9 Modal Validation of Academic Bladed Disk with DIC 63 for each individual point. Note, it was assumed the time step between each image was exactly the inverse of the frame rate or 0.2ms. For the purpose of validating DIC with SLV measurements, three mode shapes were compared via a modal assurance criterion or MAC. A MAC is a quantitative score ranging from 0 to 1 and is a measure of the similarity between two normalized mode shapes. The equation for MAC can be seen in Eq. (9.1): MAC= |σT DIC σSLV| 2 σT DIC σDIC σ T SLV σSLV (9.1) where σDIC and σSLV are simply normalized vectors of the mode shapes found via DIC and PSV, respectively, and superscript T represents the transpose operator [3]. The frequencies and corresponding MAC scores comparing the similarity between mode shapes found from DIC and SLV are reported in Table 9.1. However, it is important to note that the points taken along the blade by both measurement locations were not precisely at the same location. Therefore the MAC score was calculated by comparing third-order polynomial fits of both DIC and SLV data so that modal response along the length of the blade could be directly compared. The adjustedR 2 terms for each fit is also presented in Table 9.1, and no less than 0.966 which demonstrates each fit is a good approximation of the collected data. The MAC scores presented in Table 9.1 indicates good agreement, or greater than 0.8 MAC score, between DIC and SLV. Additionally in Table 9.1, the maximum of the (|W|) displacements are reported to present a reference to the level of Wdisplacements observed. This is an important note as often TWE measurements are typically within the nano- to micrometer range. For qualitative comparison, the normalized mode shapes of Ware presented in Fig. 9.2 and again indicate good agreement between DIC and SLV measurements of mode shape. 9.4 Conclusions The work presented demonstrates that DIC is an accurate and effective alternative method to laser vibrometry for measuring mode shapes with amplitudes on the order of nanometers with MAC scores greater than 0.95. Additionally, experimental testing time may be reduced by a linear scaling factor equal to the number of points of interest defined when using a SLV. Table 9.1 MAC score between DICandSLV Freq. (Hz) MAC max(|W|) (nm) DICR 2 adj SLVR 2 adj 1043.8 0.992 276 0.998 0.985 1131.9 0.979 109 0.986 0.994 1360.6 0.980 228 0.984 0.968 40 60 80 100 120 140 -1.2 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 Fig. 9.2 Mode shape comparison of DIC and SLV at 1131.9 Hz
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