Half Space and Quarter Space Dirichlet Problems for the Partial Differential Equation

A customary, heuristic, method, by which the Poisson integral formula for the Dirichlet problem, for the half space, for Laplace's equation is obtained, involves Green's function, and Kelvin's method of images. Although this heuristic method leads one to guess the correct result, this Poisson formula still has to be verified directly, independently of the method by which it was arrived at, in order to be absolutely certain that a solution of the Dirichlet problem for the half space, for Laplace's equation, has been actually obtained. A similar heuristic method, as seems to be generally known, could be followed in solving the Dirichlet problem, for the half space, for the equation where is a real constant. However, in Part 1, a different, labor-saving, method is used to study Dirichlet problems for the equation. This method is essentially based on what Hadamard called the method of descent. Indeed, it is shown that he who has solved the half space Dirichlet problem for Laplace's equation has already solved the half space Dirichlet problem for the equation In Part 2, the solution formula for the quarter space Dirichlet problem for Laplace's equation is obtained from the Poisson integral formula for the half space Dirichlet problem for Laplace's equation. A representation theorem for harmonic functions in the quarter space is deduced. The method of descent is used, in Part 3, to obtain the solution formula for the quarter space Dirichlet problem for the equation by means of the solution formula for the quarter space Dirichlet problem for Laplace's equation. So that, indeed, it is also shown that he who has solved the quarter space Dirichlet problem for Laplace's equation has already solved the quarter space Dirichlet problem for the " equation" For the sake of completeness and clarity, and for the convenience of the reader, the appendix, at the end of Part 3, contains a detailed proof that the Poisson integral formula solves the half space Dirichlet problem for Laplace's equation. The Bibliography for Parts 1,2, 3 is to be found at the end of Part 1.     

Pull-in Instability Analysis of Nanoelectromechanical Rectangular Plates Including the Intermolecular,Hydrostatic, and Thermal Actuations Using an Analytical Solution Methodology

The current paper presents a thorough study on the pull-in instability of nanoelectromechanical rectangular plates under intermolecular, hydrostatic, and thermal actuations. Based on the Kirchhoff theory along with Eringen's nonlocal elasticity theory, a nonclassical model is developed. Using the Galerkin method(GM), the governing equation which is a nonlinear partial differential equation(NLPDE) of the fourth order is converted to a nonlinear ordinary differential equation(NLODE) in the time domain. Then, the reduced NLODE is solved analytically by means of the homotopy analysis method. At the end, the effects of model parameters as well as the nonlocal parameter on the deflection, nonlinear frequency, and dynamic pull-in voltage are explored.     

Elastic stability of all edges simply supported,stepped and stiffened rectangular plate under Biaxial loading

In this paper using finite difference method the lower bound buckling load for simply supported (a) stepped and stiffened rectangular thin plate (b) linear and non-linear variation of thickness (c) uniformly distributed compressive forces in both directions (d) uniformly distributed compressive force in y direction and non-uniform distribution of compressive force in x-direction is discussed. The thin plate is divided into 900 rectangular meshes. The partial derivatives are approximated using central difference formula. Eight hundred and forty one equations are formed and using the program developed and the least eigenvalue is obtained. The buckling coefficients are calculated for different types of stepped and non prismatic plates and the results are presented in tables and graphs for ready use by designers. Buckling factors for some cases are presented in the form of three separate tables and compared with the values obtained by Xiang, Wei and Wang. The results are in close agreement.     

Structural response of stiffened plates in similitude under a turbulent boundary layer excitation

In structural dynamics, similitude laws usually deal with simple configurations as thin flat plates with point forces. Only recently, few papers have analyzed stiffened shells or stochastic pressure loads.This research activity extends the applicability of some similitude laws, developed for thin flat plates under a turbulent boundary layer load, to ribbed plates forced by the same wall pressure fluctuations.The work addresses the problem of designing a scaled experimental test-article and, successively, of re-modulating the measured data in order to get the structural response of an original (unscaled) configuration.Due to the complexity of the structural domain, the design of a scaled configuration leads to a distorted similitude. Then, a simple approach, to circumvent the distortion effects, is proposed.     

