Modal analysis for localization of harmonic oscillations in nonlinear pendulum arrays subjected to horizontal excitation

Abstract

This study aims to clarify the reason for the occurrence of localization of harmonic oscillations when an array of N nonlinear pendulums connected by linear torsion springs is subjected to sinusoidal displacement excitation in the horizontal direction. Modal analysis is used to derive the modal equations of motion, which form an autoparametric system when identical pendulums are connected by identical torsion springs. Van der Pol’s method is applied to the modal equations of motion, and subsequently the expressions for the frequency response curves (FRCs) of the harmonic oscillations are obtained. By using the transformation equation from the modal coordinates to the physical coordinates, the FRCs in the physical coordinates are determined. The FRCs in the modal and physical coordinates for N = 2 and 3 and the backbone curves for N = 2 are shown. The numerical simulation results for N = 10 are also compared in the modal and physical coordinates. When multiple modes appear simultaneously, the summation of these modes provides localization in the physical coordinates. The vector diagram helps to visualize the mechanism of the localization. The effect of array imperfection on the FRCs is also briefly discussed. Furthermore, experimental data confirmed that when localization occurred in a two-pendulum array, the two modes appeared simultaneously.