Research On Vibration Characteristics For Beams Plates Shells And Coupled Structures Based On The Differential Quadrature Finite Element Method

The beam,plate and shell structure are widely used in engineering,so it is of great engineering significance to study their vibration characteristics.The differential quadrature method is a numerical algorithm containing the features of fast convergence and high precision.By introducing the idea of element division from the traditional finite element method,a differential quadrature finite element method,which is suitable for free vibration analysis of complex geometry structures,is widely used.On the basis of this method,the differential quadrature method and the finite element method are combined to establish a systematic general method for analyzing the vibration characteristics of beam,plate and shell structures and their coupled structures.Firstly,the key rules in the differential quadrature method are introduced.According to the spirit of discretization of the finite element method,the differential quadrature finite element method is explained in detail from the point of view of energy principle,and the solving steps are given.Based on Timoshenko beam theory,a beam element of differential quadrature finite element method is established,the stiffness matrix and mass matrix are deduced.Numerical results of a single beam element and planar frame structures are presented.By comparing the results with the literature and the finite element results,the correctness and applicability of the proposed method are verified.Based on the Mindlin's plate theory,a plate element of differential quadrature finite element method is established,and the mapping technique is introduced to make the plate element suitable for the construction of irregular plate structures.The numerical results of arbitrarily-shaped straight and curved quadrilateral plate are given.A numerical example of a open rectangular plate,which can be seen as a sdtructure coupled by several plates,is presented and the experiment of the structure is conducted.Numerical results of two plates coupled with a certain angle are given.The method is compared with the experimental results and the experimental results.The correctness and applicability for the vibration problem of the plate and the coupled plate structures of the proposed method are verified.Based on the thin shell theory,a thin cylindrical shell and a thin spherical shell element of differential quadrature finite element method are established.The displacement vector is modified by the transformation matrix in order to contains the degree of freedom of rotatory,which is convenient for imposing the boundary conditions.Numerical examples of thin cylindrical shell and thin spherical shell under different boundary conditions and different structure dimensions are given.The correctness of this method for displacement vector modification are verified and the validity and applicability of the method to study the vibration problem of thin shell structures are verified.