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Index

2R planar manipulatordynamics, 547equations of motion, 550ideal, 547joint 2 acceleration, 288kinetic energy, 548Lagrangean, 549potential energy, 548

4-bar linkages, 311–313, 325, 326,330, 356

acceleration analysis, 318, 319concave, 316convex, 316coupler angle, 312coupler link, 312coupler point, 356–358coupler point curve, 356–360,

362, 363crank-crank, 320crank-rocker, 320crossed, 316dead positions, 321designing, 322drag-link, 320elbow-down, 316elbow-up, 316Grashoff criterion, 320input angle, 312input link, 312input variable, 312limit positions, 320non-crossed, 316output angle, 312output link, 312position analysis, 312possible configurations, 316rocker-rocker, 320spatial, 363sweep angles, 325velocity analysis, 317

ABS, 149Acceleration, 191

angular, 282, 285, 286, 288,289

body point, 274, 288, 289, 513capacity, 190centripetal, 288Coriolis, 514matrix, 282tangential, 288tilting, 149

Accelerationpower-limited, 191traction-limited, 191

Acceleration capacity, 190Ackerman

condition, 379history, 395mechanism, 428

Ackermangeometry, 381mechanism, 381steering, 379, 381

Ackerman condition, 379Ackerman, Rudolf, 395Aligning moment, 83, 85Angle

attitude, 238bank, 238camber, 35heading, 238inclination, 123, 138pitch, 238roll, 238sideslip, 35spin, 238steering, 380tilting, 123, 125tire contact, 45tireprint, 45

© Springer International Publishing AG 2017R.N. Jazar, Vehicle Dynamics, DOI 10.1007/978-3-319-53441-1

965

966 Index

ultimate, 123, 124yaw, 238

Angular acceleration, 282, 287, 289combination, 285in terms of Euler parameters,

286matrix, 282relative, 286vector, 282

Angular momentum, 520–5242 link manipulator, 527

Angular velocity, 243–245, 261, 267alternative definition, 274, 276alternative proof, 276combination, 266, 285coordinate transformation, 268decomposition, 266Euler frequency, 243instantaneous, 262instantaneous axis, 263matrix, 261, 267principal matrix, 265transformation, 266vector, 243, 261

Atan2 function, 140Attitude angle, 567, 570Axis-angle rotation, 254, 256–258

B-derivative, 269Based excitation, 751

acceleration, 758, 760frequency response, 751transmitted force, 761velocity, 758, 760

Bicycle carmode shape, 846, 847Natural frequency, 846, 847vibration, 843–846

Bicycle model, 583, 593, 601, 604,628, 689

body force components, 583camber trust, 696characteristic equation, 638coefficient matrix, 633, 690constant lateral force, 624

control variables, 596, 601,690, 692

coordinate frame, 565, 566critical speed, 622curvature response, 604, 627,

631, 693eigenvalue, 638equations of motion, 689, 690force system coefficients, 588,

606, 688free dynamics, 697free response, 637, 640, 697global sideslip angle, 586hatchback, notchback,

station, 708input vector, 596, 601, 692kinematic steering, 588lateral acceleration response,

604, 605, 616, 628, 631,693

linearized model, 628neutral distance, 623neutral steer, 620neutral steer point, 623Newton-Euler equations, 594oversteer, 620passing maneuver, 703, 704roll angle response, 693roll damping, 686roll steer, 696roll stiffness, 686rotation center, 647sideslip coefficient, 584, 685sideslip response, 604, 628slip response, 693stability factor, 620steady state conditions, 630steady-state motion, 692steady-state response, 607,

623, 624, 692step input, 633, 641, 643, 699time response, 632, 697time series, 640torque coefficient, 686transient response, 632

Index 967

understeer, 620vehicle velocity vector, 585yaw rate response, 604, 628,

693zero steer angle, 637

Brake forcebalance, 150optimal, 149

Bump steering, 407

Camber, 479angle, 35, 93, 96, 472force, 93moment, 96stiffness, 94torque, 95trail, 95trust, 93variation, 469

Camber angle, 472Car

classifications, 29flying, 158

Cartesianangular velocity, 244

Caster, 478negative, 478positive, 478

Caster angle, 478Catapults, 555Centrifugal moments, 528Centripetal acceleration response,

604, 628Centro, 344Characteristic equation, 781Chasles theorem, 290, 300Circumferential slip, 76Clutch, 190

dynamics, 186Foettinger, 190hydrodynamic, 190

Coordinate framebody, 567global, 567rim, 489

tire, 484, 580vehicle, 484, 565, 567, 580,

671wheel, 484, 579, 580wheel-body, 484, 579, 580

Coriolisacceleration, 285, 289effect, 514force, 514

Cornering stiffness, 83Couple, 508, 510Coupler point curve, 356Cresting, 156Critical speed, 622Critically-damped

vibration, 784–786Crouse angle, 567, 570Cruise angle, 570Curvature response, 604, 627, 631,

