Figure - available from: Nonlinear Dynamics
This content is subject to copyright. Terms and conditions apply.
![Relative errors obtained in the numerical calculation of the flapwise displacement in the Princeton beam experiment with an angle of α=30∘\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha =30^{\circ }$$\end{document} and a force of F3pb\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$F_3^{\text {pb}}$$\end{document}. For the calculation of the relative error, the result obtained with the largest number of elements depicted in Table 6 was used as a reference value for the respective elements. Hence, since only relative errors are shown, it should be noted that the results of element 3333c converge against an incorrect value](publication/311957858/figure/fig2/AS:941803696046115@1601554907952/Relative-errors-obtained-in-the-numerical-calculation-of-the-flapwise-displacement-in-the.gif)
Relative errors obtained in the numerical calculation of the flapwise displacement in the Princeton beam experiment with an angle of α=30∘\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha =30^{\circ }$$\end{document} and a force of F3pb\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$F_3^{\text {pb}}$$\end{document}. For the calculation of the relative error, the result obtained with the largest number of elements depicted in Table 6 was used as a reference value for the respective elements. Hence, since only relative errors are shown, it should be noted that the results of element 3333c converge against an incorrect value
Source publication
This study thoroughly examines various higher-order three and four-node beam elements for use in the absolute nodal coordinate formulation (ANCF). The paper carefully investigates which potential benefits and drawbacks the utilization of higher-order ANCF beam elements without in-slope vectors has in the case of the usage of full three-dimensional...
Similar publications
This paper presents an impact-angle-control guidance law with terminal constraints on the curvature of the missile trajectory. The formulation takes into account nonlinear kinematics and time-varying velocity, allowing for more general cases in which the flight path angle may not be small throughout the entire trajectory. The proposed optimal guida...
It has long been a goal to efficiently compute and use second order information on a function ($f$) to assist in numerical approximations. Here it is shown how, using only basic physics and a numerical approximation, such information can be accurately obtained at a cost of ${\cal O}(N)$ complexity, where $N$ is the dimensionality of the parameter s...
The mechanical generation of curves is a classic topic in kine-matics. Although the use of linkages for solving math problems through curve tracing is nowadays obsolete, the approximation of curves employing mechanisms maintains an industrial interest. The generalized Burmester points concept, discovered by Freudenstein in 1963, provides the theore...
Curves of constant width, which have a very special place in many fields such as kinematics, engineering, art, cam design and geometry, are specially discussed under this title. In this study, a system of differential equations characterizing the curves of constant width is examined. This is the system of the first order homogenous differential equ...
The Stretch-Fold-Shear (SFS) operator Sα is a functional linear operator acting on complex-valued functions of a real variable x on some domain containing [−1,1] in R. It arises from a stylized model in kinematic dynamo theory where magnetic field growth corresponds to an eigenvalue of modulus greater than 1. When the shear parameter α is zero, the...
Citations
... Ren [15] proposed a three-node-shearing beam element and assumed that the transverse stress is zero, which effectively alleviated the shear locking and Poisson locking problems. Ebel et al. [16] used a variety of polynomial bases to construct higher-order ANCF beam elements, which can describe the deformation mode of more refined elastic sections and alleviate the locking phenomenon. Yu et al. [17] proposed a high-order ANCF element using trapezoidal deformation mode and lateral high-order interpolation for a field, which can effectively alleviate the shear locking and Poisson locking problems, and can describe the section warpage and nonuniform tensile distribution. ...
A new 12-DOF beam element is proposed to simulate large deformation and large rotation based on the 24-DOF ANCF beam element proposed before. The centerline of the beam is interpolated by Hermite shape functions, and the frame of the beam is interpolated by linear shape functions. To reduce DOFs, the Lie-group method is used to normalize and orthogonalize the frame on each node of the beam. This way of using the Lie-group method keeps a linear relationship between the nodal vectors and shape functions and leads to the constant mass matrix and stiffness matrix. Therefore, the generalized elastic and inertial forces do not require Gaussian integration at each time step. To avoid singularity of the rotation, a relative rotation vector is adopted, correspondingly, the generalized-a integrator based on the Lie group is used to solve the dynamic equations. To improve the convergency speed and alleviate the shear locking and Poisson locking problems of this element, the assumed natural strain method (ANS) is adopted. To improve the calculational accuracy of axis stretching and torsion effects, the enhanced assumed strain method is adopted. The formulas presented in this paper have been successfully tested in several static and dynamic examples of other ANCF beam elements and analytic solutions.
