具有一般非对称晶胞的连续棒基声子晶体的截断共振理论
Theory of Truncation Resonances in Continuum Rod-based Phononic Crystals with Generally Asymmetric Unit Cells
摘要：
声子晶体表现出布拉格带隙，频率范围内，波的传播是禁止的。在固体连续介质中，带隙是周期性交替层引起的相消干涉的结果。在某些条件下，固有频率以结构边界附近的高振幅局部振动的形式出现在这些带隙内，称为截断共振。本文研究了一维杆形有限声子晶体的振动谱，解释了截断共振产生的原因。通过确定一个单胞对称参数，我们定义了一个家庭的有限声子棒共享相同的色散关系，但不同的截断形式。利用传递矩阵法推导出控制有限系统固有频率的特征方程的封闭形式的表达式，并破译跨越不同边界条件出现的截断共振。该分析建立了与截断共振，边界条件，和截断链的整体配置所规定的单元格的选择与本地化的振动之间的具体连接。该研究提供了工具来预测，调整和选择性地设计截断共振，以满足需要并独特地受益于这种截断共振的各种应用的需求。
英文摘要：
Phononic crystals exhibit Bragg bandgaps, frequency regions within which wave propagation is forbidden. In solid continua, bandgaps are the outcome of destructive interferences resulting from periodically alternating layers. Under certain conditions, natural frequencies emerge within these bandgaps in the form of high-amplitude localized vibrations near a structural boundary, referred to as truncation resonances. In this paper, we investigate the vibrational spectrum of finite phononic crystals which take the form of a one-dimensional rod, and explain the factors that contribute to the origination of truncation resonances. By identifying a unit cell symmetry parameter, we define a family of finite phononic rods which share the same dispersion relation, yet distinct truncated forms. A transfer matrix method is utilized to derive closed-form expressions of the characteristic equations governing the natural frequencies of the finite system and decipher the truncation resonances emerging across different boundary conditions. The analysis establishes concrete connections between the localized vibrations associated with a truncation resonance, boundary conditions, and the overall configuration of the truncated chain as dictated by unit cell choice. The study provides tools to predict, tune, and selectively design truncation resonances, to meet the demands of various applications that require and uniquely benefit from such truncation resonances.
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