Energetics and Stability of Translating Media With an Arbitrarily Varying Length

Abstract

Abstract The 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 coodinate systems is studied from the energy standpoint. The exact expressions for the rates of change of energies of the media are derived and interpreted from both the control volume and system viewpoints. The monotonic behavior of the energy of vibration of the translating media during extrusion and retraction is affected by their jerks rather than accelerations. Examples including a robotic arm through a prismatic joint and an elevator hoist cable in a high rise building validate the analysis. In particular, the results explain an inherent “unstable shortening cable behavior” in elevator industry.