多体系统动力学的高斯拘束分析方法

为研究高斯拘束原理下多体系统的动力学分析问题,针对一类多体开环链状结构,运用高斯拘束方法建立了动力学方程,讨论了动力学方程的符号推导过程,并给出了封闭形式的动力学方程解析表达式.以刚性和柔性结构为例,比较分析了不同分析方法、不同自由度下符号推导多体结构动力学方程的运算时间.分析结果表明,高斯拘束方法与传统的拉格朗日方法相比,更适合于多体结构动力学方程的符号推导,且结构自由度越高,其运算优势越明显.高斯拘束方法为一种较好的多体系统动力学分析方法.     

Flexible multibody dynamics — An object-oriented approach

An approach to the computer aided analysis of flexible multibody systems using object-oriented programming methods is presented. The aim is to support the rapid generation of specialized programs by providing an open, extensible C++ toolkit. This toolkit contains modules (C++ classes) which allow the declaration and manipulation of multibody components such as joints, bodies and actuators in an intuitive manner. New components (e.g., new finite elements) are easily introduced to extend the toolkit. The equations of motion for a multibody system consisting of these components are formulated by direct application of the principle of virtual work using symbolic techniques. It is possible to use absolute as well as relative coordinates in a problem-dependent manner.The author is now working for the Institut für Mechatronik, IMECH GmbH, Bergwerkstraße, 47445 Moers, Germany     

A Fully Symbolic Model of Multibody Systems Containing Flexible Plates

The modelling of flexible elements in mechanical systems has been investigated via several methods issuing from both the field of multi-body dynamics and the area of structural mechanics and vibration theory. As regards the multibody approach, recursive formulations in relative coordinates are quite suitable and efficient for a large variety of applications. Such a formalism is developed here for a general multibody system containing flexible plates and in such a way that its full symbolic generation is possible within the ROBOTRAN program [1].     

Screw-matrix method in dynamics of multibody systems

In the present paper the concept of screw in classical mechanics is expressed in matrix form, in order to formulate the dynamical equations of the multibody systems. The mentioned method can retain the advantages of the screw theory and avoid the shortcomings of the dual number notation. Combining the screw-matrix method with the tool of graph theory in Roberson/Wittenberg formalism. We can expand the application of the screw theory to the general case of multibody systems. For a tree system, the dynamical equations for eachj-th subsystem, composed of all the outboard bodies connected byj-th joint can be formulated without the constraint reaction forces in the joints. For a nontree system, the dynamical equations of subsystems and the kinematical consistency conditions of the joints can be derived using the loop matrix. The whole process of calculation is unified in matrix form. A three-segment manipulator is discussed as an example. This work is supported by the National Natural Science Fund.     

Simulation of planar flexible multibody systems with clearance and lubricated revolute joints

Modeling of clearance joints plays an important role in the analysis and design of multibody mechanical systems. Based on the absolute nodal coordinate formulation (ANCF), a new computational methodology for modeling and analysis of planar flexible multibody systems with clearance and lubricated revolute joints is presented. A planar absolute nodal coordinate formulation based on the locking-free shear deformable beam element is implemented to discretize the flexible bodies. A continuous contact-impact model is used to evaluate the contact force, in which energy dissipation in the form of hysteresis damping is considered. A force transition model from hydrodynamic lubrication forces to dry contact forces is introduced to ensure continuity in the joint reaction force. A comprehensive study with different lubrication force models has also been carried out. The generalized-α method is used to solve the equations of motion and several efficient methods are incorporated in the proposed model. Finally, the methodology is validated by two numerical examples.     

Modeling of multibody systems with the object-oriented modeling language Dymola

The object-oriented modeling language Dymola allows the physical modeling of large interconnected systems based on model components fromdifferent engineering domains. It generatessymbolic code for different target simulators. In this paper, a Dymola class library for the efficient generation of the equations of motion for multibody systems is presented. The library is based on aO(n) algorithm which is reformulated in an objectoriented way. This feature can also be interpreted as a bond graph oriented modeling of multibody systems. Furthermore a new algorithm for a certain class ofvariable structure multibody systems, such as systems with Coulomb friction, is presented, which allows the generation of efficient symbolic code.     

