VIBRATIONS OF STEPPED RECTANGULAR THIN PLATES ON WINKLER’S FOUNDATION

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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…     

阶梯形闭口薄壁杆的约束扭转振动

用奇异函数建立非单一材质的ｎ级阶梯形闭口薄壁杆约束扭转自由振动和强迫振动的微分方程并求得其通解,用Ｗ算子给出主振型函数的表达式及常见支承条件下杆的频率方程。     

阶梯式矩形板的振动

用奇异函数建立阶梯式矩形板自由振动和强迫振动的微分方程并求得其通解，用Ｗ算子给出振型函数的表达式及常见支承条件下板的频率方程，本文解可用于多种边界条件的板。     

Free vibration analysis of general stepped circular plates with internal elastic ring support resting on Winkler foundation by green function method

Natural frequencies are important dynamic characteristics of a structure. Therefore, the exact solution pertaining to free vibration of stepped circular plate elastically restrained against rotation, translation, and internal elastic ring support resting on an arbitrary variable elastic foundation using Green Function is presented in this paper. Thus, an accurate and direct modeling technique is introduced for modeling stepped circular plate on an arbitrary variable elastic foundation with arbitrary boundary conditions and internal elastic ring support. The effect of the translational along with rotational support flexibilities, as well as, the elastic coefficient of Winkler foundation and other parameters are assessed. Finally, some numerical examples are shown in order to present the efficiency and simplicity of the Green Function in the new formulation.     

中面单向受拉（压）的阶梯式矩形薄板的振动

用奇异函数建立ｘ＝０与ｘ＝ａ两对边简支并受面内均布拉（压）力作用、加两边为任意支承、非单一材质的ｎ级阶梯式矩形薄板自由振动和强迫振动的微分方程并求得其通解,用Ｗ算子给出振型了函数的表达式及常见支承条件下板的方程。文中给出的固有频率表达式表明,面内均布拉（压）力对固有的数值有影响。此处导出的各种情况下的影响函数,对于求解相应民政部下的阶梯式矩形薄板的静力弯曲和稳定性问题,也是适用的。     

Thick rectangular plates with free edges on elastic foundations

In this paper an accurate solution for the thick rectangular plate with free edges laid onelastic foundation is presented.The superposition method of trigonometric series is used.The method can solve this kind of plates directly and simply.Its results completely satisfythe boundary conditions of the four free edges and nicely agree with the solutions by WangKe-lin and Huang Yi.     

Large amplitude free vibration of symmetrically laminated magneto-electro-elastic rectangular plates on Pasternak type foundation

Nonlinear free vibration of symmetrically laminated magneto-electro-elastic rectangular plate resting on an elastic foundation is studied analytically. The plate is considered to be simply supported on all edges. It is also assumed that the magneto-electro-elastic body is poled along the z direction and subjected to electric and magnetic potentials between the upper and lower surfaces. To model the motion of the plate, the first order shear deformation theory along with the Gauss's equations for electrostatics and magnetostatics are used. Then equations of motion are reduced to a single nonlinear ordinary differential equation which is solved analytically by multiple scales method. The results are compared with the published results and good agreement is found. Some numerical examples are presented to investigate the effects of several parameters on the linear and nonlinear behavior of these plates.     

Radial vibrations of axisymmetrically loaded stepped pressure vessels

IntroductionUsedextensivelyinengineeringarethin-walledpressurevessels,suchasboilersinpowerplants,rotordrumsOfturbines,leadersofcylinders,andgasstoragetanks.Inthispaper,discussedareradialvibrationsofaxisymmetricallyloadedcylindricalpressurevesselswiththicknessvaried.1FreeVibration1.1DifferentialequationforfreevibrationanditsgeneralsolutionAssumethereisathin-walledPressurevesselwithtotallength1andthicknessvariedinnstepsalongaams.Foreachstep,lengthh=xi--xi--11massperunitareaml,thicknessh.,ac?fi…     

Green quasifunction method for vibration of simply-supported thin polygonic plates on Pasternak foundation

A new numerical method—Green quasifunction is proposed.The idea of Green quasifunction method is clarified in detail by considering a vibration problem of simply-supported thin polygonic plates on Pasternak foundation.A Green quasifunction is established by using the fundamental solution and boundary equation of the problem. This function satisfies the homogeneous boundary condition of the problem.The mode shape differential equation of the vibration problem of simply-supported thin plates on Pasternak foundation is reduced to two simultaneous Fredholm integral equations of the second kind by Green formula.There are multiple choices for the normalized boundary equation.Based on a chosen normalized boundary equation,a new normalized boundary equation can be established such that the irregularity of the kernel of integral equations is overcome.Finally,natural frequency is obtained by the condition that there exists a nontrivial solution in the numerically discrete algebraic equations derived from the integral equations.Numerical results show high accuracy of the Green quasifunction method.     

