A projection-based approach for the derivation of the floating frame of reference formulation for multibody systems

The reduction in the number of coordinates for flexible multibody systems is necessary in order to achieve acceptable simulation times of real-life structures and machines. The conventional model order reduction technique for flexible multibody systems is based on the floating frame of reference formulation (FFRF), using a rigid body frame and superimposed small flexible deformations. The FFRF leads to strongly coupled terms in rigid body and flexible coordinates as well as to a non-constant mass matrix. As an alternative to the FFRF, a formulation based on absolute coordinates has been proposed which uses a co-rotational strain. In this way, a constant mass matrix and a co-rotational stiffness matrix are obtained. In order to perform a reduction in the number of coordinates, by means of the component mode synthesis, e.g., the number of modes needs to be increased, such that all modes are represented in every possible rotated configuration. This approach leads to the method of generalized component mode synthesis (GCMS). The present paper shows in detail how the equations of motion of the FFRF evolve from the ones of the GCMS by considering rigid body constraint conditions and subsequently eliminating them via an appropriate null-space projection. This approach allows a straightforward, term-by-term interpretation of the FFRF mass matrix and of the generalized gyroscopic forces, which, to the same extent, cannot be deduced from former publications on the FFRF. From a practical point of view, the resulting expressions allow to calculate all inertia coefficients from the constant finite element mass matrix together with standard input data of the finite element model in the course of a preprocessing step. Then, the repeated updates of the FFRF mass matrix and of the gyroscopic forces in the course of time integration involve only simple vector matrix operations of low dimensions. In contrast to previous implementations of the FFRF, no evaluations of extra inertia integrals are required. Consequently, the present formulation can be implemented entirely independent of the related finite element code.