Definition of Non-Dimensional Strain Energy for Thin Plate in the Absolute Nodal Coordinate Formulation

2021 ◽  
Vol 45 (12) ◽  
pp. 1085-1090
Author(s):  
Min Seok Yang ◽  
Jae Wook Lee ◽  
Ji Heon Kang ◽  
Seung Yeop Lee ◽  
Kun Woo Kim
Keyword(s):  
Thin Plate ◽  
Strain Energy ◽  
Absolute Nodal Coordinate Formulation ◽  
Absolute Nodal Coordinate ◽  
The Absolute ◽  
Definition Of

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.

A novel absolute nodal coordinate formulation thin plate tire model with fractional derivative viscosity and surface integral-based contact algorithm

2018 ◽  
Vol 233 (3) ◽  
pp. 583-597
Author(s):  
Peng Lan ◽  
Yaqi Cui ◽  
Zuqing Yu
Keyword(s):  
Fractional Derivative ◽  
Contact Pressure ◽  
Thin Plate ◽  
Absolute Nodal Coordinate Formulation ◽  
Tire Model ◽  
Plate Element ◽  
Absolute Nodal Coordinate ◽  
Contact Patch ◽  
Contact Algorithm ◽  
The Absolute

A new absolute nodal coordinate formulation thin plate tire model, which includes the damping property of the rubber and an efficient tire–road contact algorithm is given. The fractional derivative viscosity constitutive model is proposed in this paper based on the complete form of the absolute nodal coordinate formulation thin plate element, which is created to describe the stress-free initially curved configuration of the tire. A new contact algorithm based on the integration of the contact pressure within the contact patch is developed. By solving the simultaneous equations of the tire geometry and road profile, the dimensionless coordinates for the boundary points of contact patch could be obtained directly. A self-adaptable Gauss integration strategy is introduced to perform the integration of the contact pressure within the varying region, so the integration could reach high precision by few integration points. The calculation of contact force is determined based on penalty method and smoothed Coulomb friction model. The application of fractional derivative viscosity on the absolute nodal coordinate formulation thin plate element is demonstrated by numerical results. A pressurized Golf tire model is given to show the feasibility of the proposed tire–ground contact algorithm.

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.

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