基于高斯原理的非理想系统动力学建模

高斯原理给出了通过求函数极值、从可能运动中鉴别出真实运动的规则, 它可以使得多体系统动力学问题不需通过求解微分(代数)方程, 而是采用求解最小值的优化方法来解决, 从而提供了一种适用于优化算法的建模思路, 因此, 如何定义恰当的高斯拘束函数是动力学优化方法得以实现的前提. 对于理想系统而言, 约束对系统的作用可以通过约束方程来体现, 故高斯拘束可表达为系统质点加速度的函数, 系统的动力学问题因此可以描述为目标函数为高斯拘束函数、优化变量为质点加速度的约束最优化问题; 当系统中需要考虑干摩擦等非理想因素时, 部分相互作用不能被所定义的约束方程所涵盖而需要采用额外的物理规律来描述, 这种相互作用破坏了原有的针对理想系统的高斯拘束函数的极值特性. 基于变分类的高斯原理, 推导并证明了目标函数以理想约束力所表达的非理想系统的极值原理, 针对目前文献中用于非理想系统的高斯原理进行了讨论, 指出其实际为文中的极值原理在非理想约束力与理想约束力无明显关联时的一种特殊表达形式, 当非理想约束力与理想约束力有明显的函数关系(如库仑摩擦定律中滑动摩擦力与法向约束力间的线性关系)时, 该形式失效; 同时根据文中的极值原理, 得到了考虑库仑摩擦时非理想的多体系统动力学问题的优化模型. 例子中分析了优化模型及相应的线性互补性模型的关系, 分析发现在满足刚体滑动问题的唯一性条件下二者互为充分必要条件, 从而证明了文中优化模型的可靠性; 并采用优化计算方法进行了动力学模拟, 模拟结果显示了将高斯原理与优化算法相结合的可行性及有效性.      

DYNAMIC MODELING OF MULTI-BODY SYSTEM BASED ON GAUSS'S PRINCIPLE

基于高斯最小拘束原理,以释放中的绳系卫星为背景,建立地球引力场内变长度大变形柔索联系的多体系统动力学模型. 利用基尔霍夫动力学比拟方法将柔索的变形转化为刚性截面沿中心线的转动,使包含刚性分体与变形体的刚柔耦合系统转化为由柔索的刚性截面与刚性分体组成的广义多刚体系统. 由于刚性截面的局部小变形沿弧坐标的积累不受限制,适合描述柔索的超大变形. 文中对此刚柔耦合多体系统导出其在地球引力场中的拘束函数,考虑各分体在空间中相对位置的几何约束条件,利用拉格朗日乘子构成以条件极值问题为特征的数学模型. 将高斯原理用于多体系统动力学的建模,其特点是以寻求函数极值的变分方法代替微分方程,通过数值计算直接得出运动规律. 其形式统一,不随系统拓扑结构的变化而改变,也无需区分树系统或非树系统.对于带控制的多体系统,动力学分析还可根据技术需要与系统的优化结合进行.     

基于高斯原理的多体系统动力学建模

基于高斯最小拘束原理,以释放中的绳系卫星为背景,建立地球引力场内变长度大变形柔索联系的多体系统动力学模型. 利用基尔霍夫动力学比拟方法将柔索的变形转化为刚性截面沿中心线的转动,使包含刚性分体与变形体的刚柔耦合系统转化为由柔索的刚性截面与刚性分体组成的广义多刚体系统. 由于刚性截面的局部小变形沿弧坐标的积累不受限制,适合描述柔索的超大变形. 文中对此刚柔耦合多体系统导出其在地球引力场中的拘束函数,考虑各分体在空间中相对位置的几何约束条件,利用拉格朗日乘子构成以条件极值问题为特征的数学模型. 将高斯原理用于多体系统动力学的建模,其特点是以寻求函数极值的变分方法代替微分方程,通过数值计算直接得出运动规律. 其形式统一,不随系统拓扑结构的变化而改变,也无需区分树系统或非树系统.对于带控制的多体系统,动力学分析还可根据技术需要与系统的优化结合进行.      

