Gear fault diagnosis method based on local mean decomposition and generalized morphological fractal dimensions

Mechanism and Machine Theory 91 (2015) 151–167

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Gear fault diagnosis method based on local mean decomposition and generalized morphological fractal dimensions Zhi Zheng, Wanlu Jiang ⁎, Zhenwei Wang, Yong Zhu, Kai Yang Hebei Provincial Key Laboratory of Heavy Machinery Fluid Energy Transmission and Control, Yanshan University, No. 438, Hebei Road, Qinhuangdao, Hebei 066004, PR China Key Laboratory of Advanced Forging & Stamping Technology and Science (Yanshan University), Ministry of Education of China, Yanshan University, No. 438, Hebei Road, Qinhuangdao, Hebei 066004, PR China

a r t i c l e

i n f o

Article history: Received 24 July 2014 Received in revised form 9 April 2015 Accepted 11 April 2015 Available online xxxx Keywords: Local mean decomposition Generalized morphological fractal dimension KFCM Gear fault diagnosis

a b s t r a c t Aiming at gear fault diagnosis, a fusion method of local mean decomposition (LMD) and generalized morphological fractal dimensions (GMFDs) is proposed. Firstly, a signal is decomposed by LMD into several product functions (PFs) which have physical meanings. Secondly, mutual information entropy value between each PF and original signal can be computed, and the PF corresponding to the maximum value is considered as containing the richest feature information of original signal, thus the PF is used as data source. Lastly, GMFDs are extracted from the data source, and some GMFDs which can quantitatively and comprehensively characterize nonlinear information of gear running states are adopted as feature vectors, hence gear faults can be diagnosed by kernel fuzzy c-means (KFCM). In order to demonstrate superiority of the proposed method, the GMFDs are extracted from signals of different lengths, ones sampled under three different working conditions of load and speed, ones without decomposition of LMD. The gear signals are tested and verified, and the result demonstrates that the proposed method is superior and can diagnose gear faults accurately. © 2015 Elsevier Ltd. All rights reserved.

1. Introduction Rotating machinery has been widely used in the fields of aeronautics, astronautics, metallurgy, petrochemical engineering and construction machinery, thus high speed, large load and other mal-conditions can lead to its high damage probability. And faults can result in equipment health deterioration and even breakdown [1,2]. As a rotating machinery, gear is also faced the problems, thus a lot of scholars and engineers have done much work on its condition monitoring and fault diagnosis [3,4]. Local mean decomposition (LMD) is a data-driven and novel self-adaptive analysis method in time–frequency domain. It was proposed by Smith in 2005 and firstly applied to electroencephalogram signal successfully [5]. The multi-component signal can be decomposed to a series of mono-components which are product functions (PFs), therefore each of them is the product of an envelope signal and a purely frequency modulated signal. The instantaneous amplitude of PF can come from an envelope signal, and the welldefined instantaneous frequency can be calculated from a purely frequency modulated signal. In essence, each PF is an amplitudemodulated and frequency-modulated signal (AM–FM signal) [6–11].

⁎ Corresponding author at: Hebei Provincial Key Laboratory of Heavy Machinery Fluid Energy Transmission and Control, Yanshan University, No. 438, Hebei Road, Qinhuangdao, Hebei 066004, PR China. Tel.: +86 335 8057073. E-mail address: [email protected] (W. Jiang).

http://dx.doi.org/10.1016/j.mechmachtheory.2015.04.009 0094-114X/© 2015 Elsevier Ltd. All rights reserved.

