Analytical solutions for planar simultaneous resonances of suspended cables involving two external periodic excitations

In the present study, nonlinear vibration behavior of suspended cables involving two external periodic excitations is investigated, and three simultaneous resonance cases are considered. Firstly, by applying the Hamilton’s variance principle, the partial differential equations (PDEs) of the in-plane and out-of-plane motions are derived, and the Galerkin method is applied to discretize PDEs of the planar motion. Then, considering the single-mode discretization only, the analytical solutions for three simultaneous resonances are obtained by using the multiple scales method. Frequency response equations are derived, and the stability analyses are determined via investigating the character of the singular points of the system. Moreover, the perturbation solutions are verified by the numerical integrations of the original equations, and acceptable agreements between the analytical solutions and the numerical ones are observed in these three cases. Through examining frequency response curves, detuning phase curves, time domain response curves, phase plane diagrams and Poincaré sections, the nonlinear vibration behavior of simultaneous resonances is illustrated. Parametric investigations of the suspended cable with different sag-to-span ratios, damping ratios and excitation phases are presented.