Linear instability mechanisms for coupled translational galloping

Abstract

A linearized coupled flexural two degree-of-freedom model, describing a lumped parameter system subjected to galloping, is analyzed. Through a perturbation approach, an approximated analytical solution for the eigenvalue problem is determined. Differently from the expressions existing in literature, the eigensolutions found here are valid both in quasi-resonant and non-resonant conditions. Discussing them allows depiction of the scenario of all the possible bifurcation mechanisms in the plane of the invariants of the aerodynamic damping matrix. In resonance conditions, both simple and double Hopf bifurcations are found, otherwise only simple Hopf bifurcations (eventually sequential) occur. In any case, both mono-modal and bimodal galloping can take place. A closed form expression for the critical velocity is derived; it coincides with the exact solution in the resonant case and presents very good agreement with the numerical solutions in quasi-resonant conditions. The critical velocities are compared with the Den Hartog velocity and the influence of the horizontal motion is thus evaluated. Numerical examples concerning technical cases highlight the accuracy of the proposed method.