Atmospheric Re-Entry Vehicle Mechanics Book

1 Classical Mechanics 1

1.1 Classical Point Mass Mechanics 1

1.1.1 Fundamental Principles 1

1.1.2 Noninertial Frames 2

1.1.3 Linear and Angular Momentum 3

1.1.4 Modeling the Forces 4

1.1.5 Conservation of Energy 4

1.1.6 Isolated System 6

1.2 Mechanics of Rigid Bodies 7

1.2.1 Linear Momentum and Angular Momentum 7

1.2.2 Center of Mass and Equations of Movement 8

1.2.3 Eulerian Frames 9

2 Topography and Gravitation 15

2.1 The Geodetic Frame of Reference 15

2.1.1 Coordinates of a Point Relating to the Earth 16

2.1.2 The Geodetic Systems 17

2.1.3 Calculation of the Geographical Latitude and Height 18

2.2 The Terrestrial Field of Gravitation 19

3 Models of Atmosphere 21

3.1 Main Parameters and Hypotheses 21

3.2 The Isothermal Exponential Model 22

3.3 Standard Models of Earth's Atmosphere 23

3.4 Martian Models 26

4 Aerodynamics 31

4.1 Aerodynamic Coefficients 33

4.1.1 Static Coefficients 34

4.1.2 Dynamic Derivatives 36

4.1.3 Axisymmetric Vehicles 37

4.2 Modes of Flow 38

4.2.1 Parameters of Similarity 38

4.2.2 Characteristics of the Main Flow Modes 39

4.3 Continuous Mode 40

4.3.1 Experimental 40

4.3.2 Numerical [and] 40

4.3.3 Approximate Analytical Method [TRU] 41

4.3.4 Continuous Coefficients by Newton's Method 43

4.4 Rarefield Mode 52

4.4.1 Free Molecular Flow 52

4.4.2 Intermediate Flow 56

4.5 Qualities of Flight 58

4.5.1 Static Stability 59

4.5.2 Gyroscopic Stability 64

4.6 Characteristics of a Family of Sphere Cones 65

4.6.1 Mach Number Influence 66

4.6.2 Influence of Angle of Attack 70

4.6.3 Aerodynamic Modeling for Trajectory Codes 74

4.7 Planetary Entry Capsule 76

5 Intertial Models 85

5.1Moments of Inertia 85

5.2 CG Offset and Principal Axis Misalignment 86

6 Changing of Reference Frame 89

6.1 Direction Cosine Matrices 89

6.1.1 Angular Velocity 91

6.1.2 Composition of Angular Velocities 94

6.1.3 Evolution of the Direction Cosine Matrices 95

6.2 Euler Angles 96

6.2.1 Euler Rotation Matrix 96

6.2.2 Evolution of Euler Angles 99

6.3 Representations with Four Parameters 103

6.3.1 Vectorial Representation 103

6.3.2 Quaternion 104

7 Exoatmospheric Phase 109

7.1 Movement of the Center of Mass 109

7.1.1 Keplerian Trajectories 109

7.1.2 Ballistic Trajectories 118

7.1.3 Influence of Earth Rotation 128

7.2 Movement Around Mass Center 136

7.2.1 Rotation Around a Principal Axis 136

7.2.2 Coning Motion 138

8 Six Degree-of-Freedom Reentry 143

8.1 General Equations of Motion 143

8.2 Solutions of General Equations 146

9 Zero Angle of Attack Reentry 149

9.1 Allen's Reentry Results 149

9.1.1 Axial Load Factor and Dynamic Pressure 151

9.1.2 Heat Flux 151

9.1.3 Thermal Energy at Stagnation Point 152

9.1.4 Duration of Reentry 154

9.2 Influence of Ballistic Coefficient and Flight Path Angle 156

9.3 Influence of Range 157

10 Decay of Initial Incidence 163

10.1 Zero Spin Rate 165

10.1.1 First Approximate Solution 166

10.1.2 Second Approximate Solution 168

10.2 Nonzero Spin 172

11 End of the Convergence of the Incidence 179

11.1 Linear Equations 180

11.2 Instantaneous Angular Movement 185

11.2.1 Epicyclic Movement 186

11.2.2 Tricyclic Movement 190

11.3 Real Angular Motion 198

12 Roll-lock-in Phenomenon 201

12.1 Association of Aerodynamic Asymmetry and CG Offset 202

12.1.1 Equilibrium on Critical Frequency 204

12.1.2 Lock-in Near Resonance 206

12.1.3 Variable Critical Frequency 209

12.1.4 Criterion for Out-of-plane Asymmetries 209

12.2 Isolated Center of Gravity Offset 216

12.3 Isolated Principal Axis Misalignment 217

12.4 Combined CG Offset and Principal Axis Misalignment 220

12.4.1 Out-of-plane Misalignment 221

12.4.2 In-plane Misalignment 226

13 Instabilities 231

13.1 Static Instabilities 231

13.2 Dynamic Instabilities 238

13.2.1 Approximate Study at Low Angle of Attack 238

13.2.2 Examples of Unstable Dynamic Behaviors 241

14 Reentry Errors 245

14.1 Zero Angle-of-attack Dispersions 245

14.1.1 Initial Conditions 245

14.1.2 Consequences of Drag Dispersions 246

14.1.3 Sensitivity to Initial Conditions 250

14.2 Nonzero Angle of Attack 251

14.2.1 Effects of Incidence on Aerodynamic Loads 251

14.2.2 Effects of Initial Angle-of-attack 253

Epilog 281

Exercises 285

Solutions 297

1 Equations of Motion Relating to a Reference Frame in Rotation 297

2 Accelerometer Measurements 299

3 Vertical and Apparent Gravity 300

4 Coriolis Force 301

5 Quaternions 302

6 Quaternions and Euler Angles 303

7 Equivalence of Changing Frame and Rotation Quaternion Operators for Euler Angle Sequence 304

8 Pendulum of Foucault 305

9 Motion of the Earth's Mass Center 311

10 Launch Windows to Mars 311

11 Energy of a Solid in Free Rotation 311

12 Total Angular Momentum of a Deformable System in Gravitation 312

13 Lunar Motion 315

14 Angular Stabilization with Inertia Wheels 315

15 Balancing Machines 316

16 Aerodynamics of Apollo Reentry Capsule in Continuous Flow 318

17 Aerodynamics of Apollo Reentry Capsule in Free Molecular Flow 320

18 Aerodynamics of Viking Reentry Capsule in Continuous Flow 321

19 Aerodynamics of Viking Reentry Capsule in Free Molecular Flow 324

20 Aerodynamics of Pathfinder Entry Capsule in Intermediate Flow 325

21 Aerodynamics of a Biconic Reentry Vehicle in Continuous Mode 327

22 Targeting Errors of Ballistic Trajectories 330

23 Stability of Free Rotational Motion of a Satellite with Spin 332

24 Allen and Eggers Reentry 333

25 Mars Atmosphere Measurement 334

26 Entry of Meteorites 336

27 Normal Load Factor Related to Incidence 338

28 Artefact in Computer Codes 339

29 Gyroscopic Stabilization of an Entry Capsule 340

30 Effect of Equilibrium Lift on CG Motion 341

31 Effects of Drag Dispersions 344

32 Wind Effects 345

33 Effect of a Sharp Stability Variation, Case of a Plane Oscillation 348

34 Mars Skip Out Trajectories 349