具有中间支承的矩形板自由振动分析

应用一般解析解来求解具有中间支承矩形板的自由振动问题。一般解析解能求解任意边界条件矩形板的振动问题,求解过程是将整块板看成是沿中间支承分开的两块板,沿支承边两块板的挠度均等于零,斜度和弯矩均相等,再由全部边界条件和连续性条件可以求解各阶频率和振型。对几种具有简支边,平夹边或自由边的混合边界矩形板进行了计算。     

混合边界矩形板的自由振动分析

应用一块板的一般解析解来求解混合边界矩形板的自由振动问题。一块板的一般解析解适用于求解任意边界条件矩形板的振动问题。混合边界矩形板的求解过程是将整块板分解为若干块板,每块板仅有均匀的边界,每两块相连的板边则具有连续性条件。利用全部边界条件和连续性条件即可求解各阶固有频率和振型。对几种具有简支边、平夹边或自由边混合边界的板做了计算,并与其它文献结果进行了对比。     

箱形正交异性矩形板结构的自由振动分析

箱形正交异性矩形板结构为由四块板组成的结构。可以用一般解析解来求解这种结构的自由振动问题。这种解能用于求解板具有任意边界。对于每块板,其上边和下边具有边界条件,而在相连的两块板边则具有连续性条件。由四块板的全部条件方程式即可求解自然频率及其振型。为求解所有频率,可利用变形的对称和反对称条件,以矩形的和正方形的板结构为例进行了计算和分析。     

转动弹性支承矩形板自由振动的一般解析解法

转动弹性支承边与简支边,平夹边均有不同,一方面板边均能防止上下移动,即其挠度均为零。而转动弹性支承边,由于在边界装有均匀分布的转动弹簧使边界弯矩受到斜度的制约而与板边的斜度成正比。采用矩形薄板自由振动横向位移函数的微分方程建立了一般性的解析解,该一般解包括三角函数和双曲线函数组成的解,它能满足4个边为任意边界条件的问题。解中的待定常数可由4边的边界条件来确定,由此得出的齐次线性代数方程系数矩阵行列式等于零可以精确地求得各阶固有频率及其振型。由于矩形板对中间轴具有对称性,利用对称和反对称条件可使求解大大简化。对于正方形板还可利用对角线的对称性而毫无遗漏地找出最低的各阶频率及其振型。以四边均为转动弹性支承方板为例进行计算和讨论。     

两对边简支另两对边任意支承的板内有任意多个点支矩形板横向自振的精确解析解

本文研究两对边简支另两对边任意支承的板内有任意多个点支矩形板的横向自由振动问题,将点支反力视为作用于板上的待定外力,求得了在待定外力作用下两对边简支矩形板动态响应的解析解,由矩形板另两对边的边界条件确定待定积分常数,利用点支处板位移为零条件决定待求的点支反力,并得到以一阶数等于点支个数的行列式表示的频率方程,由计算机求解各阶频率参数,方法有独特的优越性,本文最后给出了一个数值算例。     

离散支承矩形板的自由振动

本文研究两对边简支，另两对边离散支承矩形的自由振动问题，推导出了用支点反力表示的矩形板自由振动的振型函数，即Ｇｒｅｅｎ函数，利用支点处的位移相容性，确定支点反力和频率方程，可以求解任意阶的固有频率和振型。     

弹性矩形薄板弯曲问题的一个解法

本文建立一个关于弹性矩形薄板弯曲的解析解法。该方法适用于求解任意荷载及边界条件下矩形薄板的弯曲问题，特别是对具有边梁支承的矩形板，更显示出特有的优越性。     

一边支承三边自由矩形板振形曲线及其正交性

本文提出主振方向排序法的思想 ,建立了一边支承三边自由矩形板统一的振形函数表达式。该振形曲线基本符合板边界条件所限定的变形曲线形态 ,可以满足全部边界条件 ,又具有振形的正交性。本文阐明了主振方向排序法的物理含义 ,推导了 2种矩形板自由振动频率方程 ,计算了各 2 5个自振频率及相应的振形。     

一对边支承一对边自由的矩形板弯曲统一求解方法

根据边界条件的相近性，本文对一对边支承一对边自由的矩形板提出了一种统一的弯曲挠度表达式。该表达式切合板边界所能激发出的弯曲变形形态和角点位时所导致的非弯曲变形特点。它可以计算矩形板在任意荷载作用下和板边界发生任意支座位移时的弯曲。该方法求解思路清晰，收敛速度快，计算精度高。     

边界配点法解中厚板弯曲问题

采用边界配点法解决具有剪切变形的矩形板的弯曲问题,既有效地利用解析法分析的成果,又发挥了离散计算方法的优点,本文给出在均布荷载作用下四边简支,四边固定,两对边简支另两对边固支等边界条件的算例表明:本法的精度高,计算工作量少,能用微型机解决问题,具有工程实用价值,对于其它支承条件、荷载情况,文内提出了求解的途径。     

双参数弹性地基上受压的正交异性板的自由振动

双参数弹性地基上面内受压的正交异性矩形薄板自由振动问题可分两种情况求解;当板的四边为简支时可用双正弦级数解法来求得各阶固有频率。对其他任意边界情形,可采用分离变量法先求得各种代数多项式解以及单正弦级数解,然后建立一个适用于除四边简支外能满足四边以及四角的一般解,其中的积分常数由边界条件来确定。以四边简支和平夹的正方形板为例进行了计算和分析。这种解法简单全面,便于实际应用。全部公式同样可用来求解板的稳定性问题。此时令板的固有频率为零,两个对边压力的比或其中一个为已知即可求得各阶临界压力。     

