功能梯度压电圆板轴对称自由振动问题精确解

将功能梯度压电圆板的位移变量和电势变量写为分离变量的形式，由压电动力学平衡方程导出以位移、电势及其一阶导数为状态变量的状态方程，考虑周边固支接地的边界条件，导出了求解功能梯度压电圆板自振频率精确解的方程。将方程退化至一般的非梯度纯弹性圆板的形式，求解其自振频率，得到的结果与相应的理论解完全吻合，从而验证了本文方法的正确性。更进一步地对梯度函数沿板厚以指数形式变化的功能梯度压电圆板的自振频率进行了计算，并得到了梯度化对板自振频率的影响规律。     

轴对称von Korman位移型方程的打靶法求解

本文采用常微分方程两点边值问题的打靶法，建立了圆薄板轴对称大挠度弯曲ｖｏｎＫáｒｍáｎ位移型方程的自动求解过程．作为例子，分析了圆薄板在均布横向截荷作用下的非线性弯曲问题，给出了载荷参数大范围变化的解曲线     

轴对称von Korman位移型方程的打靶法求解

本文采用常微分方程两点边值问题的打靶法，建立了圆薄板轴对称大挠度弯曲ｖｏｎＫáｒｍáｎ位移型方程的自动求解过程．作为例子，分析了圆薄板在均布横向截荷作用下的非线性弯曲问题，给出了载荷参数大范围变化的解曲线     

上覆单相弹性层的饱和地基上刚性圆板的扭转振动分析

采用解析的方法研究了上覆单相弹层的饱和地基上刚性圆板的扭转振动。首先运用积分变换技术分别求解了单相弹性介质和饱和介质情况时的控制方程，然后按混合边值条件建立了上覆单相弹性层的饱和地基上刚性圆板扭转振动的对偶积分方程，并把对偶积分方程化为易于数值求解的第二类Fredholm积分方程，并给出了数值算例。     

多圆孔圆板问题的数值解

提出了一种求解多圆孔圆板问题的新方法。首先引入了基本特解，它由主要部分和附加部分组成。主要部分为带奇点无限平板的一个特殊弹性力学解，奇点取在内圆孔的中心处。附加部分为实心圆饭的一个特解。整个基本特解满足外圆周界为自由条件。文中把待求解取为特解系的形式，其中待定系数可用变分原理得出。最后给出了算例。     

饱和地基上含刚核弹性圆板的竖向振动分析

基于作者提出的饱和土弹性波动方程，研究了饱和地基上含刚核弹性圆板在竖向集中谐和力作用下的振动特性。首先应用Hankel积分变换求解该饱和土波动方程，然后按混合边值条件建立起一组描述含刚核弹性圆板振动的对偶积分方程，并将其化为第二类Fredholm积分方程进行数值求解。文末给出了含刚核弹性圆板在饱和地基上振动的阻抗函数随无量纲频率a0的变化曲线，并考察了土的渗透系数，弹性板含刚域的大小以及板的柔度等参数对阻抗函数的影响，得出了一些有意义的结论。     

轴对称圆薄板大挠度微分方程的数值分析方法

根据轴对称问题的特点,利用级数展开和求极限法则,证明了轴对称大挠度圆薄板在圆心处应满足的边界条件,并以圆薄板轴对称大挠度弯曲变形微分方程为基础,建立了圆心处非奇异的轴对称大挠度圆板弯曲微分方程,从而可以方便地利用现有的常微分方程数值求解方法(如变步长龙格-库塔法)对实心圆板的轴对称问题进行数值求解,又不必像摄动法那样推导复杂的公式。在数值求解轴对称圆板大挠度弯曲变形微分方程时,将非线性微分方程的求解主要归结为迭代求解圆心处三个未知边界条件的问题,即圆心处的径向膜力、圆心处的挠度、圆心处挠度的二阶导数,并提出了相应的求解方法。实例中,对于圆薄板受均布横向荷载的问题,分析了周边固支边界条件下的非线性弯曲问题,给出了中心挠度参数大范围变化时的荷载和部分边界值变化曲线,并与经典摄动解进行了对比。对比结果可见,本文方法和摄动法的解非常接近,在量纲归一化中心挠度不超过4.0时,两种方法解的相对误差均小于5.0%。另外,本文还分析了与挠度有关的液体压力作用下和集中荷载作用下周边固支圆板的非线性弯曲问题。通过算例可见:本文方法可以灵活处理不同的荷载问题;对于不同的问题,计算过程相似,不必推导复杂的计算公式,计算精度容易控制。     

圆板在物体撞击下的非线性动力响应

本文在Von Kármán大位移的意义上,利用虚位移原理伽辽金方法建立了圆板在物体撞击下的非线性动力响应的控制微分方程,在研究响应问题时,考虑了冲击载荷与圆板位移响应之间的耦合影响,文中使用时间增量法和奇异摄动理论求解问题的控制方程,获得了固支圆板非线性动力响应的近似解,并且求解了具体算例,绘出了圆板位移、应力响应曲线以及冲击力随时间的变化曲线。     

