一种O(n)算法复杂度的递推绝对节点坐标法研究

相比于传统的浮动坐标法,绝对节点坐标法(absolute nodal coordinate formulation,ANCF)在处理柔性体非线性大变形问题上具有显著优势,但是对于ANCF的求解目前主要依据拉格朗日方程等分析力学原理建立微分代数方程(differential algebraic equation,DAE)进行,其算法复杂度为O(n2)或O(n3)(n为系统自由度),且求解过程存在位置或速度的违约问题.据此,研究了一种O(n)算法复杂度的递推绝对节点坐标法(recursive absolute nodal coordinate formulation,RANCF).该方法采用ANCF描述大变形柔性体,借鉴铰接体递推算法(articulatedbody algorithm,ABA)思路建立多柔体系统逐单元的运动学和动力学递推关系,得到微分形式的系统动力学方程(ordinary differential equation,ODE).在ODE方程中,系统广义质量阵为三对角块矩阵,通过恰当的矩阵处理,可以得到逐单元求解该方程的递推算法.在此基础上,给出了RANCF算法的详细流程,并对流程中每个步骤进行了细致的算法效率分析,证明了RANCF是算法复杂度为O(n)的高效算法.RANCF方法保留了ANCF对大转动、大变形多柔体系统精确计算的优点,同时极大地提升了算法效率,特别在处理高自由度复杂多柔体系统中具有显著优势.并且该方法采用ODE求解,无DAE的违约问题,因此具有更高的算法精度.最后,在算例部分,通过MSC.ADAMS仿真软件、能量守恒测试、算法复杂度曲线对RANCF的正确性、计算精度和计算效率进行了验证.     

ANCF/CRBF平面梁闭锁问题及闭锁缓解研究

本文系统地研究了基于一致旋转场列式的绝对节点坐标 (ANCF consistentrotation-based formulation, ANCF/CRBF)平面梁单元的泊松闭锁问题及闭锁缓解技术.为了全面理解该类型单元的闭锁特性及明确单元的应用范围,文中首先开发了两种新的ANCF/CRBF刚性截面梁单元, 新单元在ANCF全参数梁的基础上,对梯度向量施加正交矩阵约束, 得到梯度与转角对时间导数之间的速度转换矩阵,从而引入转角参数. 新单元节点处完全消除了泊松闭锁和剪切效应,这是与传统ANCF/CRBF刚性截面梁单元的不同之处. 然后,对比分析了这三种ANCF/CRBF刚性截面梁单元泊松闭锁的特点.发现该类型单元对节点的横向梯度施加了运动学约束, 导致节点处截面不能变形,无法捕捉泊松效应, 但是单元内部能完全捕捉,这种不连续情况会加重单元整体的泊松闭锁问题. 并且发现对单元梯度约束的越多,闭锁问题越严重. 随后, 分别采用两种闭锁缓解技术, 弹性线方法和应变分解方法,进一步研究了单元的收敛性. 最终,通过多种静力学和动力学测试研究了泊松闭锁对ANCF/CRBF平面梁单元计算精度的影响及闭锁缓解技术在该类型单元上的缓解效果.      

基于绝对节点坐标法的平面梁有限变形下变形重构

现有的对有限变形条件下柔性结构变形重构的研究往往单纯基于曲率与应变间的几何关系, 同时忽略了被测体的纵向变形及其与弯曲变形的耦合效应. 为得到一种更加精确且能借助现有的力学工具进行应用方向扩展的变形重构方法, 以平面梁为对象, 借鉴变形重构逆有限元法的思想, 将平面梁的变形重构问题视作一类最优化问题. 首先, 通过引入绝对节点坐标法(absolute nodal coordinate formulation, ANCF)对柔性结构大变形下非线性的平面梁应变?位移关系进行精确描述, 构造了一种逆梯度缩减ANCF平面索梁单元. 然后, 对此逆ANCF单元进行改进, 在简化节点自由度的同时通过引入罚函数确保单元节点处的曲率连续性, 既保证了本问题的适定性, 也提升了最终解的精确性. 最后, 基于该单元利用Newton法构造了平面梁有限变形下变形重构问题的两种求解算法, 即逐单元算法和多单元整体算法, 以实现不同需求下的稳定求解. 数值仿真结果表明, 本方法在大变形条件下的变形重构误差小于1%, 而且在测点较少的情况下依然保持较高的精度, 同时验证了本方法的收敛性与计算效率.      

