Vehicle Dynamics 2004

 

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

 

FACHHOCHSCHULE ACHHOCHSCHU LE 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

 

1 Int Intro rodu ducti ction on 1.1

1.2

1.3

1. 1.4 4

1.4.3 1.4.3

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T oe.2.1 an and Cam bertions Angle Ang   cordin . . ding . .g .to. DIN . . .70 . 000 . . .  .. .. .. .. .. .. .. .. 1.4.2. 1.4 1d Camber Defini Definitio nsle ac accor 1. 1.4. 4.2. 2.2 2 Calc Calcul ulat atio ion n  . . . . . . . . . . . . . . . . . . . . . . Steeri Steering ng Geomet Geometry ry   . . . . . . . . . . . . . . . . . . . . . . . 1.4. 1.4.3. 3.1 1 Ki King ngpi pin n . . . . . . . . . . . . . . . . . . . . . . . 1.4.3. 1.4 .3.2 2 Caste Casterr an and d Kingpi Kingpin n Angle Angle   . . . . . . . . . . . . . . 1.4.3.3 1.4. 3.3 Disturbing Disturbing Force Force Lever Lever,, Caster Caster and King Kingpin pin Offset Offset   .

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Introd Introduc uctio tion n . . . . . . . . 2.1.1 2.1.1 Ti Tire re Deve Develop lopmen mentt   . 2.1.2 2.1.2 Ti Tire re Compo Composit sites es   . 2.1.3 2.1. 3 Forces Forces and and Torqu Torques es

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2 Th The e Tire Tire 2.1

 

Termino erminolog logy y  . . . . . . . . . . . . . . . . . 1.1.1 1.1.1 Veh ehicl iclee Dynam Dynamics ics   . . . . . . . . . 1.1. 1.1.2 2 Dri Driver  er   . . . . . . . . . . . . . . . . 1. 1.1. 1.3 3 Vehic ehicle le   . . . . . . . . . . . . . . . 1.1. 1.4 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

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2.2

2. 2.3 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 T Torqu orquee   Cambe Camberr Influe Influence nce   . . . . . . . . . . . . . . . . . . . Bore Bore Torque orque   . . . . . . . . . . . . . . . . . . . . . . Typical ypical Tire Tire Characte Characteristic risticss   . . . . . . . . . . . . . .

3 Verti ertical cal Dyn Dynami amics cs 3.1 3.1 3.2

3.3

3.4

3.5

4.1

II

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15 15 17 17 18 20 21 24 25 27 29

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Goals   . . . . . . . . . . . . . . . . . . . . . . . . . Basic Basic Tuning uning   . . . . . . . . . . . . . . . . . . . . . 3. 3.2. 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 Veh ehicl iclee Data Data   . . . . . . . . . . . . . . . . . 3. 3.4. 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 Tr Trav avel 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 Exc Excita itatio tion n . . . . . . . 3. 3.5. 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   . .

4 Lon Longit gitudi udinal nal Dyn Dynami amics cs

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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 Gra Grade de   . . . . . . . . . . . . . . . . . . . . . . . . . . . .

31 31 31 32 32 33 34 36 36 37 39 39 40 41 41 42 42 42 44 45 45 45 47 48 48 50

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 Ti Tilti lting ng Limits Limits   . . . . . . . . . . . . . . . . . . . 4.2.2 4.2.2 Fricti Friction on Limits Limits   . . . . . . . . . . . . . . . . . . Drivin Driving g an and d 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 an and d 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 an and d Brake Brake Pitch Pitch   . . . . . . . . . . . . . . . . . . . 4.4.1 4.4.1 Veh ehicl iclee 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 an and d Brakin Braking g . . . . . . . . . . . . . . . 4.4.5 4.4.5 Brake Brake Pitch Pitch Pol Polee   . . . . . . . . . . . . . . . . .

5 Lat Latera erall Dyn Dynam amics ics 5.1

5.2

5.3

Kinema Kinematic tic Appro Approac ach h . . . . . . . . . . 5.1.1 5.1.1 Kinema Kinematic tic Ti Tire re 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 Tr Trailer ailer   . . . 5. 5.1. 1.4. 4.1 1 Posi Positi tion on   . . . . . . . 5. 5.1. 1.4. 4.2 2 Vehic ehicle le   . . . . . . . . 5.1.4. 5.1 .4.3 3 Enter Entering ing a Curve Curve   . . . 5. 5.1. 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   . . . . . . .

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5.2.3 5.2. 3 Roll Rolll Center Support Supp ortand andRoll Camber Camb ers Compensa Comp 5.2.4 5. 2.4 Rol Cen ter Ro ll Axis Axi   . . .ensation . . tion . . .  5. 5.2. 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. 5.3. 3.2 2 Kine Kinema matic ticss   . . . . . . . . . . . . . . . . 5. 5.3. 3.3 3 Tire ire Forc Forces es   . . . . . . . . . . . . . . . . 5. 5.3. 3.4 4 Late Latera rall Slip Slipss   . . . . . . . . . . . . . . . 5.3.5 5.3.5 Equati Equation onss of Motion Motion   . . . . . . . . . . . 5. 5.3. 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 App Approx roxima 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. 5.3. 3.7. 7.3 3 Slip Slip Angl Angles es   . . . . . . . . . . . . . . Influence Influence of Whee Wheell Load Load on Cornering Cornering Stiffnes Stiffnesss  .

6 Drivin Driving g Be Behav havior ior o off Sin Single gle V Vehic ehicles les 6.1

6.2

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. 2.1 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 Veh ehicl iclee 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 Termin erminology ology 1.1.1 Vehicle Vehicle Dynamics The Expression ’Vehicle Dynamics’ encompasses the interaction of  • dri driver ver,, • veh vehicle icle • loa load d and and • environment environment Vehicle dynamics mainly deals with • the improvement improvement of active safety and and driving comfort as well as • the redu reduction ction of road destruc destruction. tion. In vehicle dynamics • computer calculations • test rig me measu asureme rements nts and • fiel field d tests tests are employed. employed. The intera interacti ction onss betwe between en the single single sy syste stems ms and the proble problems ms with with comput computer er ca calcu lculat 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 Driver Driver By various means of interference the driver can interfere with the vehicle:

   

driver

steering wheel   lateral dynamics gas pedal brake pedal longitudinal dynamics clutch gear shift

 

The vehicle provides the driver with some information:

vehicle

 −→    −→

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

The environment also influences the driver:

vehicle

driver

climate environment



traffic density track 

 −→

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: • Moto Motorcyc rcycles, les, • Pas Passen senger ger Cars, Cars, • Buss Busses, es, • Tr Trucks ucks

2

 

FH Regensburg, University of Applied Sciences

© Prof. Dr.-Ing. G. Rill 

• Agricultural Tractors, Tractors, • Passenger Passenger Cars with Trailer Trailer • Truck Trailer / Semitrailer, Semitrailer, • Road Tr Trains ains.. 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.  mass, inertia, center of gravity Load 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 Environment Environment The Environment influences primarily the vehicle:  Road: irregulari irregularities, ties, coefficie coefficient nt of friction friction Environment Air: Air: resi resist stan ance ce,, cros crosss wi wind nd



 −→

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 rroad oad irregularities and the interaction of tires and road as well as the calculation of aerodynamic forces and torques.

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Vehicle Dynamics

FH Regensburg, University of Applied Sciences

1.2 Wheel Wheel/Axle /Axle Sus Suspensi pension on Syst Systems ems 1.2.1 General General Remarks 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 Fig.  1.1.. E

E

G

zR

zR

ϕ2 yR

D

xR R

δS

O1

S

Q F

N1

  U2

λ

ϕ1

Q R

zB  

yB

W

X SR V

U zB

D

Q

P U

B

S

xB

Y

Z

xR

B

xR

F

D

yR

P

A

U1

δS

yR

P

G

G

zR

F

O2

N3

O

C

M

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. Fig.  1.2. Solid  Solid axles tend to tramp on rough road.

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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 Suspensio Suspension n Systems The Semi-Trailing Arm, the SLA and the Twist Beam axle suspension are suitable only for non steered axles, Fig. 1.3. 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 Stee Steering ring Syst Systems ems 1.3.1 Requireme Requirements nts 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.

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Vehicle Dynamics

FH Regensburg, University of Applied Sciences

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

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

wheel and wheel body

Q

 in k  a g l  i l n  d r a

uZ

P

δL

pinion

rack

δ1

 

steer box

δ2

L

Figure 1.4: Rack and Pinion Steering transformed by the steering box to the rack travel   uZ    =   uZ (δ L )   and then via the drag links transmitted to the wheel rotations  δ 1   =   δ 1 (uZ ),  δ 2   =   δ 2 (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 Lever Arm Steering Steering Syst System em δG

Q1

s t e  ee    r  l e  ev    e r   1 

drag link 1 P1 δ1

steer box 2  r  2   e e v e  l   l  r e e  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.

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The rotations of the steering wheel   δ L   are firstly transformed by the steering box to the rotation of the steer levers   δ G   =   δ G (δ L ). The drag links transmit this rotation to the wheel δ 1   =   δ 1 (δ G ),   δ 2   =   δ 2 (δ G ). 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 At solid axles the drag link steering system is used, Fig. 1.6. Fig.  1.6. le v e r e e r  l  e  s t  e δH

O

H

wheel and wheel body

steer box

(90o rotated)

steer link

I L δ1

 

K

δ2

drag link Figure 1.6: Drag Link Steering System

The rotations of the steering wheel  δ L  are transformed by the steering box to the rotation of the steer lever arm  δ H H    =   δ H H  (δ L )  and further on to the rotation of the left wheel,  δ 1   =   δ 1 (δ H  H ). The drag link transmits the rotation of the left wheel to the right wheel,  δ 2   =   δ 2 (δ 1 ). The steering ratio is defined by the ratio of the steer box and the kinematics of the steer link. Here the ratio δ 2   =  δ 2 (δ 1 ) given by the kinematics of the drag link can be changed separately.

1.3.5 Bus Steer Steer System System In bus usse sess th thee 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 st stee eerr syst system emss are needed, Fig. 1.7. Fig.  1.7. The rotations of the steering wheel  δ L  are transformed by the steering box to the rotation of the steer lever arm  δ H H    =  δ H H  (δ L ). Via the steer link the left lever arm is moved,  δ H  H   =  δ H  H (δ G ). This motion is transferred by a coupling link to the right lever arm. Via the drag links the left and  =  δ 1 (δ H  =  δ 2 (δ H  right wheel are rotated,  δ 1  = δ  H  )  and  δ 2  = δ  H ).

