.1 Vehicle ehicle Model Model The vehicle model drawn in Fig. 4.8 consists of five rigid bodies. The body has three degrees of freedom: Longitudinal motion xA , vertical motion zA and pitch β A. The coordinates z1 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 be described via suspension kinematics as functions of the vertical wheel motion:

x1 = x1 (z1 ) , β 1 = β 1 (z1 ) ; x2 = x2 (z2 ) , β 2 = β 2 (z2 ) .

(4.41)

The rotation angles ϕR1 and ϕR2 describe the wheel rotations relative to the wheel bodies. The forces between wheel body and vehicle body are labelled F F 1 F 1 and F F 2 F 2 . At the wheels drive torques M A1 , M A2 and brake torques M B1 , M B 2 , longitudinal forces F x1 , F x2 and the wheel

60



z

A

x

A

FF1

MA1

βA

z1 MB1

MA1

hR

ϕ

R1

MB1 Fz1

z2 MB2

MA2

Fx1 R

a1

FF2

MA2

ϕ

R2

MB2

a2

Fz2

Fx2

Figure 4.8: Plane Vehicle Model loads F z1 , F z2 apply. The brake torques are supported directly by the wheel bodies, the drive torques are transmitted by the drive shafts to the vehicle body. The forces and torques that apply to the single bodies are listed in the last column of the tables 4.1 and 4.2. The velocity of the vehicle body and its angular velocity is given by

            v0A,0 A,0 =

x˙ A 0 0

       

+

     

0 0 z˙A

     

    −   −   −              0 ˙ β A 0

ω0A,0 A,0 =

;

.

(4.42)

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

v0RK ,0 = v0R 1

1

v0RK ,0 = v0R 2

,0

2

,0

=

x˙ A 0 0

=

x˙ A 0 0

+

+

0 0 z˙A 0 0 z˙A

+

+

hR β ˙A 0 ˙A a1 β

+

˙A hR β 0 ˙A +a2 β

+

∂x 1 ∂z 1

0 z˙1

∂x 2 ∂z 2

ω0RK ,0 = 1

+

0 β ˙ 1 0

and

ω0R

1

,0

=

0 β ˙A 0

+

z˙2

0 z˙2

The angular velocities of the wheel bodies and wheels are given by

0 ˙A β 0

z˙1

0 ˙1 β 0

       

+

;

(4.43)

.

(4.44)

0 ϕ˙ R1 0

(4.45)

61

Vehicle Dynamics


as well as

ω0RK ,0 = 2

        0 ˙A β 0

0 β ˙ 2 0

+

ω0R

and

2

,0

=

            0 β ˙A 0

0 ˙2 β 0

+

+

0 ϕ˙ R2 0

(4.46)

Introducing a vector of generalized velocities

 

˙ 2 ϕ˙ R2 x˙ A z˙A β ˙A β ˙1 ϕ˙ R1 β

z =



T

(4.47)

the velocities and angular velocities (4.42 ( 4.42), ), (4.43 ( 4.43), ), (4.44), 4.44), (4.45), 4.45), (4.46) 4.46) can be written as 7

v0i =

j=1 j =1

∂v 0i z j ∂z j

7

and



ω0i =

j=1 j =1

∂ω 0i z j ∂z j

(4.48)

4.4.2 Equations Equations of of Motion Motion ∂ω i i i =1(1)5 and for The partial velocities ∂v ∂z j and partial angular velocities ∂z j for the five bodies 1(1)7 are arranged in the tables 4.1 and 4.2. With the aid of the seven generalized speeds j = 1(1)7 0

0

partial velocities ∂v 0i /∂z j bodies chassis

mA

wheel body front

mRK 1

wheel front

mR1

wheel body rear

mRK 2

wheel rear

mR2

x˙ A 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0

z˙A 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1

β ˙A 0 0 0 hR 0 a1 hR 0 a1 hR 0 a2 hR 0 a2

− − − − − −

z˙1 0 0 0 ∂x 1 ∂z 1

0 1

∂x 1 ∂z 1

0 1 0 0 0 0 0 0

ϕ˙ R1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

applied forces

z˙2 0 0 0 0 0 0 0 0 0 ∂x 2 ∂z 2

0 1

∂x 2 ∂z 2

0 1

ϕ˙ R2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

F ie 0 0 F F 1 F 1 + F F 2 F 2 mA g 0 0 F F 1 F 1 mRK 1 g F x1 0 F z1 mR1 g 0 0 F F 2 F 2 mRK 2 g F x2 0 F z2 mR2 g

−

− − −

− − −

Table 4.1: Partial Velocities and Applied Forces the partial velocities and partial angular velocities the elements of the mass matrix M and the components of the vector of generalized forces and torques Q can be calculated. 5

M (i, j ) =

  k=1

62

∂v 0k ∂z i

T

∂v 0k mk + ∂z j

5

  k=1

∂ω 0k ∂z i

T

Θk

∂ω 0k ; ∂z j

i, j = 1(1)7 ;

(4.49)



partial angular velocities ∂ω 0i /∂z j bodies chassis

ΘA

wheel body front

ΘRK 1

wheel front

ΘR1

wheel body rear

ΘRK 2

wheel rear

ΘR2

x˙ A 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

β ˙A 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0

z˙A 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

z˙1 0 0 0 0

ϕ˙ R1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0

∂β 1 ∂z 1

0 0

∂β 1 ∂z 1

0 0 0 0 0 0 0

z˙2 0 0 0 0 0 0 0 0 0 0

applied torques

M ie 0 M A1 M A2 a1 F F 1 F 1 + a2 F F 2 F 2 0 0 M B1 0 0 M A1 M B1 R F x1 0 0 M B2 0 0 M A2 M B2 R F x2 0

ϕ˙ R2 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0

∂β 2 ∂z 2

0 0

∂β 2 ∂z 2

0

−

−

−

−

−

−

−

Table 4.2: Partial Angular Velocities and Applied Torques 5

Q(i) =

  ∂v 0k ∂z i

k=1

5

T

F ke

+

  k=1

∂ω 0k ∂z i

T

M ke ;

i = 1(1)7 .

(4.50)

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

M z˙ = Q .

(4.51)

4.4.3 Equilibrium Equilibrium With the abbreviations abbreviations

m1 = mRK 1 + mR1 ;

m2 = mRK 2 + mR2 ;

mG = mA + m1 + m2

(4.52)

and

h = hR + R

(4.53)

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

Q(1) = F x1 + F x2 ; Q(2) = F z 1 + F z 2 Q(3) =

−a F

1 z1

Q(4) = F z1 Q(5) =

−m

G

+ a2 F z 2

(4.54)

g;

− h(F

∂x 1 ∂z 1

x1

+ F x2 ) + a1 m1 g

− F + F − m M − M − R F ; A1

F 1 F 1

B1

x1

1

g+

∂β 1 (M A1 ∂z 1

−a

2

m2 g ;

− R F

x1 ) ;

(4.55)

x1

63

Vehicle Dynamics


Q(6) = F z2 Q(7) =

∂x 2 ∂z 2

− F + F − m M − M − R F . F 2 F 2

A2

x2

B2

2

g+

∂β 2 (M A2 ∂z 2

x2 ) ;

− R F

(4.56)

x2

Without drive and brake forces

M A1 = 0 ;

M A2 = 0 ;

M B1 = 0 ;

M B2 = 0

(4.57)

from (4.54 (4.54), ), (4.55) 4.55) and (4.56 (4.56)) one gets the steady state longitudinal forces, the spring preloads and the wheel loads

F x01 = 0 ;

F x02 = 0 ;

b 0 F F 1 F 1 = a+b mA g ; F z01 = m1g + a+b b mA g ;

a 0 F F 2 F 2 = a+b mA g ; F z02 = m2 g + a+a b mA g .

(4.58)

4.4.4 Driving Driving and Braking Braking Ä = 0 the wheels neither slip nor Assuming that on accelerating or decelerating the vehicle x lock,



R ϕ˙ R1 = x˙ A R ϕ˙ R2 = x˙ A

˙ + ˙ R β A +

∂x 1 ∂z 1 ∂x 2 ∂z 2

−h −h

R β A

z˙1 ;

(4.59)

z˙2 .

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

β A = β Ast = const. ;

z1 = z1st = const. ;

z2 = z2st = const.

(4.60)

and (4.59 (4.59)) simplifies to

R ϕ˙ R1 = x˙ A ;

R ϕ˙ R2 = x˙ A .

(4.61)

With(4.60 With(4.60), ), (4.61 ( 4.61)) and (4.53 (4.53)) the equation of motion (4.51 (4.51)) results in

mG xÄ = F xa1 + F xa2 ; 0 = F za1 + F za2 ;

−h

Ä R (m1 + m2 ) x

∂x 1 ∂z 1

∂x 2 ∂z 2

Ä + m1 x

m2 xÄ +

+ Θ R1

∂β 1 ∂z 1

∂β 2 ∂z 2

ΘR 1

x Ä R

x Ä R

+ Θ R2

= F za1

ΘR1 x¨RA

=

x Ä R

=

ΘR2

ΘR2 x¨RA

x Ä R

=

a z1

−a F

= a F 1 F 1

∂x 1 ∂z 1

+ b F za2

a x1

− F + F M − M − R F F − F + F M − M − R F A1

a z2

a x1

B1

a F 2 F 2

A2

B2

∂x 2 ∂z 2

+

− (h

R

+ R)(F )(F xa1 + F xa2 ) ; (4.62)

∂β 1 (M A1 ∂z 1

− R F

∂β 2 (M A2 ∂z 2

− R F

a x1 ) ;

(4.63)

;

a x2

+

a x2

;

a x2 ) ;

(4.64)

where the steady state spring forces, longitudinal forces and wheel loads have been separated into initial and acceleration-dependent terms st 0 a F xi = F xi + F xi ;

64

F zist = F zi0 + F zia ;

F Fsti = F F0 i + F Fai ;

i = 1, 2 .

