Springer Vehicle Dynamics Mar 2008 ISBN 0387742433 pdf

  Vehicle Dynamics:

Theory and Application

  Reza N. Jazar Vehicle Dynamics:

Theory and Applications

  Reza N. Jazar Dept. of Mechanical Engineering Manhattan College Riverdale, NY 10471

  ISBN: 978-0-387-74243-4 e-ISBN: 978-0-387-74244-1 Library of Congress Control Number: 2007942198 2008 Springer Science+Business Media, LLC

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  Dedicated to my son, Kavosh, my daughter, Vazan, and my wife, Mojgan. Happiness is when you win a race against yourself.

Preface

  This text is for engineering students. It introduces the fundamental knowl- edge used in vehicle dynamics. This knowledge can be utilized to develop computer programs for analyzing the ride, handling, and optimization of road vehicles.

  Vehicle dynamics has been in the engineering curriculum for more than a hundred years. Books on the subject are available, but most of them are written for specialists and are not suitable for a classroom application. A new student, engineer, or researcher would not know where and how to start learning vehicle dynamics. So, there is a need for a textbook for beginners. This textbook presents the fundamentals with a perspective on future trends.

  The study of classical vehicle dynamics has its roots in the work of great scientists of the past four centuries and creative engineers in the past century who established the methodology of dynamic systems. The development of vehicle dynamics has moved toward modeling, analysis, and optimization of multi-body dynamics supported by some compliant members. Therefore, merging dynamics with optimization theory was an expected development. The fast-growing capability of accurate positioning, sensing, and calculations, along with intelligent computer programming are the other important developments in vehicle dynamics. So, a textbook help the reader to make a computer model of vehicles, which this book does.

  Level of the Book This book has evolved from nearly a decade of research in nonlinear dynamic systems and teaching courses in vehicle dynamics. It is addressed primarily to the last year of undergraduate study and the first year graduate student in engineering. Hence, it is an intermediate textbook. It provides both fundamental and advanced topics. The whole book can be covered in two successive courses, however, it is possible to jump over some sec- tions and cover the book in one course. Students are required to know the fundamentals of kinematics and dynamics, as well as a basic knowledge of numerical methods.

  The contents of the book have been kept at a fairly theoretical-practical level. Many concepts are deeply explained and their application empha- sized, and most of the related theories and formal proofs have been ex- plained. The book places a strong emphasis on the physical meaning and applications of the concepts. Topics that have been selected are of high x Preface

  broad range of topics and approaches.

  There are four special chapters that are indirectly related to vehicle dy- namics: Applied Kinematics, Applied Mechanisms, Applied Dynamics, and Applied Vibrations. These chapters provide the related background to un- derstand vehicle dynamics and its subsystems.

  Organization of the Book The text is organized so it can be used for teaching or for self-study.

  Chapter 1 “Fundamentals,” contains general preliminaries about tire and rim with a brief review of road vehicle classifications. Part I “One Dimensional Vehicle Dynamics,” presents forward vehicle dynamics, tire dynamics, and driveline dynamics. Forward dynamics refers to weight transfer, accelerating, braking, engine performance, and gear ratio design.

  Part II “Vehicle Kinematics,” presents a detailed discussion of vehicle mechanical subsystems such as steering and suspensions. Part III “Vehicle Dynamics,” employs Newton and Lagrange methods to develop the maneuvering dynamics of vehicles. Part IV “Vehicle Vibrations,” presents a detailed discussion of vehi- cle vibrations. An attempt is made to review the basic approaches and demonstrate how a vehicle can be modeled as a vibrating multiple degree- of-freedom system. The concepts of the Newton-Euler dynamics and La- grangian method are used equally for derivation of equations of motion. The RMS optimization technique for suspension design of vehicles is intro- duced and applied to vehicle suspensions. The outcome of the optimization technique is the optimal stiffness and damping for a car or suspended equip- ment.

  Method of Presentation This book uses a "fact-reason-application" structure. The "fact" is the main subject we introduce in each section. Then the reason is given as a

  "proof." The application of the fact is examined in some "examples." The "examples" are a very important part of the book because they show how to implement the "facts." They also cover some other facts that are needed to expand the subject.

  Prerequisites Since the book is written for senior undergraduate and first-year graduate- level students of engineering, the assumption is that users are familiar with matrix algebra as well as basic dynamics. Prerequisites are the fundamen- tals of kinematics, dynamics, vector analysis, and matrix theory. These basics are usually taught in the first three undergraduate years.

  • Lowercase bold letters indicate a vector. Vectors may be expressed in an n dimensional Euclidian space. Example: r , s , d , a , b , c p , q , v , w , y , z ω , α , ²
  • Uppercase bold letters indicate a dynamic vector or a dynamic ma- trix, such as force and moment. Example:
  • Lowercase letters with a hat indicate a unit vector. Unit vectors are not bolded. Example:
  • Lowercase letters with a tilde indicate a 3 × 3 skew symmetric matrix associated to a vector. Example:

  Example: r = |r| , a = |a| , b = |b| , s = |s|

  , a =

  ⎡ ⎣ a

  1

  a

  2

  a

  3

  ⎤ ⎦

  −−→ ON = a position vector from point O to point N • The length of a vector is indicated by a non-bold lowercase letter.

