Application of the DC Offset Cancellation Method

applied sciences Article

Application of the DC Offset Cancellation Method and S Transform to Gearbox Fault Diagnosis Xinghui Zhang 1, *, Jianmin Zhao 1 , Rusmir Bajri´c 2 and Liangliang Wang 1 1 2

*

The management department of Mechanical Engineering College, Shijiazhuang 050003, China; [email protected] (J.Z.); [email protected] (L.W.) Public Enterprise Elektroprivreda BiH, Coal Mines Kreka, Tuzla 75000, Bosnia and Herzegovina; [email protected] or [email protected] Correspondence: [email protected]; Tel.: +86-186-0329-2514

Academic Editor: David He Received: 31 December 2016; Accepted: 9 February 2017; Published: 20 February 2017

Abstract: In this paper, the direct current (DC) offset cancellation and S transform-based diagnosis method is verified using three case studies. For DC offset cancellation, correlated kurtosis (CK) is used instead of the cross-correlation coefficient in order to determine the optimal iteration number. Compared to the cross-correlation coefficient, CK enhances the DC offset cancellation ability enormously because of its excellent periodic impulse signal detection ability. Here, it has been proven experimentally that it can effectively diagnose the implanted bearing fault. However, the proposed method is less effective in the case of simultaneously present bearing and gear faults, especially for extremely weak bearing faults. In this circumstance, the iteration number of DC offset cancellation is determined directly by the high-speed shaft gear mesh frequency order. For the planetary gearbox, the application of the proposed method differs from the fixed-axis gearbox, because of its complex structure. For those small fault frequency parts, such as planet gear and ring gear, the DC offset cancellation’s ability is less effective than for the fixed-axis gearbox. In these studies, the S transform is used to display the time-frequency characteristics of the DC offset cancellation processed results; the performances are evaluated, and the discussions are given. The fault information can be more easily observed in the time-frequency contour than the frequency domain. Keywords: planetary gearbox; fault diagnosis; predictive maintenance; deterministic component cancellation; S transform

1. Introduction To stay competitive, many companies all over the world try to go beyond the costly industry standard of preventative maintenance by implementing predictive maintenance procedures. For the industry related to oil exploitation, nuclear power and wind power, predictive maintenance procedures are the backbone in order to improve the operation and maintenance. Additionally, those industries for operation use drive trains that usually contain a gearbox as one of the key components. Vibration analysis is the most widely-used predictive maintenance procedure in gearbox predictive maintenance. In many mechanical systems, gearboxes are widely used for power transmission. Its condition is very important for the performance of the whole mechanical system. Therefore, many companies—and in the last two decades of academia also—have paid great attention to the gearbox condition monitoring. Vibration analysis can be applied very effectively to gearbox condition monitoring and predictive maintenance. In 2006, Jardine et al. [1] reviewed the machinery fault diagnosis and prognosis in the condition-based maintenance view. The authors considered fault diagnosis as an important part of condition-based maintenance. We know that the gearbox is composed of many parts, such as gears, shafts, bearings, etc. In addition, there are many types of gearbox. The most typical are the Appl. Sci. 2017, 7, 207; doi:10.3390/app7020207

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fixed-axis gearbox and planetary gearbox. In 2011, bearing fault diagnosis work was reviewed by Randall and Antoni [2]. Later, Ray and Upadhyay [3] reviewed the signal processing techniques used in bearing fault diagnosis. This focuses mainly on the recent developments of the fault diagnosis signal processing techniques. Then, they discussed the advantages and disadvantages of these methods used in bearing fault diagnosis. For time-frequency analysis, Feng et al. [4] reviewed more than 20 major time-frequency analysis methods used in fault diagnosis since 1990 and examined their advantages and disadvantages. As a major time-frequency analysis method, multiwavelet’s development in fault diagnosis has been reviewed by Sun et al. [5]. Finally, they discussed the existing problems and future research directions. As we know, the fault mechanism of the planetary gearbox is very different from fixed-axis gearbox. Therefore, many authors pay great attention to the fault diagnosis of planetary gearboxes. Lei et al. [6] reviewed the papers emerging in the planetary gearbox fault diagnosis domain and compared the fixed-axis and planetary gearbox fault diagnosis. Finally, they pointed to the potential research topics. Through these detailed review papers, readers will have a good knowledge of gearbox fault diagnosis. Whether for the gear or the bearing, fault will produce impulse signals with a specific period. Therefore, the main object for fault diagnosis is to detect these impulse signals. Therefore, many methods were developed to detect impulse signals. Spectral kurtosis was one such effective method used for impulse signal detection. From its proposal, it becomes an important and effective way to detect the impulse signal. Recently, Wang et al. [7] reviewed the development of spectral kurtosis and pointed out its future use in fault prognosis. Nevertheless, these periodic impulse signals (PIS) produced by fault are always overwhelmed by deterministic signals produced by the regular gear mesh and structural vibration. These deterministic signals are also called discrete frequency signals or the deterministic component. In order to detect the impulse signal more easily, it is necessary to suppress the deterministic signals. There are several methods developed to subtract the deterministic component from the original signal. The most dominant were: the autoregressive (AR) model method [8–15], time synchronous averaging [16–18], adaptive noise cancellation [19,20] and the cepstrum pre-whitening method [21–23]. These four methods all have been used in the separation of the deterministic signal and PIS signal successfully. Except for these methods, a new deterministic component cancellation method was proposed, recently [24]. It is based on an iterative calculation of the signal envelope when the DC offset is cancelled before envelope calculation. The only parameter that needs to be pre-determined is the iteration number for DC offset cancellation. In the paper, a threshold of cross-correlation coefficient interval of [0.9, 0.95] was selected to judge the proper number of envelope calculations. The selected correlation coefficient interval was efficient for the examples in the paper, but this threshold method is not a proper way to acquire the most powerful PIS. Sometimes, a greater iteration number will distort the waveform of PIS and lead to the misdiagnosis of faults. In order to resolve this problem, correlated kurtosis (CK) is proposed to be used in determining when to stop the iteration for envelope calculation [25]. This indicator can detect the PIS properly. However, CK is efficient only with strong enough PIS. When the PIS is very weak, CK is inefficient due to the inability to correctly detect the PIS. Therefore, in the case of very weak PIS, it is very difficult to determine the appropriate iteration number for envelope calculation. Time frequency analysis methods, such as short time Fourier transform, the Wigner–Ville distribution and wavelet transform, were widely used in rotating machinery fault diagnosis. However, they all have some demerits. Compared to these time frequency analysis methods, the S transform can greatly avoid the existing deficiencies of the above three methods [26]. Therefore, it can be used to display the time-frequency characteristics of the separation results of the deterministic component and the random component. In addition, fault features can be extracted from the resulting signal of the S transform to track the degradation of the component of interest. From the time-frequency contour produced by the S transform, one can distinguish the different faults more easily; or it can be used as the input for the deep learning methods to intelligent fault classification.

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The above discussions show the need to use CK to optimize the iteration number of the envelope calculation with strong enough PIS. This can suppress the deterministic component effectively and properly. The S transform advantages inspire us to diagnose faults of the gearbox component by separation of the results of the deterministic component and the random component. The method is validated using the experimental datasets collected from a lab AC motor faulty bearing; a lab fixed-axis gearbox faulty bearing and gear and a lab planetary gearbox faulty sun, planet and ring gear. Experimental case studies will benefit the application of this novel method and will also provide a detailed guide for engineers in the industrial application. In addition, validation has been conducted using both intentionally-implanted bearing and gear damage data and naturally-developed distributed gear and bearing wear data under constant motor speed and various loads. Based on the above-mentioned literature and analysis, the main contributions of this paper can be summarized as follows: (1) instead of the cross-correlation coefficient, correlated kurtosis was used as a criterion to estimate the optimal iteration number of DC offset cancellation, since CK has the superior ability to detect the PIS signal and is more robust to the noise interference; (2) when PIS is very weak compared with the deterministic signal or existing bearing fault and gear fault simultaneously, even CK cannot detect weak PIS produced by the bearing fault effectively; in this circumstance, the maximum order of high-speed shaft mesh frequency was used as the iteration number of DC offset cancellation; (3) because of several existing merits, the S transform was used to combine with DC offset cancellation to display the time-frequency characteristic of various gearbox faults; it is very efficient and intuitive to find faults through time-frequency contours; in addition, it could be used for deep learning-based fault classification; (4) through the planetary gearbox case study, it was found that DC offset cancellation is less effective for those small fault frequency signals, such as the planet gear, ring gear, carrier, etc. Compared to the DC offset cancellation used in [24], the advantage of this study is that it used CK instead of the cross-correlation coefficient to determine the optimal iteration number of DC offset cancellation. According to the CK’s value, it is more corrective to select the iteration number because it can detect PIS more accurately than the cross-correlation coefficient and is not easily interfered with by impulse-like noise. However, the disadvantage of this method is that it is less effective when the PIS is very weak. For the S transform, the advantages are that it can display the fault information more fruitfully and intuitively than the frequency domain. However, for some small fault frequencies, it is difficult to display them in the time-frequency contour because this needs a greater sampling length and will consume more computational resources, especially requiring a high ability graphic processing unit. The remainder part of the paper is organized as follows. In Section 2, a novel pre-whitening method and the S transform are briefly introduced. In Section 3, the experimental case study of an implanted bearing outer race fault is presented. In Section 4, the experimental case study of bearing and gear fault diagnosis in the gearbox is presented and explained in detail. In Section 5, the experimental case study of the planetary gearbox fault diagnosis using the proposed method is presented. In Section 6, the effectiveness and existing problem of the novel pre-whitening method are discussed. Finally, Section 7 concludes the work. 2. Novel Pre-Whitening Method and S Transform 2.1. Novel Pre-Whitening Method The vibration signal acquired from a gearbox is usually complex, containing many different signal components. Generally speaking, deterministic components (also called discrete frequency) represent signals produced by gear meshing, shaft bending, shaft misalignment, etc. However, random components, also called PIS, usually represent signals produced by gear or bearing faults. Traditionally, many scholars represent random components as signals of bearing faults. In this sense, when a gear has faults, such as a chipped tooth, a cracked tooth and wear of the tooth face, the meshing

