59 Empirical Slow-Flow Identification for Structural Health Monitoring and Damage Detection 621 in Fig. 59.3a clearly illustrate damped (linear) normal-mode vibrations, whose envelope at each position along the beam decay exponentially while retaining their respective mode-shape forms at each instant; and the corresponding contour plots preserve (almost) straight vertical lines at the nodes. On the other hand, those for the VI beam highlight the VI effects near the impact instants; that is, local instantaneous distortions of the modal properties are evident for HF IMOs (cf. Fig. 59.3b). More global changes (that tend to be insensitive to local VI effects) are predominant for LF IMOs. Such distortions of modal properties seem to prevail near the impact (or structural defect) positionx9 and near each impact instant. Although representation of the dynamics in the frequency-energy domain has been proved to be a powerful tool for the global approach to NSI [3], a frequency-energy plot contains condensed dynamics information (i.e., it is an implicit representation of specific dynamics such as mode shapes, but carries explicit information mainly about the characteristic frequency and total energy of periodic orbits of the underlying Hamiltonian dynamical system). On the other hand, the spatiotemporal IMOs (or corresponding slow-flow models) can be regarded as an alternative means for global NSI with more specific dynamics information (such as changes in mode shapes and natural frequencies for respective modes) being explicit. Therefore, it will benefit an NSI analysis to implement the FEP together with such spatiotemporal IMOs. 59.3 Applications of NSI Results to Damage Identification We showed that modal characteristics or slow-flow dynamics change in the presence of certain defects (vibro-impacts) in a structure. In particular, qualitative changes seem to be prominent near the nodes of the natural modes closest to the impact position and at the instants of vibro-impacts. Study of such changes in vibration characteristics may illuminate essential features of structures with defects in applications to structural health monitoring (SHM) and damage detection (DD). Numerous methodologies have been implemented for SHM and DD – e.g., statistical pattern recognition algorithms as complementary tools for damage detection in civil structures [21]; substructural damage identification with incomplete structural information [22]; use of guided-wave techniques for estimating damage location [23]; and many other vibrationbased damage identification methods [24–29]. In this section, we present the modal assurance criterion (MAC [8]) and the coordinate modal assurance criterion (COMAC [30]) as a tool for damage identification by monitoring mode shapes instantaneously extracted from the NSI results. Indeed, noting that the spatial variations of a spatiotemporal IMO (STIMO) at each instant retain the corresponding mode shape, we can extract the instantaneous mode shape functions from the constructed STIMOs. The MAC is a popular means for diagnosing global changes in vibration characteristics of a structure, and is computed as MAC.r;s/ D j T r sj 2 T r r T s s (59.10) where and denote the discretized mode shape functions (or modal vectors) obtained from theory (or a healthy structure) and experiment (or a damaged structure), respectively; the subscripts denote the orders of the modes; and the superscript ‘T’ denotes the transpose of a vector. Since the MAC essentially computes the correlation between the mode shape functions, its results should be the Kronecker’s delta function of the two mode numbers in a healthy structure (i.e., unity along the diagonal of a MAC matrix with all zero off-diagonal elements). Unless modal analysis of the measured data was done poorly or the measured data were significantly affected by noise, MAC values of less than unity may imply the existence of nonlinearities caused by defects in the structure [8]. Indeed, the MAC values computed at the measurement points close to any nodes of a particular mode are found to be a more sensitive indicator of changes in the mode shape caused by damage [24]. Figure 59.4a, b depict the MAC matrices near the two impact instants in the initial transients. The first few impacts affect mostly the high-frequency modes (above the sixth), whereas the low-frequency modes retain their linear modal properties. This makes sense because nonlinear modal interactions caused by vibro-impacts to which high-frequency modes can be more susceptible are localized in the initial stage of the motion where high-frequency modes are more pronounced. The MAC values for most of the modes become smaller (i.e., globally low correlations between mode shapes due to vibroimpacts are recorded) after the initial VI transients, providing evidence that the effects of vibro-impacts start to prevail over the whole beam; and for some VI instants the linear structure of the beam is not even significantly. Indeed, the MAC matrix itself does not provide spatial information about where in the structure damage exists. This is because damage is typically a local phenomenon, which can be captured by high-frequency modes; on the other hand, the global behaviors are detected by low-frequency modes [24]. Noting that more energy is required to produce measurable response at higher frequencies, we find that it is more difficult to excite these higher-frequency responses of a structure. These aspects as well as the information loss from time series measurements may cause difficulties for modal-based damage
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