Research ArticleInstantaneousAmplitude-Frequency Feature Extraction for RotorFault Based on BEMD and Hilbert Transform

Chuanjin Huang 1 Haijun Song1 Wenping Lei2 Zhanya Niu3 and Yajun Meng1

1Zhengzhou Institute of Technology No 18 Ying Cai Street Hui Ji District Zhengzhou 450044 China2School of Mechanical Engineering Zhengzhou University Zhengzhou 450052 China3Henan Suda Electric Vehicles Technology Co Ltd Sanmenxia 472000 China

Correspondence should be addressed to Chuanjin Huang zzdxhcj163com

Received 26 October 2018 Revised 9 January 2019 Accepted 19 February 2019 Published 12 March 2019

Academic Editor Davood Younesian

Copyright copy 2019 Chuanjin Huang et al -is is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

-e vibration signals propagating in different directions from rotating machines can contain a variety of characteristic in-formation A novel feature extraction method based on bivariate empirical mode decomposition (BEMD) for rotor is proposed tocomprehensively extract the fault features In this work the number of signal projection directions is determined throughsimulation and the energy end condition based on the energy threshold is increased using BEMD to enhance the decompositionquality Mixed vibration signals are generated along two orthogonal directions -en the acquired vibration signal can bedecomposed into several intrinsic mode functions (IMFs) at the rotational speed using the BEMD method Furthermore theinstantaneous frequency and instantaneous amplitude of the real signals and the imaginary part of the IMF signals are obtainedusing the Hilbert transform -e fault features along two and three dimensions can be investigated providing more compre-hensive information to aid in the fault diagnosis of rotor Experimental results on oil film oscillation the oil whirl the bistability ofthe rotor and looseness and rotor rubbing composite fault indicate the effectiveness of the proposed method

1 Introduction

-e features of vibration signals form the basis of faultdiagnosis of rotating machinery A variety of common faultfeature extraction methods such as the wavelet transform(WT) [1] empirical mode decomposition (EMD) [2] andlocal mean decomposition (LMD) [3 4] are widely used forsuch diagnosis In the WT the wavelet function must bepreset which leads to limitations in the self-adaptability oftheWT-e EMD and LMDmethods adaptively decomposethe signals into a series of intrinsic mode functions (IMFs)and product functions (PFs) respectively according to thecharacteristics of the signal envelope However these tra-ditional methods are suitable only for processing real-valuedata on a single channel [5]

-e fault vibration characteristics from two orthogonalsensors which are placed in various locations on the ma-chine have been shown to be important for diagnosing

faults To improve the reliability of the diagnostic results thefeature extraction method based on multichannel signalfeatures is becoming increasingly more popular

-e orbit of each harmonic from the vibration signals isan ellipse when the rotor runs in a steady state -e faultfeature extraction methods based on homology informationsuch as the holospectrum [6] full spectrum [7] and fullvector spectrum [8] have been proposed in previous reports-e vibration signals in the orthogonal direction can form anellipse Because the homologous information incorporatesthe orthogonal direction of the vibration signal features thediagnostic results are more comprehensive and accurate [9]Fault features can be described by the homologous in-formation through round or elliptical information Un-fortunately in most cases the circle or ellipse informationcan be extracted using only a Fourier transform -ereforethe homologous information method is applicable primarilyto the analysis of stationary signals

HindawiShock and VibrationVolume 2019 Article ID 1639139 19 pageshttpsdoiorg10115520191639139

-e EMD or LMD methods can locally smooth thenonlinear signal Homologous information technologycombined with EMD and LMD [10 11] has been proposedto extract the features of the nonlinear signal However theEMD and LMD methods decompose the signals from dif-ferent directional sensors individually and separately -ecircle or ellipse information is calculated according to theone-to-one mode principle [11] When the EMD or LMD isused to decompose the orthogonal vibration signals thenumber of decomposition result is nonuniform whichcauses the information fusion to become increasinglychallenging [12] In addition the EMD or LMD method isrelatively sensitive to noise which causes the IMF group orPF group frequency to be inconsistent with other IMF or PFgroups

To improve the capacity of EMD and LMD thesemethods have been extended recently to process multivariatesignals Some examples include bivariate empirical modedecomposition (BEMD) [13] complex local mean de-composition (CLMD) [14] trivariate EMD (TEMD) [15]and multivariate EMD (MEMD) [16] However these ex-tended methods were not specifically developed for machinefault feature extraction In BEMD the local mean of thebivariate signal is calculated by projecting the signal to anumber of directions Following the same idea the TEMDand MEMD were proposed respectively for three-dimensional and n-dimensional signals -e CLMD esti-mates the local mean by projecting only a bivariate signalonto the x- and y-axes -e bivariate EMD has beenemployed to detect wind turbine mechanical and electricalfaults by decomposing the fault signals [5] Here theelectrical signal was analyzed without considering the noiseand the number of BEMD projections -e associated ex-periments indicated that the number of projection directionsaffects the decomposition results particularly when noise isincluded In addition additional false components wereidentified in [5]-eMEMD has also been employed to faultdiagnosis of rolling bearing [17] Methods based on BEMDor MEMD are used to analyze and extract signal charac-teristics in multichannels However the joint informationbetween multiple sensors was not considered in [5 17] -ismethod combined with the CLMD or MEMD and fullspectrum has been proposed to obtain joint informationamong multichannel signals [12 18] -e methods based onCLMD or BEMD simultaneously decompose the signals intomultichannels thus ensuring that the number of de-composition results is the same and is readily incorporatedHowever the instantaneous feature cannot be extracted dueto the elliptical information that was generated using theconventional Fourier transform

-e signals are analyzed as the superposition of slow andfast oscillations in the EMD and the bivariate extension andbivariate signals eg the orthogonal vibrations are analyzedas the superposition of IMFs at the rotational speed in theBEMD To extract the instantaneous vibration characteris-tics and the joint information among the multichannelsignals the fault feature extraction method based on theBEMD is proposed to decompose the multicomponentrotation signal into a series of single-component rotation

signals -e instantaneous amplitude and instantaneousfrequency of the IMFs are further obtained using the Hilberttransform (HT)

2 BriefDescription and Improvement ofBEMD

21 Brief Description of BEMD -e fundamental concept ofBEMD is that bivariate signals are made up of slow rotationsignals and fast rotation signals superimposed on the slowrotation signals [13] For a mixed signal z(t) the de-composition process based on the BEMD is as follows [5]

Step 1 determine the number of projections N andcalculate the projection directions

φn 2nπN

n isin [1 N] (1)

Step 2 project the complex-valued signal z(t) on thedirections φn

pφn(t) Re e

minusjφn z(t)1113872 1113873 j minus1

radic (2)

Step 3 extract all local maxima of pφn(t) tn

i pφn(tn

i )1113966 1113967Herein i indicates the number order of individual localmaximaStep 4 interpolate the set tn

i pφn(tn

i )1113966 1113967 by spline in-terpolation to obtain the tangent along the directionφn eφn

(t)Step 5 repeat 2ndash4 until the tangents in allN projectionsare obtainedStep 6 compute the mean of all tangents

m(t) 1N

1113944

N

n1eφn

(t) (3)

Step 7 subtract m(t) from z(t) to obtain h(t) ie

h(t) z(t)minusm(t) (4)

Step 8 perform sifting process whether the stoppingcriterion similar to the one proposed in [19] is methowever h(t) and m(t) are bivariate signals If notregard h(t) as original signal and repeat Steps 2ndash7 untilthe stopping criterion is metStep 9 record the obtained IMF and remove it fromz(t) ie

c1(t) h(t)

r1(t) z(t)minus c1(t)(5)

Step 10 take r1(t) as the original signal and repeat theabove calculation until the second IMF c2(t) is ob-tained -e remainder is then calculated as follows

r2(t) r1(t)minus c2(t) (6)

Step 11 iterate the previous calculations until acquiringall IMFs contained in z(t)After the above process z(t) can be expressed asfollows

2 Shock and Vibration

z(t) 1113944K

k1ck(t) + rK(t) (7)

where K represents the total number of IMFs

During the decomposition process the bivariate rotatingsignal is required to rotate around the zero point -e ro-tating machinery often revolves around a central movementwhich coincides with the requirements of the BEMD -epresent study takes the typical Jeffcott rigid rotor-bearingsystem dynamics equation to structure the complex z(z x+ jy) with x and y representing the vibration signalscollected by two orthogonal sensors -e obtained charac-teristics curve is shown in Figure 1 -e BEMD analysis ofthe z(t) with the parameter N of 4 is shown in Figure 2which shows that the target signal is decomposed into bi-variate rotation components at the rotation speed allowinginvestigation of the instantaneous amplitude-frequency(IAF) characteristics of the main components

22 Improved BEMD Method In the original BEMD algo-rithm the loop cutoff condition is such that all IMF com-ponents are obtained from the original signal which leads toadditional IMF components and increases the computingtime of BEMD In most cases the fault characteristics areprimarily contained in the IMFs with a higher energy andthe vibration faults corresponding to the rotational modesare limited In this study the end condition of the BEMDbased on the energy threshold is proposed based on thereasons mentioned above A ratio λ is set and the ratiobetween the signal to be decomposed and the energy of theoriginal signal is less than a specific value that serves as acriterion to stop the BEMD algorithm λ is calculated usingthe following formula

1113936nL1abs[r(L)]2

1113936nL1abs[z(L)]2

le λ L 1 2 n (8)

3 IAF Feature Extraction of the BivariateRotation Signal

As noted in Section 2 BEMD decomposes the complex-valued signal of the multiple components into the complex-valued signals of a single component ci is a complex-valuedsignal with ci cxi + jcyi cxi represents the horizontalcomponent of the vibration signal and cyi represents thevertical component of the vibration signal -e HT is aclassical method to obtain the IAF of the signal -e real andthe imaginary components of ci are transformed with the HTto obtain the IAF -e following formulae are established

cxi(t) 1π

1113946+infin

minusinfin

cxi(τ)

tminus τdτ

cyi(t) 1π

1113946+infin

minusinfin

cyi(τ)

tminus τdτ

(9)

-e corresponding resolution signal expressions are

xi(t) cxi(t) + j times cxi(t) axi(t)ejΦxi(t)

yi(t) cyi(t) + j times cyi(t) ayi(t)ejΦyi(t)

(10)

where axi and ayi are the instantaneous amplitudes of theanalytic signals of xi and yi respectivelyΦxi andΦyi are thephase functions of xi and yi respectively and

Φxi(t) arctancxi(t)

cxi(t)1113890 1113891

Φxi(t) arctancxi(t)

cxi(t)1113890 1113891

(11)

-e instantaneous frequency function is derived fromthe phase function

fxi(t) 12π

timesdΦxi(t)

dt

fyi(t) 12π

timesdΦyi(t)

dt

(12)

We define axi(t) and fxi(t) as the instantaneous am-plitude and the instantaneous frequency of cxi respectivelyand define ayi(t) and fyi(t) as the instantaneous amplitudeand the instantaneous frequency of cyi respectively

4 Experiment Verification

41 Analysis of the Oil Film Oscillation Signal -e rotor oilfilm oscillation test devices are shown in Figure 3-e signalsare collected using eddy current sensors along the orthog-onal direction to the axis -e first-order critical speed isfound to be in the range of 3200 to 3400 rpm using theresonance phenomena by increasing the speed of the rotortest rig whose speed is increased from 0 to 8331 rpm andthen decreased to 0 Certain oil film vortex and oil filmoscillation phenomena occur during the experiment -edata recording device uses the INV306 collector which has a16-bit 4-way parallel AD converter -e sampling frequencyof the collector was set to 2048Hz in the followingexperiments

-e oil film oscillation signals along x and y (where therotor speed is 6480 rpm) in the horizontal and verticaldirections are collected by eddy current sensors x and y andtheir Fourier spectra are shown in Figure 4 whereX 108Hz is the frequency of the fundamental frequencysignal Let z x+ jy -e three-dimensional time-domainwaveform and the two-dimensional plots of z are shownin Figure 5 Figures 4(b) and 4(d) indicate that the amplitudeof the 048X component is more prominent particularly inthe vertical direction and even exceeds the amplitude of thefundamental frequency signal Figure 5 indicates that theamplitude of the oscillation signals is variable but that theFourier spectra cannot distinguish this change

-e EMD is applied to decompose the signals x and y asshown in Figure 6 -e S1 and S2 columns represent thedecomposition results of x and y respectively -e numberof IMFs generated from signal x is 10 but there are only 8

Shock and Vibration 3

from signal y e mismatch in the number of IMFs leads tochallenges in information fusion In addition modal aliasingoccurs in IMF1 from signal xemode aliasing obscures thephysical meaning of the IMF components and aects thesubsequent analysis

e decomposition results of the oil lm oscillationsignal using BEMD (N 4) are shown in Figure 7 esignal z is decomposed into eight complex rotationalcomponents c1ndashc8 After decomposition the IMF numbersfrom x are the same as those from y e noise component c1is more easily separated from signal z relative to the case ofIMF1 from signal x c2 and c3 are considered fundamentaland oscillatory components respectively the frequencycomponent of c2 is 108Hz and the frequency of c3 is 52Hze two trajectories composed of c2 and c3 generated by the

BEMD method are shown in Figures 8(a) and 8(b) re-spectively while the corresponding ones of IMF2 and IMF3generated by EMD method are shown in Figures 8(c) and8(d) respectively Figure 8 indicates that the orbit ar-rangements obtained by the BEMD method are superior tothose of the EMD method

Relative to the EMDmethod the decomposition eect isimproved when using BEMD to decompose the oil lmoscillation signal However there are still two problemswhen using BEMD to decompose the oil lm oscillationsignal One is the occurrence of modal aliasing in the IMFsand the other is increased false components From Figure 6the modal aliasing occurs at the end of c4 which causes thevalue of c3 to become small at the end After decompositioneight IMF components and one residual r are obtained

001

0203

ndash2

0

2ndash2

0

2

Time (s)Real(z)

Imag

(z)

(a)

ndash2 0 2ndash2

0

2

Real(z)

Imag

(z)

(b)

Figure 1 e simulated signal of z (a) the three-dimensional time-domain waveform (b) the two-dimensional plots

001

0203

ndash2

0

2ndash2

0

2

Real

Imag

Four directionsof tangent z(t)

Tangent mean

Time (s)

(a)

001

0203

ndash2

0

2ndash2

0

2

Time (s)Real

Imag

c1

c2

(b)

ndash2 0 2ndash2

0

2

Real

Imag

c1

c2

(c)

Figure 2 (a) z(t) and its four directions of the tangent and tangential mean (b) the decomposition results of z(t) based on the BEMD (c)the planes of c1 and c2

4 Shock and Vibration

However most of the IMF components contain uselessinformation In most cases the fault characteristics are oftenincluded in higher-energy components -e projection di-rections and the energy end condition based on the energythreshold are increased in the improved BEMD method toenhance the decomposition quality

-e decomposition results of the oil film oscillationsignal are shown in Figure 9 using the improved BEMDmethod (N 16 λ 005) -e oil film oscillation signal isdecomposed into four complex rotational components c1ndashc4

and the remaining component r c1 is considered as the noisecomponent According to the Fourier analysis c2 is thefundamental signal with a frequency of 108Hz and c3 is theoscillation signal with a frequency of 52Hz It is clear thatthe number of IMFs is reduced and the fundamental fre-quency component c2 and the oscillation component c3 aremore accurately extracted from the original signal In par-ticular the value of the c4 end becomes smaller and the valueof the c3 end becomes larger relative to the values of c3 and c4shown in Figure 7 Two trajectories made up of c2 and c3 are

Rigid foundation

Motor

Motorcontroller

Eddy current probe system

DiscFlexible coupling

Bearing predstal

Oil cup

Probes

Computer

Axis

ADcard

Figure 3 -e schematic diagram and the experimental apparatus

0 01 02 03 04 05ndash200

ndash100

0

100

200

Time (s)

x (micro

m)

(a)

0 048X X 2X 3X 4X0

30

60 X 52Y 4961

Frequency (Hz)

A (micro

m)

X 108Y 5257

(b)

0 01 02 03 04 05ndash100

ndash50

0

50

100

150

Time (s)

y (microm

)

(c)

0 048X X 2X 3X 4X0

30

60

Frequency (Hz)

A (micro

m)

X 52Y 4698

X 108Y 4437

(d)

Figure 4-e oil film oscillation signal (a) the horizontal and vertical direction signals x (b) the Fourier spectrum of signal x (c) the verticaldirection signals y (d) the Fourier spectrum of signal y

Shock and Vibration 5

shown in Figure 10 From c2 and c3 three-dimensional timedomain and the trajectories the fundamental frequencysignal c2 has a small elliptic amplitude change c3 is a largeseries of amplitude conversion elliptical compositions thatcause a signicant oscillation Relative to the orbit of c3 inFigure 6 the orbit of c3 in Figure 10 is improved particularlyin the center region

Increasing the number of signal projection directionsresults in an increase in the number of projection signals

en the tangent mean which is obtained by interpolatingthe local maximum of the projected signals with a splineinterpolation is more accurate However it is meaningless tocontinue to increase the number of projection directions whenthe tangent mean is accurately tted It is found that the signaldecomposition results of N 16 are almost the same as N 1024 when considering the complex rotation componentsseparated by the original signal However the calculation timeof the BEMD algorithm with N 1024 is greatly increased

001

0203

0405

ndash100

0

100

ndash100

ndash50

0

50

100

150

Time (s)Real(z) (microm)

Imag

(z) (microm

)

(a)

ndash150 ndash100 ndash50 0 50 100 150ndash150

ndash100

ndash50

0

50

100

150

Real(z) (microm)

Imag

(z) (microm

)

(b)

Figure 5 e oil lm oscillation signal of z (a) the three-dimensional time-domain waveform (b) the two-dimensional plots

ndash80

80

IMF 1

ndash80

80

IMF 2

ndash90

90

IMF 3

ndash30

30

IMF 4

ndash1010

IMF 5

ndash10

10

IMF 6

ndash4

4

IMF 7

ndash44

IMF 8

ndash33

IMF 9

ndash44

IMF 10

0 01 02 03 04 05ndash5

1r

Time (s)(a)

ndash4

4IM

F 1

ndash6060

IMF 2

ndash60

60

IMF 3

ndash15

15

IMF 4

ndash55

IMF 5

ndash55

IMF 6

ndash5

5

IMF 7

ndash0505

IMF 8

(b)

0 01 02 03 04 0505

1r

Time (s)

Figure 6 e decomposition results of signals x and y using EMD

6 Shock and Vibration

e HT is applied to the real and imaginary parts of c2and c3 respectively and the instantaneous frequency andinstantaneous amplitude are obtained as shown in Fig-ure 11 where ax2 and ax3 represent the instantaneous

amplitude of the real parts of c2 and c3 respectively ay2and ay3 represent the instantaneous amplitude of theimaginary parts of c2 and c3 respectively fx2 and fx3represent the instantaneous frequency of the real parts of

0025

05

ndash50

5ndash5

0

5

Time (s)Real(c1)

Imag

(c1)

(a)

0025

05

ndash1000

100ndash100

0

100

Time (s)Real(c2 )

Imag

(c2)

(b)

0025

05

ndash1000

100ndash100

0

100

Time (s)Real(c3 )

Imag

(c3)

(c)

0025

05

ndash200

20ndash50

0

50

Time (s)Real(c4 )

Imag

(c4)

Modal aliasing

(d)

0025

05

ndash100

10ndash10

0

10

Time (s)Real(c5 )Im

ag(c

5)

(e)

0025

05

ndash50

5ndash5

0

5

Time (s)Real(c6 )

Imag

(c6)

(f )

0025

05

ndash50

5ndash1

0

1

Time (s)Real(c7)

Imag

(c7)

(g)

0025

05

ndash50

5ndash1

0

1

Time (s)Real(c8)

Imag

(c8)

(h)

0025

05

ndash100

10ndash5

0

5

Time (s)Real(r)

Imag

(r)

(i)

Figure 7 Decomposition results of the oil lm oscillation signal using BEMD (N 4)

ndash80 ndash40 0 40 80ndash60

ndash30

0

30

60

Real(c2)

Imag

(c2)

(a)

ndash100 ndash50 0 50 100ndash100

ndash50

0

50

100

Real(c3)

Imag

(c3)

