Advanced Dynamics: Rigid Body, Multibody, and Aerospace Applications

Preface xiii

Part I Fundamentals 1

1 Fundamentals of Kinematics 3

1.1 Coordinate Frame and Position Vector 3

1.1.1 Triad 3

1.1.2 Coordinate Frame and Position Vector 4

1.1.3 Vector Definition 10

1.2 Vector Algebra 12

1.2.1 Vector Addition 12

1.2.2 Vector Multiplication 17

1.2.3 Index Notation 26

1.3 Orthogonal Coordinate Frames 31

1.3.1 Orthogonality Condition 31

1.3.2 Unit Vector 34

1.3.3 Direction of Unit Vectors 36

1.4 Differential Geometry 37

1.4.1 Space Curve 38

1.4.2 Surface and Plane 43

1.5 Motion Path Kinematics 46

1.5.1 Vector Function and Derivative 46

1.5.2 Velocity and Acceleration 51

1.5.3 Natural Coordinate Frame 54

1.6 Fields 77

1.6.1 Surface and Orthogonal Mesh 78

1.6.2 Scalar Field and Derivative 85

1.6.3 Vector Field and Derivative 92

Key Symbols 100

Exercises 103

2 Fundamentals of Dynamics 114

2.1 Laws of Motion 114

2.2 Equation of Motion 119

2.2.1 Force and Moment 120

2.2.2 Motion Equation 125

2.3 Special Solutions 131

2.3.1 Force Is a Function of Time, F = F (t) 132

2.3.2 Force Is a Function of Position, F = F(x) 141

2.3.3 Elliptic Functions 148

2.3.4 Force Is a Function of Velocity, F = F (v) 156

2.4 Spatial and Temporal Integrals 165

2.4.1 Spatial Integral: Work and Energy 165

2.4.2 Temporal Integral: Impulse and Momentum 176

2.5 Application of Dynamics 188

2.5.1 Modeling 189

2.5.2 Equations of Motion 197

2.5.3 Dynamic Behavior and Methods of Solution 200

2.5.4 Parameter Adjustment 220

Key Symbols 223

Exercises 226

Part II Geometric Kinematics 241

3 Coordinate Systems 243

3.1 Cartesian Coordinate System 243

3.2 Cylindrical Coordinate System 250

3.3 Spherical Coordinate System 263

3.4 Nonorthogonal Coordinate Frames 269

3.4.1 Reciprocal Base Vectors 269

3.4.2 Reciprocal Coordinate Frame 278

3.4.3 Inner and Outer Vector Product 285

3.4.4 Kinematics in Oblique Coordinate Frames 298

3.5 Curvilinear Coordinate System 300

3.5.1 Principal and Reciprocal Base Vectors 301

3.5.2 Principal–Reciprocal Transformation 311

3.5.3 Curvilinear Geometry 320

3.5.4 Curvilinear Kinematics 325

3.5.5 Kinematics in Curvilinear Coordinates 335

Key Symbols 346

Exercises 347

4 Rotation Kinematics 357

4.1 Rotation About Global Cartesian Axes 357

4.2 Successive Rotations About Global Axes 363

4.3 Global Roll–Pitch–Yaw Angles 370

4.4 Rotation About Local Cartesian Axes 373

4.5 Successive Rotations About Local Axes 376

4.6 Euler Angles 379

4.7 Local Roll–Pitch–Yaw Angles 391

4.8 Local versus Global Rotation 395

4.9 General Rotation 397

4.10 Active and Passive Rotations 409

4.11 Rotation of Rotated Body 411

Key Symbols 415

Exercises 416

5 Orientation Kinematics 422

5.1 Axis–Angle Rotation 422

5.2 Euler Parameters 438

5.3 Quaternion 449

5.4 Spinors and Rotators 457

5.5 Problems in Representing Rotations 459

5.5.1 Rotation Matrix 460

5.5.2 Axis–Angle 461

5.5.3 Euler Angles 462

5.5.4 Quaternion and Euler Parameters 463

5.6 Composition and Decomposition of Rotations 465

5.6.1 Composition of Rotations 466

5.6.2 Decomposition of Rotations 468

Key Symbols 470

Exercises 471

6 Motion Kinematics 477

6.1 Rigid-Body Motion 477

6.2 Homogeneous Transformation 481

6.3 Inverse and Reverse Homogeneous Transformation 494

6.4 Compound Homogeneous Transformation 500

6.5 Screw Motion 517

6.6 Inverse Screw 529

6.7 Compound Screw Transformation 531

6.8 Plücker Line Coordinate 534

6.9 Geometry of Plane and Line 540