  B(oxyz) , B(Oxyz) , B

  1

  1

  (o

  1

  x

  1

  y

  1

  z

  1

  )

  ⎤ ⎦

  2 a

  Preface xi

  , ˆ e

  Unit System The system of units adopted in this book is, unless otherwise stated, the international system of units (SI). The units of degree (deg) or radian ( rad) are utilized for variables representing angular quantities.

  Symbols

  , θ , δ , φ

  F , M

  ˆı , ˆ j , ˆ k , ˆ e , ˆ u , ˆ n

  ˆ I ,

  ˆ J ,

  ˆ K , ˆ e

  θ

  ϕ

  −a

  , ˆ e

  ψ

  ˜ a = ⎡ ⎣

  −a

  3 a

  2

  a

  3

  −a

  1

  • An arrow above two uppercase letters indicates the start and end points of a position vector. Example:
  • Capital letter B is utilized to denote a body coordinate frame. Ex- ample:
xii Preface

  • Capital letter G is utilized to denote a global, inertial, or fixed coor- dinate frame. Example:

  G , G(XY Z) , G(OXY Z)

  • Right subscript on a transformation matrix indicates the departure frames. Example:

  R = transformation matrix from frame B(oxyz)

  B

  • Left superscript on a transformation matrix indicates the destination frame. Example:

  G

  R = transformation matrix from frame B(oxyz)

  B

  to frame G(OXY Z)

  • Capital letter R indicates rotation or a transformation matrix, if it shows the beginning and destination coordinate frames. Example:

  ⎡ ⎤ cos α − sin α 0

  G

  R = ⎣ sin α cos α ⎦

  B

  1

  • Whenever there is no sub or superscript, the matrices are shown in a bracket. Example:

  ⎡ ⎤ cos α − sin α 0

  [T ] = ⎣ sin α cos α ⎦

  1

  • Left superscript on a vector denotes the frame in which the vector is expressed. That superscript indicates the frame that the vector belongs to; so the vector is expressed using the unit vectors of that frame. Example:

  G

  r = position vector expressed in frame G(OXY Z)

  • Right subscript on a vector denotes the tip point that the vector is referred to. Example:

  G

  r = position vector of point P

  P

  expressed in coordinate frame G(OXY Z)

  • Right subscript on an angular velocity vector indicates the frame that the angular vector is referred to. Example:

  ω = angular velocity of the body coordinate frame B(oxyz)

  B

  Preface xii i

  • Left subscript on an angular velocity vector indicates the frame that the angular vector is measured with respect to. Example:

  ω = angular velocity of the body coordinate frame B(oxyz)

  G B

  with respect to the global coordinate frame G(OXY Z)

  • Left superscript on an angular velocity vector denotes the frame in which the angular velocity is expressed. Example:

  B 2

  ω B 1 = angular velocity of the body coordinate frame B

  1 G

  with respect to the global coordinate frame G, and expressed in body coordinate frame B

  2 Whenever the subscript and superscript of an angular velocity are

  the same, we usually drop the left superscript. Example:

  G

  ω ω

  

G B B

  ≡ G Also for position, velocity, and acceleration vectors, we drop the left subscripts if it is the same as the left superscript. Example:

  

B B

  v v

  P P B ≡

  • Left superscript on derivative operators indicates the frame in which the derivative of a variable is taken. Example:

  G G B

  d d d

  B G

  r r x , ,

  P P B

  dt dt dt If the variable is a vector function, and also the frame in which the vector is defined is the same frame in which a time derivative is taken, we may use the following short notation,

  G B

  d d

  G G B B

  r = , r =

  P ˙r P P ˙r P o o

  dt dt and write equations simpler. Example:

  G

  d

  

G G G

  v r = (t) =

  ˙r dt

  • If followed by angles, lowercase c and s denote cos and sin functions in mathematical equations. Example: cα = cos α , sϕ = sin ϕ
x v i Preface

  • Capital bold letter I indicates a unit matrix, which, depending on the dimension of the matrix equation, could be a 3 × 3 or a 4 × 4 unit matrix. I

  3 or I 4 are also being used to clarify the dimension of

  I . Example:

  ⎡ ⎤ 1 0 0

  I = I = ⎣ 0 1 0 ⎦

  3

  0 0 1

  • An asterisk

  F indicates a more advanced subject or example that is not designed for undergraduate teaching and can be dropped in the first reading. Contents

  Preface x

  74

  39 2.2 Parked Car on an Inclined Road . . . . . . . . . . . . . . .

