Study On Vibration Modeling Methods And Vibration Characteristics Of Rectangular Plate With Cutouts Based On The Three-demensional Elastic Theory

Composite plate structures with cutouts have been widely used in aerospace,ships and other engineering fields.The cutouts will destroy the overall continuity of the structure,and cause serious stress concentration,reduce the strength and stiffness of the overall structure,and greatly reduce its stability.At the same time,serious concentration will occur at the cutouts,which will affect the service life of the structure.The vibration characteristics of the structure with cutouts are the basis of structural design,and in the process of vibration characteristics analysis due to the diversity of material properties and the shape of the cutouts,it is necessary to apply different modeling theories and establish different analysis models,which brings certain difficulties to the parameterized design analysis of the structure.Therefore,establishing a unified vibration analysis model of the plate structure suitable for different material properties and cutout shapes and studying its vibration characteristics can provide theoretical foundation and engineering guidance for practical applications.In this paper,the following work is focuses on the vibration modeling method and three-dimensional vibration characteristic analysis of rectangular plate structure with cutouts:The rectangular plate structures are the basic structural component,the dimension of the plate in the thickness direction is much smaller than the other two dimensions.Therefore,the three-dimensional vibration analysis of the plate is often simplified to two dimensional vibration problems.The vibration solution process of rectangular plate structure under the classical thin plate theory and the first-order shear deformation theory is derived.Aiming at the limitations of the classical thin plate theory ignoring the transverse shear deformation and the low calculation accuracy,and the first-order shear deformation theory depends on human factors and the high-order frequency error of the structure,etc.,a three-dimensional vibration analysis model of rectangular plate under elastic boundary conditions is established based on the three-dimensional elastic theory.The displacement field function of rectangular plate structure adopts the form of three-dimensional Fourier cosine series plus auxiliary function,and the auxiliary function selects cosine series in order to improve the calculation efficiency in the integration process.Three sets of linear springs are uniformly arranged at the boundary of the rectangular plate structure,and different elastic boundary conditions are simulated by changing the spring stiffness values.The energy functional of rectangular plate structure is derived by using Hamilton principle,and the motion control differential equation is solved by Rayleigh Ritz method.Numerical examples are used to verify the accuracy of the three-dimensional vibration analysis model of the rectangular plate structure established in this paper.The results show that the calculation time of trigonometric series is 48.38%lower than that of polynomial function,and 83.02%lower than that of finite element analysis method with the same accuracy.On the basis of the above-mentioned three-dimensional vibration analysis model of the rectangular plate,a three-dimensional vibration analysis model of the graded functionally composite material rectangular plate with rectangular cutouts is established,and its vibration characteristics are studied.The energy functional of three-dimensional vibration analysis of rectangular plate and rectangular cutout is constructed in the unified Cartesian rectangular coordinate system,respectively.The energy functional of the structure is obtained by energy subtraction,and then the motion differential equation of graded functionally composite material perforated plate structure is expressed as a set of linear equations.The Rayleigh-Ritz method is used to calculate the extreme value of the unknown coefficient in the displacement field,and the frequency parameters and modal shapes of rectangular plate with rectangular cutouts are obtained.The model can be reduced to the solution of three-dimensional vibration problems of isotropic rectangular plates cutouts,and the model also can be applied to the solution of vibration problems of structures with multiple cutouts and structures without cutouts.In the numerical examples,the accuracy of the three-dimensional vibration analysis model established in this paper is verified through the vibration characteristics analysis of rectangular plate structure with cutouts in isotropic material and graded functionally composite material.Then,the effects of cutout size,cutout position and boundary conditions on the vibration characteristics of the structure are discussed.Based on the three-dimensional vibration analysis model of GFC rectangular plate with rectangular cutouts,the three-dimensional vibration analysis model of GFC rectangular plate with circular cutouts is established.A new calculation method is proposed for the problems such as the difficulty of solving the energy integral in different coordinate systems.The rectangular plate with circular cutout domain is discretized into four closed regions,and then the closed region is mapped into a unit square calculation domain by using the blending function method.According to the independent coordinate relationship between plate domain and cutout domain,the Jacobian coordinate transformation matrix is obtained,and then the rectangular plate with circular cutout is transformed into rectangular plate with rectangular cutout for solution.The model can be used to solve the vibration problems of rectangular plate with circular cutouts,and can also be used to solve the vibration problems of annular plate or circular plate with eccentric circular cutouts.The correctness of the method proposed in this paper is verified by the comparative analysis with the finite element analysis method and the results of reference data.Finally,parametric analysis of the free vibration characteristics and harmonic response characteristics of rectangular plate with circular cutouts is carried out.Based on the research of graded functionally composite materials,a three-dimensional vibration analysis model of laminated composite structures with rectangular cutout is established.The three-dimensional elastic theory is used to model the independent sub-layers of the laminated composite structure,the displacement field function of the independent sublayers is constructed by using the three-dimensional improved Fourier series,and the interlayer stress and displacement continuity conditions are introduced at the ply interface.The Lagrange energy functional of laminated composite plate with cutouts is derived.In the analysis of numerical examples,the influences of the boundary conditions,the laminations,and the fiber orientations on the vibration characteristics of the structure are studied.The results show that the proposed method can effectively overcome the shortcomings of inaccurate estimation of transverse deformation and interlaminar stress in two-dimensional simplified theory.