Nonlinear dynamics of a parametric analytical model for beam-cable-beam structures

A parametric model is proposed to analytically describe the nonlinear dynamics of the structural system composed by two vertical cantilever beams connected by a suspended sagged cable. Focus is made on the geometric nonlinearities that characterize the boundary interactions between the linear beams and the nonlinear Irvine cable. The closed form solution of the linear eigenproblem governing the undamped small-amplitude vibrations enables-first-the clear distinction between global modes, dominated by the beam dynamics, and local modes, dominated by the cable vibrations, and-second-the parametric assessment of some parameter combinations corresponding to integer frequency ratios (1:1, 2:1) between global and local modes. Such internal resonances open the way to different phenomena of linear and nonlinear interactions, which can sustain the transfer of mechanical energy between the interacting modes and, consequently, the onset of high amplitude local oscillations. After the reduction to a single mode basis, the qualitative and quantitative relevance of the system nonlinearities is analyzed. In particular, the effects of the global, local, hybrid nature of the modal shapes on the softening/hardening behavior of the frequency response are investigated.