  44 2.3 Accelerating Car on a Level Road . . . . . . . . . . . . . . .

  50 2.4 Accelerating Car on an Inclined Road . . . . . . . . . . . .

  55 2.5 Parked Car on a Banked Road . . . . . . . . . . . . . . . .

  65

  2.6 F Optimal Drive and Brake Force Distribution . . . . . . .

  68

  2.7 F Vehicles With More Than Two Axles . . . . . . . . . . .

  2.8 F Vehicles on a Crest and Dip . . . . . . . . . . . . . . . .

  2 Forward Vehicle Dynamics

  78

  2.8.1 F Vehicles on a Crest . . . . . . . . . . . . . . . . .

  78

  2.8.2 F Vehicles on a Dip . . . . . . . . . . . . . . . . . .

  82 2.9 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . .

  87 2.10 Key Symbols . . . . . . . . . . . . . . . . . . . . . . . . . .

  88 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

  90

  39 2.1 Parked Car on a Level Road . . . . . . . . . . . . . . . . . .

  37

  1 Tire and Rim Fundamentals

  21

  1 1.1 Tires and Sidewall Information . . . . . . . . . . . . . . . .

  1 1.2 Tire Components . . . . . . . . . . . . . . . . . . . . . . . .

  11 1.3 Radial and Non-Radial Tires . . . . . . . . . . . . . . . . .

  14 1.4 Tread . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

  17

  1.5 F Hydroplaning . . . . . . . . . . . . . . . . . . . . . . . .

  18 1.6 Tireprint . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

  20 1.7 Wheel and Rim . . . . . . . . . . . . . . . . . . . . . . . . .

  1.8 Vehicle Classifications . . . . . . . . . . . . . . . . . . . . .

  34 I One-Dimensional Vehicle Dynamics

  25

  1.8.1 ISO and FHWA Classification . . . . . . . . . . . . .

  25

  1.8.2 Passenger Car Classifications . . . . . . . . . . . . .

  28 1.8.3 Passenger Car Body Styles . . . . . . . . . . . . . .

  30 1.9 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . .

  31 1.10 Key Symbols . . . . . . . . . . . . . . . . . . . . . . . . . .

  33 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

  3 Tire Dynamics 95 i

  3.2 Tire Stiffness . . . . . . . . . . . . . . . . . . . . . . . . . .

  5.2 Successive Rotation About Global Cartesian Axes . . . . . 223

  4.4.1 Geometric Ratio Gearbox Design . . . . . . . . . . . 188

  4.4.2 F Progressive Ratio Gearbox Design . . . . . . . . . 190

  4.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205

  4.6 Key Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . 207 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209

  

II Vehicle Kinematics 217

  5 Applied Kinematics 219

  5.1 Rotation About Global Cartesian Axes . . . . . . . . . . . . 219

  5.3 Rotation About Local Cartesian Axes . . . . . . . . . . . . 225

  4.3 Gearbox and Clutch Dynamics . . . . . . . . . . . . . . . . 178

  5.4 Successive Rotation About Local Cartesian Axes . . . . . . 229

  5.6 General Transformation . . . . . . . . . . . . . . . . . . . . 241

  5.7 Angular Velocity . . . . . . . . . . . . . . . . . . . . . . . . 248

  5.8 F Time Derivative and Coordinate Frames . . . . . . . . . 257

  5.9 Rigid Body Velocity . . . . . . . . . . . . . . . . . . . . . . 267

  5.10 Angular Acceleration . . . . . . . . . . . . . . . . . . . . . . 272

  Contents xvi

  4.4 Gearbox Design . . . . . . . . . . . . . . . . . . . . . . . . . 187

  4.2 Driveline and Efficiency . . . . . . . . . . . . . . . . . . . . 173

  98

  3.5.3 F Effect of Sideslip Angle on Rolling Resistance . . 125

  3.3 Tireprint Forces . . . . . . . . . . . . . . . . . . . . . . . . . 104

  3.3.1 Static Tire, Normal Stress . . . . . . . . . . . . . . . 104

  3.3.2 Static Tire, Tangential Stresses . . . . . . . . . . . . 108

  3.4 Effective Radius . . . . . . . . . . . . . . . . . . . . . . . . 109

  3.5 Rolling Resistance . . . . . . . . . . . . . . . . . . . . . . . 114

  3.5.1 F Effect of Speed on the Rolling Friction Coefficient 119

  3.5.2 F Effect of Inflation Pressure and Load on the Rolling Friction Coefficient . . . . . . . . . . . . . . . . . . . 122

