四边任意支承条件下弹性矩形薄板弯曲问题的解析解

利用辛几何法推导出了四边为任意支承条件下矩形薄板弯曲的解析解。在分析过程中首先把矩形薄板弯曲问题表示成Hamilton正则方程，然后利用辛几何方法对全状态相变量进行分离变量，求出其本征值后，再按本征函数展开的方法求出四边为任意支承条件下矩形薄板弯曲的解析解。由于在求解过程中并不需要人为的事先选取挠度函数，而是从弹性矩形薄板弯曲的基本方程出发，直接利用数学的方法求出问题的解析解，使得这类问题的求解更加理论化和合理化。文中的最后还给出了计算实例来验证本文方法的正确性。     

四边固支矩形弹性薄板的精确解析解

利用辛几何方法本文推导出了四边固支矩形弹性薄板弯曲问题的精确解析解.由于在求解过程中不需要事先人为的选取挠度函数,而是从弹性薄板的基本方程出发,首先将矩形薄板弯曲问题表示成Hamilton正则方程,然后利用分离变量和本征函数展开的方法求出可以完全满足四边固支边界条件的精确解析解.本文中所采用的方法突破了传统的半逆法的限制,使得问题的求解更加合理化.文中还给出了计算实例来证明推导结果的正确性.     

Winkler地基上四边自由正交各向异性矩形薄板弯曲问题的辛叠加解

研究Winkler地基上正交各向异性矩形薄板弯曲方程所对应的Hamilton正则方程, 计算出其对边滑支条件下相应Hamilton算子的本征值和本征函数系, 证明该本征函数系的辛正交性以及在Cauchy主值意义下的完备性, 进而给出对边滑支边界条件下Hamilton正则方程的通解, 之后利用辛叠加方法求出Winkler地基上四边自由正交各向异性矩形薄板弯曲问题的解析解. 最后通过两个具体算例验证了所得解析解的正确性.     

固支矩形Mindlin板弯曲问题的辛求解体系

本文利用二类变量广义变分原理推出了Mindlin板弯曲问题的Hamilton体系,利用辛几何方法对全状态向量进行分离变量,得到相应的横向本征问题,在求出其本征值后,按本征函数展开法导出了原问题的辛本征通解。给出了一个承受集中载荷的四边固支矩形薄板的算例,按本文求解体系得到的解与经典解吻合较好。本文直接从Mindlin板弯曲问题出发,在其Hamilton体系内使用辛几何方法给出了的一套新的求解体系,突破了传统解法的局限性,具有一般性及较高的理论推广价值。     

矩形薄板弯曲问题的二维广义有限积分变换法

运用二维广义有限积分变换解法，本文推导出不同边界条件下矩形薄板弯曲问题的解析解．在推导过程中，选取满足边界条件的梁振型函数为广义积分变换的积分核，由此构造出广义有限积分变换对，通过对薄板弯曲问题的控制方程进行二维广义积分变换，可以将控制方程转换为易于求解的线性代数方程组．该方法无需预先选取位移函数，无需进行繁琐的叠加过程，求解过程思路清晰，说明该方法更加正确合理．最后通过计算实例对比，验证了该方法的合理性及所推导公式的正确性．     

矩形中厚板哈密顿体系的一种构造方法及典型算例分析

从矩形中厚板弯曲问题的基本方程出发,将问题导入Hamilton体系,然后利用辛几何中的分离变量和本征函数展开的方法求出了矩形中厚板典型弯曲问题的解析解.所构造的Hamilton对偶方程形式简洁,求解方便;采用的方法不必事先人为地选择挠度函数,突破了传统半逆解法的限制,使得问题的求解更加合理化,并通过计算实例证明了本文推导结果的正确性.     

