Multi-Mode Separation and Nonlinear Feature ...

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Multi-Mode Separation and Nonlinear Feature Extraction of Hybrid Gear Failures in Coal Cutters using Adaptive Nonstationary Vibration Analysis Zhixiong Li · Yu Jiang · Xuping Wang · Z Peng

Received: date / Accepted: date

Abstract Reliable condition monitoring and fault diagnosis (CMFD) is an important issue for the normal operation of coal cutter gear systems. Intrinsic deterioration indicators are always hidden in the vibration response of the gearboxes but it is often very difficult to correctly extract them due to nonlinear/chaotic nature of the vibration signal. Literature review suggests that hybrid gear faults diagnosis is a challenging task and how to extract quantitative indicators for hybrid faults detection is attracting considerable attentions. In order to address this issue, a new adaptive nonstationary vibration analysis method is proposed in this paper to extract useful quantitative indicators for hybrid gear faults decoupling detection. In this new technology, the centre frequencies of the narrow-bands of intrinsic modes contained in the vibration signal were adaptively estimated by the variational model decomposition (VMD) to determine the bandwidth of the modes. Hence the hybrid gear faults were decoupled into single faults in the form of VMD modes. Then the time and frequency features of each mode were calculated to drive the feature space. Lastly, the feature space was projected into the reproducing kernel Hilbert space (RKHS) by the spectral regression optimized kerZhixiong Li( ) · Yu Jiang School of Mechatronic Engineering; Jiangsu Key Laboratory of Mine Mechanical and Electrical Equipment; China University of Mining and Technology, Xuzhou 221110, China Fax: +61-045-1069970 E-mail: [email protected] Xuping Wang Xi’an Research Institute of Hi-Tech, Xi’an 710025, P.R. China Zhixiong Li · Z Peng School of Mechanical and Manufacturing Engineering, The University of New South Wales, Sydney, NSW 2052, Australia

nel fisher discrimination (SRKFD), where the instinct nonlinear structure in the original data can be identified and thus useful quantitative indicators can be extracted. Reliable hybrid faults decoupling detection was then achieved. Specially designed numerical simulations and experiments were conducted to evaluate the proposed VMD-SRKFD method on hybrid gear faults diagnosis of coal cutters. The performance was compared with existing techniques. The analysis results show high performance of the proposed method on quantitative hybrid faults detection in the coal cutter gear system. Keywords Coal cutters · Gear transmission systems · Nonlinear vibration · Hybrid faults · Fault decoupling

1 Introduction Currently, coal cutters are used as primary mining equipment in most of coal seams [1]. For most coal cutters, multi-stage gearboxes are adopted to reduce the driving motor speed for the cutting rollers. Harsh working environment in the coal seams accelerates the wear process of the gearboxes and the supporting bearings are the most vulnerable components in the gearboxes [2, 3]. A simple failure, such as a crack in the outer race of a bearing, would induce serious faults for the bearing and hence eventually break down the whole system [4]. Therefore, it is crucial to monitor the condition of the gearboxes and detect the undergoing faults in the early stage [5, 6]. A challenging task in fault detection in gearboxes is to detect hybrid faults [7, 8]. Hybrid gear faults often occur in practice [3]. Recent publications focus on decoupling hybrid faults [9-17]. The hybrid faults were decomposed into the form of single faults in their researches. By identifying each single fault, the hybrid

