Intrinsic Localized Modes of Harmonic Oscillations in Pendulum-Arrays Subjected to Horizontal Excitation

Abstract

The behavior of intrinsic localized modes (ILMs) is investigated for an array with N pendula which are connected with each other by weak, linear springs when the array is subjected to horizontal, sinusoidal excitation. 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 presented for N = 2 and 3 and compared with the results of the numerical simulations. Patterns of oscillations are classified according to the stable steady-state solutions of the response curves, and the patterns in which ILMs appear are discussed in detail. The influence of the connecting springs of the pendula on the appearance of ILMs is examined. Increasing the values of the connecting spring constants may affect the excitation frequency range of ILMs and cause Hopf bifurcation to occur, followed by amplitude modulated motions (AMMs) including chaotic vibrations. The influence of the imperfections of the pendula on the system response is also investigated. Bifurcation sets are calculated to examine the influence of the system parameters on the excitation frequency range of ILMs and determine the threshold value for the connecting spring constant after which ILMs do not appear. Experiments were conducted for N = 2, and the data were compared with the theoretical results in order to confirm the validity of the theoretical analysis.