Smoothing experimental data in dynamic substructuring of built up systems COUPLED SYSTEM INTERNAL DOFS UNKNOWN SUBSYSTEM COUPLING DOFS RESIDUAL SUBSYSTEM INTERNAL DOFS FRFS AT COUPLING DOFS + FRFS AT SOME INTERNAL DOFS → NOISE + IDENTIFICATION ERRORS A B Fig. 2 Scheme of the decoupling problem the coupling DoFs between subsystems Aand B, but should as well include the internal DoFs of subsystemB. However, by taking into account that the problem can be solved by considering only 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 andB; • extended interface, including also some internal DoFs (i ⊆b) of the residual substructure; • mixed interface, including some coupling DoFs (d ⊆c) and some 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 NC be the number of interface DoFs on which compatibility is enforced. The compatibility condition can be generally expressed as: BAB C BB C uAB uB ={0} (14) where each row of [BC]= [BAB C ][BB C] corresponds to a pair of matching DoFs. Note that [BC] is, in most cases, a signed Boolean matrix. Before deriving the equilibrium condition, it should be noted that the interface DoFs involved in the equilibrium condition are not necessarily the same used to enforce the compatibility condition, as long as controllability between equilibrium and compatibility DoFs is ensured. If the compatibility and the equilibrium DoFs are not the same, the approach is called non-collocated [16], as opposite to the collocated approach, i.e. same compatibility and equilibrium DoFs. The equilibrium condition for constraint forces implies that, when the connecting forces are added for a pair of matching DoFs, their sum must be zero, i.e. 95
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