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).