Flight of Uncontrolled Rockets (eBook)

Front Cover 1
The Flight of Uncontrolled Rockets 4
Copyright Page 5
Table of Contents 6
PREFACE 10
EDITOR'S NOTE 12
DEFINITIONS AND SYMBOLS 13
CHAPTER 1. EQUATIONS OF MOTION OF A ROCKET 16
1. The Rocket as a System of Changing Composition 16
2. Equations of Motion of a Particle and of a System of Constant Composition 17
3. Systems of Changing Composition. Rockets 21
4. Application of the Solidification Principle to Rockets 28
5. Equations of Motion of the Centre of Mass of a Rocket 30
6. Rotational Motion of the Rocket 31
CHAPTER 2. CALCULATION OF THE TRAJECTORY OF A ROCKET 33
7. Statement of the Problem 33
8. The Thrust 34
9. Equations for the Motion of the Mass-centre of a Rocket during Powered Flight 37
10. The Tsiolkovskii Formula 39
11. Variation in the Rocket Velocity during Powered Flight. Determination of the Arc-length of the Trajectory 43
12. Direction of the Rocket Velocity during Powered Flight 49
13. The Influence of the Drag and Gravitational Forces on the Magnitude of the Velocity of a Rocket 52
14. Calculation of the Complete Trajectory. Problems 68
CHAPTER 3. FUNDAMENTALS OF ROCKET DISPERSION 74
15. Introduction 74
16. Factors Causing Lateral Dispersion 75
17. Continuous Random Variables 77
18. The Role of the Powered and Unpowered Sections of the Trajectory in the Lateral Dispersion of a Rocket 82
19. Angular Deviation at the End of Powered Flight and Its Horizontal and Vertical Projections 84
20. Random Vectors 87
21. Angular Deviation and Its Connection with Lateral Dispersion 89
22. Range Dispersion 90
CHAPTER 4. THE DISPERSION OF FINNED ROCKETS 94
23. Introduction 94
24. Statement of the Problem 94
25. Equations of Motion 96
26. Integration of the Equations of Motion 99
27. Integration of the Equation of Motion with an Incomplete Set of Forces and Moments 101
28. Thrust Asymmetry as the Principal Factor Causing Angular Deviation of Finned Rockets 105
29. The Dispersion of Actual Rockets 106
30. Simplification of the Fundamental Formulae 109
31. The Critical Section of the Trajectory 110
32. Formulae for the Angle of Attack 6 and the Angular Deviation y 112
33. A Graphical Method. The Cornu Spiral 114
34. The Standard Formula for Calculating Angular Deviation 118
35. Angular Deviation as a Function of the Structural Parameters of the Rocket and the Length of the Launcher 121
36. Effective Launcher Length 124
37. The Stepped Acceleration Diagram. Mortar Rockets and Multi-stage Rockets 126
38. Rockets Launched from Aircraft 129
39. The Angular Deviation due to Initial Perturbations 131
40. Method of Improving the Dispersion of Finned Rockets 133
41. Integration of the Fundamental Differential Equation with a Complete Set of Forces and Moments 134
42. Formulae for the Angles 6 and tp using a Complete Set of Forces and Moments 135
43. Analysis of the Formulae. Major and Minor Forces and Moments 137
44. The Coriolis Force and Moment 141
45. The Effect of the Coriolis Force and Moment on the Angular Deviation of a Finned Rocket 145
CHAPTER 5. THE DISPERSION OF FINNED ROTATED ROCKETS 153
46. Introduction 153
47. Statement of the Problem 154
48. Rotational Thrust Moments 157
49. The Equation of Rotational Motion 161
50. Integration of the Rotational Equation of Motion 162
51. Analysis of the Rotational Motion 167
52. The Angular Deviation o! Finned Rotated Rockets 169
