A. Culla, W. D’Ambrogio, A. Fregolent and A. Schiavone gAB r +g B s =0: this holds for any pair of matching DoFs. Furthermore, if DoF k on the coupled systemAB (the residual subsystemB) does not belong to the equilibrium interface, it must be gAB k =0 (g B k =0): this holds for any DoF not involved in the equilibrium condition. Let NE be the number of interface DoFs on which equilibrium is enforced. Overall, the above conditions can be expressed as: LAB E LB E T gAB gB ={0} (15) where the matrix[LE]= [L AB E ][L B E] is a Boolean localisation matrix. In the framework of the dual formulation in the frequency domain (see Section 2.1.1), the union between the coupled systemABand the fictitious subsystem can be written (see Eq. 10) as: ⎡ ⎢⎣ ZAB [0] BAB E T [0] − ZB BB E T BAB C BB C [0] ⎤ ⎥⎦ ⎧⎪⎨ ⎪⎩ uAB uB {λ} ⎫⎪⎬ ⎪⎭ =⎧⎪⎨ ⎪⎩ f AB f B {0} ⎫⎪⎬ ⎪⎭ (16) Following the same procedure used in Section 2.1.1, it is possible to obtain the FRF of the unknown subsystemA. HA = HAB [0] [0] − HB − HAB [0] [0] − HB BAB E T BB E T × ×' BAB C BB C HAB [0] [0] − HB BAB E T BB E T ( −1 × × BAB C BB C HAB [0] [0] − HB (17) Note that [HAB] and [HB] are the FRFs at the full set of DoFs of the coupled system and the residual subsystem. 96
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