二维非线性单原子正方晶格色散摄动分析  

Perturbation analysis on the dispersion relations of a two-dimensional nonlinear single atom square lattice

作　　者：王杰[1] 朱江[1] 黄文博 何欢[1,2] WANG Jie;ZHU Jiang;HUANG Wenbo;HE Huan(State Key Laboratory of Mechanics and Control of Mechanical Structures,Nanjing University of Aeronautic and Astronautic,Nanjing 210016,China;Institute of Vibration Engineering Research,Nanjing University of Aeronautics and Astronautics,Nanjing 210016,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

出　　处：《振动与冲击》2020年第9期88-96,共9页Journal of Vibration and Shock

摘　　要：声子晶体的色散关系决定弹性波在其中的传播方式。从二维无限周期结构的波动方程出发,提出了一种分析非线性离散型声子晶体的色散关系的一阶近似摄动法。通过引入Bloch理论与小参数摄动展开法,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。以二维单原子正方晶格为例,得到了其一阶频散曲线。色散结果显示带隙及传播方向与波幅相关。最后模拟了晶格对点谐波激励的响应,以验证摄动分析有效性。The dispersion relation of phononic crystals determines the propagation mode of elastic waves.Starting from the wave equation of a two-dimensional infinite periodic structure,a first-order approximate perturbation method for analyzing the dispersion relation of nonlinear discrete phononic crystals was proposed.Based on the Bloch theory and small parametric perturbation expansion method,the first-order dispersion relations and dispersion curves were obtained to analyze the effects of impedance configuration and nonlinear coefficient on the dispersion and group velocity in different directions.Two-dimensional single-atom lattices were used as examples.Their first-order dispersion curves and group velocity contours were presented.The dispersion results reveal that the band gap and direction of propagation are related to the amplitude of waves.Finally,the response of the lattice to the point harmonic force was simulated to verify the effectiveness of the perturbation analysis.

关 键 词：声子晶体 单原子正方晶格 弱非线性 摄动法 色散关系 

分 类 号：TU311.3[建筑科学—结构工程] TU352.1