7 Tips, Tricks, and Obscure Features for Modal Parameter Estimation 71 5 10 15 20 25 30 35 40 45 50 Frequency (Hz) 100 101 102 Acceleration/Excitation Force (G/lbf) Mode Indicator Function Fig . 7. 5 Complex mode indicator function (CMIF) showing constant frequency bands for residue estimation a good group of poles from a stability, or consistency, diagram, while the second step is generally considered the easy part with not much more to do except sometimes moving the frequency range cursors and maybe changing the default selection for the residuals to include in the solution for the residues. However, there are some other, obscure features available that may lead to better results in the residue-estimation process. The frequency range to include in the residue-estimation process should naturally encompass the span of the frequencies of the selected poles, (e.g., from 10% below the lowest frequency to 10% above the highest frequency). Of course, any frequency ranges or spectral lines excluded from the pole estimation should also be excluded from the residue estimation. While the default might be to include all other spectral lines between the frequency range cursors, there may be wide valleys between the peaks not containing any modes (i.e., not containing any response that would help the cause of estimating residues). Several of these regions can be found in Fig. 7.3, and at least two more in Figs. 7.5 and 7.6. So instead of using everything, a more judicious approach might be to use only frequency bands in the vicinity of the selected poles, which are designated by the triangular markers in Fig. 7.6. There are several ways to define these bands: as a number of spectral lines, as a frequency bandwidth in Hz, or as a percentage of the frequency of each pole. For the first two options, the frequency band will have a constant width, but for the third, the frequency band will be wider for higher-frequency modes, which are typically less densely spaced, and will be narrower for lower-frequency modes, which may yield too few spectral lines. In this case, there needs to be some minimum number of spectral lines to ensure a credible solution. In the example shown in Fig. 7.5, the gray patches represent the ±0.5Hz (±10 spectral lines) frequency bands, where for closely spaced modes the bands may overlap. In the example shown in Fig. 7.6, the gray patches represent the ±5% frequency bands, where there is more overlap of the higher-frequency bands. In the example shown in Fig. 7.7, the mode at 7 Hz and the modes between 11 and 14 Hz were not well excited by whatever set of inputs were chosen for this dataset. While these frequency ranges do not necessarily contain “bad data” that should be excluded entirely, no viable modes are found in them, so including those spectral lines is not going to help get better estimates of the residues for the four selected modes designated by the triangular markers. In this case, an alternative would be to use the 5% frequency bands represented by the gray patches. 7.4 Solving for Normal Modes Residue estimation starts with the partial fraction form of a displacement-over-force FRF as in Eq. (7.2), where H is the FRF, A r are the residues, λ r are the poles, 2 N in the upper limit of the summation indicates conjugate pairs of modes, and the standard, single-term upper and lower residuals (RL and RU) have been added to account for out-of-band modes.
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