Meshing frequency modulation assisted empirical wavelet transform for fault diagnosis of wind turbine planetary ring gear

Accepted Manuscript Meshing frequency modulation assisted empirical wavelet transform for fault diagnosis of wind turbine planetary ring gear Yun Kong, Tianyang Wang, Fulei Chu PII:

S0960-1481(18)31090-5

DOI:

10.1016/j.renene.2018.09.027

Reference:

RENE 10567

To appear in:

Renewable Energy

Received Date: 12 June 2018 Revised Date:

25 August 2018

Accepted Date: 10 September 2018

Please cite this article as: Kong Y, Wang T, Chu F, Meshing frequency modulation assisted empirical wavelet transform for fault diagnosis of wind turbine planetary ring gear, Renewable Energy (2018), doi: 10.1016/j.renene.2018.09.027. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

ACCEPTED MANUSCRIPT

Meshing Frequency Modulation Assisted Empirical Wavelet

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Transform for Fault Diagnosis of Wind Turbine Planetary Ring Gear

3

Yun Kong, Tianyang Wang, Fulei Chu*

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Department of Mechanical Engineering, Tsinghua University, Beijing 100084, China

5

ABSTRACT Condition monitoring and fault diagnosis for wind turbine gearbox is significant to

6

save operation and maintenance costs. However, strong interferences from high-speed parallel gears

7

and background noises make fault detection of wind turbine planetary gearbox challenging. This

8

paper addresses the fault diagnosis for wind turbine planetary ring gear, which is intractable for

9

traditional spectral analysis techniques, since the fault characteristic frequency of planetary ring

10

gear can be resulted from the revolving planet gears inducing modulations even in healthy

11

conditions. The main contribution is to establish an adaptive empirical wavelet transform

12

framework for fault-related mode extraction, which incorporates a novel meshing frequency

13

modulation phenomenon to enhance the planetary gear related vibration components in wind

14

turbine gearbox. Moreover, an adaptive Fourier spectrum segmentation scheme using iterative

15

backward-forward search algorithm is developed to achieve adaptive empirical wavelet transform

16

for fault-related mode extraction. Finally, fault features are identified from envelope spectrums of

17

the extracted modes. The simulation and experimental results show the effectiveness of the

18

proposed framework for fault diagnosis of wind turbine planetary ring gear. Comparative studies

19

prove its superiority to reveal evident fault features and avoid the ambiguity from the planet carrier

20

rotational frequency over ensemble empirical mode decomposition and spectral kurtosis.

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Keywords: Wind turbine; planetary ring gear; fault diagnosis; meshing frequency modulation;

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adaptive Fourier spectrum segmentation; empirical wavelet transform.

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*

Corresponding author. Department of Mechanical Engineering, Tsinghua University, Beijing 100084, China. Email address: [email protected] (F. L. Chu) 1

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1 Introduction

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In recent years, power generation through wind energy has experienced a remarkable expansion to

25

address the challenges of global climate change, fossil fuels exhaustion and renewable technology

26

development [1, 2]. However, wind turbines (WTs) are frequently exposed to extreme and harsh

27

operational conditions, including the rapid variation of wind speed and external random load, which

28

result in relatively high failure rates of WTs. Moreover, failures of WTs not only cause reliability

29

and stability problems but also result in huge costs for maintenance and repair. It is reported that

30

operation and maintenance (O&M) costs for WTs can account for 10%-20% of the total energy

31

generation costs, and this percentage can reach up to 35% at the end of wind turbine lifetime [3]. A

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condition-based maintenance strategy that avoids unexpected shutdown can considerably reduce

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O&M costs and promote the reliability and safety. Therefore, extensive researches have been

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carried out about the condition monitoring and fault diagnosis technology for wind turbines.

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Wind turbine gearbox serves as one of the most important components in wind energy

36

conversion system, of which the failure rate is low but the downtime and maintenance costs account

37

for the largest proportion [4, 5]. As illustrated in Fig. 1, a typical wind turbine gearbox consists of

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one planetary gear transmission operating at the low-speed stage and two fixed-axis gear

39

transmissions operating at the intermediate stage and high-speed stage, respectively, which could

40

transmit large power in a relatively compact structure. Vibration-based fault detection technique has

41

been regarded as one of the most effective techniques for wind turbine gearbox condition

42

monitoring [6]. However, the vibration signals measured from wind turbine planetary gearbox

43

exhibit several unique features, including the complex physical configuration inducing modulation

44

characteristics, strong interferences from high-speed fixed-axis gearboxes, heavy background noises

45

and nonstationary features due to the time-varying operation conditions. Therefore, these features

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make the fault detection of wind turbine planetary gearbox a very challenging task and have

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stimulated a considerable amount of researchers [1, 2, 6].

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2

ACCEPTED MANUSCRIPT Wind Turbine Gearbox

Pinion 4

Blades Ring Gear

Wind

Pinion 2

Brake

Planet Gear

Generator Gear 3

LSS: Low speed stage Carrier Sun Gear

IMS: Intermediate stage Gear 1

48 49

IMS Fixed-axis Gearbox I

HSS: High speed stage HSS Fixed-axis Gearbox II

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LSS Planetary Gearbox

Fig. 1. Typical drivetrain configuration for a wind turbine.

For the fault diagnosis of wind turbine planetary gearbox, many signal processing approaches

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have been developed in recent years, such as statistical analysis [7-9], spectral kurtosis [10-12],

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time synchronous averaging (TSA) algorithm [13-17], wavelet transform [18-21], demodulation

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analysis via adaptive signal decomposition methods [22-24] and advanced time frequency

54

representation methods [25-27]. Among statistical analysis, Lei et al. [8] proposed two diagnostic

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parameters to detect the fault type of sun gear, including root mean square of the filtered signal and

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normalized summation of difference spectrum. Bartelmus [9] developed a new diagnostic feature to

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monitor the planetary gearbox under time-varying load conditions. Besides, spectral kurtosis was

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successfully applied to detect impulsive components induced by the natural crack of bearing in

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high-speed shafts [10, 11] and the tooth crack of planetary ring gear [12] in wind turbine gearbox.

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TSA has been known as one of the most effective techniques for gear related fault detection, but

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implementing TSA for the planetary gearbox is challenging due to the revolving planet gears

62

inducing modulation characteristics. To address this challenge, McFadden proposed to use window

63

functions and TSA to extract vibration signals at the instances where the planet gears are positioned

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directly under the transducer [13]. Further, various window functions for planetary gear vibration

65

extraction were investigated, including Hanning window [14], autocorrelation-based window [15]

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and Turkey window [16]. However, fault-related features can be inadvertently filtered out by

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improper location of the window function in TSA implementation for planetary gearboxes [17]. As

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a powerful multiscale signal analysis method, wavelet transform was applied for noise elimination

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ACCEPTED MANUSCRIPT and fault feature extraction in mechanical vibration signals [18]. Time wavelet energy spectrum

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based on continuous wavelet transform was proposed to enhance the fault signature of planet gears

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in wind turbine gearbox [19]. Tang et al. [20, 21] presented the adaptive Morlet wavelet transform

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method to denoise the vibration signals and extract features of wind turbine gearbox. However, the

73

selection of mother wavelet and transform scales remains a challenge for fault feature extraction of

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wind turbine planetary gears. Moreover, from the aspect of spectral characteristics comparison

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between the healthy and faulty vibration signals, various adaptive signal decomposition methods

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were incorporated with demodulation algorithms to realize fault detection of planetary gearbox,

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including ensemble empirical mode decomposition [22], local mean decomposition [23] and

78

intrinsic time-scale decomposition methods [24]. To address the challenge of nonstationary feature

79

extraction for wind turbine gearbox under variable speed conditions [25], Feng et al. proposed

80

advanced time frequency representation methods to reveal time-varying fault characteristic

81

frequency of planetary gears, including adaptive optimal kernel time-frequency analysis [26] and

82

iterative generalized synchrosqueezing transform [27]. Furthermore, taking into account the

83

component fault and system uncertainties simultaneously, several model-based fault reconstruction

84

schemes and advanced fault-tolerant control frameworks are developed [28, 29] and applied to

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ensure the operational reliability and stability in wind turbine system [30, 31]. These contributions

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have greatly enriched the literature about fault detection of wind turbine planetary gearbox.

