30 R. Schultz and G. Nelson of the DUT as much as possible, so it is helpful to use some DUT test response as a basis for converting from the specification octave PSDs to the narrowband PSDs needed for the controller. If available, DUT field response would be the most representative, but DUT lab system identification response can work in many cases as well. Generalized specifications are beneficial in that they are simple, averaged representations of response. However, that simplification requires we “fill in the blanks” with lab DUT-specific information to perform a meaningful test. This can be done by utilizing the DUT system identification test data (PSDs) as a set of basis functions. Those DUT PSDs contain all the DUT-specific frequency, mode shape, and damping information we want to incorporate into the MIMO test. The basis PSDs allow the generalized PSDs to be mapped to narrowband PSDs with the following process: 1. Create the set of basis PSDs for the lab DUT (a set of Ncontrol DOFs xMfrequency lines narrowband PSDs). This most easily obtained from system identification test data on the lab DUT, but may come from other sources. 2. Octave average the basis PSDs to the same frequency bands as the generalized MIMO specification PSDs. 3. Zone-average the octave-averaged basis PSDs, making the averaged basis PSDs comparable to the generalized MIMO specification PSDs (size Nzones xMfrequency bands) 4. Compute the difference between the generalized MIMO specification PSDs and average basis PSDs as a set of scale factors (size Nzones xMfrequency bands). 5. Interpolate the scale factors to narrowband frequency lines (size Nzones xMfrequency lines). This can be done with, for example, spline interpolation on each zone’s PSD scale factors. 6. Apply the narrowband scale factors to the basis PSDs for each DOF in each zone where the DOFs in the same zone are scaled the same. This results in a set of basis PSDs which are scaled to match the generalized specification in terms of location (zone) and frequency band (size Ncontrol DOFs xMfrequency lines) Note that the objective of the above process is to scale the narrowband basis PSDs such that they match the zone-averaged and octave-averaged PSDs in the specification. The interpolation of scale factors from octave to narrowband can cause the octave-averaged scaled basis PSDs to be slightly different from the desired specification PSDs, so the process can be repeated for a few iterations and get better agreement. The mapping from generalized averaged specification PSDs to testable narrowband PSDs is shown in Figure 6, Figure 7, and Figure 8. Figure 6 shows the averaged specification PSD along with the narrowband PSD it was derived from (the System A MPE PSD) in comparison with the System B basis PSD. Note the differences in spectral content between the System A MPE and System B basis PSDs. Also note how the frequency bands capture similar features in both sets. For example, the band around 1000 Hz captures three peaks in both the System A MPE and the System B basis. Figure 7 shows an example of the octave and narrowband scale factors used to scale the basis PSD to the averaged specification PSD, including each scaling iteration. Figure 8 shows the resulting mapped specification in both narrowband and octave-averaged form compared with the basis PSD and desired octave-averaged specification. Two key observations: Fig. 6 Comparison of the System A MPE narrowband PSD vs. the System B basis PSD with the octave specification PSD to show band edges .
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