基于LuGre摩擦模型的接触约束法旋转柔性梁斜碰撞研究

基于接触约束法和LuGre摩擦模型对在重力场作用下作大范围旋转运动的柔性梁系统和斜坡发生含摩擦斜碰撞的动力学问题进行研究.首先运用刚柔耦合的多体系统动力学理论对大范围运动的柔性梁进行离散化和动力学建模,在碰撞时采用冲量动量法求出跳跃速度,其次在法向上引入接触约束求解出碰撞力,在切向上采用LuGre摩擦模型分两种方式求解...     

桁架频率优化解存在性及其算法研究

给出桁架在给定固频约束时,动力不优化解存在性的基本理论。指出：桁架固频约束通常是决定其动力学优化解是否存在的“关键约束”。基此,给出了一种工程算法,只用到固频对设计变量的一阶导数,就可很快确定约束是否可能满足。反之,当提出的方法判定给定固频约束不能满足时,可以求出一个解存在的极限频率的狭窄范围及相应的设计变量范围,以供设计人员重新设计时参考。三个由简到繁的算例说明了所提出方法实用、有效。     

多体系统接触碰撞问题的牛顿-欧拉线性互补方法

基于非光滑动力学方法的多体系统接触碰撞分析是目前多体系统动力学的研究热点.本文采用牛顿-欧拉方法建立多体系统接触、碰撞问题的动力学模型,给出一种牛顿-欧拉型线性互补公式.该建模方法与目前一般采用的拉格朗日建模方法的不同之处是约束条件中除了库仑摩擦、单边约束之外还含有光滑等式约束.在建立系统动力学模型时,首先解除摩擦约束和单边约束得到原系统对应的基本系统.牛顿-欧拉方法采用最大数目坐标建立基本系统的动力学方程,由于坐标不相互独立,因此基本系统中带有等式约束,其数学模型为一组微分代数方程.借助约束雅可比矩阵,在基本系统微分代数方程中添加摩擦接触和单边约束对应的拉氏乘子,就可以得到系统全局运动的具有变拓扑结构特征的动力学方程,再结合非光滑约束互补条件便可构成完备的系统动力学模型.完备的动力学模型由动力学微分方程以及等式约束和不等式约束组成.线性互补公式采用分块矩阵形式进行推导,简化了推导过程.数值计算采用基于线性互补的时间步进算法.时间步进算法是目前流行的非光滑数值算法,其突出特点是可以免去数值积分中繁琐的事件检测过程,而数值积分过程中通过对线性互补问题的求解可以确定系统的触-离状态.通过对典型的曲柄滑块间隙机构进行数值分析,验证本文方法的有效性.     

Numerical method for dynamics of multi-body systems with two-dimensional Coulomb dry friction and nonholonomic constraints

Based on the dynamical theory of multi-body systems with nonholonomic constraints and an algorithm for complementarity problems, a numerical method for the multi-body systems with two-dimensional Coulomb dry friction and nonholonomic constraints is presented. In particular, a wheeled multi-body system is considered. Here,the state transition of stick-slip between wheel and ground is transformed into a nonlinear complementarity problem(NCP). An iterative algorithm for solving the NCP is then presented using an event-driven method. Dynamical equations of the multi-body system with holonomic and nonholonomic constraints are given using Routh equations and a constraint stabilization method. Finally, an example is used to test the proposed numerical method. The results show some dynamical behaviors of the wheeled multi-body system and its constraint stabilization effects.     

Selection of the optimum parameters for sandwich construction with honeycomb core

The minimum weight of sandwich construction which is regarded as objective function has been discussed. Under given constraint condition of the strength or the stiffness, the four optimum parameters of sandwich construction with honeycomb core (thickness of the face t_f, thickness of the honeycomb core h_c, thickness of the honeycomb wall t_s, side length of the honeycomb cell c) are evaluated. By using constraint condition of the strength, a equation of high degree is finally solved. In the constraint condition of the stiffness, the constraint optimization problem is treated as inconstraint optimization problem with the method of obtaining extreme value solution by undetermined parameter multiplication. Also, the results are discussed.     