(4.65)



Ä , the wheel forces F xa1 , F xa2 , With given torques of drive and brake the vehicle acceleration x a a F za1 , F za2 and the spring forces F F 1 ( 4.62), ), (4.63 ( 4.63)) and (4.64 (4.64)) F 1 , F F 2 F 2 can be calculated from (4.62 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 a F F 1 F 1 a F F 2 F 2 F za1 F za2

= = = =

cA1 z1a , cA2 z2a , cR1 (zAa a β Aa + z1a) , cR2 (zAa + b β Aa + z2a ) .

− −

−

(4.66)

besides the vertical motions of the wheels. a = 0 , caused by drive or brake is, if too distinct, felt as Especially the pitch of the vehicle β A annoying.



By an axle kinematics with ’anti dive’ and/or ’anti squat’ properties the drive and/or brake pitch angle can be reduced by rotating the wheel body and moving the wheel center in longitudinal direction during jounce and rebound.

4.4.5 Brake Pitch Pitch Pole Pole For real suspension systems the brake pitch pole can be calculated from the motions of the wheel contact points in the x-, z -plane, Fig. 4.9. 4.9.

pitch pole

x-, z- motion of the contact points during compression and rebound

Figure 4.9: Brake Pitch Pole Increasing the pitch pole height above the track level means a decrease in the brake pitch angle.

65

5 Late Latera rall Dyna Dynami mics cs 5.1 Kinematic Kinematic Approach Approach 5.1.1 Kinematic Kinematic Tire Tire Model Model When a vehicle drives through the curve at low lateral acceleration, low lateral forces are needed for for cour course se hold holdin ing. g. At the the whee wheels ls then then hard hardly ly late latera rall slip slip occu occurs rs.. In the the idea ideall case case,, with with vanis anishi hing ng lateral slip, the wheels only move in circumferential direction. The speed component of the contact point in the tire’s lateral direction then vanishes

vy = eyT v0P = 0 .

(5.1)

This This kinema kinematic tic constr constrain aintt equati equation on can can be used used for course course calcu calculat lation ion of slowly slowly movin moving g vehic vehicles les..

5.1.2 Ackermann Ackermann Geometry Geometry Within Within the validity validity limits of the kinematic kinematic tire model the necessa necessary ry steering steering angle of the front wheels can be constructed via given momentary turning center M , Fig. 5.1. 5.1. At slowly moving vehicles the lay out of the steering linkage is usually done according to the Ackermann geometry. Then, it holds

tan δ1 =

a R

and

tan δ2 =

a , R+s

(5.2)

where s the track width and a denotes the wheel base. Eliminating the curve radius R we get

tan δ2 =

a

or

a +s tan δ1

tan δ2 =

a tan δ1 . a + s tan δ1

(5.3)

δ2A of the actual steering angle δ2a from the Ackermann steering The deviations δ2 = δ2a angle δ2A , which follows from (5.3 ( 5.3), ), are used to judge a steering system.



−

At a rotation around the momentary pole M the direction of the velocity is fixed for every point of the vehicle. The angle β between the velocity vector v and the vehicle’s longitudinal axis is called side slip angle. The side slip angle at point P is given by

tan β P P =

x R

or

tan β P P =

x tan δ1 , a

where x denotes the distance of P to the to the inner rear wheel.

66

(5.4)


© Prof. Dr.-Ing. G. Rill δ2

δ1

v

βP P

a

x

M

δ2

βP

δ1

s

R

Figure 5.1: Ackermann Steering Geometry at a two-axled Vehicle

5.1.3 Space Requirement Requirement The The Acke Ackerm rman ann n appr approa oach ch can can also also be used used to calc calcul ulat atee the the spac spacee requ requir irem emen entt of a vehi vehicl clee duri during ng cornering, Fig. 5.2. 5.2. If the front wheels of a two-axled vehicle are steered according to the Ackermann geometry the outer point of the vehicle front runs on the maximum radius Rmax and a point on the inner side of the vehicle at the location of the rear axle runs on the minimum radius Rmin. We get 2 Rmax = (Rmin + b)2 + (a (a + f ) f )2 ,

汽车的通过宽度

(5.5)

where a, b are the wheel base and the width of the vehicle, and f specifies the distance of the vehicle front to the front axle. Hence, the space requirement

R = R − R max

min

=



(Rmin + b)2 + (a (a + f ) f )2

−R

min

,

(5.6)

can be calculated as a function of the cornering radius Rmin . The space requirement R of a typical passenger car and a bus is plotted in Fig. 5.3 versus the minimum cornering radius.



In narrow curves Rmin = 5.0 m a bus requires a space of 2.5 the width, whereas a passenger car needs only 1.5 the width.

67

Vehicle Dynamics


R

f

m a x

a

Rmin

M

b

Figure 5.2: Space Requirement 7 bus: a=6.25 m, b=2.50 m, f=2.25 m car: a=2.50 m, b=1.60 m, f=1.00 m

6 5 ] m4 [ R

∆ 3

2 1 0

0

10

20 30 R min [m]

40

50

Figure 5.3: Space Requirement of typical Passenger Car and Bus

68



5.1.4 Vehicle ehicle Model with Trailer Trailer 5.1.4.1 Position Position Fig. 5.4 shows a simple lateral dynamics model for a two-axled vehicle with a single-axled trailer. Vehicle and trailer move on a horizontal track. The position and the orientation of the

x 1

y 1

a

δ

A1

y 2 b

γ x 2

K

A2

c

y0

κ x 3 y 3

A3

x0

Figure 5.4: Kinematic Model with Trailer vehicle relative to the track fixed frame x0 , y0 , z0 is defined by the position vector to the rear axle center

r02, 02,0 = and the rotation matrix

A02 =

 

cos γ sin γ 0

    xF yF R

− sin γ

0 cos γ 0 0 1

(5.7)

 

.

(5.8)

Here, the tire radius R is considered to be constant, and xF , yF as well as γ are generalized coordinates.

69

Vehicle Dynamics


The position vector

r01, 01,0 = r02, 02,0 + A02 r21, 21,2

r21, 21,2 =

mit

and the rotation matrix

A01 = A02 A21

mit

A21 =

 

cos δ sin δ 0

    a 0 0

− sin δ cos δ 0

0 0 1

(5.9)

 

(5.10)

describe the position and the orientation of the front axle, where a = const labels the wheel base and δ the steering angle. The position vector



 −   

r03, K,2 + A23 rK 3,3 03,0 = r02, 02,0 + A02 r2K,2 with

r2K,2 K,2 = and the rotation matrix

−   

A03 = A02 A23

b 0 0

mit

rK 3,2 =

and

A23 =

 

(5.11)

c 0 0

cos κ sin κ 0

− sin κ cos κ 0

(5.12)

0 0 1

 

(5.13)

define the position and the orientation of the trailer axis, with κ labelling the bend angle between vehicle and trailer and b, c marking the distances from the rear axle 2 to the coupling point K and from the coupling point K to the trailer axis 3.

5.1.4.2 Vehicle Vehicle According to the kinematic tire model, cf. section 5.1.1, the velocity at the rear axle can only have a component in the vehicle’s longitudinal direction

v02, 02,2 = The time derivative of (5.7 ( 5.7)) results in

        vx2 0 0

v02, 02,0 = r˙02, 02,0 =

70

.

x˙ F y˙F 0

(5.14)

.

(5.15)



With the transformation of (5.14 (5.14)) into the system 0

v02, 02,0 = A02 v02, 02,2 = A02

      vx2 0 0

=

cos γ vx2 sin γ vx2 0

 

(5.16)

one gets by equalizing with (5.15 ( 5.15)) two first order differential equations for the position coordinates xF and yF

x˙ F = cos γ vx2 ,

(5.17)

y˙F = sin γ vx2 . The velocity at the front axis follows from (5.9 ( 5.9))

v01, 01,0 = r˙01, 01,0 = r˙02, 02,0 + ω02, 02,0

×A

02 r21, 21,2

.

(5.18)

Transformed into the vehicle fixed system x2 , y2 , z2

         ×                    −   

v01, 01,2 =

vx2 0 0

v02, 02,2

remains. The unit vectors

ex1,2 =

0 0 γ ˙

a 0 0

ω02, 02,2

r21, 21,2

+

cos δ sin δ 0

and

ey1,2 =

=

vx2 a γ ˙ 0

 

sin δ cos δ 0

 

define the longitudinal and lateral direction at the front axle.

.

(5.19)

(5.20)

According to (5.1 (5.1)) the velocity component lateral to the wheel must vanish,

eyT 1,2 v01, 01,2 =

− sin δ v

x2

+ cos δ a γ ˙ = 0.

(5.21)

In longitudinal direction then

exT 1,2 v01, ˙ = vx1 01,2 = cos δ vx2 + sin δ a γ

(5.22)

remains. From (5.21 (5.21)) a first order differential equation follows for the yaw angle

γ ˙ =

vx2 tan δ . a

(5.23)

71

Vehicle Dynamics


5.1.4.3 5.1.4.3 Entering Entering a Curve Curve In analogy to (5.2 (5.2)) the steering angle δ can be related to the current track radius R or with k = 1/R to the current track curvature

tan δ =

a = ak. R

(5.24)

The differential equation for the yaw angle then reads as

γ ˙ = vx2 k .

(5.25)

With the curvature gradient

k = k(t) = kC

t T

(5.26)

The entering of a curve is described as a continuous transition from a line with the curvature k = 0 into a circle with the curvature k = kC . The yaw angle of the vehicle can now be calculated by simple integration

vx2 kC t2 γ (t) = , T 2

(5.27)

where at time t = 0 a vanishing yaw angle, γ (t =0) = 0, has been assumed. The vehicle’s position then follows with ( 5.27) 5.27) from the differential equations (5.17 (5.17)) t=T

xF = vx2

  cos

t=0

vx2 kC t2 T 2



t=T

dt ,

yF = vx2

  sin

vx2 kC t2 T 2

t=0



dt .

