Intrinsic Localized Modes of Harmonic Oscillations in Nonlinear Oscillator Arrays

Abstract

Intrinsic localized modes (ILMs) are investigated in an array with N Duffing oscillators that are weakly coupled with each other when each oscillator is subjected to sinusoidal excitation. The purpose of this study is to investigate the behavior of ILMs in nonlinear multi-degree-of-freedom (MDOF) systems. In the theoretical analysis, van der Pol's method is employed to determine the expressions for the frequency response curves for fundamental harmonic oscillations. In the numerical calculations, the frequency response curves are shown for N = 2 and 3 and compared with the results of the numerical simulations. Basins of attraction are shown for a two-oscillator array with hard-type nonlinearities to examine the possibility of appearance of ILMs when an oscillator is disturbed. The influences of the connecting springs for both hard- and soft-type nonlinearities on the appearance of the ILMs are examined. Increasing the values of the connecting spring constants may cause Hopf bifurcation followed by amplitude modulated motion (AMM) including chaotic vibrations. The influence of the imperfection of an oscillator is also investigated. Bifurcation sets are calculated to show the influence of the system parameters on the excitation frequency range of ILMs. Furthermore, time histories are shown for the case of N = 10, and many patterns of ILMs may appear depending on the initial conditions.

Nonlinear vibration localisation in a symmetric system of two coupled beams

“…In the free case, a nonlinear modal analysis is carried out, and we demonstrate that localised states bifurcate from the homogeneous out-of-phase mode. The results are similar to bifurcated states calculated in smooth systems, such as in chains of Duffing oscillators [12,15,19]. In the case of externally driven vibration, if the excitation is perfectly in phase, three kinds of stable response states may result.…”

Stabilization of solitons in coupled nonlinear pendulums with simultaneous external and parametric excitations

“…The frequency response was obtained by solving the derived equation of motion using Van Der Pols method. The authors showed the effect of the linear coupling spring an added imperfection on the soliton and they proved these results on an array of Duffing oscillators [9] subject to an external excitation.…”

Nonlinear normal modes and localization in two bubble oscillators

“…Herein, we particularly focus on nonlinear localization [8] of the mutual bubble interaction in which the total vibrational energy of the system is confined to some bubbles due to the nonlinearity of the bubble oscillation even though they are equally-sized and arranged in a symmetric configuration. This symmetry breaking property is one of the distinctive feature of the localized oscillation [14,15] considered in this study. This symmetrical arrangements and equal-sized assumption have been used in numerical investigation of the effects of bubble sizes and spatial arrangement on the coupled bubble dynamics [11,12,13,16].…”