2.3.2. Rank Substructure A Finite Element Model Modes In this subsection, it is assumed that there is a given set of experimental modes for substructure C to be used in the construction of W. The objective is to determine the contribution of the transmission simulator modes, substructure A, to the singular values of W, such that transmission simulator modes can be included or excluded, depending on their contributions. Analogous to the previous subsection, define the nC unC matrix QAM W T W ˆm A. Let LAM represent a matrix of the eigenvalues of QAM sorted in descending order, and let <AM be the corresponding eigenvectors. Note that the eigenvalues of QAM are also the squares of the singular values of W. Therefore, determining the contributions of the transmission simulator modes to the eigenvalues of QAM is also equivalent to determining their contributions to the singular values of W. As before, define the expression > @^2 AM AM e W < (42) Now each row represents one of the substructure A transmission simulator modes, and each column represents one of the eigenvalues of QAM . Each column of eAM adds to the corresponding eigenvalue of QAM , so term eAMij gives the contribution of the ith transmission simulator mode to the jth eigenvalue of QAM . Normalizing with respect to the eigenvalues of QAM yields > @^2 1 AMn AM AM e L W < (43) where eAMnij gives the fractional contribution of the ith transmission simulator mode to the jth eigenvalue. Using Eqs. (42) and (43), transmission simulator modes that contribute significantly to offending eigenvalues of QAM can be omitted, or the transmission simulator mode set can be truncated such that the resulting eigenvalues of QAM are less than 1.0. 2.4. Ranking Modes based on Stiffness Approximation In all of the cases analyzed to date, it has been the mass approximation of substructure B that has been most restrictive with respect to the proper sign definiteness of the resulting matrix after subtraction of transmission simulator A. However, the stiffness approximation for substructure B, ˆkB ZC 2 WT Z A 2 W, should also be examined to verify positive semi-definiteness. Note that both ZC and ZA will in general contain 6 zeros on the diagonal corresponding to rigid body modes. The generalized inverse of the diagonal matrix ZC then has the simple form ZC † 0 0 0 ZCe 1 ª ¬ « « º ¼ » » (44) in which ZCe is a diagonal matrix of the elastic experimental frequencies for substructure C. The stiffness approximation can then be written as 121
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