# The Flight of Uncontrolled Rockets

## 1st Edition

### International Series of Monographs on Aeronautics and Astronautics

**Authors:**F. R. Gantmakher L. M. Levin

**Editors:**H. L. Dryden

**eBook ISBN:**9781483164946

**Imprint:**Pergamon

**Published Date:**1st January 1964

**Page Count:**396

## Description

International Series of Monographs on Aeronautics and Astronautics, Division VII, Volume 5: The Flight of Uncontrolled Rockets focuses on external ballistics of uncontrolled rockets. The book first discusses the equations of motion of rockets. The rocket as a system of changing composition; application of solidification principle to rockets; rotational motion of rockets; and equations of motion of the center of mass of rockets are described. The text looks at the calculation of trajectory of rockets and the fundamentals of rocket dispersion. The selection further focuses on the dispersion of finned rockets. Topics include the critical section of the trajectory; standard formula for calculating angular deviation; dispersion of actual rockets; and effective launcher length. The text also describes the dispersion of finned rotated rockets and of finned anti-tank rockets. The book also examines the effect of wind on the flight of rockets. Topics include correction to the coordinates of the point of impact for finned rockets; general effect of wind on dispersion; and general treatment of powered flight in the presence of wind. The text is important for readers interested in the ballistics of uncontrolled rockets.

