30 W. D’Ambrogio and A. Fregolent "LRU E LR E # T (gRU gR ) D " LRU E LR E # T 2 4 BRU E T BR E T 3 5 D0 (3.5) In the dual assembly, the total set of DoFs is retained, i.e. each interface DoF appears twice. Since Eq. (3.5) is always satisfied, the three-field formulation reduces to: 8 ˆˆˆˆˆ< ˆˆˆˆˆ: "ZRU 0 0 ZR #(uRU uR ) C 2 4 BRU E T BR E T 3 5 D( fRU fR ) hBRU C BR Ci ( uRU uR ) D 0 (3.1 ) (3.2) or in more compact form: 8 < : ZuCBE T Df BCuD0 (3.1 ) (3.2) To eliminate , Eq. (3.1 ) can be written: uD Z 1BE T CZ 1f (3.1 ) which substituted in Eq. (3.2) gives: BCZ 1BE T DBCZ 1f (3.6) fromwhich , to be back-substituted in Eq. (3.1 ), is found as: D BCZ 1BE T CBCZ 1f (3.7) To obtain a determined or overdetermined matrix for the generalized inversion operation, the number of rows of BC must be greater or equal than the number of rows of BE, i.e. NC NE nc (3.8) Note that, if NC >NE, Eq. (3.6) is not satisfied exactly by vector given by Eq. (3.7), but only in the minimum square sense. This implies that also Eq. (3.2) is not satisfied exactly, i.e. compatibility conditions at interface are approximately satisfied. On the contrary, equilibrium is satisfied exactly due to the introduction of the connecting force intensities as in Eq. (3.4). By substituting inEq. (3.1 ), it is obtained: ZuCBE T BCZ 1BE T CBCZ 1f Df (3.9) Finally, ucan be written as uDHf, which provides the FRF of the unknown subsystemU: u D Z 1 Z 1BE T BCZ 1BE T CBCZ 1 f (3.10) i.e., by noting that the inverse of the block diagonal dynamic stiffness matrix can be expressed as: "ZRU 0 0 ZR #DZ 1 DHD" HRU 0 0 HR # (3.11) where HRU andHR are the FRFs of the assembled structure and of the residual substructure, it is:
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