Non-linear Galloping of Sagged Cables in 1:2 Internal Resonance

The aeroelastic behaviour of a flexible elastic suspended cable driven by mean wind speed, blowing perpendicularly to the cable's plane, is investigated. By applying the Galerkin procedure to the partial differential equations of motion and using an in-plane and an out-of-plane mode as shape functions, a two-d.o.f. model is derived. The discrete equations are coupled through quadratic and cubic terms arising both from geometric and aerodynamic effects. The associated linear frequencies are assumed to be in an almost 1:2 ratio, so that internal resonance occurs. The multiple scale perturbation method is employed to obtain a set of three amplitude modulation equations, whose coefficients depend on the mean wind speed, which is assumed as control parameter. Two perturbative solutions are developed, each based on a different assumption about the order of magnitude of the static displacements, produced by steady state wind forces. Analytical results are then compared with direct numerical integrations of discrete non-linear equations. By performing a bifurcation analysis, the existence of several equilibrium branches is proved. The relative importance of geometric and aerodynamic non-linearities is discussed through simplified models. The influence on critical and postcritical behaviour of several parameters, including geometrical cable parameters, detuning and non-symmetric flow effects, is investigated. The important role played by the steady state forces is highlighted.