Non-Linear Electromechanics (eBook)

Foreword 6
Introduction 8
Contents 12
Description of electromechanical systems 16
1.1 Electrostatics and magnetostatics 16
1.2 The Lagrange-Maxwell equations 23
1.3 Stability of stationary motions of systems with quasi- cyclic coordinates and the mechanical equilibrium under the action of magnetic . eld 27
1.4 Motion of electromechanical systems due to constant external electromotive forces 38
Motion of the conducting rigid body in alternating magnetic . eld 45
2.1 Asymptotic transformation of the equations of motion for conducting rigid body in alternating magnetic . eld 45
2.2 Pendulum in the alternating magnetic .eld 54
2.3 Magnetic suspension in the .eld of alternating currents 60
Dynamics of synchronous electric machines 78
3.1 The idealized model of synchronous machine 78
3.2 Equation of the synchronous machine in the case of a powerful network 80
3.3 Systems with electromagnetic vibration absorbers 92
3.4 Optimal parameters of the magneto-electric extinguisher of small vibrations of a single- degree- of- freedom system 98
3.5 Swinging of the rotor of synchronous machine and motion of the pendulum with magneto- electric extinguishers 103
3.6 Dynamics of synchronous machine that autonomously works on load 112
3.7 Asymptotic transformation of equations when synchronous machine works on load through recti . er 122
3.8 Stationary regimes of operation of recti.er working on pure active load 125
3.9 Control of the load current 131
3.10 Work of generator on counter-electromotive force via recti . er 133
3.11 Control of excitation of brief action turbogenerators 134
3.12 Dynamics of two synchronous generators operating in parallel into a common active- inductive load 143
3.13 Equation for electromechanical processes in inductor machines 158
3.14 Some inverse problems of theory of inductor machines 169
3.15 Single-pack and unlike-pole machines, machines with open slot 170
3.16 Equation of electromechanical processes in three- phase unlike- pole machine 174
3.17 Feature of transient processes for star connection with zero wire 183
Periodic solutions in problems of excitation of mechanical oscillations 187
4.1 Special form of notation for equations of motion and their solutions 187
4.2 Integral stability criterion for periodic motions of electromechanical systems and systems with quasi- cyclic coordinates 194
4.3 Energy relationships for oscillations of current conductors 203
4.4 On the relationship between the resonant and non- resonant solutions 205
4.5 Routh’s equations which are linear in the positional coordinates 209
Oscillations caused by electromagnets 217
5.1 Equations for determining the constant components of magnetic . uxes 217
5.2 Systems with a single electromagnet 224
5.3 System with electromagnets in the di . erentiating circuit 232
5.4 Interaction e.ects. Synthesis of electromagnets for vibration excitation 238
5.5 Generating solution for the induction - .eld strength nonlinearity 241
5.6 Oscillation of the feed frequency in system with reactive electromagnet 243
5.7 Oscillations in systems with two electromagnets. Asymmetrical modes 248
5.8 Vibrations in system with collisions. In.uence of mechanical nonlinearity 253
Charged particle in electromagnetic . eld 265
6.1 Motion due to fast oscillating forces 265
6.2 Motion of the charged particle in high- frequency electromagnetic .eld 271
6.3 Particle in electromagnetic .eld with high- frequency and constant components 278
6.4 Relativistic charged particle in essentially heterogeneous crossed electric and magnetic . elds 283
6.5 Motion of the non-relativistic particle in heterogeneous magnetic . eld 292
6.6 Motion of the non-relativistic particle in perturbed axisymmetric . eld 296
Some problems of nonlinear magnetoelasticity 301
7.1 Statement of elastic-linear problems of nonlinear magnetoelasticity 301
7.2 Equilibrium of ferromagnetic bodies. Nonlinear boundary- value problems 306
7.3 Boundary-value problems for equilibrium of conductors with currents 315
7.4 Equilibrium of elastically suspended rigid bodies 320
7.5 Equilibrium of string subjected to magnet force and prescribed external load 323
7.6 The Emden-Fowler equation. Reduction to autonomous system 333
7.7 Equilibrium of ferromagnetic membrane and threads with currents 344
7.8 On the stable equilibrium forms of conductors with currents 348
Averaging quasilinear systems with several fast variables 355
A.1 Linear equation with coe.cients given by asymptotical series 355
A.2 The case when function .( t, e) is given by an asymptotic sequence 360
A.3 Separating slow variables in quasi-linear systems 362
A.4 Remarks 369
Systems integrable in the .rst approximation of the averaging method 370
Higher approximations of the averaging method for systems with discontinuous variables 377
On qualitative investigation of motion by the asymptotic methods of nonlinear mechanics 384
References 391
Index 402