(b)

ndash80 ndash40 0 40 80ndash60

ndash30

0

30

60

IMFx2

IMF y

2

(c)

ndash100 ndash50 0 50 100ndash100

ndash50

0

50

100

IMFx3

IMF y

3

(d)

Figure 8e orbits made up of (a) the real and imaginary parts of c2 (b) real and imaginary parts of c3 (c) IMFx2 and IMFy2 and (d) IMFx3and IMFy3

Shock and Vibration 7

c2 and c3 respectively and fy2 and fy3 represent the in-stantaneous frequency of the imaginary parts of c2 and c3respectively

In Figure 11 ax3 is much larger than ay3 and their phasesare separated by nearly 180deg which shows that the amplitudeand the phase of the oil lm oscillation signal in dierentdirections can vary fx2 and fy2 and fx3 and fy3 are ap-proximately the same ax2 and ay2 show little change andtheir phases are the same Since BEMD is a bivariate ex-tension of EMD like EMD BEMD also has an endpointeect ere are some uctuations in the instantaneousamplitude and instantaneous frequency due to the end eect

and the edge eect of the Hilbert transform e three-dimensional time domain of ax2 and ay2 and of ax3 and ay3 isshown in Figure 12e amplitude range of c3 is much largerthan that of c2 It is can be inferred that the main componentof oscillation is c3 with the frequency of 52Hz in the oil lmoscillation signal

e decomposition results of the oil lm oscillationsignal are shown in Figure 13 using the CLMD methodproposed in reference [14] e oil lm oscillation signal isdecomposed into four complex product functions cpf1ndashcpf4e noise component is not separated from the oil lmoscillation signal using the CLMD method Moreover the

0025

05

ndash50

5

Time (s)Real(c1 )

ndash5

0

5Im

ag(c

1)

(a)

0025

05

ndash1000

100

Time (s)Real(c2)

ndash100

0

100

Imag

(c2)

(b)

0025

05ndash100

0100

ndash100

0

100

Imag

(c3)

Real(c3 ) Time (s)

(c)

0 02505

ndash100

10

Time (s)Real(c4)

ndash20

0

20

Imag

(c4)

(d)

0025

05ndash10

010

Time (s)Real(c5)

ndash10

0

10

Imag

(c5)

(e)

0 02505

ndash50

5

Time (s)Real(c6)

ndash5

0

5

Imag

(c6)

(f )

0 02505

ndash202

Time (s)Real(c7)

ndash2

0

2

Imag

(c7)

(g)

0025

05ndash5

05

Time (s)Real(c8)

ndash5

0

5

Imag

(c8)

(h)

0025

05ndash10

010

Time (s)Real(r)

ndash5

0

5

Imag

(r)

(i)

Figure 9 Decomposition results of the oil lm oscillation signal using the improved BEMD method (N 16 λ 005)

ndash80 0 40 80ndash40Real(c2)

ndash60

ndash30

0

30

60

Imag

(c2)

(a)

ndash80

ndash60

ndash40

ndash20

0

20

40

60

80

Imag

(c3)

ndash40 0 40 80ndash80Real(c3)

(b)

Figure 10 e orbits were made up of c2 (a) and c3 (b) obtained with the improved BEMD method

8 Shock and Vibration

single component fundamental frequency signal and the oillm oscillation signal were not successfully separated Oneof the possible reasons is that in the CLMD algorithm thecomplex signal is only projected onto the x-axis and the y-axis unlike BEMD which projected on multiple directionsIn addition CLMD is a bivariate extension of LMD LMDused a moving average algorithm when tting the signalenvelope which can lter noise to a certain extent esignals other than the noise component c1 in Figure 9 areadded to obtain a ltered oil lm oscillation signal which isthen decomposed by the CLMD method and the rst twodecomposed results are shown in Figure 13 It is seen thatthe single component fundamental frequency signal and

the single component oil lm oscillation signal areseparated

cpf1 and cpf2 consisted of real part signals and imaginarypart signals both of which were composed of the product ofthe envelope signal and the pure frequency modulationfunctione envelope signal is the instantaneous amplitudeof the signal e corresponding instantaneous frequencywas obtained by deriving the inverse function of the cosinepure frequency modulation function e instantaneousamplitude and instantaneous frequency curves are shown inFigure 14 e instantaneous amplitude and frequencyobtained by the CLMD method are smoother than in Fig-ure 12e reason is mainly that the CLMDmethod uses the

ax2ay2

30

40

50

60

70A

mpl

itude

01 02 03 04 050Time (s)

(a)

fx2fy2

70

110

150

Freq

uenc

y (H

z)

01 02 03 04 050Time (s)

(b)

ax3ay3

20

70

120

Am

plitu

de

01 02 03 04 050Time (s)

(c)

fx3fy3

0

50

100

Freq

uenc

y (H

z)01 02 03 04 050

Time (s)

(d)

Figure 11 (a) e instantaneous amplitude of the real part and imaginary part of c2 (b) the instantaneous frequency of the real part andimaginary part of c2 (c) the instantaneous amplitude of the real part and imaginary part of c3 (d) the instantaneous frequency of the real partand imaginary part of c3

0 01 02 03 04 05

2050

800

20

40

60

80

Time (s)ax2

a y2

(a)

3060

90120

0

20

40

60

80

Time (s)ax3

a y3

0 01 02 03 04 05

(b)

Figure 12 e 3D time domain of instantaneous amplitude was made up of (a) ax2 and ay2 and (b) ax3 and ay3

Shock and Vibration 9

moving average ltering algorithm to obtain the signalenvelope curve However this is the result of using theCLMD algorithm after ltering out noise with BEMD If theBEMD algorithm is not used for ltering noise the in-stantaneous amplitude and instantaneous frequency curvesof the single component were not obtained by the CLMDmethod

42 Analysis ofOilWhirl Signal Based on the ImprovedBEMDMethod In the method similar to that presented in Section41 the oil whirl signals of the rotor test rig with a speedparameter of 4320 rpm are collected by two orthogonalsensors as shown in Figure 15 e gure shows the typicalwhirl phenomenon of large circles with embedded smallerones e decomposition results based on the improvedBEMD method (N 16 λ 005) are shown in Figure 16

Only three IMFs appear in Figure 16 and the singlecomponents c2 and c3 and the noise component c1 are suc-cessfully separated from the original signal e HT is appliedto the real and imaginary parts of c2 and c3 respectively andthe instantaneous frequency and instantaneous amplitude areobtained as shown in Figure 17 e three-dimensional timedomain of ax2 and ay2 and of ax3 and ay3 is shown in Figure 18Figure 17 indicates that the frequency of c2 is approximatelytwice the frequency of c3 and that ax2 is larger than ay2 Inaddition ay3 is slightly larger than ax3 but the range of changefor ax3 is greater than the range for ay3 It is inferred that c2 isthe fundamental frequency signal and that c3 is the half-frequency signal in the oil whirl signal

43 Analysis of Looseness and Rotor Rubbing Composite FaultSignal Based on the Improved BEMD Method Loose androtor rubbing composite faults are set on the testequipment shown in Figure 3 in Section 41 Loose fault isset on the nondrive end of the motor and the distancebetween the plastic rod and the shaft is xed near thesensor on the left side of the disk As the rotor speedincreases the vibration increases and the rubbing faultoccurs which is stable at around 1700 rmin e com-posite fault signals are collected by two orthogonal sen-sors as shown in Figure 19 e decomposition resultsbased on the improved BEMD method are shown inFigure 20

Figures 19(c) and 19(d) indicate that the signal com-ponent mainly contain 1X 2X (X 28Hz) and a frequencymodulated signal generated due to time-varying stinessis phenomenon is similar to that described reference [20]Four IMFs appear in Figure 20 and the single components1X 2X signals and the FM signal c2 are successfully separatedfrom the original signal e HT is applied to the real andimaginary parts of c2 c3 and c4 respectively and the in-stantaneous frequency and instantaneous amplitude areobtained as shown in Figure 21 ere are some uctuationsin the frequencies of c2 and c3 but these uctuations aredierent from the random uctuations in the above casesey have obvious regularity and are characteristic of FMsignals e frequency modulation characteristics of c2 aremore obvious than those of c3 In addition by observing theinstantaneous amplitude of c2 it is seen that c2 is still anamplitude modulation signal

minus100

0100

minus100

0

100

Time (s)Real(cpf1)

Imag

(cpf

1)

0 01 02 03 04 05

(a)

Time (s)Real(cpf2) minus100

0100

minus100

0

100

Imag

(cpf

2)

0 01 02 03 04 05

(b)

minus80 0 80minus60

0

60

Real(cpf1)

Imag(cpf1)

(c)

minus100 0 100minus100

0

100

Real(cpf2)

Imag(cpf2)

(d)

Figure 13 e rst two decomposed results of the ltered signal based on the CLMD method

10 Shock and Vibration

0025

05

ndash150

0

150ndash150

0

150

Time (s)Real(z)

Imag

(z)

(a)

ndash150 ndash75 0 75 150ndash150

ndash75

0

75

150

Real(z)

Imag

(z)

(b)

0 72 144 216 2880

40

80

Frequency (Hz)

Am

plitu

de

X 72Y 7321

X 36Y 3433

Real(z)

(c)

Frequency (Hz)0 72 144 216 288

0

40

80

X 36Y 3475

Am

plitu

de

X 72Y 6157

Imag(z)

(d)

Figure 15e oil whirl signal z (a) the 3D time-domain wave of z (b) the 2D plane of z (c) the Fourier spectrum of Real[z] (d) the Fourierspectrum of Imag[z]

30

40

50

60

70

Am

plitu

de

0 01 02 03 04 05Time (s)

ax1ay1

(a)

70

110

150

Freq

uenc

y (H

z)

0 01 02 03 04 05Time (s)

fx1fy1

(b)

0 01 02 03 04 0520

70

120

Time (s)

Am

plitu

de

ax2ay2

(c)

0 01 02 03 04 050

50

100

Time (s)

Freq

uenc

y (H

z)

fx2fy2

(d)

Figure 14 (a) e instantaneous amplitude of the real part and imaginary part of cpf1 (b) the instantaneous frequency of the real part andimaginary part of cpf1 (c) the instantaneous amplitude of the real part and imaginary part of cpf2 (d) the instantaneous frequency of the realpart and imaginary part of cpf2

Shock and Vibration 11

0025

05

ndash5

0

5ndash5

0

5

Time (s)Real(c1)

Imag

(c1)

(a)

0025

05

ndash1000

100ndash100

0

100

Time (s)

Imag

(c2)

Real(c2)

(b)

0025

05

ndash500

50ndash50

0

50

Time (s)

Imag

(c3)

Real(c3)

(c)

0025

05

ndash100

10ndash20

0

20

Time (s)

Imag

(r)

Real(r)

(d)

ndash100 ndash50 0 50 100ndash100

ndash50

0

50

100

Imag

(c2)

Real(c2)

(e)

ndash40 ndash20 0 20 40ndash40

ndash20

0

20

40Im

ag(c

3)

Real(c3)

(f )

Figure 16 e decomposition results of the oil whirl signal based on the improved BEMD method

60

90

120

Am

plitu

de

0 01 02 03 04 05Time (s)

ax2ay2

(a)

50

75

100

Freq

uenc

y (H

z)

0 01 02 03 04 05Time (s)

fx2fy2

(b)

Figure 17 Continued

12 Shock and Vibration

0 01 02 03 04 0520

35

50

Time (s)

Am

plitu

de

ax3ay3

(c)

0 01 02 03 04 0520

35

50

Time (s)

Freq

uenc

y (H

z)

fx3fy3

(d)

Figure 17 e instantaneous amplitude and frequency of c2 and c3 from the oil whirl signal obtained by the HT (a) the instantaneousamplitude of the real part and imaginary part of c2 (b) the instantaneous frequency of the real part and imaginary part of c2(c) the instantaneous amplitude of the real part and imaginary part of c3 (d) the instantaneous frequency of the real part and imaginarypart of c3

0

025

05

40

80

12040

80

120

Time (s)

X 03877Y 9582Z 826

ax2

X 01309Y 9595Z 8452

a y2

(a)

0

025

05

20

35

5020

35

50

Time (s)

X 03955Y 3762Z 3822

ax3

X 008838Y 3608Z 3733

a y3

(b)

Figure 18 e 3D time domain of the instantaneous amplitude of ax2 and ay2 (a) and ax3 and ay3 (b) from the oil whirl signal

0025

05

minus1500

150minus150

0

150

t (s)Real(z)

Imag

(z)

(a)

Imag(z)

Real(z)

minus150

0

150

minus150 0 150

(b)

Figure 19 Continued

Shock and Vibration 13

In order to further verify the correctness of the in-stantaneous amplitude-frequency characteristics of theproposed method the real and imaginary parts of thecomposite fault signal z are analyzed separately using syn-chrosqueezed wavelet transforms (SWT) proposed in ref-erence [21]-e results are shown in Figure 22 It is seen thatthe time-frequency representations of the composite faultsignal z also include the AM-FM signal and the 1X signalwhich proves the correctness of the proposed methodCompared with the SWT method the instantaneousamplitude-frequency characteristics acquired by the HTmethod are relatively straightforward

44 lte Bistable Behavior Analysis of the Fan Rotor Based onBEMD -e bistability of the rotor is a nonlinear behaviorof the rotor-bearing system which is the state in which therotor jumps from one stable state to another forming astep -e bivariate signal of the bistable behavior iscomposed of two signals collected by two displacementsensors from orthogonal locations on the experimentaldevices in literature [22] as shown in Figure 23 Literature[22] shows that the cause of the bistable behavior remainsto be further explored -is paper uses this case to il-lustrate the feasibility of BEMD to analyze nonstationarysignals

0

40

80

Am

plitu

de Real(z)

0 100 200 300 400Frequency (Hz)

(c)

Am

plitu

de

0 100 200 300 400Frequency (Hz)

0

20

40

60

80

Imag(z)

(d)

Figure 19 -e composite fault signal z (a) the 3D time domain wave of z (b) the 2D plane of z (c) the Fourier spectrum of Real[z] (d) theFourier spectrum of Imag[z]

0025

05

minus30

3minus5

0

5

Time (s)Real(c1)

Imag

(c1)

(a)

0025

05

minus200

20minus20

0

20

Time (s)Real(c2 )

Imag

(c2)

(b)

0025

05

minus200

20minus50

0

50

Time (s)Real(c2)

Imag

(c2)

(c)

0025

05

minus1000

100minus100

0

100

Time (s)Real(c3)

Imag

(c3)

(d)

0025

05

minus400

40minus30

0

30

Time (s)Real(r)

Imag

(r)

(e)

Figure 20 -e decomposition results of the composite fault signal based on the improved BEMD method

14 Shock and Vibration

-e x and y signals in the horizontal and vertical di-rections of the left and right bearings respectively from thefan rotors are collected with four displacement sensorsLetting z x+ jy the time and frequency domain plots of z areshown in Figure 24 where the fan rotor speed is 5500 rpm thesampling frequency is 2000Hz and the number of samplingpoints is 1024 -e left and right columns respectively showthe time and frequency domain plots of the vibration signalsfrom the left and right bearings of the fan rotor Bistablebehavior arises in the fan rotor and the amplitudes of the

vibration signals vary significantly in different positions anddirections Further studies are required to explain the causesof this bistability -e present study focuses on extracting thebistable behavioral signal characteristics to verify the feasi-bility of the proposed method

-e decomposition results of the bistable behavioralsignals based on the improved BEMDmethod are shown inFigure 25 c1 c2 c3 and r are separated in order from zusing the improved BEMD method c1 shows a randomarrangement and is considered the high-frequency noise

0

10

20

Am

plitu

de

ax2ay2

0 025 05Time (s)

(a)

0

152

304

Freq

uenc

y (H

z)

0 025 05Time (s)

fy2

fx2

(b)

0

10

20

30

40

Am

plitu

de

ax3ay3

0 025 05Time (s)

(c)

0

56

112

Freq

uenc

y (H

z)

fx3fy3

0 025 05Time (s)

(d)

0 025 050

50

100

Time (s)

Am

plitu

de

ax4ay4

(e)

0 025 050

28

56

Time (s)

Freq

uenc

y (H

z)

fx4fy4

(f )

Figure 21 -e instantaneous amplitude and frequency of c2 c3 and c4 obtained by the HT (a) the instantaneous amplitude of the real partand imaginary part of c2 (b) the instantaneous frequency of the real part and imaginary part of c2 (c) the instantaneous amplitude of the realpart and imaginary part of c3 (d) the instantaneous frequency of the real part and imaginary part of c3 (e) the instantaneous amplitude of thereal part and imaginary part of c4 (f ) the instantaneous frequency of the real part and imaginary part of c4

Shock and Vibration 15

signal c2 is considered to represent the extracted bistablebehavior signals -e HT is applied to the real andimaginary parts of c2 to obtain the instantaneous amplitudeand frequency of c2 from the left and right columns fromFigure 25 as shown in Figure 26 Figure 27 shows thethree-dimensional time domain of ax2 and ay2 from the leftand right columns respectively Figure 26 shows that thevibration signal amplitude on the left side of the fan de-creases from large to small opposite of the behavior ofthe right -e horizontal vibration signal amplitude on theleft side of the fan is larger than that of the vertical di-rection signal opposite of the right -is result validatesthat the vibration signals from different directions orpositions are different when the fan produces bistablebehavior In addition the time of the bistable behavior canbe determined according to the jump point of the am-plitude or frequency

5 Discussion

-e BEMD algorithm decomposes two orthogonal di-rections of vibration signals as a complex signal which is atwo-dimensional digital signal processing method thusensuring that the real and imaginary parts have the samedecomposition scale Similar to EMD the envelope mean iscritical for the decomposition effect of BEMD but the en-velope mean in BEMD is three-dimensional If the numberof projection directions of the complex signal in three-dimensional space is larger the corresponding envelope

signal is also more -us the envelope mean value is moreaccurate and the BEMD decomposition effect is betterIncreasing the number of projection directions can improvemodal aliasing Like EMD BEMD also produces falsecomponents when decomposing signals Generally speakingthe energy of the false components is low and these low-energy false components do not contain fault characteristicinformation and the introduction of the energy thresholdcriterion in the termination condition can increase thedecomposition speed of the BEMD

-e experimental results show that there is a certaindifference in the existence of vibration signal character-istics in different directions when rotating machinery failsIn addition when the number of projection directionsis increased the decomposition speed of BEMD willdecrease

6 Conclusions

We use BEMD and HT to extract the instantaneousamplitude-frequency features of rotor faults A bivariateinstantaneous feature extraction method based on the im-proved BEMD method and the HT is investigated whichextends the fault feature extraction technology to two di-mensions -e BEMD method is suitable to analyze thecomplex multicomponent bivariate signals -e mainsingle-component bivariate signals are separated from themulticomponent bivariate signals of the fan rotor bistabilityfor the oil film oscillation and the oil film vortex using the

0

2

4

6

Time (s)

Freq

uenc

y (H

z)

0 01 02 03 04 052

3

4

5

6

7

8log2 fx

(a)

0

2

4

6

Time (s)

Freq

uenc

y (H

z)

0 01 02 03 04 052

3

4

5

6

7

8log2 fy

(b)

Figure 22 -e results of the composite fault signal z based on SWT the time-frequency representation of (a) the real part of z and (b) theimaginary part of z

Left bearing predstal

Locations of sensors

Fanrotor

Right bearing predstal

Axis

Orthogonal directions

Figure 23 -e schematic diagram of the experimental apparatus

16 Shock and Vibration

Real(z) 00256

0512minus300

0300

minus400

0

400

Time (s)

Imag

(z)

00256

0512minus500

0500

minus500

0

500

Time (s)Real(z)

Imag

(z)

Imag

(z)

minus300 0 300minus500

0

500

Real(z)minus500 0 500

minus500

0

Imag

(z)

Real(z)

500

0 100 200 300 4000

100

200

300

Frequency (Hz)

Am

plitu

de Imag(z)

Frequency (Hz)0 100 200 300 400

0

200

400

Am

plitu

de

Imag(z)

0 100 200 300 4000

70

140

Frequency (Hz)

Am

plitu

de

Real(z)

0 100 200 300 4000

200

400

Frequency (Hz)

Am

plitu

de Real(z)

(a) (b)

Figure 24 -e time and frequency domain plots of the bistable behavior signals

00256

0512

minus800

80minus80

0

80

Time (s)Real(c1 )