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In order to get the length of an irregular curve, Minkowski proposed “Minkowski cover” method. The method is used to dilate the curve by forming the union of disk structure elements (SE) in different scales. Based on this, Bouligand proposed an improved method called “Minkowski–Bouligand dimension” method. The method replaced the disk that SE “Minkowski cover” method adopted with any kind of planar SE, but two-dimension of planar SE leads to quadratic computational complexity. Aiming at the shortcoming, Maragos proposed an improved method named “Morphological covering” method. It replaced two-dimension SE with onedimension one, thus computational complexity is reduced, and he firstly applied the method to fractal dimension estimation of a one-dimensional signal. If height parameter of SE is set as 0, Minkowski–Bouligand dimension is completely independent from signal covering area [12,13], thus flat SE is adopted in this study. The above fractal dimension estimation method is based morphological technique, and it can be named morphological fractal dimension (MFD). And traditional fractal dimension is often based on box-counting method, and the method is based on regular generation of grids in signal covering area, thus fractal dimension estimation is greatly dependent on relationship between grid positions and the covering area, and grid positions are not fixed relative to the signal, hence it leads to random error of fractal dimension estimation [14,15]. The single fractal dimension (FD) cannot characterize nonlinear information of a signal more comprehensively than generalized fractal dimensions (GFDs), but GFDs from box-counting method also has that kind of error. Thus Li proposed a method of generalized morphological fractal dimensions (GMFDs), and it is applied to fault diagnosis of gear successfully [16], and the result shows that Li's method is superior to the other method of GMFDs proposed by Xia [17], and there is doubt that it is also superior to traditional method of GFDs. But GMFDs directly extracted from a signal without de-noising can lead to incorrect character of rotating machinery nonlinear information. Although a signal is de-noised and then MFD is extracted, the single MFD can only characterize one-sided nonlinear information of running states. Aiming at the problems, the fusion method of LMD and GMFDs is proposed in this study. Firstly, a signal is decomposed by LMD, and the product functions (PFs) which have physical meanings can be obtained. Secondly, mutual information entropy value between each PF and original signal is computed based on mutual information theory [18,19], and the PF corresponding to the maximum value is considered to be the closest to original signal, and it can be regarded as containing the most amounts of feature information of original signal, thus the PF is selected as data source. Lastly, GMFDs are extracted from the data source, and some GMFDs which can characterize quantitatively and comprehensively nonlinear information of gear running states are adopted as feature vectors, hence kernel fuzzy c-means (KFCM) can diagnose faults effectively. The paper is organized as follows. Section 2 describes principles of LMD, KFCM and mutual information theory. In Section 3, principle of GFDs is presented. In Section 4, principle of GMFDs is detailed. Section 5 briefly depicts the fusion method. In Section 6, examples of applying GMFDs to synthetic fractal signal analysis are presented. In Section 7, GMFDs are applied to characterize nonlinear information of gear running states, and gear faults are diagnosed by KFCM. In Section 8, the conclusions are drawn. 2. Principle of LMD, KFCM and mutual information theory 2.1. Principle of LMD In traditional time-frequency analysis method of wavelet transform, there are still some limitations such as mutual constraint of time and frequency resolution, low matched-degree of kernel function and signal and so on, furthermore, selection of kernel function is another difficult point. As a new self-adaptive analysis method in time-frequency domain, LMD overcomes above limitations and can adaptively decomposed a multi-component signal of nonlinear and non-stationary signal into a set of mono-component AM– FM signals (PFs) in different time scales based on morphological features of a signal. In essence, each AM–FM signal is the product of a purely frequency modulated signal and an envelope signal [5–11]. 2.2. Principle of KFCM KFCM is developed based on FCM, and it is an unsupervised learning algorithm. With kernel function, samples in original feature space can be nonlinearly mapped into kernel space, and intrinsic different information among samples can be extracted adequately, therefore the information can be highlighted and amplified accurately in kernel space. And samples in kernel space are clustered according to their attributes, as a result, samples in the same cluster are in high similarity, and the ones in different clusters are in high dissimilarity [20,21]. Clustering quality can be evaluated by partition coefficient and partition entropy, and they are listed as follows: P¼

c X n 1X 2 u : n i¼1 j¼1 i j

E¼−

c X n 1X u lnui j : n i¼1 j¼1 i j

ð1Þ

ð2Þ

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Where c is the number of cluster centers, n is the sample number of X, and uij represents membership of xj belonging to cluster i. It demonstrates that clustering quality is better if value of P is much closer to 1 and value of E is much closer to 0. 2.3. Principle of mutual information theory In information theory, mutual information theory can measure different degrees of two signals [15,16]. Given two signals X = {xi|i = 1, 2, …, n} and Y = {yj|j = 1, 2, …, m}, mutual information entropy of the two can be written as: IðX; Y Þ ¼ H ðX Þ þ H ðY Þ−H ðX; Y Þ:

ð3Þ

Where H(X) and H(Y) are respectively entropies of X and Y, and H(X,Y) is the joint entropy of the two. Their expressions are as follows: X px ðxi Þ lgpx ðxi Þ: HðX Þ ¼ −

ð4Þ

i

HðY Þ ¼ −

    py y j lgpy y j :

X

ð5Þ

j

    X HðX; Y Þ ¼ − pxy xi ; y j lgpxy xi ; y j :

