78 W. D’Ambrogio and A. Fregolent rotational DoFs, are already embedded in each FRF of the assembled system. Furthermore [7], qualitative criteria for an appropriate selection of the internal DoFs used to replace unwanted coupling DoFs are stated. In this paper, a procedure to optimally replace coupling DoFs with internal DoFs is developed, using indicators based either on the Frequency Response Function (FRF) or on the transmissibility between internal and coupling DoFs. Such indicators are tested with satisfactory results on an assembled structure made by a cantilever column with two staggered short arms (residual substructure) coupled to a horizontal beam (unknown substructure). 8.2 Direct Decoupling Using Dual Assembly The unknown substructureU(NU DoFs) is a portion of a larger structureRU(NRU DoFs). The known portion of the assembled structure RU, defined as residual substructure R (NR DoFs), is joined to the unknown substructure through a number of couplings (see Fig. 8.1). The degrees of freedom (DoFs) can be partitioned into internal DoFs (not belonging to the couplings) of substructure U(u), internal DoFs of substructure R(r), and coupling DoFs (c). The goal is to find the FRF of the unknown substructureUstarting from the FRFs of the assembled structureRUandof the residual substructure R. The dynamic behaviour of the unknown substructure Ucan be extracted from that of the assembled structure RU by taking out the dynamic effect of the residual subsystemR. This can be accomplished by considering a negative structure, i.e. by adding to the assembled structure RUa fictitious substructure with a dynamic stiffness opposite to that of the residual substructure Rand satisfying compatibility and equilibrium conditions. The dynamic equilibrium of the assembled structure RUand of the negative substructure is expressed in block diagonal format as: "ZRU 0 0 ZR#( uRU uR ) D( fRU fR ) C( gRU gR ) (8.1) where: • ZRU, ZR are the dynamic stiffness matrices of the assembled structure RUand of the negative structure, respectively; • uRU, uR are the vectors of degrees of freedom of the assembled structure RUand of the negative structure, respectively; • fRU, fR are the external force vectors on the assembled structure RUand on the negative structure, respectively; • gRU, gR are the vectors of disconnection forces between the assembled structure and the negative structure (constraint forces associated with compatibility conditions). Equilibrium of disconnection forces and compatibility must be considered at the interface between the assembled structure RU and the negative structure: such interface includes not only the coupling DoFs between substructures U and R, but includes as well the internal DoFs of substructure R(the blue part of the structure in Fig. 8.1). However, it is not required to consider the full set of these interface DoFs, because it is sufficient that the number of interface DoFs be not less than the number of coupling DoFs nc. Therefore, several options for interface DoFs can be considered: • standard interface, including only the coupling DoFs (c) between substructures Uand R; • extended interface, including also a subset of internal DoFs (i r) of substructure R; Fig. 8.1 Scheme of the direct decoupling problem
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