693Cycloid, 488, 489

curtate, 489prolate, 489

Damper, 725linear, 726parallel, 728, 729serial, 727viscous, 726

Damping ratio, 742determination, 792

De Dion suspension, 456Deviation moments, 528Differentiating, 269

B-derivative, 269, 271G-derivative, 269, 274second, 277transformation formula, 273,

274Dipping, 161Directional

cosine, 254Directions

cosine, 229principal, 531

968 Index

Dissipation function, 819, 820Drive force

optimal, 149–151Driveline, 173, 180, 183

clutch, 182differential, 182drive shafts, 182drive wheels, 182dynamics, 173engine, 182gearbox, 182propeller shaft, 182

Dynamicsdirect, 515forward, 515indirect, 515inverse, 515Newtonian, 516

Eartheffect of rotation, 514kinetic energy, 544revolution, 544rotation, 544rotation effect, 285

Eccentric base excitation, 769, 823frequency response, 769, 774mass ratio, 772

Eccentric excitation, 763, 822acceleration, 768eccentric mass, 763eccentricity, 763frequency response, 763mass ratio, 766transmitted force, 769velocity, 768

Eccentricity, 764Efficiency, 180

convertor, 182differential, 187driveline, 183engine, 176mechanical, 184–186overall, 182thermal, 184–186

transmission, 182volumetric, 184–186

Eigenvalue, 781Eigenvalue problem, 837

characteristic equation, 837Eigenvector

first-unit, 838high-unit, 838last-unit, 838normal form, 838normalization, 838

Eigenvector problem, 838Energy

conservation, 551, 552Earth kinetic, 544kinetic, 511, 512, 515, 521,

525, 541, 725, 820mechanical, 550potential, 544, 725, 820

Engine, 173Diesel, 174dynamics, 173efficiency, 176front, 183gasoline, 174ideal, 179injection Diesel, 174maximum speed, 192performance, 173rear, 183spark ignition, 174speed, 187torque, 186, 187working range, 194, 205

Envelope, 189Euler

-Lexell-Rodriguez formula,256

angles, 239–245coordinate frame, 245equation of motion, 520, 524,

526frequencies, 243, 244, 267global rotation matrix, 240inverse matrix, 251

Index 969

local rotation matrix, 240rotation matrix, 239, 240, 251

Euler equationbody frame, 524

Eulerianviewpoint, 281

Excitationbase, 740, 751, 945eccentric, 740, 945eccentric base, 740, 945forced, 740, 741, 945harmonically, 740, 945

Flying car, 158Foettinger clutch, 190Foettinger law, 190Force, 508

body, 508centrifugal, 514conservative, 544contact, 508Coriolis, 514effective, 514external, 508function, 515generalized, 539, 541, 545, 820internal, 508moment of, 508potential, 544resultant, 508rotating, 525time varying, 515total, 508

Force system, 508equivalent, 508

Forced excitation, 741acceleration, 746fequency response, 742transmitted force, 746, 748velocity, 746

Formularelative acceleration, 286Rodriguez, 259

Four wheel steering, 409Four-wheel vehicle, 589

dynamics, 589linearized dynamics, 592

Framecentral, 518principal, 521, 524, 529, 531,

532Free dynamics, 697Free response, 637, 640, 697Free system, 836Frequency

angular, 726cyclic, 726damped natural, 784natural, 782nodal, 802ratio, 742response, 740, 742

Frequency ratio, 742Frequency response, 740Freudenstein’s equation, 314, 322Friction

adhesion, 79Burckhardt models, 81cold welding, 79De-Wit models, 81deformation, 79Kiencke and Daviss models,

81Pacejka models, 80wear, 80

Friction ellipse, 102–104Friction mechanisms, 79Friction models, 80, 104Front-engined, 183Front-wheel-drive, 183Front-wheel-steering, 379Fuel

consumption, 177Full car

mode shape, 858natural frequency, 858vibration, 853–855, 858

Functionatan2, 140dissipation, 820

970 Index

Rayleigh, 820signum, 141

G-derivative, 269Gear ratio, 187Gear reduction ratio, 182Gearbox, 186, 187, 192, 194, 195,

197, 199, 201, 205, 206,209, 210

design, 194, 195, 197, 199, 201,205, 206, 209, 210

dynamics, 186geometric, 195, 197, 199, 201,

205, 206progressive, 209, 210stability condition, 192step jump, 195

Gearbox ratio, 182Generalized

coordinate, 539, 541, 542, 545force, 539, 541, 543, 545, 547,

550Global sideslip angle, 583, 586Gough diagram, 89Grashoff criterion, 320Grip, 87

