Gyrodynamics and Its Engineering Applications (eBook)

Front Cover 1
Gyrodynamics and Its Engineering Applications 4
Copyright Page 5
Table of Contents 8
PREFACE 6
CHAPTER 1. GENERAL INTRODUCTION 12
1.1 History 12
1.2 Introduction to Vectors 17
1.3 Kinematics of a Particle 21
1.4 Newton's Laws 23
1.5 Rotating Systems 27
1.6 Illustrative Examples 35
CHAPTER 2. KINEMATICS OF RIGID BODIES 39
2.1 Products of Vectors 39
2.2 Translation and Rotation 46
2.3 Instantaneous Axis of Rotation 48
2.4 Finite and Infinitesimal Rotations 53
2.5 General Motion 57
CHAPTER 3. FRAMES OF REFERENCE 60
3.1 Transformation of Vector Components 60
3.2 Eulerian Angles 61
3.3 Moving Axes 63
3.4 Relative Motion 68
3.5 Illustrative Examples 71
CHAPTER 4. MOMENTS AND PRODUCTS OF INERTIA 76
4.1 Definitions 76
4.2 Transformation Theorems and Principal Axes 77
4.3 Mohr's Circle 85
4.4 Symmetric Bodies 86
4.5 Experimental Measurements 89
CHAPTER 5. DYNAMICAL THEOREMS 94
5.1 Linear and Angular Momentum 94
5.2 General Equations of Motion 96
5.3 Equations of Motion for Rigid Bodies 101
5.4 Kinetic Energy 104
5.5 Lagrange's Equations 107
5.6 D'Alembert's Principle 112
CHAPTER 6. MOTION or A FREE BODY 115
6.1 Introduction 115
6.3 Polhodes and Herpolhodes 121
6.4 General Solution 124
6.5 Axially-symmetric Bodies 133
CHAPTER 7. SYMMETRICAL GYROSCOPE UNDER GRAVITY 140
7.1 Introduction 140
7.2 Equations of Motion 140
7.3 Uniform Precession 145
7.4 Nutation 151
7.5 General Solution 154
CHAPTER 8. GYRODYNAMICS OF MACHINES 161
8.1 Effects of Coriolis Forces 161
8.2 Spinning Discs and Rotors 168
8.3 Rolling Wheels and Discs 177
8.4 Grinding Mills 182
CHAPTER 9. GYROSCOPIC VIBRATION ABSORBERS AND STABILIZERS 188
9.1 Introduction 188
9.2 Historical Note 194
9.3 Gyroscopic Control of Forced Vibration 195
9.4 Gyroscopic Control of Self-excited and Free Vibration 207
9.5 Stabilization of Ships 219
9.6 The Monorail 230
CHAPTER 10. THE GYRO-COMPASS 239
10.1 Introduction 239
10.2 Foucault's Directional Gyroscope 239
10.3 Principle of Action of the Gyro -compass 241
10.4 Mechanical Arrangement 245
10.5 General Motion with Pendular Control 247
10.6 The Mercury Ballistic 256
10.7 General Motion with Mercury Ballistic 258
10.8 Errors 262
10.9 
275 
CHAPTER 11. SUSPENSIONS FOR GYROSCOPES 277
11.1 Introduction 277
11.2 Rotor Drives and Bearings 277
11.3 Single and Double Suspensions 279
11.4 The Cardan Suspension and Hooke's Joint 281
11.5 Gimbal Errors 286
11.6 Dynamics of Gimbals 292
11.7 Effect of Flexibility of Rotor Shaft 297
11.8 Dynamics of a Platform supported in Gimbals 301
11.9 Dynamics of a Rotor driven by a Hooke's Joint 308
CHAPTER 12. GYRO-VERTICALS 317
12.1 Introduction 317
12.2 The Gyro-pendulum 317
12.3 Response to Vehicle Acceleration 322
12.4 Effect of Gimbal Bearing Friction 325
12.5 Response to Forced Vibration 327
12.6 Practical Gyro-verticals 330
12.7 Response to Vehicle Acceleration 334
12.8 The Tilted-axis Gyro-vertical 345
12.9 Mechanical Design 347
12.10 Schuler Tuning of a Gyro-pendulum 350
CHAPTER 13. RATE AND INTEGRATING GYROSCOPES 355
13.1 Introduction 355
13.2 Single-axis Rate Gyroscopes 355
13.3 Aircraft Turn-and-slip Indicator 361
13.4 Floated Single-axis Gyroscopes 363
13.5 A Two-axis Rate Gyroscope 369
13.6 An Integrating Gyro-accelerometer 376
13.7 A Vibratory Rate Gyroscope 380
CHAPTER 14. INERTIAL NAVIGATION 387
14.1 Principles 387
14.2 Navigation around a Fixed Sphere 392
14.3 Motion around a Rotating Sphere 396
14.4 An Inertial Navigating System 398
14.5 Instrument Errors 400
14.6 Geophysical Effects 403
14.7 Stabilized Platforms 405
CHAPTER 15. GYROSCOPIC EFFECTS IN THE WHIRLING OF SHAFTS 417
15.1 Introduction 417
15.2 Shaft with a Single Disc 420
15.3 Shaft with Several Discs 427
15.4 Shaft with Many Discs 431
15.5 Shaft with a Single Unsymmetrical Body 434
CHAPTER 16. GYROSCOPIC EFFECTS IN AIRCRAFT 437
16.1 Introduction 437
16.2 Rotors and Propellers 437
16.3 General Dynamics of Aircraft 442
16.4 Gyroscopic Coupling of Aircraft Oscillations 446
16.5 Aircraft in Steady Roll 448
16.6 Application to Missiles 451
16.7 Helicopter Rotors 453
APPENDICES 459
A. The Operator D 459
B. The Second-order Equation of Vibration Theory 462
C. Routh-Hurwitz Stability Criteria 464
PROBLEMS 466
ANSWERS TO PROBLEMS 485
INDEX 490