Supercritical Speed Stability of an Axially Moving String on an Elastic Foundation

Abstract

Abstract The stability of an axially-moving string supported by a discrete or distributed elastic foundation is examined analytically. Particular attention is directed at the distribution of the critical speeds and identifying divergence instability of the trivial equilibrium. In contrast to the unsupported axially-moving string, the elastically supported string shows considerably different stability behavior. In particular, any elastic foundation (discrete or distributed) leads to multiple critical speeds and a single region of divergence instability above the first critical speed, whereas the unsupported string has one critical speed and is stable at all supercritical speeds. Additionally, the elastically supported string critical speeds are bounded above, and the maximum critical speed is the upper bound of the divergent speed region. The analysis draws on the self-adjoint eigenvalue problem for the critical speeds and a perturbation analysis about the critical speeds. No numerical approximation is required and the identified stability phenomena apply to general elastically supported traveling strings.