Structure-preserving Analysis on Folding and Unfolding Process of Undercarriage

The main idea of the structure-preserving method is to preserve the intrinsic geometric properties of the continuous system as much as possible in numerical algorithm design. The geometric constraint in the multi-body systems, one of the difficulties in the numerical methods that are proposed for the multi-body systems, can also be regarded as a geometric property of the multi-body systems. Based on this idea, the symplectic precise integration method is applied in this paper to analyze the kinematics problem of folding and unfolding process of nose undercarriage. The Lagrange governing equation is established for the folding and unfolding process of nose undercarriage with the generalized defined displacements firstly. And then, the constrained Hamiltonian canonical form is derived from the Lagrange governing equation based on the Hamiltonian variational principle. Finally, the symplectic precise integration scheme is used to simulate the kinematics process of nose undercarriage during folding and unfolding described by the constrained Hamiltonian canonical formulation. From the numerical results, it can be concluded that the geometric constraint of the undercarriage system can be preserved well during the numerical simulation on the folding and unfolding process of undercarriage using the symplectic precise integration method.     

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.     

优化形式的刚体系统动力学模拟

基于广义坐标形式的高斯最小拘束原理来研究刚体系统的动力学问题的优化方法. 相比目前动力学建模普遍采用的质点形式的高斯最小拘束原理,广义坐标形式的高斯最小拘束原理因对所选择的广义坐标没有要求,而使得建模过程更简单及具有更强的通用性. 本文分别建立了有约束和无约束条件下问题的优化动力学模型,对问题进行了动力学数值模拟,并与用拉格朗日微分方程处理的模型进行了对比分析,从而验证了所提方法的有效性.     

Brachistochrone with limited reaction of constraint in an arbitrary force field

The paper considers brachistochronic motion of a particle along a curve y=y(x) in an arbitrary force field in the vertical plane of Cartesian coordinate system. The curve is treated as a bilateral or unilateral constraint that can be smooth or rough. The projection of the reaction force of the curve onto the normal to the curve is confined to the fixed limits. A control variable u is given as the second derivative of the function y(x) relative to the horizontal coordinate?x of the particle, i.e., u=d ² y/dx ². Applying Pontryagin??s maximum principle and singular optimal control theory, the problem is reduced to numerical solving of the corresponding two-point boundary value problem. The procedure based on the shooting method is used to solve the boundary value problem. Two examples with friction forces of the viscous friction and Coulomb friction type have been solved.     

Shape optimization of a body located in low Reynolds number flow

The purpose of this study is to perform a numerical application of the shape optimization formulation of a body located in an incompressible viscous flow field. The formulation is based on an optimal control theory in which a performance function of the fluid force is introduced. The performance function should be minimized satisfying the state equation. This problem can be transformed into the minimization problem without constraint condition by the Lagrange multiplier method and the adjoint equations using adjoint variables corresponding to the state equations. As a numerical study, the drag force minimization problem in the steady Stokes flow, which means approximated equation of the low Reynolds number Navier–Stokes equation is carried out. After that, the unsteady Navier–Stokes flow is analysed. As the minimization algorithm, the steepest descent method is successfully applied. Copyright © 2005 John Wiley & Sons, Ltd.     

考虑摩擦特性的柔性关节空间机器人自适应CMAC神经网络鲁棒控制

讨论了关节摩擦力矩影响下,具有柔性铰关节的漂浮基空间机器人系统的动力学控制问题.设计了基于高斯基函数的小脑神经网络(CMAC)鲁棒控制器和摩擦力矩补偿器.用奇异摄动理论对系统的动力学模型进行快慢变子系统分解,针对快变子系统,设计力矩微分反馈控制器来抑制机械臂关节柔性引起的振动;对于慢变子系统,设计了基于自适应CMAC神...     

GAUSS PRINCIPLE OF LEAST CONSTRAINT IN GENERALIZED COORDINATES AND ITS GENERALIZATION

当采用广义坐标描述系统的运动时,相比质点形式的高斯最小拘束原理,通过广义坐标形式的高斯最小拘束原理来建立动力学优化模型,计算效率更高. 从高斯原理的变分形式出发推导了广义坐标形式的高斯最小拘束原理,并研究了非理想约束、单边约束及刚体碰撞情形下的高斯最小拘束原理的形式. 研究认为:对刚体碰撞情形下,高斯最小拘束原理不能取代碰撞恢复定律,碰撞恢复定律以碰撞后广义速度的约束方程形式起作用.     

