超细长弹性杆的分析力学问题

超细长弹性杆作为DNA等生物大分子链的力学模型，其平衡和稳定性问题已成为力学与分子生物学交叉的研究热点．虽然在Kirchhoff动力学比拟的基础上，用分析力学方法讨论弹性杆的文章已见诸文献，但尚未形成弹性杆分析力学的严格理论．本文研究了超细长弹性杆分析力学的若干基础性问题．对杆截面的自由度、虚位移、约束方程及约束力等基本概念给出严格的定义和表达式．建立弹性杆平衡的D’Alembert-Lagrange原理、Jourdain原理和Gauss原理；从D’Alembert-Lagrange原理导出Hamilton原理．从变分原理出发导出Lagrange方程、Nielsen方程、Appell方程和Hamilton正则方程；对于受约束的弹性杆，导出了带乘子的Lagrange方程．讨论了Lagrange方程的首次积分．对于杆中心线存在尖点的情形，导出了微段杆平衡的近似方程。     

变加速动力学系统的广义高斯最小拘束原理

变加速运动在日常生活和工程问题中普遍存在.变加速动力学又称牛顿猝变动力学,因其在混沌理论和非线性动力学中的应用而获得广泛关注.高斯原理是一个具有极值性质的微分变分原理.因此,研究变加速动力学系统的广义高斯原理在理论和应用两方面都有重要意义.文章提出并研究变加速动力学系统的广义高斯原理.首先,引入急动度空间的广义高斯变分概念,将质点的达朗贝尔原理对时间求导数后与广义高斯变分点乘,并利用高斯意义下的理想约束条件,建立了变加速动力学系统的广义高斯原理.在此基础上,通过构造广义拘束函数建立并证明变加速动力学系统的广义高斯最小拘束原理,并给出原理的阿佩尔形式、拉格朗日形式和尼尔森形式.其次,研究原理对变质量力学的推广.从密歇尔斯基方程出发,将它对时间求导并与广义高斯变分点乘,建立了具有理想约束的变质量变加速动力学系统的广义高斯原理.通过构造变质量系统的广义拘束函数,建立并证明变质量力学系统变加速运动的广义高斯最小拘束原理.文中以开普勒-牛顿空间问题为例,利用所得的广义高斯最小拘束原理方法进行计算,验证了方法的有效性.     

RELATIVISTIC VARIATION PRINCIPLES AND EQUATION OF MOTION FOR VARIABLE MASS CONTROLLABLE MECHANICAL SYSTEM

ＲＥＬＡＴＩＶＩＳＴＩＣＶＡＲＩＡＴＩＯＮＰＲＩＮＣＩＰＬＥＳＡＮＤＥＱＵＡＴＩＯＮＯＦＭＯＴＩＯＮＦＯＲＶＡＲＩＡＢＬＥＭＡＳＳＣＯＮＴＲＯＬＬＡＢＬＥＭＥＣＨＡＮＩＣＡＬＳＹＳＴＥＭ￥(罗绍凯)ＬｕｏＳｈａｏｋａｉ（ＳｈａｎｇｑｉｕＴｅａｃｈｅｒｓ...     

Nielsen方程的广义梯度表示及其稳定性分析

研究Nielsen方程的广义梯度表示以及方程零解稳定性．首先给出4类广义梯度系统及其性质．其次,给出完整系统和非完整系统的Nielsen方程转化成广义梯度系统的条件;将两类Nielsen方程分别化为广义梯度系统并研究方程的零解稳定性．最后,举例验证结果的应用并通过数值模拟验证结论的准确性．     

爬壁机器人系统的Noether对称性和守恒量

爬壁机器人的运动是一种模仿壁虎爬行的运动, 爬壁机器人的运动可分解为四肢带动身体的运动, 先前的研究都是基于牛顿力学的方法. 本文采用Lagrange 力学的方法建立爬壁机器人系统的运动方程, 并运用Lie群分析方法建立该系统的Noether对称性理论, 得出爬壁机器人的运动规律. 首先, 给出非完整爬壁机器人系统的动能、势能和Lagrange函数以及所受的非完整约束, 从而建立了非完整爬壁机器人系统的Lagrange方程; 其次, 引入关于时间和广义坐标的无限小变换, 提出了非完整爬壁机器人系统的Hamilton作用量和Hamilton作用量的基本变分公式; 第三, 给出爬壁机器人系统 Noether对称性变换和广义准对称变换的定义, 判据和存在的Noether守恒量, 并提出了非保守完整系统和非保守非完整爬壁机器人系统的Noether定理; 最后, 以圆锥面上爬壁机器人为例, 对给出的守恒量直接进行积分给出圆锥面上爬壁机器人整体运动的精确解和四肢运动的数值解, 发现了该爬壁机器人的运动规律, 很好地验证了非完整爬壁机器人系统的Noether对称性理论. 本文的研究为Lie群分析方法应用于其他复杂的机器人系统以及柔性机器人系统的对称性求解提出了一种新的对称性求解方法.      

