基于等几何分析的柔性多体系统建模方法研究

近三十年来,柔性多体系统动力学取得长足进步,尤其是以绝对节点坐标方法(Absolute Nodal Coordinate Formulation, ANCF)为代表的非线性有限元已被用来处理复杂的柔性多体系统动力学问题.但绝对节点坐标方法采用斜率矢量作为广义坐标,导致系统自由度多,计算效率低.针对柔性多体系统,基于非均匀有理B样条(Non-Uniform Rational B-Splines, NURBS)曲线和曲面分别提出了Euler-Bernoulli细长梁单元和Kirchhoff-Love薄壳单元,在完全拉格朗日格式下,根据Green应变张量对单元变形进行描述,结合第二类Piola-Kirchhoff应力张量给出单元应变能公式,推导了单元的弹性力和弹性力雅可比矩阵表达式,最后通过静力学及动力学数值算例对提出的两类单元的性能进行对比和验证,为柔性多体系统建模提供了一种精确高效的新单元.     

Flexible Multibody Systems Models Using Composite Materials Components

The use of a multibody methodology to describe the large motion of complex systems that experience structural deformations enables to represent the complete system motion, the relative kinematics between the components involved, the deformation of the structural members and the inertia coupling between the large rigid body motion and the system elastodynamics. In this work, the flexible multibody dynamics formulations of complex models are extended to include elastic components made of composite materials, which may be laminated and anisotropic. The deformation of any structural member must be elastic and linear, when described in a coordinate frame fixed to one or more material points of its domain, regardless of the complexity of its geometry. To achieve the proposed flexible multibody formulation, a finite element model for each flexible body is used. For the beam composite material elements, the sections properties are found using an asymptotic procedure that involves a two-dimensional finite element analysis of their cross-section. The equations of motion of the flexible multibody system are solved using an augmented Lagrangian formulation and the accelerations and velocities are integrated in time using a multi-step multi-order integration algorithm based on the Gear method.     

On the use of absolute interface coordinates in the floating frame of reference formulation for flexible multibody dynamics

In this work a new formulation for flexible multibody systems is presented based on the floating frame formulation. In this method, the absolute interface coordinates are used as degrees of freedom. To this end, a coordinate transformation is established from the absolute floating frame coordinates and the local interface coordinates to the absolute interface coordinates. This is done by assuming linear theory of elasticity for a body’s local elastic deformation and by using the Craig–Bampton interface modes as local shape functions. In order to put this new method into perspective, relevant relations between inertial frame, corotational frame and floating frame formulations are explained. As such, this work provides a clear overview of how these three well-known and apparently different flexible multibody methods are related. An advantage of the method presented in this work is that the resulting equations of motion are of the differential rather than the differential-algebraic type. At the same time, it is possible to use well-developed model order reduction techniques on the flexible bodies locally. Hence, the method can be employed to construct superelements from arbitrarily shaped three dimensional elastic bodies, which can be used in a flexible multibody dynamics simulation. The method is validated by simulating the static and dynamic behavior of a number of flexible systems.     

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.     

A 3D Finite Element Method for Flexible Multibody Systems

An efficient finite element (FE) formulation for the simulation of multibody systems is derived from Hamilton's principle. According to the classical assumptions of multibody systems, a large rotation formulation has been chosen, where large rotations and large displacements, but only small deformations of the single bodies are taken into account. The strain tensor is linearized with respect to a co-rotated frame. The present approach uses absolute coordinates for the degrees of freedom and forms an alternative to the floating frame of reference formulation that is based on relative coordinates and describes deformation with respect to a co-rotated frame. Due to the modified strain tensor, the present formulation distinguishes significantly from standard nodal based nonlinear FE methods. Constraints are defined in integral form for every pair of surfaces of two bodies. This leads to a small number of constraint equations and avoids artificial stress singularities. The resulting mass and stiffness matrices are constant apart from a transformation based on a single rotation matrix for each body. The particular structure of this transformation allows to prevent from the usually expensive factorization of the system Jacobian within implicit time--integration methods. The present method has been implemented and tested with the FE-package NGSolve and specific 3D examples are verified with a standard beam formulation.     

A weighted residual relationship for the contact problem with Coulomb friction

In this work, a weighted residual relationship is proposed as an extension of the standard virtual work principle to deal with the large deformation contact problem with Coulomb friction. This weak form is a mixed relationship involving the displacements and the multipliers defined on the reference contact surface of the contactor and is shown to be equivalent to the strong form of the initial/boundary value contact problem. The discretization in space by means of the finite element method is carried out on the mixed relationship in a simple way in order to obtain the semi-discrete equation system. The contact tangent stiffness is derived and numerical examples are presented to assess the efficiency of the formulation.     

