A. Culla, W. D’Ambrogio, A. Fregolent and A. Schiavone COUPLED SYSTEM INTERNAL DOFS SUBSYSTEM COUPLING DOFS SUBSYSTEM INTERNAL DOFS A B FRFS AT COUPLING DOFS → NOISE + IDENTIFICATION ERRORS Fig. 1 Scheme of substructuring problem For the sake of simplicity, the explicit frequency dependence will be omitted. The equation of motion of the subsystems to be coupled can be written in a block diagonal format as: [Z]{u}={f }+{g} i.e. ZA [0] [0] ZB uA uB = f A f B + gA gB (2) The compatibility condition at the interface DoFs implies that any pair of matching DoFs uAl and u B m, i.e. DoF l on subsystemA and DoF mon subsystemB must have the same displacement, that is uAl −u B m =0. This condition can be generally expressed as: [B]{u}={0} i.e. BA BB uA uB =0 (3) where each row of [B] corresponds to a pair of matching DoFs. Note that [B] is, in most cases, a signed Boolean matrix and it can be written by distinguishing the contribution of the different subsystems. The equilibrium condition for constraint forces associated with the compatibility conditions implies that, when the connecting forces are added for a pair of matching DoFs, their sum must be zero, i.e. gAl +g B m =0: this holds for any pair of matching DoFs. Furthermore, if DoFk on subsystemA(or B) is not a connecting DoF, it must be gA k =0: this holds for any non-interface DoF. Overall, the above conditions can be expressed as: [L]T {g}={0} (4) where the matrix [L] is a Boolean localisation matrix. Note that the number of rows of [L]T is equal to the number of non-interface DoFs plus the number of pairs of interface DoFs. Eqs. (2-4) can be put together to obtain the so called 3-field formulation [12]: 92
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