Imag

(c1)

Time (s)Real(c1 )

Imag

(c1)

00256

0512

minus400

40minus40

0

40

Time (s)Real(c2) 0

02560512

minus5000

500minus500

0

500

Imag

(c2)

Time (s)Real(c2)

Imag

(c2)

00256

0512

minus5000

500minus500

0

500

Time (s)Real(r) 00256

0512

minus800

80minus80

0

80

Imag

(irc

rm

)

Time (s)Real(r)

Imag

(r)

00256

0512

minus2000

200minus200

0

200

(a) (b)

Figure 25 -e decomposition results of the bistable behavior signals based on the improved BEMD method

Shock and Vibration 17

improved BEMD method For the single-component bi-variate signal the HT is used to obtain the correspondinginstantaneous amplitude and frequency characteristics -eproposed method can examine the detailed information of asingle rotation component

Data Availability

-e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

-e authors declare that they have no conflicts of interest

Authorsrsquo Contributions

All the authors contributed to this work Chuanjin Huangconceived and designed the simulation and experiments anddrafted the manuscript Haijun Song performed the simu-lations and experiments and analyzed the data and

0 0256 05120

50

100

150

Time (s)

Freq

uenc

y (H

z)

fx2

fy2

0 0256 05120

200

400

600

Time (s)

Am

plitu

de

ax2ay2

(a)

fx2

fy2

0 0256 05120

90

180

270

360

Time (s)

Freq

uenc

y (H

z)

0 0256 05120

200

400

600

Time (s)

Am

plitu

de

ax2Data 2

(b)

Figure 26 -e instantaneous amplitude and frequency of c2 from the (a) left and (b) right columns

0

0256

0512

0100

200300

400500

0

100

200

300

400

500

Time (s)

X 04035Y 3509Z 130

X 04235Y 2265Z 3613

X 007Y 1131Z 2191

X 0105Y 3837Z 3312

ax2

a y2

Figure 27 -e three-dimensional time domain of ax2 and ay2

18 Shock and Vibration

Wenping Lei and Yajun Meng performed the experimentsand analyzed the data All the authors contributed to thewriting and discussion of the paper

Acknowledgments

-is research was funded by the Henan Provincial HigherEducation Key Research Project (Grant nos 18A460006 and19A460029) Henan High-Level Innovative Scientific andTechnological Talent Team Construction Project (Grant noC20150034) and Zhengzhou Institute of Technology In-novation Team Project (Grant no CXTD2017K1)

References

[1] R Yan R X Gao and X Chen ldquoWavelets for fault diagnosisof rotary machines a review with applicationsrdquo Signal Pro-cessing vol 96 pp 1ndash15 2014

[2] J Cheng D Yu J Tang and Y Yang ldquoApplication of frequencyfamily separation method based upon EMD and local Hilbertenergy spectrum method to gear fault diagnosisrdquo Mechanismand Machine lteory vol 43 no 6 pp 712ndash723 2008

[3] H Liu and M Han ldquoA fault diagnosis method based on localmean decomposition and multi-scale entropy for rollerbearingsrdquoMechanism andMachinelteory vol 75 pp 67ndash782014

[4] Z Zheng W Jiang Z Wang Y Zhu and K Yang ldquoGear faultdiagnosis method based on local mean decomposition andgeneralized morphological fractal dimensionsrdquo Mechanismand Machine lteory vol 91 pp 151ndash167 2015

[5] W Yang R Court P J Tavner and C J Crabtree ldquoBivariateempirical mode decomposition and its contribution to windturbine condition monitoringrdquo Journal of Sound and Vi-bration vol 330 no 15 pp 3766ndash3782 2011

[6] L Qu X Liu G Peyronne and Y Chen ldquo-e holospectrum anewmethod for rotor surveillance and diagnosisrdquoMechanicalSystems amp Signal Processing vol 3 no 3 pp 255ndash267 1989

[7] F Q Wu and G Meng ldquoCompound rub malfunctions featureextraction based on full-spectrum cascade analysis and SVMrdquoMechanical Systems and Signal Processing vol 20 no 8pp 2007ndash2021 2006

[8] Y Chen Q Gao and Z Guan ldquoSelf-loosening failure analysisof bolt joints under vibration considering the tighteningprocessrdquo Shock and Vibration vol 2017 Article ID 203842115 pages 2017

[9] L Chen J Han W Lei Y Cui and Z Guan ldquoFull-vectorsignal acquisition and information fusion for the fault pre-dictionrdquo International Journal of Rotating Machineryvol 2016 Article ID 5980802 7 pages 2016

[10] C Chen Y Meng and Y Du ldquoApplication of the full vectorspectrum based on EMD in fault diagnosis of bearingsrdquoJournal of Mechanical Strength vol 37 pp 806ndash811 2015

[11] C Huang X Wu and W Cao ldquoLMD-based on full vectorenvelope technique and its application in TRT vibration faultdiagnosisrdquo Electric Power Automation Equipment vol 35pp 168ndash174 2015 in Chinese

[12] X Zhao T H Patel and M J Zuo ldquoMultivariate EMD andfull spectrum based condition monitoring for rotating ma-chineryrdquo Mechanical Systems and Signal Processing vol 27pp 712ndash728 2012

[13] G Rilling P Flandrin P Gonalves and J M Lilly ldquoBivariateempirical mode decompositionrdquo IEEE Signal ProcessingLetters vol 14 no 12 pp 936ndash939 2007

[14] C Park D Looney M M Van Hulle and D P Mandic ldquo-ecomplex local mean decompositionrdquo Neurocomputingvol 74 no 6 pp 867ndash875 2011

[15] N Rehman and D P Mandic ldquoEmpirical mode de-composition for trivariate signalsrdquo IEEE Transactions onSignal Processing vol 58 no 3 pp 1059ndash1068 2010

[16] N Rehman and D P Mandic ldquoMultivariate empirical modedecompositionrdquo Proceedings of the Royal Society A Mathe-matical Physical and Engineering Sciences vol 466 no 2117pp 1291ndash1302 2010

[17] Y Lv R Yuan and G Song ldquoMultivariate empirical modedecomposition and its application to fault diagnosis of rollingbearingrdquo Mechanical Systems and Signal Processing vol 81pp 219ndash234 2016

[18] C Huang Y Meng and W Lei ldquoFull vector envelopetechnique based on complex local mean decomposition andits application in fault feature extraction for rotor systemrdquoJournal of Mechanical Engineering vol 52 no 7 p 69 2016in Chinese

[19] G Rilling P Flandrin and P Goncalves ldquoOn empirical modedecomposition and its algorithmsrdquo in Proceedings of IEEE-EURASIP Workshop on Nonlinear Signal and Image Pro-cessing pp 8ndash11 IEEE Trieste Italy June 2003

[20] L Yang X Chen and S Wang ldquoMechanism of fast time-varying vibration for rotorndashstator contact system with ap-plication to fault diagnosisrdquo Journal of Vibration andAcoustics vol 140 no 1 article 014501 2018

[21] I Daubechies J Lu and H-TWu ldquoSynchrosqueezed wavelettransforms an empirical mode decomposition-like toolrdquoApplied and Computational Harmonic Analysis vol 30 no 2pp 243ndash261 2011

[22] L-S Qu Holospectrum and Holobalancing Technique inMachinery Diagnosis Beijing Science Press Beijing China2007

Shock and Vibration 19

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z(t) 1113944K

k1ck(t) + rK(t) (7)

where K represents the total number of IMFs

During the decomposition process the bivariate rotatingsignal is required to rotate around the zero point -e ro-tating machinery often revolves around a central movementwhich coincides with the requirements of the BEMD -epresent study takes the typical Jeffcott rigid rotor-bearingsystem dynamics equation to structure the complex z(z x+ jy) with x and y representing the vibration signalscollected by two orthogonal sensors -e obtained charac-teristics curve is shown in Figure 1 -e BEMD analysis ofthe z(t) with the parameter N of 4 is shown in Figure 2which shows that the target signal is decomposed into bi-variate rotation components at the rotation speed allowinginvestigation of the instantaneous amplitude-frequency(IAF) characteristics of the main components

22 Improved BEMD Method In the original BEMD algo-rithm the loop cutoff condition is such that all IMF com-ponents are obtained from the original signal which leads toadditional IMF components and increases the computingtime of BEMD In most cases the fault characteristics areprimarily contained in the IMFs with a higher energy andthe vibration faults corresponding to the rotational modesare limited In this study the end condition of the BEMDbased on the energy threshold is proposed based on thereasons mentioned above A ratio λ is set and the ratiobetween the signal to be decomposed and the energy of theoriginal signal is less than a specific value that serves as acriterion to stop the BEMD algorithm λ is calculated usingthe following formula

1113936nL1abs[r(L)]2

1113936nL1abs[z(L)]2

le λ L 1 2 n (8)

3 IAF Feature Extraction of the BivariateRotation Signal

As noted in Section 2 BEMD decomposes the complex-valued signal of the multiple components into the complex-valued signals of a single component ci is a complex-valuedsignal with ci cxi + jcyi cxi represents the horizontalcomponent of the vibration signal and cyi represents thevertical component of the vibration signal -e HT is aclassical method to obtain the IAF of the signal -e real andthe imaginary components of ci are transformed with the HTto obtain the IAF -e following formulae are established

cxi(t) 1π

1113946+infin

minusinfin

cxi(τ)

tminus τdτ

cyi(t) 1π

1113946+infin

minusinfin

cyi(τ)

tminus τdτ

(9)

-e corresponding resolution signal expressions are

xi(t) cxi(t) + j times cxi(t) axi(t)ejΦxi(t)

yi(t) cyi(t) + j times cyi(t) ayi(t)ejΦyi(t)

(10)

where axi and ayi are the instantaneous amplitudes of theanalytic signals of xi and yi respectivelyΦxi andΦyi are thephase functions of xi and yi respectively and

Φxi(t) arctancxi(t)

cxi(t)1113890 1113891

Φxi(t) arctancxi(t)

cxi(t)1113890 1113891

(11)

-e instantaneous frequency function is derived fromthe phase function

fxi(t) 12π

timesdΦxi(t)

dt

fyi(t) 12π

timesdΦyi(t)

dt

(12)

We define axi(t) and fxi(t) as the instantaneous am-plitude and the instantaneous frequency of cxi respectivelyand define ayi(t) and fyi(t) as the instantaneous amplitudeand the instantaneous frequency of cyi respectively

4 Experiment Verification

41 Analysis of the Oil Film Oscillation Signal -e rotor oilfilm oscillation test devices are shown in Figure 3-e signalsare collected using eddy current sensors along the orthog-onal direction to the axis -e first-order critical speed isfound to be in the range of 3200 to 3400 rpm using theresonance phenomena by increasing the speed of the rotortest rig whose speed is increased from 0 to 8331 rpm andthen decreased to 0 Certain oil film vortex and oil filmoscillation phenomena occur during the experiment -edata recording device uses the INV306 collector which has a16-bit 4-way parallel AD converter -e sampling frequencyof the collector was set to 2048Hz in the followingexperiments

-e oil film oscillation signals along x and y (where therotor speed is 6480 rpm) in the horizontal and verticaldirections are collected by eddy current sensors x and y andtheir Fourier spectra are shown in Figure 4 whereX 108Hz is the frequency of the fundamental frequencysignal Let z x+ jy -e three-dimensional time-domainwaveform and the two-dimensional plots of z are shownin Figure 5 Figures 4(b) and 4(d) indicate that the amplitudeof the 048X component is more prominent particularly inthe vertical direction and even exceeds the amplitude of thefundamental frequency signal Figure 5 indicates that theamplitude of the oscillation signals is variable but that theFourier spectra cannot distinguish this change

-e EMD is applied to decompose the signals x and y asshown in Figure 6 -e S1 and S2 columns represent thedecomposition results of x and y respectively -e numberof IMFs generated from signal x is 10 but there are only 8

Shock and Vibration 3

from signal y e mismatch in the number of IMFs leads tochallenges in information fusion In addition modal aliasingoccurs in IMF1 from signal xemode aliasing obscures thephysical meaning of the IMF components and aects thesubsequent analysis

e decomposition results of the oil lm oscillationsignal using BEMD (N 4) are shown in Figure 7 esignal z is decomposed into eight complex rotationalcomponents c1ndashc8 After decomposition the IMF numbersfrom x are the same as those from y e noise component c1is more easily separated from signal z relative to the case ofIMF1 from signal x c2 and c3 are considered fundamentaland oscillatory components respectively the frequencycomponent of c2 is 108Hz and the frequency of c3 is 52Hze two trajectories composed of c2 and c3 generated by the

BEMD method are shown in Figures 8(a) and 8(b) re-spectively while the corresponding ones of IMF2 and IMF3generated by EMD method are shown in Figures 8(c) and8(d) respectively Figure 8 indicates that the orbit ar-rangements obtained by the BEMD method are superior tothose of the EMD method

Relative to the EMDmethod the decomposition eect isimproved when using BEMD to decompose the oil lmoscillation signal However there are still two problemswhen using BEMD to decompose the oil lm oscillationsignal One is the occurrence of modal aliasing in the IMFsand the other is increased false components From Figure 6the modal aliasing occurs at the end of c4 which causes thevalue of c3 to become small at the end After decompositioneight IMF components and one residual r are obtained

001

0203

ndash2

0

2ndash2

0

2

Time (s)Real(z)

Imag

(z)

(a)

ndash2 0 2ndash2

0

2

Real(z)

Imag

(z)

(b)

Figure 1 e simulated signal of z (a) the three-dimensional time-domain waveform (b) the two-dimensional plots

001

0203

ndash2

0

2ndash2

0

2

Real

Imag

Four directionsof tangent z(t)

Tangent mean

Time (s)

(a)

001

0203

ndash2

0

2ndash2

0

2

Time (s)Real

Imag

c1

c2

(b)

ndash2 0 2ndash2

0

2

Real

Imag

c1

c2

(c)

Figure 2 (a) z(t) and its four directions of the tangent and tangential mean (b) the decomposition results of z(t) based on the BEMD (c)the planes of c1 and c2

4 Shock and Vibration

However most of the IMF components contain uselessinformation In most cases the fault characteristics are oftenincluded in higher-energy components -e projection di-rections and the energy end condition based on the energythreshold are increased in the improved BEMD method toenhance the decomposition quality

-e decomposition results of the oil film oscillationsignal are shown in Figure 9 using the improved BEMDmethod (N 16 λ 005) -e oil film oscillation signal isdecomposed into four complex rotational components c1ndashc4

and the remaining component r c1 is considered as the noisecomponent According to the Fourier analysis c2 is thefundamental signal with a frequency of 108Hz and c3 is theoscillation signal with a frequency of 52Hz It is clear thatthe number of IMFs is reduced and the fundamental fre-quency component c2 and the oscillation component c3 aremore accurately extracted from the original signal In par-ticular the value of the c4 end becomes smaller and the valueof the c3 end becomes larger relative to the values of c3 and c4shown in Figure 7 Two trajectories made up of c2 and c3 are

Rigid foundation

Motor

Motorcontroller

Eddy current probe system

DiscFlexible coupling

Bearing predstal

Oil cup

Probes

Computer

Axis

ADcard

Figure 3 -e schematic diagram and the experimental apparatus

0 01 02 03 04 05ndash200

ndash100

0

100

200

Time (s)

x (micro

m)

(a)

0 048X X 2X 3X 4X0

30

60 X 52Y 4961

Frequency (Hz)

A (micro

m)

X 108Y 5257

(b)

0 01 02 03 04 05ndash100

ndash50

0

50

100

150

Time (s)

y (microm

)

(c)

0 048X X 2X 3X 4X0

30

60

Frequency (Hz)

A (micro

m)

X 52Y 4698

X 108Y 4437

(d)

Figure 4-e oil film oscillation signal (a) the horizontal and vertical direction signals x (b) the Fourier spectrum of signal x (c) the verticaldirection signals y (d) the Fourier spectrum of signal y

Shock and Vibration 5

shown in Figure 10 From c2 and c3 three-dimensional timedomain and the trajectories the fundamental frequencysignal c2 has a small elliptic amplitude change c3 is a largeseries of amplitude conversion elliptical compositions thatcause a signicant oscillation Relative to the orbit of c3 inFigure 6 the orbit of c3 in Figure 10 is improved particularlyin the center region

Increasing the number of signal projection directionsresults in an increase in the number of projection signals

en the tangent mean which is obtained by interpolatingthe local maximum of the projected signals with a splineinterpolation is more accurate However it is meaningless tocontinue to increase the number of projection directions whenthe tangent mean is accurately tted It is found that the signaldecomposition results of N 16 are almost the same as N 1024 when considering the complex rotation componentsseparated by the original signal However the calculation timeof the BEMD algorithm with N 1024 is greatly increased

001

0203

0405

ndash100

0

100

ndash100

ndash50

0

50

100

150

Time (s)Real(z) (microm)

Imag

(z) (microm

)

(a)

ndash150 ndash100 ndash50 0 50 100 150ndash150

ndash100

ndash50

0

50

100

150

Real(z) (microm)

Imag

(z) (microm

)

(b)

Figure 5 e oil lm oscillation signal of z (a) the three-dimensional time-domain waveform (b) the two-dimensional plots

ndash80

80

IMF 1

ndash80

80

IMF 2

ndash90

90

IMF 3

ndash30

30

IMF 4

ndash1010

IMF 5

ndash10

10

IMF 6

ndash4

4

IMF 7

ndash44

IMF 8

ndash33

IMF 9

ndash44

IMF 10

0 01 02 03 04 05ndash5

1r

Time (s)(a)

ndash4

4IM

F 1

ndash6060

IMF 2

ndash60

60

IMF 3

ndash15

15

IMF 4

ndash55

IMF 5

ndash55

IMF 6

ndash5

5

IMF 7

ndash0505

IMF 8

(b)

0 01 02 03 04 0505

1r

Time (s)

Figure 6 e decomposition results of signals x and y using EMD

6 Shock and Vibration

e HT is applied to the real and imaginary parts of c2and c3 respectively and the instantaneous frequency andinstantaneous amplitude are obtained as shown in Fig-ure 11 where ax2 and ax3 represent the instantaneous

amplitude of the real parts of c2 and c3 respectively ay2and ay3 represent the instantaneous amplitude of theimaginary parts of c2 and c3 respectively fx2 and fx3represent the instantaneous frequency of the real parts of

0025

05

ndash50

5ndash5

0

5

Time (s)Real(c1)

Imag

(c1)

(a)

0025

05

ndash1000

100ndash100

0

100

Time (s)Real(c2 )

Imag

(c2)

(b)

0025

05

ndash1000

100ndash100

0

100

Time (s)Real(c3 )

Imag

(c3)

(c)

0025

05

ndash200

20ndash50

0

50

Time (s)Real(c4 )

Imag

(c4)

Modal aliasing

(d)

0025

05

ndash100

10ndash10

0

10

Time (s)Real(c5 )Im

ag(c

5)

(e)

0025

05

ndash50

5ndash5

0

5

Time (s)Real(c6 )

Imag

(c6)

(f )

0025

05

ndash50

5ndash1

0

1

Time (s)Real(c7)

Imag

(c7)

(g)

0025

05

ndash50

5ndash1

0

1

Time (s)Real(c8)

Imag

(c8)

(h)

0025

05

ndash100

10ndash5

0

5

Time (s)Real(r)

Imag

(r)

(i)

Figure 7 Decomposition results of the oil lm oscillation signal using BEMD (N 4)

ndash80 ndash40 0 40 80ndash60

ndash30

0

30

60

Real(c2)

Imag

(c2)

(a)

ndash100 ndash50 0 50 100ndash100

ndash50

0

50

100

Real(c3)

Imag

(c3)

(b)

ndash80 ndash40 0 40 80ndash60

ndash30

0

30

60

IMFx2

IMF y

2

(c)

ndash100 ndash50 0 50 100ndash100

ndash50

0

50

100

IMFx3

IMF y

3

(d)

Figure 8e orbits made up of (a) the real and imaginary parts of c2 (b) real and imaginary parts of c3 (c) IMFx2 and IMFy2 and (d) IMFx3and IMFy3

Shock and Vibration 7

c2 and c3 respectively and fy2 and fy3 represent the in-stantaneous frequency of the imaginary parts of c2 and c3respectively

