An alternative analytical solution for water-wave motion over a submerged horizontal porous plate

This study gives an alternative analytical solution for water-wave motion over an offshore submerged horizontal porous-plate breakwater in the context of linear potential theory. The matched-eigenfunction-expansions method is used to obtain the solution. The solution consists of a symmetric part and an antisymmetric part. The symmetric part is also the solution of wave reflection by a vertical solid wall with a submerged horizontal porous plate attached to it. In comparison with previous analytical solutions with respect to finite submerged horizontal porous plates, no complex water-wave dispersion relations are included in the present solution. Thus, the present solution is easier for numerical implementation. Numerical examples show that the convergence of the present solution is satisfactory. The results of the present solution also agree well with previous results by different analytical approaches, as well as previous numerical results by different boundary-element methods. The present solution can be easily extended to the case of multi-layer submerged horizontal porous plates, which may be more significant in practice for meeting different tide levels.     

Diffusive-convective transport into a porous membrane. A comparison of theory and experiment using scanning electrochemical microscopy operated in reverse imaging mode

Molecule transfer at the interface between a single ion-selective micropore and aqueous solutions is quantitatively investigated using scanning electrochemical microscopy operated in reverse imaging mode (SECM-RIM). Accumulation of two electroactive solute molecules, acetaminophen and ferrocenylmethyltrimethylammonium, at the pore/solution interface is observed when an electrical current is passed through the pore. Slow interfacial transfer of solute relative to the solvent as the solution is driven across the membrane by electroosmosis is responsible for solute accumulation. A theoretical expression for the concentration distribution of solute molecules above an individual pore opening is obtained by analytical solution of the convective-diffusive flux equation. The fluid velocity through the pore at constant electroosmotic force is determined by fitting the theoretical expression to SECM-RIM concentration profiles and is found, as anticipated, to be independent of the solute species and the bulk solute concentration. The results provide a theoretical basis for the SECM-RIM imaging of biological membranes as well as a general method for characterizing interfacial molecule/ion transfer kinetics.     

Analytical solution for three-dimensional model predicting temperature in the welding cavity of electron beam

This paper provides an analytical solution for three-dimensional model predicting temperature in the welding cavity of electron beam. It is not easy to measure the temperature on the keyhole of electron-beam welding. Therefore it is essential to develop an analytical model that can accurately predict the temperature in the keyhole. In this study, the keyhole produced by an electron beam is assumed to be a paraboloid of revolution and the intensity of electron beam is supposed to be Gaussian profile. In order to obtain an analytical solution, the parabolic coordinate system is utilized to analyze the temperature in the keyhole and the parameter approximating convection is proposed to account for the effect of convection of molten metal. Considering the momentum balance at the bottom of the keyhole but neglecting the absorption in the plume, an analytical solution is developed for semi-infinite workpieces. As compared with other analytical solutions, the analytical solution obtained by this model provides the temperature distribution more consistent with the experimental data. The effects of various parameters on the temperature distribution in the keyhole are also discussed in this study.     

Analytical optimal solution of the extension of the depth of field using cubic-phase wavefront coding. Part I. Reduced-complexity approximate representation of the modulation transfer function

In this paper we derive an approximate analytical representation for the modulation transfer function (MTF) of an imaging system possessing a defocused cubic-phase pupil function. This expression is based on an approximation using the Arctan function and significantly reduces the computational time required to calculate the resulting MTF. We derive rigorous bounds on the minimum and average accuracy of our approximation. Using this approximate representation of the MTF, the analytical solution of the problem of calculating the extension of the depth of field for a circular aperture with a cubic phase mask becomes possible. We also comment on how one can modify our method to construct a lower-bound or an upper-bound approximate analytical expression for the MTF.     

Analytical formulae for calculating stresses in unidirectional/cross-ply unbalanced laminates

A multilayer theory has been developed for unidirectional/cross-ply unbalanced laminates subjected to surface shear tractions, using the assumption of plane strain. The analytical solution for the stress field in the laminate is obtained and it is shown that the stresses in each layer are in good agreement with the results of a three-dimensional finite element analysis. Approximate formulae for the analytical stresses are also derived for a laminate subjected to an offset distributed force at one end. The accuracy of this solution is shown by comparison with a three-dimensional finite element analysis.     

Analytical solutions for time-resolved fluorescence lifetime imaging in a turbid medium such as tissue.

An analytical solution is developed to quantify a site-specific fluorophore lifetime perturbation that occurs, for example, when the local metabolic status is different from that of surrounding tissue. This solution may be used when fluorophores are distributed throughout a highly turbid media and the site of interest is embedded many mean scattering distances from the source and the detector. The perturbation in lifetime is differentiated from photon transit delays by random walk theory. This analytical solution requires a priori knowledge of the tissue-scattering and absorption properties at the excitation and emission wavelengths that may be obtained from concurrent time-resolved reflection measurements. Additionally, the solution has been compared with the exact, numerically solved solution. Thus the presented solution forms the basis for practical lifetime imaging in turbid media such as tissue.     

