Equilibrium Solutions and Existence for Traveling, Arbitrarily Sagged Elastic Cables

The exact, closed-form, three-dimensional solutions for the steady motion of traveling, sagged, elastic cables under arbitrarily distributed and concentrated loading are developed in this paper. Three components of displacement describing two equilibrium states of an extensible traveling elastic cable are derived. These exact solutions apply to straight and sagged cables traveling under their own weight and uniformly distributed loading. The exact solutions are also used to investigate the steady motion of three-dimensional traveling cables under the uniformly distributed and concentrated loading. Traveling elastic cables with large sag can be modeled approximately through the inextensible cable model when both the loading and the translation speed are very small. A slightly sagged cable must be modeled as extensible, rather than inextensible, even though both the loading and transport speed are very small. These solutions can be applied to multispan cable structures.