Half carantiroll bar, 849, 852mode shape, 851, 852natural frequency, 851, 852vibration, 848–850

Heading angle, 567, 570Helix, 290Hermitian form, 830Homogeneous matrix, 291Hook joint, 363Hydroplaning, 21

dynamic, 21rubber, 22speed, 21, 22viscous, 21

Instant center, 344application, 348

coordinate, 351, 352motion, 351, 352number of, 347of acceleration, 355

Inverted slider-crank mechanism,338

acceleration analysis, 343application, 344coupler point curve, 362input-output, 338possible configurations,

341velocity analysis, 342

Jackknifing, 401Joint, 311

coordinate, 311prismatic, 311revolute, 311universal, 363

Kennedy theorem, 345Kinematics, 227

acceleration, 282Kinetic energy, 511, 512, 541

Earth, 544rigid body, 525rotational body, 521

Kronecker’s delta, 248, 520, 538

Lagrangeequation, 819, 820equation of motion, 538–544mechanics, 544method, 819

Lagrangean, 545, 819, 820viewpoint, 281

Lane-change maneuver, 703Langensperger, George, 395Lateral acceleration response, 604,

628, 631, 693Lateral velocity response, 605,

616, 628Law

of motion, 511

Index 971

second of motion, 511, 517third of motion, 511

Linearized model, 628oversteer, 631understeer, 631

Link, 311ground, 312

Linkage, 3114-bar, 311coupler link, 312dyad, 322, 329four-bar, 312ground link, 312input angle, 312output link, 312two-link, 322, 329

Location vector, 292, 294Longitudinal force, 74Longitudinal friction, 75Longitudinal slip, 74–76

MacPherson suspension, 457Manganic, 555Manipulator

2R planar, 547one-link, 546

Manjanic, 555Manjaniq, 555Mass center, 509, 512, 517, 518Mass moment

diagonal elements, 535matrix, 528

Matrixangular velocity, 261Euler rotation, 240global rotation, 228local rotation, 233positive definite, 835positive semidefinite, 835skew symmetric, 250, 255, 261

McPherson suspensionequivalent vibrating model, 887kinematic model, 457

MechanicsNewtonian, 516

Mechanism, 312closed loop, 312instant center, 344inversion, 338inverted slider-crank, 338open loop, 312parallel, 312pole, 344serial, 312slider-crank, 331steering, 385, 403suspension, 344trapezoidal steering, 385

Mode shape, 836Moment, 508

external, 524resultant, 508, 524total, 508

Moment of inertia, 528about a line, 538about a plane, 538about a point, 538about the origin, 538characteristic equation, 536diagonal elements, 528, 535eigenvalues, 531, 535eigenvectors, 535elements, 528frame-dependent, 529Huygens-Steiner theorem,

531matrix, 528off-diagonal elements, 529parallel-axes theorem,

529–531polar, 528principal, 529–532, 536principal axes, 521principal invariants, 536product, 529rigid body, 520, 523rotated-axes theorem,

529–531Moment of momentum, 508,

509

972 Index

Moments of inertiadetermination, 794

Momentum, 508angular, 508, 509, 520–524linear, 508, 509translational, 509

Natural frequency, 742, 782, 836determination, 794

Neutral distance, 623Neutral steer, 620, 621Neutral steer point, 623Newton

equation in body frame, 518equation of motion, 511, 517,

518, 526, 538equations of motion, 541Lagrange form, 541rotating frame, 513

Onager, 555One-eighth car model, 883, 887

absolute acceleration, 890absolute displacement, 890–

892damping ratio, 884design curve, 917equation of motion, 884excitation frequency, 888frequency response, 889, 893hard suspension, 900, 901model, 735natural frequency, 884optimal characteristics, 904optimal damping, 902optimal design chart, 904optimal design curve, 893, 904,

906optimal stiffness, 902optimal suspension, 901optimization, 893optimization strategy, 895relative displacement, 890–892soft suspensions, 900, 901step input, 914

suspension clearance, 900suspension room, 900suspension travel, 900time response, 914, 917trade-off, 909wheel travel, 899, 900working frequency range, 896

Optimizationalternative method, 912cost function, 912design curve, 920one-eighth car, 883, 893quarter car, 920RMS, 893, 920time response, 914, 917transient response, 914, 917trivial, 909vehicle suspension, 902vibration, 797–805wheel travel, 932

