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.     

Three-Dimensional Large Deformation Analysis of the Multibody Pantograph/Catenary Systems

To accurately model the nonlinear behavior of the pantograph/catenary systems, it is necessary to take into consideration the effect of the large deformation of the catenary and its interaction with the nonlinear pantograph system dynamics. The large deformation of the catenary is modeled in this investigation using the three-dimensional finite element absolute nodal coordinate formulation. To model the interaction between the pantograph and the catenary, a sliding joint that allows for the motion of the pan-head on the catenary cable is formulated. To this end, a non-generalized arc-length parameter is introduced in order to be able to accurately predict the location of the point of contact between the pan-head and the catenary. The resulting system of differential and algebraic equations formulated in terms of reference coordinates, finite element absolute nodal coordinates, and non-generalized arc-length and contact surface parameters are solved using computational multibody system algorithms. A detailed three-dimensional multibody railroad vehicle model is developed to demonstrate the use of the formulation presented in this paper. In this model, the interaction between the wheel and the rail is considered. For future research, a method is proposed to deal with the problem of the loss of contact between the pan-head and the catenary cable.     

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.     

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.     

柔性多体系统中间隙铰接副的磨损预测

通过集成磨损计算与柔性多体动力学,对柔性系统中间隙铰接副部位的磨损进行了预测.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,引入连续接触力模型计算间隙铰接副部分的接触力,并采用Archard磨损模型的迭代磨损计算程序预测磨损.为了得到在不同接触情况下的磨损系数,本文中采用了径向基神经网络处理试验数据.通过对含柔性连杆的曲柄滑块机构进行仿真计算,发现当考虑部件的柔性时,得到的间隙处的冲击力大幅降低,且预测的磨损量也略有降低,这种区别会随着仿真时间的增加而变得更加明显.     

Formulation of Three-Dimensional Joint Constraints Using the Absolute Nodal Coordinates

A wide variety of mechanical and structural multibody systems consist ofvery flexible components subject to kinematic constraints. The widelyused floating frame of reference formulation that employs linear modelsto describe the local deformation leads to a highly nonlinear expressionfor the inertia forces and can be applied to only small deformationproblems. This paper is concerned with the formulation and computerimplementation of spatial joint constraints and forces using the largedeformation absolute nodal coordinate formulation. Unlike the floatingframe of reference formulation that employs a mixed set of absolutereference and local elastic coordinates, in the absolute nodalcoordinate formulation, global displacement and slope coordinates areused. The nonlinear kinematic constraint equations and generalized forceexpressions are expressed in terms of the absolute global displacementsand slopes. In particular, a new formulation for the sliding jointbetween two very flexible bodies is developed. A surface parameter isintroduced as an additional new variable in order to facilitate theformulation of this sliding joint. The constraint and force expressionsdeveloped in this paper are also expressed in terms of generalizedCholesky coordinates that lead to an identity inertia matrix. Severalexamples are presented in order to demonstrate the use of theformulations developed in the paper.     

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.     

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.     

A new model for dry and lubricated cylindrical joints with?clearance in spatial flexible multibody systems

A new approach to model and analyze flexible spatial multibody systems with clearance of cylindrical joints is presented in this work. The flexible parts are modeled by using absolute nodal coordinate formulation (ANCF)-based elements, while the rigid parts are described by employing the natural coordinate formulation (NCF), which can lead to a constant system mass matrix for the derived system equations of motion. In a simple way, a cylindrical joint with clearance is composed of two main elements, that is, a journal inside a bearing. Additionally, a lubricant fluid can exist between these two mechanical elements to reduce the friction and wear and increase the system??s life. For the case in which the joint is modeled as a dry contact pair, a technique using a continuous approach for the evaluation of the contact force is applied, where the energy dissipation in the form of hysteresis damping is considered. Furthermore, the frictional forces developed in those contacts are evaluated by using a modified Coulomb??s friction law. For the lubricated case, the hydrodynamic theory for dynamically loaded journal bearings is used to compute the forces generated by lubrication actions. The lubricated model is based on the Reynolds equation developed for the case of journal bearings with length-to-diameter ratios up to 1. Using this approach, the misalignment of the journal inside the bearing can be studied. Finally, two demonstrative examples of application are used to provide results that support the discussion and show the validity of the proposed methodologies.     

