Nonlinear Vibrations in Mechanical and Electrical SystemsPDF|Epub|txt|kindle电子书版本网盘下载

Nonlinear Vibrations in Mechanical and Electrical SystemsPDF格式电子书版下载

注意：本站所有压缩包均有解压码： 点击下载压缩包解压工具

Ⅰ．Linear Vibrations1

1．Introduction1

2．Free vibrations1

3．Forced vibrations1

4．Subharmonics and ultraharmonics7

5．Linear systems with variable coefficients10

6．Principle of superposition for linear systems．Contrast with nonlinear systems10

Ⅱ．Free Vibrations of Undamped Systems with Nonlinear Restoring Forces13

1．Classification of problems13

2．Examples of systems governed by x + f(x) = 014

3．Integration of the equation mx + f(x) = 018

4．Geometrical discussion of the energy curves in the phase plane19

Ⅲ．Free Oscillations with Damping and the Geometry of Integral Curves27

1．The plan of this chapter27

A．Geometrical and Graphical Discussion of Integral Curves29

2．Geometrical discussion of the integral curves in a special case29

3．Liébard＇s graphical construction31

B．A Study of Singular Points36

4．Singular points and criteria for their classification36

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

6．Criteria for distinguishing the types of singularities40

7．The index of a singularity45

C．Applications Using the Notion of Singularities48

8．Free oscillations without damping48

9．Wire carrying a current and restrained by springs50

10．Elastic stability treated dynamically54

11．The pendulum with damping proportional to the square of the angular velocity59

12．The pendulum with viscous damping61

13．Description of the operation of alternating current motors66

14．Pull-out torques of synchronous motors70

Ⅳ．Forced Oscillations of Systems with Nonlinear Restoring Force81

1．Introduction81

2．Duffing＇s method for the harmonic oscillations without damping83

3．The effect of viscous damping on the harmonic solutions90

4．Jump phenomena94

5．Hunting and pull out torques of synchronous motors under oscillatory loads96

6．The perturbation method98

7．Subharmonics response103

8．Subharmonics with damping107

9．The method of Rauscher110

10．Combination tones112

11．Stability questions114

12．Résumé116

Ⅴ．Self-sustained Oscillations119

A．Free Oscillations119

1．An electrical Problem leading to free self-sustained oscillations119

2．Self-sustained oscillations in mechanical systems125

3．A special case of the van der Pol equation128

4．The basic character of self-excited oscillations128

5．Perturbation method for the free oscillation134

6．Relaxation oscillations137

7．Higher order approximations for relaxation oscillations140

B．Forced Oscillations in Self-sustained Systems147

8．A typical physical problem147

9．The method of van der Pol for the forced oscillations149

10．The method of Andronow and Witt153

11．Response curves for the harmonic oscillations155

12．Stability of the harmonic oscillations159

13．Nonharmonic response in general．Existence of stable combination oscillations for large detuning163

14．Quantitative treatment of combination oscillations for large detuning166

15．Nonexistence of combination oscillations when the detuning and the amplitude of the excitation and sufficiently small171

16．Stability and uniqueness of the combination oscillations for large detuning182

17．Description of the response phenomena for intermediate values of the detuning o．jump phenomena184

18．Subharmonic response187

Ⅵ．Hill＇s Equation and Its Application to the Study of the Stability of Nonlinear Oscillations189

1．Mechanical and electrical problems leading to Hill＇s equation189

2．Floquet theory for linear differential equations with periodic coefficients193

3．The stability problem for Hill＇s equation and the Mathieu equation198

4．The Mathieu equation202

5．Stability of the solutions of the Mathieu equation for small values of e208

6．Stability of the harmonic solutions of the Duffing equation213

7．Orbital Stability of the harmonic solutions of the Duffing equation219

Appendix Ⅰ．Mathematical Justification of the Perturbation Method223

1．Existence of the perturbation series in general223

2．Existence of the perturbation series in concrete cases227

A．Free oscillations228

B．Forced oscillations231

Appendix Ⅱ．The Existence of Combination Oscillations235

Appendix Ⅲ．The Existence of Limit Cycles in Free Oscillations of Self-sustained Systems241

1．General discussion241

2．Existence of a limit cycle243

Appendix Ⅳ．Relaxation Oscillations of the van der Pol Equation247

Appendix Ⅴ．The Criterion of Poincaré for Orbital Stability253

Appendix Ⅵ．The Uniqueness of a Limit Cycle in the Free Oscillations of a Self-sustained System259

1．General remarks259

2．The uniqueness proof260

Bibliography265

Index269