Orthogonality condition, 247Over-damped

vibration, 784–786Oversteer, 620, 621, 643

Pacejka model, 104Parallelogram suspension, 502Passing maneuver, 703, 704Pendulum

chain, 827double, 825inverted, 738oscillating, 542simple, 283, 542spherical, 545

Physicalquantityvectorial, 509

Pitch moment, 566Planar dynamics, 593, 601

attitude angle, 570body force components, 583characteristic equation, 638coefficient matrix, 633constant lateral force, 624

Index 973

control variables, 596, 601coordinate frame, 565, 566critical speed, 622crouse angle, 570curvature response, 604, 627,

631eigenvalue, 638force system coefficients, 588,

606free response, 637, 640global sideslip angle, 586heading angle, 570input vector, 596, 601kinematic steering, 588lateral acceleration response,

604, 605, 616, 628, 631linearized model, 628neutral distance, 623neutral steer, 620neutral steer point, 623Newton-Euler, 571Newton-Euler equations, 594oversteer, 620rotation center, 647sideslip coefficient, 584sideslip response, 604, 628stability factor, 620steady state conditions, 630steady-state response, 607, 623,

624steady-state turning, 604step input, 633, 641, 643time response, 632time series, 640transient response, 632understeer, 620vehicle velocity vector, 585wheel number, 568yaw rate response, 604, 628zero steer angle, 637

Plotgear-speed, 199, 201, 206power, 197, 206progressive, 209working range, 197

Pneumatic trail, 85Pole, 344Potential

energy, 512, 544field, 512force, 544function, 512kinetic, 545

Powerat wheel, 183constant, 179driveline, 183engine, 183equation, 174friction, 185ideal, 179law, 184maximum, 179peak, 178performance, 173, 174, 176,

178, 179units, 176

Power steering, 407Principal

rotation matrix, 255Principle

conservation of energy, 512superposition, 516

Quadrature, 829, 830asymmetric, 830

Quarter car, 833model, 734natural frequency, 841sprung mass, 841unsprung mass, 841

Quarter car model, 861coefficient matrix, 865dimensionless characteristics,

863equations of motion, 862frequency response, 863–865,

871history, 863invariant amplitude, 869

974 Index

invariant frequency, 868, 869main suspension, 862mathematical model, 862natural frequency, 868, 869,

872nodal amplitude, 870nodal frequency, 869, 870optimal characteristics, 930optimal design curve, 920, 925optimization, 920optimization strategy, 921resonant frequency, 869sprung mass, 862street cars, 867tire damping, 863unsprung mass, 862wheel travel, 932working frequency range, 922

Rear wheel steering, 390Rear-engined, 183Rear-wheel drive, 183Resonance, 841Resonance zone, 745Ride, 819Ride comfort, 819Rigid body

acceleration, 287angular momentum, 522, 523centroid, 281Euler equation, 524kinetic energy, 525moment of inertia, 520, 523motion composition, 260plane motion, 352principal rotation matrix, 535rotational kinetics, 520steady rotation, 526translational, 517velocity, 278, 279

Rim, 3, 5, 24, 25alloy, 25diameter, 5flange, 24hub, 24

hump, 24spider, 24width, 7

Roadbank angle, 141, 143banked, 143, 144inclination angle, 134, 138

Road pavement, 69Rodriguez

rotation formula, 256, 257,259, 293, 297

Roll angle, 566, 672Roll angle response, 693Roll axis, 463, 464Roll center, 348, 463Roll dynamics, 671

bicycle model, 682camber trust, 696coefficient matrix, 690control variables, 690, 692curvature response, 693equations of motion, 689, 690force system, 676force system coefficients, 688free dynamics, 697free response, 697hatchback, notchback,

station, 708input vector, 692lateral acceleration response,

693lateral force, 679Newton-Euler equations, 672,

675, 676passing maneuver, 703, 704roll angle response, 693roll damping, 686roll steer, 696roll stiffness, 686roll-steering angle, 679sideslip angle, 679sideslip coefficient, 685slip response, 693steady-state motion, 692steady-state response, 692

Index 975

step input, 699time response, 697tire slip coefficient, 680torque coefficient, 686two-wheel model, 682vehicle slip coefficient, 681wheel force system, 676yaw rate response, 693

Roll height, 464Roll moment, 566Roll stiffness, 465Roll torque, 464Roll-pitch-yaw

global angles, 233, 238global rotation matrix, 233,

238Rolling disc, 824Rolling friction, 63, 64, 66, 70Rolling resistance, 63–66, 68–71,

