Walter D’Ambrogio and Annalisa Fregolent tures or subsystems). This kind of problem is also known in the literature as coupling problem or subsystem addition. Many well established techniques exist, when all substructures are modeled theoretically. However, a very important issue is the possibility of combining different kind of models, some obtained from a theoretical or numerical analysis such as a Finite Element Model (FEM) and some others derived from experimental tests (Frequency Response Functions: FRFs). When the model of at least one subsystem derives from experimental tests, the process is called experimental dynamic substructuring. The subject is particularly relevant in virtual prototyping of complex systems and responds to actual industrial needs. Due to modal truncation problems, in experimental dynamic substructuring, the use of FRFs (Frequency Based Substructuring) is preferred with respect to the use of modal parameters. The main algorithm for frequency based substructuring is the improved impedance coupling [4] that involves just one matrix inversion with respect to the classical impedance coupling technique that requires three inversions. A general framework for dynamic substructuring is provided in [6, 5], where primal and dual formulation are introduced. Sometimes the opposite need arises, namely how to extract a substructure model from the assembled system. In this case one speaks of decoupling problem or subsystem subtraction. A trivial application of decoupling is mass cancellation, to get rid of the effect of the accelerometer mass on FRF measurements. Another application is joint identification. More generally, decoupling is a relevant issue for subsystems that cannot be measured separately, but only when coupled to their neighboring substructure(s) (e.g. a fixture needed for testing or subsystems that are very delicate or in operational conditions). To be more precise, the decoupling problem is defined as the identification of the dynamic behaviour of a structural subsystem, starting from the known dynamic behaviour of the assembled system, and from information about the remaining part of the structural system (residual subsystem). Substructure decoupling techniques can be classified as inverse coupling techniques or direct decoupling techniques. In inverse coupling, the equations written for the coupling problem are rearranged to isolate (as unknown) one of the substructures instead of the assembled structure. Examples of inverse coupling are impedance and mobility approaches [1, 7]. Direct decoupling consists in adding to the assembled system a fictitious subsystem, which is the negative of the residual subsystem. The technique starts from the 3-field formulation: one set of equations expressing the dynamic balance of the assembled system and, separately, of the fictitious subsystem; one set of equations enforcing compatibility at interface DoFs, one set of equations enforcing equilibrium of constraint forces at interface DoFs. To solve the problem, a primal approach or a dual approach can be used. Compatibility and equilibrium can be required either at coupling DoFs only (standard interface), or at additional internal DoFs of the residual subsystem (extended interface): as shown in [3] for the dual approach, the choice of interface DoFs determines a set of frequencies at which the decoupling problem is ill conditioned. Apparently, when using an extended interface, the problem is singular at all frequencies, although this singularity is easily removed by using standard smart inversion techniques. To circumvent this problem, in [8, 9] 48
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