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Vehicle Dynamics

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

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

I

J K

Q

d r  ra   g l i in   k 

δH

P δ1

coupl. link

L

δ2

wheel and wheel body

Figure 1.7: Bus Steer System

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

y0

yF

en   eyR ex ey Figure 1.8: Coordinate Systems with the axes   x0 ,   y0 ,   z 0   is fixed to the track. Within the vehicle fixed system the   xF -axis is

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pointing forward, the yF -axis left and the z FF  -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  e x   and e y  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 Defin Definition itions s accor according ding 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. Fig.  1.9. Thee camb Th camber er angl anglee is the the angl anglee be betw twee een n the the whee wheell cent center er plan planee and and the the tr trac ack k norm normal al.. It is po posi siti 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  e yR  in direction of  the wheel rotation axis is described in the vehicle fixed coordinate system  F , Fig. 1.11 Fig.  1.11

eyR,F    =



(1)   eyR,F 

 

(2) eyR,F 

 

(3) eyR,F 



T 

,

 

(1.1)

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Vehicle Dynamics

FH Regensburg, University of Applied Sciences

eyR

zF yF

xF e (2) yR,F

e (3) yR,F

δV

e (1) yR,F

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

tan δ   δ   =

eyR,F  (2)

eyR,F 

.

 

(1.2)

Thee re Th real al 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 ve vect ctor orss in the the dire direct ctio ion n of the wheel rotation axis  e yR  and in the direction of the track normal  e n ,

sin γ   =

T  n yR  .

 

(1.3)

(3) yR,F   .

 

(1.4)

−e  e

The wheel camber angle can be calculated by

sin γ  γ    =

  −e

On a flat horizontal road both definitions are equal.

1.4.3 Steering Steering Geomet Geometry ry 1.4.3.1 1.4.3 .1 King Kingpin pin 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 Fig.  1.12 and  and Fig. 1.13. 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 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.

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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 Cast Caster er and Kingpin Ang Angle le The current direction of the kingpin can be defined by two angles within the vehicle fixed coordinate system, Fig. 1.14 Fig.  1.14.. If the kingpin is projected into the  y F -,  z FF  -plane, the kingpin inclination angle   σ  can be read as the angle between the z FF  -axis and the projection of the kingpin. The projection of the kingpin into the   xF -,  z FF  -plane delivers the caster angle  ν  with the angle between the   z F  F -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

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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  ν    =

(1) S,F  (3) eS,F 

−e

and   tan σ   =

(2) S,F  (3) eS,F 

−e

(1)   eS,F 

with   eS,F    =

 

(2) eS,F 

 



1.4.3.3 Disturbing Forc Force e Lever Lever,, Caster and Kingpin Offset

T 

(3) eS,F 

.   (1.5)



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  where

C   d ey

 

ex

P rS

S  

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   S .

12

 

2 The Tire Tire 2.1 Intr Introduct oduction ion 2.1.1 Tire Develop Development ment 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-Borde aux-Paris (720 Miles):  50 tire deflations, 22 complete inner tube changes

1899

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

1904

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

1908

Frank Frank Seiberling Seiberling:: grooved grooved tires tires with improv improved ed road tr tractio action 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

19 1946 46 .. .

Radia Radiall Tire

Table 2.1: Mile Stones in the Development of Tires

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

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Reinfo Rei nforce rcemen ments: ts: steel, steel, rayon rayon,, nylon nylon

16% 16%

Rubber: natural/synthetic

38%

Compou Com pounds nds:: ca carbo rbon, 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, ... ... Miscellaneous Tire Mass

4% 2% 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 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 coordin coordinate ate system. The z -axis -axis is normal to the track, the  x-axis is perpendicular to the z -axis -axis and perpendicular to the wheel rotation axis  e yR . 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   tilting torque M y   rolling resistance torque M z   self aligning and bore torque

 

Fy 



Mx Fx

My 



 

F

Mz

z

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 Fig.  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 -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 -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.

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2.2 Conta Contact ct Geom Geometry etry 2.2.1 Contact Contact Point Point The current position of a wheel in relation to the fixed  x0 -,  y 0 - z 0 -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

z0 y0

road: z = z ( x , y )

a

P ey

x0

P*

b

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   =   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) ∗

 −

-direction of the body where r 0  is the undeformed tire radius and  e zB  is the unit vector in the  z -direction fixed reference frame.

P ∗ with respect to the fixed system  x0 , y0 , z 0  is determined by The position of   P  r0P    =   r0M   + rM P  , ∗

 

∗

(2.4)

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

r0P  ,0   = 0



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

∗

z 

 

∗



.

 

(2.5)

∗

 

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FH Regensburg, University of Applied Sciences

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   ,   and   ey   =   en ex . eyR en

× | × |

×

 

(2.6)

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

 

γ  γ    = arcsin eT  yR 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  e

S  zR  +

a ex + b ey  ,

 

(2.8)

where  r S   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

×  . | e ×e | . Because the unit vectors  e ezR   =

is perpendicular to  e x   and e yR scalar multiplication of (2.8 (2.8)) with en  results in

eT  n  r M P 0   =

 −

rS  eT  n  e zR

x

yR

  x   and  e y  are

  or   rS    =

 −

(2.9)

perpendicular to  e n , the

  eT  n  r M P  . eT  n  e zR 0

 

(2.10)

Now also the tire deflection can be calculated

r   =   r − r  , 0

S 

 

(2.11)

 

(2.12)

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

rM P    =

rS  ezR

 −

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 P  no longer lies on the track. calculated in the point P 0 . Therefore with an uneven track   P  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.

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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    exB 2

−

b  e 2 yR b  e 2 yR

 

−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 e zB  point in the longitudinal and vertical direction of the vehicle. The wheel rotation  L ,  R  and  F   can axis is given by  e yR . According to (2.5 (2.5)) the corresponding points on the track  L be calculated. The vectors

rRF    =   r0F 

 − r

0R

  and   rRL   =   r0L

−r

 

0R

(2.14)

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. 2.13) by the actual length   L  of the contact Of course it would be obvious to replace   LN   in (2.13) area and the unit vector  e zB  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 Whe Wheel el Loa Load d The vertical tire force   F z  can be calculated as a function of the normal tire deflection eT  r  and the deflection velocity z ˙   =  e T  r˙ n n

 z  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)

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

F zS   =   c0

z  z   +   κ (z )

2

.

 

(2.18)

The constants  c 0   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

80

8    ]    N6    k    [

   ]    N    k    [

60

   F 4

   F

40

  z

  z

20

2 0

Truck Tire: X31580 R22.5

100

0

10

20 30 ∆z [mm]

40

0

50

0

20

Figure 2.3: Tire Radial Stiffness:

40 ∆z

 

60

80

[mm]

◦ Measuremen Measurements, ts, — Approximation

F z  = 3 20 2000 N  can The radial tire stiffness of the passenger car tire at the payload of   F   can be specified with c0  = 190 190 000 000N/m. The Payload F z  = 35000 N  and  and the stiffness  c0  = 1 25 2500 00 0000N/m  of a 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 Rolli Rolling ng Radius Radius At an angular rotation of  ϕ, assuming assuming the tread particles particles stick to the track, the deflected deflected tire  x, Fig. 2.4 moves on a distance of  x Fig.  2.4..



With r 0  as unloaded and  rS   =  r 0

− 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



r   sin 0

18

ϕ.

ϕ = x = r



D



 

(2.22)

 

FH Regensburg, University of Applied Sciences

© Prof. Dr.-Ing. G. Rill 

deflected tire

rigid wheel  

Ω

Ω

rD

r0 rS ∆ϕ

vt

∆ϕ

x

x Figure 2.4: Dynamic Rolling Radius

Hence, the dynamic tire radius is given by

rD   =

  r0   sin ϕ   . ϕ





 

(2.23)

For ϕ 0 one gets the trivial solution r D   =  r 0 . 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 − 

or

2

  ϕ

= 2

−    rS  r0

1

 

(2.25)

one finally gets

rD   =   r0 remains.

 rrS  0

 −  −  1

  13 1

  =   23  r 0 +   +   13 r S 

 

(2.26)

The radius rD  depends on the wheel load F z  because of  rS   =  r S (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.

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Vehicle Dynamics

FH Regensburg, University of Applied Sciences

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

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

 

(2.28)

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

r˙M P, P,0 0 =

 d  ( rS  ezR, zR,0 0) = dt

−

  −r˙  e   −   r

S  e˙ zR,0 zR, 0  .

S  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 with the rim but does not perform rotations around the wheel rotation axis. Its time derivative is then given by ∗ e˙ zR, zR,0 0   =   ω0R,0 zR, 0 R,0 ezR,0

 

(2.31)

×

where ω 0∗R  is the angular angular velocity velocity of the wheel rim without without components components in the direction direction of the wheel rotation axis. Now (2.29 ( 2.29)) reads as

r˙M P, P,0 0 =

∗ 0R,0 R,0

  r˙ e   −   r  ω × e zR, zR,0 0

S 

 

ZR,0 ZR, 0

(2.32)

and the contact point velocity can be written as

v0P, P,0 0   =   v0M, M,0 0  +

∗ 0R,0 R,0

r˙ e − r  ω ×e zR, zR,0 0

S 

ZR,0 ZR, 0  .

 

(2.33)

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

en  v 0P    = 0 .

 

(2.34)

The tire deformation velocity is defined by this demand T  n

∗ 0M   + rS  ω 0R eT  n  e zR

 ( v r˙   =   −e  (v

×e

ZR )

.

 

(2.35)

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

vx   =   eT  x   v0P 

 

(2.36)

vy   =   eT  y   v0P 

 

(2.37)

and can be calculated.

20

 

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© Prof. Dr.-Ing. G. Rill 

2.4 Longi Longitudin tudinal al Forc Force e and Longitu Longitudinal dinal 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 Fig.  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  v x  and the bottom with the average transport velocity  v t   =  r D Ω. Depending on the speed difference v   =  r D Ω vx  the tread particle is deflected in longitudinal direction



u   = (rD Ω

−v )t.

−

 

x

(2.38)

rD Ω

v x

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   T   =

  L  , rD Ω

 

| |

(2.39)

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

| |

 t =  = T   T  the contact area The maximum deflection occurs when the tread particle leaves at  t umax   = (rD Ω

− v ) T   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

FH Regensburg, University of Applied Sciences

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. 2.6 Fig.  2.6a. a. The number  p  of the tread particles can be estimated by

 p   =

  L

 .

 

(2.42)

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

b)

L

c)

L

r0 s

     ∇

r

cxt * u

a

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. Fig. 2.6  2.6b. b. 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  (r  (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

r )2 ,

(r0

 

(2.45)

− −  r   denotes the tire deflection, Fig.   2.6 2.6c. c. With ≈ 8 r r .   (2.46)

where   r0   is the undeformed tire radius, and r r0  one gets

 

L2

0

The tire deflection can be approximated by

r   =   F  /c z

R  .

 

(2.47)

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

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

− | |

22

 

(2.48)

 

FH Regensburg, University of Applied Sciences

© Prof. Dr.-Ing. G. Rill 

The non-dimensional relation between the sliding velocity of the tread particles in longitudinal direction vxS   =  v x rD Ω  and the average transport velocity  r D 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 r 0 , 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 )

large slip values Fx = FG L

L

L Fxt  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

  1  v 1 +   p2 v

 ≥ 0   (Druck)

  1  p3 v   v <  0   (Zug) 1   p4 v

.

