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

2003 ◽  
Author(s):  
Riki Iwai ◽  
Nobuyuki Kobayashi
Keyword(s):  
High Efficiency ◽  
Absolute Nodal Coordinate Formulation ◽  
Mass Matrix ◽  
Synthesis Method ◽  
Beam Element ◽  
Component Mode Synthesis ◽  
Flexible Beam ◽  
Absolute Nodal Coordinate ◽  
New Type ◽  
The Absolute

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.

Improvement on Evaluating Axial Elastic Force in Bernoulli-Euler Beam Based on the Absolute Nodal Coordinate Formulation by Accurate Mean Axial Strain Measure

2011 ◽  
Author(s):  
Tsubasa Wago ◽  
Nobuyuki Kobayashi ◽  
Yoshiki Sugawara
Keyword(s):  
Absolute Nodal Coordinate Formulation ◽  
Axial Strain ◽  
Computing Time ◽  
Beam Element ◽  
Elastic Force ◽  
Flexible Beam ◽  
Euler Beam ◽  
Absolute Nodal Coordinate ◽  
The Mean ◽  
The Absolute

This paper presents an improved formulation of axial elastic force in three-dimensional Bernoulli-Euler beam element based on the absolute nodal coordinate formulation. An accurate measure of mean axial strain for evaluating the axial elastic forces characterizes the presented formulation. The presented formulation evaluates the mean axial strain accurately by calculating the length of deformed beam element along its neutral axis. A comparison of the conventional formulations of the axial elastic force and the presented formulation is performed in some numerical examples which contain large bending deformation of flexible beam. As a result, it is verified that the presented formulation can express large deformation accurately with smaller number of elements than the conventional formulation which calculates the mean axial strain with straight-line distance between both element nodes. Moreover, it is also verified that the presented formulation can avoid excessive increase in computing time to simulate the dynamic behavior of flexible beam.

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

1999 ◽  
Author(s):  
R. Y. Yakoub ◽  
A. A. Shabana
Keyword(s):  
Absolute Nodal Coordinate Formulation ◽  
Sparse Matrix ◽  
Mass Matrix ◽  
Numerical Approach ◽  
Absolute Nodal Coordinates ◽  
Absolute Nodal Coordinate ◽  
Velocity Transformation ◽  
Flexible Multibody ◽  
Nodal Coordinates ◽  
The Absolute

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.

Performance of Curved Beam Elements Using the Absolute Nodal Coordinate Formulation

2009 ◽  
Author(s):  
Hiroyuki Sugiyama ◽  
Hirohisa Koyama ◽  
Hiroki Yamashita
Keyword(s):  
Absolute Nodal Coordinate Formulation ◽  
Beam Element ◽  
Solution Procedure ◽  
Large Displacement ◽  
Curved Beam ◽  
Absolute Nodal Coordinate ◽  
Curved Beam Element ◽  
Strain Components ◽  
The Absolute ◽  
Longitudinal Stretch

In this investigation, a gradient deficient beam element of the absolute nodal coordinate formulation is generalized to a curved beam for the analysis of multibody systems and the performance of the proposed element is discussed by comparing with the fully parameterized curved beam element and the classical large displacement beam element with incremental solution procedures. Strain components are defined with respect to the initially curved configuration and described by the arc-length coordinate. The Green strain is used for the longitudinal stretch, while the material measure of curvature is used for bending. It is shown that strains of the curved beam can be expressed with respect to those defined in the element coordinate system using the gradient transformation and the effect of strains at the initially curved configuration is eliminated using one-dimensional Almansi strain. This property can be effectively used with non-incremental solution procedure employed for the absolute nodal coordinate formulation. Several numerical examples are presented in order to demonstrate the performance of the gradient deficient curved beam element developed in this investigation. It is shown that the use of the proposed element leads to better element convergence as compared to that of the fully parameterized element and the classical large displacement beam element with incremental solution procedures.

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

2013 ◽  
Vol 9 (2) ◽  
Author(s):  
Alexander Olshevskiy ◽  
Oleg Dmitrochenko ◽  
Chang-Wan Kim
Keyword(s):  
Finite Element ◽  
Finite Elements ◽  
Absolute Nodal Coordinate Formulation ◽  
Mass Matrix ◽  
Equations Of Motion ◽  
Test Problems ◽  
Absolute Nodal Coordinate ◽  
Highly Nonlinear ◽  
Brick Element ◽  
The Absolute

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.

