Issue 48

J. Prawin et alii, Frattura ed Integrità Strutturale, 48 (2019) 513-522; DOI: 10.3221/IGF-ESIS.48.49 517 levels. It should be mentioned here that using the conventional spectral analysis, only one or two superharmonics can be extracted reliably under noisy environment while using the proposed approach (SSA), a large number of superharmonics are extractable. In the present work, for the application of SSA on the time history response, the window length is selected as the time lag corresponding to the first zero crossing between L/4 and L/2, computed based on autocorrelation [4]. More details related to the choice of window length are not discussed here, as it deviates from the scope of the present work. However, details on the choice of the window length can be found in Prawin et.al . [4]. 0 500 1000 1500 2000 2500 ‐1.0 ‐0.5 0.0 0.5 1.0 99.5%confidence interval sample autocorrelation function lags 1 2 3 4 5 6 ‐10 0 10 20 30 40 50 60 70 Energy difference Eigenvalue Pairs Window Length=72 Window Length=390 Window Length=700 Window Length=855 Window Length=1200 (a) (b) Figure 3: (a) Autocorrelation function (b) Window Length 1 2 3 4 5 6 7 8 9 10 11 ‐20 0 20 40 60 healthy data without noise (0.01N) damaged data without noise (100N) healthy data with 10% noise (0.01N) damaged data with 10% noise (100N) Energy Number of singular values 1 2 3 4 5 6 7 8 9 10 11 ‐20 0 20 40 60 80 100 120 healthy data without noise (0.01N) damaged data without noise (100N) healthy data with 10% noise (0.01N) damaged data with 10% noise (100N) Energy Number of singular values (a) (b) Figure 4 : Singular Spectrum – (a) actual (b) residual acceleration time history Fig. 3 (a) depicts the autocorrelation of the response measured at 0.4m from the left support corresponding to 100N excitation. Fig. 3(a) shows the zero crossing of the autocorrelation function of the response at time lags around 72, 232, 387, 545, 698, 851, 1008, 1196, 1315, 1470, and 1625 and so on. Fig. 3(b) shows the plot corresponding to the energy difference between eigenpairs with varied window lengths chosen based on the time lags corresponding to zero crossing of the response. The energy difference (of eigenpairs) plot furnished in Fig. 3(b) shows zero magnitudes when the length of the window is considered as 1190 (or above). Based on this investigation, the above-automated choice of window length is chosen in the present work. Once the window length is chosen, the next stage of SSA can be performed on the response to localize the closing crack. The singular spectrum of the noise-free response and response polluted with 10% noise (i.e. node 5) obtained by SSA is given in Fig. 4 (a), while the results of the residual response is given in Fig. 4 (b). Both Figs. 4 (a) and 4 (b) furnish the results corresponding to two different excitation amplitudes of 0.01N and 100N. Fig. 4(a) concludes that the eigentriples with close singular values exist for the total actual response corresponding to both excitation amplitudes of 0.01N and 1N. While the pairwise component i.e. harmonic signal is absent in Fig. 4 (b) for the low amplitude of excitation and present for the high amplitude of excitation even with the noisy measurements. Each of the pairwise singular values in the residual signal of 100N excitation corresponds to nonlinear harmonics components i.e. super harmonic components generated due to the