Smoothing experimental data in dynamic substructuring of built up systems ⎧⎨ ⎩ [Z]{u}={f }+{g} [B]{u}={0} [L]T {g}={0} (5) 2.1.1 Dual formulation in the frequency domain [12, 11] 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 gAl +g B m =0 at a pair of interface DoFs is ensured by choosing, for instance, gAl =−λ and g B m = λ. Due to the construction of [B], the overall interface equilibrium can be ensured by writing the connecting forces in the form: {g}=−[B] T {λ} (6) where {λ}are Lagrange multipliers corresponding to connecting force intensities. The interface equilibrium condition (4) is thus written: [L]T {g}=−[L] T [B]T {λ}={0} (7) Because [B]T is the nullspace of [L]T (see for instance [4]), Eq. (7) is always satisfied and the system of equations (5) becomes: [Z]{u}+[B] T {λ}={f } [B]{u}={0} (8) In matrix notation: [Z] [B]T [B] [0] {u} {λ} = { f } {0} (9) that is: ⎡ ⎢⎣ ZA [0] BA T [0] ZB BB T BA BB [0] ⎤ ⎥⎦ ⎧⎪⎨ ⎪⎩ uA uB {λ} ⎫⎪⎬ ⎪⎭ =⎧⎪⎨ ⎪⎩ f A f B {0} ⎫⎪⎬ ⎪⎭ (10) Note that [BA] and [BB] extract the coupling DoFs among the full set of DoFs. By eliminating {λ}, it is possible to obtain a relation in the form{u}=[H]{f }, which provides the FRF of the coupled systemAB[4]: {u}= [Z]− 1 −[Z]− 1[B]T [B][Z]−1[B]T − 1 [B][Z]−1 {f } (11) In expanded notation: 93
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