Abstract
Vibration signal of gearbox systems carries the important dynamic information for fault diagnosis. However, vibration signals always show non stationary behavior and overwhelmed by a large amount of noise make this task challenging in many cases. Thus, a new fault diagnosis method combining the Hilbert empirical wavelet transform (HEWT), the singular value decomposition (SVD) and Elman neural network is proposed in this paper. Vibration signals of normal gear, gear with tooth root crack, gear with chipped tooth in width, gear with chipped tooth in length, gear with missing tooth and gear with general surface wear are collected in different speed and load conditions. HEWT, a new selfadaptive timefrequency analysis, was applied to the vibration signals to obtain the instantaneous amplitude matrices. Singular value vectors, as the fault feature vectors were then acquired by applying the SVD. Last, the Elman neural network was used for automatic gearbox fault identification and classification. Through experimental results, it was concluded that the proposed method can accurately extract and classify the gear fault features under variable conditions. Moreover, the performance of the proposed HEWTSVD method has an advantage over that of HilbertHuang transform (HHT)SVD, local mean decomposition (LMD)SVD or wavelet packet transform (WPT)PCA for feature extraction.
1. Introduction
Gears, an important and most frequently encountered components in rotating machinery, whose operation condition directly affects the whole performance of the entire system. An unexpected fault of gear may cause huge economic losses, even personal injury if not detected in time [1]. According to statistics, 80 % of transmission machinery failure was caused by the gear, and gear failure was about 10 % of rotating machinery failure. Thus, a robust monitoring system is needed to detect earlier any malfunctions.
The vibration signals acquired during the operation of a gear carry the important dynamic information of the machine: the fault features of a gear can, therefore, be obtained from the analysis of such vibration signals. [2, 3]. Generally, gearbox diagnosis includes two main steps: feature extraction and fault classification. Feature extraction is the most challenging task. If the feature extraction is incorrect or incomplete, it will inevitably lead to erroneous classification and false positives. Furthermore, the gear fault signatures are known to be weak and often covered by the nature frequency of the machine and overwhelmed by noise with obvious nonlinear and nonstationary behavior [4]. Thus, how to efficiently obtain the vital features that robustly indicate the presence of faults from the complex dynamic mechanical signals is the key to solve the problem.
The main feature extraction methods including: Timedomain, frequencydomain and timefrequency domain methods. Timedomain methods are focused on extracting statistical features of the time waveform to determine transient phenomena originate from faulty gearbox [5, 6]. Time synchronous average (TVA) has been widely used to extract the time waveform from the vibration signal synchronous with the shaft rate of rotation. Removing the harmonic families of the gearmesh components, the residual signal waveform which contains the fault feature signatures could be used for fault diagnosis. Frequencydomain methods include Fourier spectra, cepstrum analysis, and envelope spectra technique [7, 8].
Conventional methods show their limits when applied to vibration signals that are nonstationary and that have low energy of weak signals generated by faults. To deal with this problem, timefrequency (TF) analysis techniques have been developed. Among them, there is the short time Fourier transform (STFT), this technique has been widely used in rotating machinery fault diagnosis [9]. To overcome the limitation of STFT related to the Heisenberg uncertainty principle, the WignerVille distribution has been widely used. Choy et al. [10] have shown the capacity of WVD in analyzing the effects of surface pitting and wear on the vibrations of a gear transmission system. Thereafter, several studies have exploit the technique like the work done by [11].
In the last few years, the wavelet transform (WT) has drawn a lot attention and it could overcome the classical TF tools [12]. The WT is a multiresolution analysis has the advantage for characterizing signals at different localization levels in both time and frequency domains. The WPT is an advanced version of the WT which provides a complete level by level TF decomposition of signal. It may not only supply richer information but also supply more promise frequency localization information. However, the WPT is not selfadaptive, it has prescribed dyadic subdivision in time and frequency which may leads to severe damage in identifying transient vibration features that lie in transient areas of dyadic packets [13, 14].
