A two-step approach for model reduction in flexible multibody dynamics

In this work, a two-step approach for model reduction in flexible multibody dynamics is proposed. This technique is a combination of the Krylov-subspace method and a Gramian matrix based reduction approach that is particularly suited if a small reduced-order model of a system charged with many force-inputs has to be generated. The proposed methodology can be implemented efficiently using sparse matrix techniques and is therefore applicable to large-scale systems too. By a numerical example, it is demonstrated that the suggested two-step approach has very good approximation capabilities in the time as well as in the frequency domain and can help to reduce the computation time of a numerical simulation significantly.     

Stability evaluation and system identification of flexible multibody systems

This paper is concerned with the linearized stability analysis and system identification of flexible multibody systems. Two closely related stability analysis approaches are summarized. Next, these approaches are extended to provide robust system identification procedures that combine least squares techniques and Kalman filters. The singular value decomposition, a numerically stable mathematical tool, is used to improve the robustness of the algorithm. The proposed algorithm identifies a minimum order plant based on input-output data, and is applicable to both experimental measurements or numerically computed responses. The proposed approaches are computationally inexpensive and consist of purely post processing steps that can be used with any multi-physics computational multibody tool or with experimental data. Commemorative Contribution.     

The development of a sliding joint for very flexible multibody dynamics using absolute nodal coordinate formulation

In this paper, a formulation for a spatial sliding joint is derived using absolute nodal coordinates and non-generalized coordinate and it allows a general multibody move along a very flexible cable. The large deformable motion of a spatial cable is presented using absolute nodal coordinate formulation, which is based on the finite element procedures and the general continuum mechanics theory to represent the elastic forces. And the nongeneralized coordinate, which is related to neither the inertia forces nor the external forces, is used to describe an arbitrary position along the centerline of a very flexible cable. Hereby, the non-generalized coordinate represents the arc-length parameter. The constraint equations for the sliding joint are expressed in terms of generalized coordinate and nongeneralized coordinate. In the constraint equations for the sliding joint, one constraint equation can be systematically eliminated. There are two independent Lagrange multipliers in the final system equations of motion associated with the sliding joint. The development of this sliding joint is important to analyze many mechanical systems such as pulley systems and pantograph-catenary systems for high speed-trains.     

On the use of moment-matching to build reduced order models in flexible multibody dynamics

An important issue in the field of flexible multibody dynamics is the reduction of the flexible body's degrees of freedom. For this purpose, often modal reduction through projection onto a subspace spanned by some dominant eigenvectors is used. However, as in this method the dynamical boundary conditions are not taken into account, a large number of eigenmodes is required to obtain a good approximation and also the selection of the dominant modes can be quite difficult. Therefore, the authors propose an approach based on accounting for the flexible body as an input-output system in the frequency domain. The reduced order model is generated by imposing a set of interpolation conditions concerning the values and derivatives of the system's transfer function in a predefined frequency range. This procedure is known as moment-matching and can be realised through projection onto so-called Krylov-subspaces. As this technique allows the incorporation of the frequency content and the spatial distribution of the loads, in the chosen frequency range more accurate reduced order models can be obtained compared to other model reduction techniques available in structural mechanics. The calculation of the Krylov-subspaces can be implemented very efficiently, using the Arnoldi or Lanczos procedure in connection with sparse matrix techniques. The capability of the proposed technique is demonstrated by means of a numerical example.     

Analysis of dynamic strains in tibia during human locomotion based on flexible multibody approach integrated with magnetic resonance imaging technique

Bone is known to adapt to the prevalent strain environment while the variation in strains, e.g., due to mechanical loading, modulates bone remodeling, and modeling. Dynamic strains rather than static strains provide the primary stimulus of bone functional adaptation. The finite element method can be generally used for estimating bone strains, but it may be limited to the static analysis of bone strains since the dynamic analysis requires expensive computation. Direct in vivo strain measurement, in turn, is an invasive procedure, limited to certain superficial bone sites, and requires surgical implementation of strain gauges and thus involves risks (e.g., infection). Therefore, to overcome difficulties associated with the finite element method and the in vivo strain measurements, the flexible multibody simulation approach has been recently introduced as a feasible method to estimate dynamic bone strains during physical activity. The purpose of the present study is to further strengthen the idea of using the flexible multibody approach for the analysis of dynamic bone strains. Besides discussing the background theory, magnetic resonance imaging is integrated into the flexible multibody approach framework so that the actual bone geometry could be better accounted for and the accuracy of prediction improved.     

基于Hermite 插值的多体系统动力学离散变分方法

针对多体系统动力学数值仿真问题,研究基于Hermite插值的离散变分方法.首先对广义坐标和广义速度进行Hermite插值,结合Gauss数值积分方法,利用Hamilton原理和离散力学变分原理,建立了含已知导数信息和含未知导数信息的Hermite插值离散变分数学模型,求解得到精确度较高的动力学仿真结果.该方法可以在步长较大时精确保持约束方程,并保持系统总能量在一定范围内有界变化,适用于长时间仿真情况.     

