Exact Modeling of the Spatial Rigid Body Inertia Using the Finite Element Method

1998 ◽  
Vol 120 (3) ◽  
pp. 650-657 ◽  
Author(s):  
A. P. Christensen ◽  
A. A. Shabana
Keyword(s):  
Finite Element ◽  
Rigid Body ◽  
Coordinate System ◽  
Deformable Body ◽  
Equations Of Motion ◽  
Body Motion ◽  
Intermediate Element ◽  
Nodal Coordinates ◽  
Body Inertia ◽  
Isoparametric Elements

In the classical finite element literature beams and plates are not considered as isoparametric elements since infinitesimal rotations are used as nodal coordinates. As a consequence, exact modeling of an arbitrary rigid body displacement cannot be obtained, and rigid body motion does not lead to zero strain. In order to circumvent this problem in flexible multibody simulations, an intermediate element coordinate system, which has an origin rigidly attached to the origin of the deformable body coordinate system and has axes which are parallel to the axes of the element coordinate system in the undeformed configuration was introduced. Using this intermediate element coordinate system and the fact that conventional beam and plate shape functions can describe an arbitrary rigid body translation, an exact modeling of the rigid body inertia can be obtained. The large rigid body translation and rotational displacements can be described using a set of reference coordinates that define the location of the origin and the orientation of the deformable body coordinate system. On the other hand, as demonstrated in this investigation, the incremental finite element formulations do not lead to exact modeling of the spatial rigid body mass moments and products of inertia when the structures move as rigid bodies, and such formulations do not lead to the correct rigid body equations of motion. The correct equations of motion, however, can be obtained if the coordinates are defined in terms of global slopes. Using this new definition of the element coordinates, an absolute nodal coordinate formulation that leads to a constant mass matrix for the element can be developed. Using this formulation, in which no infinitesimal or finite rotations are used as nodal coordinates, beam and plate elements can be treated as isoparametric elements.

Nonlinear Finite Element Formulation for the Large Displacement Analysis of Plates

1990 ◽  
Vol 57 (3) ◽  
pp. 707-718 ◽  
Author(s):  
Bilin Chang ◽  
A. A. Shabana
Keyword(s):  
Finite Element ◽  
Rigid Body ◽  
Coordinate System ◽  
Deformable Body ◽  
Finite Element Formulation ◽  
Deformation Analysis ◽  
Element Formulation ◽  
Rectangular Plates ◽  
Shape Functions ◽  
Intermediate Element

In this investigation a nonlinear total Lagrangian finite element formulation is developed for the dynamic analysis of plates that undergo large rigid body displacements. In this formulation shape functions are required to include rigid body modes that describe only large translational displacements. This does not represent any limitation on the technique presented in this study, since most of commonly used shape functions satisfy this requirement. For each finite plate element an intermediate element coordinate system, whose axes are initially parallel to the axes of the element coordinate system, is introduced. This intermediate element coordinate system, which has an origin which is rigidly attached to the origin of the deformable body, is used for the convenience of describing the configuration of the element with respect to the deformable body coordinate system in the undeformed state. The nonlinear dynamic equations developed in this investigation for the large rigid body displacement and small elastic deformation analysis of the rectangular plates are expressed in terms of a unique set of time invariant element matrices that depend on the assumed displacement field. The invariants of motion of the deformable body discretized using the plate elements are obtained by assembling the invariants of its elements using a standard finite element procedure.

