Cheng_2012_Digital-Signal-Processing

Digital Signal Processing22(2012)

356–366

Contents lists available at SciVerse ScienceDirect

Digital Signal Processing

https://www.360docs.net/doc/f918172503.html,/locate/dsp

A rotating machinery fault diagnosis method based on local mean decomposition Junsheng Cheng a,b,?,Yi Yang a,b,Yu Yang a,b

a State Key Laboratory of Advanced Design and Manufacturing for Vehicle Body,Hunan University,Changsha,410082,PR China

b College of Mechanical and Vehicle Engineering,Hunan University,Changsha,410082,PR China

a r t i c l e i n f o a

b s t r a

c t

Article history:

Available online19October2011

Keywords:

Local mean decomposition Product functions

Modulation

Time–frequency analysis Rotating machinery

Fault diagnosis Local mean decomposition(LMD)is a novel self-adaptive time–frequency analysis method,which is particularly suitable for the processing of multi-component amplitude-modulated and frequency-modulated(AM–FM)signals.By using LMD,any complicated signal can be decomposed into a number of product functions(PFs),each of which is the product of an envelope signal and a purely frequency modulated signal from which physically meaningful instantaneous frequencies can be obtained.In fact, each PF is just a mono-component AM–FM signal.Therefore,the procedure of LMD may be regarded as the process of demodulation.While fault occurs in gear or roller bearing,the vibration signals picked up would exactly display AM–FM characteristics.So it is possible to diagnose gear and roller bearing fault by LMD.Targeting the modulation features of the gear or roller bearing fault vibration signal,a rotating machinery fault diagnosis method based on LMD is proposed.In this paper,?rstly the LMD method is introduced;secondly,the LMD method is compared with another competing time–frequency analysis approach,namely,empirical mode decomposition(EMD)method and the results show the superiority of the LMD method;?nally,the LMD method is applied to the gear and roller bearing fault diagnosis. The analysis results from the practical gearbox vibration signal demonstrate that the diagnosis approach based on LMD could identify gear and roller bearing work condition accurately and effectively.

?2011Published by Elsevier Inc.

0.Introduction

Fault feature extraction is the key step in rotating machin-ery fault diagnosis.Currently there are many techniques of fault characteristic extraction available for the detection of rotating ma-chinery faults such as frequency/spectrum analysis,time/statistical analysis,time–frequency analysis and so on,among which time–frequency analysis has become the well-accepted techniques for it can provide time and frequency domain information of a signal si-multaneously[1,2].

However,the time–frequency analysis methods such as win-dowed Fourier transform(WFT)and wavelet transform have their own limitations.WFT can display a time signal on a joint time–frequency plane,but once the window function is chosen,the size of the time–frequency window would be?xed,therefore the time and frequency resolution are same for all components that include different time scales[3].Wavelet analysis could provide local fea-tures in both time and frequency domains and has the feature of multi-scale,which enables wavelet analysis to distinguish the abrupt components of the vibration signal[4].Therefore,wavelet analysis has been widely applied to rotating machinery fault di-

*Corresponding author at:State Key Laboratory of Advanced Design and Manu-facturing for Vehicle Body,Hunan University,Changsha,410082,PR China.Fax:+86 0731********.

E-mail address:signalp@https://www.360docs.net/doc/f918172503.html,(J.Cheng).agnosis[5].However,wavelet analysis is essentially an adjustable window Fourier transformation.When a signal is decomposed by wavelet,only the rectangular time–frequency partition of time–frequency plane can be obtained and such tile partition would fail to guarantee that instantaneous frequencies of each resulting component obtained by wavelet transform own physical signi?-cance.Therefore,wavelet analysis has no self-adaptive feature in nature[6].

Empirical Mode Decomposition(EMD)is a self-adaptive signal processing method that could decompose a complicated signal into a number of intrinsic mode functions(IMFs)whose instantaneous frequencies have physical meaning.By performing Hilbert trans-form to each IMF,the corresponding instantaneous amplitude and instantaneous frequency can be calculated.Furthermore,the com-plete time–frequency distribution can be obtained[7,8].Since EMD method was developed,it has been widely used in machinery fault diagnosis and other?elds[9–11].However,there still exist many de?ciencies in EMD such as the end effects[12],mode mixing[13], IMF criterion[14],the problem of quick algorithm and envelope line and so on[15],which are still underway.In addition,some-times the unexplainable negative instantaneous frequency would appear when calculating instantaneous frequency by performing Hilbert transform to each IMF[16].