53. The Angular Deviation Expressed in Terms of the Kernel R(s, a) 171
54. Analysis of Angular Deviation Calculations for Finned Rotated Rockets 175
55. The Role of Thrust Eccentricity as a Cause of Lateral Dispersion of Finned Rockets 178
56. Arbitrary Acceleration on the Launcher. Firing from Aircraft 181
CHAPTER 6. THE DISPERSION OFFINNED ANTI-TANK ROCKETS 183
57. Introduction 183
58. Aspects of the Ballistics of Anti-tank Rockets 184
59. The Effects of Aerodynamic Forces and Moments on the Angular Deviation of Anti-tank Rockets 187
60. Integration of the Equations of Motion 189
61. Calculation of the Angular Deviation 193
62. The Maximum Effective Range 196
63. Analysis of the Angular Deviation and Lateral Dispersion of Anti-tank Rockets 197
64. The Vertical Dispersion of Anti-tank Rockets 201
65. The Calculation of Angular Deviation for Arbitrary Time-dependence of the Acceleration 203
66. The Effect of the Coriolis Force and Moment on the Dispersion of Anti-tank Rockets 208
CHAPTER 7. ROCKET MOTION IN THREE DIMENSIONS 211
67. Introduction 211
68. The Equations of Motion of the Mass-centre of a Rocket 211
69. The Equations of Rotational Motion of a Rocket 214
70. The Forces and Moments Acting on the Rocket 217
71. Linearization of the Equations of Motion 218
72. Integration of the Equations of Motion 223
73. Spin-stabilized Rockets 227
74. Non-rotated Finned Rockets 228
75. Finned Rotated Rockets 229
CHAPTER 8. SPIN-STABILIZED ROCKETS 232
76. Criteria for Stability of the Unpowered Flight of Spin-stabilized Rockets 232
77. The Condition for Stable Powered Flight 239
78. Calculation of the Angular Deviation During Powered Flight for Spin-stabilized Rockets 240
CHAPTER 9. THE EFFECT OF WIND ON THE FLIGHT OF ROCKETS 247
79. Introduction 247
80. The Physics of the Phenomenon 248
81. The Angular Deviation of Finned Rockets During Powered Flight 250
82. Correction to the Coordinates of the Point of Impact for Finned Rockets 254
83. The General Effect of Wind on Dispersion 256
84. The General Treatment of Powered Flight in the Presence of Wind 258
85. Non-rotated Finned Rockets 263
86. Rotated Finned Rockets 266
87. Spin-stabilized Rockets 266
APPENDIX 1 FORMULAE FOR THE ADDITIONAL FORCES IN THE EQUATIONS OF MOTION OF A ROCKET 269
1. Alternative Form of the Linear and Angular Momentum Equations for the Body S 269
2. Formulae for Linear Momentum, Coriolis Force and Coriolis Moment 273
3. Movement of the Centre of Mass Relative to the Body of the Rocket 275
APPENDIX 2 BRIEF NOTES ON THE AERODYNAMICS OF ROCKETS 277
1. Introduction 277
2. Elements of Similarity Theory 277
3. Method of Describing Aerodynamic Forces and Moments 280
4. The Drag 284
6. The Lateral Aerodynamic Moment 290
7. The Lateral Damping Moment 294
8. Longitudinal Aerodynamic Moments 297
9. The Effect of Aerodynamic Characteristics on the Flight of a Rocket 300
APPENDIX 3 BRITISH AND AMERICAN WORK ON THE MOTION OF UNGUIDED ROCKETS DURING THE BURNING PERIOD 303
1. Introduction 303
2. Unrotated Rockets 309
3. Finned Rotated Rockets 319
4. Spin-Stabilized Rockets 329
5. Rocket Accuracy 336
References to Appendix 3 340
TABLES of A (x) and B(x) for APPENDIX 3 341
TABLES 344
TABLE 1 345
TABLE 2 346
TABLE 3a 347
TABLE 3b 348
TABLE 4 349
TABLE 5 357
TABLE 6 359
TABLE 7 382
TABLE 8a 387
TABLE 8b 388
TABLE 8c 389
TABLE 8d 390
REFERENCES 391
INDEX 392
OTHER TITLES IN THE SERIES IN AERONAUTICS AND ASTRONAUTICS 395