Different coherent states for lightly supported coupled pendula

Abstract

In this paper we discuss and explain the phenomenon of synchronization in lightly supported mechanical systems. The investigations are focused on the models of self–excited pendula hanged on the horizontally oscillating beam, which is lightly connected with the external support. Our results are based on the Centre-of-Mass (CoM) Theorem, which can be applied to the considered systems and allows to analytically confirm the observed behaviours. We present typical dynamical solutions, including periodic and quasiperiodic oscillations, within which the oscillators synchronize. The possible synchronous configurations are analyzed and examined, depending on the number of the pendula creating the system, their parameters and the initial conditions. We discuss bifurcations between different types of solutions, determining the regions and the conditions supporting the synchronization. Our investigations exhibit, that with the increase of the size of the network, the number of co–existing attractors also increases, leading to possible multistability and new types of behaviours (e.g., the traveling phase one). The results obtained numerically match with the analytical ones obtained from the CoM Theorem, which explains the existence of particular types of dynamical configurations. The study presented in this paper involves classical lightly supported pendula systems and due to their basic character, one can expect to observe similar behaviours in other types of mechanical models.