73Rotation, 256

about global axis, 227, 232about local axis, 233, 237axis-angle, 254, 256–258direction cosines, 229, 235general matrix, 245global Euler matrix, 251global matrices, 230instantaneous axis, 263instantaneous center, 281local Euler matrix, 251local matrix, 237local versus global, 252matrix, 232nutation, 239off-center axis, 299order of, 232orthogonality condition, 247pitch, 233pole, 281precession, 239radius of, 380, 382reverse, 256roll, 233roll-pitch-yaw matrix, 238

spin, 239successive, 232, 237X-matrix, 228x-matrix, 234Y-matrix, 228y-matrix, 234yaw, 233Z-matrix, 228z-matrix, 234

Rotation matrixelement of, 247

SAE steering definition, 626Screw, 292, 300

axis, 290central, 291, 292, 294, 296coordinate, 290general, 292left-handed, 291location vector, 291, 292motion, 290parameters, 291, 298pitch, 290principal, 299right-handed, 291rotation, 290special case, 297transformation, 294, 296,

297, 299translation, 290twist, 290

Second derivative, 277Sideslip angle, 35, 567, 582Sideslip coefficient, 583, 584Sideslip response, 604, 628Sideslip stiffness, 83Slider-crank mechanism, 331

acceleration analysis, 336,337

coupler point curve, 360input angle, 331input-output, 331limit positions, 337possible configurations, 334quick return, 338

976 Index

slider position, 331velocity analysis, 335, 336

Slip moment, 86Slip ratio, 76Slip response, 693Speed equation, 186, 188Speed ratio, 182Speed span, 197Spring, 725

linear, 726massive, 732parallel, 728, 729serial, 727stiffness, 726

Stability factor, 620Stall, 133Steady state

center of rotation, 617, 618centripetal acceleration response,

604, 614curvature response, 604, 608,

611lateral velocity response, 605,

606, 616sideslip response, 604, 612stability factor, 608, 620yaw rate response, 604, 613

Steady state response, 607Steering, 379, 380, 410

4WS factor, 418Ackerman, 427Ackerman condition, 379Ackerman mechanism, 428active steer, 421autodriver, 421bicycle model, 380, 381, 421command, 404comparison, 418counter steer, 414error, 387, 427, 436four wheel, 409–419, 421front wheel, 379independent rear wheel drive,

393

inner steer angle, 379, 380,410

inner wheel, 379, 380, 391,392, 410

inner-outer relationship, 380,385

jackknifing, 401kinematic, 379, 383, 391kinematic condition, 379,

381, 421length, 418locked rear axle, 389, 390maximum radius, 383mechanism, 385, 403–405midline, 397more than two axles, 396, 397multi-link, 429offset, 407optimization, 427, 429, 431–

433, 436outer steer angle, 379, 380,

410outer wheel, 379, 380, 391,

392, 410passive steer, 421Pitman arm, 403racecars, 394radius of curvature, 417radius of rotation, 384ratio, 403rear wheel, 390reverse efficiency, 405same steer, 414self-steering wheels, 399sign convection, 414, 417sign convention, 410six-wheel vehicle, 397smart steer, 421space requirement, 383, 384,

402speed dependent, 395steer angle, 380steer by wire, 395trapezoidal, 409, 427, 428

Index 977

trapezoidal mechanism, 385,387, 427

turning center, 379, 409, 414,416, 417

turning radius, 380–382, 413,414, 419

unequal tracks, 392with trailer, 399, 401

Steering length, 418Steering mechanisms

drag link, 404lever arm, 404multi-link, 405optimization, 427, 429, 431–

433, 436parallelogram, 403Pitman arm, 403rack-and-pinion, 404steering wheel, 403tie rod, 404trapezoidal, 427

Steering ratio, 403Step input, 633, 643, 789Step jump, 195Step response, 789

overshoot, 790peak time, 790peak value, 790rise time, 790settling time, 790steady-state, 791

Step steer input, 641, 646Suspension

anti-tramp bar, 449antiroll bar, 461camber, 479camber angle, 472caster, 478center, 463Chebyshev linkage, 451De Dion, 456dead axle, 456dependent, 447double A-arm, 457double triangle, 451

double wishbone, 457equilibrium position, 471Evance linkage, 451four-bar linkage, 471Hotchkiss, 448independent, 457, 460, 461live axle, 456McPherson, 457, 887optimization, 883Panhard arm, 451rest position, 471Robert linkage, 451roll axis, 463roll center, 348, 463S shape problem, 448semi-trailing arm, 460short/long arm, 457solid axle, 447–449, 451, 454–

456spung mass, 448stabilizer, 461straight line linkages, 451swing arm, 457swing axle, 457toe, 475trailing arm, 460triangulated linkage, 451trust angle, 480twisting problem, 449unsprung mass, 448unsprung mass problem, 454vibration, 883Watt, 451with coil spring, 456

Suspension center, 463Suspension mechanism, 331, 344,

447Chapman, 344double A arm, 331double wishbone, 331dynamic requirement, 482kinematic requirement, 481,

