Stability transitions of an axially moving string subjected to a distributed follower force

The transverse vibrations of an axially moving string that is subjected to a distributed follower force are examined here. This model provides an insight into the complex dynamics of seemingly simpler systems such as silicon wafer cutting using wire saws, and aerial or marine towing, where a relatively long flexible structure is dragged through fluid. The equation of motion is derived and it includes the axial variation in the tension that arises due to acceleration and the follower force. As the exact analytical solution of this equation is difficult to determine, the approximate closed-form modal solution of a non-travelling counterpart of the system is obtained using the asymptotic technique, which is then used as a basis to obtain the numerical solution for the axially moving string. The effect of the follower force and viscous dissipation on the eigenstructure of the system is investigated. Mathematical operations such as the Hermite form and the Routh-Hurwitz criterion are applied to the characteristic polynomial to investigate the dynamic behaviour of these modes. The semi-analytical approach presented explains the 'mathematical' instability (in the absence of damping) that arises when both axial transport and follower force are simultaneously present. An unusual transition of the dynamic behaviour from the stable to the overdamped and then directly to the unstable regime is observed