Abstract
The design of smart metamaterials for vibration control is usually based on the use of Bloch theorem which considers a single cell with adequate boundary conditions. These boundary conditions correspond to the infinite repetition of the unit cell in 1D, 2D or 3D. Complex geometries and composite systems can then be designed using this approach with finite elements. The control of the elastic waves can be performed by combining Bragg’s (wave interferences), resonant’s (resonance of a component embedded in the unit cell), damping and/or active control. The energy can then be reflected, transmitted, damped, focused or confined in a specific zone of the structure. However, the practical realization of real-life 2D or 3D finite systems may lead to some situations where energy transfers are not in accordance with those predicted by the infinite system considered in the design, because of reflections on the boundary conditions of the finite structure. The behavior of the system may be simulated by full system modelling, but this is time consuming and may lead to huge calculation costs. In this paper, we propose an extension of the Bloch approach to handle finite system boundary conditions in order to be able to identify situations in which energy transfer may arise because of reflections on the border of the elastic domain. Calculations are performed on 2 cells with adequate boundary conditions. The methodology is described and validated using full finite model and experimental tests on a 2D metamaterial structure.
Highlights
A periodic medium is a material or a structural system that exhibits spatial periodicity.1 The study of periodic structures has a long history in the field of vibrations and acoustics.2 This topic has interested researchers over the years, and a growing activity on this field is observed on the last years, with the objective of designing structures exhibiting properties that conventional ones cannot possess.3–5 The methods currently used are most of the time based on those derived from wave propagation in crystals,6 where almost no dissipation occurs
The metamaterial consists in an infinite periodic bidirectional waveguide
The dispersion for the 2 cells system is plotted in red, a branch appears in the bandgap and the corresponding deformed shape is visible in the figure 11b
Summary
A periodic medium is a material or a structural system that exhibits spatial periodicity. The study of periodic structures has a long history in the field of vibrations and acoustics. This topic has interested researchers over the years, and a growing activity on this field is observed on the last years, with the objective of designing structures exhibiting properties that conventional ones cannot possess. The methods currently used are most of the time based on those derived from wave propagation in crystals, where almost no dissipation occurs. A periodic medium is a material or a structural system that exhibits spatial periodicity.. The study of periodic structures has a long history in the field of vibrations and acoustics.. The study of periodic structures has a long history in the field of vibrations and acoustics.2 This topic has interested researchers over the years, and a growing activity on this field is observed on the last years, with the objective of designing structures exhibiting properties that conventional ones cannot possess.. The classical Floquet-Bloch approach is a method commonly used for the study of periodic structures. The periodicity is defined on the borders of the domain uR = e−jkxruL and vR = e−jkyrvL where uR VR) is the displacement on the right border and uL VL) is the displacement on the left border in x
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