84 D. Day et al. Fig. 8.11 Combined beam: base model ASD and modes with highest response The combined beam model is the same overall length and mesh size as the cantilever beam, and the two blocks are joined together, with the material properties defined independently. The model is half steel, half aluminum, with equivalenced nodes at the interface to provide an ideal joint. The perturbation is performed similarly to the previous problem, except that the modulus, density, and length are only changed for the aluminum beam. For the perturbed length model, the length of the aluminum beam was increased by 0.125 in. (3.2 mm) for each simulation and the point coordinates were unchanged. The response ASD for the base model is shown in Fig. 8.11. There are several modes across the frequency range, many of which are close in frequency. Two of the modes that produce the highest response are also shown in Fig. 8.11. Examining the results for the modulus perturbation in Fig. 8.12, the general trends are like those from the cantilever beam problem. However, the error relation is clearly nonlinear. For the change in damping, a linear relationship similar to the cantilever beam was observed, although it was not one-to-one for the maximum; the plots are given in Appendix 2. For the density and length perturbations, the results were much more unpredictable than the previous example. The error plots for the density change in Fig. 8.13 are nonlinear and inconsistent, with several turning points, although some sections appear uniform. The error results for the length perturbation shown in Fig. 8.14 are also erratic. As in the previous example, the response ASD is studied to determine the source of the inconsistency between stress error and acceleration error. Fig. 8.15 shows the ASD at Point 1 for each simulation in the modulus perturbation study. The modes change gradually as the modulus changes and no new modes enter the input frequency range, but the large variation in response could contribute to the nonlinearity of the plots in Fig. 8.12. Next, we examine possible reasons for the unpredictable error relation presented earlier. In Fig. 8.14, there is a large AZRMS error caused by a 1% perturbation in length. From the second to the third perturbed model, an increase in length of 0.125 in. (3.2 mm) resulted in a 15% change in acceleration error and almost a 10% change in maximum stress error. These changes are apparent in the ASD, in Fig. 8.16. The peak acceleration response nearly doubled at 2 kHz for Point 3. Additionally, two modes near 350 Hz are nearly indistinguishable before the small change in length, yet clearly distinct after the small increase in length. These abrupt changes had a substantial impact on the resulting error relationship.
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