双层杆结构中纵向模态的超声导波

通过分析应力和位移边界条件得到了双层杆结构中纵向模态的频散方程，并进行数值求解，得到相应的频散曲线。对该结构中纵向模态的主要传播特性进行了分析。然后，利用长度伸缩型压电陶瓷片在内层为冰外层为钢的双层杆结构中进行了激励接收纵向模态的实验。实验结果与理论分析较为吻合。结果表明，双层杆结构的频散曲线可用作模态选取的理论指导，用于该结构的缺陷检测。     

Solution to the problem of Nicolai

We consider the problem of Nicolai on dynamic stability of an elastic cantilever rod loaded by an axial compressive force P and a twisting tangential torque L in continuous formulation. The problem is to find the stability region for non-equal principal moments of inertia of the rod in the space of three parameters: P, L and the parameter α for the ratio of principal moments of inertia. New governing equations and boundary conditions, which form the basis for analytical and numerical studies, are derived. An important detail of this formulation is that the pre-twisting of the rod due to the torque L is taken into account. The singular point on the stability boundary at the critical Euler force P_E is recognized and investigated in detail. For an elliptic cross-section of a uniform rod the stability region is found numerically with the use of the Galerkin method and the exact numerical approach. The obtained numerical results are compared with the analytical formulas of the asymptotic analysis.     

椭圆截面的无限长金属棒对任意平面电磁波在有损耗的均匀各向同性无限媒质中的散射

本文利用椭圆柱形坐标系,严格地分析了具有任意偏心度的椭圆截面的无限长理想导电的金属棒,放在有损耗的均匀各向同性的无限介质中,对从任意方向入射的均匀平面波的散射问题;并将散射场表示为马许函数。     

MHD flow and heat transfer over a radially stretching/shrinking disk

A steady magnetohydrodynamic (MHD) flow past a radially stretching or shrinking disk is investigated. The governing partial differential equations are transformed into a set of ordinary (similarity) differential equations by a similarity transformation. These equations along with the corresponding boundary conditions are solved numerically using the boundary value problem solver (bvp4c) in Matlab. The effects of magnetic field and suction on the shear stress and the heat transfer are analyzed and discussed. It is found that both parameters affect more in the shrinking region. The increase in the magnetic parameter results in the increase of the skin friction coefficient but decrease in the local Nusselt number.The skin friction coefficient and the local Nusselt number increase as suction increases.     

The exact response of a two-dimensional flat-layered medium to a probing pulse: An application to a layer stripping reconstruction procedure

Abstract

We consider a simple model problem that can be found in many fields of application such as, for example, reflection seismology. That is we consider an initial boundary value problem on a half-plane for a class of two-dimensional wave equations with a piecewise-constant coefficient. This coefficient describes the flat layered medium under consideration. An initial pulse located on the boundary of the half-plane is used to probe the medium. An integral representation of the solution of this problem is obtained by studying the spectral measures of some differential operators in one variable. This integral representation is exploited to obtain an ‘explicit’ formula for the solution of the problem considered evaluated at the location of the probing pulse. This ‘explicit’ formula is exploited to reconstruct the structure of the medium from its response to a probing pulse via a layer stripping procedure. Some numerical results obtained with this procedure on test problems are shown. The ‘explicit’ formula obtained can be used in several other contexts such as, for example, the study of perturbed flat layered media or the study of random flat layered media.     

