Nonlinear oscillations of cables under harmonic loading using analytical and finite element models

In cables, near resonance time-varying loading causes large amplitude oscillations mainly involving the resonant mode. A relevant contribution of higher modes may arise as a result of nonlinear coupling phenomena. In this work, analytical and finite element models are used to study the modal interactions in both planar and spatial responses to harmonic in-plane and out-of-plane loads. The aim of the investigation is to compare the two approaches by examining the effectiveness of the analytical model in describing the response with few degrees of freedom and the ability of the nonlinear finite element procedure adopted to capture the complex features of cable dynamics, albeit limited to stable oscillation branches. To this aim a finite element procedure has been developed based on the formulation of a 3 node cable element, on a refined technique for controlling viscous damping and on a step-by-step algorithm for time integration. In parallel an analytical reduced model has been derived by discretizing the two integro-differential equations of motion in the two transverse displacement components of the cable; the eigenfunctions of the linearized equations of motion are used in expanding the solution. The analyses first explore a moderately taut cable and are then extended to the behavior of a slacker cable, in order to validate the simplifying kinematic assumptions introduced in the analytical models by comparing the obtained results with those furnished by the richer finite element models.