482McPherson, 344

Symbols, xxi

978 Index

Tangential slip, 76Theorem

Chasles, 290, 300Huygens-Steiner, 531Kennedy, 345, 463parallel-axes, 529, 531rotated-axes, 529

Time derivative, 269Time response, 780

free dynamics, 697free response, 697hatchback, notchback, station,

708homogeneous, 781homogeneous solution, 781initial condition, 786, 788initial-value problem, 780non-homogeneous, 781particular solution, 781passing maneuver, 703, 704step input, 699vehicle dynamics, 632, 697

Time series, 637, 640Tire, 3, 35

adhesion friction, 79aligning moment, 37, 83, 85–

87, 98American, 9aspect ratio, 5, 9bank moment, 36bead, 14, 16belt, 14bias ply, 5bias-ply, 17blocks, 20bore torque, 37camber angle, 73, 96, 98camber arm, 96camber force, 93, 96camber moment, 96camber stiffness, 94, 100camber torque, 95camber trail, 95camber trust, 93Canadian, 10

carcass, 15circumferential slip, 76cold welding friction, 79combined force, 100combined slip, 102, 103components, 14contact angle, 45coordinate frame, 35, 37, 484,

485cords, 15cornering force, 87cornering stiffness, 83, 87critical speed, 68damping structure, 65deflection, 40deformation friction, 80diameter, 8dissipated power, 70DOT, 4, 10DOT index, 9drag force, 87dynamics, 3E-Mark, 4, 10effective radius, 43, 45equivalent radius, 45, 76equivalent speed, 75European, 10, 11force system, 35, 99forces model, 103forward force, 36forward velocity, 43friction, 77, 79friction coefficient, 75friction ellipse, 102friction stress, 61function, 19geometric radius, 43, 45grip, 87groove, 15, 20, 21height, 4, 6history, 16hydroplaning, 21hysteresis, 42inflation, 13inflation pressure, 45, 72

Index 979

inner liner, 14lateral force, 36, 83, 86, 87,

89, 91, 93, 95, 96, 98lateral load, 61lateral ratio, 83lateral stiffness, 84lateral stress, 91, 92light truck, 11load, 45load index, 5–7load rate, 5loaded height, 43longitudinal force, 36, 74longitudinal friction, 77longitudinal ratio, 83longitudinal slip, 74, 75, 100lugs, 20M&S, 4, 9maximum velocity, 78motorcycles, 70non-radial, 17–19, 65non-radiale, 98normal force, 36normal load, 58, 60, 61normal stress, 58, 60, 61, 64on a circle, 92overturning moment, 36pitch moment, 36plane, 35plus one, 13pneumatic trail, 85racecar, 69radial, 5, 17–19, 65radial displacement, 46radiale, 98radius, 8roll moment, 36rolling friction, 63, 64, 66, 70rolling radius, 43, 45rolling resistance, 63–66, 68–

71, 73rolling resistance torque, 36rubber, 15, 16SAE coordinate frame, 37section height, 4

section width, 4self aligning moment, 37shallow, 19shear stress, 61side force, 87sideslip, 35sideslip angle, 35, 73, 83, 98,

102sidewall, 4, 12, 13, 15size, 4, 6slick, 69sliding line, 85slip coefficient, 75slip models, 80, 82, 104slip moment, 86slip ratio, 74–78, 80, 82, 100,

102, 104slots, 20spare, 27speed index, 5, 8, 9spring structure, 65stiffness, 38, 40–42, 84strain, 53stress, 53, 58, 60, 61tangential slip, 76tangential stress, 61, 63tilting torque, 36tireprint, 3, 23tireprint angle, 45tireprint model, 99tireprint zone, 46tread, 15, 20, 21, 46, 47, 49tread acceleration, 54tread displacement, 49–51tread jerk, 55tread travel, 47tread velocity, 48, 49tread wear index, 12tube-type, 19tubeless, 19type index, 4UTQG index, 11vertical force, 36voids, 20wear, 23

980 Index

wear friction, 80weight, 9wheel load, 36width, 4, 5, 7yaw moment, 37

Tireprint, 23, 35, 58, 60, 99angle, 45force, 465

Toe, 475Toe-in, 475Toe-out, 475Torque, 508

at wheel, 184, 187equation, 174maximum, 179peak, 178performance, 174, 176, 187

Track, 380Traction

force, 186Traction equation, 186, 188Trailer, 135, 141Transformation

general, 245tire to vehicle frame, 491tire to wheel frame, 486, 487tire to wheel-body frame, 487,

