Symplectic analysis for elastic wave propagation in two-dimensional cellular structures  

作　　者：Xiu-Hui Hou Zi-Chen Deng Jia-Xi Zhou Tie-Quan Liu 

机构地区：[1]Department of Engineering Mechanics,Northwestern Polytechnical University, Xi'an 710072, China [2]State Key Laboratory of Structural Analysis for Industrial Equipment,Dalian University of Technology, Dalian 116024, China

出　　处：《Acta Mechanica Sinica》2010年第5期711-720,共10页力学学报（英文版）

基　　金：supported by the National Natural Science Foundation of China (10972182, 10772147, 10632030);the National Basic Research Program of China (2006CB 601202);the Doctorate Foundation of Northwestern Polytechnical University (CX200908);the Graduate Starting Seed Fund of Northwestern Polytechnical University (Z200930);the NPU Foundation for Fundamental Research;the Open Foundation of State Key Laboratory of Structural Analysis of Industrial Equipment (GZ0802)

摘　　要：On the basis of the finite element analysis, the elastic wave propagation in cellular structures is investigated using the symplectic algorithm. The variation principle is first applied to obtain the dual variables and the wave propagation problem is then transformed into two-dimensional （2D） symplectic eigenvalue problems, where the extended Wittrick-Williams algorithm is used to ensure that no phase propagation eigenvalues are missed during computation. Three typical cellular structures, square, triangle and hexagon, are introduced to illustrate the unique feature of the symplectic algorithm in higher-frequency calculation, which is due to the conserved properties of the structure-preserving symplectic algorithm. On the basis of the dispersion relations and phase constant surface analysis, the band structure is shown to be insensitive to the material type at lower frequencies, however, much more related at higher frequencies. This paper also demonstrates how the boundary conditions adopted in the finite element modeling process and the structures＇ configurations affect the band structures. The hexagonal cells are demonstrated to be more efficient for sound insulation at higher frequencies, while the triangular cells are preferred at lower frequencies. No complete band gaps are observed for the square cells with fixed-end boundary conditions. The analysis of phase constant surfaces guides the design of 2D cellular structures where waves at certain frequencies do not propagate in specified directions. The findings from the present study will provide invaluable guidelines for the future application of cellular structures in sound insulation.On the basis of the finite element analysis, the elastic wave propagation in cellular structures is investigated using the symplectic algorithm. The variation principle is first applied to obtain the dual variables and the wave propagation problem is then transformed into two-dimensional （2D） symplectic eigenvalue problems, where the extended Wittrick-Williams algorithm is used to ensure that no phase propagation eigenvalues are missed during computation. Three typical cellular structures, square, triangle and hexagon, are introduced to illustrate the unique feature of the symplectic algorithm in higher-frequency calculation, which is due to the conserved properties of the structure-preserving symplectic algorithm. On the basis of the dispersion relations and phase constant surface analysis, the band structure is shown to be insensitive to the material type at lower frequencies, however, much more related at higher frequencies. This paper also demonstrates how the boundary conditions adopted in the finite element modeling process and the structures＇ configurations affect the band structures. The hexagonal cells are demonstrated to be more efficient for sound insulation at higher frequencies, while the triangular cells are preferred at lower frequencies. No complete band gaps are observed for the square cells with fixed-end boundary conditions. The analysis of phase constant surfaces guides the design of 2D cellular structures where waves at certain frequencies do not propagate in specified directions. The findings from the present study will provide invaluable guidelines for the future application of cellular structures in sound insulation.

关 键 词：Cellular structures Symplectic analysis Dispersion relation - Phase constant surface Sound insulation 

分 类 号：O422[理学—声学] V214.6[理学—物理]