基于绝对节点坐标法的弹性线方法研究

相比于浮动坐标系法, 绝对节点坐标法(absolute nodal coordinateformulation, ANCF)在处理柔性体非线性大变形问题上具有显著优势,ANCF将单元节点坐标定义在全局坐标系下,采用斜率矢量代替节点转角坐标, 具有常数质量阵,不存在科氏离心力等优点, 然而弹性力阵为非线性项,其求解将比较耗时且占用资源. 据此, 在弹性力求解方法中,引入弹性线方法(elastic line method, ELM),该方法将格林--拉格朗日应变张量定义在中心线上,采用曲率公式来定义弯曲应变, 转角公式来定义扭转应变.同时采用有限元法对三维柔性梁位移场进行离散,求解梁单元常数质量阵、广义刚度阵、广义力阵,进而得到单元的动力学方程, 通过转换矩阵得到三维梁的动力学方程.接着从理论上指出连续介质力学方法(continuum mechanics method,CMM)和弹性线方法在求解弹性力上的不同点, 并编制动力学仿真软件.最后分别采用连续介质力学方法和弹性线方法对柔性单摆以及履带式车辆的动力学问题进行仿真分析,结果表明:弹性线方法能在保证精度的前提下有效提高计算效率.      

ANCF方法在小变形问题中的应用研究

绝对节点坐标法(Absolute Nodal Coordinate Formulation,ANCF)具有不存在小变形、小转动假设,质量矩阵为常数矩阵等优点,但由于弯曲应变与轴向应变不一致,带来剪切闭锁问题.本文基于小变形假设,利用ANCF方法得到两节点梁的刚度矩阵,进而将该方法推广到多节点梁,解得多节点梁的刚度矩阵.利用Maple软件编制求解程序,求解矩形截面梁在无约束条件下的固有频率及外伸梁的末端静挠度.通过与ABAQUS仿真结果及解析解对比发现:当梁上节点数较少时,用ANCF方法得到的结果相较于仿真结果、解析解较为"刚硬","剪切闭锁"现象较为严重.随着节点数的增加,ANCF方法得到的计算结果与仿真结果、解析解趋于一致.当梁上节点数增加到31时,对于自由模态的前4阶固有频率,ANCF方法的求解结果与解析解之间的误差均小于3‰;对端部带有集中质量的外伸梁的末端静挠度,ANCF方法的求解结果与解析解之间的误差均小于0.5‰,剪切闭锁问题得到有效解决.     

ANCF索梁单元应变耦合问题与模型解耦

索粱结构在土木工程、航空航天等领域有着广泛的应用.在各类索梁动力学建模方法中,由于绝对节点坐标方法(absolute nodal coordinate formulation,ANCF)能够描述柔性体的大变形和大转动问题,因此非常适合大变形索梁结构的动力学建模.对绝对节点坐标索梁单元的应变进行分析可知,弯曲变形会引起单元内部轴向应变的不均匀分布,即单元轴向应变与弯曲应变相互耦合.这种应变耦合效应使单元产生伪应变能,导致单元刚度增大,造成单元失真.分析不同弯曲角下的单元应变及应变能可知,弯曲变形越大,单元失真越严重.通过构造等效一维杆单元重新描述轴向应变,实现了轴向应变与弯曲应变解耦.在此基础上推导广义弹性力,得到了绝对节点坐标索梁单元的应变解耦模型.对解耦前后的两种梁模型进行静力学和动力学仿真,结果表明;解耦模型消除了单元伪应变,相比原模型表现出更好的收敛性和曲率连续性,在相同单元数目下具有更高的精度.同时由于解耦模型降低了单元刚度,因此相比原模型,速度曲线中不再有高频振动.     

两类基于局部标架梁单元的闭锁缓解方法

对于大转动、大变形柔性体的刚柔耦合动力学问题,基于李群SE(3)局部标架(local frame formulation, LFF)的建模方法能够规避刚体运动带来的几何非线性问题,离散数值模型中广义质量矩阵与切线刚度矩阵满足刚体变换的不变性,可明显地提高柔性多体系统动力学问题的计算效率. 有限元方法中,闭锁问题是导致单元收敛性能低下的主要原因, 例如梁单元的剪切以及泊松闭锁.多变量变分原理是缓解梁、板/壳单元闭锁的有效手段. 该方法不仅离散位移场,同时离散应力场或应变场, 可提高应力与应变的计算精度. 本文基于上述局部标架,研究几类梁单元的闭锁处理方法, 包括几何精确梁(geometrically exact beam formulation, GEBF)与绝对节点坐标(absolute nodal coordinate formulation, ANCF)梁单元. 其中, 采用Hu-Washizu三场变分原理缓解几何精确梁单元中的剪切闭锁,采用应变分解法缓解基于局部标架的ANCF全参数梁单元中的泊松闭锁. 数值算例表明,局部标架的梁单元在描述高转速或大变形柔性多体系统时,可消除刚体运动带来的几何非线性, 极大地减少系统质量矩阵和刚度矩阵的更新次数.缓解闭锁后的几类局部标架梁单元收敛性均得到了明显提升.      

