# A Comparison of Deep Learning Algorithms for Anomaly Detection in Discrete Mechanical Systems
## 3 Experimental Campaign  
To locate damage, both the conventional method and the machine learning algorithm rely on detection of the changes in the behavior of the structure. In particular, the methods studied in this paper use the differences in the vibration measurements between the building in its nominal configuration, called \"healthy\", and the \"damaged\" configuration as an indicator of damage. For this reason, the experimental campaign conducted on the tested structure aims to acquire raw data for both \"healthy\" and \"damaged\" structures. The experimental setup for both the cases consists of:  
* Four TE triaxial capacitive MEMS accelerometers, one per storey;
* A PCB piezotronics impact hammer;
* A National Instruments c-DAQ. The structure is excited by an impact hammer and the transversal vibrations are measured. In particular, as the impact hammer is used manually, it was checked that the peak force for each acquisition ranged between 15 N and 70 N in order to excite the structure with similar impact energies. For the \"healthy\" case, a set of 1000 records of 70 s each was recorded with a sampling frequency equal to 128 Hz. An example of the responses, limited to the first 30 s, is shown in Figure 3.  
As the vibration signals were acquired in Volts, they need to be pre-processed before being analyzed. To do this, the sensitivity of both the accelerometer and the impact hammer must be taken into account and the unit of measurement must be correctly restored (m/s\({}^{2}\) and N, respectively). Moreover, it must be kept in mind that the measured data are digitized signals and acquired over a finite window of time. For this reason, they may be affected by issues such as (i) noise, (ii) aliasing (as they are sampled), and (iii) leakage, as the signals are acquired for a finite window of time. For these reasons, an averaging procedure was conducted and two different estimators, reported in Equation (9), are considered to evaluate the inertance (acceleration/force) Frequency Response Function (FRF) for each accelerometer. \[H_{1}(f)=\frac{X(f)}{F(f)}=\frac{G_{XF}(f)}{G_{FF}(f)} \tag{9}\] \begin{table}
\begin{tabular}{c c} \hline \hline \(\alpha\) & 0.03 \\ \(\beta\) & 0.028 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Results of the least square minimization process. where \(F\) and \(X\) are, respectively, the Fourier transforms of the input force and the output vibration and \(G_{FF}\) and \(G_{XX}\) are the estimates of the auto spectra, while \(G_{XF}\) and \(G_{FX}\) are the estimates of the cross spectra, which in general are evaluated as:  
\[ \begin{split} G_{XX}(f)&=\frac{1}{N}\sum_{j=1}^{N}X_{ j}^{*}(f)\cdot X_{j}(f)\\ G_{XY}(f)&=\frac{1}{N}\sum_{j=1}^{N}X_{j}^{*}(f) \cdot Y_{j}(f)\end{split}  \tag{10}\]  
where \(X\) and \(Y\) are two general sampled signals and N is their length. It could be interesting to evaluate the cross and power spectra among the different floor accelerations. This is represented in Figure 4, where it is possible to observe the natural frequencies of the system and the ratio between the responses of the different floors corresponding to these frequencies. Figure 5 shows an example of FRFs for the case in which the structure was excited by an input force acting on the 5 th floor. Once the FRFs were computed for each accelerometer, the natural frequencies and the mode shapes were extrapolated by means of the Experimental Modal Analysis (EMA) [29, 30]. In particular, the natural frequencies of the system are reported both for the numerical and the experimental model in Table 3. The mode shapes, normalized with respect to their maximum value, are reported in Figure 6.  
Figure 3: Acceleration of each floor after an input was applied on the 5 th floor. Figure 4: Cross and power spectra for each accelerometer and different references. Figure 5: FRFs evaluated for each accelerometer, one for each floor, after an input was applied on the 5 th floor.