基于谱方法的矩形薄板自由振动分析
Free Vibration Analysis of Rectangular Thin Plates Based on Spectral Method
DOI: 10.12677/IJM.2019.81007, PDF,  被引量    国家自然科学基金支持
作者: 赵中凯, 张君华：北京信息科技大学，机电工程学院，北京；刘彦琦：北京市劳动保护科学研究所，北京
关键词: 谱方法；自由振动；切比雪夫多项式；矩形薄板；Spectral Method； Free Vibration； Chebyshev Polynomials； Rectangular Thin Plates
摘要: 提出了一种无刚度矩阵及无质量矩阵的切比雪夫谱方法，基于该方法研究了四边固支矩形薄板的自由振动特性。根据薄板自由振动的偏微分方程，采用分离变量法推导了薄板自由振动的特征频率方程，建立了薄板各物理参数与自由振动频率之间的关系式。基于切比雪夫谱方法，构造出满足四边固支矩形薄板自由振动微分方程的高阶求导矩阵，得到了薄板自由振动的特征频率矩阵方程，该方程不含刚度矩阵和质量矩阵。通过算例和有限元方法比较，验证了切比雪夫谱方法的高效性与收敛性。对比两种方法的计算结果，算例中薄板的频率变化规律以及各阶振型变化规律基本一致。
Abstract: A Chebyshev spectral method without both stiffness and mass matrices is proposed. Free vibrations of rectangular thin plates with clamped boundaries are researched by Chebyshev spectral method. Based on the partial differential equations of thin plates for free vibrations, the equations of eigen frequencies are educed by separating variables. Relationship between frequencies and physical parameters of materials is established. Referring to Chebyshev spectral method, high order derivative matrix is obtained which is a power operation of first order derivative matrix. The eigen frequency matrix equation of free vibration of thin plates is derived, but there is no stiffness matrix and no mass matrix. An example of clamped rectangular thin plate is solving by two methods. Compared with finite element method, efficiency and convergence of spectral method are verified. Comparing results by two methods, the frequencies and modes of the plate are basically consistent.
文章引用：赵中凯, 张君华, 刘彦琦. 基于谱方法的矩形薄板自由振动分析[J]. 力学研究, 2019, 8(1): 54-64. https://doi.org/10.12677/IJM.2019.81007

参考文献

[1] 李东旭. 挠性航天器结构动力学[M]. 北京: 科学出版社, 2010.
[2] 翟婉明. 车辆—轨道耦和动力学[M]. 第四版. 北京: 科学出版社, 2015.
[3] Ritz, W. (1909) Theorie der Transversalschwingungen einer quadratischen platte mit freien randern. Annalen der Physik, 333, 737-786. [Google Scholar] [CrossRef] 
[4] Mindlin, R.D. (1951) Influence of Rotatory Inertia and Shear on Flexural Motion of Isotropic Elastic Plate. Journal of Applied Mechanics, 18, 31-38.
[5] Gorman, D.J. (1982) Free Vibration Analysis of Rectangular Plates. Elsevier, New York.
[6] 钟阳, 李锐, 田斌. 四边固支矩形薄板自由振动的哈密顿解析解[J]. 应用力学学报, 2011, 28(4): 323-327.
[7] Liew, K.M. and Wang, C.M. (1993) pb-2 Rayleigh-Ritz Method for General Plate Analysis. Engineering Structure, 15, 55-60. [Google Scholar] [CrossRef] 
[8] Cheung, Y.K. and Zhou, D. (2002) Three-Dimensional Vibration Analysis of Cantilevered and Completely Free Isosceles Triangular Plates. International Journal of Solid and Structure, 39, 673-687. [Google Scholar] [CrossRef] 
[9] 陈林, 肖伟, 刘见华, 等. 基于改进傅立叶级数的矩形薄板振动特性分析[J]. 噪声与振动控制, 2018, 38(5): 21-26.
[10] Silling, S.A. (2000) Reformulation of Elasticity Theory for Discontinuities and Long-Range Forces. Journal of the Mechanics and Physics of Solids, 48, 175-209. [Google Scholar] [CrossRef] 
[11] Orszag, S.A. (1969) Numerical Methods for the Simulation of Turbulence. Physics of Fluids, 12, 250-257.
[12] Gottlieb, D. and Orszag, S.A. (1977) Numerical Analysis of Spectral Method: Theory and Applications. Capital City Press, Montpelier. [Google Scholar] [CrossRef] 
[13] Canuto, C., Hussaini, M.Y., Quarteroni, A. and Zang, T.A. (1988) Spectral Method in Fluid Dynamics. Springer, Berlin. [Google Scholar] [CrossRef] 
[14] Shen, J., Tang, T. and Wang, L.L. (2011) Spectral Methods: Algorithms, Analysis and Applications. Springer, Berlin. [Google Scholar] [CrossRef] 
[15] Chen, S. and Shen, J. (2018) Enriched Spectral Methods and Applications to Problems with Weakly Singular Solutions. Journal of Scientific Computing, 77, 1468-1489. [Google Scholar] [CrossRef] 
[16] Shen, Y.P., Zhu, Z.J., Wang, S.L. and Wang, G. (2019) Dynamic Analysis of Tapered Thin-Walled Beams Using Spectral Finite Element Method. Shock and Vibration, 2019, Article ID 2174209. [Google Scholar] [CrossRef] 
[17] Trefethen, L.N. (1996) Finite Difference and Spectral Method for Ordinary and Partial Differential Equations. Cornell University, Ithaca, New York.
[18] 刑誉峰, 刘波. 板壳自由振动的精确解[M]. 北京: 科学出版社, 2015.
[19] Fox, L. and Parker, I.B. (1968) Chebyshev Polynomials in Numerical Analysis. Oxford University Press, Oxford.
[20] Steeb, W. and Hardy, Y. (2006) Problems and Solutions in Introductory and Advanced Matrix Calculus. World Scientific Publishing Company, Singapore. [Google Scholar] [CrossRef] 

为你推荐