In Figure 11 ax3 is much larger than ay3 and their phasesare separated by nearly 180deg which shows that the amplitudeand the phase of the oil lm oscillation signal in dierentdirections can vary fx2 and fy2 and fx3 and fy3 are ap-proximately the same ax2 and ay2 show little change andtheir phases are the same Since BEMD is a bivariate ex-tension of EMD like EMD BEMD also has an endpointeect ere are some uctuations in the instantaneousamplitude and instantaneous frequency due to the end eect

and the edge eect of the Hilbert transform e three-dimensional time domain of ax2 and ay2 and of ax3 and ay3 isshown in Figure 12e amplitude range of c3 is much largerthan that of c2 It is can be inferred that the main componentof oscillation is c3 with the frequency of 52Hz in the oil lmoscillation signal

e decomposition results of the oil lm oscillationsignal are shown in Figure 13 using the CLMD methodproposed in reference [14] e oil lm oscillation signal isdecomposed into four complex product functions cpf1ndashcpf4e noise component is not separated from the oil lmoscillation signal using the CLMD method Moreover the

0025

05

ndash50

5

Time (s)Real(c1 )

ndash5

0

5Im

ag(c

1)

(a)

0025

05

ndash1000

100

Time (s)Real(c2)

ndash100

0

100

Imag

(c2)

(b)

0025

05ndash100

0100

ndash100

0

100

Imag

(c3)

Real(c3 ) Time (s)

(c)

0 02505

ndash100

10

Time (s)Real(c4)

ndash20

0

20

Imag

(c4)

(d)

0025

05ndash10

010

Time (s)Real(c5)

ndash10

0

10

Imag

(c5)

(e)

0 02505

ndash50

5

Time (s)Real(c6)

ndash5

0

5

Imag

(c6)

(f )

0 02505

ndash202

Time (s)Real(c7)

ndash2

0

2

Imag

(c7)

(g)

0025

05ndash5

05

Time (s)Real(c8)

ndash5

0

5

Imag

(c8)

(h)

0025

05ndash10

010

Time (s)Real(r)

ndash5

0

5

Imag

(r)

(i)

Figure 9 Decomposition results of the oil lm oscillation signal using the improved BEMD method (N 16 λ 005)

ndash80 0 40 80ndash40Real(c2)

ndash60

ndash30

0

30

60

Imag

(c2)

(a)

ndash80

ndash60

ndash40

ndash20

0

20

40

60

80

Imag

(c3)

ndash40 0 40 80ndash80Real(c3)

(b)

Figure 10 e orbits were made up of c2 (a) and c3 (b) obtained with the improved BEMD method

8 Shock and Vibration

single component fundamental frequency signal and the oillm oscillation signal were not successfully separated Oneof the possible reasons is that in the CLMD algorithm thecomplex signal is only projected onto the x-axis and the y-axis unlike BEMD which projected on multiple directionsIn addition CLMD is a bivariate extension of LMD LMDused a moving average algorithm when tting the signalenvelope which can lter noise to a certain extent esignals other than the noise component c1 in Figure 9 areadded to obtain a ltered oil lm oscillation signal which isthen decomposed by the CLMD method and the rst twodecomposed results are shown in Figure 13 It is seen thatthe single component fundamental frequency signal and

the single component oil lm oscillation signal areseparated

cpf1 and cpf2 consisted of real part signals and imaginarypart signals both of which were composed of the product ofthe envelope signal and the pure frequency modulationfunctione envelope signal is the instantaneous amplitudeof the signal e corresponding instantaneous frequencywas obtained by deriving the inverse function of the cosinepure frequency modulation function e instantaneousamplitude and instantaneous frequency curves are shown inFigure 14 e instantaneous amplitude and frequencyobtained by the CLMD method are smoother than in Fig-ure 12e reason is mainly that the CLMDmethod uses the

ax2ay2

30

40

50

60

70A

mpl

itude

01 02 03 04 050Time (s)

(a)

fx2fy2

70

110

150

Freq

uenc

y (H

z)

01 02 03 04 050Time (s)

(b)

ax3ay3

20

70

120

Am

plitu

de

01 02 03 04 050Time (s)

(c)

fx3fy3

0

50

100

Freq

uenc

y (H

z)01 02 03 04 050

Time (s)

(d)

Figure 11 (a) e instantaneous amplitude of the real part and imaginary part of c2 (b) the instantaneous frequency of the real part andimaginary part of c2 (c) the instantaneous amplitude of the real part and imaginary part of c3 (d) the instantaneous frequency of the real partand imaginary part of c3

0 01 02 03 04 05

2050

800

20

40

60

80

Time (s)ax2

a y2

(a)

3060

90120

0

20

40

60

80

Time (s)ax3

a y3

0 01 02 03 04 05

(b)

Figure 12 e 3D time domain of instantaneous amplitude was made up of (a) ax2 and ay2 and (b) ax3 and ay3

Shock and Vibration 9

moving average ltering algorithm to obtain the signalenvelope curve However this is the result of using theCLMD algorithm after ltering out noise with BEMD If theBEMD algorithm is not used for ltering noise the in-stantaneous amplitude and instantaneous frequency curvesof the single component were not obtained by the CLMDmethod

42 Analysis ofOilWhirl Signal Based on the ImprovedBEMDMethod In the method similar to that presented in Section41 the oil whirl signals of the rotor test rig with a speedparameter of 4320 rpm are collected by two orthogonalsensors as shown in Figure 15 e gure shows the typicalwhirl phenomenon of large circles with embedded smallerones e decomposition results based on the improvedBEMD method (N 16 λ 005) are shown in Figure 16

Only three IMFs appear in Figure 16 and the singlecomponents c2 and c3 and the noise component c1 are suc-cessfully separated from the original signal e HT is appliedto the real and imaginary parts of c2 and c3 respectively andthe instantaneous frequency and instantaneous amplitude areobtained as shown in Figure 17 e three-dimensional timedomain of ax2 and ay2 and of ax3 and ay3 is shown in Figure 18Figure 17 indicates that the frequency of c2 is approximatelytwice the frequency of c3 and that ax2 is larger than ay2 Inaddition ay3 is slightly larger than ax3 but the range of changefor ax3 is greater than the range for ay3 It is inferred that c2 isthe fundamental frequency signal and that c3 is the half-frequency signal in the oil whirl signal

43 Analysis of Looseness and Rotor Rubbing Composite FaultSignal Based on the Improved BEMD Method Loose androtor rubbing composite faults are set on the testequipment shown in Figure 3 in Section 41 Loose fault isset on the nondrive end of the motor and the distancebetween the plastic rod and the shaft is xed near thesensor on the left side of the disk As the rotor speedincreases the vibration increases and the rubbing faultoccurs which is stable at around 1700 rmin e com-posite fault signals are collected by two orthogonal sen-sors as shown in Figure 19 e decomposition resultsbased on the improved BEMD method are shown inFigure 20

Figures 19(c) and 19(d) indicate that the signal com-ponent mainly contain 1X 2X (X 28Hz) and a frequencymodulated signal generated due to time-varying stinessis phenomenon is similar to that described reference [20]Four IMFs appear in Figure 20 and the single components1X 2X signals and the FM signal c2 are successfully separatedfrom the original signal e HT is applied to the real andimaginary parts of c2 c3 and c4 respectively and the in-stantaneous frequency and instantaneous amplitude areobtained as shown in Figure 21 ere are some uctuationsin the frequencies of c2 and c3 but these uctuations aredierent from the random uctuations in the above casesey have obvious regularity and are characteristic of FMsignals e frequency modulation characteristics of c2 aremore obvious than those of c3 In addition by observing theinstantaneous amplitude of c2 it is seen that c2 is still anamplitude modulation signal

minus100

0100

minus100

0

100

Time (s)Real(cpf1)

Imag

(cpf

1)

0 01 02 03 04 05

(a)

Time (s)Real(cpf2) minus100

0100

minus100

0

100

Imag

(cpf

2)

0 01 02 03 04 05

(b)

minus80 0 80minus60

0

60

Real(cpf1)

Imag(cpf1)

(c)

minus100 0 100minus100

0

100

Real(cpf2)

Imag(cpf2)

(d)

Figure 13 e rst two decomposed results of the ltered signal based on the CLMD method

10 Shock and Vibration

0025

05

ndash150

0

150ndash150

0

150

Time (s)Real(z)

Imag

(z)

(a)

ndash150 ndash75 0 75 150ndash150

ndash75

0

75

150

Real(z)

Imag

(z)

(b)

0 72 144 216 2880

40

80

Frequency (Hz)

Am

plitu

de

X 72Y 7321

X 36Y 3433

Real(z)

(c)

Frequency (Hz)0 72 144 216 288

0

40

80

X 36Y 3475

Am

plitu

de

X 72Y 6157

Imag(z)

(d)

Figure 15e oil whirl signal z (a) the 3D time-domain wave of z (b) the 2D plane of z (c) the Fourier spectrum of Real[z] (d) the Fourierspectrum of Imag[z]

30

40

50

60

70

Am

plitu

de

0 01 02 03 04 05Time (s)

ax1ay1

(a)

70

110

150

Freq

uenc

y (H

z)

0 01 02 03 04 05Time (s)

fx1fy1

(b)

0 01 02 03 04 0520

70

120

Time (s)

Am

plitu

de

ax2ay2

(c)

0 01 02 03 04 050

50

100

Time (s)

Freq

uenc

y (H

z)

fx2fy2

(d)

Figure 14 (a) e instantaneous amplitude of the real part and imaginary part of cpf1 (b) the instantaneous frequency of the real part andimaginary part of cpf1 (c) the instantaneous amplitude of the real part and imaginary part of cpf2 (d) the instantaneous frequency of the realpart and imaginary part of cpf2

Shock and Vibration 11

0025

05

ndash5

0

5ndash5

0

5

Time (s)Real(c1)

Imag

(c1)

(a)

0025

05

ndash1000

100ndash100

0

100

Time (s)

Imag

(c2)

Real(c2)

(b)

0025

05

ndash500

50ndash50

0

50

Time (s)

Imag

(c3)

Real(c3)

(c)

0025

05

ndash100

10ndash20

0

20

Time (s)

Imag

(r)

Real(r)

(d)

ndash100 ndash50 0 50 100ndash100

ndash50

0

50

100

Imag

(c2)

Real(c2)

(e)

ndash40 ndash20 0 20 40ndash40

ndash20

0

20

40Im

ag(c

3)

Real(c3)

(f )

Figure 16 e decomposition results of the oil whirl signal based on the improved BEMD method

60

90

120

Am

plitu

de

0 01 02 03 04 05Time (s)

ax2ay2

(a)

50

75

100

Freq

uenc

y (H

z)

0 01 02 03 04 05Time (s)

fx2fy2

(b)

Figure 17 Continued

12 Shock and Vibration

0 01 02 03 04 0520

35

50

Time (s)

Am

plitu

de

ax3ay3

(c)

0 01 02 03 04 0520

35

50

Time (s)

Freq

uenc

y (H

z)

fx3fy3

(d)

Figure 17 e instantaneous amplitude and frequency of c2 and c3 from the oil whirl signal obtained by the HT (a) the instantaneousamplitude of the real part and imaginary part of c2 (b) the instantaneous frequency of the real part and imaginary part of c2(c) the instantaneous amplitude of the real part and imaginary part of c3 (d) the instantaneous frequency of the real part and imaginarypart of c3

0

025

05

40

80

12040

80

120

Time (s)

X 03877Y 9582Z 826

ax2

X 01309Y 9595Z 8452

a y2

(a)

0

025

05

20

35

5020

35

50

Time (s)

X 03955Y 3762Z 3822

ax3

X 008838Y 3608Z 3733

a y3

(b)

Figure 18 e 3D time domain of the instantaneous amplitude of ax2 and ay2 (a) and ax3 and ay3 (b) from the oil whirl signal

0025

05

minus1500

150minus150

0

150

t (s)Real(z)

Imag

(z)

(a)

Imag(z)

Real(z)

minus150

0

150

minus150 0 150

(b)

Figure 19 Continued

Shock and Vibration 13

In order to further verify the correctness of the in-stantaneous amplitude-frequency characteristics of theproposed method the real and imaginary parts of thecomposite fault signal z are analyzed separately using syn-chrosqueezed wavelet transforms (SWT) proposed in ref-erence [21]-e results are shown in Figure 22 It is seen thatthe time-frequency representations of the composite faultsignal z also include the AM-FM signal and the 1X signalwhich proves the correctness of the proposed methodCompared with the SWT method the instantaneousamplitude-frequency characteristics acquired by the HTmethod are relatively straightforward

44 lte Bistable Behavior Analysis of the Fan Rotor Based onBEMD -e bistability of the rotor is a nonlinear behaviorof the rotor-bearing system which is the state in which therotor jumps from one stable state to another forming astep -e bivariate signal of the bistable behavior iscomposed of two signals collected by two displacementsensors from orthogonal locations on the experimentaldevices in literature [22] as shown in Figure 23 Literature[22] shows that the cause of the bistable behavior remainsto be further explored -is paper uses this case to il-lustrate the feasibility of BEMD to analyze nonstationarysignals

0

40

80

Am

plitu

de Real(z)

0 100 200 300 400Frequency (Hz)

(c)

Am

plitu

de

0 100 200 300 400Frequency (Hz)

0

20

40

60

80

Imag(z)

(d)

Figure 19 -e composite fault signal z (a) the 3D time domain wave of z (b) the 2D plane of z (c) the Fourier spectrum of Real[z] (d) theFourier spectrum of Imag[z]

0025

05

minus30

3minus5

0

5

Time (s)Real(c1)

Imag

(c1)

(a)

0025

05

minus200

20minus20

0

20

Time (s)Real(c2 )

Imag

(c2)

(b)

0025

05

minus200

20minus50

0

50

Time (s)Real(c2)

Imag

(c2)

(c)

0025

05

minus1000

100minus100

0

100

Time (s)Real(c3)

Imag

(c3)

(d)

0025

05

minus400

40minus30

0

30

Time (s)Real(r)

Imag

(r)

(e)

Figure 20 -e decomposition results of the composite fault signal based on the improved BEMD method

14 Shock and Vibration

-e x and y signals in the horizontal and vertical di-rections of the left and right bearings respectively from thefan rotors are collected with four displacement sensorsLetting z x+ jy the time and frequency domain plots of z areshown in Figure 24 where the fan rotor speed is 5500 rpm thesampling frequency is 2000Hz and the number of samplingpoints is 1024 -e left and right columns respectively showthe time and frequency domain plots of the vibration signalsfrom the left and right bearings of the fan rotor Bistablebehavior arises in the fan rotor and the amplitudes of the

vibration signals vary significantly in different positions anddirections Further studies are required to explain the causesof this bistability -e present study focuses on extracting thebistable behavioral signal characteristics to verify the feasi-bility of the proposed method

-e decomposition results of the bistable behavioralsignals based on the improved BEMDmethod are shown inFigure 25 c1 c2 c3 and r are separated in order from zusing the improved BEMD method c1 shows a randomarrangement and is considered the high-frequency noise

0

10

20

Am

plitu

de

ax2ay2

0 025 05Time (s)

(a)

0

152

304

Freq

uenc

y (H

z)

0 025 05Time (s)

fy2

fx2

(b)

0

10

20

30

40

Am

plitu

de

ax3ay3

0 025 05Time (s)

(c)

0

56

112

Freq

uenc

y (H

z)

fx3fy3

0 025 05Time (s)

(d)

0 025 050

50

100

Time (s)

Am

plitu

de

ax4ay4

(e)

0 025 050

28

56

Time (s)

Freq

uenc

y (H

z)

fx4fy4

(f )

Figure 21 -e instantaneous amplitude and frequency of c2 c3 and c4 obtained by the HT (a) the instantaneous amplitude of the real partand imaginary part of c2 (b) the instantaneous frequency of the real part and imaginary part of c2 (c) the instantaneous amplitude of the realpart and imaginary part of c3 (d) the instantaneous frequency of the real part and imaginary part of c3 (e) the instantaneous amplitude of thereal part and imaginary part of c4 (f ) the instantaneous frequency of the real part and imaginary part of c4

Shock and Vibration 15

signal c2 is considered to represent the extracted bistablebehavior signals -e HT is applied to the real andimaginary parts of c2 to obtain the instantaneous amplitudeand frequency of c2 from the left and right columns fromFigure 25 as shown in Figure 26 Figure 27 shows thethree-dimensional time domain of ax2 and ay2 from the leftand right columns respectively Figure 26 shows that thevibration signal amplitude on the left side of the fan de-creases from large to small opposite of the behavior ofthe right -e horizontal vibration signal amplitude on theleft side of the fan is larger than that of the vertical di-rection signal opposite of the right -is result validatesthat the vibration signals from different directions orpositions are different when the fan produces bistablebehavior In addition the time of the bistable behavior canbe determined according to the jump point of the am-plitude or frequency

5 Discussion

-e BEMD algorithm decomposes two orthogonal di-rections of vibration signals as a complex signal which is atwo-dimensional digital signal processing method thusensuring that the real and imaginary parts have the samedecomposition scale Similar to EMD the envelope mean iscritical for the decomposition effect of BEMD but the en-velope mean in BEMD is three-dimensional If the numberof projection directions of the complex signal in three-dimensional space is larger the corresponding envelope

signal is also more -us the envelope mean value is moreaccurate and the BEMD decomposition effect is betterIncreasing the number of projection directions can improvemodal aliasing Like EMD BEMD also produces falsecomponents when decomposing signals Generally speakingthe energy of the false components is low and these low-energy false components do not contain fault characteristicinformation and the introduction of the energy thresholdcriterion in the termination condition can increase thedecomposition speed of the BEMD

-e experimental results show that there is a certaindifference in the existence of vibration signal character-istics in different directions when rotating machinery failsIn addition when the number of projection directionsis increased the decomposition speed of BEMD willdecrease

6 Conclusions

We use BEMD and HT to extract the instantaneousamplitude-frequency features of rotor faults A bivariateinstantaneous feature extraction method based on the im-proved BEMD method and the HT is investigated whichextends the fault feature extraction technology to two di-mensions -e BEMD method is suitable to analyze thecomplex multicomponent bivariate signals -e mainsingle-component bivariate signals are separated from themulticomponent bivariate signals of the fan rotor bistabilityfor the oil film oscillation and the oil film vortex using the

0

2

4

6

Time (s)

Freq

uenc

y (H

z)

0 01 02 03 04 052

3

4

5

6

7

8log2 fx

(a)

0

2

4

6

Time (s)

Freq

uenc

y (H

z)

0 01 02 03 04 052

3

4

5

6

7

8log2 fy

(b)

Figure 22 -e results of the composite fault signal z based on SWT the time-frequency representation of (a) the real part of z and (b) theimaginary part of z

Left bearing predstal

Locations of sensors

Fanrotor

Right bearing predstal

Axis

Orthogonal directions

Figure 23 -e schematic diagram of the experimental apparatus

16 Shock and Vibration

Real(z) 00256

0512minus300

0300

minus400

0

400

Time (s)

Imag

(z)

00256

0512minus500

0500

minus500

0

500

Time (s)Real(z)

Imag

(z)

Imag

(z)

minus300 0 300minus500

0

500

Real(z)minus500 0 500

minus500

0

Imag

(z)

Real(z)

500

0 100 200 300 4000

100

200

300

Frequency (Hz)

Am

plitu

de Imag(z)

Frequency (Hz)0 100 200 300 400

0

200

400

Am

plitu

de

Imag(z)

0 100 200 300 4000

70

140

Frequency (Hz)

Am

plitu

de

Real(z)

0 100 200 300 4000

200

400

Frequency (Hz)

Am

plitu

de Real(z)

(a) (b)

Figure 24 -e time and frequency domain plots of the bistable behavior signals

00256

0512

minus800

80minus80

0

80

Time (s)Real(c1 )

Imag

(c1)

Time (s)Real(c1 )

Imag

(c1)

00256

0512

minus400

40minus40

0

40

Time (s)Real(c2) 0

02560512

minus5000

500minus500

0

500

Imag

(c2)

Time (s)Real(c2)

Imag

(c2)

00256

0512

minus5000

500minus500

0

500

Time (s)Real(r) 00256

0512

minus800

80minus80

0

80

Imag

(irc

rm

)

Time (s)Real(r)

Imag

(r)

00256

0512

minus2000

200minus200

0

200

(a) (b)