A Procedure for the Inclusion of Transverse Shear Deformation in a Beam Element Based on the Absolute Nodal Coordinate Formulation

2007 ◽  
Author(s):  
Aki Mikkola ◽  
Oleg Dmitrochenko ◽  
Marko Matikainen
Keyword(s):  
Shear Deformation ◽  
Transverse Shear ◽  
Absolute Nodal Coordinate Formulation ◽  
Beam Element ◽  
Transverse Shear Deformation ◽  
High Frequencies ◽  
Absolute Nodal Coordinate ◽  
Numerical Performance ◽  
The Absolute ◽  
Shear Deformable

In this study, a procedure to account for transverse shear deformation in the absolute nodal coordinate formulation is presented. In the absolute nodal coordinate formulation, shear deformation is usually defined by employing the slope vectors in the element transverse direction. This leads to the description of deformation modes that are, in practical problems, associated with high frequencies. These high frequencies, in turn, complicate the time integration procedure burdening numerical performance. In this study, the description of transverse shear deformation is accounted for in a two-dimensional beam element based on the absolute nodal coordinate formulation without the use of transverse slope vectors. In the introduced shear deformable beam element, slope vectors are replaced by vectors that describe the rotation of the beam cross-section. This procedure represents a simple enhancement that does not decrease the accuracy or numerical performance of elements based on the absolute nodal coordinate formulation. Numerical results are presented in order to demonstrate the accuracy of the introduced element in static and dynamic cases. The numerical results obtained using the introduced element agree with the results obtained using previously proposed shear deformable beam elements.

Shear Correction for Thin Plate Finite Elements Based on the Absolute Nodal Coordinate Formulation

2009 ◽  
Author(s):  
Oleg Dmitrochenko ◽  
Aki Mikkola
Keyword(s):  
Shear Deformation ◽  
Absolute Nodal Coordinate Formulation ◽  
Beam Element ◽  
Transverse Shear Deformation ◽  
High Frequencies ◽  
Absolute Nodal Coordinate ◽  
Plate Elements ◽  
Numerical Performance ◽  
The Absolute ◽  
Shear Deformable

This study is an extension of a newly introduced approach to account transverse shear deformation in absolute nodal coordinate formulation. In the formulation, shear deformation is usually defined by employing slope vectors in the element transverse direction. This leads to the description of deformation modes that, in practical problems, may be associated with high frequencies. These high frequencies, in turn, could complicate the time integration procedure, burdening numerical performance of shear deformable elements. In a recent study of this paper’s authors, the description of transverse shear deformation is accounted for in a two-dimensional beam element, based on the absolute nodal coordinate formulation without the use of transverse slope vectors. In the introduced shear deformable beam element, slope vectors are replaced by vectors that describe the rotation of the beam cross-section. This procedure represents a simple enhancement that does not decrease the accuracy or numerical performance of elements based on the absolute nodal coordinate formulation. In this study, the approach to account for shear deformation without using transverse slopes is implemented for a thin rectangular plate element. In fact, two new plate elements are introduced: one within conventional finite element and another using the absolute nodal coordinates. Numerical results are presented in order to demonstrate the accuracy of the introduced plate element. The numerical results obtained using the introduced element agree with the results obtained using previously proposed shear deformable plate elements.

Analysis of Thin Plate Structures Using the Absolute Nodal Coordinate Formulation

2005 ◽  
Vol 219 (4) ◽  
pp. 345-355 ◽  
Author(s):  
K Dufva ◽  
A A Shabana
Keyword(s):  
Finite Element ◽  
Thin Plate ◽  
Absolute Nodal Coordinate Formulation ◽  
Mass Matrix ◽  
Plate Element ◽  
Shell Elements ◽  
Absolute Nodal Coordinate ◽  
Plate And Shell ◽  
Spatial Parameters ◽  
The Absolute

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

1999 ◽  
Author(s):  
Yoshitaka Takahashi ◽  
Nobuyuki Shimizu
Keyword(s):  
Absolute Nodal Coordinate Formulation ◽  
Mass Matrix ◽  
Equations Of Motion ◽  
Nonlinear Function ◽  
Absolute Nodal Coordinates ◽  
Large Rotation ◽  
Inertia Effects ◽  
Absolute Nodal Coordinate ◽  
Elastic Forces ◽  
The Absolute

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.

A new higher-order locking-free beam element based on the absolute nodal coordinate formulation

2017 ◽  
Vol 232 (19) ◽  
pp. 3410-3423 ◽  
Author(s):  
Haidong Yu ◽  
Chunzhang Zhao ◽  
Bin Zheng ◽  
Hao Wang
Keyword(s):  
Large Deformation ◽  
Cross Section ◽  
Absolute Nodal Coordinate Formulation ◽  
Beam Element ◽  
Higher Order ◽  
Continuity Condition ◽  
Large Rotation ◽  
Absolute Nodal Coordinate ◽  
The Absolute ◽  
The Cross

The beam elements based on the absolute nodal coordinate formulation are widely used in large deformation and large rotation problems. Some of them lead to shear and Poisson locking problems when the continuum mechanics method is employed to deduce the generalized elastic force of the element. To circumvent these locking problems, a new higher-order beam element is proposed that may capture the warping and non-uniform stretching distribution of the cross-section by introducing the trapezoidal cross-section deformation mode and increasing the order of interpolation polynomials in transverse direction. The curvature vectors are chosen as the nodal coordinates of the new element that improve the continuity condition at the element interface. Static and dynamic analyses are conducted to investigate the performance of the new element. Poisson locking phenomena may be eliminated effectively for the new element even when Poisson’s ratio is greater than zero. Meanwhile, the distortion deformation of the cross-section may be described directly. The new element has a better convergence performance compared with the spatial absolute nodal coordinate formulation beam element for that shear locking issue is eliminated. The results also show that the new element fulfills energy conservation and may be applied to the dynamics of both straight and initial curved structures with large deformation.

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