A Corotational Approach for 3D Absolute Nodal Coordinate Elements

Practical Applications ◽

Elastic Forces ◽

Linearized Model ◽

Small Deformations

The present paper deals with the modeling of three dimensional (3D), nonlinear beam finite elements which do not employ rotational parameters. The beam elements are based on the so-called absolute nodal coordinate formulation (ANCF), which has been introduced in the late 1990’s. Early implementations of 3D ANCF beam finite elements incorporated problems regarding Poisson, thickness and shear locking. In the present paper, two alternative models for the work of elastic forces are presented. The first approach, which is intended to provide reference solutions, is close to the original approach. However, the effect of Poisson and thickness locking is eliminated by proper integration of the contributing terms in the virtual work of elastic forces. In the second approach, a corotationally linearized model is developed, which is based on a simple formulation for the elastic forces. The latter model only takes into account small deformations with respect to a corotating reference frame, but it is different from the conventional floating frame of reference formulation, because it has a constant mass matrix. The second approach is intended to be advantageous in practical applications where only small deformations with respect to large rigid body motions need to be taken into account, such as in robotics or machine dynamics. Numerical results are presented, which show that the new approaches agree with the solution of static and dynamic problems using classical finite elements or analytical methods.

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.

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.

Free Undamped Vibration of Geometrically Non-Linear Cable Structures

ASME 1997 Turbo Asia Conference ◽

10.1115/97-aa-095 ◽

1997 ◽

Author(s):

Alan S. K. Kwan

Keyword(s):

Finite Elements ◽

Mass Matrix ◽

Three Dimensional ◽

Geometrically Nonlinear ◽

Theoretical Derivation ◽

Cable Structures ◽

Non Linear ◽

Membrane Models ◽

High Degree ◽

Axial Element

The stiffness relationship and the distributed mass matrix for a geometrically nonlinear three dimensional straight axial element is derived for use in prestressed cablenet structures. The justification for the use of a linearised stiffness relationship is provided through a theoretical derivation. Results using this simple element have shown a high degree of correlation with results to those available in the literature obtained with more complex curved finite elements, analogous membrane models and other techniques.

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.

Volume 6: 5th International Conference on Multibody Systems, Nonlinear Dynamics, and Control, Parts A, B, and C ◽

Improved Description of Elastic Forces for the Absolute Nodal Coordinate Based Plate Element

10.1115/detc2005-84592 ◽

2005 ◽

Author(s):

Marko K. Matikainen ◽

Aki M. Mikkola

Keyword(s):

Finite Element ◽

Continuum Mechanics ◽

Three Dimensional ◽

Plate Element ◽

Finite Element Software ◽

Elastic Forces ◽

The Absolute ◽

The Continuum

In this study, the improved description of elastic forces for the absolute nodal coordinate based plate element is introduced. The absolute nodal coordinate formulation, which utilizes global displacements and slope coordinates as nodal variables, can be used in large rotation and deformation dynamic analysis of beam and plate structures. The formulation avoids difficulties that arise when a rotation is interpolated in three-dimensional applications. In the absolute nodal coordinate formulation, a continuum mechanics approach has become the dominating procedure when elastic forces are defined. It has recently been perceived, however, that the continuum mechanics based absolute nodal coordinate elements suffer from serious shortcomings, including Poisson’s locking and poor convergence rate. These problems can be circumvented by modifying the displacement field of a finite element in the definition of elastic forces. This allows the use of the mixed type interpolation technique, leading to accurate and efficient finite element formulations. This approach has been previously applied to two- and three-dimensional absolute nodal coordinate based finite elements. In this study, the improved approach for elastic forces is extended to the absolute nodal coordinate plate element. The introduced plate element is compared in static examples to the continuum mechanics based absolute nodal coordinate plate element, as well as to commercial finite element software.

Mean-Axis Conditions for Finite Elements Undergoing Large Rotations and Translations

Volume 1A: 16th Biennial Conference on Mechanical Vibration and Noise ◽

10.1115/detc97/vib-4188 ◽

1997 ◽

Author(s):

Venkata R. Sonti ◽

Om P. Agrawal

Keyword(s):

Finite Elements ◽

Mass Matrix ◽

Three Dimensional ◽

The Body ◽

Large Rotation ◽

Mass Matrices ◽

Large Rotations ◽

The Individual ◽

Rectangular Plate Element ◽

Individual Mass

Abstract The mean-axis conditions for three types of finite elements that undergo large rotation and translation are derived. The cases include two Lagrangian planar elements (linear triangle and linear rectangle) and a three dimensional rectangular plate element. Each case includes two elements connected at the nodes. Three separate coordinate systems are used in describing the systems. The numerical values of different matrices including the mass matrix are presented for specific cases. It may be seen that the non-linear mass matrix is expressed in terms of a set of time-invariand matrices. The total mass matrix of the body i is obtained by assembling the individual mass matrices of the finite elements of body i.

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.

