An analytical-numerical study of dynamic stability of an axially moving elastic web

Abstract

This paper is devoted to a dynamic stability analysis of an axially moving elastic web, modelled as a panel (a plate undergoing cylindrical deformation). The results are directly applicable also to the travelling beam. In accordance with the dynamic approach of stability analysis, the problem of harmonic vi- brations is investigated via the study of the dependences of the system’s nat- ural frequencies on the problem parameters. Analytical implicit expressions for the solution curves, with respect to problem parameters, are derived for ranges of the parameter space where the natural frequencies are real-valued, corresponding to stable vibrations. Both axially tensioned and non-tensioned travelling panels are considered. The special cases of the non-tensioned trav- elling panel, and the tensioned stationary (non-travelling) panel are also dis- cussed, and special-case solutions given. Numerical evaluation of the obtained general analytical results is discussed. Numerical examples are given for panels subjected to two different tension levels, and for the non-tensioned panel. The results allow the development of very efficient, lightweight solvers for deter- mining the natural frequencies of travelling panels and beams. The results can also be used to help locate the bifurcation points of the solution curves, corre- sponding to points where mechanical stability is lost.