考虑几何非线性的杆系结构弯曲变形样条有限点法

研究了杆系结构考虑几何非线性的大挠度弯曲变形问题,推算并验证了一种考虑几何非线性的杆系结构弯曲变形计算新方法.以三次B样条函数为基函数,采用广义参数法,构造出梁的样条基函数,通过最小势能原理,建立了杆系结构考虑几何非线性的刚度方程,对处于弹性范围内的杆系结构的大变形弯曲问题进行了计算,提出了考虑几何非线性时杆系结构弯曲变形计算的样条有限点法.结果表明:方法不用进行单元坐标变换、划分等分数少、收敛速度快且计算精度较高,是一种较传统有限元法更简单且可行的方法.     

结构分析中的广义变分原理及其应用

本文将讨论在杆系结构中用待定乘数法建立广义变分原理以分析超静定桁架结构.同时导出一个对称矩阵,并给出这个矩阵的解法,从而求出各杆的内力.     

Cosserat生长弹性杆动力学的Gauss最小拘束原理

以自然界中具有生长、变形和运动特征的细长体为背景,用经典力学中的Gauss最小拘束原理研究生长弹性杆的动力学建模问题.在为生长弹性杆动力学建模提供新方法的同时,扩大了Gauss原理的应用范围.以Cosserat弹性杆为对象,分析弹性杆生长和变形的几何规则,表明生长应变和弹性应变是非线性耦合的;本构方程给出了截面的内力与弹性变形的线性关系;利用逆并矢,将经典力学中的Gauss原理和Gauss最小拘束原理用于生长弹性杆动力学,得到等价的两种表现形式,反映了时间和弧坐标在表述上的对称性,由此导出了封闭的动力学微分方程.给出了两种形式的最小拘束函数,表明生长弹性杆的实际运动使拘束函数取驻值,且为最小值.最后讨论了生长弹性杆的约束与条件极值等问题.     

载电流夹紧杆的非线性稳定性分析*

本文研究了载电流夹紧杆在磁场作用下的非线性稳定性,其磁场由两根无限长相互平行的刚性直导线产生.杆的自然状态在刚性导线所在的平面内,并且与两刚性导线等距.首先,在空间变形的假定下,给出了问题的数学描述,讨论了线性化问题和临界电流.其次,证明了杆的过屈曲状态总是平面的.最后,数值计算了分支解的全局响应,得到了杆过屈曲状态的挠度、内力和弯矩的分布.结果表明,载电流杆既可发生超临界屈曲,又可发生次临屈界曲,其性态依赖于杆与导线间的距离;同时,在超临界的过屈曲状态上还存在极限点型的失稳,这与通常的压杆失稳有着本质的区别.     

线性正泛函数论与超静定结构实现原理

引言回顾超静定结构力学理论,自 Castigliano 及 Maxwell 等人奠定了基础以来,在结构的分析方面,虽有较大的发展,但在基本理论方面,许多问题至今尚未探明,如1.设超静定结构系统之几何形式已定,在一定荷载作用下,系统杆件断面参数的变动必使各杆内力变动,问其变动之区域为何?     

储油罐的变位识别与罐容表标定模型

用数学建模方法试图研究解决储油罐变位识别与罐容表标定问题.首先精确推导了无变位以及纵向倾斜时罐内储油量与油位高度的函数关系,且设计了两种系统误差补偿方法.一种是基于等效思想的δ值法,另一种拟合法.其次,确定变位参数时依然使用补偿思想,将球冠体部分的储油体积和系统误差统一为整体拟合成油位高度h的三次多项式,并应用最小二乘...     

用奇异函数求具有悬臂端环板极限荷载的另一种方法

本文在文[1]的基础上给出用奇异函数求具有悬臂端环板极限荷载的另一种方法,在平衡方程中直接将rq(r)及内力M_θ用奇异函数表示,然后按奇异函数的积分规则进行求解.并以具有内悬臂和外悬臂的环板的塑性极限分析问题为例,给出计算过程和结果。     

应力函数一般解的补充

本文指出平面问题极坐标形式应力函数一般解并不完备,不能处理曲杆受任意边界分布力的问题.为此,提出两个新的应力函数,将一般解作若干补充之后,能解曲杆r=a,b上受任意分布力的问题.这是包含区域边界几何参数的新的应力函数.     

