Absolute Nodal Coordinate Formulation

1997 ◽  
Author(s):  
Ahmed A. Shabana ◽  
Hussien A. Hussien ◽  
José L. Escalona
Keyword(s):  
Finite Element ◽  
Rigid Body ◽  
Rigid Bodies ◽  
Rotation Vector ◽  
Finite Rotations ◽  
Large Rotation ◽  
Finite Element Procedure ◽  
Floating Frame ◽  
Inertia Forces ◽  
Body Inertia

Abstract There are three basic finite element formulations, which are used in multibody dynamics. These are the floating frame reference approach, the incremental method and the large rotation vector approach. In the floating frame of reference and incremental formulations, the slopes are assumed small in order to define infinitesimal rotations that can be treated and transformed as vectors. This description, however, limits the use of some important elements such as beams and plates in a wide range of large displacement applications. As demonstrated in some recent publications, if infinitesimal rotations are used as nodal coordinates, the use of the finite element incremental formulation in the large reference displacement analysis does not lead to exact modeling of the rigid body inertia when the structures rotate as rigid bodies. In this paper, a new and simple finite element procedure that employs the mathematical definition of the slope and uses it to define the element coordinates instead of the infinitesimal and finite rotations is developed for large rotation and deformation problems. By using this description and by defining the element coordinates in the global system, not only the need for performing coordinate transformation is avoided, but also a simple expression for the inertia forces is obtained. Furthermore, the resulting mass matrix is constant and it is the same matrix that appears in linear structural dynamics. It is demonstrated in this paper, that this coordinate description leads to exact modeling of the rigid body inertia when the structure rotate as rigid bodies. Nonetheless, the stiffness matrix becomes nonlinear function of time even in the case of small displacements. The method presented in this paper differs from previous large rotation vector formulations in the sense that the inertia forces, the kinetic energy, and the strain energy are not expressed in terms of any orientation coordinates, and therefore, the method does not require interpolation of finite rotations. While the use of the formulation is demonstrated using a simple planar beam element, the generalization of the method to other element types and to the three dimensional case is straightforward. Using the finite element procedure presented in this paper, beams and plates can be treated as isoparametric elements.

Application of the Absolute Nodal Coordinate Formulation to Large Rotation and Large Deformation Problems

1998 ◽  
Vol 120 (2) ◽  
pp. 188-195 ◽  
Author(s):  
A. A. Shabana ◽  
H. A. Hussien ◽  
J. L. Escalona
Keyword(s):  
Finite Element ◽  
Rigid Body ◽  
Rigid Bodies ◽  
Frame Of Reference ◽  
Finite Rotations ◽  
Large Rotation ◽  
Finite Element Procedure ◽  
Floating Frame ◽  
Inertia Forces ◽  
Body Inertia

There are three basic finite element formulations which are used in multibody dynamics. These are the floating frame of reference approach, the incremental method and the large rotation vector approach. In the floating frame of reference and incremental formulations, the slopes are assumed small in order to define infinitesimal rotations that can be treated and transformed as vectors. This description, however, limits the use of some important elements such as beams and plates in a wide range of large displacement applications. As demonstrated in some recent publications, if infinitesimal rotations are used as nodal coordinates, the use of the finite element incremental formulation in the large reference displacement analysis does not lead to exact modeling of the rigid body inertia when the structures rotate as rigid bodies. In this paper, a simple non-incremental finite element procedure that employs the mathematical definition of the slope and uses it to define the element coordinates instead of the infinitesimal and finite rotations is developed for large rotation and deformation problems. By using this description and by defining the element coordinates in the global system, not only the need for performing coordinate transformation is avoided, but also a simple expression for the inertia forces is obtained. The resulting mass matrix is constant and it is the same matrix that appears in linear structural dynamics. It is demonstrated in this paper that this coordinate description leads to exact modeling of the rigid body inertia when the structures rotate as rigid bodies. Nonetheless, the stiffness matrix becomes nonlinear function even in the case of small displacements. The method presented in this paper differs from previous large rotation vector formulations in the sense that the inertia forces, the kinetic energy, and the strain energy are not expressed in terms of any orientation coordinates, and therefore, the method does not require interpolation of finite rotations. While the use of the formulation is demonstrated using a simple planar beam element, the generalization of the method to other element types and to the three dimensional case is straightforward. Using the finite element procedure presented in this paper, beams and plates can be treated as isoparametric elements.