Figure 25 -e decomposition results of the bistable behavior signals based on the improved BEMD method

Shock and Vibration 17

improved BEMD method For the single-component bi-variate signal the HT is used to obtain the correspondinginstantaneous amplitude and frequency characteristics -eproposed method can examine the detailed information of asingle rotation component

Data Availability

-e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

-e authors declare that they have no conflicts of interest

Authorsrsquo Contributions

All the authors contributed to this work Chuanjin Huangconceived and designed the simulation and experiments anddrafted the manuscript Haijun Song performed the simu-lations and experiments and analyzed the data and

0 0256 05120

50

100

150

Time (s)

Freq

uenc

y (H

z)

fx2

fy2

0 0256 05120

200

400

600

Time (s)

Am

plitu

de

ax2ay2

(a)

fx2

fy2

0 0256 05120

90

180

270

360

Time (s)

Freq

uenc

y (H

z)

0 0256 05120

200

400

600

Time (s)

Am

plitu

de

ax2Data 2

(b)

Figure 26 -e instantaneous amplitude and frequency of c2 from the (a) left and (b) right columns

0

0256

0512

0100

200300

400500

0

100

200

300

400

500

Time (s)

X 04035Y 3509Z 130

X 04235Y 2265Z 3613

X 007Y 1131Z 2191

X 0105Y 3837Z 3312

ax2

a y2

Figure 27 -e three-dimensional time domain of ax2 and ay2

18 Shock and Vibration

Wenping Lei and Yajun Meng performed the experimentsand analyzed the data All the authors contributed to thewriting and discussion of the paper

Acknowledgments

-is research was funded by the Henan Provincial HigherEducation Key Research Project (Grant nos 18A460006 and19A460029) Henan High-Level Innovative Scientific andTechnological Talent Team Construction Project (Grant noC20150034) and Zhengzhou Institute of Technology In-novation Team Project (Grant no CXTD2017K1)

References

[1] R Yan R X Gao and X Chen ldquoWavelets for fault diagnosisof rotary machines a review with applicationsrdquo Signal Pro-cessing vol 96 pp 1ndash15 2014

[2] J Cheng D Yu J Tang and Y Yang ldquoApplication of frequencyfamily separation method based upon EMD and local Hilbertenergy spectrum method to gear fault diagnosisrdquo Mechanismand Machine lteory vol 43 no 6 pp 712ndash723 2008

[3] H Liu and M Han ldquoA fault diagnosis method based on localmean decomposition and multi-scale entropy for rollerbearingsrdquoMechanism andMachinelteory vol 75 pp 67ndash782014

[4] Z Zheng W Jiang Z Wang Y Zhu and K Yang ldquoGear faultdiagnosis method based on local mean decomposition andgeneralized morphological fractal dimensionsrdquo Mechanismand Machine lteory vol 91 pp 151ndash167 2015

[5] W Yang R Court P J Tavner and C J Crabtree ldquoBivariateempirical mode decomposition and its contribution to windturbine condition monitoringrdquo Journal of Sound and Vi-bration vol 330 no 15 pp 3766ndash3782 2011

[6] L Qu X Liu G Peyronne and Y Chen ldquo-e holospectrum anewmethod for rotor surveillance and diagnosisrdquoMechanicalSystems amp Signal Processing vol 3 no 3 pp 255ndash267 1989

[7] F Q Wu and G Meng ldquoCompound rub malfunctions featureextraction based on full-spectrum cascade analysis and SVMrdquoMechanical Systems and Signal Processing vol 20 no 8pp 2007ndash2021 2006

[8] Y Chen Q Gao and Z Guan ldquoSelf-loosening failure analysisof bolt joints under vibration considering the tighteningprocessrdquo Shock and Vibration vol 2017 Article ID 203842115 pages 2017

[9] L Chen J Han W Lei Y Cui and Z Guan ldquoFull-vectorsignal acquisition and information fusion for the fault pre-dictionrdquo International Journal of Rotating Machineryvol 2016 Article ID 5980802 7 pages 2016

[10] C Chen Y Meng and Y Du ldquoApplication of the full vectorspectrum based on EMD in fault diagnosis of bearingsrdquoJournal of Mechanical Strength vol 37 pp 806ndash811 2015

[11] C Huang X Wu and W Cao ldquoLMD-based on full vectorenvelope technique and its application in TRT vibration faultdiagnosisrdquo Electric Power Automation Equipment vol 35pp 168ndash174 2015 in Chinese

[12] X Zhao T H Patel and M J Zuo ldquoMultivariate EMD andfull spectrum based condition monitoring for rotating ma-chineryrdquo Mechanical Systems and Signal Processing vol 27pp 712ndash728 2012

[13] G Rilling P Flandrin P Gonalves and J M Lilly ldquoBivariateempirical mode decompositionrdquo IEEE Signal ProcessingLetters vol 14 no 12 pp 936ndash939 2007

[14] C Park D Looney M M Van Hulle and D P Mandic ldquo-ecomplex local mean decompositionrdquo Neurocomputingvol 74 no 6 pp 867ndash875 2011

[15] N Rehman and D P Mandic ldquoEmpirical mode de-composition for trivariate signalsrdquo IEEE Transactions onSignal Processing vol 58 no 3 pp 1059ndash1068 2010

[16] N Rehman and D P Mandic ldquoMultivariate empirical modedecompositionrdquo Proceedings of the Royal Society A Mathe-matical Physical and Engineering Sciences vol 466 no 2117pp 1291ndash1302 2010

[17] Y Lv R Yuan and G Song ldquoMultivariate empirical modedecomposition and its application to fault diagnosis of rollingbearingrdquo Mechanical Systems and Signal Processing vol 81pp 219ndash234 2016

[18] C Huang Y Meng and W Lei ldquoFull vector envelopetechnique based on complex local mean decomposition andits application in fault feature extraction for rotor systemrdquoJournal of Mechanical Engineering vol 52 no 7 p 69 2016in Chinese

[19] G Rilling P Flandrin and P Goncalves ldquoOn empirical modedecomposition and its algorithmsrdquo in Proceedings of IEEE-EURASIP Workshop on Nonlinear Signal and Image Pro-cessing pp 8ndash11 IEEE Trieste Italy June 2003

[20] L Yang X Chen and S Wang ldquoMechanism of fast time-varying vibration for rotorndashstator contact system with ap-plication to fault diagnosisrdquo Journal of Vibration andAcoustics vol 140 no 1 article 014501 2018

[21] I Daubechies J Lu and H-TWu ldquoSynchrosqueezed wavelettransforms an empirical mode decomposition-like toolrdquoApplied and Computational Harmonic Analysis vol 30 no 2pp 243ndash261 2011

[22] L-S Qu Holospectrum and Holobalancing Technique inMachinery Diagnosis Beijing Science Press Beijing China2007

Shock and Vibration 19

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from signal y e mismatch in the number of IMFs leads tochallenges in information fusion In addition modal aliasingoccurs in IMF1 from signal xemode aliasing obscures thephysical meaning of the IMF components and aects thesubsequent analysis

e decomposition results of the oil lm oscillationsignal using BEMD (N 4) are shown in Figure 7 esignal z is decomposed into eight complex rotationalcomponents c1ndashc8 After decomposition the IMF numbersfrom x are the same as those from y e noise component c1is more easily separated from signal z relative to the case ofIMF1 from signal x c2 and c3 are considered fundamentaland oscillatory components respectively the frequencycomponent of c2 is 108Hz and the frequency of c3 is 52Hze two trajectories composed of c2 and c3 generated by the

BEMD method are shown in Figures 8(a) and 8(b) re-spectively while the corresponding ones of IMF2 and IMF3generated by EMD method are shown in Figures 8(c) and8(d) respectively Figure 8 indicates that the orbit ar-rangements obtained by the BEMD method are superior tothose of the EMD method

Relative to the EMDmethod the decomposition eect isimproved when using BEMD to decompose the oil lmoscillation signal However there are still two problemswhen using BEMD to decompose the oil lm oscillationsignal One is the occurrence of modal aliasing in the IMFsand the other is increased false components From Figure 6the modal aliasing occurs at the end of c4 which causes thevalue of c3 to become small at the end After decompositioneight IMF components and one residual r are obtained

001

0203

ndash2

0

2ndash2

0

2

Time (s)Real(z)

Imag

(z)

(a)

ndash2 0 2ndash2

0

2

Real(z)

Imag

(z)

(b)

Figure 1 e simulated signal of z (a) the three-dimensional time-domain waveform (b) the two-dimensional plots

001

0203

ndash2

0

2ndash2

0

2

Real

Imag

Four directionsof tangent z(t)

Tangent mean

Time (s)

(a)

001

0203

ndash2

0

2ndash2

0

2

Time (s)Real

Imag

c1

c2

(b)

ndash2 0 2ndash2

0

2

Real

Imag

c1

c2

(c)

Figure 2 (a) z(t) and its four directions of the tangent and tangential mean (b) the decomposition results of z(t) based on the BEMD (c)the planes of c1 and c2

4 Shock and Vibration

However most of the IMF components contain uselessinformation In most cases the fault characteristics are oftenincluded in higher-energy components -e projection di-rections and the energy end condition based on the energythreshold are increased in the improved BEMD method toenhance the decomposition quality

-e decomposition results of the oil film oscillationsignal are shown in Figure 9 using the improved BEMDmethod (N 16 λ 005) -e oil film oscillation signal isdecomposed into four complex rotational components c1ndashc4

and the remaining component r c1 is considered as the noisecomponent According to the Fourier analysis c2 is thefundamental signal with a frequency of 108Hz and c3 is theoscillation signal with a frequency of 52Hz It is clear thatthe number of IMFs is reduced and the fundamental fre-quency component c2 and the oscillation component c3 aremore accurately extracted from the original signal In par-ticular the value of the c4 end becomes smaller and the valueof the c3 end becomes larger relative to the values of c3 and c4shown in Figure 7 Two trajectories made up of c2 and c3 are

Rigid foundation

Motor

Motorcontroller

Eddy current probe system

DiscFlexible coupling

Bearing predstal

Oil cup

Probes

Computer

Axis

ADcard

Figure 3 -e schematic diagram and the experimental apparatus

0 01 02 03 04 05ndash200

ndash100

0

100

200

Time (s)

x (micro

m)

(a)

0 048X X 2X 3X 4X0

30

60 X 52Y 4961

Frequency (Hz)

A (micro

m)

X 108Y 5257

(b)

0 01 02 03 04 05ndash100

ndash50

0

50

100

150

Time (s)

y (microm

)

(c)

0 048X X 2X 3X 4X0

30

60

Frequency (Hz)

A (micro

m)

X 52Y 4698

X 108Y 4437

(d)

Figure 4-e oil film oscillation signal (a) the horizontal and vertical direction signals x (b) the Fourier spectrum of signal x (c) the verticaldirection signals y (d) the Fourier spectrum of signal y

Shock and Vibration 5

shown in Figure 10 From c2 and c3 three-dimensional timedomain and the trajectories the fundamental frequencysignal c2 has a small elliptic amplitude change c3 is a largeseries of amplitude conversion elliptical compositions thatcause a signicant oscillation Relative to the orbit of c3 inFigure 6 the orbit of c3 in Figure 10 is improved particularlyin the center region

Increasing the number of signal projection directionsresults in an increase in the number of projection signals

en the tangent mean which is obtained by interpolatingthe local maximum of the projected signals with a splineinterpolation is more accurate However it is meaningless tocontinue to increase the number of projection directions whenthe tangent mean is accurately tted It is found that the signaldecomposition results of N 16 are almost the same as N 1024 when considering the complex rotation componentsseparated by the original signal However the calculation timeof the BEMD algorithm with N 1024 is greatly increased

001

0203

0405

ndash100

0

100

ndash100

ndash50

0

50

100

150

Time (s)Real(z) (microm)

Imag

(z) (microm

)

(a)

ndash150 ndash100 ndash50 0 50 100 150ndash150

ndash100

ndash50

0

50

100

150

Real(z) (microm)

Imag

(z) (microm

)

(b)

Figure 5 e oil lm oscillation signal of z (a) the three-dimensional time-domain waveform (b) the two-dimensional plots

ndash80

80

IMF 1

ndash80

80

IMF 2

ndash90

90

IMF 3

ndash30

30

IMF 4

ndash1010

IMF 5

ndash10

10

IMF 6

ndash4

4

IMF 7

ndash44

IMF 8

ndash33

IMF 9

ndash44

IMF 10

0 01 02 03 04 05ndash5

1r

Time (s)(a)

ndash4

4IM

F 1

ndash6060

IMF 2

ndash60

60

IMF 3

ndash15

15

IMF 4

ndash55

IMF 5

ndash55

IMF 6

ndash5

5

IMF 7

ndash0505

IMF 8

(b)

0 01 02 03 04 0505

1r

Time (s)

Figure 6 e decomposition results of signals x and y using EMD

6 Shock and Vibration

e HT is applied to the real and imaginary parts of c2and c3 respectively and the instantaneous frequency andinstantaneous amplitude are obtained as shown in Fig-ure 11 where ax2 and ax3 represent the instantaneous

amplitude of the real parts of c2 and c3 respectively ay2and ay3 represent the instantaneous amplitude of theimaginary parts of c2 and c3 respectively fx2 and fx3represent the instantaneous frequency of the real parts of

0025

05

ndash50

5ndash5

0

5

Time (s)Real(c1)

Imag

(c1)

(a)

0025

05

ndash1000

100ndash100

0

100

Time (s)Real(c2 )

Imag

(c2)

(b)

0025

05

ndash1000

100ndash100

0

100

Time (s)Real(c3 )

Imag

(c3)

(c)

0025

05

ndash200

20ndash50

0

50

Time (s)Real(c4 )

Imag

(c4)

Modal aliasing

(d)

0025

05

ndash100

10ndash10

0

10

Time (s)Real(c5 )Im

ag(c

5)

(e)

0025

05

ndash50

5ndash5

0

5

Time (s)Real(c6 )

Imag

(c6)

(f )

0025

05

ndash50

5ndash1

0

1

Time (s)Real(c7)

Imag

(c7)

(g)

0025

05

ndash50

5ndash1

0

1

Time (s)Real(c8)

Imag

(c8)

(h)

0025

05

ndash100

10ndash5

0

5

Time (s)Real(r)

Imag

(r)

(i)

Figure 7 Decomposition results of the oil lm oscillation signal using BEMD (N 4)

ndash80 ndash40 0 40 80ndash60

ndash30

0

30

60

Real(c2)

Imag

(c2)

(a)

ndash100 ndash50 0 50 100ndash100

ndash50

0

50

100

Real(c3)

Imag

(c3)

(b)

ndash80 ndash40 0 40 80ndash60

ndash30

0

30

60

IMFx2

IMF y

2

(c)

ndash100 ndash50 0 50 100ndash100

ndash50

0

50

100

IMFx3

IMF y

3

(d)

Figure 8e orbits made up of (a) the real and imaginary parts of c2 (b) real and imaginary parts of c3 (c) IMFx2 and IMFy2 and (d) IMFx3and IMFy3

Shock and Vibration 7

c2 and c3 respectively and fy2 and fy3 represent the in-stantaneous frequency of the imaginary parts of c2 and c3respectively

In Figure 11 ax3 is much larger than ay3 and their phasesare separated by nearly 180deg which shows that the amplitudeand the phase of the oil lm oscillation signal in dierentdirections can vary fx2 and fy2 and fx3 and fy3 are ap-proximately the same ax2 and ay2 show little change andtheir phases are the same Since BEMD is a bivariate ex-tension of EMD like EMD BEMD also has an endpointeect ere are some uctuations in the instantaneousamplitude and instantaneous frequency due to the end eect

and the edge eect of the Hilbert transform e three-dimensional time domain of ax2 and ay2 and of ax3 and ay3 isshown in Figure 12e amplitude range of c3 is much largerthan that of c2 It is can be inferred that the main componentof oscillation is c3 with the frequency of 52Hz in the oil lmoscillation signal

e decomposition results of the oil lm oscillationsignal are shown in Figure 13 using the CLMD methodproposed in reference [14] e oil lm oscillation signal isdecomposed into four complex product functions cpf1ndashcpf4e noise component is not separated from the oil lmoscillation signal using the CLMD method Moreover the

0025

05

ndash50

5

Time (s)Real(c1 )

ndash5

0

5Im

ag(c

1)

(a)

0025

05

ndash1000

100

Time (s)Real(c2)

ndash100

0

100

Imag

(c2)

(b)

0025

05ndash100

0100

ndash100

0

100

Imag

(c3)

Real(c3 ) Time (s)

(c)

0 02505

ndash100

10

Time (s)Real(c4)

ndash20

0

20

Imag

(c4)

(d)

0025

05ndash10

010

Time (s)Real(c5)

ndash10

0

10

Imag

(c5)

(e)

0 02505

ndash50

5

Time (s)Real(c6)

ndash5

0

5

Imag

(c6)

(f )

0 02505

ndash202

Time (s)Real(c7)

ndash2

0

2

Imag

(c7)

(g)

0025

05ndash5

05

Time (s)Real(c8)

ndash5

0

5

Imag

(c8)

(h)

0025

05ndash10

010

Time (s)Real(r)

ndash5

0

5

Imag

(r)

(i)

Figure 9 Decomposition results of the oil lm oscillation signal using the improved BEMD method (N 16 λ 005)

ndash80 0 40 80ndash40Real(c2)

ndash60

ndash30

0

30

60

Imag

(c2)

(a)

ndash80

ndash60

ndash40

ndash20

0

20

40

60

80

Imag

(c3)

ndash40 0 40 80ndash80Real(c3)

(b)

Figure 10 e orbits were made up of c2 (a) and c3 (b) obtained with the improved BEMD method

8 Shock and Vibration

single component fundamental frequency signal and the oillm oscillation signal were not successfully separated Oneof the possible reasons is that in the CLMD algorithm thecomplex signal is only projected onto the x-axis and the y-axis unlike BEMD which projected on multiple directionsIn addition CLMD is a bivariate extension of LMD LMDused a moving average algorithm when tting the signalenvelope which can lter noise to a certain extent esignals other than the noise component c1 in Figure 9 areadded to obtain a ltered oil lm oscillation signal which isthen decomposed by the CLMD method and the rst twodecomposed results are shown in Figure 13 It is seen thatthe single component fundamental frequency signal and

the single component oil lm oscillation signal areseparated

cpf1 and cpf2 consisted of real part signals and imaginarypart signals both of which were composed of the product ofthe envelope signal and the pure frequency modulationfunctione envelope signal is the instantaneous amplitudeof the signal e corresponding instantaneous frequencywas obtained by deriving the inverse function of the cosinepure frequency modulation function e instantaneousamplitude and instantaneous frequency curves are shown inFigure 14 e instantaneous amplitude and frequencyobtained by the CLMD method are smoother than in Fig-ure 12e reason is mainly that the CLMDmethod uses the

ax2ay2

30

40

50

60

70A

mpl

itude

01 02 03 04 050Time (s)

(a)

fx2fy2

70

110

150

Freq

uenc

y (H

z)

01 02 03 04 050Time (s)

(b)

ax3ay3

20

70

120

Am

plitu

de

01 02 03 04 050Time (s)

(c)

fx3fy3

0

50

100

Freq

uenc

y (H

z)01 02 03 04 050

Time (s)

(d)

Figure 11 (a) e instantaneous amplitude of the real part and imaginary part of c2 (b) the instantaneous frequency of the real part andimaginary part of c2 (c) the instantaneous amplitude of the real part and imaginary part of c3 (d) the instantaneous frequency of the real partand imaginary part of c3

0 01 02 03 04 05

2050

800

20

40

60

80

Time (s)ax2

a y2

(a)

3060

90120

0

20

40

60

80

Time (s)ax3

a y3

0 01 02 03 04 05

(b)

Figure 12 e 3D time domain of instantaneous amplitude was made up of (a) ax2 and ay2 and (b) ax3 and ay3

Shock and Vibration 9

moving average ltering algorithm to obtain the signalenvelope curve However this is the result of using theCLMD algorithm after ltering out noise with BEMD If theBEMD algorithm is not used for ltering noise the in-stantaneous amplitude and instantaneous frequency curvesof the single component were not obtained by the CLMDmethod