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will change. Every time the faulty tooth meshes, it produces impulse signals. If the gear has severe faults, the transmission ratio will vary also. Impulse signals induced by the faulty gear are very different with the signals induced by the normal gear. The PIS produced by the faulty gear also has a random property, such as the faulty bearing, because the gear faults usually produce impulse vibration signals. Therefore, signals produced by the gear fault also belong to the random components. It is now well established that the impact produced by the local gear fault is non-stationary in nature, and conventional signal processing approaches are inappropriate; and local tooth damage (i.e., fatigue Appl. Sci. 2017, 7, 207 4 of 22 crack, pitting, etc.) produces sharp transients in the vibration signature, which can be classified as vibration signals. Therefore, signals produced by the gear also belong to thecollected random from the non-stationary, non-linear and non-Gaussian in nature [20]. In fault this content, signals components. It is nowas: well established that the impact produced by the local gear fault is nongearbox can be represented stationary in nature, and conventional signal processing approaches are inappropriate; and local x(t) = u(t) + f (t) (1) tooth damage (i.e., fatigue crack, pitting, etc.) produces sharp transients in the vibration signature, which can be classified as non-stationary, non-linear and non-Gaussian in nature [20]. In this content,

where u(t) denotes the deterministic components and f (t) denotes random components induced by signals collected from the gearbox can be represented as: the gear or bearing fault. This is under the assumption that the rotating speed is constant or can x(t) = u(t) + f(t) (1) be transformed to be constant in the angular domain by the assistance of the tachometer signal. where u(t) denotes deterministic as: components and f(t) denotes random components induced by The envelope signal can betherepresented the gear or bearing fault. This is under the assumption that the rotating speed is constant or can be h i transformed to be constant in the angular domain by the assistance ofˆ the tachometer signal. ˆ ˆ y t = x t + j x t = u t + f t + j u t + f t ( ) ( ) ( ) ( ) ( ) ( ) ( ) The envelope signal can be represented as:

(2)

Because the squared envelope + jxˆ ( t ) =SNR y ( thas u ( t ) +than f ( t ) +the j uˆ (envelope t + fˆ ( t )  alone, the squared ) = x (at )bigger (2)envelope of  )  the hybrid signal can be calculated as: Because the squared envelope has a bigger SNR than the envelope alone, the squared envelope of the hybrid signal can be calculated| yas: (t)|2 = x (t)2 + xˆ (t)2

(3)

y (2u t ) (=t)xf( (t )t)++xˆ (uˆt )(t)2 + fˆ(t)2 + 2uˆ (t) fˆ(t) |y(t)|2 = u(t)2 + f (t)2 +

(4)

2

2

2

(3)

Then, Ming et al. [24] derived forms of 2 2the analytic 2 2 the 2squared envelope and the squared (4) y ( t ) = u ( t ) + f ( t ) + 2u ( t ) f ( t ) + uˆ ( t ) + fˆ ( t ) + 2uˆ ( t ) fˆ ( t ) envelope spectrum. They found that the primary energy of the multi-component signal was shifted to the direct current (DC) thederived envelope signal.forms Both of thethedeterministic component and the random Then, Mingoffset et al. of [24] the analytic squared envelope and the squared envelope spectrum.inThey that the energy of the multi-component signal was shifted component are reserved thefound envelope ofprimary the multi-component signal when the DC offset has not to the direct current (DC) offset of the envelope signal. Both theofdeterministic component and the been cancelled. If the DC offset is cancelled, the cross-terms different mono-components would random component are reserved in the envelope of the multi-component signal when the DC offset dominate the envelope. However, the cross-terms of the deterministic component have one fewer has not been cancelled. If the DC offset is cancelled, the cross-terms of different mono-components harmonic would than the original ones. In However, addition,the allcross-terms harmonics of cross-terms between deterministic and dominate the envelope. of the deterministic component have one fewer harmonic than the original ones. In addition, all harmonics of cross-terms between random component shift to low band direction with different distances. That is to say, one DC offset deterministic random component shift to lowof band with different That is to elimination operationand can subtract one harmonic thedirection cross-terms of the distances. deterministic component say, one DC offset elimination operation can subtract one harmonic of the cross-terms of the from the envelope signal. If we repeat this process, the deterministic component would be suppressed deterministic component from the envelope signal. If we repeat this process, the deterministic efficiently.component According to this theory, they proposed a novel deterministic component cancellation would be suppressed efficiently. According to this theory, they proposed a novel deterministic component is cancellation method. The detail method. The detail procedure illustrated in Figure 1. procedure is illustrated in Figure 1.

Figure 1. Framework for the deterministic component cancellation method.

Figure 1. Framework for the deterministic component cancellation method.

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According to [24], the detailed process can be depicted as follows: (1)

(2)

Cancel the DC offset from the original multi-component signal x(t), and calculate its envelope denoted as Hk [x(t)]; k = 1, 2, . . . , Hk [x(t)] represent the k-th envelope obtained by Hilbert transform. Specifically, H◦ [x(t)] = x(t). Calculate the correlated kurtosis of the envelope signal using the following equation:  2 M ∑tN=1 ∏m =0 y ( t − mτ) Correlated kurtosis o f M − shi f t =CK M (τ ) =   2 M +1 ∑tN=1 y(t)

(3) (4)

(5)

y(t) is the input signal. τ is the interesting period of the fault. N is the number of samples of input signal y(t). If τ = 0 and M = 1, then it gives the traditional kurtosis and can be used to detect the specific PIS. For example, if the desired fault frequency is 50 Hz and the sampling frequency is 10,000 Hz, the value of τ will be 200 samples. Determine the optimal iteration number according to the CK value. A high CK value denotes the strong PIS. When the optimal iteration number is acquired, then the squared envelope spectrum of the resulting envelope signal can be acquired as follows:

|y(t)|2 =< Hk−1 [ x (t)], Hk [ x (t)] >   n o F |y(t)|2 = F < Hk−1 [ x (t)], Hk [ x (t)] >

(6) (7)

where k denotes the optimal iteration number of the proposed method. In the original paper, the iteration number is judged by following two steps as a substitute of Steps (2) and (3). First, calculate the cross-correlation coefficient of the adjacent envelopes/signals using the following equation:

µk =

2 < Hk−1 [ x (t)], Hk [ x (t)] >

< Hk−1 [ x (t)], Hk−1 [ x (t)] > + < Hk [ x (t)], Hk [ x (t)] >

(8)

where denotes the inner product operation. According to the Cauchy–Schwarz inequality, cross-correlation coefficient µk , k = 1, 2, 3, . . . , is in interval [0, 1]. When these two signals are everywhere equal, this indicator may be 1.0. Second, a hard threshold λ is set. If µk ≥ λ, the iteration is terminated. Through this process, the deterministic component can be adaptively subtracted by cancellation of the DC offset from the envelope. 2.2. S Transform The S transform combines the separate strengths of the STFT and wavelet transforms [27] and has provided an alternative approach to process the non-stationary signals generated by mechanical systems [28]. Suppose the short time Fourier transform (STFT) of signal y(t) is as below: STFT(τ, f ) =

w∞

y(t) g(τ − t)e− j2π f t dt

(9)

−∞

where τ and f denote the time of spectral localization and Fourier frequency, respectively, and g(t) denotes a window function.

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The S transform can be derived from (9) by replacing the window function g(t) with the Gaussian function, depicted as: t2 f 2 |f| (10) g(t) = √ e− 2 2π Then, the S transform is defined as: S(τ, f ) =

w∞

( τ − t )2 f 2 |f| y(t) √ e− 2 e− j2π f t dt 2π −∞

(11)

Actually, the S transform is a special case of STFT with the Gaussian window function. If the continuous wavelet transform (CWT) is: W (τ, d) =

w∞

y(t)ω (t − τ, d)dt

(12)

−∞

where d denotes the “width” of wavelet w(t,d), and thus, it controls the resolution; and w(t,d) denotes a scaled copy of the fundamental mother wavelet. Then, the S transform is a CWT with a specific mother wavelet multiplied by the phase factor:

where the mother wavelet is:

S(τ, f ) = e− j2π f t W (τ, d)

(13)

t2 f 2 |f| ω (t, f ) = √ e− 2 e− j2π f t 2π

(14)

Note that the factor d is the inverse of the frequency f. Compared with STFT and CWT, the S transform has the following merits: (1)

(2)

The window length used in STFT is a fixed value. On the contrary, the window length used in the S transform is a function of time and frequency. In other words, the window length is adaptive. In the time domain, the window will be wider for low frequencies and will be narrower for high frequencies. This will provide better localization in the frequency domain for low frequencies and provide better localization in the time domain for higher frequencies. These characteristics are very similar to the wavelet transform. The great difference of the S transform and the wavelet transform is the different presentation of the time-frequency characteristic. The S transform represents signal features in the time-frequency contour. However, the wavelet transform represents signal features in the time-scale contour. Because the S transform uses frequency as a variable, this allows it to have a direct connection and avoids the error in the frequency estimation.

The most important merit of the S transform is the preservation of both the amplitude and phase information. However, wavelet coefficients only contain amplitude information. Because of these merits, the S transform is used for mechanical fault diagnosis combined with the novel deterministic component cancellation method. 3. Novel Proposed Method Application for Implanted Bearing Fault Diagnosis The vibration fault dataset used in this case study was obtained from the Mechanical Failures Prevention Group (MFPT), assembled and prepared on behalf of MFPT by Eric Bechhoefer [29]. Only the outer race faults are considered for evaluation and discussion in this research. The test bearings, which support the motor shaft, are radial ball bearings produced by RBC NICE with the following parameters: roller diameter 0.235 inch, pitch diameter 1.245 inch, number of elements 8 and the contact angle of 0◦ . In this case study dataset with a 25-Hz input shaft rate, a 48,828-Hz sampling frequency, a 3-s sampling duration and a 25-lbs. load are used. If we define the four bearing fault frequencies

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Appl. Sci. 2017, 7, 207 contact angle of 0°.

