292 R.A. Caldwell Jr. and B.F. Feeny Xis a measurement ensemble of displacements, and Vis a measurement ensemble of velocities. Typically the mean of the time history of each sensor is subtracted from each of the data samples of that sensor. The general form of all ensembles used in this body work is XD 2 666 4 x1.0/ x1. t/ x1..N 1/ t/ x2.0/ x2. t/ x2..N 1/ t/ : : : : : : : : : : : : xM.0/ xM. t/ xM..N 1/ t/ 3 777 5 (26.21) where Mis the number of sensors and N is the number of samples. The time history for each sensor is organized in rows, such that each column is a set of samples at each sampling interval. Starting with the SOD eigenvalue problem, as R‰ƒDS‰ and substituting in the expressions for Rand S, leads to XXT ‰ƒDVVT ‰: Chedize and Zhou [4] used the approximation V ŠXDT, and showed that XDDTXT Š XAT, where Dis a matrix that performs a simple finite difference numerical derivative. Plugging this intoS, the EVP becomes XXT ‰ƒD XAT ‰: Now noting that Ais an ensemble of accelerations, then fromMACKXD0, we have AD M 1KX. Then XXT ‰ƒDXŒM 1KX T ‰ or XXT ‰ƒDXXTKM T ‰ since Kand Mare symmetric. Assuming XXT has full rank, and is invertible, we have ‰ƒDKM T ‰: The assumption that XXT is full rank corresponds to a fully multimodal motion with N = M(or a sufficient noise level). Taking the inverse transpose of the above equation yields, ‰ƒDKM T ‰ T : Again from symmetry we have K‰ T DM‰ T ƒ: (26.22) Comparing Eq. (26.22) with Eq. (26.18) we see that the SCOD eigenvalues represent the structural eigenvalues, i.e. the modal frequencies squared, and that the SOD eigenvectors and the linear normal modes (LNMs) of the mass spring system are related as ˆD‰ T. These conclusions were reached in [4, 5]. Another relationship worth noting can be shown if we step back to equation (26.22) and let ˆDM‰, which yields, K.Mˆ/ T D M.Mˆ/ Tƒ KM T ˆ T DMM T ˆ T ƒ
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