(5.28)

At constant vehicle speed vx2 = const. (5.28) 5.28) is the parameterized form of a clothoide. From (5.24 (5.24)) the necessary steering angle can be calculated, too. If only small steering angles are necessary for driving through the curve, the tan-function can be approximated by its argument, and

δ = δ(t)

≈ ak

= a kC

t T

(5.29)

holds, i.e. the driving through a clothoide is manageable by continuous steer motion.

5.1.4.4 Trailer The velocity of the trailer axis can be received by differentiation of the position vector ( 5.11) 5.11)

v03, 03,0 = r˙03, 03,0 = r˙02, 02,0 + ω02, 02,0

×A

With

+ A02 r˙23, 23,2 .

− −   −  b

r23, 23,2 = r2K,2 K,2 + A23 rK 3,3 =

72

02 r23, 23,2

c cos κ c sin κ 0

(5.30)

(5.31)


and


  −      × −   −              − −        × −  −  − −               −       − −  −    − −  c cos κ c sin κ 0

0 0 κ˙

r˙23, 23,2 =

ω23, 23,2

c sin κ κ˙ c cos κ κ˙ 0

=

(5.32)

A23 rK 3,3

it remains, remains, if (5.30) 5.30) is transformed into the vehicle fixed frame x2 , y2 , z2

v03, 03,2 =

vx2 0 + 0

v02, 02,2

b

0 0 γ ˙

ω02, 02,2

c cos κ c sin κ 0

c sin κ κ˙ c cos κ κ˙ 0

+

r23, 23,2

=

vx2 + c sin κ (κ˙ + γ ˙) b γ ˙ c cos κ (κ˙ + γ ˙) 0

r˙23, 23,2

 

.

(5.33)

The longitudinal and lateral direction at the trailer axis are defined by the unit vectors

cos κ sin κ 0

ex3,2 =

and

sin κ cos κ 0

ey3,2 =

.

(5.34)

At the trailer axis the lateral velocity must also vanish

eyT 3,2 v03, 03,2 =

sin κ vx2 + c sin κ (κ˙ + γ ˙ ) + cos κ

b γ ˙

c cos κ (κ˙ + γ ˙)

= 0 . (5.35)

In longitudinal direction

exT 3,2 v03, ˙ + γ ˙ ) + sin κ 03,2 = cos κ vx2 + c sin κ (κ

b γ ˙

c cos κ (κ˙ + γ ˙)

= vx3 (5.36)

remains.

When (5.23 (5.23)) is inserted into (5.35 ( 5.35), ), one gets a differential equation of first order for the bend angle

κ˙ =

−

vx2 a



a sin κ + c



b cos κ + 1 c

  tan δ

.

(5.37)

The differential equations (5.17 ( 5.17)) and (5.23 (5.23)) describe position and orientation within the x0 , y0 plane. The position of the trailer relative to the vehicle follows from (5.37 ( 5.37). ).

5.1.4.5 Course Calculations For a given set of vehicle parameters a, b, c, and predefined time functions of the vehicle speed, vx2 = vx2 (t) and the steering angle, δ = δ (t) the course of vehicle and trailer can be calculated by numerical integration of the differential equations (5.17 ( 5.17), ), (5.23 ( 5.23)) and (5.37 (5.37). ). If the steering angle is slowly increased at constant driving speed, then the vehicle drives figure which is similar to a clothoide, Fig. 5.5.

73

Vehicle Dynamics


front axle rear axle trailer axle

20 ] m [

10

0

-30

-20

-10

0

10 [m]

20

30

40

50

60

30 front axle steer angle

δ

] 20 d a r G [

10 0

0

5

10

15 [s]

20

25

30

Figure 5.5: Entering a Curve

5.2 Steady Steady State State Cornerin Cornering g 5.2.1 Cornering Cornering Resistanc Resistance e In a body fixed reference frame B , Fig. 5.6, 5.6, the velocity state of the vehicle can be described by

v0C,B =

 

v cos β v sin β 0

 

und

ω0F,F =

    0 0 ω

.

(5.38)

where β denotes the side slip angle of the vehicle at the center of gravity. The angular velocity of a vehicle cornering with constant velocity v on an flat horizontal horizontal road is given by

ω=

v , R

(5.39)

where R denotes the radius of curvature. In the body fixed reference frame linear and angular momentum result in

m

− 

m

74

 

v2 sin β R v2 cos β R

= F x1 cos δ

− F

sin δ + F x2 ,

(5.40)

= F x1 sin δ + F y1 cos δ + F y2 ,

(5.41)

y1

0 = a1 (F x1 sin δ + F y1 cos δ )

−a

2

F y2 ,

(5.42)



Fx2

a2

Fy2

a1

C

ω

R β v

xB

Fx1

yB

Fy1

δ

Figure 5.6: Cornering Resistance where m denotes the mass of the vehicle, F x1 , F x2 , F y1 , F y2 are the resulting forces in longitudinal and vertical direction applied at the front and rear axle, and δ specifies the average steer angle at the front axle. The engine torque is distributed by the center differential to the front and rear axle. Then, in steady state condition it holds

F x1 = k F D

und

F x2 = (1

− k) F

D

,

(5.43)

where F D is the driving force and by k different driving conditions can be modelled:

k=0 0
Rear Wheel Drive All Wheel Drive Front Wheel Drive

F x1 = 0, F x2 = F D F x1 k = 1 k F x2 F x1 = F D , F x2 = 0

−

75

Vehicle Dynamics


If we insert (5.43 (5.43)) into (5.40 (5.40)) we get



 −

k cos δ + (1 k) F D

−

k sin δ F D +

sin δ F y1

=

cos δ F y1 +

a1k sin δ F D + a1 cos δ F y1

−

mv 2 sin β , R mv2 cos β , R

−

F y2 =

(5.44)

a2 F y2 = 0 .

This equations can be resolved for the drive force

a2 cosβ sin δ sin β cosδ mv 2 a1 + a2 F D = . k + (1 k) cos δ R

−

−

(5.45)

The drive force vanishes, if

a2 cosβ sin δ = sin β cosδ a1 + a2

a2 tan δ = tan β a1 + a2

or

(5.46)

holds. This corresponds with the Ackermann geometry. But the Acke Ackerma rmann nn geomet geometry ry holds holds only only for small small latera laterall accel accelera eratio tions. ns. In real real drivin driving g situa situatio tions ns the side slip angle of a vehicle at the center of gravity is always smaller then the Ackermann side slip angle. Then, due to tan β < a a+a tan δ a drive force F D > 0 is needed to overcome the ’cornering resistance’ of the vehicle. 2

1

2

5.2.2 Overturning Overturning Limit The overturning hazard of a vehicle is primarily determined by the track width and the height of the center of gravity. With trucks however, also the tire deflection and the body roll have to be respected., Fig. 5.7. The balance of torques at the already inclined vehicle delivers for small angles α1

(F zL zL

− F

zR zR )

s = m ay (h1 + h2 ) + m g [(h [(h1 + h2)α1 + h2 α2 ] , 2

 1, α  1 2

(5.47)

where ay indicates the lateral acceleration and m is the sprung mass. On a left-hand tilt, the right tire raises K F zR = 0

(5.48)

and the left tire carries all the vehicle weight K F zL = mg.

(5.49)

Using (5.48 (5.48)) and (5.49 (5.49)) one gets from (5.47 ( 5.47))

s 2 = g h1 + h2

aK y

76

K 1

−α −

h2 α2K . h1 + h2

(5.50)



α1

α2

m ay h2 mg

h1

F yL

F yR

FzL

s/2

s/2

FzR

Figure 5.7: Overturning Hazard on Trucks The vehicle turns over, when the lateral acceleration ay rises above the limit aK y K Roll of axle and body reduce the overturning limit. The angles αK 1 and α2 can be calculated from the tire stiffness cR and the body’s roll stiffness.

On a straight-ahead drive, the vehicle weight is equally distributed to both tires stat stat F zR = F zL =

1 mg. 2

(5.51)

F

(5.52)

With K stat F zL = F zL +

z

and the relations (5.49 ( 5.49), ), (5.51) 5.51) one gets for the increase of the wheel load at the overturning limit

F

z

=

1 mg. 2

(5.53)

The resulting tire deflection then follows from

F

z

= cR

r ,

(5.54)

where cR is the radial tire stiffness.

77

Vehicle Dynamics


Because the right tire simultaneously rebounds for the same amount, for the roll angle of the axle

2

r = s α

K 1

2

αK 1 =

or

r s

=

mg . s cR

(5.55)

holds. In analogy to (5.47 (5.47)) the balance of torques at the body delivers

cW α2 = m ay h2 + m g h2 (α1 + α2 ) ,

∗

(5.56)

where cW names the roll stiffness of the body suspension. Accordingly, at the overturning limit ay = aK y

α2K

aK mgh2 mgh2 y = + α1K g cW mgh2 cW mgh2

−

(5.57)

−

holds. Not allowing the vehicle to overturn already at aK y = 0 demands a minimum of roll min = mgh2 . stiffness cW > cW With (5.55 (5.55)) and (5.57 ( 5.57)) the overturning condition (5.50 ( 5.50)) reads as

aK s y (h1 + h2 ) = g 2

−

1 (h1 + h2 ) ∗ cR

−

aK 1 y h2 ∗ g cW 1

− −h

2

1 ∗ cW

1

−1 c ∗ ,

(5.58)

R

where, for abbreviation purposes, the dimensionless stiffnesses

cR ∗ cR = mg s

and

∗ cW =

cW m g h2

(5.59)

1 ∗ cR

(5.60)

have been used. Resolved for the normalized lateral acceleration

aK y g

=

s 2 h1 + h2 +

h2 ∗ cW

−

−1

remains.

000 kg . The radial stiffness of one At heavy trucks, a twin tire axle can be loaded with m = 13 000 800 000 N/m and the track with can be set to s = 2 m. The values h1 = 0.8 m and tire is cR = 800 h2 = 1.0 m hold at maximal load. This values deliver the results shown in Fig. 5.8 Even at a ∗ 0.5 g . rigid body suspension cW the vehicle turns over at a lateral acceleration of ay The roll angle of the vehicle then solely results from the tire deflection.