ð6Þ

i; j

Where px(xi), py(yj) and pxy(xi,yj) are respectively the probability distribution functions of X, Y and XY. And they are subject to     m n m ∑ px ðxi Þ ¼ 1, ∑ py y j ¼ 1 and ∑ ∑ pxy xi y j ¼ 1. n

i¼1

j¼1

i¼1 j¼1

The bigger mutual information entropy value is, the closer the correlation of X and Y is, and vice versa. 3. Principle of GFDs FD can characterize nonlinear information of a signal based on its morphological features, thus it can characterize nonlinear information of rotating machinery qualitatively and quantitatively [22]. GFDs can characterize nonlinear information more comprehensively than FD, and GFDs are also the most used analysis way in multi-fractal theory. The main analysis ways of GFDs include methods of box-counting, fixed radius and fixed mass, and box-counting method is in common use among them [21–25]. In box-counting method, a signal area is covered by a lot of boxes of length ε, and box number is N(ε), and Pi(ε) is the probability of points located in ith box. Given parameter q, generalized information entropy can be computed according to N X q lg ½pi ðε Þ i¼1

K q ðεÞ ¼

:

1−q

ð7Þ

The GFDs are written as:

Dq ¼

lim lgK q ðεÞ

ε→0

lgð1=ε Þ

:

ð8Þ

GFDs are obtained by box-counting method, and position uncertainty of grids can lead to large random error of values of GFDs, thus GFDs cannot better characterize nonlinear information of rotating machinery. Aiming at the problems, GMFDs proposed by Li are introduced [16]. 4. Principle of GMFDs 4.1. Basic mathematical morphological transforms Mathematical morphology was proposed by Matheron and Serra in 1960s, and its two basic morphological operators are respectively erosion and dilation [27–34].

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Let f(n) be a one-dimensional signal whose domain is F = {0,1,2,…,N − 1}, and let g(m) be a one-dimensional signal called SE whose domain is G = {0,1,2,…M − 1} G = {0, 1, 2, …, M − 1}, where M and N are respectively point numbers of signals, M b b N. Erosion and dilation are respectively defined as follows: f Θg ðnÞ ¼ minf f ðn þ mÞ−g ðmÞg; m∈f0; 1; 2; …; M−1g; n∈f0; 1; 2; …; N−1g:

ð9Þ

f ⊕g ðnÞ ¼ maxf f ðn−mÞ þ g ðmÞg; m∈f0; 1; 2; …; M−1g; n∈f0; 1; 2; …; N−1g:

ð10Þ

Where Θ and ⊕ respectively stand for erosion operator and dilation operator. Erosion operator can oppress negative impulses and reserve positive ones, and dilation operator performs oppositely [26–33]. 4.2. Principle of Minkowski–Bouligand dimension Let g(m) denote unit SE, and εg is SE in ε scale, and it can be obtained by dilating g for ε − 1 times, thus it can be defined as: εg ðmÞ ¼ g ⊕g⊕⋯⊕g : |fflfflfflfflfflffl{zfflfflfflfflfflffl} ε−1

ð11Þ

times

Let f(n) be a one-dimensional signal, and multi-scale dilation and erosion of f(n) by εg can be respectively written as: f ⊕εg ðnÞ ¼ f ⊕g ⊕g⊕…⊕gðnÞ: |fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl}

ð12Þ

f Θεg ðnÞ ¼ f Θg ΘgΘ…Θg ðnÞ: |fflfflfflfflfflffl{zfflfflfflfflfflffl}

ð13Þ

ε−1 times

ε−1

times

So cover area Ag(ε) in ε (1 ≤ ε ≤ N/2) scale is computed by Ag ðε Þ ¼

N X ½ f ⊕εg ðnÞ−f Θεg ðnÞ:

ð14Þ

n¼1

According to [13], Ag(ε) can be written as:     Ag ðεÞ 1 ¼ D lg lg þ c: M ε ε2

ð15Þ

DM is Minkowski–Bouligand dimension, thus it can be obtained according to

DM ¼ lim

h i lg Ag ðε Þ=ε2

ε→0

lgð1=εÞ

:

ð16Þ

Fit a straight line by least squares to the graph of lg[Ag(ε)/ε2] versus lg(1/ε), and the slope of the line is the approximate estimation of Minkowski–Bouligand dimension of f(n). Minkowski–Bouligand dimension has two attractive properties: (1) if f(n) is shifted, DM is not changed. (2) If flat SE (height parameter is 0) is adopted, DM remains invariant with respect to any affine scaling of amplitude of f(n). 4.3. Principle of GMFDs The principle of GMFDs is similar to that of box-counting method, thus normalized measure ui(ε) obtained by “morphological covering” method can be written as: ui ðε Þ ¼

f ⊕εg ðnÞ−f Θεg ðnÞ : N X ½ f ⊕εg ðnÞ−f Θεg ðnÞ

ð17Þ

n¼1

Where n = 1, 2,…, N. In Eq. (17), the numerator shows differences of result of a signal respectively eroded and dilated in ε scale, and the denominator is the signal covering area in ε scale.

