矩形扁壳弯曲问题的一般解

本文建立了四边挠度为零的矩形扁壳弹性弯曲问题的一般解析解.以四边位移为零的固支矩形扁壳为例求解了对称变形问题。     

自由正交异性矩形厚板的动态稳定

对在一条边上作用着均匀分布的非保守跟随力的四边自由正交异性矩形厚板的动态稳定进行了分析，通过把位移和剪力展成重傅立叶级数解，把微分方程简化成了代数方程。计算表明厚度的微小变化会引起颤振载荷明显的减小。这个明显减小是因为存在剪切变形。     

变系数曲线支承矩形厚板自由振动的傅里叶分析

本文给出了变系数曲线支承的Ａｍｂａｒｓｕｍｉａｎ矩形厚板自由振动问题的级数解，将位移和剪力在板域内展成重傅里叶级数，将其导数在边界上展成单傅里叶级数，通过傅里叶变换将控制微分方程和边界条件转化成关于位移级数的系数的一组无穷线性代数方程，最终将板的自由振动问题转化为矩阵特征值问题。     

不同支承条件矩形中厚板的精确解

本文在Ｒｅｉｓｓｎｅｒ板理论中和学应力变量方法得到挠度函数ｗ（ｘ，ｙ）和应力函数ψ（ｘ，ｙ）解。在这些解中，选择一些三角级数和多项式作为问题的挠度作为函烽和应力函数，从而得到一些矩形厚板问题的解，例如矩形悬臂厚板，三边固定一边自由矩形厚板等。这里不需要繁锁也叠加。     

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．     

四边自由各向异性矩形中厚板弯曲解析解

将基于阿穆巴诸米扬各向异性中厚板理论的控制方程、不同支撑下建立的板与支撑的协调方程以及板的自由边界条件,通过对称性分解成中心对称问题和中心反对称问题的叠加。利用傅立叶三角级数求解混合奇偶阶偏微分方程组,求得包括板的内力和支撑反力表达式的弯曲解析解。本文方法取消了直法线假设,克服了数学上的求解困难,去除了数值法的弊端,得出的结果更贴近实际。用该方法不但可以研究四边自由各向异性中厚板的弯曲特性和振动特性,而且还可分析不同边界约束下各向异性矩形中厚板的静动力特性。     

相邻两边固支其余两边自由矩形正交各向异性薄板弯曲的辛叠加解

应用辛叠加方法,求出相邻两边固支其余两边自由矩形正交各向异性薄板在均匀荷载下弯曲问题的解析解。先将原方程转换成Hamilton正则方程;用辛方法计算出一边滑支对边简支条件下对应Hamilton算子的辛本征值和辛本征函数系,证明该辛本征函数系的辛正交性,进而在Cauchy主值意义下证明了它的完备性;根据辛本征函数系的完备性,得到了对应Hamilton正则方程的通解,再应用叠加方法计算出原弯曲问题的解析解;最后通过两个具体矩形薄板的数值算例,验证了所得辛叠加解的正确性。     

各向异性矩形薄板弯曲问题的一般解

给出了各向异性矩形薄板弯曲问题微分方程的一般解。可以求解任意载荷作用下各种边界的弯曲问题。以四边固支的正方形板为例进行了数值计算。     

矩形板耦合热冲击问题的摄动解

本文通过对薄板耦合热弯曲问题的完备方程的无量纲化,引出了关于薄板的无量纲热弹性耦合系数,并以此系数为摄动参数,运用奇异摄动方法,导出了其摄动方程,得到了关于矩形薄板耦合热冲击问题的一致有效的渐近解。然后,在对该解讨论及计算的基础上,获得了一些关于该类问题的规律性的结论。     

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。     

EXACT SOLUTIONS FOR FREE IN-PLANE VIBRATIONS OF RECTANGULAR PLATES

All possible exact solutions are successfully obtained in terms of 10 sets of distinct eigensolutions for the free in-plane vibration of isotropic rectangular plates. The plates have simply supported condition at two opposite edges and any combination of classical boundary conditions at the other two edges. The exact solutions are validated through both mathematical proof and comparisons with the solutions of differential quadrature method. Some unusual phenomena are revealed in free in-plane vibrations of rectangular plates due to one of the eigenvalues being zero. This work constitutes an improved version of very recent corresponding work by the same authors [Int. J. Mech. Sci., 2009, 51: 246-255]. Both the solution forms and solving procedures in the previous work are substantially simplified. Some new results are also given, which are useful for validation purpose in future.     

