As mentioned previously, in order for ˆmB to be a physically meaningful representation of the mass of substructure B, it must be positive definite, meaning its eigenvalues must be positive. This then implies that the eigenvalues of ˆmA W T W must be less than 1.0, which implies that the singular values of W must be less than 1.0. This condition may or may not be satisfied, depending on several factors. In the event that this is not satisfied, one would like to determine which modes of the subcomponents are causing the mass matrix to become negative. The following subsections present methods for ranking the contribution of each subcomponent mode to the negative mass or stiffness. These methods are loosely based on the Effective Independence method for ranking sensor locations in vibration testing [7]. 2.3.1. Rank Substructure C Experimental Modes In this subsection, it is assumed that there is a given set of transmission simulator modes to be used in the construction of W. The objective is to determine the contribution of the experimental modes of substructure C to the singular values of W, such that experimental modes can be included or excluded, depending on their contributions to singular values greater than 1.0. Define the nA unA matrix QCM WW T . Let L CM represent a diagonal matrix of the eigenvalues of QCM sorted in descending order, and let <CM be the corresponding eigenvectors. Note that the eigenvalues of QCM are the squares of the singular values of W. Therefore, determining the contributions of the experimental modes to the eigenvalues of QCM is equivalent to determining their contributions to the singular vales of W. Define the expression eCM W T< CM ª¬ º¼ ^2 (40) where > @^2 represents a term-by-term square. Each row represents one of the substructure C experimental modes, and each column represents one of the eigenvalues of QCM . It was shown in [7] that each column of eCM adds to the corresponding eigenvalue of QCM . Therefore, term eCMij gives the contribution of the ith experimental mode of substructure C to the jth eigenvalue of QCM . If eCM is normalized with respect to the eigenvalues of QCM eCMn W T< CM ª¬ º¼ ^2 L CM 1 (41) then term eCMnij gives the fractional contribution of the ith experimental mode of substructure C to the jth eigenvalue. Using Eqs. (40) and (41), experimental modes that contribute significantly to offending eigenvalues of QCM can be identified for possible omission from the C mode set. However, it is important to note that while eCMij gives the contribution if the ith C mode to the jth eigenvalue of QCM for the current mode set, deleting this mode does not mean that the corresponding eigenvalue will be reduced by this amount. As a mode is deleted, the matrix QCM must be recomputed for the new mode set, which in general may have different eigenvalues with a different distribution over the remaining modes. Therefore, once identified modes are deleted from the C mode set, the eigenvalues of the new matrix QCM must be calculated to make sure they are less than 1.0. 120
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