108 C. Beale et al. modal responses back to physical space at a larger set of response DOFs. A pseudo-inverse is used to develop transformation matrix, which implies that the technique is susceptible to the selection of the measured a-set DOF and the presence of noise in the measured responses. Two systems, one simple and one complex, are used to demonstrate expansion and investigate the effects of gauge placement (DOF selection) and noise on the accuracy of the expansion results with and without regularization. While these are contrived systems implemented as finite element models, they exhibit characteristics similar to the systems encountered in practice making them proper candidates for investigating the factors that affect the accuracy of the expansion results. The simple system, representative of a tuning fork, is used to demonstrate how gauge placement affects the conditioning of the expansion transformation matrix and accuracy of the expansion results. The complex system consists of three coupled components and is analyzed with a fixed-set of gauge locations (a-set DOF) where one component is entirely unmeasured. This represents typical field instrumentation configurations where certain components cannot be fully instrumented due to space limitations, cabling limitations, or other practical considerations. This complex system is used to examine the effects of noise on expansion results, and to demonstrate how regularization techniques can be used to improve expansion results when there is noise on the a-set DOF responses, which is typical of real measurements. These models demonstrate several important characteristics of expansion problems. First, the test engineer must choose the locations of the a-set DOFs (measured gauge locations) carefully to maximize the information which is used in the expansion process. Next, the quality of the measurements should be high to avoid noise propagation and amplification issues during the expansion process. If noise is present and non-trivial, regularization techniques such as Tikhonov regularization or singular value regularization can be used to significantly improve expansion results by minimizing the noise amplification effects in the expansion process. 11.2 Theory Generally, expansion involves a transformation from the set of known responses (a-set DOFs) to a larger set of unknown responses (n-set DOFs), via a transformation matrix such as {xn(t)}=[T] {xa(t)}, (11.1) where the known and unknown responses are {xa} and {xn}, respectively, and the transformation matrix is [T]. This transformation matrix can take many forms and in a fundamental sense is simply a mechanism for relating response at a few points to the response at many points, often via some shape-based spatial relationship. The a-set andn-set responses can be time histories, as shown in Eq. 11.1, frequency domain quantities such as linear spectra and cross- power spectral density (CPSD) matrices, or mode shape vectors. Expansion of linear spectra is identical to Eq. 11.1, except the response quantities are linear spectra vectors: {Xn (ω)}=[T] {Xa (ω)}, (11.2) where the known and unknown linear spectra are Xa(ω) andXn(ω), respectively. Expansion of mode shape matrices from the a-set DOF, [Ua], to the n-set DOF, [Un], is also similar: [Un] =[T] [Ua] . (11.3) Expansion of CPSDs at the a-set DOF, [Saa], to the n-set DOF, [Snn], is simply an outer product of Equation 11.2 because a CPSD is an outer product of linear spectra vectors: [Snn (ω)] =[T] [Saa (ω)] [T] T. (11.4) 11.2.1 SEREP Expansion The SEREP method [1] uses a transformation matrix developed from the mode shape vectors of the system. During the transformation, a modal projection is performed on the a-set responses, {xa}, to estimate the modal responses, {p}, using a
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