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.

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.

A Finite Element Formulation of Flexible Slider Crank Mechanism Using Local Coordinates

1994 ◽  
Author(s):  
Behrooz Fallahi ◽  
S. Lai ◽  
C. Venkat
Keyword(s):  
Rigid Body ◽  
Coordinate System ◽  
High Speed ◽  
Displacement Vector ◽  
Lagrange Equation ◽  
Local Coordinate System ◽  
Body Motion ◽  
Elastic Displacement ◽  
Global Coordinate System ◽  
Element Formulation

Abstract The need for higher productivity has lead to the design of machines operating at higher speeds. At high speed the rigid body assumption is no longer valid and the links should be considered flexible. In this work a method which is based on Modified Lagrange Equation for modeling flexible mechanism is presented. The method posses a more computational efficiency for not requiring the transformation from the local coordinate system to the global coordinate system. Also an approach using the homogeneous coordinate for element matrices generation is presented. The approach leads to a formalism where the displacement vector is expressed as a product of two matrices and a vector. The first matrix is a function of rigid body motion. The second matrix is a function of rigid body configuration. The vector is a function of elastic displacement. This formal separation helps to facilitate the generation of element matrices using symbolic manipulations.

A Finite Element Formulation of a Flexible Slider Crank Mechanism Using Local Coordinates

1995 ◽  
Vol 117 (3) ◽  
pp. 329-335 ◽  
Author(s):  
Behrooz Fallahi ◽  
S. Lai ◽  
C. Venkat
Keyword(s):  
Rigid Body ◽  
Coordinate System ◽  
Displacement Vector ◽  
Lagrange Equation ◽  
Local Coordinate System ◽  
Body Motion ◽  
Elastic Displacement ◽  
Global Coordinate System ◽  
Element Formulation ◽  
Crank Mechanism

The need for higher manufacturing throughput has lead to the design of machines operating at higher speeds. At higher speeds, the rigid body assumption is no longer valid and the links should be considered flexible. In this work, a method based on the Modified Lagrange Equation for modeling a flexible slider-crank mechanism is presented. This method possesses the characteristic of not requiring the transformation from the local coordinate system to the global coordinate system. An approach using the homogeneous coordinate for element matrices generation is also presented. This approach leads to a formalism in which the displacement vector is expressed as a product of two matrices and a vector. The first matrix is a function of rigid body motion. The second matrix is a function of rigid body configuration. The vector is a function of the elastic displacement. This formal separation helps to facilitate the generation of element matrices using symbolic manipulators.

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.

An efficient finite element formulation for nonlinear analysis of clustered tensegrity

2016 ◽  
Vol 33 (1) ◽  
pp. 252-273 ◽  
Author(s):  
Liang Zhang ◽  
Qiang Gao ◽  
Yin Liu ◽  
Hongwu Zhang
Keyword(s):  
Finite Element ◽  
Nonlinear Analysis ◽  
Stiffness Matrix ◽  
Body Motion ◽  
Finite Element Formulation ◽  
Force Density ◽  
Element Formulation ◽  
Content Type ◽  
Tangent Stiffness ◽  
Tangent Stiffness Matrix

Purpose – The purpose of this paper is to propose an efficient finite element formulation for nonlinear analysis of clustered tensegrity that consists of classical cables, clustered cables and bars. Design/methodology/approach – The derivation of the finite element formulation is based on the co-rotational approach, which decomposes a geometrically nonlinear deformation into a large rigid body motion and a small-strain deformation. A tangent stiffness matrix of a clustered cable is proposed and the Newton-Raphson scheme is employed to solve the nonlinear equation. Findings – The derived tangent stiffness matrix, including an additional stiffness terms that describes the slide effect of pulleys, can regress to the stiffness matrix of a classical cable, which is convenient for the implementation of finite element procedure. Two typical numerical examples show that the proposed formulation is accurate and requires less iteration than the force density method. Originality/value – The co-rotational formulation of a clustered cable is originally proposed, although some mature methods, such as the TL, Force Density and Dynamic Relaxation method, have been applied to nonlinear analysis of clustered tensegrity. The proposed co-rotational formulation proved efficient.

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