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     

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.     

A study of moderately thick quadrilateral plate elements based on the absolute nodal coordinate formulation

Finite element analysis using plate elements based on the absolute nodal coordinate formulation (ANCF) can predict the behaviors of moderately thick plates subject to large deformation. However, the formulation is subject to numerical locking, which compromises results. This study was designed to investigate and develop techniques to prevent or mitigate numerical locking phenomena. Three different ANCF plate element types were examined. The first is the original fully parameterized quadrilateral ANCF plate element. The second is an update to this element that linearly interpolates transverse shear strains to overcome slow convergence due to transverse shear locking. Finally, the third is based on a new higher order ANCF plate element that is being introduced here. The higher order plate element makes it possible to describe a higher than first-order transverse displacement field to prevent Poisson thickness locking. The term “higher order” is used, because some nodal coordinates of the new plate element are defined by higher order derivatives. The performance of each plate element type was tested by (1) solving a comprehensive set of small deformation static problems, (2) carrying out eigenfrequency analyses, and (3) analyzing a typical dynamic scenario. The numerical calculations were made using MATLAB. The results of the static and eigenfrequency analyses were benchmarked to reference solutions provided by the commercially available finite element software ANSYS. The results show that shear locking is strongly dependent on material thickness. Poisson thickness locking is independent of thickness, but strongly depends on the Poisson effect. Poisson thickness locking becomes a problem for both of the fully parameterized element types implemented with full 3-D elasticity. Their converged results differ by about 18 % from the ANSYS results. Corresponding results for the new higher order ANCF plate element agree with the benchmark. ANCF plate elements can describe the trapezoidal mode; therefore, they do not suffer from Poisson locking, a reported problem for fully parameterized ANCF beam elements. For cases with shear deformation loading, shear locking slows solution convergence for models based on either the original fully parameterized plate element or the newly introduced higher order plate element.     

Effect of the centrifugal forces on the finite element eigenvalue solution of a rotating blade: a comparative study

In this study, the effect of the centrifugal forces on the eigenvalue solution obtained using two different nonlinear finite element formulations is examined. Both formulations can correctly describe arbitrary rigid body displacements and can be used in the large deformation analysis. The first formulation is based on the geometrically exact beam theory, which assumes that the cross section does not deform in its own plane and remains plane after deformation. The second formulation, the absolute nodal coordinate formulation (ANCF), relaxes this assumption and introduces modes that couple the deformation of the cross section and the axial and bending deformations. In the absolute nodal coordinate formulation, four different models are developed; a beam model based on a general continuum mechanics approach, a beam model based on an elastic line approach, a beam model based on an elastic line approach combined with the Hellinger–Reissner principle, and a plate model based on a general continuum mechanics approach. The use of the general continuum mechanics approach leads to a model that includes the ANCF coupled deformation modes. Because of these modes, the continuum mechanics model differs from the models based on the elastic line approach. In both the geometrically exact beam and the absolute nodal coordinate formulations, the centrifugal forces are formulated in terms of the element nodal coordinates. The effect of the centrifugal forces on the flap and lag modes of the rotating beam is examined, and the results obtained using the two formulations are compared for different values of the beam angular velocity. The numerical comparative study presented in this investigation shows that when the effect of some ANCF coupled deformation modes is neglected, the eigenvalue solutions obtained using the geometrically exact beam and the absolute nodal coordinate formulations are in a good agreement. The results also show that as the effect of the centrifugal forces, which tend to increase the beam stiffness, increases, the effect of the ANCF coupled deformation modes on the computed eigenvalues becomes less significant. It is shown in this paper that when the effect of the Poisson ration is neglected, the eigenvalue solution obtained using the absolute nodal coordinate formulation based on a general continuum mechanics approach is in a good agreement with the solution obtained using the geometrically exact beam model.     

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.     

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.     

