The non-stationary delay of establishment nonlinear vibrations in the system of two connected oscillators. Part 1. General situation. The formation of reduced system.
Authors:   Syktyvkar State University, Oktyabrskiy prosp. 55, Syktyvkar 167001, Russia, Ivanov A.P. , Shavrov V.G. ,   Kotelnikov Institute of Radioengineering and Electronics of Russian Academy of Sciences, Mokhovaya 11-7, Moscow 125009, Russia, Shcheglov V.I. ,   Kotelnikov Institute of Radioengineering and Electronics of Russian Academy of Sciences, Mokhovaya 11-7, Moscow 125009, Russia
Published: April 2021

The effect of non-stationary delay excitation of large-amplitude chaotic vibrations in the system of two connected nonlinear and linear oscillators is investigated. The short description of two real physical systems which allow the excitation of chaotic vibrations having non-stationary delay is described. The first system consist of normal magnetized ferrite plate having magnetoelastic properties. The second system consist of electro-dynamic resonator with normal magnetized ferrite disc. When the input power exceeds the threshold level the vibrations in both systems acquire chaotic character. It is shown that the vibrations in both systems may be described on the basis of the same model consist of two connected oscillators. The first of these oscillators is nonlinear and the second is linear. For this model the system of two second order differential equations is proposed. The nonlinear properties in these equations are described by the expansion of power row by two variable quantities. The first equation contains the cube nonlinearity and linear and nonlinear connection with second equation. The second equation contains only linear connection with first equation. The excitation of vibrations is realized by the switching the external sinusoidal force on the first oscillator. It is shown that in this system by sufficient level of external signal it is excited the chaotic vibrations having large amplitude. The beginning of this vibrations excitation is occurred by jump after the some delay time after the beginning of excitation force action. By the reason of very high complexity of initial system the main task of this paper is the signing out of whole system the maximum simplify core which preserves the properties of high-amplitude chaotic vibrations with corresponding delay time. This signing is carried out by exclusion of secondary components of both equations. As the first step of simplification it is investigated the possibility of exclusion of external excitation of system. In this case the excitation of vibrations is realized by assignment of initial displacement or initial velocity of anyone oscillator. For the realization of non-subside vibrations the dissipative components in both equations are equated to zero. It is shown that by proper choice of initial displacement the character of vibrations every oscillators as a whole is similar to its for the system with external excitation. The comparison of dependencies of delay time from the level of external excitation and initial displacement is executed. It is shown that in both cases the dependencies on the whole are in geometry similar and are described by the low of inverse proportion. The character of vibrations in both sides away the interval of delay realization is investigated. It is shown that in both cases of external excitation and initial displacement the character of vibrations remains the same. As a result of executed comparison it is made the intermediate conclusion about equivalency of excitation system from two oscillators also as external excitation so as initial displacement. It is established that the reason of equivalency is the identity of dynamic potential in both cases. With the purpose of determination of possibility to further simplification of connected equation system it is investigated the role of correlation between own frequencies both oscillators. It is shown that the excitation of high-amplitude chaos with corresponding delay by proper level of initial displacement is realized in continuous diapason of correlation both oscillators right until five relative units. It is established that the dependence of threshold level of initial displacement from ratio of frequency of second oscillator to the frequency of first oscillator is continuous and has clear determined increase quadratic character. The dependence of delay time from initial displacement by different correlations of both frequencies is investigated. It is shown that over anyone values of this correlation the delay time by increasing of initial displacement beginning from appointed critical meaning decreases by the low of inverse proportion. The critical meaning of displacement along the increasing of frequency correlation is also increased. With the purpose of determination of possibility to further simplification of system it is investigated the role of item of first equation which determines the linear connection of first oscillator with second oscillator. It is shown that the exclusion of linear connection by small correlation ratio of frequencies determines the decreasing of critical meaning of displacement and by large ratio of frequencies maintains former. In all of these cases the high-amplitude chaos is remained and delay time may reach very large values. With the purpose of further simplification of system it is investigated the role of own frequency of first oscillator. The dependence of delay time from initial displacement by equality to zero of first oscillator frequency and different frequencies of second oscillator are investigated. It is shown that in this case the critical meaning of displacement decreases more when the frequency of second oscillator decreases. In all of these cases when the item which correspond to frequency of first oscillator is absent the general character of high-amplitude chaos and corresponding delay is conserved. In the course of examination it was established that the exclusion of other items from the first equation break the possibility of realization of high-amplitude chaos with corresponding delay. The exclusion of potential item from second equation break its vibration character so it is not admissible. So in present stage of investigation the further simplification of system is determined as inexpedient. As a result of carried out the investigation from initial whole system of equation for two oscillators it is selected the core which is responsible for excitation of high-amplitude chaos with corresponding delay. The equation for displacement of first oscillator contains the second derivative by time, the item with cube nonlinearity and item of nonlinear connection of first oscillator with second oscillator. The equation for displacement of second oscillator contains the second derivative by time, the potential item which correspond to own frequency of second oscillator and item of linear connection of second oscillator with first oscillator. In this case the excitation of system may be realized by assignment of initial displacement or initial velocity of anyone oscillator.