Inertia forces and shape integrals in the floating frame of reference formulation

Modeling and analysis of complex dynamical systems can be effectively performed using multibody system (MBS) simulation software. Many modern MBS packages are able to efficiently and reliably handle rigid and flexible bodies, often offering a wide choice of different formulations. Despite many advances in modeling of flexible systems, the most widely used formulation remains the well-established floating frame of reference formulation (FFRF). Although FFRF usually allows inclusion of only small elastic deformations, this assumption is adequate for many industrial applications. In addition, FFRF is computationally efficient if implemented with appropriate model order reduction techniques and effective handling of system inertia terms by utilization of so-called inertia shape integrals. However, derivation of the system of equations of motion for FFRF bodies is a complex and often error-prone task. The main goal of this paper is to provide a reliable, detailed, universal and clear set of inertia terms within the FFRF. The paper concentrates on detailed derivation of the inertia forces with focus on accurate determination and exploitation of the inertia shape integrals. Two standard methods are employed, namely the Lagrangian and Virtual Work approaches. Additionally, the introduced derivations are executed without selection of specific rotational parameters. Direct application of Euler parameters and Euler angles is presented. It is found that the derived expressions are well suited for direct implementation and simplify derivation of force components.