Contact-impact formulation for multi-body systems using component mode synthesis

The efficiency and accuracy are two most concerned issues in the modeling and simulation of multi-body systems involving contact and impact. This paper proposed a formulation based on the component mode synthesis method for planar contact problems of flexible multi-body systems. A flexible body is divided into two parts： a contact zone and an un-contact zone. For the un-contact zone, by using the fixed-interface substructure method as reference, a few low-order modal coordinates are used to replace the nodal coordinates of the nodes, and meanwhile, the nodal coordinates of the local impact region are kept unchanged, therefore the total degrees of freedom （DOFs） are greatly cut down and the computational cost of the simulation is significantly reduced. By using additional constraint method, the impact constraint equations and kinematic constraint equations are derived, and the Lagrange equations of the first kind of flexible multi-body system are obtained. The impact of an elastic beam with a fixed half disk is simulated to verify the efficiency and accuracy of this method.     

Contact dynamics of elasto-plastic thin beams simulated via absolute nodal coordinate formulation

Under the frame of multibody dynamics, the contact dynamics of elasto-plastic spatial thin beams is numerically studied by using the spatial thin beam elements of absolute nodal coordinate formulation (ANCF). The inter-nal force of the elasto-plastic spatial thin beam element is derived under the assumption that the plastic strain of the beam element depends only on its longitudinal deformation. A new body-fixed local coordinate system is introduced into the spatial thin beam element of ANCF for efficient con-tact detection in the contact dynamics simulation. The linear isotropic hardening constitutive law is used to describe the elasto-plastic deformation of beam material, and the classical return mapping algorithm is adopted to evaluate the plastic strains. A multi-zone contact approach of thin beams previ-ously proposed by the authors is also introduced to detect the multiple contact zones of beams accurately, and the penalty method is used to compute the normal contact force of thin beams in contact. Four numerical examples are given to demonstrate the applicability and effectiveness of the pro-posed elasto-plastic spatial thin beam element of ANCF for flexible multibody system dynamics.     

Deployment dynamics of a simplified spinning IKAROS solar sail via absolute coordinate based method

The spinning solar sail of large scale has been well developed in recent years. Such a solar sail can be considered as a rigid-flexible multibody system mainly composed of a spinning central rigid hub, a number of flexible thin tethers, sail membranes, and tip masses. A simplified interplanetary kite-craft accelerated by radiation of the Sun (IKAROS) model is established in this study by using the absolute-coordinate-based (ACB) method that combines the natural coordinate formulation (NCF) describing the central rigid hub and the absolute nodal coordinate formulation (ANCF) describing flexible parts. The initial configuration of the system in the second-stage deployment is determined through both dynamic and static analyses. The huge set of stiff equations of system dynamics is solved by using the generalized-alpha method, and thus the deployment dynamics of the system can be well understood.     

A new subregion mesh method for the investigation of the elastic-plastic impact in flexible multibody systems

Impact processes between flexible bodies often lead to local stress concentration and wave propagation of high frequency. Therefore, the modeling of flexible multi-body systems involving impact should consider the local plastic deformation and the strict requirements of the spa-tial discretization. Owing to the nonlinearity of the stiffness matrix, the reduction of the element number is extremely important. For the contact-impact problem, since different regions have different requirements regarding the element size, a new subregion mesh method is proposed to reduce the number of the unnecessary elements. A dynamic model for flexible multibody systems with elastic-plastic contact impact is established based on a floating frame of reference formulation and complete Lagrange incremental nonlin-ear finite-element method to investigate the effect of the elastic-plastic deformation as well as spatial discretization. Experiments on the impact between two bodies are carried out to validate the correctness of the elastic-plastic model. The proposed formulation is applied to a slider-crank system with elastic-plastic impact.     

