充液系统液体-多体耦合动力响应分析

提出了充液系统的液体-多体耦合力学模型，基于ALE有限元法和多体系统动力学理论，发展了液体-多体耦合动力响应分析的一种有效方法. 对于液体子系统，将其运动分解为随同贮箱的大位移运动和相对贮箱的大幅晃动，引入贮箱固连参考系中的任意拉格朗日-欧拉(ALE)运动学描述，建立了贮箱固连非惯性参考系中液体的ALE有限元方程，对液体有限元方程的缩聚大大减少了液体子系统的计算规模. 为了计及液体阻尼的影响，引入了液体修正的Rayleigh阻尼，避免了伪阻尼力的出现. 对于多体子系统，应用多体系统动力学理论建立动力学方程. 在此基础上详细导出了液体-多体耦合动力学方程，并采用预估-多重校正算法(PMA)和时间步长控制算法进行迭代求解，既保证了迭代收敛，又提高了计算效率. 所给算例成功求解了液体运输车辆系统的液体-多体耦合动力响应，深入分析了有关参数对系统动力响应的影响，获得了一些结论.     

基于绝对节点坐标的多柔体系统动力学高效计算方法

绝对节点坐标法已经被广泛应用于柔性多体系统的动力学研究之中, 但是其计算效率问题尚未得到很好的解决. 基于绝对节点坐标方法计算弹性力及其对广义坐标的偏导数矩阵(Jacobi矩阵), 通常是基于第二类Piola-Kirchhoff应力张量来完成, 计算效率不高.根据虚功原理并采用第一类Piola-Kirchhoff应力张量的方法直接推导得到了弹性力及其Jacobi矩阵的解析表达式. 基于不同方法所得的数值算例结果对比研究表明, 该方法可使计算效率大大提高.      

带约束多体系统动力学方程的隐式算法

研究了带约束多体系统隐式算法,用子矩阵的形式推导出了多体系统正则方程的Ｊａｃｏｂｉ矩阵,它适用于多种隐式算法并给出了隐式Ｒｕｎｇｅ－Ｋｕｔｔａ算法,最后用一算例表明了隐式算法的计算效率和精度明显优于算法。     

On the numerical solution of tracked vehicle dynamic equations

In this investigation, the solution of the nonlinear dynamic equations of the multibody tracked vehicle systems are obtained using different procedures. In the first technique, which is based on the augmented formulation that employes the absolute Cartesian coordinates and Lagrange multipliers, the generalized coordinate partitioning of the constraint Jacobian matrix is used to determine the independent coordinates and the associated independent differential equations. An iterative Newton-Raphson algorithm is used to solve the nonlinear constraint equations for the dependent variables. The numerical problems encountered when one set of independent coordinates is used during the simulation of large scale tracked vehicle systems are demonstrated and their relationship to the track dynamics is discussed. The second approach employed in this investigation is the velocity transformation technique. One of the versions of this technique is discussed in this paper and the numerical problems that arise from the use of inconsistent system of kinematic equations are reported. In the velocity transformation technique, the tracked vehicle system is assumed to consist of two kinematically decoupled subsystems; the first subsystem consists of the chassis, the rollers, the sprocket and the idler, while the second subsystem consists of the track which is represented as a closed kinematic chain that consists of rigid links connected by revolute joints. It is demonstrated that the use of one set of recursive equations leads to numerical difficulties because of the change in the track configuration. Singular configurations can be avoided by repeated changes in the recursive equations. The sensitivity of the predictor-corrector multistep numerical integration schemes to the method of formulating the state equations is demonstrated. The numerical results presented in this investigation are obtained using a planner tracked vehicle model that consists of fifty four rigid bodies.     

具有奇异位置的多体系统动力学方程的隐式算法

本文研究了在运动过程中具有奇异位置的多体系统动力学方程的隐式算法，给出了隐式算法所用的Ｊａｃｏｂｉ矩阵，并建立了该矩阵中各子矩阵间的计算关系，提高了计算效率，计算结果表明隐式算法的计算速度和精度明显优于显式算法。     

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.     

