VIBRATIONS OF STEPPED RECTANGULAR THIN PLATES ON WINKLER’S FOUNDATION

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The general solution for axial symmetrical bending of nonhomogeneous circular plates resting on an elastic foundation

In this paper,a new method,the exact analytic method,is presented on the basis of stepreduction method.By this method,the general solution for the bending of nonhomogenouscircular plates and circular plates with a circular hole at the center resting,on an elastfcfoundation is obtained under arbitrary axial symmetrical loads and boundary conditions.The uniform convergence of the solution is proved.This general solution can also be applieddirectly to the bending of circular plates without elastic foundation.Finally,it is onlynecessary to solve a set of binary linear algebraic equation.Numerical examples are givenat the end of this paper which indicate satisfactory results of stress resultants anddisplacements can be obtained by the present method.     

VIBRATIONS OF ONE-WAY RECTANGULAR STEPPEDTHIN PLATES ON WINKLER'S FOUNDATION

I.IntroductionUsedextensivelyillengineeringareone-wayrectallgularsteppedthinplates.suchaslongandnarrow"platesofl'Oundationbaseincivilengineering,high-c"a}'surface.retainingwall,undergroundstructurefi-ame,baseplatesofsluice.shipyardando\-ertlowdam.Theyshouldnotonlymeetrequirementsofstrength,stiffnessalldsteadinessofstructuresbutalsosavematerialstotileftlll.Thus,steppedstructuresarecolnmonl}'adoptedinengineerings.Sofar.vibratio,nsofsuChplateshavenotbeendiscussedinanyliterature.Inthispaper.disc…     

非均匀Winkler弹性地基上变厚度矩形板自由振动的DTM求解

针对非均匀Winkler弹性地基上变厚度矩形板的自由振动问题,通过一种有效的数值求解方法——微分变换法（DTM）,研究其无量纲固有频率特性。已知变厚度矩形板对边为简支边界条件,其他两边的边界条件为简支、固定或自由任意组合。采用DTM将非均匀Winkler弹性地基上变厚度矩形板无量纲化的自由振动控制微分方程及其边界条件变换为等价的代数方程,得到含有无量纲固有频率的特征方程。数值结果退化为均匀Winker弹性地基上矩形板以及变厚度矩形板的情形,并与已有文献采用的不同求解方法进行比较,结果表明,DTM具有非常高的精度和很强的适用性。最后,在不同边界条件下分析地基变化参数、厚度变化参数和长宽比对矩形板无量纲固有频率的影响,并给出了非均匀Winkler弹性地基上对边简支对边固定变厚度矩形板的前六阶振型。     

弹性地基加肋功能梯度板自由振动分析的无网格法

基于一阶剪切变形理论和移动最小二乘近似研究Winkler弹性地基上加肋功能梯度板的固有频率。假设功能梯度板的材料性质沿厚度方向按幂函数连续变化,基于物理中面和移动最小二乘近似分别推导功能梯度板和肋条的动能和势能,再通过引入位移协调条件,建立板和肋条节点参数转换关系,叠加两者的总能量,然后利用Hamilton原理推导加肋功能梯度板自由振动控制方程。采用完全转换法施加边界条件。通过将本文的计算结果与有限元以及文献的结果对比,验证方法的收敛性以及准确性。     

Static and free vibration analysis of straight and circular beams on elastic foundation

Static and free vibration analyses of straight and circular beams on elastic foundation are investigated. The Timoshenko beam theory is adopted in the derivation of the governing equation. Ordinary differential equations in scalar form obtained in the Laplace domain are solved numerically using the complementary functions method. The static and free vibration analyses of beams on elastic foundation are analyzed through various examples.     

FREE VIBRATION OF ANISOTROPIC RECTANGULAR PLATES BY GENERAL ANALYTICAL METHOD

According to the differential equation for transverse displacement function of anisotropic rectangular thin plates in free vibration, a general analytical solution is established. This general solution, composed of the composite solutions of trigonometric function and hyperbolic function, can satisfy the problem of arbitrary boundary conditions along four edges. The algebraic polynomial with double sine series solutions can also satisfy the problem of boundary conditions at four corners. Consequently, this general solution can be used to solve the vibration problem of anisotropic rectangular plates with arbitrary boundaries accurately. The integral constants can be determined by boundary conditions of four edges and four corners. Each natural frequency and vibration mode can be solved by the determinate of coefficient matrix from the homogeneous linear algebraic equations equal to zero. For example, a composite symmetric angle ply laminated plate with four edges clamped has been calculated and discussed.     

