关于具有大参数a~2/R_0h的r>0等厚圆环薄壳轴对称问题的二次渐近解

在这篇文章中,根据Love-Kirchhoff假设的薄壳理论,导出了r>0等厚圆环薄壳力矩理论轴对称问题的基本方程.对具有大参数a~2/R_0h的r>0等厚圆环薄壳,给出了二次渐近解.本文也给出了当边缘远离圆环薄壳顶点时的边缘问题的二次渐近解.它们的误差都是在Love-Kirchhoff假设的薄壳理论的允许误差范围之内.     

偏心圆柱薄壳的分析

本文推导了偏心圆柱薄壳小挠度时的近似方程，并利用解析方法求解了该方程，得出了偏心圆柱薄壳的应力、位移与偏心距之间的关系。     

薄壳理论中的Schrdinger方程

本文是文[50]和[51]的继续.在本文中:(1)将常曲率弹性薄壳的小挠度问题的Love-Kirchhoff方程化归为Schr?dinger方程的求解,并特别指出了它在轴对称问题中的形式:(2)作为小挠度的例子,求得了等厚度球形薄壳在中面力和轴对称外场联合作用下的振动问题的通解,其中的轴对称外场与文[50]不同,它现在是空间位置的函数,而不再是时间的函数;(3)将扁壳大挠度问题的von Kármán-ΒЛасов方程化归为AKNS方程的形式,其一维问题成为简单的Schr(?)dinger方程的本征值问题,从而使非线性问题成为可解的线性问题.     

a/r_2>1等厚圆环薄壳轴对称问题

本文给出了α/r_2>1等厚圆环薄壳轴对称问题力矩理论复变量方程的一致有效渐近解.     

自重作用下的圆环薄壳

本文得到了等厚度圆环薄壳在自重荷载下一种简化形式的Новожилов方程,用Fourier级数求得了方程的特解.利用文[2]已有的齐次解结果,从而给出了问题的一般解.作为结果的应用,文中给出了两个算例.     

旋转薄壳自由振动中的转点问题及其奇异摄动解

在用渐近方法求解任意旋转薄壳(圆柱壳和球壳除外)自由振动的微分方程组时,在一定的参数范围内,存在转(向)点问题。其中,对于存在唯一简单转点的情况(此时,该转点具有枝点奇性),至今未获解决。本文解决了这一问题。文中仅讨论截顶旋转壳。此时,转点是唯一的奇点。最后,对于两端固定的任意旋转薄壳,给出了频率方程。     

椭圆环壳在一般荷载下的轴对称问题

本文给出了在任意分布荷载下轴对称椭圓环壳的简化复变量方程。该方程准确度在薄壳理论误差范围内,并消除了全部经线极值奇点。得到了问题的等价的积分方程组,用数值积分方法给出了数值解。计算了膨胀节、液压圆环壳和半椭圆形密封环的算例,与准确解和实验结果作了比较。     

薄壳小挠度屈曲方程的加权解法

基于薄壳小挠度屈曲方程,提出了一种求临界载荷解析表达式的加权解法.在复杂边界条件下,小挠度屈曲方程的解析解仍然是一个难点.以轴对称屈曲问题为例,从方程中找出临界载荷的影响因素,加权平均得到临界载荷,再用特例解确定影响系数.这一方法利用特殊问题的已知解来求一般问题的解析解,简化了求解过程,拓宽了解决问题的范围;其计算结果与利用Algor有限元程序得到的数值解一致.     

薄壳稳定的变分原理

本文按照易曲物体的形变理论来确定薄壳的内力和内矩,应变能以及外力的功。从而根据虚位移原理求得临载荷的能量准则,并导出稳定问题的平衡方程和边界条件。     

旋转壳的抗扭刚度

本文列出了旋转壳在包括扭转在内的轴对称变形下的一般平衡方程,并证明了旋转对称壳内的剪应力独立于壳内其它薄膜和弯曲应力.本文求解了只考虑薄膜应力的扭转问题,也求解了考虑弯曲扭应力在内的扭转问题,并指出了在薄壳中,抗扭刚度的主要部份来源于薄膜应力.     

1:1内共振条件下矩形薄板的全局分叉和多脉冲混沌动力学

首次利用广义Melnikov方法研究了一个四边简支矩形薄板的全局分叉和多脉冲混沌动力学．矩形薄板受面外的横向激励和面内的参数激励．利用von Krmn模型和Galerkin方法得到一个二自由度非线性非自治系统用来描述矩形薄板的横向振动．在1∶1内共振条件下,利用多尺度方法得到一个四维的平均方程．通过坐标变换把平均方程化为标准形式,利用广义Melnikov方法研究该系统的多脉冲混沌动力学,并且解释了矩形薄板模态间的相互作用机理．在不求同宿轨道解析表达式的前提下,提供了一个计算Melnikov函数的方法．进一步得到了系统的阻尼、激励幅值和调谐参数在满足一定的限制条件下,矩形薄板系统会存在多脉冲混沌运动．数值模拟验证了该矩形薄板的确存在小振幅的多脉冲混沌响应．     

窄域上2D弱阻尼KdV方程的blow-up的研究

得到了2D的弱阻尼KdV方程在窄域上blow-up时间估计.     

