30 A. Singh et al. Table 4.2 Modes of a single beam (half of the S4 assembly) vs. model frequencies Mode Description Experimental frequency (Hz) Single beam model frequency (Hz) Percent error (%) 1 1st Bending 177.87 177.29 −0.33 2 2nd Bending 497.84 498.09 0.05 3 1st Stiff Bending 576.00 576.84 0.14 4 2nd Stiff Bending 979.84 988.44 0.88 5 Torsion 1474.67 1471.22 −0.23 6 – 1556.46 – – 7 Torsion 1585.58 1595.47 0.62 Table 4.3 Single beam material property updating Nominal Optimal Elastic Modulus 29,000 (ksi) 27,245 (ksi) (−6.05%) Poisson Ratio 0.29 0.29 (0.00%) 4.3.1 Single Beam Calibration Prior to calibrating the whole-joint models with linear springs, a finite element model for a single beam was used to calibrate the material properties for steel. Table 4.2 lists the adjusted elastic modulus and Poisson ratio for the single beam and shows the resulting agreement between the experimental natural frequencies and those of the model. All model frequencies were within 1% of the experimental frequencies, with the highest error in a second stiff bending mode. The densities of the model were calibrated by measuring the mass of the experimental beams and dividing by the volume of the FEM to ensure that the FEM has correct mass. The first four elastic bending modes were used to tune the elastic modulus and the fifth and seventh modes (torsion) were used to tune the Poisson ratio. Although, Mode 6 for the single beam was identified by the experimental setup, it was not matched within the FEM and thus was not used in the calibration of the beam. These properties were then used in all subsequent modeling (Table 4.3). 4.3.2 Whole-Joint Spring Calibration Linear model updating was performed for all four candidate models, and after using a Monte Carlo Simulation to minimize the objective function in Eq. 4.5, a set of values was found for each of the six spring constants. The springs on either end of the S4 beam were assumed to be identical. Tables 4.4 and 4.5 depict the percent error natural frequencies of the assembly after optimization and the spring stiffnesses for each of the cases respectively. The S4 Beam has several different types of mode shapes, and each is influenced by different springs depending on how the joint is loaded. The mode shapes shown in Table 4.4 can be used to deduce these differences. For example, Modes 1 and 5 involve opening of the joint and hence are most sensitive to the Z-direction translational stiffness, whereas Mode 4 is completely insensitive to the joint stiffness. While the overall agreement was similar for each candidate model, a few differences are noted between the results obtained using RBE3 and RBAR elements. Most notably, the models with RBE3s were not able to capture Mode 2 as accurately; this mode is sensitive to the axial stiffness of the joint, loading it in the fashion of the lap joints that have been studied in many prior works [6, 12]. The RBE3 models have very high values for Tx and yet they still under-predict the frequency of this mode. On the other hand, the RBE3 models do slightly better at predicting the frequencies of Mode 1, and this might have been expected since the RBE3 doesn’t artificially rigidize the interface and Mode 1 would tend to be sensitive to this because it bends the interface region. In comparing the results with the reduced and full interfaces, one can see that the reduced interface typically required higher spring stiffnesses than the full interface (e.g. consider RY in Table 4.5). This result makes sense, as reducing the interface area effectively decreases the stiffness of the joint region, and so the spring constants must be increased to compensate. To get a sense of how sensitive the natural frequencies are to the spring constants found in each case, the constants for the full interface RBAR case were used in the three other models and the natural frequencies were computed. As expected, the natural frequencies of modes 3 and 4 didn’t change significantly. However, the models gave frequency errors ranging from 1 to 7% for the other modes. For conciseness, these results are presented in the Appendix.
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