锚杆支护圆形隧洞弹塑性解

该文推导了锚杆支护前后圆形隧洞的应力、位移和塑性区半径表达式。首先考虑静水压力状态下,岩石表现为弹脆塑性材料模型,采用Mohr-Coulomb线性屈服准则以及非关联流动法则,推导了无支护状态下圆形隧洞的弹塑性解析新解。在此基础上,通过均匀化方法,从宏观尺度将锚杆高密度支护模式下的岩石和锚杆复合体考虑成均匀连续、强度参数增强的等效材料,定义能够反映隧洞支护参数的锚杆密度因子,推导出等效弹性模量,等效粘聚力和等效内摩擦角的表达式,然后通过该文得到的隧洞在无支护状态下的解析解可以得到锚杆支护圆形隧洞解析解。最后将该文结果与前人研究结果及数值结果进行了比较分析,对比了锚杆支护前后圆形隧洞的塑性区半径、塑性区位移和塑性区应力,结果表明锚杆支护对隧洞的加固效果明显,该文结果可为地下工程中圆形隧洞的稳定性分析提供参考。     

分数阶Duffing振子的亚谐共振

研究了含分数阶微分项的Duffing振子的亚谐共振,利用平均法得到了系统的一阶近似解。提出了亚谐共振时等效线性阻尼和等效线性刚度的概念,分析了分数阶微分项的系数和阶次对系统动力学特性的影响。建立了亚谐共振定常解的幅频曲线的解析表达式,并得到了亚谐共振周期响应的存在条件和稳定性判断准则。最后进行了数值解和解析解的比较,证明了解析结果的准确性,并通过数值仿真研究了分数阶微分项的参数对亚谐共振解的存在条件、稳定性条件和系统幅频曲线的影响。     

Analytical stress around mode II crack parallel to principle axis in orthotropic composite plate

Stress analyses for orthotropic composite materials containing a through crack under remote shear loads (Mode II) are conducted. By employing the complex theory, a harmonic differential equation was established for the orthotropic plates with axes normal to the three orthogonal planes of material symmetry. An analytical complex function was introduced by following the Westergaard approach. Stress around a mode II crack in the orthotropic composite plate is deduced to have an analytical form. In addition, the analytical solution for a mode II crack was examined in the case of isotropic materials. It demonstrated that the analytical solution obtained is correct for the mode II cracked orthotropic composite plates.     

Asymptotic analysis and finite element calculation of a rubber notch contacting with a rigid wedge

Summary In this paper, the strain energy function given by Knowles-Sternberg in 1973 is used, and the contact problem of a rubber notch with a rigid wedge is analyzed. The basic equations of the deformation field near the notch corner are derived and solved. An analytical solution is obtained for the expanding sector while the numerical solution is given for two shrinking sectors. A special interesting case is that a half rubber-like plane contacts with a rigid wedge, for which the completely analytical solutions are obtained for both expanding sector and shrinking sectors. The analysis of this paper is also valid for the contact problem of a rubber wedge with a rigid wedge. To verify the analytical results, a finite element program of nonlinear elasticity is made with which the same problem is calculated, and the results are consistent with the analytical results very well.     

Interaction between a point heat source and a nonuniformly coated circular inclusion

This paper investigates the interaction between a point heat source and a nonuniformly coated circular inclusion under plane deformation. Based on the complex variable theory and conformal mapping in conjunction with the analytical continuation method, the general expressions of the temperature, displacements, and stresses for three dissimilar media are derived explicitly in a series form. Numerical results are provided for some particular examples to investigate the effect of material combinations on the interfacial stresses. The derived analytical solution can also be used as a fundamental solution for the boundary element method as well as for other physical models, including welding and cutting.     

Enhanced joint pricing and lotsizing problem in a two-echelon supply chain with logit demand function

In retail supply chains, ordering and pricing policies for the retailer and shipment plan for the supplier are the most important decisions. These policies are often conducted either individually or sequentially with poor overall performance for the whole supply chain resulting to extra inventory and other deficiencies. In this paper, an integrated marketing-inventory model in a two-echelon supply chain model is developed involving discount promotion, customer behaviour more realistically and operations aspects to determine optimal ordering, shipping and pricing quantities simultaneously. An efficient analytical solution procedure and a Particle Swarm Optimisation solution algorithm are also developed. Finally, a number of numerical tests are conducted to approve the interesting theoretical results of the analytical approach.     

On the modeling of tilted fixed-guided flexible beams under tension

We derive explicit solutions for a fixed-guided slender suspension beam that is initially straight and tilted with respect to the moving direction of its sliding end. The beam experiences substantial axial forces during the tension, resulting in a nonlinear boundary value problem. We consider sliding end displacements in the direction that cause longitudinal tension along the beam. We first propose an exact approach, leading to analytical solutions for various physical variables such as the transverse force and deflection profile, in terms of the axial force and the positive real solution of a third-order algebraic equation. We also propose an alternative approximate solution based on a second-order equation, which provides closed-form analytical solutions for the physical variables. We also introduce analytical validation techniques for the underlying assumptions. Consistency with nonlinear finite-element analysis is also addressed. Moreover, the results of the approximate method are represented by dimensionless formulas, generating charts to predict solutions for arbitrarily assigned beam parameters. Magnitudes of the normal and shear stress values are also included to consider the effects of yield and shear strengths as the limiting factors at large deflection conditions.     

