Two Simple Triangular Plate Elements Based on the Absolute Nodal Coordinate Formulation

Author(s):  
Oleg Dmitrochenko ◽  
Aki Mikkola

In this paper, two triangular plate elements based on the absolute nodal coordinate formulation (ANCF) are introduced. Triangular elements employ the Kirchhoff plate theory and can, accordingly, be used in thin plate problems. As usual in ANCF, the introduced elements can exactly describe arbitrary rigid body motion when their mass matrices are constant. Previous plate developments in the absolute nodal coordinate formulation have focused on rectangular elements that are difficult to use when arbitrary meshes need to be described. The elements introduced in this study have overcome this problem and represent an important addition to the absolute nodal coordinate formulation. The two elements introduced are based on Specht’s and Morley’s shape functions, previously used in conventional finite element formulations. The numerical solutions of these elements are compared with previously proposed rectangular finite element and analytical results.

Author(s):  
Oleg Dmitrochenko ◽  
Aki Mikkola

In this paper, triangular finite elements based on the absolute nodal coordinate formulation are introduced. Triangular elements employ the Kirchhoff plate theory and can, accordingly, be used in thin plate bending problems. These elements can exactly describe arbitrary rigid body motion while their mass matrices are constant. Previous plate developments in the absolute nodal coordinate formulation have focused on rectangular elements that are difficult to use when arbitrary meshes need to be described. The elements introduced in this study have overcome this problem and represent an important addition to the absolute nodal coordinate formulation. The two elements introduced are based on Specht’s and Morley’s shape functions. The numerical solutions of these elements are compared with results obtained using the previously proposed rectangular finite element and analytical results.


1999 ◽  
Vol 122 (4) ◽  
pp. 498-507 ◽  
Author(s):  
Marcello Campanelli ◽  
Marcello Berzeri ◽  
Ahmed A. Shabana

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]


Author(s):  
Abdel-Nasser A. Mohamed

In this work, a new three-dimensional fully parameterized triangular plate element based on the absolute nodal coordinate formulation (ANCF) is introduced. This plate element has 12 coordinates per node; therefore, it can be used in thick plate applications. The proposed 12 shape functions are obtained by adding three shape functions to the nine shape functions that were previously used with the ANCF thin triangular plate element. Unlike the existing ANCF thin triangular plate element, which allows only the use of classical Kirchoff's plate theory, the fully parameterized ANCF triangular plate element proposed in this work allows for the use of a general continuum mechanics approach and also allows for a straight forward implementation of general nonlinear constitutive equations. Moreover, all deformation modes including thickness deformation can be captured using the fully parameterized ANCF triangular plate element proposed in this paper. The numerical results obtained in this investigation show that in case of negligible deformation, the fully parameterized ANCF triangular plate element behaves like a rigid body. Moreover, it is found that there is a good agreement between the solutions obtained using the proposed fully parameterized ANCF triangular plate element and the theoretical model in the case of small deformations. Furthermore, it is shown that the results of the proposed element agree well with the results obtained using the existing fully parameterized ANCF rectangular plate element when large deformation conditions are applied. The twist behavior of the proposed element is verified by comparison with the results obtained using a conventional nonlinear rectangular plate element.


Author(s):  
Marcello Berzeri ◽  
Marcello Campanelli ◽  
A. A. Shabana

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.


Author(s):  
Alexander Olshevskiy ◽  
Oleg Dmitrochenko ◽  
Chang-Wan Kim

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.


Author(s):  
Oleg Dmitrochenko ◽  
Aki Mikkola

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.


2003 ◽  
Vol 126 (3) ◽  
pp. 478-487 ◽  
Author(s):  
Hiroyuki Sugiyama ◽  
Ahmed A. Shabana

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.


Author(s):  
K Dufva ◽  
A A Shabana

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.


Author(s):  
Imad M. Khan ◽  
Kurt S. Anderson

In this paper, we investigate the absolute nodal coordinate finite element (FE) formulations for modeling multi-flexible-body systems in a divide-and-conquer framework. Large elastic deformations in the individual components (beams and plates) are modeled using the absolute nodal coordinate formulation (ANCF). The divide-and-conquer algorithm (DCA) is utilized to model the constraints arising due to kinematic joints between the flexible components. We develop necessary equations of the new algorithm and present numerical examples to test and validate the method.


Author(s):  
Daniel Garci´a-Vallejo ◽  
Kimmo S. Kerkka¨nen ◽  
Aki M. Mikkola

In this paper, the applicability of the absolute nodal coordinate formulation for the modeling of belt-drive systems is studied. A successful and effective analyzing method for belt-drive systems requires the exact modeling of the rigid body inertia during an arbitrary rigid body motion, accounting of shear deformation, description of nonlinear deformations and a simple as well as realistic description of the contact. The absolute nodal coordinate formulation meets the challenge and is a promising approach for the modeling of belt-drive systems. In this study, a recently proposed two-dimensional shear deformable beam element based on the absolute nodal coordinate formulation has been modified to obtain a belt-like element. The belt-like element allows the user to control the axial and bending stiffness through the use of two parameters. In this study, the interaction between the belt and the pulleys is modeled using an elastic approach in which the contact is accounted for by the inclusion of a set of external forces that depend on the penetration between the belt and pulley.


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