二维非线性离散声子晶体色散摄动分析  

Perturbation Analysis for Dispersion Relations of Two-Dimensional Periodic Structure

作　　者：吴添 朱江[1] 黄文博 何欢[1,2] 惠旭龙 刘小川[3] WU Tian;ZHU Jiang;HUANG Wenbo;HE Huan;XI Xulong;LIU Xiaochuan(State Key Laboratory of Mechanics and Control of Mechanical Structures,Nanjing University of Aeronautics&Astronautics,Nanjing,210016,China;Institute of Vibration Engineering Research,Nanjing University of Aeronautics&Astronautics,Nanjing,210016,China;Aviation Key Laboratory of Science and Technology on Structures Impact Dynamics,Aircraft Strength Research Institute of China,Xi’an,710065,China)

机构地区：[1]南京航空航天大学机械结构力学及控制国家重点实验室,南京210016 [2]南京航空航天大学振动工程研究所,南京210016 [3]中国飞机强度研究所结构冲击动力学航空科技重点实验室,西安710065

出　　处：《南京航空航天大学学报》2020年第6期997-1006,共10页Journal of Nanjing University of Aeronautics & Astronautics

基　　金：江苏高校优势学科建设工程资助项目。

摘　　要：弹性波在声子晶体中的传播方式可由其色散关系决定,从二维无限周期结构的波动方程出发,通过引入Bloch理论与小参数摄动展开法,提出了一种分析非线性离散型声子晶体色散关系的一阶近似摄动法。得到了一阶近似的色散关系与频散曲线,以分析不同方向上的阻抗配置与非线性系数对频散及群速度的影响。分别以二维单原子网格、二维双原子二自由度网格和二维双原子四自由度网格为例,得到了它们的一阶频散曲线,色散结果显示带隙及传播方向与波幅相关。同时结合数值积分解验证了其在有限波幅谐波激励下解的精度。The propagation mode of the elastic wave in phononic crystals is determined by its dispersion relation. Starting from the wave equation of two-dimensional infinite periodic structure and introducing the Bloch theory and small parametric perturbation expansion method,a first-order approximate perturbation method for analyzing the dispersion relationship of nonlinear discrete phononic crystals is proposed. And the first-order dispersion relations and dispersion curves are obtained to analyze the effects of impedance configuration and nonlinear coefficient pair frequency on dispersion and group velocity in different directions.Two-dimensional single-atom lattices,two-dimensional two-atom two-degree-of-freedom grids,and twodimensional two-atom four-degree-of-freedom grids are used as examples. Their first-order dispersion curves are obtained. Dispersion results are displayed. Dispersion results reveal that band gap and direction of propagation are related to theamplitude of the wave. Finally,combining with the numerical integral result to verify the solution’s accuracy under harmonic excitation of finite amplitude.

关 键 词：声子晶体 离散周期结构 弱非线性 摄动法 色散关系 

分 类 号：TU311.3[建筑科学—结构工程] TU352.