On effects of viscous damping of harmonically varying axially moving string

In this study, the stability of an axially moving string under the influence of viscous damping in terms of transverse vibrations has been examined. Mathematically, an axially moving string can be expressed as a linear-homogenous partial differential equation with the initial and boundary conditions. The axial speed of string is taken to be time-varying, sinusoidal, and small compared to wave velocity. In order to approximate the exact solutions of the initial-boundary value problem, a Fourier-expansion method, together with the two timescales perturbation method, has been used. General resonance case and the detuning case have been studied in detail. The total energy of an infinite-dimensional coupled system has been obtained. Under certain values of the damping parameter, this total mechanical energy is either obtained to be bound or unbound. In addition, it turned out that there is a possibility of mode-truncation depending on the certain values of the damping parameter. (C) 2020 The Authors. Published by IASE.