## Non-Linear ElectromechanicsThis is the first book to view problems of electromechanics through the lens of analytical mechanics. It offers fundamental results in the theory of non-linear electromechanical systems and includes examples of industrial applications. |

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### Contents

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13 Stability of stationary motions of systems with quasicyclic coordinates and the mechanical equilibrium under the action of magnetic ﬁeld | 12 |

14 Motion of electromechanical systems due to constant external electromotive forces | 23 |

Motion of the conducting rigid body in alternating magnetic ﬁeld | 31 |

22 Pendulum in the alternating magnetic ﬁeld | 40 |

23 Magnetic suspension in the ﬁeld of alternating currents | 46 |

Dynamics of synchronous electric machines | 64 |

53 System with electromagnets in the differentiating circuit | 220 |

54 Interaction effects Synthesis of electromagnets for vibration excitation | 226 |

55 Generating solution for the induction ﬁeld strength nonlinearity | 229 |

56 Oscillation of the feed frequency in system with reactive electromagnet | 231 |

electromagnets Asymmetrical modes | 236 |

58 Vibrations in system with collisions Inﬂuence of mechanical nonlinearity | 241 |

Charged particle in electromagnetic ﬁeld | 253 |

62 Motion of the charged particle in highfrequency electromagnetic ﬁeld | 259 |

32 Equation of the synchronous machine in the case of a powerful network | 67 |

33 Systems with electromagnetic vibration absorbers | 79 |

34 Optimal parameters of the magnetoelectric extinguisher of small vibrations of a singledegreeoffreedom system | 85 |

35 Swinging of the rotor of synchronous machine and motion of the pendulum with magnetoelectric extinguishers | 90 |

36 Dynamics of synchronous machine that autonomously works on load | 99 |

37 Asymptotic transformation of equations when synchronous machine works on load through rectiﬁer | 109 |

38 Stationary regimes of operation of rectiﬁer working on pure active load | 112 |

39 Control of the load current | 118 |

310 Work of generator on counterelectromotive force via rectiﬁer | 120 |

311 Control of excitation of brief action turbogenerators | 121 |

312 Dynamics of two synchronous generators operating in parallel into a common activeinductive load | 130 |

313 Equation for electromechanical processes in inductor machines | 145 |

314 Some inverse problems of theory of inductor machines | 156 |

315 Singlepack and unlikepole machines machines with open slot | 157 |

316 Equation of electromechanical processes in threephase unlikepole machine | 161 |

317 Feature of transient processes for star connection with zero wire | 170 |

Periodic solutions in problems of excitation of mechanical oscillations | 175 |

42 Integral stability criterion for periodic motions of electromechanical systems and systems with quasicyclic coordinates | 182 |

43 Energy relationships for oscillations of current conductors 191 | 191 |

44 On the relationship between the resonant and nonresonant solutions | 193 |

45 Rouths equations which are linear in the positional coordinates | 197 |

Oscillations caused by electromagnets | 205 |

52 Systems with a single electromagnet | 212 |

highfrequency and constant components | 266 |

64 Relativistic charged particle in essentially heterogeneous crossed electric and magnetic ﬁelds | 271 |

65 Motion of the nonrelativistic particle in heterogeneous magnetic ﬁeld | 280 |

66 Motion of the nonrelativistic particle in perturbed axisymmetric ﬁeld | 284 |

Some problems of nonlinear magnetoelasticity | 289 |

72 Equilibrium of ferromagnetic bodies Nonlinear boundaryvalue problems | 294 |

73 Boundaryvalue problems for equilibrium of conductors with currents | 303 |

74 Equilibrium of elastically suspended rigid bodies | 308 |

75 Equilibrium of string subjected to magnet force and prescribed external load | 311 |

76 The EmdenFowler equation Reduction to autonomous system | 321 |

77 Equilibrium of ferromagnetic membrane and threads with currents | 332 |

78 On the stable equilibrium forms of conductors with currents | 336 |

Averaging quasilinear systems with several fast variables | 343 |

A2 The case when function ξτεisgivenbyan asymptotic sequence | 348 |

A3 Separating slow variables in quasilinear systems | 350 |

A4 Remarks | 357 |

Systems integrable in the ﬁrst approximation of the averaging method | 358 |

Higher approximations of the averaging method for systems with discontinuous variables | 367 |

On qualitative investigation of motion by the asymptotic methods of nonlinear mechanics | 374 |

References | 383 |

394 | |

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### Common terms and phrases

adiabatic invariant amplitude anchor angular velocity asymptotic averaged equations axis body circuit coeﬃcients component condition conductors const constant coordinates corresponding currents curve damper contour denotes dependence derivative determined diﬀerent diﬀerential dimensionless displacement electric electromagnetic electromagnetic forces electromechanical energy equal equations of motion equilibrium position excitation expression fast variables ferromagnetic ﬁnd ﬁrst approximation ﬁxed ﬂux linkage free ring frequency Hamilton function inductor inequality inﬂuence integral integral curve interval introduce Khodzhaev K.Sh Let us consider linear load magnetic ﬁeld matrix mechanical mechanical equilibrium mutual induction nonlinear obtain oscillating system particle phase ponderomotive forces potential problem quasi-cyclic quasi-periodic functions rectiﬁer relationship respect ring rotation rotor slow variables small parameter solution stability stator suﬃcient symmetric synchronous machine tion values vector velocity vibrations voltage winding zero