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Given a parameter q, generalized morphological information entropy is presented as: N X q lg ½μ i ðε Þ

K q ðεÞ ¼ α ðεÞ

i¼1

:

1−q

ð18Þ

Where α(ε) is a coefficient, it is defined as: 2

α ðεÞ ¼

lgAg ðεÞ=ε : lgN

ð19Þ

If the area covered signal is calculated by box-counting method, N(ε) that is the number of boxes multiply by ε2 that is the area of a box is Ag(ε), thus Eq. (19) can be rewritten as:

α ðεÞ ¼

lgAg ðεÞ=ε2 lg½Nðε Þ ¼ ¼ 1: lgNðεÞ lg½Nðε Þ

ð20Þ

Eq. (20) ensures consistency of Eqs. (18) and (7). The partition function χ(ε, q) is defined as:

χ ðε; qÞ ¼

" N X

# α ðε Þ q

ui ðε Þ

:

ð21Þ

i¼1

Thus GMFDs can be calculated by Eq. (22)

Dq ¼

1 lgχ ðε; qÞ lim : q−1 ε→0 lgε

ð22Þ

Like the process of Eq. (16), fit a straight line by least squares to the graph of lg χ(ε, q) versus lg(ε), and slope of the line is the approximate estimation of GMFDs of f(n). When q = 0, Eq. (22) can be rewritten as:

Dq ¼ lim ε→0

h i lg Ag ðεÞ=ε2 lgð1=ε Þ

:

ð23Þ

Eq. (23) is identical with Eq. (16), and Dq becomes Minkowski–Bouligand dimension, thus calculation of GMFDs is proved to be valid. 5. Fault diagnosis method based on LMD, GMFDs and KFCM Firstly, LMD is applied to adaptively decompose a signal into several PFs from high frequency band to the low one based on morphological features of a signal. Secondly, mutual information entropy value between each PF and original signal is computed based on mutual information theory, and the PF corresponding to maximum value is considered as containing the richest feature information of original signal, thus it can be adopted as data source. Lastly, GMFDs are extracted from the data source, and some GMFDs which can characterize nonlinear information of gear running states quantitatively and comprehensively are selected as feature vectors, and gear faults can be diagnosed with KFCM effectively. Flow chart is shown in Fig. 1.

Fig. 1. Flow chart of the proposed method.

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Fig. 2. WCF of W0.6(t).

6. Synthetic fractal signal analysis In order to verify effectiveness and accuracy of fractal dimension estimation based on morphological technique, a typical synthetic fractal signal named Weierstrass Cosine Function (WCF) is adopted, and it is defined as:

WH

∞ X

γ

−kH

  k cos 2πγ t ; 0bHb1:

ð24Þ

k¼0

Where γ N 1. Ideally, fractal dimension of WH(t) is D = 2 − H. Discretization is adopted to process the signal, where sample frequency is set as 1000 Hz, sample time is set as 1 s, and γ is set as 5, 0 ≤ k ≤ 20. Figs. 2 and 3 display time domain waveforms of W0.6(t) (D = 1.4) and W0.3(t) (D = 1.7). It is known from two figures that waveform of W0.6(t) is more complicated than that of W0.3(t).

6.1. Minkowski–Bouligand dimension Minkowski–Bouligand dimensions of W0.6(t) and W0.3(t) are respectively computed, where scale range is set as 1–500. Fig. 4 shows log–log plot of W0.6(t), and slope of the line is 1.4396 which is the actual value of Minkowski–Bouligand dimension of W0.6(t), and error between theoretical value 1.4 and actual value is 0.0283. Similarly, error between theoretical value 1.7 and actual value of Minkowski–Bouligand dimension of W0.3(t) 1.7039 is 0.0055. Both of the errors are very small, the result illustrates that accuracy of Minkowski–Bouligand dimension is very high, and it is effective. Because of discretization process of WCF synthetic signal, parameter k becomes finite, which generates the errors.

6.2. Generalized morphological fractal dimensions

(t)

GMFDs of W0.5(t) are computed, where scale range is set as 1– 500, parameter q range is set as [−–40:2:40], and Fig. 5 demonstrates the result. It can be concluded from Fig. 5 that errors between values of GMFDs and theoretical value 1.5 are also very small, and accuracy of GMFDs is also very high, hence it is effective.

