Generalized Component Mode Synthesis for the Spatial Motion of Flexible Bodies With Large Rotations About One Axis

By far the most common approach to describe flexible multi-body systems in industrial practice is the floating frame of reference formulation (FFRF) very often combined with the component mode synthesis (CMS) in order to reduce the number of flexible degrees of freedom. As a result of the relative formulation of the flexible deformation with respect to the reference frame, the mass matrix and the quadratic velocity vector are state-dependent, i.e. non-constant. This requires an evaluation of both the mass matrix and the quadratic velocity vector in every integration step, representing a considerable numerical cost. One way to avoid the state-dependency is to use an absolute formulation as proposed in [2], which was extended in [4] for the use of the same shape functions as used in the classical CMS approach. In this approach, referred to as generalized component mode synthesis (GCMS), the total absolute displacements are approximated directly. Consequently, the mass matrix is constant, there is no quadratic velocity vector and the stiffness matrix is a co-rotated constant matrix. However, it was shown that when using the same shape functions as in the classical CMS approach, nine times the number of degrees of freedom are necessary to describe the same deformation shapes as in the CMS. Even though the integration times of the CMS and GCMS are of the same order, as presented in [5], in technical systems the majority of components are constrained to motions with only one single large rotation. Therefore, in this work the GCMS is formulated for large rotations around a fixed spatial axis. This allows to reduce the number of necessary flexible shape functions to three times the number of CMS shape functions and, consequently, further increases numerical efficiency compared to the GCMS for arbitrary large rotations. A piston engine composed of three flexible bodies, two of which rotate, is used as a test example for the planar formulation. It is compared to the GCMS and a classical FFRF with CMS. The results agree very well, while the GCMS for planar rotations is about three times faster than the other formulations.