Theory of Oscillators (eBook)

Front Cover 1
Theory of Oscillators 4
Copyright Page 5
Table of Contents 6
PREFACE TO THE SECOND RUSSIAN EDITION 14
NOTE FROM THE ENGLISH EDITOR 15
INTRODUCTION 16
Chapter I. LINEAR SYSTEMS 34
§ 1. A linear system without friction (harmonic oscillator) 34
§ 2. The concept of the phase plane. Representation on the phase plane of the totality of motions of a harmonic oscillator 37
§ 3. Stability of a state of equilibrium 44
§ 4. Linear oscillator in the presence of friction 48
§ 5. Oscillator with small mass 69
§ 6. Linear systems with "negative friction" 83
§ 7. Linear system with repulsive force 97
Chapter II. NON-LINEAR CONSERVATIVE SYSTEMS 107
§ 1. Introduction 107
§ 2. The simplest conservative system 108
§ 3. Investigation of the phase plane near states of equilibrium 112
§ 4. Investigation of the character of the motions on the whole phase plane 121
§ 5. Dependence of the behaviour of the simplest conservative system upon a parameter 131
§ 6. The equations of motion 151
§ 7. General properties of conservative systems 158
Chapter III. NON-CONSERVATIVE SYSTEMS 179
§ 1. Dissipative systems 179
§ 2. Oscillator with Coulomb friction 184
§ 3. Valve oscillator with a . characteristic 190
§4. Theory of the clock. Model with impulses 201
§ 5. Theory of the Clock. Model of a "recoil escapement" without impulses 215
§ 6. Properties of the simplest self-oscillating systems 232
§ 7. Preliminary discussion of nearly sinusoidal self-oscillations 233
Chapter IV. DYNAMIC SYSTEMS WITH A FIRST ORDER DIFFERENTIAL EQUATION 242
§ 1. Theorems of existence and uniqueness 242
§ 2. Qualitative character of the curves on the t, x plane depending on the form of the function f(x) 245
§ 3. Motion on the phase line 246
§ 4. Stability of the states of equilibrium 248
§ 5. Dependence of the character of the motions on a parameter 251
§ 6. Periodic motions 267
§ 7. Multivibrator with one RC circuit 279
Chapter V. DYNAMIC SYSTEMS OF THE SECOND ORDER 286
§ 1. Phase paths and integral curves on the phase plane 287
§ 2. Linear systems of the general type 290
§ 3. Examples of linear systems 300
§ 4. States of equilibrium and their stability 305
§ 5. Example: states of equilibrium in the circuit of a Voltaic arc 314
§ 6. Limit cycles and self-oscillations 320
§ 7. Point transformations and limit cycles 324
§ 8. Poincaré's indices 333
§ 9. Systems without closed paths 338
§ 10. The behaviour of the phase paths near infinity 357
§ 11. Estimating the position of limit cycles 365
§ 12. Approximate methods of integration 373
Chapter VI. FUNDAMENTALS OF THE QUALITATIVE THEORY OF DIFFERENTIAL EQUATIONS OF THE SECOND ORDER 384
§ 1. Introduction 384
§ 2. General theory of the behaviour of paths on the phase plane. Limit paths and their classification 386
§ 3. Qualitative features of the phase portrait on the phase plane. Singular paths 396
§ 4. Coarse systems 407
§ 5. Effect of a parameter variation on the phase portrait 438
Chapter VII. SYSTEMS WITH A CYLINDRICAL PHASE SURFACE 452
§ 1. Cylindrical phase surface 452
§ 2. Pendulum with constant torque 455
§ 3. Pendulum with constant torque. The non-conservative case 460
§ 4. Zhukovskii's problem of gliding flight 469
Chapter VIII. THEMETHOD OF THE POINT TRANSFORMATIONS IN PIECE-WISE LINEAR SYSTEMS 476
§ 1. Introduction 476
§ 2. A valve generator 479
§ 3. Valve generator (the symmetrical case) 494
§ 4. Valve generator with a Biassed . characteristic 501
§ 5. Valve generator with a two-mesh RC circuit 513
§ 6. Two-position automatic pilot for ships' controller 534
§ 7. Two-position automatic pilot with delay 550
§ 8. Relay Operated Control Systems 569
§ 9. Oscillator with square-law friction 588
§ 10. Steam-engine 592
Chapter IX. NON-LINEAR SYSTEMS WITH APPROXIMATELY SINUSOIDAL OSCILLATIONS 616
§ 1. Introduction 616
§ 2. Van der Pol's method 618
§ 3. Justification of Van der Pol's method 626
§ 4. Application of Van der Pol's method 634
§ 5. Poincaré's method of perturbations 646
§ 6. Application of Poincaré's method 656
§ 7. A valve generator with a segmented characteristic 659
§ 8. The effect of grid currents on the performance of a valve oscillator 665
§ 9. The bifurcation or branch theory for a self-oscillating system close to a linear conservative system 668
§ 10. Application of branch theory in the investigation of the modes of operation of a valve generator 670
Chapter X. DISCONTINUOUS OSCILLATIONS 678
§ 1. Introduction 678
§ 2. Small parameters and stability of states of equilibrium 682
§ 3. Small parasitic parameters and discontinuous oscillations 692
§ 4. Discontinuous oscillations in systems of the second order 703
§ 5. Multivibrator with one RC circuit 713
§ 6. Mechanical discontinuous oscillations 723
§ 7. Two electrical generators of discontinuous oscillations 729
§ 8. Frühhauf's circuit 735
§ 9. A multivibrator with an inductance in the anode circuit 745
§ 10. The "Universal" circuit 757
§ 11. The blocking oscillator 763
§ 12. Symmetrical multivibrator 783
§ 13. Symmetrical multivibrator (with grid currents) 791
Chapter XI. COMMENTS ON MORE RECENT WORKS 822
APPENDIX: BASIC THEOREMS OF THE THEORY OF DIFFERENTIAL EQUATIONS 828
REFERENCES 834
INDEX 846