一种不规范的斯特姆-刘维本征值问题

由于端点系有集中质量的弹性杆的振动问题的边界条件不是斯特姆－刘维本征值问题所要求的边界条件，分离变数所得到的函数系{Xn(x)}不是正交函数系，故求解十分困难，本文提出一种变换，将其边界条件化为第一类，第二类或第三类边界条件，此时{Xn(x)}为正交完备函数系，问题大为简化，最后给出了几种不同端点情况下此类问题的解。     

环形激光二极管抽运棒状激光器中瞬态温度和热应力分析

直接从激光二极管发光强度的角分布出发,采用光线追迹方法获得激光棒内的热沉积分布,在此基础上采用热传导模型和热力模型,比较了不同抽运功率、不同棒半径下达到稳态温度分布的时间,并且对稳态和瞬态热应力进行了详细模拟计算。结果表明,采用环形激光二极管阵列侧面抽运的棒状激光器中的热效应问题十分严重,不同的抽运结构参量下,温度分布不同;达到稳态所需时间随棒半径增大而增加,而不受抽运功率的影响;抽运功率越大,棒内温差增大,热应力也越大;热破坏主要集中于激光棒中心区域和表面区域。     

Large-Time Behavior of Solutions to the Equations of a One-Dimensional Viscous Polytropic Ideal Gas in Unbounded Domains

The large-time behavior of solutions to the initial and initial boundary value problems for a one-dimensional viscous polytropic ideal gas in unbounded domains is investigated. Using a special cut-off function to localize the problem, we derive a local representation for the specific volume. With the help of the local representation, and certain new estimates for the temperature and the stress, and the weighted energy estimates, we prove that in any bounded interval, the specific volume is pointwise bounded from below and above for all t≥ 0 and a generalized solution is convergent as time goes to infinity. Received: 21 July 1997/ Accepted: 21 June 1998     

Inverse scattering by a continuation method with initial guesses from a direct imaging algorithm

A novel continuation method is presented for solving the inverse medium scattering problem of the Helmholtz equation, which is to reconstruct the shape of the inhomogeneous medium from boundary measurements of the scattered field. The boundary data is assumed to be available at multiple frequencies. Initial guesses are chosen from a direct imaging algorithm, multiple signal classification (MUSIC), along with a level set representation at a certain wavenumber, where the Born approximation may not be valid. Each update via recursive linearization on the wavenumbers is obtained by solving one forward and one adjoint problem of the Helmholtz equation.     

PHYSICAL AND NUMERICAL MODELLING OF THE DYNAMIC BEHAVIOR OF A FLY LINE

The planar equations of motion for a tapered fly line subjected to tension, bending, aerodynamic drag, and weight are derived. The resulting theory describes the large non-linear deformation of the line as it forms a propagating loop during fly casting. A cast is initiated by the motion of the tip of the fly rod that represents the boundary condition at one end of the fly line. At the opposite end, the boundary condition describes the equations of motion of a small attached fly (point mass with air drag). An efficient numerical algorithm is reviewed that captures the initiation and propagation of a non-linear wave that describes the loop. The algorithm is composed of three major steps. First, the non-linear initial-boundary-value problem is transformed into a two-point boundary-value problem, using finite differencing in time. The resulting non-linear boundary-value problem is linearized and then transformed into an initial-value problem in space. Example results are provided that illustrate how an overhead cast develops from initial conditions describing a perfectly laid out back cast. The numerical solutions are used to explore the influence of two sample effects in fly casting, namely, the drag created by the attached fly and the shape of the rod tip path.     

Investigation of non-Newtonian liquid jet motion

The development of a non-self-similar liquid-liquid jet of a non-Newtonian reactive liquid is considered with in the framework of the boundary layer theory. The analytic expressions are obtained on the basis of the local similarity method for variation of integral parameters of jet flow along its axis, which agree fairly well with the numerical solution of a non-self-similar problem.     

