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The Construction and Analysis of Mathematical Model of the Nonlinear Processes in Parametrical Resonator with Asymmetry its Internal Structure and Harmonious External Influence


L.V. Cherckesova

Researches of parametrical interactions of electromagnetic wave fields of considerable intensity with the nonlinear environment are actual today. This article is devoted to simulation, modelling and the analysis of the nonlinear processes occurring in oscillatory resonant nonlinear-parametrical zone system (pazone system, NPS) in the presence of intensive harmonious external influence, at a deviation of physical parameters of system on some quantity. The modelling is fulfilled with the assistance of nonlinear electric circuits, in which is easy to obtain, to describe and to research every possible phenomenon, conditions and processes, connected by resonance nonlinear-parametrical interactions. The general equations for the analysis of similar oscillatory systems in any arbitrary zone of instability with number n are received. Communication of the constructed model with the classical equations of oscillatory systems with parametrical excitation is demonstrated. It is established, that factors of asymmetry of system – distinction and difference in coils and nonidentity of magnetic cores – lead to expansion the area of instability that causes of increase in intensity of influence for excitation of oscillations in a concrete zone of instability. Asymmetry of nonlinear elements leads to appearance of the basic harmonic of excitation in oscillations of system and influences, both on elastic force, and on force of a friction (attenuation, dissipation of the system). Nonidentity of parameters – factor of asymmetry of system causes expansion of area of instability of oscillations and generates necessity of increase in demanded intensity of external influence for excitation of oscillations in the higher zones of instability. The decision of constructed mathematical model allows receiving amplitude, frequency and phase characteristics of oscillations of nonlinear system. The analysis of experimental curves shows that the phenomenon of sharp change and delaying of phase of parametrical oscillations takes place. At increase in intensity of influence, the system passes from one zone of instability in another. There is taking away, selecting of energy of the system and deformation of curve of oscillations that leads to change and a delay of a phase of parametrical oscillations. This result shows the facts that at energy, that taking away and selecting in oscillatory system (in the moment of phase changing) speed of increasing of reactance of system is positive. At an energy investment in oscillatory system (in the moment of phase changing) speed of increase of reactance of system is negative. Presence of asymmetry of system causes occurrence of the "not complete" period of parametrical oscillations. The more asymmetry of system, the is more shown this effect (occurrence of the "not complete" period) as there is compulsory "imposing" oscillations of a new phase and change of their period more intensively. The analysis of results shows that oscillations in the higher zones of instability have multifrequency character. Asymmetry of system causes expansion of a spectrum of oscillations, increase in amplitude of even harmonics and appearance of nonbasic components of a spectrum. In an instability zone, the basic harmonic has the maximum amplitude (energy). Results of the spent researches show coincidence of results of calculation to experiment. Some difference of settlement curves from experimental on amplitude (on intensity of attenuation in the end of the period of oscillations) is caused by that in mathematical model the active losses connected with a hysteresis and vortical currents in magnetic cores are not considered. In real resonant system under the influence of active losses in a oscillation contour fade faster. With increasing in number of a considered zone of oscillations accuracy of coincidence of results increases, that testifies to defining influence on oscillatory process of frequency of excitation. Thus, the general equations for the analysis oscillatory resonant nonlinear-parametrical zone systems are received. The equations describe the energy correlation in asymmetric system with dissipation (with attenuation, damping, or fading) in any arbitrary zone of instability of oscillations and allow estimating the influence of nonidentity parameters on amplitude-and-frequency characteristics of nonlinear system. It is necessary to notice that the similar results representing practical interest for calculations NPS, for any arbitrary zones of instability have not been received earlierю
June 24, 2020
May 29, 2020

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