7 of 22 In this case study dataset with a 25-Hz input shaft rate, a 48,828-Hz sampling contact angle of 0°. In this case study dataset with a 25-Hz input shaft rate, a 48,828-Hz sampling frequency, a 3-s sampling duration and a 25-lbs. load are used. If we define the four bearing fault frequency, a 3-s sampling duration and a 25-lbs. load are used. If we define the four bearing fault frequencies as ball pass frequency inner race (BPFI), ball pass frequency outer race (BPFO), ball spin frequencies as ball passinner frequency inner race ball passouter frequency outer race (BPFO), ball spin as ball pass frequency race (BPFI), ball (BPFI), pass frequency race (BPFO), ball spin frequency frequency (BSF) and fundamental train frequency (FTF), fault frequencies could be calculated frequency (BSF) and fundamental train frequency (FTF), fault frequencies could be calculated (BSF) and fundamental train frequency (FTF), fault frequencies could be calculated according to the according to the geometric parameters [30]. Four fault frequencies under the shaft rate of 25 Hz are: according parameters to the geometric parameters [30]. Four fault under of (81.12 25 Hz Hz), are: geometric [30]. Four fault frequencies underfrequencies the shaft rate of 25the Hzshaft are: rate BPFO BPFO (81.12 Hz), BPFI (118.88 Hz), BSF (63.86 Hz) and FTF (10.14 Hz). BPFO(118.88 (81.12 Hz), BSF BPFI(63.86 (118.88 Hz), and FTF (10.14 Hz). BPFI Hz) andBSF FTF(63.86 (10.14Hz) Hz). According to the framework in Figure 1, the DC offset is gradually cancelled. First, the process According to to the theframework frameworkin inFigure Figure1,1,the theDC DCoffset offsetisisgradually gradually cancelled. First, process According cancelled. First, thethe process is is repeated 100 times, and CK and cross-correlated coefficient μ are used to judge the termination of is repeated times, and cross-correlated coefficient μ are used to judge termination of repeated 100100 times, and CKCK andand cross-correlated coefficient µ are used to judge the the termination of the the iteration indicators. These two indicators, variation vs. iteration number, are illustrated in Figure 2. the iteration indicators. These indicators, variation vs. iteration number, are illustrated in Figure 2. iteration indicators. These twotwo indicators, variation vs. iteration number, are illustrated in Figure 2. The The value of μ increases, as can be seen, with the iteration number, but up to the 100th iteration. The value of μ increases, as can be seen, with the iteration number, but up to the 100th iteration. value of µ increases, as can be seen, with the iteration number, but up to the 100th iteration. However, However, it cannot attain the threshold defined in [24], which is [0.9, 0.95]. For the CK value, However, it cannot attain the threshold defined [24],0.95]. which [0.9, the CK value, it cannot attain the threshold defined in [24], which in is [0.9, Foristhe CK0.95]. value,For it increases at the it increases at the beginning up to a high value, and then, it decreases. After the 15th iteration, it increasesupattothe beginning up to a high value, and then, decreases. the 15thtoiteration, beginning a high value, and then, it decreases. After the it 15th iteration,After it continues increase. it continues to increase. Two local optimal values were noticed during the 100 iteration process. it continues to increase. optimal values were noticedprocess. during Iteration the 100 iteration process. Two local optimal valuesTwo werelocal noticed during the 100 iteration 5 is the first one Iteration 5 is the first one and the second one Iteration 85. Figure 3 illustrates the squared envelope Iteration 5 is theone firstIteration one and85. the Figure second3one Iteration 85.squared Figure 3envelope illustrates theitssquared envelope and the second illustrates the and spectrum of the and its spectrum of the DC offset cancellation after the fifth iteration. The squared envelope and its andoffset its spectrum of theafter DC offset cancellation the fifth envelope iteration. and The its squared envelope its DC cancellation the fifth iteration. after The squared spectrum of DCand offset spectrum of DC offset cancellation after the 100th iteration are shown in Figure 4. Figure 4 illustrates spectrum of after DC offset cancellation the 100th shown in Figurethat 4. Figure 4 illustrates cancellation the 100th iterationafter are shown in iteration Figure 4. are Figure 4 illustrates normal envelope that normal envelope signals are distorted by an excess of DC offset cancellation. Moreover, the PIS that normal envelope are of distorted bycancellation. an excess of Moreover, DC offset cancellation. the PIS signals are distorted bysignals an excess DC offset the PIS of theMoreover, squared envelope of the squared envelope signal after the fifth DC offset cancellation is stronger than the original signal. of the squared envelope signal after the fifth DC offset cancellation is stronger than the original signal. signal after the fifth DC offset cancellation is stronger than the original signal. Iteration 5 is the local Iteration 5 is the local optimum of the CK value. However, CK can detect the PIS signal when it is Iteration 5ofisthe theCK local optimum of theCK CKcan value. However, CK canwhen detect PIS signal is optimum value. However, detect the PIS signal it the is more intensewhen than itthe more intense than the other components. If the PIS are weak or not contained in the envelope signal, more components. intense than the other the PIS are weak orenvelope not contained inCK thewould envelope other If the PIScomponents. are weak or If not contained in the signal, notsignal, work, CK would not work, and its high value cannot denote the strong PIS. In this case, after the fifth CK would work, and its high denote strong PIS.iteration, In this case, after thesignal fifth and its high not value cannot denote the value strong cannot PIS. In this case,the after the fifth the envelope iteration, the envelope signal is corrupted by excess DC offset cancellation operation. Actually, iteration, thebyenvelope corrupted operation. by excessActually, DC offset operation. is corrupted excess DCsignal offset is cancellation thecancellation CK values after the fifthActually, iteration the CK values after the fifth iteration cannot denote the strength of PIS. the CK denote values the afterstrength the fifthofiteration cannot denote the strength of PIS. cannot PIS.

Figure 2. 2. (a) Cross-correlation coefficient variation vs. iteration number; (b) correlated kurtosis (CK) Figure 2. (a) Cross-correlation coefficient variation vs. iteration number; (b) correlated kurtosis (CK) value vs. value vs. iteration number. number. value vs. iteration number.

Figure 3. (a) Squared envelope signal after the fifth fifth DC offset offset cancellation; (b) (b) squared envelope envelope Figure (a) Squared Squared envelope envelope signal after Figure 3. 3. (a) signal after the the fifth DC DC offset cancellation; cancellation; (b) squared squared envelope spectrum after the fifth DC offset cancellation. spectrum spectrum after after the the fifth fifth DC DC offset offset cancellation. cancellation.

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Figure 4. Squared envelope signal after the 100th DC offset Figure 4. (a) (a) offset Figure 4. (a) Squared Squared envelope envelope signal signal after after the the 100th 100th DC DC Figure 4. after (a) Squared envelope signal after the 100th DC offset offset spectrum the 100th DC offset cancellation. spectrum after the 100th DC offset cancellation. spectrum after after the the 100th 100th DC spectrum DC offset offset cancellation. cancellation.

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cancellation; squared cancellation; (b) (b) squared envelope envelope cancellation; (b) squared cancellation; (b) squared envelope envelope

In In order order to to observe observe the the effect effect of of DC DC offset offset cancellation cancellation from from the the time-frequency time-frequency domain domain and and In order to the the effect and observe of DC offset cancellation fromis the time-frequency domain compare it with traditional envelope analysis, the S transform applied to the squared compare it with the traditional envelope analysis, the S transform is applied to the squared envelope envelope compare it with the traditional envelope analysis, the S transform is applied tosignal. the squaredresults envelope signal signal after after the the DC DC offset offset cancellation cancellation and and the the envelope envelope signal signal of of the the original original signal. The The results are are signal after the DC offset and the signal of The results are cancellation envelope the original signal. shown in Figures 5 and 6. From the local amplification of these two time-frequency contours, the shown in Figures 5 and 6. From the local amplification of these two time-frequency contours, the shown in Figures 5 and 6. From the local amplification of these two time-frequency contours, the obvious obvious BPFO BPFO can can be be seen seen in in these these figures. figures. However, However, the the time-frequency time-frequency contour contour of of the the envelope envelope of of obvious BPFO BPFO can canbe beseen seenininthese thesefigures. figures. However, the time-frequency contour of the envelope of However, the time-frequency contour of the envelope of the the the original original signal signal shown shown in in Figure Figure 5a 5a does does not not have have an an obvious obvious BPFO BPFO contour, contour, whereas whereas the the timetimethe original signal shown in Figure 5a does not have an obvious BPFO contour, whereas the timeoriginal signal shown in Figure 5a does not have an obvious BPFO contour, whereas the time-frequency frequency frequency contour contour of of the the squared squared envelope envelope signal signal after after the the fifth fifth DC DC offset offset cancellation cancellation has has an an obvious obvious frequency of the squared envelope signal the fifth DCcancellation offset cancellation has an obvious contour ofcontour the squared envelope signal after the after fifth DC offset has an obvious BPFO BPFO contour. This demonstrates that the DC offset cancellation suppresses the deterministic BPFO contour. This demonstrates that the DC offset cancellation suppresses the deterministic BPFO contour. This demonstrates that the DC offset cancellation suppresses the deterministic contour. This demonstrates that the DC offset cancellation suppresses the deterministic components components components and and enables enables PIS PIS clearly clearly enough. enough. components andclearly enablesenough. PIS clearly enough. and enables PIS

Figure 5. (a) Time-frequency contour of envelope signal the fault signal; contour of the envelope signal of theof original bearingbearing fault signal; local Figure 5. 5. (a) (a)Time-frequency Time-frequency contour of the the envelope signal of the original original bearing fault(b) signal; Figure 5.amplification (a) Time-frequency contour of the envelope signal of the original outer bearing fault signal; (b) local of (a) between 0 and 1000 Hz. BPFO, ball pass frequency race. amplification of (a) between 0 and 1000 Hz. 1000 BPFO, ball passball frequency outer race. (b) local amplification of (a) between 0 and Hz. BPFO, pass frequency outer race. (b) local amplification of (a) between 0 and 1000 Hz. BPFO, ball pass frequency outer race.

Figure 6. (a) Time-frequency contour of the square envelope signal after the fifth DC offset cancellation; Figure Figure 6. 6. (a) (a) Time-frequency Time-frequency contour contour of of the the square square envelope envelope signal signal after after the the fifth fifth DC DC offset offset (b) local amplification of (a) between 0 and 1000 Hz. Figure 6. (a) Time-frequency contour of the square envelope signal after the fifth DC offset cancellation; (b) local amplification of (a) between 0 and 1000 Hz. cancellation; (b) local amplification of (a) between 0 and 1000 Hz. cancellation; (b) local amplification of (a) between 0 and 1000 Hz.

