Green Quasifunction Method For Vibration Of Thin Plates

Compared with the traditional method of numerical analysis such as finite element method, boundary element method and meshless method, etc, Green quasifunction method has its own advantages. In recent years, some domestic and international scholars have obtained many important fruits in study on Green quasifunction method. Green quasifunction method was first proposed by Rvachev, and applied to solve the problem of boundary value of Poisson equation. Yuan Hong, etc. popularized this method in various kinds of arithmetic operators, and applied it to solve the bending problem of polygonic elastic plate under different boundary conditions. On the basis of their work, application of Green quasifunction method to the free vibration problem of the simply-supported thin polygonic plates are presented in this dissertation, which enriches theory of Green quasifunction and extends applications of Green quasifunction method.At the beginning of the dissertation, the advantages and disadvantages of finite element method and boundary element method are briefly introduced. Then the dissertation summarizes characteristic and superiority of Green quasifunction method and explains the essence of Green quasifunction method. Various kinds of analytical and numerical methods of the plate vibration at present are briefly reviewed. Recent research and development of Green quasifunction method are introduced.Chapter 2 surveys the application of Green quasifunction method to all kinds of arithmetic operators such as Laplace operator, Helmholtz operator and biharmonic operator, etc. These arithmetic operators are widely used in the Mechanics and Physics .Chapter 3 deals Green quasifunction method of the free vibration problem of the simply-supported thin polygonic plates, and the free vibration problem of the simply-supported thin polygonic plates on Winkler foundation. Utilizing intermediate variable, the mode shape differential equation of the free vibration problem of simply-supported polygonic thin plates is divided into two simultaneous differential equations of lower order which couple each other. Through the change of variables, the original problem with inhomogeneous boundary condition is reduced to the boundary value problem of homogeneous boundary condition. Green quasifunction is constructed using basic solution,which satisfies the homogeneous boundary condition. According to Green's formulation, and using Green quasifunction and boundary conditions, differential equation is transformed into integral equation. By choosing suitable boundary normalized equation in according to R-function theory, the irregularity of integral kernel in integral equation can be eliminated. Then, numerical method is applied to discrete the integral equation, and suitable mathematical method is used to treat the irregularity in formulae. By meshing the whole plate into different quantities of grid, the convergence of Green quasifunction method in analyzing the free vibration problem of simply-supported thin polygonic plates and plates on Winkler foundation is confirmed. The dissertation also calculates the natural frequency of simply-supported plates with different shape, and examines the validity and the feasibility of Green quasifunction method.For the free vibration problem of simply-supported thin polygonic plate on Pasternak foundation, the basic principle of Green quasifunction method is similar with that in analyzing the problem of thin plate on Winkler foundation, except an additional term. In order to examine the validity and the feasibility of Green quasifunction method in analyzing the free vibration problem of simply-supported plate on the Pasternak foundation model, several numerical examples are calculated. Through meshing the different quantities of grid to plate, the convergence of Green quasifunction method is also confirmed.Examples analyzed show that Green quasifunction method is effective in solving the free vibration problem of thin polygonic plates. The method is of high accuracy. With increasing of the quantities of meshing grid, the result of Green quasifunction method can be quickly convergent to the exact solution, which verifies a very high convergence of the method. Green quasifunction method is not insensitive to the shape of meshing grid , especially being suitable to analyzing the free vibration problem of all kinds of plates with arbitrary shapes. Therefore, Green quasifunction method is an attractive and prospective method in solving the free vibration problem of plates with complicated boundary conditions and arbitrary shapes.