Localization phenomena in annular-type oscillator arrays

Abstract

This paper investigates intrinsic localized modes (ILMs) in a system where N nonlinear oscillators are arranged in a circle and connected by weak springs when the system is subjected to sinusoidal, harmonic excitation torque. These oscillators consist of point masses at the tip of pinned, massless rigid rods, nonlinear springs, and dashpots. 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 for N=3 are shown in the cases of both soft-spring and hard-spring nonlinear arrays and compared with the results of the numerical simulations. As a result, in the case of the soft-spring nonlinear arrays, eight stable oscillation patterns may appear when the connecting spring constants are comparatively small, and ILMs appear in six patterns depending on the initial conditions and the excitation frequency. The number of the oscillation patterns decreases as the connecting spring constants are increased. Comparatively, small values of the connecting spring constants may cause Hopf bifurcation to appear followed by amplitude modulated motions (AMMs), including regular ILMs, moving ILMs, and chaotic vibrations. In the case of the hard-spring nonlinear arrays, similar phenomena are observed except for moving ILMs. ILMs do not appear when the values of connecting springs exceed specific values in the both cases.