A general method for impact dynamic analysis of a planar multi-body system with a rolling ball bearing joint

Impact affects the dynamic characteristics of mechanical multi-body systems and damages those rotating parts, such as the joint rolling element bearings, which are high-precision, defect intolerant components. Based on multi-body dynamic theory, Hertzian contact theory, and a continuous contact model, this study proposed a modelling method that can describe the dynamic behaviour of planar mechanical multi-body systems containing a rolling ball bearing joint under impact. In this method, the rigid bodies and bearing joint were connected according to their joint force constraints; the impact constraint between the multi-body system and the target rigid body was constructed using a continuous contact force model. Based on this method, the reflection relationship between the external impacts of the mechanical multi-body system and the variation law governing the dynamic load on the rolling bearing joint were revealed. Subsequently, an impact multi-body system, which was composed of a sliding–crank mechanism containing a rolling ball bearing joint and the target rigid body with an elastic support, was analysed to explore the dynamic response of such a complex discontinuous dynamic system andthe relevant relationship governing the dynamic load on the rolling bearing joint. In addition, a multi-body dynamic simulation software was used to build a virtual prototype of the impact slider–crank system. Compared with the theoretical model, the prototype had an additional deep groove ball bearing. That is to say, the prototype model took account of the specific geometric structural characteristics and the complex contact relationship of the inner and outer races, rolling balls, and bearing cage. Finally, the effectiveness of the theoretical method proposed in this study was verified by comparative analysis of the results. The results suggested that the external impact of a mechanical multi-body system was prone to induce sudden changes in the equivalent reaction force on its bearing joint and the dynamic load carried on its rolling balls. This study provided an effective method for exploring the distribution characteristics of dynamic loads on rolling ball bearing joints under working impact load conditions. Moreover, it offered support for the parameter optimisation of geometric structure, performance evaluation, and dynamic design of the rolling ball bearings.     

平面棱柱铰内考虑摩擦效应的接触分析

研究了多体系统中平面棱柱铰内的摩擦接触分析。在忽略铰内的碰撞效应的前提下,根据接触力系与约束反力的等效关系,以及各个角点接触力自身满足的互补关系,得到了接触力系关于摩擦力的线性互补方程。结合库仑摩擦定律,确定了铰内角点处的接触反力以及摩擦合力;另外,由接触力曲线可以得到铰内的接触形式,以及各个接触形式发生改变的时刻。通过ADAMS算例验证,这些接触形式转换的时刻就是铰内发生碰撞的时刻,为高效率地研究铰内碰撞提供了可能。     

Shape Optimization of Body Located in Incompressible Viscous Flow Based on Optimal Control Theory

This paper presents a numerical method of shape optimization of a body located in an incompressible viscous flow described by the Stokes and Oseen equations. The purpose of this study is to find the optimal shape that minimizes the fluid forces subjected to the body. The formulation of the shape optimization is based on the optimal control theory. The first thing that should be carried out in the optimal control theory is to define a performance function, which expresses the optimal shape. In this study, the fluid forces minimization problem is treated, i.e. fluid forces are directly used in the performance function. The performance function must be minimized subject to the basic equation. The optimal shape, which minimizes the fluid force, is pursued in this paper. This problem can be transformed into the minimization problem without constraint conditions by the Lagrange multiplier. As a numerical example, drag force minimization problems of a body located in low Reynolds number flows are carried out.     

THE SOLUTION FOR THE GENERALIZED RICCATI ALGEBRAIC EQUATIONS OF LINEAR EQUALITY CONSTRAINT SYSTEM

I.IntroductionTheestablishmentofanalogytheorybetweencomputationalstructuralmechanicsalldoptimalcontrolisbasedonthesubstructuralchaintheoryinstructuralmechanicsandthelillearquadratic(LQ)controlprobleminoptimalcontrolllJ.Whendealillgwiththecontrolproblenlbyapplyingtheabovethe.ry,wecanrealizetheadvantagesofmethodsinstrLlcturalInechanicsl2'3].Howevermostworkhasbeenlimitedtothenon-constraintsystemandlittleworkhasbeendonefortheconstraintsystemsofLlr.InRef.I4],basedonthegeneralizedvariationalprin…