## 4 Numerical Application: Simply Supported Beam  
In order to initially evaluate the SAE's efficiency to characterize vibration data, numerical simulations are performed using a simply supported beam model (see Figure 4a). The beam is modeled with twelve Euler-Bernoulli elements (two degrees of freedom per node - one translation and one rotation) with the following properties: total length = 12 m; [PERSON]'s modulus = 210 GPa; cross-section area = 0.04 m\({}^{2}\); specific mass = 7850 kg m\({}^{-3}\) and moment of inertia = 5.33 \(\times\) 10\({}^{-4}\) m\({}^{4}\). The numerical integration is performed by using [PERSON]'s method with a time step of 0.001 s. The excitation force \(F(t)\) is a typical random vector with 10 kN of maximum magnitude, which is randomly set for every dynamic simulation carried out and is applied at node 10. \(F(t)\) (kN) is simulated as shown in Equation (1):  
\[F(t)=10\times V\sim D(0,1) \tag{1}\]  
where \(V\sim D(0,1)\) is a vector with random values extracted from a uniform distribution from 0 to 1. Figure 4b,c present a typical response of the system for a random excitation force \(F(t)\), respectively. Three structural damage scenarios were simulated: healthy structure, damage level #1, and damage level #2. The first and second damage levels were defined by reducing in 10% and 20%, respectively, the Young's modulus of element #4 from the undamaged structural model (depicted in Figure 4 by the gray element). The vertical dynamic responses of points 2 (Ac1), 4(Ac2), 6(Ac3) and 8(Ac4) were recorded in terms of accelerations and were used to evaluate the SAE-based damage detection methodology. Each vibration signal lasts 5 s and has 5001 discrete time samples (sampling frequency of 1000 Hz). The structure's natural frequencies in all assumed conditions are given in Table 1, merely for information purposes. Figure 3: SHM strategy using SAE. Three hundred different numerical simulations for each structural stage were generated by varying the excitation, totaling 900-time histories per measurement point (\(N\) = 900 available dynamic signals). Since the excitation is random, the structure's dynamic responses are also random. The proposed SHM approach is evaluated by investigating the signals from one point at a time. Thus, the entire dataset is organized in four input matrices [900 \(\times\) 5001] (900 dynamic time histories composed of 5001 time samples), corresponding to the responses measured in positions Ac1, Ac2, Ac3 and Ac4. Since the SVM classifies the data category, it is also necessary to indicate the damage class of the signals. For this reason, a target matrix [900 \(\times\) 3] is assembled, specifying in its rows the respective signal category through the binary code: (1 0 0)--healthy beam; (0 1 0)--damage level #1; and (0 0 1)--damage level #2. It is worth mentioning that the target matrix is only used in the SVM training phase. During the evaluation phase, these labels have no direct participation in the classification. They are only applied to compare the results obtained by the SVM with the actual structure degradation level. Before being processed by SAE, the signal matrices are normalized within the range [\(-\)1;1]. The goal of normalization is to change the values of numeric columns in the dataset to use a common scale, without distorting differences in the ranges of values or losing information [45]. Afterwards, the input matrices--and consequently, the target matrices--were randomly subdivided into other two matrices, allocating 50% of the signals for the training phase--matrix \(\mathbf{A}\) [450 \(\times\) 5001]--and the remaining 50% of data for the evaluation phase--matrix \(\mathbf{C}\) [450 \(\times\) 5001]. In order to guarantee that all structural scenarios are equally represented in the SAE and SVM models, the training and evaluation matrices should have approximately the same number of samples from each damage level (balanced dataset). For further information about this subject, see [PERSON] and [PERSON] (2002) [46].