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4. Novel Proposed Method Application for Naturally-Developed Fault Diagnosis 4. Novel Proposed Method Application for Naturally-Developed Fault Diagnosis For the novel method application, experimental signals are collected from a lab fixed-axis For the novel method application, experimental signals collected from afor labtesting fixed-axis gearbox gearbox test rig with naturally-developed bearing and gear are faults. The dataset high-speed test rig with naturally-developed bearing and gear faults. The dataset for testing high-speed (HS) (HS) shaft gear and bearings is from an end of life test case with an overall duration of 548 h. Detail shaft bearings is from an of life test caseThe with an overall duration 548 h. Detail end end ofgear life and test case information canend be found in [31]. intermediate speed (IS)ofshaft and its gears of life test case information can be found in [31]. The intermediate speed (IS) shaft and its gears and and bearings and the low speed (LS) shaft and its gear and bearings are from an ended implanted bearings the low speedfor (LS) and gear and bearings are from test test case. and Wear-in duration theshaft IS and LSits shaft-related components lastedan10ended h withimplanted different load case. Wear-in duration for the IS and LS shaft-related components lasted 10 h with different load and and speed combinations. speedFigure combinations. 7 illustrates a schematic of the experimental gearbox test rig. It includes a two-stage fixed7 illustrates a schematicasynchronous of the experimental gearbox testthe rig.gearbox It includes axis Figure gearbox, a 4-kW three-phase motor for driving and aa two-stage magnetic fixed-axis gearbox, a 4-kW three-phase asynchronous motor for driving the gearbox and a magnetic powder brake for loading. The motor speed can be adjusted for different values. The NI data powder brake for loading. motor can be adjusted different values. The NI data acquisition system consists ofThe four IEPE speed accelerometers (Dytranfor 3056B4—Dytran Instruments Inc., acquisition system consists of four IEPE accelerometers (Dytran 3056B4—Dytran Instruments Inc., Fraser, MN, USA), PXI-1031 mainframe, PXI-4472B data acquisition cards and LabVIEW software Fraser, MN, USA), PXI-1031 mainframe, PXI-4472B data acquisition cards and LabVIEW software (National Instrument Inc., Austin, TX, USA). In addition, a tachometer and torque sensor is installed (National Instrument Inc., Austin, USA). Inload addition, a tachometer and torque sensorofisthe installed in in the input shaft for acquiring theTX, speed and information. The internal structure gearbox the input shaft for acquiring the speed load information. structure of the gearbox is depicted in Figure 7. The gearbox hasand three shafts supportedThe by internal rolling element bearings. The LS is depicted in Figure 7. The gearbox has three shafts supported by rolling element bearings. The LS shaft gear has 81 teeth and meshes with the IS shaft gear with 18 teeth. The IS shaft gear with 64 teeth shaft gear has 81 teeth and meshes with the IS shaft gear with 18 teeth. The IS shaft gear with 64 teeth meshes with the HS shaft with 35 teeth. Therefore, the overall gear ratio of the gearbox equals 8.22:1. meshes with the HS datasets shaft with 35 teeth. Therefore, the overall ratio sampling of the gearbox equals a8.22:1. In this case, study with a 20-Hz input shaft rate, agear 20-kHz frequency, 12-s In this case, study datasets with a 20-Hz input shaft rate, a 20-kHz sampling frequency, a 12-s sampling sampling duration, a 199-Nm and 405-Nm loads are used. duration, a 199-Nm and 405-Nm loads are used.

Figure 7. A schematic of the experimental gearbox test rig. Figure 7. A schematic of the experimental gearbox test rig.

After the was dismounted in order to check the bearings’ condition. It was It found thetest, test,the thegearbox gearbox was dismounted in order to check the bearings’ condition. was that all bearings have faults at different levels. However, only the HS shaft and LS shaft bearings have found that all bearings have faults at different levels. However, only the HS and LS shaft obvious Both HSfaults. shaft Both bearings had the outer had race the fault shown Figure 8a,b,inand the right bearings faults. have obvious HS shaft bearings outer raceinfault shown Figure 8a,b, bearing addition had in theaddition inner race fault ball race fault fault shown in Figure 9a,b,shown respectively. Faults of and theinright bearing had theand inner and ball fault in Figure 9a,b, the IS shaft bearing were not obvious faults. The left LS shaft bearing had the inner race fault and ball respectively. Faults of the IS shaft bearing were not obvious faults. The left LS shaft bearing had the fault in Figure 10a,b, The10a,b, HS and IS shaft areThe supported byshaft SKF are 6205supported bearings innershown race fault and ball faultrespectively. shown in Figure respectively. HS and IS and the 6205 LS shaft by SKF 6208 fundamental faultsThe frequencies of SKF 6205frequencies under a 1-Hz by SKF bearings and thebearings. LS shaft The by SKF 6208 bearings. fundamental faults of rotating BPFO (3.585 speed Hz), BPFI (5.415(3.585 Hz), BSF Hz) and SKF 6205speed underare a 1-Hz rotating are BPFO Hz),(2.357 BPFI (5.415 Hz),FTF BSF(0.398 (2.357Hz). Hz) For and SKF FTF 6208, BPFO (5.423 (3.578 Hz), BSF (2.337 and Hz), FTF (0.398 Hz). Therefore, the (0.398they Hz).are For SKF (3.578 6208, Hz), they BPFI are BPFO Hz), BPFIHz) (5.423 BSF (2.337 Hz) and FTF actual fault frequencies can be acquired through these fundamental frequencies multiplied by the (0.398 Hz). Therefore, the actual fault frequencies can be acquired through these fundamental rotating frequency. The by bearing types and the characteristic frequencies 20-Hz input shaft rate are frequencies multiplied the rotating frequency. The bearing types andfor thea characteristic frequencies concerned summarized in Table 1. for a 20-Hzand input shaft rate are concerned and summarized in Table 1.

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(b) (b) (b)

(a) (a) (a)

Figure 8. High High speed (HS) (HS) shaft bearing outer race faults: (a) (a) left bearing; bearing; (b) right bearing. Figure Figure 8. 8. High speed speed (HS) shaft shaft bearing bearing outer outer race race faults: faults: (a) left left bearing; (b) (b) right right bearing. bearing. Figure 8. High speed (HS) shaft bearing outer race faults: (a) left bearing; (b) right bearing.

(a) (a) (a)

(b) (b) (b)

Figure Figure 9. 9. HS HS right right bearing bearing faults: faults: (a) (a) inner inner race race fault; fault; (b) (b) ball ball fault. fault. Figure 9. HS right bearing faults: (a) inner race fault; (b) ball fault.

(a) (a) (a)

(b) (b) (b)

Figure 10. speed (LS) left bearing inner race fault; (b) fault. Figure 10. 10. Low Low left bearing bearing faults: faults: (a) (a) fault; (b) (b) ball ball fault. fault. Figure Low speed speed (LS) (LS) left (a) inner inner race race fault; Figure 10. Low speed (LS) left bearing faults: faults: (a) inner race fault; (b) ball ball fault. Table 1. frequencies bearings BPFO: pass outer BPFI: Table 1. 1. Characteristic Characteristic frequencies frequencies of of bearings bearings (Hz). (Hz). BPFO: BPFO: ball ball pass pass frequency frequency outer outer race; race; BPFI: BPFI: ball ball Table Table 1. Characteristic Characteristic frequencies of of bearings (Hz). (Hz). BPFO: ball ball pass frequency frequency outer race; race; BPFI: ball ball pass frequency inner race; BSF: ball spin frequency; FTF: fundamental train frequency. pass frequency inner race; BSF: ball spin frequency; FTF: fundamental train frequency. pass pass frequency frequency inner inner race; race; BSF: BSF: ball ball spin spin frequency; frequency; FTF: FTF: fundamental fundamental train train frequency. frequency. Bearing Bearing Type Type Bearing Type Bearing Type 6205 6205 6205 6208 6205 6208 6208 6208

BPFO BPFO BPFO BPFO 71.70 71.70 71.70 8.70 71.70 8.70 8.70 8.70

BPFI BPFI BPFI BPFI 108.30 108.30 108.30 13.19 108.30 13.19 13.19 13.19

BSF BSF BSF BSF 47.17 47.17 47.17 5.68 47.17 5.68 5.68 5.68

FTF FTF FTF FTF 7.96 7.96 7.96 0.96 7.96 0.96 0.96 0.96

It It is is very very difficult difficult to to diagnose diagnose the the bearing bearing fault fault of of the the LS LS shaft shaft because because of of the the low low speed speed and and It is very difficult to diagnose the bearing fault of the LS shaft because of the low speed and requirement of special techniques. In this case, only HS bearing faults under loads of 199 Nm and requirement ofdifficult special to techniques. In this case, only HS bearing faults under of loads of 199 Nm and It is very diagnose the bearing fault of the LS shaft because the low speed requirement of special techniques. In this case, only HS bearing faults under loads of 199 Nm and 405 Nm Nm are are considered. considered. Except the the bearing bearing faults, theHS HS shaft shaft gear gear with 35 teeth teeth had a199 slight tooth 405 Except faults, the HS with 35 had slight requirement of special techniques. In this case, only faults under loads ofa Nmtooth and 405 Nm are considered. Except the bearing faults, the HSbearing shaft gear with 35 teeth had a slight tooth wear fault. The IS shaft gear with 64 teeth had a chipped tooth fault, and the IS shaft gear with wearNm fault. The IS shaftExcept gear with 64 teethfaults, had athe chipped tooth fault, and the IS shaft gear tooth with 405 are considered. the bearing HS shaft gear with 35 teeth had a slight wear fault. The IS shaft gear with 64 teeth had a chipped tooth fault, and the IS shaft gear with 18 had aa severe wear fault shown Figure 11. aa gear with fixed 18 teeth teeth had severe tooth wear64 fault shown in Figuretooth 11. For For gear transmission with fixed axes, wear fault. The IS shafttooth gear with teeth had ain chipped fault, andtransmission the IS shaft gear with 18 axes, teeth 18 teeth had a severe tooth wear fault shown in Figure 11. For a gear transmission with fixed axes, the gear meshing frequency (GMF) can be calculated by the following formula: GMF = N × Z, where the gear meshing can calculated the formula: N × Z, where had a severe toothfrequency wear fault(GMF) shown in be Figure 11. Forby gearfollowing transmission withGMF fixed= the gear the gear meshing frequency (GMF) can be calculated bya the following formula: GMF =axes, N × Z, where N N means means the the rotating rotating speed speed of of the the test test gear gear and and Z Z represents represents the the number number of of the the test test teeth. teeth. According According N means the rotating speed of the test gear and Z represents the number of the test teeth. According

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meshing frequency (GMF) can be calculated by the following formula: GMF = N × Z, where N means of the test gear and Z represents the number of the test teeth. According to this, 11 ofthe 22 GMF of the HS and IS shaft gears is determined as 700 Hz, and the GMF of the IS and LS shaft gears is to this, the GMF of theHz. HS and IS shaft gears is determined as 700 Hz, and the GMF of the IS and LS determined as 196.93 shaft gears is determined as 196.93 Hz. the rotating Appl. Sci. 2017, speed 7, 207

(a)

(b)

Figure Figure 11. 11. (a) (a) HS HS shaft shaft gear gear tooth tooth wear; wear; (b) (b) intermediate intermediate speed speed (IS) (IS) shaft shaft gears, gears, tooth tooth chip chip and and wear. wear.

