## 2 Materials and Methods
They consist of a connection of different layers where each layer has a defined number of neurons. A neuron is similar to a computing block defined by an activation function, a set of weights and biases, an input, and an output. For more complex problems, a number of hidden layers can be inserted. Data are propagated through the ANN and the output of each layer represents the input of the next layer. The input to an ANN usually comprises the features and the targets. The feature data are used to predict the target data. In the case of mass operators, the features are the accelerations while the targets are the stresses. Such an ANN architecture can be described as a feedforward neural network [13]. If \(p\) is considered to be the input to a neuron and \(b\) the neuron's bias, then the output of that neuron is \(a=f(w\cdot p+b)\), where \(f\) represents the neuron activation function and \(w\) is a weighting factor. While \(f\) is chosen with regard to the problem to be solved, \(w\) and \(b\) are both parameters that will be calculated based on a learning rule during the training [13]. During training,the network's neurons are first initialized, i.e., a random set of weights and biases is attributed to each neuron and an activation function needs to be assigned to each neuron in the layer. Then, the training data are forward-propagated through the network; each neuron applies its random weights and biases and its activation function to the input and produces an output, which is further propagated until the data reach the output layer. Afterward, an error function \(E\) is evaluated, usually the mean squared error (MSE) between the calculated outputs \(Y_{i}\) and the target values \(T_{i}\):  
\[E=\ MSE=\frac{1}{n}\sum\
olimits_{i=1}^{n}(T_{i}-Y_{i})^{2}\ \text{with}\ n=\text{number of data points}. \tag{2}\]  
Finally, the error is back-propagated through the network in order to identify the neurons that are responsible for the error. The latter are then adapted to minimize the error, specifically their weights and biases are altered, while the connections of the neurons producing a low error are reinforced in this process [14]. The recurrent ANN is capable of exhibiting a dynamic behavior where the output of one layer can also be used as the input for a preceding layer. This makes it then possible for the neural network to create a temporary memory and process sequences of inputs [15]. This is particularly relevant as vibration tests are performed using frequency sweep where, for example, the frequency increases with time. In this study, four different neural network models are compared to each other:  
* A frequency-dependent ANN (see Figure 3a): a feedforward ANN with the frequency values as additional feature data as in [3]. Thus, it is ensured that the data are associated with the corresponding frequency. * A pretrained ANN: a feedforward ANN trained in two steps, in order to give special attention to the natural frequencies, which represent the most critical frequencies during the vibration test with respect to accelerations and stresses. * A nonlinear autoregressive exogenous (NARX) model (see Figure 3b): a recurrent ANN to depict the sequence nature of the input data, taking into account the last time step before making a prediction about the next one. For a nonlinear autoregressive exogenous (NARX) model, besides  
Figure 2: Example of the artificial neural network (ANN) structure with 3 neurons in the input layer, 5 hidden neurons in 1 hidden layer, and 1 output layer and the connectivity between neurons. the external feature sequence \(u_{t}\), \(u_{t-1}\), \(u_{t-2}\), \(u_{t-3}\), \(\ldots\), the targets \(y_{t}\) of the network are also used as features, while a delayed version of them \(y_{t-1}\), \(y_{t-2}\), \(y_{t-3}\), \(\ldots\) is fed back into a feedforward network, according to [16] by \(y_{t}\ =\ f(y_{t-1}\), \(y_{t-2}\), \(y_{t-3}\), \(\ldots\), \(u_{t}\), \(u_{t-1}\), \(u_{t-2}\), \(u_{t-3}\), \(\ldots\)).