Non-linear strain-displacement equations exactly representing large rigid-body motions. Part III. Analysis of TM shells with constraints

The precise representation of arbitrarily large rigid-body motions in the displacement patterns of curved Timoshenko-Mindlin-type (TM) shell elements has been considered in Part I of the present work. In Part II it has been developed an enhanced mixed finite element formulation that allows using load increments that are much larger than possible with existing geometrically exact displacement-based shell element formulations. In this paper the developed formulation is employed to solve frictionless contact problems for TM shells undergoing finite deformations and interacting with rigid bodies. The contact conditions are incorporated into the assumed stress-strain TM shell formulation by applying a perturbed Lagrangian procedure with the fundamental unknowns consisting of 6 displacements and 11 strains of the bottom and top surfaces of the shell, 11 conjugate stress resultants and the Lagrange multiplier, associated with a nodal contact force, through using the non-conventional technique. The efficiency and accuracy of the proposed finite element formulation are demonstrated by means of several numerical examples.     

Geometric nonlinear effects on the planar dynamics of a pivoted flexible beam encountering a point-surface impact

Flexible-body modeling with geometric nonlinearities remains a hot topic of research by applications in multibody system dynamics undergoing large overall motions. However, the geometric nonlinear effects on the impact dynamics of flexible multibody systems have attracted significantly less attention. In this paper, a point-surface impact problem between a rigid ball and a pivoted flexible beam is investigated. The Hertzian contact law is used to describe the impact process, and the dynamic equations are formulated in the floating frame of reference using the assumed mode method. The two important geometric nonlinear effects of the flexible beam are taken into account, i.e., the longitudinal foreshortening effect due to the transverse deformation, and the stress stiffness effect due to the axial force. The simulation results show that good consistency can be obtained with the nonlinear finite element program ABAQUS/Explicit if proper geometric nonlinearities are included in the floating frame formulation. Specifically, only the foreshortening effect should be considered in a pure transverse impact for efficiency, while the stress stiffness effect should be further considered in an oblique case with much more computational effort. It also implies that the geometric nonlinear effects should be considered properly in the impact dynamic analysis of more general flexible multibody systems.     

Flexible Multibody Dynamics: Review of Past and Recent Developments

In this paper, a review of past and recent developments in the dynamics of flexible multibody systems is presented. The objective is to review some of the basic approaches used in the computer aided kinematic and dynamic analysis of flexible mechanical systems, and to identify future directions in this research area. Among the formulations reviewed in this paper are the floating frame of reference formulation, the finite element incremental methods, large rotation vector formulations, the finite segment method, and the linear theory of elastodynamics. Linearization of the flexible multibody equations that results from the use of the incremental finite element formulations is discussed. Because of space limitations, it is impossible to list all the contributions made in this important area. The reader, however, can find more references by consulting the list of articles and books cited at the end of the paper. Furthermore, the numerical procedures used for solving the differential and algebraic equations of flexible multibody systems are not discussed in this paper since these procedures are similar to the techniques used in rigid body dynamics. More details about these numerical procedures as well as the roots and perspectives of multibody system dynamics are discussed in a companion review by Schiehlen [79]. Future research areas in flexible multibody dynamics are identified as establishing the relationship between different formulations, contact and impact dynamics, control-structure interaction, use of modal identification and experimental methods in flexible multibody simulations, application of flexible multibody techniques to computer graphics, numerical issues, and large deformation problem. Establishing the relationship between different flexible multibody formulations is an important issue since there is a need to clearly define the assumptions and approximations underlying each formulation. This will allow us to establish guidelines and criteria that define the limitations of each approach used in flexible multibody dynamics. This task can now be accomplished by using the absolute nodal coordinate formulation which was recently introduced for the large deformation analysis of flexible multibody systems.     

A Deformation Field for Euler–Bernoulli Beams with Applications to Flexible Multibody Dynamics

The deformation field commonly used for Euler–Bernoulli beamsin structural dynamics is investigated to determine its suitability foruse in flexible multibody dynamics. It is found that the traditionaldeformation field fails to produce an elastic rotation matrix that iscomplete to second-order in the deformation variables. A completesecond-order deformation field is proposed along with the equationsneeded to incorporate the beam model into a graph-theoretic formulationfor flexible multibody dynamics [1]. This beam modeland formulation have been implemented in a symbolic computer programcalled DynaFlex that can use Taylor, Chebyshev, or Legendrepolynomials as the basis functions in a Rayleigh–Ritz discretizationof the beam's deformation variables. To demonstrate the effects of the proposed second-order deformationfield on the response of a flexible multibody system,two examples are presented.     

Analysis of Large Flexible Body Deformation in Multibody Systems Using Absolute Coordinates

To consider large deformation problems in multibody system simulations afinite element approach, called absolute nodal coordinate.formulation,has been proposed. In this formulation absolute nodal coordinates andtheir material derivatives are applied to represent both deformation andrigid body motion. The choice of nodal variables allows a fullynonlinear representation of rigid body motion and can provide the exactrigid body inertia in the case of large rotations. The methodology isespecially suited for but not limited to modeling of beams, cables andshells in multibody dynamics.This paper summarizes the absolute nodal coordinate formulation for a 3D Euler–Bernoulli beam model, in particular the definition of nodal variables, corresponding generalized elastic and inertia forces and equations of motion. The element stiffness matrix is a nonlinear function of the nodal variables even in the case of linearized strain/displacement relations. Nonlinear strain/displacement relations can be calculated from the global displacements using quadrature formulae.Computational examples are given which demonstrate the capabilities of the applied methodology. Consequences of the choice of shape.functions on the representation of internal forces are discussed. Linearized strain/displacement modeling is compared to the nonlinear approach and significant advantages of the latter, when using the absolute nodal coordinate formulation, are outlined.     

