# RUL Prediction of Rolling Bearings Based on a DCAE and CNN
## 3 Degradation Modeling of Rolling Bearings
#### Experimental Data  
In order to verify the effectiveness of the method, the PHM2012 bearing dataset was selected as the experimental data. The PHM2012 dataset was provided by FEMTO-ST Research Institute. The experiment was carried out on the PRONOSTIA test bench, which realized the accelerated degradation experiment of the bearing under different working conditions. The test bench is shown in Figure 6. The dataset was designed for three different working conditions according to the load force and speed, as shown in Table 1.  
Figure 5: The process of the rolling bearing degradation modeling method. Figure 6: Overview of PRONOSTIA. This article used 7 bearingss under working condition 1 to conduct experiments and re-divide the dataset. The signal sampling frequency was 25.6 kHz, and sampling was performed every 10 s. Each sampling contained 2560 sampling points, and the sampling time was 0.1 s. Figure 7 shows the original vibration data of each bearing in the horizontal direction under working condition 1. The degradation of some bearings was complicated. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Working Conditions** & **Load Force** & **Speed** \\ \hline
1 & 4000 N & 1800 r/min \\
2 & 4200 N & 1650 r/min \\
3 & 5000 N & 1500 r/min \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Three working conditions of the PHM2012 dataset. Figure 7: Original waveforms of seven bearings under working condition 1: (**a**) Bearing1_1; (**b**) Bearing1_2; (**c**) Bearing1_3; (**d**) Bearing1_4; (**e**) Bearing1_5; (**f**) Bearing1_6; and (**g**) Bearing1_7. The specific steps of modeling the bearing degradation state are explained with Bearing1_3 described as an example:  
Step 1: The sampling point of Bearing1_3 is 2375, and each sampling point corresponds to 2560 vibration values. Taking the 2560 vibration values as one sample, there are 2375 samples in total. After segmented processing, the original data are directly input into the DCAE model. Step 2: DCAE model training. The detailed structure diagram of the DCAE is shown in Figure 8, where C1-C12 represents the convolution layer, P1-P4 represents the pooling layer, and U1-U4 represents the upsampling layer. The coding part contains 6 convolutional layers, and the number of convolution kernels in each layer is 16, 32, 64, 128, 32 and 1. The decoding part contains 6 convolutional layers, and the number of convolution kernels in each layer is 32, 128, 64, 32, 16 and 1. The size of all convolution kernels is 3 1. The model learning rate is set to 0.001, the ReLU function is selected as the activation function after each convolutional layer, and the optimizer selects RMSProp. Step 3: SOM network training. The output after layer C6 is selected as the input of the SOM network, 80 features extracted from the DCAE are input into the SOM network as a sample, and the number of nodes in the input layer of the SOM network is consistent with the number of features. Each neuron in the topological layer is represented by a vector composed of the weights between the neurons in the input layer connected to it; that is, the dimension is consistent with the input data. The training steps of the SOM network are as follows:  
Step 1: Set the number of neurons at the topological layer as \(d\) and the maximum training times as \(N\). Usually, the number of neurons at the topological layer is \(d=5\sqrt{M}\), and \(M\) is the input sample number. Step 2: Select the 80-dimension sample \(x_{t}=\{x_{1t},x_{2t},\ldots,x_{80t}\}\) at the \(t\)th time point of the training data, where \(x_{t}\) is the 80-dimension feature extracted from the training set data at each time point. \(t=1,2,\ldots,p\), \(p\) is the number of time points, \(x_{it}\) is the \(j\)th neuron at the \(t\)th time point of the input layer and \(j=1,2,\ldots,80\). Input the samples of all \(p\) time points into the SOM network, and the number of nodes in the input layer of the network is the same as the number of features.