Fig. 3. WCF of W0.3(t).

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Fig. 4. The log–log plot of W0.6(t).

Fig. 5. Diagram of GMFDs.

7. Application to gear fault signals 7.1. Experimental scheme In order to verify the validity of the fusion method of LMD and GMFDs, the measurement is performed on a mechanical fault simulator (MFS–MG) produced by Spectra Quest incorporation. Gear fault simulator of MFS–MG, missing tooth fault gear and broken

Fig. 6. Fault gears simulator system.

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Fig. 7. Three state signal waveforms of 8192 points.

tooth fault gear are shown in Fig. 6. Under three working conditions of load and speed, signals are measured at sample frequency 50 kHz and sample time is set as 2 s, where the conditions are at a speed of 826 r/min and load of 5.7 in-lbs, 472 r/min and 5.7 inlbs, 826 r/min and 3.8 in-lbs.

7.2. Selection of data source There are signals of four different lengths for each state of normal, missing tooth and broken tooth fault, and lengths are respectively 8192 points (about 0.1638 s), 16,384 points (about 0.3277 s), 32,768 points (about 0.6554 s) and 65,536 points (about 1.3107 s). LMD decomposes a signal into some PFs. The mutual information entropy of each PF and original signal is computed, and the bigger the entropy is, the closer their relationship is, thus the PF corresponding to the maximum entropy values is selected as data source. Moreover, the PF is regarded as containing the most varsity of feature information of original signal. When gear becomes fault, fault component often causes impact, thus impact energy is generated. There is no doubt that impact energy is full of feature information, thus many researchers take advantage of it to diagnose faults of rotating machinery [35,36]. In order to verify effectiveness of the selection method of data source, energy ratio is proposed. Energy ratio of each PF and original signal is computed, and the PF corresponding to the maximum value of energy ratios is considered as containing the most amounts of feature information. If energy ratio value of the selected PF is also the maximum, the selection method is proved to be effective.

Table 1 Energy ratio and entropy values between normal signal of 8192 points and the first three PFs. Parameters

PF1

PF2

PF3

Mutual information entropy Energy ratio

0.2463 10.18%

0.5025 53.43%

0.3761 18.16%

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Table 2 Energy ratio and entropy values between missing tooth fault signal of 8192 points and the first three PFs. Parameters

PF1

PF2

PF3

Mutual information entropy Energy ratio

0.1933 23.63%

0.3820 28.6%

0.4516 35.27%

Table 3 Energy ratio and entropy values between broken tooth fault signal of 8192 points and the first three PFs. Parameters

PF1

PF2

PF3

Mutual information entropy Energy ratio

0.1375 5.82%

0.5823 75.32%

0.2361 14.5%

The decomposition of three state signals of 8192 points by LMD is taken as the example under 826 r/min and 5.7 in-lbs. The original signals are shown in Fig. 7. Mutual information entropies and energy ratios are respectively computed between each PF and original signal. Conclusions are obtained as follows. (1) For normal signal of different lengths, the first three PFs correspond to the three maximum values of mutual information entropies and energy ratios, and PF2 corresponds to both of the maxima of mutual information entropy and energy ratio values, thus PF2 are used as data source. (2) Likewise, it is the same situation for missing tooth fault and broken fault signal of different lengths, thus the first three PFs correspond to the three maximum values of mutual information entropies and energy ratios, and PF3 and PF2 are used as data source. Tables 1–3 list the values of entropies and energy ratios. The time domain waveforms of the three PFs are shown in Fig. 8.

Fig. 8. Three state data source waveforms of 8192 points.

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Fig. 9. Diagrams of GMFDs extracted from original signals of different lengths under working condition of 826 r/min and 5.7 in-lbs.

