Partition method for impact dynamics of flexible multibody systems based on contact constraint

The impact dynamics of a flexible multibody system is investigated. By using a partition method, the system is divided into two parts, the local impact region and the region away from the impact. The two parts are connected by specific boundary conditions, and the system after partition is equivalent to the original system. According to the rigid-flexible coupling dynamic theory of multibody system, system＇s rigid-flexible coupling dynamic equations without impact are derived. A local impulse method for establishing the initial impact conditions is proposed. It satisfies the compatibility con- ditions for contact constraints and the actual physical situation of the impact process of flexible bodies. Based on the contact constraint method, system＇s impact dynamic equa- tions are derived in a differential-algebraic form. The contact/separation criterion and the algorithm are given. An impact dynamic simulation is given. The results show that system＇s dynamic behaviors including the energy, the deformations, the displacements, and the impact force during the impact process change dramatically. The impact makes great effects on the global dynamics of the system during and after impact.     

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。     

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.     

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.     

含摩擦与碰撞平面多刚体系统动力学线性互补算法

基于接触力学理论和线性互补问题的算法, 给出了一种含接触、碰撞以及库伦干摩擦, 同时具有理想定常约束(铰链约束) 和非定常约束(驱动约束) 的平面多刚体系统动力学的建模与数值计算方法. 将系统中的每个物体视为刚体, 但考虑物体接触点的局部变形, 将物体间的法向接触力表示成嵌入量与嵌入速度的非线性函数,其切向摩擦力采用库伦干摩擦模型. 利用摩擦余量和接触点的切向加速度等概念, 给出了摩擦定律的互补关系式; 并利用事件驱动法, 将接触点的黏滞-滑移状态切换的判断及黏滞状态下摩擦力的计算问题转化成线性互补问题的求解. 利用第一类拉格朗日方程和鲍姆加藤约束稳定化方法建立了系统的动力学方程, 由此可降低约束的漂移, 并可求解该系统的运动、法向接触力和切向摩擦力, 还可以求解理想铰链约束力和驱动约束力. 最后以一个类似夯机的平面多刚体系统为例, 分析了其动力学特性, 并说明了相关算法的有效性.      

Dynamic simulation of liquid-filled flexible multibody systems via absolute nodal coordinate formulation and SPH method

A new computation method is proposed to study the coupled dynamics of a partially liquid-filled flexible multibody system, where the liquid is modeled by using the Smoothed Particle Hydrodynamics (SPH) method and the flexible bodies are described by using the Absolute Nodal Coordinate Formulation (ANCF). Extra virtual particles are introduced and embedded in the liquid neighboring the rigid or flexible boundaries in order to prevent field particles from penetrating the boundary and force them to follow the deformation of flexible boundary. The interaction forces between the liquid and the flexible multibody system are transmitted by the virtual particles. The domain decomposition is used to improve the efficiency of interaction detection in SPH computation. A predictor-corrector scheme is used to solve the governing equations of liquid discretized by SPH particles. The generalized-alpha method based on sparse matrix storage skill is used to solve a huge set of dynamic equations of the multibody system. The OpenMP+OpenACC based parallel computation skills are embedded in the iteration processes to speed up the computation efficiency. Finally, three numerical examples are given to validate the proposed computation method.     

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.     

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.     

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.      

含摩擦滑移铰平面多刚体系统动力学的数值算法

基于LuGre摩擦模型和线性互补问题(LCP)的数值算法,给出了具有双边约束含摩擦滑移铰平面多体系统动力学的数值算法.首先,根据滑移铰的特点,当间隙充分小时,将其视为双边约束,给出了滑移铰中滑道作用于滑块上的法向接触力的互补关系;LuGre摩擦模型能有效地描述机械系统中的黏滞与滑移运动,将该模型用于描述滑块与滑道间的摩擦力.其次,结合Baumgarte约束稳定化方法,应用第一类Lagrange方程,建立了该多体系统的动力学方程,给出了Lagrange乘子与滑移铰中作用于滑块上的法向接触力的关系式.然后,将滑块与滑道间多种接触状态的判断以及作用于滑块上的法向接触力的计算转换为线性互补问题的求解,并用常微分方程的数值算法求解该多体系统的动力学方程.最后,通过数值仿真算例揭示了滑移铰中滑块的黏滞与滑移现象,以及滑块在滑道内的多种接触状态;另外,在文中分别采用Coulomb干摩擦模型和LuGre摩擦模型,对算例中的某些工况进行了数值仿真,并且分别用本文方法得到的数值仿真结果与已有方法得到的数值仿真结果对比,表明了本文给出的方法的有效性.      

