## 4 Artificial Neural Networks (ANNs)  
Neural networks are composed of neurons that are connected to each other in a way similar to that in which neurons are connected in a nervous system. What enables this type of system to work properly and in the way that it is intended to is the applicability of the connections that constitute the network [59]. The strong capacity of a neural network can be related to its ability to associate different memories, its high performance in terms of adaptation, and the scope for self-organization and non-linear mapping. All of these indicators are important parameters for solving non-complex problems. ANN is thus a technique that can be widely used in FD [43]. The vibrational spectra that characterize different fault conditions can be processed using ANNs. The training of ANNs can be performed using extracted numerical data; thus, the prediction of damage presence and severity can be completed [60]. After generating a database of simulated signals in healthy and damaged conditions, two types of classification  
Figure 11: Points representing each object location from the simulated model of the test bench (tangential shaft surface). The first step aimed at the extraction of the displacements from each point, while the second step aimed at the extraction of the vibrations after a double integration. can be performed: binary and multi-class. The determination of the characteristics of a system, in this case healthy and damaged states, should be well performed so that a consistent classification can be made. This procedure can save time and effort when identifying a small set of features [56]. Damage identification was initially the main target of this study, based on ANNs using the classification method followed by damage quantification via the multi-class classification approach. To address this problem, the ANN approach applied here can be activated to identify whether a system has sustained damage or not using pattern recognition. The network is composed of three major elements: input, hidden, and output layers. The Neural Network Pattern Recognition Tool (nprtool), in MATLAB, was used to construct the architecture. It is an effective tool when the data can be inaccurate or incomplete [61]. Additionally, the hidden layer (referred to as neurons) uses a sigmoid activation function. In this way, the specific output is produced by the mathematical functions raised from the sigmoid function. Then, a linear function is used by the output. In the hidden layer of signal processing nodes, the number of neurons can be equal to one or more, depending on the ANN architecture characteristics and its number of objects [62]. Hyper-parameters are known by their characteristics consisting of the number of hidden layers and neurons in the data frame. Each neuron of layer \(L\) is connected to each neuron of layer \(L+1\). Then, a value can be assigned to the neurons according to the weight matrix. Hyper-parameters are applied to create, visualize, and train two-layer feedforward networks to solve data classification problems. For the network to achieve the target value, a relative non-linearity can be introduced inside the function. In this way, the output function can be predicted and, after that, an error that occurred after training can be constantly pursued by the network. The propagation of the prediction goes back to the networks, using the Back Propagation (BP) learning algorithm. This happens when the prediction is wrong and the stochastic gradient descendant method is used to find the gradient error [34]. The first part of the analysis aims at the characterization of damage (classification), which is represented in Figure 12. The second set of analyses aim at the definition of damage severity (multi-class classification), which is represented in Figure 13.  
Figure 12: Example of a Neural Network Pattern Recognition for damage presence (classification). \(W\) is the weight and \(b\) is the bias, which are the learnable parameters of the ANN model.