A New Plate Element Based on the Absolute Nodal Coordinate Formulation

Mass Matrix ◽

Body Motion ◽

Deformation Analysis ◽

Plate Element ◽

Large Rotation ◽

Shell Elements ◽

Plates And Shells ◽

Abstract In this investigation, a method for the finite rotation and large deformation analysis of plates is presented. The method, which is based on the absolute nodal coordinate formulation, leads to a plate element capable of representing exact rigid body motion. In this method, continuity conditions on all the displacement gradients are imposed. Therefore, non-smoothness of the plate mid-surface at the nodal points is avoided. By developing such a plate element, a constant mass matrix is obtained, and as a consequence, the centrifugal and Coriolis forces are equal to zero. Generalization of the formulation to the case of shell elements is discussed. Numerical results are presented in order to demonstrate the use of the proposed method in the large rotation and deformation analysis of plates and shells.

Proceedings of the Institution of Mechanical Engineers Part K Journal of Multi-body Dynamics ◽

Analysis of Thin Plate Structures Using the Absolute Nodal Coordinate Formulation

10.1243/146441905x50678 ◽

2005 ◽

Vol 219 (4) ◽

pp. 345-355 ◽

Cited By ~ 21

Author(s):

K Dufva ◽

A A Shabana

Keyword(s):

Finite Element ◽

Thin Plate ◽

Mass Matrix ◽

Plate Element ◽

Shell Elements ◽

Plate And Shell ◽

Spatial Parameters ◽

The absolute nodal coordinate formulation can be used in multibody system applications where the rotation and deformation within the finite element are large and where there is a need to account for geometrical non-linearities. In this formulation, the gradients of the global positions are used as nodal coordinates and no rotations are interpolated over the finite element. For thin plate and shell elements, the plane stress conditions can be applied and only gradients obtained by differentiation with respect to the element mid-surface spatial parameters need to be defined. This automatically reduces the number of element degrees of freedoms, eliminates the high frequencies due to the oscillations of some gradient components along the element thickness, and as a result makes the plate element computationally more efficient. In this paper, the performance of a thin plate element based on the absolute nodal coordinate formulation is investigated. The lower dimension plate element used in this investigation allows for an arbitrary rigid body displacement and large deformation within the element. The element leads to a constant mass matrix and zero Coriolis and centrifugal forces. The performance of the element is compared with other plate elements previously developed using the absolute nodal coordinate formulation. It is shown that the finite element used in this investigation is much more efficient when compared with previously proposed elements in the case of thin structures. Numerical examples are presented in order to demonstrate the use of the formulation developed in this paper and the computational advantages gained from using the thin plate element. The thin plate element examined in this study can be efficiently used in many applications including modelling of paper materials, belt drives, rotor dynamics, and tyres.

Study on Elastic Forces of the Absolute Nodal Coordinate Formulation for Deformable Beams

Volume 7A: 17th Biennial Conference on Mechanical Vibration and Noise ◽

10.1115/detc99/vib-8203 ◽

1999 ◽

Author(s):

Yoshitaka Takahashi ◽

Nobuyuki Shimizu

Keyword(s):

Mass Matrix ◽

Equations Of Motion ◽

Nonlinear Function ◽

Absolute Nodal Coordinates ◽

Large Rotation ◽

Inertia Effects ◽

Elastic Forces ◽

Abstract There are three basic finite element formulations which are used in the dynamics of flexible beams. These are the floating frame of reference approach, the finite segment method and the large rotation vector approach. Recently, the absolute nodal coordinate formulation was proposed by A.A.Shabana et al. In this procedure, there is no need to transform the element matrices since the equations of motion are defined in terms of absolute nodal coordinates. The mass matrix becomes constant, whereas the stiffness matrix becomes nonlinear function of time, even in case of linear elastic problems. One possible method to avoid such cumbersome of the absolute nodal coordinate formulation in calculating clastic forces is to assume the infinitesimal deformation theory against beams undergoing large rotation. In this paper, a new formulation to calculate the elastic forces and add the rotary inertia effects in the expression of the inertia forces. This formulation is based on the assumption that the deformations within each element remain very small. The expression of the resulting clastic force is simple, and the need for performing coordinate transformation is avoided. As the method assumes that the deformation of the beam from a selected beam axis is very small, a large number of finite elements is required for large deformation problems. However, the formulation has been found to be efficient for large rotation and medium deformation problems. Numerical examples are demonstrated for this formulation by using planar flexible pendulum problems.