42 Analysis ofOilWhirl Signal Based on the ImprovedBEMDMethod In the method similar to that presented in Section41 the oil whirl signals of the rotor test rig with a speedparameter of 4320 rpm are collected by two orthogonalsensors as shown in Figure 15 e gure shows the typicalwhirl phenomenon of large circles with embedded smallerones e decomposition results based on the improvedBEMD method (N 16 λ 005) are shown in Figure 16

Only three IMFs appear in Figure 16 and the singlecomponents c2 and c3 and the noise component c1 are suc-cessfully separated from the original signal e HT is appliedto the real and imaginary parts of c2 and c3 respectively andthe instantaneous frequency and instantaneous amplitude areobtained as shown in Figure 17 e three-dimensional timedomain of ax2 and ay2 and of ax3 and ay3 is shown in Figure 18Figure 17 indicates that the frequency of c2 is approximatelytwice the frequency of c3 and that ax2 is larger than ay2 Inaddition ay3 is slightly larger than ax3 but the range of changefor ax3 is greater than the range for ay3 It is inferred that c2 isthe fundamental frequency signal and that c3 is the half-frequency signal in the oil whirl signal

43 Analysis of Looseness and Rotor Rubbing Composite FaultSignal Based on the Improved BEMD Method Loose androtor rubbing composite faults are set on the testequipment shown in Figure 3 in Section 41 Loose fault isset on the nondrive end of the motor and the distancebetween the plastic rod and the shaft is xed near thesensor on the left side of the disk As the rotor speedincreases the vibration increases and the rubbing faultoccurs which is stable at around 1700 rmin e com-posite fault signals are collected by two orthogonal sen-sors as shown in Figure 19 e decomposition resultsbased on the improved BEMD method are shown inFigure 20

Figures 19(c) and 19(d) indicate that the signal com-ponent mainly contain 1X 2X (X 28Hz) and a frequencymodulated signal generated due to time-varying stinessis phenomenon is similar to that described reference [20]Four IMFs appear in Figure 20 and the single components1X 2X signals and the FM signal c2 are successfully separatedfrom the original signal e HT is applied to the real andimaginary parts of c2 c3 and c4 respectively and the in-stantaneous frequency and instantaneous amplitude areobtained as shown in Figure 21 ere are some uctuationsin the frequencies of c2 and c3 but these uctuations aredierent from the random uctuations in the above casesey have obvious regularity and are characteristic of FMsignals e frequency modulation characteristics of c2 aremore obvious than those of c3 In addition by observing theinstantaneous amplitude of c2 it is seen that c2 is still anamplitude modulation signal

minus100

0100

minus100

0

100

Time (s)Real(cpf1)

Imag

(cpf

1)

0 01 02 03 04 05

(a)

Time (s)Real(cpf2) minus100

0100

minus100

0

100

Imag

(cpf

2)

0 01 02 03 04 05

(b)

minus80 0 80minus60

0

60

Real(cpf1)

Imag(cpf1)

(c)

minus100 0 100minus100

0

100

Real(cpf2)

Imag(cpf2)

(d)

Figure 13 e rst two decomposed results of the ltered signal based on the CLMD method

10 Shock and Vibration

0025

05

ndash150

0

150ndash150

0

150

Time (s)Real(z)

Imag

(z)

(a)

ndash150 ndash75 0 75 150ndash150

ndash75

0

75

150

Real(z)

Imag

(z)

(b)

0 72 144 216 2880

40

80

Frequency (Hz)

Am

plitu

de

X 72Y 7321

X 36Y 3433

Real(z)

(c)

Frequency (Hz)0 72 144 216 288

0

40

80

X 36Y 3475

Am

plitu

de

X 72Y 6157

Imag(z)

(d)

Figure 15e oil whirl signal z (a) the 3D time-domain wave of z (b) the 2D plane of z (c) the Fourier spectrum of Real[z] (d) the Fourierspectrum of Imag[z]

30

40

50

60

70

Am

plitu

de

0 01 02 03 04 05Time (s)

ax1ay1

(a)

70

110

150

Freq

uenc

y (H

z)

0 01 02 03 04 05Time (s)

fx1fy1

(b)

0 01 02 03 04 0520

70

120

Time (s)

Am

plitu

de

ax2ay2

(c)

0 01 02 03 04 050

50

100

Time (s)

Freq

uenc

y (H

z)

fx2fy2

(d)

Figure 14 (a) e instantaneous amplitude of the real part and imaginary part of cpf1 (b) the instantaneous frequency of the real part andimaginary part of cpf1 (c) the instantaneous amplitude of the real part and imaginary part of cpf2 (d) the instantaneous frequency of the realpart and imaginary part of cpf2

Shock and Vibration 11

0025

05

ndash5

0

5ndash5

0

5

Time (s)Real(c1)

Imag

(c1)

(a)

0025

05

ndash1000

100ndash100

0

100

Time (s)

Imag

(c2)

Real(c2)

(b)

0025

05

ndash500

50ndash50

0

50

Time (s)

Imag

(c3)

Real(c3)

(c)

0025

05

ndash100

10ndash20

0

20

Time (s)

Imag

(r)

Real(r)

(d)

ndash100 ndash50 0 50 100ndash100

ndash50

0

50

100

Imag

(c2)

Real(c2)

(e)

ndash40 ndash20 0 20 40ndash40

ndash20

0

20

40Im

ag(c

3)

Real(c3)

(f )

Figure 16 e decomposition results of the oil whirl signal based on the improved BEMD method

60

90

120

Am

plitu

de

0 01 02 03 04 05Time (s)

ax2ay2

(a)

50

75

100

Freq

uenc

y (H

z)

0 01 02 03 04 05Time (s)

fx2fy2

(b)

Figure 17 Continued

12 Shock and Vibration

0 01 02 03 04 0520

35

50

Time (s)

Am

plitu

de

ax3ay3

(c)

0 01 02 03 04 0520

35

50

Time (s)

Freq

uenc

y (H

z)

fx3fy3

(d)

Figure 17 e instantaneous amplitude and frequency of c2 and c3 from the oil whirl signal obtained by the HT (a) the instantaneousamplitude of the real part and imaginary part of c2 (b) the instantaneous frequency of the real part and imaginary part of c2(c) the instantaneous amplitude of the real part and imaginary part of c3 (d) the instantaneous frequency of the real part and imaginarypart of c3

0

025

05

40

80

12040

80

120

Time (s)

X 03877Y 9582Z 826

ax2

X 01309Y 9595Z 8452

a y2

(a)

0

025

05

20

35

5020

35

50

Time (s)

X 03955Y 3762Z 3822

ax3

X 008838Y 3608Z 3733

a y3

(b)

Figure 18 e 3D time domain of the instantaneous amplitude of ax2 and ay2 (a) and ax3 and ay3 (b) from the oil whirl signal

0025

05

minus1500

150minus150

0

150

t (s)Real(z)

Imag

(z)

(a)

Imag(z)

Real(z)

minus150

0

150

minus150 0 150

(b)

Figure 19 Continued

Shock and Vibration 13

In order to further verify the correctness of the in-stantaneous amplitude-frequency characteristics of theproposed method the real and imaginary parts of thecomposite fault signal z are analyzed separately using syn-chrosqueezed wavelet transforms (SWT) proposed in ref-erence [21]-e results are shown in Figure 22 It is seen thatthe time-frequency representations of the composite faultsignal z also include the AM-FM signal and the 1X signalwhich proves the correctness of the proposed methodCompared with the SWT method the instantaneousamplitude-frequency characteristics acquired by the HTmethod are relatively straightforward

44 lte Bistable Behavior Analysis of the Fan Rotor Based onBEMD -e bistability of the rotor is a nonlinear behaviorof the rotor-bearing system which is the state in which therotor jumps from one stable state to another forming astep -e bivariate signal of the bistable behavior iscomposed of two signals collected by two displacementsensors from orthogonal locations on the experimentaldevices in literature [22] as shown in Figure 23 Literature[22] shows that the cause of the bistable behavior remainsto be further explored -is paper uses this case to il-lustrate the feasibility of BEMD to analyze nonstationarysignals

0

40

80

Am

plitu

de Real(z)

0 100 200 300 400Frequency (Hz)

(c)

Am

plitu

de

0 100 200 300 400Frequency (Hz)

0

20

40

60

80

Imag(z)

(d)

Figure 19 -e composite fault signal z (a) the 3D time domain wave of z (b) the 2D plane of z (c) the Fourier spectrum of Real[z] (d) theFourier spectrum of Imag[z]

0025

05

minus30

3minus5

0

5

Time (s)Real(c1)

Imag

(c1)

(a)

0025

05

minus200

20minus20

0

20

Time (s)Real(c2 )

Imag

(c2)

(b)

0025

05

minus200

20minus50

0

50

Time (s)Real(c2)

Imag

(c2)

(c)

0025

05

minus1000

100minus100

0

100

Time (s)Real(c3)

Imag

(c3)

(d)

0025

05

minus400

40minus30

0

30

Time (s)Real(r)

Imag

(r)

(e)

Figure 20 -e decomposition results of the composite fault signal based on the improved BEMD method

14 Shock and Vibration

-e x and y signals in the horizontal and vertical di-rections of the left and right bearings respectively from thefan rotors are collected with four displacement sensorsLetting z x+ jy the time and frequency domain plots of z areshown in Figure 24 where the fan rotor speed is 5500 rpm thesampling frequency is 2000Hz and the number of samplingpoints is 1024 -e left and right columns respectively showthe time and frequency domain plots of the vibration signalsfrom the left and right bearings of the fan rotor Bistablebehavior arises in the fan rotor and the amplitudes of the

vibration signals vary significantly in different positions anddirections Further studies are required to explain the causesof this bistability -e present study focuses on extracting thebistable behavioral signal characteristics to verify the feasi-bility of the proposed method

-e decomposition results of the bistable behavioralsignals based on the improved BEMDmethod are shown inFigure 25 c1 c2 c3 and r are separated in order from zusing the improved BEMD method c1 shows a randomarrangement and is considered the high-frequency noise

0

10

20

Am

plitu

de

ax2ay2

0 025 05Time (s)

(a)

0

152

304

Freq

uenc

y (H

z)

0 025 05Time (s)

fy2

fx2

(b)

0

10

20

30

40

Am

plitu

de

ax3ay3

0 025 05Time (s)

(c)

0

56

112

Freq

uenc

y (H

z)

fx3fy3

0 025 05Time (s)

(d)

0 025 050

50

100

Time (s)

Am

plitu

de

ax4ay4

(e)

0 025 050

28

56

Time (s)

Freq

uenc

y (H

z)

fx4fy4

(f )

Figure 21 -e instantaneous amplitude and frequency of c2 c3 and c4 obtained by the HT (a) the instantaneous amplitude of the real partand imaginary part of c2 (b) the instantaneous frequency of the real part and imaginary part of c2 (c) the instantaneous amplitude of the realpart and imaginary part of c3 (d) the instantaneous frequency of the real part and imaginary part of c3 (e) the instantaneous amplitude of thereal part and imaginary part of c4 (f ) the instantaneous frequency of the real part and imaginary part of c4

Shock and Vibration 15

signal c2 is considered to represent the extracted bistablebehavior signals -e HT is applied to the real andimaginary parts of c2 to obtain the instantaneous amplitudeand frequency of c2 from the left and right columns fromFigure 25 as shown in Figure 26 Figure 27 shows thethree-dimensional time domain of ax2 and ay2 from the leftand right columns respectively Figure 26 shows that thevibration signal amplitude on the left side of the fan de-creases from large to small opposite of the behavior ofthe right -e horizontal vibration signal amplitude on theleft side of the fan is larger than that of the vertical di-rection signal opposite of the right -is result validatesthat the vibration signals from different directions orpositions are different when the fan produces bistablebehavior In addition the time of the bistable behavior canbe determined according to the jump point of the am-plitude or frequency

5 Discussion

-e BEMD algorithm decomposes two orthogonal di-rections of vibration signals as a complex signal which is atwo-dimensional digital signal processing method thusensuring that the real and imaginary parts have the samedecomposition scale Similar to EMD the envelope mean iscritical for the decomposition effect of BEMD but the en-velope mean in BEMD is three-dimensional If the numberof projection directions of the complex signal in three-dimensional space is larger the corresponding envelope

signal is also more -us the envelope mean value is moreaccurate and the BEMD decomposition effect is betterIncreasing the number of projection directions can improvemodal aliasing Like EMD BEMD also produces falsecomponents when decomposing signals Generally speakingthe energy of the false components is low and these low-energy false components do not contain fault characteristicinformation and the introduction of the energy thresholdcriterion in the termination condition can increase thedecomposition speed of the BEMD

-e experimental results show that there is a certaindifference in the existence of vibration signal character-istics in different directions when rotating machinery failsIn addition when the number of projection directionsis increased the decomposition speed of BEMD willdecrease

6 Conclusions

We use BEMD and HT to extract the instantaneousamplitude-frequency features of rotor faults A bivariateinstantaneous feature extraction method based on the im-proved BEMD method and the HT is investigated whichextends the fault feature extraction technology to two di-mensions -e BEMD method is suitable to analyze thecomplex multicomponent bivariate signals -e mainsingle-component bivariate signals are separated from themulticomponent bivariate signals of the fan rotor bistabilityfor the oil film oscillation and the oil film vortex using the

0

2

4

6

Time (s)

Freq

uenc

y (H

z)

0 01 02 03 04 052

3

4

5

6

7

8log2 fx

(a)

0

2

4

6

Time (s)

Freq

uenc

y (H

z)

0 01 02 03 04 052

3

4

5

6

7

8log2 fy

(b)

Figure 22 -e results of the composite fault signal z based on SWT the time-frequency representation of (a) the real part of z and (b) theimaginary part of z

Left bearing predstal

Locations of sensors

Fanrotor

Right bearing predstal

Axis

Orthogonal directions

Figure 23 -e schematic diagram of the experimental apparatus

16 Shock and Vibration

Real(z) 00256

0512minus300

0300

minus400

0

400

Time (s)

Imag

(z)

00256

0512minus500

0500

minus500

0

500

Time (s)Real(z)

Imag

(z)

Imag

(z)

minus300 0 300minus500

0

500

Real(z)minus500 0 500

minus500

0

Imag

(z)

Real(z)

500

0 100 200 300 4000

100

200

300

Frequency (Hz)

Am

plitu

de Imag(z)

Frequency (Hz)0 100 200 300 400

0

200

400

Am

plitu

de

Imag(z)

0 100 200 300 4000

70

140

Frequency (Hz)

Am

plitu

de

Real(z)

0 100 200 300 4000

200

400

Frequency (Hz)

Am

plitu

de Real(z)

(a) (b)

Figure 24 -e time and frequency domain plots of the bistable behavior signals

00256

0512

minus800

80minus80

0

80

Time (s)Real(c1 )

Imag

(c1)

Time (s)Real(c1 )

Imag

(c1)

00256

0512

minus400

40minus40

0

40

Time (s)Real(c2) 0

02560512

minus5000

500minus500

0

500

Imag

(c2)

Time (s)Real(c2)

Imag

(c2)

00256

0512

minus5000

500minus500

0

500

Time (s)Real(r) 00256

0512

minus800

80minus80

0

80

Imag

(irc

rm

)

Time (s)Real(r)

Imag

(r)

00256

0512

minus2000

200minus200

0

200

(a) (b)

Figure 25 -e decomposition results of the bistable behavior signals based on the improved BEMD method

Shock and Vibration 17

improved BEMD method For the single-component bi-variate signal the HT is used to obtain the correspondinginstantaneous amplitude and frequency characteristics -eproposed method can examine the detailed information of asingle rotation component

Data Availability

-e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

-e authors declare that they have no conflicts of interest

Authorsrsquo Contributions

All the authors contributed to this work Chuanjin Huangconceived and designed the simulation and experiments anddrafted the manuscript Haijun Song performed the simu-lations and experiments and analyzed the data and

0 0256 05120

50

100

150

Time (s)

Freq

uenc

y (H

z)

fx2

fy2

0 0256 05120

200

400

600

Time (s)

Am

plitu

de

ax2ay2

(a)

fx2

fy2

0 0256 05120

90

180

270

360

Time (s)

Freq

uenc

y (H

z)

0 0256 05120

200

400

600

Time (s)

Am

plitu

de

ax2Data 2

(b)

Figure 26 -e instantaneous amplitude and frequency of c2 from the (a) left and (b) right columns

0

0256

0512

0100

200300

400500

0

100

200

300

400

500

Time (s)

X 04035Y 3509Z 130

X 04235Y 2265Z 3613

X 007Y 1131Z 2191

X 0105Y 3837Z 3312

ax2

a y2

Figure 27 -e three-dimensional time domain of ax2 and ay2

18 Shock and Vibration

Wenping Lei and Yajun Meng performed the experimentsand analyzed the data All the authors contributed to thewriting and discussion of the paper

Acknowledgments

-is research was funded by the Henan Provincial HigherEducation Key Research Project (Grant nos 18A460006 and19A460029) Henan High-Level Innovative Scientific andTechnological Talent Team Construction Project (Grant noC20150034) and Zhengzhou Institute of Technology In-novation Team Project (Grant no CXTD2017K1)

References

[1] R Yan R X Gao and X Chen ldquoWavelets for fault diagnosisof rotary machines a review with applicationsrdquo Signal Pro-cessing vol 96 pp 1ndash15 2014

[2] J Cheng D Yu J Tang and Y Yang ldquoApplication of frequencyfamily separation method based upon EMD and local Hilbertenergy spectrum method to gear fault diagnosisrdquo Mechanismand Machine lteory vol 43 no 6 pp 712ndash723 2008

[3] H Liu and M Han ldquoA fault diagnosis method based on localmean decomposition and multi-scale entropy for rollerbearingsrdquoMechanism andMachinelteory vol 75 pp 67ndash782014

[4] Z Zheng W Jiang Z Wang Y Zhu and K Yang ldquoGear faultdiagnosis method based on local mean decomposition andgeneralized morphological fractal dimensionsrdquo Mechanismand Machine lteory vol 91 pp 151ndash167 2015

[5] W Yang R Court P J Tavner and C J Crabtree ldquoBivariateempirical mode decomposition and its contribution to windturbine condition monitoringrdquo Journal of Sound and Vi-bration vol 330 no 15 pp 3766ndash3782 2011

[6] L Qu X Liu G Peyronne and Y Chen ldquo-e holospectrum anewmethod for rotor surveillance and diagnosisrdquoMechanicalSystems amp Signal Processing vol 3 no 3 pp 255ndash267 1989

[7] F Q Wu and G Meng ldquoCompound rub malfunctions featureextraction based on full-spectrum cascade analysis and SVMrdquoMechanical Systems and Signal Processing vol 20 no 8pp 2007ndash2021 2006

[8] Y Chen Q Gao and Z Guan ldquoSelf-loosening failure analysisof bolt joints under vibration considering the tighteningprocessrdquo Shock and Vibration vol 2017 Article ID 203842115 pages 2017

[9] L Chen J Han W Lei Y Cui and Z Guan ldquoFull-vectorsignal acquisition and information fusion for the fault pre-dictionrdquo International Journal of Rotating Machineryvol 2016 Article ID 5980802 7 pages 2016

[10] C Chen Y Meng and Y Du ldquoApplication of the full vectorspectrum based on EMD in fault diagnosis of bearingsrdquoJournal of Mechanical Strength vol 37 pp 806ndash811 2015

[11] C Huang X Wu and W Cao ldquoLMD-based on full vectorenvelope technique and its application in TRT vibration faultdiagnosisrdquo Electric Power Automation Equipment vol 35pp 168ndash174 2015 in Chinese

[12] X Zhao T H Patel and M J Zuo ldquoMultivariate EMD andfull spectrum based condition monitoring for rotating ma-chineryrdquo Mechanical Systems and Signal Processing vol 27pp 712ndash728 2012

[13] G Rilling P Flandrin P Gonalves and J M Lilly ldquoBivariateempirical mode decompositionrdquo IEEE Signal ProcessingLetters vol 14 no 12 pp 936ndash939 2007

[14] C Park D Looney M M Van Hulle and D P Mandic ldquo-ecomplex local mean decompositionrdquo Neurocomputingvol 74 no 6 pp 867ndash875 2011

[15] N Rehman and D P Mandic ldquoEmpirical mode de-composition for trivariate signalsrdquo IEEE Transactions onSignal Processing vol 58 no 3 pp 1059ndash1068 2010

