[论文解读] Active elastic metamaterials with equidistant solely resonant bandgaps

Hasan B. Al Ba’ba’a|arXiv (Cornell University)|Nov 15, 2023

Acoustic Wave Phenomena Research参考文献 3被引用 5

一句话总结

本文提出了两种主动弹性超材料设计，通过在接地弹性元件中引入频率相关刚度，实现了无限系列等间距、仅由局域共振形成的带隙，且所有带隙宽度相同。带隙通过传递矩阵法（TMM）分析设计，并通过有限元频率响应仿真验证，实现了多谐波下精确、可调制的波衰减，且不受布里渊散射干扰。

ABSTRACT

Elastic metamaterials are man-made structures with properties that transcend naturally occurring materials. One predominant feature of elastic metamaterials is locally resonant bandgaps, i.e., frequency ranges at which wave propagation is blocked. Locally resonant bandgaps appear at relatively low frequency and arise from the existence of periodically placed mechanical local resonators. Typically, elastic metamaterials exhibit both locally resonant and Bragg-scattering bandgaps, which can generally be different in width and frequency ranges. This paper proposes two designs of active elastic metamaterials that only exhibit locally resonant bandgaps, which are infinite in number, evenly spaced in the frequency spectrum, and identical in width. The mathematical model is established using the transfer matrix method and synthesis of locally resonant bandgaps is achieved via an active elastic support with carefully designed frequency-dependent stiffness. A single unit cell of each proposed metamaterials is thoroughly studied, and its dispersion relation is derived analytically, along with the periodically repeating bandgap limits and widths. Following the dispersion analysis and bandgap parametric studies, finite arrays of the proposed metamaterials are considered, and their frequency response is calculated to verify the analytical predictions from dispersion analyses.

研究动机与目标

• 设计仅表现出局域共振带隙（排除布里渊散射效应）的主动弹性超材料。
• 实现无限多个在频率上等间距分布且宽度相同的带隙。
• 利用传递矩阵法（TMM）建立数学框架，以合成具有无限多个局域共振频率的色散行为。
• 通过有限数组的有限元频率响应分析验证分析预测。
• 为未来利用主动调制的频率相关刚度元件设计复杂色散行为提供支持。

提出的方法

• 采用传递矩阵法（TMM）推导主动超材料单胞的色散关系。
• 为接地共振器设计具有频率依赖性的刚度元件，以在规则间隔处产生无限序列的局域共振频率。
• 利用传递矩阵方法建立受驱波的色散关系，参数源自质量与刚度分布。
• 通过求解色散方程并分析波数的实部与虚部，解析定义带隙的边界与宽度。
• 在MATLAB中使用两节点线性杆单元的有限元法（FEM）模拟有限阵列（20个单元）的频率响应。
• 通过对比色散分析所得的带隙位置与宽度，以及模拟频率响应中的振幅跌落，验证分析预测。

Figure 1: Schematics of a typical discrete diatomic lattice (left) and EMM (right), showing the inertial and stiffness properties for the self-repeating unit cell. Typical dispersion relations of both cases are shown in the figure for reference. Bragg and locally-resonant (LR) bandgaps are observed

实验结果

研究问题

• RQ1 能否设计一种主动弹性超材料，使其仅表现出局域共振带隙，而不受布里渊散射贡献？
• RQ2 是否可以通过主动刚度调制，实现无限序列的等间距、相同宽度的局域共振带隙？
• RQ3 频率相关刚度函数如何影响不同参数区域下带隙的间距、宽度与稳定性？
• RQ4 分析色散预测与有限元模型动态响应之间的匹配程度如何？
• RQ5 所提出的设训方法能否推广至合成弹性超材料中其他复杂色散行为？

主要发现

• 第一种设计在共振频率 Ω = nπ（n ≥ 1）处产生等间距带隙，零频带隙宽度为其他带隙的一半。
• 第二种设计在 Ω = (2n+1)π/2（从 3π/2 开始）处生成带隙，所有带隙宽度相同，并关于其反共振频率对称。
• 随着阻尼参数 γ 增大，第一种设计的带隙宽度收敛于 π，第二种设计收敛于 2π，且两者宽度均随 γ 单调增加。
• 在第一种设计中，当 γ → 0 时，零频带隙消失；而在第二种设计中，无论 γ 取值如何，均不存在此类带隙，证实了低频共振的缺失。
• 20个单元阵列的有限元仿真结果与分析预测高度一致：位移响应中的剧烈振幅跌落与预测的带隙位置完全对应。
• 波数的虚部在局域共振频率处发散，证实了带隙的存在，并验证了分析所得的带隙边界。

Figure 2: ( top ) Schematic of an elastic rod with a unit-cell length of $\ell$ and an equally-spaced elastic foundation elements $k_{\text{d}}$ . The continuous elastic rod has a Young’s modulus and density of $E$ and $\rho$ , respectively. ( bottom ) Frequency response of the frequency-dependent s