Three Dimensional Absolute Nodal Coordinate Formulation for Beam Elements: Implementation and Applications

Journal of Mechanical Design ◽

10.1115/1.1410099 ◽

2000 ◽

Vol 123 (4) ◽

pp. 614-621 ◽

Cited By ~ 165

Author(s):

Refaat Y. Yakoub ◽

Ahmed A. Shabana

Keyword(s):

High Speed ◽

Mass Matrix ◽

Three Dimensional ◽

Large Rotation ◽

Incremental Methods ◽

Beam Elements ◽

High Speed Forming ◽

This part of these two companion papers demonstrates the computer implementation of the absolute nodal coordinate formulation for three-dimensional beam elements. Two beam elements that relax the assumptions of Euler-Bernoulli and Timoshenko beam theories are developed. These two elements take into account the effect of rotary inertia, shear deformation and torsion, and yet they lead to a constant mass matrix. As a consequence, the Coriolis and centrifugal forces are identically equal to zero. Both beam elements use the same interpolating polynomials and have the same number of nodal coordinates. However, one of the elements has two nodes, while the other has four nodes. The results obtained using the two elements are compared with the results obtained using existing incremental methods. Unlike existing large rotation vector formulations, the results of this paper show that no special numerical integration methods need to be used in order to satisfy the principle of work and energy when the absolute nodal coordinate formulation is used. These results show that this formulation can be used in manufacturing applications such as high speed forming and extrusion problems in which the element cross section dimensions significantly change.

Performance of the Incremental and Non-Incremental Finite Element Formulations in Flexible Multibody Problems

Journal of Mechanical Design ◽

10.1115/1.1289636 ◽

1999 ◽

Vol 122 (4) ◽

pp. 498-507 ◽

Cited By ~ 35

Author(s):

Marcello Campanelli ◽

Marcello Berzeri ◽

Ahmed A. Shabana

Keyword(s):

Finite Element ◽

Multibody Systems ◽

Flexible Multibody Systems ◽

Large Displacement ◽

Body Motion ◽

Flexible Multibody ◽

The Absolute ◽

Finite Element Formulations

Many flexible multibody applications are characterized by high inertia forces and motion discontinuities. Because of these characteristics, problems can be encountered when large displacement finite element formulations are used in the simulation of flexible multibody systems. In this investigation, the performance of two different large displacement finite element formulations in the analysis of flexible multibody systems is investigated. These are the incremental corotational procedure proposed in an earlier article (Rankin, C. C., and Brogan, F. A., 1986, ASME J. Pressure Vessel Technol., 108, pp. 165–174) and the non-incremental absolute nodal coordinate formulation recently proposed (Shabana, A. A., 1998, Dynamics of Multibody Systems, 2nd ed., Cambridge University Press, Cambridge). It is demonstrated in this investigation that the limitation resulting from the use of the infinitesmal nodal rotations in the incremental corotational procedure can lead to simulation problems even when simple flexible multibody applications are considered. The absolute nodal coordinate formulation, on the other hand, does not employ infinitesimal or finite rotation coordinates and leads to a constant mass matrix. Despite the fact that the absolute nodal coordinate formulation leads to a non-linear expression for the elastic forces, the results presented in this study, surprisingly, demonstrate that such a formulation is efficient in static problems as compared to the incremental corotational procedure. The excellent performance of the absolute nodal coordinate formulation in static and dynamic problems can be attributed to the fact that such a formulation does not employ rotations and leads to exact representation of the rigid body motion of the finite element. [S1050-0472(00)00604-8]

A Numerical Approach to Solving Flexible Multibody Systems Using the Absolute Nodal Coordinate Formulation

Volume 7A: 17th Biennial Conference on Mechanical Vibration and Noise ◽

10.1115/detc99/vib-8204 ◽

1999 ◽

Author(s):

R. Y. Yakoub ◽

A. A. Shabana

Keyword(s):

Sparse Matrix ◽

Mass Matrix ◽

Numerical Approach ◽

Absolute Nodal Coordinates ◽

Velocity Transformation ◽

Flexible Multibody ◽

Nodal Coordinates ◽

Abstract By utilizing the fact that the absolute nodal coordinate formulation leads to a constant mass matrix, a Cholesky decomposition of the mass matrix can be used to obtain a constant velocity transformation matrix. This velocity transformation can be used to express the absolute nodal coordinates in terms of the generalized Cholesky coordinates. In this case, the inertia matrix associated with the Cholesky coordinates is the identity matrix, and therefore, an optimum sparse matrix structure can be obtained for the augmented multibody equations of motions. The implementation of a computer procedure based on the absolute nodal coordinate formulation and Cholesky coordinates is discussed in this paper. A flexible four-bar linkage is presented in this paper in order to demonstrate the use of Cholesky coordinates in the simulation of the small and large deformations in flexible multibody applications. The results obtained from the absolute nodal coordinate formulation are compared to those obtained from the floating frame of reference formulation.

Three-Dimensional Solid Brick Element Using Slopes in the Absolute Nodal Coordinate Formulation

Journal of Computational and Nonlinear Dynamics ◽

10.1115/1.4024910 ◽

2013 ◽

Vol 9 (2) ◽

Cited By ~ 25

Author(s):

Alexander Olshevskiy ◽

Oleg Dmitrochenko ◽

Chang-Wan Kim

Keyword(s):

Finite Element ◽

Finite Elements ◽

Mass Matrix ◽

Equations Of Motion ◽

Test Problems ◽

Highly Nonlinear ◽

Brick Element ◽

The present paper contributes to the field of flexible multibody systems dynamics. Two new solid finite elements employing the absolute nodal coordinate formulation are presented. In this formulation, the equations of motion contain a constant mass matrix and a vector of generalized gravity forces, but the vector of elastic forces is highly nonlinear. The proposed solid eight node brick element with 96 degrees of freedom uses translations of nodes and finite slopes as sets of nodal coordinates. The displacement field is interpolated using incomplete cubic polynomials providing the absence of shear locking effect. The use of finite slopes describes the deformed shape of the finite element more exactly and, therefore, minimizes the number of finite elements required for accurate simulations. Accuracy and convergence of the finite element is demonstrated in nonlinear test problems of statics and dynamics.

