多体系统动力学方程符号推导

符号演算又名计算机代数,是近二十年来蓬勃发展的计算机科学。用符号演算导出复杂的多体系统动力学非线性微分方程组,可以提高计算效率,计算精度,减轻人的劳动並为设计方案的选择提供快速分析的依据。本文介绍用FORTRAN-77在VAX/11—780上符号推导以6自由度机器人为例的多体系统动力学方程。文中方法可应用于航天器、机构等动力学方程推导。     

DYNAMIC MODELING OF NONIDEAL SYSTEM BASED ON GAUSS'S PRINCIPLE$^{\bf 1)}$

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.     

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.      

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.     

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.     

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.      

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.     

弹性杆-刚性体系统动力学分析的广义逆矩阵方法

将多刚体系统的广义逆矩阵方法推广到含弹性杆与刚性体的混合系统的动力学分析中,建立了以节点坐标表示的基于全局惯性坐标系的刚体-柔性体混合系统动力学方程.首先以两端节点坐标为变量推导了弹性杆的动力学方程,以刚性体节点坐标为变量推导了刚性体节点速度约束方程和刚性体动力学方程,最后得到弹性杆与刚性体混合系统的动力学方程和速度约束方程.本方法在同一个惯性坐标系对刚柔多体系统进行描述,具有方法简洁、便于计算建模的特点.论文最后给出两个数值算例,检验了方法的有效性.     

多体系统Lagrange方程数值算法的研究进展

Lagrange方法是建立多体系统动力学方程的普遍方法之一,其方程的形式为常微分方程组或微分 - 代数方程组,数值计算与数值分析是研究多体系统动力学特性的重要方法.本文简要介绍了多体系统动力学方程的第一、二类Lagrange方程和修正的Lagrange方程的基本形式及这些方程的正则形式,着重介绍了正则方程在数值计算中的特点,就多体系统Lagrange方程的隐式算法、辛算法和多体系统动力学特性的数值分析方法(包括数值仿真、Poincar'e映射和Lyapunov指数的计算方法)的研究现状进行了综述.     

Dynamic recursive decoupling method for closed-loop flexible mechanical systems

In this paper, a new method for the dynamic analysis of a closed-loop flexible kinematic mechanical system is presented. The kinematic and force models are developed using absolute reference, joint relative, and elastic coordinates as well as joint reaction forces. This recursive formulation leads to a system of loosely coupled equations of motion. In a closed-loop kinematic chain, cuts are made at selected auxiliary joints in order to form spanning tree structures. Compatibility conditions and reaction force relationships at the auxiliary joints are adjoined to the equations of open-loop mechanical systems in order to form closed-loop dynamic equations. Using the sparse matrix structure of these equations and the fact that the joint reaction forces associated with elastic degrees of freedom do not represent independent variables, a method for decoupling the joint and elastic accelerations is developed. Unlike existing recursive formulations, this method does not require inverse or factorization of large non-linear matrices. It leads to small systems of equations whose dimensions are independent of the number of elastic degrees of freedom. The application of dynamic decoupling method in dynamic analysis of closed-loop deformable multibody systems is also discussed in this paper. The use of the numerical algorithm developed in this investigation is illustrated by a closed-loop flexible four-bar mechanism.     

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.     

Reliability Index and Failure Probability

Abstract

Numerical algorithms for the solution of nonlinear algebraic equation systems are discussed. Special application to the mechanism and multibody system kinematic analysis, as well as to the problems of constraint stabilization during dynamics simulation is regarded. Special attention is paid to the approaches of a separate solution of the differential equations and constraint stabilization. Numerical procedures that are effective additions to the well-known algorithms based on the Newton-Raphson method are presented. The problems of loss of precision and achievement of large unreal increments of the varying parameters are discussed. The traditional Newton-Raphson method is modified by applying a step reduction procedure that is developed numerically for the symbolic form of kinematic and dynamic equations. An optimization method for stabilization of constraints using the mass matrix of dynamic equations is suggested. According to the objective function defined the stabilization procedure provides minimal deviations of the parameters and their velocities with respect to the solution of the differential equations. No generalized coordinate partitioning is required either for solution of the dynamic equations or for stabilization of the constraints. Several examples of kinematic analysis of single and four contour plane mechanisms and constraint stabilization are solved, and the results are compared. The advantages of the algorithms developed are tested with a high-degree of initial deviation from the real solution. It is also shown that the step correction algorithm could provide admissible solution even when, in many cases, the classical approaches are not reliable. An example of the direct and inverse kinematic problem solutions of the four-degrees-of-freedom spatial platform is presented.     

Symbolic generation of large multibody system dynamic equations using a new semi-explicit Newton/Euler recursive scheme

Summary The aim of this paper is to show that multibody systems with a large number of degrees of freedom can be efficiently modelled, taking conjointly advantage of a recursive formulation of the equations of motion and of the symbolic generation capabilities.Recursive schemes are widely used in the field of multibody dynamics since they avoid the explosion of the number of arithmetical operations in case of large multibody models. Within the context of our field of applications (railway dynamics simulation), explicit integration schemes are still prefered and thus oblige us to compute the generalized accelerations at each time step. To achieve this, we propose a new formulation of the well-known Newton/Euler recursive method, whose efficiency will be compared with a so-called O(N) formulation.A regards the symbolic generation, often decried due to the size of the equations in case of large systems, we have recently implemented recursive multibody formalisms in the symbolic programme ROBOTRAN [1]. As we shall explain, the recursive nature of these formalisms is particularly well-suited to symbolic manipulation.All these developments have been successfully applied in the field of railway dynamics, and in particular allowed us to analyse the dynamic behaviour of several railway vehicles. Some typical results related to a completely non-conventional bogie will be presented before concluding.     

