118 Y. F. Xu et al. Step4. Estimate ˜yi ˜x associated with measured modes in Ni half-scan periods using the demodulation method. Step5. Estimate ζi based on Eq. (13.23) usingωiζi obtained by solving the optimization problem in Eq. (13.22). Step6. Express Qiφi ˜x as ˜yi,1 ˜x eωiζi(t−t1) with 0 ≤t −t1 ≤T andωiζi obtained in Step 5. 13.2.4 Baseline-Free Structural Damage Identification Local damage of a structure can cause prominent anomalies in its curvature mode shapes in neighborhoods of the damage, and the damage can be identified by comparing the curvature mode shapes with those of the associated undamaged structure [22]. However, the curvature mode shapes of the undamaged structure that can be considered as baselines are usually unavailable in practice. When the undamaged structure is geometrically smooth and made of materials without mass and/or stiffness discontinuities, the curvatures of the undamaged structure can be well approximated by those from polynomials that fit mode shapes of the damaged structure with properly determined orders. In previous works [15, 16, 23], a curvature damage index (CDI) was proposed, which consists of the difference between a curvature mode shape of a damaged structure and that from a polynomial fit: δi (x) = φ i (x) −φ p i (x) 2 (13.24) where a prime denotes spatial differentiation with respect to the arc lengths of a scan path at x, andφ p i is the corresponding mode shape from the polynomial that fits φi. Since mode shapes corresponding to multiple modes can be measured in one scan, CDIs corresponding to multiple modes can be obtained in the scan, and damage regions can be identified in neighborhoods with consistently large CDI values associated with the measured modes. Note that use of δi corresponding to rigid-body modes of a structure should be excluded in damage identification as their curvature mode shapes are zero, and one should use δi corresponding to elastic modes of the structure in damage identification. An auxiliary CDI associated with δi corresponding to various measured modes can be defined to assist identification of the neighborhoods; it can be expressed by ˜δ (x) = ˆδi (˜x) (13.25) where ˆδi is a normalized CDI associated with the i-th mode of the structure with the maximum unit amplitude and denotes summation of ˆδi over all measured modes. Since boundary distortions would occur in curvature free response shapes of a structure associated with its free response shapes obtained from the demodulation method [17], similar distortions would occur in curvature mode shapes here. Hence, boundary regions are excluded in normalization of δi in ˜δ and presenting them. Neighborhoods with consistently large values of δi associated with measured modes can be identified in those with large values of ˜δ. By nondimensionalizing s so that it ranges between −1 and 1, a polynomial that fits φi with an order r can be expressed by φ p i (˜ s) = r q=0 aq˜s q (13.26) where ˜s denotes the nondimensionalizeds, aq are coefficients of the polynomial. As pointed out in Ref. [17], an increase of r in the polynomial in Eq. (13.26) can improve the level of approximation of φ p i to φi. To determine a proper order of the polynomial fit, the modal assurance criterion (MAC) value between a mode shape of the damaged structure and that from a polynomial that fits the mode shape, which is defined by MAC φi,φ p i = φ Hi φ p i 2 φ Hi φi φ pH i φ p i ×100% (13.27) where the superscript H denotes matrix Hermitization, is used. A proper order for the polynomial fit is two plus the minimum order with which MAC φi,φ p i is greater than 90% [23]. Two is added here in order to preserve smoothness of a curvature mode shape from the polynomial fit, since calculation of a curvature incurs second-order differentiation that reduces the order of a polynomial by two.
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