  3.5.4 F Effect of Camber Angle on Rolling Resistance . . 127

  4.1 Engine Dynamics . . . . . . . . . . . . . . . . . . . . . . . . 165

  3.6 Longitudinal Force . . . . . . . . . . . . . . . . . . . . . . . 127

  3.7 Lateral Force . . . . . . . . . . . . . . . . . . . . . . . . . . 135

  3.8 Camber Force . . . . . . . . . . . . . . . . . . . . . . . . . . 145

  3.9 Tire Force . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

  3.10 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

  3.11 Key Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . 159 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

  4 Driveline Dynamics 165

5.5 F Euler Angles . . . . . . . . . . . . . . . . . . . . . . . . . 231

  5.12 F Axis-angle Rotation . . . . . . . . . . . . . . . . . . . . . 282

  8.2 Independent Suspension . . . . . . . . . . . . . . . . . . . . 465

  7.6 F Steering Mechanism Optimization . . . . . . . . . . . . . 424

  7.7 F Trailer-Truck Kinematics . . . . . . . . . . . . . . . . . . 434

  7.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . 447

  7.9 Key Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . 449 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 451

  8 Suspension Mechanisms 455

  8.1 Solid Axle Suspension . . . . . . . . . . . . . . . . . . . . . 455

  8.3 Roll Center and Roll Axis . . . . . . . . . . . . . . . . . . . 470

  7.4 Steering Mechanisms . . . . . . . . . . . . . . . . . . . . . . 403

  8.4 F Car Tire Relative Angles . . . . . . . . . . . . . . . . . . 478

  8.4.1 F Toe . . . . . . . . . . . . . . . . . . . . . . . . . . 479

  8.4.2 F Caster Angle . . . . . . . . . . . . . . . . . . . . . 482

  8.4.3 F Camber . . . . . . . . . . . . . . . . . . . . . . . 483

  8.4.4 F Trust Angle . . . . . . . . . . . . . . . . . . . . . 483

  8.5 Suspension Requirements and Coordinate Frames . . . . . . 485

  8.5.1 Kinematic Requirements . . . . . . . . . . . . . . . . 485

  7.5 F Four wheel steering. . . . . . . . . . . . . . . . . . . . . . 409

  7.3 F Vehicle with Trailer . . . . . . . . . . . . . . . . . . . . . 398

  5.13 F Screw Motion . . . . . . . . . . . . . . . . . . . . . . . . 288

  6.5 Coupler Point Curve . . . . . . . . . . . . . . . . . . . . . . 356

  5.14 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301

  5.15 Key Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . 304 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305

  6 Applied Mechanisms 309

  6.1 Four-Bar Linkage . . . . . . . . . . . . . . . . . . . . . . . . 309

  6.2 Slider-Crank Mechanism . . . . . . . . . . . . . . . . . . . . 332

  6.3 Inverted Slider-Crank Mechanism . . . . . . . . . . . . . . . 339

  6.4 Instant Center of Rotation . . . . . . . . . . . . . . . . . . . 346

  6.5.1 Coupler Point Curve for Four-Bar Linkages . . . . . 356

  7.2 Vehicles with More Than Two Axles . . . . . . . . . . . . . 395

  6.5.2 Coupler Point Curve for a Slider-Crank Mechanism . 360

  6.5.3 Coupler Point Curve for Inverted Slider-Crank Mech- anism . . . . . . . . . . . . . . . . . . . . . . . . . . 362

  6.6 F Universal Joint Dynamics . . . . . . . . . . . . . . . . . 363

  6.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372

  6.8 Key Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . 373 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374

  7 Steering Dynamics 379

  7.1 Kinematic Steering . . . . . . . . . . . . . . . . . . . . . . . 379

  Contents xvii

  8.5.3 Wheel, wheel-body, and tire Coordinate Frames . . . 487

  665

  10.4 Two-wheel Rigid Vehicle Dynamics . . . . . . . . . . . . . . 609

  10.5 Steady-State Turning . . . . . . . . . . . . . . . . . . . . . . 620

  10.6 F Linearized Model for a Two-Wheel Vehicle . . . . . . . . 631

  10.7 F Time Response . . . . . . . . . . . . . . . . . . . . . . . 635

  10.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . 655

  10.9 Key Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . 657 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 659

  11.1 F Vehicle Coordinate and DOF . . . . . . . . . . . . . . . . 665

  8.6 F Caster Theory . . . . . . . . . . . . . . . . . . . . . . . . 497

  11.2 F Equations of Motion . . . . . . . . . . . . . . . . . . . . 666

  11.3 F Vehicle Force System . . . . . . . . . . . . . . . . . . . . 671

  11.3.1 F Tire and Body Force Systems . . . . . . . . . . . 671

  11.3.2 F Tire Lateral Force . . . . . . . . . . . . . . . . . . 674

  11.3.3 F Body Force Components on a Two-wheel Model . 677

  11.4 F Two-wheel Rigid Vehicle Dynamics . . . . . . . . . . . . 684

  Contents xviii

  10.3.3 Two-wheel Model and Body Force Components . . . 601

  10.3.2 Tire Lateral Force . . . . . . . . . . . . . . . . . . . 600

  10.3.1 Tire Force and Body Force Systems . . . . . . . . . 597

  9.3 Rigid Body Rotational Dynamics . . . . . . . . . . . . . . . 530

  8.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . 508

  8.8 Key Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . 510 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 512