Finite element formulation for dynamics of planar flexible multi-beam system

In some previous geometric nonlinear finite element formulations, due to the use of axial displacement, the contribution of all the elements lying between the reference node of zero axial displacement and the element to the foreshortening effect should be taken into account. In this paper, a finite element formulation is proposed based on geometric nonlinear elastic theory and finite element technique. The coupling deformation terms of an arbitrary point only relate to the nodal coordinates of the element at which the point is located. Based on Hamilton principle, dynamic equations of elastic beams undergoing large overall motions are derived. To investigate the effect of coupling deformation terms on system dynamic characters and reduce the dynamic equations, a complete dynamic model and three reduced models of hub-beam are prospected. When the Cartesian deformation coordinates are adopted, the results indicate that the terms related to the coupling deformation in the inertia forces of dynamic equations have small effect on system dynamic behavior and may be neglected, whereas the terms related to coupling deformation in the elastic forces are important for system dynamic behavior and should be considered in dynamic equation. Numerical examples of the rotating beam and flexible beam system are carried out to demonstrate the accuracy and validity of this dynamic model. Furthermore, it is shown that a small number of finite elements are needed to obtain a stable solution using the present coupling finite element formulation.     

A general three-dimensional L-section beam finite element for elastoplastic large deformation analysis

In this paper, the development of a general three-dimensional L-section beam finite element for elastoplastic large deformation analysis is presented. We propose the generalized interpolation scheme for the isoparametric formulation of three-dimensional beam finite elements and the numerical procedure is developed for elastoplastic large deformation analysis. The formulation is general and effective for other thin-walled section beam finite elements. To show the validity of the formulation proposed, a 2-node three-dimensional L-section beam finite element is implemented in an analysis code. As numerical examples, we first perform elastic small and large deformation analyses of a cantilever beam structure subjected to various tip loadings, and elastoplastic large deformation analysis of the same structure under reversed cyclic tip loading. We then analyze the failures of simply supported beam structures of different lengths and slenderness ratios under elastoplastic large deformation. The same problems are solved using refined shell finite element models of the structures. The numerical results of the L-section beam finite element developed here are compared with the solutions obtained using shell finite element analyses. We also discuss the numerical solutions in detail.     

中心刚体-柔性梁刚柔耦合动力学模型降阶研究

有限单元法被广泛的采用来描述柔性体的弹性变形,然而有限元节点坐标数目庞大,将会给动力学方程求解带来巨大的计算负担.如何降低柔性体的自由度,是当前柔性多体系统动力学研究的一个重要命题.本文以中心刚体-柔性梁系统为例,采用Krylov方法和模态方法进行降价.然后分别采用有限元全模型、Krylov降阶模型和模态降阶模型,对中心刚体-柔性梁进行刚-柔耦合动力学仿真.仿真结果表明,与采用模态降阶方法相比,采用Krylov模型降阶方法只需要较低的自由度,就可以得到与采用有限元方法完全一致的结果.说明Krylov模型降阶方法能够有效的用于柔性多体系统的模型降价研究.     

Curve-induced distortion of polynomial space curves,flat-mapped extension modeling,and their impact on ANCF thin-plate finite elements

The Absolute Nodal Coordinate Formulation (ANCF) is a relatively new nonlinear finite element type that uses Hermite splines for shape functions. In this investigation, the ANCF is examined as a possible tool for use in modeling the media in flexible media transport systems, such as printers, copy machines, and roll-to-roll systems. However, it is demonstrated using an example of a thin plate-type ANCF finite element that these elements can suffer from significant membrane locking, which can be problematic for paper or paper-like media. One source of this locking is identified to be a property of all parametric curves that are composed of polynomials. The property is that for parametric polynomial curves, changes in the state of curvature of the curve cause changes in the distribution of points along the curve. This property is labeled Curve-Induced Distortion (CID) by the authors of this paper. CID can cause axial and membrane strain distortion in elements, causing them to be overly stiff. A new solution method is proposed to directly counteract CID in finite elements that use cubic Hermite curves for shape functions, specifically for modeling problems in which bending occurs primarily around one axis, such as paper in printing and media transport machinery. This method is labeled Flat-Mapped Extension Modeling (FMEM). FMEM is a mixed field method that uses a 1D Hermite polynomial kinematically linked to the 3D Hermite curve to represent the axial displacement field. FMEM significantly reduces the effect of CID in the ANCF element tested here. This investigation demonstrates using a single ANCF plate element type that the ANCF’s accuracy can be significantly improved by FMEM with only a small increase in computational cost. It is shown with this plate-element example that without correcting CID, the ANCF element tested is computationally much slower than contemporary methods like the co-rotational formulation for similar accuracy. But with FMEM, the ANCF is significantly faster than the co-rotational formulation for similar accuracy.     

