Analytical Mechanics of Aerospace Systems.pdf

A NALYTICAL M ECHANICS

A EROSPACE S YSTEMS

Hanspeter Schaub

and

John L. Junkins

January 1, 2002

Contents

Preface

I BASIC MECHANICS

1 Particle Kinematics 3

1.1 Particle Position Description . . . . . . . . . . . . . . . . . . . . 3

1.1.1 Basic Geometry . . . . . . . . . . . . . . . . . . . . . . . . 3

1.1.2 Cylindrical and Spherical Coordinate Systems . . . . . . . 6

1.2 Vector Dierentiation . . . . . . . . . . . . . . . . . . . . . . . . 8

1.2.1 Angular Velocity Vector . . . . . . . . . . . . . . . . . . . 8

1.2.2 Rotation about a Fixed Axis . . . . . . . . . . . . . . . . 10

1.2.3 Transport Theorem . . . . . . . . . . . . . . . . . . . . . 11

1.2.4 Particle Kinematics with Moving Frames . . . . . . . . . 15

2 Newtonian Mechanics 25

2.1 Newton's Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.2 Single Particle Dynamics . . . . . . . . . . . . . . . . . . . . . . . 29

2.2.1 Constant Force . . . . . . . . . . . . . . . . . . . . . . . . 29

2.2.2 Time-Varying Force . . . . . . . . . . . . . . . . . . . . . 32

2.2.3 Kinetic Energy . . . . . . . . . . . . . . . . . . . . . . . . 34

2.2.4 Linear Momentum . . . . . . . . . . . . . . . . . . . . . . 35

2.2.5 Angular Momentum . . . . . . . . . . . . . . . . . . . . . 35

2.3 Dynamics of a System of Particles . . . . . . . . . . . . . . . . . 38

2.3.1 Equations of Motion . . . . . . . . . . . . . . . . . . . . . 38

2.3.2 Kinetic Energy . . . . . . . . . . . . . . . . . . . . . . . . 41

2.3.3 Linear Momentum . . . . . . . . . . . . . . . . . . . . . . 43

2.3.4 Angular Momentum . . . . . . . . . . . . . . . . . . . . . 45

2.4 Dynamics of a Continuous System . . . . . . . . . . . . . . . . . 47

2.4.1 Equations of Motion . . . . . . . . . . . . . . . . . . . . . 47

2.4.2 Kinetic Energy . . . . . . . . . . . . . . . . . . . . . . . . 49

2.4.3 Linear Momentum . . . . . . . . . . . . . . . . . . . . . . 50

2.4.4 Angular Momentum . . . . . . . . . . . . . . . . . . . . . 51

2.5 The Rocket Problem . . . . . . . . . . . . . . . . . . . . . . . . . 52

iii

CONTENTS

3 Rigid Body Kinematics 63

3.1 Direction Cosine Matrix . . . . . . . . . . . . . . . . . . . . . . . 64

3.2 Euler Angles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

3.3 Principal Rotation Vector . . . . . . . . . . . . . . . . . . . . . . 78

3.4 Euler Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

3.5 Classical Rodrigues Parameters . . . . . . . . . . . . . . . . . . . 91

3.6 Modied Rodrigues Parameters . . . . . . . . . . . . . . . . . . . 96

3.7 Other Attitude Parameters . . . . . . . . . . . . . . . . . . . . . 103

3.7.1 Stereographic Orientation Parameters . . . . . . . . . . . 103

3.7.2 Higher Order Rodrigues Parameters . . . . . . . . . . . . 105

3.7.3 The (w,z) Coordinates . . . . . . . . . . . . . . . . . . . . 106

3.7.4 Cayley-Klein Parameters . . . . . . . . . . . . . . . . . . 107

3.8 Homogeneous Transformations . . . . . . . . . . . . . . . . . . . 107

4 Eulerian Mechanics 115

4.1 Rigid Body Dynamics . . . . . . . . . . . . . . . . . . . . . . . . 115

4.1.1 Angular Momentum . . . . . . . . . . . . . . . . . . . . . 115

4.1.2 Inertia Matrix Properties . . . . . . . . . . . . . . . . . . 118

4.1.3 Euler's Rotational Equations of Motion . . . . . . . . . . 123

4.1.4 Kinetic Energy . . . . . . . . . . . . . . . . . . . . . . . . 124

4.2 Torque-Free Rigid Body Rotation . . . . . . . . . . . . . . . . . . 128

4.2.1 Energy and Momentum Integrals . . . . . . . . . . . . . . 128

4.2.2 General Free Rigid Body Motion . . . . . . . . . . . . . . 133

4.2.3 Axisymmetric Rigid Body Motion . . . . . . . . . . . . . 135

4.3 Momentum Exchange Devices . . . . . . . . . . . . . . . . . . . . 137

4.3.1 Spacecraft with Single VSCMG . . . . . . . . . . . . . . . 138

4.3.2 Spacecraft with Multiple VSCMGs . . . . . . . . . . . . . 143

4.4 Gravity Gradient Satellite . . . . . . . . . . . . . . . . . . . . . . 145