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However, the reported literature have rarely referred to the fault diagnosis of wind turbine

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planetary ring gear. There remain several unsolved challenges for the fault detection of wind turbine

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planetary ring gear. Firstly, the planetary gear related vibration components are generally

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overwhelmed by the interferential vibrations from high-speed parallel gears and heavy background

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noises. Thus it is hard to separate and extract them from the mixed signals measured from wind

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turbine gearbox. Secondly, the fault characteristic frequency fcfr of planetary ring gear equals to the

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planet carrier rotational frequency fc multiplied by the number of planet gears Np [32], which may

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also be induced by the involving planet gears inducing modulations even in a healthy wind turbine 4

ACCEPTED MANUSCRIPT planetary gearbox [33, 34]. Thus, based on the identification of fault characteristic frequency in the

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modulation sidebands and envelope spectrums, typical spectral analysis techniques fail to

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distinguish the faulty planetary ring gear from the healthy one in wind turbine gearbox. Recently,

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Wang et al. [35] have proposed a meshing resonance based filtering algorithm for fault diagnosis of

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wind turbine planetary ring gear, which exploits the meshing resonance phenomenon to highlight

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the planetary gear related vibrations. However, this filtering algorithm replies on the prescribed

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subdivision scheme in the frequency domain, which originates from the idea of Kurtogram [36] but

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lacks an adaptive Fourier spectrum segmentation scheme for the optimal filter band determination.

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To address the preceding challenges, a novel meshing frequency modulation assisted empirical

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wavelet transform (MFM-EWT) framework is proposed in this paper for the fault diagnosis of wind

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turbine planetary ring gear. Within our method, the meshing frequency modulation (MFM)

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phenomenon is pointed out, which can facilitate us to determine the so-called meshing modulation

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regions for enhancement and separation of the planetary gear related vibration signals in wind

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turbine gearbox. In order to determine the so-called meshing modulation regions, a fully flexible

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Fourier spectrum segmentation scheme originated from empirical wavelet transform (EWT) is

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exploited and an indicator MFMindex is developed to evaluate the significance level of meshing

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frequency modulation. The essence of EWT is to determine the Fourier spectrum segments and then

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design a series of empirical wavelet filters to decompose the signal into several modes. Thus, we

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present an MFMindex assisted iterative backward-forward search algorithm to determine the

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Fourier spectrum segments for EWT adaptively. As a result, several modes could be extracted by

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the proposed MFM-EWT framework. Finally, the envelope spectrum analysis of the fault-related

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modes is conducted to achieve a diagnostic conclusion about the wind turbine planetary ring gear.

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The remainder of the paper is organized as follows. Section 2 presents the meshing frequency

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modulation phenomenon in wind turbine gearbox, and MFMindex is constructed to evaluate the

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significance level of meshing frequency modulation. In Section 3, the meshing frequency

120

modulation assisted empirical wavelet transform is proposed for fault diagnosis of wind turbine 5

ACCEPTED MANUSCRIPT planetary ring gear, including the empirical wavelet transform theory, the MFMindex assisted

122

adaptive Fourier spectrum segmentation scheme and overall algorithmic procedures of the proposed

123

MFM-EWT framework. Section 4 and Section 5 verify the effectiveness of the proposed framework

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with simulated signals and experimental vibration signals in a wind turbine gearbox test bench,

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respectively. Furthermore, comparative studies with ensemble empirical mode decomposition and

126

spectral kurtosis are conducted in section 5.2. Finally, conclusions are drawn in Section 6.

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2 Meshing frequency modulation phenomenon in wind turbine gearbox

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According to previous researches in [37, 38], a novel meshing frequency modulation phenomenon

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has been overlooked in the gearbox vibration signals, where the alternating gear meshing behavior

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may also induce weak impulsive components modulated by the meshing frequency. Based on this

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phenomenon, recent researches have demonstrated that the meshing frequency modulation

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phenomenon can be exploited to distinguish the resonance frequency bands induced by the gear and

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bearing faults for compound fault diagnosis [39] and detect the planet bearing fault [40]. Besides,

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this phenomenon has also been exploited to indicate and highlight the vibration components related

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to planetary gears in wind turbine gearbox [35]. However, these works do not provide a clear and

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illustrative explanation about the meshing frequency modulation phenomenon using experimental

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vibration signals.

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To explain the meshing frequency modulation phenomenon clearly, an experimental vibration

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signal measured from a faulty wind turbine gearbox is analyzed and illustrated in Fig. 2. The

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experiment details about the wind turbine gearbox test bench can be referred in Section 5. From the

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Fourier spectrum in Fig. 2(a), we can observe that the frequency components in wind turbine

142

gearbox are rather complicated and the modulation sidebands around the planetary meshing

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frequency fp,m are overwhelmed, due to the interferences from the high-speed gear meshing

144

components and unwanted noises. As shown in the enlarged spectrum in Fig. 2(a), the frequency

6

ACCEPTED MANUSCRIPT spacing of fp,m could be clearly revealed, which demonstrates that the planetary meshing frequency

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modulates the frequency band indicated by the shaded area. In other words, the meshing frequency

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modulation phenomenon occurs in this specific frequency band which is referred to as the so-called

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meshing modulation region hereafter. Moreover, the raw vibration signal is band-pass filtered from

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the indicated meshing modulation region and the envelope spectrum of the filtered signal is

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illustrated in Fig. 2(b). Obviously, the dominant frequencies are the planetary meshing frequency

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fp,m and its harmonics in the envelope spectrum, which further demonstrates that the modulation

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frequency of the filtered signal is the planetary meshing frequency. These observations consistently

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verify the meshing frequency modulation phenomenon occurred in the wind turbine gearbox.

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Although the meshing frequency modulation phenomenon has not been rigorously testified by gear

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dynamics, our experimental findings still provide reliable evidences for this phenomenon, which

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motivates us to achieve fault diagnosis of planetary gears in wind turbine gearbox.

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Fig. 2. An illustrative explanation for the meshing frequency modulation phenomenon with a faulty

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wind turbine gearbox test bench. (a) Fourier spectrum of the vibration signal; (b) envelope spectrum

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of the filtered signal from the meshing modulation region indicated by the shaded area in (a).

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Because the meshing frequency modulation is highly relevant to one certain meshing frequency,

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such as the planetary meshing frequency fp,m in the above illustrative case, the vibration components

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in the corresponding meshing modulation regions are supposed to be highly relevant to the certain

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meshing gears. According to this idea, the meshing frequency modulation phenomenon has a

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potential to facilitate vibration extraction of one certain pair of meshing gears and free of

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interferences from other meshing gear pairs in complex multistage gearboxes. It is worth noting that

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ACCEPTED MANUSCRIPT the meshing modulation regions carry plenty of vibration information related to one specified pair

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of meshing gears, thus the meshing modulation regions could be regarded as a feasible and

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promising supplement to reflect the health condition of meshing gears. In other words, the health

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state for meshing gears can be reflected by not only the modulation sidebands around the meshing

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frequency but also the spectral components in the meshing modulation regions, which will provide

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a new approach for gear fault detection in multistage gearboxes. Therefore, in this paper, we attempt

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to exploit this phenomenon for enhancement and separation of the planetary gear related vibration

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signals and further detect the planetary ring gear fault in wind turbine gearbox.