... However, the focus of this paper is on the reduction of computational complexity in dynamic equations to enhance computational efficiency, without employing methods to address the locking problem. Nevertheless, the locking issues can be effectively alleviated using the approaches outlined in the works of Nachbagauer et al. [68], Bauchau et al. [69], Gerstmayr et al. [70], Ebel et al. [71], and Patel and Shabana [72]. ...
The viscoelastic dynamic model of flexible multibody coupled with large rotation and deformation can be described by the Absolute Nodal Co-ordinate Formulation (ANCF). However, with the increase of degrees of freedom, the computational cost of viscoelastic multibody systems will be very high. In addition, for non-proportionally viscoelastic flexible multibody systems, the orthogonality and superposition of complex modes only exist in the state space. In this investigation, a systematical procedure of model reduction method for viscoelastic flexible multibody systems de-scribed by ANCF is proposed based on the complex modal synthesis method. Firstly, the whole motion process of the system is divided into a se-ries of quasi-static equilibrium configurations. Then the dynamic equation is locally linearized based on the Talor expansion to obtain the constant tangent stiffness matrix and damping matrix. The initial modes and modal coordinates need to be updated for each subinterval. A modal selection criterion based on the energy judgment is proposed to ensure the energy conservation and accuracy by the minimum number of truncations. Finally, three numerical examples are carried out as verification. Simulation results indicate that the method proposed procedure reduces the system scale and improves the computational efficiency under the premise of ensuring the simulation accuracy.
... С учетом (19) получим ...
... Рис. 1. Неструктурированные шестигранные сетки для модели пружины и пневматического актуатора (в том числе в разрезе) (а), эталонная шестигранная для задачи деформирования куба (б), перемещения вдоль осей Ozи Oy для теста Принстон (в, г) Пример 2. Перейдем к тесту, осуществленному в Принстонском университете, США (далее тест Принстон) [10,19] о деформировании прямоугольной балки длиной 0.508 м вдоль оси Ox, шириной 3.2024×10 −3 м вдоль Oy и высотой 12.377×10 −3 м вдоль Oz. Параметры линейно-упругого материала E = 2.07 × 10 11 Па, ν = 0.3. ...
... На рисунке 1в представлены результаты перемещения конца балки вдоль оси Oz, где присутствует наибольшая жесткость конструкции на изгиб и, как следствие, наиболее явно проявляются эффекты сдвигового запирания. При расчетной шестигранной сетке, содержащей 700 слоев ячеек вдоль оси Ox (где каждый слой состоит из 8 ячеек, равномерно распределенных в плоскости Oyz), расхождение полученных МКЭ в терминах ANCF значений u 1 z с данными u [19] z , представленными в статье [19] (на основе ячеек типа 3843 и других элементов более высокого порядка), не превышает 1.5%. Для сетки, состоящей из 300 слоев, расхождение значений u 2 z с данными u [19] z не превышает 12%, при этом для 150 слоев максимальное отличие в результатах u 3 z достигает 40%. ...
We consider a finite element approach to solving the problem of elasticity theory in terms of absolute nodal coordinate formulation (ANCF), in which large body displacements are described in a global reference frame without using any local coordinate system. The main feature of the method is the absence of gyroscopic effects and, as a result, constancy of the mass matrix and the vector of the generalized force of gravity. In contrast to the traditional ANCF approach, the sets of nodal degrees of freedom of the finite element are formed only on the basis of absolute coordinates of nodes, which allows us to solve the problem using, in particular, unstructured hexahedral meshes. To construct the stiffness matrix, we apply a second-order automatic differentiation algorithm, which ensures its symmetrical form (Hessian matrix) and is analytically accurate in calculating the derivative. This approach also makes it possible to carry out calculations for models of hyperelastic materials without the corresponding Piola-Kirchhoff tensor. It has been shown that within the framework of discretization of the equation of motion, along with the well-known Newmark numerical integration scheme, it is possible to use the HTT-α scheme, which is unconditionally stable, second-order accurate and dissipative for high frequencies. We present several examples of solving static and dynamic elasticity problems for compressible and incompressible models of hyperelastic materials, in which the functions of internal energy density of the body are specified in terms of the deformation gradient.
... Compared with the eight-node solid element (or the twenty-node hexahedron element), the ANCF beam elements can be computationally more efficient (Obrezkov et al., 2021). In addition, with the cross-sectional higher-order polynomial (Matikainen et al., 2014;Shen et al., 2014;Ebel et al., 2017;Orzechowski and Shabana, 2016) or warping function (Tang et al., 2022) interpolations on the cross sections, the in-plane and out-of-plane warping deformations can be obtained computational effectively while needing fine cross-sectional mesh via dozens of solid elements on the section. ...