Implicit and explicit integration in the solution of the absolute nodal coordinate differential/algebraic equations

This investigation is concerned with the use of an implicit integration method with adjustable numerical damping properties in the simulation of flexible multibody systems. The flexible bodies in the system are modeled using the finite element absolute nodal coordinate formulation (ANCF), which can be used in the simulation of large deformations and rotations of flexible bodies. This formulation, when used with the general continuum mechanics theory, leads to displacement modes, such as Poisson modes, that couple the cross section deformations, and bending and extension of structural elements such as beams. While these modes can be significant in the case of large deformations, and they have no significant effect on the CPU time for very flexible bodies; in the case of thin and stiff structures, the ANCF coupled deformation modes can be associated with very high frequencies that can be a source of numerical problems when explicit integration methods are used. The implicit integration method used in this investigation is the Hilber–Hughes–Taylor method applied in the context of Index 3 differential-algebraic equations (HHT-I3). The results obtained using this integration method are compared with the results obtained using an explicit Adams-predictor-corrector method, which has no adjustable numerical damping. Numerical examples that include bodies with different degrees of flexibility are solved in order to examine the performance of the HHT-I3 implicit integration method when the finite element absolute nodal coordinate formulation is used. The results obtained in this study show that for very flexible structures there is no significant difference in accuracy and CPU time between the solutions obtained using the implicit and explicit integrators. As the stiffness increases, the effect of some ANCF coupled deformation modes becomes more significant, leading to a stiff system of equations. The resulting high frequencies are filtered out when the HHT-I3 integrator is used due to its numerical damping properties. The results of this study also show that the CPU time associated with the HHT-I3 integrator does not change significantly when the stiffness of the bodies increases, while in the case of the explicit Adams method the CPU time increases exponentially. The fundamental differences between the solution procedures used with the implicit and explicit integrations are also discussed in this paper.     

Computational dynamics: theory and applications of multibody systems

Multibody system dynamics is an essential part of computational dynamics a topic more generally dealing with kinematics and dynamics of rigid and flexible systems, finite elements methods, and numerical methods for synthesis, optimization and control including nonlinear dynamics approaches. The theoretical background of multibody dynamics is presented, the efficiency of recursive algorithms is shown, methods for dynamical analysis are summarized, and applications to vehicle dynamics and biomechanics are reported. In particular, the wear of railway wheels of high-speed trains and the metabolical cost of human locomotion is analyzed using multibody system methods.     

Computer Implementation of the Absolute Nodal Coordinate Formulation for Flexible Multibody Dynamics

Deformable components in multibody systems are subject to kinematic constraints that represent mechanical joints and specified motion trajectories. These constraints can, in general, be described using a set of nonlinear algebraic equations that depend on the system generalized coordinates and time. When the kinematic constraints are augmented to the differential equations of motion of the system, it is desirable to have a formulation that leads to a minimum number of non-zero coefficients for the unknown accelerations and constraint forces in order to be able to exploit efficient sparse matrix algorithms. This paper describes procedures for the computer implementation of the absolute nodal coordinate formulation' for flexible multibody applications. In the absolute nodal coordinate formulation, no infinitesimal or finite rotations are used as nodal coordinates. The configuration of the finite element is defined using global displacement coordinates and slopes. By using this mixed set of coordinates, beam and plate elements can be treated as isoparametric elements. As a consequence, the dynamic formulation of these widely used elements using the absolute nodal coordinate formulation leads to a constant mass matrix. It is the objective of this study to develop computational procedures that exploit this feature. In one of these procedures, an optimum sparse matrix structure is obtained for the deformable bodies using the QR decomposition. Using the fact that the element mass matrix is constant, a QR decomposition of a modified constant connectivity Jacobian matrix is obtained for the deformable body. A constant velocity transformation is used to obtain an identity generalized inertia matrix associated with the second derivatives of the generalized coordinates, thereby minimizing the number of non-zero entries of the coefficient matrix that appears in the augmented Lagrangian formulation of the equations of motion of the flexible multibody systems. An alternate computational procedure based on Cholesky decomposition is also presented in this paper. This alternate procedure, which has the same computational advantages as the one based on the QR decomposition, leads to a square velocity transformation matrix. The computational procedures proposed in this investigation can be used for the treatment of large deformation problems in flexible multibody systems. They have also the advantages of the algorithms based on the floating frame of reference formulations since they allow for easy addition of general nonlinear constraint and force functions.     

