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.     

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.     

A referenced nodal coordinate formulation

Multibody System Dynamics - We develop a referenced nodal coordinate formulation (RNCF) to study the dynamics of flexible bodies undergoing large-distance travels and/or high-speed rotations. RNCF...     

Three-dimensional formulation of rigid-flexible multibody systems with flexible beam elements

Multibody systems generally contain solids with appreciable deformations and which decisively influence the dynamics of the system. These solids have to be modeled by means of special formulations for flexible solids. At the same time, other solids are of such a high stiffness that they may be considered rigid, which simplifies their modeling. For these reasons, for a rigid-flexible multibody system, two types of formulations coexist in the equations of the system. Among the different possibilities provided in the literature on the material, the formulation in natural coordinates and the formulation in absolute nodal coordinates are utilized in this paper to model the rigid and flexible solids, respectively. This paper contains a mixed formulation based on the possibility of sharing coordinates between a rigid solid and a flexible solid. The global mass matrix of the system is shown to be constant and, in addition, many of the constraint equations obtained upon utilizing these formulations are linear and can be eliminated.     

A parallel Hamiltonian formulation for forward dynamics of closed-loop multibody systems

This paper presents a novel recursive divide-and-conquer formulation for the simulation of complex constrained multibody system dynamics based on Hamilton’s canonical equations (HDCA). The systems under consideration are subjected to holonomic, independent constraints and may include serial chains, tree chains, or closed-loop topologies. Although Hamilton’s canonical equations exhibit many advantageous features compared to their acceleration based counterparts, it appears that there is a lack of dedicated parallel algorithms for multi-rigid-body system dynamics based on the Hamiltonian formulation. The developed HDCA formulation leads to a two-stage procedure. In the first phase, the approach utilizes the divide and conquer scheme, i.e., a hierarchic assembly–disassembly process to traverse the multibody system topology in a binary tree manner. The purpose of this step is to evaluate the joint velocities and constraint force impulses. The process exhibits linear \(O(n)\) (\(n\) – number of bodies) and logarithmic \(O(\log_{2}{n})\) numerical cost, in serial and parallel implementations, respectively. The time derivatives of the total momenta are directly evaluated in the second parallelizable step of the algorithm. Sample closed-loop test cases indicate very small constraint violation errors at the position and velocity level as well as marginal energy drift without any additional form of constraint stabilization techniques involved in the solution process. The results are comparatively set against more standard acceleration based Featherstone’s DCA approach to indicate the performance of the HDCA algorithm.     

A new locking-free formulation for planar,shear deformable,linear and quadratic beam finite elements based on the absolute nodal coordinate formulation

Many widely used beam finite element formulations are based either on Reissner’s classical nonlinear rod theory or the absolute nodal coordinate formulation (ANCF). Advantages of the second method have been pointed out by several authors; among the benefits are the constant mass matrix of ANCF elements, the isoparametric approach and the existence of a consistent displacement field along the whole cross section. Consistency of the displacement field allows simpler, alternative formulations for contact problems or inelastic materials. Despite conceptional differences of the two formulations, the two models are unified in the present paper.     

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.     

绝对节点坐标法下斜率不连续问题处理方法讨论

Shabana提出的绝对节点坐标法,引入节点斜率坐标作为节点自由度描述转动.对于由梁板壳及块体组成的组合结构,在结构节点处相交单元的节点斜率自由度不连续,这给组合结构的建模和分析带来特殊的困难.本文讨论了文献中研究斜率不连续问题时的处理办法.在简要介绍绝对节点坐标法后,详细地讨论了经典折梁算例和截面呈阶梯变化的直梁算例中斜率不连续问题.对这两个算例,本文采用约束函数法和现有文献中的转换坐标方法,计算了在结构节点处相交杆件的轴向应变,对比这些数值结果,本文指出现有文献中的转换坐标办法,忽视了斜率自由度和转角自由度的差别,从而不能正确给出斜率不连续处相交杆件的轴向应变,需要进一步研究.     

Three- and four-noded planar elements using absolute nodal coordinate formulation

This paper investigates two new types of planar finite elements containing three and four nodes. These elements are the reduced forms of the spatial plate elements employing the absolute nodal coordinate approach. Elements of the first type use translations of nodes and global slopes as nodal coordinates and have 18 and 24 degrees of freedom. The slopes facilitate the prevention of the shear locking effect in bending problems. Furthermore, the slopes accurately describe the deformed shape of the elements. Triangular and quadrilateral elements of the second type use translational degrees of freedom only and, therefore, can be utilized successfully in problems without bending. These simple elements with 6 and 8 degrees of freedom are identical to the elements used in conventional formulation of the finite element method from the kinematical point of view. Similarly to the famous problem called “flying spaghetti” which is used often as a benchmark for beam elements, a kind of “flying lasagna” is simulated for the planar elements. Numerical results of simulations are presented.     

Flexible multibody dynamics using coordinate reduction improved by dynamic correction

This paper is concerned with the efficient dynamic analysis of flexible multibody systems using a robust coordinate reduction technique. Unlike conventional static correction, the formulation is derived by dynamic correction that considers the inertia effect. In this formulation, the constraint and fixed-interface normal modes, which are representative modes in the typical coordinate reduction, are corrected by considering the truncated modal effect with the residual flexibility. Therefore, the proposed method can offer a more precise reduced system without increasing the dimension, which consequently leads to a more accurate and efficient flexible multibody simulation. We implement here the proposed method under augmented formulations of the floating reference frame approach, and test its performance with numerical examples.     