Recently,a new self-adaptive time–frequency analysis method, local mean decomposition(LMD,as de?ned in Section1)was put forward by Jonathan S.Smith and has been used to analyze

1051-2004/$–see front matter?2011Published by Elsevier Inc. doi:10.1016/j.dsp.2011.09.008

J.Cheng et al./Digital Signal Processing22(2012)356–366357

electroencephalogram signal[16].LMD can self-adaptively decom-

pose a complicated multi-component signal into a set of prod-

uct functions(PFs),each of which is the product of an enve-

lope signal from which instantaneous amplitude of the PF can

be got and a purely frequency modulated signal from which a

well-de?ned instantaneous frequency could be calculated.There-

fore,each resulting PF component is,in fact,a mono-component

amplitude-modulated and frequency-modulated(AM–FM)signal.

Furthermore,the complete time frequency distribution of the orig-

inal signal could be obtained by assembling the instantaneous am-

plitude and instantaneous frequency of all PF components.

Since a multi-component AM–FM signal could be decomposed

into a set of mono-component AM–FM signals by LMD,LMD is

suitable for processing of the multi-component AM–FM signal.

When gear or roller bearing fault occurs in the rotating machin-

ery,it is generally the case that the vibration signals measured

by sensor present AM–FM feature[17,18].For this kind of signal,

the demodulation analysis is the most common method.There-

fore it is possible to apply LMD to the feature extraction of gear

and roller bearing fault vibration signals because the decomposi-

tion process of LMD is exactly the process of demodulation.Fur-

thermore the modulation feature could be extracted effectively by

applying spectra analysis to instantaneous amplitude of each PF.In

this paper,LMD is introduced into simulation signals analysis and

the analysis results demonstrate the validity of LMD.In addition,

a comparison is made to another competing self-adaptive decom-

position approach,EMD.Furthermore,we apply the methodology

to the analysis of gearbox and roller bearing vibration signals.The

practical signal analysis results show that LMD can be applied to

the rotating machinery fault diagnosis effectively.

This paper is organized as follows.In Section1we give LMD.

Comparison of simulation signals analysis between LMD and EMD

is given in Section2,which shows that superior results can be

obtained by LMD.The analysis results from gear and roller bearing

fault vibration signals are given in Section3.Finally,we offer the

conclusion in Section4.

1.LMD analysis method

The nature of LMD is to demodulating AM–FM signals.By using

LMD a complicated signal can be decomposed into a set of product

functions,each of which is the product of an envelope signal and

a purely frequency modulated signal.Furthermore,the completed

time–frequency distribution of the original signal can be derived.

For any signal x(t),it can be decomposed as follows[16]:

(1)Determine all local extrema n i of the original signal x(t),

and then the mean value m i of two successive extrema n i and n i+1

can be calculated by

m i=n i+n i+1

(1)

All mean value m i of two successive extreme are connected by straight lines,and then local mean function m11(t)can be formed by using moving averaging to smooth the local means m i.

(2)A corresponding envelope estimate a i is given by

a i=|n i?n i+1|

(2)

Similarly,the envelope estimate a i is smoothed in the same way and corresponding envelope function a11(t)is formed.

(3)The local mean function m11(t)is subtracted from the orig-inal signal x(t)and the resulting signal h11(t)is given by

h11(t)=x(t)?m11(t)(3)

(4)h11(t)can be amplitude demodulated by dividing it by en-velope function a11(t)s11(t)=h11(t)/a11(t)(4) Ideally,s11(t)is a purely frequency modulated signal,namely, the envelope function a12(t)of s11(t)should satisfy a12(t)=1.If a12(t)=1,then s11(t)is regarded as the original signal and the above procedure needs to be repeated until a purely frequency modulated signal s1n(t)that meets?1 s1n(t) 1is derived.In other words,envelope function a1(n+1)(t)of the resulting s1n(t) should satisfy a1(n+1)(t)=1.Therefore

???

h11(t)=x(t)?m11(t)

h12=s11(t)?m12(t)

h1n(t)=s1(n?1)(t)?m1n(t)

(5)

in which,

???

s11(t)=h11(t)/a11(t)

s12(t)=h12(t)/a12(t)

s1n(t)=h1n(t)/a1n(t)

(6) where the objective is that

lim

n→∞

a1n(t)=1(7)

In practice,a variationδcan be determined in advance.If1?δ a1(n+1)(t) 1+δand?1 s1n(t) 1,then iterative process would be stopped.