Huang et al. [15], have proposed a selfadaptive TF analysis called HHT, it can overcome the problem of the prescribed dyadic subdivision in WT and WPT. In [16], Peng Z. K. et al. compared the HHT and WT, and then verified that the HHT has better resolution and computational efficiency in timefrequency domain. EMD and Hilbert spectrum were used to diagnose rotating machines faults [17, 18]. Several adaptive methods based on EMD have been developed such as the LMD [19], intrinsic timescale decomposition (ITD) [20]. These methods however, all have the same drawbacks, like the mode mixing, distorted components and the end effects phenomena. Recently, Daubechies et al. [21] proposed a wavelet based timefrequency reallocation method called Synchrosqueezed wavelet transform. This method was successfully applied in gear fault diagnosis [22].
In recent years, Gilles, [23] developed the empirical wavelet transform (EWT) [24, 25]. The uniqueness of this method is in building an adaptive wavelet filter bank capable of extracting amplitude modulatedfrequency modulated (AMFM) components of a signal. It is demonstrated to be superior to EMD [23]. Merainani et al. [26], proposed the Hilbert empirical wavelet transform (HEWT) and did a comparison with HHT. The technique has shown a good results in gear tooth crack, tooth pitting and rolling bearing diagnosis [2628]. This combination leads to selforganizing TF plane which is very beneficial for fault feature extraction. Consequently, the HEWT is used in this research. Nevertheless, the instantaneous amplitude signals of HEWT are always complex and too large, thus, in order to enhance the robustness of the classification both principal component analysis (PCA) and SVD can be employed for dimensionality reduction. Moreover, the singular values have great stability, so that they change little when the matrix elements change. Hence, the HEWT together with SVD is proposed for gear fault feature extraction.
Using solely advanced signal processing techniques for fault diagnosis require a great deal of expertise to apply them successfully. Techniques are required that can automatically make decisions on the health of running machines. After extracting the feature vectors, intelligent classification techniques can be used to identify the fault modes. These include the Elman neural network [29]. In this study, Elman neural network is employed for automatic gearbox identification and classification.
This paper is organized as follows: Section 2 introduces the HEWTSVD. Section 3, describes the experimental setup and experimental results. Conclusion is offered in Section 4.
2. Brief theory review
2.1. Timefrequency signal decomposition based on HEWT
The HEWT is a merger of the empirical wavelet transform and Hilbert transform [26]. The EWT is used to extract adaptive modes from the vibration signal. Then, the instantaneous amplitude and frequency are performed for each mode using HT.
2.1.1. Empirical wavelet transform
The idea of EWT is to construct a set of $N$ wavelet filters (one Low pass and ($N1$) band pass filters) capable of extracting Amplitude ModulatedFrequency Modulated components that is AMFM components (modes) ${f}_{k}\left(t\right)$ of an input signal $f\left(t\right)$ by adaption from the processed signal [23] such as:
For the adaptation process, the wavelet filter bank is based on the Fourier supports detected from the information contained in the processed signal spectrum by finding the local maxima then taking support boundaries ${\omega}_{i}$ as the middle between successive maxima [23].
As it shown in Fig. 1, the Fourier support [0, $\pi $] is partitioned into $N$ contiguous segments. Each segment is denoted as ${\mathrm{\Lambda}}_{n}=\left[{\omega}_{n1},{\omega}_{n}\right]$, thus ${\bigcup}_{n=1}^{N}{\mathrm{\Lambda}}_{n}=\left[0,\mathrm{}\mathrm{}\pi \right]$. Centered around each ${\omega}_{n}$, there is transient phase ${T}_{n}$ with width $2{\tau}_{n}$.
