## 4 Severity Curves Derived from the Calculated Damage Deflections
### Tests Performed Involving Laboratory Experiments  
To prove the accuracy of the developed algorithm, we also conducted laboratory studies on steel cantilever beams affected by transverse cracks of known location, depth, and width. The tests consist of measuring the natural frequencies in the intact and damaged state. Because the accuracy of RFS calculated with Equation (11) is relevant for damage detection, in this section we compare these RFS with those obtained from measurements. The laboratory setup consists of a rigid structure including a vise in which the beam is fastened, an excitation device, and the data acquisition system. The experimental setup is presented in Figure 11 and described in detail in [27]. The excitation system involves a speaker and amplifier which are controlled using AudioDope software. The beam is excited at specific frequencies, and the vibration response is acquired. To acquire the signals, we use a data acquisition system consisting of a Kistler 8772 accelerometer which transmits the signal through the analog-to-digital conversion module NI9234 to the compact chassis NIcDAQ-9175. This module is connected to a second laptop, on which the LabVIEW software is installed. The acceleration signal is acquired and is subsequently processed to extract the natural frequencies with high accuracy using the procedure described in [37; 38]. The Python code implementing the procedure is available in [39]. The experimental study was carried out on four S355 JR steel cantilever beams of dimensions 1 \(\times\) 0.05 \(\times\) 0.005 m, at first in an intact state and later in a damaged state, by generating transverse cracks of width \(w\) = 2 mm by saw cutting. One end of the beams is fixed in a vise (see Figure 12). At least five natural frequency readings were made for each test and the arithmetic mean was considered. For each reading, the first six natural frequencies of the beam were extracted, and the obtained values are listed and compared with the natural frequencies  
Figure 11: Experimental setup. Figure 12: Test specimen with generated transverse crack mounted on the experimental stand. obtained from FEM in Table 6. From the compared values the small differences can be observed. A transverse open crack was generated on each beam analyzed above, thus resulting, for a single beam, six natural frequency values corresponding to the six transverse vibration modes. We present in Table 7 the crack dimensions for each damage scenario and the measured natural frequencies. In this table, we also included the severities for the four cracks, derived using the method described in the current research. We calculate now the RFSs with the frequencies taken from Tables 6 and 7, which are found from experiments. To this aim we use Equation (11). We also calculate the RFSs analytically (see Section 2). In this case, the input data consists of the deflections obtained from FE simulations; these permit calculation of the severity with Equation (1). With the calculated severity we find the frequency drop with Equation (3) and, finally, the RFSs with Equation (11). The comparison between the RFSs obtained with experimental results and those calculated is presented in Figure 13.  
From the diagrams represented in this figure, it can be observed that there is a good fit between the compared values. Note that once the severity curves are known and plotted as in Figure 6, the severity can be easily determined for any damage depth. Thus, the FE simulations are no longer required when the procedure is applied. By using the method described in the current paper, by employing Equation (11), we have generated training data for developing a damage detection neural network similar to the one presented in [13]. The training data consist of the RFS values for the six transverse vibration modes. After the network was trained, we successfully determined the position and depth of the cracks for the four experimental cases by considering the RFS values. The errors obtained are presented in Table 8.