VIBRATION LOCALIZATION IN DUAL-SPAN, AXIALLY MOVING BEAMS .1. FORMULATION AND RESULTS

An investigation of the vibration localization phenomenon in dual-span, axially moving beams is presented. The effects of a tension difference among the spans, also referred to as disorder, on the natural modes of free vibration are studied in terms of inter-span coupling and transport speed. The equations governing the transverse vibration of the two-span, axially moving beam are derived through Hamilton's principle and solution methods are developed. Results demonstrate that normal mode localization indeed occurs for both stationary and translating disordered two-span beams, especially for small inter-span coupling. The occurrence of localization is characterized by a peak deflection much greater in one span than in the other. In the stationary disordered case, localization becomes more pronounced as inter-span coupling decreases, i.e., as the span axial tension increases. In the axially moving disordered case, the transport speed has a significant influence on localization and, generally speaking, localization becomes stronger with increasing speed. For a moving beam with identical spans, the two loci of each pair of natural frequencies may exhibit one or more crossing(s) (depending on the value of tension) when plotted against the axial transport speed. These crossing become veerings when the beam is disordered, and localization is strongest at those speeds at which the eigenvalue veerings occur.