电磁接触问题的变分原理与有限元求解

电磁接触耦合作用的力学分析的难点是必须考虑电磁场以及由此引起的电磁力与可移动接触边界间的耦合作用，属于强非线性问题。本文给出接触面区域电磁场分析的处理条件，并进一步建立了两类变分方程，一类是电磁分析的变分泛函，其考虑了接触区域对结构电磁场的影响；另一类是二维电磁力学接触分析的参数变分原理，可以方便地对接触问题进行求解。数值结果验证了本文的理论与算法。     

传热与接触两类问题耦合作用的有限元分析

考虑传热接触耦合作用的热力学分析问题大量存在于工程中,分析的难点是必须考虑热与可移动接触边界间的耦合作用。针对这类问题的求解,该文给出了接触边界上热交换与温度边界条件,并在此基础上建立了两类变分方程,一类是热力学变分泛函,其考虑了接触区域对结构热传导的影响;另一类是二维热弹性接触分析的参数变分原理,可以方便地对接触问题进行求解。文中给出有限元分析的离散公式,并进一步给出两类问题耦合分析的迭代算法,其中接触分析的惩罚因子是可以消除的,数值结果验证了该文的理论与算法。     

薄板与圆柱薄壳大挠度计算的混合法

板壳几何非线性问题主要是从两个方面进行分析的:一是“微分方程法”,即求解非线性(对于薄板是指冯·卡门)微分方程组;二是“能量法”,即由总能量泛函的极值或驻值条件给出问题的解,也就是依据最小势能原理或最小余能原理来求解。本文提出一组介于“能量法”与“微分方程法”之间的混合方程:这就是,采用板、圆柱壳的中曲面势能泛函的极值方程以及挠曲平衡微分方程,共同作为分析其几何非线性问题的控制方程组。这组混合方程既有能量的概念又有平衡的概念——控制方程组中既有积分方程又有微分方程,这与传统的计算途径有所不同而具有一定的优越性。文中从变分原理证明了所提出的混合方程组的极值解即为该问题的真实解,并给出“中面自变函数u、v的泛函π的极值原理”。     

摩擦接触问题的比例边界等几何B可微方程组方法

摩擦接触问题是计算力学领域最具挑战性的问题之一,接触系统的泛函具有非线性、非光滑的特点,导致接触算法的收敛性与精确性难以保证.因此将比例边界等几何分析(scaled boundary isogeometric analysis,SBIGA)与B可微方程组(B differential equation,BDE)相结合,提出了求解二维摩擦接触问题的比例边界等几何B可微方程组方法.在比例边界等几何坐标变换的基础上,通过虚功原理推导了关于边界控制点变量的接触平衡方程,表示成B可微方程组形式的接触条件可被严格满足,求解B可微方程组的算法的收敛性有理论保证.此比例边界等几何B可微方程组方法 (SBIGA-BDE)只需在接触体边界进行等几何离散,使问题降低一维,能精确描述接触边界,并可通过节点插入算法进行真实接触区域的识别.此外,由于几何建模和数值分析使用相同的基函数,节约了划分网格的时间.以赫兹接触问题和悬臂梁摩擦接触问题为例,通过与解析解及数值计算软件ANSYS计算结果进行对比,验证了该方法求解二维摩擦接触问题的有效性及高精度等特点.     

俯仰运动圆柱贮箱中液体的非线性晃动

首次对储仰运动圆柱贮箱中液体的有限幅值晃动问题进行了解析研究。首先建立了描述俯仰和／或偏航运动贮箱中液体晃动的非线性偏微分方程组,而后提出了相应的变分原理,建立了压力体积分形式的Ｌａｇｒａｎｇｅ函数,通过变分方程,最终得到措述俯仰和／或偏航运动圆柱贮箱中液体晃动的非线性动力学微分方程组,该动力学方程组自然满足液体自由表面的运动学和动力学办界条件。而后动用多尺度法求解了所得的动力学方程组,对非线性液     

