Dynamics of Particles and Rigid Bodies

List of Figures xiii Preface xxiii Acknowledgement xxvii Introduction xxix About the Companion Website xliii 1 Kinematics of Particles 1 1.1 Inertial Frames 1 1.2 Rotating Frames 2 1.3 Rotation Matrices 4 1.4 Velocity of a Particle in a Three-dimensional Space 8 1.5 Acceleration of a Particle in a Three-dimensional Space 14 Exercises 21 2 Dynamics of Particles: Vectorial Approach 27 2.1 Newton’s Second Law of Dynamics 27 2.2 Stiffness and Viscous Damping 37 2.3 Dry Friction 40 2.4 Dynamics of a System of Particles 43 2.5 Newton’s Law of Gravitation 47 Exercises 50 Reference 54 3 Dynamics of Rigid Bodies: Vectorial Approach 55 3.1 Center of Mass 55 3.2 Mass Moment of Inertia 57 3.3 Parallel Axis Theorem 61 3.4 Rotation of the Inertia Matrix 65 3.4.1 The Principal Axes 66 3.5 Planar Motion of Rigid Bodies 69 3.5.1 Moment about an Inertial Point 72 3.5.2 Moment about a Moving Point on the Body 73 3.5.3 Moment about the Center of Mass or a Fixed Point on the Body 73 3.6 Non-planar Rigid-body Motion 83 3.6.1 Euler Rotational Equations 85 Exercises 94 Reference 101 4 System Constraints and Virtual Displacement 103 4.1 Constraints 103 4.1.1 Classification of Constraints 104 4.2 Actual and Virtual Displacements 110 4.3 Virtual Work 113 Exercises 115 Reference 116 5 Dynamics of Particles: Analytical Approach 117 5.1 The Brachistochrone Problem 117 5.2 Lagrange’s Equation for a Conservative System 123 5.3 Lagrange’s Equation for Non-conservative Systems 131 5.3.1 Viscous Damping 134 5.4 Lagrange’s Equations with Constraints 141 5.4.1 Physical Interpretation of Lagrange Multipliers 146 5.5 Cyclic Coordinates 151 5.6 Advantages and Disadvantages of the Analytical Approach 154 Exercises 155 References 159 6 Dynamics of Rigid Bodies: Analytical Approach 161 6.1 Kinetic Energy of a Rigid Body 161 6.2 Lagrange’s Equation Applied to Rigid Bodies 166 Exercises 176 7 Momentum 183 7.1 Linear Momentum 183 7.2 Collision 186 7.3 Angular Momentum of Particles 192 7.3.1 Angular Impulse 195 7.4 Angular Momentum of Rigid Bodies (Planar Motion) 199 7.4.1 Angular Momentum about an Axis Passing through the Center of Mass 199 7.4.2 Angular Momentum about an Axis Passing through a Fixed Point on the Body 201 7.4.3 Angular Momentum about an Axis Passing through an Arbitrary Inertial Point 201 7.5 Angular Momentum of Rigid Bodies (Non-planar Motion) 205 7.5.1 Angular Momentum about a Set of Axes Located at the Center of Mass 205 7.5.2 Angular Momentum about a Set of Axes Located at a Fixed Point 206 7.5.3 Angular Momentum about a Set of Axes Located at an Arbitrary Inertial Point 206 7.5.4 Conservation of Angular Momentum for Rigid Bodies 206 7.6 Generalized Momenta 213 Exercises 219 8 Motion of Charged Bodies in an Electric Field 227 8.1 Electrostatics 227 8.1.1 Electrostatic Forces 227 8.1.2 Electric Field 229 8.1.3 Electric Flux 232 8.1.4 Electrostatic Potential Energy 234 8.1.5 Electric Potential (Voltage) 235 8.1.6 Capacitance 237 8.1.7 Motion in an Electric Field 239 8.2 Electromagnetism 247 8.2.1 Electromagnetic Force 247 8.2.2 Forces on a Current-carrying Conductor 253 8.2.3 Electromagnetic Coupling 255 8.2.4 Ampere’s Law 257 8.2.5 Faraday’s Law of Induction 262 8.3 Lagrangian Formulation for Electrical Elements 268 8.3.1 Capacitor 268 8.3.2 Inductor 269 8.3.3 Resistor 269 8.4 Maxwell’s Equations 273 8.4.1 Maxwell’s First Equation 273 8.4.2 Maxwell’s Second Equation 273 8.4.3 Maxwell’s Third Equation 274 8.4.4 Maxwell’s Fourth Equation 274 8.5 Lagrangian Formulation of the Lorentz Force 275 Exercises 279 References 284 9 Introduction to Analysis Tools 285 9.1 Basic Definitions 285 9.2 Equilibrium Solutions of Dynamical Systems 287 9.3 Stability and Classification of Equilibrium Solutions 288 9.4 Phase-plane Representation of the Dynamics 296 9.4.1 Conservative Systems 296 9.4.2 Non-conservative Systems 303 9.5 Bifurcation of Equilibrium Solutions 308 9.5.1 Static Bifurcations 308 9.5.2 Dynamic (Hopf) Bifurcation 315 9.6 Basins of Attraction 323 Exercises 324 References 326 Index 327