中性线修正型变截面梁类构件压电控制

传统绝对节点坐标法(absolute nodal coordinate formulation,ANCF)在变截面梁类构件建模过程中常以几何中位线等效构造单元中性线,难以对变截面单元位移场状态进行精确描述.为解决此类问题,本文以中细型变截面梁类构件为研究对象,深入考虑变截面结构几何因素及复合材料属性对变截面梁类构件中性...     

基于绝对节点坐标法的弹性线方法研究

相比于浮动坐标系法,绝对节点坐标法(absolute nodal coordinate formulation, ANCF)在处理柔性体非线性大变形问题上具有显著优势, ANCF将单元节点坐标定义在全局坐标系下,采用斜率矢量代替节点转角坐标,具有常数质量阵,不存在科氏离心力等优点,然而弹性力阵为非线性项,其求解将比较耗时且占用资源.据此,在弹性力求解方法中,引入弹性线方法 (elastic line method, ELM),该方法将格林–拉格朗日应变张量定义在中心线上,采用曲率公式来定义弯曲应变,转角公式来定义扭转应变.同时采用有限元法对三维柔性梁位移场进行离散,求解梁单元常数质量阵、广义刚度阵、广义力阵,进而得到单元的动力学方程,通过转换矩阵得到三维梁的动力学方程.接着从理论上指出连续介质力学方法 (continuum mechanics method, CMM)和弹性线方法在求解弹性力上的不同点,并编制动力学仿真软件.最后分别采用连续介质力学方法和弹性线方法对柔性单摆以及履带式车辆的动力学问题进行仿真分析,结果表明:弹性线方法能在保证精度的前提下有效提高计算效率.     

柔性对多体系统中铰接副磨损的影响

通过集成柔性多体动力学与磨损计算程序,提出了一种用于对柔性多体系统中间隙铰接副部位的磨损进行了预测的方法.基于绝对节点坐标方法(ANCF)建立了柔性部件的多体动力学模型,引入Lankanrani和Nikravesh提出的连续接触力模型计算间隙铰接副部分的法向接触力,采用Lu Gre摩擦模型计算切向摩擦力,并利用基于Archard模型的迭代计算程序计算磨损.为了提高计算效率,引入了并行计算策略.最后,通过对一个含柔性连杆的曲柄滑块机构机构进行仿真计算,发现当考虑部件的柔性时,得到的间隙处的冲击力会大幅降低,预测的磨损量也随之降低,并且随着机构柔性的增强,这种效果更为明显.     

Nonlinear formulation for flexible multibody system with large deformation

In this paper, nonlinear modeling for flexible multibody system with large deformation is investigated. Absolute nodal coordinates are employed to describe the displacement, and variational motion equations of a flexible body are derived on the basis of the geometric nonlinear theory, in which both the shear strain and the transverse normal strain are taken into account. By separating the inner and the boundary nodal coordinates, the motion equations of a flexible multibody system are assembled. The advantage of such formulation is that the constraint equations and the forward recursive equations become linear because the absolute nodal coordinates are used. A spatial double pendulum connected to the ground with a spherical joint is simulated to investigate the dynamic performance of flexible beams with large deformation. Finally, the resultant constant total energy validates the present formulation. The project supported by the National Natural Science Foundation of China (10472066, 10372057). The English text was polished by Yunming Chen.     

Analysis of Thin Beams and Cables Using the Absolute Nodal Co-ordinate Formulation

The purpose of this paper is to present formulations for beam elements based on the absolute nodal co-ordinate formulation that can be effectively and efficiently used in the case of thin structural applications. The numerically stiff behaviour resulting from shear terms in existing absolute nodal co-ordinate formulation beam elements that employ the continuum mechanics approach to formulate the elastic forces and the resulting locking phenomenon make these elements less attractive for slender stiff structures. In this investigation, additional shape functions are introduced for an existing spatial absolute nodal co-ordinate formulation beam element in order to obtain higher accuracy when the continuum mechanics approach is used to formulate the elastic forces. For thin structures where bending stiffness can be important in some applications, a lower order cable element is introduced and the performance of this cable element is evaluated by comparing it with existing formulations using several examples. Cables that experience low tension or catenary systems where bending stiffness has an effect on the wave propagation are examples in which the low order cable element can be used. The cable element, which does not have torsional stiffness, can be effectively used in many problems such as in the formulation of the sliding joints in applications such as the spatial pantograph/catenary systems. The numerical study presented in this paper shows that the use of existing implicit time integration methods enables the simulation of multibody systems with a moderate number of thin and stiff finite elements in reasonable CPU time.     

