A Sound Derivation of the Nonlinear Differential Equations of Motion of a Travelling String

Abstract

Axially moving materials, such as high-speed magnetic tapes, belts and band saws, have been discussed since 1897. In this paper the nonlinear differential equations, which describe the string's plane motion (lateral and longitudinal), are developed by two different methods: direct method (Newton's second law) and Hamilton's principle. The control volume method is presented briefly. The equations are stated in two different coordinates systems. Comparison between the equations developed by the different methods and coordinates systems shows that they are the same. The coupling between the lateral and longitudinal vibrations is of the second order, hence linearization (to the first order) leads to uncoupled equations.