Direct decoupling of substructures using primal and dual formulation 2.3.2 Singularity using non collocated extended interface For the sake of simplicity, a special case of non collocated approach is considered where compatibility is enforced at DoFs c and i and equilibrium only at DoFs c. In this case, Eq. (30) becomes: ⎡ ⎣ ˆHAB cc ˆHAB ic⎤ ⎦− ⎡ ⎣ ˆHB cc ˆHB ic⎤ ⎦ =⎡ ⎣ ˆHAB cc ˆHAB ic⎤ ⎦ ˆZB cc − ˆZAB cc ˆHB cc It can be noticed that [ ˆHB] cc is singular at resonances of the residual substructure B with coupling DoFs grounded. Therefore, the use of non collocated approach does not prevent from this kind of singularity. 3 Primal vs dual formulation In this section, the expressions of the FRF of the unknown subsystemA provided by primal formulation and dual formulation are compared to establish whether they are the same or not, and under which conditions. The FRF of subsystemAprovided by the primal formulation, Eq. (13), can be rewritten in compact form as: [HA] P = [LE] T [Z][LC] + [LE] T (13) where the subscript Pstands for primal. If NC =NE, Eq. (13) becomes: [HA] P = [LE] T [Z][LC] −1 [LE] T (32) Premultiplying by [LE] T [Z][LC], one obtains: [LE] T [Z][LC][HA] P =[LE] T (33) The FRF of the unknown subsystemA provided by dual formulation, Eq. (24), can be rewritten in compact form for NC =NE: [HA] D =[H]−[H][BE] T [BC][H][BE] T − 1 [BC][H] (34) where the subscript Dstands for dual. Premultiplying by [Z], one obtains: [Z][HA] D =[I] −[BE] T [BC][H][BE] T − 1 [BC][H] (35) Premultiplying by LT E, one obtains: [LE] T [Z][HA] D =[LE] T −[LE] T [BE] T [BC][H][BE] T − 1 [BC][H] (36) 59
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