The empirical wavelets are defined as band pass filters on each ${\mathrm{\Lambda}}_{n}$ and based on ${\mathrm{\Lambda}}_{n}$ a wavelet tight frame can be defined. Hence, inspired by the Meyer’s and LittlewoodPaley wavelets, a wavelet tight frame $B=\left\{{\left\{{\varphi}_{n}\left(t\right)\right\}}_{n=1}^{N1},\mathrm{}\mathrm{}{\left\{{\psi}_{n}\left(t\right)\right\}}_{n=1}^{N1}\right\}$ is defined. And for arbitrary $n>$ 0, their Fourier transforms, i.e. the empirical scaling function ${\widehat{\varphi}}_{n}\left(\omega \right)$ and the empirical wavelets ${\widehat{\psi}}_{n}\left(\omega \right)$ are given by the expressions of (Eqs. (2) and 3), respectively:
The function $\beta \left(x\right)$ is given as follows:
Note that the most used function that satisfies this property is:
Fig. 1Fourier axis segmentation and EWT wavelets construction
Once the tight frame set of empirical wavelets is built, the definition of EWT, ${\mathcal{W}}_{f}^{\epsilon}\left(n,t\right)$ can be defined in the same way as the classical wavelet transform. The detail coefficients are obtained by the inner products of the input signal with the empirical wavelets [23]:
The approximation coefficients are obtained by the inner product with the scaling function:
So, the signal is decomposed into various empirical modes ${f}_{k}$, which is given by:
The EWT is invertible and the signal can be reconstructed as follows:
As the initial goal of the EWT is to get a decomposition as depicted in Eq. (1). Comparing the Eq. (1) with (11), we can deduce that each mode ${f}_{k}$ in Eq. (1) corresponds to Eqs. (9) and (10).
2.1.2. Hilbert transform
Hilbert Transform can be seen like a convolution of signal with $1/\pi t$.
For every extracted mode $f\left(t\right)$, the analytical signal $z\left(t\right)$ associated with $f\left(t\right)$ is given by:
where: $\stackrel{~}{f}\left(t\right)$ is the Hilbert Transform of $f\left(t\right)$, defined as:
Thus, the instantaneous amplitude and frequency are Eqs. (14) and (15) respectively:
The instantaneous amplitude of each mode is considered as its envelope signal. It’s a time varying signal that gives beneficial information on investigating intrinsic characteristics of the signal. For this raison, the envelope signal $A\left(t\right)$ is computed to support our analysis. However, the envelope signals still too large and complex to be taken as the fault features. Thus, we further propose the use of SVD for dimension reduction to improve the robustness of the fault features.
2.2. Singular value decomposition on the HEWT
The singular value decomposition is a matrix transformation algorithm that decomposes any given matrix $M$ ($m\times n$) into three matrices $U$, $\mathrm{\Sigma}$ and $V$ as follows:
where $U$ ($m\times m$) and $V$ ($n\times n$) are orthogonal matrices and $\mathrm{\Sigma}$ is an ($m\times n$) diagonal matrix of singular values, ${\sigma}_{1}\ge {\sigma}_{2}\cdots \ge {\sigma}_{n}$. The matrix $\mathrm{\Sigma}$ is represented as:
Fig. 2Feature extraction process using the proposed approach
The columns of the orthogonal matrix $U$ are called the left singular vectors and the columns of the matrix $V$ are called the right singular vectors of the matrix $M$. An important property of $U$ and $V$ is that they are mutually orthogonal [30].
As seen in Fig. 2, after decomposing the signal using EWT, the analytical signal of each extracted mode is obtained using HT (see Eq. (12)). Subsequently, the instantaneous amplitude ${A}_{i}\left(t\right)$ ($i=$1,…, $n$) is obtained using Eq. (14). The feature matrix $M$ is therefore constructed from the instantaneous amplitude ${A}_{i}\left(t\right)$, and the fault feature vector ($\sigma \left(M\right)=\left[{\sigma}_{1},{\sigma}_{2},\dots ,{\sigma}_{n}\right]$) by applying the SVD.
