Spatial Discretization of Axially Moving Media Vibration Problems

Spatial discretization of axially moving media eigenvalue problems is examined from the perspectives of moving versus stationary system basis functions, configuration space versus state space form discretization, and subcritical versus supercritical speed conver-. gence. The moving string eigenfunctions, which have previously been shown to give excellent discretization convergence under certain conditions, become linearly dependent and cause numerical problems as the number of terms increases. This problem does not occur in a discretization of the state space form of the eigenvalue problem, although convergence is slower, not monotonic, and not necessarily from above. Use of the moving string basis at supercritical speeds yields strikingly poor results with either the configuration or state space discretizations. The stationary system eigenfunctions provide reliable eigenvalue predictions across the range of problems examined. Because they have known exact solutions, the moving string on elastic foundation and the traveling, tensioned beam are used as illustrative examples. Many of the findings, however, apply to more complex moving media problems, including nontrivial equilibria of nonlinear models.