488wheel to tire frame, 485, 487wheel to wheel-body frame,

489wheel-body to vehicle frame,

493Transformation matrix

elements, 248Transient response

free dynamics, 697free response, 697hatchback, notchback, station,

708passing maneuver, 703, 704step input, 699vehicle dynamics, 632, 697

Transmission ratio, 182, 183, 187Transmission ratios, 192

Trapezoidal steering, 385, 387Tread, 20, 21

grooves, 20lugs, 20slots, 20voids, 20

Trebuchet, 553Trigonometric equation, 140Trochoid, 489Trust angle, 480Turning center, 409, 414, 416, 417Two-wheel vehicle, 583, 588, 593,

601, 604, 628, 689body force components, 583camber trust, 696characteristic equation, 638coefficient matrix, 633, 690constant lateral force, 624control variables, 596, 601,

690, 692coordinate frame, 565, 566critical speed, 622curvature response, 604, 627,

631, 693eigenvalue, 638equations of motion, 689, 690force system coefficients, 588,

606, 688free dynamics, 697free response, 637, 640, 697global sideslip angle, 586hatchback, notchback,

station, 708input vector, 596, 601, 692kinematic steering, 588lateral acceleration response,

604, 605, 616, 628, 631,693

linearized model, 628neutral distance, 623neutral steer, 620neutral steer point, 623Newton-Euler equations, 594oversteer, 620passing maneuver, 703, 704

Index 981

roll angle response, 693roll damping, 686roll steer, 696roll stiffness, 686rotation center, 647sideslip coefficient, 584, 685sideslip response, 604, 628slip response, 693stability factor, 620steady state conditions, 630steady-state motion, 692steady-state response, 607, 623,

624, 692step input, 633, 641, 643, 699time response, 632, 697time series, 640torque coefficient, 686transient response, 632understeer, 620vehicle velocity vector, 585yaw rate response, 604, 628,

693zero steer angle, 637

Under-dampedvibration, 784–786

Understeer, 620, 621, 641Unit system, xxUniversal joint, 363, 365–367, 369–

371double, 369history, 370, 371speed ratio, 367

Vecface, 510Vecfree, 510Veclane, 510Vecline, 510Vecpoface, 510Vecpoint, 510Vecpolane, 510Vecpoline, 510Vecporee, 510Vector

axis, 509

bounded, 510characteristics, 509definition, 509direction, 509, 510end point, 509free, 510length, 509line, 510line of action, 509, 510plane, 510point, 510point-free, 510point-line, 510point-plane, 510requirements, 509sliding, 510start point, 509surface, 510types, 509vecface, 510vecfree, 510vecline, 510vecpoface, 510vecpoint, 510vecpolane, 510vecpoline, 510vecporee, 510

Vehicle, 29accelerating, 126, 128, 129,

131–133, 135braking, 125classifications, 29curb weight, 32FHWA classifications, 29gross weight, 33ISO classifications, 29longitudinal dynamics, 115,

116, 119–121, 123–126,128, 129, 131, 133, 135,141, 143, 146, 149–152,154, 156, 158, 159, 161,163, 168

mass center, 150mass center position, 116,

117, 119, 120, 168

982 Index

maximum acceleration, 128,129, 133, 135

more than two axles, 152, 154on a banked road, 141, 143on a crest, 156, 158, 159on a dip, 161, 163on a level pavement, 115on an inclined pavement, 121,

125optimal brake force, 146, 149optimal drive force, 146, 150,

151passenger car classifications,

32, 34size classifications, 32stall, 133weight classifications, 32wheel loads, 116wheel locking, 150with a trailer, 135, 141

Vehicle dynamics180 deg quick turn, 602aligning moment, 566attitude angle, 567, 570bank moment, 566bicycle model, 583, 585, 593,

601, 604, 628, 682body force components, 583body force system, 577camber trust, 696characteristic equation, 638coefficient matrix, 633, 690coefficients matrix, 596, 601constant lateral force, 624control variables, 596, 601, 690,

692critical speed, 622crouse angle, 567, 570curvature response, 604, 627,

631, 693direct, 633eigenvalue, 638equations of motion, 585, 689,

690force system, 566, 676

force system coefficients, 587,588, 606, 688

forward, 633forward force, 566four-wheel-steering, 597free dynamics, 697free response, 637, 640, 697front-wheel-steering, 630general motion, 676hatchback, notchback,

station, 708heading angle, 567, 570indirect, 633input vector, 596, 601, 692inputs vector, 601inverse, 633Lagrange method, 573lateral acceleration response,

604, 605, 616, 628, 631,693

lateral force, 566, 582, 583,586, 591, 679

lateral moment, 566linearized model, 628, 630longitudinal force, 566neutral, 620, 621neutral distance, 623neutral steer, 620neutral steer point, 623Newton-Euler, 571Newton-Euler equations, 594,