柔性多体系统的递推组集建模与仿真软件的实现

简要地阐述多体系统动力学单向递推组集建模方法，介绍根据这种方法开发而成的仿真软件系统ＤＡＦＭＢ及它的功能与特点。通过双摆算例指出本文提出的模型在计算效率与计算精度方面优于笛卡尔坐标的微分－代数方程。     

A Variational Approach to Dynamics of Flexible Multibody Systems

Abstract

This paper presents a variational formulation of constrained dynamics of flexible multibody systems, using a vector-variational calculus approach. Body reference frames are used to define global position and orientation of individual bodies in the system, located and oriented by position of its origin and Euler parameters, respectively. Small strain linear elastic deformation of individual components, relative to their body reference frames, is defined by linear combinations of deformation modes that are induced by constraint reaction forces and normal modes of vibration. A library of kinematic couplings between flexible and/or rigid bodies is defined and analyzed. Variational equations of motion for multibody systems are obtained and reduced to mixed differential-algebraic equations of motion. A space structure that must deform during deployment is analyzed, to illustrate use of the methods developed     

Comparison of Various Techniques for Modelling Flexible Beams in Multibody Dynamics

The modelling of flexible elements in mechanical systems has been widely investigated through several methods issuing from both the area of structural mechanics and the field of multibody dynamics. As regards the latter discipline, beside the problem of the generation of the multibody equations of motion, the choice of a spatial discretization method for modelling flexible elements has always been considered as a critical phase of the modelling. Although this subject is abundantly tackled in the open-literature, the latter probably lacks an objective comparison between the most commonly used approaches.This contribution presents an extensive investigation of several discretization techniques of flexible beams, in a pure multibody context. In particular, it is shown that shape functions based on power series monomials are very suitable and versatile to model beams being part of a multibody system and thus constitutes an interesting alternative to finite element analysis. For this purpose, a symbolic multibody program, in which various discretization techniques were implemented, was generalized to compute the equations of motion of a general multibody system containing flexible beams.     

Optimization of Vehicle Suspension Systems for Improved Comfort of Road Vehicles Using Flexible Multibody Dynamics

Methods that account for the flexibility of multibody systems extend the range of applications to areas such as flexible robots, precision machinery, vehicle dynamics or space satellites. The method proposed here for flexible multibody models allows for the representation of complex-shaped bodies using general finite-element discretizations which deform during the dynamic loading of the system, while the gross rigid body motion of these bodies is still captured using fixed-body coordinate frames. Components of the system for which the deformations are relatively unimportant are represented with rigid bodies. This method is applied to a road vehicle where flexibility plays an important role in its ride and handling dynamic behavior. Therefore, for the study of the limit behavior of the vehicles, the use of flexible multibody models is of high importance. The design process of these vehicles, very often based on intuition and experience, can be greatly enhanced through the use of generalized optimization techniques concurrently with multibody codes. The use of sparse matrix system solvers and modal superposition, to reduce the number of flexible coordinates, in a computer simulation, assures a fast and reliable analysis tool for the optimization process. The optimum design of the vehicle is achieved through the use of an optimization algorithm with finite-differencesensitivities, where the characteristics of the vehicle components are the design variables on which appropriate constraints are imposed. The ride optimization is achieved by finding the optimum of a ride index that results from a metric that accounts for the acceleration in several key points in the vehicle properly weighted in face of their importance for the comfort of the occupant. Simulations with different road profiles are performed for different speeds to account for diverse ride situations. The results are presented and discussed in view of the different methods usedwith emphasis on models and algorithms.     

Finite element modeling of contact conditions in multibody system dynamics

In this paper a new method is developed for the dynamic analysis of contact conditions in flexible multibody systems undergoing a rolling type of motion. The relative motion between the two contacting bodies is treated as a constraint condition describing their kinematic and geometric relations. Equations of motion of the system are presented in a matrix form making use of Kane's equations and finite element method. The method developed has been implemented in a general purpose program called DARS and applied to the simulation and analysis of a rotating wheel on a track. Both the bodies are assumed flexible and discretized using a three dimensional 8-noded isoparametric elements. The time variant constraint conditions are imposed on the nodal points located at the peripheral surfaces of the bodies under consideration. The simulation is carried out under two different boundary conditions describing the support of the track. The subsequent constraint forces associated with the generalized coordinates of the system are computed and plotted. The effects of friction are also discussed.     