The use of SVD presents its own advantages. It is able to expresses the feature matrix $M$ in the form of several values (singular values), so it is endowed a dimension reduction strategy. Furthermore, the singular values have a good stability. In other words, when the feature matrix element changes, a large variance of its singular values does not occur.
2.3. Elman neural network
The Elman neural network is generally divided into four layers, including input layer, hidden layer, association layer and output layer. The basic structure of the network is seen in Fig. 3. The association between context layer and hidden layer, makes the neural networks sensitive to the history of input data. The context neurons can be treated as the memory units, so, this internal feedback improves significantly the capacity of the network to deal with dynamic information, overcoming the drawback of the feedforward network. To this end, the Elman neural network compared with other neural networks, is robust in fault classification.
The transfer function of hidden layer is a non linear function, which is generally using a Sigmoid function. The activation function of the output layer neuron is a linear function. The mathematical model of Elman neural network is analyzed as follows:
where $y$, $x$, $u$, ${x}_{c}$ are separately representing the output vector of the network, the output vector of the hidden layer, input vector of the network and the feedback state vector. Then ${\omega}_{3}$, ${\omega}_{2}$, ${\omega}_{1}$ are separately representing connection weight matrix from the hidden layer to the output layer, from the input layer to the hidden layer and from the context layer to the hidden layer. $f\left(\cdot \right)$ is the transfer function of the hidden layer and $g\left(\cdot \right)$ denotes the activity function of the output neurons.
Fig. 3Structure of Elman neural network
3. Experimental result and discussion
In order to test the effectiveness of the proposed method, experiments were conducted on the dataset provided by the laboratory of contact and structure mechanics at INSA Lyon, France [25]. Experiments of the proposed method and detailed comparison with another widely used methods HHTSVD, LMDSVD and WPT together with PCA is also conducted as follows.
3.1. Experimental system description
Gear vibration testing experimental apparatus is shown in Fig. 4. The rotation motion of the equipment is generated by an electric dcmotor controlled in rotational speed with a nominal speed of 3600 RPM. Torque will be transmitted to the gearbox through the coupling where several pinion fault configurations were assembled. After the reducer outputting, through gear coupling, the torque will be transferred to a magnetic powder brake capable of generating different resistive torques [31].
Fig. 4a) Experimental gearbox test rig, b) structure of the single stage gear in the gearbox
In order to verify the effectiveness of the proposed method, six pinions with different fault states were considered. The first one is referred as Good (G), whereas the others have several different types of faults: a Tooth Root Crack (TRC), a Chipped Tooth in Length (CTL), a Chipped Tooth in Width (CTW), a Missing Tooth (MT) and General Surface Wear (GSW) as shown in Fig. 5. Three pinions are simultaneously mounted on the input shaft of the gearbox, the engagement of each of them is done by a simple axial movement of the wheel on the output shaft (Fig. 3(b)). To record vibration signals, two accelerometers with a sensitivity 100 mV/g was mounted radially, one vertically and the other horizontally on the bearing case of the output shaft. The time sampling frequency of the accelerometer channels is 125 kHz. The cutoff frequency of the antialiasing filter is 27 kHz. The acquisition duration is 30 s.
Fig. 5Six pinion states: a) normal, b) tooth root crack, c) chipped tooth in length, d) chipped tooth in width, e) missing tooth, f) general surface wear
a)
b)
c)
d)
e)
f)
The accelerometer signals have been recorded for four different operating modes: under the state of loading 8 N.m: 900 RPM, under the state of loading 11 N.m: 900 RPM, under the state of loading 5 N.m: 1200 RPM, under the state of loading 8 N.m: 1200 RPM.
Fig. 6 shows the acceleration vibration signals collected from pinions with different fault modes for an operating speed 1200 RPM and under the state of loading 5 N.m.