受压对称迭层矩形板的自由振动分析

根据受压的对称迭层矩形板自由振动的微分方程可以求得各种解析解来求解各种边值问题。迭层板有两种,一种是正交铺设,其方向与坐标轴平行,属正交异性板,当板的四边为简支时,可用双正弦级数来求解自由振动的各阶频率及其振型以及均匀受压的各阶临界载荷及其屈型。另一种是角铺设,属各向异性板,当两相邻边为自由,另两边为简支或固支时可用复数级数来求解其最低频率及其振型以及最低临界载荷及其屈型。此时其特征方程的根为两对复根,且可表成三角级数和双曲线级数,以满足边界条件。另外用代数多项式和双正弦级数组成的解来满足角点条件。在算例中计算了若干板受压或不受压的振动频率和临界载荷,并与其他文献进行了对比。     

正交异性矩形薄板自由振动的一般解析解

本文建立了正交异性矩形薄板自由振动横向位移函数微分方程的一般解。可用来求解任意边界矩形板的振动问题。作为例题,计算了复合材料悬臂层合板的频率。其结果与试验值基本吻合。     

Closed and approximate analytical solutions for rectangular Mindlin plates

Summary Basing on the Nádai-Lévy and the Vlasov-Kantorovich methods closed and approximate analytical solutions of Mindlin's plate equations in the case of rectangular plates are discussed. For elastic, homogeneous and isotropic plates three unknowns of the governing two-dimensional boundary value problem are formulated as series of products of functions depending on a single coordinate. Specifying the functions for one of the in-plane coordinate directions the governing partial differential equations for a special type of boundary conditions and the principle of virtual displacements for the general case yield a set of ordinary differential equations. The analytical solution of these equations provides expressions for functions depending on the other in-plane coordinate. For plates with simply supported edges for one of the coordinate directions and for arbitrary homogeneous boundary conditions for the other one the Nádai-Lévy method provides a closed or exact solution in the sense that the infinite series for displacements and stress resultants can be truncated to obtain any desired accuracy. In the general case of nonsimply supported edges the iterative Vlasov-Kantorovich method yields an approximate analytical solution. Both methods are nonsensitive to a reduction of the thickness with respect to accuracy and represent the boundary layer solutions in terms of exponential functions. Applications to rectangular plates with various types of boundary conditions are presented.     

Vibration of clamped moderately thick general cross-ply plates using a generalized Navier approach

A hitherto unavailable analytical solution to the free vibration problem of general cross-ply laminated rigidly clamped rectangular plates, incorporating first-order shear deformation, and rotatory and in-plane inertias into the formulation, is presented. A recently developed boundary continuous displacement-based generalized Navier solution technique is used to solve the five highly coupled linear second-order partial differential equations with constant coefficients, and the associated geometric boundary conditions. The assumed solution functions are in the form of double Fourier series, which satisfy the rigidly clamped boundary conditions a priori in a manner similar to the conventional Navier method. Convergence characteristics of the natural frequencies of both symmetric and antisymmetric cross-ply plates are numerically established. Other numerical results presented herein include (i) comparison with the corresponding available first-order shear deformation theory-based Galerkin and classical lamination theory-based boundary-discontinuous analytical solutions, and (ii) study of the effects of thickness and aspect ratio on the natural frequencies.     

An accurate solution method for the static and dynamic deflections of orthotropic plates with general boundary conditions

Extending the previous work on isotropic beams and plates by the second author [Li WL, et al. An exact series solution for the transverse vibration of rectangular plates with general elastic boundary supports. J Sound Vib 2009;321:254–69], this paper describes an accurate analytical method for calculating the static deflections and modal characteristics of orthotropic plates with general elastic boundary supports. The displacement function is expressed as a 2-D Fourier cosine series supplemented with several terms in the form of 1-D series. The series expansions for all the relevant derivatives can be directly obtained through term-by-term differentiations of the displacement series. Thus, a classical solution can be derived by letting the series exactly satisfy the governing differential equation at every field point and all the boundary conditions at every boundary point, respectively. Several numerical examples are presented to demonstrate the excellent accuracy and convergence of the current solutions.     

各向异性板平面应力问题的一般解析解

根据各向异性矩形板平面应力问题的基本方程精确地求得了位移函数的一般解。一般解包括三角函数和双曲线函数组成的解,还有代数多项式解。前者能满足四个边为任意边界条件问题;后者能满足四个角为任意边界条件等问题。这些边界条件用以决定一般解中的积分常数。因此一般解可用以求解任意边界条件下的平面应力问题。以四边为均匀分布和非均匀分布位移的对称角铺设复合材料板为例进行了计算和分析     

Fourier analysis of thick cross-ply Levy type clamped doubly-curved panels

A hitherto unavailable Levy type analytical solution to the problem of deformation of a finite-dimensional general cross-ply thick doubly-curved panel of rectangular plan-form, modeled using a higher order shear deformation theory (HSDT), is presented. A solution methodology, based on a boundary-discontinuous generalized double Fourier series approach is used to solve a system of five highly coupled linear partial differential equations, generated by the HSDT-based laminated shell analysis, with the C4-type simply supported boundary condition prescribed on two opposite edges, while the remaining two edges are subjected to the SS3-type constraint. The numerical accuracy of the solution is ascertained by studying the convergence characteristics of the deflection and moment of a cross-ply spherical panel, and also by comparison with the available FSDT (first-order shear deformation theory) based analytical solution. Additionally, numerical results pertaining to flat symmetric and antisymmetric cross-ply plates with the same boundary conditions have also been reproduced. Hitherto unavailable important numerical results presented include sensitivity of the predicted response quantities of interest to shell geometry (cylindrical and spherical), lamination, lamina material property, and thickness effects, as well as their interactions. Comparison with their SS2 counterparts demonstrates the effect of end clamping on the deflections and moments of thin to thick singly- and doubly-curved cross-ply panels.