Linear and nonlinear dynamics of a beam-cable-beam model

Many modern engineering applications can be smartly faced using structural cable elements. From the mechanical viewpoint, cables are characterized by high flexibility, extreme slenderness and low damping. Consequently, predicting, describing and mitigating different linear and nonlinear phenomena remain primary interests for the theoretical and applied research in cable dynamics. The present paper illustrates a parametric, fully analytic structural model composed by two vertical cantilever beams connected by a suspended and geometrically nonlinear Irvine cable. First, focus is made on parametrically analyzing the closed-form solution of the integral-differential eigenproblem governing the undamped small-amplitude vibrations. The analyses disclose a rich variety of linear eigenfunctions, ranging from global modes (dominated by the beam dynamics) to local modes (dominated by the cable dynamics). In certain regions of the parameter space, where a strict veering of the related nearly-resonant frequencies occurs, these eigenfunctions give rise to hybridization processes. Second, the nonlinear behaviour of the beam-cable-beam system is analyzed by means of a reduced two degrees-of-freedom model. Focus is made on the effects of the quadratic nonlinearities, which govern the onset of dynamic bifurcations in 1:2 superharmonic resonance conditions between a local and a global mode. An autoparametric excitation is found responsible for the birth of high-amplitude local vibrations, following a period-doubling bifurcation.