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faults diagnosis was achieved. In order to realize the decoupling of hybrid faults, the wavelet transform [9, 10], empirical mode decomposition (EMD) [11], order tracking [12], and blind source separation (BSS) algorithms [14-16], were adopted in literature. The EMD is recognized as a powerful tool for analysing nonlinear and non-stationary signals and hence is extensively employed for mechanical vibration analysis [18]. The EMD is able to adaptively decompose a signal into different modes of unknown but separate spectral bands, i.e. the intrinsic mode functions (IMFs) [19]. However, the EMD suffers from mode mixing and overestimation problems due to noise and sampling. To address the noise issue, recent works proposed the variational approach to EMD [20, 21]. The candidate modes can be extracted variationally but in a recursive decomposition way. That is, like the EMD, the variational EMD decomposes the IMF one by one, which still leads to the mode overestimation problem. In order to solve both the noise and overestimation, a full adaptive VMD approach was proposed to decompose the candidate modes concurrently [19]. Experiments on artificial and real data demonstrated that the VMD is robust to sampling and noise. In [11] the EMD was used to decompose hybrid gearbox faults signal into several IMFs and the correlation measurement was adopted to select useful IMFs. Each selected IMF corresponded to a decoupled single fault. The correlation measurement acted in the work as a tool to response the model overestimation problem of the EMD but essentially cannot solve the IMF overestimation. In contrast, the VMD is able to avoid the overestimation problem in the mode decomposition processing. Hence, it is reasonable to evaluate the performance of the VMD based approach for hybrid faults decoupling diagnosis. On the other hand, unfortunately, most current works on hybrid faults decoupling detection seldom reported quantitative fault indicators. The quantitative indicators are extremely useful for practical application of fault diagnosis. However, intrinsic deterioration indicators are always hidden in the vibration response of the gearboxes but it is often very difficult to correctly extract them [22]. Existing fault indicators often adopt the root mean square (RMS) and kurtosis of the vibration signal. Their fault detection ability is very limited. A promising way for quantitative indicators extraction is information fusion [22]. Popular methods include the principal component analysis (PCA) and linear discriminant analysis (LDA) while their application limits to linear structure and Gaussian distribution data. The nonnegative matrix decomposition (NMF) is considered to be the most promising method of data fusion, but its negative restriction may lead to partial characteriza-

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tion problem [23]. Alternatively, the manifold learning is able to effectively excavate the essence of the nonlinear high-dimensional data structure and seize the key features of the data [23]. Representative algorithms include locally linear embedding (LLE), ISOMAP, and Laplacian Eigenmap. Existing studies have shown that these algorithms are a very effective for nonlinear dimension reduction. However, the eigenvalue decomposition significantly increases their computational complexity [24]. Recently the spectral regression (SR) is proposed to solve this problem [25]. SR adopts the leastsquares regression, instead of eigenvalue decomposition, to greatly reduce the computational complexity in the data fusion. In practice, it is very important to combined prior knowledge of the data into the manifold mining process. However, very limited work has reported this problem in gear fault diagnosis. It is worth extracting quantitative fault indicators using the SR based model. In order to address the hybrid faults detection in coal cutter gearboxes using quantitative indicators, this paper presents a new VMD based SRKFD approach. Taking advantages of strong adaptability of the VMD, the raw vibration signal of the gearbox was firstly decomposed into principal modes. Each mode represented a kind of single fault vibration response. The hybrid faults in the gearboxes can be correctly decoupled into single fault modes by the VMD analysis. Since the output principal modes were controllable, the overestimation problem in the EMD was effectively resolved. Then various features of the extracted modes were calculated to form the original feature space. Lastly, the SRKFD model was proposed to nonlinearly project the feature space into a low dimensional quantitative indicator space. By analyzing the quantitative indicators of each decoupled VMD mode, the hybrid faults were identified. Since the quantitative indicator space kept the most important information of the original feature space, it provided acceptable fault identification performance. The specific contribution of this work is that a new VMD-SRKFD method is proposed to provide quantitative indicators the hybrid faults decoupling diagnosis of coal cutter gearboxes. Both simulations and experiments are used to demonstrate this contribution.

2 The proposed hybrid faults diagnosis method 2.1 VMD-SRKFD approach In this work the VMD-SRKFD approach is presented for hybrid gear faults diagnosis. The VMD is recently

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proposed as an entirely non-recursive variational decomposition method to concurrently recover undergoing modes in a signal [19]. Given a gear vibration time series x = [x1 , x2 , ..., xm ], it contains k essential modes U = [u1 , u2 , ..., uk ] ∈ Rm and the corresponding instantaneous frequencies Ω = [ω1 , ω2 , ..., ωk ] ∈ Rm . The goal of the VMD is to find the optimal U and Ω subjecting to the following constrain [19]: 

