Modal Analysis for Localization Phenomena in Two-Pendulum Arrays Subjected to Vertical Excitation

Abstract

When an array of two identical pendula connected by a linear, torsion spring is subjected to vertical excitation, the two pendula may oscillate with different amplitudes. This is referred to as a localization phenomenon. In this paper, modal analysis is performed to investigate the cause of the phenomenon. The equations of motion for the two pendula are derived in the physical coordinate. First, in the linearized equations of motion, the natural frequencies and the corresponding modal vectors are determined. Then, the equations of motion in the physical coordinates are transformed to those in the modal coordinates. Van der Pol’s method is employed to obtain the frequency response curves for the principal parametric resonance in the modal coordinates, and the response curves for the physical coordinates are determined using the coordinate transformation. These theoretical response curves are validated by being compared to the time histories and their FFT results. The numerical results show that as the excitation frequency decreases, a branch of the response curves for the first mode undergoes a pitchfork bifurcation, which induces the second mode owing to the nonlinear coupling with the first mode, and the two modes appear simultaneously at a specific excitation frequency range. This is the reason why localization phenomena are observed in the physical coordinates.