四边简支各向同性矩形厚板的插值矩阵法解

针对强厚度矩形板四边简支情况,论文根据状态变量法思想,基于三维弹性理论基本方程,以3个位移分量及3个应力分量按双三角级数展开,将三维弹性力学控制方程转化为常微分方程边值问题.尽管一些各向异性弹性矩形厚板早已由状态空间法获得分析解,可是各向同性厚板的分析解至今难以获得,因为状态空间解法中特征方程有重根问题而不易于收敛.论文提出采用插值矩阵法直接对常微分方程进行求解,获得各向同性矩形厚板在四边简支边界条件下三维理论的位移和应力解,并与有限元精细结果进行比较,证明了本文解的准确性.     

THE METHOD OF THE RECIPROCAL THEOREM OF FORCED VIBRATION FOR THE ELASTIC THIN RECTANGULAR PLATES(Ⅱ)—RECTANGULAR PLATES WITH TWO ADJACENT CLAMPED EDGES

In this paper,applying the method of reciprocal theorem,we give the distributions ofthe amplitude of bending moments along clamped edges and the amplitude of deflectionsalong free edges of rectangular plates with two adjacent clamped edges under harmonicdistributed and concentrated loads.     

变截面梁板弯曲问题的一般解答

本文利用广义函数研究了截面呈阶梯形变化的梁板弯曲问题,直接导出了挠度通用公式。研究结果表明,不论梁板边界的约束条件怎样,均可归结为求解一个二元一次线性代数方程组的问题,与传统方法相比要简捷得多。     

正交各向异性厚板的边界元解法

本文利用 Hormander 算子法和平面波分解法导出了计入剪切变形的正交各向异性厚板的基本解。建立了计入剪切变形的正交各向异性厚板的边界积分方程。文中详细地讨论了基本解的数值计算,并用边界元法分析了一些算例。     

NONLINEAR DYNAMICS OF A FGM PLATE WITH TWO CLAMPED OPPOSITE EDGES AND TWO FREE EDGES

This paper deals with the nonlinear forced vibration of FGM rectangular plate with a boundary of two edges clamped opposite and the other two free. The plate is subjected to transversal and in-plane excitations. The present research treats the material properties of the FGM plates as temperature-dependent and graded continuously throughout the thickness direction, following the volume fraction of the constituent materials according to the power law. The temperature is assumed to be constant in the plane and varied only in the thickness direction of the plate. In the framework of geometrical nonlinearity the plate is modeled and the equations of motion are obtained on Hamilton＇s principle. With the help of Galerkin discretization, the nonlinear ordinary differential equations describing transverse vibration of the plate are proposed. By the numerical method, the nonlinear dynamical responses of the FGM plate with two clamped opposite and two free edges are analyzed.     

CHARACTERISTIC EQUATIONS AND CLOSED-FORM SOLUTIONS FOR FREE VIBRATIONS OF RECTANGULAR MINDLIN PLATES

The direct separation of variables is used to obtain the closed-form solutions for the free vibrations of rectangular Mindlin plates. Three different characteristic equations are derived by using three different methods. It is found that the deflection can be expressed by means of the four characteristic roots and the two rotations should be expressed by all the six characteristic roots,which is the particularity of Mindlin plate theory. And the closed-form solutions,which satisfy two of the three governing equations and all boundary conditions and are accurate for rectangular plates with moderate thickness,are derived for any combinations of simply supported and clamped edges. The free edges can also be dealt with if the other pair of opposite edges is simply supported. The present results agree well with results published previously by other methods for different aspect ratios and relative thickness.     

A NEW STATE SPACE SOLUTION FOR RECTANGULAR THICK LAMINATES WITH CLAMPED EDGES Hu Wenfeng Liu Yihua

通过把固支边上的边界位移函数作为状态变量引入状态方程,得到了含固支边矩形单层与叠层厚板的精确解. 在求解过程中,将非齐次状态方程的求解变为齐次状态方程的求解,省略了求解待定常数的中间过程,使求解过程变得简单. 所得到的解能够严格满足固支边界条件,在同一材料层内不需作分层处理,因而更加精确. 此外,对固支边的应力提出了新的计算方法,能够得到更精确的边界应力. 算例表明,本文解比现有精确解收敛快,与有限元解吻合的更好,尤其是在固支边处体现得更加明显.