Forced vibrations in the system of two connected oscillators in conditions of cubic nonlinearity and quadratic connection. Abstract

Zhurnal Radioelektroniki - Journal of Radio Electronics. eISSN 1684-1719. 2020. No. 8
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DOI https://doi.org/10.30898/1684-1719.2020.8.7

UDC 537.874; 537.624

FORCED VIBRATIONS IN THE SYSTEM OF TWO CONNECTED OSCILLATORS IN CONDITIONS OF CUBIC NONLINEARITY AND QUADRATIC CONNECTION

A. P. Ivanov¹, V. G. Shavrov², V. I. Shcheglov²

¹Syktyvkar State University, Oktyabrskiy prosp. 55, Syktyvkar 167001, Russia

²Kotelnikov Institute of Radioengineering and Electronics of Russian Academy of Sciences, Mokhovaya 11-7, Moscow 125009, Russia

The paper is received on July 31, 2020

Abstract. The auto-modulation and chaotic vibrations in the system of two connected oscillators stimulated by external periodic force are investigated. It is supposed that one of oscillators has cubic nonlinearity and the connection between oscillators is quadratic. Two examples of tasks which lead to this model are described. The first is the task about excitation of power hypersound in ferrite plate having magnetoelastic properties. The second is the task about excitation of electromagnetic vibrations in ferrite disc placed in electro-dynamic resonator. For both tasks the main motion equations are described and it is shown that both systems came to the same simplified equations system for two connected oscillators. We found the system from two reduced equations vibration type. The first of these equations corresponds to magnetic oscillator and the second equation corresponds to elastic or electro-dynamic oscillator. It is shown that the first equation of this system contains the nonlinearity of third degree but the second remains linear. The connection between equations in the first equation is quadratic and in the second is linear. We investigated the behaviour of system in broad interval of excitation amplitude. We found the developments of vibrations in time, parametric portraits for variable quantity and its derivative and also the spectral characteristics of excited vibrations. On an example of middle level of excitation we investigated the typical variety of vibrations. The parametrical portrait for first oscillator has the appearance of very spread “horizontal figure-of-eight” and for second oscillator has the appearance of combination from several rings which are applied one to another with small displacement. The spectral composition of excited vibrations is investigated. It is shown that for the first oscillator the spectrum of both displacement and its derivative has the form of several (two or three) wide, very indented ā stripes with a pronounced maximum in the middle of the stripe. For the second oscillator, the vibration spectrum has a well-ordered form near the main frequency of excitation. The classification of the main regimes of oscillations observed in a wide range of excitation amplitudes is performed. We found two main regimes: regime ¹1 – large-harmonic regular; regime ¹2 – large-harmonic quasi-chaotic. It is shown that as the amplitude of excitation increases, starting from the regular one, there is a rotation of the regular and chaotic regimes.

A model imitation of parametrical portrait formation is performed. It is shown that the formation of the “horizontal eight” of the parametric portrait is due to the presence of high-order odd harmonics in the oscillation spectrum. We carried out an analytical investigation of free oscillations, described by an equation containing the second derivative in combination with a variable in the third power, which describes cubic nonlinearity. It is shown that the frequency of free vibrations of this system is proportional to their amplitude specified by the initial displacement. The regions of the formation of a chaotic regime for various levels of excitation are considered for three main cases: the first is the absence of a connection between the oscillators, the second is the presence of a linear connection, and the third is the presence of a quadratic connection. It is shown that in the absence of connection, the regions of chaotic and regular regimes rotate in an almost correct order. An increase in the level of excitation leads to a gradual increase in the numbers of excited harmonics. In the case of a linear connection, the character of the rotation of the regions of both regimes, in general, doesn’t change, however, the whole picture is stretched towards higher values of excitation. In the case of a quadratic connection, the rotation is not observed. Starting from a rather low level of excitation, the oscillations become chaotic and regularization does not occur further. The criterion of chaotic regime forming s found. It is shown that the main reason of chaos formation is successive change of harmonics when the excitation level is increased. We investigated the amplitude of vibrations by different level of excitation. It is shown that as the level of excitation increases, the amplitude of the oscillations increases, and the increase in the amplitude follows the dependence of the root type. Using the equation for the first oscillator in the absence of coupling as an example, it is shown that the reason for the slowdown in the growth of the oscillation amplitude is cubic nonlinearity.

Key words: nonlinear vibrations, connected oscillators, cubic nonlinearity.

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For citation:

Ivanov A.P. Shavrov V.G., Shcheglov V.I. Forced vibrations in the system of two connected oscillators in conditions of cubic nonlinearity and quadratic connection. Zhurnal Radioelektroniki - Journal of Radio Electronics. 2020. No.8. https://doi.org/10.30898/1684-1719.2020.8.7 (In Russian)