Nonstructural geometric discontinuities in finite element/multibody system analysis

Abstract

Existing multibody system (MBS) algorithms treat articulated system components that are not rigidly connected as separate bodies connected by joints that are governed by nonlinear algebraic equations. As a consequence, these MBS algorithms lead to a highly nonlinear system of coupled differential and algebraic equations. Existing finite element (FE) algorithms, on the other hand, do not lead to a constant mesh inertia matrix in the case of arbitrarily large relative rigid body rotations. In this paper, new FE/MBS meshes that employ linear connectivity conditions and allow for arbitrarily large rigid body displacements between the finite elements are introduced. The large displacement FE absolute nodal coordinate formulation (ANCF) is used to obtain linear element connectivity conditions in the case of large relative rotations between the finite elements of a mesh. It is shown in this paper that a linear formulation of pin (revolute) joints that allow for finite relative rotations between two elements connected by the joint can be systematically obtained using ANCF finite elements. The algebraic joint constraint equations, which can be introduced at a preprocessing stage to efficiently eliminate redundant position coordinates, allow for deformation modes at the pin joint definition point, and therefore, this new joint formulation can be considered as a generalization of the pin joint formulation used in rigid MBS analysis. The new pin joint deformation modes that are the result of C 0 continuity conditions, allow for the calculations of the pin joint strains which can be discontinuous as the result of the finite relative rotation between the elements. This type of discontinuity is referred to in this paper as nonstructural discontinuity in order to distinguish it from the case of structural discontinuity in which the elements are rigidly connected. Because ANCF finite elements lead to a constant mass matrix, an identity generalized mass matrix can be obtained for the FE mesh despite the fact that the finite elements of the mesh are not rigidly connected. The relationship between the nonrational ANCF finite elements and the B-spline representation is used to shed light on the potential of using ANCF as the basis for the integration of computer aided design and analysis (I-CAD-A). When cubic interpolation is used in the FE/ANCF representation, C 0 continuity is equivalent to a knot multiplicity of three when computational geometry methods such as B-splines are used. C 2 ANCF models which ensure the continuity of the curvature and correspond to B-spline knot multiplicity of one can also be obtained. Nonetheless, B-spline and NURBS representations cannot be used to effectively model T-junctions that can be systematically modeled using ANCF finite elements which employ gradient 810 A.M. Hamed et al.coordinates that can be conveniently used to define element orientations in the reference configuration. Numerical results are presented in order to demonstrate the use of the new formulation in developing new chain models.

Ideal Compliant Joints and Integration of Computer Aided Design and Analysis

“…Using ANCF finite elements, mechanical joints between finite elements can be formulated using linear connectivity conditions (Vallejo et al, 2003;Hamed et al, 2011;. This is an important feature that can be exploited to significantly reduce the number of nonlinear algebraic constraint equations and the associated Lagrange multipliers in many MBS applications.…”

“…can be effective in solving a class of small deformation problems Hamed et al, 2011;Shabana, 2010 Dimtrochenko and Pogorelov, 2003;Abbas et al, 2010;Dufva et al, 2005Dufva et al, , 2007Schwab and Meijaard, 2010;Shabana and Mikkola, 2003;Tian et al, 2009;Yakoub and Shabana, 2001;Omar and Shabana, 2001). ANCF finite elements can be used to model both structural and nonstructural discontinuities in a straightforward manner Hamed et al, 2011).…”

“…ANCF finite elements can be used to model both structural and nonstructural discontinuities in a straightforward manner Hamed et al, 2011). Using ANCF finite elements, the absolute position vectors and their gradients are considered nodal coordinates.…”

“…This approach leads to complex nonlinear constraints that are adjoined to the system equations of motion using the technique of Lagrange multipliers. Nonetheless, as demonstrated in previous publications, a linear finite element mesh of chains that consist of flexible links can be developed in many MBS applications (Vallejo et al, 2003;Hamed et al, 2011;. This can be achieved using ANCF finite elements which allow for developing an FE mesh that includes one or more chains with flexible links, and each of the links can have independent relative finite rotations.…”

“…In addition to the high nonlinearity of the inertia forces, the joint constraints equations developed using the FFR formulation are also highly nonlinear because of the use of orientation parameters (Korkealaakso et al, 2009). Furthermore, rigid triads and unit vectors are often used in the formulation of the joints in the rigid body and FFR algorithms, and as a result, these ideal joint formulations do not capture modes of deformation.While the FFR formulation will remain an effective approach for modeling small deformation problems because it allows for filtering out high frequency modes, the more general large rotation large deformation finite element absolute nodal coordinate formulation (ANCF)can be effective in solving a class of small deformation problems Hamed et al, 2011;Shabana, 2010 Dimtrochenko and Pogorelov, 2003;Abbas et al, 2010;Dufva et al, 2005Dufva et al, , 2007Schwab and Meijaard, 2010;Shabana and Mikkola, 2003;Tian et al, 2009;Yakoub and Shabana, 2001;Omar and Shabana, 2001). ANCF finite elements can be used to model both structural and nonstructural discontinuities in a straightforward manner Hamed et al, 2011).…”