→∞

≈

∗ = 5 the overturning limit lies at ay 0.45 g and so reaches At a normalized roll stiffness of cW already 90% of the maximum. The vehicle will then turn over at a roll angle of α 10◦ .

≈

78

≈



overturning limit a y /g

=αK +αK

roll angle

0.6

α

1

2

20

0.5 15 0.4 0.3

10

0.2 5 0.1 0

0

0 10 20 normalized roll stiffness stiffness c W *

0 10 20 normalized normalized roll roll stiffness c W *

Figure 5.8: Tilting Limit for a Truck at Steady State Cornering

5.2.3 Roll Support Support and Camber Compens Compensation ation When a vehicle drives through a curve with the lateral acceleration ay , centrifugal forces are delivered to the single masses. At the even roll model in Fig. 5.9 these are the forces 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 roll torque is generated, that rolls the body with the angle αA and leads to a opposite deflection of the tires z1 = z2 .

−

b /2

b/2 zA

mA a y

αA

SA

yA FF1

FF2 h0

z2 mR a y r0

mR a y

S2 Q2 Fy2

z1

α2

y2 F y2

α1

S1 Q1 F z1

y1 F y1

Figure 5.9: Plane Roll Model

79

Vehicle Dynamics


At steady state cornering, the vehicle is balanced. With the principle of virtual work

δW = 0

(5.61)

the equilibrium position can be calculated. At the plane vehicle model in Fig. 5.9 the suspension forces F F 1 F 1 , F F 2 F 2 and tire forces F y 1 , F z 1 , F y2 , F z 2 , are approximated by linear spring elements with the constants cA and cQ , cR . The work W of these forces can be calculated directly or using W = V via the potential V . At small deflections with linearized kinematics one gets

−

W =

−m a y −m a ( y + h α + y ) − m a ( y + h α + y ) − c z − c z (5.62) − c (z − z ) − c (y + h α + y + r α ) − c (y + h α + y + r α ) − c z + α +z − c z − α +z , where the abbreviation h = h − r has been used and c describes the spring constant of the A

y

R

y

A

A

1 2

2 A 1

1 2

S

1

1 2

Q

A

1 2

R



R

1 2

R

y

A

R

A

2

2

2

0

A

2

1

2 A 2

2

b 2

A

A

1

A

R

1

0



0

1

2

1 2

2

R

0

1 2



Q

A

b 2

A

A

0

2

S



A

2

0

2

2

2

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 z1 , ∂z

α1 =

∂α α1 ∂z

∂y z − ∂z

y2 =

and

2,

α2 =

α − ∂α ∂z

(5.63)

2

the work W can be described as function of the position vector

y = [ yA , zA , αA , z1 , z2 ]T .

(5.64)

W = W ( W (y)

(5.65)

Due to principle of virtual work (5.61 ( 5.61)) leads to

δW =

∂W δy = 0 . ∂y

(5.66)

Because of δy = 0 a system of linear equations in the form of



Ky = b

(5.67)

results from (5.66 (5.66). ). The matrix K and the vector b are given by

K =

80

    −

2 cQ

0

2 cQ h0

0

2 cR

0

2 cQ h0

0

cα

∂y Q c ∂z Q

cR

∂y Q c ∂z Q

cR

∂y ∂z

cQ

−

cR Q b c + h0 ∂y∂z cQ 2 R

Q b c + h0 ∂y∂z cQ 2 R

b 2 R

Q

∂y Q 0 ∂z cQ

− c −h

∗ + cS + cR cA

−c

S

Q

∂y ∂z

cQ

cR b 2 R

∂y Q 0 ∂z cQ

− c −h −c

S

∗ cA + cS + cR

    

(5.68)


and

b =

  − 


mA + 2 mR 0 (m1 + m2 ) hR mR ∂y/∂z mR ∂y/∂z

−

The following abbreviations have been used:

∂y Q ∂y ∂α = + r0 , ∂z ∂z ∂z

∗ cA

= cA + cQ

 ∂y ∂z

  

ay .

(5.69)

2

,

cα =

2 cQ h20

+ 2 cR

 b 2

2

.

(5.70)

The system of linear equations (5.67 ( 5.67)) can be solved numerically, e.g. with MATLAB. Thus the influence of axle suspension and axle kinematics on the roll behavior of the vehicle can be investigated. a)

b)

αA

γ 1

γ 2 roll center

γ 1

αA

roll center

0

γ 2

0

Figure 5.10: Roll Behavior at Cornering: a) without and b) with Camber Compensation If the wheels only move vertically to the body at bound and rebound, then, at fast cornering the wheels are no longer perpendicular to the track Fig. 5.10 a.

因为转向的侧倾，侧倾角为 0，内侧轮的侧倾角是如何为0的？

The camber angles γ 1 > 0 and γ 2 > 0 result in an unfavorable pressure distribution in the contact area, which leads to a reduction of the maximally transmittable lateral forces. At more sportive vehicles thus axle kinematics are employed, where the wheels are rotated around the longitudinal axis at bound and rebound, α1 = α1 (z1 ) and α2 = α2 (z2 ). With this, a 0. Fig. 5.10 b. By the rotation of ”camber compensation” compensation” can be achieved with γ 1 0 and γ 2 the wheels around the longitudinal axis on jounce, the wheel contact points are moved outwards, i.e against the lateral force. By this a ’roll support’ is achieved, that reduces the body roll.

≈

≈

5.2.4 Roll Center Center and and Roll Axis The ’roll center’ can be constructed from the lateral motion of the wheel contact points Q1 and Q2, Fig. 5.10. 5.10. The line through the roll center at the front and rear axle is called ’roll axis’, Fig. 5.11. 5.11.

81

Vehicle Dynamics


roll axis roll center front

roll center rear

Figure 5.11: Roll Axis

5.2.5 5.2.5 Wheel Wheel Loads Loads The roll angle of a vehicle during cornering depends on the roll stiffness of the axle and on the position of the roll center. Different axle layouts at the front and rear axle may result in different roll angles of the front and rear part of the chassis, Fig. 5.12. 5.12. -TT

+TT

PR0+∆P PF0+∆P

PF0-∆P

PR0+∆PR

PR0-∆P PF0+∆PF

PR0-∆PR

PF0-∆PF

Figure 5.12: Wheel Loads for a flexible and a rigid Chassis On most passenger cars the chassis is rather stiff. Hence, front an rear part of the chassis are forced via an internal torque to an overall chassis roll angle. This torque affects the wheel loads and generates different wheel load differences at the front and rear axle. Due to the digressive influence of the wheel load to longitudinal and lateral tire forces the steering tendency of a vehicle can be affected.

82



5.3 Simple Handling Handling Model Model 5.3.1 Modelling Modelling Concep Conceptt The main vehicle motions take place in a horizontal plane defined by the earth-fixed axis x0 and y0, Fig. 5.13. The tire forces at the wheels of one axle are combined to one resulting force. Tire x0 a2

y0

a1 Fy2 x2 y2

C

γ β

yB

xB

Fy1

y1 x1

δ

Figure 5.13: Simple Handling Model torques, the rolling resistance and aerodynamic forces and torques applied at the vehicle are left out of account.

5.3.2 Kinematics Kinematics The vehicle velocity at the center of gravity can easily be expressed in the body fixed frame xB , yB , zB

vC,B =

 

v cos β v sin β 0

 

,

(5.71)

where β denotes the side slip angle, and v is the magnitude of the velocity.

83

Vehicle Dynamics


For the calculation of the lateral slips, the velocity vectors and the unit vectors in longitudinal and lateral direction of the axles are needed. One gets

ex and

1

,B

=

      cos δ sin δ 0

ex

2

,B

=

,

1 0 0

ey

1

,

=

,B

ey

2

,B

−   

sin δ cos δ 0

=

0 1 0

 

  ,

,

v01,B 01,B =

v02,B 02,B =

 

 

v cos β v sin β + β + a1 γ ˙ 0

v cos β v sin β a2 γ ˙ 0

−

 

 

(5.72)

,

(5.73)

˙ 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.

5.3.3 5.3.3 Tire Tire Forces Forces Unlike with the kinematic tire model, now small lateral motions in the contact points are permitted. At small lateral slips, the lateral force can be approximated by a linear approach

F y = cS sy

(5.74)

where cS is a constant depending on the wheel load F z and the lateral slip sy is defined by (2.51). 2.51). Because the vehicle is neither accelerated nor decelerated, the rolling condition is fulfilled at every wheel rD Ω = exT v0P . (5.75) Here rD is the dynamic tire radius, v0P the contact point velocity and ex the unit vector in longitudinal direction. With the lateral tire velocity

vy = eyT v0P

(5.76)

and the rolling condition (5.75 ( 5.75)) the lateral slip can be calculated from

sy =

T y 0P T x 0P

−e v , |e v |

(5.77)

with ey labelling the unit vector in the tire’s lateral direction. So, the lateral forces can be calculated from

F y1 = cS 1 sy1 ; F y2 = cS 2 sy2 .

84

(5.78)



5.3.4 5.3.4 Lateral Lateral Slips Slips With (5.73 (5.73), ), the lateral slip at the front axle follows from (5.77 ( 5.77): ):

sy 1 =

+sin δ (v cos β ) cos δ (v sin β + β + a1 γ ˙) . cos δ (v cos β ) + sin δ (v sin β + β + a1 γ ˙)

−

|

|

(5.79)

The lateral slip at the rear axle is given by

sy 2 =

β − a γ ˙ − v sin | v cos β | . 2

(5.80)

˙ , the side slip angle β and the steering angle δ are considered The yaw velocity of the vehicle γ to be small a1 γ ˙ v ; a2 γ ˙ v (5.81)

|

|| | | || | | β |  1 and | δ |  1 .

