超材料(metamaterials)的應用，在這幾年引起廣泛的討論，其中聲學超材料(acoustic metamaterials)領域中的負等效質量密度(mass density)、負等效體積模數(bulk modulus)的探討，更成為學者們重要的研究課題之一。利用系統的直線運動來探討負等效質量密度，可以達到濾波的效果，但本文不同的地方在於，我們利用轉動運動來探討負等效質量慣性矩(mass moment of inertia)，也可達到濾波的效果。在本文中，我們建立離散系統和連續體的模型，離散系統的部份我們利用轉動和頻散方程式等效的概念去設計，而連續體模型的部分，我們將具有旋轉效應的微極彈性理論，導入我們的設計中，除此之外，我們將模型的材料組成，設計成自然界中方便取得的材料。最後，我們將設計完後的模型進行波傳分析，以及求解模型的負等效質量慣性矩，並分析其與帶隙(band gap)的關係。

The application of metamaterials has drawn much attention, particularly in the field of acoustic metamaterials. The properties of negative mass density and negative bulk modulus have great application in practice, because it is the basic concept of filters. Generally speaking, using the models of a straight-line motion can verify the filter’s effect. Unlike previous researches, we design a model with rotational effect to prove the existence of filters. We can utilize rotational motion to devise a model which is in virtue of the similarity between mass density and mass moment of inertia. In this thesis, we propose discrete and continuum models to analyze the behavior of wave propagation and the negative effective material properties. In one-dimensional (1D) discrete model, we use the equivalence of dispersive equation to design and analyze the negative effective mass moment of inertia. We also design a two-dimensional (2D) discrete model with negative Poisson’s ratio and negative effective stiffness. We exercise the elastic theory of Cosserat in our continuum models. Moreover, in order to manufacture the models conveniently and cost-effectively, we design it with natural materials. Last but not least, aside from analyzing the behavior of wave propagation and negative effective mass moment of inertia, we also study the relation of band gap between them.

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