A continuum model is presented that describes the three dimensional nonlinear response of an elastic cable excited by small tangential oscillations of one support. An asymptotic form of the model is derived for suspensions with small equilibirium curvature and coupled inplane/out- of-plane nonlinear oscillations are captured through a two degree-of-freedom approximation. A Perturbation analysis of the discrete model reveals the key roles played by quadratic nonlinearities and internal resonances. It is noted that the quadratic nonlinearities may excite a two-to-one internal resonance whenever the natural frequency of the in-plane mode is approximately twice that of the out-of-plane mode. The perturbation analysis is used to determine the existence and stability of periodic motions which develop for conditions near parametric and external resonance.
Cable are versatile structural elements capable of transmitting forces, carrying payloads and conducting signals across large distances. In offshore and ocean engineering applications, cables are used in the mooring of ships, platforms, buoys, and instruments, as tethers and umbilicals, and as optical and electrical signal transmission lines. Being both lightweight and flexible, cables are easily excited into sustained oscillations by relatively small forces. In applications, such oscillations may increase cable drag forces, promote cable fatigue and degrade the performance and positioning of any attached instruments. The study of cable dynamics has enjoyed a long and rich history (Irvine, 1981) and key results of the linear theory of suspended cables are reviewed by Irvine and Caughey (1974) and Triantafyllou (1984, 1987). In the linear theory, the three-dimensional response of a planar suspended cable is the Super-position of two decoupled motions: one lying in the plane defined by the cable equilibrium and one normal to this plane. For geometrically nonlinear cable response, finite cable stretching may couple in-plane and out-of-plane motions (Luongo et al., 1984).