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.     

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.     

Geometrically non-linear formulation of flexible multibody systems in terms of beam elements: Geometric stiffness

The occurrence of strong deflections and major axial forces in many applications involving flexible multibodies entails including non-linear terms coupling deformation-induced axial and transverse displacements in the motion equation. The formulations, including such terms, are known as geometrically non-linear formulations. The authors have developed one such formulation that preserves higher-order terms in the strain energy function. By expressing such terms as a function of selected elastic coordinates, three stiffness matrices and two non-linear vectors of elastic forces are defined. The first matrix is the conventional constant-stiffness matrix, the second is the classical geometric stiffness matrix and the third is a second-order geometric stiffness matrix. The aim of this work is to define the third matrix and the two non-linear vectors of elastic forces by using the finite-element method.     

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.     

Developments of multibody system dynamics: computer simulations and experiments

It is an exceptional success when multibody dynamics researchers Multibody System Dynamics journal one of the most highly ranked journals in the last 10 years. In the inaugural issue, Professor Schiehlen wrote an interesting article explaining the roots and perspectives of multibody system dynamics. Professor Shabana also wrote an interesting article to review developments in flexible multibody dynamics. The application possibilities of multibody system dynamics have grown wider and deeper, with many application examples being introduced with multibody techniques in the past 10 years. In this paper, the development of multibody dynamics is briefly reviewed and several applications of multibody dynamics are described according to the author’s research results. Simulation examples are compared to physical experiments, which show reasonableness and accuracy of the multibody formulation applied to real problems. Computer simulations using the absolute nodal coordinate formulation (ANCF) were also compared to physical experiments; therefore, the validity of ANCF for large-displacement and large-deformation problems was shown. Physical experiments for large deformation problems include beam, plate, chain, and strip. Other research topics currently being carried out in the author’s laboratory are also briefly explained. Commemorative Contribution.     

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.     

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.     

A method for improving dynamic solutions in flexible multibody dynamics

This paper presents a method for improving dynamic solutions that are obtained from the dynamic simulation of flexible multibody systems. The mode-acceleration concept in linear structural dynamics is utilized in the proposed method for improving accuracy in the postprocessing stage. A theoretical explanation is made on why the proposed method improves the dynamic solutions in the context of the mode-acceleration method. A mode-acceleration equation for each flexible body is defined and the load term in the right hand side of the equation is represented as a combination of space-dependent and time-dependent terms so that efficient computation of dynamic solutions can be achieved. The load term is obtained from dynamic simulation of a flexible multibody system and a finite element method is used to compute dynamic solutions by quasi-static analyses. Numerical examples show the effectiveness of the proposed method.     

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.     

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.     

Discussion on the optimization problem formulation of flexible components in multibody systems

This paper is dedicated to the structural optimization of flexible components in mechanical systems modeled as multibody systems. While most of the structural optimization developments have been conducted under (quasi-)static loadings or vibration design criteria, the proposed approach aims at considering as precisely as possible the effects of dynamic loading under service conditions. Solving this problem is quite challenging and naive implementations may lead to inaccurate and unstable results. To elaborate a robust and reliable approach, the optimization problem formulation is investigated because it turns out that it is a critical point. Different optimization algorithms are also tested. To explain the efficiency of the various solution approaches, the complex nature of the design space is analyzed. Numerical applications considering the optimization of a two-arm robot subject to a trajectory tracking constraint and the optimization of a slider-crank mechanism with a cyclic dynamic loading are presented to illustrate the different concepts.     

多体系统中典型铰的摩擦力计算模型

在铰内间隙很小的前提下,多体系统中铰仍具有运动学约束作用.但由于铰内接触形式与系统状态相关,铰内摩擦力与约束反力之间具有复杂的函数关系.本文在假设铰内接触为刚性接触的前提下,基于分布接触反力与点接触反力之间的等效关系,给出了几种典型铰内摩擦力的计算模型,并通过数值算例验证了所提模型的正确性.     

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.     