求解一类非线性互补问题的松弛two-sweep模系矩阵分裂迭代法

本文构造了求解一类非线性互补问题的松弛two-sweep模系矩阵分裂迭代法. 理论分析建立了新方法在系数矩阵为正定矩阵或H₊矩阵时的收敛性质．数值实验结果表明新方法是行之有效的, 并且在最优参数下松弛two-sweep模系矩阵分裂迭代法在迭代步数和时间上均优于传统的模系矩阵分裂迭代法和two-sweep模系矩阵分裂迭代法.     

一种模糊模式识别新方法及其在脑电图信号识别中的应用

本文给出一种确定模糊子集隶属函数的新方法——STF 隶属函数确定法.该法将随机性与模糊性联系起来,使得对相当广泛的一类问题,可借助数理统计方法确定模糊概念的隶属函数.本文还给出一种从一个模糊概念的不同方面综合考察该模糊概念的方法.提出了隶属优势的概念,在此基础上给出了一种模糊模式识别的新方法——最大隶属优势准则模式识别法.应用这种方法对137例脑电图进行了计算机识别,得到了较好的效果.     

An inverse eigenvalue method for frequency isolation in spring–mass systems

The action of external vibrating forces on mechanical structures can cause severe damages when resonance occurs. The removal of natural frequencies of the structure from resonance bands is therefore of great importance. This problem is called frequency isolation problem and is the subject of this paper. A new inverse eigenvalue method is proposed and applied to spring–mass systems, which have generated much interest in the literature as prototypes of vibrating structures. The novelty of the method lies in using the zeros of the frequency response function at the last mass as control variables in an optimization problem to minimize the impact of redesign. Numerically accurate algorithms for computing the sensitivity with respect to the control variables are presented, which form the basis of an efficient multidimensional search strategy to solve the frequency isolation problem. Copyright © 2001 by John Wiley & Sons, Ltd.     

Plastic limit analysis and optimum design of structures for uncertain conditions.

At the application of the plastic analysis and design methods the control of the plastic behaviour of the structures is an important requirement. Since the plastic limit analysis provides no information about the magnitude of the plastic deformations and residual displacements accumulated before the adaptation of the structure, therefore complementary strain energy of the residual forces could be considered an overall measure of the plastic performance of structures and the plastic deformations should be controlled by introducing a bound for magnitude of this energy. If the design uncertainties (manufacturing, strength, geometrical) are expressed by the calculation of the complementary strain energy of the residual forces a reliability based extended plastic limit design problem is formed. In this research, due to the uncertainties the bound for the complementary strain energy of the residual forces is given randomly and a general approach to evaluate the limit load capacity of structures for uncertain conditions is presented. The aim of this study is to evaluate limit load capacity of structures with limited residual strain energy on the probabilistically given conditions. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A general solution of equations of equilibrium in linear elasticity

A general solution of equations of equilibrium in linear elasticity is presented in cylindrical coordinates in terms of three harmonic functions describing an arbitrary displacement field. The structure of this solution is similar to the general solution given by Love (Kelvin’s solution) in spherical coordinates. Galerkin vector representation of our solution leads to an integral connecting the harmonic functions. The connections to Papkovich–Neuber and Muki’s general representations are also provided. Suitable choices of the harmonic functions in our new representation yield general solutions for axisymmetric deformations due to Love, Boussinesq and Michell. Some unbounded deformations induced by singular forces are tabulated in terms of the scalar harmonic functions to justify the simple nature of our representation. Exact solution of the half-space boundary value problem is also provided to demonstrate the power of our approach. The stress components computed via our solution are also listed (see the Appendix).     

Geometric multigrid for an implicit-time immersed boundary method

The immersed boundary (IB) method is an approach to fluid-structure interaction that uses Lagrangian variables to describe the deformations and resulting forces of the structure and Eulerian variables to describe the motion and forces of the fluid. Explicit time stepping schemes for the IB method require solvers only for Eulerian equations, for which fast Cartesian grid solution methods are available. Such methods are relatively straightforward to develop and are widely used in practice but often require very small time steps to maintain stability. Implicit-time IB methods permit the stable use of large time steps, but efficient implementations of such methods require significantly more complex solvers that effectively treat both Lagrangian and Eulerian variables simultaneously. Several different approaches to solving the coupled Lagrangian-Eulerian equations have been proposed, but a complete understanding of this problem is still emerging. This paper presents a geometric multigrid method for an implicit-time discretization of the IB equations. This multigrid scheme uses a generalization of box relaxation that is shown to handle problems in which the physical stiffness of the structure is very large. Numerical examples are provided to illustrate the effectiveness and efficiency of the algorithms described herein. These tests show that using multigrid as a preconditioner for a Krylov method yields improvements in both robustness and efficiency as compared to using multigrid as a solver. They also demonstrate that with a time step 100–1000 times larger than that permitted by an explicit IB method, the multigrid-preconditioned implicit IB method is approximately 50–200 times more efficient than the explicit method.     

