138 R. Schultz et al. Fig. 12.8 Coupled mode expansion (mode 66, 2608 Hz) from left to right: real structural mode, imaginary structural mode, real acoustic mode, and imaginary acoustic mode Fig. 12.9 Candidate a-DOF locations on the structural component (left) and acoustic component (right) Several uncertainties in the expansion method were discovered during this study. One uncertainty in particular was how to deal with the complex modes. Should the modes in the [Ua] and [Un] matrices be left as-is, fully complex? Or, should the real and imaginary portions be expanded separately? Both methods were tried, and it was found that leaving the modes as-is, fully complex, was needed for accurate transient expansion. It is possible that taking only the real or only the imaginary parts may be useful if measurements at the a-DOF are only in one component. That is, if measurements are only on the structure and the expansion is to some acoustic DOF, then the [Ua] matrix may benefit by being forced to be purely real or imaginary depending on the coupled nature of each mode. This is a possible avenue for future research. Expansion of the transient response to a forced impulse was calculated by using all modes but the rigid body modes, and calculating the transformation matrix with the complex mode shapes. The expansion results at two n-DOF are compared to the actual simulated responses in terms of the time response in Figs. 12.10 and 12.11. Two of the structure and two of the acoustic DOFs are shown for brevity; the results from these DOFs is representative of overall quality of the expansion. Although differences are seen in both comparison plots, the expanded time responses follow the same general trends as the actual time responses. The magnitude of the linear spectra of the time responses were also compared and are shown in Figs. 12.12 and 12.13. The spectra comparison plot shows that the expanded responses deviate more at higher frequencies, which could be an indicator of modal truncation. Perhaps the addition of more modes, or a more logical selection of modes could improve the expansion results.
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