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.     

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.     

IMPROVEMENT OF EXPLICIT ALGORITHMS STABILITY WITH VISCO-ELASTIC ARTIFICIAL BOUNDARY ELEMENTS 1)

黏弹性人工边界单元是目前常用的处理半无限空间波动问题的数值模拟方法,可有效吸收计算区域内产生的外行波动.黏弹性人工边界单元具有与内部介质不同的质量密度、刚度和阻尼,受其影响,对整体模型进行显式时域逐步积分时,在边界区域易发生失稳现象,影响整体系统显式积分的计算效率. 针对该问题目前尚无行之有效的解决方法.本文针对二维黏弹性人工边界单元,建立可代表整体系统典型特征的侧边子系统和角点子系统,利用传递矩阵谱半径分析方法,基于传统中心差分格式,推导得到局部子系统稳定性条件的解析解.在此基础上通过研究解析解中各物理参数对稳定性条件的影响,给出通过增加人工边界单元的质量密度,以改善采用黏弹性人工边界单元时显式算法稳定性的方法.均匀和成层半空间波动问题算例分析表明,将内部单元质量密度设置为人工边界单元质量密度的上限,可以在保证黏弹性人工边界计算精度的前提下,有效改善整体系统显式时域逐步积分的数值稳定性,大幅提高计算效率.     

Geometry and differentiability requirements in multibody railroad vehicle dynamic formulations

The dynamic equations of multibody railroad vehicle systems can be formulated using different sets of generalized coordinates; examples of these sets of coordinates are the absolute Cartesian and trajectory coordinates. The absolute coordinate based formulations do not require introducing an intermediate track coordinate system since all the absolute coordinates are defined in the global system. On the other hand, when the trajectory coordinates are used, a track coordinate system that follows the motion of a body in the railroad vehicle system is introduced. This track coordinate system is defined by the track geometry and the distance traveled by the body along the track centerline. The configuration of the body with respect to the track coordinate system is defined using five coordinates; two translations and three Euler angles. In this paper, the formulations based on the absolute and trajectory coordinates are compared. It is shown that these two sets of coordinates require different degrees of differentiability and smoothness. When an elastic contact formulation is used to study the wheel/rail dynamic interaction, there are significant differences in the order of the derivatives required in both formulations. In fact, as demonstrated in this study, in the absence of a contact constraint formulation, higher order derivatives with respect to geometric parameters are still required when the equations are formulated using the trajectory coordinates. The formulation of the constraints used in the analysis of the wheel/rail contact is discussed and it is shown that when the absolute coordinates are used, only third order derivatives need to be evaluated. The relationship between the track frame used in railroad vehicle dynamics and the Frenet frame used in the theory of curves to describe the curve geometry is also discussed in this paper. Based on the analysis presented in this paper, the advantages and drawbacks of a hybrid method which employs both the absolute and trajectory coordinates and planar contact conditions in order to reduce the number of contact constraints and relax the differentiability requirements are discussed. In this method, the absolute coordinates are used to formulate the equations of motion of the railroad vehicle system. The absolute coordinate solution can be used to determine the trajectory coordinates and their time derivatives. Using the trajectory coordinates, the motion of the body in the vehicle with respect to the track coordinate system can be predicted and used in the formulation of the planar contact model.     

受约束多体系统一种新的违约校正方法

本文针对受约束多体系统的违约问题提出了一种新的违约校正方法．同以往的违约校正方法相比，本方法具有物理意义明确，计算工作量小，校正效果明显的优点．本文的方法对动力学方程的破坏较小，能同任何积分方法配合使用，可有效地将约束方程的违约控制在给定的精度范围内．数值仿真的结果表明了本方法的有效性．     

基于李群局部标架的多柔体系统动力学建模与计算

多柔体系统动力学主要研究由多个具有运动学约束、存在大范围相对运动的柔性部件构成的动力学系统的建模、计算和控制.多柔体系统不仅具有柔体大变形导致的几何非线性,更具有大范围刚体运动引起的几何非线性,其非线性程度远高于计算结构力学所研究的几何非线性问题.本文基于李群局部标架(local frame of Lie group,...     

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.     

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。