(5.82)

Because the side slip angle always labels the smaller angle between speed vector and vehicle longitudinal axis, instead of v sin β v β the approximation

≈

v sin β

≈ |v| β

(5.83)

has to be used. Respecting (5.81 (5.81), ), (5.82) 5.82) and (5.83 (5.83), ), from (5.79 (5.79)) and (5.80 (5.80)) then follow

sy 1 =

−β − |av| γ ˙ + |vv| δ

(5.84)

−β + |av| γ˙ .

(5.85)

and

sy2 =

1

2

5.3.5 Equations Equations of of Motion Motion To derive the equations of motion, the velocities, angular velocities and the accelerations are needed. For small side slip angles β

 1, (5.71 ( 5.71)) can be approximated by vC,B =

The angular velocity is given by

 ||  

ω0F,B =

v v β 0 0 0 γ ˙

   

.

(5.86)

.

(5.87)

85

Vehicle Dynamics


If the vehicle accelerations are also expressed in the vehicle fixed frame xF , yF , zF , one finds at constant vehicle speed v = const and with neglecting small higher order terms

aC,B = ω0F,B

×v

+ v˙ C,B =

C,B

The angular acceleration is given by

ω˙ 0F,B = where the substitution

 

0 ˙ v γ + γ ˙ + v β 0

||

 

.

(5.88)

    0 0 ω˙

(5.89)

γ ˙ = ω

(5.90)

was used. The linear momentum in the vehicle’s lateral direction reads as

m (v ω + v β ˙ ) = F y1 + F y2 ,

||

(5.91)

where, due to the small steering angle, the term F y1 cos δ has been approximated by F y1 and m describes the vehicle mass. With (5.90 (5.90)) the angular momentum delivers

Θ ω˙ = a1 F y1

−a

2 F y 2

,

(5.92)

where Θ names the inertia of vehicle around the vertical axis. With the linear description of the lateral forces (5.78 ( 5.78)) and the lateral slips (5.84 ( 5.84), ), (5.85) ( 5.85) one gets from (5.91 (5.91)) and (5.92 (5.92)) two coupled, but linear first order differential equations

− − || || − −

˙ = cS 1 β m v ω˙ =

a1 v ω+ δ v v

β

a1 cS 1 Θ

β

||



cS 2 + m v

||

a2 β + β + ω v

||

−

v ω v

||

(5.93)

−  || || ||    −      −  | || | ||  | | | |               −  | |   | |   a1 v ω+ δ v v

−

−

a2 cS 2 Θ

β + β +

a2 ω v

,

(5.94)

which can be written in the form of a state equation

    −      β ˙ ω˙ x˙

=

cS 1 + cS 2 m v

a2 cS 2

||

−a

1 cS 1

Θ

a2 cS 2 a1 cS 1 m v v

a21 cS 1 + a22 cS 2 Θ v

A

v v

β + ω x

v cS 1 v m v

v a1 cS 1 v Θ

δ

. (5.95)

u

B

If a system can be, at least approximatively, described by a linear state equation, then, stability, steady state solutions, transient response, and optimal controlling can be calculated with classic methods of system dynamics.

86



5.3.6 5.3.6 Stabilit Stability y 5.3.6.1 Eigenvalues Eigenvalues The homogeneous state equation

x˙ = A x

(5.96)

describes the eigen-dynamics. If the approach

xh(t) = x0 eλt

(5.97)

is inserted into (5.96 (5.96), ), then the homogeneous equation remains

(λ E

− A) x

= 0.

(5.98)

− A| = 0 .

(5.99)

0

Non-trivial solutions x0 = 0 one gets for



det λ E

|

The eigenvalues λ provide information about the stability of the system.

5.3.6.2 Low Speed Speed Approximation Approximation The state matrix

  − 

Av→0 =

approximates at v

cS 1 + cS 2 m v

||

 −  − | || | ||   − ||

a2 cS 2 a1 cS 1 m v v a21 cS 1

v v

(5.100)

a22 cS 2

+ Θ v

0

→ 0 the eigen-dynamics of vehicles at low speeds.

The matrix (5.100 (5.100)) has the eigenvalues

λ1v

→0

=

−

cS 1 + cS 2 m v

and

||

λ2v

→0

=

−

a21 cS 1 + a22 cS 2 . Θ v

||

(5.101)

The eigenvalues are real and, independent from the driving direction, always negative. Thus, vehicles at low speed possess an asymptotically stable driving behavior!

5.3.6.3 High Speed Approximation Approximation At highest driving velocities v

→ ∞, the state matrix can be approximated by

Av→∞ =

 

0 a2 cS 2

−a

Θ

 −| |   v v

1 cS 1

.

(5.102)

0

87

Vehicle Dynamics


Using (5.102 (5.102)) one receives from (5.99 (5.99)) the relation

λv2→∞ + with the solutions

λ1,2v

→∞

=

v a2 cS 2 a1 cS 1 = 0 v Θ

−

||

 ± −

v a2 cS 2 a1 cS 1 . v Θ

−

||

(5.103)

(5.104)

When driving forward with v > 0, the root argument is positive, if

a2 cS 2

−a

1 cS 1

< 0

(5.105)

holds. Then however, one eigenvalue is positive and the system is unstable. Two zero-eigenvalues λ1 = 0 and λ2 = 0 one gets for

a1 cS 1 = a2 cS 2 .

(5.106)

The driving behavior is then indifferent. Slight parameter variations however can lead to an unstable behavior. With

a2 cS 2

−a

1 cS 1

> 0 or a1 cS 1 < a2 cS 2

这是汽车稳定 (5.107) 的条件

and v > 0 the root argument in (5.104 ( 5.104)) becomes negative. The eigenvalues are then imaginary, and disturbances lead to undamped vibrations. To avoid instability, high-speed vehicles have to satisfy the condition ( 5.107). 5.107). The root argument in (5.104 ( 5.104)) changes at backward driving its sign. A vehicle showing stable driving behavior at forward driving becomes unstable at fast backward driving!

5.3.7 Steady Steady State Soluti Solution on 5.3.7.1 Side Slip Angle and Yaw Yaw Velocity Velocity With a given steering angle δ = δ0 , after a certain time, a stable system reaches steady state. With xst = const. or x˙ st = 0, the state equation (5.95 ( 5.95)) is reduced to a linear system of equations

A xst =

−B u .

(5.108)

With the elements from the state matrix A and the vector B one gets from (5.108 ( 5.108)) two equations to determine the steady state side slip angle β st st and the steady state angular velocity ωst at a constant given steering angle δ = δ0

|v| (c |v| (a

S 1

88

+ cS 2 ) β st st + (m v v + a1 cS 1 a2 cS 2 ) ωst = v cS 1 δ0 ,

(5.109)

+ (a21 cS 1 + a22 cS 2 ) ωst = v a1 cS 1 δ0 ,

(5.110)

1 cS 1

||

−a

2 cS 2 ) β st st

−



−m |v| |v| and the second with −Θ |v|. The

where the first equation has been multiplied by solution can be derived from

β st st =

v cS 1 δ0

m v v + a1 cS 1 a2 cS 2

v a1 cS 1 δ0

a21 cS 1 + a22 cS 2

|v| (c

S 1

|v| (a

1 cS 1

||

+ cS 2 )

−a

−

(5.111)

m v v + a1 cS 1 a2 cS 2

||

−

a21 cS 1 + a22 cS 2

2 cS 2 )

and

|v| (c |v| (a

ωst =

|v| (c

S 1

|v| (a

1 cS 1

S 1

1 cS 1

+ cS 2 )

v cS 1 δ0

−a

2 cS 2 )

+ cS 2 )

v a1 cS 1 δ0

(5.112)

m v v + a1 cS 1 a2 cS 2

||

−

a21 cS 1 + a22 cS 2

−a

2 cS 2 )

The denominator results in

detD = v

||



cS 1 cS 2 (a1 + a2 )2 + m v v (a2 cS 2

||

−a

1 cS 1 )

For a non vanishing denominator detD = 0 steady state solutions exist



a m v |v | − v c (a + a ) |v| a + a + m v |v| a c − a c δ c c (a + a ) 1

.

(5.113)

1

a2

β st st =



S 2

1

2

2 S 2

2

S 1 S 2

1 S 1

1

0

,

(5.114)

2

v

ωst =

(5.115) a2 cS 2 a1 cS 1 δ0 . a1 + a2 + m v v cS 1 cS 2 (a1 + a2 ) At forward driving vehicles v > 0 the steady state side slip angle, starts with the kinematic

−

||

value v→0 β st =

v a2 δ0 v a1 + a2

v →0 ωst =

and

||

v δ0 a1 + a2

(5.116)

and decreases with increasing speed. At speeds larger then

vβ st = st=0



a2 cS 2 (a1 + a2 ) a1 m

(5.117)

the side slip angle changes the sign. Using the kinematic value of the yaw velocity equation ( 5.115) 5.115) can be written as

ωst =

v a1 + a2

1 1 +

||  v v

δ0 ,

(5.118)

v 2 vch

89

Vehicle Dynamics


where

vch =



cS 1 cS 2 (a1 + a2 )2 m (a2 cS 2 a1 cS 1 )

(5.119)

−

is called the ’characteristic’ speed of the vehicle. Because the rear wheels are not steered, higher slip angles at the rear axle can only be reached by slanting the car. steady state side slip angle

radius of curvrature

2

200

0 150 ] g e d [ β

-2 ] m 100 [ r

-4 -6

a1*c S1 /a2*c S2 = 0.66667 a1*cS1 /a2*c S2 = 1 a1*cS1 /a2*c S2 = 1.3333

-8 -10

0

10

m=700 kg; kg ; Θ=1000 kg m2; 这里是重心位置在中间

20 v [m/s]

30

a1 =1. =1.2 m; a2 =1. =1.3 m;

a1*cS1 /a2*cS2 = 0.66667 a1*cS1 /a2*cS2 = 1 a1*cS1 /a2*cS2 = 1.3333

50

0 0

40

10

cS 1 = 80000 N m;

20 v [m/s]

cS 2

30

40

110770 N m 不足 = 73846 N m 中性 55385 N m 过度

Figure 5.14: Steady State Cornering

In Fig. 5.14 the side slip angle β , and the driven curve radius R are plotted versus the driving speed v . The steering angle has been set to δ0 = 1.4321◦ , in order to let the vehicle drive a circle 0. The actually driven circle radius R has been calculated with the radius R0 = 100 m at v via v ωst = . (5.120)

→

R

Some concepts for an additional steering of the rear axle were trying to keep the vehicle’s side slip angle to zero by an appropriate steering or controlling. Due to numerous problems production stage could not yet be reached.

