In this post, we will see the book Mechanics of Gyroscopic Systems by A. Yu. Ishlinskii.

# About the book

This book discusses a fairly wide range of problems in mechanics connected with the practical application of gyroscopes.

The classical studies of A.N. Krylov and B. V. Bulgakov on the theory of gyroscopes are insufficient for solving the problems encountered in the development of new gyroscopic systems. Stricter standards of accuracy have made it necessary to take into account factors formerly neglected and to explain previously undetected experimental facts. New problems in kinematics, the applied theory of elasticity, the theory of oscillations and stability, and the theory of gyroscopes proper have thus arisen

Several new papers on the theory of gyroscopic systems have been published by the author since this book was written (the present monograph is a second slightly revised edition of the book which was first printed in 1952 in a limited issue). Three of them are given here as appendixes.

Translated from Russian under Israel Program for Scientific Translations Jerusalem 1965

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You can get the book here.

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# Contents

FOREWORD. 1

## Chapter I. GEOMETRY AND KINEMATICS OF GYROSCOPIC SYSTEMS 5

§ 1. Geometry of gimbal suspension systems. Determination of a

ship’s pitch and roll angles and its course. Gimbal error. Bicardan suspensions. 5

§ 2. Relative rotation of two stabilized systems during ship’s rolling 12

§ 3. Stabilization errors caused by inaccurate mounting of the

gimbal systems (geometry of two bicardan suspensions) 17

§ 4. Horizontal stabilization errors of combinations of different types of gimbal systems 24

§ 5. Variation of the polar coordinates of a fixed point caused by

horizontal stabilization errors (analytical treatment) 30

§ 6. Geometric determination of the stabilization errors by the theory of infinitesimal rotations of a rigid body 35

§ 7. Variation of the ship’s roll and pitch angles and of its course caused by a finite rotation of the ship about an arbitrary axis 40

## Chapter II. ORIENTATION OF GYRO-CONTROLLED OBJECTS 46

§ 1. The orientation accuracy of an object launched from an

inclined base 46

§ 2. Deviation of a self- guiding missile from the specified direction during flight 55

§ 3. Some general considerations on methods for solving problems on the geometry of stabilization systems 62

§ 4. Nonholonomic motions of gyroscopic systems 67

## Chapter III. PHENOMENA CONNECTED WITH THE ELASTICITY OF GYRO-SYSTEM ELEMENTS 75

§ 1. Elastic deformations of the gyro rotor under the influence of centrifugal forces 75

§ 2. Deformation of the gyro housing 79

§ 3. The rigidity of the gimbal rings 82

§ 4. Discontinuous motion of insufficiently rigid kinematic transmissions 86

§ 5. Influence of the rigidity of the gyroscopic-system elements on the frequency of nutations 93

§ 6 The damping of gyroscopic and other devices mounted on objects moving at high accelerations 97

## Chapter IV. LINEAR THEORY OF GYROSCOPIC SYSTEMS 105

§ 1. The equations of gyroscopic systems 105

§ 2. Theory of the gyrovertical with aerodynamic suspension and its possible improvements 115

§ 3. Gyrovertical with auxiliary gyro 134

§ 4. Theory of the gyroscopic heel equalizer 148

§ 5. The gyroscopic frame 169

## Chapter V. NONLINEAR PROBLEMS IN THE THEORY OF GYROSCOPES 178

§ 1. Sliding motions of gyroscopic systems aera 178

§ 2. Energy method for investigating the stability of gyroscopic systems 192

§ 3. Forced oscillations of a gyroscopic frame (monoaxial stabilizer) 203

§ 4. Behavior of a directional gyro on a rolling base 211

## Chapter VI. VARIOUS PROBLEMS IN GYRO-SYSTEM MECHANICS 221

§ 1. Application of probability methods to determining the errors of a gyrohorizon with contact correction during rolling 221

§ 2. The effect of the ship’s yaw on the accuracy of the gyrohorizon readings 232

§ 3. The gyro Top Bow 237

§ 4. The errors of the gyroscopic apparent-velocity meter 243

§ 5. Precessional oscillations of a gyroscope acted upon by a load 245

§ 6. Influence of vibrations on the accuracy of gyroscopic instrument readings 250

§ 7. Theory of follow-up systems 251

APPENDIX I. THEORY OF COMPLEX GYROSCOPIC STABILIZATION SYSTEMS 261

APPENDIX II. THEORY OF THE GYROHORIZONCOMPASS 275

APPENDIX III. DETERMINING THE POSITION OF A MOVING OBJECT BY GYROS AND ACCELEROMETERS 288

BIBLIOGRAPHY 304

LIST OF RUSSIAN ABBREVIATIONS 311