which agrees with Eq. (22). Hence, this analysis has shown that the MCFS substructure modal coupling technique is equivalent to approximating the transmission simulator mass and stiffness matrices using a SEREP TAM representation for the measured degrees of freedom and then removing them from a FEM model for C. The accuracy of the approach is dependent upon the accuracy of the approximation ICA | ˆI CA . 2.3. Ranking Modes based on Mass Approximation Given a set of experimental modes for substructure C and a set of analytical modes for transmission simulator A, it is desirable to determine how each mode contributes to negative mass and stiffness in the estimate for B. Once the modes that contribute most have been identified the problem can be remedied as discussed in Section 3, sometimes by simply removing them from the model for C. From the previous analysis, the approximation for the mass matrix of substructure B is given by ˆmB I ˆmA , where the transmission simulator mass is approximated by ˆmA W T W. Matrix W has dimension n A unC and was previously shown to have the form W IAm † I Cm IAm T I Am ª¬ º¼ 1 I Am T I Cm (36) Prior to ranking modes, it is interesting to develop a better understanding of what matrix W represents. It was previously illustrated that for the MCFS synthesis method to be accurate, the experimental modes at the measurement locations for substructure C, ICm, must lie in the range space of the transmission simulator modes at the same measurement locations, IAm. As mentioned previously, if this is the case then one can write the modes of C as linear combinations of the modes of A using ICm IAm J, where J is the corresponding coefficient matrix. Premultiplying this expression by the generalized inverse of IAm gives † † Am Cm Am Am I I I I J J (37) and since † Am Cm W I I , W and J can be interchanged and one can write, ICm IAm W (38) A least squares solution for W gives back the expression in Eq. (36). If this solution is then substituted into Eq. (38), the result is ˆI Cm IAm IAm T I Am ª¬ º¼ 1 I Am T I Cm PAm ICm (39) which is consistent with Eq. (5). Therefore, the use of W in the MCFS synthesis approach produces an approximation of the substructure C experimental modes at the measurement locations, ˆICm, which minimizes the norm of the error, e ICm ˆI Cm. Matrix W then just represents the linear combination of transmission simulator modes that produces the best fit to the experimental modes. 119
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