A formal procedure and invariants of a transition from conventional finite elements to the absolute nodal coordinate formulation

In this paper, finite elements based on the absolute nodal coordinate formulation (ANCF) are studied. The formulation has been developed by various authors for the dynamical simulation of large-displacement and large-rotation problems in flexible multibody dynamics. This study introduces a procedure to track the general geometrical properties of ANCF elements back to their prototypes in the conventional finite-element method (FEM), which deals with small-displacement problems. In this study, it is shown that each known ANCF element can be derived from a conventional FEM using a universal transform. Moreover, some important static and dynamic properties of the elements in small-displacement problems are automatically preserved. In the past, the authors of each newly proposed ANCF element have made unnecessary efforts to show the consistency of the above mentioned properties.     

Developments of multibody system dynamics: computer simulations and experiments

It is an exceptional success when multibody dynamics researchers Multibody System Dynamics journal one of the most highly ranked journals in the last 10 years. In the inaugural issue, Professor Schiehlen wrote an interesting article explaining the roots and perspectives of multibody system dynamics. Professor Shabana also wrote an interesting article to review developments in flexible multibody dynamics. The application possibilities of multibody system dynamics have grown wider and deeper, with many application examples being introduced with multibody techniques in the past 10 years. In this paper, the development of multibody dynamics is briefly reviewed and several applications of multibody dynamics are described according to the author’s research results. Simulation examples are compared to physical experiments, which show reasonableness and accuracy of the multibody formulation applied to real problems. Computer simulations using the absolute nodal coordinate formulation (ANCF) were also compared to physical experiments; therefore, the validity of ANCF for large-displacement and large-deformation problems was shown. Physical experiments for large deformation problems include beam, plate, chain, and strip. Other research topics currently being carried out in the author’s laboratory are also briefly explained. Commemorative Contribution.     

Fast update of elastic body simulation under impact load by efficient reduction method

A large number of eigenmodes are required for the realistic simulation of the object under arbitrary dynamic load such as impact load while only a few lower modes are adequate for the simulation of free vibration with initial deformation. For the accurate dynamic simulation under impact, a large amount of computer resources are required to obtain considerable number of eigenpairs. The present study presents an efficient dynamic simulation algorithm and simulation results of elastically deformable body under impact. We employ a modal analysis technique that pre-computes the eigenvalues and the corresponding eigenvectors in the framework of finite element method. For low latency time of real-time simulation, eigenmodes are computed from a reduced system constructed from the original one. To construct reduced system, energy estimation technique in the element level is proposed in the present study. Through a number of examples, it is demonstrated that the proposed method saves computational cost effectively and provides reliable real time results of the deformable solids under impact.     

