Normal modes and boundary layers for a slender tensioned beam on a nonlinear foundation

In this paper, the nonlinear normal modes (NNMs) of a thin beam resting on a nonlinear spring bed subjected to an axial tension is studied. An energy-based method is used to obtain NNMs. In conjunction with a matched asymptotic expansion, we analyze, through simple formulas, the local effects that a small bending stiffness has on the dynamics, along with the secular effects caused by a symmetric nonlinearity. Nonlinear mode shapes are computed and compared with those of the unperturbed linear system. A double asymptotic expansion is employed to compute the boundary layers in the nonlinear mode shape due to the small bending stiffness. Satisfactory agreement between the theoretical and numerical backbone curves of the system in the frequency domain is observed.