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 引用本文: 薛坚, 牛牧青, 张文勇, 陈立群. 二元复合材料板的自由振动: 半解析法. 力学学报, 2022, 54(7): 2041-2049.
Xue Jian, Niu Muqing, Zhang Wenyong, Chen Liqun. Free vibration of a binary composite plate: A semi-analytical approach. Chinese Journal of Theoretical and Applied Mechanics, 2022, 54(7): 2041-2049.
 Citation: Xue Jian, Niu Muqing, Zhang Wenyong, Chen Liqun. Free vibration of a binary composite plate: A semi-analytical approach. Chinese Journal of Theoretical and Applied Mechanics, 2022, 54(7): 2041-2049.

## FREE VIBRATION OF A BINARY COMPOSITE PLATE: A SEMI-ANALYTICAL APPROACH

• 摘要: 二元复合材料板是超材料板结构中常见的单元之一. 针对由材料参数相差两个量级的基体和嵌入体组成的二元复合材料板, 提出结构自由振动的半解析模型, 并对其振动特性进行了研究. 基于区域分解法和二元材料的分布, 将二维平板分解成两个子区域. 通过在振型函数中附加区域试函数, 来描述复合材料板面内刚度突变引起局部位移和转角的非光滑性. 基于二元复合材料板的基本边界条件和两子区连接处的变形协调条件, 构造了新的振型函数. 基于经典薄板理论, 利用带特殊试函数的里兹法, 求得不同几何构型下二元复合材料板的固有频率和振型, 并研究了嵌入体的尺寸和位置对结构振动特性的影响规律. 通过收敛分析并与有限元仿真结果对比, 验证了本文方法的准确性. 研究结果表明: 传统的全局试函数在分析具有振动局部化的模态时会得到不准确的结果, 而附加区域试函数可以显著提高里兹法的收敛速度以及结果的准确性; 嵌入体位置对低阶固有频率的作用不明显, 却能显著改变低阶振型节线的分布和振动局部化发生的区域.

Abstract: Binary composite plate is one of the common elements in metamaterial plate structure. A semi-analytical model of the free vibration of the structure is proposed for the binary composite plate composed of a matrix and an embedded body with different material parameters, and its vibration characteristics are studied. The plate is decomposed into two sub-regions based on the domain decomposition method and the distribution of binary materials. The non-smoothness of the local displacement and strain caused by the sudden change of stiffness in the composite plate is described by adding a local trial function to the mode shape function. Based on the essential boundary conditions of the binary composite plate and the condition of compatibility for the displacement at the joint of the two sub-regions, a new mode shape function is constructed. Based on the classical thin plate theory, the Ritz method with special trial functions is used to calculate the natural frequencies and modes of the binary material plate under different geometric configurations. The influence of the size and location of the embedded body on the vibration characteristics of the structure is investigated. The accuracy of this method is verified by the convergence analysis and the finite element simulation results. The results show that the classical global trial function will lead to inaccurate results when analyzing the modes with vibration localization, while the additional local trial function can significantly improve the convergence speed of the Ritz method and the accuracy of the results; the effect of the embedded body position on the low-order natural frequencies is not obvious, but it can significantly change the distribution of the low-order mode shape nodal lines and the region where vibration localization occurs.

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