Internal resonance and non-linear response of a cable under periodic excitation

Abstract

The coupled non-linear equations of motion of a sagged cable in its first symmetric mode of in-plane and out-of-plane oscillations are solved by the method of multiple scales for its forced vibration response. The cases of both internal and external resonances are considered. A uniform lateral load is assumed to act along with an in-plane harmonic component, similar to a situation under vortex-induced oscillation. The quadratic non-linearity terms in the equations of motion are shown to affect the cable behaviour significantly when the in-plane frequency is about twice the out-of-plane frequency of oscillation. A stability analysis is performed on the steady state solutions. The effect of cable sag on these solutions and their stability is studied indirectly through the internal resonance parameter. The influence of the lateral load component, with and without internal resonance, on the stability regions is discussed.