4.4.1 Gravity Gradient Torque . . . . . . . . . . . . . . . . . . 145

4.4.2 Rotational - Translational Motion Coupling . . . . . . . . 148

4.4.3 Small Departure Motion about Equilibrium Attitudes . . 149

5 Generalized Methods of Analytical Dynamics 159

5.1 Generalized Coordinates . . . . . . . . . . . . . . . . . . . . . . . 159

5.2 D'Alembert's Principle . . . . . . . . . . . . . . . . . . . . . . . . 162

5.2.1 Virtual Displacements and Virtual Work . . . . . . . . . . 163

5.2.2 Classical Developments of D'Alembert's Principle . . . . . 164

5.2.3 Holonomic Constraints . . . . . . . . . . . . . . . . . . . . 170

5.2.4 Newtonian Constrained Dynamics of N Particles . . . . . 177

5.2.5 Lagrange Multiplier Rule for Constrained Optimization . 178

5.3 Lagrangian Dynamics . . . . . . . . . . . . . . . . . . . . . . . . 182

5.3.1 Minimal Coordinate Systems and Unconstrained Motion . 183

5.3.2 Lagrange's Equations for Conservative Forces . . . . . . . 187

5.3.3 Redundant Coordinate Systems and Constrained Motion 190

5.3.4 Vector-Matrix Form of the Lagrangian Equations of Motion195

CONTENTS

6 Advanced Methods of Analytical Dynamics 203

6.1 The Hamiltonian Function . . . . . . . . . . . . . . . . . . . . . . 203

6.1.1 Some Special Properties of The Hamiltonian . . . . . . . 203

6.1.2 Relationship of the Hamiltonian to Total Energy and Work

Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203

6.1.3 Hamilton's Canonical Equations . . . . . . . . . . . . . . 203

6.1.4 Hamilton's Principal Function and the Hamilton-Jacobi

Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . 203

6.2 Hamilton's Principles . . . . . . . . . . . . . . . . . . . . . . . . . 203

6.2.1 Variational Calculus Fundamentals . . . . . . . . . . . . . 204

6.2.2 Path Variations versus Virtual Displacements . . . . . . 204

6.2.3 Hamilton's Principles from D'Alembert's Principle . . . . 204

6.3 Dynamics of Distributed Parameter Systems . . . . . . . . . . . . 204

6.3.1 Elementary DPS: Newton-Euler Methods . . . . . . . . . 204

6.3.2 Energy Functions for Elastic Rods and Beams . . . . . . . 204

6.3.3 Hamilton's Principle Applied for DPS . . . . . . . . . . . 204

6.3.4 Generalized Lagrange's Equations for Multi-Body DPS . 204

7 Nonlinear Spacecraft Stability and Control 205

7.1 Nonlinear Stability Analysis . . . . . . . . . . . . . . . . . . . . . 206

7.1.1 Stability Denitions . . . . . . . . . . . . . . . . . . . . . 206

7.1.2 Linearization of Dynamical Systems . . . . . . . . . . . . 210

7.1.3 Lyapunov's Direct Method . . . . . . . . . . . . . . . . . 212

7.2 Generating Lyapunov Functions . . . . . . . . . . . . . . . . . . . 219

7.2.1 Elemental Velocity-Based Lyapunov Functions . . . . . . 221

7.2.2 Elemental Position-Based Lyapunov Functions . . . . . . 227

7.3 Nonlinear Feedback Control Laws . . . . . . . . . . . . . . . . . . 233

7.3.1 Unconstrained Control Law . . . . . . . . . . . . . . . . . 233

7.3.2 Asymptotic Stability Analysis . . . . . . . . . . . . . . . . 236

7.3.3 Feedback Gain Selection . . . . . . . . . . . . . . . . . . . 242

7.4 Lyapunov Optimal Control Laws . . . . . . . . . . . . . . . . . . 247

7.5 Linear Closed-Loop Dynamics . . . . . . . . . . . . . . . . . . . . 253

7.6 Reaction Wheel Control Devices . . . . . . . . . . . . . . . . . . 258

7.7 Variable Speed Control Moment Gyroscopes . . . . . . . . . . . . 260

7.7.1 Control Law . . . . . . . . . . . . . . . . . . . . . . . . . 261

7.7.2 Velocity Based Steering Law . . . . . . . . . . . . . . . . 264

7.7.3 VSCMG Null Motion . . . . . . . . . . . . . . . . . . . . 269

II CELESTIAL MECHANICS

283

8 Classical Two-Body Problem 285

8.1 Geometry of Conic Sections . . . . . . . . . . . . . . . . . . . . . 286

8.2 Relative Two-Body Equations of Motion . . . . . . . . . . . . . . 294

8.3 Fundamental Integrals . . . . . . . . . . . . . . . . . . . . . . . . 296

8.3.1 Conservation of Angular Momentum . . . . . . . . . . . . 296

Plik z chomika:

Inne pliki z tego folderu:

Inne foldery tego chomika: