# Identification of Damage in Beams by Modal Curvatures Using Acoustic Beamformers
## 3 Direct Problem
### Sensitivity of Modal Quantities to Local Damage and Generation of the Replica Vector  
To investigate the effects of damage on the modal properties of the beam, a total of eight damage scenarios are investigated. The cases under consideration include one beam with a notch located at \(s^{D}=0.25\) (cases A), below sensor #2, and a second one with notch at \(s^{D}=0.4375\) (cases B), between sensors #3 and #4. In both cases the notch has a fixed extension \(b^{D}=1\) mm. A total of four increasing stiffness reductions are considered for each of the two locations. The reference labels of the different damage scenarios are reported in Table 1, along with locations, residual heights of the cross-section, and percent stiffness reductions. The cases under study correspond to four damage depths, starting from a small reduction of the height of the cross-section (12.5%), associated with the damage depth of 2.5 mm. As a first step, natural frequencies are observed. The frequencies of the first three modes (\(i=1\), 2, 3) of undamaged (\(U\)) and damaged (\(D\)) beams in the eight damage scenarios are reported in Table 2, along with their percent reduction \(\Delta f_{i}=(f_{i}^{U}-f_{i}^{D})/f_{i}^{U}\) with respect to the undamaged state. Despite the fact that the reduction of the bending stiffness is significant even in the mildest damage cases D1.A and D1.B (\(\beta=33\%\)), the variation of frequencies is rather small, ranging from 0.25% to 1.23%, dep \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline  & **D1.A** & **D2.A** & **D3.A** & **D4.A** & **D1.B** & **D2.B** & **D3.B** & **D4.B** \\ \hline \(s^{D}\) & 0.25 & 0.25 & 0.25 & 0.25 & 0.4375 & 0.4375 & 0.4375 \\ \hline \(h^{D}\) & 0.875 & 0.750 & 0.625 & 0.500 & 0.875 & 0.750 & 0.625 & 0.500 \\ \hline \(\beta\) & 33.0 & 57.8 & 75.6 & 87.5 & 33.0 & 57.8 & 75.6 & 87.5 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Labels of the damage scenarios with location (\(s^{D}\)), nominal height of the damaged cross-section (\(H^{D}\)), and percent stiffness reduction (\(\beta\)). Figure 1: Schematic representation of the beam longitudinal view (**left**) and of its cross section (**right**). location of the damage. Only in the most severe cases D4.A and D4.B, associated with a stiffness reduction of almost 90%, the frequency reductions are in fact significant, ranging from 3.54% to 12.60%. Figure 2 reports, for cases A, the variation due to damage of modal displacements (\(\Delta\phi_{i}=\phi_{iD}-\phi_{iUl}\), a, b, c) and modal curvatures (\(\Delta\chi_{i}=\chi_{iD}-\chi_{iUl}\), d, e, f) of the first three modes as a function of the nondimensional longitudinal abscissa \(s=S/L\) (Figure 1). Modal displacements are the amplitudes of vertical displacements of the \(i\)-th mode measured on the top flange. Both modal amplitudes and modal curvatures are normalized with respect to the mass distribution. It is obvious that the differences between undamaged and damaged modal displacements (Figure 2a-c) are distributed all over the entire beam length, while differences between modal curvatures (Figure 2d-f) tend to concentrate near the notch, especially for low-order modes. In all cases, the variation of modal displacements and curvatures increases monotonically with the increase of damage. It should be noted that the change of modal curvature at damage location always remains a global maximum [19; 20; 33], although it perceptually decreases when the mode order increases, when compared to other local maxima. Conversely, in modal displacements variations, for higher-order modes, global maxima do not necessarily appear where damage is located, but can be observed elsewhere. Similar results are obtained for the damage position B but are not reported for the sake of brevity. These observations hint at the fact that modal curvatures could be more effective in detecting damage than other modal quantities, such as mode shapes and natural frequency variations.