Modal interactions in the nonlinear dynamics of a beam–cable–beam

Quadratic and cubic modal interactions characterize the geometrically nonlinear dynamics of a parametric analytical model composed by two cantilever beams connected by a suspended shallow cable. The natural frequencies and modes of the linearized model are determined exactly, by solving the integral–differential eigenproblem governing the undamped free oscillations. Interesting phenomena of linear cable–beam interaction (frequency veering and modal hybridization) are recognized in the spectrum. Global and local modes are distinguished by virtue of the two localization factors measuring the modal kinetic energy stored in the beams and cable, respectively. The localization level is also put in relation to the magnitude of the quadratic and cubic nonlinearities. Therefore, the exact linear eigensolution is employed to formulate a nonlinearly coupled two-degrees-of-freedom model, defined in the reduced space of the modal amplitudes corresponding to a global and a local mode. The modal interactions between the two modes are analyzed, with focus on the autoparametric excitation mechanisms that can be favored by the occurrence of integer frequency ratios (1:2 and 2:1). Such internal resonance conditions enable significant transfers of mechanical energy—essentially governed by the quadratic coupling terms—from the small amplitudes of the externally excited global mode to the high amplitudes of the autoparametrically excited local mode. Different regimes of periodic and quasi-periodic oscillations are identified.