Comparison of Different Approaches to Include Connection Elements into Frequency-Based Substructuring

Dynamic substructuring (DS) is a research field that has gained a great deal of attention in both science and industry. The aim of DS techniques is to provide engineers in structural vibrations and sound practical solutions for analyzing the dynamic behavior of complex systems. This paper addresses the singularity problem that occurs when flexible joints are implemented as substructures into the Lagrange Multiplier Frequency-Based Substructuring (LM-FBS) coupling process. For illustration, we use rubber bushings from an automotive application. Considering the rubber isolators to exhibit hysteretic damping, we assume that only the property of the dynamic stiffness of material is given. To avoid singularity appearing in the admittance when inverting the impedance of a massless joint, we compare three different approaches to include rubber bushings in the framework of LM-FBS. One method consists in including the dynamic stiffness of material directly in the space of the interface constraints and add it to the assembled interface flexibility of the LM-FBS equation. This corresponds to a relaxation of the interface compatibility condition. In the second method, the rubber bushing is treated as a substructure by adding small masses to the equation of the joint. As a result, we obtain a nonsingular total dynamic stiffness matrix that can be included in the coupling process. The third method describes a novel extension of the LM-FBS approach, based on a solution for singular problems. If the applied forces are self-equilibrated with respect to the rigid body modes, a solution for the singular dynamic stiffness matrix exists. The methods are outlined, both mathematically and conceptually, based on a notation commonly used in LM-FBS. They facilitate the integration of connecting elements together with experimental or numerical determined system dynamics of substructures in order to predict the assembled system behavior.