  

III Vehicle Dynamics 519

  9 Applied Dynamics 521

  9.1 Force and Moment . . . . . . . . . . . . . . . . . . . . . . . 521

  9.2 Rigid Body Translational Dynamics . . . . . . . . . . . . . 528

  9.4 Mass Moment of Inertia Matrix . . . . . . . . . . . . . . . . 542

  10.3 Force System Acting on a Rigid Vehicle . . . . . . . . . . . 597

  9.5 Lagrange’s Form of Newton’s Equations of Motion . . . . . 554

  9.6 Lagrangian Mechanics . . . . . . . . . . . . . . . . . . . . . 561

  9.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . 571

  9.8 Key Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . 574 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 575

  10 Vehicle Planar Dynamics 583

  10.1 Vehicle Coordinate Frame . . . . . . . . . . . . . . . . . . . 583

  10.2 Rigid Vehicle Newton-Euler Dynamics . . . . . . . . . . . . 589

11 F Vehicle Roll Dynamics

11.5 F Steady-State Motion . . . . . . . . . . . . . . . . . . . . 688

  Contents xix

  11.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . 710

  11.8 Key Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . 712 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 715

  

IV Vehicle Vibration 727

  12 Applied Vibrations 729

  12.1 Mechanical Vibration Elements . . . . . . . . . . . . . . . . 729

  12.2 Newton’s Method and Vibrations . . . . . . . . . . . . . . . 738

  12.3 Frequency Response of Vibrating Systems . . . . . . . . . . 744

  12.3.1 Forced Excitation . . . . . . . . . . . . . . . . . . . 745

  12.3.2 Base Excitation . . . . . . . . . . . . . . . . . . . . . 756

  12.3.3 Eccentric Excitation . . . . . . . . . . . . . . . . . . 768

  12.3.4 F Eccentric Base Excitation . . . . . . . . . . . . . 775

  12.3.5 F Classification for the Frequency Responses of One- DOF Forced Vibration Systems . . . . . . . . . . . . 781

  12.4 Time Response of Vibrating Systems . . . . . . . . . . . . . 786

  12.5 Vibration Application and Measurement . . . . . . . . . . . 799

  12.6 F Vibration Optimization Theory . . . . . . . . . . . . . . 804

  12.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . 816

  12.8 Key Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . 818 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 821

  13 Vehicle Vibrations 827

  13.1 Lagrange Method and Dissipation Function . . . . . . . . . 827

  13.2 F Quadratures . . . . . . . . . . . . . . . . . . . . . . . . . 838

  13.3 Natural Frequencies and Mode Shapes . . . . . . . . . . . . 845

  13.4 Bicycle Car and Body Pitch Mode . . . . . . . . . . . . . . 853

  13.5 Half Car and Body Roll Mode . . . . . . . . . . . . . . . . . 858

  13.6 Full Car Vibrating Model . . . . . . . . . . . . . . . . . . . 864

  13.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . 875

  13.8 Key Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . 876 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 878

  14 Suspension Optimization 883

  14.1 Mathematical Model . . . . . . . . . . . . . . . . . . . . . . 883

  14.2 Frequency Response . . . . . . . . . . . . . . . . . . . . . . 890

  14.3 RMS Optimization . . . . . . . . . . . . . . . . . . . . . . . 894

  14.4 F Time Response Optimization . . . . . . . . . . . . . . . . 918

  14.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . 924

  14.6 Key Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . 925 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 927

  15 931 xx Contents

  15.1 Mathematical Model . . . . . . . . . . . . . . . . . . . . . . 931

  15.2 Frequency Response . . . . . . . . . . . . . . . . . . . . . . 933

15.3 F Natural and Invariant Frequencies . . . . . . . . . . . . . 938

  15.4 F RMS Optimization . . . . . . . . . . . . . . . . . . . . . 953

  15.5 F Optimization Based on Natural Frequency and Wheel Travel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 964

  15.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . 970

  15.7 Key Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . 971 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 973

  References 977

  A Frequency Response Curves 983 B Trigonometric Formulas 989 C Unit Conversions

  993 Index

  997

1 Tire and Rim Fundamentals

  We introduce and review some topics about tires, wheels, roads, vehicles, and their interactions. These subjects are needed to understand vehicle dynamics better.

1.1 Tires and Sidewall Information

  Pneumatic tires are the only means to transfer forces between the road and the vehicle. Tires are required to produce the forces necessary to control the vehicle, and hence, they are an important component of a vehicle.

Figure 1.1 illustrates a cross section view of a tire on a rim to show the dimension parameters that are used to standard tires.

  Tireprint width

  h T , Section height

  Sidewall Pan width

  w T , Section width

FIGURE 1.1. Cross section of a tire on a rim to show tire height and width.