Volume 5: 19th Biennial Conference on Mechanical Vibration and Noise, Parts A, B, and C ◽

A New Type Component Mode Synthesis Beam Element Based on the Absolute Nodal Coordinate Formulation

10.1115/detc2003/vib-48353 ◽

2003 ◽

Author(s):

Riki Iwai ◽

Nobuyuki Kobayashi

Keyword(s):

High Efficiency ◽

Mass Matrix ◽

Synthesis Method ◽

Beam Element ◽

Component Mode Synthesis ◽

Flexible Beam ◽

New Type ◽

This paper establishes a new type component mode synthesis method for a flexible beam element based on the absolute nodal coordinate formulation. The deformation of the beam element is defined as the sum of the global shape function and the analytical clamped-clamped beam modes. This formulation leads to a constant and symmetric mass matrix as the conventional absolute nodal coordinate formulation, and makes it possible to reduce the system coordinates of the beam structure which undergoes large rotations and large deformations. Numerical examples show that the excellent agreements are examined between the presented formulation and the conventional absolute nodal coordinate formulation. These results demonstrate that the presented formulation has high accuracy in the sense that the presented solutions are similar to the conventional ones with the less system coordinates and high efficiency in computation.

Usability of finite elements based on the absolute nodal coordinate formulation for deformation analysis of the Achilles tendon

International Journal of Non-Linear Mechanics ◽

10.1016/j.ijnonlinmec.2020.103662 ◽

2021 ◽

Vol 129 ◽

pp. 103662

Author(s):

Leonid Obrezkov ◽

Pernilla Eliasson ◽

Ajay B. Harish ◽

Marko K. Matikainen

Keyword(s):

Finite Elements ◽

Achilles Tendon ◽

Deformation Analysis ◽

Three Dimensional Absolute Nodal Coordinate Formulation for Beam Elements: Theory

Journal of Mechanical Design ◽

10.1115/1.1410100 ◽

2000 ◽

Vol 123 (4) ◽

pp. 606-613 ◽

Cited By ~ 243

Author(s):

Ahmed A. Shabana ◽

Refaat Y. Yakoub

Keyword(s):

Coordinate System ◽

Mass Matrix ◽

Local Coordinate System ◽

Three Dimensional ◽

Deformation Analysis ◽

Large Element ◽

Beam Elements ◽

Highly Nonlinear

The description of a beam element by only the displacement of its centerline leads to some difficulties in the representation of the torsion and shear effects. For instance such a representation does not capture the rotation of the beam as a rigid body about its own axis. This problem was circumvented in the literature by using a local coordinate system in the incremental finite element method or by using the multibody floating frame of reference formulation. The use of such a local element coordinate system leads to a highly nonlinear expression for the inertia forces as the result of the large element rotation. In this investigation, an absolute nodal coordinate formulation is presented for the large rotation and deformation analysis of three dimensional beam elements. This formulation leads to a constant mass matrix, and as a result, the vectors of the centrifugal and Coriolis forces are identically equal to zero. The formulation presented in this paper takes into account the effect of rotary inertia, torsion and shear, and ensures continuity of the slopes as well as the rotation of the beam cross section at the nodal points. Using the proposed formulation curved beams can be systematically modeled.

A Simple Absolute Nodal Coordinate Formulation for Thin Beams With Large Deformations and Large Rotations

Journal of Computational and Nonlinear Dynamics ◽

10.1115/1.4028610 ◽

2015 ◽

Vol 10 (6) ◽

Cited By ~ 11

Author(s):

Hui Ren

Keyword(s):

Cross Sections ◽

Mass Matrix ◽

Current Approach ◽

Time Step ◽

Elastic Line ◽

Large Rotation ◽

The Cross ◽

Strain Tensors

A simple but effective formulation of beams with large deformation and large rotation is derived from the principles of continuum mechanics. Proper assumptions are imposed, and the beam strain tensors are formulated from the Green strain tensors. The mass matrix is constant, and the elastic forces and the stiffness matrix entries are polynomials of the generalized coordinates, so numerical quadratures are not required in each time step of simulation, which makes the current approach much more superior in numerical efficiency than other formulations. The shape of the cross sections can be arbitrary, either uniform or nonuniform, and the beam can be either straight or curved. The generalized free of traction assumption ensures the strains in the cross section and the beam strains are independent, which resolves the Possion's locking issue and renders this approach can be accurately applied to general composite material beams. The elastic line approach (ELA) in the absolute nodal coordinate formulation (ANCF) can be derived from the current formulation.