Nonlinear excitation of hypersound vibrations in ferrite plate in conditions of combine influence in two frequencies. Part 2. Variation of constant field
Authors:   Syktyvkar State University of Sorokin, Oktyabrsky prosp. 55, Syktyvkar 167001, Russia, Vlasov V.S. , Pleshev D.A. ,   Syktyvkar State University of Sorokin, Oktyabrsky prosp. 55, Syktyvkar 167001, Russia, Shavrov V.G. ,   Kotel’nikov Institute of Radio Engineering and Electronics RAS, Mokhovaya 11-7, Moscow 125009, Russia, Shcheglov V.I. ,   Kotel’nikov Institute of Radio Engineering and Electronics RAS, Mokhovaya 11-7, Moscow 125009, Russia
Published: December 2021

The task about nonlinear excitation of hypersound vibrations in ferrite plate in conditions of combine influence in two frequencies is investigated. The thickness of plate is chosen so as the frequency of its elastic resonance was equal to difference of two components of alternating field. The most attention is given to properties of excited elastic vibrations by changing the value of applied constant field. The system of nonlinear motion equations of magnetization and elastic displacement is described. For solving of this system, the numerical method Runge-Cutta is applied. The results of this calculation are the time-evolvent of vibrations, dependencies magnetic end elastic vibrations amplitudes and also the spectra of vibrations in permanent conditions after end of relaxation processes. The influence of constant field value on character of vibrations is investigated. The comparison of amplitude-frequency characteristics of magnetic and elastic vibrations in linear and nonlinear regimes by the thickness of plate correspond to resonance on central and different frequencies is carried out. It is shown that by thickness which correspond to resonance on different frequency the characteristic has large indented skirting. It is found the large variety of dependencies of elastic vibrations character from field. In condition of stabilization so as after the completion of relaxation establishment processes it is found five more character regimes of vibrations: regime №1 – small-amplitude chaos; regime №2 – regular vibrations; regime №3 – non-symmetrical doubling of period; regime №4 – symmetrical doubling of period; regime №5 – irregular beating. For each of regimes the evolution of vibrations along time are obtained and the corresponding frequency spectra are found. The localization of regimes along the value of constant field by difference excitation level is investigated. It is found that the region of large-amplitude regimes (№2–№5) is limited in both sides by regions of small-amplitude regime №1. It is found that inside of region where large-amplitude regimes take place the most extended regime is generalized regime with doubling of period which is the sum of regimes №3 and №4 which occupies about 79%. The next on prevalence it the regime №5 – irregular beating which occupies 13% of whole region. The most exceptional is the regime №2 – regular vibrations which occupies only 8% from whole interval. It is found the extremely large critical intending of dependence of elastic vibrations amplitude from constant field. It is found that the structure of amplitude-field characteristic is very sensual to excitation level and plate thickness and the degree of sensitivity reach part of percent. From the comparison of localization on field by magnetic and elastic characteristics it is found that elastic characteristics as a whole are displaced down along field relatively magnetic characteristics. In this case the low-field falling of elastic characteristics is localized on the same falling of magnetic characteristics but the high-field falling of elastic characteristics is localized slightly lower of the field value which correspond to ferromagnetic resonance on central frequency. It is found that the reason of this displacement in this case is elastic resonance of plate on difference frequency. The card of regimes on the plane "alternating field – constant field” along broad interval of varying of both fields is constructed. It is found that by the constant field the large-amplitude regimes on the card occupy the “curved-linear trapezium” which axis lies along the coordinate “alternating field” and transverse width along coordinate “constant field” by increasing of alternating field is increased. It is found that outside of both sides by constant field of this “trapezium” the low-amplitude regime №1 – “small-amplitude chaos” is exited. The middle of “trapezium” occupies the strip directed along coordinate “alternating field” which is filled by regime №5 – “irregular beating”. Along both sides from this strip also right until the boundaries of “trapezium” the regimes №3 and №4 – “doubling of period” are excided. The reasons and necessary conditions of chaotic character of elastic vibrations are investigated. It is found that the necessary condition of chaos is excitation simultaneous in two frequencies. It is found that the large indented jumping character of dependence elastic vibrations amplitude from constant field has as first-reason the chaotic character magnetic vibrations. The character of jumps in dependence elastic vibrations amplitude from constant field is investigated. It is shown that by two-frequencies excitation the increasing of step along field causes large increasing of indenting of characteristics. It is found that the same behavior of amplitude-field dependencies of vibrations reveals its fractal character. The some comments about nature of jumps are proposed. It is found that jumps are determined by non-permanent character exactly magnetic vibrations which are shown only by two-frequency excitation on enough large level and have fractal character. As an analogy is mentioned that the line (skirting) of elastic vibrations field dependence on the plane “amplitude-field” is similar to chaotic trajectory of time-evolvent of vibrations on the plane “amplitude-time” for different oscillators which show chaotic vibrations. It is mentioned that the same behavior of trajectories on the plane “coordinate-potential” take place in cases when potential has dynamic character. The supposition is proposed that in investigated here task about two-frequency excitation of magnetostriction transducer may by constructed the function which can play the role of dynamic potential.