Direct decoupling of substructures using primal and dual formulation To obtain a determined or overdetermined matrix for the generalized inversion operation, the following condition must be satisfied: number of columns of [LE] ≥number of columns of [LC] i.e. (NAB+NB−NE) ≥(NAB+NB−NC) ⇒ NC ≥NE ≥nc (12) where it is worth to recall that nc ia the number of coupling DoFs. Note that, if NC >NE, Eq. (10) is not satisfied exactly by vector {q} given by Eq. (11), but only in the minimum square sense. This implies that also Eq. (6) is not satisfied exactly, i.e. equilibrium conditions at interface are approximately satisfied. On the contrary, compatibility is satisfied exactly due to the unique choice of {q}. From Eq. (11), the FRF of the unknown subsystemAcan be written as: HA =⎛ ⎝ LAB E LB E T ZAB [0] [0] − ZB LAB C LB C ⎞ ⎠ + LAB E LB E T (13) HA = LAB E T ZAB LAB C − L B E T ZB LB C + LAB E T LB E T (14) With the primal formulation, the columns of [HA] corresponding to the equilibrium interface DoFs appear twice. Furthermore, when using an extended interface, [HA] contains some meaningless rows and columns: those corresponding to the internal DoFs of the residual substructure B. Obviously, only meaningful and independent entries are retained. 2.2 Dual formulation In the dual formulation, the total set of DoFs is retained, i.e. each interface DoF is present as many times as there are substructures connected through that DoF. The equilibrium condition g(r) l +g (s) m =0 at a pair of equilibrium interface DoFs is ensured by choosing, for instance, g(r) l =−λand g (s) m = λ. Due to the construction of [BE], the overall interface equilibrium can be ensured by writing the connecting forces in the form: gAB gB =−⎡ ⎣ BAB E T BB E T ⎤ ⎦{ λ} (15) where {λ}are Lagrange multipliers corresponding to connecting force intensities. The interface equilibrium condition (6) is thus written: 53
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