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An intuitive idea to exploit the meshing frequency modulation phenomenon for fault diagnosis

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of wind turbine planetary ring gear is finding the meshing modulation regions related to the

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planetary meshing frequency modulation. In order to locate the meshing modulation regions, an

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indicator called MFMindex is developed to evaluate the significance level of meshing frequency

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modulation quantitively. The MFMindex is defined on the envelope spectrums, which can be

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derived by Hilbert transform and FFT algorithm. Supposing that the instantaneous amplitude and

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envelope spectrum of signal x(t) of length N are denoted by IAmp(t) and ES(f ), respectively,

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MFMindex is defined as follows,

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1/2

ES ( f ) =

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I Amp (t ) = ( x(t )) 2 + (Hilbert{x(t )}) 2  ,

2 DFT{I Amp (t )} , N N

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H MFMindex(x, f mesh ) = ∑  max ES ( f )  f − kf mesh ≤δ f  k =1 

2

(1) (2)

∑

ES 2 ( f ) ,

0 < f ≤ f upper

(3)

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where fmesh and fupper are the meshing frequency and the upper limit of frequency considered in the

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envelope spectrum, respectively. In this paper, the harmonic order NH and the allowable frequency

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deviation δf are set as 3 and 1 Hz, respectively. Based on the constructed indicator, MFMindex is

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capable to quantitively evaluate the significance level of meshing frequency and its harmonics in

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the interval (0, fupper] of envelope spectrum. The significant meshing frequency modulation

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phenomenon will produce prominent amplitude peaks at meshing frequency and its harmonics in

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the envelope spectrum. Thus the MFMindex will be large if the meshing frequency modulation is 8

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significant, otherwise it will have a very small value. As a result, the MFMindex could be deemed as a quantitive indicator for evaluation of meshing

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frequency modulation phenomenon and assist to locate the meshing modulation regions for

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enhancement of planetary gear related vibrations in wind turbine gearbox.

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3 Meshing frequency modulation assisted empirical wavelet transform

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In order to locate the meshing modulation regions and extract the planetary gear related vibrations

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for fault detection in wind turbine gearbox, a fully flexible and adaptive Fourier spectrum

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segmentation scheme is preferred, which could overcome the limitations of other prescribed and

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nonadaptive spectrum subdivision schemes. In this section, a meshing frequency modulation

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assisted empirical wavelet transform method is proposed, which incorporates the fully flexible tight

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wavelets originated from empirical wavelet transform and the MFMindex assisted adaptive Fourier

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spectrum segmentation scheme.

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3.1 Empirical wavelet transform

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Empirical wavelet transform is a recently proposed approach by Gilles [41], which designs fully

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flexible empirical wavelets to extract different modes of the analyzed signal. Within this approach,

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extracting all modes of a signal is assumed to appropriately segment the Fourier spectrum and apply

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filtering operation to each detected Fourier support by constructing a family of empirical wavelets.

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For the clarity of empirical wavelet transform illustration, we consider a normalized Fourier axis

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with a periodicity of 2π and the frequency limit is restricted within [0, π] to respect the Shannon

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criteria. As a result, it is assumed that the initial Fourier supports [0, π] is segmented into N

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contiguous Fourier segments Λ n = [ωn −1 , ωn ] so that

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boundary limits between each segment (where ω1 = 0 and ω N = π ). To make empirical wavelets

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defined on the segmental support Λ n physically realizable, the transition phase centered around

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the boundary limit ωn is defined, of which the width 2τ n is simply chosen to be proportional to ωn :

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N n=1

Λ n = [ 0, π ] , where we denote

ωn to be the

ACCEPTED MANUSCRIPT 214

τ n = γωn .

Consequently, the fully flexible empirical wavelets are constructed as bandpass filters on each

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Fourier support Λ n . In specific, according to the idea of the construction of both Littlewood-Paley

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and Meyer’s wavelets [42], the empirical scaling function and the empirical wavelets are defined by

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the following Eqs. (4) and (5), respectively. 1,    π  1 φn (ω )= cos  β  ω − (1 − γ ) ωn )   , (     2  2γωn 0, 

ω ≤ (1 − γ ) ωn

(1 − γ ) ωn ≤ ω ≤ (1 + γ ) ωn , others

(4)

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1, (1 + γ ) ωn ≤ ω ≤ (1 − γ ) ωn +1    π  1 cos  2 β  2γω ( ω − (1 − γ ) ωn +1 )   , (1 − γ ) ωn +1 ≤ ω ≤ (1 + γ ) ωn +1   n +1    , ψ n (ω )=     π  1 ( ω − (1 − γ ) ωn )  , (1 − γ ) ωn ≤ ω ≤ (1 + γ ) ωn sin  β      2  2γωn 0, others 

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(5)

where the function β (ω ) is defined as follows [42]: β (ω ) = ω 4 (35 − 84ω + 70ω 2 − 20ω 3 ) ,

(6)

 ωn +1 − ωn   should be satisfied to ensure that the set  ωn +1 + ωn 

{φ (t ),{ψ (t )} }

and the condition γ < min n 

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consisting of the empirical scaling function and all empirical wavelets is a tight frame of L2( ). An

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illustrative example for empirical wavelet filter bank is provided in Fig. 3.

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φ1(ω) ψ1(ω) ψ2(ω)

ψn(ω)

ψN-1(ω)

N

1

n

n =1

ψN(ω)

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2τn=2γωn 2τn+1=2γωn+1

0.5

0

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0

0.1π 0.2π

ωn (0.4π)

ωn+1 (0.7π) 0.85π

π

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Fig. 3. An illustrative example for empirical wavelet filter bank in Fourier spectrum with boundary

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limits set as [0.1π, 0.2π, 0.4π, 0.7π, 0.85π]. The shaded areas depict the transition phases.

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After building a tight frame set of empirical wavelets, the empirical wavelet transform Wx (n, t )

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can be achieved in the same way as the classical wavelet transform. In specific, the detail

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ACCEPTED MANUSCRIPT 228

coefficients are defined by the inner product between the analyzed signal x(t) and the empirical

229

wavelet ψ n (t ) as follows:

(

)

Wx (n, t )= x(t ),ψ n (t ) = ∫ x(τ )ψ n (τ − t )dτ = F −1 X (ω )ψ n (ω ) ,

(7)

and the approximation coefficients are given by the inner product with the scaling function φ1 (t ) as

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follows:

(

)

Wx (0, t )= x(t ), φ1 (t ) = ∫ x(τ )φ1 (τ − t )dτ = F −1 X (ω )φ1 (ω ) ,

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(8)

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where X(ω) and F-1 stand for the Fourier transform of signal x(t) and the inverse Fourier transform,

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respectively. The reconstruction of signal x(t) is obtained by

where * is the convolution operator.

