Supercritical Stability of an Axially Moving Beam Part I: Model and Equilibrium Analysis

Axially moving material problems consider the dynamic response, vibration and stability of long, slender members which are in a state of translation. This study focuses on the response of axially moving beam-like elements at translation speeds that exceed the classical “critical speed stability limit”. A non-linear model for an axially moving beam is derived that accounts for the initial beam curvature induced by supporting pulleys or wheels. Presently, the model is used to determine steady responses at critical and supercritical translation speeds. The properties of the equilibrium problem are examined using an approximate linear solution and an exact, non-linear solution. The deficiency of the linear solution is illustrated by its inability to capture essential features of the equilibrium problem particularly near and above the critical speed. In this high-speed region, the translating beam undergoes large overall buckling deformations leading to multiple and bifurcated equilibrium states. The stability of the equilibria is assessed in Part II.