Coupling dynamic analysis of spacecraft with multiple cylindrical tanks and flexible appendages

This paper is mainly concerned with the coupling dynamic analysis of a complex spacecraft consisting of one main rigid platform, multiple liquid-filled cylindrical tanks,and a number of flexible appendages. Firstly, the carrier potential function equations of liquid in the tanks are deduced according to the wall boundary conditions. Through employing the Fourier–Bessel series expansion method, the dynamic boundaries conditions on a curved free-surface under a low-gravity environment are transformed to general simple differential equations and the rigid-liquid coupled sloshing dynamic state equations of liquid in tanks are obtained. The state vectors of rigid-liquid coupled equations are composed with the modal coordinates of the relative potential function and the modal coordinates of wave height. Based on the Bernoulli–Euler beam theory and the D'Alembert's principle, the rigid-flexible coupled dynamic state equations of flexible appendages are directly derived, and the coordinate transform matrixes of maneuvering flexible appendages are precisely computed as time-varying. Then, the coupling dynamics state equations of the overall system of the spacecraft are modularly built by means of the Lagrange's equations in terms of quasi-coordinates. Lastly, the cou-pling dynamic performances of a typical complex spacecraft are studied. The availability and reliability of the presented method are also confirmed.     

Discrete time transfer matrix method for dynamics of multibody system with flexible beams moving in space

In this paper,by defining new state vectors and developing new transfer matrices of various elements moving in space,the discrete time transfer matrix method of multi-rigid-flexible-body system is expanded to study the dynamics of multibody system with flexible beams moving in space.Formulations and numerical example of a rigidflexible-body three pendulums system moving in space are given to validate the method.Using the new method to study the dynamics of multi-rigid-flexible-body system moving in space,the global dynamics equations of system are not needed,the orders of involved matrices of the system are very low and the computational speed is high,irrespective of the size of the system.The new method is simple,straightforward,practical,and provides a powerful tool for multi-rigid-flexible-body system dynamics.     

DYNAMIC ANALYSIS OF FLEXIBLE-LINK AND FLEXIBLE-JOINT ROBOTS

The dynamic modeling and simulation of an N-flexible-link and N-flexible joint robot is reported. Each flexible joint is modeled as a linearly elastic torsional spring and the approach of assumed modes is adopted to describe the deformation of the flexible-link. The complete governing equations of motion of the flexible-link-joint robots are derived via Kane's method. An illustrative example is given to validate the algorithm presented and to show the effects of flexibility on the dynamics of robots.     

Modeling and analysis of rigid multibody systems with driving constraints and frictional translation joints

An approach is proposed for modeling and anal- yses of rigid multibody systems with frictional translation joints and driving constraints. The geometric constraints of translational joints with small clearance are treated as bilat- eral constraints by neglecting the impact between sliders and guides. Firstly, the normal forces acting on sliders, the driv- ing constraint forces （or moments） and the constraint forces of smooth revolute joints are all described by complementary conditions. The frictional contacts are characterized by a set- valued force law of Coulomb＇s dry friction. Combined with the theory of the horizontal linear complementarity problem （HLCP）, an event-driven scheme is used to detect the transi- tions of the contact situation between sliders and guides, and the stick-slip transitions of sliders, respectively. And then, all constraint forces in the system can be computed easily. Secondly, the dynamic equations of multibody systems are written at the acceleration-force level by the Lagrange multiplier technique, and the Baumgarte stabilization method is used to reduce the constraint drift. Finally, a numerical example is given to show some non-smooth dynamical behaviors of the studied system. The obtained results validate the feasibility of algorithm and the effect of constraint stabilization.     

Dynamic response of liquid-filled rectangular tank with elastic appendages under pitching excitation

Nonlinear dynamics of liquid-filled rectangular tank with elastic appendages are studied. Based on the assumption of ideal fluid, the coupling dynamic equations of rigid tank, elastic appendages and liquid fuel are derived using H-O principle. In the case of pitch excitation, the modified potential function and wave height function are introduced to describe the moving boundary of fluid, then Galerkin's method is used to discretize the dynamic equations into ordinary differential equations. The natural frequencies of the coupling system are formulated in liquid depth, the length of the tank, etc. The formulae are confirmed by numerical simulations, which also show that the effects of liquid and elastic appendages on the attitude angular of rigid.     

