Three-dimensional non-linear coupling and dynamic tension in the large-amplitude free vibrations of arbitrarily sagged cables

This paper presents a model formulation capable of analyzing large-amplitude free vibrations of a suspended cable in three dimensions. The virtual work-energy functional is used to obtain the non-linear equations of three-dimensional motion. The formulation is not restricted to cables having small sag-to-span ratios, and is conveniently applied for the case of a specified end tension. The axial extensibility effect is also included in order to obtain accurate results. Based on a multi-degree-of-freedom model, numerical procedures are implemented to solve both spatial and temporal problems. Various numerical examples of arbitrarily sagged cables with large-amplitude initial conditions are carried out to highlight some outstanding features of cable non-linear dynamics by accounting also for internal resonance phenomena. Non-linear coupling between three- and two-dimensional motions, and non-linear cable tension responses are analyzed. For specific cables, modal transition phenomena taking place during in-plane vibrations and ensuing from occurrence of a dominant internal resonance are observed. When only a single mode is initiated, a higher or lower mode can be accommodated into the responses, making cable spatial shapes hybrid in some time intervals.