New approach to beating phenomenon in coupled nonlinear oscillatory chains

Abstract

We propose an adequate analytical approach to the description of nonlinear vibration with strong energy exchange between weakly coupled oscillators. The fundamental notion of the limiting phase trajectory (LPT) corresponding to full energy transfer is introduced. In a certain sense this is alternative to the nonlinear normal mode (NNM) characterized by conservation of energy. Many well-known approximations based on NNMs turn out to be valid for the case of weak energy exchange, and the proposed approach can be used for the description of nonlinear processes with strong energy exchange between weakly coupled oscillators or oscillatory chains. Such a description is formally similar to that of a vibro-impact process and can be considered as generating such a process when dealing with a description of other processes with intensive energy transfer. We consider subsequently two weakly coupled conservative, damped and forced nonlinear oscillators as well as a system of two weakly coupled oscillatory chains. It is shown that two dynamic transitions occur in the system under consideration when the nonlinearity parameter increases. The first, well-known, transition corresponds to the bifurcation of one of the cooperative nonlinear normal modes (NNMs) and leads to its instability and the appearance of two stable normal modes with predominant excitation of one of oscillators. The second transition is characterized by the instability of the the limiting phase trajectory (LPT), corresponding to complete energy transfer between the oscillators. As a result of such a transition the complete energy transfer is broken, and energy localization on one of oscillators (whereas it was initially concentrated on it) becomes possible instead of beating. This motion with the most intensive energy transfer turns out to be as important as NNMs for the explanation of dynamic transformations and the description of closely related regimes.

Stationary and nonstationary dynamics of the system of two harmonically coupled pendulums

“…Используя дополнительный интеграл, можно редуцировать фазовое пространство системы до двумер-ного, введя относительные амплитуды и фазы колебаний маятников. Однако физически значимыми являются не абсолютные значения фаз, а только их разность [21][22][23][24][25][26][27], поэтому описание динамики системы может быть представлено в терминах «угловых» переменных θ и Δ = δ 1 − δ 2 :…”

“…Анализ устойчивости нормальных колебаний и нахожде-ние аналитических решений в окрестности положения равновесия системы двух связанных маятников с малой или произвольной линейной связью представлены в работах [21][22][23]. При наличии сильных амплитудных модуляций весьма эффективным оказывается подход, рассматривающий как нелинейные нормальные моды (ННМ), так и предельные фазовые траектории (ПФТ) [3,4,[24][25][26][27][28][29]. Так, в этом случае удается получить аналитическое описа-ние сложных нестационарных режимов в терминах негладких функций.…”

Non-stationary resonance dynamics of weakly coupled pendula

“…Therefore only isolated and predominantly numerical results were obtained in this field [11,12]. The recently developed concept of Limiting Phase Trajectories (LPTs) allowed for a systematic approach to description of non-stationary resonance regimes [13][14][15][16][17][18], including coupled pendula dynamics [19]. This concept introduces a fundamental non-stationary process of new type which corresponds to maximum possible energy exchange between the oscillators (in particular, pendula) or clusters of oscillators.…”

Localization of Low-Frequency Oscillations in Single-Walled Carbon Nanotubes

“…From the mathematical point of view, the solution like a breather exists only in the infinite systems with a continuous spectrum, while the nanoscale objects can be rather considered as finite ones. In such a case the problem of definition of nonlinear localized excitations has significant pecularities.It was recently shown [20][21][22] that the finite systems of weakly coupled oscillators exhibit strongly modulated non-stationary oscillations characterized by the maximum possible energy exchange between the groups of the oscillators or the maximum energy transfer from the external source of energy to the system [23]. The solutions describing these processes are referred to as Limiting Phase Trajectories (LPTs).…”

“…It was recently shown [20][21][22] that the finite systems of weakly coupled oscillators exhibit strongly modulated non-stationary oscillations characterized by the maximum possible energy exchange between the groups of the oscillators or the maximum energy transfer from the external source of energy to the system [23]. The solutions describing these processes are referred to as Limiting Phase Trajectories (LPTs).…”