Dynamics of flexible mechanical systems with contact-impact and plastic deformations

A computer based formulation for the analysis of mechanical systems is investigated as a feasible method to predict the impact response of complex structural systems. A general methodology for the dynamic analysis of rigid-flexible multibody systems using a number of redundant Cartesian coordinates and the method of the Lagrange multipliers is presented. The component mode synthesis is then used to reduce the number of flexible degrees of freedom. In many impact situations, the individual structural members are overloaded giving rise to plastic deformations in highly localized regions, called plastic hinges. This concept is used by associating revolute nonlinear actuators with constitutive relations corresponding to the collapse behavior of the structural components. The contact of the system components is described using a continuous force model based on the Hertz contact law with hysteresis damping. The effect and importance of structural damping schemes in flexible bodies are also addressed here. Finally, the validity of this methodology is assessed by comparing the results of the proposed models with those obtained in different experimental tests where: a beam collides transversally with a rigid block; a torque box impacts a rigid barrier.     

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.     

Control structure interaction in the nonlinear analysis of flexible mechanical systems

The effect of the control structure interaction on the feedforward control law as well as the dynamics of flexible mechanical systems is examined in this investigation. An inverse dynamics procedure is developed for the analysis of the dynamic motion of interconnected rigid and flexible bodies. This method is used to examine the effect of the elastic deformation on the driving forces in flexible mechanical systems. The driving forces are expressed in terms of the specified motion trajectories and the deformations of the elastic members. The system equations of motion are formulated using Lagrange's equation. A finite element discretization of the flexible bodies is used to define the deformation degrees of freedom. The algebraic constraint equations that describe the motion trajectories and joint constraints between adjacent bodies are adjoined to the system differential equations of motion using the vector of Lagrange multipliers. A unique displacement field is then identified by imposing an appropriate set of reference conditions. The effect of the nonlinear centrifugal and Coriolis forces that depend on the body displacements and velocities are taken into consideration. A direct numerical integration method coupled with a Newton-Raphson algorithm is used to solve the resulting nonlinear differential and algebraic equations of motion. The formulation obtained for the flexible mechanical system is compared with the rigid body dynamic formulation. The effect of the sampling time, number of vibration modes, the viscous damping, and the selection of the constrained modes are examined. The results presented in this numerical study demonstrate that the use of the driving forees obtained using the rigid body analysis can lead to a significant error when these forces are used as the feedforward control law for the flexible mechanical system. The analysis presented in this investigation differs significantly from previously published work in many ways. It includes the effect of the structural flexibility on the centrifugal and Coriolis forces, it accounts for all inertia nonlinearities resulting from the coupling between the rigid body and elastic displacements, it uses a precise definition of the equipollent systems of forces in flexible body dynamics, it demonstrates the use of general purpose multibody computer codes in the feedforward control of flexible mechanical systems, and it demonstrates numerically the effect of the selected set of constrained modes on the feedforward control law.     

Continuous contact force models for impact analysis in multibody systems

One method for predicting the impact response of a multibody system is based on the assumption that the impacting bodies undergo local deformations and the contact forces are continuous. In a continuous analysis, the integration of the system equations of motion is carried out during the period of contact; therefore, a model for evaluating the contact forces is required. In this paper, two such contact force models are presented, both Hertzian in nature and based upon the direct-central impact of two solid particles.At low impact velocities, the energy dissipation during impact can be represented by material damping. A model is constructed based on the general trend of the Hertz contact law in conjuction with a hysteresis damping function. The unknown parameters are determined in terms of a given coefficient of restitution and the impact velocity. When local plasticity effects are the dominant factor accounting for the dissipation of energy at high impact velocities, a Hertzian contact force model with permanent indentation is constructed. Utilizing energy and momentum considerations, the unknown parameters in the model are again evaluated. The two particle models are generalized to an impact analysis between two bodies of a multibody system.     