Nonlinear dynamics of a cable–pulley system using the absolute nodal coordinate formulation

It is reasonable to develop models and to investigate the dynamic behaviour of systems composed of cables since cable vibration can have an important effect on the motion of these mechanical systems. This paper deals with the application of the nonlinear formulation for flexible body dynamics called the absolute nodal coordinate formulation (ANCF). It is used for modelling the systems composed of cables, pulleys, other rigid bodies and a motor with prescribed motion. The ANCF was chosen as a suitable approach, which that can allow to consider a detailed interaction of the cable and the pulley with its nonlinear dynamical behaviour. The ANCF uses absolute positions of nodes (reference vectors) and slopes (reference vector derivations) as a set of nodal coordinates. An in-house modelling tool in the MATLAB system was created based on the proposed modelling methodology and two case studies were performed. A simple system containing a pulley and a cable with two attached bodies was used in order to test the simulation tool based on the proposed modelling methodology with respect to different parameters. A more complex mechanical system composed of a driven weight joined with a motor by a cable led over a pulley was numerically and also experimentally investigated. The comparison of obtained numerical and experimental results shows sufficient agreement and proves that the proposed modelling approach can be used for dynamic analyses of such systems.     

Use of Cholesky Coordinates and the Absolute Nodal Coordinate Formulation in the Computer Simulation of Flexible Multibody Systems

In a previous publication, procedures that can be used with the absolute nodal coordinate formulation to solve the dynamic problems of flexible multibody systems were proposed. One of these procedures is based on the Cholesky decomposition. By utilizing the fact that the absolute nodal coordinate formulation leads to a constant mass matrix, a Cholesky decomposition is used to obtain a constant velocity transformation matrix. This velocity transformation is used to express the absolute nodal coordinates in terms of the generalized Cholesky coordinates. The inertia matrix associated with the Cholesky coordinates is the identity matrix, and therefore, an optimum sparse matrix structure can be obtained for the augmented multibody equations of motion. The implementation of a computer procedure based on the absolute nodal coordinate formulation and Cholesky coordinates is discussed in this paper. Numerical examples are presented in order to demonstrate the use of Cholesky coordinates in the simulation of the large deformations in flexible multibody applications.     

INTEGRATION OF NON-UNIFORM RATIONAL B-SPLINES GEOMETRY AND RATIONAL ABSOLUTE NODAL COORDINATES FORMULATION FINITE ELEMENT ANALYSIS

This investigation is intended to develop a computer procedure for the integration of NURBS geometry and the rational absolute nodal coordinate formulation （RANCF） finite element analysis. A linear transformation is given that can be used to convert the NURBS curve to RANCF cable element mesh retaining the same geometry and the same degree of continuity, including the discussion of continuity control and mesh refinement. The green strain tensor is used to establish the nonlinear dynamic equations with numerical examples to demonstrate the use of the procedure in the dynamic analysis of flexible bodies.     

Development of flexible telescopic boom model using absolute nodal coordinate formulation sliding joint constraints with LuGre friction

In this investigation, a modeling procedure of a telescopic boom of cranes is developed using the absolute nodal coordinate formulation together with the sliding joint constraints. Since telescopic booms are extracted and retracted under various operating conditions, the overall length of the boom changes dynamically, leading to the time-variant vibration characteristics. For modeling the telescopic structure of booms, a special care needs to be exercised since the location of the sliding contact point moves along the deformable axis of the flexible boom and the solution to a moving boundary problem is required. This issue indeed makes the modeling of the telescopic boom difficult, despite the significant needs for the analysis. It is, therefore, the objective of this investigation to develop a modeling procedure for the flexible telescopic boom by considering the sliding contact condition with the dynamic frictional effect. To this end, the sliding joint constraint developed for the absolute nodal coordinate formulation is employed for describing relative sliding motion between flexible booms, while flexible booms are modeled using the beam element of the absolute nodal coordinate formulation, which allows for modeling the large rotation and deformation of the structure.     

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.     

稳定节点积分伽辽金无网格法的应力计算方法研究

应力计算是基于稳定节点积分的伽辽金无网格法的重要组成部分.该文着重研究稳定节点积分伽辽金无网格法的应力计算方法,对稳定节点积分方法的变分一致条件进行了讨论.证明当节点代表域内的应变采用非局郎光滑应变时,相应的应力在节点代表域内为常数,稳定节点积分伽辽金无网格离散方程是变分一致的.文中提出了三种节点应力计算方法,研究表明,基于位移梯度的节点应力计算方法不满足变分一致性要求,而采用光滑应变的节点应力计算方法和一致形心应力计算方法满足变分一致性要求.典型数值算例的误差分析表明,满足变分一致性不一定确保得到更为精确的结果.而基于光滑应变的一致形心应力计算方法总是较其它两种方法更为精确.