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N N   x(t )=Wx (0, t ) ∗ φ1 (t ) + ∑ Wx (n, t ) ∗ψ n (t )=F −1  Wx (0, ω )φ1 (ω ) + ∑ Wx (n, ω )ψ n (ω )  , n =1  n =1 

(9)

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As presented above, the fully flexible Fourier spectrum segmentation scheme is achieved by

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empirical wavelets whose Fourier supports could be specified flexibly. When applied to extract

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different modes of the analyzed signal, the essential issue for empirical wavelet transform is how to

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segment the Fourier spectrum into a certain amount of compact Fourier supports

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Moreover, the Fourier spectrum segmentation scheme determines the adaptability of empirical

240

wavelet transform. Several empirical techniques have been suggested to adaptively detect the

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boundary limits for Fourier segments in references. A simple empirical method is to firstly detect all

242

local maxima in the Fourier spectrum of the analyzed signal and then define the boundary limits ωn

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as the center location or the location of the smallest minima between two consecutive maxima [41].

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Other representative methods are fine to coarse histogram segmentation algorithm and scale-space

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representation method to detect meaningful minima automatically [43, 44].

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3.2 MFMindex assisted adaptive Fourier spectrum segmentation scheme

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Obviously, the reported adaptive Fourier spectrum segmentation schemes heavily rely on the

248

selection of local maxima or minima in Fourier spectrum. These schemes can be applied to analyze

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signals when the signal-to-noise ratio (SNR) is high and the modes of the signal could be visually

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identified [45-47], such as the vibration signal of faulty bearings with a relatively high SNR.

N n=1

Λ n = [ 0, π ] .

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ACCEPTED MANUSCRIPT However, when applied to fault diagnosis of the complex wind turbine gearbox, these previous

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adaptive Fourier spectrum segmentation algorithms for empirical wavelet transform will suffer a lot,

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due to the complex modulation characteristics, heavy noises and unwanted gear meshing vibration

254

interferences in the vibration signal. Therefore, an MFMindex assisted adaptive Fourier spectrum

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segmentation scheme is proposed to automatically detect the Fourier segments for empirical

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wavelet transform. The proposed scheme is potential to make empirical wavelet transform adaptive

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and to enhance and extract the planetary gear related vibration components in wind turbine gearbox,

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because it incorporates the meshing frequency phenomenon to highlight the planetary gear related

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vibrations with the assistance of MFMindex. The MFMindex assisted adaptive Fourier spectrum

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segmentation scheme is implemented using the following iterative backward-forward search

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algorithm, which is illustrated in Fig. 4.

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Step 2: Iterative backward search operation

Step 1: Determination of initial Fourier segments

ω1

ω2

ω*p

ωp+1

0

ω1

ω2

ω*p

ωp+1

...

ωn-1

ωn

π

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0

...

ωn-1

ωn

π

0

ω1

ω2

ω*p

ωn1

0

ω1

ω2

ω*p

... ω*h1 .......

0

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ω*p

ωn1

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0 ω1 ...

ω*l1 ωn3

EP

Step 3: Iterative forward search operation

ω*l2

...

ω*l1 ωn3

ω*p

ωn1

ω

*

h1

ω*h1

...

...

ωnn

ω*h1

ωnn

π

ωn1 ω*h2

π

...

Step 4: Fourier boundary detection for EWT

0 ω*l2

π

ω*l1

ω*p

ω*h1

ω*l1

ω*p

ω*h1

ω*h2

π

1 0.5

ωnn

π

.......

0

* 0 ω l2

ω*h2

π

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Fig. 4. Iterative backward-forward search algorithm for Fourier spectrum segmentation.

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3.2.1 Determination of initial Fourier segments

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Since the meshing frequency modulation phenomenon in the wind turbine gearbox locates in the

266

relatively high frequency band [35], one can search an appropriate and coarse Fourier spectrum

267

segment [ωp*, π], which consists of the upper cutoff frequency π and a lower cutoff frequency ωp*.

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The lower cutoff frequency ωp* could be optimized among some possible frequency candidates

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ACCEPTED MANUSCRIPT 269

ωn∈{ π/M, 2π/M,…, (M-1)π/M} along the normalized frequency axis. M is the number of possible

270

frequency candidates, which should be appropriately determined to achieve a sufficiently fine

271

search performance. In this paper, the number of possible frequency candidates M is recommended

272

to select according to the following criterion, log 2 ( Fs f ch ) 

(10)

,

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M = 2

where Fs and fch are the sampling frequency and the characteristic frequency of interest (i.e., the

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planetary meshing frequency fp,m in the later sections), respectively. If three harmonics 3fch of the

275

characteristic frequency are considered in final envelope spectrums, 6fch should be covered in the

276

segmental Fourier spectrum supports so that the lower cutoff frequency candidates can shorten as

277

ωn∈{ π/M, 2π/M,…, π−2π×6fch/Fs} to accelerate the search process. Here, 2π×6fch/Fs is the

278

normalized frequency in terms of radians. Due to the FFT and convolution theorem applied in the

279

EWT implementation, the calculation time for the initial search process is fast. Therefore, the

280

Fourier spectra of candidate empirical wavelets in the determination of initial Fourier segments are

281

expressed as follows:

282

1, (1 + γ 1 )ωn ≤ ω ≤ (1 − γ 1 ) π    π  1 cos  2 β  2γ π ( ω − (1 − γ 1 ) π )   , (1 − γ 1 ) π ≤ ω ≤ (1 + γ 1 ) π   1  ψ n (ω )=   , ωn ∈{π M , 2π M ,..., π − 2π × 6 f ch Fs },   π  1 ( ω − (1 − γ 1 )ωn )  , (1 − γ 1 )ωn ≤ ω ≤ (1 + γ 1 )ωn sin  β      2  2γ 1ωn 0, others 

285 286

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π − ωn where γ 1 = . Then, combining Eqs. (7) and (11) could obtain Wx (n, t ) . π + ωn

(11)

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Moreover, the lower cutoff frequency ωp* for the initial Fourier spectrum segmentation could be optimized according to the maximal MFMindex, which is formulated as follows: ω *p = arg max MFMindex(Wx (n, t ), f ch ), ωn ∈ {π M , 2π M ,..., π − 2π × 6 f ch Fs } . ωn

(12)

287

As a result, the frequency support [0, π] is divided into two segments [0, ωp*] and [ωp*, π]. The

288

corresponding procedures are illustrated in Step 1 in Fig. 4. To further divide these two initial

289

Fourier segments, iterative backward and forward search operations are required.

13

ACCEPTED MANUSCRIPT 3.2.2 Iterative backward search operation

291

Iterative backward search operation could not only further divide the initial Fourier segment [ωp*,

292

π], but also locate the more precise meshing modulation regions for enhancement of the planetary

293

gear related vibration components in wind turbine gearbox. The first backward search operation is

294

conducted to find an optimized upper cutoff frequency ωh1*, given the initial boundary limit ωp*.

295

Possible frequency candidates for the upper cutoff frequency ωh1 are ωn∈{ωp*+2π×6fch/Fs,

296

ωp*+2π×6fch/Fs+π/M,…, π} along the normalized frequency axis. Therefore, the Fourier spectra of

297

candidate empirical wavelets in the first backward search operation are expressed as follows:

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1, (1 + γ 2 )ω *p ≤ ω ≤ (1 − γ 2 )ωn   π  1  ω − (1 − γ 2 ) ωn )   , (1 − γ 2 ) ωn ≤ ω ≤ (1 + γ 2 ) ωn ( cos  β      2  2γ 2ωn ψ n (ω )=  ,    π  1 ω − (1 − γ 2 ) ω *p )   , (1 − γ 2 ) ω *p ≤ ω ≤ (1 + γ 2 ) ω *p sin  β  * ( 2 2 γ ω  2 p     0, others 

(13)

ωn ∈{ω *p + 2π × 6 fch Fs , ω *p + 2π × 6 f ch Fs + π M ,..., π},

299 300

where γ 2 =

ωn − ω *p . Then, combining Eqs. (7) and (13) could obtain Wx (n, t ) . ωn + ω *p

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Moreover, the upper cutoff frequency ωh1* for the first backward search operation could be optimized according to the maximal MFMindex, which is formulated as follows: ωh*1 = arg max MFMindex(Wx (n, t ), f ch ), ωn ∈ {ω *p + 2π × 6 f ch Fs , ω *p + 2π × 6 fch Fs + π M ,..., π} .

