OUR NICHE IN THE COSMOS 1
1-1 Introduction 1
1-2 Why History? 3
1-3 Importance of Mathematics in the Development of Mechanics 3
1-4 Our Sources from Antiquity: Getting the Message from There to Here 4 *
1 -4.1 Invention of Writing 5
1-4.2 Hieroglyphics 6
1-4.3 Cuneiform 7
1-4.4 Ancient Egyptian Papyri 7
1-4.5 Mesopotamian Clay Tablets 9
1-5 Ancient Egyptian Astronomy and Mathematics 9
1 -5.1 Ancient Egyptian Astronomy 10
1-5.2 Ancient Egyptian Mathematics 11
1-6 Mesopotamian Astronomy and Mathematics 14
1-6.1 Mesopotamian Astronomy 15
1-6.2 Mesopotamian Mathematics 15
1-7 Mathematics of the Mayans, Indians, Arabs, and Chinese 16
1-8 The First Great Engineering Society 19
1-9 Adverse Criticism of Ancient Egyptian and Mesopotamian Mathematics 24
1-10 Evolution through the Hellenic Era 29
1-11 The Unification of Celestial and Terrestrial Motion 31
1-11.1 Celestial Motion 31
1-11.2 Terrestrial Motion 44 "
1-11.3 Unification 45
1-12 Variational Principles in Dynamics 47
1-13 The Internationalism of Dynamics 52
1-14 Our Niche in the Cosmos 53
2 DESIGN, MODELING, AND FORMULATION OF EQUATIONS OF MOTION 55
2-1 Introduction 55
2-2 Design and Modeling 56
2-2.1 The Design Process 56
2-2.2 The Modeling Process 57
2-2.3 Our More Modest Goals 58
2-3 Direct and Indirect Approaches for Formulation
of Equations of Motion 59
3 KINEMATICS 68
3-1 Introduction 68
3-2 Position, Velocity, and Acceleration 69
3-3 Plane Kinematics of Rigid Bodies 75
3-3.1 The General Motion of a Rigid Body 75
3-3.2 Types of Plane Motion of a Rigid Body 76
3-3.3 Angular Displacement, Angular Velocity,and Angular Acceleration 77
3-3.4 A Cautionary Note about Finite Rotations 83
3-4 Time Rate of Change of Vector in Rotating Frame 85
3-5 Kinematic Analysis Utilizing Intermediate Frames 90
3-6 Generalizations of Kinematic Expressions 108
Problems for Chapter 3 111
4 MOMENTUM FORMULATION FOR SYSTEMS 01 PARTICLES 135
4-1 Introduction 135
4-2 The Fundamental Physics 136
4-2.1 Newton‘s Laws of Motion 136
4-2.2 A Particle 137
4-2.3 Linear Momentum and Force 138
4-2.4 Inertial Reference Frames 139
4-2.5 The Universal Law of Gravitation 140
4-3 Torque and Angular Momentum for a Particle 141
4-4 Formulation of Equations of Motion: Examples 144
4-4.1 Problems of Particle Dynamics of the First Kind 145
4-4.2 Problems of Particle Dynamics of the Second Kind 151
Problems for Chapter 4 163
5 VARIATIONAL FORMULATION FOR SYSTEMS OF PARTICLES 179
5-1 Introduction 179
5-2 Formulation of Equations of Motion 180
5-3 Work and State Functions 181
5-3.1 Work 182
5-3.2 Kinetic State Functions 183
5-3.3 Potential State Functions 185
5-3.4 Energy and Coenergy 189
5-4 Generalized Variables and Variational Concepts 190
5-4.1 Generalized Coordinates 190
5-4.2 Admissible Variations, Degrees of Freedom, Geometric Constraints, and Holonomicity 195
5-4.3 Variational Principles in Mechanics 201
5-4.4 Generalized Velocities and Generalized Forces
for Holonomic Systems 205
5-5 Equations of Motion for Holonprnic Mechanical Systems
via Variational Principles 213
5-6 Work-Energy Relation 238
5-7 Nature of Lagrangian Dynamics 241
Problems for Chapter 5 243
6 DYNAMICS OF SYSTEMS CONTAINING RIGID BODIES 268
6-1 Introduction 268
6-2 Momentum Principles for Rigid Bodies 269
6-2.1 Review of Solids in Equilibrium and Particle Dynamics 270
6-2.2 Models of Rigid Bodies 271
6-2.3 Momentum Principles for Extended Bodies:
The Newton-Euler Equations 272
6-2.4 Momentum Principles for Rigid Bodies Modeled
as Systems of Particles 273
6-2.5 Momentum Principles for Rigid Bodies Modeled as Continua 275
6-3 Dynamic Properties of Rigid Bodies 279 *
6-3.1 The Inertia Tensor 279
6-3.2 Parallel-Axes Theorem 290
6-3.3 Principal Directions and Principal Moments of Inertia 296
6-3.4 Uses of Mass Symmetry 298
6-4 Dynamics of Rigid Bodies via Direct Approach 303
6-5 Lagrangian for Rigid Bodies 308
6-5.1 Kinetic Coenergy Function for Rigid Body 308
6-5.2 Potential Energy Function for Rigid Body 310
6-6 Equations of Motion for Systems Containing Rigid Bodies
in Plane Motion 311