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uk ,ωk 2 k (1) X    s.t. u = x k  k

where t denotes time, ∗ denotes convolution operation and δ denotes the Dirac distribution. Referring to [19], the VMD algorithm can be briefly summarized as below. (1) Initialize U, Ω and other parameters; (2) Update U and Ω; (3) Check the terminal condition. Once the optimal U is obtained, we can calculate the feature space F for each mode. In this work, the energy, entropy, kurtosis, skewness, root mean square (RMS), crest factor (CF), frequency center (FC), root mean square frequency (RMSF), standard deviation frequency (STDF) [26], fractal dimensions D0 , D1 and D2 [27] are selected. These features carry certain information about the gear health state; however, neither of them can be used as an efficient fault indicator for hybrid gear fault detection. In order to discover the intrinsic link between the features, the SRKFD model is employed here to extract quantitative indicators by fuse these features. The Fisher discrimination can be express as [24] J(W) = arg max

WT SB W WT SW W

(2)

where, W is the discrimination coefficient matrix, SB is the inter-class covariance matrix of the feature space F, and SW is the intra-class covariance matrix. In order to preserve the nonlinear structure of F, the feature space is projected into the reproducing kernel Hilbert space (RKHS) H to yield the kernel Fisher discrimination (KFD): J(V) = arg max

φ VT SB V φ VT SW V

(3)

φ where, V is the coefficient matrix of W in RKHS, SB is the inter-class covariance matrix of new feature space φ H, SW is the intra-class covariance matrix, and φ is the kernel function K(x, y) = hφ(x), φ(y)i. Eq. (3) involves the eigenvalue decomposition to solve V: φ φ SB V = λSW V

(4)

where λ is the eigenvalue. In order to avoid massive computation consumption, The SR framework is introduce to transform Eq. (4) into [24] KLKV = λKKV

(5)

where K denotes the kernel matrix (Kij = K(xi , xj )) and L is the weight matrix. Let KV=y, we can derive Ly = λy

(6)

According to [24], y can be calculated from L, so Eq. (6) is transformed as the following linear equation (K + εI)V = y

(7)

where ε is a constant. Decompose Eq. (7) by Cholesky decomposition: (K + εI)V = RT R −→ RV = (RT )−1 y = Z

(8)

where R is the Cholesky lower of (K + εI). Hence, the solution to Eq. (7) is equal to the following regularized regression [58]:  m X  2   min (φ(xi ) − zi )2 + ε kφk2   i=1 (9) m X     φ(x) = V K(x, x ) i i  i=1

where zi and Vi denote the i th component of Z and V, respectively. Hence, V can be solved by the regression algorithm and then the new feature space (quantitative indicators) can be obtained.

2.2 An overview of the hybrid faults decoupling diagnosis In this work a new VMD-SRKFD approach is proposed for hybrid gear faults detection. Since the VMD is a full adaptive signal analysis method, inheritably, the new approach can perform the adaptive data mining on the gear vibration signals. For practical vibration analysis, the proposed VMD-SRKFD is able to provide (1) correct decomposition of principal mode of the vibration signals without overestimation and (2) nonlinear feature extraction to form useful and effective quantitative fault indicators. Fig. 1 shows the block diagram of the overview of the proposed hybrid faults decoupling approach. In this study, the hybrid faults vibration signal was firstly decomposed into k VMD modes. The mode number k was controlled by prior knowledge to approximate the hybrid faults number. By doing so, the hybrid faults each mode can be decoupled into equal number VMD

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Fig. 1 Block diagram of the overview of the proposed VMD-SRKFD approach for hybrid faults diagnosis of coal cutter gears.

modes. Then the feature space for each mode was calculated for the SRKFD model. Lastly, the quantitative indicators were extracted by SRKFD with different dimensions (usually in one dimension (1-D), 2-D or 3-D but sometimes in 4-D or even 5-D [28]). By determine the fault identification threshold values or using an intelligent classifier (i.e. a support vector machine), the fault patterns can be recognized.