  The section height, tire height, or simply height, h , is a number that

  T

  must be added to the rim radius to make the wheel radius. The section width, or tire width, w , is the widest dimension of a tire when the tire is

  T not loaded.

  Tires are required to have certain information printed on the tire sidewall.

Figure 1.2 illustrates a side view of a sample tire to show the important information printed on a tire sidewall.

1 Tire and Rim Fundamentals

  We introduce and review some topics about tires, wheels, roads, vehicles, and their interactions. These subjects are needed to understand vehicle dynamics better.

1.1 Tires and Sidewall Information

  Pneumatic tires are the only means to transfer forces between the road and the vehicle. Tires are required to produce the forces necessary to control the vehicle, and hence, they are an important component of a vehicle.

Figure 1.1 illustrates a cross section view of a tire on a rim to show the dimension parameters that are used to standard tires.

  Tireprint width

  h T , Section height

  Sidewall Pan width

  w T , Section width

FIGURE 1.1. Cross section of a tire on a rim to show tire height and width.

  The section height, tire height, or simply height, h , is a number that

  T

  must be added to the rim radius to make the wheel radius. The section width, or tire width, w , is the widest dimension of a tire when the tire is

  T not loaded.

  Tires are required to have certain information printed on the tire sidewall.

Figure 1.2 illustrates a side view of a sample tire to show the important information printed on a tire sidewall.

  2

1. Tire and Rim Fundamentals

  

1

R15 5/ 60

  1

  96H I N P2 CA NA DA M DOT FA 7C D & E F S A D E 3 R 9 E

  4 M

  6 IN S

  7 E F O R C E L S

  11 E 1 D

  5 2 L B E 4 P 5 IA U T D A

  1 A H

  3 6 R R 9 5 1 R M S

6 A

  2 5/ 1 P2 ES PR X T i Ps R4 S U

  IR E A+

  8

  2 FIGURE 1.2. Side view of a tire and the most important information printed on a tire sidewall.

  The codes in Figure 1.2 are: 1 Size number.

  2 Maximum allowed inflation pressure.

  3 Type of tire construction.

  4 M&S denotes a tire for mud and snow.

  5 E-Mark is the Europe type approval mark and number.

  6 US Department of Transport (DOT) identification numbers.

  7 Country of manufacture.

  8 Manufacturers, brand name, or commercial name. The most important information on the sidewall of a tire is the size number, indicated by 1 . To see the format of the size number, an example is shown in Figure 1.3 and their definitions are explained as follows.

  P Tire type. The first letter indicates the proper type of car that the tire is made for. P stands for passenger car. The first letter can also be ST for special trailer, T for temporary, and LT for light truck.

  215 Tire width. This three-number code is the width of the unloaded

  1. Tire and Rim Fundamentals

  3 P 215 / 60 R 15 96 H

  P Passenger car

  215 Tire width [mm]

  60 Aspect ratio [%] R

  Radial

  15 Rim diameter [in]

  96 Load rating H

  Speed rating

FIGURE 1.3. A sample of a tire size number and its meaning.

  60 Aspect ratio. This two-number code is the ratio of the tire section height to tire width, expressed as a percentage. Aspect ratio is shown by s T . h

  

T

  s T = × 100 (1.1) w

  

T

  Generally speaking, tire aspect ratios range from 35, for race car tires, to 75 for tires used on utility vehicles.

  R Tire construction type. The letter R indicates that the tire has a radial construction. It may also be B for bias belt or bias ply, and D for diagonal.

15 Rim diameter. This is a number in [ in] to indicate diameter of the rim that the tire is designed to fit on.

  96 Load rate or load index. Many tires come with a service description at the end of the tire size. The service description is made of a two-digit number (load index) and a letter (speed rating). The load index is a rep- resentation of the maximum load each tire is designed to support.

Table 1.1 shows some of the most common load indices and their load- carrying capacities. The load index is generally valid for speeds under

  210 km/ h ( ≈ 130 mi/ h).

  H Speed rate. Speed rate indicates the maximum speed that the tire can sustain for a ten minute endurance without breaking down.

Table 1.2 shows the most common speed rate indices and their meanings.

  Example 1 Weight of a car and load index of its tire.

  For a car that weighs 2 tons = 2000 kg, we need a tire with a load index higher than 84. This is because we have about 500 kg per tire and it is in a

  4

1. Tire and Rim Fundamentals

  Index Maximum load Index Maximum load 45 kg ≈ 99 lbf

  ≈ 1565 lbf 126 1700 kg ≈ 3748 lbf 97 730 kg

  ≈ 1323 lbf 120 1400 kg ≈ 3086 lbf 91 615 kg

  ≈ 1356 lbf 121 1450 kg ≈ 3197 lbf 92 630 kg

  ≈ 1389 lbf 122 1500 kg ≈ 3368 lbf 93 650 kg

  ≈ 1433 lbf 123 1550 kg ≈ 3417 lbf 94 670 kg

  ≈ 1477 lbf 124 1600 kg ≈ 3527 lbf 95 690 kg

  ≈ 1521 lbf 125 1650 kg ≈ 3690 lbf 96 710 kg

  ≈ 1609 lbf 127 1750 kg ≈ 3858 lbf 98 750 kg

  ≈ 1235 lbf 118 1320 kg ≈ 2910 lbf 89 580 kg

  ≈ 1653 lbf 128 1800 kg ≈ 3968 lbf 99 775 kg

  ≈ 1709 lbf · · · · · · 199 13600 kg

  ≈ 30000 lbf Example 2 Height of a tire based on tire numbers.