Slope discontinuities in the finite element absolute nodal coordinate formulation: gradient deficient elements

In this paper, the treatment of the slope discontinuities in the finite element absolute nodal coordinate formulation (ANCF) is discussed. The paper explains the fundamental problems associated with developing a constant transformation that accounts for the slope discontinuities in the case of gradient deficient ANCF finite elements. A procedure that allows for the treatment of slope discontinuities in the case of gradient deficient finite elements which do not employ full parameterization is proposed for the special case of commutative rotations. The use of the proposed procedure leads to a constant orthogonal element transformation that describes the element initial configuration. As a consequence, one obtains in the case of large deformation and commutative rotations, a constant mass matrix for the structures. In order to achieve this goal, the concept of the intermediate finite element coordinate system is invoked. The intermediate finite element coordinate system used in this investigation serves to define the element reference configuration, follows the rotation of the structure, and maintains a fixed orientation relative to the structure coordinate system. Since planar rotations are always commutative, the procedure proposed in this investigation is applicable to all planar gradient deficient ANCF finite elements.     

First order sensitivity analysis of flexible multibody systems using absolute nodal coordinate formulation

Design sensitivity analysis of flexible multibody systems is important in optimizing the performance of mechanical systems. The choice of coordinates to describe the motion of multibody systems has a great influence on the efficiency and accuracy of both the dynamic and sensitivity analysis. In the flexible multibody system dynamics, both the floating frame of reference formulation (FFRF) and absolute nodal coordinate formulation (ANCF) are frequently utilized to describe flexibility, however, only the former has been used in design sensitivity analysis. In this article, ANCF, which has been recently developed and focuses on modeling of beams and plates in large deformation problems, is extended into design sensitivity analysis of flexible multibody systems. The Motion equations of a constrained flexible multibody system are expressed as a set of index-3 differential algebraic equations (DAEs), in which the element elastic forces are defined using nonlinear strain-displacement relations. Both the direct differentiation method and adjoint variable method are performed to do sensitivity analysis and the related dynamic and sensitivity equations are integrated with HHT-I3 algorithm. In this paper, a new method to deduce system sensitivity equations is proposed. With this approach, the system sensitivity equations are constructed by assembling the element sensitivity equations with the help of invariant matrices, which results in the advantage that the complex symbolic differentiation of the dynamic equations is avoided when the flexible multibody system model is changed. Besides that, the dynamic and sensitivity equations formed with the proposed method can be efficiently integrated using HHT-I3 method, which makes the efficiency of the direct differentiation method comparable to that of the adjoint variable method when the number of design variables is not extremely large. All these improvements greatly enhance the application value of the direct differentiation method in the engineering optimization of the ANCF-based flexible multibody systems.     

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．     

An energy preserving/decaying scheme for nonlinearly constrained multibody systems

Several numerical time integration methods for multibody system dynamics are described: an energy preserving scheme and three energy decaying ones, which introduce high-frequency numerical dissipation in order to annihilate the nondesired high-frequency oscillations. An exhaustive analysis of these four schemes is done, including their formulation, and energy preserving and decaying properties by taking into account the presence of nonlinear algebraic constraints and the incrementation of finite rotations. A new energy preserving/decaying scheme is developed, which is well suited for either stiff or nonstiff nonlinearly constrained multibody systems. Examples on a series of test cases show the performance of the algorithms.     

Order-(n+m) direct differentiation determination of design sensitivity for constrained multibody dynamic systems

With the complexity and large dimensionality of many modern multibody dynamic applications, the efficiency of the sensitivity evaluation methods used can greatly impact the overall computation cost and as such can greatly limit the usefulness of the sensitivity information. Most current direct differentiation approaches suffer from prohibitive computational cost, which may be as great as O(n⁴+n²m²+nm³) (for systems with n generalized coordinates and m algebraic constraints). This paper presents a concise and computationally efficient sensitivity analysis scheme to facilitate such sensitivity calculations. A unique feature of this scheme is its use of recursive procedures to directly embed the algebraic constraint relations in forming and simultaneously solving a minimal set of equations. This results in far fewer operations than more traditional, or so-called O(n), counterparts. The algorithm determines the derivatives of generalized accelerations in O(n+m) operations overall. The resulting equations are exact to integration accuracy and enforce constraints exactly at both the velocity and acceleration levels.     

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.     

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.