The stability of such pipes conveying fluid can be qualitatively compared with that of a column subjected to a non-conservative tangential follower force at the end, but with the additional presence of a gyroscopic effect. The equation of motion of such axially moving systems is also equivalent to that of pipes conveying fluid there, the gyroscopic effect occurs due to the axial motion of fluid along with the transverse vibration of the pipe. Moreover, the total mechanical energy of such systems is non-constant, and they are hence non-conservative. The transverse vibrations of such systems have non-constant spatial phase and any disturbances in the system travel at different speeds in the upstream and downstream directions. The presence of these acceleration terms renders the dynamics of such systems complex. This motion gives rise to terms corresponding to the centripetal and Coriolis (gyroscopic) components of acceleration in the equation of motion. Also, as such continuous structures vibrate in the transverse direction, which is superposed on a rigid body translation in the axial direction, an element on the system travels in a curved path with respect to a fixed frame of reference. A basic characteristic which distinguishes these systems from their corresponding non-travelling counterparts is their velocity-dependent natural frequencies. An unusual transition of the dynamic behaviour from the stable to the overdamped and then directly to the unstable regime is observed.Īxially moving continuous structures occur in several engineering and technological systems such as in chain and belt drives, magnetic tape, elevator cables, band saws and high-speed fibre winding. 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. 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 effect of the follower force and viscous dissipation on the eigenstructure of the system is investigated. 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 equation of motion is derived and it includes the axial variation in the tension that arises due to acceleration and the follower force. 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 transverse vibrations of an axially moving string that is subjected to a distributed follower force are examined here.
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