 

  ≥   0, (3.40)

 −

 =  p 3   =  d  and  p2   =  p 4   = 0. A linear damper with the constant  d  is described by  p1  = p For the spring characteristics the approach

F FF  (x) =   M g   +

  F R   1  p5 x x xR 1  p5 xR

− − ||

(3.41)

is used, where  M g   marks the spring preload. With parameters within the range  0   p5   <   1, one gets differently progressive characteristics. characteristics. The special case p5   = 0 describes a linear spring  c  = F   F R /xR . All spring characteristics run through the operating point  xR , F R . with the constant c = 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 Profi Profile le 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) =  v F  t .

 ∗

 

(3.42)

The irregularities of the track can thus be written as time function  z R   =  z R (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 Fig.  3.6 shows  shows the first track profile  z S  S  (x). Profiles number two and three are generated from the first by multiplication with the factors  3  and 5, z SS   (x) = 3 z SS   (x), z SS   (x) = 5 z SS   (x). 1

2

40

∗

1

3

∗

1

 

FH Regensburg, University of Applied Sciences

© Prof. Dr.-Ing. G. Rill 

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 The values, arranged in table 3.2, table  3.2, describe  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]   [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 0 00 0 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 )5, which describe the nonlinear spring-damper characteristics, The five parameters,  p i ,  i = 1(1 are calculated by minimizing merit functions.

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

GK 1   =

  1 tE  t0

−

                    tE

t0

z  z¨ B g

2

comfort

+

F zD F zS 

2

 .

 

(3.43)

safety

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

 −

(M  +  + m) g.

41

 

Vehicle Dynamics

FH Regensburg, University of Applied Sciences

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

  1 tE  t0

GK 2   =

−

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

z  z¨ B g

t0

2

+

comfort

P D P S  S 

2

safety

 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 Paramet Parameter er 3.4.5.1 Linear Characteristics Judging the driving comfort and safety after the criteria  G K 1  and restricting to linear characteristics, with  p 1   =  p 3  and  p 2   =  p 4   =  p 5   = 0, one gets the results arrayed in table3.3 table 3.3.. The spring optima opt imall parame parameter ter parts parts in merit merit functi function on road roa d load load   p1   p2   p3   p4   p5   comfort safety 1   +   35 357 766 0 35766 0 2   +   357 35763 0 35763 0 3   +   357 35762 0 35762 0 1 2 3

  −   202 20298   −   203 20300   −   199 19974

0 20298 0 0 20300 0 0 19974 0

0 0.00 0.002 2886 886 0 0.02 0.025 5972 972 0 0.07 0.072 2143 143

0.00 .00266 2669 0.02 .02401 4013 0.06 .06670 6701

0 0.00 0.003 3321 321 0 0.02 0.029 9889 889 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 values in table 3.2 table 3.2.. One gets:cempty  = 225 000 000N/m  and cloaded  = 400 000 000N/m. As ex expe pect cted ed th thee resu result ltss are are almo almost st inde indepe pend nden entt from from the the tr trac ack. k. The The op opti tima mall val alue ue of the the da damp mpin 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. 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  p 5   = 0, appear, Fig. 3.8. Fig.  3.8. At  At moderate track irregularities, one gets strongly progressive springs.

42

 

FH Regensburg, University of Applied Sciences

© Prof. Dr.-Ing. G. Rill 

optim ptimal al para parame mete terr road roa d load load   p1   p2

 

p3

 

p4   p5

parts arts in merit erit fun functio ction n   co comfort safety

1   +   16 1618 182 2 0.00 0.000 0 20 2002 028 8 1. 1.31 316 6 0.967 0.9671 1 0.00 0.0002 0265 65 0.00 0.0011 1104 04 2   +   52 5217 170 0 2.68 2.689 9 57 5789 892 2 1. 1.17 175 5 0.698 0.6983 3 0.00 0.0090 9060 60 0.01 0.0127 2764 64 3   +   187 1875 5 3. 3.048 048 31 31177 1773 3 4.29 4.295 5 0.00 0.0000 00 0.0408 0.040813 13 0.050 0.050069 069 1 2 3

1396 961 1 0.000 0.000   −   13   −   16 1608 081 1 0.808 0.808   −   99 9942 42 0. 0.22 227 7

17 1725 255 5 0. 0.33 337 7 0.920 0.9203 3 0.00 0.0008 0819 19 0.00 0.0034 3414 14 27 2770 703 3 0. 0.45 454 4 0.656 0.6567 7 0.01 0.0129 2947 47 0.03 0.0312 1285 85 64 6434 345 5 0. 0.71 714 4 0.0 0.000 000 0 0.06 0.0609 0992 92 0.09 0.0902 0250 50

Table 3.4: Nonlinear Spring and Damper Characteristics optimized via  G K 1

The dampers are digressive and differ in jounce and rebound. In co comp mpar 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 no note ted, 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. Fig.  3.7. body accelerations [m/s2 ]

10 5 0 -5 -10 0

2

4

 [s]

6

8

Figure 3.7: Body Accelerations optimized via G K 1   (

· · · 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:  G K 1

43

 

Vehicle Dynamics

FH Regensburg, University of Applied Sciences

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 table  3.5.. opti optima mall para parame mete terr

part partss in meri meritt func functi tion on   p4   p5   co comfort safety s. travel

road roa d load load   p1   p2   p3

1   +   6872 68727 7 0 6872 68727 7 0 0 0.00 0.0038 3854 54 0 0.0 .003 0367 673 3 0. 0.00 0063 6339 39 2   +   6866 68666 6 0 6866 68666 6 0 0 0.03 0.0346 4657 57 0 0.0 .033 3302 025 5 0. 0.05 0570 7097 97 3   +   7288 72882 2 0 7288 72882 2 0 0 0.09 0.0989 8961 61 0 0.0 .094 9443 431 1 0. 0.14 1487 8757 57 1 2 3

  −   3533 35332 2 3565 35656 6    −−    3748 37480 0

0 3533 35332 2 0 0 0.00 0.0044 4417 17 0 0.0 .004 0470 701 1 0. 0.00 0066 6638 38 0 3565 35656 6 0 0 0.04 0.0400 0049 49 0 0.0 .042 4250 507 7 0. 0.05 0591 9162 62 0 3748 37480 0 0 0 0.11 0.1121 2143 43 0 0.1 .116 1672 722 2 0. 0.15 1552 5290 90

Table 3.5: Linear Spring and Damper Characteristics optimized via  G K 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 wa rdss to p5 0.6. Star Startin ting g wi with th the the line linear ar pa para rame mete ters rs fr from om ta tabl blee 3.5, an optimizati optimization on via criteria criteria

 ≤

optimal parameter road roa d load load   p1

 

p2

 

p3

 

p4   p5

parts in merit function   comfort safety s. trav ravel

1   +   175530 175530 12.89 12.89 102997 102997 3.437 3.437 0.4722 0.4722 0.001747 0.001747 0.0020 0.002044 44 0.005769 0.005769 2   +   204674 204674 5.505 5.505 107498 107498 1.234 1.234 0.6000 0.6000 0.015877 0.015877 0.0185 0.018500 00 0.050073 0.050073 3   +   327864 327864 4.844 4.844 152732 152732 1.165 1.165 0.5140 0.5140 0.064980 0.064980 0.0683 0.068329 29 0.116555 0.116555 1 2 3

 −   −   − 

66391 66391 5.244 5.244 50353 50353 2.082 2.082 0.584 0.5841 1 0.00 0.00238 2380 0 0.00394 0.003943 3 0.00559 0.005597 7 37246 37246 0.601 0.601 37392 37392 0.101 0.101 0.545 0.5459 9 0.02 0.02452 4524 4 0.03315 0.033156 6 0.05971 0.059717 7 89007 89007 1.668 1.668 68917 68917 0.643 0.643 0.361 0.3614 4 0.08 0.08500 5001 1 0.10287 0.102876 6 0.12504 0.125042 2

Table 3.6: Nonlinear Spring and Damper Characteristics optimized via  G K 2

GK 2  delivers the results arranged in table 3.6. table  3.6. A vehicle with  G K 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. Fig.  3.9.

44

 

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© Prof. Dr.-Ing. G. Rill 

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

2

4

 [s]

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

6

8

(- - - GK 1 , — G K 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 table 3.6 lie  lie even below the values of the linear model from table 3.3, table  3.3, where  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 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 Dynam Dynamic ic For Force ce Ele Elements ments 3.5.1 System System Respons Response e in the Frequenc Frequency y Domain 3.5.1.1 First H Harmonic armonic O Oscillation scillation 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

FH Regensburg, University of Applied Sciences

 ≤ f  ≤ f   and amplitudes A  ≤ A  ≤ A i

E E  

min

 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 t tS i + T i , is approximated by harmonic functions as good as possible

tS i

 ≤ ≤

F  F ((t)

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

    ≈   



first harmonic approximation

measured or calculated

 

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

tSi



− F F ((t)

2

dt

  −→

  M inimu inimum m.

 

(3.47)

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

     

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

tSi tSi+T i

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

tSi

 

− F F ((t) − F F ((t)

2

2

sin(2π sin(2 π f i t) dt   = 0 (3.48)

cos(2π cos(2 π f i t) dt   = 0

with the solutions

αi   =

    −     −         −  −    

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

β i   =

,

 

(3.49)

  F  F    cos cos    dt2 dt   sin  cos dt2 dt   F  F  2  sin sin  sin   dt cos    sin  sin cos   dt   sin cos   dtcos 

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

 

 sin cos  cos   dt   = 0 ;

holds, and as solution

 2 αi   = T i

 

 ≤ t ≤ t

  sin2 dt   =

 

  F  F    sin 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  F    cos 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

 

FH Regensburg, University of Applied Sciences

© Prof. Dr.-Ing. G. Rill 

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

xe (t) =   A  sin(2  sin(2π π f t) ,

 

(3.52)

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

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

  2π   A  cos(2  cos(2π π f t) , T 

 

(3.53)

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

xe (t) =   A  sin(2  sin(2π π h(t))

 

(3.54)

the time derivative results in

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

 

(3.55)

 h (t)  delivers a period, that fades linear in time, i.e: Now we demand, that the function  h(   1   1 h˙ (t) =  =  , T ( T (t)  p q t

 

(3.56)

−

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

h(t) =

  −   1q  ln(  ln( p  p − q t) +   C 

 

(3.57)

 h((t  = 0) = 0  fixes the integration constant follows. The initial condition h  1  ln  ln p  p . q 

C  C    =

 

(3.58)

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

xe (t) =   A   sin

2π   p   ln , q   p q t



delivering linear fading period durations.

 

(3.59)



−

The important zero values for determining the period duration lie at

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

−

−

and

tn   =

 p (1 q 

−e

−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   =

 p

−t

−n q

 e q 

n =

 p (1 e−(n+1) q q 

−

−q

−n q

− 1+ e

) , n  = 0,   1,   2, .

