180 G. Adkins et al. Table 17.5 Estimated damping ratios of the first torsion and OOP modes for all parts Orientation Mode UR DB TH ABS Solid TOR 1 Damping Ratio 0.012 ±0.004 0.015 ±0.002 0.009 ±0.004 ABS Solid OOP 1 Damping Ratio 0.010 ±0.002 0.020 ±0.004 0.009 ±0.001 ABS Lattice TOR 1 Damping Ratio NA NA NA ABS Lattice OOP 1 Damping Ratio 0.012 ±0.001 0.0078 ±0.0007 0.0075 ±0.0006 Steel TOR 1 Damping Ratio 0.0016 ±0.0003 0.0011 ±0.0006 0.00025 ±0.00007 Steel OOP 1 Damping Ratio 0.00025 ±0.00005 0.0017 ±0.0001 0.0015 ±0.0001 0.0276 Amplitude Phase 0.98 phi(W) [rad] 0.7812 0.5825 0.3837 0.185 –0.01375 –0.2125 –0.4112 –0.61 –0.8088 –1.008 –1.206 –1.405 –1.604 –1.803 –2.001 –2.2 0.02591 0.02423 0.02254 0.02085 0.01916 0.01747 0.01579 0.0141 0.01241 0.01072 0.009037 0.00735 0.005662 0.003975 0.002288 0.0006 A(W) [mm] Fig. 17.11 Typical DIC images of the amplitude (left) and phase (right) of the first torsion mode 17.4.2 Simulations Ten model input parameters present a significant challenge for inverse analysis. In fact, as the following results show, such a problem would be ill-posed and have non-uniqueness of solutions for the specific test structures under consideration. To reduce the model input dimension, a Taguchi orthogonal array subset of the parameter space is simulated in ABAQUS and the relative contribution of each input parameter to the frequencies of each mode class is evaluated. The Taguchi subset consists of 128 parameter sets with each parameter having 2 levels. This yields a factor resolution of 5, so primary factor effects are only confounded with 4-factor interactions or higher. The coefficients of determination for each parameter are presented in Fig. 17.12. For all mode shape classes, only three of the input parameters are non-negligible: density and two elastic moduli. This creates the previously discussed non-uniqueness issue. Only the two stress moduli are considered in the inverse analyses, since density can be measured empirically (Table 17.6). This reduces the model calibration problem to two dimensions. Broyden’s method [32] is used to optimize the simulated modal frequencies to the experimental modal frequencies, with respect to the input elastic and shear moduli. Resulting calibrated values for solid and lattice parts are presented in Table 17.7. Ranges are provided which correspond to the uncertainty of the experimental data. Percent comparisons of the midpoints are also calculated, relative to the Upright orientation.
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