Synchronization and sommerfeld effect as typical resonant patterns

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Synchronization and Sommerfeld Effect as Typical Resonant Patterns Kovriguine D.A. Abstract This paper presents results of theoretical studies inspired by the problem of reducing the noise and vibrations by using hydraulic absorbers as dampers to dissipate the energy of oscillations in railway electric equipments. The results of experimental trials over these problem and some theoretical calculations, discussed in the text, are demonstrated the ability to customize the damping properties of hydraulic absorbers to save an electric power and protect the equipment itself due to utilizing the synchronous modes of rotation of the rotors. Key words: Synchronization; resonance, stability, rotor vibrations; dampers. Introduction The phenomenon of the phase synchronization, had being

first physically described by Huygens, was intensively studied mathematically only since the mid 20-th century, in parallel with significant advances in electronics [1-4]. Fundamental results on the synchronization in terms of the qualitative theory of differential equations and bifurcation theory prove the resonance nature of this phenomenon [5, 6]. Now the application of this theory is widely used to solve pressing practical problems in a wide range of activities, from microelectronics to power supply [7-9]. Now the research interest in advanced fields of the synchronization theory is concentrated, apparently due to the rapid development of new technologies, on studying complex systems with chaotic dynamics, discrete objects and systems with time delay variables. However, in

the traditional areas of human activity such as, for instance, energy and transport, there is also noticeable growth of attention in this phenomenon focused on the searching effective ways to save the energy and integrity of power units. Progressive developments in the scientific researches are constantly improving and expanding in our understanding over the synchronization phenomenon, as a consistent coherent dynamic process. This one occurs usually due to very small, almost imperceptible bonds between the individual elements of the system, which, nevertheless, cause a qualitative change in the dynamical behavior of the object. The basic equation of the theory of phase synchronization of a pair of oscillators or rotators reads , where is a small frequency (or angular velocity)

detuning, is the depth of the phase modulation, is the time. This one being a very simple equation has the general solution in the following form , where is an arbitrary constant of integration. From this solution follows a simple stability criterion for the stable phase synchronization: . It shows that the phase mismatch must be small, or, accordingly, the parameter of modulation must be sufficiently large, otherwise the synchronization may be destroyed. A more detailed mathematical study of this problem, referred to a two-rotor system based on an elastic base, turns out that the reduced model is incomplete. Namely, one draws some surprising attention to that the model lacks any description of that element of the system which provides the coupling between the rotors. More

detailed studies lead to the following structure of the refined model: , , where describes a measure of the amplitude of oscillations of the elastic foundation. This additional equation appears as a result of the phase modulation of the angular velocity of rotors due to the elastic vibrations of the base. So that, the perturbed rotors, in turn, cause the resonant excitation of vibrations of the base, described by the first equation. In the study of the refined model one can explain that the stable synchronization requires the same condition: . But, one more necessary condition is required, namely, the coefficient of the resonant excitation of vibrations of the base should not exceed the rate of energy dissipation , i. e. . The last restriction significantly alters the stability