... Among various locking alleviation methods compared in Patel and Shabana (2018) and Obrezkov et al. (2022b), the enhanced continuum mechanics generally showed the best results via many benchmarks but limited the material models that can be employed to those that satisfy the elastic tensor split rule (Obrezkov et al., 2022b), such as the Saint Venant-Kirchhoff model. However, the enhanced continuum mechanics cannot fully cure the locking for some ANCF beam elements, as shown in the so-called Princeton beam experiment (Ebel et al., 2017;Tang et al., 2022;Obrezkov et al., 2022b). It is important to note that the enhanced continuum mechanics and strain split approaches over-predicted the deformation of a pre-curved cantilever beam in Patel and Shabana (2018). ...
... By increasing the order of the polynomial expansion in the transverse directions, significant improvements in both convergence rate and accuracy were observed. This enhancement applies to both beam elements and plate elements, as evidenced by studies such as Shen et al. (2014), Ebel et al. (2017), Orzechowski and Shabana (2016), Patel and Shabana (2018) and Matikainen et al. (2014), just name a few. These higher-order beam elements can also capture the displacement deviation away from the original planar plane, i.e., the cross-sectional warping deformation (Shen et al., 2014;Orzechowski and Shabana, 2016). ...
... This is due to the fact that slopes instead of infinitesimal rotations are used as nodal coordinates. In regard to the development of the ANCF beam elements, generally there are two main research directions, i.e., the gradient deficient elements [26][27][28] and the fully parameterized elements [29][30][31][32][33][34][35]. The former is parameterized as a centerline and therefore in its nodal coordinates that has only the gradients of the position vector with respect to the beam axis. ...
... Nonetheless, for the transverse low-order beam elements (see [29][30][31]) the cross-sectional deformation modes are very simple and will always be flat [37], which are similar to the Timoshenko beam model. But by the transverse higher-order beam elements (see [32][33][34][35]), the cross-sectional warping and distortion with more complex modes can be captured, where that proposed by the authors of [33] has been applied to some challenging nonlinear dynamics problems [38][39][40][41][42]. ...
A sufficiently accurate prediction of mechanics behavior for thin-walled structures has always received much attention in the development of beam theory. This paper proposes a novel modeling approach for thin-walled beams in the framework of the absolute nodal coordinate formulation, in which cross-sectional warping and distortion effects can be explicitly captured by using the Taylor-like polynomials as the description of the kinematics of beam cross-section. Meanwhile, the linear Lagrange interpolation and the cubic Hermite interpolation are respectively adopted for the off-axis vectors of the position coordinates, where it is found that the thin-walled beam elements constructed from the latter have a faster convergence rate. Elastic forces and the Jacobian of the thin-walled composite beams are derived on the basis of nonlinear continuum mechanics, which will therefore contain a strong coupling between various deformations. Furthermore, their efficient calculation scheme is presented by separating the axial and transverse variables of shape functions and by expressing the vector and matrix quantities in terms of their components. By comparison to finite shell element, the accuracy and characteristic features of the new elements are illustrated.
... Типичными примерами являются лента транспортера, бурильная колонна в криволинейной скважине, шина автомобиля. Примеры стержневых, пластинчатых и оболочечных конечных элементов в формализме ANCF приведены в работах [3][4][5][6]. ...
A method for modeling the dynamics of flexible bodies allowing an arbitrary gross motion, large elastic displacements and small deformations is considered. Equations of motion are derived in absolute nodal coordinates by finite element method using the known Craig-Bampton approach. The method is intended for modeling elastic solids, shells and rods included in a multibody system. Examples of practical use in modeling the dynamics of a car tire, air spring, mechanical springs are given.
... The other issue is that the adaptive ANCF curved beam element doesn't have elemental shape functions with the same interpolation orders of each direction, which will result in Poisson locking and aggravate the coupling degree of axial and transverse deformation when calculating transient response problems. [32][33][34][35][36] So the rotating REF model has pseudo strain and low computational efficiency when calculating nonlinear response. ...
Radial tires are widely used in transportation machinery, such as large aircraft, new energy vehicles, and engineering vehicles. The ring on the elastic foundation (REF) model is often used for dynamic analysis of radial tires, but the influence of elastic boundary on the dynamic characteristics of radial tires is ignored in this model. In this paper, based on the absolute nodal coordinate formulation (ANCF) and plane stress theory, a rotating thick ring on the elastic foundation (RTREF) model is established to explore the influence of elastic boundary position and other factors on dynamic characteristics. Because existing rectangular and triangular elements in ANCF can’t accurately describe radial/circumferential mesh of a thick ring, a novel ANCF-annular-sector element (AASE) is proposed for the RTREF model. By analyzing the influences of elastic boundary position, thick wall, and rotational speed on the vibration characteristics and nonlinear forced transient response of the RTREF model, it is demonstrated that even if the ring wall is thin, the influence of elastic boundary positions on the dynamic characteristics of the model can’t be ignored.
