Walter D’Ambrogio and Annalisa Fregolent singular only when all data are known without errors or noise. In any case, the use of smart inversion techniques, e.g. the truncated SVD, allows to deal with the problem. 2. Because, as shown in [3] and recalled in appendix, [ ˆHAB] is singular at the resonant frequencies of the coupled structure ABwith interface DoFs grounded. 3. Because [ ˆHB] is singular at the resonant frequencies of the residual substructure Bwith interface DoFs grounded. Note that the resonant frequencies of the residual substructure B, with interface DoFs grounded are a subset of the resonant frequencies of the coupled structure AB with interface DoFs grounded. (This is also apparent from Fig. 2, showing that the unknown subsystemAand the residual subsystemBare independent one of another when interface DoFs are grounded.) Therefore, the interface flexibility matrix is twice singular at the resonant frequencies of the residual subsystemB with coupling DoFs grounded. It should be noted that, differently from what could be expected from previous statements, the interface flexibility matrix is not singular at the resonant frequencies of the unknown subsystemAwith interface DoFs grounded. In fact these are cancelled by the frequencies at which the determinant of [ ˆZB] −[ ˆZAB]=−[ ˆZA] tends to infinity, that (see Appendix) are the resonant frequencies of the unknown subsystemAwith interface DoFs grounded. In cases 2 and 3, if noise is present as usual, the problem becomes ill conditioned but smart inversion techniques are not able to remove ill conditioning. INTERNAL DOFS UNKNOWN SUBSYSTEM COUPLING DOFS RESIDUAL SUBSYSTEM INTERNAL DOFS A B Fig. 2 Structure with (extended) interface DoFs grounded A particular case is that of standard interface where Eq. (30) reduces to: ˆHAB cc − ˆHB cc = ˆHAB cc ˆZB cc − ˆZAB cc ˆHB cc (31) and singularity occurs only, but twice, at the resonant frequencies of the residual substructure Bwith coupling DoFs grounded. 58
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