7 A Modal Model to Simulate Typical Structural Dynamic Nonlinearity 75 The Full Modal Filter and the SMAC modal filter each had certain modes on which they excelled. The SMAC modal filter was better at removing extra modes than the Full Modal Filter in general. The Single Modal Filter was much worse than the other two. Band-pass filtering was very helpful for improving the Hilbert Transform results. Without band-pass filtering the Hilbert Transform results were much more oscillatory and nebulous, usually due to other modes that were not completely filtered. Band-pass filtering was not helpful if the pass band was too narrow (e.g. 5 % of the resonant frequency). It could be tailored to each mode for optimum results, but generally a band-pass of 30–50 % of the resonant frequency was robust. The Hilbert Transform is required for the FREEVIB and Iwan approaches to obtain frequency and damping variation as a function of amplitude. If the modal filtered response was not uni-modal, these functions of frequency and damping were oscillatory, hampering the fitting. The Hilbert Transform tends to have some early-time oscillations of frequency and damping at the very high amplitudes. The spurious oscillations at the high amplitude can be problematic, since that is the region most important to the nonlinear model. 7.6.4 Nonlinear Pseudo-Modal Model Forms We contend that all three methods would give satisfactory nonlinear simulations with enough user interaction. For these studies, the Iwan model and the RFS frequency fit model gave satisfying simulation results. The FV model was not quite as good, but with some user interaction these results could be improved. We did not iterate on the FV fits near as much as the Iwan fits. Six parameters were chosen for each mode for each method to put them on equal footing. This was initially based on the Iwan approach, which uses a linear spring and damper and a four parameter Iwan. With this many parameters, any of the model forms could be used for this nonlinear hardware. The Iwan model is the most utilized in simulating structural dynamic joint nonlinearities in the recent past. Its form appeared to be a very good representation for the nonlinearities where damping increased and frequency decreased with amplitude, which seems to be typical of many joints. We considered the simulations with the Iwan models very good. The disadvantages we noted with the Iwan were: (1) a great deal of user interaction (and iteration) was required to get good Iwan parameters; (2) the understanding of those parameters is complex compared to FREEVIB stiffness and damping or cubic springs and damping; (3) the inability to simulate constant damping with softening or decreased damping with stiffening; (4) a strong dependence on the quality of the Hilbert Transform results. The Feldman FREEVIB model is relatively easy to understand. We modified the nonparametric approach by fitting cubic polynomials to the functions of frequency and damping vs. amplitude to keep it on the same six parameter footing as the other two model forms. FREEVIB requires free response ringdown data, which works well with impact testing as was performed on this hardware. Simulation results are very dependent on the results of the Hilbert Transform and the modal filter. We found that parameters for mode 9 were not well quantified using FV, causing significant over-prediction for this mode. However, if we removed the band-pass filtering which allowed us to include earlier time data while still maintaining the free-vibration requirement, we could get better results. Additionally, using the FMF (which performed better for mode 9, see Fig. 7.7) instead of the SMAC modal filter for this particular mode also resulted in a better prediction of the measured data. We surmise that capturing the most nonlinear response early in time and eliminating non-target modes is critically important to FV. We did not include the improved results since the data processing (different modal filter and/or exclusion of band pass filtering) was inconsistent with the other two nonlinear models. The cubic springs and dampers as a function of amplitude are easy for an engineer to understand. The RFS approach in the frequency domain focused on the frequency lines around each resonance gave results on par with the best results from iterative Iwan fits. The RFS frequency approach also has other significant advantages over the other models because it does not require the Hilbert Transform or extensive user interaction. Because of the ease of understanding the polynomial type nonlinearity, elimination of Hilbert Transform step, low user interaction and the final quality of the nonlinear simulation, RFS with cubic nonlinearities emerged as our favored approach. Notice This manuscript has been authored by Sandia Corporation under Contract No. DE-AC04-94AL85000 with the U.S. Department of Energy. The United States Government retains and the publisher, by accepting the article for publication, acknowledges that the United States Government retains a non-exclusive, paid-up, irrevocable, world-wide license to publish or reproduce the published form of this manuscript, or allow others to do so, for United States Government purposes.
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