虽然4WS是为了消除汽车的侧偏角，但是因为各种原因，这是不可能达到的。

5.3.7.2 Steering Tendency Tendency After reaching reaching the steady state solution, solution, the vehicle moves moves in a circle. circle. When inserting inserting ( 5.120) 5.120) into (5.115 (5.115)) and resolving for the steering angle, one gets

a1 + a2 v 2 v a2 cS 2 a1 cS 1 δ0 = + m . R R v cS 1 cS 2 (a1 + a2 )

||

90

−

(5.121)



The first term is the Ackermann steering angle, which follows from ( 5.2) 5.2) with the wheel 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, 0 the second expression in (5.121 because at v ( 5.121)) becomes neglectably small.

→

At higher speeds, depending on the value of a2 cS 2 a1 cS 1 and the driving direction (forward: v > 0, backward: v < 0), the necessary steering angle differs from the Ackermann-steering angle. The difference is proportional to the lateral acceleration

−

v2 ay = . R

(5.122)

At v > 0 the steering tendency of a vehicle is defined by the position of the center of gravity a1 , a2 and the cornering stiffnesses at the axles cS 1 , cS 2 . The various steering tendencies are arranged in the table 5.1.

•

understeer

δ0 > δ0A

or

a1 cS 1 < a2 cS 2

or

a1 cS 1 <1 a2 cS 2

•

neutral

δ0 = δ0A

or

a1 cS 1 = a2 cS 2

or

a1 cS 1 =1 a2 cS 2

•

oversteer

δ0 < δ0A

or

a1 cS 1 > a2 cS 2

or

a1 cS 1 >1 a2 cS 2

Table 5.1: Steering Tendency Tendency of a Vehicle at Forward Driving

5.3.7.3 5.3.7.3 Slip Angles Angles

˙ st ˙ st = 0 and the relation (5.120 With the conditions for a steady state solution β (5.120), ), the st = 0, ω equations of motion (5.91 ( 5.91)) and (5.92 (5.92)) can be dissolved for the lateral forces F y1st =

a2 v2 m , a1 + a2 R

or

2

F y2st =

a1 v m a1 + a2 R

a1 F y2st = . a2 F y1st

(5.123)

With the linear tire model (5.74 ( 5.74)) one gets

F yst1 = cS 1 sst y1

and

F yst2 = cS 2 sst y2 ,

(5.124)

st where sst 5.123) and (5.124 (5.124)) yA and syA label the steady state lateral slips at the axles. From ( 5.123) now follows 1

2

F yst2 cS 2 sst a1 y2 = st = st a2 F y1 cS 1 sy1

or

sst a1 cS 1 y2 = st . a2 cS 2 sy 1

(5.125)

91

Vehicle Dynamics


That means, at a vehicle with understeer tendency ( a1 cS 1 < a2 cS 2 ) during steady state cornerst ing the slip angles at the front axle are larger then the slip angles at the rear axle, sst y 1 > sy 2 . So, the steering tendency can also be determined from the slip angle at the axles.

5.3.8 Influence Influence of Wheel Load on Cornering Cornering Stiffness Stiffness With identical tires at the front and rear axle, given a linear influence of wheel load on the raise of the lateral force over the lateral slip, lin cS 1 = cS F z 1

lin cS 2 = cS F z 2 .

and

(5.126)

holds. The weight of the vehicle G = m g is distributed over the axles according to the position of the center of gravity

F z1 =

a2 G and a1 + a2

.F z 2 =

a1 G a1 + a2

(5.127)

With (5.126 (5.126)) and (5.127 (5.127)) one gets

a2 G a1 + a2

(5.128)

a1 G. a1 + a2

(5.129)

lin a1 cS 1 = a1 cS

and lin a2 cS 2 = a2 cS

A vehicle with identical tires would thus be steering neutrally at a linear influence of wheel load on the slip stiffness, because of lin lin a1 cS (5.130) 1 = a2 cS 2 The fact that the lateral force is applied behind the center of the contact area at the caster offset a1 |vv| nL and a2 a2 + |vv| nL to a stabilization of the distance, leads, because of a1 driving behavior, independent from the driving direction.

→ −

→

1

1

At a real tire, a digressive influence of wheel load on the tire forces is observed, Fig. 5.15. 5.15. According to (5.92 (5.92)) the rotation of the vehicle is stable, if the torque from the lateral forces F y1 and F y2 is aligning, i.e. a1 F y1 a2 F y2 < 0 (5.131)

−

holds. At a vehicle with the wheel base a = 2.45 m the axle loads F z1 = 4000 N and F z 2 = 3000 N deliver the position of the center of gravity a1 = 1.05 m and a2 = 1.40 m. At equal slip on front and rear axle one receives from the table in 5.15 F y1 = 2576 N and F y2 = 2043 N . With this, 257.55 . The value is significantly the condition (5.131 (5.131)) delivers delivers 1.05 2576 1.45 2043 = 257. negative and thus stabilizing.

∗

−

∗

−

Vehicles with a1 < a2 have a stable, i.e. understeering driving behavior. If the axle load at the rear axle is larger than at the front axle ( a1 > a2 ), a stable driving behavior can generally only be achieved with different tires.

92



6

5 α

4 ] N k [

3

y

F

2

1

0 0

1

2

3

4

Fz [kN]

5

6

7

8

F z [N ] N ]

F y [N ] N ]

0 1000 2000 3000 4000 5000 6000 7000 8000

0 758 1438 2043 2576 3039 3434 3762 4025

Figure 5.15: Lateral Force F y over Wheel Load F z at different Slip Angles At increasing lateral acceleration the vehicle is more and more supported by the outer wheels. At a sufficiently rigid vehicle body the wheel load differences can differ, because of different kinematics (roll support) or different roll stiffnesses Due to the digressive influence of wheel load, the deliverable lateral force at an axle decreases with increasing wheel load difference. If the wheel load is split more strongly at the front axle than at the rear axle, the lateral force potential at the front axle decreases more than at the rear axle and the vehicle becomes more stable with increasing lateral force, i.e. more understeering.

93

6 Dri Drivin ving g Behavior Behavior of Single Single Vehi Vehicl cles es 6.1 Standard Standard Driving Driving Maneuve Maneuvers rs 6.1.1 Steady Steady State Corneri Cornering ng The steering tendency of a real vehicle is determined by the driving maneuver called steady state cornering. The maneuver is performed quasi-static. The driver tries to keep the vehicle on a circle with the given radius R. He slowly increases the driving speed v and, with this, because of ay = vR , the lateral acceleration, until reaching the limit. Typical results are displayed in Fig. 6.1. 2

80

4

60

] 2 g e d [ e l g 0 n a p i l s e -2 d i s

] g e d [ e40 l g n a r e e20 t s

0

-4

4

6

] 3 g e d [ e l g 2 n a l l o r

] N k [ 4 s d a o 3 l l e e h 2 w

5

1

1 0

0

0.2

0.4

0.6

lateral acceleration [g]

0.8

0 0

0.2

0.4

0.6

0.8

lateral acceleration [g]

Figure 6.1: Steady State Cornering: Rear-Wheel-Driven Car on R = 100 m The vehicle is under-steering and thus stable. The inclination in the diagram steering angle over lateral velocity decides, according to (5.121 ( 5.121)) with (5.122 (5.122), ), about the steering tendency and stability behavior.

94



The The nonl nonlin inea earr influ influen ence ce of the the whee wheell load load on the the tire tire perf perfor orma manc ncee is here here used used to desi design gn a vehi vehicl clee that is weakly stable, but sensitive to steer input in the lower range of lateral acceleration, 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 overturning torque is intercepted by according wheel load differences between the outer and inner wheels. With a sufficiently rigid frame the use of a anti roll bar at the front axle allows to increase the wheel load difference there and to decrease it at the rear axle accordingly. The digressive influence of the wheel load on the tire properties, cornering stiffness and maximally possible lateral force is thus stressed more strongly at the front axle and the vehicle becomes more under-steering and stable at increasing lateral acceleration, until, in the limit situation, it drifts out of the curve over the front axle. Problems occur at front driven vehicles, because, due to the traction, the front axle cannot be relieved at will. Having a sufficiently large test site, the steady state cornering maneuver can also be carried out at constant speed. There the steering wheel is slowly turned until the vehicle reaches the limit range. That way also weakly motorized vehicles can be tested at high lateral accelerations.

6.1.2 6.1.2 Step Steer Steer Input Input The dynamic response of a vehicle is often tested with a step steer input. Methods for the calculation and evaluation of an ideal response, as used in system theory or control technics, can not be used with a real car, for a step input at the steering wheel is not possible in practice. In Fig. 6.2 a real steering angle gradient is displayed. 40 ] g e d [

30

e l g n a

20

g n i r e e t s

10

0 0

0.2

0.4 0.6 time [s]

0.8

1

Figure 6.2: Step Steer Input Not the angle at the steering wheel is the decisive factor for the driving behavior, but the steer angle at the wheels, which can differ from the steering wheel angle because of elasticities, friction influences and a servo-support. At very fast steering movements also the dynamic raise of tire forces plays an important role. In practice, a step steer input is usually only used to judge vehicles subjectively subjectively.. Exceeds in yaw velocity, roll angle and especially sideslip angle are felt as annoying.

