Issue 48

J. Prawin et alii, Frattura ed Integrità Strutturale, 48 (2019) 513-522; DOI: 10.3221/IGF-ESIS.48.49 514 monitoring and found to be very effective. A detailed review of earlier works of SSA on structural health monitoring applications can be found in Prawin et.al, [4]. In most of the earlier works on SSA in SHM applications, the noisy components are neglected during the reconstruction and transformation back to the actual response. Oliveria et al., [5] recently reported that the discontinuity related damage sensitive features are found in these ignored residual noise components of the actual response. In view of this, we process further the residual (noisy) components obtained from SSA to reliably check whether any higher order or superharmonic resonances present or buried in the residual signal. Further, the proposed SSA based decomposition technique can reliably extract all the higher order harmonic components in contrast to the existing approaches which are based on only first few harmonics and the damage detection can be enhanced with consideration of more number of super harmonics. S IGNAL DECOMPOSITION USING SINGULAR SPECTRUM ANALYSIS he breathing crack is usually assessed by the spectrum of the response of the structure measured at a particular location. The presence of super harmonics (i.e. higher order harmonics) apart from the fundamental excitation harmonic confirms the nonlinear behaviour induced by closing crack when the cracked structure is subjected to harmonic excitation. Earlier researchers [6-8] basically employed the ratio of the power spectrum amplitude of the response at second or third order harmonic to the first order/linear/excitation harmonic as damage index for closing crack localization. Few other researchers [6-8] have used the spatial curvature of the ratio of the power spectrum of super- harmonics to excitation harmonic as damage index due to the fact that the second spatial derivative magnifies the cracks/discontinuity in the response. Further, most of the existing techniques consider only one or two higher order harmonic frequencies for closing crack localization and the rich damage sensitive features present in the higher order harmonics have been ignored. The basic reason behind this is that the amplitude of super harmonics (i.e. nonlinear harmonics) are of very less order in magnitude when compared to linear fundamental excitation harmonic. These higher order nonlinear harmonic components also often get buried in noise and difficult to conclude whether it is noise or nonlinear component as both are having matched (low) energy levels. Therefore, the extraction of these harmonics under noisy environment and damage detection at its incipient stage is highly challenging. In the proposed closing crack localization technique, the cracked structure i.e. with closing crack is subjected to harmonic excitation with a particular single frequency; therefore the response is also harmonic in nature. As mentioned earlier, the response of the cracked structure, i.e. with closing crack vibrates at the excitation frequency as well as the higher order super harmonics of the excitation frequency. In contrast, the healthy structure (i.e., without breathing crack) vibrates only at excitation frequency due to single tone harmonic excitation. Since the response of the structure is harmonic in nature, the decomposition of the time history data of the cracked structure, i.e., structural or structural component with closing/breathing crack at a particular location, by SSA contains only harmonic components (i.e. both linear excitation and nonlinear super harmonic harmonics) and noise. Harmonic signals basically exhibit two eigen triples with close singular values [1-5]. Therefore, this property helps in easy interpretation and identification of the harmonic signal components during decomposition of the measured response by SSA. The number of pairwise eigenvalues indicates the number of harmonic components present in the response. Since the structure is excited by a single tone harmonic load, there exists only one pairwise eigenvalue for the structure without a closing crack. In contrast, there will be more than one pairwise eigenvalues exist for the structure with closing crack due to the presence of super harmonics. These harmonic components (i.e. pairwise singular values) are present in descending order (i.e. decreasing) of their energy corresponding to each frequency due to the property of SSA. The first pairwise eigenvalue will be always the linear excitation harmonic component and it always exhibits dominant energy for both healthy and cracked structure (i.e. structure with breathing crack). This can be easily verified by the Fourier power spectrum of the respective harmonic component. The subsequent pairwise singular values (i.e. present with low energy levels) represent the nonlinear superharmonic components. The contribution of higher energy in the case of fundamental excitation harmonic is clearly justified by the fact that the amplitude of fundamental harmonic is of two or three order higher in magnitude when compared to the amplitude of nonlinear superharmonic. In the present work, the linear components can be isolated by the first pairwise singular value. It should be mentioned here even though there is no theoretical evidence, extensive studies carried out on closing crack problems reveal that the energy content of the dominant pairwise eigenvalues corresponding to linear response constitutes 99% of the total energy [4, 7,9]. However, to be on the safer side, we prefer to use the first pairwise singular value for extracting the linear fundamental excitation harmonic component. In order to reliably extract the low energy nonlinear harmonics, we apply SSA again on the residual (noisy) components obtained from SSA after ignoring linear harmonic components. This is in contrast to the earlier works [1-3] where the noisy T