Vibration-based gearbox fault diagnosis using deep neural networks

2.1. Restricted Boltzmann machine

The restricted Boltzmann machine is a generative stochastic artificial neural network with two layers as shown in Fig. 1, which can learn a probability distribution over its set of inputs. The standard RBM has binary-valued hidden and visible units, and consists of a matrix of weights associated with the connection between hidden unit and visible unit. Given these, an energy function of the configuration ($v$, $h$) is defined as following [18]:

where $\mathbf{v}$ and $\mathbf{h}$ denote the visible and the hidden neurons, $\mathbf{b}$ and $\mathbf{c}$ stand for their offsets, and $\theta =\{\mathbf{W},\mathbf{b},\mathbf{c}\}$ is network parameters. To accommodate the real-valued input data, Salakhutdinov et al. [28] proposed Gaussian-Bernoulli RBM (GRBM), where the binary visible neurons can be replaced by the Gaussian ones. The energy function is redefined as the following:

where ${\sigma }_i$ is the standard deviation associated with Gaussian visible neuron $v_i$. The statistical parameters for the fault diagnosis are real-valued, so Eq. (2) is selected as the energy function in this paper.

Fig. 1A restricted Boltzmann machine

The probability that the network assigns to every possible pair of a visible and a hidden vector is given via this energy function as the following:

where $Z$ is called as “partition function” and defined as the sum of $e^{-E(v,h|\theta )}$ over all possible configurations.

The network assigns probability to a visible vector, $v$, is given by summing over all possible hidden vectors:

By adjusting $\theta =\{\mathbf{W},\mathbf{b},\mathbf{c}\}$ to lower the energy of a training sample and to raise the energy of other samples, the probability that the network assigns to the training sample can be raised, especially those which have low energies and then make a big contribution to the partition function.

A standard approach to estimate the parameters of a statistical model is maximum-likelihood estimation, which maximizes the likelihood by using the training data to train the parameters $\theta =\{\mathbf{W},\mathbf{b},\mathbf{c}\}$. The likelihood is defined as:

where $S$ represents the set of samples and $n_s$ is the size of $S$. Maximizing the likelihood is the same as maximizing the log-likelihood given by:

Gradient descent method is usually employed to find the maximum likelihood parameters analytically. The derivative of the log probability of a training data $\mathbf{v}$ with respect to $\theta$ is given by:

Because there are no direct connections between the hidden units in an RBM, it is very easy to calculate the first item of Eq. (7). Given a randomly selected training data (real-valued), $\mathbf{v}$, the binary state of each hidden unit, $h_j$, is set to 1 with probability:

where $s\left(\right)$ is a sigmoid function. Similarly, given a hidden vector $\mathbf{h}$, $v_i$ is set to 1 with probability:

where $N(\cdot )$ expresses normal distribution function.

However, it is much more difficult to get the second item. It can be done by starting at any random state of the visible units and performing alternating Gibbs sampling for a very long time. An iteration of alternating Gibbs sampling consists of updating all of the hidden units in parallel using Eq. (8) followed by updating all of the visible units in parallel using Eq. (9).

The algorithm performs Gibbs sampling and is used inside a gradient descent procedure to compute weight, which is updated as the following [29]:

(1) Take a training sample $\mathbf{v}$, compute the probabilities of the hidden units and sample a hidden activation vector $\mathbf{h}$ from this probability distribution.

(2) Compute the outer product of $\mathbf{v}$ and $\mathbf{h}$ and call this the positive gradient.

(3) From $\mathbf{h}$, sample a reconstruction ${\mathbf{v}}^{\text{'}}$ of the visible units, then resample the hidden activations ${\mathbf{h}}^{\text{'}}$ from this. (Gibbs sampling step).

(4) Compute the outer product of ${\mathbf{v}}^{\text{'}}$ and ${\mathbf{h}}^{\text{'}}$ and call this the negative gradient.

(5) Update the weight: $W_{i,j}=W_{i,j}+\mathrm{\Delta }{W}_{i,j}$. $\mathrm{\Delta }{W}_{i,j}=\in (\mathbf{v}{\mathbf{h}}^T-\mathbf{v}\text{'}\mathbf{h}{\text{'}}^T)$ is expressed as: the positive gradient minus the negative gradient, the result of which times some learning rate.

The update rule for the biases $\mathbf{b}$ and $\mathbf{c}$ is defined analogously.