(14)

EP

ωn

As a result, the frequency support [ωp*, π] is divided into two segments [ωp*, ωh1*] and [ωh1*, π].

302

Then, replace ωp* with ωh1* and repeat the backward search operation iteratively can achieve more

303

Fourier segments along the backward direction. Finally, [ωp*, π] is divided into Fourier segments as

304

[ωp*, ωh1*]⋃[ωh1*, ωh2*]⋃…⋃[ωhn*, π]. The corresponding procedures are shown in Step 2 in Fig. 4.

305

3.2.3 Iterative forward search operation

306

Similarly, the iterative forward search operation could further divide the initial Fourier segment [0,

307

ωp*] and enhance the planetary gear related vibration components in the low frequency bands. As

308

illustrated in Step 3 in Fig. 4, the first forward search operation is conducted to find an optimized

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ACCEPTED MANUSCRIPT lower cutoff frequency ωl1*, given the initial boundary limit ωp*. Possible frequency candidates for

310

the lower cutoff frequency ωl1 are ωn∈{ π/M,…, ωp*-2π×6fch/Fs-π/M, ωp*-2π×6fch/Fs} along the

311

normalized frequency axis. Therefore, the Fourier spectra of candidate empirical wavelets in the

312

first forward search operation are expressed as follows:

314 315

where γ 3 =

ω *p − ωn . Then, combining Eqs. (7) and (15) could obtain Wx (n, t ) . ω *p + ωn

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(15)

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1, (1 + γ 3 ) ωn ≤ ω ≤ (1 − γ 3 ) ω *p   π  1  ω − (1 − γ 3 ) ω *p )   , (1 − γ 3 ) ω *p ≤ ω ≤ (1 + γ 3 ) ω *p cos  β  * (    2  2γ 3ω p   ψ n (ω )=   ,  π  1  sin  β  ( ω − (1 − γ 3 ) ωn )  , (1 − γ 3 ) ωn ≤ ω ≤ (1 + γ 3 ) ωn   2  2γ 3ωn   0, others  * ωn ∈ {π M ,..., ω p − 2π × 6 f ch Fs − π M , ω *p − 2π × 6 fch Fs },

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Moreover, the lower cutoff frequency ωl1* for the first forward search operation could be optimized according to the maximal MFMindex, which is formulated as follows: ωl*1 = arg max MFMindex(Wx (n, t ), f ch ), ωn ∈ {π M ,..., ω *p − 2π × 6 f ch Fs − π M , ω *p − 2π × 6 fch Fs }. ωn

(16)

As a result, the frequency support [0, ωp*] is divided into two segments [0, ωl1*] and [ωl1*, ωp*].

317

Then, replace ωp* with ωl1* and repeat the forward search operation iteratively can obtain more

318

Fourier segments along the forward direction. Finally, [0, ωp*] is divided into Fourier segments as

319

[0, ωln*]⋃…⋃[ωl2*, ωl1*]⋃[ωl1*, ωp*]. The iterative procedures are also illustrated in Step 3 in Fig. 4.

320

3.2.4 MFMindex assisted empirical wavelet transform

321

After the iterative backward-forward search algorithm, the MFMindex assisted Fourier spectrum

322

segmentation scheme is achieved for adaptive empirical wavelet transform for mode extraction.

323

Supposing that the final results of Fourier boundary detection are [0, ωln*]⋃…⋃[ωl2*, ωl1*]⋃[ωl1*,

324

ωp*]⋃[ωp*, ωh1*]⋃[ωh1*, ωh2*]⋃…⋃[ωhn*, π], the empirical wavelet transform could be conducted to

325

extract several modes of the analyzed signal and these modes result in higher MFMindex. For

326

instance in Fig. 4, six modes are obtained by the MFMindex assisted empirical wavelet transform.

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15

328

turbine planetary ring gear

329

In order to realize the fault diagnosis of wind turbine planetary ring gear using the meshing

330

frequency modulation assisted empirical wavelet transform framework, the extracted modes should

331

be followed by envelope demodulation analysis to reveal the fault features of planetary ring gear.

332

The complete algorithmic procedures for the MFM-EWT framework are summarized as follows,

333

and the corresponding flowchart is illustrated in Fig. 5.

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ACCEPTED MANUSCRIPT 3.3 MFM assisted empirical wavelet transform for fault diagnosis of wind

Step 1 Acquire vibration signals of wind turbine gearbox with a high sampling frequency.

335

Step 2 Implement the MFMindex assisted adaptive Fourier spectrum segmentation using the iterative backward-forward search algorithm and obtain the Fourier boundary detection results.

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Step 3 Conduct adaptive empirical wavelet transform with the detected Fourier segments.

338

Step 4 Conduct the Hilbert demodulation analysis on the extracted modes by MFM-EWT and

339

identify the fault features for wind turbine planetary ring gear.

It should be noted that, unlike the wavelet packet transform which uses the pre-defined

341

decomposition level and follows the fixed dyadic Fourier partition scheme, both the determination

342

on the number of modes and the Fourier spectrum segmentation scheme are data-driven in the

343

proposed MFM-EWT framework. Moreover, the MFM-EWT framework incorporates the merits of

344

meshing frequency modulation phenomenon, which has specific physical meaning to the gearbox

345

structure parameters and operating parameters and has a potential to enhance the planetary gear

346

related vibration components in wind turbine gearbox. Therefore, the proposed approach has a high

347

potential for health condition assessment of the wind turbine gearbox. Hereinafter, the MFM-EWT

348

framework is applied to diagnosis wind turbine planetary ring gear fault using both the simulated

349

and experimental vibration signals.

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Fig. 5. Flowchart of meshing frequency modulation assisted empirical wavelet transform

352

framework.

17

ACCEPTED MANUSCRIPT

4 Simulation validations

354

In this section, a synthetic signal is modeled to simulate a faulty wind turbine planetary gearbox

355

vibration signal, considering the interferences from the fixed-axis meshing gears and background

356

noises. Moreover, the synthetic signal is employed to verify the effectiveness of our proposed

357

MFM-EWT framework. In specific, the synthetic signal model [32] consists of the faulty planetary

358

gearbox vibration xplanetary(t), the interferences from fixed-axis gearbox vibration xfix(t) and

359

background noises n(t), as expressed in Eq. (17). x(t ) = xplanetary (t ) + xfix (t ) + n(t ) .

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(17)

As for the faulty planetary gearbox vibration xplanetary(t) in Eq. (18), it considers not only the

361

amplitude modulation (AM) and frequency modulation (FM) components related to planetary

362

gearbox but also the repetitive transients induced by gear fault.