672normal force, 566oversteer, 620, 621overturning moment, 566passing maneuver, 703, 704path of motion, 575pitch angle, 566, 672pitch moment, 566pitch rate, 566, 672planar, 565principal method, 575rear-wheel-steering, 601rigid vehicle, 565, 671roll angle, 566, 672

Index 983

roll angle response, 693roll damping, 686roll dynamics, 671, 672, 676roll moment, 566roll rate, 566, 672roll rigid vehicle, 676roll steer, 696roll stiffness, 686roll-steering angle, 679rotation center, 647SAE steering definition, 626second-order equations, 657sideslip angle, 567, 679sideslip coefficient, 584, 685sideslip coefficients, 583sideslip response, 604, 628six DOF, 675slip response, 693stability factor, 620steady state conditions, 630steady-state motion, 692steady-state response, 607, 623,

624, 692steady-state turning, 604steer angle, 584step input, 633, 641, 643, 699step steer input, 646tilting torque, 566time response, 632, 641, 643,

697time series, 637, 640tire force system, 577tire lateral force, 582tire slip coefficient, 680torque coefficient, 686traction force, 566transient response, 632, 697two-wheel model, 583, 585, 593,

601, 604, 628, 682understeer, 620, 621vehicle load, 566vehicle slip coefficient, 681vehicle velocity vector, 585vertical force, 566wheel force system, 676

wheel frame, 579wheel number, 568yaw angle, 566, 672yaw moment, 566yaw rate, 566, 672yaw rate response, 604, 628,

693zero steer angle, 637

Vehicle vibration, 819alternative optimization, 912antiroll bar, 849, 852base excited model, 883bicycle car, 843, 846, 847body pitch, 843body roll, 848–850bounce, roll, and pitch, 853dissipation function, 820driver, 833excitation frequency, 888frequency response, 889full car, 853–855half car, 848–850Lagrange equation, 820Lagrange method, 820McPherson suspension, 887mode shape, 836, 851, 852,

858natural frequenc, 858natural frequency, 836, 851,

852one-eighth model, 883optimal design curve, 893optimization, 883optimization strategy, 895quadrature, 829quarter car, 833, 861, 862sprung mass, 883time response, 914, 917wheel travel, 899, 900working frequency range, 896

Velocitybody point, 513

Vibration1/8 car model, 735absorber, 797

984 Index

amplitude, 742angular frequency, 726angular lag, 742application, 792base excitation, 740, 751, 945beating, 749characteristic equation, 837cyclic frequency, 726damping ratio, 742discrete model, 733displacedspring, 887dynamic amplitude, 745eccentric base excitation, 740,

945eccentric excitation, 740, 945eigenvalue problem, 837eigenvector problem, 838equilibrium position, 733Equivalent system, 735excitation, 727forced, 727, 745forced excitation, 740, 945Frahm absorber, 798–805Frahm damper, 798–805free, 786, 788free system, 836frequency ratio, 742frequency response, 740, 742,

746harmonic, 727initial condition, 786, 788isolator, 797lumped model, 733measurement, 792mechanical, 725natural frequency, 742Newton’s method, 733nontrivial solution, 837optimization theory, 797–805orthogonality functions, 749periodic, 727phase, 742quarter car model, 734random, 727resonance zone, 745

rest position, 837ride comfort, 819stable, 734static amplitude, 745steady-state solution, 740step input, 789tilted spring, 885–887transient, 727transmitted force, 748, 761trivial solution, 837two-DOF base excited, 737unstable, 734vehicle, 819work of a harmonic force, 788

Virationcharacteristic equation, 781characteristic parameters,

781critically-damped, 784damped natural frequency,

784eigenvalues, 781forced, 781forced classification, 775free, 781initial-value problem, 780natural frequency, 782, 783over-damped, 784time response, 780, 782transient response, 782under-damped, 784

Virtualdisplacement, 541work, 541

Wheel, 24, 25angular velocity, 43camber angle, 481coordinate frame, 481, 484,

485degrees-of-freedom, 481flange, 24forward velocity, 43history, 27non-steerable, 482

Index 985

spider, 24spin, 481steer angle, 481steerable, 482wire spoke, 27

Wheel number, 568Wheel travel, 899

lower, 900upper, 900

Wheel-bodycoordinate frame, 484, 485

Wheelbase, 380Windshield wiper, 322

double-arm opposing, 322

double-arm parallel, 322sweep angles, 325

Work, 511, 512, 515virtual, 541

Work-energy principle, 512

Yaw moment, 566Yaw rate response, 604, 628, 693Yaw velocity, 389Yoke joint, 363

Zero steer input, 637Zero velocity point, 281