[16] N Rehman and D P Mandic ldquoMultivariate empirical modedecompositionrdquo Proceedings of the Royal Society A Mathe-matical Physical and Engineering Sciences vol 466 no 2117pp 1291ndash1302 2010

[17] Y Lv R Yuan and G Song ldquoMultivariate empirical modedecomposition and its application to fault diagnosis of rollingbearingrdquo Mechanical Systems and Signal Processing vol 81pp 219ndash234 2016

[18] C Huang Y Meng and W Lei ldquoFull vector envelopetechnique based on complex local mean decomposition andits application in fault feature extraction for rotor systemrdquoJournal of Mechanical Engineering vol 52 no 7 p 69 2016in Chinese

[19] G Rilling P Flandrin and P Goncalves ldquoOn empirical modedecomposition and its algorithmsrdquo in Proceedings of IEEE-EURASIP Workshop on Nonlinear Signal and Image Pro-cessing pp 8ndash11 IEEE Trieste Italy June 2003

[20] L Yang X Chen and S Wang ldquoMechanism of fast time-varying vibration for rotorndashstator contact system with ap-plication to fault diagnosisrdquo Journal of Vibration andAcoustics vol 140 no 1 article 014501 2018

[21] I Daubechies J Lu and H-TWu ldquoSynchrosqueezed wavelettransforms an empirical mode decomposition-like toolrdquoApplied and Computational Harmonic Analysis vol 30 no 2pp 243ndash261 2011

[22] L-S Qu Holospectrum and Holobalancing Technique inMachinery Diagnosis Beijing Science Press Beijing China2007

Shock and Vibration 19

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However most of the IMF components contain uselessinformation In most cases the fault characteristics are oftenincluded in higher-energy components -e projection di-rections and the energy end condition based on the energythreshold are increased in the improved BEMD method toenhance the decomposition quality

-e decomposition results of the oil film oscillationsignal are shown in Figure 9 using the improved BEMDmethod (N 16 λ 005) -e oil film oscillation signal isdecomposed into four complex rotational components c1ndashc4

and the remaining component r c1 is considered as the noisecomponent According to the Fourier analysis c2 is thefundamental signal with a frequency of 108Hz and c3 is theoscillation signal with a frequency of 52Hz It is clear thatthe number of IMFs is reduced and the fundamental fre-quency component c2 and the oscillation component c3 aremore accurately extracted from the original signal In par-ticular the value of the c4 end becomes smaller and the valueof the c3 end becomes larger relative to the values of c3 and c4shown in Figure 7 Two trajectories made up of c2 and c3 are

Rigid foundation

Motor

Motorcontroller

Eddy current probe system

DiscFlexible coupling

Bearing predstal

Oil cup

Probes

Computer

Axis

ADcard

Figure 3 -e schematic diagram and the experimental apparatus

0 01 02 03 04 05ndash200

ndash100

0

100

200

Time (s)

x (micro

m)

(a)

0 048X X 2X 3X 4X0

30

60 X 52Y 4961

Frequency (Hz)

A (micro

m)

X 108Y 5257

(b)

0 01 02 03 04 05ndash100

ndash50

0

50

100

150

Time (s)

y (microm

)

(c)

0 048X X 2X 3X 4X0

30

60

Frequency (Hz)

A (micro

m)

X 52Y 4698

X 108Y 4437

(d)

Figure 4-e oil film oscillation signal (a) the horizontal and vertical direction signals x (b) the Fourier spectrum of signal x (c) the verticaldirection signals y (d) the Fourier spectrum of signal y

Shock and Vibration 5

shown in Figure 10 From c2 and c3 three-dimensional timedomain and the trajectories the fundamental frequencysignal c2 has a small elliptic amplitude change c3 is a largeseries of amplitude conversion elliptical compositions thatcause a signicant oscillation Relative to the orbit of c3 inFigure 6 the orbit of c3 in Figure 10 is improved particularlyin the center region

Increasing the number of signal projection directionsresults in an increase in the number of projection signals

en the tangent mean which is obtained by interpolatingthe local maximum of the projected signals with a splineinterpolation is more accurate However it is meaningless tocontinue to increase the number of projection directions whenthe tangent mean is accurately tted It is found that the signaldecomposition results of N 16 are almost the same as N 1024 when considering the complex rotation componentsseparated by the original signal However the calculation timeof the BEMD algorithm with N 1024 is greatly increased

001

0203

0405

ndash100

0

100

ndash100

ndash50

0

50

100

150

Time (s)Real(z) (microm)

Imag

(z) (microm

)

(a)

ndash150 ndash100 ndash50 0 50 100 150ndash150

ndash100

ndash50

0

50

100

150

Real(z) (microm)

Imag

(z) (microm

)

(b)

Figure 5 e oil lm oscillation signal of z (a) the three-dimensional time-domain waveform (b) the two-dimensional plots

ndash80

80

IMF 1

ndash80

80

IMF 2

ndash90

90

IMF 3

ndash30

30

IMF 4

ndash1010

IMF 5

ndash10

10

IMF 6

ndash4

4

IMF 7

ndash44

IMF 8

ndash33

IMF 9

ndash44

IMF 10

0 01 02 03 04 05ndash5

1r

Time (s)(a)

ndash4

4IM

F 1

ndash6060

IMF 2

ndash60

60

IMF 3

ndash15

15

IMF 4

ndash55

IMF 5

ndash55

IMF 6

ndash5

5

IMF 7

ndash0505

IMF 8

(b)

0 01 02 03 04 0505

1r

Time (s)

Figure 6 e decomposition results of signals x and y using EMD

6 Shock and Vibration

e HT is applied to the real and imaginary parts of c2and c3 respectively and the instantaneous frequency andinstantaneous amplitude are obtained as shown in Fig-ure 11 where ax2 and ax3 represent the instantaneous

amplitude of the real parts of c2 and c3 respectively ay2and ay3 represent the instantaneous amplitude of theimaginary parts of c2 and c3 respectively fx2 and fx3represent the instantaneous frequency of the real parts of

0025

05

ndash50

5ndash5

0

5

Time (s)Real(c1)

Imag

(c1)

(a)

0025

05

ndash1000

100ndash100

0

100

Time (s)Real(c2 )

Imag

(c2)

(b)

0025

05

ndash1000

100ndash100

0

100

Time (s)Real(c3 )

Imag

(c3)

(c)

0025

05

ndash200

20ndash50

0

50

Time (s)Real(c4 )

Imag

(c4)

Modal aliasing

(d)

0025

05

ndash100

10ndash10

0

10

Time (s)Real(c5 )Im

ag(c

5)

(e)

0025

05

ndash50

5ndash5

0

5

Time (s)Real(c6 )

Imag

(c6)

(f )

0025

05

ndash50

5ndash1

0

1

Time (s)Real(c7)

Imag

(c7)

(g)

0025

05

ndash50

5ndash1

0

1

Time (s)Real(c8)

Imag

(c8)

(h)

0025

05

ndash100

10ndash5

0

5

Time (s)Real(r)

Imag

(r)

(i)

Figure 7 Decomposition results of the oil lm oscillation signal using BEMD (N 4)

ndash80 ndash40 0 40 80ndash60

ndash30

0

30

60

Real(c2)

Imag

(c2)

(a)

ndash100 ndash50 0 50 100ndash100

ndash50

0

50

100

Real(c3)

Imag

(c3)

(b)

ndash80 ndash40 0 40 80ndash60

ndash30

0

30

60

IMFx2

IMF y

2

(c)

ndash100 ndash50 0 50 100ndash100

ndash50

0

50

100

IMFx3

IMF y

3

(d)

Figure 8e orbits made up of (a) the real and imaginary parts of c2 (b) real and imaginary parts of c3 (c) IMFx2 and IMFy2 and (d) IMFx3and IMFy3

Shock and Vibration 7

c2 and c3 respectively and fy2 and fy3 represent the in-stantaneous frequency of the imaginary parts of c2 and c3respectively

In Figure 11 ax3 is much larger than ay3 and their phasesare separated by nearly 180deg which shows that the amplitudeand the phase of the oil lm oscillation signal in dierentdirections can vary fx2 and fy2 and fx3 and fy3 are ap-proximately the same ax2 and ay2 show little change andtheir phases are the same Since BEMD is a bivariate ex-tension of EMD like EMD BEMD also has an endpointeect ere are some uctuations in the instantaneousamplitude and instantaneous frequency due to the end eect

and the edge eect of the Hilbert transform e three-dimensional time domain of ax2 and ay2 and of ax3 and ay3 isshown in Figure 12e amplitude range of c3 is much largerthan that of c2 It is can be inferred that the main componentof oscillation is c3 with the frequency of 52Hz in the oil lmoscillation signal

e decomposition results of the oil lm oscillationsignal are shown in Figure 13 using the CLMD methodproposed in reference [14] e oil lm oscillation signal isdecomposed into four complex product functions cpf1ndashcpf4e noise component is not separated from the oil lmoscillation signal using the CLMD method Moreover the

0025

05

ndash50

5

Time (s)Real(c1 )

ndash5

0

5Im

ag(c

1)

(a)

0025

05

ndash1000

100

Time (s)Real(c2)

ndash100

0

100

Imag

(c2)

(b)

0025

05ndash100

0100

ndash100

0

100

Imag

(c3)

Real(c3 ) Time (s)

(c)

0 02505

ndash100

10

Time (s)Real(c4)

ndash20

0

20

Imag

(c4)

(d)

0025

05ndash10

010

Time (s)Real(c5)

ndash10

0

10

Imag

(c5)

(e)

0 02505

ndash50

5

Time (s)Real(c6)

ndash5

0

5

Imag

(c6)

(f )

0 02505

ndash202

Time (s)Real(c7)

ndash2

0

2

Imag

(c7)

(g)

0025

05ndash5

05

Time (s)Real(c8)

ndash5

0

5

Imag

(c8)

(h)

0025

05ndash10

010

Time (s)Real(r)

ndash5

0

5

Imag

(r)

(i)

Figure 9 Decomposition results of the oil lm oscillation signal using the improved BEMD method (N 16 λ 005)

ndash80 0 40 80ndash40Real(c2)

ndash60

ndash30

0

30

60

Imag

(c2)

(a)

ndash80

ndash60

ndash40

ndash20

0

20

40

60

80

Imag

(c3)

ndash40 0 40 80ndash80Real(c3)

(b)

Figure 10 e orbits were made up of c2 (a) and c3 (b) obtained with the improved BEMD method

8 Shock and Vibration

single component fundamental frequency signal and the oillm oscillation signal were not successfully separated Oneof the possible reasons is that in the CLMD algorithm thecomplex signal is only projected onto the x-axis and the y-axis unlike BEMD which projected on multiple directionsIn addition CLMD is a bivariate extension of LMD LMDused a moving average algorithm when tting the signalenvelope which can lter noise to a certain extent esignals other than the noise component c1 in Figure 9 areadded to obtain a ltered oil lm oscillation signal which isthen decomposed by the CLMD method and the rst twodecomposed results are shown in Figure 13 It is seen thatthe single component fundamental frequency signal and

the single component oil lm oscillation signal areseparated

cpf1 and cpf2 consisted of real part signals and imaginarypart signals both of which were composed of the product ofthe envelope signal and the pure frequency modulationfunctione envelope signal is the instantaneous amplitudeof the signal e corresponding instantaneous frequencywas obtained by deriving the inverse function of the cosinepure frequency modulation function e instantaneousamplitude and instantaneous frequency curves are shown inFigure 14 e instantaneous amplitude and frequencyobtained by the CLMD method are smoother than in Fig-ure 12e reason is mainly that the CLMDmethod uses the

ax2ay2

30

40

50

60

70A

mpl

itude

01 02 03 04 050Time (s)

(a)

fx2fy2

70

110

150

Freq

uenc

y (H

z)

01 02 03 04 050Time (s)

(b)

ax3ay3

20

70

120

Am

plitu

de

01 02 03 04 050Time (s)

(c)

fx3fy3

0

50

100

Freq

uenc

y (H

z)01 02 03 04 050

Time (s)

(d)

Figure 11 (a) e instantaneous amplitude of the real part and imaginary part of c2 (b) the instantaneous frequency of the real part andimaginary part of c2 (c) the instantaneous amplitude of the real part and imaginary part of c3 (d) the instantaneous frequency of the real partand imaginary part of c3

0 01 02 03 04 05

2050

800

20

40

60

80

Time (s)ax2

a y2

(a)

3060

90120

0

20

40

60

80

Time (s)ax3

a y3

0 01 02 03 04 05

(b)

Figure 12 e 3D time domain of instantaneous amplitude was made up of (a) ax2 and ay2 and (b) ax3 and ay3

Shock and Vibration 9

moving average ltering algorithm to obtain the signalenvelope curve However this is the result of using theCLMD algorithm after ltering out noise with BEMD If theBEMD algorithm is not used for ltering noise the in-stantaneous amplitude and instantaneous frequency curvesof the single component were not obtained by the CLMDmethod

42 Analysis ofOilWhirl Signal Based on the ImprovedBEMDMethod In the method similar to that presented in Section41 the oil whirl signals of the rotor test rig with a speedparameter of 4320 rpm are collected by two orthogonalsensors as shown in Figure 15 e gure shows the typicalwhirl phenomenon of large circles with embedded smallerones e decomposition results based on the improvedBEMD method (N 16 λ 005) are shown in Figure 16

Only three IMFs appear in Figure 16 and the singlecomponents c2 and c3 and the noise component c1 are suc-cessfully separated from the original signal e HT is appliedto the real and imaginary parts of c2 and c3 respectively andthe instantaneous frequency and instantaneous amplitude areobtained as shown in Figure 17 e three-dimensional timedomain of ax2 and ay2 and of ax3 and ay3 is shown in Figure 18Figure 17 indicates that the frequency of c2 is approximatelytwice the frequency of c3 and that ax2 is larger than ay2 Inaddition ay3 is slightly larger than ax3 but the range of changefor ax3 is greater than the range for ay3 It is inferred that c2 isthe fundamental frequency signal and that c3 is the half-frequency signal in the oil whirl signal

43 Analysis of Looseness and Rotor Rubbing Composite FaultSignal Based on the Improved BEMD Method Loose androtor rubbing composite faults are set on the testequipment shown in Figure 3 in Section 41 Loose fault isset on the nondrive end of the motor and the distancebetween the plastic rod and the shaft is xed near thesensor on the left side of the disk As the rotor speedincreases the vibration increases and the rubbing faultoccurs which is stable at around 1700 rmin e com-posite fault signals are collected by two orthogonal sen-sors as shown in Figure 19 e decomposition resultsbased on the improved BEMD method are shown inFigure 20

Figures 19(c) and 19(d) indicate that the signal com-ponent mainly contain 1X 2X (X 28Hz) and a frequencymodulated signal generated due to time-varying stinessis phenomenon is similar to that described reference [20]Four IMFs appear in Figure 20 and the single components1X 2X signals and the FM signal c2 are successfully separatedfrom the original signal e HT is applied to the real andimaginary parts of c2 c3 and c4 respectively and the in-stantaneous frequency and instantaneous amplitude areobtained as shown in Figure 21 ere are some uctuationsin the frequencies of c2 and c3 but these uctuations aredierent from the random uctuations in the above casesey have obvious regularity and are characteristic of FMsignals e frequency modulation characteristics of c2 aremore obvious than those of c3 In addition by observing theinstantaneous amplitude of c2 it is seen that c2 is still anamplitude modulation signal

minus100

0100

minus100

0

100

Time (s)Real(cpf1)

Imag

(cpf

1)

0 01 02 03 04 05

(a)

Time (s)Real(cpf2) minus100

0100

minus100

0

100

Imag

(cpf

2)

0 01 02 03 04 05

(b)

minus80 0 80minus60

0

60

Real(cpf1)

Imag(cpf1)

(c)

minus100 0 100minus100

0

100

Real(cpf2)

Imag(cpf2)

(d)

Figure 13 e rst two decomposed results of the ltered signal based on the CLMD method

10 Shock and Vibration

0025

05

ndash150

0

150ndash150

0

150

Time (s)Real(z)

Imag

(z)

(a)

ndash150 ndash75 0 75 150ndash150

ndash75

0

75

150

Real(z)

Imag

(z)

(b)

0 72 144 216 2880

40

80

Frequency (Hz)

Am

plitu

de

X 72Y 7321

X 36Y 3433

Real(z)

(c)

Frequency (Hz)0 72 144 216 288

0

40

80

X 36Y 3475

Am

plitu

de

X 72Y 6157

Imag(z)

(d)

Figure 15e oil whirl signal z (a) the 3D time-domain wave of z (b) the 2D plane of z (c) the Fourier spectrum of Real[z] (d) the Fourierspectrum of Imag[z]

30

40

50

60

70

Am

plitu

de

0 01 02 03 04 05Time (s)

ax1ay1

(a)

70

110

150

Freq

uenc

y (H

z)

0 01 02 03 04 05Time (s)

fx1fy1

(b)

0 01 02 03 04 0520

70

120

Time (s)

Am

plitu

de

ax2ay2

(c)

0 01 02 03 04 050

50

100

Time (s)

Freq

uenc

y (H

z)

fx2fy2

(d)

Figure 14 (a) e instantaneous amplitude of the real part and imaginary part of cpf1 (b) the instantaneous frequency of the real part andimaginary part of cpf1 (c) the instantaneous amplitude of the real part and imaginary part of cpf2 (d) the instantaneous frequency of the realpart and imaginary part of cpf2

Shock and Vibration 11

0025

05

ndash5

0

5ndash5

0

5

Time (s)Real(c1)

Imag

(c1)

(a)

0025

05

ndash1000

100ndash100

0

100

Time (s)

Imag

(c2)

Real(c2)

(b)

0025

05

ndash500

50ndash50

0

50

Time (s)

Imag

(c3)

Real(c3)

(c)

0025

05

ndash100

10ndash20

0

20

Time (s)

Imag

(r)

Real(r)

(d)

ndash100 ndash50 0 50 100ndash100

ndash50

0

50

100

Imag

(c2)

Real(c2)

(e)

ndash40 ndash20 0 20 40ndash40

ndash20

0

20

40Im

ag(c

3)

Real(c3)

(f )

Figure 16 e decomposition results of the oil whirl signal based on the improved BEMD method

60

90

120

Am

plitu

de

0 01 02 03 04 05Time (s)

ax2ay2

(a)

50

75

100

Freq

uenc

y (H

z)

0 01 02 03 04 05Time (s)

fx2fy2

(b)

Figure 17 Continued

12 Shock and Vibration

0 01 02 03 04 0520

35

50

Time (s)

Am

plitu

de

ax3ay3

(c)

0 01 02 03 04 0520

35

50

Time (s)

Freq

uenc

y (H

z)

fx3fy3

(d)

Figure 17 e instantaneous amplitude and frequency of c2 and c3 from the oil whirl signal obtained by the HT (a) the instantaneousamplitude of the real part and imaginary part of c2 (b) the instantaneous frequency of the real part and imaginary part of c2(c) the instantaneous amplitude of the real part and imaginary part of c3 (d) the instantaneous frequency of the real part and imaginarypart of c3

0

025

05

40

80

12040

80

120

Time (s)

X 03877Y 9582Z 826

ax2

X 01309Y 9595Z 8452

a y2

(a)

0

025

05

20

35

5020

35

50

Time (s)

X 03955Y 3762Z 3822

ax3

X 008838Y 3608Z 3733

a y3

(b)

Figure 18 e 3D time domain of the instantaneous amplitude of ax2 and ay2 (a) and ax3 and ay3 (b) from the oil whirl signal

0025

05

minus1500

150minus150

0

150

t (s)Real(z)

Imag

(z)

(a)

Imag(z)

Real(z)

minus150

0

150

minus150 0 150

(b)

Figure 19 Continued

Shock and Vibration 13

In order to further verify the correctness of the in-stantaneous amplitude-frequency characteristics of theproposed method the real and imaginary parts of thecomposite fault signal z are analyzed separately using syn-chrosqueezed wavelet transforms (SWT) proposed in ref-erence [21]-e results are shown in Figure 22 It is seen thatthe time-frequency representations of the composite faultsignal z also include the AM-FM signal and the 1X signalwhich proves the correctness of the proposed methodCompared with the SWT method the instantaneousamplitude-frequency characteristics acquired by the HTmethod are relatively straightforward

