Supercritical stability of an axially moving beam part II: Vibration and stability analyses

Abstract

This paper focuses on the stability of axially moving beam-like materials (e.g., belts, bands, paper and webs) which translate at speeds near to and above the so-called "critical speed stability limit." In the companion paper, a theoretical model for an axially moving beam was presented which accounted for geometrically non-linear beam deflections and the initial beam curvature generated by supporting wheels and pulleys. In that paper, analysis of steady response revealed that the beam possesses multiple, non-trivial equilibrium states when translating at supercritical speeds. The equations of motion are presently linearized about these equilibria and their stability is predicted from the eigenvalue problem for free response. Asymptotic and numerical solutions to the eigenvalue problem are presented for the respective cases of small and arbitrary equilibrium curvature. The solutions illustrate that the translating beam has multiple stable equilibrium states in the supercritical speed regime. The solutions confirm that the critical speed behavior for axially moving materials is extremely sensitive to system imperfections, such as initial curvature.