(5)Envelope signal a1(t),namely,instantaneous amplitude function,can be derived by multiplying together the successive envelope estimate functions that are acquired during the iterative process described above.

a1(t)=a11(t)a12(t)···a1n(t)=

q=1

a1q(t)(8)

where q is the times of the iterative process.

(6)Multiplying envelope signal a1(t)by the purely frequency modulated signal s1n(t)the?rst product function PF1of the origi-nal signal can be obtained.

PF1(t)=a1(t)s1n(t)(9) PF1contains the highest frequency oscillations of the orig-inal signal.Meantime,it is a mono-component AM–FM signal, whose instantaneous amplitude is exactly the envelope signal a1(t) and instantaneous frequency is de?ned from the purely frequency modulated signal s1n(t)as

f1(t)=

d[arccos(s1n(t))]

d t

(10)

(7)Subtract the?rst PF component PF1(t)from the original sig-nal x(t)and we have a new signal u1(t),which becomes the new original signal and the whole of the above procedure is repeated, i.e.up to k times,until u k becomes monotonic function

???

u1(t)=x(t)?PF1(t)

u2(t)=u1(t)?PF2(t)

u k(t)=u k?1(t)?PF k(t)

(11)

Thus,the original signal x(t)was decomposed into k-product and a monotonic function u k

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Fig.1.Simulation signal x (t )and its two components.

x (t )=

k p =1

PF p (t )+u k (t )

(12)

where p is the number of the product function.

Furthermore,the corresponding complete time–frequency dis-tribution could be obtained by assembling the instantaneous am-plitude and instantaneous frequency of all PF components.2.Applications to simulation signals

In LMD method,the ?rst step is to determine the local ex-trema of the original signal.However,from a practical point of view the two ends of the signal may be neither the local max-ima nor the local minima that means so-called end effects would also appear in LMD that is the same as the EMD approach.There-fore,the end effect should be restrained.Here in order to compare the decomposition e?ciencies of LMD and EMD,?rstly the two de-composition methods in which end effect is not solved on purpose are carried out respectively to analyze the same simulation signal.

Here we consider a multi-component AM–FM signal x (t )

x (t )= 1+0.5cos

πt

100

cos

πt

+2cos

πt

+4sin

πt

2500

sin

6πt 50

t =0,1,...,600(13)

Fig.1shows the simulation signal and its two components.Af-ter setting the variation δ=10?5,LMD is applied to decompose

the signal.When the iterative times reaches 22,the iterative stop-ping condition 1?δ a 1(n +1)(t ) 1+δand ?1 s 1n (t ) 1are satis?ed that means the ?rst PF component PF 1(t )has been suc-cessfully separated from the original x (t ).Continuing the iterative process 5times and the second PF component PF 2(t )is then ac-quired.The decomposition result by LMD is shown in Fig.2,from which we know that the two PF components exactly correspond to the two AM–FM components in the origin signal.Therefore LMD is a self-adaptive decomposition method according to signal itself and each physically meaningful PF component re?ects the nature of the signal.

Replacing LMD method by EMD,energy difference tracking method is used to determine IMF component in IMF “sifting”pro-cess [14].The ?rst two IMF components c 1(t )and c 2(t )shown in Fig.3are obtained after the “sifting”process continues 262and 153times,https://www.360docs.net/doc/f918172503.html,pare Fig.2and Fig.3,it is sug-gested that although end effect are needed to restrain in both LMD and EMD,the purely frequency modulated signal from which a physically meaningful instantaneous frequency can be derived is calculated by certain division in LMD,which results in the

faster

Fig.2.LMD decomposition results of simulation signal x (t )

Fig.3.EMD decomposition results of simulation signal x (t )

Fig.4.The instantaneous amplitude a 1(t )of the ?rst PF component PF 1(t )obtained by LMD.

compute speed and less iterative times.So the end effect is not obvious in LMD.By contrast,in EMD each IMF component from which a well-de?ned instantaneous frequency also can be derived is acquired by continuous “sifting”that would lead to more itera-tive times.Therefore distinct end effect and greater residue r 2(t )appear.The analysis results show that the LMD is superior to EMD under the circumstances that end effect is not solved.

