Walter D’Ambrogio and Annalisa Fregolent can be accomplished by adding to the coupled systemAB a fictitious subsystem with a dynamic stiffness opposite to that of the residual subsystemBand satisfying compatibility and equilibrium conditions. The dynamic equilibrium of the coupled systemABand of the fictitious subsystem can be expressed in block diagonal format as: ZAB [0] [0] − ZB uAB uB = f AB f B + gAB gB (1) where: • [ZAB], [ZB] are the dynamic stiffness matrices of the coupled systemAB and of the residual subsystemB, respectively; • {uAB)}, {uB)} are the vectors of degrees of freedom of the coupled systemAB and of the residual subsystemB, respectively; • {f AB}, {f B} are the external force vectors on the coupled systemABand on the fictitious subsystem, respectively; • {gAB}, {gB}are the vectors of connecting forces between the coupled system and the fictitious subsystem, and viceversa (constraint forces associated with compatibility conditions). According to this point of view, the interface between the coupled systemAB and the fictitious subsystem should not only include all the coupling DoFs between subsystems Aand B, but should as well include all the internal DoFs of subsystem B. However, by taking into account that the problem can be solved by considering just coupling DoFs, the number of interface DoFs should be greater than or equal to the number of coupling DoFs nc. Therefore, three options for interface DoFs can be considered: • standard interface, including only the coupling DoFs (c) between subsystems A and B; • extended interface, including also a subset of internal DoFs (i ⊆b) of the residual substructure; • mixed interface, including a subset of coupling DoFs (d ⊆c) and a subset of internal DoFs (i ⊆b) of the residual substructure. The compatibility condition at the (standard, extended, mixed) interface DoFs implies that any pair of matching DoFs uAB l and u B m, i.e. DoF l on the coupled systemAB and DoF mon subsystemB must have the same displacement, that is uAB l −u B m =0. Let the number of interface DoFs on which compatibility is enforced be denoted as NC. The compatibility condition can be generally expressed as: BAB C BB C uAB uB ={0} (2) where each row of [BC]= [BAB C ][BB C] corresponds to a pair of matching DoFs. Note that [BC] has size NC×(NAB+NB) and is, in most cases, a signed Boolean matrix. 50
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