80 D. Kiracofe et al. Fig. 9.5 Results of Driving-point velocity FRF comparison between an intact metastructure (Fig. 9.2a), slightly damaged metastructure (Fig. 9.2b), and heavily damaged metastructure (Fig. 9.2c). Bandgaps are boxed Unlikehi,j measured by the roving-hammer or roving-sensor technique, hj corresponds to a diagonal entry of Hin Eq. (9.2). Though φk,j can be equal or close to zero, hj can be used to identify all natural frequencies that can be identified using H. Driving-point frequency response functions hj corresponding to the intact, slightly damaged, and heavily damaged metastructures are shown in Fig. 9.5a, b and c, respectively. It can be seen that the extra frequency response function magnitudes exist in the neighborhood of the damage, i.e., P15, in the bandgaps, and such peaks can indicate the location of the damage. Based on the unique advantage of driving-point frequency response functions, a damage index (DI) is formulated; the DI associated with a bandgap of hj ranging between ω1 and ω2 is expressed by δj = ω2 ω1 h 2 j (ω)dω (9.4) The location of damage can be identified at neighborhoods with highδj values. The DI δj corresponding to the intact, slightly damaged, and heavily damaged metastructures is shown in Fig. 9.6, with ω1 and ω2 corresponding to the first bandgap. With δj corresponding to the intact metastructure, one can identify the anomaly of δj corresponding to the two damaged metastructures that indicates the location of the damage. The DI δj corresponding to the heavily damaged metastructure has higher values at the neighborhood of the damage, i.e., P15, than δj corresponding to the slightly damaged metastructure. Effectiveness of δj for identifying the anomaly caused by damage can be further visualized by comparing δj corresponding to the three metastructures, which is shown in Fig. 9.7, and it is verified that more severe damage leads to a higher level of anomaly in hj and a higher δj value. These observations can be further verified by DI results associated with the second bandgap shown in Figs. 9.8 and 9.9, where local anomalies can be observed in the neighborhood of the damage. As compared to a DI for the first bandgap, the DI for the second bandgap shows a much high sensitivity for the heavily damaged structure.
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