2.2. Deep Boltzmann machine

A deep Boltzmann machine (DBM) [28] is undirected graphical models with bipartite connections between adjacent layers of hidden units, which is a network of symmetrically coupled stochastic units. Similar to RBMs, this binary-binary DBM can be easily extended to modeling dense real-valued count data. For real-valued cases, Cho et al. [30] proposed a Gaussian-Bernoulli deep Boltzmann machine (GDBM) which used the Gaussian neurons in the visible layer of the DBM. Fig. 2(b) presents a three-hidden-layer DBM, whose energy is defined as Eq. (10), where $L$ is the number of hidden layers:

Salakhutdinov et al. [28] introduced a greedy and layer-by-layer pretraining algorithm by learning a stack of modified RBMs for DBM model, where contrastive divergence learning [25] works well and the modified RBM is good at reconstructing its training data. In this modified RBM with tied parameters, the conditional distributions over the hidden and visible states are defined as Eq. (11) and Eq. (12):

where $s\left(\right)$ is a sigmoid function. When a stack of more than two RBMs is greedily being trained, the modification only needs to be used for the first and the last RBM in the stack. For all the intermediate RBMs, simply halve their weights in both directions when composing them to form a deep Boltzmann machine. It should be noted that there are two special cases: the last and the first hidden layers for the above equation. For the last hidden layer (i.e., $l=L$), we set $N_{L+1}=$0. As for the first hidden layer (i.e., $l=$1), parameters for Eq. (12) should be set as:

Fig. 2DBN&DBM

a) Deep belief network

b) Deep Boltzman machine

2.3. Deep belief networks

Deep belief networks (DBNs) [19] can be viewed as another greedy, layer-by-layer unsupervised learning algorithm that consists of learning a stack of RBMs one layer at a time. The top two layers form a restricted

Boltzmann machine which is an undirected graphical model, but the lower layers form a directed generative model (see Fig. 2(a)). The training algorithm for DBNs proceeds as follows. Let $X$ be a matrix of inputs, and regarded as a set of feature vectors.

(1) Train a restricted Boltzmann machine on $X$ to obtain its weight matrix, $W$, and use this as the weight matrix between the lower two layers of the network.

(2) Transform $X$ by the RBM to produce new data $X\text{'}$.

(3) Repeat this procedure with $X\leftarrow X\text{'}$ for the next pair of layers, until the top two layers of the network are reached.

(4) Fine-tune all the parameters of this deep architecture with respect to the supervised criterion.

2.4. Stacked Auto-encoders

The Auto-encoder is trained to encode the input $X$ into some representation $C\left(X\right)$ so that the input can be reconstructed from that representation [29]. Hence the target output of the auto-encoder is the auto-encoder input itself. If there is one linear hidden layer and the mean squared error criterion is used to train the network, then the $k$ hidden units learn to project the input in the span of the first $k$ principal components of the data. Auto-encoders have been used as building blocks to build and initialize a deep multi-layer neural network [15, 30, 31]. The training procedure is similar to the one for Deep Belief Networks. The principle is exactly the same as the one previously proposed for training DBNs, but auto-encoders instead of RBMs are used as the following [20]:

(1) Train the first layer as an auto-encoder to minimize some forms of reconstruction errors of the raw input.

(2) The outputs of hidden units on the auto-encoder are used as input for another layer, which is also trained to be an auto-encoder.

(3) Iterate as step (2) to initialize the desired number of additional layers.

(4) Take the last hidden layer output as input to a supervised layer and initialize its parameters (either randomly or by supervised training, keeping the rest of the network fixed).

(5) Fine-tune all the parameters of this deep architecture with respect to the supervised criterion. Alternately, unfold all the auto-encoders into a very deep auto-encoder and fine-tune the global reconstruction error, as in [32].

3.1. Frequency-domain feature extraction

For a vibration signal of the gearbox, $x\left(t\right)$, its spectral representation $X\left(f\right)$ can be calculated by Eq. (14):

where the “^” stands for the Fourier transform, $t$ is the time and $f$ is the frequency.

The time domain signal was multiplied by a Hanning window to obtain the FFT spectrum. The spectrum can be divided into multiple bands, and the root mean square value (RMS) for each band keeps track of the energy in the spectrum peaks. RMS value is evaluated with Eq. (15), where $M$ is the number of samples of each frequency band:

Fig. 3 and Fig. 4 present the FFT spectrum and its RMS representation of a vibration signal, respectively. It is obvious that the root mean square (RMS) values keep track of the energy in the spectrum peaks. To reduce the number of input data, the spectrum was split in multiple bands and the RMS value of each band is used as feature representation in the spectrum domain.