An application of the Fourier method of separation of variables to constructing exactly solvable deformations of partial differential operators

As is known, in mathematical physics there are differential operators with constant coefficients whose fundamental solutions can be constructed explicitly; such operators are said to be exactly solvable. In this paper, the problem of adding lower-order terms with variable coefficients to exactly solvable operators in such a way that the new operators (deformations) admit constructing fundamental solutions in explicit form is posed. This problem is directly related to Hadamard’s problem of describing differential operators satisfying the Huygens’ principle. On the basis of the Fourier method of separation of variables and the method of gauge-equivalent operators, an effective method for finding exactly solvable deformations depending on one variable is constructed. An application of such deformations to constructing Huygens’ differential operators associated with the cone of real symmetric positive-definite matrices is suggested. __________ Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 38, Suzdal Conference-2004, Part 3, 2006.     

A penalty approximation for a unilateral contact problem in non-linear elasticity

Using Ball's approach to non-linear elasticity, and in particular his concept of polyconvexity, we treat a unilateral three-dimensional contact problem for a hyperelastic body under volume and surface forces. Here the unilateral constraint is described by a sublinear function which can model the contact with a rigid convex cone. We obtain a solution to this generally non-convex, semicoercive Signorinin problem as a limit of solutions of related energy minimization problems involving friction normal to the contact surface where the friction coefficient goes to infinity. Thus we extend an approximation result of Duvaut and Lions for linear-elastic unilateral contact problems to finite deformations and to a class of non-linear elastic materials including the material models of Ogden and of Mooney-Rivlin for rubberlike materials. Moreover, the underlying penalty method is shown to be exact, that is a sufficiently large friction coefficient in the auxiliary energy minimization problems suffices to produce a solution of the original unilateral problem, provided a Lagrange multiplier to the unilateral constraint exists.     

Deformations of a thick-valued,slowly rotating cylinder under gravitational load

An analytic method of calculating the deformations of an infinitely long thick-walled cylinder under a periodically varying gravitational load is described. The cylinder material is linearly viscoelastic. The solution employs the elasto-viscoelastic analogy. The corresponding elastic problem is solved by Muskhelishvili's method. The nonhomogeneity of the equilibrium equations is eliminated by applying the theorem of body forces with a potential function. As an example the maximum displacements of the inner channel are calculated for a cylinder of incompressible material in the cases of slow continuous rotation and periodic rotation through 180°.Ordzhonikidze Moscow Aviation Institute. Translated from Mekhanika Polimerov, No. 4, pp. 727–733, July–August, 1970.     

A mathematical analysis of the elasto-plastic plane stress problem of a power-law material

In this paper a generic study on the plane stress problem ofa power-law material undergoing infinitesimal deformations iscarried out, and a general solution for the stress and strainfields is derived using a stress function method and analyticfunction theory. Hencky's deformation theory and von Mises'yield criterion are used, and a differential transformationis invoked in the analysis. As an example, the closed-form solutionof the pure bending problem of a thin beam of power-law materialis obtained by applying the general solution directly.     

用复变函数法求解弹性力学空间轴对称问题的方法及解例

本文证明了空间轴对称问题的Love应力函数可用两个适当选择的复变量广义解析函数[1]表示,并导出了应力分量、位移分量及边界条件的复变函数表达式.为了表明文中所述方法的可行性以及检验所得公式的正确性,本文用幂级数法求解了含有球形空腔的圆柱体在周围受压及两端受拉时的解答,与用其他方法得到的该问题的解答完全一致.最后本文还求解了一个锥体在侧面受均匀剪力时的解答,同时通过把常体力化为表面力以后还求解了锥体在重力作用下的解答.     

A priori error analysis of two force-based atomistic/continuum models of a periodic chain

The force-based quasicontinuum (QCF) approximation is a non-conservative atomistic/continuum hybrid model for the simulation of defects in crystals. We present an a priori error analysis of the QCF method, applied to a one-dimensional periodic chain, that is valid for an arbitrary interaction range, large deformations, and takes coarse-graining into account. Our main tool in this analysis is a new concept of atomistic stress. Moreover, we formulate a new atomistic/continuum coupling mechanism based on coupling stresses instead of forces and extend the a priori analysis to this new method. We show that the new method has several theoretical advantages over the original QCF method.