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xplanetary (t ) = 0.1 × [1 + Aring cos(2π fcf r t )]cos[2π f p,m t + Bring sin(2π fcf r t )] +

0.3 × [1 + Acarrier cos(2π f c t )]cos[2π f p,m t + Bcarrier sin(2π f c t + ϕ1 )] +

∑A

− β ring ( t − m fcf r −T0,ring )

m ,ring

e

n ,mesh

e

m =0 N −1

∑A n =0

cos  2π f r,ring (t − m fcf r − T0,ring )  u (t − m fcf r − T0,ring ) + ,

− β mesh ( t − n f p,m −T0,mesh )

(18)

cos  2π f r,mesh (t − n f p,m − T0,mesh )  u (t − n f p,m − T0,mesh )

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M −1

where the first term represents the AM-FM components of planetary gear meshing vibrations

364

induced by ring gear fault, in which the carrier frequency is the planetary meshing frequency fp,m

365

and the modulation frequency is the fault characteristic frequency fcfr of planetary ring gear. The

366

second term in Eq.(18) represents the AM-FM components induced by the planet carrier revolution,

367

in which the modulation frequency is the rotational frequency fc of the planet carrier. As for the

368

notations, Aring and Bring are the modulation amplitudes of the AM and FM parts respectively when

369

the modulation frequency is fcfr, Acarrier and Bcarrier are the corresponding counterparts when the

370

modulation frequency is fc. Moreover, the third term in Eq.(18) represents the repetitive transients

371

induced by the localized ring gear fault, where Am,ring, βring, fr,carrier and T0,ring are the amplitude of the

372

mth impulse, the structural damped characteristic frequency, the excited structure resonance

373

frequency and the initial arrival moment of the impulse series, respectively. Correspondingly, the

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ACCEPTED MANUSCRIPT 374

fourth term in Eq.(18) reflects the repetitive transients induced by the meshing frequency

375

modulation phenomenon, where An,mesh, βmesh, fr,mesh and T0,mesh are the associated counterparts of

376

Am,ring, βring, fr,carrier and T0,ring, respectively. Note that the fault characteristic frequency fcfr of

377

planetary ring gear is threefold of the carrier rotational frequency, i.e., 3fc. As for the interferential signal xfix(t) from the fixed-axis gearbox, it is modeled as the AM-FM

379

components of meshing vibration, in which the carrier frequency is the gear meshing frequency

380

ffix,mesh and the modulation frequency is the rotational frequency fdrive of the drive gear. xfix (t ) = 2 × [1 + Afix cos(2π f drivet )]cos[2π f fix,mesh t + Bfix sin(2π f drive t )] + 0.8 × [1 + Afix cos(2π f drivet )]cos[6π f fix,mesh t + Bfix sin(2π f drive t )]

SC

1.2 × [1 + Afix cos(2π f drivet )]cos[4π f fix,mesh t + Bfix sin(2π f drive t )] + ,

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(19)

where Afix and Bfix are the modulation amplitudes of the AM and FM component, respectively. The

382

unwanted background noise n(t) is simulated by white gaussian noise with variance σ (0.05). All the

383

simulation parameters in the synthetic signal model are listed in Table 1. The sampling frequency

384

and duration of the synthetic signal is 16384 Hz and 20 seconds, respectively.

385

Table 1. Simulation parameters in the synthetic signal model.

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fp,m

Bring

Afix

1.35 Hz fc 0.45 Hz

35.4 Hz Bcarrier 0.5 fr,ring 7000 Hz fr,mesh 7000 Hz

0.8

0.5 Bfix 0.5 ffix,mesh 200 Hz fdrive 20 Hz

βring

EP

1 Acarrier 0.6 Am,ring 0.4 An,mesh 0.5

fcfr

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Aring

1200

βmesh

T0,ring 0.3 s T0,mesh 0.01 s

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1200

ϕ1 π/3

386

The time domain waveform and amplitude spectrum of the synthetic signal is illustrated in Fig.

387

6(a) and (b), respectively. As we can observe, no fault related impulses can be revealed in Fig. 6(a)

388

and the planetary gear related vibration components are completely overwhelmed by the

389

interferences from the fixed-axis gear related meshing vibration in Fig. 6(b). Further, the traditional

390

modulation sidebands analysis and envelope spectrum analysis are profoundly carried out, as

391

illustrated in Fig. 6(c) and (d). As observed in the modulation sidebands in Fig. 6(c), the planetary

392

meshing frequency fp,m are modulated by the carrier rotational frequency fc, which originates from

19

ACCEPTED MANUSCRIPT the modulations induced by the revolving planet gears. Though the sideband spacing of fault

394

characteristic frequency fcfr (3fc) can be found in Fig. 6(c), it may also be interpreted as the third

395

harmonics of the carrier rotational frequency. Similarly in the envelope spectrum in Fig. 6(d), it is

396

uncertain whether the frequency 3fc results from the third harmonics of fc or the fault characteristic

397

frequency fcfr. Therefore, traditional spectral analysis techniques based on the modulation sidebands

398

and envelope spectrum would encounter the dilemma that the fault characteristic frequency fcfr

399

cannot be distinguished from the third harmonics of the carrier rotational frequency 3fc, and thus

400

cannot provide sufficient evidence to achieve the fault diagnosis of planetary ring gear.

TE D f p,m +6f c

f p,m+2f c f p,m +3f c

f p,m+f c

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f p,m -2f c f p,m -f c

Amplitude

f p,m

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402

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Amplitude

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Fig. 6. The simulation signal: (a) time domain waveform; (b) amplitude spectrum of the raw signal

404

in (a); (c) modulation sidebands around the planetary meshing frequency; (d) Hilbert envelope

405

spectrum of the raw signal in (a).

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To address the aforementioned dilemma, the proposed MFM-EWT framework is applied to

407

analyze the synthetic signal. Based on the meshing frequency modulation phenomenon and the

408

MFMindex, the iterative backward-forward search algorithm is implemented to adaptively

409

determine the Fourier spectrum segments for EWT. The results of MFMindex assisted Fourier

410

boundary detection are illustrated in Fig. 7(a), where two Fourier segments including [fl1, fp] and [fp,

20

ACCEPTED MANUSCRIPT fh1] are automatically determined. In Step 1 for initial Fourier segments, the detected Fourier

412

boundary fp is 6944 Hz, which reaches a good agreement with the predetermined resonance

413

frequency fr,ring (7000 Hz) and proves the effectiveness of the proposed search algorithm. Besides,

414

the first implementations in the iterative backward (Step 2) and iterative forward (Step 3) search

415

processes achieve the maximal MFMindex at the first candidate frequency, and their maximal

416

MFMindex values are greater than the maximum MFMindex in Step 1, as illustrated in Fig. 7(a).

417

Therefore, the subsequent iterative search operations are omitted to reduce computation complexity.

418

As a result, the empirical wavelet filter bank constructed by Fourier segments [0, fl1], [fl1, fp], [fp, fh1]

419

and [fh1, Fs/2] is illustrated in Fig. 7(b). Finally, two extracted modes (mode 2 and 3) by MFM-EWT

420

and their corresponding envelope spectrums are illustrated in Fig. 7(c) and (d). The envelope

421

spectrums reveal the fault characteristic frequency fcfr of planetary ring gear and its multiples (k×fcfr)

422

significantly. Moreover, the ambiguity resulting from multiples of carrier rotational frequency are

423

greatly alleviated. Therefore, our proposed MFM-EWT framework shows an excellent performance

424

in the fault feature identification for planetary ring gear and greatly alleviate the interferences from

425

multiples of planet carrier rotational frequency.