Fig. 3Original frequency representation

Fig. 4Frequency representation using RMS values

3.2. Time-domain feature extraction

The time-domain signal collected from a gearbox usually changes when damage occurs in a gear or bearing. Both its amplitude and distribution may be different from those of the time-domain signal of a normal gear or bearing. Root mean square value reflects the vibration amplitude and energy in time domain. Standard deviation, skewness and kurtosis may be used to represent time series distribution of the signal in time domain.

Fig. 5Gearbox fault diagnosis based on deep neural networks

Four time-domain features, namely, standard deviation, mean value, skewness and kurtosis are calculated. They are defined as follows.

(1) Mean value:

(2) Standard deviation:

(3) Skewness:

(4) Kurtosis:

To sum up, the vector of the features of the preprocessed signal is formed as follows: $N_{RMS}$ RMS values, standard deviation, skewness, kurtosis, rotation frequency and applied load measurements, which are used as input parameters for the deep neural networks. In this paper, $N_{RMS}$ is set to 251.

5.1. Data set I

The data set I of vibration signal includes different basic fault patterns as defined in Table 1 for the gearbox diagnosis experiments. 11 patterns with 3 different load conditions (300, 600, and 900 rpm) and 3 different input speeds (zero, small, and great) were applied during the experiments. For each pattern, load and speed condition, we repeated the tests for 5 times. Each time, the vibration signals were collected with 24 durations, each duration covered 0.4096 sec. The sampling frequency for the vibration signals was set for 50 kHz and 10 kHz, respectively.

Fig. 6Pseudo-code of DNN-based classifier

Fig. 7a) Fault simulator setup; b) Internal configuration of the gearbox

a)

b)

Data set I was obtained from the measurements of a vertically allocated accelerometer in the gearbox fault diagnosis experimental platform shown in Fig. 7. Fig. 7(a) shows the fault simulator setup of the gearbox. A motor (SIEMENS, 3, 2.0 HP) through a coupling is used, whose speed is controlled by a frequency inverter (DANFOSS VLT 1.5 kW). An electromagnetic torque load is used, which is controlled by a torque controller (TDK-Lambda, GEN 100-15-IS510). The vibration signals of the gearbox were collected by an accelerometer (PCB ICP 353C03). Fig. 7(b) shows the internal configuration of the gearbox, which is a two-stage transmission of the gearbox. The parameters of all components on the gearbox are listed here: Input helical gear: $Z_1=$ 30, modulus = 2.25, impact angle = 20°, and helical angle = 20°; Two intermediate helical gears: $Z_2=Z_3=$45; and the output gear: $Z_4=$ 80. The faulty components used in the experiments include gears $Z_1$, $Z_2$, $Z_3$, and $Z_4$, bearing 1 and house 1 are labeled in Fig. 7(b). Based on the above experimental platform of gearbox fault diagnosis, 11880 vibration signals (i.e., [$x_1^{\left(I\right)}\left(t\right)$, …, $x_{11880}^{\left(I\right)}\left(t\right)$]) corresponding to 11 condition patterns (i.e., [$B_1$, $B_2$, …, $B_{11}$]) have been recorded.

Table 1Condition patterns of the gearbox configuration

Faulty pattern	$B_1$	$B_2$	$B_3$	$B_4$	$B_5$	–
Faulty component	Gear $Z_1$	Gear $Z_2$	Gear $Z_3$	Gear $Z_3$	Gear $Z_4$	–
Faulty detail	Worn tooth	Chaffing tooth	Pitting tooth	Worn tooth	Chipped tooth	–
Faulty photo						–
Faulty pattern	B6	B7	B8	B9	B10	$B_{11}$
Faulty component	Gear Z4	Bearing 1	Bearing 1	Bearing 1	House 1	N/A
Faulty detail	Root crack tooth	Inner race fault	Outer race fault	Ball fault	Eccentric	N/A
Faulty photo						N/A

5.2. Data set II

In data set I, each vibration signal only includes information of one fault component, which has only a kind of fault. However, there are usually two or more fault components in the real-world rotating mechanical system. In order to evaluate whether the proposed approach is applicable in fault diagnosis of industrial reciprocating machinery, data set II is constructed, where each fault pattern includes two or more basic faults. Firstly, some basic faults are defined in Table 2 and Table 3, which include 11 kinds of basic gear faults and 8 kinds of bear faults, respectively. 12 combined fault patterns are defined in Table 4.