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

Studies on the Stiffness Properties of the Absolute Nodal Coordinate Formulation for Three-Dimensional Beams

10.1115/detc2003/vib-48325 ◽

2003 ◽

Cited By ~ 3

Author(s):

Jussi T. Sopanen ◽

Aki M. Mikkola

Keyword(s):

Cross Section ◽

Three Dimensional ◽

Beam Element ◽

Element Formulation ◽

Numerical Examples ◽

Elastic Forces ◽

Nodal Coordinates ◽

The objective of this study is to investigate the accuracy of elastic force models that can be used in the absolute nodal coordinate finite element formulation for the analysis of threedimensional beams. The elastic forces of the absolute nodal coordinate formulation can be derived using a continuum mechanics approach. This study investigates the accuracy and usability of such an approach for the three-dimensional absolute nodal coordinate beam element. This study also presents an improvement proposal for the use of a continuum mechanics approach in deriving the expression of the elastic forces of the beam element. The improvement proposal is verified using several numerical examples. Numerical examples show that the proposed elastic force model of the beam element agrees with analytical results as well as with solutions obtained using existing finite element formulation. The results also imply that the beam element does not suffer from the phenomenon called shear locking. In the beam element under investigation, global displacements and slopes are used as the nodal coordinates, which resulted in a large number of nodal degrees of freedom. This study provides a physical interpretation of the nodal coordinates used in the absolute nodal coordinate beam element. It is shown that the beam element based on the absolute nodal coordinate formulation relaxes the assumption of the rigid cross-section and is capable of representing a distortional deformation of the cross-section.

Development of Elastic Forces for a Large Deformation Plate Element Based on the Absolute Nodal Coordinate Formulation

Journal of Computational and Nonlinear Dynamics ◽

10.1115/1.1961870 ◽

2005 ◽

Vol 1 (2) ◽

pp. 103-108 ◽

Cited By ~ 34

Author(s):

Aki M. Mikkola ◽

Marko K. Matikainen

Keyword(s):

Finite Element ◽

Dynamic Analysis ◽

Continuum Mechanics ◽

Three Dimensional ◽

Plate Element ◽

Elastic Forces ◽

The Absolute ◽

The Continuum

Dynamic analysis of large rotation and deformation can be carried out using the absolute nodal coordinate formulation. This formulation, which utilizes global displacements and slope coordinates as nodal variables, make it possible to avoid the difficulties that arise when a rotation is interpolated in three-dimensional applications. In the absolute nodal coordinate formulation, a continuum mechanics approach has become the dominating procedure when elastic forces are defined. It has recently been perceived, however, that the continuum mechanics based absolute nodal coordinate elements suffer from serious shortcomings, including Poisson’s locking and poor convergence rate. These problems can be circumvented by modifying the displacement field of a finite element in the definition of elastic forces. This allows the use of the mixed type interpolation technique, leading to accurate and efficient finite element formulations. This approach has been previously applied to two- and three-dimensional absolute nodal coordinate based finite elements. In this study, the improved approach for elastic forces is extended to the absolute nodal coordinate plate element. The introduced plate element is compared in static examples to the continuum mechanics based absolute nodal coordinate plate element, as well as to commercial finite element software. A simple dynamic analysis is performed using the introduced element in order to demonstrate the capability of the element to conserve energy.

Nonlinear Dynamics of Rigid and Flexible Multibody Systems

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

10.1115/detc99/vib-8248 ◽

1999 ◽

Author(s):

Evtim V. Zakhariev

Keyword(s):

Finite Elements ◽

Multibody Systems ◽

Virtual Work ◽

Space Station ◽

Three Dimensional ◽

Numerical Procedure ◽

Numerical Approach ◽

Dynamic Equations ◽

Dynamics Modeling ◽

Lagrange Equations

Abstract In the present paper a unified numerical approach for dynamics modeling of multibody systems with rigid and flexible bodies is suggested. The dynamic equations are second order ordinary differential equations (without constraints) with respect to a minimal set of generalized coordinates that describe the parameters of gross relative motion of the adjacent bodies and their small elastic deformations. The numerical procedure consists of the following stages: structural decomposition of elastic links into fictitious rigid points and/or bodies connected by joints in which small force dependent relative displacements are achieved; kinematic analysis; deriving explicit form dynamic equations. The algorithm is developed in case of elastic slender beams and finite elements achieving spatial motion with three translations and three rotations of nodes. The beam elements are basic design units in many mechanical devices as space station antennae and manipulators, cranes and etc. doing three dimensional motion which large elastic deflections could not be neglected or linearised. The stiffness coefficients and inertia mass parameters of the fictitious joints and links are calculated using the numerical procedures of the finite element theory. The method is called finite elements in relative coordinates. Its equivalence with the procedures of recently developed finite segment approaches is shown, while in the treatment different results are obtained. The approach is used for solution of some nonlinear static problems and for deriving the explicit configuration space dynamic equations of spatial flexible system using the principle of virtual work and Euler-Lagrange equations.