Issue 48

J. Prawin et alii, Frattura ed Integrità Strutturale, 48 (2019) 513-522; DOI: 10.3221/IGF-ESIS.48.49 515 components are usually neglected during the reconstruction of the response. SSA has been performed repeatedly twice on the response in the proposed closing crack localization technique, to isolate the linear components from the actual response in the first instance and to identify the presence or absence of the higher order harmonic resonances from the residual/noisy part of the signal in the second instance. It should be mentioned here that the residual components contain the nonlinear components. The higher order harmonic components, induced by the nonlinear behaviour of the closing crack, are usually buried in the residual component and it is easily detected by the same pairwise eigenvalue concept mentioned earlier in the proposed technique. This clearly demonstrates the ability of the SSA in decomposing the linear, nonlinear harmonic and noise components from the total response. Damage diagnostic scheme based on singular spectrum analysis The steps involved in closing crack localization based on the signal decomposition technique are 1. Decompose the acceleration time history response obtained at various spatial locations across the structure using SSA. 2. The linear and nonlinear harmonic components are identified through pairwise eigenvalues concept. The first pair indicates the linear component and the rest are nonlinear harmonic components generated by the closing crack. However, these nonlinear harmonics are buried in the noisy components and difficult to extract reliably at this stage 3. Reconstruct the residual components (eliminating first Eigen pair component) by grouping and diagonal averaging of SSA technique. 4. Decompose the residual time history response obtained from step-3 using SSA. The nonlinear components are identified by pairwise eigenvalue concept. 5. The cumulative sum of the peak amplitudes of each of the nonlinear harmonic component is considered as damage index. The damage index is defined as follows 1 nonlinear component) nf peak,k k DI( i ) A (sup erharmonic    (1) where DI (i) indicates the damage index of the i th degree of freedom (or sensor node) and nf, peak,k A represent the number of superharmonic frequencies and the peak Fourier power spectrum amplitude of the selected ‘k’ th nonlinear harmonic component. The damage index is computed for each sensor location. The sensor node exhibiting the higher magnitude of magnitude index is the true spatial location of the breathing crack N UMERICAL INVESTIGATIONS simple beam like structures such as simply supported beam and cantilever beam, simulated with closing cracks at varied spatial locations are chosen as numerical examples; to test the SSA based closing crack localization technique. Since we have also carried out experimental investigations using the cantilever beam, and the numerical investigations carried out on both the beams are similar in nature, we present only the results of the simply supported beam in this paper to avoid repetition or redundancy. However, the experimental studies on the cantilever beam are presented later in this paper. Steel cracked simply supported beam given in Fig. 1 is chosen in the present work for numerical investigation. The span of the beam is 0.7m and has an area of 4e-4m2 and Moment of Inertia as 0.667e-8m4. The natural frequencies of the underlying linear healthy beam are found to be 89.671Hz, 354.689Hz, 799.16Hz, 1419.46Hz, 1711.46Hz and 2218.17Hz. The finite element model of the beam considers standard one-dimensional Euler-Bernoulli beam elements with two nodes per beam element and each node have three degrees of freedom; longitudinal displacement, translational displacement and bending rotation. Heavy side step function, widely preferred by the researchers is used in the present work to model the opening-closing behaviour of the breathing crack. The damaged element stiffness matrix with the bilinear behaviour induced by the rotations ( ,  i j   ) at the nodes of the respective damaged element [6-9] is given by                    1 0 i j i j d i j c i j i j H θ θ ,θ θ K K H θ θ K : H θ θ ,θ θ (2) A