95

Vehicle Dynamics


0.6

12

0.5

10

] g [ n0.4 o i t a r e0.3 l e c c a0.2 l a r e t a0.1 l

] s / g e d [ y t i c o l e v w a y

8 6 4 2

0

0

3

1

2.5

0.5 ] g e 0 d [ e l g-0.5 n a p i l -1 s e d i s-1.5

2

] g e d [ 1.5 e l g n a 1 l l o r

0.5 0 0

2

4

-2

0

2

[t]

4

Figure 6.3: Step Steer: Passenger Car at v = 100 km/h The vehicle behaves dynamically dynamically very well, Fig. 6.3. 6.3. Almost no exceeds at roll angle and lateral acceleration. Small exceeds at yaw velocity and sideslip angle.

6.1.3 Driving Driving Straight Straight Ahead Ahead 6.1.3.1 Random Road Profile Profile The irregularities of a track are of stochastic nature. Fig. 6.4 shows a country road profile in different scalings. To To limit the effort at the stochastic description of a track, one usually employs simplifying models. Instead of a fully two-dimensional description either two parallel tracks are evaluated

z = z (x, y )

→

z1 = z1 (s1) ,

and

z2 = z2 (s2)

(6.1)

or one uses uses an isotro isotropic pic track. track. At an isotro isotropic pic track track the statis statistic tic proper propertie tiess are direct direction ion-independent. Then a two-dimensional track with its stochastic properties can be described by a single random process z = z (x, y ) z = z (s) ; (6.2)

→

96



0.05 0.04 0.03 0.02 0.01 0 -0.01 -0.02 -0.03 -0.04 -0.05 0

10

20

30

40

50

60

70

80

90

100

0

1

2

3

4

5

Figure 6.4: Track Irregularities A normally distributed, stationary and ergodic random process z = z (s) is completely characterized by the first two expectation values, mean value s



1 s→∞ 2s

mz = lim

z (s) ds

(6.3)

−s

and correlating function

1 Rzz (δ) = lim s→∞ 2s

s



z (s) z (s

−s

− δ) ds

(6.4)

. A vanishing mean value mz = 0 can always be achieved by an appropriate coordinate transformation. The correlation function is symmetric,

Rzz (δ ) = Rzz ( δ )

−

and

1 s→∞ 2s

Rzz (0) (0) = lim

(6.5)

s

   z ( s)

2

ds

(6.6)

−s

describes the squared average of zs .

Stochastic track irregularities are mostly described by power spectral densities (abbreviated by psd). Correlating function and the one-sided power spectral density are linked by the Fouriertransformation ∞

Rzz (δ) =



S zz cos(Ωδ) dΩ zz (Ω) cos(Ωδ

(6.7)

0

where Ω denotes the space circular frequency. With ( 6.7) 6.7) follows from (6.6 (6.6)) ∞

Rzz (0) =



S zz zz (Ω) dΩ .

(6.8)

0

97

Vehicle Dynamics


The psd thus gives information, how the square average is compiled from the single frequency shares. The power spectral densities of real tracks can be approximated by the relation 1

S zz zz (Ω) = S 0

  Ω Ω0

−w

(6.9)

Where the reference frequency is fixed to Ω0 = 1 m−1. The reference psd S 0 = S zz zz (Ω0 ) acts as a measurement for unevennes and the waviness w indicates, whether the track has notable irregularities in the short or long wave spectrum. At real tracks reference-psd and waviness lie within the range

1 10−6 m3

∗

≤ S ≤ 100 ∗ 10 0

−6

m3

and

6.1.3.2 6.1.3.2 Steering Steering Activity Activity A straightforward drive upon an uneven uneven track makes continuous steering corrections necessary. necessary. 90 km/h are displayed in Fig. 6.5. The histograms of the steering angle at a driving speed of v = 90km/h 6.5.

-6

highway: S0=1*10 =1*10

3

-5

m ; w=2

country road: S0=2*10

1000

1000

500

500

0

-2

0

[deg] 2

0

-2

0

3

m ; w=2

[deg] 2

Figure 6.5: Steering Activity on different Roads The track quality is reflected in the amount of steering actions. The steering activity is often used to judge a vehicle in practice.

6.2 Coach Coach with different different Loading Loading Conditions Conditions 6.2.1 6.2 .1 Da Data ta At trucks and coaches the difference between empty and laden is sometimes very large. In the table 6.1 all relevant data of a travel coach in fully laden and empty condition are arrayed. 1

cf.: M. Mitschke: Dynamik der Kraftfahrzeuge (Band B), Springer-Verlag, Berlin 1984, S. 29.

98


vehicle

kg ] mass [kg]


inertias [kg m2 ]

center of gravity [m]

empty

12 500 500

−3.800 | 0.000 | 1.500

fully laden

18 000 000

−3.860 | 0.000 | 1.600

12 500 0 0 0 15 5 00 0 0 0 0 155 000 15 400 0 250 0 20 0 55 0 0 250 0 202 160

Table 6.1: Data for a Laden and Empty Coach

The coach has a wheel base of a = 6.25 m. The front axle with the track width sv = 2.046 m has a double wishbone single wheel suspension. The twin-tire rear axle with the track widths soh = 2.152 m and sih = 1.492 m is guided by two longitudinal links and an a-arm. The airsprings are fitted to load variations via a niveau-control.

6.2.2 Roll Steer Steer Behavior Behavior ] 10 m c [ l 5 e v a r t 0 n o i s n -5 e p s u -10 s

-1

0 steer angle [deg]

1

Figure 6.6: 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 steer effect, Fig. 6.6.

6.2.3 Steady Steady State Corneri Cornering ng Fig. 6.7 shows the results of a steady state cornering on a 100 m-Radius. The fully occupied vehicle is slightly more understeering than the empty one. The higher wheel loads cause greater tire aligning torques and increase the digressive wheel load influence on the increase of the lateral forces. Additionally roll steering at the rear axle occurs. In the limit range both vehicles can not be kept on the given radius. Due to the high position of the center of gravity the maximal lateral acceleration is limited by the overturning hazard. At

99

Vehicle Dynamics

FH Regensburg, University of Applied Sciences steer angle δ

LW

[deg]

vehicle course

250

200

200

150

150

] 100 m [

100

50

50

0 0

0.1 0.2 0.3 0.4 lateral acceleration a y [g]

-100

wheel loads [kN] 100

100

50

50

0

0


0

0 [m]

100

wheel loads [kN]

0


Figure 6.7: Steady Steady State Cornering: Cornering: Coach Coach - - empty, empty, — fully occupied occupied the empty vehicle, the inner front wheel lift off at a lateral acceleration of ay 0.35 g . vehicle is fully occupied, this effect occurs already at ay

≈

≈ 0.4 g . If the

6.2.4 6.2.4 Step Steer Steer Input Input The results of a step steer input at the driving speed of v = 80 km/h can be seen in Fig. 6.8. 6.8. To achieve comparable acceleration values in steady state condition, the step steer input was done at the empty vehicle with δ = 90 Grad and at the fully occupied one with δ = 135 Grad. The steady state roll angle is at the fully occupied bus 50% larger than at the empty one. By the niveau-control the air spring stiffness increases with the load. Because the damper effect remains unchange, the fully laden vehicle is not damped as well as the empty one. The results are higher exceeds in the lateral acceleration, the yaw speed and sideslip angle.

6.3 Differen Differentt Rear Axle Concepts Concepts for a Passenge Passengerr Car A medium-sized passenger car is equipped in standard design with a semi-trailing rear axle. By accordingly changed data this axle can easily be transformed into a trailing arm or a single wishbone axis.

100


© Prof. Dr.-Ing. G. Rill yaw velocity

lateral acceleration a y [g] 0.4

ω Z [deg/s]

10 8

0.3

6 0.2 4 0.1 0

2 0

2

4

6

roll angle

α [deg]

8

0

0

2

4

side slip angle

6

8

β [deg]

8 2 6

1

4

0 -1

2

-2 0

0

2

4

[s] 6

8

0

2

4

[s] 6

8

Figure Figure 6.8: Step Steer: Steer: - - Coach empty empty, — Coach Coach fully occupied occupied 10

] m c 5 [ n o i t o 0 m l a -5 c i t r e-10 v

-5

0 lateral motion [cm]

5

Figure Figure 6.9: 6.9: Rear Rear Axle Kinematics: Kinematics: — Semi-T Semi-Trailin railing g Arm, - - Single Single Wishbone, ishbone, Arm

· · · Trailing

The semi-trailing axle realized in serial production represents, according to the roll support, Fig. 6.9, a compromise between the trailing arm and the single wishbone. The influences on the driving behavior at steady state cornering on a 100 m radius are shown in Fig. 6.10. Substituting the semi-trailing arm at the standard car by a single wishbone, one gets, without adaption of the other system parameters, a vehicle, which oversteers in the limit range. The single wishbone causes, compared to the semi-trailing arm a notably higher roll support. This increases the wheel load difference at the rear axle, Fig. 6.10. 6.10. Because the wheel load difference is simultaneously reduced at the front axle, the understeer tendency is reduced. In the limit range, this even leads to oversteer behavior.

101

Vehicle Dynamics


steer angle δ

LW

[deg]

roll angle α [Grad]

100

5 4 3

50

2 1

0 0

0.2

0.4

0.6

0.8

0 0

wheel loads front front [kN] 6

4

4

2

2


0.4

0.6

0.8

wheel loads rear [kN]

6

0 0

0 .2

0 0

0 .2 0.4 0.6 lateral acceleration a y [g]

0.8

Figure 6.10: Steady Steady State Cornering, — Semi-Trailing Semi-Trailing Arm, - - Single Wishbone, Arm

· · · Trailing

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.

