The non-stationary delay of establishment nonlinear vibrations in the system of two connected oscillators. Part 2. The influence of oscillators to each other.
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, hcheglov V.I. ,   Kotelnikov Institute of Radioengineering and Electronics of Russian Academy of Sciences, Mokhovaya 11-7, Moscow 125009, Russia
Published: April 2021

This work is the continuation of investigation of non-stationary delay of establishment nonlinear vibrations in the system of two connected oscillators. The physical foundation of this system is the excitation of power hypersound in ferrite plate having magnetoelastic properties and also excitation of intensive electromagnetic vibrations in ferrite disc placed in electro-dynamic resonator. The most object of investigation in this work is the question about influence of oscillators one to another. The investigation is carried out on the basis of received in first part of this work the simplified system of two connected equations of second order. In this system the equation for first oscillator is nonlinear and the equation for second oscillator is linear. It is proposed two kinds of simplified systems: oscillatory and relaxation. The distinction between these systems consist of oscillatory and relaxation character of equation for second oscillator. It is investigated the development of vibrations in time for both systems which take place after the establishment of initial displacement of first oscillator. In both cases it is found the existence of considerable length time interval of small-amplitude delay after which the large-amplitude chaos is exited. Owing to similarity of vibrations in both systems the further investigation is made only on relaxation system as it is more simply and contains less free parameters. For the relaxation system it is investigated the dependence of frequency of own vibrations of first oscillator when the connection with second oscillator is absent from the value of initial displacement. It is shown that this dependence has linear character which correspond to the cubic structure of potential item of first equation. It is investigated the role of potential of first oscillator in formation of character of vibrations and in the first turn – the large-amplitude chaos. It is shown that in the process of vibrations the potential of first oscillator is dynamical. In this case in the both sides from zero it has two symmetrical displaced minima which alternate with each other in the cycle of vibrations of second oscillator. As a mechanism of this behaviour it is proposed the model of “jumping” potential which in the process of vibrations performs the “jumps” from one extreme position to other. It is found that first oscillator in the process of large-amplitude chaos follows to the position of minimum of dynamic potential which is determined by the displacement of second oscillator. It is found that in the time-interval of initial delay the displacement of first oscillator remains monotonous and changes its sign one-two units but never does not exceed the value of initial displacement. In this case the displacement of second oscillator also being monotonous and remain the constant sign gradually increases to the value which exceed initial displacement to one-two orders. In the end of delay the displacement of first oscillator by sudden leap increase to five orders and the displacement of second oscillator also sharp increases to 5-10 times and both oscillators changes to regime of large-amplitude chaotic vibrations. The quality picture of development of vibrations is proposed. In this picture the first oscillator has some initial displacement. Owing to the connection between oscillators this displacement is passed to the second oscillator and after this action return to the first oscillator prompting the increasing of its displacement. Further the process repeats by ringing character causing the more considerable avalanche-similar displacement of second oscillator. It is investigated the influence on the whole character of vibrations of main parameters of system so as the parameter of linear connection of second oscillator and also the parameters of potentiality (cubic nonlinearity) and nonlinear connection of first oscillator. It is shown that the increasing of linear connection of second oscillator brings to decreasing of delay time as a law of inverse ratio of this parameter. It is investigated the dependencies of chaotic amplitude for both oscillators from the coefficient of linear connection of second oscillator. It is shown that both dependencies has monotonous increasing character and in the whole investigated interval of varying of connection remain similar to each other. The influence on the character of vibrations the parameter of potentiality which correspond to cubic nonlinearity of first oscillator is investigated. It is shown that the delay time does not depend from the potentiality parameter so as this parameter ensures only the variation of “opening” the branches of potential and does not change its main structure. So when the parameter of potentiality is increasing the amplitude of chaotic vibrations of first oscillator decreases what is determined by decreasing of opening of potential branches. It is found that in this case the dependencies of chaotic amplitude of both oscillators do not similar and its relation when parameter of potentiality increases is gradually decrease. The influence on the character of vibrations the parameter of nonlinear connection of first oscillator is investigated. It is shown that by the increasing of this parameter the delay time decreases as a law of inverse ratio of this parameter. It is found the general empirical formula which describes the dependence of delay time from parameters of linear connection of second oscillator and nonlinear connection of first oscillator containing six charactering coefficients. The dependencies of chaotic amplitude of first and second oscillators from the parameter of nonlinear connection of first oscillator is investigated. It is shown that by the increasing of parameter of nonlinear connection the amplitude of chaotic vibrations of first oscillator increase similar to the dependency by the increasing the parameter of connection of second oscillator. It is found the decreasing character of dependency of chaotic amplitude from the absolute value of parameter of nonlinear connection. As a possible reason of this decreasing it is proposed the negative character of nonlinear connection parameter. It is proposed some considerations which concern to the further development of this work. As a most important task it is proposed the construction of the model which describes the duration of initial delay. It is proposed some possible directions and described preliminary considerations about constructing of this model.