Amplitude modulated dynamics and bifurcations in the resonant response of a structure with cyclic symmetry

Abstract

Periodic structures with cyclic symmetry are often used as idealized models of physical systems and one such model structure is considered. It consists ofn identical particles, arranged in a ring, interconnected by extensional springs with nonlinear stiffness characteristics, and hinged to the ground individually by nonlinear torsional springs. These cyclic structures that, in their linear approximations, are known to possess pairwise double degenerate natural frequencies with orthogonal normal modes, are studied for their forced response when nonlinearities are taken into account. The method of averaging is used to study the nonlinear interactions between the pairs of modes with identical natural frequencies. The external harmonic excitation is spatially distributed like one of the two modes and is orthogonal to the other mode. A careful bifurcation analysis of the amplitude equations is undertaken in the case of resonant forcing. The response of the structure is dependent on the amplitude of forcing, the excitation frequency, and the damping present. For sufficiently large forcing, the response does not remain restricted to the directly excited mode, as both the directly excited and the orthogonal modes participate in it. These coupled-mode responses arise due to pitchfork bifurcations from the single-mode responses and represent traveling wave solutions for the structure. Depending on the amount of damping, the coupled-mode responses can undergo Hopf bifurcations leading to complicated amplitude-modulated motions of the structure. The amplitude-modulated motions exhibit period-doubling bifurcations to chaotic amplitude-modulations, multiple chaotic attractors as well as “crisis”. The existence of chaotic amplitude dynamics is related to the presence of Sil'nikov-type conditions for the averaged equations.