ˆkB ZC I ZC † WT Z A ZA WZC † ª¬ º¼ Z C (45) Defining WK ZA WZC † , Eq. (45) becomes ˆk B ZC I WK T W K ª¬ º¼ ZC (46) For ˆkB to be positive semi-definite, the matrix I WK T W K must be positive definite. This condition has the same form as that used in the mass approximation. The singular values of WK must therefore be less than 1.0. As in the case of mass, modes can now be ranked with respect to the substructure B stiffness approximation. 2.4.1. Rank Substructure C Experimental Modes As in the previous case of mass, it is assumed that there is a given set of transmission simulator modes to be used in the construction of WK . The objective is to determine the contribution of the experimental modes of substructure C to the singular values of WK . Define the nA unA matrix QCK WK WK T . Let L CK represent a matrix of the eigenvalues of QCK sorted in descending order, and let <CK be the corresponding eigenvectors. Analogous to the mass problem, define the expression eCK WK T< CK ª¬ º¼ ^2 (47) Each row represents one of the substructure C experimental modes, and each column represents one of the eigenvalues of QCK . Each column of CK e adds to the corresponding eigenvalue of QCK . Therefore, term eCKij gives the contribution of the ith experimental mode of substructure C to the jth eigenvalue of QCK . The metric can also be normalized as was done for eAM and CM e and used to determine which modes contribute most to the offending eigenvalues of the stiffness matrix. 2.4.2. Rank Substructure A Finite Element Model Modes One can also rank the modes of the transmission simulator A to the offending eigenvalues of the stiffness matrix. The derivation follows the same form as those presented previously with QAK WK T W K ˆk A, resulting in the following metric. eAK WK<AK > @ ^2 (48) 3. Numerical Examples This section applies the proposed metrics to two different systems. The first is an assembly of beams which was studied in a prior publication [8]. Many of the results presented there are repeated here because they are important to illustrate the issues that can arise. The 122
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