  A tire has the size number P 215/60R15 96H. The aspect ratio 60 means the height of the tire is equal to 60% of the tire width. To calculate the tire height in [ mm], we should multiply the first number (215) by the second number (60) and divide by 100. h T = 215 ×

  60 100

  = 129 mm (1.2)

  ≈ 1279 lbf 119 1360 kg ≈ 3074 lbf 90 600 kg

  ≈ 1201 lbf 117 1285 kg ≈ 2833 lbf 88 560 kg

  · · · · · · 100 800 kg ≈ 1764 lbf 71 345 kg

  ≈ 908 lbf 107 975 kg ≈ 2149 lbf 78 425 kg

  ≈ 761 lbf 101 825 kg ≈ 1819 lbf 72 355 kg

  ≈ 783 lbf 102 850 kg ≈ 1874 lbf 73 365 kg

  ≈ 805 lbf 103 875 kg ≈ 1929 lbf 74 375 kg

  ≈ 827 lbf 104 900 kg ≈ 1984 lbf 75 387 kg

  ≈ 853 lbf 105 925 kg ≈ 2039 lbf 76 400 kg

  ≈ 882 lbf 106 950 kg ≈ 2094 lbf 77 412 kg

  ≈ 937 lbf 108 1000 kg ≈ 2205 lbf 79 437 kg

Table 1.1 - Maximum load-carrying capacity tire index.

  ≈ 963 lbf 109 1030 kg ≈ 2271 lbf 80 450 kg

  ≈ 992 lbf 110 1060 kg ≈ 2337 lbf 81 462 kg

  ≈ 1019 lbf 111 1090 kg ≈ 2403 lbf 82 475 kg

  ≈ 1047 lbf 113 1120 kg ≈ 2469 lbf 83 487 kg

  ≈ 1074 lbf 113 1150 kg ≈ 2581 lbf 84 500 kg

  ≈ 1102 lbf 114 1180 kg ≈ 2601 lbf 85 515 kg

  ≈ 1135 lbf 115 1215 kg ≈ 2679 lbf 86 530 kg

  ≈ 1163 lbf 116 1250 kg ≈ 2806 lbf 87 545 kg

  1. Tire and Rim Fundamentals

  5

Table 1.2 - Maximum speed tire index.

  Index Maximum speed Index Maximum speed B 50 km/ h P 150 km/ h

  ≈ 31 mi/ h ≈ 93 mi/ h C 60 km/ h Q 160 km/ h

  ≈ 37 mi/ h ≈ 100 mi/ h D 65 km/ h R 170 km/ h

  ≈ 40 mi/ h ≈ 106 mi/ h E 70 km/ h S 180 km/ h

  ≈ 43 mi/ h ≈ 112 mi/ h F 80 km/ h T 190 km/ h

  ≈ 50 mi/ h ≈ 118 mi/ h G 90 km/ h U 200 km/ h

  ≈ 56 mi/ h ≈ 124 mi/ h J 100 km/ h H 210 km/ h

  ≈ 62 mi/ h ≈ 130 mi/ h K 110 km/ h V 240 km/ h

  ≈ 68 mi/ h ≈ 150 mi/ h L 120 km/ h W 270 km/ h

  ≈ 75 mi/ h ≈ 168 mi/ h M 130 km/ h Y 300 km/ h

  ≈ 81 mi/ h ≈ 188 mi/ h N 140 km/ h Z +240 km/ h

  ≈ 87 mi/ h ≈ +149 mi/ h Example 3 Alternative tire size indication.

  If the load index is not indicated on the tire, then a tire with a size number such as 255/50R17 100V may also be numbered by 255/50V R17. Example 4 Tire and rim widths.

  The dimensions of a tire are dependent on the rim on which it is mounted. For tires with an aspect ratio of 50 and above, the rim width is approxi- mately 70% of the tire’s width, rounded to the nearest 0.5 in. As an example, a P 255/50R16 tire has a design width of 255 mm = 10.04 in however, 70% of 10.04 in is 7.028 in, which rounded to the nearest 0.5 in, is 7 in. Therefore, a P 255/50R16 tire should be mounted on a 7 × 16 rim.