Amplitude

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Amplitude Amplitude

426

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MFMindex

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Fig. 7. The results of synthetic signal using the MFM-EWT framework. (a) Adaptive Fourier 21

ACCEPTED MANUSCRIPT spectrum boundary detection using MFMindex; (b) empirical wavelet filter bank in frequency

430

domain, which are constructed with the detected Fourier spectrum boundaries in (a); (c)-(d) the

431

second and third mode extracted by MFM-EWT and their envelope spectrums, respectively.

432

5 Case study: fault diagnosis of wind turbine planetary ring gear

433

In this section, experiments are conducted in a wind turbine gearbox test bench and serve as a case

434

study to validate the effectiveness of the proposed MFM-EWT framework for wind turbine

435

planetary ring gear fault detection. To further demonstrate the superiority of our framework,

436

comparative studies with two advanced fault diagnosis methods are conducted, including ensemble

437

empirical mode decomposition and spectral kurtosis.

438

5.1 Fault diagnosis using the MFM-EWT framework

439

The vibration measurement experiments are conducted in the wind turbine gearbox test bench at the

440

machine dynamics and fault diagnostics laboratory of Tsinghua Univerisity. The drivetrain

441

configuration of wind turbine gearbox test bench is illustrated in Fig. 8(a), including the drive motor,

442

frequency converter, two symmetrically installed multistage gearbox with same physical parameters,

443

loading motor and loader for applying operational load to the drivetrain. The multistage gearbox

444

consists of one stage of planetary gear transmission and two stages of fixed-axis gear transmission,

445

which is configurated with the same physical parameters as a wind turbine gearbox in real

446

applications (see Fig. 1 for the drivetrain configuration of a typical wind turbine gearbox). The left

447

multistage gearbox in Fig. 8(a) serves as a speed increasing gearbox to simulate the operation

448

condition of wind turbine gearbox. Detailed physical parameters and characteristic frequencies of

449

the wind turbine gearbox test bench can be referred to Ref. [19, 35].

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The planetary ring gear with a localized fault, as depicted in Fig. 8(b), are assembled in the wind

451

turbine gearbox test bench, and the accelerometers are installed at the casing of planetary gearbox to

452

measure vibration signals (see Fig. 8(c) for the location of accelerometers). In our vibration

453

measurements, the drive motor rotates at 1514 RPM and the oil pressure from the pump into the 22

ACCEPTED MANUSCRIPT loader is 0.35 MPa. In such operation condition, the characteristic frequencies for wind turbine

455

planetary gearbox are listed in Table 2. Note that the fault characteristic frequency fcfr of planetary

456

ring gear is N times the carrier rotational frequency fc, i.e., Nfc, where the number of planet gears N

457

is 3 in our experiment case.

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Fig. 8. Wind turbine gearbox test bench: (a) drivetrain configuration; (b) planetary ring gear with a

460

localized fault; (c) location of accelerometers in vibration measurement.

461

Table 2. Characteristic frequencies for wind turbine planetary gearbox.

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Rotational frequency

Fault characteristic frequency

Meshing frequency

Carrier Sun gear Ring gear Planet gear

fc (0.448) fs (2.53) -fp (0.694)

-fcfs (2.082) fcfr (1.344) fcfp (1.142)

fp,m (35.39)

TE D

Unit: Hz

The vibration signal was collected with a sampling frequency of 16384 Hz and the recording

463

time is 30 seconds. The time domain waveform, amplitude spectrum and envelope spectrum of the

464

faulty wind turbine gearbox vibration signal are illustrated in Fig. 9. No periodic impulses can be

465

observed in Fig. 9(a) and the kurtosis value of the raw signal is merely 2.9984, which demonstrates

466

that impulsive features induced by planetary ring gear fault are overwhelmed by strong

467

interferences, such as the high-speed fixed-axis gear meshing vibrations and random noises. From

468

the modulation sidebands around the planetary meshing frequency fp,m in Fig. 9(c), the dominant

469

component is the frequency of the first lower sideband fp,m - fc, which is slightly removed from the

470

planetary meshing frequency. This asymmetry of modulation sidebands has been predicted in Ref.

471

[33, 34]. Besides, the revolutions of planet gears will produce modulation sidebands spaced at

472

multiples of the carrier rotational frequency fc around the meshing frequency fp,m, even in the

473

healthy state of planetary gearbox [33, 34]. Thus, the frequency spacing of 3fc (fcfr) of modulation

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23

ACCEPTED MANUSCRIPT sidebands in Fig. 9(c) could not provide sufficient evidence to the faulty ring gear. As with the

475

envelope spectrum in Fig. 9(d), one of the dominant component locates at the carrier rotational

476

frequency fc. Although the fault characteristic frequency fcfr (3fc) and its second harmonic 2fcfr can

477

be observed in Fig. 9(d), it may be produced by multiples of the carrier rotational frequency fc.

478

Therefore, traditional spectral analysis techniques fail to reveal the fault signatures of wind turbine

479

planetary ring gear, due to the fact that the revolving planet gears inducing modulations result in

480

similar modulation sidebands in the healthy or faulty condition of planetary ring gear and the

481

ambiguity between fault characteristic frequency fcfr and multiples of carrier rotational frequency in

482

envelope spectrums.

484

Amplitude(m/s 2 )

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Amplitude(m/s 2 )

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Fig. 9. The vibration signal measured from the wind turbine gearbox test bench with faulty

486

planetary ring gear: (a) time domain waveform; (b) amplitude spectrum; (c) modulation sidebands

487

around the planetary meshing frequency; (d) envelope spectrum.

488

In order to extract the fault signatures of wind turbine planetary ring gear reliably and get rid of

489

the interferences from the revolving planet gear inducing modulations, the proposed MFM-EWT

490

method is applied to analyze the experimental vibration signals. The meshing frequency modulation

491

phenomenon is employed to highlight and enhance the planetary gear related vibrations. MFMindex

24

ACCEPTED MANUSCRIPT assisted adaptive Fourier spectrum segmentation for EWT is performed using the iterative

493

backward-forward search algorithm, where the upper frequency fupper for MFMindex calculation in

494

Eq. (3) is considered as 5fmesh. The results are illustrated in Fig. 10(a), where ten maxima of

495

MFMindex are detected in the search process. Accordingly, equipped with 0 and Nyquist frequency,

496

ten frequencies corresponding to the maximal MFMindex of each searching step in ascending order,

497

are determined as the final boundary limits of Fourier spectrum segments. The resulting empirical

498

wavelet filter bank in frequency domain is illustrated in Fig. 10(b). Finally, four representative

499

modes (mode 8, 9, 10 and 11) extracted by MFM-EWT and their corresponding envelope spectrums

500

are illustrated in Fig. 10(c)-(f). It is obviously found that the dominant frequencies are the fault

501

characteristic frequency fcfr of planetary ring gear and its multiples (2fcfr and 3fcfr) in the envelope

502

spectrums. Additionally, the ambiguity resulting from multiples of carrier rotational frequency and

503

other interferential frequencies are greatly alleviated. Therefore, the proposed MFM-EWT approach

504

achieves adaptive empirical wavelet transform for fault-related mode extraction and successful

505

identification of fault signatures for wind turbine planetary ring gear, while greatly alleviating the

506

interferences from the revolving planet gear inducing modulations and other unwanted components.

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Amplitude

507

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MFMindex

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508

25

Mode 9

Amplitude(m/s 2 )

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Amplitude(m/s 2 )

Mode 11

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Amplitude(m/s 2 )

Mode 8

Amplitude(m/s 2 ) Amplitude(m/s 2 )

Mode 10 Amplitude(m/s 2 ) Amplitude(m/s 2 )

509

ACCEPTED MANUSCRIPT

510

Fig. 10. The results of wind turbine gearbox vibration signal using the proposed MFM-EWT

512

framework. (a) adaptive Fourier spectrum boundary detection using MFMindex; (b) empirical

513

wavelet filter bank in frequency domain, which are constructed with the detected Fourier spectrum

514

boundaries in (a); (c)-(f) the extracted mode 8, 9, 10 and 11 by MFM-EWT and their corresponding

515

envelope spectrums, respectively.

