SUPERCRITICAL SPEED STABILITY OF THE TRIVIAL EQUILIBRIUM OF AN AXIALLY-MOVING STRING ON AN ELASTIC FOUNDATION

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 the divergence instability of the trivial equilibrium. The elastically supported string shows unique stability behavior that is considerably different from unsupported axially-moving string. 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. Neither numerical solution nor spatial discretization, which can produce substantially incorrect results, are required. The system falls in the class of dispersive gyroscopic continua, and its behavior provides a useful comparison for general gyroscopic systems. The analytical findings also serve as a benchmark for approximate methods applied to gyroscopic continua.