Steady-state bifurcations and dynamical stability of a heavy whirling cable acted on by aerodynamic drag

In this article the problem of whirling a heavy flexible cable subject to aerodynamic drag is considered. The top end is moved in a circular path and the bottom end is free. This simple system is found to exhibit remarkably rich bifurcation behaviour as the angular speed, air drag and radius of the circular path are varied. Bifurcation diagrams for the quasi–stationary configurations (cable shapes that appear stationary relative to a reference frame rotating with the same angular speed as the cable) are described. A linearized stability analysis of these quasi–stationary solutions is also described.It is shown that there is an exchange of stability at the turning points of the quasi–stationary bifurcation curves, and a transition to unstable behaviour at sufficiently large angular frequency. There is also unstable behaviour between undistinguished frequencies along the solution branches, the size and number of these regions of instability on each branch increasing as the circular path radius is increased. It is demonstrated that these unstable regions are removed for sufficiently large air drag.