Table 2 Comparison of Predicted and Measured Modal Frequencies Modal Frequency (Hz) Mode Shape FEA Model Experimental Average Percent Difference A 483 504.5 4.35 B 652 666.25 2.16 C 1138 1109.5 2.54 D 1082 1147.5 5.88 E 1875 1973.5 5.12 F 2303 2230 3.22 In order to determine the modal frequencies and shapes that are summarized in Table 1 and Table 2, the Complex Mode Indicator Function (CMIF) method was used to analyze the data. This method utilized all six of the vertical sensor reference channels to analyze the data, and also made it possible to identify repeated roots in the data. Other common methods for modal parameter estimation, such as peak-pick, do not allow for the detection of repeated roots. The CMIF method uses the Singular Value Decomposition (SVD) of the frequency response function matrix to identify peak frequencies (natural frequencies) in the singular value matrix and their corresponding modal vectors. This method was utilized for all four modal tests to determine and animate the mode shapes. It was predicted that an increase in bolt preload level should result in some increase in each of the system’s modal frequencies. This hypothesis was investigated by comparing the frequencies at which each of the fixture's first six modes (named A through F) occurred for each bolt preload level and impact amplitude. In a linear system, modal properties should not depend on impact amplitude, but at low preloads it was expected that the assumption of linearity would be poorer than at high preloads. The modal frequencies of the first six modes for the system for each of the four tests are listed in Table 3. Table 3 Modal Frequencies as a Function of Bolt Preload and Impact Amplitude Modal Frequencies (Hz) 1000N Bolt Preload 20000N Bolt Preload Mode Shape Low Amplitude High Amplitude Low Amplitude High Amplitude Predicted A 498 498 511 511 483 B 655 654 678 678 652 C 1036 1013 1195 1194 1138 D 1147 1146 1149 1148 1082 E 1973 1973 1975 1973 1875 F 2193 2152 2289 2286 2303 As expected, there is an increase in modal frequency for each mode when the bolt preload is increased, although this increase is nearly negligible for modes C and D. This result is most likely due to the fact that little motion along the axis of the bolts is present in these modes. The large jump in frequency for mode C when the preload is increased is particularly interesting. This mode is the third mode of vibration for the low preload value tests, but this mode becomes the fourth mode of vibration for the tests with the high preload value. This switch is due to geometric stiffening in the bolts as the preload is increased. Mode C exhibits the most motion along the bolt axis, and is, therefore, the most affected mode due to the effects of bolt stiffening. It was initially believed that the nonlinearity due to impact amplitude could be observed by examining modal frequencies. It was expected that the nonlinearities would be larger for the low preload value than the high preload value because the higher preload level increases coupling thereby allowing less relative motion. A comparison of modal frequencies from impacts of different amplitudes is shown in Table 4. Although the changes in modal frequency for the low preload level were greater, the difference between the two preload levels is nearly negligible. This does not indicate that the two cases have the same amount of nonlinearity. As discussed earlier, the literature suggests that the largest changes in modal frequency due to nonlinearity normally occur at higher frequency modes. Another method for characterizing nonlinearity was needed based on these results. 571
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