The lumped-mass method is applied to study the propagation of elastic waves in two-dimensional binary periodic systems, i.e., periodic soft rubber/epoxy and vacuum/epoxy composites, for which the conventional methods fail or converge very slowly. A comprehensive study is performed for the two-dimensional binary locally resonant phononic crystals, which are composed of periodic soft rubber cylinders immersed in epoxy host. Numerical simulations predict that subfrequency gaps also appear because of the high contrast of mass density and elastic constant of the soft rubber. The locally resonant mechanism in forming the subfrequency gaps is thoroughly analyzed by studying the two-dimensional model and its quasi-one-dimensional mechanical analog. The rule used to judge whether a resonant mode in the phononic crystals can result in a corresponding subfrequency gap or not is found.
Flexural vibration in Timoshenko beams with periodically attached local resonators is studied theoretically and experimentally. The existence of a low frequency flexural vibration gap is indicated by the complex band structure calculated with transfer matrix theory for an infinite beam, as well as the frequency response function calculated with the finite element method for a finite Timoshenko beam with finite local resonators. This finite Timoshenko beam was manufactured and vibration experiments generated an experimental frequency response function curve showing a vibration gap as expected. The existence of low frequency gaps in Timoshenko beams with local resonators provides a method of flexural vibration control of beams.
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