A geometrically exact beam element based on the absolute nodal coordinate formulation

In this study, Reissner’s classical nonlinear rod formulation, as implemented by Simo and Vu-Quoc by means of the large rotation vector approach, is implemented into the framework of the absolute nodal coordinate formulation. The implementation is accomplished in the planar case accounting for coupled axial, bending, and shear deformation. By employing the virtual work of elastic forces similarly to Simo and Vu-Quoc in the absolute nodal coordinate formulation, the numerical results of the formulation are identical to those of the large rotation vector formulation. It is noteworthy, however, that the material definition in the absolute nodal coordinate formulation can differ from the material definition used in Reissner’s beam formulation. Based on an analytical eigenvalue analysis, it turns out that the high frequencies of cross section deformation modes in the absolute nodal coordinate formulation are only slightly higher than frequencies of common shear modes, which are present in the classical large rotation vector formulation of Simo and Vu-Quoc, as well. Thus, previous claims that the absolute nodal coordinate formulation is inefficient or would lead to ill-conditioned finite element matrices, as compared to classical approaches, could be refuted. In the introduced beam element, locking is prevented by means of reduced integration of certain parts of the elastic forces. Several classical large deformation static and dynamic examples as well as an eigenvalue analysis document the equivalence of classical nonlinear rod theories and the absolute nodal coordinate formulation for the case of appropriate material definitions. The results also agree highly with those computed in commercial finite element codes.     

Definition of the Elastic Forces in the Finite-Element Absolute Nodal Coordinate Formulation and the Floating Frame of Reference Formulation

The equivalence of the finite-element formulations used inflexible multibody dynamics is the focus of this investigation. Thisequivalence will be used to address several fundamental issues related tothe deformations, flexible body coordinate systems, and the geometriccentrifugal stiffening effect. Two conceptually different finite-elementformulations that lead to exact modeling of the rigid body dynamics will beused. The first one is the absolute nodal coordinateformulation in which beams and plates can be treated as isoparametricelements. This formulation leads to a constant and symmetric mass matrix andhighly nonlinear elastic forces. In this study, it is demonstrated thatdifferent element coordinate systems which are used for the convenience ofdescribing the element deformations lead to similar results as the elementsize is reduced. In particular, two element frames are used;the pinned and the tangent frames. The pinned frame has one ofits axes passing through two nodes of the element, while the tangent frame isrigidly attached to one of the ends of the element. Numerical resultsobtained using these two different frames are found tobe in good agreement as the element size decreases. The relationshipbetween the coordinates used in the absolute nodal coordinate formulationand the floating frame of reference formulation is presented. Thisrelationship can be used to obtain the highly nonlinear expression of thestrain energy used in the absolute nodal coordinate formulation from thesimple energy expression used in the floating frame of referenceformulation. It is also shown that the source of the nonlinearityis due to the finite rotation of the element. The result of the analysispresented clearly demonstrates that the instability observedin high-speed rotor analytical models due to the neglect of the geometriccentrifugal stiffening is not a problem inherent to a particular finite-element formulation. Such a problem can only be avoided by considering the known linear effect of the geometric centrifugal stiffening or by using a nonlinear elastic model as recently demonstrated. Fourier analysis of the solutions obtained in this investigation also sheds new light on the fundamental problem of the choice of the deformable body coordinate system in the floating frame of reference formulation. Another method forformulating the elastic forces in the absolute nodal coordinate formulationbased on a continuum mechanics approach is also presented.     

Reduction of system matrices of planar beam in ANCF by component mode synthesis method

A method of reducing the system matrices of a planar flexible beam described by an absolute nodal coordinate formulation (ANCF) is presented. In this method, we focus that the bending stiffness matrix expressed by adopting a continuum mechanics approach to the ANCF beam element is constant when the axial strain is not very large. This feature allows to apply the Craig–Bampton method to the equation of motion that is composed of the independent coordinates when the constraint forces are eliminated. Four numerical examples that compare the proposed method and the conventional ANCF are demonstrated to verify the performance and accuracy of the proposed method. From these examples, it is verified that the proposed method can describe the large deformation effects such as dynamic stiffening due to the centrifugal force, as well as the conventional ANCF does. The use of this method also reduces the computing time, while maintaining an acceptable degree of accuracy for the expression characteristics of the conventional ANCF when the modal truncation number is adequately employed. This reduction in CPU time particularly pronounced in the case of a large element number and small modal truncation number; the reduction can be verified not only in the case of small deformation but also in the case of a fair bit large deformation.     