3.1 Discussions In the numerical simulation, without loss of the generality, we present a simple example to illustrate the proposed VMD based fault diagnosis method. The simulated gearbox vibration signal was x = m1 + m2 + m3 + η = [1 + α cos(2πfr1 t)] cos(2πfr1 t)+ [1 + β cos(2πfr2 t)] cos(2πfr2 t)+

(10)

[1 + γ cos(2πfr3 t)] cos(2πfr3 t) + η

3 Validation and results In this work both numerical simulations and experimental validations were conducted to evaluate the performance of the proposed method. The numerical simulation aimed to illustrate the advantage of the VMD model against EMD in principal modes extraction of the artificial vibration signal. Then the real data collected from a gearbox tester was used to examine the performance of the VMD-SRKFD on bearing fault identification.

where three different vibration modes m1 , m2 and m3 with different frequencies f1 , f2 and f3 were modulated by three different shaft rotating frequency fr1 , fr2 and fr3 respectively in the gear meshing movement, and η was the noise. Hence, there were three principal modes in x. The physic meaning of Eq. (10) was that there were three gear faults with the faults frequencies f1 , f2 and f3 respectively in the gearbox vibration signal x. The VMD was adopted to extract the fault modes and its performance was compared with the EMD. In the simulation, the parameters of x were as: α =1.0, β =1/4, γ =1/16, fr1 =2 Hz, f1 =5 Hz, fr2 =4 Hz, f2 =35 Hz, fr3 =7.2 Hz, fr3 =300 Hz, the signal-noise ratio (SNR) of η was 30 dB, and the mode number K of the VMD was 3. Fig. 2 shows the vibration signal

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x and the original three modes contained in x, Fig. 3 presents the mode extraction results by the VMD and Fig. 4 gives the time-frequency distributions of the extracted three modes.

It can be noticed in Fig. 3 that the three vibration modes m1 , m2 and m3 were correctly recovered by the VMD approach. The extracted modes tracked the original modes well. It can be also seen in Figs. 4 that the time-frequency distributions of the extracted modes matched well with that of the original modes. In Figs. 4(a) and (b) the period information about 0.5 s (1/fr1 ) and 0.1 s (1/2f1 ) were observed. In Figs. 4(c) and (d) the period information about 0.25 s (1/fr2 ) and 0.0143 s (1/2f2 ) were presented. In Figs. 4(e) and (f) the frequency centres located at 300 Hz (f3 ). These observations clear demonstrated the characteristics of the vibration modes m1 , m2 and m3 . The VMD was able to decompose the correct modes from the vibration signal x contaminated by noise. In order to convince the advantage of the VMD approach in mode extraction, its performance was compared with the EMD method. Fig. 5 shows the mode extraction results via EMD. It can be seen in the figure that the EMD decomposed x into eight IMF modes and the properties of the most modes were not corresponding to the original modes. Thus, the EMD failed to recover the original modes from x. The fault identification efficacy may be influenced by redundant/wrong modes information. To verify this viewpoint, the SRKFD was employed to evaluate the pattern recognition on the three modes m1 , m2 and m3 . By randomly changing the noise SNR in Eq. (10), there 50 samples in total were prepared for x. Then the VMD was applied to each sample to extract the three modes. The energy, entropy, kurtosis, skewness, root mean square (RMS), crest factor (CF), frequency center (FC), root mean square frequency (RMSF), standard deviation frequency (STDF), fractal dimensions D0 , D1 and D2 . Hence, there were 12 feature values for each sample and the overall feature space for all samples was F50×12 . Lastly, the SRKFD was used to fuse the high dimensional feature vector into one dimension, i.e. F50×12 −→ H50×1 . By doing so, clear identification on different modes can be observed using one dimensional feature value. Fig. 6 gives the identification results. It can be seen in the figure that the three modes m1 , m2 and m3 can be correctly recognized by the SRKFD fused feature. According to Fig. 6, if we choose the threshold value 2.0 as the boundary for m1 and m2 , and 1.0 for m2 and m3 , the mode identification rate for the three modes was 91.33%.

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The mode identification of VMD-SRKFD was compared with EMD-SRKFD. Fig. 7 shows the identification results via EMD methods. Comparing with Fig. 6, it can be noticed in Fig. 7 that the decomposed eight IMFs did not match well with the threshold values. The second mode m2 was failed to be identified. In addition, due to overestimation of the modes in x by the EMD, it was difficult to determine which IMF provided the correct information. Actually, in Fig. 5 it demonstrated that none of the IMF modes correctly corresponded to m1 , m2 and m3 . Hence, the mode detection results in Fig. 7 were significantly influenced by the useless IMF modes and the identification rate was very low. As a result, the EMD was not suitable for the hybrid fault recognition in this study.

