热弹性材料齐次状态向量方程和复合材料层合面板的精确解

首先以修正的H-R(Hellinger-Reissner)变分原理,先推导出热弹性材料的非齐次状态方程,再利用热平衡方程和热传导方程变量间的对偶关系,通过增加非齐次状态向量矩阵的维数,将非齐次状态方程转化为可以独立求解的齐次状态向量方程;根据对偶理论,将热传导方程并入材料的本构关系中,得到新的修正的H-R变分原理,从而可直接推导出热弹性材料的齐次状态向量方程.齐次状态向量方程的导出可大大简化热弹性体稳态温度问题的求解,最后以一算例验证文中方法的准确性和可靠性.     

弹性理论中广义变分原理的研究及其在有限元计算中的应用

本文的目的在于说明怎样系统地建立各种广义变分原理,怎样合理地使用各种广义变分原理来改进有限元计算的成效。为了易于说明问题,本文只局限于弹性理论的各种广义变分原理,但其推广并不困难。本文指出,广义变分原理的泛函,可以系统地采用拉格朗日乘子法,把一般有条件的变分原理化为无条件的变分原理来唯一地决定的。拉格朗日所代表的物理量,可以通过变分求极值或驻值的过程求得,从而消除了在建立广义变分原理的泛函时,人们经常陷入的象猜谜一样的困境。本文也指出：我们同样可以用拉格朗日乘子法,把一般有多个条件的变分原理,化为条件个数较少的变分原理。我们称变分条件减少了的变分原理为各级不完全的广义变分原理。凡是全部变分条件都消除了的变分原理,称为完全的广义变分原理,或简称广义变分原理;实际上是完全无条件的变分原理。本文建立了弹性小位移变形理论中的各级不完全的广义位能原理和各级不完全的广义余能原理,包括从最小位能原理和最小余能原理分别导出的最完全的广义变分原理,且并证明了这两个弹性力学广义变分原理的泛函是等同的。在这些广义变分原理中,包括了HELLINGER REISSNER（1950）[1],[2],胡海昌－鹫津久一郎[3],[4]的广义变分原理。本文也建立了弹性大移位变形中的位能原理和余能原理,并建立了有关位能余能的各级不完全的广义变分原理,包括从大位移变形的最小余能原理分别导出的弹性力学广义变分原理,并且也证明了大位移变形情况下,这两个弹性力学的广义变分原理也是同等的。本文除了列举广义变分原理在有限元法上的众所周知的应用外,还补充三个比较重要的应用范围。     

插齿刀加工少齿数齿轮方法的研究

据包络法原理,得到了插齿刀加工少齿数齿轮精确的齿廓曲线方程,包括过渡曲线方程和渐开线方程,为这种新的少齿数齿轮加工方法提供理论基础;通过Pro/E软件的建模,验证了所求方程的正确性,并观察得到了一些齿形特点;最后通过ADAMS软件对少齿数齿轮副进行了传动动力学仿真研究,结果表明：传动处于微小变载状态、较平稳及传动效率高,也进一步证明了所求方程的正确性和合理性。     

The Stressed State of the “Boundary Layer” Type in Cylindrical Shells Investigated according to a Nonclassical Theory

Two variants of a refined theory for calculating the stress–strain state in the boundary zones of cylindrical shells are presented. The relevant mathematical models are based on equations of the three-dimensional elasticity theory and the increase over the thickness of the shell in the orders of the polynomials that approximate the sought displacements. The Lagrange variational principle is applied to the value of the shell’s total energy functional defined more exactly with respect to the classical Kirchhoff–Love theory. The formulated boundary problems allow determination with different degrees of accuracy of additional stressed states of the “boundary layer” type. The calculated results obtained in this work are compared with the results obtained according to the classical theory. It has been established that the above stresses make a significant contribution to the total stressed state of the shell and should be considered when designing and testing machine structures for strength and longevity.     

A symplectic pseudospectral method for nonlinear optimal control problems with inequality constraints

A symplectic pseudospectral method based on the dual variational principle and the quasilinearization method is proposed and is successfully applied to solve nonlinear optimal control problems with inequality constraints in this paper. Nonlinear optimal control problem is firstly converted into a series of constraint linear-quadratic optimal control problems with the help of quasilinearization techniques. Then a symplectic pseudospectral method based on dual variational principle for solving the converted constrained linear-quadratic optimal control problems is developed. In the proposed method, inequality constraints which can be functions of pure state, pure control and mixed state-control are transformed into equality constraints with the help of parameteric variables. After that, state variables, costate variables and parametric variables are interpolated locally at Legendre-Gauss-Lobatto points. Finally, based on the parametric variational principle and complementary conditions, the converted problem is transformed into a standard linear complementary problem which can be solved easily. Numerical examples show that the proposed method is of high accuracy and efficiency.     