It can be seen that the TRC and the CTL faults (Fig. 6(b, d) respectively) do not significantly affect the vibration signal. On the other hand, a significant increase in the time signal energy is caused by the CTW, MT and GSW faults (Fig. 6(c, e, f) respectively) and as for localized defects, the presence of a repetitive shock waves at every revolution period. Similar observations were noted from the vibration signals of the other operating speed and load conditions.
Fig. 6Pinion vibration signals of gearbox with different faults: a) G, b) TRC, c) CTL, d) CTW, e) MT, f) GSW
3.2. Comparison between the proposed HEWTSVD method with HHTSVD, LMDSVD and WPT in tandem with PCA for feature extraction
To verify the robustness of the proposed method, three widely used methods, HHTSVD [32], LMDSVD and WPT together with PCA are also implemented and their performances are compared with that of HEWTSVD for gear feature extraction under variable conditions.
In four different operating modes, vibration signals of a normal gearbox and five kinds of pinion faults are divided into 15 groups of data. Thus, for each operating mode, we end up with 90 samples.
Since the experiment data we used are collected from test bench in laboratory setup, the vibration signals, therefore, present a low noise level, while in real word application, tremendous noise level may corrupt the vibration signals. Thus, strong background Gaussian white noises (GWN) with SNR of 0.5 dB was added to the vibration signals as seen in Fig. 7.
To begin with, we first analyze the various signals shown in Fig. 7 by the HEWT method. An example of the HEWT application for one certain fault and operating modes (gearbox with tooth root crack fault under the operating speed 1200 RPM and 5 N.m of load) is shown in Fig. 8, from which we can display the Fourier spectrum segmentation, the extracted modes and their envelopes.
It can be seen in Fig. 8(a), that the whole spectrum is adaptively divided into six regions. Consequently, six different modes are obtained in total. The analytical signal, thereafter, was computed for each mode by HT. The feature matrix can therefore be constructed for fault identification. Finally, the singular value vector which is the fault feature vector can be obtained by conducting SVD.
Fig. 7Vibration signals of gearbox with different faults corrupted with GWN of SNR = 0.5 dB: a) G, b) TRC, c) CTL, d) CTW, e) MT, f) GSW
Fig. 8HEWT results: a) Fourier spectrum segmentation, b) the extracted modes, c) envelopes of modes
Table 1, partly gives the fault feature values obtained by HEWTSVD for an operating speed 1200 RPM and under the state of loading 5 N.m, each vector contains the first three singular values.
To make a comparison between the proposed method with HHTSVD, LMDSVD and WPTPCA, they are all implemented on the same data sets.
The EMD (resp. LMD) decomposes the vibration signals into a set of IMFs (resp. product functions (PFs)). As their number is defined by the characteristic and complexity of the analyzed signal, only the first 6 IMFs (resp. PFs) were taken. To perform the HHT, the IMFs are used to get the analytical signal using HT. Next, the feature matrix was constructed and the fault feature vectors were obtained using SVD.
When using WPTPCA, the signals are decomposed by four layers sym1 wavelet packet decomposition, and signal energy of 16 frequency bands are obtained. Subsequently, PCA is used to extract principal components from the wavelet packet energy feature for dimensionality reduction.
