An intelligent bearing fault diagnosis method based on the AFEEMD and 1D CNNs

2.1. AFEEMD

The empirical mode decomposition (EMD) can decompose the signal $x\left(t\right)$ into$m$ IMFs: $IM{F}_1$, $IM{F}_2$,…, $IM{F}_n$ and a residue signal $r$. The IMFs include different frequency bands and imply a distinct time characteristic scale. It can be expressed as:

In the process of decomposing complex vibration signals, the ensemble EMD (EEMD) is used to eliminate the mode mixing problem. The EEMD adds the white noise in the data analysis, and it can clearly separate the natural scale of signals. The complete ensemble empirical mode decomposition (CEEMD) algorithm [7] adds positive and negative pairs of white noise to the observed signal, which can better eliminate noise residuals. Based on the CEEMD algorithm, the generated adaptive noise is added to the signal in pairs. Fast ensemble empirical mode decomposition (FEEMD) [8] optimizes stopping criteria and improve the EEMD efficiency. To better eliminate the noise residuals and improve the EEMD efficiency, the AFEEMD is finally proposed. It has been proved that this approach can work effectively [3].

2.2. 1D CNNs

The 1D CNNs model is a typical feedforward end-to-end neural networks, which essentially extracts the characteristics of input data by establishing multiple filters. A typical 1D CNNs model includes an input layer, several convolutional layers, several pooling layers, several fully-connected layers and an output layer.

(1) Convolutional layer.

In this paper, the collected vibration signals are time series, so 1D convolutional operation is used to construct the 1D CNNs, which is more suitable for feature extraction of vibration signals. The forward propagation (FP) from convolution layer $l-1$ to the layer $l$ can be described as follows:

where $W_i^{l-1}$ and $b_i^l$ represent the weights and biases of the $i$-th filter kernel in layer $l-1$, respectively; the notation $\mathrm{*}$ denotes the 1D Convolutional operation; $S^{l-1}\left(k\right)$ is the $k$-th local region in layer $l-1$; $y_i^l\left(k\right)$is the input of the $k$-th neuron in frame $i$ of layer $l+1$.

(2) Activation operation.

After the convolutional layers, the activation function is utilized to obtain nonlinear feature relationship of the input signal. Recently, rectified linear unit (ReLU) activation function can improve the representation ability and accelerate the convergence of 1D CNNs. ReLU is given by:

where $y_i^l\left(k\right)$ is the output value of convolution layers and $a_i^l\left(k\right)$ is the activation value of $y_i^l\left(k\right)$.

(3) Overlapping max-pooling layer.

The pooling operation is a self-sampling process, which achieves spatial invariance. It can reduce parameters and improve model generalization performance. A common pooling method is to take the maximum or average value of all neurons for a local feature. The max-pooling operation can be expressed as:

where $a_i^{l-1}\left(t\right)$ is the value of $t$-th neuron in the $i$-th frame of layer $l-1$; $m$ is the width of the pooling region, and $C_i^l\left(k\right)$ represents the result of the pooling operation.

Different from the traditional pooling operation, this paper uses the overlapping max-pooling, that is, the step size of the window movement is smaller than the size of the window itself, which can improve the prediction accuracy and alleviate overfitting to a certain extent.

4.1. Bearing test rig and data acquisition

In this paper, the bearing signals are investigated to demonstrate the effectiveness of the proposed method. The public bearing data collected from the CWRU are analyzed [9]. The test rig is shown in Fig. 2, which includes a 2-hp motor, a torque transducer, and a dynamometer.

Fig. 2Experimental setup used for bearing fault diagnosis

In this case study, the vibration signals collected from the driven end bearing are used. There are four types data under different working conditions, including the normal (N), inner fault (IF), ball fault (BF), and outer fault (OF). The sampling frequency is 48 kHz, and the fault samples are the 0.18 mm damage severity. To further prove the generalization performance of this method, the training dataset collected from working condition at the 1hp load and 1772 rpm motor speed, and the testing dataset is 2 hp load and 1750 rpm. The former all has the 1840 samples, the latter is 360 samples. The detail description of the dataset is shown in Table 1.

Each raw sample has 1024 sampling points. Then it is decomposed by AFEEMD, and the IMFs component is obtained. The all 18 statistical features in the time domain and frequency domain of the first six IMFs are respectively calculated to form feature input sample which comprehensively expresses most of the fault information. Hence, the dimension of each preprocessing sample is 108, which is used as the input of the 1D CNNs. The hyper parameters of the 1D CNNs are listed in Table 2. The epoch number is 100, and the dropout is 0.5. To verify the effectiveness of the proposed method, the average accuracy, standard deviation, and computing time are used as the evaluation indexes.

4.2. Results

In order to evaluate the stability of proposed method, six trials are carried out. For comparison, the BPNN is utilized to analyze the same dataset. The structure of the BPNN is 108-40-10-4, and the epoch number is 500. Table 3 lists the results of the testing accuracy, standard deviation, and time. It can be observed from Tables 3 that the average testing accuracy is 98.24 % by using the proposed method, which is higher than 81.62 % by using the BPNN. It proves that the proposed method can achieve high accuracy and possess good generalization performance. The standard deviation of the proposed method is 0.33 %, and it is smaller than the 1.00 % of BPNN, which shows a more stable performance. Due the feature vector as the input of 1D CNNs, it reduces the training time and complexity of the network. The computing time of the proposed method is 22.5 s, which is the same as the traditional BPNN of 21.67 s.

The Fig. 3 shows the confusion matrix of the two methods. It contains the information for classification and misclassification samples. The horizontal axis represents actual condition of classification, and the ordinate axis represents predicted condition for samples. From Fig. 3, we can find that condition 3 has the lowest accuracy, namely, there are three misclassified samples with ball fault.

Fig. 3Condition classification confusion matrix of the proposed methods for the first trial

Table 4 displays fault diagnosis result of each category for the first trial using the proposed method and BPNN. Only the classification accuracy of bearing outer fault through the proposed approach is the same as the BPNN. The proposed method performs better in other conditions and obtains the highest overall testing accuracy of 98.61 %. It achieves the end-to-end feature learning based on the AFEEMD.