小参数摄动法与保辛

应用数学与力学经常使用小参数摄动近似．在物理与力学中有大量保守体系的分析．保守体系的特点是保辛．本文指出小参数摄动法保辛的问题应予考虑．位移法摄动是保辛的,而辛矩阵的加法摄动则未能保辛．数值例题给出了对比．     

动力学积分保辛的数学根据

用最小作用量变分原理来解释保辛,对于连续时间系统、离散时间系统、有限元法、结构力学、最优控制和动力学计算等,可以通用的.     

约束动力系统的分析结构力学积分

约束保守系统导出了微分代数方程,其数值求解总是采用差分法.微分-代数方程的约束带来Lagrange参变函数转变而得的微分方程,有其指标问题,扩大了求解的规模.虽然已经注意差分的保辛,但沿切面积分再投影,仍带来许多问题.本文运用分析结构力学的方法,以节点处的独立位移为未知数且严格满足节点的约束条件,再将有限元近似用于区段作用量函数,在区段内部用简单插值求解.则按分析结构力学的理论,不但达到了积分的保辛且区段内部的约束条件也可在变分原理的意义下近似满足.数值结果满意.     

三类保辛摄动及其数值比较

摄动法近似应当保辛．本文指出,有限元位移法自动保辛,有限元混合能表示也保辛．摄动法的刚度阵Taylor级数展开能证明保辛;混合能的Taylor级数展开摄动也证明了保辛．但传递辛矩阵的Taylor级数展开摄动却不能保辛．辛矩阵只能在乘法群下保辛,故传递辛矩阵的保辛摄动必须采用正则变换的乘法．虽然刚度阵加法摄动、混合能矩阵加法摄动与传递辛矩阵正则变换乘法摄动都保辛,但这3种摄动近似并不相同．最后通过数值例题给出了对比．     

非线性动力学系统最优控制问题的保辛求解方法

基于对偶变量变分原理提出了求解非线性动力学系统最优控制问题的一种保辛数值方法.以时间区段一端状态和另一端协态作为混合独立变量,在时间区段内采用拉格朗日插值近似状态变量与协态变量,然后利用对偶变量变分原理并将非线性最优控制问题转化为非线性方程组的求解,最终得到求解非线性动力学系统最优控制问题的保辛数值方法.数值实验验证了本文算法在求解精度与求解效率上的有效性.     

某型自卸车车架的受力分析及优化

自卸汽车车架是整个车辆的安装基础,本文主要针对某型自卸汽车车架进行受力分析,通过采用材料力学、弹性力学、结构力学等力学理论对车架纵梁进行受力分析简化处理,并进行车架的优化.     

非线性最优控制系统的保辛摄动近似求解

非线性两端边值问题是在非线性最优控制计算中遇到的主要困难, 通常将其转化为线性两端边值问题的迭代求解．因此, 很有必要发展求解线性时变非齐次方程的两端边值问题的精确、高效算法. 本文通过引入区段混合能的概念, 将问题转化为区段的混合能矩阵及向量的求解, 进一步给出了它们的保辛摄动算法. 该算法具有很强的并行性, 高效而精确. 本文还指出经典的 Riccati 变换方法是该方法的一个特例. 数值算例验证了本文方法的有效性.     

管道机器人三轴差动式驱动单元的设计研究

设计了由三轴差速机构、管径适应机构组成的管道机器人三轴差动式驱动单元,对驱动单元在直管、弯管的差速特性与力学特性进行了理论分析．三轴差动式驱动单元在直管中运动时差速机构不起作用;在弯管运动时,根据管道拓扑环境实现自主差速,且行走轮运动状态为纯滚动,无寄生功率产生．所设计的管道机器人三轴差动式驱动单元为机械自适应型差动式管道机器人的理论研究奠定了基础．     

时间-空间混和有限元

分析动力学与分析结构力学在数学理论上是一致的.振动与结构力学问题,其实只是一个符号之差.分析力学方法对两方面可通用.双曲型偏微分方程与椭圆型偏微分方程也是差一个符号.虽然性质不同,但分析上有共同之处.本文提出在有限元分析方面,不用对时间、空间分别离散而是组成混和的时空混和有限元网格.数值结果表明,时空混和有限元是有前途的.     

