398 D. DiMaio • Define the controlled acceleration level for each FRF generation. • Repeat the frequency sweep for several levels of acceleration up to the maximum amplifier capacity or control limits. The overall measurement process has been rather long. Indeed, although this modal testing process seems trivial, there are many external factors that are likely to lead to measurements issues. For instance, the way the structure is suspended, the shaker tip is positioned and attached are two sources of measurement dispersion. Data Post-processing Most of the modes of interest in this study show some nonlinearities, the advantage of performing the measurements with acceleration control is that most individual FRFs generated are linearized and can be analysed using linear modal analysis tools. In the case of force control, a complex nonlinear post-processing is often required as the FRFs generated can be highly distorted for high levels of force. This is why, for the convenience of measurements and data post-processing, the nonlinear identification is performed using the amplitude control technique. Figure 35.13 shows the regeneration of the experimental FRF. Figure 35.14 shows the overlap of several regenerated FRFs. Another nonlinearity indicator is the normalized modal damping coefficient defined as the ratio of the modal damping coefficient of the linearized FRF of a given level of vibration by the one of the lowest level of vibrations. Figure 35.15 shows the results of this type of analysis (Fig. 35.16). This graph, in Fig. 35.15, does not give relevant nonlinear information. Indeed, the modal damping is supposed to be a parameter affected by a nonlinearity induced by a friction joint. Only mode 8 shows an important increase in modal damping although most other modes also show a slight increase in damping. However, it can be noticed that mode 4 shows a decrease in modal damping, which was not expected but can be explained as seen further. This is partly due to the curve fitting process performed using ICATS, where the damping coefficient is sensitive to the fitting quality. Then, it has been decided to study the evolution of peak level instead of modal damping coefficient, although those parameters are similar. The normalized peak evolution seems to be a more relevant nonlinearity indicator in the case of our flange structure. Indeed, the modes showing a nonlinear trend can be observed here. Modes M6, M7 and M8 show a clear decrease in peak level with increasing vibration amplitude. On the other hand, modes M12 and M4 show an increase in peak level which will be further explained. Finally, the last nonlinearity indicator is the normalized Natural Frequency (NF) evolution of each linearized FRF. Indeed, a nonlinearity caused by a bolted joint is often associated with a shift in frequency due to a decrease in joint stiffness with an increasing level of vibrations (as it has been explained in the literature review [2]). However, the actual change is frequency was rather small and data are not reported in this paper. The NF shift can be clearly observed here for mode M4, M6 and M8, although its normalized value is relatively low (less than 1 % in the tested frequency range). From all those nonlinear indicators analysis, an actual nonlinear criticality ranking can be established and compared with the prediction. However, only the four most critical modes will be retained as the other modes (the least nonlinear) are not easy to compare with the data acquired. It is important to note that all modes could not be tested properly for several reasons previously explained. However, from the measurements done on similar structures [12], those four most critical modes seem relevant (Table 35.2). Mode 7 (B4) data was probably wrong. Indeed, a “jump” phenomena appears on the high vibration level FRFs that probably comes from a control issue. Then, it has been inserted in the most nonlinear modes with caution. Some further measurements would have been necessary for this mode. Although the ranking was not exactly similar using the FEM Fig. 35.13 FRF curve fitting process in ICATS for mode 10 (the green curve being the experimental data and the red one being the regenerated linearized FRF)
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