Dynamics of a nonlinear periodic structure with cyclic symmetry

The free and forced motions of a nonlinear periodic structure with cyclic symmetry are studied. The structure consists of a number of identical linear flexural members coupled by means of nonlinear stiffnesses of the third degree. It is found that this system can only possess n "similar" nonlinear modes of free oscillation, and that no other modes are possible. Moreover, there exist pairs of nonlinear modes with mutually orthogonal nodal diameters having, in general, distinct "backbone" curves. A multiple-scales averaging analysis is used to study the nonlinear interaction between a pair of modes with orthogonal nodal diameters. As a result of this analysis, it is found that all pairs of nonliner modes along with all their linear combinations are orbitally unstable, and the only possible orbitally stable periodic motions are free travelling waves, that propagate through the structure in the clockwise and anti-clockwise directions. Under harmonic forcing, a bifuraction of a stable branch of forced travelling waves from a branch of forced normal mode motions is detected, and "jump" phenomena between branches of periodic solutions are observed. The analytical results are in agreement with experimental observations of an earlier work, and, in addition, are verified by numerical simulations.