Using Floquet-Bloch Theory to Research Vibration Characteristics of Composite Material with Trusslike Periodic Micro-structures

摘要

Trusslike periodic structures have been considered as a promising alternative to lightweight materials.With careful design of rod size and truss topology within the trusslike periodic unit cell, the periodic structures can also be used to control propagation of elastic waves due to their well-known band gap property. Therefore, this kind of lightweight materials can be designed not only to bear a heavy load but also to insulate vibration.Here, we address the vibration characteristics of composite materials with trusslike periodic micro-structures, i.e.micro parallelepiped unit cells.In order to enhance the effect of insulation of vibration, a kind of high damping polymer materials is filled into the trusslike periodic structures, in this way the whole trusslike periodic structures will present structure damping characteristics, which will result in temporal and spatial attenuations of the elastic waves as they propagate through the trusslike periodic structures.It should be pointed out that the computational method for analyzing the fundamental wave propagation characteristics in undamped trusslike periodic structures have been well-documented by using Floquet-Bloch theory, meanwhile, the analysis method of damped Bloch waves in periodic elastic materials has also been studied recently, which mainly considers temporal attenuations of the elastic waves regardless of spatial attenuations of the elastic waves.Focusing on the insulation of vibration, we present an algorithm for analyzing damped Bloch waves in trusslike periodic structures filled with a high damping polymer material, which mainly considers spatial attenuations of the elastic waves by extending the wave number vector from real numbers to complex numbers.As a result, the algorithm needs to iteratively solve the eigenvalue problem for natural frequencies of unit cells to find the damping coefficients or attenuation coefficients corresponding to each Bloch mode.Meanwhile, for the sake of improving the damping coefficients, drawing inspiration from a mechanical vibration absorber, we embed a heavy sphere into the center of micro parallelepiped unit cell.According to our previous studies, it has been proved that embedding a heavy sphere into the unit cells can greatly improve the damping characteristics of trusslike periodic structures. The paper aims at researching the effect of different parameters of the embedded spheres, including dimension parameters and material parameters, on the vibration characteristics of trusslike periodic structures. The results of numerical examples show when the radius and density of spheres are chosen properly, the band-gap starting and cut-off frequency can drop sharply, the band-gap range can become wider, and the effect of vibration alleviation can be improved much better within a given frequency range.