On the boundary conditions for axially moving beams

Axially moving beam-typed structures are of technical importance and present in a wide class of engineering problem. As the axial speed of a beam may significantly affect the dynamic characteristics of the structure even at a low velocity, it is important to accurately predict the dynamic characteristics and stability of such structures. In most previous studies, the net energy flux through the left-end and right-end boundaries of the finite beam over two simple supports has been implicitly assumed to be zero by completely ignoring the effects of its left (incoming) and right (outgoing) semi-infinite beam parts or by applying fixed boundary conditions for its longitudinal vibration, which seems to be very non-realistic from the physical point of view. Thus, this paper investigates the effects of the continuously incoming and outgoing semi-infinite beam parts on the dynamic characteristics and stability of an axially moving beam by using the spectral element method. The spectral element model is formulated from the equations of motion derived by using the Hamiltonʹs principle extended for the systems of changing mass. It is numerically shown that the effects of the continuously incoming and outgoing semi-infinite beam parts should be taken into account for further accurate prediction of the dynamic characteristics and stability for such axially moving beams.