Evaluation of Joint Modeling Techniques Using Calibration and Fatigue Assessment of a Bolted Structure 43 a) b) Fig. 14 Model 1a initial study MAC plot, free-free test 1 (a) and free-free test 2 (b) Table 4 Model 1a free-free comparison (free-free test 1) Test mode Test 1 Freq (Hz) Test 2 Freq (Hz) Model 1a Mode Model 1a Freq (Hz) Freq Err 1% Freq Err 2% MAC 1 MAC 2 1 193.6 233.2 1 291.6 50.6 25.0 0.980 0.896 2 356.4 307.3 2 404.0 13.4 31.4 0.949 0.857 3 371.2 407.7 3 448.4 20.8 10.0 0.969 0.986 4 1201.5 1212.7 4 991.4 −17.5 −18.2 0.953 0.935 5 1307.3 1259.5 6 1230.2 −5.9 −2.3 0.992 0.962 6 1517.2 1520.0 7 1507.6 −0.6 −0.8 0.999 0.973 7 1565.8 1644.5 5 1130.8 −27.8 −31.2 0.989 0.964 Table 5 Model 1 interface Tied Joint stiffness parameters Parameter Initial Model 1a Modified Model 1b Units kx 6.13e6 5.00e7 lbf/in ky 1.00e6 9.50e6 lbf/in kz 1.00e6 9.50e6 lbf/in krx 1.00e7 5.00e6 in·lbf/rad kry 1.00e7 1.40e6 in·lbf/rad krz 1.00e7 1.00e8 in·lbf/rad Interface contact size Full surface (1.25 ×1.25) 1.125 (diameter) in 5.2 Free-Free Comparison Taking Model 1b as the baseline calibrated model, mode shapes and frequencies were calculated for each of the other three models outlined in Sect. 4, and results were compared to test data, summarized in Table 7. Comparing the results, MAC values were similar between all cases, though slightly lower for Model 4b. Frequency results were close as well, except that Model 2b and 4b had larger error in modes 1 and 3. Model 3b matched best with the calibrated Model 1b and the test. Results from the calibration study show that a solid mesh representation of the bolt may not be required for structural dynamic models, since the single Tied Joint model (Model 3) was able to provide almost identical results. In addition, the two-spring model (Model 2) does not provide any additional advantages. Furthermore, the Tied Data model (Model 4) is the least accurate in terms of frequency and may not be suitable, especially if a model of the fastener is required. Overall, the study indicates that modeling bolted joints with a single Tied Joint spring element appears to be a low fidelity yet accurate technique that exhibits similar dynamic behavior to the a fully detailed bolt with a Tied Joint interface.
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