Vibration localization in dual-span, axially moving beams: Part II: Perturbation analysis

Abstract

An analytical study of the effect of small irregularities on the dynamics of nearly periodic, dual-span, stationary and axially moving beams with cyclic symmetry is presented. Emphasis is placed on utilizing a perturbation approach to extend and interpret the numerical results obtained in the companion paper (Part I). The perturbation approach provides additional insights into the phenomenon of vibration localization, i.e., into the sensitivity of the system dynamics to the tension disorder, the inter-span coupling, as well as the transport speed. Analytical perturbation results are obtained in the limiting cases of weak and strong inter-span coupling. Findings show that the degree of vibration localization depends primarily upon the relative strengths of disorder and inter-span coupling; namely, for strong inter-span coupling (i.e., for small axial band tension) the disordered system behavior is well predicted as a perturbation of that of the corresponding ordered coupled system, in which case a small tension disorder has a minimal effect on the dynamic response. On the other hand, for weak inter-span coupling (i.e., for large axial band tension), the disordered system behavior is well predicted as a perturbation of that of the disordered decoupled system, in which case the natural modes of the coupled system undergo large changes for small tension disorder.