Modeling Method and Application of Rational Finite Element Based on Absolute Nodal Coordinate Formulation

In multibody system dynamics, the absolute nodal coordinate formulation(ANCF)uses power functions as interpolating polynomials to describe the displacement field. It can get accurate results for flexible bodies that undergo large deformation and large rotation. However, the power functions are irrational representation which cannot describe the complex shapes precisely, especially for circular and conic sections. Different from the ANCF representation,the rational absolute nodal coordinate formulation(RANCF) utilizes rational basis functions to describe geometric shapes, which allows the accurate representation of complicated displacement and deformation in dynamics modeling. In this paper, the relationships between the rational surface and volume and the RANCF finite element are provided, and the generalized transformation matrices are established correspondingly. Using these transformation matrices, a new four-node three-dimensional RANCF plate element and a new eight-node three-dimensional RANCF solid element are proposed based on the RANCF. Numerical examples are given to demonstrate the applicability of the proposed elements. It is shown that the proposed elements can depict the geometric characteristics and structure configurations precisely, and lead to better convergence in comparison with the ANCF finite elements for the dynamic analysis of flexible bodies.     

Dynamic analysis of a rigid-flexible inflatable space structure coupled with control moment gyroscopes

The vibration generated by the inflatable structure after deployment has a great impact on the performance of the payloads. In this paper, the influence of the control moment gyroscopes (CMGs) on the dynamic responses and characteristics of an inflatable space structure is studied, based on the flexible multibody dynamics in a combination of the absolute nodal coordinate formulation (ANCF) and the natural coordinate formulation (NCF). Firstly, the ANCF and NCF are used to accurately describe the large deformations and large overall motions of flexible inflatable tubes and rigid satellites, respectively. Then, instead of modeling gyroscopic flexible bodies, this paper pioneers a rigid body dynamic model of the CMG in detail by using the NCF modeling scheme, which can be attached to and coupled with any flexible bodies without any assumptions. Then, the orbital dynamic equations of the inflatable space structure coupled with distributed CMGs are obtained by considering the effects of Coriolis force, centrifugal force, and gravity gradient through coordinate transformation. The dynamic characteristics of the inflatable space structure are also analyzed by deriving the eigenvalue problem of a flexible multibody system. Finally, the accuracy of the CMG dynamic model is verified via a classic heavy top example. Several numerical examples are presented to study the influence of the magnitudes and directions of the rotor angular momentum of the CMGs on the dynamic responses and characteristics of the inflatable space structure.

     

完备化ANCF薄板单元及在钢板弹簧动力学建模中的应用

绝对节点坐标方法已在多体系统动力学研究中广泛应用, 但常用来描述板壳类结构的薄板单元, 由于梯度不完备而无法直接用于带有初始弯曲参考构型的柔性体变形描述. 为避免全参数板单元建立车辆钢板弹簧模型时存在的严重截面闭锁问题, 拟采用薄板单元用于板簧建模. 为此, 探索了将现有绝对节点坐标薄板单元纳入一般连续介质力学弹性力表达的方法, 采用中面上单位法向量作为单元厚度方向的梯度向量, 从而得到了完备化的薄板单元及其描述初始弯曲构型时消除初应变的方法. 在此基础上通过定义簧片的未变形构型, 在钢板弹簧中引入可控的预应力, 实现对钢板弹簧装配过程的准确模拟. 通过数值算例验证了本方法的正确性. 建立了车辆钢板弹簧模型, 通过建立在簧片上的局部坐标系实现接触点的跨单元搜索, 并采用惩罚函数法和平滑化的库伦摩擦模型施加簧片间的接触力. 引入参考节点的概念建立了整合车身与吊耳及其机构运动关系的刚柔耦合模型.}}      

On the Use of Implicit Integration Methods and the Absolute Nodal Coordinate Formulation in the Analysis of Elasto-Plastic Deformation Problems