径向基点插值法在旋转柔性梁动力学中的应用

将无网格径向基点插值法用于旋转柔性梁的动力学分析. 利用无网格方法对柔性梁的变形场进行离散,考虑梁的纵向拉伸变形和横向弯曲变形,并计入横向弯曲变形引起的纵向缩短,即非线性耦合项,运用第二类拉格朗日方程推导得到系统刚柔耦合动力学方程. 将无网格径向基点插值法的仿真结果有限元法和假设模态法进行比较分析,说明假设模态法的局限性,并表明其作为一种柔性体离散方法在刚柔耦合多体系统动力学的研究中具有可推广性,并讨论了径向基形状参数的影响. 同时运用3 种求解系统动力学方程的方法:纽马克方法、4阶龙格库塔法、亚当姆斯预报校正法,并比较各方法的计算效率, 结果表明纽马克方法最快.      

Partition method and experimental validation for impact dynamics of flexible multibody system

The impact problem of a flexible multibody system is a non-smooth, high-transient, and strong-nonlinear dynamic process with variable boundary. How to model the contact/impact process accurately and efficiently is one of the main difficulties in many engineering applications. The numerical approaches being used widely in impact analysis are mainly from two fields: multibody system dynamics (MBS) and computational solid mechanics (CSM). Approaches based on MBS provide a more efficient yet less accurate analysis of the contact/impact problems, while approaches based on CSM are well suited for particularly high accuracy needs, yet require very high computational effort. To bridge the gap between accuracy and efficiency in the dynamic simulation of a flexible multibody system with contacts/impacts, a partition method is presented considering that the contact body is divided into two parts, an impact region and a non-impact region. The impact region is modeled using the finite element method to guarantee the local accuracy, while the non-impact region is modeled using the modal reduction approach to raise the global efficiency. A three-dimensional rod-plate impact experiment is designed and performed to validate the numerical results. The principle for how to partition the contact bodies is proposed: the maximum radius of the impact region can be estimated by an analytical method, and the modal truncation orders of the non-impact region can be estimated by the highest frequency of the signal measured. The simulation results using the presented method are in good agreement with the experimental results. It shows that this method is an effective formulation considering both accuracy and efficiency. Moreover, a more complicated multibody impact problem of a crank slider mechanism is investigated to strengthen this conclusion.     

柔性梁刚柔耦合碰撞动力学及变形传播研究

运用柔性多体系统刚柔耦合动力学理论,研究了作大范围回转运动柔性梁的碰撞动力学问题.考虑柔性梁的横向变形,以及横向变形引起的纵向缩短项即非线性耦合变形项.采用基于Hertz接触理论及非线性阻尼理论的非线性弹簧阻尼模型来求解碰撞过程中产生的碰撞力,运用第二类拉格朗日方程建立了系统的刚柔耦合碰撞动力学方程.编制仿真软件进行动力学仿真计算,得到了碰撞力和系统动力学响应,对比分析了不同动力学模型对系统动力学响应的影响.同时研究了碰撞导致的柔性梁横向变形传播的波动特性.     

平面柔性多体系统正碰撞动力学建模理论研究

针对目前柔性多体系统碰撞动力学建模方法存在的不足,对影响碰撞动力学仿真的主要因素如柔性体建模和碰撞初始条件进行分析,建立起基于变约束的柔性体碰撞动力学方程。首先,为了解决子结构法在处理碰撞界面搜索时面临的难题,引入多体系统柔性体有限元描述方法,推导出凸形柔性体接触点间法向位移约束的二阶导数形式。其次,从碰撞引起的接触界面速度不连续机理出发,结合连续介质力学间断面理论,给出碰撞瞬时由物体本身物理性质决定的接触位置处速度跳跃公式。最后对两弹性圆盘低速碰撞问题进行数值仿真。结果表明本文提出的改进方法符合力学基本原理,仿真结果满足收敛性要求。     

