Stability analyses for axially moving strings in nonlinear free and aerodynamically excited vibrations

Nonlinear free vibration and stability problems are investigated for axially moving strings in transverse motions. The equation of nonlinear, free motion is derived and discretized using the Galerkin’s method. The method of multiple time scale is adopted to obtain the approximate response. It is pointed out that the motion stays stable for transportations with speed less than the linear critical speed. For taut strings that move with large transport speeds, the stability of the equilibrium configuration under steady aerodynamic excitation is explicitly analyzed. Based on the Routh–Hurwitz criterion, the condition for Hopf bifurcation is presented with multiple parameters for transverse motions perturbed in the vicinity of the equilibrium configurations.