 

(3.62)

e )

(1

− 47

 

Vehicle Dynamics

FH Regensburg, University of Applied Sciences

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

−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 pa q /p can be calculated from (3.63 rameters q  and   and the relation  q/p (3.63))

  1  f E  E  q   q   =   ln , N  f 0

1

 −  

 

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  N +1 +1   =

 1

−(N +1) +1) q

−e

 

q/p

(3.65)

seconds.

3.5.2 Hydro-Mo Hydro-Mount unt 3.5.2.1 3.5.2 .1 Princ Principle iple and Mod Model el 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  __ 2T

cF uF MF

ring channel chamber 2

dF  __  2

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

48

dF  __  2

 __  cT 2

 

FH Regensburg, University of Applied Sciences

© Prof. Dr.-Ing. G. Rill 

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  x e   +   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  A K ,   AR   of chamber and ring channel the generalized fluid mass is given by

AK 

M FF     =

2

M F R .

 

(3.67)

AR

     −

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

   

cF  (xe

F FF  (xe

−u

F )

=

0

cF  (xe

−u −u

F )

  F )

+   sF 

  sF 

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

(xe

F 

e

f 

e

f 

F 

F 

(3.68)

F 

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

 

| − |≤ ∗

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

M FF   u¨F    =

  − 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))   damping constant 100 000 N/m   fluid stiffness sF    = 0.0002 mm   clearance in membrane bearing

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 Dynam Dynamic ic Wh Wheel eel 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

a1

a2

 

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 -plane read as

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

− F 

z2

− mg,

 

(4.1)  

(4.2)

a2 + (F  ( F x1 + F x2 ) h ,

 

(4.3)

where  v˙  indicates the vehicle’s acceleration,  m  is the mass of the vehicle,  a 1 + 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

FH Regensburg, University of Applied Sciences

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

  a2 a1 + a2

  h m  v˙ , a1 + a2

 

(4.5)

1 F z 2   =   m g   a +   h m  ˙v . a1 + a2 a1 + a2

 

(4.6)

F z1   =   m g

The static parts

F zst1   =  m g

−

  a2 , a1 + a2

 =  m g F zst2  = m

  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   =

−m g a   +h a 1

2

v˙  , g

F zdyn 2   = +m g

  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 Grade Grade

 

v 

z 

x  F x 

1

mg

F z 

1

h 

a 1

F x 

2  2 

a 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

52

1

(4.9)

F  a  + (F  ( F   + F  ) h ,

0 =   F  a z1

− m g   sin α , − m g   cos α ,

−

z2

2

x1

x2

 

FH Regensburg, University of Applied Sciences

© Prof. Dr.-Ing. G. Rill 

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

F z 1   =   m g   cos α

 a 2

− h tan α − a  + a

  h m  ˙v , a1 + a2

 

(4.10)

 a  + h tan α   h 1 + a1 + a2 m  ˙v , a1 + a2

 

(4.11)

1

F z2   =   m g   cos α

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 Aerodynamic Aerodynamic 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 Fig.  4.3..

FD1

FD2 FAR h mg  

Fx1 a1

Fz1

Fx2  

a2

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 A AR R)   , a1 + a2

 

(4.13)

F z2   =   F D2 + m g

  a1   h + (m  ˙v  + F A AR R)   . 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

FH Regensburg, University of Applied Sciences

4.2 Maxim Maximum um Acc Accelera eleration 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

  ≤   ah   cos α − sin α   and   vg˙   ≥ − ah   cos α − sin α , 2

1

 

(4.15)

which can be combined to

−   ah   cos α   ≤

  v˙  + sin α g

1

 ≤

  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   a2  a 1   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 L Limits imits 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  has been assumed at front and rear axle.

 µ

In the limit case

F x1   =

± µ F 

z1

  and   F x2   =

± µ F 

z2

 

(4.19)

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

m  v˙ max   =

  ± µ   (F 

z 1  +

F z 2 )

− m g   sin α .

 

(4.20)

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

54

≥ tan α.

 

FH Regensburg, University of Applied Sciences

© Prof. Dr.-Ing. G. Rill 

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



1 +   F    +   F  mg mg D1

D2

−

AR   F AR mg

 ≤

  v˙ g

 ≤

  µ



1 +   F    +   F  mg mg

D2

D1

−

AR   F AR   .   (4.22) mg

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

4.3 Dri Drivin ving g and Bra Brakin king 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  ˙vR WD   =   µ m g



v˙ R WD   a1 +   h 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

  1

−

µ a1 .   h a1 + a2 µ 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 a2 ,   h a1 + a2 1+µ a1 + a2

 

(4.25)

where the subscript   F WD  denotes front wheel drive. Depending on the parameter  µ , a 1 ,  a 2   and h  the accelerations may be limited by the tilting condition   vg˙   ah .

 ≤

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 0.4 a2 /(a1+a2) 0.5. For µ =  µ  = 1 and h =  h  = 0.55 this results load distribution is in the range of   0.  0..77  v/g ˙   0.64. Front wheel driven passenger cars in maximum accelerations in between  0 usually cover the range  0  0..55   a2 /(a1 + a2 )   0.60  which produces accelerations in the range  0.45  v/g ˙ 0.49. Hence, rear wheel driven vehicles can accelerate much faster than front of  0. 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 /   (a (a1+a2)

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

4.3.2 Braking 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  ˙vF WB   =

  −µ m g



  a2 a1 + a2

−

  h v˙ F WB a1 + a2 g



 

(4.26)

where the subscript   F WB   indicates front wheel braking. The maximum deceleration is then given by

v˙ F WB   = g

µ a2 .   (4.27)   h a1 + a2 1 µ a1 + a2 If only the rear axle is braked (F x1 = 0, F x2 = µ F z2 ) one gets the maximal deceleration v˙ R WB  = g

 −

 −

 

− −  

µ a1 ,   h a1 + a2 1+ µ a1 + a2

 

(4.28)

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   vg˙  . 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  a 2 /(a1 + a2 )  usually covers the range from 0.4 to 0.  0.6. If only the front axle is braked then decelerations from  v/g v/g  ˙   = 0.51 to  v/g v/g = ˙  = 0.77 can be achieved. This is pretty much compared to the deceleration range of a braked rear axle v/g  ˙   = 0.49  to  v/g 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

© Prof. Dr.-Ing. G. Rill 

0 range of load distribution

-0.2

  g    /     v

FWB

.

-0.4

-0.6

-0.8

RWB -1

0

0.2

0.4

0.6

0.8

1

a2 /   (a (a1+a2)

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

4.3.3 Optimal Optimal Distribution Distribution of Drive and Brake Forces 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 (4.19)) the longitudinal forces are limited by longitudinal forces   F x1   and   F x2 . According to (4.19 wheel load and friction.

F x1   and  F x2  is achieved, when front and rear axle have the same The optimal combination of   F  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

F x1 mg  = and

F x2  = mg

 ± ν µ

 ± ν µ

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

 ± ν µ ,

 

 

(4.32)

(4.33)

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

F x1 mg

 =

  v˙ g



a2

  v˙

h

 − g

  h



(4.34)

a1 + a2

57

 

Vehicle Dynamics

FH Regensburg, University of Applied Sciences

and

F x2   v˙  = mg g



a1   v˙  + h g



  h . a1 + a2

 

(4.35)

remain. Depending on the acceleration  ornow deceleration  v˙ >  0  v˙ <  0  the longitudinal forces that grant the same skiddesired resistance at both axles can be calculated. Fig.4.6 Fig. 4.6 shows  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      i     n     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

v/g = ˙  = the tilting limits  v/g lifting axle.

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

 

FH Regensburg, University of Applied Sciences

and

  F x2 mg =  v˙ d g

d



© Prof. Dr.-Ing. G. Rill 

a1  v˙  + 2 h g



  h . a1 + a2

 

(4.37)

For  v/g v/g = ˙  = 0 the initial gradient remains as

d F x2 d F x1



= 0

  a1 . a2

 

(4.38)

4.3.4 Different Different Distribution Distributions s of Brake Brake 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. Fig.  4.7.

Fx1 /mg

Fx1 /mg     g     m      /      2     x

     F

constant distribution

Fx1 /mg     g     m      /

limitation

     2     x

     F

 

    g     m      /

reduction

     2     x

     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 relev ant 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 an optimal distribution distribution of of brake forces independent from loading conditions.

4.3.5 Anti-Lock-Systems Anti-Lock-Systems 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.

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Vehicle Dynamics

FH Regensburg, University of Applied Sciences

˙  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  sL   = 0 the rolling condition is fulfilled. Then of  s rD  Ω˙ = x ¨

 

(4.39)

¨  is the vehicle’s acceleration. According holds, where  r D  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˙ | ≤   r1 |x¨| .

 

(4.40)

D

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 Br Brake ake Pi Pitc tch h 4.4.1 Vehicle ehicle Model Model The vehicle model drawn in Fig. 4.8 Fig.  4.8  consists of five rigid bodies. The body has three degrees of freedom: Longitudinal motion  x A , vertical motion  z A  and pitch  β A. The coordinates  z 1   and z 2  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   =  x 1 (z 1 ) , β 1   =  β 1 (z 1 ) ; x2   =  x 2 (z 2 ) , β 2   =  β 2 (z 2 ) .

(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

 

FH Regensburg, University of Applied Sciences

© Prof. Dr.-Ing. G. Rill 

z

A

x

A

F MA1

F1

βA

z1 MB1

MA1

hR

ϕ

R1

MB1 Fz1

 

z2

 

MB2

MA2

Fx1

a1

FF2

MA2

R

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  4.1  and 4.2.  4.2. The velocity of the vehicle body and its angular velocity is given by

v0A, A,0 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   =

x˙ A 0 0

,0   =

x˙ A 0 0

2

     

 +

 +

   

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) (4.45)

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Vehicle Dynamics

FH Regensburg, University of Applied Sciences

as well as

ω0RK  ,0   = 2

          0 ˙A β  0

0 β ˙ 2 0

 +

  and   ω0R

2

,0   =

              0 β ˙A 0

 +

 +

0 ϕ˙ R2 0

(4.46) (4.46)

Introducing a vector of generalized velocities

˙ 2   ϕ˙ R2   x˙ A   z ˙A   β ˙A   β ˙1   ϕ˙ R1   β 

z   z   =

0 ˙2 β  0

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  j=1 =1

7

∂v 0i z  j   and   ω0i   = ∂z   jj

 j=1  j =1

∂ω 0i z  j ∂z  j j

 

(4.48)

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

0

partial velocities  ∂v  ∂ v0i /∂z   jj bodies chassis

mA

wheel body front

mRK 1

wheel front

mR1

wheel body rear

mRK 2

wheel rear

mR2

 

x˙ A   z ˙A   β ˙A   1 0 0 0 0 0 0 1 0 1 0 hR 0 0 0 0 1 a1 0 hR 1 0 0 0 0 1 a1 10 0 1 0 0