Finite Element Incremental Approach and Exact Rigid Body Inertia

1996 ◽  
Vol 118 (2) ◽  
pp. 171-178 ◽  
Author(s):  
A. A. Shabana
Keyword(s):  
Finite Element ◽  
Rigid Body ◽  
Rigid Bodies ◽  
Simple Expression ◽  
Shape Functions ◽  
Large Rotation ◽  
Nodal Coordinates ◽  
Inertia Properties ◽  
Element Shape ◽  
Body Inertia

In the dynamics of multibody systems that consist of interconnected rigid and deformable bodies, it is desirable to have a formulation that preserves the exactness of the rigid body inertia. As demonstrated in this paper, the incremental finite element approach, which is often used to solve large rotation problems, does not lead to the exact inertia of simple structures when they rotate as rigid bodies. Nonetheless, the exact inertia properties, such as the mass moments of inertia and the moments of mass, of the rigid bodies can be obtained using the finite element shape functions that describe large rigid body translations by introducing an intermediate element coordinate system. The results of application of the parallel axis theorem can be obtained using the finite element shape functions by simply changing the element nodal coordinates. As demonstrated in this investigation, the exact rigid body inertia properties in case of rigid body rotations can be obtained using the shape function if the nodal coordinates are defined using trigonometric functions. The analysis presented in this paper also demonstrates that a simple expression for the kinetic energy can be obtained for flexible bodies that undergo large displacements without the need for interpolation of large rotation coordinates.

Finite Element Incremental Approach and Exact Rigid Body Inertia

1995 ◽  
Author(s):  
A. A. Shabana
Keyword(s):  
Finite Element ◽  
Rigid Body ◽  
Shape Function ◽  
Rigid Bodies ◽  
Shape Functions ◽  
Mass Moment ◽  
Nodal Coordinates ◽  
Inertia Properties ◽  
Element Shape ◽  
Body Inertia

Abstract In the dynamics of multibody systems that consist of interconnected rigid and deformable bodies, it is desirable to have a formulation that preserves the exactness of the rigid body inertia. As demonstrated in this paper, the incremental finite element approach, which is often used to solve large rotation problems, does not lead to the exact inertia of simple structures when they rotate as rigid bodies because the physical nodal coordinates can not be used to describe large rotations in the case of beams and plates. Nonetheless, the exact inertia properties, such as the mass moments of inertia and the moments of mass, of the rigid bodies can be obtained using the finite element shape functions that describe large rigid body translations by introducing an intermediate element coordinate system. The results of application of the parallel axis theorem can be obtained using the finite element shape functions by simply changing the element nodal coordinates. A simple rigid body rotation, however, can cause a significant error if the element shape function and the nodal coordinates are used to evaluate the inertia properties of bodies that undergo large rigid body rotations. As demonstrated in this investigation, the exact rigid body inertia properties in case of rigid body rotations can be obtained using the shape function if the nodal coordinates are defined using trigonometric functions that lack a physical meaning. Linearization of the nodal coordinate vector can lead to different results when different methods are used to define the rigid body inertia. For example, the calculation of the mass moment of inertia using position coordinates only leads to results which are different from those obtained using energy expressions or the laws of motion.

An Element Independent Corotational Procedure for the Treatment of Large Rotations

1986 ◽  
Vol 108 (2) ◽  
pp. 165-174 ◽  
Author(s):  
C. C. Rankin ◽  
F. A. Brogan
Keyword(s):  
Finite Element ◽  
Rigid Body ◽  
Displacement Field ◽  
Rigid Body Motion ◽  
Rotation Vector ◽  
Body Motion ◽  
Large Rotation ◽  
Total Displacement ◽  
Large Rotations ◽  
Finite Element Formulations

A new corotational procedure is developed which enables existing finite element formulations to be used in problems that contain arbitrarily large rotations. Through the use of a nonsingular large rotation vector, the contribution of the rigid body motion of the element to the total displacement field is removed before element computations are performed, with the result that almost any element can be easily upgraded to handle large rotations. This paper contains a derivation of the theory, an outline of the implementation into the STAGS code, and a demonstration of performance for problems involving large rotations and moderate strains.

Computer-Aided Modelling of Cloth Deformation

1996 ◽  
Author(s):  
Y. F. Zhao ◽  
S. T. Tan ◽  
T. N. Wong ◽  
W. J. Chen
Keyword(s):  
Finite Element Method ◽  
Exact Solution ◽  
Finite Element ◽  
Rigid Body ◽  
Present Method ◽  
Geometric Constraint ◽  
Computationally Efficient ◽  
Large Rotation ◽  
Developable Surfaces ◽  
Computer Aided

Abstract A constrained finite element method for modelling cloth deformation is developed. The bending deformation and the geometric constraint of developable surfaces of the cloth objects are considered. The representation of large rotation and the motion of rigid body are described using the current coordinates with the geometric constraint. The effectiveness of the present method is verified by comparing the thread deformation with the exact solution of catenary. Several examples are given to show that the proposed method converges quickly and is thus computationally efficient.