3.2 Experimental validation In order to highlight the performance of the proposed VMD-SRKFD approach, in this work the gear hybrid faults experiments was carried out on the gearbox testrig (shown in Fig. 8). The configuration of the gearbox tester is as Driver motor → Coupler → Gear Z26 (26 teeth) → Gear Z64 (64 teeth) → Gear Z40 (40 teeth) → Gear Z85 (85 teeth) → Coupler → Brake. The hybrid faults were posed to Gear Z40 and Gear Z26. Four accelerometers were mounted on the gearbox body close to the shafts. The vibration signals were collected under 500 rpm (i.e. the driving frequency of DC motor fd is 8.33 Hz) in the experiment. The faulty frequency of the gear Z40 is f40 = 0.4fd = 3.38 Hz and the faulty frequency of the gear Z26 is f26 = fd = 8.33 Hz. The meshing frequency of the gear Z40 is fm40 = 135.28 Hz and the meshing frequency of the gear Z26 is fm26 = 216.58 Hz. The sample sampling frequency was 10,000 Hz. Fig. 9 shows the gear faults in the experiments. The vibration signals of the gearbox were recorded under four different conditions, including (1) normal condition, (2) hybrid faults of worn gear Z26 tooth and spalled gear Z40 tooth, (3) hybrid faults of cracked gear Z26 tooth and broken gear Z40 tooth, and (4) hybrid faults of worn gear Z40 tooth and broken gear Z26 tooth. Here the condition of hybrid faults of cracked gear Z26 tooth and broken gear Z40 tooth was used to illustrate the mode extraction of VMD. Fig. 10 shows the time and frequency spectra of the original vibration. Then the VMD was applied to decompose the original signal into two principal modes. Figs. 11 and 12 provide the decomposition results.

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It can be seen in Fig. 10 that evident impulsive components caused by the hybrid gear faults were observed in the time spectrum and the meshing frequencies of the two faulty gears clearly appeared in the frequency spectrum. The fault frequency about the gear Z40 was also observed in the frequency spectrum. However, the fault frequency about the gear Z26 was hardly found. After the VMD processing, it can be clear noticed in Fig. 11 that the fault frequency about the gear Z40 was fully demodulated from the original vibration sig-

nal. Dominating frequency components located at the fault frequency f40 and its harmonics. Encouragingly, it can be seen in Fig. 12 that the fault frequency about the gear Z26 was fully demodulated from the original vibration signal. Dominating frequency components located at the fault frequency f26 and its harmonics. In addition, comparing Fig. 11 with Fig. 12, one can be noticed that the fault information about the hybrid faults was satisfactorily separated into two individual modes. That is, the cracked gear Z26 tooth was decomposed into mode 1 by the VMD and the broken gear Z40 tooth was decomposed into mode 2. Similar observations can

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Fig. 4 The time-frequency distributions of: (a) m1 , (b) extracted mode 1, (c) m2 , (d) extracted mode 2, (e) m3 , and (f) extracted mode 3.

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Fig. 8 The configuration of the experiment platform.

Fig. 9 The gear faults in the experimental tests.

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be also found for the other two hybrid fault conditions. These results indicated that the hybrid faults were decoupled into two single faults in the forms of two VMD principal modes. Hence, by identifying the two modes using the SRKFD approach, the hybrid faults can be detected. In the experiments 80 samples were prepared for each condition and hence there were 320 samples in total. The VMD was firstly adopted to decompose each sample into two principal modes. In order to extract intrinsic features of each mode, the wavelet package transform [29] was employed to decompose each mode signal into eight sub-wavelet bands at level 3. Then the energy, entropy, kurtosis, skewness, RMS, CF, FC, RMSF,

STDF, D0 , D1 and D2 were calculated for each subwavelet band. Hence, for each mode, there were 96 feature values and its feature space was F80×96 . Lastly, the SRKFD was applied to each separated mode to fuse its high dimensional feature space into a low dimension space, i.e. F80×96 −→ H80×N , where N denotes the extracted feature dimension. Here different values of N were investigated. Figs. 13-16 show the distributions of the extracted features with N = 1, 2, 3, 4, respectively.