空间多体机械系统耦合动力学的键合图法

为提高多能域耦合空间多体系统动力学分析的效率及可靠性,提出了键合图法．从能量守恒的基本原理出发,讨论了键合图中的多通口元件MTF所具有的特性．以此为基础阐述了建立空间多体机械系统键合图模型的一般方法及其动力学原理．将运动副约束反力视做未知势源加在系统键合图模型相应的0-结处,有效地解决了微分因果关系及非线性结型结构所带来的十分困难的代数问题,推导出便于计算机自动生成的系统状态方程及运动副约束反力方程的统一公式,实现了计算机自动建模与仿真．实际算例表明了所述方法的有效性．     

Optimization of a complex flexible multibody systems with composite materials

The paper presents a general optimization methodology for flexible multibody systems which is demonstrated to find optimal layouts of fiber composite structures components. The goal of the optimization process is to minimize the structural deformation and, simultaneously, to fulfill a set of multidisciplinary constraints, by finding the optimal values for the fiber orientation of composite structures. In this work, a general formulation for the computation of the first order analytical sensitivities based on the use of automatic differentiation tools is applied. A critical overview on the use of the sensitivities obtained by automatic differentiation against analytical sensitivities derived and implemented by hand is made with the purpose of identifying shortcomings and proposing solutions. The equations of motion and sensitivities of the flexible multibody system are solved simultaneously being the accelerations and velocities of the system and the sensitivities of the accelerations and of the velocities integrated in time using a multi-step multi-order integration algorithm. Then, the optimal design of the flexible multibody system is formulated to minimize the deformation energy of the system subjected to a set of technological and functional constraints. The methodologies proposed are first discussed for a simple demonstrative example and applied after to the optimization of a complex flexible multibody system, represented by a satellite antenna that is unfolded from its launching configuration to its functional state.     

Composite rigid body formalism for flexible multibody systems

The paper describes the extension of the composite rigid body formalism for the flexible multibody systems. The extension has been done in such a way that all advantages of the formalism with respect to the coordinates of large motion of rigid bodies are extended to the flexible degrees of freedom, e.g. the same recursive treatment of both coordinates and no appearance of O(n ³) computational complexity terms due to the flexibility. This extension has been derived for both open loop and closed loop systems of flexible bodies. The comparison of the computational complexity of this formalism with other known approaches has shown that the described formalism of composite rigid body and the residual algorithm based on it are more efficient formalisms for small number of bodies in the chains and deformation modes than the usual recursive formalism of articulated body inertia.     

Task-level approaches for the control of constrained multibody systems

This paper presents a task-level control methodology for the general class of holonomically constrained multibody systems. As a point of departure, the general formulation of constrained dynamical systems is reviewed with respect to multiplier and minimization approaches. Subsequently, the operational space framework is considered and the underlying symmetry between constrained dynamics and operational space control is discussed. Motivated by this symmetry, approaches for constrained task-level control are presented which cast the general formulation of constrained multibody systems into a task space setting using the operational space framework. This provides a means of exploiting task-level control structures, native to operational space control, within the context of constrained systems. This allows us to naturally synthesize dynamic compensation for a multibody system, that properly accounts for the system constraints while performing a control task. A set of examples illustrate this control implementation. Additionally, the inclusion of flexible bodies in this approach is addressed.     

多柔体系统动力学研究进展与挑战

首先回顾多体系统动力学的学科发展和学术交流情况,然后系统概述了多柔体系统动力学方程数值算法、多柔体系统接触/碰撞动力学与柔性空间结构展开动力学三个方面的研究进展及值得关注的若干问题,最后给出了开展多柔体系统动力学研究的若干建议.     

基于等几何分析的柔性多体系统建模方法研究

近三十年来,柔性多体系统动力学取得长足进步,尤其是以绝对节点坐标方法(Absolute Nodal Coordinate Formulation, ANCF)为代表的非线性有限元已被用来处理复杂的柔性多体系统动力学问题.但绝对节点坐标方法采用斜率矢量作为广义坐标,导致系统自由度多,计算效率低.针对柔性多体系统,基于非均匀有理B样条(Non-Uniform Rational B-Splines, NURBS)曲线和曲面分别提出了Euler-Bernoulli细长梁单元和Kirchhoff-Love薄壳单元,在完全拉格朗日格式下,根据Green应变张量对单元变形进行描述,结合第二类Piola-Kirchhoff应力张量给出单元应变能公式,推导了单元的弹性力和弹性力雅可比矩阵表达式,最后通过静力学及动力学数值算例对提出的两类单元的性能进行对比和验证,为柔性多体系统建模提供了一种精确高效的新单元.     

Three-dimensional formulation of rigid-flexible multibody systems with flexible beam elements

Multibody systems generally contain solids with appreciable deformations and which decisively influence the dynamics of the system. These solids have to be modeled by means of special formulations for flexible solids. At the same time, other solids are of such a high stiffness that they may be considered rigid, which simplifies their modeling. For these reasons, for a rigid-flexible multibody system, two types of formulations coexist in the equations of the system. Among the different possibilities provided in the literature on the material, the formulation in natural coordinates and the formulation in absolute nodal coordinates are utilized in this paper to model the rigid and flexible solids, respectively. This paper contains a mixed formulation based on the possibility of sharing coordinates between a rigid solid and a flexible solid. The global mass matrix of the system is shown to be constant and, in addition, many of the constraint equations obtained upon utilizing these formulations are linear and can be eliminated.     

