In this paper, an analytical matrix method is presented to drive closed-form characteristic equations for natural frequencies of finite monoatomic and diatomic metamaterials with various boundary conditions. Here, we extend the matrix method introduced by Louck for monoatomic lattice chains. The proposed method is used to calculate the vibration frequencies of the monoatomic metamaterials with fixed–fixed, fixed-free and free-free boundary conditions. In addition, the natural frequencies of fixed–fixed diatomic metamaterials are calculated. The existence of band gaps in the frequencies of the metamaterials is numerically shown.

中文翻译：

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一种计算有限一维声学超材料固有频率的分析方法

本文提出了一种解析矩阵方法来驱动具有各种边界条件的有限单原子和双原子超材料的固有频率的封闭形式特征方程。在这里，我们扩展了Louck引入的单原子晶格链的矩阵方法。该方法用于计算具有固定，固定，自由和自由边界条件的单原子超材料的振动频率。此外，还计算了固定-固定双原子超材料的固有频率。数字地显示了超材料频率中带隙的存在。