Figs.4–7give the instantaneous amplitudes and instantaneous frequencies of the ?rst two PF components derived from LMD,respectively,which show that the instantaneous amplitude and in-stantaneous frequency re?ect the true information of the original.

On the other hand,instantaneous amplitude and instantaneous frequency of each IMF component shown in Fig.3also can be

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Fig.5.The instantaneous frequency f 1(t )of the ?rst PF component PF 1(t )obtained by

LMD.

Fig.6.The instantaneous amplitude a 2(t )of the second PF component PF 2(t )ob-tained by

LMD.

Fig.7.The instantaneous frequency f 2(t )of the second PF component PF 2(t )ob-tained by

LMD.

Fig.8.The instantaneous amplitude a 1(t )of the ?rst IMF component c 1(t )obtained by EMD.

calculated by performing Hilbert transform to IMFs,which are il-lustrated in Figs.8–11.It is clear from the ?gures that notable distortion has been caused in instantaneous amplitude and instan-taneous frequency derived from EMD,which can be explained by the fact of end effect of EMD and associate Hilbert transform,from which we know that LMD outperforms EMD method.

Now we consider the situation that LMD and EMD are used,re-spectively,to decompose the same simulation signal after the end effect has been restrained effectively by using the support vector regression machines [12].The decomposition results are show in Figs.12and 13respectively and the corresponding instantaneous amplitude and instantaneous frequency of each component are demonstrated in Figs.14–21,from which we know that both LMD and EMD can give excellent decomposition results when the end effect is solved.However the difference of decomposition

e?ciency

Fig.9.The instantaneous frequency f 1(t )of the ?rst IMF component c 1(t )obtained by

EMD.

Fig.10.The instantaneous amplitude a 2(t )of the second IMF component c 2(t )ob-tained by

EMD.

Fig.11.The instantaneous frequency f 2(t )of the second IMF component c 2(t )ob-tained by EMD.

between two approaches still exists.In EMD method,relatively greater ?uctuation has been generated in instantaneous amplitude and instantaneous frequency of each IMF component because of the cubic spline envelope error resulted from too many “sifting”times.On the contrast,less ?uctuation appear in instantaneous amplitude and instantaneous frequency of each PF component de-rived from LMD,which can be explained by the fact of less iter-ation times and the iteration process in LMD https://www.360docs.net/doc/f918172503.html,pare the analysis results before and after end effect is restrained,it is clear that the LMD is superior to EMD.

3.Application to gear fault diagnosis

When the gear with faults is operating,the corresponding vi-bration signals always present the feature of AM–FM.To extract fault feature of gear modulation signal,demodulation analysis has been established as the widespread method,among which the Hilbert transform has been widely used in gear fault diagnosis as one of the most common demodulation methods for it has quick algorithm and could extract envelop of the fault vibration signal effectively [6].However,there are two main problems in demod-ulation approach based on Hilbert transform.On the one hand,owing to the inevitable window effect of Hilbert transform,the demodulation results display non-instantaneous response charac-teristic,that is,at the two ends of the modulation signal which has been demodulated as well as the middle part of the mod-ulation signal with break amplitude would produce modulation again,which makes the amplitude ?uctuates in an exponential

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Fig.12.LMD decomposition results of simulation signal x (t )after end effect is re-

strained.

Fig.13.EMD decomposition results of simulation signal x (t )after end effect is re-

strained.

Fig.14.The instantaneous amplitude a 1(t )of the ?rst PF component PF 1(t )obtained by LMD after end effect is

restrained.

Fig.15.The instantaneous frequency f 1(t )of the ?rst PF component PF 1(t )obtained by LMD after end effect is

restrained.Fig.16.The instantaneous amplitude a 2(t )of the second PF component PF 2(t )ob-tained by LMD after end effect is

restrained.

Fig.17.The instantaneous frequency f 2(t )of the second PF component PF 2(t )ob-tained by LMD after end effect is

restrained.

