3 Spectral Element Based Optimization Scheme for Damage Identification 25 -150 -200 -250 -300 -350 -400 -450 -500 -550 4 4.05 4.1 4.15 4.2 4.3 4.4 4.45 4.5 4.25 4.35 Lateral Disp. dB (m/N) a x 104 Frequency (Hz) -150 -200 -250 -300 -350 -400 -450 -500 -550 4 4.05 4.1 4.15 4.2 4.3 4.4 4.45 4.5 4.25 4.35 Lateral Disp. dB (m/N) b x 104 Frequency (Hz) Pristine Damage 1 Damage 2 Damage 3 Pristine Damage 1 Damage 2 Damage 3 Fig. 3.4 Frequency responses for damage scenarios considered in (a) case study 2, and (b) case study 3 Table 3.2 Damage identification results Case study 1 Case study 2 Case study 3 Actual reduction (%) Estimated reduction (%) Actual width Estimated width Actual location Estimated location Damage 1 25 25 1 1 150 150 Damage 2 50 50 3 3 250 250 Damage 3 75 75 5 5 350 350 a completely different effect. While the shifts in FRFs monotonically increase with crack width and severity, damage location has dissimilar effects on different parts of the FRF. For instance, damage location 1 results in a larger shift for the resonance peak located initially at 44,115 Hz, however, for the 40,080 Hz peak, damage location 2 has the most profound effect. At low frequency ranges, the effect of damage location on the FRF can be interpreted through the mode shape associated with each fundamental frequency. However, this task gets more complicated at the high frequency ranges typically encountered with impedance-based SHM. The trends of FRFs dependence on each of the damage defining parameters can be utilized to develop a smart optimization scheme, where the search direction is updated based on the relative shifts in resonance and anti-resonance frequencies. Such an optimization algorithm is currently under development (Table 3.2). Finally, for each damage scenario in the three case studies presented in this section, the inverse problem is addressed. Damage identification is carried out by minimizing the objective function in Eq. 3.15 with FRFs replacing impedance signatures, the results are summarized in Table 3.2. Due to their monotonic effects on FRFs, very few iterations where needed to determine the optimal values of damage width and relative stiffness reduction. Finding the optimal value for damage location is relatively more complicated, and more iterations are required. The trends discussed earlier can be implemented in the optimization algorithm so as to improve its efficiency. 3.5 Summary Although modelling is not needed for the basic damage detection when using impedance based SHM, modelling provides valuable insights to this technique. Modeling becomes a necessity when performing more advanced SHM tasks such as damage identification and remaining life estimation (prognosis). In this study, a single impedance measurement is used for damage detection and identification in a beam. Spectral element method is utilized to calculate the high frequency structural impedance of the damaged beam. Damage defining parameters, which are damage location, width and severity, are then updated through an optimization scheme. The proposed technique is computationally efficient, as it requires solving a very small system of equations with three optimization parameters.
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