In the general theory of continuum mechanics, the state of rotation and deformation of material points can be uniquely defined from the displacement field by using the nine independent components of the displacement gradients. For this reason, the use of the absolute rotation parameters as nodal coordinates, without relating them to the displacement gradients, leads to coordinate redundancy that leads to numerical and fundamental problems in many existing large rotation finite element formulations. Because of this fundamental problem, special measures that require modifications of the numerical integration methods were proposed in the literature in order to satisfy the principle of work and energy. As demonstrated in this paper, no such measures need to be taken when the finite element absolute nodal coordinate formulation is used since the principle of work and energy are automatically satisfied. This formulation does not suffer from the problem of coordinate redundancy and ensures the continuity of stresses and strains at the nodal points. In this study, the use of the implicit integration methods with the consistent Lagrangian elasto-plastic tangent moduli is examined when the absolute nodal coordinate formulation is used. The performance of different numerical integration methods in the dynamic analysis of large elasto-plastic deformation problems is investigated. It is shown that all these methods, in the case of convergence, yield a solution that satisfies the principle of work and energy without the need of taking any special measures. Semi-implicit integration methods, however, can lead to numerical difficulties in the case of very stiff problems due to the linearization made in these methods in order to avoid the iterative Newton--Raphson procedure. It is also demonstrated that the use of the consistent Lagrangian-plastic tangent moduli derived in this investigation using the absolute nodal coordinate formulation leads to better convergence of the iterative Newton--Raphson procedure used in the implicit integration methods.     

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.     

Efficient modeling of cable-pulley system with friction based on arbitrary-Lagrangian-Eulerian approach

In conventional modeling of a cable-pulley system, the cable must be finely meshed with Lagrangian elements for valid contact detections with pulleys, leading to extremely low efficiency. The sliding joint method based on the arbitrary-LagrangianEulerian(ALE) formulation still lacks an efficient cable element, and in particular, modeling of friction between a sliding joint and the cable has not been studied. This paper presents efficient multi-body modeling of a cable-pulley system with friction. A variablelength cable element with a node movable along the cable, which is described with ALE,is developed to mesh the cable. A transitional cable element is then proposed to model the contact part of the cable by fixing its two nodes to the two corresponding locations of the pulley. Friction of the cable-pulley is derived as a simple law of tension decay and embedded in the multi-body system modeling. It is simplified as a generalized friction force acting only on the arc-length coordinate. This approach can use a rough mesh on the cable, and is free of contact detections, thus significantly saving computation time.Several examples are presented to validate the proposed method, and show its effectiveness in real engineering applications.     

On the formulation of the planar ANCF triangular finite elements

In this paper, new planar isoparametric triangular finite elements (FE) based on the absolute nodal coordinate formulation (ANCF) are developed. The proposed ANCF elements have six coordinates per node: two position coordinates that define the absolute position vector of the node and four gradient coordinates that define vectors tangent to coordinate lines (parameters) at the same node. To shed light on the importance of the element geometry and to facilitate the development of some of the new elements presented in this paper, two different parametric definitions of the gradient vectors are used. The first parametrization, called area parameterization, is based on coordinate lines along the sides of the element in the reference configuration, while the second parameterization, called Cartesian parameterization, employs coordinate lines defined along the axes of the structure (body) coordinate system. The fundamental differences between the ANCF parameterizations used in this investigation and the parametrizations used for conventional finite elements are highlighted. The Cartesian parameterization serves as a unique standard for the triangular FE assembly. To this end, a transformation matrix that defines the relationship between the area and the Cartesian parameterizations is introduced for each element in order to allow for the use of standard FE assembly procedure and define the structure (body) inertia and elastic forces. Using Bezier geometry and a linear mapping, cubic displacement fields of the new ANCF triangular elements are systematically developed. Specifically, two new ANCF triangular finite elements are developed in this investigation, namely four-node mixed-coordinate and three-node ANCF triangles. The performance of the proposed new ANCF elements is evaluated by comparison with the conventional linear and quadratic triangular elements as well as previously developed ANCF rectangular and triangular elements. The results obtained in this investigation show that in the case of small and large deformations as well as finite rotations, all the elements considered can produce correct results, which are in a good agreement if appropriate mesh sizes are used.