多柔体系统数值分析的模型降噪方法

多柔体系统的动力学方程通常是一组刚性微分方程, 目前普遍采用的刚性微分方程数值解法主要通过数值阻尼滤除系统响应中的高频分量, 其求解效率难以令人满意. 为了降低多柔体系统动力学方程的刚性, 从而可采用ODE45等常规微分方程求解器进行求解, 研究了在建模过程中滤除高频振荡分量的方法. 在以当前时刻为起点的短时间内对柔性体的应力进行均匀化, 用均匀化后的应力计算柔性体的变形虚功率, 由此得到的系统动力学方程的解中不含过高频率的弹性振动, 并且可以通过调节均匀化时间区间的长度参数控制滤波的范围. 数值算例表明: 这种模型降噪方法的计算效率和精度均不低于刚性微分方程求解器, 并且在刚性微分方程求解器失效的情况下模型降噪方法仍有良好的精度和效率. 本文所提的模型降噪方法可成为求解多柔体系统动力学方程的新途径.      

多刚体系统动力学的旋量-矩阵方法

本文将经典力学中的旋量概念以矩阵形式表示,用以建立多刚体系统的动力学方程。这种旋量-矩阵方法能保留旋量融矢量与矢量矩于一体的优点,却避免以往对偶数记法的缺点。结合 Roberson/wittenburg的图论工具,旋量-矩阵方法的应用范围可扩大到一般多刚体系统。对于树形系统,利用旋量通路矩阵推导各个由第i铰联结的全部外侧刚体组成的第i子系统的动力学方程,可避免出现铰的约束反力,对于非树系统,则利用回路矩阵导出各子系统动力学方程及运动学相容条件,全部计算过程统一为矩阵运算,以操作机器人作为具体算例。     

挠性多体系统动力学

本文以递推形式给出了挠性多体系统的动力学方程。此方程具有如下特点:(1)对整个系统而言,方程数目多,但递推的每一步只需要解低维的方程组,从而避免了高维矩阵的求逆,减少了计算量。(2)对不同类型的问题,方程形式具有很高的相似性,便于程序编制,各个需要求逆的矩阵都是对称正定的,这样减少了计算量和计算机内存单元。(3)这种方程形式处理非树形和带有约束条件的系统很方便。     

A velocity transformation method for the nonlinear dynamic simulation of railroad vehicle systems

The solution of the constrained multibody system equations of motion using the generalized coordinate partitioning method requires the identification of the dependent and independent coordinates. Using this approach, only the independent accelerations are integrated forward in time in order to determine the independent coordinates and velocities. Dependent coordinates are determined by solving the nonlinear constraint equations at the position level. If the constraint equations are highly nonlinear, numerical difficulties can be encountered or more Newton–Raphson iterations may be required in order to achieve convergence for the dependent variables. In this paper, a velocity transformation method is proposed for railroad vehicle systems in order to deal with the nonlinearity of the constraint equations when the vehicles negotiate curved tracks. In this formulation, two different sets of coordinates are simultaneously used. The first set is the absolute Cartesian coordinates which are widely used in general multibody system computer formulations. These coordinates lead to a simple form of the equations of motion which has a sparse matrix structure. The second set is the trajectory coordinates which are widely used in specialized railroad vehicle system formulations. The trajectory coordinates can be used to obtain simple formulations of the specified motion trajectory constraint equations in the case of railroad vehicle systems. While the equations of motion are formulated in terms of the absolute Cartesian coordinates, the trajectory accelerations are the ones which are integrated forward in time. The problems associated with the higher degree of differentiability required when the trajectory coordinates are used are discussed. Numerical examples are presented in order to examine the performance of the hybrid coordinate formulation proposed in this paper in the analysis of multibody railroad vehicle systems.     