7.3. Extraction of GMFDs under working condition of 826 r/min and 5.7 in-lbs 7.3.1. Extraction of GMFDs based on original signals GMFDs are extracted directly from each state signals of different lengths without decomposition of LMD, where value range of parameter q is set as [−40:2:40], (0 0 0) is selected as unit SE, and scale ranges of SE are set as 2–35. Each state has four samples, and each sample is constituted of GMFDs. In order to improve validity and superiority of the proposed method, value range of parameter q, unit SE scale ranges are all set [−40:2:40], (0 0 0) and 2–35 for the other two working conditions, and each state also have four samples. Fig. 9 depicts four samples of each state signals of four different lengths. Normal signal is close to random noises, thus morphological features of normal signal are uncertain and irregular, which makes values of normal state GMFDs big. When gear becomes fault, periodic impulses make morphological features of fault signal characteristic, hence values of fault state GMFDs are small. In Fig. 9, normal state GMFDs overlap with those of broken tooth fault state, thus values of normal state GMFDs in the whole range of q are not the maximum, which contradicts the above physical principle. When q = 0, the single MFD is called Minkowski–Bouligand dimension. For Minkowski–Bouligand dimensions of each state signals of different lengths, there are overlaps or those of normal state that are not the maximum. Under working condition of 826 r/min and 5.7 in-lbs, the conclusions are listed as follows. (1) GMFDs extracted in the whole range of q cannot characterize quantitatively and comprehensively nonlinear information of gear running states. (2) Minkowski–Bouligand dimensions of each state cannot characterize quantitatively and comprehensively nonlinear information of gear running states.

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Fig. 10. Diagrams of GMFDs extracted from PFs of different lengths under condition of 826 r/min and 5.7 in-lbs.

7.3.2. Extraction of GMFDs based on PFs The GMFDs are extracted from PFs, and Fig. 10 displays four samples of each state data source of four different lengths. In Fig. 10, one state GMFDs do not overlap with those of adjacent state and values of normal state GMFDs are the maximum with the help of LMD when q N 0. With the help of LMD, Minkowski–Bouligand dimensions of each state do not overlap with each other, and Minkowski–Bouligand dimension value of normal state is the maximum. Under working condition of 826 r/min and 5.7 in-lbs, conclusions are obtained as follows. (1) GMFDs extracted in q N 0 from signal decomposed by LMD can characterize quantitatively and comprehensively nonlinear information of gear running states. (2) Minkowski–Bouligand dimensions can characterize quantitatively and comprehensively nonlinear information of gear running states. (3) LMD can filter noises and extract much feature information. 7.4. Extraction of GMFDs under working condition of 472 r/min and 5.7 in-lbs 7.4.1. Extraction of GMFDs based on original signals Fig. 11 demonstrates four samples of each state original signals of four different lengths. In Fig. 11, there are overlaps between adjacent state GMFDs, thus values of normal state GMFDs extracted in the whole range of q are not the maximum. Likewise, it is the same situation for Minkowski–Bouligand dimensions of each state as those in Section 7.3.1.

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Fig. 11. Diagrams of GMFDs extracted from original signals of different lengths under working condition of 472/min and 5.7 in-lbs.

Besides the conclusion (2) in Section 7.3.1, another conclusion can be obtained that GMFDs extracted in q N 0 directly from a signal cannot characterize quantitatively and comprehensively nonlinear information of gear running states under working condition of 472 r/min and 5.7 in-lbs. 7.4.2. Extraction of GMFDs based on PFs Based on the selection method, PF2, PF3 and PF2 and are selected as data source for normal state, missing tooth fault and broken tooth fault signals of different lengths. Fig. 12 displays four samples of each state data source of four different lengths. In Fig. 12, one state GMFDs do not overlap with other state ones and values of normal state GMFDs are the maximum with the help of LMD when q N 0. Situation of Minkowski–Bouligand dimensions of each state is similar with those in Section 7.3.2. Besides conclusion (2) and (3) in Section 7.3.2, the conclusion can be obtained that GMFDs extracted in q N 0 from a signal decomposed by LMD can characterize quantitatively and comprehensively nonlinear information of gear running states under working condition of 472 r/min and 5.7 in-lbs. 7.5. Extraction of GMFDs under working condition of 826 r/min and 3.8 in-lbs 7.5.1. Extraction of GMFDs based on original signals Fig. 13 shows four samples of each state original signals of four different lengths. In Fig. 13, GMFDs of normal state extracted in the whole range of q either overlaps with others or their values are not the maximum. Similarly, the situation of Minkowski–Bouligand dimensions of each state is the same as those in Section 7.3.1.

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Fig. 12. Diagrams of GMFDs extracted from PFs of different lengths under condition of 472/min and 5.7 in-lbs.

Besides the conclusion (2) in Section 7.3.1, other conclusion can be obtained that GMFDs extracted in q N 0 directly from a signal cannot characterize quantitatively and comprehensively nonlinear information of gear running states under working condition of 826 r/min and 3.8 in-lbs.