热环境中旋转运动功能梯度圆板的强非线性固有振动

研究热环境中旋转运动功能梯度圆板的非线性固有振动问题．针对金属－陶瓷功能梯度圆板，考虑几何非线性、材料物理属性参数随温度变化以及材料组分沿厚度方向按幂律分布的情况，应用哈密顿原理推得热环境中旋转运动功能梯度圆板的非线性振动微分方程．考虑周边夹支边界条件，利用伽辽金法得到了横向非线性固有振动方程，并确定了静载荷引起的静挠度．用改进的多尺度法求解强非线性方程，得出非线性固有频率表达式．通过算例，分析了旋转运动功能梯度圆板固有频率随转速、温度等参量的变化情况．结果表明，非线性固有频率随金属含量的增加而降低；随转速和圆板厚度的增大而升高；随功能梯度圆板表面温度的升高而降低．     

功能梯度与均匀圆板静动态解之间的相似转换关系

基于经典板理论,研究了功能梯度材料圆板的轴对称弯曲、屈曲和自由振动解与相应的均匀材料圆板解之间的转换关系.通过消去拉-弯耦合项得到了以挠度函数表示的功能梯度圆板的弯曲、屈曲和自由振动控制方程.分析功能梯度圆板与均匀圆板的控制方程之间的相似性,得到了功能梯度材料圆板与均匀圆板的解之间解的相似转换关系,在假定FGM圆板的材料性质沿厚分别以幂函数和指数函数的度变规律后,给出了相应的转换系数的解析表达式.该系数集中反映了功能梯度圆板的材料非均匀性.在已知均匀材料圆板轴对称解的条件下,可将功能梯度材料圆板轴对称问题的求解转化为相似转换系数的计算问题.这一方法可为非均匀板的求解提供了十分便捷有效的途径,而且便于工程应用.     

Analysis of the transpiration cooling of a thin porous plate in a hot laminar convective flow

This paper deals with the asymptotic and numerical analysis for the steady-state transpiration cooling of a thin porous flat plate in a laminar hot convective flow, taking into account the streamwise heat conduction through the plate. For high conductivity plates, a regular perturbation analysis has been carried out, yielding a three-term asymptotic solution for the distribution of plate temperature. In the limit of a very poorly conducting plate, a singular perturbation technique, based on matched asymptotic expansions, is employed to solve the governing equations. We also solved the equations numerically using a quasilinearization technique. The numerical results are in good agreement with the asymptotic solution close to the asymptotic limits studied.     

Bending of corrugated thick rectangular plates

An approximate analytical solution describing the bending of hinged corrugated thick plates is obtained using the second variant of the boundary shape perturbation method and taking into account the first three approximations. The effect of the shape of the boundary surfaces in the zone of maximum external load on the magnitude and nonlinear variation of the displacements and stresses throughout the thickness of a corrugated thick plate depending on the corrugation amplitude and spatial frequency is analyzed. The results are compared to the exact solution for a flat plate     

Shear deformation effect in plates stiffened by parallel beams

In this paper a general solution for the analysis of shear deformable stiffened plates subjected to arbitrary loading is presented. According to the proposed model, the arbitrarily placed parallel stiffening beams of arbitrary doubly symmetric cross section are isolated from the plate by sections in the lower outer surface of the plate, taking into account the arising tractions in all directions at the fictitious interfaces. These tractions are integrated with respect to each half of the interface width resulting two interface lines, along which the loading of the beams as well as the additional loading of the plate is defined. Their unknown distribution is established by applying continuity conditions in all directions at the interfaces. The utilization of two interface lines for each beam enables the nonuniform distribution of the interface transverse shear forces and the nonuniform torsional response of the beams to be taken into account. The analysis of both the plate and the beams is accomplished on their deformed shape taking into account second-order effects. The analysis of the plate is based on Reissner’s theory, which may be considered as the standard thick plate theory with which all others are compared, while the analysis of the beams is performed employing the linearized second order theory taking into account shear deformation effect. Six boundary value problems are formulated and solved using the analog equation method (AEM), a BEM based method. The solution of the aforementioned plate and beam problems, which are nonlinearly coupled, is achieved using iterative numerical methods. The adopted model permits the evaluation of the shear forces at the interfaces in both directions, the knowledge of which is very important in the design of prefabricated ribbed plates. The effectiveness, the range of applications of the proposed method and the influence of shear deformation effect are illustrated by working out numerical examples with great practical interest.     