Table 1Fault feature values obtained by HEWTSVD for an operating speed 1200 RPM and under the state of loading 5 N.m
Condition  Fault feature values  
1  2  3  4  5  6  7  8  
Normal  22.3618  22.5440  22.4297  22.8140  22.3895  22.2406  22.3029  22.4069 
6.5069  6.6871  6.5681  6.7349  6.5326  6.4637  6.4964  6.5661  
5.1798  5.2598  4.9472  5.5414  4.8536  5.0332  4.8663  5.3470  
Tooth root crack  30.5701  32.2918  31.7623  32.1969  31.9797  30.9457  30.9196  32.5190 
9.3210  10.0070  9.7765  9.8322  9.8691  9.3443  9.3615  10.0657  
6.1924  6.5807  6.4116  6.4888  6.6530  6.3118  6.2887  6.7280  
Chipped tooth in length  51.0280  51.5032  50.1618  50.9716  51.2508  51.2586  51.4757  53.2674 
19.5267  19.5201  19.2042  19.4590  19.3184  19.2715  19.4209  20.0547  
9.5368  10.2388  9.7723  9.9054  9.8470  10.0134  9.9200  10.5689  
Chipped tooth in width  40.2331  40.5439  40.1559  40.3599  40.5140  40.1301  40.2482  39.6597 
12.7980  12.9040  13.0148  12.7087  13.1321  12.6266  12.7598  12.6037  
8.7177  8.8491  8.7624  8.5900  8.7750  8.6759  8.4879  8.5784  
Missing tooth  78.4453  78.5593  78.1124  79.9563  79.1634  79.0539  79.2209  78.9756 
31.3720  31.3594  31.0467  32.1410  32.0537  32.1315  32.1755  31.9143  
19.2667  20.5065  19.5896  20.3227  20.5552  19.7883  19.1645  20.4280  
General surface wear  120.6759  121.2908  123.6672  124.6340  124.3797  124.1565  125.1376  124.1526 
37.4148  37.5389  38.2903  38.7183  38.0071  38.0113  39.3841  38.7904  
24.4425  26.3756  27.3797  27.3329  27.5331  26.9715  27.4802  27.0742 
Fig. 9 shows the projections of the first three singular values obtained by HEWTSVD of the normal gearbox and five kinds of pinion faults in four different operating modes in the threedimensional plan view, the classification results for different faults of the gearbox can be seen from the figures. Obviously, for one certain fault mode and whatever the operating mode, the singular values extracted by HEWTSVD have a high degree of coincidence. In the same context, comparing the singular value clusters of the normal gearbox with the five fault modes for different operating modes shown in Figs. 9(ad), one can see that, even though the signals are noisy, the mode separability is satisfactory, thus a good classification and it looks better for the operating speed 1200 RPM and under the state of loading 5 N.m shown in Fig. 9(c).
The results obtained by HHTSVD for the operating speed 900 RPM and under the state of loading 8 and 11 N.m respectively shown in Fig. 10(a and b) and those obtained by LMDSVD for the operating speed 1200 RPM and under the state of loading 5 respectively shown in Fig. 11(c) are almost comparable to the results obtained by HEWTSVD shown in Fig. 9. A noticeable separability between faults and a good coincidence of the singular value vectors. However, under the remaining operating modes, the classification effect is worst between normal gear and gear with TRC and between gears with CTW and CTL as shown in Figs. 10(c, d) and Figs. 11(a, d) and between normal gear and gear with CTW as shown in Fig. 11(b) from which, the clustering gap is very small or even inexistent that may lead to the misclassification.
Fig. 9Scatter plots of the first three singular values obtained by HEWTSVD for different faults: a) load 8 N.m, speed 900 RPM, b) Load 11 N.m, speed 900 RPM, c) Load 5 N.m, speed 1200 RPM, d) Load 8 N.m, speed 1200 RPM
Fig. 10Scatter plots of the first three singular values obtained by HHTSVD for different faults: a) load 8 N.m, speed 900 RPM, b) load 11 N.m, speed 900 RPM, c) load 5 N.m, speed 1200 RPM, d) load 8 N.m, speed 1200 RPM
Fig. 11Scatter plots of the first three singular values obtained by LMDSVD for different faults: a) load 8 N.m, speed 900 RPM, b) load 11 N.m, speed 900 RPM, c) load 5 N.m, speed 1200 RPM, d) load 8 N.m, speed 1200 RPM
Fig. 12Scatter plots of the first three principal components obtained by WPTPCA for different faults: a) load 8 N.m, speed 900 RPM, b) load 11 N.m, speed 900 RPM, c) load 5 N.m, speed 1200 RPM, d) load 8 N.m, speed 1200 RPM
On the other hand, comparison between Figs. 12(a, c, d), Figs. 10(a, c, d) and Figs. s11(a, c, d) shows that: utilizing the PCA in the process of fault feature extraction for wavelet packet frequency band energy, the effect is significantly better than the results obtained by HHTSVD and LMDSVD. Nevertheless, there exist major fluctuations for all states, the clustering intervals between the faults states are relatively small. Moreover, the effect of classification is worst between gears with TRC and CTW as shown in Fig. 12(b).