On acoustic radiation by a rigid object in a fluid flow

A problem of sound radiation by an absolutely rigid object, moving with respect to the surrounding fluid, is considered on the basis of the Lighthill's equation for aerodynamic sound. An integral representation of the radiated acoustic field is utilized, where the field is characterized as the sum of three fields, generated by a volume distribution of monopoles and by distributions of monopoles and dipoles on the surface of the rigid object. It is shown that, due to a discontinuity of Lighthill's stress tensor on the rigid boundary, a layer of surface divergence of hydrodynamic stresses on the boundary must be taken into account when evaluating the volume integral over Lighthill's quadrupole sources. When the contribution of the surface divergence is included in the solution of Lighthill's equation, amplitudes of the monopole and dipole sound radiated by the rigid object are shown to depend on the potential components of the normal velocity and the pressure on the rigid surface. The obtained solution is compared with Curle's solution for this problem, which establishes that the sound radiation by a rigid object is determined by the force exerted by the object upon the fluid. Both solutions are applied to two known problems of sound scattering and radiation by a rigid sphere in variable pressure and velocity fields. It is shown that predictions based on the obtained solution are equivalent to the results known from literature, whereas Curle's solution gives predictions contradicting the known results. It is also shown that the Ffowcs Williams and Hawkings equation, which coincides with Curle's equation for an immoveable rigid object, does not lead to the correct predictions as well.     

变系数瞬态热传导问题边界元格式的特征正交分解降阶方法

提出了一种基于边界元法求解变系数瞬态热传导问题的特征正交分解(POD)降阶方法,重组并推导出变系数瞬态热传导问题适合降阶的边界元离散积分方程,建立了变系数瞬态热传导问题边界元格式的POD降阶模型,并用常数边界条件下建立的瞬态热传导问题的POD降阶模态,对光滑时变边界条件瞬态热传导问题进行降阶分析.首先,对一个变系数瞬态热传导问题,建立其边界域积分方程,并将域积分转换成边界积分;其次,离散并重组积分方程,获得可用于降阶分析的矩阵形式的时间微分方程组;最后,用POD模态矩阵对该时间微分方程组进行降阶处理,建立降阶模型并对其求解.数值算例验证了本文方法的正确性和有效性.研究表明:1)常数边界条件下建立的低阶POD模态矩阵,能够用来准确预测复杂光滑时变边界条件下的温度场结果;2)低阶模型的建立,解决了边界元法中采用时间差分推进技术求解大型时间微分方程组时求解速度慢、算法稳定性差的问题.     

Stress and mixed boundary conditions for two-dimensional dodecagonal quasi-crystal plates

For plate bending and stretching problems in two-dimensional (2D) dodecagonal quasi-crystal (QC) media, the reciprocal theorem and the general solution for QCs are applied in a novel way to obtain the appropriate stress and mixed boundary conditions accurate to all order. The method developed by Gregory and Wan is used to generate necessary conditions which the prescribed data on the edge of the plate must satisfy in order that it should generate a decaying state within the plate; these decaying state conditions are obtained explicitly for axisymmetric bending and stretching of a circular plate when stress or mixed conditions are imposed on the plate edge. They are then used for the correct formulation of boundary conditions for the interior solution. For the stress data, our boundary conditions coincide with those obtained in conventional forms of plate theories. More importantly, appropriate boundary conditions with a set of mixed edge-data are obtained for the first time. Furthermore, the corresponding necessary conditions for transversely isotropic elastic plate are obtained directly, and their isotropic elastic counterparts are also obtained.      

弹性压扭直杆的Greenhill公式对精确模型的推广

将圆截面Kirchhoff弹性压扭直杆的Greenhill公式推广到精确模型.基于平面截面假定,在弯扭的基础上增加了拉压和剪切变形,将弹性杆的位形表达为截面的弧坐标历程.由弹性杆精确模型的平衡微分方程,得到了两端受力螺旋作用时对应于直线平衡状态的特解,导出了线性化扰动方程及其通解,再根据两端为铰支时的边界条件以及积分常数存在非零解的条件导出弹性直杆精确模型的Greenhill公式.结果表明,由力螺旋表示的稳定域为一对称的封闭区域,拉压和剪切对稳定性的影响取决于拉压柔度与剪切柔度之差、抗弯刚度和杆长这三个因素.     

