Limit-cycle oscillations of a heavy whirling cable subject to aerodynamic drag

In this paper a simple experimental system consisting of a length of cable, fixed to the edge of a rotating disc at its upper end, and free at its lower end or with a point mass (drogue) attached there, is described. This system exhibits a rich variety of bifurcation behaviours as the length of cable, angular speed of the fixed end, mass of the drogue and elasticity of the cable is varied. Bifurcation diagrams for the quasistationary configurations (cable shapes that appear stationary with respect to the rotating reference frame) are described. Linearized stability analyses of these quasistationary balloons are compared with solutions to the full time-dependent equations of motion. It is shown that there is an exchange of stability at the turning points of the quasi-stationary bifurcation curves, and that Hopf bifurcations occur at otherwise undistinguished points of these curves. It is shown that limit-cycle oscillations of the system occur at angular speeds corresponding to points on the bifurcations curves in the neighbourhood of the Hopf bifurcation points. These oscillations have been observed experimentally.