00 1 0 0 1

− − − − −0h

z ˙1   ϕ˙ R1   z ˙2   ϕ˙ R2 0 0 0 0 0 0 0 0 0 0 0 0 ∂x 0 0 0 ∂z 0 0 0 0 0 0 0 1 ∂x 0 0 0 ∂z 0 0 0 0 0 0 0 1 1

1

1

1

00 0 0 0 0

R

a2 hR 0 a2

−

 

00 0 0 0 0

∂x 2 ∂z 2

0 1

∂x 2 ∂z 2

0 1

00 0 0 0 0

applied forces

F ie 0 0 F FF 1 1 + F FF 2 2 mAg 0 0 F FF 1 1 mRK 1 g F x1 0 F z1 mR1 g  

−

− − −

00 F FF 2 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 ) =

∂v 0k ∂z i

T 

  k=1

62

mk

∂v 0k ∂z  j

5

+

∂ω 0k ∂z i

  k=1

T 

Θk

∂ω 0k ∂z  j

;   i, j  = 1(1)7 ;   (4.49)

 

FH Regensburg, University of Applied Sciences

© Prof. Dr.-Ing. G. Rill 

partial angular velocities  ∂ω  ∂ ω0i /∂z   jj

  x˙ A   z ˙A   β ˙A   z ˙1   ϕ˙ R1   z ˙2   ϕ˙ R2 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0

bodies chassis

ΘA

00 0 0 0 0 0 0 0 0 0 0

wheel body front

ΘRK 1

wheel front

ΘR1

wheel body rear

ΘRK 2

wheel rear

ΘR2

00 0 0 0 0 0 0 0 0 0 0

01 0 0 1 0 0 1 0 0 1 0

00 0 0 1 0 0 0 0 0 0 0

01 ∂β  ∂z 1 0 0 ∂β 1 ∂z 1

0 0 0 0 0 0 0

00 0 0 0 0 0

 

applied torques

 

−M  − A1

00 0 0 0 0 0 0 0 0 1 0

∂β 2 ∂z 2

0 0 ∂β 2 ∂z 2

0

M ie 0 M A2 a1 F FF 1 1 + a2 F FF 2 2 0

−

M A1

−

M A2

−

M 0B1 0 0 M B1 R F x1 0 0 M B2 0 0 M B2 R F x2 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   M  z ˙   =   Q .

 

(4.51)

4.4.3 Equilibrium Equilibrium With the abbreviations abbreviations

m1   =  m RK 1 + mR1 ;   m2   =  m RK 2 + mR2 ;   mG   =  m A + 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

−m

Q(3) =

a2 F z 2

  −a F 

1 z 1 +

(4.54)

g ;

− h(F 

  ∂x 1 ∂z 1

x1  +

− F   + F  − m =   M  − M  − R F   ;

Q(4) =   F z1 Q(5)

G

A1

F F 1  1

B1

x1

F x2 ) + a1 m1 g

1

g +   ∂β  (M A1 ∂z 1

1

−a

2

m2 g ;

− R F 

x1 ) ;

(4.55)

x1

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Vehicle Dynamics

FH Regensburg, University of Applied Sciences

Q(6) =   F z2 Q(7)

  ∂x 2 ∂z 2

− F   + F  − m =   M  − M  − R F   . F F 2  2

A2

B2

x2

2

(M A g +   ∂β  ∂z 2

2

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   a 0 0 F F    F F 2 F 1 1   = a+b  m A g ; F 2   = a+b  m A g ; F z01   =   m1g  + a  +b b  m A g ;   F z02   =   m2 g +  a+a b  mA g .

(4.58)

4.4.4 Driving Driving and Braking Braking ¨A = 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

1 ˙A +   ∂x R  β  ∂z 1   ∂x 2  ˙ R β A  + ∂z 2

−h −h

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   =  β Ast  =  const. ;   z 1  = z   =  z 1st  =  const. ;   z 2  = z   =  z 2st  =  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 ¨ xA   =   F xa1 + F xa2 ; ¨ A + Θ R1 x¨RA  + ΘR2 R (m1 + m2 ) x

−h

∂x 1 ∂z 1

ΘR1 m1 ¨ xA +   ∂β  ∂z 1

1

x ¨A R

ΘR1 x¨RA ∂x 2 ∂z 2

m2 ¨ xA +   ∂β  ΘR2 ∂z 2

2

x ¨A R

ΘR2 x¨RA

0 =   F za1 + F za2 ; x ¨A   = a F za1 + b F za2 R

  =   F za1

 −

a F  F 1 1

  ∂x 1 ∂z 1

a x1

− F   + F   +   =   M  − M  − R F   ; F   +   =   F  − F   +   =   M  − M  − R F   ; A1

a z2

A2

B2

R  +

  ∂x 2 ∂z 2

a x2

a x2

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

B1

a F  F 2 2

− (h

a x1 ) ;

(4.63)

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   =   F xi 0  + F xi a  ;   F zi st   =   F zi 0  + F zi a  ;   F Fsti   =   F F0 i  + F Fa i ;   i = 1, 2 . F xi

64

 

(4.65)

 

FH Regensburg, University of Applied Sciences

© Prof. Dr.-Ing. G. Rill 

¨A , 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  ( 4.62), ), ((4.63 4.63)) and (4.64 (4.64)) F 1 1 ,  F F  F 2 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  F 1 1 a F F  F 2 2 a F z1 F za2

       

=   cA1 z 1a  , =   cA2 z 2a  , = cR1 (z   (z Aa a β Aa  + z 1a) , = cR2 (z   (z Aa  + b β Aa  + z 2a ) .

 −  −

−

(4.66)

besides the vertical motions of the wheels. a Especially the pitch of the vehicle   β A = 0 , caused by drive or brake is, if too distinct, felt as 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, -plane, Fig. 4.9 Fig. 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 La Late tera rall Dy Dyna nami mics cs 5.1 Kinem Kinematic atic Appr Approach oach 5.1.1 Kinematic Kinematic Tir Tire e Mode Modell When a vehicle drives through the curve at low lateral acceleration, low lateral forces are needed for for cour course se ho hold ldin ing. g. At the the whee wheels ls then then ha hard rdly ly late latera rall sl slip ip oc occu curs rs.. In the the idea ideall case case,, wi with th 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   =   eT  y   v0P    = 0 .

 

(5.1)

Thiss kinema Thi kinematic tic constr constrain aintt equati equation on ca can n be used used for course course ca calcu lculat lation ion of slowly slowly movin moving g vehic vehicles les..

5.1.2 Ackerma Ackermann nn Geometry Geometry Within the validity Within 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 Fig. 5.1.. At slowly moving vehicles the lay out of the steering linkage is usually done according to the Ackermann geometry. Then, it holds

  a  a   and   tan δ 2   =  , R R + s

tan δ 1   =

 

(5.2)

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

tan δ 2   =

 

a

a +s tan δ 1

or

  tan δ 2   =

  a   tan δ 1 . a + s   tan δ 1

 

(5.3)

δ 2A of the actual steering angle   δ 2a   from the Ackermann steering The deviations δ 2   =   δ 2a angle δ 2A , which follows from (5.3 (5.3), ), are used to judge a steering system.

 

 −

At a rotation around the momentary pole  M  the  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 PP     =

 x  x   tan δ 1 ,   or   tan β P  P    = a R

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

66

 

(5.4)

 

FH Regensburg, University of Applied Sciences

© 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 Space Requiremen Requirementt The Acke The Ackerm rman ann n appr approa oach ch can can also also be us used ed to calc calcul ulat atee the the spac spacee requ requir irem emen entt of a vehi vehicl clee du duri ring ng cornering, Fig. 5.2 Fig. 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

R2max   = (Rmin + b)2 + (a (a + f )2 ,

 

(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 Fig.  5.3 versus  versus the minimum cornering radius.



In narrow curves  R min   = 5.0  m  a bus requires a space of   2. 2 .5  the width, whereas a passenger car needs only  1  1..5  the width.

67

 

Vehicle Dynamics

FH Regensburg, University of Applied Sciences

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

 

FH Regensburg, University of Applied Sciences

© Prof. Dr.-Ing. G. Rill 

5.1.4 Vehicle Vehicle Model Model with Trailer Trailer 5.1.4.1 Pos Position ition Fig.   5.4 shows 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  x 0 ,  y 0 ,  z 0  is defined by the position vector to the rear axle center

r02, 02,0   = and the rotation matrix

A02   =

 

    xF  yF  R

cos γ  sin γ    0 sin γ    cos γ    0 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.

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Vehicle Dynamics

FH Regensburg, University of Applied Sciences

The position vector

r01, 01,0   =   r02, 02,0   +   A02 r21, 21,2   mit   r21, 21,2   = and the rotation matrix

A01   =   A02 A21   mit   A21   =

 

    a 0 0

cos δ  sin δ    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  the steering angle. The position vector

r03, 03,0   =   r02, 02,0 + A02 with

−b K,2 2  = r2K,

and the rotation matrix

  00



r2K, K,2 2  + A23 rK 3,3

−c

  and   rK 3,2   =

A03   =   A02 A23   mit   A23   =

 



  00

 

cos κ sin κ   0 sin κ   cos κ   0 0 0 1

 −

 

(5.11)

 

(5.12)

 

(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   to the trailer axis  3.

5.1.4.2 Ve Vehicle hicle According to the kinematic tire model, cf. section  5.1.1, the  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)

 

FH Regensburg, University of Applied Sciences

© Prof. Dr.-Ing. G. Rill 

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 , y 2 , z 2

v01, 01,2   =

remains. The unit vectors

vx2 0 0 v02, 02,2

0 0 γ  ˙

a 0 0

ω02, 02,2

r21, 21,2

+

=

vx2 a  γ  ˙ 0

 .

 

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

ex1,2   =

cos δ  sin δ  0

  and   ey1,2   =

sin δ  cos δ  0

 

(5.19)

(5.20)

define the longitudinal and lateral direction at the front axle.

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

eT  01,2   = y 1,2 v01,

  − sin δ v

x2   +

cos δ a  ˙γ   γ   = 0 .

 

(5.21)

 

(5.22)

In longitudinal direction then

γ   γ   =   vx1 eT  01,2   = cos δ vx2   + sin δ a  ˙ x1,2 v01, remains. From (5.21 (5.21)) a first order differential equation follows for the yaw angle

γ  γ   ˙  =

  vx2   tan δ . a

 

(5.23)

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FH Regensburg, University of Applied Sciences

5.1.4.3 5.1.4 .3 Ente Entering ring a Cu Curve rve 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

 a tan δ   δ   = R   =   a k .

 

(5.24)

The differential equation for the yaw angle then reads as

γ  γ   ˙   =   vx2 k .

 

With the curvature gradient

t T 

k   =   k(t) =   kC 

(5.25)

 

(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 =  k  = k  k C . The yaw angle of the vehicle can now be calculated by simple integration 2

γ (t)

=   vx2

kC 

T 

t  , 2

 

(5.27)

 t  = 0  a vanishing yaw angle,  γ (t =0) = 0, has been assumed. where at time t = 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   dt , T  2

t=T 

  



vx2 kC  t2 sin   dt . T  2

yF    =   vx2

t=0

 

(5.28)

At constant vehicle speed  v x2   =  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   =   δ (t)

 ≈   a k   =   a k

C 

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

02 r23, 23,2   +   A02  r˙23, 23,2  .

