弹性矩形薄板受迫振动的功的互等定理法(Ⅰ)——四边固定的矩形板和三边固定的矩形板

本文将功的互等定理法(MRT)推广于求解在简谐干扰力作用下矩形板的稳态响应.给出了各种边界条件矩形板的一系列封闭解并提供了一些有实用价值的图表.功的互等定理法(MRT)是求解在各种简谐干扰力作用下的矩形板稳态响应的一个简便、通用的方法.本文包括三部分:(Ⅰ)四边固定的矩形板和三边固定的矩形板;(Ⅱ)二邻边固定的矩形板;(Ⅲ)悬臂矩形板.我们准备分三次陆续发表它们.     

求解厚矩形板弯曲问题的功的互等定理法

在本文中,功的互等定理法(RTM)被推广于求解基于Reissner理论的厚矩形板弯曲问题。首先,本文绘出了厚矩形板弯曲的基本解;其次,给出了三边固定一边自由在均布载荷作用下厚矩形板弯曲的精确解析解;最后,我们分析了本文解的数值结果。     

弹性半空间地基上四边自由矩形板稳态振动的解析解

采用双重Fourier变换,分析得到弹性半空间地基受竖向稳态荷载作用下的积分变换解．与四边自由矩形板的振动解析解相结合,得出弹性半空间地基上四边自由矩形板稳态振动的解析解．还给出算例及参数影响分析．     

中厚悬臂板的固有振动和稳定

本文由设定两个位移函数，应用最小二乘法和能量法，得到中厚悬臂矩形板固有振动和稳定的Ｒｅｉｓｓｎｅｒ近似解。     

弹性矩形薄板受迫振动的功的互等定理法(Ⅲ)—悬臂矩形板

本文应用功的互等定理法给出了在均布谐载和在任意点受集中谐载作用下悬臂矩形板受迫振动的稳态解,并给出了有关弯矩和挠度幅值的图表.     

文克尔地基上阶梯式单向矩形薄板的振动

用奇异函数建立文克尔地基上阶梯式单向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算子给出主振型函数的表达式及常见支承条件下板的频率方程，用广义函数讨论板在不同形式载荷作用下的强迫振动响应·     

矩形中厚板和夹层板的后屈曲

本文研究了矩形Ｒｅｉｓｓｎｅｒ中厚板和夹层板的后屈曲特性。首先将矩形中厚板和夹层板的基本方程和边界条件表述成统一的无量纲形式。对不同的边界条件，特别是不对称边界条件，文中发展了一种应用于非线性分析的混合Ｆｏｕｒｉｅｒ级数求解新方法，获得了级数形式的精确解。非线性偏微分方程化为无穷元非线性代数方程组，数值计算中截取有限项进行迭代求解。     

双参数地基上Reisner板弯曲问题的边界积分方程

本文应用广义函数的Ｆｏｕｒｉｅｒ积分变换，导出了双参数地基上Ｒｅｉｓｓｎｅｒ板弯曲问题的两个基本解·在此基础上，从虚功原理出发，依据胡海昌导出的Ｒｅｉｓｓｎｅｒ板弯曲理论，推导出适用于任意形状、任意荷载、任意边界条件情形的三个边界积分方程，为边界元法在这一问题中的应用提供了理论基础·文中给出了固支、简支、自由三类边界的算例，并与解析解比较，均得到满意的结果·     

矩形薄板弹性振动的一般解析解

本文建立了矩形薄板弹性横向自由振动位型函数微分方程的一般解,可以求解任意边界矩形薄板的振动问题.以四边自由矩形板为例求解了板的频率及其振型.     

正交各向异性功能梯度层合板稳态热传导问题的精确解

对轴对称正交各向异性功能梯度层合圆板稳态热传导问题进行精确分析.假设材料热传导率沿板厚方向按指数函数形式梯度分布,从正交各向异性功能梯度圆板稳态热传导的基本方程出发,利用分离变量法,获得了在上、下表面作用任意热分布情况下的精确解.通过数值算例的分析,指出材料性质的梯度变化、板厚边界条件等分析了对温度场分布的影响.所获得的精确结果,可以作为评价其它近似方法的标准解答.     

A symplectic analytical wave propagation model for damping and steady state forced vibration of orthotropic composite plate structure

An analytical wave propagation model is proposed in this paper for damping and steady state forced vibration of orthotropic composite plate structure by using the symplectic method. By solving an eigen-problem derived in the symplectic dual system of free bending vibration of orthotropic rectangular thin plates, the wave shape of plate is obtained in symplectic analytical form for any combination of simple boundary conditions along the plate edges. And then the specific damping capacity of wave mode is obtained symplectic analytically by using the strain energy theory. The steady state forced vibration of built-up plates structure is calculated by combining the wave propagation model and the finite element method. The vibration of the uniform plate domain of the built-up plates structure is described using symplectic analytical waves and the connector with discontinuous geometry or material is modeled using finite elements. In the numerical examples, the specific damping capacity of orthotropic rectangular thin plate with three different combinations of boundary condition is first calculated and analyzed. Comparisons of the present method results with respect to the results from the finite element method and from the Rayleigh–Ritz method validate the effectiveness of the present method. The relationship between the specific damping capacity of wave mode and that of modal mode is expounded. At last, the damped steady state forced vibration of a two plates system with a connector is calculated using the hybrid solution technique. The availability of the symplectic analytical wave propagation model is further validated by comparing the forced response from the present method with the results obtained using the finite element method.     