Data set II was obtained from the measurements of a vertically accelerometer on another gearbox fault diagnosis experimental platform. Fig. 8 indicates the internal configuration of the gearbox and positions for accelerometers, which is a two-stage transmission of the gearbox with 3 shafts and 4 gears. The parameters of all components on this gearbox are as follows: Input gear: $Z_1=$ 27, modulus = 2, and $\mathrm{\Phi }$ of pressure = 20; Two intermediate gears: $Z_2=Z_3=$ 53; and the output gear: Z₄= 80. The faulty components used in the experiments include gears $Z_1$, $Z_2$, $Z_3$, and $Z_4$, bearing $B_1$, $B_2$, $B_3$, and $B_4$ as labeled in Fig. 8(a). The conditions of the test are described in the Table 5, where 4 different load conditions and 5 different input speeds were applied for each fault pattern during the experiments. For each pattern with different load and speed condition, we repeated tests for 5 times. Each time, the vibration signals were collected with 10 durations, each duration covered 0.4096 sec.

Based on the above experimental platform for gearbox fault diagnosis, data set II has 12000 vibration signals (i.e., [$x_1^{\left(II\right)}\left(t\right)$, …, $x_{12000}^{\left(II\right)}\left(t\right)$]) corresponding to 12 combined condition patterns (i.e., [$C_1$, $C_2$, …, $C_{12}$]) to be recorded.

Fig. 8a) Internal configuration of the gearbox; b) Positions for accelerometers

a)

b)

5.3. Data set III

One or two test cases cannot fully reflect the reliability and robustness of an algorithm. Although some classifiers are effective for some special data sets, they get easily stuck in “apparent local minima or plateaus” in some other cases, resulting in a disability to classify fault patterns effectively. To further validate the reliability and robustness of the DNN, a fault condition pattern library has been constructed, which has 55 kinds of condition patterns based on the fundamental patterns described in Table 2 and Table 3. Each condition pattern holds more than one basic gearbox fault.

To challenge the proposed approaches, we have generated a large number of data sets. Each data set includes $N$ kinds of condition patterns. Here three kinds of $N$’s value are considered for these data sets: 12, 20 and 30, respectively. It is obvious that bigger value of $N$ means the classification and identification of faults are more difficult. For each size of $N$, 20 different data sets were generated, where each one involves unique combination of condition patterns that are randomly selected from the above mentioned pattern library.

Here each data set is collected from the measurements of a vertically accelerometer on the gearbox fault diagnosis experimental platform shown in Fig. 8, whose test conditions and generating method are the same as that of data set II. Each data set has 12000 vibration signals (i.e., [$x_1^{\left(i\right)}\left(t\right)$,…, $x_{12000}^{\left(i\right)}\left(t\right)$]) corresponding to each combination of condition patterns (i.e., [$C{P}_1$, $C{P}_2$, …, $C{P}_N$]). Here $i$ expresses $i$th data set, and [$C{P}_1$, $C{P}_2$,…, $C{P}_N$] is a combination randomly selected from the pattern library. 60 different data sets are generated in total to further evaluate the performance of the proposed approaches.

6.1. Parameters tuning

As mentioned above, the training of DNN includes two stages: pre-training and fine-tuning. At the stage of fine-tuning, the DNN is usually treated as a feed-forward neural network (FFNN) by using supervised training. FFNN is also typically used in supervised learning to make a prediction or classification. To evaluate the performance of DNN, a comparison study between FFNN with DNN is presented for gearbox fault diagnosis. The net parameters are set as Table 6-7. Based on different training parameters, five typical FFNNs are defined in Table 8. Four parameters (nn.n, nn.unit, nn.epoch1 and nn.epoch2) are fine-tuned based on data set I and data set II as follows.

6.1.3. Epochs of training

The epochs of training also influence the performance of FFNN-based and DNN-based classifier. If the epoch of training is too long, it will be possible to lead to “overfitting”; or even worse, it will possibly result in a lack of training. nn.epoch₁ and nn.epoch₂ represent the epochs of training in the pre-training and fine-tuning stage of DNN, respectively. Table 11 presents the experiment results of varying pre-training epochs (nn.epoch₁ = 1 to 10), where nn.epoch₂ = 100. As shown in Table 11, when nn.epoch₁ is equal to 1, good classification accuracy can be obtained. If the pre-training epochs get longer, better results cannot be obtained.

Fig. 9 and Fig. 10 present the convergence process of error rate for data set I and data set II, respectively. For data set I, after only 20 epochs of fine-tuning, the error becomes very small. Fig. 9 shows that the error rate is lower than 0.1 after 50 epochs of fine-tuning for data set II. Compared with the FFNN starting from random initialization (FFNN_Scheme1~5), Fig. 9 and Fig. 10 also show DNN (DBN, DBM,RBM and SAE) obviously reduce “overfitting” phenomenon for gearbox fault diagnosis. nn.epoch₁ and nn.epoch₂ are set to 1 and 100 for the following experiment evaluations, respectively.

