*2.4. Signal Analysis*

The elastic wave signal comparison in the initial stage showed that they were not exactly the same (Figure 6). There may be several reasons for this: the piezoelectric transducers may not have been positioned precisely in the same place (the central axis of the bolt); the thickness of the wax layer was not the same; the signals were affected by the closest components of the connection (washers, brackets on the pipe, weld thickness, etc.). Despite these differences, an attempt was made to train the ANNs for the purpose of force prediction. It was even decided to perform signal normalization within the range ±0.9 (with respect to the initial stage), but this did not significantly affect the results of the predicted force values.

**Figure 6.** Excitation signal and the received responses measured at the S2 to S5 bolts in the P1 connection.

As a result of the length of each signal, it was necessary to determine the parameters sensitive to load changes in the connections. Thus, the measured signals were transformed into the domain of the principal components [8,12]. This allows one to compute the linear transformation:

$$\mathbf{y} = \mathbf{W} \cdot \mathbf{s} \tag{1}$$

which maps data from a high-dimensional space **s** ⊂ R*<sup>N</sup>* to a lower dimensional space **y** ⊂ R*<sup>K</sup>* of the principal components, without much loss of information. In this way, the elastic wave signals, each containing 10,002 points, after decimation at the rate *r* = 2, were reduced to only 12 principal components.

On the basis of the preliminary test results [11], it turns out, however, that the principal components alone may not be sufficient for proper training of the diagnosis system. The point is for the ANNs to acquire generalization capabilities that enable the prediction of force values in bolts in which no control measurements were carried out. In other words, the obtained values of the estimated forces in the screw, which did not participate in the training of ANNs, differed from the nature of the work of the other screws, for which these forces were physically measured using washer sensors. Therefore, an attempt was made to determine other parameters from the signals that could improve the generalization abilities of ANNs. The pattern database was extended to include data related to the time and amplitude of the occurrence of subsequent maxima in the response signals (Figure 7a). These arrival times were expressed as the distance between the maximum excitation amplitude and the local extremes of the response amplitudes. Their values were determined for all signals and analyzed connections. To illustrate the nature of the changes in arrival times, the value *t*<sup>0</sup> obtained without a load

was subtracted from each of them (Δ*t* = *ti* − *t*0, where *i* = {1, 2, ... , 7}). An example of the obtained time variations in the case of the P1 connection is shown in Figure 7b. Examples in the literature indicate that the relationship between ToF changes and load is most often almost linear [18,19]. In this particular case, with the type of sensors used and the established excitation signal parameters, these relationships are non-linear.

**Figure 7.** Arrival time and amplitudes: (**a**) an example of how they were determined; (**b**) differences of arrival time in the P1 connection.

Moreover, although this type of graph does not indicate this, along with the increase in the axial force, the determined arrival times of the individual peaks in the signal were generally constantly reduced. It was expected that the tensioning of the connection would cause screw elongation and a longer wave path would increase the related propagation time. Although there are examples in the literature where the impact of stress on the propagation speed was analyzed [16], a wider discussion of this issue, which is very interesting, is not the main subject of this study.
