74 A. M. Steenhoek et al. Fig. 8.1 (a) The assembled system consisting of the base-frame, the bridge including the active vibration source and the L-shape, and (b) the digital equivalent of the assembled system in the VIBES software for (live) data processing Fig. 8.2 Assembly of system A and system B Dynamic Substructuring of the FRFs and application of the blocked forces to the system assembly. The results are then compared with validation measurements of the complete assembly. 8.2 Methodology To allow assessment of the components individually and to combine them in a modular fashion, three methods are considered, i.e. the Virtual Point Transformation, Dynamic Substructuring and Source Characterization with Transfer Path Analysis using Blocked Forces. The combination of these methods ensures compatibility between component structures and allows for coupling and response synthesis, i.e. all required ingredients given the goal for modular modeling. Consider an assembly consisting of two systems (A and B), as depicted in Fig. 8.2 above. When constituting this system in a “dual representation”, all local component DoFs are explicitly considered. We can describe the equations of motion for both systems independently in the coupled situation, by means of Eq. (8.1). u=Yf EoM (8.1) In this notation the force vector f contains all forces (external and interface) forces acting on the components and the system dynamics are governed by the system’s FRFs Y. This system of equations can now be used to illustrate the methods mentioned above. 8.2.1 Virtual Points For a component model obtained by measurement, all nu > nq measured displacements u around an interface can be transformed to the virtual point by means of a kinematic relation between the measured DoFs u and the virtual DoFs, here denoted by q. This relation is governed by the so-called Interface Displacement Mode (IDM) matrix Ru, such that: u=Ruq. A similar relation Rf can be set up for the nf >nm forces f and virtual point forces/moments, denoted by m. The inverted IDM matrices (Ru)+ and (Rf)+ are the actual transformation matrices that convert a nu ×nf measured FRF matrix
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