280 Y.F. Xu et al. However, model-based methods could have problems due to inaccuracy of the model, environmental and other nonstationary effects on measurements, and lack of data in suitable frequency ranges [4]. In fact, it is difficult to construct models of most existing structures with high accuracy. Hence, methods that only analyze MSs or operating deflection shapes (ODSs) without the aid of a model can be good alternatives to model-based methods to locate damage, and they are non-model-based ones [5]. Since MSs are not sensitive to damage of small extent, curvatures of MSs, or curvature mode shapes (CMSs), are used to locate damage [6]. Differences between CMSs of a damaged beam and an undamaged one are localized in the region of damage and increase as the damage size increases [7]. A gapped-smoothing method was used to locate delamination in a composite beam by inspecting the smoothnesses of CMSs [8], and the method was extended to use broad-band ODS data [9]. It was applied to locate damage in a beam using a global fitting method, where generic MSs were used to fit measured MSs of a damaged beam [10], and the global fitting method was extended to ODS data to conduct damage detection on beams and plates [11]. A crucial aspect of the damage detection methods using CMSs is calculation of spatial derivatives of MSs. Optimal spatial sampling intervals were proposed for CMSs to avoid undersampling and oversampling of MS measurements, both of which have adverse effects on the quality of damage detection [12]. A novel Laplacian scheme was developed and experimentally validated in [13] to locate a delamination zone in a composite beam using associated modal curvatures with multiple resolutions. Besides CMSs, wavelet transforms of MSs can also be used in damage detection, since they are sensitive to localized abnormalities in MSs and can be presented in multiple scales. Cracks were identified in beams using a “symmetrical 4” wavelet; the position of a crack was accurately detected with the aid of a beam model [14]. Damage in the form of cracks in beams and thickness reductions in plates was identified using continuous wavelet transform (CWT), which was manifested as peaks in associated CWT coefficients [15]. The CWT of differences between MSs of a damaged beam and those of the associated undamaged one can be used to locate cracks with high sensitivities [16]. Beams with horizontal embedded cracks are studied in this work; they are similar to composite beams with delaminations. Natural frequencies of beams and plates will decrease if delaminations occur; the larger a delamination, the larger the reductions of the natural frequencies [17]. Free vibration analysis of a laminated beam was studied using a layerwise theory. Effects of the lamination angle, location, and size, and the number of delaminations on the natural frequencies of beams were addressed in [18]. A generalized variational principle was used to formulate equations of motion and associated boundary conditions for the free vibration of a delaminated composite beam; the coupling effect of longitudinal and bending vibrations was shown to be significant for the calculated natural frequencies and MSs [19]. Modal tests were conducted in [20] using polyvinylidene fluoride film sensors and piezoceramic patches with sine sweep actuation; backpropagation neural network models were developed using results from the beam theory and used to predict a delamination size. A spatial wavelet analysis was used in [21] to process static deformation profiles of cantilever beams to numerically and experimentally locate delaminations; the deformation profiles obtained by dense measurements were smoothed before applying the spatial wavelet analysis. Two non-model-based crack identification methods are proposed in this work. CMSs are presented in multiple resolutions in order to reduce the adverse effects of measurement noise. The relationship between CWTs and CMSs is shown. Polynomials of MS-dependent orders are used to fit MSs of a damaged beam, which can be considered as MSs of the associated undamaged one; the fitted MSs are virtually extended beforehand beyond the boundaries of the beam in order to enhance the similarity of the CMSs obtained from the resulting polynomials to those of the associated undamaged ones near the boundaries. Differences between MSs of the damaged beam and those from the resulting polynomials are used to yield two damage indices: the curvature damage index (CDI) and the CWT damage index (CWTDI) with a Gaussian wavelet function. A cantilever beam made of acrylonitrile butadiene styrene (ABS) with an embedded horizontal crack is modeled, and the crack tips can be successfully located using the two proposed damage indices; they are located near the peaks of CDIs and CWTDIs with the second-order Gaussian wavelet, and the valleys of CWTDIs with the third-order Gaussian wavelet. The proposed methods can not only identify embedded horizontal cracks but also edge cracks and slant cracks. 31.2 Damage Identification Using CDIs and CWTDIs The finite element (FE) model of a cantilever beam made of steel with an elastic modulus ED210GPa and Poisson’s ratio D0:3, with an embedded horizontal crack, is constructed using FE software. The dimensions of the beam and the crack are shown in Fig. 31.1. An analytical model of the undamaged beam with the same material properties and dimensions as the damaged beam is also constructed for comparison purposes. In this section, MSs of the damaged and undamaged beams, denoted by fd and fu, respectively, are used to illustrate the proposed methodologies. The j-th MSs of the damaged and undamaged beams are denoted by fd;j and fu;j, respectively, and the MSs are normalized so that their maximum absolute values are one. White noise is added to the MSs with a signal-to-noise ratio (SNR) of 60 to simulate measurement noise.
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