7.5.2. Extraction of GMFDs based on PFs PF2, PF3 and PF3 are selected as data source for normal state, missing tooth fault and broken tooth fault signals of different lengths by using the selection method. Four samples of each state data source of four different lengths are shown in Fig. 14. In Fig. 14, there are no overlaps among all three state GMFDs and values of normal state GMFDs are the maximum with the help of LMD when q N 0. Although signals are decomposed by LMD, there are still overlaps between Minkowski–Bouligand dimension of normal state and that of broken tooth fault state. Conclusion (2) in Section 7.3.1 and (3) in Section 7.3.2 can also be obtained, and other conclusions can be obtained that GMFDs extracted in q N 0 from a signal decomposed by LMD can characterize quantitatively and comprehensively nonlinear information of gear running states under working condition of 826 r/min and 3.8 in-lbs. Because LMD can filter noises and extract much feature information of each state, GMFDs that extract from signals decomposed by LMD can contain much clear nonlinear information of gear running states, thus influences of speed, load and signal length on GMFDs are analyzed based on Figs. 10, 12 and 14. It can be known from Figs. 10, 12 and 14 that no matter if the signal length is long or not, values of GMFDs extracted in q N 0 of the same state change a little bit. It can be known that although speed decreases from of 826 r/min to 472 r/min, values of GMFDs extracted in q N 0 of the same state change a little bit by comparing Figs. 10 and 12.

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Fig. 13. Diagrams of GMFDs extracted from original signals of different lengths under condition of 826/min and 3.8 in-lbs.

It can be concluded that although load decreases from 5.7 in-lbs to 3.8 in-lbs, values of GMFDs extracted in q N 0 of the same state change a little bit by comparing Figs. 10 and 14. Figs. 10 and 12 suggest that values of Minkowski–Bouligand of the same state change a little bit when speed decreases from of 826 r/min to 472 r/min. Figs. 10 and 14 illustrate that there are overlaps between Minkowski–Bouligand dimension of broken tooth fault and that of missing tooth fault when load decreases from 5.7 in-lbs to 3.8 in-lbs. Figs. 10, 12 and 14 indicate that no matter if the signal length is long or not, values of Minkowski–Bouligand of the same state change a little bit under the same working condition. The conclusions can be drawn from above as follows. (1) GMFDs which work effectively are a little bit sensitive to speed, load and signal length. (2) Minkowski–Bouligand of each state are sensitive to load and are a little bit sensitive to speed and signal length. 7.6. Fault diagnosis based on KFCM GMFDs extracted in q N 0 from signals decomposed by LMD under three working conditions of 826 r/min and 5.7 in-lbs, 472 r/min and 5.7 in-lbs and 826 r/min and 3.8 in-lbs can characterize quantitatively and comprehensively nonlinear information of gear running states. Thus GMFDs extracted above working conditions of load and speed are adopted as samples, and each running state has 15 samples. The fault diagnosis result can be obtained by KFCM, and results under the three working conditions are listed in Tables 4, 5 and 6. In order to show effectiveness and superiority of the proposed method clearly, Table 7 lists all diagnosis results. In Table 4, all samples can be diagnosed correctly for the signals of 32,768 and 65,536 points, and 4 samples of normal state are falsely diagnosed as broken tooth fault ones for the signal of 8192 points, and 3 samples of missing tooth fault state are falsely diagnosed as normal state ones for the signal of 16,384 points.

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Fig. 14. Diagrams of GMFDs extracted from PFs of different lengths under condition of 826/min and 3.8 in-lbs.

In Tables 5 and 6, all samples can be diagnosed correctly for all signals of different lengths. Conclusion can be achieved from Tables 4 and 5 that a low speed of 472 r/min can make GMFDs of each state perform a little bit better under same working condition of load of 5.7 in-lbs. It is can be concluded from Tables 4 and 6 that GMFDs of each state, which are extracted under working condition of small load of 3.8 in-lbs and the same speed of 826 r/min, can work a little bit effectively.