With

−b − c   cos κ

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

 72

c   sin κ 0

−

 

 



(5.30)

(5.31)

 

FH Regensburg, University of Applied Sciences

and

© Prof. Dr.-Ing. G. Rill 

   −     × −   −               −  −        × −  −  − −                  −         − −   −    − −  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)

A r

23 K 3,3

it remains, remains, if (5.30) 5.30) is transformed into the vehicle fixed frame  x2 , y 2 , z 2

v03, 03,2  =

vx2 0 + 0

v02, 02,2

b

0 0 γ  ˙

c   cos κ c   sin κ 0

c   sin κ  ˙κ c   cos κ  ˙κ 0

+

r23, 23,2

ω02, 02,2

=

vx2 + c   sin κ (κ˙ + γ  ˙) b  ˙γ  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   =

sin κ cos κ 0

  and   ey3,2   =

 .

 

(5.34)

At the trailer axis the lateral velocity must also vanish

eT  03,2   = y3,2 v03,

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

b  ˙γ  γ 

c   cos κ (κ˙ + γ  ˙ )   = 0 .   (5.35)

In longitudinal direction

˙ + γ  ˙ )   + sin κ eT  03,2   = cos κ vx2 + c   sin κ (κ x3,2 v03,

b  ˙γ  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˙   =

  − va

a   sin κ + c

x2



b



c

  cos κ + 1

tan δ    .

 

(5.37)

 

The differential equations (5.17 (5.17)) and (5.23 (5.23)) describe position and orientation within the  x 0 ,  y 0 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   =  v x2 (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. Fig.  5.5.

73

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Vehicle Dynamics

FH Regensburg, University of Applied Sciences

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 Stea Steady dy St State ate Cor Cornerin nering g 5.2.1 Cornering Cornering Resistan Resistance ce In a body fixed reference frame B , Fig. 5.6 Fig. 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   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 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

 

v2   sin β    =   F x1 cos δ  F y1 sin δ  +  + F x2 , R   v2   cos β    =   F x1 sin δ  +  + F y1 cos δ  +  + F y2 , R

−

0 =   a1   (F x1 sin δ  +  + F y1 cos δ )

−a

2

F y2 ,

 

(5.40)

 

(5.41)  

(5.42)

74

 

FH Regensburg, University of Applied Sciences

© Prof. Dr.-Ing. G. Rill 

Fx2

a2

Fy2

a1

ω

C 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  =  k F D   und   F x2   = (1

− k)   F   ,

 

D

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

k  = 0

F x1  = 0,   F x2   =  F D   F x1   k 0  < k <  1   All Wheel Drive = 1 k F x2 k  = 1   Front Wheel Drive   F x1  = F   =  F D ,   F x2  = 0  

Rear Wheel Drive

 

−

(5.43)

75

 

Vehicle Dynamics

FH Regensburg, University of Applied Sciences

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



 −

k cos δ  +  + (1 ( 1 k) F D

 −

  sin δ F y1

mv 2 =   sin β , R   mv2

 

 −

(5.44)

R   cos β , k sin δ  + cos δ  +    = a1k sin δ F D   +   a1 cos δ F y1   a2 F y2   = 0 . F D

F y1

F y2

 −

This equations can be resolved for the drive force

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

−

 

−

(5.45)

The drive force vanishes, if 

a2   a2 cosβ   sin δ   δ   = sin β cosδ  cosδ    or tan δ  δ    = tan β  a1 + a2 a1 + a2

 

(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 lat latera erall 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   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. Fig.  5.7. The balance of torques at the already inclined vehicle delivers for small angles  α1 zL (F zL

− F 

zR zR )  s

y  (h1  + h2 ) +   m g [(h [(h1 + h2)α1 + h2 α2 ] , 2   =   m a  (h

 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   = 0 F zR

 

(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

K  1  α

−  −

  h2 K  2 h1 + h2 α   .

 

(5.50)

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FH Regensburg, University of Applied Sciences

α2

© Prof. Dr.-Ing. G. Rill 

α1

m ay h2 mg

h1

F F yR s/2

yL

FzL

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  calculated 1   and  α 2   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 zR F   

stat zL =   F   

 1 = 2  m g .

 

(5.51)

With stat K   +   =   F zL F zL

 F 

 

z

(5.52)

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    =   12 m g . z

 

(5.53)

 

(5.54)

The resulting tire deflection then follows from

F    =   c r , z

where cR  is the radial tire stiffness.

R

77

 

Vehicle Dynamics

FH Regensburg, University of Applied Sciences

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

2

r   =   s α

K  1  

or   αK  1   =

 2

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 (  (α α1 + α2 ) ,

 ∗

 

(5.56)

 

(5.57)

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

α2K  

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

 −

 −

holds. Not allowing the vehicle to overturn already at   aK  y   = 0   demands a minimum of roll min stiffness c W   > cW    =  mg  mgh h2 . 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 c∗W  1

 −   −   h

2

1

1

 − 1 c ∗   ,

c∗W 

 

(5.58)

R

where, for abbreviation purposes, the dimensionless stiffnesses

  cR c∗R   = m g s

and   c∗W    =

  cW  m g h2

(5.59)

−  c1

(5.60)

have been used. Resolved for the normalized lateral acceleration K  ay

g

 =

s 2   h2 h1 + h2 + ∗ cW  1

∗ R

 −

remains.

 m =  = 13 00 0000 kg . The radial stiffness of one At heavy trucks, a twin tire axle can be loaded with  m 800 000 N/m  and the track with can be set to  s  s =  = 2 m. The values h1   = 0.8 m and tire is c R  = 800 h2   = 1.0 m  hold at maximal load. This values deliver the results shown in Fig.  5.8  5.8 Even  Even at a ∗ a y   0.5 g . rigid body suspension  c W  the vehicle turns over at a lateral acceleration of   a The roll angle of the vehicle then solely results from the tire deflection.

 → ∞

 ≈

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

 ≈

≈

78

 

FH Regensburg, University of Applied Sciences

© Prof. Dr.-Ing. G. Rill 

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 normalize d roll roll stiffness c W *

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

5.2.3 Roll Support Support and Camb Camber er Compensatio Compensation n 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 Fig. 5.9 these  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  α A  and leads to a opposite deflection of the tires z 1   = z 2 .

−

b/2

b/2 zA

mA a y

 

αA

SA

yA FF1

FF2 h0

z2 mR a y r0

 

 

S2 Q2 Fy2

z1

α2

mR a y

y2 F y2

Figure 5.9: Plane Roll Model

α1

S1 Q1 F z1

y1 F y1

79

 

Vehicle Dynamics

FH Regensburg, University of Applied Sciences

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

δW    = 0 δW 

 

(5.61)

the equilibrium position can be calculated. At the plane vehicle model in Fig. 5.9 Fig.  5.9 the  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   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 anti roll bar, converted to the vertical displacement of the wheel centers. A

y

R

y

A

A

1 2

2 A 1

1 2

S 

1

1 2

Q

A

1 2

R

A

R

 1 2



1

2

R

y

A

R

A

2

2

2 A 2 2

2

0

 b 2

A

A

1

0

2

A

R

1

2

1

 1 2

R



0

 1 2

Q

A

A



0

A

2

0

2

2

2

 b 2

A

0

2



S 

The kinematics of the wheel suspension are symmetrical. With the linear approaches

y1   =

  ∂y  z 1 , ∂z 

  ∂α   α1   and   y2   = ∂z 

α1   =

  − ∂y  z   , ∂z 

α2   =

2

  − ∂α  α ∂z 

2

 

(5.63)

 W  can be described as function of the position vector the work  W  y   = [ yA , z A , αA , z 1 , z 2 ]T  .

 

(5.64)

Due to

W   =  W   W ((y)

 

(5.65)

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

δW    = δW 

  ∂W    δy   = 0 . ∂y

 

(5.66)

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

 

K y  = b

 

(5.67)

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

K   =

  

2 cQ

 

2 cR

2 cQ h0   0 ∂y Q  c ∂z Q

 −

2 cQ h0

0

0

  cR

   

0

 

 

cR

Q b c + h0 ∂y∂z   cQ 2 R

−

 

∂y Q b  c c + h R 0 ∂z Q 2

cα

Q

∂y ∂z   cQ

∂y Q  c ∂z Q

  c∗A + cS  + cR

∂y Q  c ∂z Q

cR b 2 R

∂y Q 0 ∂z   cQ

  − c −h   −c

S 

Q

 

cR

2b cR

 − −

h0 ∂y∂z   cQ

  −c

S 

 

∗  + cS  + cR cA

   

(5.68)

80

 

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and

b =

© Prof. Dr.-Ing. G. Rill 

     −   −  mA + 2 mR 0 (m1 + m2 ) hR

ay  .

 

(5.69)

mR ∂y/∂z  mR ∂y/∂z 

The following abbreviations have been used:

∂α   ∂y ∂y Q  ,   + r0  = ∂z  ∂z  ∂z 

c∗A   =   cA + cQ

∂y ∂z 

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 Fig.  5.10 a. 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.

Q–O§‘nv— 0ÿQ–O§‘nv— O§P>›Òf/Y‡O N:0v—ÿ˜

At more sportive vehicles thus axle kinematics are employed, where the wheels are rotated around the longitudinal axis at bound and rebound,  α1   =  α 1 (z 1 )  and  α 2   =  α 2 (z 2 ). With this, a 0. Fig. 5.10 ”camber compensation” compensation” can be achieved with γ 1 0  and γ 2 Fig. 5.10 b. By the rotation of  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 Roll Roll Axis The ’roll center’ can be constructed from the lateral motion of the wheel contact points  Q1  and Q2, Fig. 5.10 Fig. 5.10.. The line through the roll center at the front and rear axle is called ’roll axis’, Fig.  5.11.  5.11.

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roll axis roll center front

roll center rear

Figure 5.11: Roll Axis

5.2.5 Wheel Wheel Loa Loads ds 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 Fig.  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.

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5.3 Simple Handl Handling ing M Model odel 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. Fig.  5.13. The  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 , z B

vC,B   =

 

v   cos β  v   sin β  0

 

 ,

 

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

(5.71)

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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

1

cos δ  sin δ    ,

,B   =

   

ey

1

ex

2

,B   =

1 0 0

cos δ 

,B   =



0

and

− sin δ 

 ,

ey

2

,B   =

0

  0 1 0

 ,

v01,B 01,B   =

  ,

v   cos β  v   sin β  +  + a1  γ  ˙

v02,B 02,B   =

 



0

v   cos β  v   sin β  a2  γ  ˙ 0

−

 

 

(5.72)

 

(5.73)

  ,

˙ where  a 1   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 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 =   eT    (5.75) x  v 0P   . 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 T  vy   =   ey   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 .

