Walter D’Ambrogio and Annalisa Fregolent HA = HAB [0] [0] − HB − HAB [0] [0] − HB ⎡ ⎣ BAB E T BB E T ⎤ ⎦× ×⎛ ⎝ BAB C BB C HAB [0] [0] − HB ⎡ ⎣ BAB E T BB E T ⎤ ⎦ ⎞ ⎠ + × × BAB C BB C HAB [0] [0] − HB (24) With the dual formulation, the rows and the columns of [HA] corresponding to all the 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. In Eq. (24), the product of the three matrices to be inverted can be defined as interface flexibility matrix. The interface flexibility matrix can be rewritten as: BAB C BB C HAB [0] [0] − HB ⎡ ⎣ BAB E T BB E T ⎤ ⎦ = = BAB C HAB BAB E T − BB C HB BB E T (25) It can be noticed that BAB C HAB BAB E T = ˆHAB where[ ˆHAB] is a subset of the FRF matrix of the coupled structure: pre-multiplication by [BAB C ] extracts rows at compatibility DoFs, and post-multiplication by [BAB E ] extracts columns at the equilibrium DoFs. Similarly, BB C HB BB E T = ˆHB where [ ˆHB] is the FRF of the residual structure at the same DoFs as above. Therefore, the interface flexibility matrix becomes: BAB C HAB BAB E T − BB C HB BB E T = ˆHAB − ˆHB (26) Note that, whenever compatibility and equilibrium DoFs are the same, [ ˆHAB] and [ ˆHB] can be seen as the inverse of the condensed dynamic stiffness matrices of the coupled structure [ ˆZAB] and the residual structure [ ˆZB], respectively. In this case, the interface flexibility matrix can be rewritten as: 56
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