THE METHOD OF THE RECIPROCAL THEOREM OF FORCED VIBRATION FOR THE ELASTIC THIN RECTANGULAR PLATES(Ⅰ)—RECTANGULAR PLATES WITH FOUR CLAMPED EDGES AND WITH THREE CLAMPED EDGES

In this paper the method of the reciprocal theorem (MRT) is extended to solve the steady state responses of rectangular plater under harmonic disturbing forces. A series of the closed solutions of rectangular plates with various boundary conditions are given and the tables and figures which have practical value are provided.MRT is a simple, convenient and general method for solving the steady stale responses of rectangular plates under various harmonic disturbing forces.The paper contains three parts: (I) rectangular plates with four damped edges and with three clamped edges; (II) rectangular plates with two adjacent clamped edges; (III) cantilever plates.We arc going to publish them one after another.

UNIFORMLY VALID ASYMPTOTIC SOLUTIONS OF THE NONLINEAR UNSYMMETRICAL BENDING FOR ORTHOTROPIC RECTANGULAR THIN PLATE OF FOUR CLAMPED EDGES WITH VARIABLE THICKNESS

By using “the method of modified two-variable”,“the method of mixing perturbation” and introducing four small parameters,the problem of the nonlinear unsymmetrical bending for orthotropic rectangular thin plate with linear variable thickness is studied.And the uniformly valid asymptotic solution of Nth-order for ε_1 and Mth-order for ε_2 of the deflection functions and stress function are obtained.

BENDING OF RECTANGULAR THIN PLATES WITH FREE EDGES LAID ON TENSIONLESS WINKLER FOUNDATION

In this paper, the bending problem of rectangular thin plates with free edges laid on tensionless Winkler foundation has been solved by employing Fourier series with supplementary terms. By assuming proper form of series for deflection, the basic differential equation with given boundary conditions can be transformed into a set of infinite algebraic equations. Because the boundary of contact region cannot bedetermined in advance, these equations are weak nonlinear ones. They can be solved by using iterative procedures.

Free Vibration Analysis of Rectangular Plate with Cutouts under Elastic Boundary Conditions in Independent Coordinate Coupling Method

Based on Kirchhoff plate theory and the Rayleigh-Ritz method,the model for free vibration of rectangular plate with rectangular cutouts under arbitrary elastic boundary conditions is established by using the improved Fourier series in combination with the independent coordinate coupling method(ICCM).The effect of the cutout is taken into account by subtracting the energies of the cutouts from the total energies of the whole plate.The vibration displacement function of the hole domain is based on the coordinate system of the hole domain in this method.From the continuity condition of the vibration displacement function at the cutout,the transition matrix between the two coordinate systems is constructed,and the mass and stiffness matrices are completely obtained.As a result,the calculation is simplified and the computational efficiency of the solution is improved.In this paper,numerical examples and modal experiments are presented to validate the effectiveness of the modeling methods,and parameters related to influencing factors of the rectangular plate are analyzed to study the vibration characteristics.

Element-free Galerkin method for free vibration of rectangular plates with interior elastic point supports and elastically restrained edges

The element-free Galerkin method is proposed to solve free vibration of rectangular plates with finite interior elastic point supports and elastically restrained edges.Based on the extended Hamilton＇s principle for the elastic dynamics system,the dimensionless equations of motion of rectangular plates with finite interior elastic point supports and the edge elastically restrained are established using the element-free Galerkin method.Through numerical calculation,curves of the natural frequency of thin plates with three edges simply supported and one edge elastically restrained,and three edges clamped and the other edge elastically restrained versus the spring constant,locations of elastic point support and the elastic stiffness of edge elastically restrained are obtained.Effects of elastic point supports and edge elastically restrained on the free vibration characteristics of the thin plates are analyzed.

BENDING OF UNIFORMLY LOADED RECTANGULAR PLATES WITH TWO ADJACENT EDGES CLAMPED, ONE EDGE SIMPLY SUPPORTED AND THE OTHER EDGE FREE

In this paper, an exact solution for an uniformly loaded rectangular plate with two adjacent edges clamped, one edge simply supported and the other edge free, was given by using the concept of generalized simply supported edges and superposition method. The numerical results were given for the deflections along the free edge and bending moments along the clamped edges of a square plate.