Problems for Chapter 6 334
7 DYNAMICS OF ELECTRICAL AND ELECTROMECHANICAL SYSTEMS 366
7-1 Introduction 366
7-2 Formulation of Equations of Motion for Electrical Networks 369
7-3 Constitutive Relations for Circuit Elements 371
7-3.1 Passive Elements 371
7-3.2 Active Electrical Elements 376
7-4 Hamilton‘s Principle and Lagrange‘s Equations
for Electrical Networks 380
7-4.1 Generalized Charge Variables 380
7-4.2 Generalized Flux Linkage Variables 382
7-4.3 Work Expressions 383
7-4.4 Summary of Lumped-Parameter Offering of Variational Electricity 386
7-4.5 Examples 386
7-5 Constitutive Relations for Transducers 407
7-5.1 Ideal Movable-Plate Capacitor 408
7-5.2 Electrically Linear Movable-Plate Capacitor 410
7-5.3 Ideal Movable-Core Inductor 412
7-5.4 Magnetically Linear Movable-Core Inductor 413
7-6 Hamilton‘s Principle and Lagrange‘s Equations for
Electromechanical Systems 415
7-6.1 Displacement-Charge Variables Formulation 416
7-6.2 Displacement-Flux Linkage Variables Formulation 417
7-6.3 Examples 419
7-7 Another Look at Lagrangian Dynamics 428
Problems for Chapter 7 429
8 VIBRATIOIV OF LINEAR LUMPED-PARAMETER SYSTEMS 439
8-1 Introduction 439
4r 8-2 Single-Degree-of-Freedom First-Order Systems 440
8-2.1 Free Response 441
8-2.2 Step Response 444
8-2.3 Ramp Response 446
8-2.4 Harmonic Response 449
8-2.5 Summary of Responses for Single-Degree-of-Freedom
First-Order Systems 459
8-3 Single-Degree-of-Freedom Second-Order Systems 460
8-3.1 Free Response 461
8-3.2 Natural Frequency via Static Deflection 467
8-3.3 Logarithmic Decrement 468
‘8-3.4 Energy Loss of Free Vibration 471
8-3.5 Harmonic Response 472
8-3.6 Summary of Responses for Single-Degree-of-Freedom
Second-Order Systems 498
8-4 Two-Degree-of-Freedom Second-Order Systems 500
8-4.1 Natural Modes of Vibration 501
8-4.2 Response to Initial Conditions 514 *
8-4.3 Harmonic Response 527
8-5 Stability of Nonlinear Systems 541
K Problems for Chapter 8 557
9 DYNAMICS OF CONTINUOUS SYSTEMS 576
9-1 Introduction 576
9-2 Equations of Motion 578
9-2.1 Longitudinal Motion of System Containing Rod 579
9-2.2 Twisting Motion of System Containing Shaft 586
9-2.3 Electric Transmission Line 589
9-2.4 Flexural Motion of System Containing Beam 594
9-2.5 Summaries 602
9-3 Natural Modes of Vibration 607
9-3.1 Method of Separation of Variables 608
9-3.2 Time Response 610
9-3.3 Eigenfunctions for Second-Order Systems 612
9-3.4 Eigenfunctions for Fourth-Order Systems 620
9-3.5 General Solutions for Free Undamped Vibration 633
9-4 Response to Initial Conditions 636
9-4.1 An Example: Release of Compressed Rod 636
9-4.2 An Example: Shaft Stopped after Rotation 647
9-4.3 An Example: Sliding-Free Beam Initially Bent 650
9-5 Response to Harmonic Excitations 660
9-5.1 An Example: Specified Harmonic Motion of Boundary
9-5.2 An Example: Distributed Harmonic Force 662
9-5.3 An Example: Harmonic Force on Boundary 665
9-6 Summaries 672
Problems for Chapter 9 673
BIBLIOGRAPHY
Historical 684
Astronomy 686
Design, Systems, and Modeling 686
Elementary Dynamics 686
Intermediate/Advanced Dynamics ‘ 686
Hamilton‘s Law of Varying Action and Hamilton‘s Principle
Electrical and Electromechanical Systems 687
Vibration 687
APPENDIX A FINITE ROTATION 688
A-l Change in Position Vector Due to Finite Rotation 688
A-2 Finite Rotations Are Not Vectors 690
A-3 Do Rotations Ever Behave as Vectors? 692
A-3.1 Infinitesimal Rotations Are Vectors 692
A-3.2 Consecutive Finite Rotations about a Common Axis
Are Vectors 692
APPENDIX B GENERAL KINEMATIC ANALYSIS
All Angular Velocities Defined with Respect to Fixed Reference Frame (Case 1) 694
Each Angular Velocity Defined with Respect to
Immediately Preceding Frame (Case 2) 698
APPENDIX C MOMENTUM PRINCIPLES FOR SYSTEMS OF PARTICLES
C-l Asserted Momentum Principles 705
C-2 Principles for Single Particle 706
C-3 Principles for System of Particles 707
C-3.1 Asserted System Momentum Principles 708
C-3.2 System Momentum Principles Derived from Particle Momentum Principles 709
C-3.3 Conditions on Internal Forces 711