不可压缩平面问题的位移-压力混合重心插值配点法

引入人工压力变量,将弹性本构方程以应力、应变和压力表达,建立求解不可压缩平面弹性问题的位移-压力方程和不可压缩条件方程的耦合偏微分方程组。利用张量积型重心Lagrange插值近似二元函数,得到计算插值节点处偏导数的偏微分矩阵。采用配点法离散不可压缩弹性控制方程,利用偏微分矩阵直接离散弹性力学控制方程为矩阵形式方程组。利用插值公式离散位移和应力边界条件,将离散边界条件与离散控制方程组合为新的方程组,得到求解弹性问题的过约束线性代数方程组;利用最小二乘法求解线性方程组,得到弹性力学问题位移数值解。数值算例验证了所提方法的数值计算精度为10-14~10-10。     

含摩擦与碰撞平面多刚体系统动力学线性互补算法

基于接触力学理论和线性互补问题的算法, 给出了一种含接触、碰撞以及库伦干摩擦, 同时具有理想定常约束(铰链约束) 和非定常约束(驱动约束) 的平面多刚体系统动力学的建模与数值计算方法. 将系统中的每个物体视为刚体, 但考虑物体接触点的局部变形, 将物体间的法向接触力表示成嵌入量与嵌入速度的非线性函数,其切向摩擦力采用库伦干摩擦模型. 利用摩擦余量和接触点的切向加速度等概念, 给出了摩擦定律的互补关系式; 并利用事件驱动法, 将接触点的黏滞-滑移状态切换的判断及黏滞状态下摩擦力的计算问题转化成线性互补问题的求解. 利用第一类拉格朗日方程和鲍姆加藤约束稳定化方法建立了系统的动力学方程, 由此可降低约束的漂移, 并可求解该系统的运动、法向接触力和切向摩擦力, 还可以求解理想铰链约束力和驱动约束力. 最后以一个类似夯机的平面多刚体系统为例, 分析了其动力学特性, 并说明了相关算法的有效性.     

滚动体几何相似条件在轴承边界元法中的应用

滚动轴承作为多物体接触系统,具有高度的非线性。为了求解滚动轴承的三维载荷分布,考虑所有滚动体的形状和边界条件相同,利用几何相似条件通过初始位置滚动体的离散模型,得到其他滚动体的离散数据。将所有滚动体看作是一个物体,轴承系统可以用3个物体无摩擦接触边界元法进行模拟计算。考虑全部滚动体作为一个整体的特殊性,叙述了边界积分方程和耦合的矩阵方程的建立,编制了基于滚动体几何相似条件的轴承边界元法Fortran源程序。利用该程序对轧机四列圆锥滚子轴承进行数值模拟,得到轴承的接触压力和载荷,滚动体接触宽度和接触滚动体个数,并说明了方法的有效性。     

THE FRICTIONLESS SLIDING OF AN OBJECT ALONG A ROPE

提出了物体在无摩擦情况下沿一段柔性绳索下滑运动力学问题的分析方法,利用牛顿第二定律建立耦合运动方程组,采用龙格-库塔（Runge-Kutta）方法编写程序对已建方程组进行求解,用Matlab 中的曲线拟合命令对方程组的数值解进行公式拟合,用得到的物体运动公式结合耦合运动方程组可求出物体下滑各时段绳索的内力. 提出的分析方法对物体在不同初始状态下的计算结果表明：绳索内力随物体的简谐振动呈周期性变化;物体下滑高度越高,经过相同位置时速度、加速度和绳索内力就越大.     

一种新的三维动力接触约束单元

本文应用广义变分原理，利用拉氏乘子法和罚函数法计入接触约束条件修正，建立了分析三维动力接触问题的一般有限元分析的理论模式；推导了处理这类问题的一种新的接触约束单元的接触刚度矩阵；采用增量求解模型解，研编了实施程序３ＤＤＣＦ；处理了方程中的病态问题、碰撞及能量释放条件问题。进行了算例考核，还首次给出了一个三维动力接触问题的算例结果。     

阶梯式Timoshenko梁自由振动的DCE解

本文基于微分容积法和区域叠加技术提出了微分容积单元法（Differential Cubature Element method，以下简称DCE方法），并用之求解阶梯式变截面Timoshenko梁的自由振动问题。根据梁的变截面情况将其划分为几个单元，在每个单元内应用微分容积法将梁的控制微分方程和边界约束方程离散成为一组关于该单元内配点位移的线性代数方程组，将这些方程组写在一起并在各单元之间应用连续性条件和平衡条件得到一组关于整个域内各点位移的齐次线性代数方程组，这是一广义特征值问题，由子空间迭代法求解该特征问题便可求得系统的自振动频率。数值算例表明，本方法能稳定收敛、并有较高的数值精度和计算效率。     