## Table of Contents

Contents

Preface

Editor's Note

Definitions and Symbols

Chapter 1. Equations of Motion of A Rocket

1. The Rocket As A System of Changing Composition

2. Equations of Motion of A Particle and of A System of Constant Composition

3. Systems of Changing Composition. Rockets

4. Application of the Solidification Principle to Rockets

5. Equations of Motion of the Center of Mass of A Rocket

6. Rotational Motion of the Rocket

Chapter 2. Calculation of the Trajectory of A Rocket

7. Statement of the Problem

8. The Thrust

9. Equations for the Motion of the Mass-Center of A Rocket During Powered Flight

10. The Tsiolkovskii Formula

11. Variation in the Rocket Velocity During Powered Flight. Determination of the Arc-Length of the Trajectory

12. Direction of the Rocket Velocity During Powered Flight

13. The Influence of the Drag and Gravitational Forces On The Magnitude of the Velocity of A Rocket

14. Calculation of the Complete Trajectory. Problems

Chapter 3. Fundamentals of Rocket Dispersion

15. Introduction

16. Factors Causing Lateral Dispersion

17. Continuous Random Variables

18. The Role of the Powered and Unpowered Sections of the Trajectory in the Lateral Dispersion of A Rocket

19. Angular Deviation At the End of Powered Flight and Its Horizontal and Vertical Projections

20. Random Vectors

21. Angular Deviation and Its Connection With Lateral Dispersion

Chapter 4. The Dispersion of Finned Rockets

23. Introduction

24. Statement of the Problem

25. Equations of Motion

26. Integration of the Equations of Motion

27. Integration of the Equations of Motion With An Incomplete Set of Forces and Moments

28. Thrust Asymmetry As The Principal Factor Causing Angular Deviation of Finned Rockets

29. The Dispersion of Actual Rockets

30. Simplification of the Fundamental Formulae

31. The Critical Section of the Trajectory

32. Formulae For The Angle of Attack δ and The Angular Deviation ψ

33. A Graphical Method. The Cornu Spiral

34. The Standard Formula For Calculating Angular Deviation

35. Angular Deviation As A Function of the Structural Parameters of the Rocket and the Length of the Launcher

36. Effective Launcher Length

37. The Stepped Acceleration Diagram. Mortar Rockets and Multi-Stage Rockets

38. Rockets Launched From Aircraft

39. The Angular Deviation Due To Initial Perturbations

40. Methods of Improving The Dispersion of Finned Rockets

41. Integration of the Fundamental Differential Equation With A Complete Set Of Forces and Moments

42. Formulae For The Angles D and \P Using A Complete Set of Forces and Moments

43. Analysis of the Formulae. Major and Minor Forces and Moments

44. The Coriolis Force and Moment

45. The Effect of the Coriolis Force and Moment On The Angular Deviation of A Finned Rocket

Chapter 5. The Dispersion of Finned Rotated Rockets

46. Introduction

47. Statement of the Problem

48. Rotational Thrust Moments

49. The Equation of Rotational Motion

50. Integration of the Rotational Equation of Rotation

51. Analysis of the Rotational Motion

52. The Angular Deviation of Finned Rotated Rockets

53. The Angular Deviation Expressed In Terms of the Kernel R(s,σ)

54. Analysis of Angular Deviation Calculations For Finned Rotated Rockets

55. The Role of Thrust Eccentricity As A Cause of Lateral Dispersion of Finned Rockets

56. Arbitrary Acceleration On The Launcher. Firing From Aircraft

Chapter 6. The Dispersion of Finned Anti-Tank Rockets

57. Introduction

58. Aspects of the Ballistics of Anti-Tank Rockets

59. The Effects of Aerodynamic Forces and Moments On The Angular Deviation of Antitank Rockets

60. Integration of the Equations of Motion

61. Calculation of the Angular Deviation

62. The Maximum Effective Range

63. Analysis of the Angular Deviation and Lateral Dispersion of Anti-Tank Rockets

64. The Vertical Dispersion of Anti-Tank Rockets

65. The Calculation of Angular Deviation For Arbitrary Time-Dependence of the Acceleration

66. The Effect of the Coriolis Force and Moment On The Dispersion of Anti-Tank Rockets

Chapter 7. Rocket Motion In Three Dimensions

67. Introduction

68. The Equations of Motion of the Mass-Centre of A Rocket

69. The Equations of Rotational Motion of A Rocket

70. The Forces and Moments Acting On the Rocket

71. Linearization of the Equations of Motion

72. Integration of the Equations of Motion

73. Spin-Stabilized Rockets

74. Non-Rotated Finned Rockets

75. Finned Rotated Rockets

Chapter 8. Spin-Stabilized Rockets

76. Criteria For The Stability of the Unpowered Flight of Spin-Stabilized Rockets

77. The Condition For Stable Powered Flight

78. Calculation of the Angular Deviation During Powered Flight For Spin-Stabilized Rockets

Chapter 9. The Effect of Wind On The Flight of Rockets

79. Introduction

80. The Physics of the Phenomenon

81. The Angular Deviation of Finned Rockets During Powered Flight

82. Corrections to the Coordinates of the Point of Impact For Finned Rockets

83. The General Effect of Wind On Dispersion

84. The General Treatment of Powered Flight in the Presence of Wind

85. Non-Rotated Finned Rockets

86. Rotated Finned Rockets

87. Spin-Stabilized Rockets

Appendix 1. Formulae For the Additional Forces in the Equations of Motion of A Rocket

1. An Alternative Form of the Linear and Angular Momentum Equations For the Body S.

2. Formulae For Linear Momentum, Coriolis Force and Coriolis Moment

3. Movement of the Centre of Mass Relative to the Body of the Rocket

Appendix 2. Brief Notes On the Aerodynamics of Rockets

1. Introduction

2. Elements of Similarity Theory

3. Methods of Describing Aerodynamic Forces and Moments

4. The Drag

5. The Lift Force

6. The Lateral Aerodynamic Moment

7. The Lateral Damping Moment

8. Longitudinal Aerodynamic Moments

9. The Effect of Aerodynamic Characteristics On The Flight of A Rocket

Appendix 3. British and American Works On The Motion of Unguided Rockets During The Burning Period

1. Introduction

2. Unrotated Rockets

3. Finned Rotated Rockets

4. Spin-Stabilized Rockets

5. Rocker Accuracy

References To Appendix 3

Tables of A(X) and B(X) For Appendix 3

Tables :

Table 1

Table 2

Table 3 a

Table 3 b

Table 4

Table 5

Table 6

Table 7

Table 8 a

Table 8 b

Table 8 c

Table 8 d

References

Index

Other Titles in the Series In Aeronautics and Astronautics

## Details

- No. of pages:
- 396

- Language:
- English

- Copyright:
- © Pergamon 1964

- Published:
- 1st January 1964

- Imprint:
- Pergamon

- eBook ISBN:
- 9781483164946

## About the Author

### F. R. Gantmakher

### L. M. Levin

## About the Editor

### H. L. Dryden

### Affiliations and Expertise

Case Institute of Technology, Cleveland, Ohio