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.

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.

Finite element analysis of the geometric stiffening effect. Part 2: Non-linear elasticity

2005 ◽  
Vol 219 (2) ◽  
pp. 203-211 ◽  
Author(s):  
D García-Vallejo ◽  
H Sugiyama ◽  
A A Shabana
Keyword(s):  
Finite Element ◽  
Finite Elements ◽  
Elasticity Theory ◽  
Linear Elasticity ◽  
Critical Speed ◽  
Absolute Nodal Coordinate Formulation ◽  
Linear Elasticity Theory ◽  
Rotating Beams ◽  
Absolute Nodal Coordinate ◽  
Non Linear

In the first part of this paper, the relationship between the number of finite elements used to model the dynamics of rotating beams and the critical speed at which an incorrect solution is obtained when using linear elasticity theory is discussed. The increase in the number of finite elements leads to an increase in the critical speed when linear elasticity is used and no measures are taken, as recommended in the literature, to account for the effect of the coupling between the bending and axial displacements. In this part of the paper, a non-linear finite element model based on the absolute nodal coordinate formulation is used to study the dynamics of rotating beams. It is shown that, when the non-linear elasticity theory is used, a stable solution is always obtained regardless of the number of finite elements used. Numerical results of various simulations are presented in order to compare the solution of a three-dimensional rotating beam that is obtained using the absolute nodal coordinate formulation with the results previously reported in the literature. A finite element numerical study of the dynamics of a helicopter rotor blade is also presented in this investigation. It is shown that, when the finite element absolute nodal coordinate formulation is used in the analysis of helicopter blades, the problem of ill-conditioning that characterizes many of the existing formulations is not encountered.

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.

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.

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

1999 ◽  
Vol 122 (4) ◽  
pp. 498-507 ◽  
Author(s):  
Marcello Campanelli ◽  
Marcello Berzeri ◽  
Ahmed A. Shabana
Keyword(s):  
Finite Element ◽  
Multibody Systems ◽  
Absolute Nodal Coordinate Formulation ◽  
Flexible Multibody Systems ◽  
Large Displacement ◽  
Body Motion ◽  
Absolute Nodal Coordinate ◽  
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]

Elastic Forces and Coordinate Systems in the Finite Element Formulations

1999 ◽  
Author(s):  
Marcello Berzeri ◽  
Marcello Campanelli ◽  
A. A. Shabana
Keyword(s):  
Finite Element ◽  
Absolute Nodal Coordinate Formulation ◽  
Frame Of Reference ◽  
Coordinate Systems ◽  
Absolute Nodal Coordinate ◽  
Floating Frame Of Reference ◽  
Elastic Forces ◽  
Floating Frame ◽  
The Absolute ◽  
Finite Element Formulations

Abstract The equivalence of the elastic forces of finite element formulations used in flexible multibody dynamics is the focus of this investigation. Two conceptually different finite element formulations that lead to exact modeling of the rigid body dynamics will be used. These are the floating frame of reference formulation and the absolute nodal coordinate formulation. It is demonstrated in this study that different element coordinate systems, which are used for the convenience of describing the element deformations in the absolute nodal coordinate formulation, lead to similar results as the element size is reduced. The equivalence of the elastic forces in the absolute nodal coordinate and the floating frame of reference formulations is shown. The result of this analysis clearly demonstrates that the instability observed in high speed rotor analytical models due to the neglect of the geometric centrifugal stiffening is not a problem inherent to a particular finite element formulation but only depends on the beam model that is used. Fourier analysis of the solutions obtained in this investigation also sheds new light on the fundamental problem of the choice of the deformable body coordinate system in the floating frame of reference formulation. A new method is presented and used to obtain a simple expression for the elastic forces in the absolute nodal coordinate formulation. This method, which employs a nonlinear elastic strain-displacement relationship, does not result in an unstable solution when the angular velocity is increased.

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.

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.

Application of Plasticity Theory and Absolute Nodal Coordinate Formulation to Flexible Multibody System Dynamics

2003 ◽  
Vol 126 (3) ◽  
pp. 478-487 ◽  
Author(s):  
Hiroyuki Sugiyama ◽  
Ahmed A. Shabana
Keyword(s):  
Finite Element ◽  
Absolute Nodal Coordinate Formulation ◽  
Plasticity Theory ◽  
Position Vector ◽  
Element Formulation ◽  
Absolute Nodal Coordinate ◽  
Flexible Multibody ◽  
The Absolute ◽  
Return Algorithm ◽  
Finite Element Formulations

The objective of this investigation is to develop a nonlinear finite element formulation for the elastic-plastic analysis of flexible multibody systems. The Lagrangian plasticity theory based on J2 flow theory is used to account for the effect of plasticity in flexible multibody dynamics. It is demonstrated that the principle of objectivity that is an issue when existing finite element formulations using rate-type constitutive equations are used is automatically satisfied when the stress and strain rate are directly calculated in the Lagrangian descriptions using the absolute nodal coordinate formulation employed in this investigation. This is attributed to the fact that, in the finite element absolute nodal coordinate formulation, the position vector gradients can completely define the state of rotation and deformation within the element. As a consequence, the numerical algorithm used to determine the plastic deformations such as the radial return algorithm becomes much simpler when the absolute nodal coordinate formulation is used as compared to existing finite element formulations that employ incrementally objective algorithms. Several numerical examples are presented in order to demonstrate the use of the formulations presented in the paper.

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