An alternative model for elastic thermal stresses in two materials joined by a graded layer

The elastic stress distribution resulting from thermal mismatch in a film/graded layer/substrate system is a critical issue. Recently by a three-variable method Hsueh and Lee formulated a closed-form analytical solution [Hsueh CH, Lee S. Modeling of elastic thermal stresses in two materials joined by a graded layer. Composites: Part B 2003;34:747–52] in order to overcome the complexity of the traditional analytical models. The paper is devoted to developing an alternative analytical model for graded beams by Zhang’s two-variable method for multilayered beams. To illustrate applications of the present solution, specific results are calculated for the GaAs/graded GaAs–Si/Si systems. The transition phenomenon between tensile stress and compressive stress in the graded beam is investigated in detail for different graded parameters or different thickness. The present results agree well with existing analytical results. The differences between the present two-variable model and Hsueh’s three-variable model are also discussed.     

两黏性颗粒间切向阻力的渐近解析解

从分析两黏性颗粒的相对切向运动着手,化二阶变系数非齐次液桥流体压力微分方程为欧拉方程,解得具有相对运动的不等径颗粒间液桥流体压力和切向黏性阻力的渐近解析解,并与Goldman意义上的近似解和其他文献中的数值解进行对比。结果表明:利用这些解析解可直接定义黏性颗粒力学模型,也可分析不同参数条件下液桥流体压力与颗粒间切向阻力的变化规律。     

Three-dimensional analytical solution for transient guided wave propagation in liquid-filled pipe systems

The objective of this study is to investigate the three-dimensional (3-D) analytical solution for transient guided wave propagation in liquid-filled pipe systems using the eigenfunction expansion method (EEM). The eigenfunctions corresponding to finite liquid-filled pipe systems with a traction-free lateral boundary and rigid smooth end boundaries are obtained. Additionally, the orthogonality of the eigenfunctions is proved in detail. Subsequently, the exact 3-D analytical transient response of finite liquid-filled pipe systems to external body forces is constructed using the EEM, based on which, the approximate 3-D analytical transient response of the systems to external surface forces is derived. Furthermore, the analytical solution for transient guided wave propagation in finite liquid-filled pipe systems is extended explicitly and concisely to infinite liquid-filled pipe systems. Several numerical examples are given to illustrate the analysis of the spatial and frequency distributions of the radial and axial displacement amplitudes of various guided wave modes; the numerical examples also simulate the transient displacement of the pipe wall and the transient pressure of the internal liquid from the present solution. The present solution can provide some theoretical guidelines for the guided wave nondestructive evaluation of liquid-filled pipes and the guided wave technique for downhole data transfer.     

Flow in a pipe driven by suction at an accelerating wall

Summary The axisymmetric flow of a viscous fluid in a pipe driven by suction at the pipe wall and acceleration of the wall is examined using a similarity form of solution: the particular combination of the two effects chosen is one in which there is a simple analytical solution. The temporal stability of solutions (both this analytical one and other numerical ones) is investigated for various values of the Reynolds numberR, and stable and unstable parts of the solution branches are identified. A transcritical bifurcation is found and also a Hopf bifurcation. We consider also the corresponding nonlinear diffusion problem and present some solutions. Finally we examine the continuation from the present problem to the flow in a pipe driven by suction only.     

Vibration Performance,Stability and Energy Transfer of Wind Turbine Tower via Pd Controller

In this paper, we studied the vibration performance, energy transfer and stability of the offshore wind turbine tower system under mixed excitations. The method of multiple scales is utilized to calculate the approximate solutions of wind turbine system. The proportional-derivative controller was applied for reducing the oscillations of the controlled system. Adding the controller to single degree of freedom system equation is responsible for energy transfers in offshore wind turbine tower system. The steady state solution of stability at worst resonance cases is studied and examined. The offshore wind turbine system behavior was studied numerically at its different parameters values. Furthermore, the response and numerical results were obtained and discussed. The stability is also analyzed using technique of phase plane and equations of frequency response. In addition, the numerical results and comparison between analytical and numerical solutions were obtained with MAPLE and MATLAB algorithms.     

A distributed parameter model for particles in the spray drying process

A multicomponent particle drying model has been developed to describe the compositions and temperatures of spray-dried particles using distributed drying kinetics. This means that component migration is expected to occur within the particles, thus temperature and moisture content gradients through the particles are present and might not be uniform. Particles are divided into a number of shells and may be spray dried from a solution of several solutes dissolved in water, such as sugars and proteins. The model simulates the drying kinetics within a concurrent spray dryer and also accounts for the effects of the solubilities of the solids, as well as surface activity of the components. The model has been successfully compared with an analytical solution for evaporation alone, to test the modelling of the external heat and mass-transfer processes. The model has also been compared with an analytical solution for internal diffusion in a sphere, which has tested the modelling of the internal heat and mass-transfer processes.