An Augmented Formulation for Mechanical Systems with Non-Generalized Coordinates: Application to Rigid Body Contact Problems

In computational multibody algorithms, the kinematic constraintequations that describe mechanical joints and specified motiontrajectories must be satisfied at the position, velocity andacceleration levels. For most commonly used constraint equations, onlyfirst and second partial derivatives of position vectors with respect tothe generalized coordinates are required in order to define theconstraint Jacobian matrix and the first and second derivatives of theconstraints with respect to time. When the kinematic and dynamicequations of the multibody systems are formulated in terms of a mixedset of generalized and non-generalized coordinates, higher partialderivatives with respect to these non-generalized coordinates arerequired, and the neglect of these derivatives can lead to significanterrors. In this paper, the implementation of a contact model in generalmultibody algorithms is presented as an example of mechanical systemswith non-generalized coordinates. The kinematic equations that describethe contact between two surfaces of two bodies in the multibody systemare formulated in terms of the system generalized coordinates and thesurface parameters. Each contact surface is defined using twoindependent parameters that completely define the tangent and normalvectors at an arbitrary point on the body surface. In the contact modeldeveloped in this study, the points of contact are searched for on lineduring the dynamic simulation by solving the nonlinear differential andalgebraic equations of the constrained multibody system. It isdemonstrated in this paper that in the case of a point contact andregular surfaces, there is only one independent generalized contactconstraint force despite the fact that five constraint equations areused to enforce the contact conditions.     

DYNAMICS ANALYSIS OF STOCHASTIC SPATIAL FLEXIBLE MULTIBODY SYSTEM 1)

轻质、高精度的柔性多体系统被广泛应用于实际工程领域中.由于实际设计公差、制造误差及环境温度等多种不确定因素的存在,使得柔性多体系统的结构参数(物理参数和几何参数)表现出随机性.具有随机结构参数的动力学模型能够客观地反映出真实系统的动力学行为,且结构参数的不确定性对空间柔性多体系统动力学响应的影响是不容忽视的.针对具有多个随机参数的空间柔性多体系统,提出了一种基于广义alpha算法的非侵入式随机柔性多体系统动力学计算方法.采用绝对节点坐标公式(absolute node coordinate formulation, ANCF)来描述柔性体, 推导建立多体系统动力学模型.利用混沌多项式展开(polynomial chaos expansion, PCE)法构建系统随机动力学方程的代理模型,然后将随机响应面法(stochastic response surface method, SRSM)嵌入广义-alpha方法中,分别采用改进抽样的回归方法(regression method of improved sampling, RMIS)和单项求容积法则(Monte Carlo simulation, MCR)来确定样本点.将数值计算结果与蒙特卡洛模拟(Monte Carlo simulation, MCS)结果进行对比, 验证了所提算法的有效性.在相同的定积分精度的条件下,根据单项求容积法则确定的样本点的计算结果稳定性更强, 且其计算效率更高.     

随机空间柔性多体系统动力学分析

轻质、高精度的柔性多体系统被广泛应用于实际工程领域中.由于实际设计公差、制造误差及环境温度等多种不确定因素的存在,使得柔性多体系统的结构参数(物理参数和几何参数)表现出随机性.具有随机结构参数的动力学模型能够客观地反映出真实系统的动力学行为,且结构参数的不确定性对空间柔性多体系统动力学响应的影响是不容忽视的.针对具有多个随机参数的空间柔性多体系统,提出了一种基于广义alpha算法的非侵入式随机柔性多体系统动力学计算方法.采用绝对节点坐标公式(absolute node coordinate formulation, ANCF)来描述柔性体, 推导建立多体系统动力学模型.利用混沌多项式展开(polynomial chaos expansion, PCE)法构建系统随机动力学方程的代理模型,然后将随机响应面法(stochastic response surface method, SRSM)嵌入广义-alpha方法中,分别采用改进抽样的回归方法(regression method of improved sampling, RMIS)和单项求容积法则(Monte Carlo simulation, MCR)来确定样本点.将数值计算结果与蒙特卡洛模拟(Monte Carlo simulation, MCS)结果进行对比, 验证了所提算法的有效性.在相同的定积分精度的条件下,根据单项求容积法则确定的样本点的计算结果稳定性更强, 且其计算效率更高.      

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.