# Identification of Damage in Beams by Modal Curvatures Using Acoustic Beamformers
## 3 Direct Problem
### Comparison between Numerical and Experimental Results  
Experiments were carried out using the setup of Figure 3. The steel beam has free-free end conditions to avoid uncertainties in the constraint behavior that can arise from friction and other imperfections. This setup enables a better evaluation of the different estimators. The response was measured in damaged and undamaged conditions using seven strain gauges equally spaced along the beam axis and two accelerometers, whose mass (12 g each) was included in the FE model. The beam was dynamically excited by an impulse force generated by an instrumented hammer by hitting point #8 or #9, and repeating the test ten times per each point of application of the forcing function. The applied impulse force is capable of exciting frequencies up to 2500/3000 Hz, which enabled the determination of the first three natural frequencies. Accelerometers were placed at points #8 and #9 for the sake of modal curvature normalization. The frequency response function measured at the abscissae \(s_{i}=id_{1}\) (Figure 1), with \(i=1...7\), for a point force applied at the abscissa \(s_{j}\) where \(j\) can be equal to 8 or 9, is \(H_{ij}(\omega)=U_{i}(\omega)/P_{j}(\omega)\), where \(U_{i}(\omega)\) and \(P_{j}(\omega)\) are the Fourier transforms of the response output and of the force input, respectively. Note that we can express the frequency response function in terms of curvatures as  
\[H_{ij}(\omega)=\sum_{k=1}^{\infty}\frac{\chi_{k}(s_{i})\phi_{k}(s_{j})/(\rho L \omega^{2}||\phi_{k}||^{2})}{1-(\omega_{k}^{2}+i\eta_{k}\omega_{k}^{2})/ \omega^{2}} \tag{8}\]  
Figure 2: Variation of modal displacements (**a**–**c**) and modal curvatures (**d**–**f**) in the four damage scenarios of location A, for the first (**a**,**d**), second (**b**,**e**) and third (**c**,**f**) mode. where \(\omega_{i}\) is the \(i\)-th natural frequency, and \(\chi_{i}\) stands for the second spatial derivative of the \(i\)-th mode shape \(\phi_{i}\), that is, the modal curvature. The comparison of numerical and experimental quantities requires using the same normalization: numerical modal curvatures are derived by a standard modal analysis [34], where modes are normalized with respect to the mass distribution. For the experimental results, as evident in Equation (8), it is possible to adopt the same normalization, provided that the acceleration where the force is applied is known as already observed in [16]. This is the reason why two accelerometers are placed at points #8 and #9 (see Figure 3) and their response is used for normalization. The geometrical properties of the real system differ slightly from the nominal ones, and were therefore measured in the laboratory with a caliber. This applies in particular to the base \(B\), the height \(H^{U}\), and the residual heights \(H^{D}\). To further improve the match between the experimental and numerical data, the beams were also weighted to evaluate their mass density \(\rho\). [PERSON]'s modulus was determined by minimizing the difference between the experimental and numerical frequencies of the undamaged beam. All these geometrical and mechanical properties are reported in Table 3, including the stiffness reductions \(\beta\), and Table 4. The largest value of residual height under consideration (D1.A and D1.B) relates to the smallest damage that can be accurately saw cut with our equipment. Tables 5 and 6 report the experimental values of the first three natural frequencies (\(f_{e}\)) of the undamaged and damaged beams, together with the related percent changes due to damage (\(\Delta f_{e}\)). The experimental frequencies are averaged over ten repetitions. Their percent coefficient of variation (standard deviation/average) is extremely limited, always smaller than 0.02%. The expected monotonic reduction of frequency due to increasing damage is evident. The tables also report the values of frequencies (\(f_{n}\)) and their percent variations (\(\Delta f_{n}\)) obtained from a model where damage depths were updated according to Tables 3 and 4.