In Fig. 13 it can be seen that the 1-D features isolated

Hybrid Gear Failures Detection

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the normal gear from the faulty gear conditions effectively while a certain overlap appeared between different fault types. For the gear Z26 faults in Fig. 13(a), if choose the threshold values as 18, 25 and 31, the four gear conditions can be identified. Similarly, the threshold values can be set as 6.5, 10 and 14 for the gear Z40 faults in Fig. 13(b). In Fig. 14 it observed good separability between different gear conditions in the 2-D feature distribution map. Similar observation can also found in the 3-D feature distribution map in Fig. 15. So by judging the location of the feature values of a sample, one can assess its health state. The location was hence treated as a quantitative indicator. In Fig. 16 the 4-D feature distribution grouped the ex-

tracted features into four different colour hyperplanes in space. The health condition of the gear can be assessed by observing its hyperplane in the 4-D feature space. In order to recognize the fault patterns in an intelligent manner, the support vector machine (SVM) was employed to identify the hybrid faults. The goal is to recognize different modes of separate single faults from the hybrid gear faults. Table 1 lists the hybrid gear faults identification results, where the proposed VMDSRKFD was compared with the EMD-SRKFD. It can be seen in table 1 that the fault pattern recognition rate of the proposed VMD-SRKFD was much better than that of the EMD-SRKFD. Both the two approaches can successfully distinguish the normal con-

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150 S

Fig. 14 The distribution of the extracted 2-D features by the SRKFD.

dition from the faulty ones. The highest identification rate of the EMD-SRKFD was 95.0% for normal gear Z26 and 92.5% for normal gear Z40 while the highest identification rate of the VMD-SRKFD was 98.75% for both the two. It can also be noticed that with the increase of the indicator dimension the fault pattern recognition rate increased. The best recognition rate of the VMD-SRKFD for each VMD decomposed single fault mode was beyond 86.26%. Since the proposed method provided satisfactory fault pattern recognition performance, it can be used as an efficient tool in the hybrid gear faults identification. 3.3 Discussions The purpose of this work is to deal with hybrid gear faults. The challenging task here is how to decouple

the hybrid faults into different single ones and then how to establish quantitative indicators to assess the faults. We introduced the VMD-SRKFD approach to address the challenge. On the one hand, in the numerical simulation it can be seen in Figs. 3 and 4 that the newly proposed approach was able to correctly decouple the hybrid modes into individual modes. Quantitative indicators were calculated for fault pattern recognition in Figs. 6 and 7. Satisfactory fault identification rate was achieved. Comparing the VMD-SRKFD with the EMD-SRKFD, since the EMD method was not able to correctly separate the multiply modes contained in the original vibration signal, the fault identification rate was very low using the EMD-SRKFD. Moreover, the simulation results also demonstrated the overestimation problem of the EMD, which significantly increased the difficulty and amount of the data analysis in the

Hybrid Gear Failures Detection

13 (a)

25 Normal gear Z26 Worn gear Z26 tooth Cracked gear Z26 tooth Broken gear Z26 tooth

Feature value

20

15

10

5 20 15 10 Fe at ur

5

ev

alu

0

e

12

10

14

22

20

18

16

24

26

Feature value (b)

Normal gear Z40 Worn gear Z40 tooth Spalled gear Z40 tooth Broken gear Z40 tooth

Feature value

8 6 4 2 0 7 6 5

14

4 Fe

12 10

3

at

ur e

va lu e

8 6

2 1

4 2

Fig. 15 The distribution of the extracted 3-D features by the SRKFD.

Fig. 16 The distribution of the extracted 4-D features by the SRKFD.

Feature value

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Table 1 The hybrid gear faults identification results

