Intermodal resonance of vibrating suspended cables

The weakly nonlinear free vibrations of a single suspended cable, or a coupled system of suspended cables, may be classified as gravity modes (no tension variations to leading order) and elasto-gravity modes (tension and vertical displacement equally important). It was found earlier [12] that the gravity mode (probably the most common type of vibration of relatively inelastic spans) does not exist for particular values of the problem parameter. The reason is that for these parameter values the 1st and 2nd harmonic are in resonance. The true nature of this resonance has now been established and analysed in detail by an application of the Lindstedt-Poincaré technique. The leading order 1st and 2nd harmonic can only exist together with each other. As a result, the tension, albeit of 2nd harmonic, does not vanish at leading order and the mode is not anymore truly dominated by gravity alone. The analysis is worked out here in detail for a single span. It is conjectured that designing the suspended cable with parameter values right at this resonance will delay or hinder the occurrence of galloping.