## 4 Severity Curves Derived from the Calculated Damage Deflections
### Tests Performed with FEA  
Damage detection using modal parameters requires accurate algorithms to detect the slightest frequency changes in structures. For determining the accuracy of the described method used for detecting transverse cracks, we have performed FEM modal simulations  
Figure 6: Severity evolution with the crack depth for closed and open cracks. using the ANSYS software for the same cantilever beam described in Section 2. The beam is successively affected by closed and open transverse cracks with different depths and located in different slices of the beam. As a first example, we present in Figure 7 the natural frequencies obtained from simulation and with Equation (3) for the beam with a transverse crack that has the parameters: \(x=604\) mm, \(a=1\) mm, and \(w=0\) mm. When using the analytical approach, we calculate the frequencies with Equation (3) in which we consider the measured frequency of the intact beam and the severity deduced from the deflections achieved by static analysis. At a first look, the frequencies in Figure 7 fit, but it is difficult to evaluate the accuracy of the method. However, one can observe that the differences between the natural frequencies obtained involving the analytical method and the FEM results are small. To trace a relevant conclusion, we represent in Figure 8 the difference between the predicted frequencies and those obtained from simulation. Before being represented, the differences are normalized, according to the mathematical relation  
\[\epsilon=\frac{f_{i-D}(FEM)-f_{i-D}(a,x)}{f_{i-U}} \tag{13}\]  
One can observe that absolute normalized differences are extremely small, between \(-0.000019\) and \(+0.000125\). Because the frequency differences are normalized, the error does  
Figure 8: The normalized differences between the natural frequencies determined analytically and using FEM for the cantilever beam with a transverse closed crack that has the parameters \(x=604\) mm and \(a=1\) mm. Figure 7: Comparison of the natural frequencies obtained for the cantilever beam with a transverse closed crack that has the parameters \(x=604\) mm and \(a=1\) mm. not increase with the mode number. By calculating the normalized differences for more damage scenarios, we found out that the errors are comparable or smaller. This proves that the results obtained with the DS-SHC method are reliable. A second example considers also closed cracks, thus \(w\) = 0 mm. The damage scenarios are defined in ANSYS, for three crack depths \(a\) = 0.2, 1, and 1.6 mm. For each crack depth, the position of the crack is \(x\) = 125 mm and \(x\) = 489 mm, respectively. After we defined the damage scenarios, we determined the severity \(\gamma(a)_{FEM}\) using relation 12 for the first six weak-axis vibration modes. Relative to the crack's main dimensions, \(a\), \(x\), and \(w\), the damage scenarios are noted as C(\(a\),\(x\),\(w\)). The severity values \(\gamma(a)_{FEM}\) are compared, in Figure 9, with the calculated ones \(\gamma(a)\). Figure 9: The normalized differences obtained between the natural frequencies determined analytically and by means of FEM for the cantilever beam with a transverse crack: (**a**) Damage scenario C(0.2, 125, 0); (**b**) Damage scenario C(0.2, 489, 0); (**c**) Damage scenario C(1, 125, 0); (**d**) Damage scenario C(1, 489, 0); (**e**) Damage scenario C(1.6, 125, 0); (**f**) Damage scenario C(1.6, 489, 0). A third example considers open cracks. We performed FEM simulations for defined damage scenarios that involve cracks with widths \(w\) of 0.5, 1, and 2 mm. The results for all damage scenarios are presented in [36].