Energetics and Stability of Translating Media with an Arbitrarily Varying Length

Abstract

The linear dynamics of a class of translating media with an arbitrarily varying length is investigated. The tension in the media arising from their longitudinal accelerations is incorporated. The dynamic stability of the continuous media relative to the inertial and moving coordinate systems is studied from the energy standpoint. The exact expressions for the rates of change of energies of media are derived and interpreted from both control volume and system viewpoints. The stability analyses relative to the inertial and moving coordinate systems result in the same predictions. Examples including a robotic arm through a prismatic joint and an elevator cable in a high-rise building illustrate the analysis. In particular, the results explain an inherent “unstable shortening cable behavior” encountered in elevator industry. [S0739-3717(00)00503-1]