Heat Transfe for Power Law Non-Newtonian Fluids

We present a theoretical analysis for heat transfer in power law non-Newtonian fluid by assuming that the thermal diffusivity is a function of temperature gradient. The laminar boundary layer energy equation is considered as an example to illustrate the application. It is shown that the boundary layer energy equation subject to the corresponding boundary conditions can be transformed to a boundary value problem of a nonlinear ordinary differential equation when similarity variables are introduced. Numerical solutions of the similarity energy equation are presented.     

Nonlinear size-dependent longitudinal vibration of carbon nanotubes embedded in an elastic medium

In this paper, we study the longitudinal linear and nonlinear free vibration response of a single walled carbon nanotube (CNT) embedded in an elastic medium subjected to different boundary conditions. This formulation is based on a large deformation analysis in which the linear and nonlinear von Kármán strains and their gradient are included in the expression of the strain energy and the velocity and its gradient are taken into account in the expression of the kinetic energy. Therefore, static and kinetic length scales associated with both energies are introduced to model size effects. The governing motion equation along with the boundary conditions are derived using Hamilton's principle. Closed-form solutions for the linear free vibration problem of the embedded CNT rod are first obtained. Then, the nonlinear free vibration response is investigated for various values of length scales using the method of multiple scales.     

Numerical-experimental hybrid method for stress separation in digital gradient sensing method

A numerical-experimental hybrid method for the stress separation in the digital gradient sensing (DGS) method is proposed in this study. In the proposed hybrid method, boundary conditions for a local finite element model, that is, nodal force along boundaries are inversely determined from experimental values obtained by the digital gradient sensing method. The hybrid method follows two stages. In stage 1, the DGS method measures the Cartesian stress gradient components directly and, subsequently, the sum in Cartesian stresses at all interesting points on the surface; stress sum are used to compute the unknown boundary conditions for the local model. In stage 2, the individual stress components are calculated by the direct finite element method using the computed boundary conditions from stage 1. The effectiveness is demonstrated by applying the proposed method to a stress concentration problem involving concentrated load acting on an edge of a large planar sheet. The individual stress components thus determined are summed and compared with analytical stress sum, confirming the effectiveness and accuracy of the proposed technique.     

Nonlinear effects of non-Newtonian fluids on natural convection in a porous medium

This paper investigates the rheological effects of non-Newtonian fluids on the natural convection mechanism in a porous medium. A non-Newtonian behavior of power law fluid with a yield stress, saturating a porous medium, in which yield stress is temperature dependent, is considered. The cases of constant temperature boundary and constant heat flux boundary, along the heated vertical cylinder, are analyzed. The approximate similarity solutions in a closed form are shown, from which the velocity and temperature profiles are determined. The numerical solutions for a constant temperature boundary are also shown and discussed.     

Thermal Stress Analysis of a Continuous and Pulsed End-Pumped Nd:YAG Rod Crystal Using Non-Classic Conduction Heat Transfer Theory

In this paper, temperature distribution in the continuous and pulsed end-pumped Nd:YAG rod crystal is determined using nonclassical and classical heat conduction theories. In order to find the temperature distribution in crystal, heat transfer differential equations of crystal with consideration of boundary conditions are derived based on non-Fourier’s model and temperature distribution of the crystal is achieved by an analytical method. Then, by transferring non-Fourier differential equations to matrix equations, using finite element method, temperature and stress of every point of crystal are calculated in the time domain. According to the results, a comparison between classical and nonclassical theories is represented to investigate rupture power values. In continuous end pumping with equal input powers, non-Fourier theory predicts greater temperature and stress compared to Fourier theory. It also shows that with an increase in relaxation time, crystal rupture power decreases. Despite of these results, in single rectangular pulsed end-pumping condition, with an equal input power, Fourier theory indicates higher temperature and stress rather than non-Fourier theory. It is also observed that, when the relaxation time increases, maximum amounts of temperature and stress decrease.