*3.3. Force Prediction Using Load/Elongation Increments*

The concept of identifying forces in bolts using load or elongation increments consists of the fact that the load change in the tested connection is introduced. This is accompanied by the simultaneous measurement of elastic wave propagation and the elongation in one of the screws. Then, the obtained elongation increment is added to the other parameters provided in the input vector to the ANN.

To show the potential of the proposed approach, preliminary simulations were performed on the data related to the P3 connection. For this purpose, the database was divided into relatively small increments. It turned out that better identification results were obtained when these increments were not uniform, but variable with a random Gaussian distribution. Thus, the database used for ANN training consisted of elongation increments of the S1 bolt and the parameters of signals (taken from the encoder) measured at the following bolts: S4 for learning; S2 for testing; S5 for validation; and S3 for prediction.

The results of the preliminary tests are presented in Figure 12a together with the results corresponding to the ANN trained directly using the elongation of the S1 screw (Figure 12b). It can be seen that there were some differences in the accuracy of testing, validation, and prediction of the axial forces. The error values calculated on this basis are collected in Table 5. Despite the clearly superior results in the case of training the ANN with the direct elongation of the S1 screw, the results of both approaches remained in good agreement with the results of the experimental measurements of the F2 and F5 axial forces in the S2 and S5 screws. Therefore, in future studies, this approach will also be extended to other connections.

**Figure 12.** The results of force identification in the P3 connection based on: (**a**) load/elongation increments; (**b**) direct elongation of the bolt S1.


**Table 5.** Comparison of the learning, testing, and validation errors: P3 and P3 increments.

As a summary of the approaches analyzed so far, the results of axial force prediction in the S3 bolt of the P3 connection (which was not equipped with a force sensor) are summarized in Figure 13a. There, we can see the results obtained for the ANNs trained on the basis of data from a single connection (P3 with elongation, P3inwith elongation increments), two connections (P2P3), and three connections (P2P3P4). The results related to P3 and P2P3P4 are similar to each other (which is clearly seen in Figure 13b), while P3in and P2P3 slightly differ from them.

**Figure 13.** The result of the prediction forces with respect to the P3 connection: (**a**) comparison of the studied input vector scenarios; (**b**) details of (**a**).