First, in order to ensure stable signal, the data acquisition system is adjusted to use the data First, in order to ensure stable signal, the data acquisition system is adjusted to use the data sample rate based on the tachometer signal. The adjusted sampling rate is a synchronous sampling sample rate based on the tachometer signal. The adjusted sampling rate is a synchronous sampling rate rate and donates the resampled vibration signal. During the experiment, the load of 199 Nm was and donates the resampled vibration signal. During the experiment, the load of 199 Nm was applied applied to the output shaft (LS shaft), and the resampled signal is used. An accelerometer was to the output shaft (LS shaft), and the resampled signal is used. An accelerometer was mounted on mounted on the top of the gearbox casing (as shown in Figure 7), and the vibration signals were the top of the gearbox casing (as shown in Figure 7), and the vibration signals were collected from collected from the accelerometer at position S2. Figure 12a illustrates the correlation coefficient the accelerometer at position S2. Figure 12a illustrates the correlation coefficient variation vs. the variation vs. the iteration number of DC offset cancellation, while Figure 12b illustrates the CK value iteration number of DC offset cancellation, while Figure 12b illustrates the CK value variation vs. variation vs. the iteration number of DC offset cancellation. Figure 12c illustrates the squared the iteration number of DC offset cancellation. 12c illustrates the squared envelopeand signal envelope signal after the 100th iteration. It can beFigure seen that both the correlation coefficient CK after the 100th iteration. It can be seen that both the correlation coefficient and CK cannot be used cannot be used to determine the optimal iteration number for DC offset cancellation in the multi-fault to determine the optimal DC offset cancellation multi-fault mode case. mode case. Especially for iteration the weaknumber PIS, CKfor will not work. However, in as the mentioned in Section 2.1, Especially forcancellation the weak PIS, CK willcan notsubtract work. However, as mentioned in Section 2.1, one one DC offset operation one harmonic deterministic component. InDC thisoffset case, cancellation operation can subtract deterministic component. this case, HS gear HS shaft gear mesh frequency andone itsharmonic harmonics dominate the frequencyInspectrum, as shaft shown in mesh frequency and its harmonics dominate the frequency spectrum, as shown in Figure 13. It can Figure 13. It can be seen that there are mainly five harmonics of the gear mesh frequency. Therefore, be seen that there are mainlyDC five harmonics of thecan gear mesh frequency. Therefore, theoretically, theoretically, only five-times offset cancellation suppress the deterministic component of the only five-times DC offset cancellation can suppress the deterministic component of the original signal. original signal. Therefore, we conduct five iteration DC offset cancellation for the time synchronous Therefore, conduct five iteration DCand offset cancellation forcontour the time synchronous signal. The we squared envelope spectrum its time-frequency are illustrated insignal. FigureThe 14. squared envelope spectrum and its time-frequency contour are illustrated in Figure 14. After checking After checking the squared envelope spectrum carefully, it is very hard to find the bearing fault the squared envelope carefully, it is very hard to find the bearingspectrum fault frequencies. Instead frequencies. Instead ofspectrum the bearing fault frequencies, the squared envelope is dominated by of the fault rate frequencies, the squared envelope spectrum is dominated by the HSHowever, and IS shaft the HSbearing and IS shaft and its harmonics because of the gear chip fault and wear fault. in ratetime-frequency and its harmonics because of the gear chip fault and wear fault. However, theBSF time-frequency the contour of the squared envelope signal, weak BPFO, BPFI in and information contour of the squared envelope weak BPFI and BSFthe information can be seen. the can be seen. On the contrary, nosignal, bearing faultBPFO, information from envelope spectrum andOn timecontrary, no bearing fault information from the envelope spectrum and time-frequency contour of the frequency contour of the original signal shown in Figure 15 can be seen. Similarly, the envelope original signal shown in Figure 15 can be seen. Similarly, the envelope spectrum of the original signal spectrum of the original signal is also dominated by the HS and IS shaft rate and its harmonics. is also dominated by the HS and IS shaft rate and its harmonics.

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Figure 12. (a) coefficient vs. number;(b) (b)CK CKvalue value vs. iteration number; Figure 12. coefficient iteration CK value Figure 12. Correlation (a) Correlation coefficientvs. vs. iteration iteration number; vs.vs. iteration number; (c) Squared Squared envelope signal after the 100th DC offset cancellation. (c) envelope signal after the 100th (c) Squared envelope signal after the 100thDC DCoffset offset cancellation. cancellation.

Figure 13. Frequency spectrum of the time synchronous signal; GMF, gear meshing frequency. Figure 13. Frequency spectrum of the time synchronous signal; GMF, gear meshing frequency. Figure 13. Frequency spectrum of the time synchronous signal; GMF, gear meshing frequency.

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Figure 14. (a) Squared envelope spectrum after the fifth DC offset cancellation; (b) time-frequency

Figure 14. (a) envelope the fifth fifthDC DCoffset offsetcancellation; cancellation; time-frequency Figure 14. Squared (a) Squared envelopespectrum spectrum after after the (b) (b) time-frequency contour of the squared envelope signal after the fifth DC offset cancellation. contour of squared the squared envelope signalafter afterthe the fifth fifth DC contour of the envelope signal DCoffset offsetcancellation. cancellation. Figure 14. (a) Squared envelope spectrum after the fifth DC offset cancellation; (b) time-frequency contour of the squared envelope signal after the fifth DC offset cancellation.

Figure 15. (a) Envelope spectrum of the original signal; (b) time-frequency contour of the original signal. Figure 15. (a) Envelope spectrum of the original signal; (b) time-frequency contour of the original signal.

Figure 15. (a) Envelope spectrum of the original signal; (b) time-frequency contour of the original signal. Second, in order to ensure a stable signal, the data acquisition system is adjusted to use the data Second, in order to ensure a stable signal, the data acquisition system is adjusted to use the data sample rate15. based on the spectrum tachometer signal. The adjusted sampling rate is a synchronous sampling Figure (a) Envelope of the original signal; (b) acquisition time-frequency contour of the original to signal. Second, in order to a stable signal, the data system adjusted use the data sample rate based on ensure the tachometer signal. The adjusted sampling rate is a is synchronous sampling rate and donates the resampled vibration signal. During the experiment, the load of 405 Nm was rate and donates the resampled vibration signal. During the experiment, the load of 405 Nm was sample rate based on the tachometer signal. The adjusted sampling rate is a synchronous sampling applied to theinoutput (LS shaft), and resampled signal is used.system An accelerometer was Second, order shaft to ensure a stable signal, the data acquisition is adjusted to usemounted the data applied to the output shaft (LS shaft), and resampled signal is used. An accelerometer was mounted rate and donates the resampled vibration signal. During thevibration experiment, the load of sampling 405from Nm was on the top of based the gearbox (as shown in Figure 7), andsampling the were collected sample rate on thecasing tachometer signal. The adjusted ratesignals is a synchronous on the top of the gearbox casing (as shown in Figure 7),signal and the vibration signals were collected from applied to the output shaft (LS shaft), and resampled is used. An accelerometer was mounted the position S2. Figure 16 illustrates the frequency spectrumthe of the time rateaccelerometer and donates at the resampled vibration signal. During the experiment, load of synchronous 405 Nm was the accelerometer at position S2. Figure 16 illustrates the frequency spectrum of the time synchronous on the top with oftothe gearbox casing (as shown Figure andTherefore, vibration signals were collected signal the dominant six harmonics ofin the HS-IS 7), GMF. the DC offset cancellation applied the output shaft (LS shaft), and resampled signal isthe used. Anfor accelerometer was mounted from signal with the dominant six harmonics of the HS-IS GMF. Therefore, for the DC offset cancellation operation, wethe can repeatcasing six Figure times to16delete the deterministic component. the synchronous squared on the top of gearbox (as shown in Figure 7), the vibration signalsFinally, from the accelerometer at position S2. illustrates theand frequency spectrum ofwere the collected time operation, we can repeat six times to delete the deterministic component. Finally, the squared envelope spectrum and relevant time-frequency contour can be acquired as shown in Figure 17. the accelerometer at position S2. Figure 16 illustrates the frequency spectrum of the time synchronous signal with the dominant harmonics of the HS-IS GMF.can Therefore, foras theshown DC offset cancellation envelope spectrum andsix relevant time-frequency contour be acquired in Figure 17. Similarly, the squared envelope spectrum is dominated by the HS and IS shaft rate and its harmonics. signal with the dominant six harmonics of the HS-IS GMF. Therefore, for the DC offset cancellation operation, we can repeat six times to delete the deterministic Finally, the its squared envelope Similarly, the squared envelope spectrum is dominated by thecomponent. HS and IS shaft rate and harmonics. Information about bearing cannot be found in the squared envelope spectrum. operation, we canthe repeat six fault timesfrequencies to delete can the deterministic component. Finally, the squared the spectrum and relevant time-frequency contour be acquired as shown in Figure 17. Similarly, Information about the bearing fault frequencies cannot be found in the squared envelope spectrum. However, weak BPFIand andrelevant BSF cantime-frequency be found in the time-frequency contour as byshown the S transform. In envelope spectrum contour can be acquired in Figure 17. However, weak BPFI andisBSF can be found in HS the and time-frequency contour byharmonics. the S transform. In squared envelope spectrum dominated by the IS shaft rate and its Information order to compare with the traditional analysis, the and envelope the time Similarly, the squared envelope spectrum envelope is dominated by the HS IS shaftspectrum rate and itsofharmonics. to compare with the traditional the envelope the timeweak aboutorder the bearing fault frequencies cannot beenvelope found inanalysis, the squared envelope spectrum spectrum.ofHowever, synchronous signalthe is given in Figure 18a. The time-frequency contour the envelope signal of the Information about bearing fault frequencies cannot be found in the of squared envelope spectrum. synchronous signalfound is given in Figure 18a. The time-frequency contour of the envelope signaltoofcompare the BPFItime andsynchronous BSF can the time-frequency thethat Scontour transform. signal isin illustrated Figurein18b. Ittime-frequency can beby seen it is very to find any However, weakbeBPFI and BSF can beinfound thecontour by difficult theInS order transform. In time synchronous signal is illustrated in Figure 18b. It can be seen that it is very difficult to find any with bearing the traditional envelope analysis, envelope spectrum of the time synchronous is given information fromthe this time-frequency contour. order tofault-related compare with the traditional envelope analysis, the envelope spectrum ofsignal the time bearing fault-related information from this time-frequency contour. in Figure 18a. The time-frequency contour of the envelope signal synchronous signal synchronous signal is given in Figure 18a. The time-frequency contourofofthe the time envelope signal of the time synchronous signal is illustrated in Figure 18b. It can be seen that it is very difficult to find any is illustrated in Figure 18b. It can be seen that it is very difficult to find any bearing fault-related bearing fault-related information from this time-frequency contour. information from this time-frequency contour.