... Moreover, there are many other advantages using elements with higher-order interpolate functions, which can represent the complex physics boundary, and describe the strain field with large gradient. Besides, the higher-order continuity of the interpolation function also allows to introduce higher-order gradients, which is an effective way to alleviate the Poisson locking problem of the ANCF elements [5,15]. ...
... An efficient ANCF element with high precision is essential to the dynamic calculation, so the research on new ANCF beam elements and shell elements attracts considerable attention. Plenty of beam and shell elements based on ANCF have been proposed [5,9,11]. These elements have been applied to many challenging engineering fields such as vehicles [7,16], deployable space structure [4,10], and multi-filed coupling [3,19]. ...
The computational efficiency has been largely hindering the application of the absolute nodal coordinate formulation (ANCF). To improve the computational efficiency of the absolute nodal coordinate formulation, the collocation strategy is employed to establish the ANCF beam element. A novel ANCF beam element based on multi-node collocation (ANCF_C) is proposed. Zero points of the second-order derivative of pth-order Legendre polynomial and the boundary points in the element domain are used as nodes to discretize the beam node-wisely. Accordingly, the th-order Lagrange interpolation is employed for the longitudinal displacement interpolation. The elastic force and stiffness matrix are deduced based on the enhanced continuum mechanics formulation (ECMF) to avoid the locking problem. By using p-point Lobatto quadrature for the numerical integration of the elastic force and the mass matrix, the quadrature points coincide with the discretized nodes in the element. The performance of the ANCF_C is verified by both static and dynamic examples.
... Here, the higher-order three-nodded element with the second-order interpolation in longitudinal and thickness directions denoted 3363 is used. It does not require any modifications to demonstrate good performance even for complicated loading cases [26] and allows the use of all material laws based on the 3D elasticity. ...
... As mentioned above, in this work, we consider 3363 beam elements [26]. The vectors of nodal coordinates related to this element are presented as follows: ...
Experimental results have revealed the sophisticated Achilles tendon (AT) structure, including its material properties and complex geometry. The latter incorporates a twisted design and composite construction consisting of three subtendons. Each of them has a nonstandard cross-section. All these factors make the AT deformation analysis computationally demanding. Generally, 3D finite solid elements are used to develop models for AT because they can discretize almost any shape, providing reliable results. However, they also require dense discretization in all three dimensions, leading to a high computational cost. One way to reduce degrees of freedom is the utilization of finite beam elements, requiring only line discretization over the length of subtendons. However, using the material models known from continuum mechanics is challenging because these elements do not usually have 3D elasticity in their descriptions. Furthermore, the contact is defined at the beam axis instead of using a more general surface-to-surface formulation. This work studies the continuum beam elements based on the absolute nodal coordinate formulation (ANCF) for AT modeling. ANCF beam elements require discretization only in one direction, making the model less computationally expensive. Recent work demonstrates that these elements can describe various cross-sections and materials models, thus allowing the approximation of AT complexity. In this study, the tendon model is reproduced by the ANCF continuum beam elements using the isotropic incompressible model to present material features.
... It has eight nodes with a full set of three position vector gradients at each node for a total of 96 degrees of freedom per element. While other more accurate ANCF elements exist [4,5], these particular elements were selected since they provide consistent steps of increasing complexity for the comparisons. ...
For multibody dynamics simulation using the Absolute Nodal Coordinate Formulation, multiple strategies are reported in the linear elastic material literature for calculating the generalized internal force and its Jacobian matrix. When examining the presentation of these strategies, which are all sound, it is difficult to assess which method is more efficient. We seek to clarify this issue by reporting the results of a comprehensive study that included five different ANCF solution strategies discussed in the literature. To increase the relevance of the study, we first extended these methods to incorporate a linear viscoelastic material model to account for damping effects within the elements. A beam, a shell, and a hexahedral element are each examined to provide a broader comparison. Both simple hand calculations and actual timing comparisons on a multi-core CPU architecture are investigated. For the simple beam element, only small differences manifest among the methods studied. However, for the shell and hexahedral elements, we noticed pronounced performance and storage cost differences among the methods.