具有非线性homologous变形约束的桁架结构形态分析

研究了具有非线性homologous变形约束条件的桁架结构形态分析问题。在已有的线性homologous变形约束桁架形态分析的基础上,将结构的节点分成三类:homologous变形约束节点,形状可变节点和边界点。运用Moore-Penrose广义逆矩阵性质,将基础方程组解的存在条件表示为包含形状可变节点未知坐标的非线性方程组,为采用Newton-Raphson方法求解非线性方程组,对AA (A为任意矩阵,A 为A的Moore-Penrose广义逆矩阵)求偏导数,找到了满足保型要求的形态,给出的桁架算例说明了本文方法的有效性。     

热-动力学耦合多体系统建模与降阶

在轨运行的卫星天线受到太阳辐射热冲击后容易出现热致振动或指向不准确等问题, 严重时会导致航天器失效. 本文提出了一种基于改进模态综合法的刚柔热耦合多体系统建模与降阶方法. 采用绝对节点坐标法单元形函数对柔性天线的位移场与温度场进行统一离散插值, 避免了两种物理场网格不匹配带来的映射误差与效率问题, 并使用绝对节点坐标参考节点描述中心刚体. 在系统方程中考虑了热流输入和表面自热辐射. 针对绝对节点坐标法切线刚度阵高度非线性的特点, 利用一阶泰勒展开对系统动力学和传热学方程进行了分段线性化, 在线性化区间内切线刚度矩阵为常数矩阵, 避免了每个时间步上的弹性力及其雅克比矩阵的迭代计算, 并使得基于模态的降阶手段得以应用. 利用改进的模态综合方法划分子结构并缩减系统自由度. 相邻子结构之间通过约束方程保证连接精度和连续性. 通过纯导热半圆形薄板、薄板的热膨胀、柔性太阳能电池板和刚柔热耦合抛物线天线四个数值算例验证本文所提出方法的有效性. 结果表明, 本文提出的方法在保证仿真精度的前提下缩减了系统规模, 提高了仿真计算效率.      

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.     

Control structure interaction in the nonlinear analysis of flexible mechanical systems

The effect of the control structure interaction on the feedforward control law as well as the dynamics of flexible mechanical systems is examined in this investigation. An inverse dynamics procedure is developed for the analysis of the dynamic motion of interconnected rigid and flexible bodies. This method is used to examine the effect of the elastic deformation on the driving forces in flexible mechanical systems. The driving forces are expressed in terms of the specified motion trajectories and the deformations of the elastic members. The system equations of motion are formulated using Lagrange's equation. A finite element discretization of the flexible bodies is used to define the deformation degrees of freedom. The algebraic constraint equations that describe the motion trajectories and joint constraints between adjacent bodies are adjoined to the system differential equations of motion using the vector of Lagrange multipliers. A unique displacement field is then identified by imposing an appropriate set of reference conditions. The effect of the nonlinear centrifugal and Coriolis forces that depend on the body displacements and velocities are taken into consideration. A direct numerical integration method coupled with a Newton-Raphson algorithm is used to solve the resulting nonlinear differential and algebraic equations of motion. The formulation obtained for the flexible mechanical system is compared with the rigid body dynamic formulation. The effect of the sampling time, number of vibration modes, the viscous damping, and the selection of the constrained modes are examined. The results presented in this numerical study demonstrate that the use of the driving forees obtained using the rigid body analysis can lead to a significant error when these forces are used as the feedforward control law for the flexible mechanical system. The analysis presented in this investigation differs significantly from previously published work in many ways. It includes the effect of the structural flexibility on the centrifugal and Coriolis forces, it accounts for all inertia nonlinearities resulting from the coupling between the rigid body and elastic displacements, it uses a precise definition of the equipollent systems of forces in flexible body dynamics, it demonstrates the use of general purpose multibody computer codes in the feedforward control of flexible mechanical systems, and it demonstrates numerically the effect of the selected set of constrained modes on the feedforward control law.     

A comparative study of joint formulations: application to multibody system tracked vehicles

This paper is focused on the dynamic formulation of mechanical joints using different approaches that lead to different models with different numbers of degrees of freedom. Some of these formulations allow for capturing the joint deformations using a discrete elastic model while the others are continuum-based and capture joint deformation modes that cannot be captured using the discrete elastic joint models. Specifically, three types of joint formulations are considered in this investigation; the ideal, compliant discrete element, and compliant continuum-based joint models. The ideal joint formulation, which does not allow for deformation degrees of freedom in the case of rigid body or small deformation analysis, requires introducing a set of algebraic constraint equations that can be handled in computational multibody system (MBS) algorithms using two fundamentally different approaches: constrained dynamics approach and penalty method. When the constrained dynamics approach is used, the constraint equations must be satisfied at the position, velocity, and acceleration levels. The penalty method, on the other hand, ensures that the algebraic equations are satisfied at the position level only. In the compliant discrete element joint formulation, no constraint conditions are used; instead the connectivity conditions between bodies are enforced using forces that can be defined in their most general form in MBS algorithms using bushing elements that allow for the definition of general nonlinear forces and moments. The new compliant continuum-based joint formulation, which is based on the finite element (FE) absolute nodal coordinate formulation (ANCF), has several advantages: (1) It captures modes of joint deformations that cannot be captured using the compliant discrete joint models; (2) It leads to linear connectivity conditions, thereby allowing for the elimination of the dependent variables at a preprocessing stage; (3) It leads to a constant inertia matrix in the case of chain like structure; and (4) It automatically captures the deformation of the bodies using distributed inertia and elasticity. The formulations of these three different joint models are compared in order to shed light on the fundamental differences between them. Numerical results of a detailed tracked vehicle model are presented in order to demonstrate the implementation of some of the formulations discussed in this investigation.