11 Use of Continuous Scanning LDV for Diagnostics 133 Fig. 11.8 RMSE between ODS and polynomial fitting (the polynomial order is expressed as sidebands number) Expanding out trigonometrically, the following representation of the vibration signal is derived: vz.t/ D p;qX m;nD0 VRn;mcos !˙n x ˙m y t C p;qX m;nD0 VIn;msinŒ.!˙n x ˙m y /t (11.9) For simplicity, we assume that the ODS is entirely Real so that the Imaginary part is negligible. In order to deal with the ODS as displacement we have to integrate the equation: fxzgD p;qX m;nD0 XRn;mZ cosŒKt dt D p;qX m;nD0 XRn;mKsinŒKt Ccost (11.10) where K D ! ˙n x ˙m y. The ODSs calculated in Eq. (11.1) were used for generating the polynomial functions as presented in Eq. (11.6). The polynomial order was identified by minimizing the error, less than 1 %, between the ODS and its polynomial form. The polynomial order was calculated for every simulated ODS and the Root Mean Square Error (RMSE) plot is presented in Fig. 11.8. The higher the ODS complexity the higher the polynomial order. The polynomial function obtained from the calculated ODS was used for simulating the LDV output signals, one from the undamaged and one from the damaged component. The RMSE was calculated between the two sets of sidebands and plotted as shown in Fig. 11.9. It is interesting to notice that even for the smallest damage severity, which is the alteration of one single element in the FE model, the change of spectral sidebands is very clear. This is visible in Fig. 11.9 from the plots labelled Damage 1 and Damage A1, respectively. As the damage severity increases as the change of the spectral sidebands shows more clearly. However, a noise free test data is rather impossible. The next set of simulations regards the same ODS data sets but noise is added for creating a more realistic measurement environment. The LDV output signal was added with broadband white noise. The assessment of the noise on the sidebands is presented in Fig. 11.10 which shows how the sidebands with and without noise were correlated. The calculation was carried out as follows. Every pair of sideband set, with and without noise, were correlated and averaged by the number of sidebands. So a single averaged number was calculated for every excitation frequency. Clearly, the sidebands near the resonances were less sensitive to the noise and the degree of correlation was rather high for the given level of noise. Despite the addition of noise, signs of sidebands changes are still present in the error plots as shown in Fig. 11.11. Clearly, if Damage 1 in Fig. 11.9 presented sings of changes for frequencies between 20 and 70 Hz this is not visible any longer in the plot of Damage 1 in Fig. 11.11.
RkJQdWJsaXNoZXIy MTMzNzEzMQ==