516

5.2 Comparisons with other advanced fault diagnosis methods

517

In this section, comparative studies with two advanced fault diagnosis methods are conducted to

518

verify the superiority of the MFM-EWT framework for planetary ring gear fault detection in wind

519

turbine, including ensemble empirical mode decomposition (EEMD) and spectral kurtosis (SK).

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EEMD, as an improved version of EMD which is a powerful adaptive signal decomposition

521

method for nonlinear and nonstationary signal, preserves the strong scale separation capability of

522

EMD and largely eliminates the mode mixing problem of EMD [48]. It has been widely applied in

523

fault diagnosis of rotating machinery [49]. Thus, EEMD is applied to analyze the experiment

524

signals for comparison. In this paper, the amplitude of added noise and number of ensembles in

525

EEMD are set as 0.2 times the standard deviation of the raw signal and 200, respectively [48]. The

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first six intrinsic mode functions (IMFs) of EEMD are selected for further envelope spectrum

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IMFs contain no diagnostic information for planetary ring gear fault. These IMFs and their

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envelope spectrums are illustrated in Fig. 11. It can be concluded from Fig. 11 that no clear periodic

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impulses related to gear fault can be detected in the IMFs extracted by EEMD. Besides, compared

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with the MFM-EWT framework in Fig. 10(c)-(f), all envelope spectrums except IMF 1 fail to reveal

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the fault characteristic frequency fcfr and no visible fault features can be identified due to

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unexpected noises and interferential frequencies in Fig. 11. The main reason is that the spectral

534

content of these IMFs are widespread and the impulses induced by planetary ring gear fault are

535

overwhelmed by strong interferences from high-speed parallel gears and heavy noises. Moreover,

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the calculation costs for EEMD is tremendous, because of the expensive searching process for local

537

extrema of signal amplitude and a large number of iterations required to achieve the definition of an

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IMF. Specifically, just one implementation of EMD within EEMD requires at least 300 seconds

539

while the execution time for one EWT with determined Fourier segments costs no more than 1.2

540

seconds in our experimental case, where the comparison is performed under Windows 7 operating

541

system and MATLAB 2016b running on a computer equipped with an Intel Core i7 CPU at 3.20

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GHz and 12 GB of RAM. Therefore, EEMD has limited capability to identify fault signatures of the

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planetary ring gear in wind turbine gearbox, and it suffers greatly from interferential frequencies

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and the time-consuming computational expense. IMF 2 Amplitude(m/s 2 ) Amplitude(m/s 2 )

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IMF 1 Amplitude(m/s 2 ) Amplitude(m/s 2 )

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IMF 4 Amplitude(m/s 2 ) Amplitude(m/s 2 ) IMF 6 2 Amplitude(m/s 2 ) Amplitude(m/s )

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IMF 5 Amplitude(m/s 2 ) Amplitude(m/s 2 )

IMF 3 Amplitude(m/s 2 ) Amplitude(m/s 2 )

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Fig. 11. The results of wind turbine gearbox vibration signal using EEMD. (a)-(f) the extracted

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IMFs by EEMD and their corresponding envelope spectrums, respectively. Spectral kurtosis technique, as a strong benchmark for fault-induced impulse feature extraction

551

[36], has been applied in a wide range of applications for fault diagnosis of rotating machines [50].

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Moreover, it has been reported that the application of SK could successfully detect the planetary

553

ring gear fault in wind turbine gearbox in Ref. [12]. Therefore, SK combined with envelope

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spectrum analysis is further applied to analyze the experiment signals for comparison. The results

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using SK technique are illustrated in Fig. 12. Fast kurtogram of the raw signal in Fig. 12(a)

556

indicates that the most impulsive frequency band with maximal spectral kurtosis value locates in the

557

region with center frequency 896 Hz and bandwidth 256 Hz. However, the dominant frequency of

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the envelope spectrum in Fig. 12(c) is the rotational frequency fc of planet carrier and no useful

559

diagnostic information at the fault characteristic frequency fcfr can be revealed. Therefore, spectral

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kurtosis technique is ineffective to identify the periodic impulse features induced by the planetary

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ring gear fault in wind turbine gearbox, due to the fact that the impulse features repeat at a very low

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frequency and are overwhelmed by strong interferences from the high-speed parallel gears.

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Amplitude(m/s 2 ) Amplitude(m/s 2 )

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Fig. 12. The results of wind turbine gearbox vibration signal using SK technique: (a) kurtogram of

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the raw signal (the filter band indicated by the orange arrow is determined by the maximum

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kurtosis); (b) the filtered signal via spectral kurtosis; (c) envelope spectrum of the filtered signal.

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The preceding comprehensive comparative studies demonstrate that the MFM-EWT framework

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has superior performance over EEMD and spectral kurtosis techniques at feature extraction and

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identification for the planetary ring gear fault in wind turbine gearbox.

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6 Conclusions

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In this paper, a novel meshing frequency modulation assisted empirical wavelet transform

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framework is proposed for fault diagnosis of wind turbine planetary ring gear. The main

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contributions are summarized as follows. Firstly, a meshing frequency modulation phenomenon is

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pointed out and an indicator MFMindex is formulated to evaluate the significance level of meshing

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frequency modulation quantitively, which is beneficial to alleviate the interferences from the

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high-speed parallel gear meshing vibrations in wind turbine multi-stage gearbox. Secondly, the

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MFMindex assisted adaptive Fourier spectrum segmentation scheme is developed using an iterative

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backward-forward search algorithm such that the adaptive empirical wavelet transform is achieved

579

for fault-related mode extraction. Moreover, it is obtained from the numerical and experimental

580

results that the MFM-EWT framework could successfully extract and identify the fault features of

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wind turbine planetary ring gear. Finally, compared with ensemble empirical mode decomposition

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and spectral kurtosis techniques, MFM-EWT framework exhibits superior performance to reveal

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evident fault features and avoid the ambiguity from the planet carrier rotational frequency for fault

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diagnosis of wind turbine planetary ring gear. It should be pointed out that the meshing frequency modulation phenomenon was

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experimentally tested and proved in the wind turbine gearbox test bench. Therefore, the dynamical

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modeling and mechanism explanation for meshing frequency modulation phenomenon should be

588

further investigated in the future work. Moreover, the meshing frequency modulation assisted

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empirical wavelet transform framework for adaptive mode extraction should be extended to fault

590

diagnosis for other critical components and even compound fault diagnosis for multiple components

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in wind turbine gearbox.

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Acknowledgments

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This research work was partially supported by National Natural Science Foundation of China

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under Grant No. 51335006 and 51605244. The valuable comments and suggestions from the editor

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and anonymous reviewers are highly appreciated.

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Highlights


A novel meshing frequency modulation phenomenon is pointed out in the vibration signal of

wind turbine gearbox.


An adaptive Fourier spectrum segmentation scheme is developed for empirical wavelet




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transform. A meshing frequency modulation assisted empirical wavelet transform method is proposed to

extract fault-related modes for fault diagnosis of wind turbine gearbox.


Simulation and experimental validations are carried out to demonstrate the feasibility of the

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proposed fault diagnosis method for wind turbine planetary ring gear.