On the Weakly Damped Vibrations of a String Attached to a Spring Mass Dashpot System

Abstract

In this paper, we consider an initial-boundary value problem for a homogeneous string (or wave) equation. One end of the string is assumed to be fixed and the other end of the string is attached to a spring-mass-dashpot system, where the damping generated by the dashpot is assumed to be small. This problem can be regarded as a simple model describing oscillations of flexible structures such as overhead power transmission lines. A semigroup approach will be used to show the well-posedness of the problem as well as the asymptotic validity of formal approximations of the solution on long timescales. A multiple timescales perturbation method will be used to construct asymptotic approximations of the solution. Although the problem is linear the construction of these approximations is far from being elementary because of the complicated, non-classical boundary condition.