kr T T ZC 2 0 0 ZA 2 ª ¬ « « º ¼ » » T ZC 2 WT Z A 2 W (9) since the matrices in Eqs. (8) and (9) actually represent approximations for the experimental based substructure B mass and stiffness matrices using the modal coordinates of C as a basis (see eq. (6) above). The synthesized equation of motion for experimental based substructure B is then ˆ ˆ 0 B C B C m q k q (10) with ˆmB IC W T W (11) and ˆk B ZC 2 WT Z A 2 W (12) In order for the experimental substructure B to be physically realistic, the mass matrix and the stiffness matrix must be positive definite, and positive semi-definite, respectively. The following section presents an alternative derivation, which shows that the MCFS substructure modal uncoupling technique is equivalent to approximating the transmission simulator mass and stiffness matrices using a SEREP TAM representation [5] for the measured degrees of freedom and then removing the approximated transmission simulator mass and stiffness from a finite element model for C. The approach will only be accurate if the modes of the C system at all points on the transmission simulator can be accurately represented using the modes of the transmission simulator as a basis. The metrics for ranking the contributions of the subcomponent modes to negative mass and stiffness are presented in Sections 2.3 and 2.4 respectively. 2.2. Alternate Derivation of MCFS Uncoupling by Decomposition An alternative approach for deriving the relationships for the experimentally based substructure B mass and stiffness matrices is based on the more physically intuitive decomposition of substructure C. In practice, one does not have a finite element model for C, which is the reason that one is trying to perform experimental substructure uncoupling. However, if the finite element model for C were known, its equation of motion in physical coordinates could be written as, C C C C C M x K x F (13) where the displacement vector can be partitioned into degrees of freedom associated with substructure A and substructure B as xC xCA T x CB T ª ¬ º ¼ T . The physical mass matrix MC can be transformed to modal coordinates using 115
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