Non-linear dynamics of flexible multibody systems

In this work we set to examine several important issues pertinent to currently very active research area of the finite element modeling of flexible multibody system dynamics. To that end, we first briefly introduce three different model problems in non-linear dynamics of flexible 3D solid, a rigid body and 3D geometrically exact beam, which covers the vast majority of representative models for the particular components of a multibody system. The finite element semi-discretization for these models is presented along with the time-discretization performed by the mid-point scheme. In extending the proposed methodology to modeling of flexible multibody systems, we also present how to build a systematic representation of any kind of joint connecting two multibody components, a typical case of holonomic contraint, as a linear superposition of elementary constraints. We also indicate by a chosen model of rolling contact, an example of non-holonomic constraint, that the latter can also be included within the proposed framework. An important aspect regarding the reduction of computational cost while retaining the consistency of the model is also addressed in terms of systematic use of the rigid component hypothesis, mass lumping and the appropriate application of the explicit-implicit time-integration scheme to the problem on hand. Several numerical simulations dealing with non-linear dynamics of flexible multibody systems undergoing large overall motion are presented to further illustrate the potential of presented methodology. Closing remarks are given to summarize the recent achievements and point out several directions for future research.     

Linear dynamics of flexible multibody systems

We consider mechanical systems where the dynamics are partially constrained to prescribed trajectories. An example for such a system is a building crane with a load and the requirement that the load moves on a certain path.Enforcing this condition directly in form of a servo constraint leads to differential-algebraic equations (DAEs) of arbitrarily high index. Typically, the model equations are of index 5, which already poses high regularity conditions. If we relax the servo constraints and consider the system from an optimal control point of view, the strong regularity conditions vanish, and the solution can be obtained by standard techniques.By means of the well-known \(n\)-car example and an overhead crane, the theoretical and expected numerical difficulties of the direct DAE and the alternative modeling approach are illustrated. We show how the formulation of the problem in an optimal control context works and address the solvability of the optimal control system. We discuss that the problematic DAE behavior is still inherent in the optimal control system and show how its evidences depend on the regularization parameters of the optimization.     

A low order,torsion deformable spatial beam element based on the absolute nodal coordinate formulation and Bishop frame

Multibody System Dynamics - Heretofore, the Serret–Frenet frame has been the ubiquitous choice for analyzing the elastic deformations of beam elements. It is well-known that this frame is...     

A conserving formulation of a simple shear- and torsion-free beam for multibody applications

Multibody System Dynamics - Many engineering fields such as aerospace, robotics, and computer graphics, have applications that contain elements amenable to be modeled as slender beams with...     

Spatial joint constraints for the absolute nodal coordinate formulation using the non-generalized intermediate coordinates

In this investigation, a systematic procedure that can be used for modeling joint constraints for the absolute nodal coordinate formulation is developed. To this end, the non-generalized intermediate coordinates are introduced to derive a mapping between the generalized gradient coordinates and the non-generalized rotation parameters. With this mapping, a wide variety of joint constraints can be defined for the absolute nodal coordinate formulation in terms of the non-generalized reference coordinates and, therefore, existing well-developed constraint libraries formulated for the rigid body reference coordinates can be directly employed without significant modifications in existing codes. Furthermore, in order to define a rigid surface at the joint definition point, a set of orthonormality conditions is imposed on the gradient coordinates. This leads to not only accurate modeling of interface to mechanical joint, but also a simpler definition of the joint coordinate system obtained by the orthonormal gradient vectors. For this reason, a simpler form of constraint Jacobian and quadratic velocity vectors can be obtained as compared to those of the existing approach which requires the use of highly nonlinear joint coordinate system. A systematic procedure for eliminating the non-generalized coordinates and the dependent Lagrange multipliers associated with the coordinate mapping equations from the equations of motion is presented. As a result, a standard augmented form of the equations of motion can be obtained in terms of the generalized coordinates only. Several numerical examples are presented in order to demonstrate the use of the joint constraint formulation developed in this investigation.     

Improved bushing models for general multibody systems and vehicle dynamics

The development and computational implementation, on a multibody dynamics environment, of a constitutive relation to model bushing elements associated with mechanical joints used in the models of road and rail vehicles is presented here. These elements are used to eliminate vibrations in vehicles, due to road irregularities, to allow small misalignment of axes, to reduce noise from the transmission, or to decrease wear of the mechanical joints. Bushings are made of a special rubber, used generally in energy dissipation, which presents a nonlinear viscoelastic relationship between the forces and moments and their corresponding displacements and rotations. In the methodology proposed here a finite element model of the bushing is developed in the framework of the finite element code ABAQUS to obtain the constitutive relations of displacement/rotation versus force/moment for different loading cases. The bushing is modeled in a multibody code as a nonlinear restrain that relates the relative displacements between the bodies connected with the joint reaction forces, and it is represented by a matrix constitutive relation. The basic ingredients of the multibody model are the same vectors and points relations used to define kinematic constraints in any multibody formulation. One particular, and relevant, characteristic of the formulation now presented is its ability to represent standard kinematic joints, clearance, and bushing joints just by defining appropriate constitutive relations. Spherical, revolution, cylindrical, and translational bushing joints are modeled, implemented, and demonstrated through the simulation of two multibody models of a road vehicle, one with perfect kinematic joints for the suspension sub-systems, and other with bushing joints. The tests conducted include an obstacle avoidance maneuver and a vehicle riding over bumps. It is shown that the bushing models for vehicle multibody models proposed here are accurate and computationally efficient so that they can be included in the vehicle models leading reliable simulations.