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…     

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.     

多层正交异性矩形薄板弯曲振动稳定解析解

构造了带有补充项的双重正弦傅里叶级数通解来求解各种边界条件的多层正交各向异性矩形薄板的弯曲、振动和稳定问题.将坐标轴取在中性面上,求出用挠度表示的应力表达式,然后由横截面上每单位宽度的应力合成板的内力;再将层合板的内力代入板的平衡方程中得到板的控制方程,将多层板的物理参数折算为等价的单层板物理参数;最后联立控制方程与边界条件,求得未知量的系数并代入本文的通解中.本文的通解不需要叠加即可求解各种边界条件的板的弯曲、振动和稳定问题;现有的对于单层板的研究都可以用本文的方法拓展到多层板领域;对于复杂边界条件的板,也可以使用该通解分析.     

Construction of rectangular displacement-based element for thick/thin plates

In this paper, a method of constructing displacement-based element for thick/thin plates is developed by using the technique of generalized compatibility, and a rectangular displacement based element with 12 degrees of freedom for thick/thin plates is presented. This method enjoys a good accuracy with simple formulation and is free of shear locking as the thickness of the plate approaches zero. The project supported by National Natural Science Foundation of China through Grant No. 59208075     

THEORETIC SOLUTION OF RECTANGULAR THIN PLATE ON FOUNDATION WITH FOUR EDGES FREE BY SYMPLECTIC GEOMETRY METHOD

The theoretic solution for rectangular thin plate on foundation with four edges free is derived by symplectic geometry method. In the analysis proceeding, the elastic foundation is presented by the Winkler model. Firstly, the basic equations for elastic thin plate are transferred into Hamilton canonical equations. The symplectic geometry method is used to separate the whole variables and eigenvalues are obtained simultaneously. Finally, according to the method of eigen function expansion, the explicit solution for rectangular thin plate on foundation with the boundary conditions of four edges frees are developed. Since the basic elasticity equations of thin plate are only used and it is not need to select the deformation function arbitrarily. Therefore, the solution is theoretical and reasonable. In order to show the correction of formulations derived, a numerical example is given to demonstrate the accuracy and convergence of the current solution.     

任意边界条件下双模量矩形薄板的弯曲

将双模量板等效为两个各向同性小矩形板组成的层合板,假定该层合板的中性面即为两个小矩形板的交界面。根据中性面上应力为零且薄板全厚度上应力的代数和为零,推导了双模量矩形薄板的中性面位置。本文采用严宗达提出的带补充项的双重正弦傅里叶级数通解,该通解可以适用于任意边界条件的矩形薄板且不需要叠加或者重新构造。联立边界条件和控制方程,求得通解中的待定系数并代入到通解中,即可得到任意边界条件下双模量矩形薄板的弯曲解析解。与有限元结果比较,本文结果符合工程精度要求。     

THE METHOD OF THE RECIPROCAL THEOREM OF FORCED VIBRATION FOR THE ELASTIC THIN RECTANGULAR PLATES (III)—CANTILEVER RECTANGULAR PLATES

In this paper, applying the method of the reciprocal theorem, we give the stationary solutions of the forced vibration of cantilever rectangular plates under uniformly distributed harmonic load and concentrated harmonic load acting at any point of the plates, the figures and tables of number value of bending moment and the deflection amplitudes as well.     

SOLVING VIBRATION PROBLEM OF THIN PLATESUSING INTEGRAL EQUATION METHOD

ＳＯＬＶＩＮＧＶＩＢＲＡＴＩＯＮＰＲＯＢＬＥＭＯＦＴＨＩＮＰＬＡＴＥＳＵＳＩＮＧＩＮＴＥＧＲＡＬＥＱＵＡＴＩＯＮＭＥＴＨＯＤ￥(许明田，程德林)ＸｕＭｉｎｇｔｉａｎ；ＣｈｅｎｇＤｅｌｉｎ（Ｄｅｐａｒｔｍｅｎｔ．ｏｆＭａｔｈｅｍａｔｉｃｓａｎｄＰｈｙｓｉ...     

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     

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 semi- infinite elastic foundation.Some computational results and the analysis on the influence of parameters were presented.