A node-based smoothed point interpolation method for dynamic analysis of rotating flexible beams

We proposed a mesh-free method, the called node-based smoothed point interpolation method (NS-PIM), for dynamic analysis of rotating beams. A gradient smoothing technique is used, and the requirements on the consistence of the displacement functions are further weakened. In static problems, the beams with three types of boundary conditions are analyzed, and the results are compared with the exact solution, which shows the effectiveness of this method and can provide an upper bound solution for the deflection. This means that the NS-PIM makes the system soften. The NS-PIM is then further extended for solving a rigid-flexible coupled system dynamics problem, considering a rotating flexible cantilever beam. In this case, the rotating flexible cantilever beam considers not only the transverse deformations, but also the longitudinal deformations. The rigid-flexible coupled dynamic equations of the system are derived via employing Lagrange’s equations of the second type. Simulation results of the NS-PIM are compared with those obtained using finite element method (FEM) and assumed mode method. It is found that compared with FEM, the NS-PIM has anti-ill solving ability under the same calculation conditions.     

考虑剪切效应的旋转FGM楔形梁刚柔耦合动力学建模与仿真

本文对一类中心刚体-柔性梁系统在大范围转动下的刚柔耦合动力学问题进行了研究. 柔性梁为功能梯度材料(functionally graded materials, FGM)楔形变截面梁,材料体积分数在梁轴向呈幂律分布变化. 以弧长坐标来描述柔性FGM梁的几何位移关系,分别使用倾角和拉伸应变变量描述柔性梁的横向弯曲和纵向拉伸变形,并计及剪切效应. 采用假设模态法离散变形场,运用第二类拉格朗日方程进行方程推导,得到系统考虑剪切效应的刚柔耦合动力学模型. 基于全新的刚柔耦合动力学建模理论,研究不同轴向材料梯度分布的FGM楔形梁,通过数值仿真计算,分析讨论不同的转速、梯度分布规律以及变截面参数对系统动力学特性的影响. 结果表明,剪切效应对大高跨比的FGM楔形梁的变形影响较为明显,不容忽略;材料梯度分布规律和截面参数的选取均会对旋转FGM楔形梁的动力学响应和频率产生较大影响. 本文提出的考虑剪切效应的倾角刚柔耦合动力学模型是对以往非剪切模型的进一步完善,可应用于工程中的 Timoshenko梁结构的动力学问题求解.      

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.     

基于径向基点插值法的旋转Mindlin板高次刚柔耦合动力学模型

将无网格径向基点插值法(radial point interpolation method, RPIM)用于中心刚体?旋转柔性板的动力学分析. 基于浮动坐标系方法和一阶剪切变形理论即Mindlin板理论, 考虑剪切变形的影响, 并计入板面内变形的非线性耦合变形项, 采用径向基点插值法描述板的变形场, 保留动能中有关非线性耦合变形项的所有高阶量, 通过构造高阶形函数避免了径向基点插值法出现剪切闭锁的现象, 建立了既能处理薄板问题又能处理中厚板问题的作大范围运动矩形板的高次刚柔耦合动力学模型. 高阶形函数可通过添加高阶多项式的方式获得, 静力学算例表明径向基点插值法中添加15项多项式可基本消除剪切闭锁. 将零次近似模型、一次近似模型和高次模型的仿真结果对比, 说明零次近似模型的缺陷, 同时说明高次模型有更广的适用范围, 可分析大变形问题. 将径向基点插值法的仿真结果与有限元法和假设模态法进行比较分析, 说明本文方法的正确性, 也表明无网格径向基点插值法作为一种柔性体离散方法在刚柔耦合多体系统动力学的研究中具有可推广性.