# Identification of Damage in Beams by Modal Curvatures Using Acoustic Beamformers
## 3 Direct Problem
### Comparison between Numerical and Experimental Results
The observed experimental frequency variations are smaller than 0.8% for a 30% variation in stiffness reduction, and reach 10% only for a stiffness reduction of 87%. Furthermore, it should be noted that while the numerical frequencies of the undamaged beam (Table 2) compare well to the experimental ones, those of the damaged model do so to a lesser extent. In fact, in the majority of cases, the resulting frequency changes experimentally observed are smaller than the analytical ones. This calls for further investigations of the appropriate damage models. We recall here that the numerical values of the modal curvatures were averaged over a length of 6 mm, that is, the length of the grid of the strain gauges used in the experiments. Figure 4 (cases A) and Figure 5 (case B) report numerical and experimental modal curvatures of the first, second and third modes and their variations in the four damage states. Experimental modal curvatures are averaged over the ten test repetitions. Note that modal curvatures present a higher coefficient of variation, which can reach 10%. Figure 4 shows that a remarkable variation of curvature is detected at sensor #2, where damage is located. However, variations take place all along the beam span, with a number of nodes increasing along with the mode order, as is apparent also in Figure 2d-f. On the contrary, when damage is located between sensors #3 and #4, as reported in Figure 5 (case B), the observed curvature variations are considerably smaller in value than those in case A, and are distributed all along the beam axis. No remarkable peak can be observed. The variation of modal curvatures increases monotonically with the increase in damage  
Figure 3: Schematic representation of the experimental setup (**a**) and image of the beam in laboratory conditions (**b**). and with the mode order. In all cases considered, the pattern of experimental curvatures is quite similar to the numerical one, although the experimental values and their variation below the cut (#2) appear to be smaller than the numerical ones. This is similar to what was observed for frequency variation. Regarding case B, which fit the numerical results to a lesser extent, the coefficient of variation of their modal curvatures was generally higher than in case A, which leads to greater uncertainty in the definition of the experimental curvature. \begin{table}
\begin{tabular}{c c c c c c c} \hline  & \(f_{e1}\) & \(\Delta f_{e1}\) & \(f_{e2}\) & \(\Delta f_{e2}\) & \(f_{e3}\) & \(\Delta f_{e3}\) \\ \hline U & 297.7 & & 809.8 & & 1584.5 & \\ \hline D1.B & 295.5 & 0.73 & 808.3 & 0.19 & 1579.0 & 0.35 \\ \hline D2.B & 287.8 & 3.33 & 803.5 & 0.78 & 1564.1 & 1.29 \\ \hline D3.B & 282.1 & 5.23 & 799.0 & 1.34 & 1549.6 & 2.20 \\ \hline D4.B & 267.6 & 10.12 & 790.0 & 2.45 & 1521.5 & 3.97 \\ \hline  & \(f_{n1}\) & \(\Delta f_{n1}\) & \(f_{n2}\) & \(\Delta f_{n2}\) & \(f_{n3}\) & \(\Delta f_{n3}\) \\ \hline U & 297.5 & & 811.2 & & 1582.6 & \\ \hline D1.B & 295.1 & 0.83 & 808.9 & 0.28 & 1578.7 & 0.26 \\ \hline D2.B & 288.1 & 3.16 & 802.6 & 1.06 & 1567.5 & 0.96 \\ \hline D3.B & 279.1 & 6.19 & 794.9 & 2.01 & 1553.5 & 1.85 \\ \hline D4.B & 265.7 & 10.68 & 784.3 & 3.32 & 1534.2 & 3.06 \\ \hline \end{tabular}
\end{table} 
Table 6: Case B: experimental (\(f_{ei}\)) and numerical (\(f_{ni}\)) natural frequencies [Hz] of the first three modes (\(i=1,\,2,\,3\)), and their percent variation (\(\Delta f_{ei},\,\Delta f_{ni}\)) due to damage. \begin{table}
\begin{tabular}{c c c c c c c} \hline  & \(f_{e1}\) & \(\Delta f_{e1}\) & \(f_{e2}\) & \(\Delta f_{e2}\) & \(f_{e3}\) & \(\Delta f_{e3}\) \\ \hline U & 298.8 & & 813.52 & & 1589.8 & \\ \hline D1.A & 298.2 & 0.21 & 810.5 & 0.38 & 1584.6 & 0.32 \\ \hline D2.A & 296.8 & 0.68 & 797.8 & 1.98 & 1561.3 & 1.79 \\ \hline D3.A & 294.1 & 1.58 & 784.5 & 3.57 &Figure 4: Case A: experimental (e) and numerical (n) modal curvatures (first (**a**), second (**b**) and third (**c**) mode) and their variations due to damage (first (**d**), second (**e**) and third (**f**) mode).