Hall effects on hydromagnetic flow on an oscillating porous plate

In this paper, an analysis is made on the unsteady flow of an incompressible electrically conducting viscous fluid bounded by an infinite porous flat plate. The plate executes harmonic oscillations at a frequency n in its own plane. A uniform magnetic field Ho is imposed perpendicular to the direction of the flow. It is found that the solution also exists for blowing at the plate. The temperature distribution is also obtained by taking viscous and Joule dissipation into account. The mean wall temperature θo（O） decreases with the increase in the Hall parameter m. It is found that no temperature distribution exists for the blowing at the plate.     

刚塑性矩形板在冲击载荷作用下的一个新的界限

通过加权残数法，给出了考虑有限变形时刚塑性板在冲击载荷下的能量守恒方程。广义屈服条件由Mij和Nij表示，通过将非光滑屈服函数用其内切和外接光滑函数代替，从而得到板位移的上、下界限解。采用此方法对一矩形板受横向冲击载荷进行了分析，计算结果与有关文献符合较好。     

HYDROELASTIC VIBRATION OF A CIRCULAR CONTAINER BOTTOM PLATE USING THE GALERKIN METHOD

The free vibration of a flexible thin plate placed into a circular hole and elastically connected to the rigid bottom slab of a circular cylindrical container filled with fluid having a free surface is studied. The liquid is assumed to be incompressible, inviscid and irrotational. The effect of the free surface wave is also taken into account in the analysis. First of all, the exact expression of velocity potential of the liquid movement is derived by a combination of the superposition method and the method of separation of variables. With the help of the Fourier–Bessel series expansion, part of the unknown coefficients in the solution is determined by the consistency condition between the liquid movement and the plate vibration, in the form of integrals associated with the dynamic deflection of the plate. Then, the Galerkin method is applied to derive the eigenfrequency equation of the fluid–plate interaction. Finally, the effects of various parameters and the free surface wave on eigenfrequencies of the fluid–plate system are discussed. As a consequence, the accuracy of the nondimensional added virtual mass incremental (NAVMI) factor solution has also been evaluated by comparing with the more accurate Galerkin solution. It is shown that the proposed method is also applicable to the vibration analysis of circular plates in contact with an infinite liquid by only taking a finite but larger size of liquid to replace the infinite liquid in the computation.     

Forced and mixed convection boundary layer flow along a flat plate with variable viscosity and variable Prandtl number: new results

The steady laminar boundary layer flow along a flat plate is studied taking into account the variation of fluid viscosity and fluid Prandtl number with temperature. In the forced convection case the plate moves with constant velocity and its temperature varies in power law with x. In the mixed convection case the plate temperature is constant and the fluid moves upwards due to an external free stream and due to buoyancy forces. The results are obtained with the direct numerical solution of the boundary layer equations. The study concerns the wall heat transfer, the wall shear stress and velocity and temperature profiles across the boundary layer. The results of the present work are different from those existing in the literature, which have been obtained with the assumption of constant Pr number.     

Formation of ice in plane stagnation flow

Summary The problem of unsteady formation of ice on an instantaneously cooled plate normal to an otherwise isothermal flow of liquid is studied analytically, taking into account the mutual interaction between the flow and the ice-formation process. Short-time solution is obtained by asymptotic series expansions, while long-time solution, based on first-order deviations from steady conditions, is derived by Laplace transforms. Results are presented in the form of thickness variations and distortions of velocity profiles by the advance of the ice front, and discussed in terms of two physical parameters.Fluid Mechanics and Heat Transfer Laboratory.     

Mixed Convection in Water Near the Density Extremum Along a Vertical Plate with Sinusoidal Surface Temperature Variation Embedded in a Porous Medium

A steady laminar boundary layer flowing along a vertical plate immersed in a Darcy–Brinkman porous medium saturated with water at 4°C is studied. The plate temperature varies sinusoidally along the plate between 0 and 8°C where the density of water varies parabolically and is almost symmetrical at about 4°C. Except for the existence of the buoyancy force, it is assumed that either the plate moves upwards or the ambient water moves upwards (moving stream). The results are obtained with the direct numerical solution of the boundary layer equations taking into account the temperature dependence of water thermophysical properties (ρ, μ and c _p). Results are presented for the wall temperature gradient and the wall shear stress along the plate for free convection and mixed convection. Temperature and velocity profiles are also presented.     

Multi-modal approach for large natural flexural vibrations of thermally stressed plates

By means of Berger's approximation, suitable for plates with immovable edges, the geometrically nonlinear problem is considerably simplified. Thermally loaded plates with polygonal planform under hard-hinged support conditions are considered, taking into account the effect of shear in transverse isotropy. The class of symmetric vibrations about the flat plate position is represented by a homogeneous and coupled set of Duffing-oscillators as a result of a multi-mode expansion. A unifying non-dimensional closed-form solution for the corresponding nonlinear natural vibration periods is given, which is independent of the special planform. The individual shape of the plate enters the transformation into real time through the linear natural frequencies, or, equivalently, through the linear eigen-values of an effectively prestressed membrane of the same planform.