17 An Experimental Investigation of Feature Availability in Nominally Identical Structures for Population-Based SHM 189 Fig. 17.6 FRFs of tails B1 and B2. The effect of the added mass of 104 g on peak 184 Hz is shown 180 182 184 186 188 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Frequency [Hz] Amplitude [m/s2/N] B1 normal B1 − 104g B2 normal B2 − 104g There were in total 75 FRFs recorded (from 75 locations) for the A1, A2, and 65 FRFs for the B1 and B2 structures. From all those FRF locations, there is a subset of 44 which are identical for all structures. A feature in this work is defined as a set of data measured or derived from measured data which can be used to distinguish the different states of the structure. By different states, one may consider ‘healthy’ and ‘damaged’. Figure 17.6 shows an example of a similar pattern appearing in both B1 and B2 structures: when a mass is added, the peak at 184 Hz drops both in frequency and amplitude. Essentially, this common pattern is a potential feature, and its suitability can be tested with the help of am appropriate classifier. Any approach would be acceptable, but here outlier analysis is the chosen method. The approach has been described in detail in [10] and it involves the fusion of a multivariate feature into a single quantity called the Mahalanobis squared distance, which is then compared against a threshold. The threshold is calculated through a Monte Carlo approach, and can be inclusive or exclusive if the potential outlier is used in its calculation or not. The Mahalanobis squared distance is calculated by Eq. (17.1). D D.x x/ TS 1 .x x/ (17.1) As only single high average FRFs were measured for each state, the potential feature (the frequency lines of interest) was copied 1,000 times and then polluted with normally distributed noise of a standard deviation equal to the 4 % of the maximum value of the original feature. In this particular feature the size of the dimensions used is 50 (50 points around the peak). The FRFs originating from the intact structures are used as the normal data, and their mean and standard deviation is calculated to be used with Eq. (17.1). Then, the Mahalanobis distance from the features created from the FRFs with the added mass are tested against the threshold in order to declare whether there is significant deviation from normality, therefore damage or not. Figure 17.7 shows the Mahalanobis distance created from the feature which was extracted from the FRFs (located at point 65) of structures B1 and B2 around the mode 184 Hz. The threshold in this case was calculated at 112.67. It is clear that this feature could be used for both structures in order to indicate the presence of added mass. What is important, is that this feature can be selected just in one of the two nominally identical structures (B1 or B2) and it will be able to indicate the presence of damage (added mass here) in the other. The location of the FRF which is shown in Fig. 17.6 is not in the subset of the common sensor positions among all of the structures, therefore a direct comparison of how well this particular feature performs in tails A1 and A2 is not possible. However, the frequency lines (which correspond to the feature) can be tested with outlier analysis in all of the rest of the sensor locations, both for A1 and A2. In order to check easily the performance of the feature, the Mahalanobis squared distance can be normalised by the threshold (which always depends on the dimension). In this way, every feature which produces a mean normalised discordancy value greater than 2 will be considered adequate and successful. Out of the 60 FRFs that there were tested for structures A1 and A2, 47 produced adequate features for both the structures. In the case of B1 and B2 this number was a bit higher at 57. Due to some problems presented during the experimental tests of the structure A2, the FRFs numbering 60–75 were not considered reliable and therefore they are excluded from this study here. Out of the 47 FRFs which produced adequate features for A1 and A2, 33 also work for B1 and B2. This result mainly means that a feature which was ‘blindly’ selected to indicate the added mass in structures B1 and B2 also works for both A1 and A2. Since the aforementioned feature was selected by visually checking both the FRFs from the B1 and B2, it may be interesting to explore what would happen if one selects features simply on only one of the four structures. To investigate this, structure A1 was arbitrarily chosen, and in the same spirit, the FRF from sensor location 20 was also arbitrarily chosen.
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