Figure 16. Frequency spectrum of the time synchronous signal Figure 16. Frequency spectrum of the time synchronous signal

Figure Frequencyspectrum spectrum of of the signal Figure 16.16. Frequency the time timesynchronous synchronous signal.

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Figure Figure 17. (a) (a) Squared Squared envelope envelope spectrum spectrum after after the the sixth sixth DC offset cancellation; cancellation; (b) (b) time-frequency time-frequency contour of the squared envelope signal after the sixth DC offset cancellation contour of the squared envelope signal after the sixth DC offset cancellation.

Figure Figure 18. 18. (a) (a) Envelope Envelope spectrum spectrum of of the the time time synchronous synchronous signal; signal; (b) (b) time-frequency time-frequency contour contour of of the the envelope signal of the original signal. envelope signal of the original signal.

5. Novel Proposed Method Application for Planetary Gearbox Fault Diagnosis 5. Novel Proposed Method Application for Planetary Gearbox Fault Diagnosis In In this this section, section, for for the the novel novel method method application, application, experimental experimental signals signals are are collected collected from from aa lab lab planetary planetary gearbox gearbox test test rig rig with with implanted implanted wear wear tooth tooth faults faults on on the the sun, sun, planet planet and and ring ring gear. gear. Figure Figure 19 19 illustrates schematicof of experimental planetary gearbox test rig. It includes a single-stage illustrates aaschematic thethe experimental planetary gearbox test rig. It includes a single-stage planetary planetary gearbox, a 4-kW three-phase asynchronous motor for driving the gearbox and a magnetic gearbox, a 4-kW three-phase asynchronous motor for driving the gearbox and a magnetic powder powder brake for loading. The motor speed can be adjusted for different values. The load can be brake for loading. The motor speed can be adjusted for different values. The load can be adjusted adjusted by a brake controller through the current of the magnetic powder brake connected to the by a brake controller through the current of the magnetic powder brake connected to the output output shaft (carrier). The NI data acquisition system consists of four IEPE accelerometers (Dytran shaft (carrier). The NI data acquisition system consists of four IEPE accelerometers (Dytran 3056B4, 3056B4, Dytran Instruments Inc., Fraser, MN, USA), PXI-1031 mainframe, PXI-4472B data acquisition Dytran Instruments Inc., Fraser, MN, USA), PXI-1031 mainframe, PXI-4472B data acquisition cards and cards and LabVIEW software (National Instrument Inc., Austin, TX, USA). In addition, a tachometer LabVIEW software (National Instrument Inc., Austin, TX, USA). In addition, a tachometer and torque and torque sensor is installed in the input shaft for acquiring the speed and load information. sensor is installed in the input shaft for acquiring the speed and load information. The single-stage The single-stage planetary gearbox consists of the sun gear, planet gear and fixed ring gear. The sun planetary gearbox consists of the sun gear, planet gear and fixed ring gear. The sun gear is connected gear is connected to the input shaft and rotates around its own center. All planet gears mesh to the input shaft and rotates around its own center. All planet gears mesh simultaneously with simultaneously with the sun gear and ring gear. These planet gears not only rotate around their own the sun gear and ring gear. These planet gears not only rotate around their own centers, but also centers, but also revolve around the center of the sun gear, and vibration signals picked up by a fixed revolve around the center of the sun gear, and vibration signals picked up by a fixed sensor attached sensor attached to the planetary gearbox housing differ significantly from that of fixed-axis/parallel to the planetary gearbox housing differ significantly from that of fixed-axis/parallel gear systems [32]. gear systems [32]. Compared to the fixed-axis gearbox test rig in Section 4, all of the settings are the Compared to the fixed-axis gearbox test rig in Section 4, all of the settings are the same, except the same, except the gearbox type. Single-stage planetary gearbox gear parameters are listed in Table 2. gearbox type. Single-stage planetary gearbox gear parameters are listed in Table 2. Table 2. Gear parameters of the single-stage planetary gearbox. Table 2. Gear parameters of the single-stage planetary gearbox.

Gear Gear number Teeth Teeth number

Sun Sun 13 13

Planet Planet 64 64

Ring Ring 146 146

Planet Number Planet 3 Number 3

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Figure 19. 19. A A schematic schematic of of the the experimental experimental planetary planetary gearbox gearbox test test rig. rig. Figure

Wear gear faults of the planetary gearbox were implanted in one tooth of the ring gear, planet gear faults of the planetary were in one tooth of the ring 20 gear, planet gear Wear and sun gear, respectively, in ordergearbox to validate theimplanted proposed analysis method. Figure illustrates gear and sun gear, respectively, in order to validate the proposed analysis method. Figure 20 the implanted gear faults. illustrates the implanted gear faults.

(a)

(b)

(c)

Figure 20. Implanted Implanted wear fault: (a) (a) ring ring gear; gear; (b) (b) planet planet gear; (c) sun gear.

Tooth wear faults faults belong belong to to localized localized fault, fault, and Tooth wear and their their fault fault frequencies frequencies can can be be calculated calculated through through the following equations [33]. The meshing frequency for the stationary ring gear f mesh = fcarrier × Zring the following equations [33]. The meshing frequency for the stationary ring gear f mesh = f carrier × equals the product of theofplanet carrier rotating frequency fcarrier fand the number of ring gear teeth Zring equals the product the planet carrier rotating frequency carrier and the number of ring gear Z ring. Relative rotating frequency with respect to the planet carrier is frelative = fmesh/Z, where Z is the teeth Zring . Relative rotating frequency with respect to the planet carrier is f relative = f mesh /Z, where total number teeth ofofthe gearofofthe interest. The characteristic frequency offrequency faulty planet gear isplanet fplanet1 Z is the total of number teeth gear of interest. The characteristic of faulty =gear fmeshis /Zfplanet, where Z planet is the number of planet gear teeth. If the damage exists on both sides of the planet1 = f mesh /Zplanet , where Zplanet is the number of planet gear teeth. If the damage exists gear tooth, theof damaged tooth also planet meshesgear withtooth the other on both sides the gearplanet tooth,gear the damaged also mating meshesgear. with The the characteristic other mating frequency of the faulty planet gear of becomes fplanet2 = 2 × gear fmesh/Z planet. The characteristic frequency of the gear. The characteristic frequency the faulty planet becomes f planet2 = 2 × f mesh /Zplanet . The faulty sun gear equals the rotating frequency of the sun rotating gear multiplied by of thethe number of characteristic frequency of relative the faulty sun gear equals the relative frequency sun gear planet gearsby fsunthe = Nnumber planet × fmesh/Zsun, where Zsun is the number of sun gear teeth and Nplanet is the number multiplied of planet gears f sun = Nplanet × f mesh /Zsun , where Zsun is the number of of planet gears. The characteristic frequency the faulty is fring = Nplanet × fmesh/Zring sun gear teeth and N is the number of of planet gears.ring Thegear characteristic frequency of. Similarly, the faulty planet for the distributed damage case, the characteristic frequency of the faulty ring gear is f ring = fmesh/Zring. ring gear is f =N ×f /Z . Similarly, for the distributed damage case, the characteristic ring

planet

mesh

ring

frequency of the faulty ring gear is f ring = f mesh /Zring . 5.1. Sun Gear Fault Experiment 5.1. Sun Fault Experiment For Gear this experiment, we used a sun gear wear fault mode to test the proposed method while the remaining gears were all in good Similar to the case studymethod in Section 4, For this experiment, we used acondition, sun gear undamaged. wear fault mode to test the proposed while in order to ensure a stable signal, the data acquisition system is adjusted to use the data sample rate the remaining gears were all in good condition, undamaged. Similar to the case study in Section 4, based ontothe tachometer Thedata adjusted sampling rateisisadjusted a synchronous sampling rate rate and in order ensure a stable signal. signal, the acquisition system to use the data sample donates the resampled vibration signal. During the experiment, the mean rotating frequency of the based on the tachometer signal. The adjusted sampling rate is a synchronous sampling rate and input shaft connecting the sun gear of the planetary gearbox was 20.0889 Hz, while a load of 405 Nm donates the resampled vibration signal. During the experiment, the mean rotating frequency of the was applied to the output shaft connecting the planet carrier. An accelerometer was mounted on the top of the gearbox casing (as shown in Figure 19), and the vibration signals were collected from the

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accelerometerat atposition positionS3. S3.According Accordingto tothe theplanetary planetarygearbox gearboxconfiguration configurationand andits itsrunning runningspeed, speed, accelerometer the characteristic characteristic frequencies frequenciesare are calculated calculated and andlisted listedin in Table Table3. 3. the input shaft connecting the sun gear of the planetary gearbox was 20.0889 Hz, while a load of 405 Nm was applied to the output shaft connecting the planetfrequencies carrier. An(Hz). accelerometer was mounted on the Table3. 3.Characteristic Characteristic Table frequencies (Hz). top of the gearbox casing (as shown in Figure 19), and the vibration signals were collected from the RotatingFrequency Frequency LocalDamage Damage Rotating accelerometerMeshing at position S3. According to the planetary gearbox Local configuration and its running speed, Meshing Frequency Frequency Sun Carrier Planet Sun Ring Sun Carrier Planet Sun Ring the characteristic frequencies are calculated and listed in Table 3. 239.8183 239.8183

20.0889 20.0889

1.6425 1.6425

3.7472 7.4944 7.4944 55.3427 55.3427 4.9278 4.9278 3.7472

Table 3. Characteristic frequencies (Hz).

