The Simplest 3- and 4-Noded Fully-Parameterized ANCF Plate Elements

2012 ◽  
Author(s):  
Oleg Dmitrochenko ◽  
Marko Matikainen ◽  
Aki Mikkola
Keyword(s):  
Inertial Frame ◽  
Absolute Nodal Coordinate Formulation ◽  
Equations Of Motion ◽  
Strain Tensor ◽  
Large Rotation ◽  
Plate Elements ◽  
Highly Nonlinear ◽  
Elastic Forces ◽  
Rotation Parameters ◽  
Lagrange Strain

In this research, the simplest kinematical models of triangular and rectangular plate finite elements using the absolute nodal coordinate formulation (ANCF) are presented. The ANCF is the finite-element large-displacement-and-rotation approach, which uses the inertial-frame nodal position vectors and their derivatives (slopes) only, without employing any rotation parameters or their equivalent. As a consequence, the kinematics of the elements becomes linear, simplifying the inertia part of the equations of motion, which is also linear. In contrary, due to the need for employing the Green-Lagrange strain tensor, the elastic forces normally appear in a more complicated highly-nonlinear manner than in other large-rotation formulations. In this research, to reduce the computational burden, two new plate elements are proposed that are the simplest possible triangular and rectangular elements in the fully-parameterized ANCF: they employ transverse slopes only, without using longitudinal slopes.

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.

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.

Exact formulations of non-linear planar and spatial Euler–Bernoulli beams with finite strains

2011 ◽  
Vol 226 (5) ◽  
pp. 1225-1236 ◽  
Author(s):  
M H Abedinnasab ◽  
H Zohoor ◽  
Y-J Yoon
Keyword(s):  
Strain Field ◽  
Dynamic Models ◽  
Equations Of Motion ◽  
Strain Tensor ◽  
Bernoulli Beam ◽  
Cross Sectional ◽  
Elastic Deformations ◽  
Non Linear ◽  
Lagrange Strain ◽  
Euler Bernoulli Beam

Using Hamilton’s principle, exact equations of motion for non-linear planar and spatial Euler–Bernoulli beams are derived. In the existing non-linear Euler–Bernoulli beam formulations, some elastic terms are dropped by differentiation from the incomplete Green–Lagrange strain tensor followed by negligible elastic deformations of cross-sectional frame. On the other hand, in this article, the exact strain field concerning considerable elastic deformations of cross-sectional frame is used as a source in differentiations. As a result, the achieved closed-form equations are exact and more accurate than formerly reported equations in the literature. Moreover, the applicable dynamic model of inextensional beams which is fully accurate, yet simple has been shown. The planar and inextensional dynamic models have been compared with the existing dynamic models in the literature, and the proposed dynamic models demonstrate significant improvements in the numerical results. Finally, experiments on the carbon fibre rods verify the model presented for inextensional beams.

scholarly journals Assessment of Linearization Approaches for Multibody Dynamics Formulations

2017 ◽  
Vol 12 (4) ◽  
Author(s):  
Francisco González ◽  
Pierangelo Masarati ◽  
Javier Cuadrado ◽  
Miguel A. Naya
Keyword(s):  
Mechanical System ◽  
Multibody Dynamics ◽  
Equations Of Motion ◽  
Differential Algebraic Equations ◽  
Algebraic Equations ◽  
Practical Applications ◽  
Linearization Methods ◽  
Highly Nonlinear ◽  
Wide Range ◽  
The Way

Formulating the dynamics equations of a mechanical system following a multibody dynamics approach often leads to a set of highly nonlinear differential-algebraic equations (DAEs). While this form of the equations of motion is suitable for a wide range of practical applications, in some cases it is necessary to have access to the linearized system dynamics. This is the case when stability and modal analyses are to be carried out; the definition of plant and system models for certain control algorithms and state estimators also requires a linear expression of the dynamics. A number of methods for the linearization of multibody dynamics can be found in the literature. They differ in both the approach that they follow to handle the equations of motion and the way in which they deliver their results, which in turn are determined by the selection of the generalized coordinates used to describe the mechanical system. This selection is closely related to the way in which the kinematic constraints of the system are treated. Three major approaches can be distinguished and used to categorize most of the linearization methods published so far. In this work, we demonstrate the properties of each approach in the linearization of systems in static equilibrium, illustrating them with the study of two representative examples.

NONLINEAR ANALYSIS OF FUNCTIONALLY GRADED CIRCULAR PLATES UNDER DIFFERENT LOADS AND BOUNDARY CONDITIONS

2008 ◽  
Vol 08 (01) ◽  
pp. 131-159 ◽  
Author(s):  
RECEP GUNES ◽  
J. N. REDDY
Keyword(s):  
Nonlinear Analysis ◽  
Strain Tensor ◽  
Homogenization Method ◽  
Functionally Graded ◽  
Gradient Material ◽  
Graphical Form ◽  
Circular Plates ◽  
Geometrically Nonlinear Analysis ◽  
Steady State Heat Transfer ◽  
Lagrange Strain

Geometrically nonlinear analysis of functionally graded circular plates subjected to mechanical and thermal loads is carried out in this paper. The Green–Lagrange strain tensor in its entirety is used in the analysis. The locally effective material properties are evaluated using homogenization method which is based on the Mori–Tanaka scheme. In the case of thermally loaded plates, the temperature variation through the thickness is determined by solving a steady-state heat transfer (i.e. energy) equation. As an example, a functionally gradient material circular plate composed of zirconium and aluminum is used and results are presented in graphical form.

