Absolute coordinate formulation and generalized component mode synthesis with rigid body coordinates

Abstract

The dynamics of linearly-elastic multibody (MB) systems is conventionally modeled via the floating frame of reference formulation (FFRF); however, its equations of motion (EOMs) involve significantly more nonlinear terms and quantities than alternative formulations, such as the absolute coordinate formulation (ACF) and generalized component mode synthesis (GCMS). This large number of operations required makes computer implementations of the FFRF laborious as well as error-prone and introduces more complexity in general. These issues associated with the FFRF, and the fact that the formulations are mathematically equivalent as shown by the authors, render the ACF and its relatives appealing alternatives due to their simplistic equation structures. To make these alternatives even more appealing, this contribution proposes an improved ACF and GCMS, which (i) reduces the nonlinearity in the EOMs compared to their standard versions and (ii) eliminates the necessity to calculate the rigid body (RB) motion from the global nodal displacement field to obtain the flexible part of the degrees of freedom (DOFs) and the rotation matrix. The proposed EOMs feature a constant mass matrix, a corotated stiffness matrix in the flexible part, and a “small” nonlinear stiffness matrix in the RB rotation part. Moreover, attaching the moving reference frame to the center of mass of the underlying rigid body and employing linearized Tisserand and rotation matrix constraints eliminates coupling terms within the mass matrix and yields implementation-friendly EOMs to analyze the dynamics of linear-elastic flexible MB systems.