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-ofvariables 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.     

Free axisymmetric vibration of FGM circular plates

Based on the three-dimensional theory, this work presents a direct displacement method to investigate the free axisymmetric vibration of transversely isotropic circular plates, whose material is functionally graded and properties obey the exponential law along the thickness direction of the plate. Under two boundary conditions, the solution satisfies all basic equations and the Corresponding boundary condition at every point. Thus, it is three-dimensional exact. Numerical examples are presented and compared with previous works. The present method can also be extended to the case of arbitrary distribution of the material properties along the thickness direction of the plate.     

Adomian-modified decomposition method for large-amplitude vibration analysis of stepped beams with elastic boundary conditions

The main objective of this paper is to apply an Adomian modified decomposition method for solving large amplitude vibration analysis of stepped beams with various general and elastic boundary conditions. Damaged or imperfect supports of beams can be modeled by using elastic boundary conditions composing of translational and rotational springs. For the beams subjected to dynamic severe loading, it is important to include the nonlinear term of axial stretching force developed by the large vibration amplitude in the governing equation for more accurate design. By using the method, the convergence studies for linear and nonlinear vibration analyses of stepped beams are shown for determining an appropriate number of terms in the solutions. The accuracy of the present results is validated numerically by comparing with some available results in the literature. New results of nonlinear frequency ratios of stepped beams with different boundary conditions are presented and discussed in detail. Aspects of step ratio, step location, boundary conditions, vibration amplitudes, etc., which have significant impact on linear and nonlinear frequencies of such beams are taken under investigation.     

Free vibration analysis of a plate on foundation with completely free boundary by finite integral transform method

The theoretical solutions of eigenfrequencies and vibration modes of a rectangular thin plate on an elastic foundation with completely free boundary are derived by using a double finite cosine integral transform method. In the analysis procedure, the elastic foundation is regarded as a Winkler elastic foundation model. Because the basic dynamic elasticity equations of the thin plate on elastic foundation are only used, it is not needed to select the deformation function arbitrarily. Therefore, the solution developed in the present paper is more reasonable and more accurate. To prove the correctness of the solutions, numerical results obtained using the present solutions are compared with those in the literatures.     

Free axisymmetric vibration of transversely isotropic laminated circular plates

Based on the three dimensional elastic theory, the state equation of the axisymmetric free vibration of transversely isotropic circular plates is established. Taking the advantage of finite Hankel transform, two exact solutions are derived for two boundary conditions, i. e. the rigid-slipping boundary and elastic simply supported boundary. Finally, numerical results are presented and compared with those of FEM. The project is supported by the National Natural Science Foundation of China and the Zhejiang Provincial Natural Science Foundation.     

General solution for large deflection problems of nonhomogeneous circular plates on elastic foundation

In this paper, a new method is presented based on [1]. It can be used to solve the arbitrary nonlinear system of differential equations with variable coefficients. By this method, the general solution for large deformation of nonhomogeneous circular plates resting on an elastic foundation is derived. The convergence of the solution is proved. Finally, it is only necessary to solve a set of nonlinear algebraic equations with three unknowns. The solution obtained by the present method has large convergence range and the computation is simpler and more rapid than other numerical methods.Numerical examples given at the end of this paper indicate that satisfactory results of stress resullants and displacements can be obtained by the present method. The correctness of the theory in this paper is, confirmed.     

Dynamics of thin periodic plates resting on a periodically inhomogeneous Winkler foundation

Summary The aim of the paper is to investigate the dynamic response of thin elastic plates having periodic substructure in planes parallel to the plates midplane and interacting with a Winkler foundation. The main goal of the analysis is to describe the effect of substructure size on the plates dynamics. For this purpose, the method proposed in [4, 5] is used. Two special cases are analysed: a plate band with a constant thickness interacting with a periodically inhomogeneous Winkler foundation, and a plate band with a periodically variable thickness interacting with a homogeneous Winkler foundation. The physical correctness conditions of the model are also discussed. Received 14 July 1998; accepted for publication 7 January 1999     

弹性地基上含孔隙功能梯度材料板的自由振动和动力响应

为研究弹性地基上含孔隙的材料特性沿厚度呈Sigmoid函数变化的功能梯度材料(S-FGM)板的振动特性,本文基于改进的Voigt模型,分别建立了孔隙为均匀分布和非均匀分布两种类型的功能梯度材料的物性参数模型.根据复合材料薄板理论导出了弹性地基上含孔隙的功能梯度材料板的运动方程,用伽辽金法寻求四边简支边界条件下板自由振动...     

