AMPLITUDE-MODULATED DYNAMICS AND BIFURCATIONS IN THE RESONANT RESPONSE OF A STRUCTURE WITH CYCLIC SYMMETRY

Periodic structures with cyclic symmetry are often used as idealized m odels of physical systems and one such model structure is considered. It consists of n identical particles, arranged in a ring, interconnect ed by extensional springs with nonlinear stiffness characteristics, an d hinged to the ground individually by nonlinear torsional springs. Th ese cyclic structures that, in their linear approximations, are known to possess pairwise double degenerate natural frequencies with orthogo nal normal modes, are studied for their forced response when nonlinear ities 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 dis tributed like one of the two modes and is orthogonal to the other mode . A careful bifurcation analysis of the amplitude equations is underta ken in the case of resonant forcing. The response of the structure is dependent on the amplitude of forcing, the excitation frequency, and t he damping present. For sufficiently large forcing, the response does not remain restricted to the directly excited mode, as both the direct ly excited and the orthogonal modes participate in it. These coupled-m ode responses arise due to pitchfork bifurcations from the single-mode responses and represent traveling wave solutions for the structure. D epending on the amount of damping, the coupled-mode responses can unde rgo Hopf bifurcations leading to complicated amplitude-modulated motio ns of the structure. The amplitude-modulated motions exhibit period-do ubling 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 t he averaged equations.