A. Culla, W. D’Ambrogio, A. Fregolent and A. Schiavone uA uB =⎛ ⎜⎝ ZA [0] [0] ZB −1 − ZA [0] [0] ZB −1 BA T BB T × ×⎛ ⎝ BA BB ZA [0] [0] ZB −1 BA T BB T ⎞ ⎠ −1 × × BA BB ZA [0] [0] ZB −1⎞ ⎟⎠ f A f B (12) i.e., by introducing the FRFs [HA] and[HB] at the full set of DoFs instead of [ZA]−1 and [ZB]−1: HAB = HA [0] [0] HB − HA [0] [0] HB BA T BB T × ×' BA BB HA [0] [0] HB BA T BB T ( −1 × × BA BB HA [0] [0] HB (13) With the dual formulation, the rows and columns corresponding to the coupling DoFs appear twice in [HAB]. Obviously, only independent entries are retained. 2.2 Subtraction of subsystems using the dual domain decomposition [5] The coupled structural system is assumed to be made by an unknown subsystem (A) and a residual subsystem (B) joined through a number of couplings (see Fig. 2). It is required to find the FRF of the unknown substructure A starting from the FRF of the coupled systemAB. The subsystemAcan be extracted from the coupled systemABby cancelling the dynamic effect of the residual subsystemB. This can be accomplished by adding to the coupled systemABa fictitious subsystem with a dynamic stiffness opposite to that of the residual subsystemBand satisfying compatibility and equilibrium conditions. According to this point of view, the interface between the coupled systemABand the fictitious subsystem should not only include 94
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