  For tires with aspect ratio 45 and below, the rim width is 85% of the tire’s section width, rounded to the nearest 0.5 in. For example, a P 255/45R17 tire with a section width of 255 mm = 10.04 in, needs an 8.5 in rim because 85% of 10.04 in is 8.534 in

  ≈ 8.5 in. Therefore, a P 255/45R17 tire should

  1 be mounted on an 8 × 17 rim.

2 Example 5 Calculating tire diameter and radius.

  We are able to calculate the overall diameter of a tire using the tire size numbers. By multiplying the tire width and the aspect ratio, we get the tire height. As an example, we use tire number P 235/75R15. h = 235 × 75%

  T

  = 176.25 mm (1.3) ≈ 6.94 in

  6

1. Tire and Rim Fundamentals

  tire’s unloaded diameter D = 2R and radius R.

  D = 2 × 6.94 + 15 = 28.88 in (1.4)

  ≈ 733.8 mm R = D/2 = 366.9 mm (1.5) Example 6 Speed rating code.

  Two similar tires are coded as P 235/70HR15 and P 235/70R15 100H. Both tires have code H

  ≡ 210 km/ h for speed rating. However, the second tire can sustain the coded speed only when it is loaded less than the specified load index, so it states 100H ≡ 800 kg 210 km/ h.

  Speed ratings generally depend on the type of tire. Off road vehicles usu- ally use Q-rated tires, passenger cars usually use R-rated tires for typical street cars or T -rated for performance cars. Example 7 Tire weight.

  The average weight of a tire for passenger cars is 10 − 12 kg. The weight of a tire for light trucks is 14

  − 16 kg, and the average weight of commercial truck tires is 135 − 180 kg.

  Example 8 Effects of aspect ratio.

  A higher aspect ratio provides a softer ride and an increase in deflection under the load of the vehicle. However, lower aspect ratio tires are normally used for higher performance vehicles. They have a wider road contact area and a faster response. This results in less deflection under load, causing a rougher ride to the vehicle.

  Changing to a tire with a different aspect ratio will result in a different contact area, therefore changing the load capacity of the tire. Example 9 F BMW tire size code.

  BMW, a European car, uses the metric system for sizing its tires. As an example, T D230/55ZR390 is a metric tire size code. T D indicates the BMW TD model, 230 is the section width in [ mm], 55 is the aspect ratio in percent, Z is the speed rating, R means radial, and 390 is the rim diameter in [ mm]. Example 10 F "MS," "M + S," "M/S," and "M&S" signs.

  The sign "M S,"and "M + S," and "M/S," and "M &S" indicate that the tire has some mud and snow capability. Most radial tires have one of these signs. Example 11 F U.S. DOT tire identification number.

  The US tire identification number is in the format "DOT DN ZE ABCD 1309." It begins with the letters DOT to indicate that the tire meets US fed- eral standards. DOT stands for Department of Transportation. The next two characters, DN , after DOT is the plant code, which refers to the man-

  1. Tire and Rim Fundamentals

  7

  The next two characters, ZE, are a letter-number combination that refers to the specific mold used for forming the tire. It is an internal factory code and is not usually a useful code for customers.

  The last four numbers, 1309, represents the week and year the tire was built. The other numbers, ABCD, are marketing codes used by the man- ufacturer or at the manufacturer’s instruction. An example is shown in Figure 1.4.

  

DOT DNZE ABCD 1309

FIGURE 1.4. An example of a US DOT tire identification number.

  DN is the plant code for Goodyear-Dunlop Tire located in Wittlich, Ger- many. ZE is the tire’s mold size, ABCD is the compound structure code, 13 indicates the 13th week of the year, and 09 indicates year 2009. So, the tire is manufactured in the 13th week of 2009 at Goodyear-Dunlop Tire in Wittlich, Germany. Example 12 F Canadian tires identification number.

  In Canada, all tires should have an identification number on the sidewall. An example is shown in Figure 1.5.

  DOT B3CD E52X 2112

FIGURE 1.5. An example of a Canadian DOT tire identification number.

  This identification number provides the manufacturer, time, and place that the tire was made. The first two characters following DOT indicate the manufacturer and plant code. In this case, B3 indicates Group Michelin located at Bridgewater, Nova Scotia, Canada. The third and fourth charac- ters, CD, are the tire’s mold size code. The fifth, sixth, seventh, and eighth characters, E52X, are optional and are used by the manufacturer. The final four numbers, 2112, indicates the manufacturing date. For example, 2112 indicate the twenty first week of year 2012. Finally, the maple leaf sign or the flag sign following the identification number indicates that the tire is manufactured in Canada. It also certifies that the tire meets Transport Canada requirements. Example 13 F E-Mark and international codes.

  All tires sold in Europe after July 1997 must carry an E-mark. An ex- ample is shown by 5 in Figure 1.2. The mark itself is either an upper or lower case "E" followed by a number in a circle or rectangle, followed by a further number. An "E" indicates that the tire is certified to com-

  8