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.

Dynamics and Stability of a Travelling String Subjected to an Axial Follower Force

The transverse vibrations of an axially moving string are investigated when it is subjected to a distributed axial follower force. The key characteristic of such a force is that it is, at every location, always oriented parallel to the instantaneous slope of the string at that location. First, the equation of motion governing the transverse vibration under the action of such an axial distributed follower force is derived. The model accounts for the effects of both time-varying travelling speed and axial variation in the string’s tension. The stable and unstable regimes due to parametric excitation are obtained using Floquet analysis and compared with a commonly-used model. The effect of the follower force on stability is investigated. The forced response of the string that is moving at constant speed is obtained where the excitation arises due to transverse oscillations of the support(s).

Stability of Cracked Fluid-Conveying Pipeline on Elastic Foundation under Distributed Follower Forces

The differential equation of fluid-conveying pipes considering distributed follower force and elastic foundation is established. The equation is discreted and solved by Galerkin method and the frequency characteristic values are solved by bending moment transfer method. The effects of crack location and elastic foundation stiffness to the form of instability of the pipes under the distributed follower force are analyzed. Results show that the elastic foundation stiffness can enforce the stability of the pipes effectively, and the effects are more obvious when the crack location is closer to the middle of the pipe.

The Mode and Nature Frequency of Pipes Conveying Fluid with Distributed Follower Force on Two-Parameter Foundation

The vibration differential equation of pipes conveying fluid with distributed follower force on two-parameter foundation is derived, the mode and nature frequency of clamped-clamped pipes with distributed follower force are calculated by complex modal. The relations between the nature frequency and the velocity of foundation liner stiffness and shear stiffness and mass ratio are studied. The numerical calculation shows that these parameters have a great influence to the nature frequency of the system and have certain effects to the characters vibrations of pipes.