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 rows of [BC] ≥number of rows of [BE] i.e. NC ≥NE ≥nc (21) which is the same as (12). Note that, if NC >NE, Eq. (19) is not satisfied exactly by vector {λ} given by Eq. (20), but only in the minimum square sense. This implies that also Eq. (5) is not satisfied exactly, i.e. compatibility conditions at interface are approximately satisfied. On the contrary, equilibrium is satisfied exactly due to the introduction of the connecting force intensities {λ}as in Eq. (15). Substituting {λ}inEq. (4∗∗), it is obtained: ZAB [0] [0] − ZB uAB uB +⎡ ⎣ BAB E T BB E T ⎤ ⎦× ×⎛ ⎝ BAB C BB C ZAB [0] [0] − ZB − 1⎡ ⎣ BAB E T BB E T ⎤ ⎦ ⎞ ⎠ + × × BAB C BB C ZAB [0] [0] − ZB − 1 f AB f B = f AB f B (22) Finally, {u} can be written in the form{u} =[H]{f }, which provides the FRF of the unknown subsystemA: uAB uB = ZAB [0] [0] − ZB − 1 − ZAB [0] [0] − ZB − 1⎡ ⎣ BAB E T BB E T ⎤ ⎦× ×⎛ ⎝ BAB C BB C ZAB [0] [0] − ZB − 1⎡ ⎣ BAB E T BB E T ⎤ ⎦ ⎞ ⎠ + × × BAB C BB C ZAB [0] [0] − ZB − 1 f AB f B (23) i.e., by noting that the inverted dynamic stiffness matrices [ZAB]−1 and [ZB]−1 are equal to the FRF matrices [HAB] and[HB] at the full set of DoFs: 55
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