Dynamics of flexible multibody systems with tree topologies

The dynamic equations of flexible multibody systems with tree topological configuration are derived by using the Jourdain's principle. The independent joint coordinates are introduced to describe the large displacements of the bodies, and the modal coordinates are used to describe small deformations of flexible bodies based on the consistent mass finite element method and normal vibration mode analysis. The minimum differential equations are developed, which are compatible with the equations of multi-rigid body systems or structural dynamics. The stiff problem in the numerical integration is thus alleviated effectively. The method used in this paper can be extended to deal with systems with other topological configurations. Finally, the validity and feasibility of the presented mathematical model are demonstrated by a numerical example of a manipulator with two elastic links. The project supported by National Natural Science Foundation of China     

A nonlinear finite-element approach for kineto-static analysis of multibody systems

Methods that treat rigid/flexible multibody systems undergoing large motion as well as deformations are often accompanied with inefficiencies and instabilities in the numerical solution due to the large number of state variables, differences in the magnitudes of the rigid and flexible body coordinates, and the time dependencies of the mass and stiffness matrices. The kineto-static methodology of this paper treats a multibody mechanical system to consist of two collections of bulky (rigid) bodies and relatively flexible ones. A mixed boundary condition nonlinear finite element problem is then formulated at each time step whose known quantities are the displacements of the nodes at the boundary of rigid and flexible bodies and its unknowns are the deformed shape of the entire structure and the loads (forces and moments) at the boundary. Partitioning techniques are used to solve the systems of equations for the unknowns, and the numerical solution of the rigid multibody system governing equations of motion is carried out. The methodology is very much suitable in modelling and predicting the impact responses of multibody system since both nonlinear and large gross motion as well as deformations are encountered. Therefore, it has been adopted for the studies of the dynamic responses of ground vehicle or aircraft occupants in different crash scenarios. The kineto-static methodology is used to determine the large motion of the rigid segments of the occupant such as the limbs and the small deformations of the flexible bodies such as the spinal column. One of the most dangerous modes of injury is the amount of compressive load that the spine experiences. Based on the developed method, a mathematical model of the occupant with a nonlinear finite element model of the lumbar spine is developed for a Hybrid II (Part 572) anthropomorphic test dummy. The lumbar spine model is then incorporated into a gross motion occupant model. The analytical results are correlated with the experimental results from the impact sled test of the dummy/seat/restraint system. With this extended occupant model containing the lumbar spine, the gross motion of occupant segments, including displacements, velocities and accelerations as well as spinal axial loads, bending moments, shear forces, internal forces, nodal forces, and deformation time histories are evaluated. This detailed information helps in assessing the level of spinal injury, determining mechanisms of spinal injury, and designing better occupant safety devices.     

动力刚化与多体系统刚—柔耦合动力学

首先指出当前柔性多体系统动力学的大量工程研究背景,在回顾柔性多体系统动力学研究进展后指出动力刚化的现象揭示了刚－柔耦合的零次建模方法的局限,认为进一步深入进行柔性多体系统刚－柔耦合动力学的研究是多体系统动力学研究的新阶段,文末提出了刚－柔耦合动力学的研究任务。     

多柔体系统动力学建模与优化研究进展

多柔体系统是由柔性部件和运动副组成的力学系统,在航空、航天、车辆、机械与兵器等众多工程领域具有广泛的应用前景, 其典型的代表包括柔性机械臂、直升机旋翼、卫星的可展开天线、太阳帆航天器等. 近年来,随着工程技术的发展,多柔体系统动力学问题日益突出,尤其是含变长度柔性部件的多柔体系统,不仅涉及其动力学 建模与计算,还涉及其动力学优化设计. 事实上,部件柔性对多柔体系统的动力学行为影响很大,直接影响到优化结果,因此需要发展基于多柔体系统动力学的优化设计方法. 本文首先阐述了多柔体系统动力学优化的研究背景及意义,简要回顾了多柔体系统动力学建模的3类方法:浮动坐标方法、几何 精确方法和绝对节点坐标方法,并介绍了含变长度柔性部件的多柔体系统动力学建模方法. 系统概述了多柔体系统动力学响应优化、动力学特性优化和动力学灵敏度分析3个方面的研究进展,并从尺寸优化、形状优化和 拓扑优化 3 个方面综述了多柔体系统部件优化的研究进展. 本文最后提出了在多柔体系统动力学优化研究中值得关注的若干问题.