Algebraic structures and poisson integrals of relativistic dynamical equations for rotational systems

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修正的偶应力线弹性理论及广义线弹性体的有限元方法

以含偶应力的弹性理论为基础,考虑小变形情况下变形体的平动变形和旋转变形,提出关于偶应力与曲率张量的线性本构关系,建立一般弹性体的线性模型。为满足有限单元C1连续性要求,考虑转角为独立变量,利用罚方法引入约束条件,构造一般弹性体的约束变分形式。应用8节点48个自由度的实体等参元,建立一般弹性体力学响应分析的有限元方程。对悬臂梁的静力和动力分析表明,一般弹性体模型较之经典弹性力学更适合结构分析;较之Timoshenko梁模型,一般弹性体模型能够计及结构尺度对结构动力特性和动力响应造成的显著影响。     

Kinetic and potential energy in the first law of thermodynamics

Kinetic and potential energy are included in the first law of thermodynamics in quite a contradictory way. Whereas in thermodynamics the total energy is understood as the sum of internal, kinetic and potential energy, the total energy in continuum mechanics incorporates only internal and kinetic energy, the potential energy being part of the work. The Gibbs ' fundamental equation is also occasionally extendend to contain a term for the potential energy. Some serious contradictions may result from this. As is first shown, kinetic and potential energy do not have any influence on the internal energy as long as relativistic effects are excluded. The Gibbs' fundamental equation therefore describes exchange processes between the “internal variables? of a system and its surroundings. Proceeding from this result one obtains a general definition of heat in open systems, including electromagnetic reactions, surfaceeffects and variable mole numbers. Exchange processes between the “external variables? of a system and its surroundings and hence also the influence of kinetic and potential energy are described by another independent equation, i.e. the energy equation of mechanics. Addition of both equations leads to the heat definition which is usually but under some further neglects given in textbooks. This definition has considerable disadvantages compared to the one derived before. In particular it is no longer possible to realize how mechanical and thermal energy are transformed into each other, which may give rise to errors.     

PARAMETRIC VARIATIONAL PRINCIPLE BASED ELASTIC-PLASTIC ANALYSIS OF COSSERAT CONTINUUM

A new algorithm is developed based on the parametric variational principle for elastic-plastic analysis of Cosserat continuum. The governing equations of the classic elastic-plastic problem are regularized by adding rotational degrees of freedom to the conventional translational degrees of freedom in conventional continuum mechanics. The parametric potential energy principle of the Cosserat theory is developed, from which the finite element formulation of the Cosserat theory and the corresponding parametric quadratic programming model are constructed. Strain localization problems are computed and the mesh independent results are obtained.     

基于参数变分原理的Cosserat连续体弹塑性分析

基于参数变分原理,提出了Cosserat模型弹塑性计算的算法,给出了基于Cosserat理论的参数最小势能原理,基于所提出的变分方程,建立了Cosserat理论弹塑性分析的参数二次规划模型,进一步将算法应用于平面应变软化问题计算中,获得的结果具有良好的非网格依赖性.     

A Generalized Theory of Stress and Strain Measures in the Classical Continuum Mechanics

A generalized theory of stress and strain tensor measures in the classical continuum mechanics is discussed: the main axioms of the theory are proposed, the general formulas for new tensor measures are derived, arid an energy conjugate theorem is formulated to distinguish the complete Lagrangian class of measures. As a subclass, a simple Lagrangian class of energy conjugate measures of stresses and finite strains is constructed in which the families of holonomic and corotational measures are distinguished. The characteristics of holonomic and corotational measures are studied by comparing the tensor measures of the simple Lagrangian class with one another and with logarithmic measures. For the simple Lagrangian class and its families, their completeness and closure are shown with respect to the choice of a generating pair of energetically conjugate measures. The applications of the new tensor measures in modeling the properties of plasticity, viscoelasticity, and shape memory are mentioned.     

On the invariant theory of nonholonomic systems with constraints of non-Chetaev type

This paper deals with the theory of the differential invariant and integral invariant for a nonholonomic system with constraints of non-Chetaev type. It gives the restricted conditions of virtual displacement in velocity space for nonholonomic constraints of non-Chetaev type and extends the Jourdain principle and the canonical equation for the system. It presents and proves generalized Noether theorem, and gives generalized energy integral and cyclic integral for the system. Finally, the basic integral variants for the system are extended and the integral invariant of Poincaré-Cartan and the universal integral invariant of Poincaré for the system are obtained.     