According to to the the framework framework of of the the proposed proposed method, method, the the squared squared envelope envelope signal signal after after DC DC According offset cancellation is acquired. For this case, the CK value shows that the first envelope has the biggest offset cancellation is acquired. For this case, the CK value shows that the first envelope has the biggest Rotating Frequency Local Damage Meshing PIS, as as shown shown inFrequency Figure 21. 21. The The squared squared envelope envelope spectrum spectrum after after the the first first DC DC offset offset cancellation cancellation and and PIS, in Figure Sun Carrier Planet Sun Ring theenvelope envelopespectrum spectrumof ofonly onlythe thetime timesynchronous synchronoussignal signalare areillustrated illustratedin inFigure Figure22. 22.Compared Comparedto to the 239.8183 20.0889 1.6425 3.7472 7.4944 55.3427 4.9278 the envelope spectrum of the original signal, the amplitudes of the sun gear fault frequency and its the envelope spectrum of the original signal, the amplitudes of the sun gear fault frequency and its harmonics are are much much higher higher compared compared to to the the envelope envelope spectrum spectrum of of the the original original signal. signal. However, However, itit harmonics According to the framework of the proposed method, thesun squared envelope signalare after DChigher offset canalso also benoticed noticed that thenoise noiseamplitudes amplitudes unrelated tothe the sun gearfault faultfrequency frequency are also higher can be that the unrelated to gear also cancellation is acquired. For this case, the CK value shows that the first envelope has the biggest PIS, compared to the envelope spectrum of the original signal. compared to the envelope spectrum of the original signal. as shown in Figure 21. The squared envelope spectrum after the first DC offset cancellation and The time-frequency time-frequency contour contour of of the the squared squared envelope envelope signal signal using using the the SS transform transform isis shown shownthe in The in envelope spectrum of only the time synchronous signal are illustrated in Figure 22. Compared to Figure 23. In contrast, the time-frequency contour of the original envelope signal using the Figure 23. In contrast, the time-frequency contour of the original envelope signal using the envelope of the original signal, the thelocal sun amplifications, gear fault frequency and its Stransform transform shownin inFigure Figure 24.From From thesetwo twoamplitudes figuresand andof their local amplifications, thesun sun gear Sthe isisspectrum shown 24. these figures their the gear harmonics are much higher compared to the envelope spectrum ofdomain the original signal. However, it the can fault frequency frequency in the the frequency domain and the time-frequency time-frequency domain can be be seen. seen. However, the fault in frequency domain and the can However, also be noticed that the noise amplitudes unrelated to the sun gear fault frequency are also higher time-frequency contour of the squared envelope signal has less useful sun gear fault information than time-frequency contour of the squared envelope signal has less useful sun gear fault information than compared to the envelope of the envelope original the time-frequency time-frequency contourspectrum of the theoriginal original envelopesignal. signal. the contour of signal.

iterationnumber. Figure 21. 21. CK CKvalue value variation variation vs. vs. iteration number. Figure

Figure 22.(a) (a) Squaredenvelope envelope spectrumafter after thefirst first DCoffset offset cancellation;(b) (b) envelopespectrum spectrum Figure Figure 22. 22. (a)Squared Squared envelopespectrum spectrum afterthe the firstDC DC offsetcancellation; cancellation; (b)envelope envelope spectrum of theoriginal original timesynchronous synchronous signal. of of the the original time time synchronous signal. signal.

The time-frequency contour of the squared envelope signal using the S transform is shown in Figure 23. In contrast, the time-frequency contour of the original envelope signal using the S

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transform is shown in Figure 24. From these two figures and their local amplifications, the sun gear fault frequency in the frequency domain and the time-frequency domain can be seen. However, the time-frequency contour of the squared envelope signal has less useful sun gear fault information than Appl. Sci. 2017, 7, 207 contour of the original envelope signal. 17 of 22 the time-frequency Appl. Sci. 2017, 7, 207 17 of 22

Figure 23. (a) Time-frequency contour of the squared squared envelope envelope signal using the the S transform; transform; (b) (b) local local Figure 23. (a) Time-frequency Time-frequency contour Figure (a) of the squared envelope signal using the SS transform; (b) local amplification of (a). amplification of amplification (a).

Figure 24. (a) Time-frequency contour of the original envelope signal using the S transform; (b) local Figure contour Figure 24. 24. (a) (a) Time-frequency Time-frequency contour of of the the original original envelope envelope signal signal using using the the SS transform; transform; (b) (b) local local amplification of (a). amplification of (a). amplification of (a).

5.2. Planet Planet Gear Gear Fault Fault Experiment Experiment 5.2. 5.2. Planet Gear Fault Experiment For this this experiment, experiment, we we analyzed analyzed one one planet planet gear gear wear wear fault fault mode mode to to test test the the proposed proposed method method For For this experiment, we analyzed one planet gear wear fault mode to test the proposed method while the the remaining remaining gears gears were were all all in in good good condition, condition, undamaged. undamaged. Similar Similar to to the the case case study study in in Section Section 4, 4, while while thetoremaining gearssignal, were all in good condition, undamaged. Similar to the case studyrate in in order ensure a stable the data acquisition system is adjusted to use the data sample in order to ensure a stable signal, the data acquisition system is adjusted to use the data sample rate Sectionon 4, the in order to ensure a stable thesampling data acquisition is adjusted to use rate the data based tachometer signal. Thesignal, adjusted rate is is aasystem synchronous sampling and based on the tachometer signal. The adjusted sampling rate synchronous sampling rate and sample rate based on the tachometer signal. The adjusted sampling rate is a synchronous sampling donates the the resampled resampled vibration vibration signal. signal. During During the the experiment, experiment, the the mean mean rotating rotating frequency frequency of of the the donates rate and donates the resampled vibration signal. During the experiment, the mean rotating frequency input shaft connecting the sun gear of the planetary gearbox was 20.0111Hz, while a load of 405 Nm input shaft connecting the sun gear of the planetary gearbox was 20.0111Hz, while a load of 405 Nm of the input shaft the sun gear of planetary gearbox was 20.0111Hz, while a load of was applied to the theconnecting output shaft shaft connecting thethe planet carrier. An accelerometer accelerometer was mounted mounted on the the was applied to output connecting the planet carrier. An was on 405 Nm was applied to the output shaft connecting the planet carrier. An accelerometer was mounted top of of the the gearbox gearbox casing casing (as (as shown shown in in Figure Figure 19), 19), and and the the vibration vibration signals signals were were collected collected from from the the top on the top of the gearbox casing (as shown in Figure 19), gearbox and the vibration signals were collected from accelerometer at position S3. According to the planetary configuration and its running speed, accelerometer at position S3. According to the planetary gearbox configuration and its running speed, the characteristic accelerometerfrequencies at position are S3. calculated Accordingand to the planetary configuration and its running the listed in Table Tablegearbox 4. the characteristic frequencies are calculated and listed in 4. speed, the characteristic frequencies are calculated and listed in Table 4. Table 4. Characteristic frequencies (Hz). Table 4. Characteristic frequencies (Hz). Table 4. Characteristic frequencies (Hz). Rotating Frequency Local Damage Local Damage Meshing Frequency Rotating Frequency Meshing Frequency Sun Carrier Planet Sun Ring Rotating Sun Frequency Carrier Planet Local Damage Sun Ring Meshing238.8748 Frequency 20.0111 1.6361 3.7324 7.4648 55.1249 4.9083 238.8748 20.0111 Carrier 1.6361 3.7324Planet 7.4648 55.1249 Sun Sun 4.9083 Ring 238.8748 20.0111 1.6361 3.7324 7.4648 55.1249The 4.9083 Similarly, we we can apply apply the the DC DC offset cancellation cancellation operation to this this signal. signal. CK value value shows shows Similarly, can offset operation to The CK that the signal only after the first DC offset cancellation iteration has the strongest PIS. The CK value that the signal only after the first DC offset cancellation iteration has the strongest PIS. The CK value variation vs. the iteration number can be seen in Figure 25. Though the CK achieves highest value at variation vs. the iteration number can be seen in Figure 25. Though the CK achieves highest value at the 10th 10th iteration, iteration, after after Iteration Iteration 6, 6, the the envelope envelope signal signal is is corrupted corrupted seriously. seriously. Therefore, Therefore, the the first first the iteration is optimal. The comparison of the squared envelope spectrum and original envelope iteration is optimal. The comparison of the squared envelope spectrum and original envelope spectrum is is shown shown in in Figure Figure 26. 26. It It can can be be seen seen that that there there is is no no difference difference in in essence, essence, except except the the spectrum amplitude. Because Because of of the the very very small small planet planet gear gear fault fault frequencies, frequencies, it it is is very very difficult difficult to to display display the the amplitude.

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Similarly, we can apply the DC offset cancellation operation to this signal. The CK value shows that the signal only after the first DC offset cancellation iteration has the strongest PIS. The CK value variation vs. the iteration number can be seen in Figure 25. Though the CK achieves highest value at the 10th iteration, after Iteration 6, the envelope signal is corrupted seriously. Therefore, the first iteration is optimal. The comparison of the squared envelope spectrum and original envelope spectrum is shown in Figure 26. It can be seen that there is no difference in essence, except the amplitude. Because Appl. Sci.very 2017small , 7, 207 planet gear fault frequencies, it is very difficult to display the time-frequency contour 18 of 22 of the Appl. Sci. 2017 , 7, 207 18 of 22 using the S transform.

Figure Figure 25. 25. CK CK value value variation variation vs. vs. iteration iteration number. number. Figure 25. CK value variation vs. iteration number.

Figure Figure 26. 26. (a) (a) Squared Squared envelope envelope spectrum spectrum after after the the first first DC DC offset offset cancellation; cancellation; (b) (b) envelope envelope spectrum spectrum Figure 26. (a) Squared envelope spectrum after the first DC offset cancellation; (b) envelope spectrum of the original signal. of the original signal. of the original signal.

5.3. 5.3. Ring Ring Gear Gear Fault Fault Experiment Experiment 5.3. Ring Gear Fault Experiment For this experiment, wear fault mode to test thethe proposed method while the For this experiment,we weused useda aring ringgear gear wear fault mode to test proposed method while For this experiment, we used a ring gear wear fault mode to test the proposed method while the remaining gears were all all in good condition, undamaged. Similar to to the the remaining gears were in good condition, undamaged. Similar thecase casestudy studyininSection Section 4, 4, remaining gears were all signal, in goodthe condition, undamaged. Similar to the case study insample Section 4, in order to ensure a stable data acquisition system is adjusted to use the data in order to ensure a stable signal, the data acquisition system is adjusted to use the data sample rate rate in order tothe ensure a stable signal. signal, the data acquisition system is adjusted to use the data sample rate based tachometer based on on the tachometer signal. The The adjusted adjusted sampling sampling rate rate is is aa synchronous synchronous sampling sampling rate rate and and based on theresampled tachometer signal. signal. The adjusted sampling rate is a synchronous sampling rateofand donates donates the the resampled vibration vibration signal. During During the the experiment, experiment, the the mean mean rotating rotating frequency frequency of the the donates the resampledthe vibration signal. During the gearbox experiment, the mean rotating frequency of the input input shaft shaft connecting connecting the sun sun gear gear of of the the planetary planetary gearbox was was 22.3889 22.3889 Hz, Hz, while while aa load load of of 405 405 Nm Nm input shaft connecting the sun gear of the planetary gearbox was 22.3889 Hz, while a load of 405 Nm was was applied applied to to the the output output shaft shaft connecting connecting the the planet planet carrier. carrier. An An accelerometer accelerometer was was mounted mounted on on the the was applied to the output (as shaft connecting the planet carrier. An accelerometer was mounted on the top top of of the the gearbox gearbox casing casing (as shown shown in in Figure Figure 19), 19), and and the the vibration vibration signals signals were were collected collected from from the top of the gearbox casing (asAccording shown in Figure 19), and the vibration signals were collected from the accelerometer accelerometer at at position position S3. S3. According to to the the planetary planetary gearbox gearboxconfiguration configuration and and its itsrunning runningspeed, speed, accelerometer at position S3. According to the planetary gearbox configuration and its running speed, the characteristic frequencies are calculated and listed in Table 5. the characteristic frequencies are calculated and listed in Table 5. the characteristic frequencies are calculated and listed in Table 5. Table 5. Characteristic frequencies (Hz). Table 5. Characteristic frequencies (Hz). Rotating Frequency Local Damage Meshing Frequency Sun Frequency Carrier PlanetLocal Damage Sun Rotating Meshing Frequency 267.2586 22.3889 1.8305 4.1759 8.3518 61.6751 Sun Carrier Planet Sun 267.2586 22.3889 1.8305 4.1759 8.3518 61.6751