ANCF Consistent Rotation-Based Finite Element Formulation

2015 ◽  
Vol 11 (1) ◽  
Author(s):  
Ahmed A. Shabana
Keyword(s):  
Rotation Vector ◽  
Element Formulation ◽  
General Description ◽  
Centrifugal Forces ◽  
Large Rotation ◽  
Plate And Shell ◽  
Shell Models ◽  
Deformation Modes ◽  
Inertia Forces ◽  
The Continuum

In this technical brief, a consistent rotation-based formulation is proposed using the absolute nodal coordinate formulation (ANCF) kinematic description. The proposed formulation defines a unique rotation field, employs one interpolation, captures shear deformations, does not suffer from the redundancy problem encountered when using large rotation vector formulations, allows for systematically describing curved geometry, and leads to elastic force definitions that eliminate high-frequency modes associated with the deformation of the cross section. The drawback of this formulation, as it is the case with the large rotation vector formulations, is the nonlinearity of the inertia forces including nonzero Coriolis and centrifugal forces. Furthermore, the formulation does not capture deformation modes that can be captured using the more general ANCF finite elements. Nonetheless, the proposed method is consistent with the continuum mechanics general description, can be related to computational geometry methods, and can be used to develop beam, plate, and shell models without violation of basic mechanics principles.

A Family of Butterfly Flexural Joints: Q-LITF Pivots

2011 ◽  
Author(s):  
Xu Pei ◽  
Jingjun Yu ◽  
Shusheng Bi ◽  
Guanghua Zong
Keyword(s):  
Finite Element Analysis ◽  
Finite Element ◽  
Rigid Body ◽  
Rigid Bodies ◽  
The Other ◽  
Element Analysis ◽  
Leaf Type ◽  
Center Shift ◽  
Body Model ◽  
Rigid Body Model

The Leaf-type Isosceles-Trapezoidal Flexural (LITF) pivot consists of two compliant beams and two rigid-bodies. For a single LITF pivot, the range of motion is small while the center-shift is relatively large. The capability of performance can be improved greatly by the combination of four LITF pivots. Base on the pseudo-rigid-body model (PRBM) of a LITF pivot, a method to construct the Quadri-LITF pivots is presented by regarding a single LITF pivot (or double-LITF pivot) as a the configurable flexure module. Ten types of Q-LITF pivots are synthesized. Compared with the single LIFT pivot, the stroke becomes larger, and stiffness becomes smaller. Four of them have the increased center-shift. The other four have the decreased center-shift. One of the quadruple LITF pivots is selected as the examples to explain the proposed method. The comparison between PRBM and Finite Element Analysis (FEA) result shows the validity and effectiveness of the method.

Assembly of Finite‐Element Helicopter Subsystems with Large Relative Rotations

1990 ◽  
Vol 35 (2) ◽  
pp. 60-68
Author(s):  
Jon‐Shen Fuh ◽  
Brahmananda Panda ◽  
David A. Peters
Keyword(s):  
Finite Element ◽  
Rigid Body ◽  
Large Angle ◽  
Finite Element Models ◽  
Coordinate Systems ◽  
Large Rotations ◽  
Finite Element Procedure ◽  
Moving Coordinate ◽  
Rigid Body Motions ◽  
Small Rotation

A general finite‐element procedure is presented for modeling rotorcraft undergoing elastic deformations in addition to large rigid body motions with respect to inertial space. Special attention is given to the coupling of the rotor and fuselage subsystems subject to large relative rotations. Initially, the rotor and fuselage subsystems are assembled separately as small‐rotation finite‐element models in a moving coordinate system. In order to handle large rigid body rotations, the coordinate systems are tied to the structure using one of several alternate constraint methods. Finally, the equations which allow large rotations are constrained together using a rotating to nonrotating transformation which allows rotor azimuth angle as a degree of freedom. The resulting system of equations, which has not been implemented, is applicable to both helicopter trim and large angle maneuver analyses.

Flexible Multibody Approaches for Dynamical Simulation of Beam Structures in Drilling

2014 ◽  
Author(s):  
Gennady Mikheev ◽  
Dmitry Pogorelov ◽  
Oleg Dmitrochenko ◽  
Raju Gandikota
Keyword(s):  
Finite Element ◽  
Degrees Of Freedom ◽  
Equations Of Motion ◽  
Element Model ◽  
Drill String ◽  
Rigid Bodies ◽  
Nonlinear Finite Element ◽  
Beam Structures ◽  
Modal Approach ◽  
Large Rotation

Two approaches for simulation of dynamics of complex beam structures such as drill strings are considered. In the first approach, the drill string is presented as a set of uniform beams connected via force elements. The beams can undergo arbitrary large displacements as absolutely rigid bodies but its flexible displacements due to elastic deformations are assumed to be small. Flexibility of the beams is simulated using the modal approach. Thus, each beam has at least twelve degrees of freedom: six coordinates define position and orientation of a local frame and six modes are used for modeling flexibility. The second approach is dynamic simulation of the drill string using nonlinear finite element model. The proposed beam finite element uses Cartesian coordinates of its nodes and node rotation angles around axis of Cartesian coordinate system as generalized coordinates. The nonlinear finite element is developed based on method of large rotation vectors. Rotation angles in the nodes can be arbitrary large. Equations of motion of beam structure are derived in the paper. The number of degrees of freedom is decreased by factor two as compared with the modal approach. Thereby, computational efficiency under simulation of dynamics of long drill strings is considerably increased. The features of creating the models and numerical methods as well as results obtained by applying both approaches are discussed in the paper.

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