Condition

VMD-SRKFD identification

EMD-SRKFD identification

1-D

2-D

3-D

4-D

1-D

2-D

3-D

4-D

Normal gear Z26

95.5%

96.25%

97.5%

98.75%

87.5%

88.75%

91.25%

95.0%

Worn gear Z26

88.75%

91.25%

93.75%

95.0%

57.5%

61.25%

63.75%

66.25%

Cracked gear Z26

85.0%

87.5%

89.5%

91.25%

51.25%

55.0%

57.5%

61.25%

Broken gear Z26

78.75%

81.25%

83.75%

86.25%

53.75%

56.25%

58.75%

62.5%

Normal gear Z40

93.75%

96.25%

97.5%

98.75%

83.75%

87.5%

90.0%

92.5%

Worn gear Z40

86.25%

88.75%

90.0%

91.25%

51.25%

53.75%

57.5%

60.0%

Spalled gear Z40

83.75%

85.0%

87.5%

88.75%

55.0%

58.75%

60.0%

63.75%

Broken gear Z40

85.0%

86.25%

88.75%

91.25%

53.75%

57.5%

61.25%

65.0%

fault identification processing. Similar analysis results were also observed in the experimental validation in Figs 11 and 12 and table 1. As a result, the proposed VMD-SRKFD method is suitable for vibration analysis of multi-mode signals. On the other hand, in the experiments the quantitative indicators was investigated in one dimension (1-D), 2D, 3-D and 4-D in Figs. 13-16. It can be seen in Fig. 13 that the 1-D indicators provided acceptable threshold values for the fault identification. In addition, by analyzing Figs. 13-16 it can be noticed that with the increase of feature dimension the separability increased in visual with the cost of computation consumption. A compromise between visualization and computation was achieved by the 2-D quantitative indicators. This is because the calculation complexity of 2-D quantitative indicators was relatively simpler to 3-D and 4-D ones and the fault identification results of the 2-D quantitative indicators in table 1 showed a good identification rate. Nevertheless, the fault identification performance of the proposed the VMD-SRKFD is better than the EMD-SRKFD. This is because the EMD failed to extract the correct fault modes from the raw vibration signals and therefore the hybrid faults modes mixed with each other. Interference information may cause misdiagnosis results. Since the proposed method correctly decoupled the hybrid gear faults into multi-mode representations and provided efficient fault detection quantitative, it is reasonable to draw the conclusion that the proposed VMDSRKFD method is capable for hybrid faults diagnosis on coal cutter gearboxes. The limitation of the proposed method is that it requires prior knowledge about the fault mechanism to check out the validity of the VMD decompositions while for practical applications it

is difficult to request the operators to understand such knowledge.

4 Conclusions Due to harsh working environment, the gear transmission systems of the coal cutters are prone to damages. Once failure happens to the gears, it is difficult to repair the components underground. Hence, reliable condition monitoring and fault diagnosis (CMFD) is an important issue for coal cutters. Literature review indicates that the diagnosis of hybrid faults in coal cutter gearboxes is a challenging task. In order to address this problem, a new method is proposed based on the adaptive data mining technology in this work. In this new method, the variational mode decomposition (VMD) is adopted to adaptively decompose the hybrid faults into multi-mode representations and each mode corresponds to a single fault of the hybrid faults. Then the nonlinear features of the representations are efficiently extracted by the spectral regression based kernel fisher discrimination (SRKFD) model. As a result, the hybrid faults can be reliably recognized by classifying the extracted nonlinear features. Both simulations and experiments were carried out to evaluate the proposed VMD-SRKFD method for hybrid gear faults in a coal cutter. The results demonstrated that (a) the proposed method is reasonable to separate multiply faults modes from the raw vibration signal while the EMD suffered from overestimation problem, (b) useful fault features can be extracted for effective fault pattern recognition, and (c) quantitative indicators can be obtained to assess the gear fault conditions by the proposed method when extracting the fault features in the low dimensional space. Since the proposed VMD-SRKFD method

Hybrid Gear Failures Detection

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provide efficient detection performance for hybrid gear faults, the new technique is very suitable for practical application in coal cutters. The findings in this work can be extended to the condition monitoring of general rotating machines. Future work will investigate the performance of the proposed method in other components of the coal cutters.

11. Li, R., Yu, D., Chen, X., Liu, J.: A compound fault diagnosis method for gearboxes based on chirplet path pursuit and EEMD. Journal of Vibration and Shock 33, 51–56 (2014)

Acknowledgements This research was funded by the National Natural Sciences Foundation of China (NSFC) (No. 51505475), the Fundamental Research Funds for the Central Universities (No. 2015XKMS018) and the Priority Academic Program Development of Jiangsu Higher Education Institutions.

13. Li, Z.: A novel solution for the coupled faults isolation in gear pairs using the conception of frequency tracking, Electrical and Electronic Engineering 20, 69–72 (2014)

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