152 J.D. Schoneman and M.S. Allen Fig. 14.10 Displacement at x D44:29mm of the beam using Abaqus/Explicit results, a coupled five mode NLROM, and a diagonal five mode NLROM. Vertical dash/dot lines correspond to linear natural frequencies of modes included in the NLROM; dotted lines indicate linear natural frequency of modes not included in the NLROM The requirement to include such high-frequency effects also demonstrates the futility of random response and other long-time-history prediction through full-order simulation: Not only are long time histories required, but extremely small time increments are necessary to accurately predict nonlinear response. In this case, the explicit integration method, often considered too expensive for structural dynamics problems, had the best combination of accuracy and efficiency relative to the Newmark/HHT method with no numerical damping. To obtain a quantitative understanding of the effects of modal energy transfer across multiple configurations of the beam, two sets of nonlinear reduced order models were developed: The first featured a full set of modal coupling terms between the modes, while the second was “diagonalized” and featured quadratic and cubic terms of single modes only. Comparing the displacement, energy, and stress results of these two NLROMs at a single design point showed that, as expected, the coupled NLROM closely matched full-order Abaqus results, while the diagonalized ROM overpredicted both stress and displacement levels. Using the efficient analysis capabilities of the NLROMs, a grid of 360 design points was evaluated for maximum stress and displacement RMS over a 7.5 s time history. This “parameter sweep,” though not an effective means of optimization in general, provided a general overview of the topology of this type of design space, showing largely convex behavior without the prevalence of local minima that would require a global optimization technique. Finally, the difference in response between the coupled and diagonal NLROMs was investigated. For this structure, no discernible pattern or preferential behavior was observable in the response contours. Examining a single solution for which nonlinear energy transfer was a major factor did show the prevalence of modal interaction in that particular example, but was at too high a response level to be of any practical use. The results here are inconclusive as to the feasibility of leveraging nonlinear energy transfer within a structure to reduce the response levels. A key difficulty in attempting to maximize the internal energy transfer within a structure is that the primary mechanism governing the transfer seems to be the ratios between linear natural frequencies; there is no apparent method to modify these natural frequency ratios without also modifying the stiffness of the structure with respect to a dynamic forcing. In this case, modifying the natural frequency ratios by shifting the support also led to a reduction in overall stiffness as the support moved away from the 34 % location corresponding to maximum bending stiffness.
RkJQdWJsaXNoZXIy MTMzNzEzMQ==