Table 4 Clustering quality based on different PF under working condition of 826 r/min and 5.7 in-lbs. Signal

Index

Signal of 8192 points Signal of 16,384 points Signal of 32,768 points Signal of 65,536 points

P

E

0.9606 0.9732 0.9802 0.9815

0.0690 0.0537 0.0364 0.0326

Table 5 Clustering quality based on different PF under working condition of 472 r/min and 5.7 in-lbs. Signal Signal of 8192 points Signal of 16,384 points Signal of 32,768 points Signal of 65,536 points

Index

P

E

0.9816 0.9898 0.9944 0.9979

0.0370 0.0255 0.0127 0.0072

166

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Table 6 Clustering quality based on different PF under working condition of 826 r/min and 3.8 in-lbs. Signal

Index

Signal of 8192 points Signal of 16,384 points Signal of 32,768 points Signal of 65,536 points

P

E

0.9820 0.9829 0.9881 0.9979

0.0342 0.0335 0.0202 0.0070

Table 7 All diagnosis results. Working condition

Works or not

Signal processing

826 r/min and 5.7 in-lbs 472 r/min and 5.7 in-lbs 826 r/min and 3.8 in-lbs

With LMD

Without LMD

Yes Yes Yes

No No No

Thus conclusions can also be drawn from above, and they are listed as follows. (1) If GMFDs can be used as feature vectors, and diagnostic accuracy is a little bit higher with increment of signal length. (2) LMD can extract much feature information as much as possible and filter noises. (3) GMFDs extracted under working conditions of low speed and small load can make diagnostic accuracy a little bit better. 8. Conclusions Aiming at the fault diagnosis of gear, a fusion method of LMD and GMFDs is proposed. The normal, missing tooth and broken tooth fault signals of gear are tested and verified, and conclusions of this study are summarized as follows. (1) If GMFDs of normal state do not overlap with those of other state and their values are the maximum, the GMFDs of each state can be adopted as good feature vectors for diagnosis. (2) GMFDs which can work effectively are much sensitive to characteristic morphological features of a signal and a little bit sensitive to signal length, load and speed. (3) Noises can be filtered and feature information can be extracted as much as possible by LMD, which can facilitate gear fault diagnosis. (4) Minkowski–Bouligand of each state are sensitive to load and are a little bit sensitive to speed and signal length. (5) The selection method based on mutual information theory is proposed, by which the PF containing the richest morphological features can be effectively selected as data source. Acknowledgments Supported by National Natural Science Foundation of China (No. 51475405), National Key Basic Research Program of China (973 Program) (No.2014CB046405) and Hebei Province Natural Science Foundation of China (No. E2013203161). References [1] J. Cheng, K. Zhang, Y. Yang, An order tracking technique for the gear fault diagnosis using local mean decomposition method, Mech. Mach. Theory 55 (2012) 67–76. [2] D.J. Bordoloi, R. Tiwari, Optimum multi-fault classification of gears with integration of evolutionary and SVM algorithms, Mech. Mach. Theory 73 (2014) 49–60. [3] J. Rafiee, M.A. Rafiee, P.W. Tse, Application of mother wavelet functions for automatic gear and bearing fault diagnosis, Expert Syst. Appl. 37 (6) (2010) 4568–4579. [4] H. Zheng, Z. Li, X. Chen, Gear fault diagnosis based on continuous wavelet transform, Mech. Syst. Signal Process. 16 (2) (2002) 447–457. [5] J.S. Smith, The local mean decomposition and its application to EEG perception data, J. R. Soc. Interface 2 (5) (2005) 443–454. [6] B.J. Chen, Z.J. He, X.F. Chen, H.R. Cao, G.G. Cai, Y.Y. Zi, A demodulating approach based on local mean decomposition and its applications in mechanical fault diagnosis, Meas. Sci. Technol. 22 (5) (2011) 055704. [7] Y.X. Wang, Z.J. He, J.W. Xiang, Y.Y. Zi, Application of local mean decomposition to the surveillance and diagnostics of low-speed helical gearbox, Mech. Mach. Theory 47 (1) (2012) 62–73. [8] J.S. Cheng, Y. Yang, Y. Yang, A rotating machinery fault diagnosis method based on local mean decomposition, Digital Signal Processing 22 (2) (2012) 356–366. [9] Y.X. Wang, Z.J. He, Y.Y. Zi, A demodulation method based on improved local mean decomposition and its application in rub-impact fault diagnosis, Meas. Sci. Technol. 20 (2) (2009) 025704. [10] L.F. Deng, R.Z. Zhao, Fault feature extraction of a rotor system based on local mean decomposition and Teager energy kurtosis, J. Mech. Sci. Technol. 28 (4) (2014) 1161–1169. [11] Y.X. Wang, Z.J. He, Y.Y. Zi, A comparative study on the local mean decomposition and empirical mode decomposition and their applications to rotating machinery health diagnosis, J. Vib. Acoust. 132 (2) (2010) 0210101–02101010. [12] P. Maragos, A. Potamianos, Fractal dimensions of speech sounds: computation and application to automatic speech recognition, J. Acoust. Soc. Am. 105 (3) (1999) 1925–1932.

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