6.9.1 Moment 540

6.9.2 Angle and Distance 541

6.9.3 Plane and Line 541

6.10 Screw and Plücker Coordinate 545

Key Symbols 547

Exercises 548

7 Multibody Kinematics 555

7.1 Multibody Connection 555

7.2 Denavit–Hartenberg Rule 563

7.3 Forward Kinematics 584

7.4 Assembling Kinematics 615

7.5 Order-Free Rotation 628

7.6 Order-Free Transformation 635

7.7 Forward Kinematics by Screw 643

7.8 Caster Theory in Vehicles 649

7.9 Inverse Kinematics 662

Key Symbols 684

Exercises 686

Part III Derivative Kinematics 693

8 Velocity Kinematics 695

8.1 Angular Velocity 695

8.2 Time Derivative and Coordinate Frames 718

8.3 Multibody Velocity 727

8.4 Velocity Transformation Matrix 739

8.5 Derivative of a Homogeneous Transformation Matrix 748

8.6 Multibody Velocity 754

8.7 Forward-Velocity Kinematics 757

8.8 Jacobian-Generating Vector 765

8.9 Inverse-Velocity Kinematics 778

Key Symbols 782

Exercises 783

9 Acceleration Kinematics 788

9.1 Angular Acceleration 788

9.2 Second Derivative and Coordinate Frames 810

9.3 Multibody Acceleration 823

9.4 Particle Acceleration 830

9.5 Mixed Double Derivative 858

9.6 Acceleration Transformation Matrix 864

9.7 Forward-Acceleration Kinematics 872

9.8 Inverse-Acceleration Kinematics 874

Key Symbols 877

Exercises 878

10 Constraints 887

10.1 Homogeneity and Isotropy 887

10.2 Describing Space 890

10.2.1 Configuration Space 890

10.2.2 Event Space 896

10.2.3 State Space 900

10.2.4 State–Time Space 908

10.2.5 Kinematic Spaces 910

10.3 Holonomic Constraint 913

10.4 Generalized Coordinate 923

10.5 Constraint Force 932

10.6 Virtual and Actual Works 935

10.7 Nonholonomic Constraint 952

10.7.1 Nonintegrable Constraint 952

10.7.2 Inequality Constraint 962

10.8 Differential Constraint 966

10.9 Generalized Mechanics 970

10.10 Integral of Motion 976

10.11 Methods of Dynamics 996

10.11.1 Lagrange Method 996

10.11.2 Gauss Method 999

10.11.3 Hamilton Method 1002

10.11.4 Gibbs–Appell Method 1009

10.11.5 Kane Method 1013

10.11.6 Nielsen Method 1017

Key Symbols 1021

Exercises 1024

Part IV Dynamics 1031

11 Rigid Body and Mass Moment 1033

11.1 Rigid Body 1033

11.2 Elements of the Mass Moment Matrix 1035

11.3 Transformation of Mass Moment Matrix 1044

11.4 Principal Mass Moments 1058

Key Symbols 1065

Exercises 1066

12 Rigid-Body Dynamics 1072

12.1 Rigid-Body Rotational Cartesian Dynamics 1072

12.2 Rigid-Body Rotational Eulerian Dynamics 1096

12.3 Rigid-Body Translational Dynamics 1101

12.4 Classical Problems of Rigid Bodies 1112

12.4.1 Torque-Free Motion 1112

12.4.2 Spherical Torque-Free Rigid Body 1115

12.4.3 Axisymmetric Torque-Free Rigid Body 1116

12.4.4 Asymmetric Torque-Free Rigid Body 1128

12.4.5 General Motion 1141

12.5 Multibody Dynamics 1157

12.6 Recursive Multibody Dynamics 1170

Key Symbols 1177

Exercises 1179

13 Lagrange Dynamics 1189

13.1 Lagrange Form of Newton Equations 1189

13.2 Lagrange Equation and Potential Force 1203

13.3 Variational Dynamics 1215

13.4 Hamilton Principle 1228

13.5 Lagrange Equation and Constraints 1232

13.6 Conservation Laws 1240

13.6.1 Conservation of Energy 1241

13.6.2 Conservation of Momentum 1243

13.7 Generalized Coordinate System 1244

13.8 Multibody Lagrangian Dynamics 1251

Key Symbols 1262

Exercises 1264

References 1280

A Global Frame Triple Rotation 1287

B Local Frame Triple Rotation 1289

C Principal Central Screw Triple Combination 1291

D Industrial Link DH Matrices 1293

E Trigonometric Formula 1300

Index 1305