Known from the above analysis, using HEWT with SVD for fault feature extraction under different operating modes is much better than HHTSVD, LMDSVD and WPT in tandem with PCA.
Since the instantaneous amplitude matrices of the vibration signals of different fault in the pinions are different, adaptively extracted by the HEWT, thus, they can be used as the feature of diagnostic fault. As the goal is to extract more efficient and more accurate fault features for fault recognition and classification, dimensionality reduction was done by the SVD, thus fault can be accurately identified.
It was obvious in the scatter plots that the various fault singles have a large difference and no overlapping portions in Figs. 9(ad), from which we suggest that the various faults can be distinguished well.
3.3. State classification based on Elman neural network
Here, the fault feature vectors obtained by HEWT–SVD are used to identify and classify the gear states by using the Elman neural network.
The data sets are divided into training dataset with 168 groups (7 groups of data for each fault status) while the remaining data set (192 groups) are used to test the effectiveness and the accuracy of the classifier. State classification results are partly shown in Table 2, from which we can see that even under variable operating modes and even the signals are noisy the actual outputs of the Elman neural network are consistent with the target outputs. Therefore, combining HEWTSVD with Elman neural network is effective for gear fault diagnosis under variable operating modes.
Table 2State classification results based on Elman neural network.
Condition  Operating condition  Target output  Actual output of the network  
Normal  (900 RPM, 8 N.m)  (1 0 0 0 0 0)  1  1.502 E7  1.090 E13  0.0071  0  4.647 E8 
(900 RPM, 11 N.m)  (1 0 0 0 0 0)  0,9999  3,524 E5  0  4,876 E7  5,220 E5  4,668 E9  
(1200 RPM, 5 N.m)  (1 0 0 0 0 0)  1  1.726 E7  1.388 E13  2.279 E6  0  4.019 E8  
(1200 RPM, 8 N.m)  (1 0 0 0 0 0)  0,9998  4,778 E6  3,896 E14  3,791 E14  1,719 E5  3,492 E8  
Tooth root crack  (900 RPM, 8 N.m)  (0 1 0 0 0 0)  0,0083  0,9997  0,0002  0  0  8,203 E7 
(900 RPM, 11 N.m)  (0 1 0 0 0 0)  0  0,9998  3,226 E6  5,551 E17  2,428 E6  1,163 E9  
(1200 RPM, 5 N.m)  (0 1 0 0 0 0)  0  0,9993  2,013 E7  5,551 E17  0,0340  1,205 E9  
(1200 RPM, 8 N.m)  (0 1 0 0 0 0)  0  0,9997  3,226 E6  5,551 E17  2,428 E6  1,163 E9  
Chipped tooth in length  (900 RPM, 8 N.m)  (0 0 1 0 0 0)  0  9,813 E6  0,9999  0  0  2,7538 E6 
(900 RPM, 11 N.m)  (0 0 1 0 0 0)  0  4,469 E5  0,9999  0  8,479 E5  4,360 E5  
(1200 RPM, 5 N.m)  (0 0 1 0 0 0)  0  6,750 E6  0,9999  0  0  3,274 E6  
(1200 RPM, 8 N.m)  (0 0 1 0 0 0)  0  4,066 E5  0,9999  0  8,210 E5  4,748 E5  
Chipped tooth in width  (900 RPM, 8 N.m)  (0 0 0 1 0 0)  9,707 E5  3,133 E7  1,937 E14  1  0  3,527 E8 
(900 RPM, 11 N.m)  (0 0 0 1 0 0)  8,410 E7  3,138 E5  0  0,9999  3,382 E5  5,475 E9  
(1200 RPM, 5 N.m)  (0 0 0 1 0 0)  0,0005  2,139 E12  5,866 E7  0,9996  0  4,369 E9  
(1200 RPM, 8 N.m)  (0 0 0 1 0 0)  0,0003  1,951 E12  6,525 E7  0,9998  0  4,280 E9  