Poisson modes and general nonlinear constitutive models in the large displacement analysis of beams

Most existing formulations for structural elements such as beams, plates and shells do not allow for the use of general nonlinear constitutive models in a straightforward manner. Furthermore, such structural element models, due to the nature of the generalized coordinates used, do not capture some Poisson modes such as the ones that couple the deformation of the cross section of the structural element and stretch and bending. In this paper, beam models that employ general nonlinear constitutive equations are presented using finite elements based on the nonlinear absolute nodal coordinate formulation. This formulation relaxes the assumptions of the Euler–Bernoulli and Timoshenko beam theories, and allows for the use of general nonlinear constitutive models. The finite elements based on the absolute nodal coordinate formulation also allow for the rotation as well as the deformation of the cross section, thereby capturing Poisson modes which can not be captured using other beam models. In this investigation, three different nonlinear constitutive models based on the hyper-elasticity theory are considered. These three models are based on the Neo–Hookean constitutive law for compressible materials, the Neo–Hookean constitutive law for incompressible materials, and the Mooney–Rivlin constitutive law in which the material is assumed to be incompressible. These models, which allow capturing Poisson modes, are suitable for many materials and applications, including rubber-like materials and biological tissues which are governed by nonlinear elastic behavior. Numerical examples that demonstrate the implementation of these nonlinear constitutive models in the absolute nodal coordinate formulation are presented. The results obtained using the nonlinear and linear constitutive models are compared in this study. These results show that the use of nonlinear constitutive models can significantly enhance the performance and improve the computational efficiency of the finite element models based on the absolute nodal coordinate formulation. The results also show that when linear constitutive models are used in the large deformation analysis, singular configurations are encountered and basic formulas such as Nanson’s formula are no longer valid. These singular deformation configurations are not encountered when the nonlinear constitutive models are used.     

Panel flutter analysis of plate element based on the absolute nodal coordinate formulation

In this study, aeroelastic analysis of a plate subjected to the external supersonic airflow is carried out. A 3-D rectangular plate element of variable thickness based on absolute nodal coordinate formulation (ANCF) has been developed for the structural model. In the approach to the problem, a continuum mechanics approach for the definition of the elastic forces within the finite element is considered. Both shear strain and transverse normal strain are taken into account. Linearized first-order potential (piston) theory is coupled with the structural model to account for pressure loading. Aeroelastic equations using ANCF are derived and solved numerically. Values of critical dynamic pressure are obtained by a modal approach, in which the mode shapes are obtained by ANCF. All the formulations and the computations are built up in a FORTRAN 90 computer program after it was confirmed by Mathematica^?, ver. 5. The results of free vibration analysis and flutter are compared with the available references and reasonable good agreement has been found. However, some results indicate that the known problem of locking (ANCF with uniform thickness) still persist in the current developed formulation.     

Study of the Centrifugal Stiffening Effect Using the Finite Element Absolute Nodal Coordinate Formulation

The finite element absolute nodal coordinate formulation is usedin this investigation to study the centrifugal stiffening effect onrotating two-dimensional beams. It is demonstrated that the geometricstiffening effect can be automatically accounted for in the above mentionedfinite element formulation by using an expression for the elastic forcesobtained with a general continuum mechanics approach. The Hill equation thatgoverns the vibration of the rotating beam is obtained in terms of a set ofgeneralized coordinates that describe the beam displacements and slopes.Under the assumption of small deformation, the Hill equation is linearized,and the complete solution is obtained and used to demonstrate analyticallythat such a solution does not exhibit instabilities as the angular velocityof the beam increases. The results obtained using this finite elementprocedure are compared with the results reported in the literature.     

Simulation of a viscoelastic flexible multibody system using absolute nodal coordinate and fractional derivative methods

This paper employs a new finite element formulation for dynamics analysis of a viscoelastic flexible multibody system. The viscoelastic constitutive equation used to describe the behavior of the system is a three-parameter fractional derivative model. Based on continuum mechanics, the three-parameter fractional derivative model is modified and the proposed new fractional derivative model can reduce to the widely used elastic constitutive model, which meets the continuum mechanics law strictly for pure elastic materials. The system equations of motion are derived based on the absolute nodal coordinate formulation (ANCF) and the principle of virtual work, which can relax the small deformation assumption in the traditional finite element implementation. In order to implement the viscoelastic model into the absolute nodal coordinate, the Grünwald definition of the fractional derivative is employed. Based on a comparison of the HHT-I3 method and the Newmark method, the HHT-I3 method is used to solve the equations of motion. Another particularity of the proposed method based on the ANCF method lies in the storage of displacement history only during the integration process, reducing the numerical computation considerably. Numerical examples are presented in order to analyze the effects of the truncation number of the Grünwald series (fading memory phenomena) and the value of several fractional model parameters and solution convergence aspects. An erratum to this article can be found at     

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.     

