148 J.D. Schoneman and M.S. Allen 14.3.3 NLROM Creation and Validation With a full-order model and its associated random response solutions obtained, NLROM validation can take place. An NLROM including the first five modes of the structure was used for all of the following computations. In constructing the NLROM, load levels causing a 50 % thickness displacement in each mode were sufficient to meet the nonlinear activation criteria mentioned in Sect. 14.3.1. (At a displacement of 50 % the beam thickness, the nonlinear response was approximately 80 % of the linear response for each mode.) Both a fully-coupled NLROM (as per Eq. (14.9)) and a diagonalized NLROM (Eq. (14.11)) are used in the validation plots below. The NNMs of the coupled and uncoupled NLROMs were computed in order to examine any immediate differences between the two nonlinear models. A frequency-energy plot of the comparison is shown in Fig. 14.5. The “backbone” curves of the two models are very similar, however, there is an internal resonance visible in the NNM of the coupled NLROM, which corresponds to an interaction between the first and fourth normal modes of the structure. Next, the direct integration results are compared to further evaluate the NLROMs. Each model was integrated in MATLAB using a fixed time-step HHT routine with no numerical damping. An integration sample rate of 48 kHz was necessary to achieve convergence with the Abaqus results. (The average sampling rate that each Abaqus routine used can be inferred from Table 14.2.) Figure 14.6 compares the NLROMs with Abaqus/Explicit results using the vertical displacement at a single node of the beam. The coupled NLROM matches Abaqus very well; the diagonal NLROM proves to be surprisingly accurate, although it does overpredict the response by 15 % relative to the full-order solution. Note that the result for the linear case with this loading yields an RMS response of 0.76 mm, more than twice the nonlinear solution. The stresses in the various models were also compared over all of the integration points in the full-order model (the stress at these locations is available from the ROMs). In Fig. 14.7, contour plots of stress are shown as functions of position vs. frequency. At right, the standard deviation and mean of the stress fields are compared. The largest differences in the stress autospectra are seen in the higher frequencies; for example, near 2000 and 3000 Hz, the diagonal NLROM predicts peak stress near the linear natural frequencies, whereas the other two models predict that these frequencies stiffen due to the large response of the lower frequency modes. This presumably causes the peak stress, near 1000 Hz, to be too large in the diagonal NLROM. 10−1 600 550 500 450 400 350 300 100 Diag. NLROM Coupled NLROM 1st Linear Mode 101 Energy [N. mm] Frequency vs. Energy for NNM 1 Frequency [Hz] 102 103 Fig. 14.5 Comparison of the beam’s 1st NNM computed from a coupled 5-mode NLROM and an uncoupled NLROM
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