局域共振型声学超材料薄板带隙特性的能量解法  

A variational method for band-gap analysis of metamaterial plates with local resonators

作　　者：米永振 杨浩森 雷博 郑辉[1] MI Yongzhen;YANG Haosen;LEI Bo;ZHENG Hui(State Key Laboratory of Mechanical System and Vibration,Shanghai Jiao Tong University,Shanghai 200240;Vehicle Engineering Institute,Chongqing University of Technology,Chongqing 400054)

机构地区：[1]上海交通大学机械系统与振动国家重点实验室,上海200240 [2]重庆理工大学车辆工程学院,重庆400054

出　　处：《声学学报》2020年第3期404-414,共11页Acta Acustica

基　　金：国家自然科学基金面上项目(51975352)资助。

摘　　要：研究局域共振型声学超材料的带隙特性时,面临的首要问题是寻求一种带隙特性的高效解法。为此,本文以局域共振型声学超材料薄板为研究对象,提出了一种基于能量泛函变分原理及正交多项式级数展开的带隙计算方法。该方法通过合并基体板及共振子的动、势能及外力功建立元胞的能量泛函,经过变分运算得到元胞的振动控制方程,进而引入第一类Chebyshev正交多项式作为基函数将控制方程进行离散。为克服无法直接对离散控制方程中的广义坐标施加周期边界条件的困难,该方法将连续边界条件弱化到若干选定的配点上以离散方式实现,并将对应的一组线性约束条件通过拉格朗日乘子法施加到系统中。仿真结果表明,该方法具有较高的计算精度和效率,并可应用于"局域共振子-夹层板"等复合结构的带隙特性研究。本文方法为局域共振型薄壁超材料的带隙求解提供了新的思路,丰富了声学超材料的振-声学理论,并为薄壁超材料的工程设计和优化提供了技术支撑。To establish an efficient solution method is the foremost issue when studying the band-gap properties of acoustic metamaterials.For this purpose,this paper presents a variational method for band-gap analysis of metamaterial plates with local resonators.This method is based on the Lagrangian functional of a unit cell and orthogonal polynomial expansions.Specifically,the Lagrangian functional is built by combining energy terms of the plate and the local resonator,to which the variational operation is performed to derive the governing equations of a unit cell.These equations are discretized to a set of generalized coordinates by introducing the Chebyshev orthogonal polynomials of the first kind as admissible functions.To overcome the difficulty that the Periodic Boundary Condition(PBC) cannot be directly applied to these generalized coordinates,the proposed method weakens the PBC to a number of pre-selected collocation points,thus allowing the continuous boundary conditions to be imposed in a more straightforward,though discrete,way.The linear constraints given by this procedure is enforced by Lagrange multipliers method.Numerical examples show that the proposed method is of good accuracy and efficiency and accommodates to composite structures like double-leaf metamaterial plates with local resonators.It can also be extended to band-gap analysis of metamaterial plates with other configurations,providing technical support for the design and optimization of their vibration and noise reduction characteristics.

关 键 词：带隙特性 声学超材料板 基函数 局域共振 声学超材料 

分 类 号：TB34[一般工业技术—材料科学与工程]