The Dynamics of a Deformable Body Experiencing Large Displacements

1988 ◽  
Vol 55 (3) ◽  
pp. 676-680 ◽  
Author(s):  
W. P. Koppens ◽  
A. A. H. J. Sauren ◽  
F. E. Veldpaus ◽  
D. H. van Campen
Keyword(s):  
Rigid Body ◽  
Mass Matrix ◽  
Deformable Body ◽  
Equations Of Motion ◽  
Body Motion ◽  
General Description ◽  
Large Displacements ◽  
Displacement Fields ◽  
Cpu Time ◽  
Time Savings

A general description of the dynamics of a deformable body experiencing large displacements is presented. These displacements are resolved into displacements due to deformation and displacements due to rigid body motion. The former are approximated with a linear combination of assumed displacement fields. D’Alembert’s principle is used to derive the equations of motion. For this purpose, the rigid body displacements and the displacements due to deformation have to be independent. Commonly employed conditions for achieving this are reviewed. It is shown that some conditions lead to considerably simpler equations of motion and a sparser mass matrix, resulting in CPU time savings when used in a multibody program. This is illustrated with a uniform beam and a crank-slider mechanism.

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.

Full Beam Formulation for Coupled Elastic and Rigid Body Motion

1991 ◽  
Author(s):  
C. Venkatakrishnan ◽  
B. Fallahi ◽  
H. Y. Lai
Keyword(s):  
Finite Element ◽  
Rigid Body ◽  
Coordinate System ◽  
Numerical Solution ◽  
Body Motion ◽  
Finite Element Formulation ◽  
Rotary Inertia ◽  
Element Formulation ◽  
Rotating Beam ◽  
Moving Coordinate

Abstract The need for higher operating speeds has led to the study of flexibility in mechanisms. In most of the previous works, rotary inertia, normal, tangential and coriolis terms are neglected. These assumptions are valid at lower speeds and for slender links. In this paper, a procedure to include all inertia terms in a local moving coordinate system is introduced. It is shown that the inertia terms lead to the introduction of three element matrices in the finite element formulation. The proposed approach is used to model the rotating beam problem. The results of a numerical solution is reported and validated.

The Global Motion of a Rigid Body Subject to Periodic Body-Fixed Torques

1995 ◽  
Author(s):  
X. Tong ◽  
B. Tabarrok
Keyword(s):  
Rigid Body ◽  
Equations Of Motion ◽  
Melnikov Method ◽  
The Body ◽  
Body Motion ◽  
Heteroclinic Orbits ◽  
Coordinate Frame ◽  
Global Motion ◽  
Stable And Unstable Manifolds ◽  
Body Subject

Abstract In this paper the global motion of a rigid body subject to small periodic torques, which has a fixed direction in the body-fixed coordinate frame, is investigated by means of Melnikov’s method. Deprit’s variables are introduced to transform the equations of motion into a form describing a slowly varying oscillator. Then the Melnikov method developed for the slowly varying oscillator is used to predict the transversal intersections of stable and unstable manifolds for the perturbed rigid body motion. It is shown that there exist transversal intersections of heteroclinic orbits for certain ranges of parameter values.

Seismic Analysis of Structures With Coulomb Friction

1983 ◽  
Vol 105 (2) ◽  
pp. 171-178 ◽  
Author(s):  
V. N. Shah ◽  
C. B. Gilmore
Keyword(s):  
Rigid Body ◽  
Coulomb Friction ◽  
Seismic Event ◽  
Seismic Analysis ◽  
Equations Of Motion ◽  
Relative Displacement ◽  
Rigid Body Motion ◽  
Body Motion ◽  
Superposition Method ◽  
Modal Superposition Method

A modal superposition method for the dynamic analysis of a structure with Coulomb friction is presented. The finite element method is used to derive the equations of motion, and the nonlinearities due to friction are represented by pseudo-force vector. A structure standing freely on the ground may slide during a seismic event. The relative displacement response may be divided into two parts: elastic deformation and rigid body motion. The presence of rigid body motion necessitates the inclusion of the higher modes in the transient analysis. Three single degree-of-freedom problems are solved to verify this method. In a fourth problem, the dynamic response of a platform standing freely on the ground is analyzed during a seismic event.