 

(5.78)

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

s  = y1

  +sin δ ( δ (v   cos β )

− cos δ δ ((v   sin β  +  + a  γ  ˙)  . |   cos δ ( δ (v   cos β ) + sin δ  δ ((v   sin β  +  + a  γ  ˙)| 1

 

(5.79)

1

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 yaw velocity of the vehicle  γ    and the steering angle  δ  are  are considered 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  v   sin β  v β  the  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  γ ˙   +  |vv| δ 

and

sy2   =

1

 

 a   −β   β   + |v|  γ˙ . 2

 

(5.84)

(5.85)

5.3.5 Equations Equations o off 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)

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If the vehicle accelerations are also expressed in the vehicle fixed frame  x F ,  y F ,  z F  F , one finds at constant vehicle speed  v =  v  = const  const  and with neglecting small higher order terms

aC,B   =   ω0F,B

0 ˙ v  γ  ˙  +  + v  β 

vC,B   + v˙ C,B   =

×

The angular acceleration is given by

ω˙ 0F,B   = where the substitution

   



0

 .

 

(5.88)



||

0 0 ω˙

 

γ  γ   ˙  = ω

(5.89)

 

(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  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  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

β 

  a1  v ω + δ  v v

 +

  cS 2 m v

 v   a1 β  v ω + v δ 

  a1 cS 1 ω˙   = Θ

β  +  +

  a2 ω v

 v ω v

  a2 β  +  + v ω

  a2 cS 2 Θ

 ,

 

 

(5.93)

(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

a2 cS 2 a1 cS 1 m v v

 v v

a21 cS 1 + a22 cS 2 Θ v

a1 cS 1

Θ

A

 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.

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5.3.6 Stability Stability 5.3.6.1 Eigenv Eigenvalues alues 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)

 −  −  | || | ||   − ||

(5.100)

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 Sp Speed eed Appr Approximation oximation The state matrix

Av→0   =

approximates at  v

  − 

cS 1 + cS 2 m v

||

0

a2 cS 2 a1 cS 1 m v v a21 cS 1 +

 

 v v

a22 cS 2

Θ v

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

The matrix (5.100 (5.100)) has the eigenvalues

λ1v

→0

 =

  − c m +|vc| S 1

S 2

  and   λ2v

→0

 =

 − a

2 1 cS 1  +

a22 c S 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 Appro Approximation ximation At highest driving velocities  v

 → ∞, the state matrix can be approximated by

Av→∞   =

 

0 a2 cS 2

 

a1 cS 1   Θ

−

 −| |   v v

0

.

 

(5.102)

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

λ2v→∞   + 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   =  a 2 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

 

(5.107)

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 Solution Solution 5.3.7.1 Side Slip Angle and Y Yaw aw V Velocity elocity

 =  δ 0 , after a certain time, a stable system reaches steady state. With a given steering angle   δ  = δ  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 sstt   and the steady state angular velocity   ωst   at a constant given steering angle  δ =  δ = δ 0

|v| (c

S 1  + cS 2 ) β sstt   +

(m v v + a1 cS 1 a2 cS 2 ) ωst   =   v cS 1 δ 0 ,

||

−

2

|v| (a

1 cS 1

−a

2 cS 2 ) β st   +

 

(5.109)

 

(5.110)

2

(a1 cS 1 + a2 cS 2 ) ωst   =   v a1 cS 1 δ 0 ,

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

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

v cS 1 δ 0   m v v + a1 cS 1 a2 cS 2

||

v a1 cS 1 δ 0

β sstt   =

S 1  + cS 2 )

|v| (a

1 cS 1

−a

2 a2 cS 2

2 a1 cS 1 +

 

|v| (c

−

(5.111)

  m v v + a1 cS 1 a2 cS 2

||

2 cS 2 )

−

a21 cS 1 + a22 cS 2

 

and

|v| (c |v| (a

ωst   =

S 1  + cS 2 )

1 cS 1

−a

2 cS 2 )

|v| (c

S 1  + cS 2 )

|v| (a

1 cS 1

 

v cS 1 δ 0   v a1 cS 1 δ 0

(5.112)

  m v v + a1 cS 1 a2 cS 2

||

−a

2 cS 2 )

−

a21 cS 1 + a22 cS 2

 

The denominator results in

detD   = v

 | |



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

||

−a

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

   a a  −   m v |v |  v c  (  (a a  + a )  = |v| a  + a   +   m v |v|   a c − a c δ   , c c  (  (a a  + a ) S 2

1

1

2

2 S 2

2

S 1 S 2

 

ωst   =

 

(5.113)

1

2

β sstt



1 cS 1 )   .

1 S 1

1

 

0

(5.114)

2

v

  (5.115)   a2 cS 2 a1 cS 1 δ 0 . a1 + a2   +   m v v cS 1 cS 2 (  (a 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   v v →0   = δ 0   and   ωst δ 0 v a1 + a2 a1 + a2

 

||

(5.116)

and decreases with increasing speed. At speeds larger then

vβ sstt=0   =

 

a2 cS 2 (  (a 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 v 1 + v

 | |

δ 0 ,  v 2 vch

 

 

(5.118)

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where

vch   =

 

 (a   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

-8

50

a1*cS1 /a2*c S2 = 1.3333 -10

0

10

m=700 kg kg;; Θ=1000 kg m2;

20 v [m/s]

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

30

a1 =1 =1..2 m; a2 =1 =1..3 m;

0 0

40

 

10

cS 1  = 80 000 000 N m;

 

20 v [m/s]

30

40

110770 N m cS 2   = 73846 N m 55385 N m

Figure 5.14: Steady State Cornering In Fig. 5.14 Fig.  5.14 the  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 δ 0  = 1.4321◦ , in order to let the vehicle drive a circle with the radius R 0  = 100 m 100  m  at  v   0. The actually driven circle radius  R  has been calculated via   (5.120) ωst   =   v   .

 →

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.

4WSf/N:Nƒm N:Nƒm‹Œdl} ‹Œdl}‘fv—O ‘fv—O§PO› §PO› ÿOFf/VàN: OFf/VàN: 4WSf/ yÍSVàÿ Vàÿ‘Ùf/N ‘Ùf/NSï•ý‘¾R0v—0 Sï•ý‘¾R0v—0

5.3.7.2 Steering T Tendency endency 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 δ 0   =   + m R

a2 cS 2

−a

1 cS 1

 (a a1 + a2 ) R v cS 1 cS 2 (

||

 .

 

(5.121)

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The first term is the Ackermann steering angle, which follows from (5.2 ( 5.2)) with the wheel base a  = a  =  a 1 + a2  and the approximation for small steering angles  tan δ 0 δ 0 .

≈

The Ackermann-steering angle provides a good approximation for slowly moving vehicles, because at v 0 the second expression in (5.121 (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. table  5.1.

A 0

•   understeer   δ    > δ  0

•   neutral •   oversteer

  or   a1 cS 1  < a2 cS 2   or

  a1 cS 1 <  1 a2 cS 2

1 S 1 δ 0   =  δ 0A   or   a1 cS 1   =  a 2 cS 2   or   a c = 1 a2 cS 2

 

  δ 0   < δ 0A   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 A Angles ngles

˙ 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 st st F yst1   =   cS 1 sst y 1   and   F y 2   =   cS 2 sy 2  ,

 

(5.124)

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

2

F yst2 cS 2 sst a1 y2 a2 = F yst1 = cS 1 sst y1

  or

sst a1 cS 1 y2 a2 cS 2 = sst y1 .

 

(5.125)

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Vehicle Dynamics

FH Regensburg, University of Applied Sciences

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 Stiffnes Stiffness s 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 clin S 1   =   cS  F z 1   and   cS 2   =   cS  F z 2  .

 

(5.126)

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

F z1   =

  a2   a1 G   and   .F z 2   = G a1 + a2 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)

a1 clin S 1   =   a1 cS  and

a2 clin S 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 a1 clin   (5.130) S 1   =   a2 cS 2 The fact that the lateral force is applied behind the center of the contact area at the caster offset a1   a1  |vv| n L   and  a 2   a2  +  |vv|  n L  to a stabilization of the distance, leads, because of   a 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.

 a   = 2.45 45 m  m  the axle loads  F z1  = 4000 N  4000 N   and F z 2  = 3000 N  3000 N  At a vehicle with the wheel base  a deliver the position of the center of gravity a1   = 1.05 m and a2   = 1.40 m. At equal slip on front 2576 N   and F y2  = 2043 N  2043 N . With this, and rear axle one receives from the table in 5.15 in  5.15 F y1  = 2576 N   1..05 2576 1.45 2043 = 257 257..55 . The value is significantly the condition (5.131 (5.131)) delivers delivers 1 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.

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FH Regensburg, University of Applied Sciences

© Prof. Dr.-Ing. G. Rill 

6

F z  [N   [ N ]]

5 α

4    ]    N    k    [ 3   y

   F

2

1

0 0

1

2

3

4

Fz  [kN]

5

6

7

8

 

F y  [N   [ N ]]

0 1000

0 758

2 30 00 00 0 4000 5000 6000 7000 8000

1 24 03 48 3 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 Drivin Driving g Behavior Behavior of Single Single V Vehi ehicl cles es 6.1 Stand Standard ard Dri Driving ving Man Maneuve euvers rs 6.1.1 Steady Steady State Cornering Cornering 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. 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 5

   ] 3   g

   ]    N

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

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

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]

 R  = 100 m Figure 6.1: Steady State Cornering: Rear-Wheel-Driven Car on  R = 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.

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FH Regensburg, University of Applied Sciences

© Prof. Dr.-Ing. G. Rill 

The no The nonl nlin inea earr in influ fluen ence ce of the the whee wheell load load on the the tire tire pe perf rfor orma manc ncee is here here us used ed 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 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 6.2 a real steering angle gradient is displayed. 40    ]   g   e    d    [     e    l   g   n   a     g   n    i   r   e   e    t   s

30

20

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.

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Vehicle Dynamics

FH Regensburg, University of Applied Sciences

0.6

12

0.5

10

   ]   g    [   n0.4   o    i    t   a   r 0.3   e    l   e   c   c   a0.2    l   a   r   e    t   a0.1    l

   ]   s    /

8

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

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

2

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

  e    d    i   s-1.5

  o   r

0.5 0 0

2

4

-2

0

2

[t] 

4

 v =  = 100 km/h Figure 6.3: Step Steer: Passenger Car at  v The vehicle behaves dynamically dynamically very well, Fig. 6.3 Fig. 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 P Profile rofile The irregularities of a track are of stochastic nature. Fig.  Fig.   6.4 shows 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   =   z (x, y)

 →

  z 1   =   z 1 (s1) ,   and   z 2   =   z 2 (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 dir direct ection ion-independent. Then a two-dimensional track with its stochastic properties can be described by a single random process

z   z   =   z (x, y)

 →

  z   z   =   z (s) ;

 

(6.2)

96

 

FH Regensburg, University of Applied Sciences

© Prof. Dr.-Ing. G. Rill 

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

 z   =  z (s)  is completely characA normally distributed, stationary and ergodic random process  z   terized by the first two expectation values, mean value s

1 mz   = sl→∞ im 2s

z (s) ds

   

 

(6.3)

−s

and correlating function

1 s→∞ 2s

Rzz (δ ) = lim

s

z (s) z (s

−s

− δ ) ds

 

(6.4)

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

Rzz (δ ) =   Rzz ( δ )

−

and

 

(6.5)

s

Rzz (0 (0)) = lim 1 s→∞ 2s

z (s)

  

2

ds

 

(6.6)

−s

describes the squared average of  z   z s .

