General Method for Modeling Slope Discontinuities and T-Sections Using ANCF Gradient Deficient Finite Elements

2010 ◽  
Vol 6 (2) ◽  
Author(s):  
Ahmed A. Shabana
Keyword(s):  
Finite Elements ◽  
Absolute Nodal Coordinate Formulation ◽  
Mass Matrix ◽  
Constant Mass ◽  
Centrifugal Forces ◽  
Linear Transformations ◽  
Complete Set ◽  
Coordinate Lines ◽  
General Method ◽  
Slope Discontinuities

Slope discontinuities and T-sections can be modeled in a straight forward manner using fully parameterized absolute nodal coordinate formulation (ANCF) finite elements that have a complete set of gradient vectors. Linear transformations that define the element connectivity can always be obtained and used to preserve ANCF desirable features that include constant mass matrix and zero Coriolis and centrifugal forces in the case of spinning structures. The objective of this paper is to develop a general method that allows for modeling slope discontinuities and T-sections using gradient deficient ANCF finite elements that do not have a complete set of coordinate lines and gradient vectors. Linear connectivity conditions that preserve all the ANCF desirable features including the constant mass matrix are developed at the nodes of slope discontinuities. At these nodes of discontinuity, one can always define a complete set of independent coordinate lines that lie on the structure. These coordinate lines can be used to define a complete set of independent gradient vectors at these nodes. Since the proposed method is based on linear coordinate transformations, the method can be implemented in a preprocessor computer program. The application of the proposed general method is demonstrated using ANCF gradient deficient beam element example.

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.

Use of the Finite Element Absolute Nodal Coordinate Formulation in Modeling Slope Discontinuity

2003 ◽  
Vol 125 (2) ◽  
pp. 342-350 ◽  
Author(s):  
Ahmed A. Shabana ◽  
Aki M. Mikkola
Keyword(s):  
Finite Element ◽  
Rigid Body ◽  
Coordinate System ◽  
Absolute Nodal Coordinate Formulation ◽  
Mass Matrix ◽  
Global Coordinate System ◽  
Absolute Nodal Coordinate ◽  
Coriolis Forces ◽  
The Absolute ◽  
Slope Discontinuities

A large rigid body rotation of a finite element can be described by rotating the axes of the element coordinate system or by keeping the axes unchanged and change the slopes or the position vector gradients. In the first method, the definition of the local element parameters (spatial coordinates) changes with respect to a body or a global coordinate system. The use of this method will always lead to a nonlinear mass matrix and non-zero centrifugal and Coriolis forces. The second method, in which the axes of the element coordinate system do not rotate with respect to the body or the global coordinate system, leads to a constant mass matrix and zero centrifugal and Coriolis forces when the absolute nodal coordinate formulation is used. This important property remains in effect even in the case of flexible bodies with slope discontinuities. The concept employed to accomplish this goal resembles the concept of the intermediate element coordinate system previously adopted in the finite element floating frame of reference formulation. It is shown in this paper that the absolute nodal coordinate formulation that leads to exact representation of the rigid body dynamics can be effectively used in the analysis of complex structures with slope discontinuities. The analysis presented in this paper also demonstrates that objectivity is not an issue when the absolute nodal coordinate formulation is used due to the fact that this formulation automatically accounts for the proper coordinate transformations.

Modeling of Slope Discontinuities in Flexible Body Dynamics Using the Finite Element Method

2002 ◽  
Author(s):  
Ahmed A. Shabana ◽  
Aki M. Mikkola
Keyword(s):  
Finite Element ◽  
Coordinate System ◽  
Absolute Nodal Coordinate Formulation ◽  
Mass Matrix ◽  
Global Coordinate System ◽  
Absolute Nodal Coordinate ◽  
Coriolis Forces ◽  
Body Dynamics ◽  
Definition Of ◽  
Slope Discontinuities

A large rigid body rotation of a finite element can be described by changing the definition of the axes of the element coordinate system or by keeping the axes unchanged and change the slopes or the position vector gradients. In the first method, the definition of the local element parameters (spatial coordinates) changes with respect to a body or a global coordinate system. The use of this method will always lead to a nonlinear mass matrix and non-zero centrifugal and Coriolis forces. The second method, in which the axes of the element coordinate system do not rotate with respect to the body or the global coordinate system, leads to a constant mass matrix and zero centrifugal and Coriolis forces when the absolute nodal coordinate formulation is used. This important property remains in effect even in the case of flexible bodies with slope discontinuities. The concept employed to accomplish this goal resembles the concept of the intermediate element coordinate system previously adopted in the finite element floating frame of reference formulation. It is shown in this paper that the absolute nodal coordinate formulation that leads to exact representation of the rigid body dynamics can be effectively used in the analysis of complex structures with slope discontinuities.