求解集中载荷作用下悬臂板的挠曲方程

应用功的互等定理,求解了悬臂厚矩形板在集中载荷作用下的挠曲面方程;同时也通过编程计算给出了具有实际价值的计算结果,进一步证明了应用功的互等法求解厚矩形板的正确性和优越性.     

Modified mixed Ritz-DQ formulation for free vibration of thick rectangular and skew plates with general boundary conditions

Recently, the present authors proposed a simple mixed Ritz-differential quadrature (DQ) methodology for free and forced vibration, and buckling analysis of rectangular plates. In this technique, the Ritz method is first used to discretize the spatial partial derivatives with respect to a coordinate direction of the plate. The DQ method is then employed to analogize the resulting system of ordinary or partial differential equations. The mixed method was shown to work well for vibration and buckling problems of rectangular plates with simple boundary conditions. But, due to the use of conventional Ritz method in one coordinate direction of the plate, the geometric boundary conditions of the problem can only be satisfied in that direction. Therefore, the conventional mixed Ritz-DQ methodology may encounter difficulties when dealing with rectangular plates involving adjacent free edges and skew plates. To overcome this difficulty, this paper presents a modified mixed Ritz-DQ formulation in which all the natural boundary conditions are exactly implemented. The versatility, accuracy and efficiency of the proposed method for free vibration analysis of thick rectangular and skew plates are tested against other solution procedures. It is revealed that the proposed method can produce highly accurate solutions for the natural frequencies of thick rectangular plates involving adjacent free edges and skew plates using a small number Ritz terms and DQ sampling points.     

New benchmark solutions for free vibration of clamped rectangular thick plates and their variants

It is of significance to explore benchmark analytic free vibration solutions of rectangular thick plates without two parallel simply supported edges, because the classic analytic methods are usually invalid for the problems of this category. The main challenge is to find the solutions meeting both the governing higher order partial differential equations (PDEs) and boundary conditions of the plates, i.e., to analytically solve associated complex boundary value problems of PDEs. In this letter, we extend a novel symplectic superposition method to the free vibration problems of clamped rectangular thick plates, with the analytic frequency solutions obtained by a brief set of equations. It is found that the analytic solutions of clamped plates can simply reduce to their variants with any combinations of clamped and simply supported edges via an easy relaxation of boundary conditions. The new results yielded in this letter are not only useful for rapid design of thick plate structures but also provide reliable benchmarks for checking the validity of other new solution methods.     

文克尔地基上阶梯式矩形薄板的振动

用奇异函数建立文克尔地基上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算子给出振型函数的表达式及常见支承条件下板的频率方程,用广义函数的富里叶展开,讨论板在不同载荷作用下的强迫振动响应     

On the natural frequency of transversely isotropic magneto-electro-elastic plates in contact with fluid

In this paper, the vibration characteristics of transversely isotropic magneto-electro-elastic (MEE) rectangular plates in contact with fluid are investigated. The mathematical formulation on the determination of added virtual mass for water-contacting MEE rectangular plates with uniform thickness is performed. Based on the recently proposed differential equation governing the dynamical responses of the MEE rectangular plates, a fluid-structure interaction model is established and analyzed. First of all, the added virtual mass incremental (AVMI) factor of the system is calculated by using the proposed approach and the added virtual mass can then be obtained, furthermore, the natural frequencies of the MEE rectangular plates in contact with fluid with different boundary conditions are also investigated. It is noted that the natural frequencies based on the proposed method are very useful for those engineers or researchers who are engaged in the vibration analysis and design of the MEE plate in contact with fluid.     

Exact solutions for free flexural vibration of Lévy-type rectangular thick plates via third-order shear deformation plate theory

In this paper, exact closed-form solutions in explicit forms are presented for transverse vibration analysis of rectangular thick plates having two opposite edges hard simply supported (i.e., Lévy-type rectangular plates) based on the Reddy’s third-order shear deformation plate theory. Two other edges may be restrained by different combinations of free, soft simply supported, hard simply supported or clamped boundary conditions. Hamilton’s principle is used to derive the equations of motion and natural boundary conditions of the plate. Several comparison studies with analytical and numerical techniques reported in literature are carried out to demonstrate accuracy of the present new formulation. Comprehensive benchmark results for natural frequencies of rectangular plates with different combinations of boundary conditions are tabulated in dimensionless form for various values of aspect ratios and thickness to length ratios. A set of three-dimensional (3-D) vibration mode shapes along with their corresponding contour plots are plotted by using exact transverse displacements of Lévy-type rectangular Reddy plates. Due to the inherent features of the present exact closed-form solution, the present findings will be a useful benchmark for evaluating the accuracy of other analytical and numerical methods, which will be developed by researchers in the future.