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 zzzz (Ω) cos(Ωδ  cos(Ωδ ) dΩ

 

(6.7)

0

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

S zz zz (Ω) dΩ .

Rzz (0) =

  0

 

(6.8)

97

 

Vehicle Dynamics

FH Regensburg, University of Applied Sciences

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

 Ω S zz (Ω) =   S 0 Ω0

−w

 

(6.9)

Where the reference frequency is fixed to  Ω 0   = 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 Stee Steering ring Acti Activity vity A straightforward drive upon an uneven uneven track makes continuous steering corrections necessary. necessary. The histograms of the steering angle at a driving speed of  v  = 90 Fig. 6.5.. 90km/h km/h are displayed in Fig. 6.5 -6

3

country road: S0=2*10-5  m3; w=2

highway: S 0=1*1 =1*10 0  m ; w=2 1000

1000

500

500

0

-2

0

[deg] 2

0

-2

0

[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 Coac Coach h with differ different ent Loading Loading Conditi Conditions ons 6.2.1 6.2 .1 Data Data At trucks and coaches the difference between empty and laden is sometimes very large. In the table 6.1 table  6.1 all  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

 

FH Regensburg, University of Applied Sciences

 [kg]]   center of gravity [m  [ m] mass [kg

vehicle empty

 

12 50 5000

0000 fully laden   18 00

  −3.800 | 0.000 | 1.500   −3.860 | 0.000 | 1.600

© Prof. Dr.-Ing. G. Rill 

 [kg m2 ] inertias [kg

 

12 500 0 0 155 000

0 0

0 0 155 000 15 400 0 250 0 200 550 0 250 0 202 160

Table 6.1: Data for a Laden and Empty Coach

The coach has a wheel base of   a a   = 6.25 25 m  m. The front axle with the track width  s v   = 2.046 046 m  m has a double wishbone single wheel suspension. The twin-tire rear axle with the track widths soh   = 2.152 152 m  m   and   shi   = 1.492 492 m  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 Cornering Cornering Fig. 6.7 Fig.  6.7 shows  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

  [deg]

vehicle course

LW

250

200

200

150

150

     ]     m100      [

100

50

50

0 0

-100

0.1 0.2 0.3 0.4 lateral acceleration a y  [g] wheel loads [kN] 100

50

50

0

0.1 0.2 0.3 0.4 lateral acceleration a y  [g]

100

wheel loads [kN]

100

0

0 [m]

0

0

0.1 0.2 0.3 0.4 lateral acceleration a y  [g]

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  a y vehicle is fully occupied, this effect occurs already at  ay 0.35 g .

 ≈

 ≈   0.4 g  . If the

6.2.4 Step Steer Steer Input Input  v   = 80 km/h can be seen in Fig. 6.8 The results of a step steer input at the driving speed of  v Fig.  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 Diff Differen erentt Rear Axle Concep Concepts ts for a Pas Passenge sengerr 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

 

FH Regensburg, University of Applied Sciences

© Prof. Dr.-Ing. G. Rill  yaw velocity

lateral acceleration a y  [g]

ω Z  [deg/s]

10

0.4

8

0.3

6 0.2

4

0.1 0

2 0

2

4

6

8

0

0

roll angle   α [deg]

2

4

side slip angle

6

8

β  [deg]

8 2 6

1

4

0 -1

2

-2 00

2

4  [s] 6

8

0

2

4  [s] 6

8

Figure Figu re 6.8: Step Steer: Steer: - - Coach empty empty, — Coac Coach h fully occupied occupied 10

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

-5

0 lateral motion [cm]

5

 ··  · · ·   Trailing

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

The semi-trailing axle realized in serial production represents, according to the roll support, Fig. 6.9, Fig.  6.9, a  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. 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

FH Regensburg, University of Applied Sciences

steer angle δ

 [deg]

roll angle α  [Grad]

LW

100

5 4 3

50

2 1

0 0

0.2

0.4

0.6

0.8

0 0

6

4

4

2

2

0.2 0.4 0.6 0.8 lateral acceleration a y  [g]

0.4

0.6

0.8

wheel loads rear [kN]

wheel loads front front [kN] 6

0 0

0.2

0 0

0.2 0.4 0.6 0.8 lateral acceleration a y  [g]

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 Diff Differen erentt Influenc Influences es on Comfo Comfort rt 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

 

FH Regensburg, University of Applied Sciences

© Prof. Dr.-Ing. G. Rill 

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 mode models. ls.

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 Simulatio Simulation n Resul Results ts  v =  = 80 km/h over a country The vehicle, a Ford Mondeo, occupied by two persons, drives with v road. The thereby occurring accelerations at the driver’s seat rail and the wheel load variations are displayed in Fig. 6.12 Fig.  6.12.. The peak values of the accelerations and the maximal wheel load variations are arranged in the tables 6.2 tables  6.2 and  and 6.3  6.3 for  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.

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Vehicle Dynamics

FH Regensburg, University of Applied Sciences

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

x ¨min   -0.7192 -0.7133 ¨max   +0.6543 +0.6100 x y¨min   -1.4199 -1.2873 y¨max   +1.3991 +1.2529 zz  ¨ min   -4.1864 -3.9986 min zz  ¨ max max   +3.0623 2.7769

-0.7403

-0.5086

-0.7328

+0.6695

+0.5092

+0.6886

-1.4344

-0.7331

-1.5660

+1.3247

+0.8721

+1.2564

-4.1788

-3.6950

-4.2593

+3.1176

+2.8114

+3.1449

Table 6.2: Peak Acceleration Values

F z   standard standard – friction friction – seat model model – engine engine moun mounts ts – comfort comfort bushing bushingss



front left

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

max z

min z

F    = F    − F  z

104

 

FH Regensburg, University of Applied Sciences

© 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

body lateral acceleration [m/s 2 ]

5

5

0

0

0

500  

1000

-5

0

[m]

wheel load front left [kN] 6

5

5

4

4

3

3

2

2

1

1

0

500  

1000 [m]

wheel load front right [kN]

6

0 0

500  

1000

0

[m]

wheel load rear left [kN] 6

5

5

4

4

3

3

2

2

1

1 0

500  

1000 [m]

500  

1000 [m]

wheel load rear right [kN]

6

0

1000  [m]

body vertical acceleration [m/s 2 ]

-5

500

[m]

0

0

500  

1000 [m]

Figure 6.12: Road Profile, Accelerations and Wheel Loads

105

 

Index Ackermann Geometry, 66 Geometry,  66 Ackermann Steering Angle, 66 Angle, 66,,  91 Aerodynamic Forces, 53 Forces,  53 Air Resistance, 53 Resistance,  53 All Wheel Drive, 75 Drive,  75 Angular Wheel Velocity, 27 Velocity,  27 Anti Dive, 65 Dive, 65 Anti Roll Bar, 80 Bar,  80 Anti Squat, 65 Squat,  65 Anti-Lock-Systems, 59 Anti-Lock-Systems,  59 Axle Kinematics, 65 Kinematics, 65 Double Wishbone, 10 Wishbone,  10 McPherson, 10 McPherson,  10 Multi-Link, 10 Multi-Link,  10 Axle Load, 52 Load,  52 Axle Suspension Suspension Rigid Axle, 4 Axle,  4 Twist Beam, 5 Beam,  5 Bend Angle, 73 Angle, 73 Brake Pitch Angle, 60 Angle, 60 Brake Pitch Pole, 65 Pole,  65

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

Camber Angle, 9, Angle,  9, 16  16 Camber Compensation, Compensation, 79,  79, 81  81 Camber Slip, 26 Slip,  26 Caster Angle, 11 Angle,  11 Caster Offset, 12 Offset, 12 Characteristic Speed, 90 Speed,  90 Climbing Capacity, 54 Capacity,  54 Comfort, 31 Comfort,  31 Contact Geometry, 15 Geometry,  15 Contact Point, 16 Point,  16 Contact Point Velocity, 20 Velocity, 20

Kingpin,  10 Kingpin, 10 Kingpin Angle, 11 Angle,  11 Kingpin Inclination, 11 Inclination, 11

Cornering Resistance, 74 Resistance, 74,, 76

Kingpin Offset, 12 Offset,  12

Generalized Fluid Mass, 49 Mass,  49 Grade, 52 Grade,  52 Hydro-Mount, 48 Hydro-Mount,  48

i

 

Vehicle Dynamics

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

FH Regensburg, University of Applied Sciences

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

ii

 

FH Regensburg, University of Applied Sciences

Longitudinal Force, 14 Force,  14,, 22,  22, 23 Longitudinal Force Characteristics Characteristics,,  23 Longitudinal Force Distribution, 23 Distribution, 23 Longitudinal Slip, 23 Slip,  23 Longitudinal Velocity, 20 Velocity,  20 Normal Force, 14 Force, 14 Pneumatic Trail, 24 Trail,  24 Radial Damping, 18 Damping, 18 Radial Direction, 16 Direction, 16 Radial Stiffness, 78 Stiffness,  78 Rolling Resistance, Resistance,  14 Self Aligning Torque, Torque,  14, 24  14,  24 Sliding Velocity, 24 Velocity, 24 Static Radius, 16, Radius,  16, 19  19 Tilting Torque, 14 Torque,  14 Transport Velocity, 19 elocity, 19 Tread Deflection, 21 Deflection,  21 Tread Particles, Particles, 21  21  19 Undeformed Radius, Radius, 19 Vertical Force, 17 Force, 17 tire composites, 13 composites,  13 Tire Development, 13 Development,  13 Tire Model Kinematic, 66 Kinematic,  66 Linear, 91 Linear,  91 TMeasy, 29 TMeasy,  29 Toe Angle, 9 Angle,  9 Track, 32 Track,  32 Track Curvature, 72 Curvature,  72 Track Normal, 16, Normal,  16, 17  17 Track Radius, 72 Radius, 72 Track Width, 66 Width,  66,,  78 Trailer, 69 Trailer,  69,,  72 Turning Center, 66 Center, 66 Understeer, 91 Understeer,  91 Vehicle,  2 Vehicle, 2 Vehicle Comfort, 31 Comfort, 31 Vehicle Data, 41 Data, 41 Vehicle Dynamics, 1 Dynamics,  1 Vehicle Model,  Model,   31 31,,   39,  39,   51,  51,   60 60,,   69 69,,   79,  79,   83 83,, 102 Virtual Work, 80 Work,  80

© Prof. Dr.-Ing. G. Rill 

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