Eigenvalue and stability analysis for axially moving strings in transverse vibrations

Eigenvalue and stability problems are investigated for axially moving strings in transverse motions. The Hamilton's canonical equations are derived, and the modal functions are obtained through a symplectic eigenvalue analysis. It is pointed out that the transverse motion does not possess divergence instability at the critical speed. Eigenvalues and eigenfunctions are obtained for the string traveling with viscous damping. It is found out that the motion is asymptotically stable around the bifurcation point of eigenvalues. For nonlinear free vibrations, the Galerkin's method and the method of multiple time scale are adopted to obtain the approximate response. The stability of equilibrium is analyzed explicitly. The transverse vibration of the string subject to aerodynamic excitation is determined using the method of multiple scales. The conditions for the Hopf bifurcation and development of stable limit cycles are presented.