Finite Element Analysis of the Geometric Stiffening Effect Using the Absolute Nodal Coordinate Formulation

2005 ◽  
Author(s):  
Daniel Garci´a-Vallejo ◽  
Hiroyuki Sugiyama ◽  
Ahmed A. Shabana
Keyword(s):  
Finite Element ◽  
Finite Elements ◽  
Linear Elasticity ◽  
Absolute Nodal Coordinate Formulation ◽  
Coupling Effect ◽  
Mass Matrix ◽  
Element Analysis ◽  
Linear Elasticity Theory ◽  
Absolute Nodal Coordinate ◽  
The Absolute

In this paper, the limitations of the linear elasticity finite element solutions in describing the coupling between the extensional and bending displacements are discussed. The fact that incorrect unstable solutions are obtained for models with more than one finite element using the linear elasticity theory motivates the analytical study of the rotating beam presented in this paper. It is shown, as documented in the literature, that the instability of the incorrect solution is directly related to the singularity of the stiffness matrix, and such an instability occurs when the angular velocity reaches the first bending fundamental frequency of the beam. The increase in the number of finite elements only leads to an increase of the critical speed. Crucial in the analysis presented in this paper is the fact that the mass matrix and the form of the elastic forces obtained using the absolute nodal coordinate formulation remain the same under orthogonal coordinate transformation. The absolute nodal coordinate formulation, in contrast to conventional finite element formulations, does account for the effect of the coupling between bending and extension. A similar concept can be incorporated into the finite element floating frame of reference formulation in order to introduce coupling between the axial and bending displacements. Nonetheless, when the linear theory of elasticity is used and no special measures are taken to account for the coupling effect as proposed in the literature, there always exists a critical velocity regardless of the number of finite elements used.

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.

Structural and Continuum Mechanics Approaches for a 3D Shear Deformable ANCF Beam Finite Element: Application to Buckling and Nonlinear Dynamic Examples

2013 ◽  
Vol 9 (1) ◽  
Author(s):  
Karin Nachbagauer ◽  
Johannes Gerstmayr
Keyword(s):  
Finite Element ◽  
Finite Elements ◽  
Continuum Mechanics ◽  
Degrees Of Freedom ◽  
Mass Matrix ◽  
Finite Element Software ◽  
High Order Convergence ◽  
Dynamics Problems ◽  
Transverse Slope ◽  
Shear Deformable

For the modeling of large deformations in multibody dynamics problems, the absolute nodal coordinate formulation (ANCF) is advantageous since in general, the ANCF leads to a constant mass matrix. The proposed ANCF beam finite elements in this approach use the transverse slope vectors for the parameterization of the orientation of the cross section and do not employ an axial nodal slope vector. The geometric description, the degrees of freedom, and a continuum-mechanics-based and a structural-mechanics-based formulation for the elastic forces of the beam finite elements, as well as their usage in several static problems, have been presented in a previous work. A comparison to results provided in the literature to analytical solution and to the solution found by commercial finite element software shows accuracy and high order convergence in statics. The main subject of the present paper is to show the usability of the beam finite elements in dynamic and buckling applications.

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.

Analysis of the Local Deformations in Machine Joints

1979 ◽  
Vol 21 (1) ◽  
pp. 25-32 ◽  
Author(s):  
M. Burdekin ◽  
N. Back ◽  
A. Cowley
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Finite Elements ◽  
Pressure Distribution ◽  
Surface Finish ◽  
The Finite Element Method ◽  
Theoretical Formulation ◽  
Apparent Area ◽  
Finish Material ◽  
General Method

This paper presents a general method for calculating the pressure distribution and the deformations in machine joints. This method assumes that the components of the joint are connected through finite elements which are defined as a function of the surface finish, material and pressure at the apparent area of contact. The system so established is solved in an iterative manner using the finite-element method, obtaining, as a final result, the pressure distribution at the contacting surfaces of the components and the deformations of the surrounding body. To prove the validity and precision of the theoretical formulation, several examples of joints are considered where the correlation between the calculated and measured deflections is shown to be good.

Computation of Exterior Acoustics Problems in Two Dimensions by Trefftz-Type Finite Elements

2015 ◽  
Vol 712 ◽  
pp. 17-22
Author(s):  
Małgorzata Stojek
Keyword(s):  
Finite Elements ◽  
Cross Section ◽  
Radiation Condition ◽  
Two Dimensions ◽  
Least Square ◽  
Fe Method ◽  
Complete Set ◽  
Least Square Procedure

The paper deals with the application of the so-called T-type finite elements [1] to the calculation of the exterior acoustic problems in two dimensions. The method is based on the use of asuitably truncated T-complete set of Trefftz functions over individual subdomains linked by means ofa least square procedure. The vertex singularities and the Sommerfeld radiation condition are readilyincorporated in the trial functions. In order to show the performance of the approach two examples ofcomputations for infinite cylinders (of circular and square cross section) are presented and comparedwith those obtained by means of h-adaptive FE method [2].

Sign in / Sign up

Export Citation Format

Share Document