Ring 5.4916 Ring 5.4916

The ring gear fault frequency is not always calculated in the same way. Theoretically, its fault The ring gear fault frequency always calculated same way.gears Theoretically, its from fault . However, becauseisofnot manufacturing error, in thethe three planet are different frequency is fmesh because of manufacturing error, the fthree planet gears are different from frequency fmesh. However, each other.isTherefore, the ring gear fault frequency may be 1/3 mesh. Therefore, it is very difficult to each other.the Therefore, theTring gear faultcase, frequency be 1/3 mesh. Therefore, it is verycancellation difficult to determine parameter in CK. In this we can may continue to fapply the first DC offset

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Table 5. Characteristic frequencies (Hz).

Meshing Frequency 267.2586

Rotating Frequency Sun

Carrier

22.3889

1.8305

Local Damage Planet 4.1759

8.3518

Sun

Ring

61.6751

5.4916

The ring gear fault frequency is not always calculated in the same way. Theoretically, its fault frequency is f mesh . However, because of manufacturing error, the three planet gears are different from each other. Therefore, the ring gear fault frequency may be 1/3 f mesh . Therefore, it is very difficult to determine the parameter T in CK. In this case, we can continue to apply the first DC offset cancellation to the signal. The resulting squared envelope spectrum and the original signal envelope spectrum can be seen in Figure 27. Similarly, it is possible to find the second order ring gear fault frequency in both envelope spectrums. There is no difference, except the value of the amplitude. Appl. Sci. 2017 , 7, 207 19 of 22

Figure 27. (a) Squared envelope spectrum after the first DC offset cancellation; (b) the envelope Figure 27. (a) Squared envelope spectrum after the first DC offset cancellation; (b) the envelope spectrum of the original signal. spectrum of the original signal.

6. Discussion 6. Discussion In this paper, three experimental case studies are conducted for the novel deterministic In this paper, three experimental case studies are conducted for the novel deterministic component component cancellation method application in detail. Not only amplitude-frequency characteristics, cancellation method application in detail. Not only amplitude-frequency characteristics, but also the but also the time-frequency characteristics are analyzed by the S transform. The methods’ application time-frequency characteristics are analyzed by the S transform. The methods’ application in those in those experimental case studies reveals issues that need to be further discussed and investigated: experimental case studies reveals issues that need to be further discussed and investigated: (1) The cross-correlation coefficient indicator should be investigated more. The region [0.9, 0.95] is (1) The cross-correlation coefficient indicator should be investigated more. The region [0.9, 0.95] is not appropriate for the illustrated cases in this paper. Therefore, there is a need for further study not appropriate for the illustrated cases in this paper. Therefore, there is a need for further study in order to determine the optimal iteration number and a more robust region. in order to determine the optimal iteration number and a more robust region. (2) The CK indicator takes advantage of when the PIS are more powerful than the other signal (2) components. The CK indicator takesstudy advantage of when the PIS are powerful than outer the other The case in Section 3 validates themore implanted bearing racesignal fault components. The case study in Section 3 validates the implanted bearing outer race fault diagnosis, with only bearing fault mode present, and confirms the powerful PIS characteristic of diagnosis, with only bearing fault mode present, and confirms the powerful PIS characteristic of implanted faults. It is confirmed that the CK gives excellent results in determining the optimal implanted faults. of It is confirmed that the CKHowever, gives excellent in determining theDC optimal iteration number DC offset cancellation. when results corrupted by the excess offset iteration number of DC offset cancellation. However, when corrupted by the excess DC offset cancellation operation, the envelope signal analysis will not give a good result because the PIS cancellation operation, thethe envelope signalin analysis will a good resultfaults because PIS are are also corrupted. For case study Section 4, not all give of the bearing arethe naturally also corrupted. For the case study in Section 4, all of the bearing faults are naturally developed. developed. Simultaneously, there are gear chip faults and wear faults. Therefore, the PIS Simultaneously, there are gearfaults chip faults andweak. wear faults. produced these produced by these bearing are very In thisTherefore, case, thethe CKPIS gives a poorbyresult. bearing faults are very weak. In this case, the CK gives a poor result. Theoretically, every DC Theoretically, every DC offset cancellation can suppress one harmonic of the deterministic offset cancellation can suppress one harmonic of the deterministic component of the original component of the original signal. For the fixed-axis gearbox, the deterministic component is signal. the For HS-IS the fixed-axis gearbox, the and deterministic component is mainly the HS-IS gear mesh mainly gear mesh frequency its harmonics. This is illustrated in Figures 13 and 16 frequency and its harmonics. This is illustrated in Figures 13 and 16 in Section 4. Therefore, in Section 4. Therefore, it is possible to determine the optimal iteration number by the number it is possible determine the optimal iteration number by the number of HS-IS gear mesh of HS-IS gear to mesh harmonics. This is very useful for a fixed-axis gearbox. Whereas there is harmonics. This is useful forsquared a fixed-axis gearbox. Whereas is another problem, no another problem, novery matter if the envelope spectrum afterthere DC offset cancellation or the envelope spectrum of originals signal is used, they are both dominated by the shaft rate and harmonics related to the HS and IS shaft. No information about bearing fault frequencies can be found. Fortunately, the time-frequency contour of the squared envelope signal after DC offset cancellation by the S transform can reveal some information about the bearing fault frequencies. This demonstrates the iterative DC offset cancellation’s effectiveness for PIS enhancement. For

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(3)

(4)

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matter if the squared envelope spectrum after DC offset cancellation or the envelope spectrum of originals signal is used, they are both dominated by the shaft rate and harmonics related to the HS and IS shaft. No information about bearing fault frequencies can be found. Fortunately, the time-frequency contour of the squared envelope signal after DC offset cancellation by the S transform can reveal some information about the bearing fault frequencies. This demonstrates the iterative DC offset cancellation’s effectiveness for PIS enhancement. For the implanted bearing outer race fault case study in Section 3, even though both the squared envelope spectrum after DC offset cancellation and the envelope spectrum of the original signal can reveal the bearing fault information effectively, however, the time-frequency contour of the squared envelope signal after DC offset cancellation has good time-frequency characteristics for revealing the bearing fault information, in comparison to the case without DC offset cancellation. Concerning planetary gearbox fault diagnosis, the cross-correlation coefficient has a similar problem as the previous case studies. The CK has the correct assessment for the iteration number for the sun gear fault and the planet gear fault. However, the CK does not take advantage of the ring gear fault, because of the uncertainty for the fault frequency. Besides the iteration number determined by the CK value, several tests for the iteration number between one and ten show that only the first DC offset cancellation gives the best performance. In addition, the mesh frequency always has high order harmonics greater than 10 for these data, and the number of harmonics cannot be used to determine the optimal iteration number. The squared envelope spectrums after the first DC offset cancellation do not have obvious superiority with respect to the original envelope analysis for the spectrum’s efficient evaluation. This may be due to the different structures of the planetary and fixed-axis gearbox. The GMF of the fixed-axis gearbox is related to the rotating frequency of the gear and the gear teeth; however, for the planetary gearbox, the GMF depends on the gearbox rotating frequency and its configuration, which is more complex than the configuration of the fixed-axis gearbox. Therefore, how to revise the DC offset cancellation method to fit the planetary gearbox fault diagnosis is a problem which needs further investigation and explorations. The time-frequency contour by the S transform needs more computational capacity. Therefore, only short time-frequency characteristics can be displayed. However, for the planet gear fault and the ring gear fault, their fault frequencies are very small. This will lead to the large space between the time-frequency lines. Usually, it is not available to display the characteristics of the small fault frequencies. This problem also needs further investigation and explorations.

7. Conclusions In order to detect the PIS signal produced by the gearbox fault, DC offset cancellation was used to process the fault signal. Through a comparative study based on an implanted bearing fault case, the CK was demonstrated to be superior to the cross-correlation coefficient to determine the iteration number of DC offset cancellation. For the implanted bearing fault case, the PIS produced by the fault are very strong. Therefore, detection of the PIS using CK is more effective. However, for the naturally-developed bearing fault compound with gear fault, the PIS are very weak for the CK to detect. In this circumstance, the CK loses its power and cannot find the proper iteration number for DC offset cancellation. The order of the high-speed shaft gear mesh frequency was used as the iteration number because every DC offset cancellation can suppress one harmonic mesh frequency. This is very useful for PIS extraction. All of above results are demonstrated on a fixed-axis gearbox. For a planetary gearbox, DC offset cancellation has a limited effect on the fault diagnosis enhancement, especially for those low fault frequency parts, such as the planet gear, ring gear, etc. In this paper, because of the merits, the S transform was used to display the time-frequency characteristics of the DC offset results. This made us observe the fault information from the time-frequency contour which is more intuitive. However, for the low fault frequency parts, it is very difficult for the time-frequency contour to display them because of the long samples. At the end of this paper, we had a detailed discussion about the DC

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offset cancellation and the S transform-based application. In the future, these time-frequency contours will be used for deep learning-based fault classification. Acknowledgments: The research is partially supported by the China NSFC under Grant Number 71401173 and by the Natural Science Foundation of Hebei Province under Grant Number E2015506012. Author Contributions: J.Z. and L.W. conceived of and designed the experiments. X.Z. performed the experiments. X.Z. and R.B. analyzed the data. X.Z. wrote the paper. Conflicts of Interest: The authors declare no conflict of interest.

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