A nonlinear visco-elastic constitutive model for large rotation finite element formulations

Although all known materials have internal damping that leads to energy dissipation, most existing large deformation visco-elastic finite element formulations are based on linear constitutive models or on nonlinear constitutive models that can be used in the framework of an incremental co-rotational finite element solution procedure. In this investigation, a new nonlinear objective visco-elastic constitutive model that can be implemented in non-incremental large rotation and large deformation finite element formulations is developed. This new model is based on developing a simple linear relationship between the damping forces and the rates of deformation vector gradients. The deformation vector gradients can be defined using the decomposition of the matrix of position vector gradients. In this paper, the decomposition associated with the use of the tangent frame that is equivalent to the QR decomposition is employed to define the matrix of deformation gradients that enter into the formulation of the viso-elastic constitutive model developed in this investigation. Using the relationship between the deformation gradients and the components of the Green–Lagrange strain tensor, it is shown that the damping forces depend nonlinearly on the strains and linearly on the classical strain rates. The relationship between the damping forces and strains and their rates is used to develop a new visco-elastic model that satisfies the objectivity requirements and leads to zero strain rates under an arbitrary rigid body displacement. The linear visco-elastic Kelvin–Voigt model frequently used in the literature can be obtained as a special case of the proposed nonlinear model when only two visco-elastic coefficients are used. As demonstrated in this paper, the use of two visco-elastic coefficients only leads to viscous coupling between the deformation gradients. The model developed in this investigation can be used in the framework of large deformation and large rotation non-incremental solution procedure without the need for using existing co-rotational finite element formulations. The finite element absolute nodal coordinate formulation (ANCF) that allows for straightforward implementation of general constitutive material models is used in the validation of the proposed visco-elastic model. A comparison with the linear visco-elastic model is also made in this study. The results obtained in this investigation show that there is a good agreement between the solutions obtained using the proposed nonlinear model and the linear model in the case of small deformations.     

Riccati discrete time transfer matrix method for elastic beam undergoing large overall motion

An efficient method for dynamics simulation for elastic beam with large overall spatial motion and nonlinear deformation, namely, the Riccati discrete time transfer matrix method (Riccati-DT-TMM), is proposed in this investigation. With finite segments, continuous deformation field of a beam can be decomposed into many rigid bodies connected by rotational springs. Discrete time transfer matrices of rigid bodies and rotational springs are used to analyze the dynamic characteristic of the beam, and the Riccati transform is used to improve the numerical stability of discrete time transfer matrix method of multibody system dynamics. A predictor-corrector method is used to improve the numerical accuracy of the Riccati-DT-TMM. Using the Riccati-DT-TMM in dynamics analysis, the global dynamics equations of the system are not needed and the computation time required increases linearly with the system’s number of degrees of freedom. Three numerical examples are given to validate the method for the dynamic simulation of a geometric nonlinear beam undergoing large overall motion.     

一种基于Delaunay背景图改进的网格变形方法

针对Delaunay网格变形方法中因计算网格点在背景网格中映射不够精细导致的大变形失效问题,提出了一种改进后的网格变形方法,旨在进一步提高大变形情形时变形后的网格质量。该方法将原始的Delaunay网格变形方法中的背景网格远场边界进行加密,增加了映射背景网格单元的数量,改善了计算网格在背景网格中的映射精细程度,从而提升了变形后网格的质量。通过一正方形网格变形基础算例和30P30N三段翼型流场网格变形算例分别进行了测试与验证,结果表明该改进方法可以在保证计算效率的前提下,显著提升大变形时变形后的网格质量。与原始的Delaunay网格变形方法相比较,改进后的网格变形方法变形能力较强,所生成网格质量较高。     

Nonlocal strain gradient finite element analysis of nanobeams using two-variable trigonometric shear deformation theory

In the present paper, a new trigonometric two-variable shear deformation beam nonlocal strain gradient theory is developed and applied to investigate the combined effects of nonlocal stress and strain gradient on the bending, buckling and free vibration analysis of nanobeams. The model introduces a nonlocal stress field parameter and a length scale parameter to capture the size effect. The governing equations derived are solved employing finite element method using a 3-nodes beam element, developed for this purpose. The predictive capability of the proposed model is shown through illustrative examples for bending, buckling and free vibration of nanobeams. Comparisons with other higher-order shear deformation beam theory are also performed to validate its numerical implementation and assess its accuracy within the nonlocal context.