变厚度扁锥壳的非线性固有频率

借助于变厚度圆薄板非线性动力学变分方程和协调方程,给出了变厚度扁薄锥壳的非线性动力学变分方程和协调方程·　假设薄膜张力由两项组成,将协调方程化为两个独立的方程,选取变厚度扁锥壳中心最大振幅为摄动参数,采用摄动变分法,将变分方程和微分方程线性化·　对周边固定的圆底变厚度扁锥壳的非线性固有频率进行了求解;一次近似得到了变厚度扁锥壳的线性固有频率,三次近似得到了变厚度扁锥壳的非线性固有频率,且绘出了固有频率与静载荷、最大振幅、变厚度参数的特征曲线图·　为动力工程提供了有价值的参考·     

窄域上2D弱阻尼KdV方程的局部吸引子

得到了窄域上2D的非自共轭且非扇形的弱阻尼KdV方程的局部吸引子的存在性.     

Specific quantum mechanical solution of difference equation of hyperbolic type

Difficulties connected to solving difference equations of hyperbolic type were analyzed in this work and discussed in detail. The results are compared to those of the standard wave equation and certain similarities were established. The method of solving the equation is generalized by means of kernel expanded into separable polynomials. The analysis was inspired by some new ideas concerning quantization of time. Two examples are given: excitons and phonons in thin crystalline films. The advanced methodology of Green’s function method and the application of this new methodology resulted in a set of interesting conclusions concerning thin film properties. The significance of the obtained spatial dependence of exciton concentration was discussed and it was concluded, on the basis of the found spatial dependence of exciton concentration, that such boundary conditions of a thin molecular film which lead to high exciton concentrations can be determined. It was also concluded that thin films possess high superconductive properties, that physical characteristics of thin films are spatially dependent and that the spatial dependence can be the basis for widening the field of application of nanostructures.     

Junctions of singularly degenerating domains with different limit dimensions. II

This paper is aimed at finding asymptotic formulas for solutions to the mixed boundary problem for the Poisson equation in a domain obtained by joining singularly degenerating domains. In this paper, which is the second part of the work (the first part was published in No. 18), the main attention is given to three-dimensional problems in which a thin plate or a periodic family of thin rods is joined to a massive body (the distances between the rods are comparable with the diameters of their cross-sections). The distinctive feature of such problems is that an integral equation arises as one of the limit problems. Bibliography: 48 titles. Translated from Trudy Seminara imeni l. G. Petrovskogo, No. 20, pp. 155–195, 1997.     

Application of the discrete Fourier transform to solving the Cauchy integral equation

The uniquely solvable system of the Cauchy integral equation of the first kind and index 1 and an additional integral condition is treated. Such a system arises, for example, when solving the skew derivative problem for the Laplace equation outside an open arc in a plane. This problem models the electric current from a thin electrode in a semiconductor film placed in a magnetic field. A fast and accurate numerical method based on the discrete Fourier transform is proposed. Some computational tests are given. It is shown that the convergence is close to exponential.     

A justification for the thin film approximation of Stokes flow with surface tension

In the free boundary problem of Stokes flow driven by surface tension, we pass to the limit of small layer thickness. It is rigorously shown that in this limit the evolution is given by the well-known thin film equation. The main techniques are appropriate scaling and uniform energy estimates in Sobolev spaces of sufficiently high order, based on parabolicity.     

Locating the first nodal linein the Neumann problem

The location of the nodal line of the first nonconstant Neumann eigenfunction of a convex planar domain is specified to within a distance comparable to the inradius. This is used to prove that the eigenvalue of the partial differential equation is approximated well by the eigenvalue of an ordinary differential equation whose coefficients are expressed solely in terms of the width of the domain. A variant of these estimates is given for domains that are thin strips and satisfy a Lipschitz condition.

     

Homogenization of a boundary-value problem with a nonlinear boundary condition in a thick junction of type 3:2:1

We consider a boundary-value problem for the Poisson equation in a thick junction Ω_ε, which is the union of a domain Ω₀ and a large number of ε-periodically situated thin curvilinear cylinders. The following nonlinear Robin boundary condition ∂_νu_ε + εκ(u_ε)=0 is given on the lateral surfaces of the thin cylinders. The asymptotic analysis of this problem is performed as ε → 0, i.e. when the number of the thin cylinders infinitely increases and their thickness tends to zero. We prove the convergence theorem and show that the nonlinear Robin boundary condition is transformed (as ε → 0) in the blow-up term of the corresponding ordinary differential equation in the region that is filled up by the thin cylinders in the limit passage. The convergence of the energy integral is proved as well. Using the method of matched asymptotic expansions, the approximation for the solution is constructed and the corresponding asymptotic error estimate in the Sobolev space H¹(Ω_ε) is proved. Copyright © 2007 John Wiley & Sons, Ltd.