Supercritical stability of axially moving materials.

Abstract

Axially moving material problems consider the dynamic response, vibration and stability of long, slender members which are in a state of translation. Examples of axially moving materials include magnetic tape recording devices, belt and chain drives, band saws, overhead conveyer systems, and the like. This dissertation focuses on the response of beam-like axially moving materials that translate at speeds that exceed the classical "critical speed stability limit". The steady and dynamic response of such materials are examined using both theoretical and experimental methods. A nonlinear model for an axially moving beam is derived that accounts for the initial beam curvature. The properties of the steady response are examined using a linear solution and a nonlinear solution. The deficiency of the linear solution is illustrated by its inability to capture essential features of the equilibrium problem particularly at translation speeds near and above the critical speed. In this high-speed region, the translating beam undergoes large buckling deformations leading to multiple equilibrium states. The equations of motion are linearized about these equilibria and their stability is predicted from the eigenvalue problem. The numerical solutions to the eigenvalue problem illustrate that the translating beam has multiple stable equilibrium states in the super-critical speed regime. The solutions confirm that the critical speed behavior for axially moving materials is extremely sensitive to system imperfections, such as initial curvature. The single beam model is extended to describe the nonlinear response of a continuous band circulating about two rotating wheels. For the steady band response, closed-form analysis leads to elliptic integral solutions for the equilibrium band geometry and tension which are evaluated over a wide range of band/wheel designs. The results demonstrate that, in contrast to the single beam model, the equilibrium geometry of the band/wheel system is independent of the translation speed. The equations of motion are linearized about the equilibrium solution and stability of the equilibrium is predicted from numerical analysis of the free response. The results show that the band/wheel model has a single equilibrium state that remains stable for all translation speeds. High-speed stability is investigated in the laboratory using a band/wheel system test stand. Measurements of both steady and dynamic response are in good agreement with theoretical results and confirm the theoretical predictions of band/wheel stability.