Fig.18.The instantaneous amplitude a 1(t )of the ?rst IMF component c 1(t )ob-tained by EMD after end effect is

restrained.

Fig.19.The instantaneous frequency f 1(t )of the ?rst IMF component c 1(t )obtained by EMD after end effect is

restrained.

Fig.20.The instantaneous amplitude a 2(t )of the second IMF component c 2(t )ob-tained by EMD after end effect is restrained.

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Fig.21.The instantaneous frequency f2(t)of the second IMF component c2(t)ob-tained by EMD after end effect is

restrained.

Fig.22.The sketch of No.3fan gearbox.

attenuation way that resulted in the increasing of demodulation error.On the other hand,Hilbert demodulation method is appli-cable for mono-component AM–FM signal while in practice most gear fault vibration signals are multi-component AM–FM signals. For this kind of problem,the signals are usually decomposed into single component AM–FM signals by band-pass?lter and then de-modulated to extract frequencies and amplitudes information in conventional demodulation methods.However,both the number of the carrier frequency components and the magnitude of the carrier frequency are hard to be determined in practice,so the selection of central frequency of band-pass?lter carries great subjectivity that would bring demodulation error and make it impossible to extract the characteristics of gear fault vibration signal.

Targeting these problems of Hilbert transform,a gear fault diag-nosis method based on LMD is presented in this paper.Essentially LMD can separate a multi-component signal into a number of sig-nal component AM–FM signals(PF components),from which the instantaneous amplitude and instantaneous frequency of each PF component could be calculated.Furthermore,modulation infor-mation can be extracted by performing spectrum analysis to the instantaneous amplitude of each PF component,and thus the fault feature of gear vibration signals would be extracted effectively.

Fig.22gives the sketch of No.3fan gearbox of a petrochemical engineering company,in which the teeth number of the gear1,2, 3,and4are11,31,23and53,respectively.The rotation speed n I of the shaft I is980r/min,namely,the rotating frequency is approx-imately16.3Hz.Consequently,the rotating frequency of the shaft II and III are approximately5.8Hz and2.5Hz,respectively.The piezoelectricity acceleration sensor has been?xed on the shell of the gearbox.The self-developed fan condition monitoring system could monitor the root mean square value and peak value of gear-box vibration signal on line.On June24th,2008,root mean square value is observed to be greater than the threshold value that sug-gests the abnormality has appeared in vibration signal.To identify the work condition of gearbox,LMD method is used to analyze

the

Fig.23.The vibration acceleration signal of the fan

gearbox.

Fig.24.The amplitude spectrum of vibration acceleration signal.

perpendicular vibration acceleration signal on9:00AM on moni-toring point A showed in Fig.22.Figs.23and24give the vibration acceleration signal(the sample frequency is1024Hz)picked up and the corresponding amplitude spectrum.From Fig.24it can be known that the frequency components of the signal is complicated and the main frequency component include66Hz,83Hz,117Hz, 133Hz,182Hz,199Hz,215Hz,232Hz,298Hz,314Hz,347Hz, 364Hz,380Hz,397Hz,413Hz,430Hz,446Hz and479Hz re-spectively,in which133Hz is exactly the mesh frequency of gear 3and gear4.The other frequencies seem to be the sideband of the mesh frequency of gear1and gear2(179Hz)for the difference between the successive frequencies is16Hz or17Hz,which is approximately equal to the rotating frequency of gear1.However, the mesh frequency179Hz and its sideband can’t be found in the spectrum shown in Fig.24.Hence it is hard to estimate whether and where fault has occurred in the gearbox.So the further study is needed.

Applying LMD method to the vibration signal shown in Fig.23,?ve PFs and a residue are acquired shown in Fig.25.Fig.26gives the instantaneous amplitudes of each PF component and Figs.27–31give the corresponding amplitude spectrum of the instanta-neous amplitudes.From Fig.27we know that there are obvious spectral lines in multiple frequencies of rotation frequency f I and f III that is also true in Fig.28.While in Fig.29and Fig.31obvi-ous spectral lines can be found in multiple frequencies of rotation frequency f I and f II.Meantime,we can easily?nd the spectral lines in multiple frequencies of rotation frequency f II and f III in Fig.30.All of these suggest that it is possible that local faults have occurred in four gears of three running https://www.360docs.net/doc/f918172503.html,ter the fan gear-box was examined and the faults of glue and wear in different degree have been found in four gears,especially severe wear have appeared in gear1and gear2,which can be shown in Figs.32 and33.After replacing the fault gears the vibration signal come-back normal,which con?rmed the validity of the diagnosis method proposed.

