## 3 Experimental Results and Analysis
### Testing for Damage Assessment (O2)  
Even though detecting damage (notch) occurrence based on signals from testing with procedure P2 is definitely feasible, assessing the different damage levels is less trivial. Recall that damage detection proved possible at two frequency regions (40 and 50 KHz- see Section 3.2). At these regions, spikes from signals associated to -C1 and -C2 slabs differ only slightly. For an answer as to whether damage level (one or two notches) may be assessed via P2, a final series of tests has been carried out. Four -C1 and another four -C2 slabs were tested following P2 and the associated power spectra were plotted in Figure 9. Since four slabs of each damage level are available, ranges of spectrum magnitudes at 40 and 50 KHz for both damage levels may be formed. These ranges indicate the variation of spectral magnitude for signals associated to slabs with one or two notches. At both frequencies, an obvious trend for signals from more damaged (-C2) slabs having spectra of lower magnitude than their counterparts from less damaged (-C1) slabs is obvious. Spectra from -C1 slabs range between \(-35.5\) to \(-40.5\) dB at 40 KHz and \(-26\) to \(-32.5\) dB at 50 KHz. Spectra from -C2 slabs range between \(-40\) to \(-43\) dB at 40 KHz and \(-32\) to \(-36.5\) dB at 50 KHz. In both frequency areas, there is one -C1 slab whose magnitude overlaps that corresponding to a -C2 slab. Hence, since most of -C1 slabs have spectra of larger magnitude than those from -C2, procedure P2 provides indicative (rather than definitive) answers on the damage level of the tested slab. These conclusions are based on the (rather empirical) estimation of the magnitude of damping values at each frequency found in [28]. Various approaches (for instance using dedicated equipment to analyze mode shape) may help to characterize differences induced by the damage level to the slab  
Figure 9: Power spectrum of response of type-C1 and -C2 slab tested following procedure P2. Evidence that the damage level is only slightly short of being detectable at 40 and 50 KHz is given in insets. vibration modes. In the current study, quantitative damping data have been obtained by the recorded signals by investing effort in signal modeling procedures, as follows:  
1. Response signals recorded by testing each -C1 or -C2 slab using procedure P2 were treated as stochastic time-series sequences. This was quite realistic, since the recorded signals had considerable noise due to electromagnetic parasitic phenomena related to the contact-less recording principle. Furthermore, given that damage may be assessed by means of two frequency regions (around 40,000 and 50,000 Hz), it was beneficial to perform high-pass filtering of the raw signals, allowing for a lower number of frequency components to be modeled. One simple means of doing so, was by differencing the raw signals (that is, forming the differences of each signal value at instant \(t\) minus that at \(t-1\) for all \(t\)) as many times as necessary for effectively undermining lower frequency (below 40 KHz) components. Naturally, a traditional high pass filter may also be applied. Here, the raw signals recorded from testing with -C1 and -C2 slabs (with their spectral characteristics shown in Figure 9) were differenced six times, for obtaining signals with insignificant frequency content below 40 KHz. 2. Each signal was modeled by means of stochastic output-only AutoRegressive representations with constant coefficients (CC-AR), identified via standard algorithms. These may be coded in various programming languages, or even be found in software packages such as MATLAB(r). 3. For each identified AR representation, poles at 40 and 50 KHz were monitored with respect to their damping. Characteristic changes in damping values (at these frequencies) between different data sets lead to damage assessment conclusions about the slabs generating these data. The CC-AR representation is a standard yet effective means of modeling dependencies among values of a stochastic sequence.