Energetics and Conserved Functional of Axially Moving Materials Undergoing Transverse Nonlinear Vibration

Abstract

Axially moving materials can represent many engineering devices such as power transmission belts, elevator cables, plastic films, magnetic tapes, paper sheets, textile fibers, band saws, aerial cable tramways, and crane hoist cables @1–3#. Energetics of axially moving materials is of considerable interest in the study of axially moving materials. The total mechanical energy associated with axially moving materials is not constant when the materials travel between two supports. It is a fundamental feature of free transverse vibration of axially moving materials, while the total energy is constant for an undamped non-translating string or beam. Chubachi @4# first discussed periodicity of the energy transfer in an axially moving string. Miranker @5# analyzed energetics of an axially moving string, and derived an expression for the time rate of change of the string energy. Barakat @6# considered the energetics of an axially moving beam and found that energy flux through the supports can invalidate the linear theories of both the axially moving string and beam at sufficiently high transporting speed. Tabarrok, Leech and Kim @7# showed that the total energy of a travelling beam without tension is periodic in time. Wickert and Mote @8# pointed out that Miranker’s expression represents the local rate of change only because it neglected the energy flux at the supports, and they presented the temporal variation of the total energy related to the local rate of change through the application of the onedimensional transport theorem. They also calculated the temporal variation of energy associated with modes of moving strings and beams. Renshaw @9# examined the change of the total mechanical energy of two prototypical winching problems, which provided strikingly different examples of energy flux at a fixed orifice of an axially moving system. Lee and Mote @10,11# presented a generalized treatment of energetics of translating continua, including axially moving strings and beams. They considered the case that there were nonconservative forces acting on two boundaries. Renshaw, Rahn, Wickert and Mote @12# examined the energy of axially moving strings and beams from both Lagrangian and Eulerian views. Their studies indicated that Lagrangian and Eulerian energy functionals are not conserved for axially moving continua. Zhu and Ni @13# investigated energetics of axially moving strings and beams with arbitrarily varying lengths. Although both the Eulerian and Lagrangian functionals for the total mechanical energy of axially moving materials are generally not constant, there do exist alternative functionals that are con-