4.Application to roller bearing fault diagnosis

When operating a roller bearing with local faults impulse is created,the high-frequency shock vibration is then generated and the amplitude of vibration signal is modulated by the impulse force.In order to extract the vibration signal characteristic of the rolling bearing with fault,it is necessary to have the vi-bration signal demodulated.However,the roller bearing fault vi-

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Fig.25.LMD decomposition results of the vibration acceleration signal of the fan

gearbox.

Fig.26.The instantaneous amplitude of each PF component shown in Fig.25.

bration signal is a multi-component amplitude-modulated signal,which should be decomposed into the sum of single-component amplitude-modulated signal before demodulation.LMD method is adopted to decompose the roller bearing vibration signal in this paper.Fig.34shows the time domain waveform of vibration ac-celeration signal of the rolling bearing with out-race fault.In this experiment [19],the tested bearing is the 6311-type roller bear-ing.The sampling frequency is 4096Hz and the rotary frequency is 25Hz,and by calculation the out-race fault characteristic fre-quency is 76Hz.The vibration acceleration signal is decomposed by LMD method and 4PF components and one residue shown in Fig.35are obtained.Fig.36gives the instantaneous amplitudes of each PF component.By applying Hilbert transform to the instan-taneous amplitude of the ?rst PF component,the corresponding amplitude spectrum shown in Fig.37can be obtained.The obvious spectral line of the out-race fault characteristic frequency (76Hz)and its double (152Hz)could be found in the amplitude spectrum shown in Fig.37,which means the roller bearing vibration signal

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Fig.27.The amplitude spectrum of instantaneous amplitude a 1(t )

Fig.28.The amplitude spectrum of instantaneous amplitude a 2(t )

Fig.29.The amplitude spectrum of instantaneous amplitude a 3(t )

Fig.30.The amplitude spectrum of instantaneous amplitude a 4(t )

Fig.31.The amplitude spectrum of instantaneous amplitude a 5(t )

Fig.32.The worn gear 1in the fan

gearbox.

Fig.33.The worn gear 2in the fan

gearbox.

Fig.34.The vibration acceleration signal of the roller bearing with out-race fault.

has been amplitude modulated by out-race fault characteristic fre-quency that is exactly the characteristics of vibration signal when there are signs in out-race fault and accords with the actual work-ing condition of the roller bearing.

Fig.38shows the time domain waveform of vibration accelera-tion signal of the rolling bearing with inner-race fault.The rotary frequency is 20Hz and by calculation the inner-race fault char-acteristic frequency is 99.2Hz.The vibration acceleration signal is decomposed by LMD method and 5PF components and one residue shown in Fig.39are obtained.Fig.40gives the instan-taneous amplitudes of each PF component.By applying Hilbert transform to the instantaneous amplitude of the ?rst PF and the second components,the corresponding amplitude spectra shown in Figs.41and 42can be obtained,respectively.The obvious spec-tral line of the inner-race fault characteristic frequency (99.2Hz)and the double rotary frequency (40Hz)could be found in the amplitude spectra shown in Fig.42and Fig.42,which means the roller bearing vibration signal has been amplitude modulated by inner-race fault characteristic frequency that is exactly the charac-teristics of vibration signal when the roller bearing has inner-race fault and accords with the actual working condition of the roller bearing.

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Fig.35.LMD decomposition results of the vibration acceleration signal of the roller bearing with out-race

fault.

Fig.36.The instantaneous amplitude of each PF component shown in Fig.

35.

Fig.37.The amplitude spectrum of instantaneous amplitude a 1(t )shown in Fig.36.