过渡单元应用于结构静力分析

过渡单元的研究是结构力学问题在计算机配合下求解的进一步发展提出的新课题。在实际工程问题中,很少是单一形态的,大多数情况是相当复杂的。过渡单元是用以适应同一力学状态的构件。不同单元形态的过渡,也可以适应不同力学状态的构件的过渡。它可以比较合理地模拟过渡性构件。虽然也可以用通常的方法——使用加强约束在两种单元的交接面上,但这种近似可能导致不可靠的结果。为克服此缺欠,而提出由一种单元形态自然过渡到另一种单元形态的模拟方式——利用过渡单元。 过渡单元问题是多种多样的,如在所列文献中有轴对称体——壳过渡元,三维体——壳过渡元等等。在这篇文章中主要阐述三维体——折板元的过渡问题,在拦河坝,坝体应力计算问题中有着广泛的实用价值。     

时滞与界带

基于分析结构力学提出的界带分析方法,将子结构间的分界面延拓为有一定宽度的分界带/分界域,从而可以用于分析计算结构的非局部效应.分析动力学中存在弹性时滞积分问题,在理论方面仍存在许多有待明确的问题.通过分析结构力学与分析动力学间的模拟关系,可将界带分析方法用到弹性时滞积分问题中去,基于辛体系建立相应的分析计算方法.数值算例验证该套理论算法的可行性,也为进一步开展深入研究打下了理论基础.     

Meshless Conservative Scheme for Multivariate Nonlinear Hamiltonian PDEs

For multivariate nonlinear Hamiltonian equations, we propose a meshless conservative method by using radial basis approximation. Based on the method of lines, we first discretize the Hamiltonian functional using radial basis function interpolation, and then obtain a finite-dimensional semi-discrete Hamiltonian system. Moreover, we define a discrete symplectic form and verify that it is an approximation to the continuous one and is conserved with respect to time. For time discretization, two conservative methods (symplectic method and energy-conserving method) are employed to derive the full-discretized system. Approximation errors together with conservation properties including symplecticity and energy are discussed in detail. Finally, we present several numerical examples to illustrate that our method is accurate and effective when processing nonlinear Hamiltonian equations with scattered nodes. Besides, the numerical results also confirm the excellent conservation properties of the proposed method.     

Generalized Thermodynamics of Maximum Work in Finite Time

We consider thermodynamic behaviour of thermal machines founded on kinetic rather than static origins. Their models, which are formulated for finite time transitions, simplify to models of classical thermodynamics in the limiting case of an infinite duration. An extended exergy is derived as a finite-time extension of the classical thermodynamic work delivered from a system of a body and its environment. With this quantity enhanced bounds can be determined for active continuous and cascade processes, in which there is an indirect energy exchange between two sybsystems through the working fluid of an engine, a refrigerator or a heat pump. These bounds refer to systems with finite exchange area or with a finite contact time. An economic framework of this theory is outlined.For both continuous and discrete processes, nonlinear thermodynamic models are derived from a combination of the energy balance and transfer equations. These models serve as constraints in the problem of work optimization. Variational and optimal control approaches are developed which are analogous to those found in analytical mechanics. Variational calculus is used along with some aspect of the canonical transformation theory to maximize work and discuss the role of a finite process intensity and of a finite duration.The optimality of a definite irreversible process for a finite-time transition of a controlled fluid is pointed out as well as a connection between the process duration, optimal dissipation and the optimal process intensity measured in terms of a hamiltonian, a dissipative quantity. It is shown that limits of the classical availability theory should be replaced by stronger limits which are obtained for finite time processes, and which are closer to reality. A hysteretic property of the generalized exergy describes a decrease of the maximum work received from an engine system and an increase of work added to a heat pump system, the features which are particularly important in high-rate regions of thermodynamic processes. For an infinite sequence of infinitesimal thermal machines, an optimal temperature strategy is obtained in the form similar to that known in the theory of simulated annealing.     

Analytical Approximation Methods for the Stabilizing Solution of the Hamilton–Jacobi Equation

In this paper, two methods for approximating the stabilizing solution of the Hamilton–Jacobi equation are proposed using symplectic geometry and a Hamiltonian perturbation technique as well as stable manifold theory. The first method uses the fact that the Hamiltonian lifted system of an integrable system is also integrable and regards the corresponding Hamiltonian system of the Hamilton–Jacobi equation as an integrable Hamiltonian system with a perturbation caused by control. The second method directly approximates the stable flow of the Hamiltonian systems using a modification of stable manifold theory. Both methods provide analytical approximations of the stable Lagrangian submanifold from which the stabilizing solution is derived. Two examples illustrate the effectiveness of the methods.      

矩阵方程组A_1XB_1=C_1,A_2XB_2=C_2的迭代算法

矩阵方程组的求解在结构设计、参数识别、生物学、电学、分子光谱学、固体力学、自动控制理论、振动理论、有限元、线性最优控制等领域都有着重要应用。本文从解线性代数方程组的共轭梯度法中受到启示,不是采用传统的矩阵分解的方法,而是采用迭代算法给出了求矩阵方程组A1XB1=C1,A2XB2=C2的解、极小范数解及其最佳逼近解的方法。     

Hamilton系统的保辛 守恒积分算法

给出了Hamilton系统基于辛矩阵乘法的显式时不变正则变换和时变正则变换．引入含参变量的近似Hamilton系统,并以近似Hamilton系统为基础进行辛矩阵乘法的正则变换．正则变换保证了数值积分的保辛性质,而通过调整引入的参变量可保证能量在积分格点上守恒．实现了Hamilton系统即保辛又保能量的算法．