Advanced Dynamics: Rigid Body, Multibody, and Aerospace Applications

A thorough understanding of rigid body dynamics as it relates to modern mechanical and aerospace systems requires engineers to be well versed in a variety of disciplines. This book offers an all-encompassing view by interconnecting a multitude of key areas in the study of rigid body dynamics, including classical mechanics, spacecraft dynamics, and multibody dynamics. In a clear, straightforward style ideal for learners at any level, Advanced Dynamics builds a solid fundamental base by first providing an in-depth review of kinematics and basic dynamics before ultimately moving forward to tackle advanced subject areas such as rigid body and Lagrangian dynamics. In addition, Advanced Dynamics: Is the only book that bridges the gap between rigid body, multibody, and spacecraft dynamics for graduate students and specialists in mechanical and aerospace engineering Contains coverage of special applications that highlight the different aspects of dynamics and enhances understanding of advanced systems across all related disciplines Presents material using the author's own theory of differentiation in different coordinate frames, which allows for better understanding and application by students and professionals Both a refresher and a professional resource, Advanced Dynamics leads readers on a rewarding educational journey that will allow them to expand the scope of their engineering acumen as they apply a wide range of applications across many different engineering disciplines.

Table of Contents:
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