Postcritical behavior of cables undergoing two simultaneous galloping modes

A nonlinear two degree-of-freedom model, describing a flexible elastic suspended cable undergoing galloping oscillations, is analyzed. By using a perturbative approach, the critical conditions occuring for different values of the aerodynamic coefficients are described. Two different type of critical conditions, corresponding to simple or double Hopf bifurcations are found. The nonlinear postcritical behavior of single taut strings in 1:1 primary internal resonance is studied through the multiple scale perturbation method. In the double Hopf bifurcation case the influence of the detuning between the critical eigenvalues on the postcritical behavior is illustrated. It is found that quasi-periodic motions, which are likely to occur in the linear field when the two critical frequencies are incommensurable, are really unstable in the nonlinear range. Therefore, the postcritical behavior of the string consists of stable periodic motions for any detuning values.