110 P. Villalpando et al. Fig. 12.3 Identification of modal properties in the structure. (a) Modal frequencies, (b) Modal shapes (NUM: Numerical model, E.CH: Earthquake Chile, E.US: Earthquake United States, NOISE: Colored Noise) MACD T N;i E;i 2 T N;i N;i T E;i E;i (12.9) MSF D T N;i E;i T E;i E;i ; (12.10) where i is the ith modal shape, and the superscripts N, E refer to responses obtained numerically and experimentally respectively, a value MAC D 0 indicates no correlation between modal shapes, while a MAC D 1 shows a complete correlation. The comparison between numerical and experimental modal shapes are shown in Fig. 12.3b. The MAC found for all modal shapes is above 95 % except for modes 5 and 6 identified using noise records. 12.4 Construction of the Database 12.4.1 Characteristic Vector The characteristic vector is formed from the modal properties of the structure (frequency and modal shape) and compares the normal condition versus the damage one, as shown in the expression (12.11). Xj D 8 ˆ ˆ< ˆ ˆ: X j Dj Uj 2 !D !U !U 2 9 > >= > >; (12.11) where: j is the j-th modal shape as a column vector; and !represents the natural frequency. Superscripts Dand U, refer to the damaged conditions and undamaged conditions to the structure, respectively. The characteristic vector of the numerical model is formed with information from the theoretical modal properties of the structure. The characteristic vector of the tested structure is determined from an identification process that considers the different types of record evaluated.
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