6.4 Differen Differentt Influences Influences on Comfort Comfort and Safety 6.4.1 Vehicle ehicle Model Model Ford motor company uses the vehicle dynamics program VeDynA (Vehicle Dynamic Analysis) for comfort calculations. The theoretical basics of the program – modelling, generating the equations of motion, and numeric solution – have been published in the book ”G.Rill: Simulation von Kraftfahrzeugen, Vieweg 1994” Through program extensions, adaption to different operating systems, installation of interfaces to other programs and a menu-controlled in- and output, VeDynA has been subsequently subsequently develdeveloped to marketability by the company TESIS GmbH in Munich. At the tire model tmeasy (tire model easy to use), as integrated in VeDynA, the tire forces are calculated dynamically with respect to the tire deformation. For every tire a contact calculation

102



is made. The local inclination of the track is determined from three track points. From the statistic characteristics of a track, spectral density and waviness, two-dimensional, irregular tracks are calculated.

Figure 6.11: Car Model The vehicle model is specially distinguished by the following details: • • • •

nonlinea nonlinearr elastic elastic kinema kinematics tics of of the wheel wheel suspe suspensio nsions, ns, friction-a friction-affe ffected cted and and elastica elastically lly suspend suspended ed dampers dampers,, fully elastic elastic motor motor suspen suspension sion by static static and and dynamic dynamic force force elements elements (rubber elements and/or hydro-mounts, integrate integrated d passenge passenger-se r-seat at models. models.

Beyond this, interfaces to external tire- and force element models are provided. A specially developed integration procedure allows real-time simulation on a PC.

6.4.2 Simulation Simulation Results Results The vehicle, a Ford Mondeo, occupied by two persons, drives with v = 80 km/h over a country road. The thereby occurring accelerations at the driver’s seat rail and the wheel load variations are displayed in Fig. 6.12. 6.12. The peak values of the accelerations and the maximal wheel load variations are arranged in the tables 6.2 and 6.3 for the standard car and several modifications. It can be seen, that the damper friction, the passengers, the engine suspension and the compliance of the wheel suspensions, (here:represented by comfort bushings) influence especially the accelerations and with this the driving comfort. At fine tuning thus all these influences must be respected.

103

Vehicle Dynamics


accelera acceleration tion standard standard – friction friction – seat seat model model – engine engine mounts mounts – comfort comfort bushing bushingss

x ¨min ¨max x y¨min y¨max z¨min z¨max

-0.7192

-0.7133

-0.7403

-0.5086

-0.7328

+0.6543 +0.6100

+0.6695

+0.5092

+0.6886

-1.4199

-1.2873

-1.4344

-0.7331

-1.5660

+1.3991 +1.2529

+1.3247

+0.8721

+1.2564

-4.1864

-3.9986

-4.1788

-3.6950

-4.2593

+3.0623

2.7769

+3.1176

+2.8114

+3.1449

Table 6.2: Peak Acceleration Values

F

z

front left

standard standard – friction friction – seat seat model model – engine engine mount mountss – comfort comfort bushing bushingss 2.3830

2.4507

2.4124

2.3891

2.2394

front right 2.4208

2. 2.3856

2.4436

2.3891

2.4148

rear left

2. 2.1450

2.2616

2.1600

2.1113

2.1018

rear right

2.3355

2. 2.2726

2.3730

2.2997

2.1608

Table 6.3: Wheel Load Variations

104

max z

min z

F = F − F z


© Prof. Dr.-Ing. G. Rill body longitudinal acceleration [m/s 2 ]

road profil [m] 0.1

5

0.05

0

0

-0.05

-0.1

0

500

1000

-5

0

500

[m]

body lateral acceleration [m/s 2 ]

body vertical acceleration [m/s 2 ] 5

5

0

0

-5

0

500

1000

-5

0

500

[m]

wheel load front left [kN]

wheel load front right [kN] 6

5

5

4

4

3

3

2

2

1

1 0

500

1000

0

0

500

[m] wheel load rear left [kN]

wheel load rear right [kN] 6

5

5

4

4

3

3

2

2

1

1 0

500

1000 [m]

1000 [m]

6

0

1000 [m]

6

0

1000 [m]

0

0

500

1000 [m]

Figure 6.12: Road Profile, Accelerations and Wheel Loads

105

Index

Ackermann Geometry, 66 Ackermann Steering Angle, 66, 66, 91 Aerodynamic Forces, 53 Air Resistance, 53 All Wheel Drive, 75 Angular Wheel Velocity, 27 Anti Dive, 65 Anti Roll Bar, 80 Anti Squat, 65 Anti-Lock-Systems, 59 Axle Kinematics, 65 Double Wishbone, 10 McPherson, 10 Multi-Link, 10 Axle Load, 52 Axle Suspension Suspension Rigid Axle, 4 Twist Beam, 5 Bend Angle, 73 Brake Pitch Angle, 60 Brake Pitch Pole, 65 Camber Angle, 9, 16 Camber Compensation, Compensation, 79, 81 Camber Slip, 26 Caster Angle, 11 Caster Offset, 12 Characteristic Speed, 90 Climbing Capacity, 54 Comfort, 31 Contact Geometry, 15 Contact Point, 16 Contact Point Velocity, 20 Cornering Resistance, 74, 74, 76

Cornering Stiffness, 24, 91 Damper Characteristic, 40 Disturbing Force Lever, 12 Down Forces, 53 Downhill Capacity, 54 Drag Link, 6, 7 Drive Pitch Angle, 60 Driver, 2 Driving Maximum Acceleration, Acceleration, 55 Driving Comfort, 35 Driving Safety, 31 Dynamic Axle Load, 52 Dynamic Force Elements, 45 Dynamic Wheel Loads, 51 Eigenvalues, 33, 33, 87 Environment, 3 First Harmonic Oscillation, 45 Fourier–Approximation, 46 Free Vibrations, 34 Frequency Frequency Domain, 45 Friction, 54 Front Wheel Drive, 55, 75 Generalized Fluid Mass, 49 Grade, 52 Hydro-Mount, 48 Kingpin, 10 Kingpin Angle, 11 Kingpin Inclination, 11 Kingpin Offset, 12

i

Vehicle Dynamics

Lateral Acceleration, Acceleration, 78, 91 Lateral Force, 84 Lateral Slip, 84, 85 Load, 3 Maximum Acceleration, 54, 54, 55 Maximum Deceleration, 54, 54, 56 Merit Function, 37, 41 Optimal Brake Force Distribution, 57 Optimal Damper, Damper, 42 Optimal Damping, 34, 34, 36 Optimal Drive Force Distribution, 57 Optimal Parameter, Parameter, 42 Optimal Spring, 42 Optimization, 38 Oversteer, 91 Overturning Limit, 76 Parallel Tracks, Tracks, 96 Pinion, 6 Power Spectral Density, Density, 97 Preload, 32 Quarter Car Model, 36, 39 Rack, 6 Random Road Profile, 40, 40, 96 Rear Wheel Drive, 55, 55, 75 Referencies Hirschberg, W., 29 Rill, G., 29 Weinfurter, H., 29 Road, 15 Roll Axis, 81 Roll Center, Center, 81 Roll Steer, 99 Roll Stiffness, 78 Roll Support, 79, 79, 81 Rolling Condition, 84 Safety, 31 Side Slip Angle, 66 Sky Hook Damper, Damper, 36 Space Requirement, 67 Spring Characteristic, 40

ii


Spring Rate, 33 Stability, 87 State Equation, 86 Steady State Cornering, 74, 94, 99 Steer Box, 6, 7 Steer Lever, 7 Steering Activity, 98 Steering Angle, 72 Steering System Drag Link Steering, 7 Lever Arm, 6 Rack and Pinion, 6 Steering Tendency, 82, 90 Step Steer Input, 95, 100 Suspension Suspension Model, 31 Suspension Spring Rate, 33 Sweep-Sine, 47 System Response, Response, 45 Tilting Condition, 54 Tire Bore Slip, 28 Bore Torque, 14, 27, 27, 28 Camber Angle, 16 Camber Influence, 25 Characteristics, 29 Circumferential Direction, 16 Contact Area, 14 Contact Forces, 14 Contact Length, 22 Contact Point, 15 Contact Torques, 14 Cornering Stiffness, 25 Deflection, 16, 22 Deformation Velocity, 20 Dynamic Offset, 24 Dynamic Radius, 19 Lateral Direction, 16 Lateral Force, 14 Lateral Force Distribution, 24 Lateral Slip, 24 Lateral Velocity, 20 Linear Model, 84 Loaded Radius, 16, 19


Longitudinal Force, 14, 14, 22, 22, 23 Longitudinal Force Characteristics, Characteristics, 23 Longitudinal Force Distribution, 23 Longitudinal Slip, 23 Longitudinal Velocity, 20 Normal Force, 14 Pneumatic Trail, 24 Radial Damping, 18 Radial Direction, 16 Radial Stiffness, 78 Rolling Resistance, Resistance, 14 Self Aligning Torque, Torque, 14, 24 Sliding Velocity, 24 Static Radius, 16, 19 Tilting Torque, 14 Transport Velocity, 19 Tread Deflection, 21 Tread Particles, 21 Undeformed Radius, 19 Vertical Force, 17 tire composites, 13 Tire Development, 13 Tire Model Kinematic, 66 Linear, 91 TMeasy, 29 Toe Angle, 9 Track, 32 Track Curvature, 72 Track Normal, 16, 17 Track Radius, 72 Track Width, 66, 66, 78 Trailer, 69, 69, 72 Turning Center, 66


Waviness, 98 Wheel Base, 66 Wheel Load, 14 Wheel Loads, 51 Wheel Suspension Suspension Central Control Arm, 5 Double Wishbone, 4 McPherson, 4 Multi-Link, 4 Semi-Trailing Semi-Trailing Arm, 5, 100 Single Wishbone, 100 SLA, 5 Trailing Arm, 100 Yaw Angle, 72 Yaw Velocity, 84

Understeer, 91 Vehicle, 2 Vehicle Comfort, 31 Vehicle Data, 41 Vehicle Dynamics, 1 Vehicle Model, 31, 31, 39, 51, 60, 60, 69, 69, 79, 83, 83, 102 Virtual Work, 80

iii

Vehicle Dynamics 2004

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