矩形板自由振动的Chebyshev谱方法  

Spectral-Chebyshev approach for free vibration of rectangular plates

作　　者：刘涛[1,2] 王青山 秦斌 王艾伦 LIU Tao;WANG Qing-shan;QIN Bin;WANG Ai-lun(State Key Laboratory of High Performance Complex Manufacturing,Central South University,Changsha 410083,China;Light Alloy Research Institute,Central South University,Changsha 410083,China;School of Traffic&Transportation Engineering,Central South University,Changsha 410075,China)

机构地区：[1]中南大学高性能复杂制造国家重点实验室,湖南长沙410083 [2]中南大学轻合金研究院,湖南长沙410083 [3]中南大学交通运输工程学院,湖南长沙410075

出　　处：《塑性工程学报》2021年第5期145-152,共8页Journal of Plasticity Engineering

基　　金：国家自然科学基金资助项目(52075554)。

摘　　要：提出一种分析矩形板结构在不同边界下的自由振动特性的Chebyshev谱方法。基于一阶剪切变形理论,考虑了矩形板结构面内和面外位移建立其控制方程和边界条件方程,将矩形板结构的位移函数进行谱展开成有限项Chebyshev多项式,再根据Gauss-Lobatto点进行离散得到结构的矩阵形式特征方程,求解特征方程获得矩形板自由振动的频率和振型。将计算的矩形板结构振动特性与相关文献和有限元软件计算结果对比,充分验证了所提出Chebyshev谱方法的快速收敛性和计算精确性。结果表明,当矩形板厚度较厚时,其面内振动频率会出现在前几阶固有频率之中;当矩形板的面积增大时,其固有频率参数降低。此方法可进一步发展为通用数值计算方法,指导结构参数化设计。A spectral-Chebyshev approach was proposed to analyze the free vibration characteristics of rectangular plates under various boundary conditions. Based on the first-order shear deformation theory,considering the in-plane and out of plane displacements of rectangular plates,the governing equation and boundary condition equation of the rectangular plate structure were established. The displacement function of the rectangular plate structure was spectral expanded into a finite term Chebyshev polynomial,and then the characteristic equation in matrix form of the structure was obtained by discretization of Gauss-Lobatto points,and the frequency and vibration mode of free vibration of rectangular plates were obtained by solving the characteristic equation. The fast convergence and calculation accuracy of the spectral-Chebyshev method proposed were fully verified by comparing the vibration characteristic of rectangular plates structure with the results of relevant literature and finite element software calculation. The results show that when the thickness of the rectangular plate is thick,the in-plane vibration frequency appears in the first several natural frequencies;when the area of the rectangular plate increases,the natural frequency parameters decrease. This method can be further developed into a general numerical calculation method to guide the parametric design of structures.

关 键 词：Chebyshev谱方法 矩形板 自由振动 一阶剪切变形理论 面内振动 

分 类 号：O326[理学—一般力学与力学基础]