The non-stationary delay of establishment nonlinear vibrations in the system of two connected oscillators. Part 3. Determinating equation.
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

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 task 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 foundation of this work is received in the first part of this work the simplified system of two connected differential second order equations having vibration character. As a main step of simplification of the task it is made the substitution of the vibration of second oscillator by auxiliary function having harmonic dependency from the time. In this case the dynamic potential acquired the character of periodic “jumps” in forward and backward directions. By this circumstance the introduced function was named as “jumping”. As a result of introduction of jumping function it was the replacement of two equations of initial system by only one “determination” equation which maintains the properties of the large-amplitude chaos and non-stationary delay. On the basis of determination equation it is investigated the development of vibrations and founded its derivative in the time and the parametrical portrait is constructed. It is shown that the development of vibrations feels large delay in time which value surpass the period of jumping function about some times. When the delay is over it is developed by the sudden jump the non-stationary vibrations having the amplitude which surpass the initial displacement more then three orders. The parametrical portrait has the form of horizontal eight which is typical for chaotic vibrations. The spectrum composition of large-amplitude chaos is investigated. It is shown that in the whole spectrum is predominated the line which is correspond to frequency of jumping function. When this function is removed it is remained only clean chaos. The width of spectrum of this chaos exceeds the frequency of jumping function on two orders. The influence of initial displacement on delay time is investigated. It is shown that the delay time is depended on the value of initial displacement in the frame of inverse proportion law. The influence of jumping function amplitude on the delay time is investigated. It is found the critical value of this amplitude. It is shown that in the lower level of critical value the large-amplitude chaos is absent. It is shown that when the amplitude of jumping function increases the delay time is decreases. But on the general smooth dependence it is displaced some sharp maxima which correspond to sharp increasing of delay time on the value about one order and more. As a possible explanation of this dependence character it is proposed the hypothesis about its resonance character. The influence of jumping function frequency on the delay time is investigated. It is shown that when this frequency is increased the delay time also increases right until its critical value. Above the level of critical value the large-amplitude chaos is absent. It is noted the analogy of this phenomenon with the excitation of electric vibration circuit by sinusoidal force. In this case on the low frequency the current in the circuit vibrates in the same phase with excitation force and after the overcoming the resonance frequency the vibrations becomes in opposite phase. On the received dependence it if found some narrow maxima which corroborate the hypothesis about its resonance character. It is found that the determination equation contains inertial and potential parts for the characteristics of which is introduced the inertial and potential parameters as a coefficients by second derivative and item of cubic nonlinearity. The influence of inertial parameter on the delay time is investigated. It is found the critical value of this parameter lower of which the large-amplitude chaos is absent. The influence of potential parameter on the delay time is investigated. It is found that the excitation of large-amplitude chaos is possible along unlimited range of this parameter without some critical value. The influence of phase displacement of jumping function in comparison with initial moment of excitation on the delay time is investigated. It is found very strong dependence which is differed by small and large delay times. The small value of delay time is characterized by absence of sharp maxima on the dependence of delay time from initial displacement. It is shown that in this case the dependence of delay time from of phase displacement has very wide maximum when the dependence of jumping function is negative. In this region the large-amplitude chaos is absent. It is found that by large delay times so as in region of sharp maxima on the dependence of delay time from initial displacement the influence of phase displacement of jumping function is sharp increased. It is proposed some comments about possible development of this work. As a general tasks it is mentioned the construction of model idea about duration of delay time, more detailed clarification of nature of critical character of some parameters and more detailed investigation of phase character of described phenomena.