Boundary-layer eigen solutions for multi-field coupled equations in the contact interface

The dissipative equilibrium dynamics studies the law of fluid motion under constraints in the contact interface of the coupling system. It needs to examine how con- straints act upon the fluid movement, while the fluid movement reacts to the constraint field. It also needs to examine the coupling fluid field and media within the contact in- terface, and to use the multi-scale analysis to solve the regular and singular perturbation problems in micro-phenomena of laboratories and macro-phenomena of nature. This pa- per describes the field affected by the gravity constraints. Applying the multi-scale anal- ysis to the complex Fourier harmonic analysis, scale changes, and the introduction of new parameters, the complex three-dimensional coupling dynamic equations are transformed into a boundary layer problem in the one-dimensional complex space. Asymptotic analy- sis is carried out for inter and outer solutions to the perturbation characteristic function of the boundary layer equations in multi-field coupling. Examples are given for disturbance analysis in the flow field, showing the turning point from the index oscillation solution to the algebraic solution. With further analysis and calculation on nonlinear eigenfunctions of the contact interface dynamic problems by the eigenvalue relation, an asymptotic per- turbation solution is obtained. Finally, a boundary layer solution to multi-field coupling problems in the contact interface is obtained by asymptotic estimates of eigenvalues for the G-N mode in the large flow limit. Characteristic parameters in the final form of the eigenvalue relation are key factors of the dissipative dynamics in the contact interface.     

Effective numerical methods for elasto-plastic contact problems with friction

Several effective numerical methods for solving the elasto-plastic contact problems with friction are presented. First, a direct substitution method is employed to impose the contact constraint conditions on condensed finite element equations, thus resulting in a reduction by half in the dimension of final governing equations. Second, an algorithm composed of contact condition probes and elasto-plastic iterations is utilized to solve the governing equation, which distinguishes two kinds of nonlinearities, and makes the solution unique. In addition, Positive-Negative Sequence Modification Method is used to condense the finite element equations of each substructure and an analytical integration is introduced to determine the elasto-plastic status after each time step or each iteration, hence the computational efficiency is enhanced to a great extent. Finally, several test and practical examples are presented showing the validity and versatility of these methods and algorithms. The Project Supported by National Natural Science Foundation of China.     

On the Computer Formulations of the Wheel/Rail Contact Problem

In this investigation, four nonlinear dynamic formulations that can be used in the analysis of the wheel/rail contact are presented, compared and their performance is evaluated. Two of these formulations employ nonlinear algebraic kinematic constraint equations to describe the contact between the wheel and the rail (constraint approach), while in the other two formulations the contact force is modeled using a compliant force element (elastic approach). The goal of the four formulations is to provide accurate nonlinear modeling of the contact between the wheel and the rail, which is crucial to the success of any computational algorithm used in the dynamic analysis of railroad vehicle systems. In the formulations based on the elastic approach, the wheel has six degrees of freedom with respect to the rail, and the normal contact forces are defined as function of the penetration using Hertzs contact theory or using assumed stiffness and damping coefficients. The first elastic method is based on a search for the contact locations using discrete nodal points. As previously presented in the literature, this method can lead to impulsive forces due to the abrupt change in the location of the contact point from one time step to the next. This difficulty is avoided in the second elastic approach in which the contact points are determined by solving a set of algebraic equations. In the formulations based on the constraint approach, on the other hand, the case of a non-conformal contact is assumed, and nonlinear kinematic contact constraint equations are used to impose the contact conditions at the position, velocity and acceleration levels. This approach leads to a model, in which the wheel has five degrees of freedom with respect to the rail. In the constraint approach, the wheel penetration and lift are not permitted, and the normal contact forces are calculated using the technique of Lagrange multipliers and the augmented form of the system dynamic equations. Two equivalent constraint formulations that employ two different solution procedures are discussed in this investigation. The first method leads to a larger system of equations by augmenting all the contact constraint equations to the dynamic equations of motion, while in the second method an embedding procedure is used to obtain a reduced system of equations from which the surface parameter accelerations are systematically eliminated. Numerical results are presented in order to examine the performance of various methods discussed in this study.     

