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Ⅰ．Linear Vibrations 1

1．Introduction 1

2．Free vibrations 1

3．Forced vibrations 1

4．Subharmonics and ultraharmonics 7

5．Linear systems with variable coefficients 10

6．Principle of superposition for linear systems．Contrast with nonlinear systems 10

Ⅱ．Free Vibrations of Undamped Systems with Nonlinear Restoring Forces 13

1．Classification of problems 13

2．Examples of systems governed by x + f(x) = 0 14

3．Integration of the equation mx + f(x) = 0 18

4．Geometrical discussion of the energy curves in the phase plane 19

Ⅲ．Free Oscillations with Damping and the Geometry of Integral Curves 27

1．The plan of this chapter 27

A．Geometrical and Graphical Discussion of Integral Curves 29

2．Geometrical discussion of the integral curves in a special case 29

3．Liébard＇s graphical construction 31

B．A Study of Singular Points 36

4．Singular points and criteria for their classification 36

5．Special cases of dv/dx = (ax + bv)/(cx + dv) 38

6．Criteria for distinguishing the types of singularities 40

7．The index of a singularity 45

C．Applications Using the Notion of Singularities 48

8．Free oscillations without damping 48

9．Wire carrying a current and restrained by springs 50

10．Elastic stability treated dynamically 54

11．The pendulum with damping proportional to the square of the angular velocity 59

12．The pendulum with viscous damping 61

13．Description of the operation of alternating current motors 66

14．Pull-out torques of synchronous motors 70

Ⅳ．Forced Oscillations of Systems with Nonlinear Restoring Force 81

1．Introduction 81

2．Duffing＇s method for the harmonic oscillations without damping 83

3．The effect of viscous damping on the harmonic solutions 90

4．Jump phenomena 94

5．Hunting and pull out torques of synchronous motors under oscillatory loads 96

6．The perturbation method 98

7．Subharmonics response 103

8．Subharmonics with damping 107

9．The method of Rauscher 110

10．Combination tones 112

11．Stability questions 114

12．Résumé 116

Ⅴ．Self-sustained Oscillations 119

A．Free Oscillations 119

1．An electrical Problem leading to free self-sustained oscillations 119

2．Self-sustained oscillations in mechanical systems 125

3．A special case of the van der Pol equation 128

4．The basic character of self-excited oscillations 128

5．Perturbation method for the free oscillation 134

6．Relaxation oscillations 137

7．Higher order approximations for relaxation oscillations 140

B．Forced Oscillations in Self-sustained Systems 147

8．A typical physical problem 147

9．The method of van der Pol for the forced oscillations 149

10．The method of Andronow and Witt 153

11．Response curves for the harmonic oscillations 155

12．Stability of the harmonic oscillations 159

13．Nonharmonic response in general．Existence of stable combination oscillations for large detuning 163

14．Quantitative treatment of combination oscillations for large detuning 166

15．Nonexistence of combination oscillations when the detuning and the amplitude of the excitation and sufficiently small 171

16．Stability and uniqueness of the combination oscillations for large detuning 182

17．Description of the response phenomena for intermediate values of the detuning o．jump phenomena 184

18．Subharmonic response 187

Ⅵ．Hill＇s Equation and Its Application to the Study of the Stability of Nonlinear Oscillations 189

1．Mechanical and electrical problems leading to Hill＇s equation 189

2．Floquet theory for linear differential equations with periodic coefficients 193

3．The stability problem for Hill＇s equation and the Mathieu equation 198

4．The Mathieu equation 202

5．Stability of the solutions of the Mathieu equation for small values of e 208

6．Stability of the harmonic solutions of the Duffing equation 213

7．Orbital Stability of the harmonic solutions of the Duffing equation 219

Appendix Ⅰ．Mathematical Justification of the Perturbation Method 223

1．Existence of the perturbation series in general 223

2．Existence of the perturbation series in concrete cases 227

A．Free oscillations 228

B．Forced oscillations 231

Appendix Ⅱ．The Existence of Combination Oscillations 235

Appendix Ⅲ．The Existence of Limit Cycles in Free Oscillations of Self-sustained Systems 241

1．General discussion 241

2．Existence of a limit cycle 243

Appendix Ⅳ．Relaxation Oscillations of the van der Pol Equation 247

Appendix Ⅴ．The Criterion of Poincaré for Orbital Stability 253

Appendix Ⅵ．The Uniqueness of a Limit Cycle in the Free Oscillations of a Self-sustained System 259

1．General remarks 259

2．The uniqueness proof 260

Bibliography 265

Index 269