区段混合能的偏微分方程

结构力学与最优控制模拟关系的共同基础就是分析力学,表明在结构力学与最优控制理论的架构内也应有分析力学的整套理论。传统分析力学总是考虑连续时间、同时的状态、不变维数的体系。并且物性为即时响应的。但结构力学有限元要考虑离散坐标、不同坐标状态、而且变动维数、时滞的体系。根据区段变形能只与其两端位移有关,就可通过数学分析得到Lagrange括号与Poisson括号等内容。区段变形能就是作用量,满足Hamilton-Jacobi方程。但还有区段混合能的表示,本文证明它满足雷同的偏微分方程。它们在离散体系时还有偏差分方程。本文进一步给出了其与Riccati方程的关系。     

Lie symmetries and conserved quantities of rotational relativistic systems

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On the Use of the Continuum Mechanics Method for Describing Interactions in Discrete Systems with Rotational Degrees of Freedom

Elastic interactions in a system of two body-points possessing both translational and rotational degrees of freedom are studied for the most general case of motion in 3D space. The continuum mechanics method is used as a theoretical foundation for describing the interactions. A definition of strain measures for the discrete system is given by analogy with that in continuum mechanics. Constitutive equations for force and moment vectors are derived based on the energy balance equation. Several new interaction potentials are suggested.     

Bi-variable damage model for fatigue life prediction of metal components

Based on the theory of continuum damage mechanics,a bi-variable damage mechanics model is developed,which,according to thermodynamics,is accessible to derivation of damage driving force,damage evolution equation and damage evolution criteria. Furthermore,damage evolution equations of time rate are established by the generalized Drucker’s postulate. The damage evolution equation of cycle rate is obtained by integrating the time damage evolution equations,and the fatigue life prediction method for smooth specimens under repeated loading with constant strain amplitude is constructed. Likewise,for notched specimens under the repeated loading with constant strain amplitude,the fatigue life prediction method is obtained on the ground of the theory of conservative integral in damage mechanics. Thus,the material parameters in the damage evolution equation can be obtained by reference to the fatigue test results of standard specimens with stress concentration factor equal to 1,2 and 3.     

GAS-KINETIC UNIFIED ALGORITHM FOR BOLTZMANN MODEL EQUATION IN ROTATIONAL NONEQUILIBRIUM AND ITS APPLICATION TO THE WHOLE RANGE FLOW REGIMES

基于过去开展稀薄自由分子流到连续流气体运动论统一算法框架,采用转动惯量描述气体分子自旋运动,确立含转动非平衡效应各流域统一玻尔兹曼模型方程.基于转动能量对分布函数守恒积分,得到计及转动非平衡效应气体分子速度分布函数方程组,使用离散速度坐标法对分布函数方程所依赖速度空间离散降维;应用拓展计算流体力学有限差分方法,构造直接求解分子速度分布函数的气体动理论数值格式;基于物面质量流量通量守恒与能量平衡关系,发展计及转动非平衡气体动理论边界条件数学模型及数值处理方法,提出模拟各流域转动非平衡效应玻尔兹曼模型方程统一算法.通过高、低不同马赫数1：5～25氮气激波结构与自由分子流到连续流全飞行流域不同克努森数（9×10^-4～10）Ramp制动器、圆球、尖双锥飞行器、飞船返回舱外形体再入跨流域绕流模拟研究,将计算结果与有关实验数据、稀薄流DSMC模拟值等结果对比分析,验证统一算法模拟自由分子流到连续流再入过程高超声速绕流问题的可靠性与精度.     

转动非平衡玻尔兹曼模型方程统一算法与全流域绕流计算应用

基于过去开展稀薄自由分子流到连续流气体运动论统一算法框架,采用转动惯量描述气体分子自旋运动,确立含转动非平衡效应各流域统一玻尔兹曼模型方程.基于转动能量对分布函数守恒积分,得到计及转动非平衡效应气体分子速度分布函数方程组,使用离散速度坐标法对分布函数方程所依赖速度空间离散降维;应用拓展计算流体力学有限差分方法,构造直接求解分子速度分布函数的气体动理论数值格式;基于物面质量流量通量守恒与能量平衡关系,发展计及转动非平衡气体动理论边界条件数学模型及数值处理方法,提出模拟各流域转动非平衡效应玻尔兹曼模型方程统一算法.通过高、低不同马赫数1:5～25氮气激波结构与自由分子流到连续流全飞行流域不同克努森数（9×10-4～10）Ramp制动器、圆球、尖双锥飞行器、飞船返回舱外形体再入跨流域绕流模拟研究,将计算结果与有关实验数据、稀薄流DSMC模拟值等结果对比分析,验证统一算法模拟自由分子流到连续流再入过程高超声速绕流问题的可靠性与精度.