薄板弯曲问题的神经网络方法

为发展神经网络方法在求解薄板弯曲问题中的应用,基于Kirchhoff板理论,提出一种采用全连接层求解薄板弯曲四阶偏微分控制方程的神经网络方法。首先在求解域、边界中随机生成数据点作为神经网络输入层的参数,由前向传播系统求出预测解;其次计算预测解在域内及边界处的误差,利用反向传播系统优化神经网络系统的计算参数;最后,不断训练神经网络使误差收敛,从而得到薄板弯曲的挠度精确解。以不同边界、荷载条件的三角形、椭圆形、矩形薄板为例,利用所提方法求解其偏微分方程,与理论解或有限元解对比,讨论了影响神经网络方法收敛的因素。研究表明,本文方法对求解薄板弯曲问题的四阶偏微分控制方程具有一定的适应性,其收敛性受多种条件影响。相比有限元,该方法收敛速度较慢,在复杂的边界条件下收敛性不佳,然而其不基于变分原理,无需计算刚度矩阵,便可获得较高的计算精度。     

弹性地基上矩形薄板问题的Hamilton正则方程及解析解

利用辛算法求出弹性地基上矩形薄板问题的解析解,将弹性地基视为双参数弹性地基,直接从弹性矩形薄板的控制方程推导出了问题的Hamilton正则方程,为求出任意边界条件下问题的理论解奠定了基础,并且通过算例验证了文中所采用方法的正确性．     

四边固支矩形薄板自由振动的哈密顿解析解

在哈密顿体系中利用辛几何方法求解了四边固支矩形薄板自由振动问题的解析解。首先,从基本方程出发,将问题表示成Hamilton正则方程,然后利用辛几何方法导出本征值问题,从而得到本征函数解,使之满足边界条件;再由方程组有非零解的条件,最终推导出四边固支矩形薄板的自振频率方程,得到频率的解析解。计算了不同长宽比情况下四边固支矩形薄板的频率,结果与已有文献完全一致。该解法有望推广至更多尚未得到解析解的矩形板的振动问题。     

弯曲波对含多圆孔薄板的散射与动应力集中

采用复变函数法和多极坐标方法,研究了弯曲波对含有多圆孔薄板的散射问题。通过板的弯曲波动方程和内力方程的推导,求出在入射弯曲波条件下该问题的一般解的函数逼近序列和边界条件的表达式。用展开正交函数的方法将待解的问题归结为对一组无穷代数方程组的求解。最后,给出了含3圆孔薄板的孔边动应力集中系数的结果,并分析了孔间距和波数对动应力分布的影响。     

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.     

弹性矩形板问题的Hamilton正则方程

为了采用辛算法求出弹性矩形板问题的解析解，中直接从弹性矩形板的控制方程出发推导了弹性矩形板，其中包括弹性矩形薄板和厚板问题以及弹性地基上矩形薄板和厚板问题的Hamilton正则方程，为利用辛几何方法求出任意边界条件下这类问题的理论解奠定了基础．     

On new symplectic superposition method for exact bending solutions of rectangular cantilever thin plates

A novel superposition method based on the symplectic geometry approach is presented for exact bending analysis of rectangular cantilever thin plates. The basic equations for rectangular thin plate are first transferred into Hamilton canonical equations. By the symplectic geometry method, the analytic solutions to some problems for plates with slidingly supported edges are derived. Then the exact bending solutions of rectangular cantilever thin plates are obtained using the method of superposition. The symplectic superposition method developed in this paper is completely rational compared with the conventional analytical ones because the predetermination of deflection functions, which is indispensable in existing methods, is dispelled.     

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.     

Exact bending solutions of orthotropic rectangular cantilever thin plates subjected to arbitrary loads

Exact bending solutions of orthotropic rectangular cantilever thin plates subjected to arbitrary loads are derived by using a novel double finite integral transform method. Since only the basic elasticity equations for orthotropic thin plates are used, the method presented in this paper eliminates the need to predetermine the deformation function and is hence completely rational thus more accurate than conventional semi-inverse methods, which presents a breakthrough in solving plate bending problems as they have long been bottlenecks in the history of elasticity. Numerical results are presented to demonstrate the validity and accuracy of the approach as compared with those previously reported in the literature     

On new symplectic approach for exact bending solutions of moderately thick rectangular plates with two opposite edges simply supported

The symplectic geometry method is introduced for exact bending solutions of moderately thick rectangular plates with two opposite edges simply supported. The basic equations for the plates are first transferred into Hamilton canonical equations. The whole state variables are then separated. Using the method of eigenfunction expansion in the symplectic geometry, typical examples for plates with selected boundary conditions are solved and exact bending solutions obtained. Since only the basic elasticity equations of the plates are used, this method eliminates the need to pre-determine the deformation function and is hence more reasonable than conventional methods. Numerical results were presented to demonstrate the validity and accuracy of this approach as compared to those reported in other literatures.