THE BENDING OF A THICK RECTANGULAR PLATE WITH THREE CLAMPED EDGES AND ONE FREE EDGE

The exact solution of the bending of a thick rectangular plate with three clamped edges and one free edge under a uniform transverse load is obtained by means of the concept of generalized simply-supported boundary[1] in Reissner's theory of thick plates. The effect of the thickness h of a plate on the bending is studied and the applicable range of Kirchhoffs theory for bending of thin plates is considered.

Closed-form solutions for free vibration of rectangular FGM thin plates resting on elastic foundation

This article presents closed-form solutions for the frequency analysis of rectangular functionally graded material (FGM) thin plates subjected to initially in-plane loads and with an elastic foundation. Based on classical thin plate theory, the governing differential equations are derived using Hamilton's principle. A neutral surface is used to eliminate stretching-bending coupling in FGM plates on the basis of the assumption of constant Poisson's ratio. The resulting governing equation of FGM thin plates has the same form as homogeneous thin plates. The separation-of-variables method is adopted to obtain solutions for the free vibration problems of rectangular FGM thin plates with separable boundary conditions, including, for example, clamped plates. The obtained normal modes and frequencies are in elegant closed forms, and present formulations and solutions are validated by comparing present results with those in the literature and finite element method results obtained by the authors. A parameter study reveals the effects of the power law index n and aspect ratio a/b on frequencies.

Double Symplectic Eigenfunction Expansion Method of Free Vibration of Rectangular Thin Plates

The free vibration problem of rectangular thin plates is rewritten as a new upper triangular matrix differential system. For the associated operator matrix, we find that the two diagonal block operators are Hamiltonian. Moreover, the existence and completeness of normed symplectic orthogonal eigenfunction systems of these two block operators are demonstrated. Based on the completeness, the general solution of the free vibration of rectangular thin plates is given by double symplectie eigenfunction expansion method.

EXACT SOLUTION FOR RECTANGULAR SLAB WITH THREE EDGES SIMPLY-SUPPORTED AND OTHER FREE

In this paper,we give all-sided pastic analysis of the rectangular slab with three edges simply-supported and other free.Here we discuss the following four cases:(1)The uniformly distributedload over the area a slab.(2).A concentrated load act at midpoint of free edges slab.(3)A concen-trated load act at the center a slab.(4)The line load act along free edge of slab.

THE APPLICATION OF WEINSTEIN-CHIEN′S METHOD——THE UPPER AND LOWER LIMITS OF FUNDAMENTAL FREQUENCY OF RECTANGULAR PLATES WITH EDGES ARE THE MIXTURE OF SIMPLY SUPPORTED PORTIONS AND CLAMPED PORTIONS

In this paper, the method of relaxed boundary conditions is applied to rectangular plates with edges which are a sort of the mixture of simply supported portions and clamped portions, so that the lower limit of fundamental frequency of such plates is evaluated. A kind of polynomial satisfying the displacement boundary conditions is designed, os that it is enabled to evaluate the upper limit of fundamental frequency by Ritz' method. The practical calculation examples solved by these methods have given satisfactory results. At the end of this paper, it is pointed out that the socalled exact solution of such plates usually evaluated by the force superposition method is essentially a kind of lower limit of solution, if the truncated error of series which occurs in actual calculation is considered.

Mixed finite element and differential quadrature method for free and forced vibration and buckling analysis of rectangular plates

This paper presents a combined application of the finite element method （FEM） and the differential quadrature method （DQM） to vibration and buckling problems of rectangular plates. The proposed scheme combines the geometry flexibility of the FEM and the high accuracy and efficiency of the DQM. The accuracy of the present method is demonstrated by comparing the obtained results with those available in the literature. It is shown that highly accurate results can be obtained by using a small number of finite elements and DQM sample points. The proposed method is suitable for the problems considered due to its simplicity and potential for further development.

Analytical solutions of steady vibration of free rectangular plate on semi-infinite elastic foundation

The method of double Fourier transform Was employed in the analysis of the semi-infinite elastic foundation with vertical load. And an integral representations for the displacements of the semi-infinite elastic foundation was presented. The analytical solution of steady vibration of an elastic rectangle plate with four free edges on the semi-infinite elastic foundation was also given by combining the analytical solution of the elastic rectangle plate with the integral representation for displacements of the semiinfinite elastic foundation. Some computational results and the analysis on the influence of parameters were presented.

