The energetics and the stability of axially moving strings undergoing planar motion

Free coupled planer vibration of an axially moving string is investigated from the point of view of energetics. The time-rate of the total mechanical energy associated with the vibration is calculated for axially accelerating strings with ends moving in a prescribed way. The result shows that the energy is not conserved for a string moving in a constant axial speed and constrained by two fixed ends. For such a string, it is proved that there exists a conserved quantity that remains a constant during the coupled planar vibration. An approximate conserved quantity is derived from the conserved quantity in the neighborhood of the straight equilibrium configuration. The approximate conserved quantity is applied to verify the Lyapunov stability of the straight equilibrium configuration.