72 L.G. Horta et al. 1 2 3 4 5 6 7 8 9 10111213141516 11.57 19.38 31.60 34.13 52.78 74.53 103.20 156.04 223.14 223.23 230.36 230.79 232.33 232.50 234.47 243.86 Model Mode Number Identified Model Frequencies (Hz) Baseline Model Frequencies (Hz) 10.07 16.88 38.25 73.02 114.87 133.51 142.08 143.97 158.75 224.62 265.46 306.74 323.60 370.14 408.22 416.32 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Fig. 7.5 Orthogonality of test versus LS-DYNA predicted modes Frequency response functions from the vibration tests with 34 accelerometers were analyzed using the MATLAB SOCIT (System Observer Controller Identification Toolbox) [6] and the Frequency Domain Identification Toolbox [7]. Frequency data were analyzed to determine the experimental mode shapes and frequencies used to compare with analysis. The results of this analysis are reported next. 7.4.2 Comparison of Modal Test Results with Analysis Figure 7.5 shows orthogonality values between the identified vibration mode shapes and the LS-DYNA predicted mode shapes. Readers should note that although orthogonality is used again, in this case the comparisons use vibration modes instead of impact shapes, as described earlier. Figure 7.5 also lists the frequencies of the identified modes and the predicted frequencies from LS-DYNA. Circled in red are the two target modes based on pre-test predictions. All orthogonality results are weighted using the reduced mass matrixMsuch that the orthogonality matrix is computed as 0D˚T dM˚t,where ˚d are the mode shapes from an implicit solution from LS-DYNA and ˚t are the mode shapes from the modal test. Pairing of test and analysis modes was based on orthogonality values (e.g. modes with orthogonality values near 1.0 have similar shapes). In some cases, one analysis mode matched several experimental modes, which is probably caused by an insufficient number of sensors. For our purposes, if an analysis mode appears to be similar to an identified mode(s) (i.e. high orthogonality value), then it is assumed that the mode exists but was not well identified. As shown in Fig. 7.5, the first analysis target mode at 73 Hz appears to be similar (i.e. high orthogonality value) to identified modes at 53, 34, and 32 Hz. Furthermore, the LS-DYNA predicted mode at 408 Hz appears to be similar to identified modes in the 230–234 Hz range. Similarly, the second target mode at 144 Hz appears to be similar to an identified mode at 74.5 Hz. Based on these results, it should be apparent that the LS-DYNA model is consistently over-predicting the frequencies of the fuselage vibration modes and therefore changes to the model should seek to reduce all predicted frequencies. Post-test examination of the baseline model revealed that frequency results were very sensitive to the boundary condition. In particular, the simply supported boundary condition, which was initially represented as a pin condition along the line of nodes in contact with a tube placed under the fuselage, had to be modified so translations were only constrained at four points. After revising the boundary condition the LS-DYNA model produced the results shown in Fig. 7.6. The two model critical frequencies have now dropped to 62 and 105 Hz. Animations of these two critical modes showed that motion occurred primarily on the floor. Although the pre-test analysis showed that exciting the fuselage at the top was adequate, during testing it was difficult to get enough energy into these
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