10 System Identification to Estimate the Nonlinear Modes of a Gong 129 100 101 10 2 RMS Center Velocity [m/s] 547.9 548 548.1 548.2 548.3 548.4 548.5 548.6 Frequency [Hz] Fig. 10.8 NNM frequency vs center point RMS velocity for the (0,1) mode of the gong, measured using force appropriation at higher frequencies. However, the measured data show that the (0,0) mode is excited more significantly in these higher frequency bands. Similarly, the response of the (0,1) mode can be seen to differ significantly from the linear FRF prediction, showing that the system is exhibiting significant nonlinearity, and that any nonlinear model for this system should be one which significantly affects the response in the aforementioned frequency bands. As described in Sect. 10.2.1, it was assumed that the nonlinearities in the system could be adequately modeled by quadratic and cubic functions of the modal displacements. In an initial effort, only the measurements in the frequency range that was excited (i.e. 225–245 Hz) were used in the identification. However, this produced a model that was inaccurate at the harmonics of the forcing and so four frequency bands of interest were identified in the figures above, specifically those which showed significant response in Figs. 10.9 and 10.10 and these were used to define the input data for RFS. The measurements from these four frequency bands were used to populate Eq. (10.10), and the least squares equation was solved for the nonlinear stiffness terms. Considering nonlinearity in all 13 modes, with coefficients up to cubic and considering all possible nonlinear coupling terms, there were over 600 terms to identify. Initially, numerical ill-conditioning was a challenge, yet after scaling each signal by its RMS value, it was possible to obtain a reliable fit to the measurements. To evaluate the accuracy of the model found through nonlinear system identification, the restoring force of the system was constructed using the identified parameters, along with the measured displacement, velocity, etc. : : : , The sum of these measured terms multiplied by their respective polynomial terms reproduces the left hand side in Eq. (10.10), and this was compared with the measured right hand side to see how well the identified parameters reconstruct the response. Figure 10.11 contains plots of the restoring force reconstructed for the (0,0) mode compared to the measured restoring force. It also contains the same reconstruction obtained using only the linear terms, which were identified from the linear FRFs and are identical to those used to reconstruct the FRFs used to compute the linear response in Figs. 10.9 and 10.10. It is interesting to note that the linear RFS prediction matches the measurements quite well, even though the true linear response to the measured forcing in Figs. 10.9 and 10.10 showed significant error. This stems from the fact that only the forcing is available as an input when using the FRFs to compute the response (i.e. in Figs. 10.9 and 10.10), whereas the RFS method uses the measured acceleration, velocity and displacement as well and these implicitly contain the effect of the nonlinearity. This seems to be a deficiency of the RFS method and deserves further investigation. Setting that issue aside, one can use the responses in Fig. 10.11 to evaluate the accuracy of the identified nonlinear model. In order to more easily do this, the error between the measured restoring force and that reproduced by the nonlinear model was computed and is shown in Fig. 10.12. In this figure one can see that the nonlinear model reduces the error near significantly near the first two frequency bands, while leaving it roughly the same at the two higher frequency bands. The same plots for the (0,1) mode can be seen in Figs. 10.13 and 10.14.
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