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.     

Comparison of different formulations of 2D beam elements based on Bond Graph technique

Bond Graphs are well suited for modelling multibody systems. In this paper modelling of planar flexible beams undergoing large overall motions are studied based on finite element (FE) technique. Two well-known approaches are used – the co-rotational (CR) and absolute nodal coordinate (ANC) formulation. Two ANC formulations are analyzed – one in which elastic forces is described using classical beam theory in a local coordinate frame, and another based on a global continuum mechanics approach. Starting from these classical formulations velocity formulations are developed and used to develop Bond Graph FE components. The effect of gravity has been considered as well. These components can be put in libraries and used for systematic Bond Graph flexible body model development. It is shown that Bond Graph technique is capable of dealing with different flexible body formulations and can be used as a general approach in parallel to other modelling approaches. Models are developed and simulations are performed using the object oriented environment of BondSim. Owing to the object oriented approach, transformation from one to the other model is relatively simply. The results are illustrated by suitable examples and they confirm accuracy of the developed models. It was shown that the CR approach offers much better performance than the both ANC formulations.     

Design of a planar multibody dynamic system with ANCF beam elements based on an element-wise stiffness evaluation procedure

The absolute nodal coordinate formulation (ANCF) has been widely applied for large deformation analysis in flexible multibody dynamics. Although the formulation led to stable solutions for time integration under large rotations and deformations, excessive time consumption was recorded. The nonlinear relationship between the deformation and the internal force accounted for repeated adjustment to the force equilibrium state as the structure deformed. In this research, an equivalent model of the ANCF beam structure was constructed. The stiffness evaluation method was applied in an element-wise manner. In this model, the irrelevant parts were separated from those that relate to the displacements and design parameters enabling efficient updates of internal forces to achieve force equilibrium. Therefore, by using this model, optimization problems, in which displacements as well as design parameters keep changing can be efficiently approached. To verify the proposed method, two examples of optimization problems related to a free-falling pendulum and a slider-crank mechanism are demonstrated.     

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.     

Slope discontinuities in the finite element absolute nodal coordinate formulation: gradient deficient elements

In this paper, the treatment of the slope discontinuities in the finite element absolute nodal coordinate formulation (ANCF) is discussed. The paper explains the fundamental problems associated with developing a constant transformation that accounts for the slope discontinuities in the case of gradient deficient ANCF finite elements. A procedure that allows for the treatment of slope discontinuities in the case of gradient deficient finite elements which do not employ full parameterization is proposed for the special case of commutative rotations. The use of the proposed procedure leads to a constant orthogonal element transformation that describes the element initial configuration. As a consequence, one obtains in the case of large deformation and commutative rotations, a constant mass matrix for the structures. In order to achieve this goal, the concept of the intermediate finite element coordinate system is invoked. The intermediate finite element coordinate system used in this investigation serves to define the element reference configuration, follows the rotation of the structure, and maintains a fixed orientation relative to the structure coordinate system. Since planar rotations are always commutative, the procedure proposed in this investigation is applicable to all planar gradient deficient ANCF finite elements.     

Large Deflection Analysis of a Thin Plate: Computer Simulations and Experiments

Many previous studies have conducted computer-aided simulations ofelastic bodies undergoing large deflections and deformations, but therehave not been many attempts to validate their numerical results. Thesubject of this paper is a thin clamped plate undergone large vibrationdue to attached end-point weight. The main aim of this paper is to showthe validity of the absolute nodal coordinate formulation (ANCF) bycomparing to the real experiments. Large oscillations of thin plates arestudied in the paper with taking into account effects of an attachedend-point weight and aerodynamic damping forces. The physicalexperiments are carried out using a high-speed camera and dataacquisition system. For numerical modeling of the plate, the absolutenodal coordinate formulation is used.     

基于拉格朗日描述的罐车内液体晃动模拟

基于拉格朗日描述的柔性多体系统动力学理论,采用绝对节点坐标有限元方法描述液体大变形运动,开展铁路液罐车内液体晃动模拟研究.本方法能够模拟液体自由表面的连续性变化,并适用于研究具有复杂外形容器的内部液体晃动问题.基于流体力学牛顿体基础理论,推导液体粘性方程和满足体积不可压缩的条件方程;采用基于绝对节点坐标方法描述的实体单元进行液体网格划分;采用罚函数方法描述液体与罐体之间的接触关系,组建液体-罐体耦合多体系统动力学方程.仿真计算液罐车内液体的横向和纵向晃动行为,发现液体自由表面形状呈非线性变化,不同断面处的高度和形状不同.     

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.