Spatial joint constraints for the absolute nodal coordinate formulation using the non-generalized intermediate coordinates

Abstract

In this investigation, a systematic procedure that can be used for modeling joint constraints for the absolute nodal coordinate formulation is developed. To this end, the non-generalized intermediate coordinates are introduced to derive a mapping between the generalized gradient coordinates and the non-generalized rotation parameters. With this mapping, a wide variety of joint constraints can be defined for the absolute nodal coordinate formulation in terms of the non-generalized reference coordinates and, therefore, existing well-developed constraint libraries formulated for the rigid body reference coordinates can be directly employed without significant modifications in existing codes. Furthermore, in order to define a rigid surface at the joint definition point, a set of orthonormality conditions is imposed on the gradient coordinates. This leads to not only accurate modeling of interface to mechanical joint, but also a simpler definition of the joint coordinate system obtained by the orthonormal gradient vectors. For this reason, a simpler form of constraint Jacobian and quadratic velocity vectors can be obtained as compared to those of the existing approach which requires the use of highly nonlinear joint coordinate system. A systematic procedure for eliminating the non-generalized coordinates and the dependent Lagrange multipliers associated with the coordinate mapping equations from the equations of motion is presented. As a result, a standard augmented form of the equations of motion can be obtained in terms of the generalized coordinates only. Several numerical examples are presented in order to demonstrate the use of the joint constraint formulation developed in this investigation.