Fig. 9The error rate on data set I for different classifiers

Fig. 10The error rate on data set II for different classifiers

6.2. Performance evaluations

Table 12 presents the classification accuracy by using 8 different classifiers for data set I and II. Compared with FFNN, DBN, DBM, RBM and SAE have better classification performance, especially for data set II.

One or two test cases cannot reflect the reliability and robustness of an algorithm. To further evaluate the performance of DNN, we constructed data set III. Firstly, we consider these data sets (#1-#20), where each one has 12 kinds of different condition patterns (CP = 12, where CP expresses the number of condition patterns included in a data set). Table 13 indicates experiment results by using 8 different classifiers for them. As shown in Table 13, the least classification accuracy among the four DNNs is 92.8 % of SAE for the 15th data set; each of the mean classification accuracy is larger than 98.0 %.

To further challenge the proposed classifiers, we add fault condition pattern included in a data set. Table 14 and 15 present the experiment results of 20 data sets, respectively. Each data set has 20 and 30 kinds of condition patterns respectively (CP = 20 or 30). More condition patterns mean that it is more difficult to obtain good results. As shown in Table 14 and 15, DBN, DBM, RBM and SAE still have good performance for these cases.

Among test cases of 21st-40th data set, DBN, DBM, RBM and SAE have larger than 90 % of mean classification accuracy; the least one is 77.6 % of DBN for the 35th data set. Among test cases of 41st-60th data set, DBN, DBM, RBM and SAE have larger than 84% of mean classification accuracy; the least one is 54.6 % of DBN for the 48th data set.

6.3. Comparison and analysis

To verify Y. Bengio’s opinion [17, 18] that the gradient-based training of supervised multi-layer neural networks (starting from random initialization) gets easily stuck in “apparent local minima or plateaus”, three multi-layer neural networks (FFNN_Scheme1, FFNN_Scheme2, FFNN_Scheme4) are used to classify the same data set for gearbox fault diagnosis. Their classification results are also indicated in Table 13, 14 and 15, respectively. In addition, SVM is employed to compare with the proposed approaches. The algorithm SVM is applied by using the LibSVM [33]. The parameters for SVM are chosen as $C=$1 and core (kernel) given by a radial basis function where $\gamma =$0.5. These parameters were found through a cross search, aiming at the best model for the SVM.

As shown in Table 13, among 20 test cases CP = 12, three FFNN-based classifiers (FFNN_Scheme1, FFNN_Scheme2, and FFNN_Scheme4) have 5 test cases with bad classification accuracy, even smaller than 70 % for them (#1, #4，#7, #13 and #20), although it is effective for other 14 test cases whose classification accuracies are larger than 90 %. Table 14 indicates that FFNN_Scheme1, FFNN_Scheme2, and FFNN_Scheme4have 4 test cases with bad classification accuracy (#29, #33, #34 and #35). Table 15 indicates that FFNN_Scheme1, FFNN_Scheme2, and FFNN_Scheme4 have 5 test cases (#48, #53, #54, #55 and #57) is bad. This also verifies the negative observations that gradient-based training of multi-layer neural networks (starting from random initialization) gets easily stuck in “apparent local minima or plateaus” in some cases. They don’t have good robustness for gearbox faults diagnosis. Corresponding to the four DNN-based classifiers, they are able to obtain good classification accuracy for 62 data sets. So, we can draw the following conclusions that the DNN-based classifiers are able to avoid falling into “apparent local minima or plateaus” and are reliable and robust for gearbox fault diagnosis. Compared with FFNN-based classifiers and SVM, DBN, DBM,RBM and SAEhave overwhelming superiority in the items of reliability and robustness for gearbox fault diagnosis.

As for the comparison between SAE, RBM, DBM and DBN, Fig.11 indicates the mean classification accuracy data set with different kinds of condition patterns (CP = 12, 20 and 30) for SAE, RBM, DBM and DBN, respectively. As shown in Fig. 11, four deep neural networks have almost equal classification accuracy for the data set with CP = 12; in the case of CP = 20 and 30, RBM and DBM are slightly better than SAE and DBN. However, the classification accuracy of SAE, RBM, DBM and DBN need to be further enhanced for the data set that includes condition patterns more than 20 kinds.

Fig. 11Comparison between SAE, RBM, DBM and DBN