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Contents

Preface ......................................................................................................... xi

1. Newtonian Mechanics ........................................................................... 1

1.1 Introduction ............................................................................................... 1

1.2 Fundamentals ........................................................................................... 11.3 Space and Time ....................................................................................... 2

1.4 Mass ......................................................................................................... 3

1.5 Force ........................................................................................................ 5

1.6 Work and Power ....................................................................................... 5

1.7 Kinematics of a Point ................................................................................ 6

1.8 Kinetics of a Particle ................................................................................. 11

1.9 Impulse ..................................................................................................... 12

1.10 Kinetic Energy .......................................................................................... 13

1.11 Potential Energy ....................................................................................... 13

1.12 Corioliss Theorem .................................................................................... 14

1.13 Newton's Laws for a Group of Particles .................................................... 15

1.14 Conservation of Momentum ..................................................................... 17

1.15 Energy for a Group of Particles ................................................................ 17

1.16 The Principle of Virtual Work .................................................................... 18

1.17 DAlemberts Principle .............................................................................. 19

2. Lagranges Equations ........................................................................... 21

2.1 Introduction ............................................................................................... 21

2.2 Generalized Co-Ordinates ........................................................................ 23

2.3 Proof of Lagranges Equations ................................................................. 25

2.4 The Dissipation Function .......................................................................... 27

2.5 Kinetic Energy .......................................................................................... 29

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2.6 Conservation Laws ................................................................................... 31

2.7 Hamiltons Equations ................................................................................ 33

2.8 Rotating Frame of Reference and Velocity-Dependent Potentials ........... 35

2.9 Moving Co-Ordinates ................................................................................ 39

2.10 Non-Holonomic Systems .......................................................................... 41

2.11 Lagranges Equations for Impulsive Forces .............................................. 43

3. Hamiltons Principle .............................................................................. 46

3.1 Introduction ............................................................................................... 46

3.2 Derivation of Hamiltons Principle ............................................................. 47

3.3 Application of Hamiltons Principle ........................................................... 49

3.4 Lagranges Equations Derived from Hamiltons Principle ......................... 51

3.5 Illustrative Example .................................................................................. 52

4. Rigid Body Motion in Three Dimensions ............................................ 55

4.1 Introduction ............................................................................................... 55

4.2 Rotation .................................................................................................... 55

4.3 Angular Velocity ........................................................................................ 58

4.4 Kinetics of a Rigid Body ............................................................................ 59

4.5 Moment of Inertia ...................................................................................... 61

4.6 Eulers Equation for Rigid Body Motion .................................................... 64

4.7 Kinetic Energy of a Rigid Body ................................................................. 65

4.8 Torque-Free Motion of a Rigid Body ........................................................ 67

4.9 Stability of Torque-Free Motion ................................................................ 72

4.10 Eulers Angles ........................................................................................... 754.11 The Symmetrical Body ............................................................................. 76

4.12 Forced Precession .................................................................................... 80

4.13 Epilogue .................................................................................................... 83

5. Dynamics of Vehicles ........................................................................... 85

5.1 Introduction ............................................................................................... 85

5.2 Gravitational Potential .............................................................................. 855.3 The Two-Body Problem ............................................................................ 88

5.4 The Central Force Problem ...................................................................... 90

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5.5 Satellite Motion ......................................................................................... 93

5.6 Effects of Oblateness ............................................................................... 100

5.7 Rocket in Free Space ............................................................................... 103

5.8 Non-Spherical Satellite ............................................................................. 106

5.9 Spinning Satellite ...................................................................................... 107

5.10 De-Spinning of Satellites ......................................................................... 107

5.11 Stability of Aircraft ..................................................................................... 109

5.12 Stability of a Road Vehicle ........................................................................ 118

6. Impact and One-Dimensional Wave Propagation ............................... 125

6.1 Introduction ...............................................................................................125

6.2 The One-Dimensional Wave .................................................................... 125

6.3 Longitudinal Waves in an Elastic Prismatic Bar ....................................... 128

6.4 Reflection and Transmission at a Boundary ............................................. 130

6.5 Momentum and Energy in a Pulse ........................................................... 132

6.6 Impact of Two Bars ................................................................................... 133

6.7 Constant Force Applied to a Long Bar ..................................................... 136

6.8 The Effect of Local Deformation on Pulse Shape ..................................... 1386.9 Prediction of Pulse Shape during Impact of Two Bars ............................. 141

6.10 Impact of a Rigid Mass on an Elastic Bar ................................................. 145

6.11 Dispersive Waves ..................................................................................... 149

6.12 Waves in a Uniform Beam ........................................................................ 155

6.13 Waves in Periodic Structures ................................................................... 161

6.14 Waves in a Helical Spring ......................................................................... 163

7. Waves in a Three-Dimensional Elastic Solid ...................................... 172

7.1 Introduction ...............................................................................................172

7.2 Strain ........................................................................................................172

7.3 Stress .......................................................................................................176

7.4 Elastic Constants ...................................................................................... 177

7.5 Equations of Motion .................................................................................. 178

7.6 Wave Equation for an Elastic Solid .......................................................... 179

7.7 Plane Strain .............................................................................................. 184

7.8 Reflection at a Plane Surface ................................................................... 186

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7.9 Surface Waves (Rayleigh Waves) ............................................................ 189

7.10 Conclusion .................................................................................................. 192

8. Robot Arm Dynamics ............................................................................ 194

8.1 Introduction ...............................................................................................194

8.2 Typical Arrangements ............................................................................... 194

8.3 Kinematics of Robot Arms ........................................................................ 197

8.4 Kinetics of a Robot Arm ............................................................................ 223

9. Relativity ................................................................................................ 235

9.1 Introduction ...............................................................................................235

9.2 The Foundations of the Special Theory of Relativity ................................ 235

9.3 Time Dilation and Proper Time ................................................................. 240

9.4 Simultaneity ..............................................................................................241

9.5 The Doppler Effect .................................................................................... 242

9.6 Velocity .....................................................................................................246

9.7 The Twin Paradox .................................................................................... 249

9.8 Conservation of Momentum ..................................................................... 250

9.9 Relativistic Force ......................................................................................252

9.10 Impact of Two Particles ............................................................................ 254

9.11 The Relativistic Lagrangian ...................................................................... 256

9.12 Conclusion ................................................................................................258

Problems ......................................... ........................................................................ 261

AppendicesAppendix 1 Vectors, Tensors and Matrices .....................................................272

Appendix 2 Analytical Dynamics ......................................................................281

Appendix 3 Curvilinear Co-Ordinate Systems .................................................288

Bibliography ............................................................................................... 297

Index ............................................................................................................ 299

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