On Energy Estimates for Damped String-Like Equation Considering Dirichlet, Neumann and Robin Boundary Conditions

Abstract

This article provides detailed construction of energy estimates of the viscous damping aspects for axially moving string, which is modeled by a linear homogeneous sting-like equation, will be studied. The nine different boundary conditions are considered for the axially moving continua. The problem at hand describes the damped vertical vibrations of string-like equations, for example, a conveyor belt system and a band-saw blade. In this work, the velocity and coefficient of damping are kept positive and fixed. The stability of the system substantially depends upon change in boundary and subsequently boundary conditions. Also a decay in oscillatory energy is observed in all the considered cases of boundary conditions due to viscous damping. In some cases, the belt energy may increase or may decrease due to variations in different parameters . This exposes the uncertainty in these cases. Keywords: Belt Conveyor, String, Axially translating, Viscous damping