Surrogate POD Models for Parametrized Sheet Metal Forming Applications

2013 ◽  
Vol 554-557 ◽  
pp. 919-927 ◽  
Author(s):  
Hamdaoui Mohamed ◽  
Guénhaël Le Quilliec ◽  
Piotr Breitkopf ◽  
Pierre Villon
Keyword(s):  
Finite Element ◽  
Sheet Metal ◽  
Sheet Metal Forming ◽  
Metal Forming ◽  
Optimization Problems ◽  
Strain Tensor ◽  
Forming Limit ◽  
Work Piece ◽  
Displacement Fields ◽  
Lagrange Strain

The aim of this work is to present a POD (Proper Orthogonal Decomposition) based surrogate approach for sheet metal forming parametrized applications. The final displacement field for the stamped work-piece computed using a finite element approach is approximated using the method of snapshots for POD mode determination and kriging for POD coefficients interpolation. An error analysis, performed using a validation set, shows that the accuracy of the surrogate POD model is excellent for the representation of finite element displacement fields. A possible use of the surrogate to assess the quality of the stamped sheet is considered. The Green-Lagrange strain tensor is derived and forming limit diagrams are computed on the fly for any point of the design space. Furthermore, the minimization of a cost function based on the surrogate POD model is performed showing its potential for solving optimization problems.

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.

Tilt-Wing Rotorcraft Dynamic Analysis Using Multibody Formulation

1992 ◽  
Author(s):  
Patrick J. O’Heron ◽  
Parviz E. Nikravesh ◽  
Ara Arabyan ◽  
Donald L. Kunz
Keyword(s):  
Flight Control ◽  
High Speed ◽  
Equations Of Motion ◽  
Flight Path ◽  
Minimal Set ◽  
Near Wake ◽  
Highly Nonlinear ◽  
Nonlinear Transient ◽  
Nonlinear Transient Dynamics ◽  
Rotor Assembly

Abstract A model is presented that can be used to simulate the highly nonlinear transient dynamics associated with advanced rotorcraft conversion processes. Multibody equations of motion of the fuselage, the tilting wing, and the rotor assembly are derived using a minimal set of coordinates. An enhanced aerodynamics model is employed to account for unsteadiness and nonlinearity in the near-wake aerodynamics, with a dynamic uniform inflow to compute the far-wake aerodynamics, and a flight control system is employed to compute the blade pitch settings that are necessary to achieve a desired flight path. The model is subjected to a demanding flight path simulation to illustrate that it can perform vertical take-off, hover, tilt-wing conversion, and high-speed forward flight maneuvers effectively.

scholarly journals Re-Evaluating the Classical Falling Body Problem

Mathematics ◽  
2020 ◽  
Vol 8 (4) ◽  
pp. 553 ◽  
Author(s):  
Essam R. El-Zahar ◽  
Abdelhalim Ebaid ◽  
Abdulrahman F. Aljohani ◽  
José Tenreiro Machado ◽  
Dumitru Baleanu
Keyword(s):  
Differential Equations ◽  
Body Problem ◽  
Inertial Frame ◽  
Analytic Solution ◽  
Equations Of Motion ◽  
Three Dimensional ◽  
Three Dimensions ◽  
Rotating Frame ◽  
Dimensional Model ◽  
Three Dimensional Model

This paper re-analyzes the falling body problem in three dimensions, taking into account the effect of the Earth’s rotation (ER). Accordingly, the analytic solution of the three-dimensional model is obtained. Since the ER is quite slow, the three coupled differential equations of motion are usually approximated by neglecting all high order terms. Furthermore, the theoretical aspects describing the nature of the falling point in the rotating frame and the original inertial frame are proved. The theoretical and numerical results are illustrated and discussed.

Absolute Nodal Coordinate Formulation With Vector-Strain Transformation for High Aspect Ratio Wings

2020 ◽  
Vol 16 (1) ◽  
Author(s):  
Keisuke Otsuka ◽  
Yinan Wang ◽  
Kanjuro Makihara
Keyword(s):  
Aspect Ratio ◽  
High Aspect Ratio ◽  
Degrees Of Freedom ◽  
Absolute Nodal Coordinate Formulation ◽  
Lattice Method ◽  
Absolute Nodal Coordinate ◽  
Aeroelastic Analysis ◽  
Strain Transformation ◽  
Highly Nonlinear ◽  
Analytical Approaches

Abstract High aspect ratio wings are potential candidates for use in atmospheric satellites and civil aircraft as they exhibit a low induced drag, which can reduce the fuel consumption. Owing to their slender and light weight configuration, such wings undergo highly flexible aeroelastic static and dynamic deformations that cannot be analyzed using conventional linear analysis methods. An aeroelastic analysis framework based on the absolute nodal coordinate formulation (ANCF) can be used to analyze the static and dynamic deformations of high aspect ratio wings. However, owing to the highly nonlinear elastic force, the statically deformed wing shape during steady flight cannot be efficiently obtained via static analyses. Therefore, an ANCF with a vector-strain transformation (ANCF-VST) was proposed in this work. Considering the slender geometry of high aspect ratio wings, the nodal vectors of an ANCF beam element were transformed to the strains. In this manner, a constant stiffness matrix and reduced degrees-of-freedom could be generated while capturing the highly flexible deformations accurately. The ANCF-VST exhibited superior convergence performance and accuracy compared to those of analytical approaches and other nonlinear beam formulations. Moreover, an aeroelastic analysis flow coupling the ANCF-VST and an aerodynamic model based on the unsteady vortex lattice method was proposed to perform the static and dynamic analyses successively. The proposed and existing aeroelastic frameworks exhibited a good agreement in the analyses, which demonstrated the feasibility of employing the proposed framework to analyze high aspect ratio wings.

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