摩擦约束有限变形弹塑性广义变分不等原理

在弹塑性接触问题中,从变分原理出发来研究接触问题,可以将摩擦力纳入问题的能量泛函以便于问题的求解,尤其是数值解。研究了带摩擦约束的有限变形弹塑性接触问题,利用拉格朗日乘子法,证明了有限变形弹塑性接触问题的广义变分不等原理,导出了率形式的能量泛函,为摩擦约束有限变形弹塑性接触问题的数值解提供了理论基础。     

一种LED光源矩形斑照明的优化算法及其仿真

微分方程求解法是设计自由曲面最常用的一种方法之一,此方法是通过建立照明源与被照明面之间的几何关系,来实现自由曲面的数值求解。通常会基于照明光源的光分布与被照明面的照明要求,建立一组偏微分方程,方程组的未知数为自由曲面的每一个节点的坐标和曲率值。通过能量守恒原理,解出所需要的自由曲面每个节点的具体坐标值。在现有微分方程理论的基础上,通过分割自由曲面和变步长的龙格-库塔算法,大幅改善了矩形照明斑的形状、能量利用率及均匀性等,对矩形照明斑质量实现了优化。     

含裂纹板断裂韧度厚度效应的理论与应用

从理论角度对断裂韧度厚度效应进行了研究。在线弹性断裂力学与含裂纹板的二维位移场的基础上,给出分离变量型的具有待定函数的三维位移场的表达式,进而通过几何方程与物理方程获取三维应变场和三维应力场。进一步,通过虚位移原理,使用变分方法建立待定函数所应满足的支配方程与边界条件,从而确定待定函数。基于上述分析,建立一个断裂韧度与试样厚度关系的理论表达式。最后通过两种试验材料LY12CZ(L-T)与TC4(L-T)进行了验证。     

反求工程中基于图像灰度信息的三维曲面重构

基于物体表面图像灰度信息的三维几何形状反求(由明暗恢复形状),已成为工业界和学术界研究的热点问题之一。首先给出描述该问题的数学物理模型,并利用变分原理将其转化为一个泛函的极小化问题,然后应用有限差分思想和非线性最小二乘方法,将该泛函极小化问题归结为求解线性方程组。最后利用该算法产生的迭代公式对合成图像进行计算,并应用曲面造型软件CAXA(制造工程师)进行曲面重构,从而验证了该方法的有效性和可行性。     

OPTIMAL CONTROL FOR VIBRATING SYSTEMS

A linear regulator problem for mechanical vibrating systems is studied in the second-order formulation. We exploit the second-order form of the differential equations involved, and solve the problem without the traditional use of a Riccati equation. In its natural representation, the optimal control problem gives rise to minimisation of a functional depending on second derivatives. The Euler–Lagrange equations and the transversality conditions developed lead to a linear fourth-order differential equation that determines the optimal control. The results are demonstrated by examples.     

悬臂输送管道流-固耦合动力学系统的直接解法

根据变分原理导出了输送管道(流—固耦合问题)自由振动的变分方程,采用直接解法求出了输送管道自由振动的固有频率和极限流速。     

含裂纹板断裂韧度厚度效应的理论研究

从理论角度对断裂韧度厚度效应进行研究。在线弹性断裂力学与含裂纹板的二维位移场的基础上，给出分离变量型的具有待定函数的三维位移场的表达式，进而通过几何方程与物理方程获取三维应变场和三维应力场。进一步，通过虚位移原理，使用变分方法建立待定函数所应满足的支配方程与边界条件，从而确定待定函数。基于上述分析，建立一个断裂韧度与试样厚度关系的理论表达式。最后通过两种试验材料LY12CZ（L-T）与TC4（L-T）进行验证。     

Non-equilibrium theory of hydrodynamic lubrication

A new variational method is developed for the solution of the Navier Stokes equations which describe the unsteady flow between two infinite plates approaching or receding from each other symmetrically. The partial differential equations of the system are reduced to ordinary non-linear differential equations by using similarity transformations containing the Reynolds number as a parameter. The application of the governing principle of dissipative processes reduces the non-linear differential equation to an algebraic equation which can be solved for the required value of R and thus the flow characteristics can be studied. The velocity functions are obtained for various values of R ranging from 0.01 to 1.0.     