双向弹簧夹层假定的弹性地基上双层板的解

建立了轴对称条件下,层间有双向弹簧夹层的弹性地基上双层板力学模型,应用Hankel(汉克尔)变换法推演得到了任意轴对称荷载作用下的Winkler(文克勒)地基、双参数地基和弹性半空间体地基上无限大双层板的一般解析解,给出了双层板的挠度、弯矩、剪力,以及层间反力和位移的计算公式.进而,利用该力学模型的解,分析了层间条件对双层板挠度、弯矩的影响规律,计算了上、下层板的中性轴位置,讨论了层间双向弹簧系数的取值方法.结果表明:1)随着层间竖向弹簧参数增大,上层板挠度和弯曲应力减小,下层板挠度和弯曲应力增大;随着层间水平摩阻参数增大,上、下层板的挠度和弯矩均减小;2)当双层板的剪切和压缩效应系数分别取2/3,3/5时,双层板的剪切和压缩效应可较好地被考虑;3)上、下层板的中性轴位置是变化的,它随着距荷载圆中心点的距离增大而向上、下层板各自中面趋近.     

环形线负载驱动变曲率反射镜技术

从薄板弹性理论出发,对可实现曲率变化的环形线负载驱动模型进行分析,给出了基于该模型的大镜厚比变曲率反射镜的形变方程.以较小的驱动力实现较大的中心形变为目标,利用MATLAB软件对不同反射镜厚度、驱动环半径下的反射镜形变情况进行模拟计算,结果表明,反射镜厚度范围在2~4 mm之间、驱动环半径数值在反射镜有效半径1/2处最佳.以此为依据,设计并研制了口径为100 mm、厚度为3 mm的铍青铜环形线负载驱动变曲率反射镜结构及原型样片,给出了变曲率反射镜整体结构前10阶的振动模态分析结果.完成装配后,反射镜原型样片的面形精度接近λ/30(λ为波长).对该结构进行极限曲率变化和面形精度保持的验证实验,通过对变曲率反射镜结构进行改进,环形线负载驱动能够实现超过30个波长(632.8 nm)的中心形变,且面形精度的变化与反射镜中心矢高的变化呈弱相关.     

An improved high-throughput liquid chromatographic/tandem mass spectrometric method for terbinafine quantification in human plasma, using automated liquid-liquid extraction based on 96-well format plates

A fully automated high-throughput liquid chromatography/tandem mass spectrometry (LC-MS/MS) method was developed for terbinafine quantification in human plasma. The plasma samples were treated by liquid-liquid extraction (LLE) in 2.2 mL 96-deepwell plates. Terbinafine and the internal standard (IS) N-methyl-1-naphthalenemethylamine were extracted from human plasma by LLE, using a mixture of methyl t-butyl ether (MTBE)-hexane (70:30, v/v) as the organic solvent. All liquid transfer steps, including preparation of calibration standards and quality control samples, as well as the addition of the IS, were performed automatically by using robotic liquid handling workstations. After vortexing, centrifugation and freezing, the supernatant organic solvent was evaporated and reconstituted in a small volume of a reconstitution solution. Sample analysis was performed by reversed-phase LC-MS/MS, with positive ion electrospray ionization, using multiple reaction monitoring (MRM). The method had a very short sample preparation time and a chromatographic run time of 2.2 min. It was proved to have excellent sensitivity, specificity, accuracy as well as inter- and intraday precision for the quantification of terbinafine in human plasma. The calibration curve was linear for the range of concentrations 5.0-2000.0 ng/mL. The proposed method was applied to the rapid and reliable determination of terbinafine in a bioequivalence study after per os administration of 250 mg tablet formulations of terbinafine.     

Bending problems of rectangular Reissner plate with free edges laid on tensionless Winkler foundations

In this paper,an analytical method for solving the bending problems of rectangularReissner plate with free edges under arbitrary loads laid on tensionless Winkler foundationsis proposed.By assuming proper form of Fourier series with supplementary terms,whichmeet derivable conditions,for deflection and shear force functions,the basic differentialequations with given boundary conditions can be transformed into a set of simple infinitealgebraic equations.For common Winkler foundations,this set of equations can be solveddirectly and for tensionless Winkler foundations,it is a set of weak nonlinear algebraicequations,the solution of which can be obtained easily by using iterative procedures.     

Thermal-buckling of thin annular plates under multiple loads

On the basis of Von Kármán equations,the thermal-buckling of thin annular platessubjected to a field of non-uniform axisymmetric temperature and a variety of boundaryconditions is discussed.The linearized problem is analyzed and stability boundaries whichcharacterize instability of a plate are obtained by means of numerical and analysis methods.