5.Conclusions

LMD is a novel time–frequency analysis method,the main inno-vation of which is that a mono-component AM–FM signal is con-sidered as a product of an envelope signal and a purely frequency modulated signal.By de?ning local mean function and

envelope

Fig.38.The vibration acceleration signal of the roller bearing with inner-race fault.

estimate function,purely frequency modulated signal from which a physical meaning instantaneous frequency could be calculated and the corresponding envelope signal may be got in repeating smoothing process,and then PF component can be derived by mul-tiplying the envelope signal and purely frequency modulated sig-nal.Since each PF component is a mono-component AM–FM sig-nal in fact,LMD is suitable for the processing non-stationary and non-linear signals,especially for multi-component AM–FM signals.Most importantly,LMD is a self-adaptive decomposition method

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Fig.39.LMD decomposition results of the vibration acceleration signal of the roller bearing with inner-race

fault.

Fig.40.The instantaneous amplitude of each PF component shown in Fig.39.

according to signal itself,and thus the resulting PF component has certain physical meaning,which can re?ect the nature of the sig-nal.

In this paper LMD approach is introduced and the comparison between LMD and EMD was carried out.The analysis results from simulation signal demonstrate the superiority of the LMD method that is as follows:

(1)The end effect is not obvious in LMD approach because of faster algorithm speed and less iterative times.While in EMD IMF component is derived by repeating“sifting”that would lead to more iterative times.Therefore distinct end effect would ap-pear in EMD.Furthermore,the more iterative times needed in EMD method would produce more envelope errors,which would result in the?uctuation of the instantaneous frequencies and amplitudes.

(2)After the end effect is resolved,the resulting instantaneous frequency by LMD is closer to the truth and has less false fre-quency components that is because the instantaneous frequency is derived from purely frequency modulated signal without the recourse to the Hilbert transform.By contrast,the EMD“sifting”process is designated to produce an IMF that can be Hilbert trans-

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Fig.41.The amplitude spectrum of instantaneous amplitude a 1(t )shown in Fig.

40.

Fig.42.The amplitude spectrum of instantaneous amplitude a 2(t )shown in Fig.40.

formed to generate an analytic signal from which the instanta-neous frequency can be obtained,which would lead to the un-explainable negative frequency.

(3)The LMD iteration process which uses smoothed local means and local magnitudes avoids the cubic spline approach used in EMD.Since the cubic spline approach may bring the envelope error,which can in?uence on the precision of the instantaneous frequency and amplitude,the instantaneous frequency and ampli-tude of the original signal from the LMD PFs can be more accurate than those from the EMD IMFs.

Based upon the analysis above and to target the modulation feature of gear and roller bearing vibration signal,a rotating ma-chinery fault diagnosis method based on LMD is put forward.The practical analysis results from gearbox vibration signal illustrated that the method proposed could be applied to gear and roller bear-ing fault diagnosis effectively.

It should be noted that since LMD was developed recently,there are many theoretical problems to require further exploration such as iterative stopping condition,which would in?uence the orthogonality between PF components and the obtaining of physi-cal signi?cance instantaneous frequency.Theoretically,the iterative stopping condition should meet lim n →∞a 1n (t )=1,which is di?-cult to satisfy in practice.Therefore further study about the itera-tive stopping condition is needed.With the improvement of these problems,the application of LMD would be wider.Acknowledgments

The supports for this research under Chinese National Sci-ence Foundation Grants (No.51075131,No.51175158)and Hunan Provincial Natural Science Foundation of China (No.11JJ2026)are deeply appreciated.References

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Junsheng Cheng ,male,was born in October,1968.He received the Ph.D.in manufacturing engineering and automation from Hunan Univer-sity in 2005.He is a Professor in College of Mechanical and Vehicle Engineering,Hunan University.His primary research interests include me-chanical fault diagnosis,dynamics signal processing,vibration and noise control.His research has been published in “Mechanical System and Signal Processing”,“Signal Processing”,“Mechanism and Machine Theory”and other journals.

Yi Yang ,female,was born in April,1984.She received the Bachelor de-gree in manufacturing engineering and automation from Hunan University in 2009.She is a Master candidate in College of Mechanical and Vehicle Engineering,Hunan University.Her primary research interests include me-chanical fault diagnosis.

Yu Yang ,female,was born in April,1971.She received the Ph.D.in manufacturing engineering and automation from Hunan University in 2005.She is a Professor in College of Mechanical and Vehicle Engineering,Hunan University.Her primary research interests include mechanical fault diagnosis,dynamics signal processing.Her research has been published in “Journal of Sound and Vibration”,“Measurement”and other journals.