Flexible Multibody Dynamics Based on a Fully Cartesian System of Support Coordinates

1993 ◽  
Vol 115 (2) ◽  
pp. 294-299 ◽  
Author(s):  
N. Vukasovic ◽  
J. T. Celigu¨eta ◽  
J. Garci´a de Jalo´n ◽  
E. Bayo
Keyword(s):  
Multibody Dynamics ◽  
Reference Frames ◽  
Deformable Body ◽  
Equations Of Motion ◽  
Flexible Multibody Dynamics ◽  
Body Motion ◽  
Cartesian System ◽  
Flexible Multibody ◽  
Local Reference ◽  
Rigid Multibody

In this paper we present an extension to flexible multibody systems of a system of fully cartesian coordinates previously used in rigid multibody dynamics. This method is fully compatible with the previous one, keeping most of its advantages in kinematics and dynamics. The deformation in each deformable body is expressed as a linear combination of Ritz vectors with respect to a local frame whose motion is defined by a series of points and vectors that move according to the rigid body motion. Joint constraint equations are formulated through the points and vectors that define each link. These are chosen so that a minimum use of local reference frames is done. The resulting equations of motion are integrated using the trapezoidal rule combined with fixed point iteration. An illustrative example that corresponds to a satellite deployment is presented.

Substructure analysis of complex systems using rigid body information of components

2002 ◽  
Vol 216 (10) ◽  
pp. 811-817 ◽  
Author(s):  
W S Hwang ◽  
D H Lee
Keyword(s):  
Finite Element ◽  
Complex Systems ◽  
Rigid Body ◽  
Computer Aided Design ◽  
Element Model ◽  
Rigid Body Motion ◽  
Body Motion ◽  
Element Analysis ◽  
Design Data ◽  
Substructure Analysis

Frequency response function (FRF) based substructure analysis can predict the response of complex systems using the FRFs of substructures. It combines the FRFs of each substructure derived from finite element analysis or experiments depending on the situation. In general, the substructure with the excitation is separated from the others by rubber bushes to prevent the transmission of vibration from the source to the main structure. In this case, the substructure with the excitation shows rigid body motion up to the mid-frequency region. This paper presents a new FRF-based substructure analysis that uses the FRFs from the rigid body information not from the complex finite element model of the substructure with rigid body motion. The rigid body information including the mass, the moment of inertia and the coordinates of the mass centre comes from the computer-aided design data. Since the mechanism of this technique is very similar to the finite element formation, it can be applied to complex systems with ease. Through a simple example of a ladder structure and a practical example of the interior noise in a car, the accuracy and efficiency of this approach is proven.

Hamiltonian Nodal Position Finite Element Method for Cable Dynamics

2017 ◽  
Vol 09 (08) ◽  
pp. 1750109 ◽  
Author(s):  
Huaiping Ding ◽  
Zheng H. Zhu ◽  
Xiaochun Yin ◽  
Lin Zhang ◽  
Gangqiang Li ◽  
...  
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Rigid Body ◽  
Rigid Body Motion ◽  
Body Motion ◽  
Frequency Mode ◽  
Integration Scheme ◽  
Long Period ◽  
Nodal Position ◽  
Element Method

This paper developed a new Hamiltonian nodal position finite element method (FEM) to treat the nonlinear dynamics of cable system in which the large rigid-body motion is coupled with small elastic cable elongation. The FEM is derived from the Hamiltonian theory using canonical coordinates. The resulting Hamiltonian finite element model of cable contains low frequency mode of rigid-body motion and high frequency mode of axial elastic deformation, which is prone to numerical instability due to error accumulation over a very long period. A second-order explicit Symplectic integration scheme is used naturally to enforce the conservation of energy and momentum of the Hamiltonian finite element system. Numerical analyses are conducted and compared with theoretical and experimental results as well as the commercial software LS-DYNA. The comparisons demonstrate that the new Hamiltonian nodal position FEM is numerically efficient, stable and robust for simulation of long-period motion of cable systems.

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