On Laplace Transform and (In) Stability of Externally Damped Axially Moving String

Abstract

This paper examines an (in) stability of an axially moving string system under the effect of external (viscous) damping. The string is taken to be fixed at both ends and general initial conditions are taken into consideration. The belt (string) speed is assumed to be non-constant harmonically varying about a relatively large means speed. The external damping is also considered to be small. Mathematically, the transverse vibrations of damped axially moving string system are modeled as second order linear homogeneous partial differential equation with variable coefficients. The approximate-analytic solution of the given initial-boundary value problem has been obtained by the application of two timescales perturbation method in conjunction of with Laplace transform method. It is found out that there are infinitely many values of resonant frequency parameter that gives rise to internal resonance in the system. However, in this study only non-resonant and the fundamental resonant cases has been studied. It turned out that the mode-response and the energy of system exhibits stability under certain values of damping parameter and mode-truncation for those parametric values is not problematic.