A linearized input–output representation of flexible multibody systems for control synthesis

In this paper, a linearized input–output representation of flexible multibody systems is proposed in which an arbitrary combination of positions, velocities, accelerations, and forces can be taken as input variables and as output variables. The formulation is based on a nonlinear finite element approach in which a multibody system is modeled as an assembly of rigid body elements interconnected by joint elements such as flexible hinges and beams. The proposed formulation is general in nature and can be applied for prototype modeling and control system analysis of mechatronic systems. Application of the theory is illustrated through a detailed model development of an active vibration isolation system for a metrology frame of a lithography machine.     

On the influence of model reduction techniques in topology optimization of flexible multibody systems using the floating frame of reference approach

Employing the floating frame of reference formulation in the topology optimization of dynamically loaded components of flexible multibody systems seems to be a natural choice. In this formulation the deformation of flexible bodies is approximated by global shape functions, which are commonly obtained from finite element models using model reduction techniques. For topology optimization these finite element models can be parameterized using the solid isotropic material with penalization (SIMP) approach. However, little is known about the interplay of model reduction and SIMP parameterization. Also securing the model reduction quality despite major changes of the design during the optimization has not been addressed yet. Thus, using the examples of a flexible frame and a slider-crank mechanism this work discusses the proper choice of the model reduction technique in the topology optimization of flexible multibody systems.     

Experimental validation of rigid-flexible coupling dynamic formulation for hub–beam system

The aim of this paper is to compare the accuracy of the absolute nodal coordinate formulation and the floating frame of reference formulation for the rigid-flexible coupling dynamics of a three-dimensional Euler–Bernoulli beam by numerical and experimental validation. In the absolute nodal coordinate formulation, based on geometrically exact beam theory and considering the torsion effect, the material curvature of the beam is derived, and then variational equations of motion of a three-dimensional beam are obtained, which consist of three position coordinates, two slope coordinates, and one rotational coordinate. In the floating frame of reference formulation, the displacement of an arbitrary point on the beam is described by the rigid-body motion and a small superimposed deformation displacement. Based on linear elastic theory, the quadratic terms of the axial strain are neglected, and the curvatures are simplified to the first order. Considering both the linear damping and the quadratic air resistance damping, the equations of motion of the multibody system composed of air-bearing test bed and a cantilevered three-dimensional beam are derived based on the principle of virtual work. In order to verify the results of the computer simulation, two experiments are carried out: an experiment of hub–beam system with large deformation and a dynamic stiffening experiment. The comparison of the simulation and experiment results shows that in case of large deformation, the frequency result obtained by the floating frame of reference formulation is lower than that obtained by the experiment. On the contrary, the result obtained by the absolute nodal coordinate formulation agrees well with that obtained by the experiment. It is also shown that the floating frame of reference formulation based on linear elastic theory cannot reveal the dynamic stiffening effect. Finally, the applicability of the floating frame of reference formulation is clarified.     

柔性梁线接触碰撞的动力学建模和实验研究

研究作旋转运动的柔性梁的线接触正碰撞问题.基于Goldsmith的线接触撞击力模型,分别用基于小变形的混合坐标法和基于大变形的绝对坐标法建立了柔性梁的动力学方程,考虑了几何非线性效应.在此基础上,进一步考虑非线性阻尼项的影响,将Hunt,Crossley的阻尼模型推广到线接触问题.介绍了柔性梁线接触碰撞的实验方法.计算结果显示,在考虑阻尼的情况下,计算结果与实验结果吻合很好.比较了混合坐标法和绝对坐标法的撞击力计算结果,与实验结果对比表明,绝对坐标法更适用于大变形的撞击问题.     

Surface Smoothing Procedures in Computational Contact Mechanics

This work addresses the problems arising in the finite element simulation of contact problems undergoing large deformation. The frictional contact problem is formulated in the continuum framework, introducing the interface laws for the normal and tangential stress components in the contact area. The variational formulation is presented, considering different methods to enforce the contact constraints. The spatial discretization within the finite element method is applied, as well as the temporal discretization required to solve the three sources of nonlinearities: geometric, material and frictional contact. The discretization of contact surfaces is discussed in detail, including different surface smoothing procedures. This numerical strategy allows to solve the difficulties associated with the discontinuities in the contact surface geometry introduced by finite element discretization, which leads to nonphysical oscillations of the contact force for large sliding problems. The geometrical accuracy of different interpolation methods is evaluated, paying particular attention to the Nagata patch interpolation recently proposed. In this framework, the Node-to-Nagata contact elements are developed using the augmented Lagrangian method to regularize the variational frictional contact problem. The techniques used to search for contact in case of large deformations are discussed, including self-contact phenomena. Several numerical examples are presented, comprising both the contact between deformable and rigid obstacles and the contact between deformable bodies. The results show that the accuracy and robustness of the numerical simulations is improved when the contact surface is smoothed with Nagata patches.     

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.