Missing tooth  (900 RPM, 8 N.m)  (0 0 0 0 1 0)  1,963 E7  6,033 E5  3,774 E15  1,110 E16  0,9999  2,308 E8 
(900 RPM, 11 N.m)  (0 0 0 0 1 0)  7,698 E5  3,288 E5  5,662 E15  4,884 E15  0,9991  2,609 E8  
(1200 RPM, 5 N.m)  (0 0 0 0 1 0)  0  1,017 E8  3,951 E10  0,0005  0,9928  7,962 E10  
(1200 RPM, 8 N.m)  (0 0 0 0 1 0)  0  6,293 E08  1,030 E10  2,068 E05  0,9999  6,262 E10  
General surface wear  (900 RPM, 8 N.m)  (0 0 0 0 0 1)  0  1,591 E10  5,182 E5  0  0  0,9999 
(900 RPM, 11 N.m)  (0 0 0 0 0 1)  0  4,087 E5  7,941 E13  1,959 E14  0  0,9999  
(1200 RPM, 5 N.m)  (0 0 0 0 0 1)  0  4,116 E7  6,221 E6  0  7,648 E5  0,9999  
(1200 RPM, 8 N.m)  (0 0 0 0 0 1)  0  3,756 E7  3,877 E6  0  7,646 E5  0,9999 
In order to compare the classification effect, another classification technique Back Propagation (BP) neural network is applied in classification. Comparison results are shown in Table 3 in which 5 examples are given to calculate the mean value of classification accuracy. As shown in the table, the Elman neural network has an advantage over BP in classification accuracy. Actually, both classifiers have classification accuracies higher than 0.9, because the obtained feature vectors have good separability. This gives confirmatory evidence about the effectiveness of the proposed feature extraction method.
Table 3Classification results of Elman and back propagation neural networks
Sequence  Test samples  Classification accuracy  
Elman  BP  
1  192  0.9947  0.9895 
2  192  0.9895  0.9843 
3  192  1  0.9791 
4  192  0.9895  0.9947 
5  192  0.9947  0.9843 
Mean value  0.9936  0.9863 
4. Conclusions
Gearbox fault feature extraction and classification have always been unsatisfactory. This constitutes the principal motivation to develop a new approach to overcome this problem. Our new approach consists of combining HEWT, SVD and Elman neural network. Experimental datasets of normal gearbox and five pinion faults and under different operating modes were processed by the HEWT. The instantaneous amplitude matrices namely the feature matrices were decomposed by SVD to get the singular value communality. Their dimensions were hence reduced and more stable features were obtained. This is followed by using the Elman neural network for fault identification and classification according to the extracted feature vectors.
In this paper, the performance of the proposed HEWTSVDElman method is shown to have an advantage over that of HHTSVD, LMDSVD and WPTPCA for feature extraction and Elman neural network with BP for classification accuracy under different operating modes.
It is worth mentioning that in this study we only applied the proposed method for gearbox fault diagnosis, future experiments should be done on similar project to verify the effectiveness of this method.
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The authors would like to thank the laboratory of contact and structure mechanics (LAMCOS) and in particular Pr. Didier RÉMOND for sharing their valuable experimental data with us.