On the dynamic bending of layered plates

The dynamic bending of layered metallic plates is studied. Their layers are reinforced with wires, and there are interfacial layers between the contact surfaces. A general approach to solving the corresponding dynamic problems is proposed. It is the principle of virtual work. A system of ordinary differential constitutive equations is obtained and used to determine the amplitude coefficients of the series of coordinate functions that approximates the deflections. The constitutive equations are reduced to Volterra equations of the second kind. Algorithms are developed to calculate the coefficients of the constitutive equations depending on the coordinate functions and on the reinforcement pattern. Possible methods to set up coordinate functions and solve integral equations are indicated     

A frictional receding contact plane problem between a functionally graded layer and a homogeneous substrate

This paper investigates the plane problem of a frictional receding contact formed between an elastic functionally graded layer and a homogeneous half space, when they are pressed against each other. The graded layer is assumed to be an isotropic nonhomogeneous medium with an exponentially varying shear modulus and a constant Poisson’s ratio. A segment of the top surface of the graded layer is subject to both normal and tangential traction while rest of the surface is devoid of traction. The entire contact zone thus formed between the layer and the homogeneous medium can transmit both normal and tangential traction. It is assumed that the contact region is under sliding contact conditions with the Coulomb’s law used to relate the tangential traction to the normal component. Employing Fourier integral transforms and applying the necessary boundary conditions, the plane elasticity equations are reduced to a singular integral equation in which the unknowns are the contact pressure and the receding contact lengths. Ensuring mechanical equilibrium is an indispensable requirement warranted by the physics of the problem and therefore the global force and moment equilibrium conditions for the layer are supplemented to solve the problem. The Gauss–Chebyshev quadrature-collocation method is adopted to convert the singular integral equation to a set of overdetermined algebraic equations. This system is solved using a least squares method coupled with a novel iterative procedure to ensure that the force and moment equilibrium conditions are satisfied simultaneously. The main objective of this paper is to study the effect of friction coefficient and nonhomogeneity factor on the contact pressure distribution and the size of the contact region.     

The Boubaker polynomials and their application to solve fractional optimal control problems

In this paper, we focus on Boubaker polynomials in fractional calculus area and obtain the operational matrix of Caputo fractional derivative and the operational matrix of the Riemann–Liouville fractional integration for the first time. Also, a general formulation for the operational matrix of multiplication of these polynomials has been achieved to solve the nonlinear problems. Then, these matrices are applied to solve fractional optimal control problems directly. In fact, the functions of the problem are approximated by Boubaker polynomials with unknown coefficients in the constraint equations, performance index and conditions. Thus, a fractional optimal control problem converts to an optimization problem, which can then be solved easily. Convergence of the algorithm is proved. Numerical results are given for several test examples to demonstrate the applicability and efficiency of the method.     

Forward and inverse dynamics of nonholonomic mechanical systems

This paper deals with the forward and the inverse dynamic problems of mechanical systems subjected to nonholonomic constraints. The intrinsically dual nature of these two problems is identified and utilised to develop a systematic approach to formulate and solve them according to an unified framework. The proposed methodology is based on the fundamental equations of constrained motion which derive from Gauss’s principle of least constraint. The main advantage arising from using the fundamental equations of constrained motion is that they represent an effective method capable to derive the generalised acceleration of a mechanical system, constrained in general by a set of nonholonomic constraints, together with the generalized constraint forces (forward dynamics). When the constraint equations are used to represent the desired behaviour of the mechanical system under study, the generalised constraint forces deriving from the fundamental equations of constrained motion provide the control actions which reproduce the specified motion for the system (inverse dynamics). This approach is systematically extended to underactuated mechanical systems introducing a new method named underactuation equivalence principle. The underactuation equivalence principle is founded on the key idea that the underactuation property of a mechanical system can be mathematically represented using a particular set of nonholonomic constraint equations. Two simple case-studies are reported to exemplify the proposed methodology. In the first case-study the computation of the generalised constraint forces relative to the revolute joint constraints of a physical pendulum is illustrated. In the second case-study the calculation of the control action which solves the swing-up problem for an inverted pendulum is described.