VIBRATIONS OF STEPPED ONE-WAY THIN RECTANGULAR PLATES SUBJECTED TO IN-PLANE TENSILE/ COMPRESSIVE FORCE IN Y-DIRECTION ON WINKLER'S FOUNDATION

Differential equations of free/forced vibrations of n_step one_way thin rectangular plates subjected to in_plane tensile/compressive force in y_direction on Winkler's foundation are established by using singular functions, their general solutions solved for, expression of vibration mode function and frequency equation on usual supports derived with W operator. Influence functions for various cases deduced here may also be used to solve problems of static buckling or stability for beams and plates in relevant circumstances.

A FREE RECTANGULAR PLATE ON ELASTIC FOUNDATION

In the theory of elastic thin plates, the bending of a rectangular plate on the elastic foundation is also a difficult problem. This paper provides a rigorous solution by the method of superposition. It satisfies the differential eguation, the boundary conditions of the edges and the free corners. Thus we are led to a system of infinite simultaneous eguations. The problem solved is for a plate with a concentrated load at its center. The reactive forces from the foundation should be made to be in equilibrium with the concentrated force to see whether our calculation is correct or not.

TRANSVERSE VIBRATION OF RECTANGULAR PLATES ELASTICALLY SUPPORTED AT POINTS ON EDGES

This paper studies transverse vibration of rectangular plates with two opposite edges simply supperted other two edges arbitrarily supported and free edges elaslically supported at points,A highly accurate solution is presented for calculating inherent frequencies and mode shape of rectangular platen elaslically supported at points. The number and location of these points on free edges may be completely arbitrary. This paper uses impulse function to represent reaction and moment at points. Fourter series is used to expand the impulse function along the edges. Characteristic equations satisfying all boundary conditions are given.Inherent frequencies and mode shape with any accutacy can be gained.

Nonlinear Free Vibrations of C-C-SS-SS Symmetrically Laminated Carbon Fiber Reinforced Plastic (CFRP) Rectangular Composite Plates

The purpose of this paper is to apply the theoretical model developed in References [1] 08D0C9EA79F9BACE118C8200AA004BA90B02000000080000000E0000005F005200650066003400310032003700320039003100350035000000 -[6] 08D0C9EA79F9BACE118C8200AA004BA90B02000000080000000E0000005F005200650066003400310032003700320039003100360030000000 in order to analyze the geometrically nonlinear free dynamic response of C-C-SS-SS rectangular CFRP symmetrically laminated plates so as to investigate the effect of nonlinearity on the nonlinear resonance frequencies, the nonlinear fundamental mode shape and associated bending stress patterns at large vibration amplitudes. Various values of the plate aspect ratio and the amplitude of vibrations will be considered, and useful numerical data also are provided.

Analysis of the Behavior of a Square Plate in Free Vibration by FEM in Ansys

In the realization of mechanical structures, achieving stability and balance is a problem commonly encountered by engineers in the field of civil engineering, mechanics, aeronautics, biomechanics and many others. The study of plate behavior is a very sensitive subject because it is part of the structural elements. The study of the dynamic behavior of free vibration structures is done by modal analysis in order to calculate natural frequencies and modal deformations. In this paper, we present the modal analysis of a thin rectangular plate simply supported. The analytical solution of the differential equation is obtained by applying the method of separating the variables. We are talking about the exact solution of the problem to the limit values. However, numerical methods such as the finite element method allow us to approximate these functions with greater accuracy. It is one of the most powerful computational methods for predicting dynamic response in a complex structure subject to arbitrary boundary conditions. The results obtained by MEF through Ansys 15.0 are then compared with those obtained by the analytical method.

一类矩形中厚板模型屈曲与振动问题的辛解析解

从实际力学问题出发抽象出一类矩形中厚板模型,并用辛体系方法进行求解。首先,通过适当的变换将该模型转化为Hamilton系统,得到了Hamilton算子的本征值和本征函数系,并证明了本征函数系的完备性,进而给出对边简支情形的一般解,最后对6种不同边界条件下的Mindlin板和Pasternak型双参数弹性地基的矩形中厚板的振动及屈曲问题进行了数值模拟。数据对比显示了本文模型的正确性和辛方法的有效性。

Pasternak 地基上自由边矩形板弯曲问题的 Fourier 级数解

应用正交函数系级数展开法分析结构的理论，以带附加补充项的Ｆｏｕｒｉｅｒ级数作为挠度函数模式，求解了Ｐａｓｔｅｒｎａｋ地基上自由边矩形板的弯曲问题。文中给出了算例。