FURTHER STUDY ON LARGE AMPLITUDE VIBRATION OF CIRCULAR SANDWICH PLATES

ＦＵＲＴＨＥＲＳＴＵＤＹＯＮＬＡＲＧＥＡＭＰＬＩＴＵＤＥＶＩＢＲＡＴＩＯＮＯＦＣＩＲＣＵＬＡＲＳＡＮＤＷＩＣＨＰＬＡＴＥＳＤｕＧｕｏｊｕｎ（杜国君）ＣｈｅｎＹｉｎｇｊｉｅ（陈英杰）Ａｂｓｔｒａｃｔ：Ｉｎｔｈｉｓｐａｐｅｒ，ａｓｏｌｕｔｉｏｎｏｆａｘｉ...     

Vibrations of stepped one-way thin rectangular plates subjected to in-plane tensile/compressive force in y-direction on Winkler' s foundation

ＩｎｔｒｏｄｕｃｔｉｏｎＶｉｂｒａｔｉｏｎｓｏｆｓｔｅｐｐｅｄｔｈｉｎｒｅｃｔａｎｇｕｌａｒｐｌａｔｅｓｈａｖｅｂｅｅｎｒｅｓｅａｒｃｈｅｄｂｙｍｅｃｈａｎｉｃａｌｗｏｒｋｅｒｓ.ＥｒｃｏｌｉａｎｄＬａｕｒａａｎａｌｙｚｅｄｖｉｂｒａｔｉｏｎｓｏｆｔｗｏ＿ｓｔｅｐｔｈｉｎｒｅｃｔａｎｇｕｌａｒｐｌａｔｅｓ[1],ＺｈａｎｇＹｉｎｇｓｈｉｅｔａｌｓｔｕｄｉｅｄｖｉｂｒａｔｉｏｎｓｏｆｎ＿ｓｔｅｐｏｎｅ＿ｗａｙｔｈｉｎｒｅｃｔａｎｇｕｌａｒｐｌａｔｅｓｏｎＷｉｎｋｌｅｒ’ｓｆｏｕｎｄａｔｉｏｎ[1],ｏｆｎ…     

Three-dimensional exact solutions for the free vibration of laminated transversely isotropic circular,annular and sectorial plates with unusual boundary conditions

New state space formulations for the free vibration of circular, annular and sectorial plates are established by introducing two displacement functions and two stress functions. The state variables can be separated into two independent catalogues and two kinds of vibrations can be readily found. Expanding the displacements and stresses in terms of Bessel functions in the radial direction and trigonometric functions in the circumferential direction, we obtained the exact frequency equation for the free vibration for some uncommon boundary conditions. Numerical results are presented and compared with those of FEM to demonstrate the reliability of the proposed method. A parametric investigation is also performed.     

NONLINEAR VIBRATION FOR MODERATE THICKNESS RECTANGULAR CRACKED PLATES INCLUDING COUPLED EFFECT OF ELASTIC FOUNDATION

IntroductionThe moderate thickness plates on elastic foundation are a kind of important structure instructural engineering. The mechanic characters of the plates on elastic foundation withdifferent boundary conditions have been received considerable atten…     

Symplectic system based analytical solution for bending of rectangular orthotropic plates on Winkler elastic foundation

This paper analyses the bending of rectangular orthotropic plates on a Winkler elastic foundation.Appropriate definition of symplectic inner product and symplectic space formed by generalized displacements establish dual variables and dual equations in the symplectic space.The operator matrix of the equation set is proven to be a Hamilton operator matrix.Separation of variables and eigenfunction expansion creates a basis for analyzing the bending of rectangular orthotropic plates on Winkler elastic foundation and obtaining solutions for plates having any boundary condition.There is discussion of symplectic eigenvalue problems of orthotropic plates under two typical boundary conditions,with opposite sides simply supported and opposite sides clamped.Transcendental equations of eigenvalues and symplectic eigenvectors in analytical form given.Analytical solutions using two examples are presented to show the use of the new methods described in this paper.To verify the accuracy and convergence,a fully simply supported plate that is fully and simply supported under uniformly distributed load is used to compare the classical Navier method,the Levy method and the new method.Results show that the new technique has good accuracy and better convergence speed than other methods,especially in relation to internal forces.A fully clamped rectangular plate on Winkler foundation is solved to validate application of the new methods,with solutions compared to those produced by the Galerkin method.