Linear and nonlinear dynamics of suspended cable considering bending stiffness

This paper reports a systematic investigation on the linear and nonlinear dynamics of a suspended cable, taking bending stiffness into account. Firstly, the linear dynamics features, for example, eigen frequencies and modes for in-plane and out-of-plane motions, are formulated. Secondly, parametrical studies are conducted to explore the effect of bending stiffness on the natural frequencies and mode shapes of the symmetrical/antisymmetrical in-plane and out-of-plane modes. Then, the three-to-one internal resonance between the first- and third-order in-plane symmetrical modes is analyzed by applying directly the method of multiple scales dealing with the nonlinear partial differential equation and boundary conditions. Finally, the frequency-response curves and force-response curves are obtained through solving the modulation equations using the Newton-Raphson method and the pseudo-arclength scheme. The results show that the bending stiffness plays a considerable role in changing the natural frequencies and mode shapes, shifting the conditions for the occurring of nonlinear interaction, saddle-node bifurcation and Hopf bifurcation of suspended cables