考虑刚-柔-热耦合的板结构多体系统的动力学建模

对热载荷作用下中心刚体与大变形薄板多体系统的动力学建模问题进行研究.基于Kirchhoff假设,从格林应变和曲率与绝对位移的非线性关系式出发,推导了非线性广义弹性力阵,用绝对节点坐标法建立了大变形矩形薄板的有限元离散的动力学变分方程.为了考虑刚体姿态运动、弹性变形和温度变化的相互耦合作用,推导了热流密度与绝对节点坐标之间的关系式.引入系统的运动学约束方程,建立了中心刚体-矩形板多体系统的考虑刚-柔-热耦合的热传导方程和带拉格朗日乘子的第一类拉格朗日动力学方程.为了有效地提高计算效率,将改进的中心差分法和广义-α法相结合,求解热传导方程和动力学方程,差分后的方程通过牛顿迭代法耦合求解.对刚-柔耦合和刚-柔-热三者耦合两种模型的仿真结果进行比较表明,刚体运动对温度梯度和热变形的影响显著.此外,本文建模方法考虑了几何非线性项,因此也考虑了热膨胀引起的轴向变形对横向变形的影响.     

具有奇异位置的多体系统动力学方程的改进算法

多体系统进行数值仿真时,很多选择了微分代数混合方程作为多体系统动力学数学模型.本文在现有的约束稳定化理论基础上,提出了针对具有奇异位置的多体系统动力学方程的改进算法.算法通过修正速度违约和控制稳定项,讨论了具有奇异位置的微分代数混合方程的数值仿真问题并给出了稳定项中相关系数的建议值,从而有效克服了求解混合方程时因为构型奇异给计算造成的困难.算例分别采用改进算法与ADAMS软件进行仿真,计算结果的比较表明了改进算法的有效性.本文给出的基于能量守恒的能量差曲线也证明了改进算法的有效性.     

含有分叉的受控多体系统传递矩阵法

多体系统传递矩阵法是解决受控多体系统动力学问题的有效方法,针对含有分叉的受控多体系统动力学问题,分别建立了控制激励元件、控制反馈元件及其含有分叉的受控多体系统总体传递矩阵．实例计算了无控和受控多体系统的运动响应与牛顿欧拉方法比较,计算表明了受控多体系统传递矩阵法解决分叉控制系统运动的有效性．     

舰炮发射动力学模块化建模方法

在舰炮元件动力学方程的基础上,使用数值积分方法,并推广振动理论传递矩阵法,形成了舰炮发射动力学模块化建模方法——舰炮发射动力学离散时间传递矩阵法．该方法可模拟得到舰炮多体系统动力响应,仿真结果证明了该方法的有效性．     

非完整约束多体系统时间离散变分积分法

基于连续Galerkin方法,给出非完整约束下多体系统时间离散的变分数值积分方法.首先对非完整多体系统Hamilton正则方程的弱形式进行时间离散,得到变分积分公式,然后讨论该积分方法对能量及约束的保持,最后以蛇板为例对该方法进行数值验证和比较.     

Transfer matrix method for linear multibody system

A new method for linear hybrid multibody system dynamics is proposed in this paper. This method, named as transfer matrix method of linear multibody system (MSTMM), expands the advantages of the traditional transfer matrix method (TMM). The concepts of augmented eigenvector and equation of motion of linear hybrid multibody system are presented at first to find the orthogonality and to analyze the responses of the hybrid multibody system using modal method. If using this method, the global dynamics equation is not needed in the study of linear hybrid multibody system dynamics. The MSTMM has a small size of matrix and higher computational speed, and can be applied to linear multi-rigid-body system dynamics, linear multi-flexible-body system dynamics and linear hybrid multibody system dynamics. This method is simple, straightforward, practical, and provides a powerful tool for the study on linear hybrid multibody system dynamics. This method can be used in the following: (1) Solve the eigenvalue problem of linear hybrid multibody systems. (2) Obtain the orthogonality of eigenvectors of linear hybrid multibody systems. (3) Realize the accurate analysis of the dynamics response of linear hybrid multibody systems. (4) Find the connected parameters between bodies used in the computation of linear hybrid multibody systems. A practical engineering system is taken as an example of linear multi-rigid-flexible-body system, the dynamics model, the transfer equations and transfer matrices of various bodies and hinges; the overall transfer equation and overall transfer matrix of the system are developed. Numerical example shows that the results of the vibration characteristics and the response of the hybrid multibody system received by MSTMM and by experiment have good agreements. These validate the proposed method.     

含干摩擦多体系统Lagrange方程的数值算法

利用第一类Lagrange方程建立了固定约束面含干摩擦的多体系统动力学方程,将摩擦力的广义力用矩阵形式描述．利用增广法。将微分-代数方程转换成常微分方程,并用矩阵形式给出,提高了计算效率．最后用算例说明该方法的有效性．