B A u p u p b i i b Fluid Structure Fig. 1 Substructuring into two components Note that, contrary to what was done in the previous section, here the partitioning is not done according to the physics. Thus a component can include both acoustic and mechanical degrees of freedom. The representation of the blocks in the matrices are provided in equations below for completeness : M= ⎡ ⎢⎢ ⎢⎢ ⎣ Mbb s 0 −Kbb sf T Mbb f Mbi s 0 −Kbi sf T Mbi f Mib s 0 −Kib sf T Mib f Mii s 0 −Kii sf T Mii f ⎤ ⎥⎥ ⎥⎥ ⎦ , (23) K=⎡ ⎢⎢ ⎢⎣ Kbb s Kbb sf 0 Kbb f Kbi s Kbi sf 0 Kbi f Kib s Kib sf 0 Kib f Kii s Kii sf 0 Kii f ⎤ ⎥⎥ ⎥⎦ . (24) The partitioning of the dofs follows accordingly qb T qi T = ub T pb T ui T pi T . (25) In literature, different approaches have been used for the construction of the Craig-Bampton reduction basis. In [10], the authors use a partitioning approach which is different from the above, namely, they separate the structural and acoustic dofs and apply the standard Craig-Bampton method to this kind of partitioning while omitting some coupling terms in the constraint modes block of the basis. In the same work, they also propose a different approach which is similar to the ideas proposed in the previous section, namely, they use different right and left projection bases. These bases include different normal modes and constraint modes for right and left bases representations. In the current work, we follow the same ideas as in [10] and [11] for the constraint and vibration modes, besides we propose to add some pseudo-vectors to the basis or replace the fixed interface modes with these pseudo-vectors. These pseudo-vectors represent the global behaviour so they are expected to provide important information on how the neighbouring components Umut Tabak and Daniel J. Rix 270
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