Buckling analysis of laminated plates using the extended Kantorovich method and a system of first-order differential equations

The extended Kantorovich method using multi-term displacement functions is applied to the buckling problem of laminated plates with various boundary conditions. The out-of-plane displacement of the buckled plate is written as a series of products of functions of parameter x and functions of parameter y. With known functions in parameter x or parameter y, a set of governing equations and a set of boundary conditions are obtained after applying the variational principle to the total potential energy of the system. The higher order differential equations are then transformed into a set of first-order differential equations and solved for the buckling load and mode. Since the governing equations are first-order differential equations, solutions can be obtained analytically with the out-of-plane displacement written in the form of an exponential function. The solutions from the proposed technique are verified with solutions from the literature and FEM solutions. The bucking loads correspond very well to other available solutions in most of the comparisons. The buckling modes also compare very well with the finite element solutions. The proposed solution technique transforms higher-order differential equations to first-order differential equations, and they are analytically solved for out-of-plane displacement in the form of an exponential function. Therefore, the proposed solution technique yields a solution which can be considered as an analytical solution.     

复合材料结构振动控制方法研究

从压电材料的非线性本构关系出发,应用Hamilton原理推导含有压电材料复合材料结构的变分方程,并证明对于具有二次本构关系的压电材料可以直接由二次本构关系得到其变分方程,推导出含有二次本构关系压电材料的复合材料结构的有限元动力方程,并提出利用压电材料的非线性进行复合材料结构振动控制的一种新方法.     

横向剪切干涉测量中准确的相位恢复算法

提出一种剪切干涉测量中实现由相位差分准确高效恢复被测波前相位的算法.该算法是基于被测波前相位与其差分完全的点对点对应关系,及最小二乘算法原理.首先由被测波前相位与其差分完全的点对点对应关系及最小二乘原理,建立一特殊的线性等式,被测相位能通过解此等式直接获得.由于该线性等式的系数矩阵为稀疏矩阵,能转化为一新的小矩阵,以降低计算机存储空间和计算量;同时,由于该矩阵为正定矩阵,Choleski因式化分解方法能用来实现该线性等式方便的解.进行了计算机数值分析和相关试验测试,结果表明:该恢复算法可行,且计算精度高、计算复杂性低;可实现由相位差分准确恢复被测波前相位,同时具有良好的噪声误差传输特性.     

基于微分几何的欠驱动机器人动力学建模和控制

应用位形流形最小嵌入模型,对带有二阶非完整约束的欠驱动机器人动力学建模和控制进行研究。采用位形流形最小嵌入模型,简化了动力学方程,为欠驱动机器人动力学建模和控制的进一步研究奠定了基础。首先用一个齐次线性方程组表示被动关节的存在,给出基于嵌入模型的动力学方程解的结构。通过引入计算转矩控制法,给出另一个和主动关节及控制输入有关的齐次线性方程组,分析控制输入和这两个方程组解之间的关系,得到一个与控制输入及这两个方程组的解相关的欠定线性方程组,求解该欠定线性方程组,得到欠驱动机器人改进的动力学方程,进而可以运用全驱动机器人控制法来实现最小嵌入模型的控制,解决了由全驱动机器人控制法实现欠驱动机器人关节空间控制的问题。最后以平面二杆欠驱动机器人为例,验证了所得理论的可行性和有效性。     

Deformations in hollow structural section (HSS) members subjected to cold-bending

The relation between the radius of bend and the distortions in cold-bent HSS steel members is investigated. The variational principle of the total potential energy is adopted to predict this relationship, accounting for both geometric and material non-linearities. A method is proposed to model the cold-bending rolling process. Based on the results from the analytical solution empirical relations in the form of best-fit equations are deduced for ready use. Results are compared to those from 108 tests on 27 different HSS profiles.     

一种充分局部化的子结构并行有限元方法的变分原理

局部化的有限元分裂内联法(以下简称L-FETI),是一种适合于机群系统的并行有限元方法,它将一个连续结构系统分解为多个独立的子域,各子域之间通过分区框架实现耦合。本文将分区框架视为无体积、无惯性的独立连接结构,在区域分解界面上引入局部化的界面位移和界面内力,直接从能量原理出发,构造了分区系统的势能泛函,完成了L-FETI方法的变分格式推导,说明了L-FETI方法在处理区域网格匹配问题和区域网格不匹配问题时所具有的一致性并引入惯性力,导出了基于L-FETI方法的动力学方程。     

复合补偿驱动直线伺服单元研究

依据动圈电动机的基本原理，研究开发了一种大推力复合补偿驱动的直线伺服单元，其推力常数为８２Ｎ／Ａ，最大驱动电流为１０Ａ，最大推力８２０Ｎ。给出了其基本原理、运动微分方程；介绍了复合补偿驱动磁路和磁路方程。给出了一设计实例和具体的性能参数，由它构成的直线驱动伺服系统，实际已应用于中凸变椭圆活塞ＣＮＣ车床。