44 lte Bistable Behavior Analysis of the Fan Rotor Based onBEMD -e bistability of the rotor is a nonlinear behaviorof the rotor-bearing system which is the state in which therotor jumps from one stable state to another forming astep -e bivariate signal of the bistable behavior iscomposed of two signals collected by two displacementsensors from orthogonal locations on the experimentaldevices in literature [22] as shown in Figure 23 Literature[22] shows that the cause of the bistable behavior remainsto be further explored -is paper uses this case to il-lustrate the feasibility of BEMD to analyze nonstationarysignals

0

40

80

Am

plitu

de Real(z)

0 100 200 300 400Frequency (Hz)

(c)

Am

plitu

de

0 100 200 300 400Frequency (Hz)

0

20

40

60

80

Imag(z)

(d)

Figure 19 -e composite fault signal z (a) the 3D time domain wave of z (b) the 2D plane of z (c) the Fourier spectrum of Real[z] (d) theFourier spectrum of Imag[z]

0025

05

minus30

3minus5

0

5

Time (s)Real(c1)

Imag

(c1)

(a)

0025

05

minus200

20minus20

0

20

Time (s)Real(c2 )

Imag

(c2)

(b)

0025

05

minus200

20minus50

0

50

Time (s)Real(c2)

Imag

(c2)

(c)

0025

05

minus1000

100minus100

0

100

Time (s)Real(c3)

Imag

(c3)

(d)

0025

05

minus400

40minus30

0

30

Time (s)Real(r)

Imag

(r)

(e)

Figure 20 -e decomposition results of the composite fault signal based on the improved BEMD method

14 Shock and Vibration

-e x and y signals in the horizontal and vertical di-rections of the left and right bearings respectively from thefan rotors are collected with four displacement sensorsLetting z x+ jy the time and frequency domain plots of z areshown in Figure 24 where the fan rotor speed is 5500 rpm thesampling frequency is 2000Hz and the number of samplingpoints is 1024 -e left and right columns respectively showthe time and frequency domain plots of the vibration signalsfrom the left and right bearings of the fan rotor Bistablebehavior arises in the fan rotor and the amplitudes of the

vibration signals vary significantly in different positions anddirections Further studies are required to explain the causesof this bistability -e present study focuses on extracting thebistable behavioral signal characteristics to verify the feasi-bility of the proposed method

-e decomposition results of the bistable behavioralsignals based on the improved BEMDmethod are shown inFigure 25 c1 c2 c3 and r are separated in order from zusing the improved BEMD method c1 shows a randomarrangement and is considered the high-frequency noise

0

10

20

Am

plitu

de

ax2ay2

0 025 05Time (s)

(a)

0

152

304

Freq

uenc

y (H

z)

0 025 05Time (s)

fy2

fx2

(b)

0

10

20

30

40

Am

plitu

de

ax3ay3

0 025 05Time (s)

(c)

0

56

112

Freq

uenc

y (H

z)

fx3fy3

0 025 05Time (s)

(d)

0 025 050

50

100

Time (s)

Am

plitu

de

ax4ay4

(e)

0 025 050

28

56

Time (s)

Freq

uenc

y (H

z)

fx4fy4

(f )

Figure 21 -e instantaneous amplitude and frequency of c2 c3 and c4 obtained by the HT (a) the instantaneous amplitude of the real partand imaginary part of c2 (b) the instantaneous frequency of the real part and imaginary part of c2 (c) the instantaneous amplitude of the realpart and imaginary part of c3 (d) the instantaneous frequency of the real part and imaginary part of c3 (e) the instantaneous amplitude of thereal part and imaginary part of c4 (f ) the instantaneous frequency of the real part and imaginary part of c4

Shock and Vibration 15

signal c2 is considered to represent the extracted bistablebehavior signals -e HT is applied to the real andimaginary parts of c2 to obtain the instantaneous amplitudeand frequency of c2 from the left and right columns fromFigure 25 as shown in Figure 26 Figure 27 shows thethree-dimensional time domain of ax2 and ay2 from the leftand right columns respectively Figure 26 shows that thevibration signal amplitude on the left side of the fan de-creases from large to small opposite of the behavior ofthe right -e horizontal vibration signal amplitude on theleft side of the fan is larger than that of the vertical di-rection signal opposite of the right -is result validatesthat the vibration signals from different directions orpositions are different when the fan produces bistablebehavior In addition the time of the bistable behavior canbe determined according to the jump point of the am-plitude or frequency

5 Discussion

-e BEMD algorithm decomposes two orthogonal di-rections of vibration signals as a complex signal which is atwo-dimensional digital signal processing method thusensuring that the real and imaginary parts have the samedecomposition scale Similar to EMD the envelope mean iscritical for the decomposition effect of BEMD but the en-velope mean in BEMD is three-dimensional If the numberof projection directions of the complex signal in three-dimensional space is larger the corresponding envelope

signal is also more -us the envelope mean value is moreaccurate and the BEMD decomposition effect is betterIncreasing the number of projection directions can improvemodal aliasing Like EMD BEMD also produces falsecomponents when decomposing signals Generally speakingthe energy of the false components is low and these low-energy false components do not contain fault characteristicinformation and the introduction of the energy thresholdcriterion in the termination condition can increase thedecomposition speed of the BEMD

-e experimental results show that there is a certaindifference in the existence of vibration signal character-istics in different directions when rotating machinery failsIn addition when the number of projection directionsis increased the decomposition speed of BEMD willdecrease

6 Conclusions

We use BEMD and HT to extract the instantaneousamplitude-frequency features of rotor faults A bivariateinstantaneous feature extraction method based on the im-proved BEMD method and the HT is investigated whichextends the fault feature extraction technology to two di-mensions -e BEMD method is suitable to analyze thecomplex multicomponent bivariate signals -e mainsingle-component bivariate signals are separated from themulticomponent bivariate signals of the fan rotor bistabilityfor the oil film oscillation and the oil film vortex using the

0

2

4

6

Time (s)

Freq

uenc

y (H

z)

0 01 02 03 04 052

3

4

5

6

7

8log2 fx

(a)

0

2

4

6

Time (s)

Freq

uenc

y (H

z)

0 01 02 03 04 052

3

4

5

6

7

8log2 fy

(b)

Figure 22 -e results of the composite fault signal z based on SWT the time-frequency representation of (a) the real part of z and (b) theimaginary part of z

Left bearing predstal

Locations of sensors

Fanrotor

Right bearing predstal

Axis

Orthogonal directions

Figure 23 -e schematic diagram of the experimental apparatus

16 Shock and Vibration

Real(z) 00256

0512minus300

0300

minus400

0

400

Time (s)

Imag

(z)

00256

0512minus500

0500

minus500

0

500

Time (s)Real(z)

Imag

(z)

Imag

(z)

minus300 0 300minus500

0

500

Real(z)minus500 0 500

minus500

0

Imag

(z)

Real(z)

500

0 100 200 300 4000

100

200

300

Frequency (Hz)

Am

plitu

de Imag(z)

Frequency (Hz)0 100 200 300 400

0

200

400

Am

plitu

de

Imag(z)

0 100 200 300 4000

70

140

Frequency (Hz)

Am

plitu

de

Real(z)

0 100 200 300 4000

200

400

Frequency (Hz)

Am

plitu

de Real(z)

(a) (b)

Figure 24 -e time and frequency domain plots of the bistable behavior signals

00256

0512

minus800

80minus80

0

80

Time (s)Real(c1 )

Imag

(c1)

Time (s)Real(c1 )

Imag

(c1)

00256

0512

minus400

40minus40

0

40

Time (s)Real(c2) 0

02560512

minus5000

500minus500

0

500

Imag

(c2)

Time (s)Real(c2)

Imag

(c2)

00256

0512

minus5000

500minus500

0

500

Time (s)Real(r) 00256

0512

minus800

80minus80

0

80

Imag

(irc

rm

)

Time (s)Real(r)

Imag

(r)

00256

0512

minus2000

200minus200

0

200

(a) (b)

Figure 25 -e decomposition results of the bistable behavior signals based on the improved BEMD method

Shock and Vibration 17

improved BEMD method For the single-component bi-variate signal the HT is used to obtain the correspondinginstantaneous amplitude and frequency characteristics -eproposed method can examine the detailed information of asingle rotation component

Data Availability

-e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

-e authors declare that they have no conflicts of interest

Authorsrsquo Contributions

All the authors contributed to this work Chuanjin Huangconceived and designed the simulation and experiments anddrafted the manuscript Haijun Song performed the simu-lations and experiments and analyzed the data and

0 0256 05120

50

100

150

Time (s)

Freq

uenc

y (H

z)

fx2

fy2

0 0256 05120

200

400

600

Time (s)

Am

plitu

de

ax2ay2

(a)

fx2

fy2

0 0256 05120

90

180

270

360

Time (s)

Freq

uenc

y (H

z)

0 0256 05120

200

400

600

Time (s)

Am

plitu

de

ax2Data 2

(b)

Figure 26 -e instantaneous amplitude and frequency of c2 from the (a) left and (b) right columns

0

0256

0512

0100

200300

400500

0

100

200

300

400

500

Time (s)

X 04035Y 3509Z 130

X 04235Y 2265Z 3613

X 007Y 1131Z 2191

X 0105Y 3837Z 3312

ax2

a y2

Figure 27 -e three-dimensional time domain of ax2 and ay2

18 Shock and Vibration

Wenping Lei and Yajun Meng performed the experimentsand analyzed the data All the authors contributed to thewriting and discussion of the paper

Acknowledgments

-is research was funded by the Henan Provincial HigherEducation Key Research Project (Grant nos 18A460006 and19A460029) Henan High-Level Innovative Scientific andTechnological Talent Team Construction Project (Grant noC20150034) and Zhengzhou Institute of Technology In-novation Team Project (Grant no CXTD2017K1)

References

[1] R Yan R X Gao and X Chen ldquoWavelets for fault diagnosisof rotary machines a review with applicationsrdquo Signal Pro-cessing vol 96 pp 1ndash15 2014

[2] J Cheng D Yu J Tang and Y Yang ldquoApplication of frequencyfamily separation method based upon EMD and local Hilbertenergy spectrum method to gear fault diagnosisrdquo Mechanismand Machine lteory vol 43 no 6 pp 712ndash723 2008

[3] H Liu and M Han ldquoA fault diagnosis method based on localmean decomposition and multi-scale entropy for rollerbearingsrdquoMechanism andMachinelteory vol 75 pp 67ndash782014

[4] Z Zheng W Jiang Z Wang Y Zhu and K Yang ldquoGear faultdiagnosis method based on local mean decomposition andgeneralized morphological fractal dimensionsrdquo Mechanismand Machine lteory vol 91 pp 151ndash167 2015

[5] W Yang R Court P J Tavner and C J Crabtree ldquoBivariateempirical mode decomposition and its contribution to windturbine condition monitoringrdquo Journal of Sound and Vi-bration vol 330 no 15 pp 3766ndash3782 2011

[6] L Qu X Liu G Peyronne and Y Chen ldquo-e holospectrum anewmethod for rotor surveillance and diagnosisrdquoMechanicalSystems amp Signal Processing vol 3 no 3 pp 255ndash267 1989

[7] F Q Wu and G Meng ldquoCompound rub malfunctions featureextraction based on full-spectrum cascade analysis and SVMrdquoMechanical Systems and Signal Processing vol 20 no 8pp 2007ndash2021 2006

[8] Y Chen Q Gao and Z Guan ldquoSelf-loosening failure analysisof bolt joints under vibration considering the tighteningprocessrdquo Shock and Vibration vol 2017 Article ID 203842115 pages 2017

[9] L Chen J Han W Lei Y Cui and Z Guan ldquoFull-vectorsignal acquisition and information fusion for the fault pre-dictionrdquo International Journal of Rotating Machineryvol 2016 Article ID 5980802 7 pages 2016

[10] C Chen Y Meng and Y Du ldquoApplication of the full vectorspectrum based on EMD in fault diagnosis of bearingsrdquoJournal of Mechanical Strength vol 37 pp 806ndash811 2015

[11] C Huang X Wu and W Cao ldquoLMD-based on full vectorenvelope technique and its application in TRT vibration faultdiagnosisrdquo Electric Power Automation Equipment vol 35pp 168ndash174 2015 in Chinese

[12] X Zhao T H Patel and M J Zuo ldquoMultivariate EMD andfull spectrum based condition monitoring for rotating ma-chineryrdquo Mechanical Systems and Signal Processing vol 27pp 712ndash728 2012

[13] G Rilling P Flandrin P Gonalves and J M Lilly ldquoBivariateempirical mode decompositionrdquo IEEE Signal ProcessingLetters vol 14 no 12 pp 936ndash939 2007

[14] C Park D Looney M M Van Hulle and D P Mandic ldquo-ecomplex local mean decompositionrdquo Neurocomputingvol 74 no 6 pp 867ndash875 2011

[15] N Rehman and D P Mandic ldquoEmpirical mode de-composition for trivariate signalsrdquo IEEE Transactions onSignal Processing vol 58 no 3 pp 1059ndash1068 2010

[16] N Rehman and D P Mandic ldquoMultivariate empirical modedecompositionrdquo Proceedings of the Royal Society A Mathe-matical Physical and Engineering Sciences vol 466 no 2117pp 1291ndash1302 2010

[17] Y Lv R Yuan and G Song ldquoMultivariate empirical modedecomposition and its application to fault diagnosis of rollingbearingrdquo Mechanical Systems and Signal Processing vol 81pp 219ndash234 2016

[18] C Huang Y Meng and W Lei ldquoFull vector envelopetechnique based on complex local mean decomposition andits application in fault feature extraction for rotor systemrdquoJournal of Mechanical Engineering vol 52 no 7 p 69 2016in Chinese

[19] G Rilling P Flandrin and P Goncalves ldquoOn empirical modedecomposition and its algorithmsrdquo in Proceedings of IEEE-EURASIP Workshop on Nonlinear Signal and Image Pro-cessing pp 8ndash11 IEEE Trieste Italy June 2003

[20] L Yang X Chen and S Wang ldquoMechanism of fast time-varying vibration for rotorndashstator contact system with ap-plication to fault diagnosisrdquo Journal of Vibration andAcoustics vol 140 no 1 article 014501 2018

[21] I Daubechies J Lu and H-TWu ldquoSynchrosqueezed wavelettransforms an empirical mode decomposition-like toolrdquoApplied and Computational Harmonic Analysis vol 30 no 2pp 243ndash261 2011

[22] L-S Qu Holospectrum and Holobalancing Technique inMachinery Diagnosis Beijing Science Press Beijing China2007

Shock and Vibration 19

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shown in Figure 10 From c2 and c3 three-dimensional timedomain and the trajectories the fundamental frequencysignal c2 has a small elliptic amplitude change c3 is a largeseries of amplitude conversion elliptical compositions thatcause a signicant oscillation Relative to the orbit of c3 inFigure 6 the orbit of c3 in Figure 10 is improved particularlyin the center region

Increasing the number of signal projection directionsresults in an increase in the number of projection signals

en the tangent mean which is obtained by interpolatingthe local maximum of the projected signals with a splineinterpolation is more accurate However it is meaningless tocontinue to increase the number of projection directions whenthe tangent mean is accurately tted It is found that the signaldecomposition results of N 16 are almost the same as N 1024 when considering the complex rotation componentsseparated by the original signal However the calculation timeof the BEMD algorithm with N 1024 is greatly increased

001

0203

0405

ndash100

0

100

ndash100

ndash50

0

50

100

150

Time (s)Real(z) (microm)

Imag

(z) (microm

)

(a)

ndash150 ndash100 ndash50 0 50 100 150ndash150

ndash100

ndash50

0

50

100

150

Real(z) (microm)

Imag

(z) (microm

)

(b)

Figure 5 e oil lm oscillation signal of z (a) the three-dimensional time-domain waveform (b) the two-dimensional plots

ndash80

80

IMF 1

ndash80

80

IMF 2

ndash90

90

IMF 3

ndash30

30

IMF 4

ndash1010

IMF 5

ndash10

10

IMF 6

ndash4

4

IMF 7

ndash44

IMF 8

ndash33

IMF 9

ndash44

IMF 10

0 01 02 03 04 05ndash5

1r

Time (s)(a)

ndash4

4IM

F 1

ndash6060

IMF 2

ndash60

60

IMF 3

ndash15

15

IMF 4

ndash55

IMF 5

ndash55

IMF 6

ndash5

5

IMF 7

ndash0505

IMF 8

(b)

0 01 02 03 04 0505

1r

Time (s)

Figure 6 e decomposition results of signals x and y using EMD

6 Shock and Vibration

e HT is applied to the real and imaginary parts of c2and c3 respectively and the instantaneous frequency andinstantaneous amplitude are obtained as shown in Fig-ure 11 where ax2 and ax3 represent the instantaneous

amplitude of the real parts of c2 and c3 respectively ay2and ay3 represent the instantaneous amplitude of theimaginary parts of c2 and c3 respectively fx2 and fx3represent the instantaneous frequency of the real parts of

0025

05

ndash50

5ndash5

0

5

Time (s)Real(c1)

Imag

(c1)

(a)

0025

05

ndash1000

100ndash100

0

100

Time (s)Real(c2 )

Imag

(c2)

(b)

0025

05

ndash1000

100ndash100

0

100

Time (s)Real(c3 )

Imag

(c3)

(c)

0025

05

ndash200

20ndash50

0

50

Time (s)Real(c4 )

Imag

(c4)

Modal aliasing

(d)

0025

05

ndash100

10ndash10

0

10

Time (s)Real(c5 )Im

ag(c

5)

(e)

0025

05

ndash50

5ndash5

0

5

Time (s)Real(c6 )

Imag

(c6)

(f )

0025

05

ndash50

5ndash1

0

1

Time (s)Real(c7)

Imag

(c7)

(g)

0025

05

ndash50

5ndash1

0

1

Time (s)Real(c8)

Imag

(c8)

(h)

0025

05

ndash100

10ndash5

0

5

Time (s)Real(r)

Imag

(r)

(i)

Figure 7 Decomposition results of the oil lm oscillation signal using BEMD (N 4)

ndash80 ndash40 0 40 80ndash60

ndash30

0

30

60

Real(c2)

Imag

(c2)

(a)

ndash100 ndash50 0 50 100ndash100

ndash50

0

50

100

Real(c3)

Imag

(c3)

(b)

ndash80 ndash40 0 40 80ndash60

ndash30

0

30

60

IMFx2

IMF y

2

(c)

ndash100 ndash50 0 50 100ndash100

ndash50

0

50

100

IMFx3

IMF y

3

(d)

Figure 8e orbits made up of (a) the real and imaginary parts of c2 (b) real and imaginary parts of c3 (c) IMFx2 and IMFy2 and (d) IMFx3and IMFy3

Shock and Vibration 7

c2 and c3 respectively and fy2 and fy3 represent the in-stantaneous frequency of the imaginary parts of c2 and c3respectively

In Figure 11 ax3 is much larger than ay3 and their phasesare separated by nearly 180deg which shows that the amplitudeand the phase of the oil lm oscillation signal in dierentdirections can vary fx2 and fy2 and fx3 and fy3 are ap-proximately the same ax2 and ay2 show little change andtheir phases are the same Since BEMD is a bivariate ex-tension of EMD like EMD BEMD also has an endpointeect ere are some uctuations in the instantaneousamplitude and instantaneous frequency due to the end eect

and the edge eect of the Hilbert transform e three-dimensional time domain of ax2 and ay2 and of ax3 and ay3 isshown in Figure 12e amplitude range of c3 is much largerthan that of c2 It is can be inferred that the main componentof oscillation is c3 with the frequency of 52Hz in the oil lmoscillation signal

e decomposition results of the oil lm oscillationsignal are shown in Figure 13 using the CLMD methodproposed in reference [14] e oil lm oscillation signal isdecomposed into four complex product functions cpf1ndashcpf4e noise component is not separated from the oil lmoscillation signal using the CLMD method Moreover the

0025

05

ndash50

5

Time (s)Real(c1 )

ndash5

0

5Im

ag(c

1)

(a)

0025

05

ndash1000

100

Time (s)Real(c2)

ndash100

0

100

Imag

(c2)

(b)

0025

05ndash100

0100

ndash100

0

100

Imag

(c3)

Real(c3 ) Time (s)

(c)

0 02505

ndash100

10

Time (s)Real(c4)

ndash20

0

20

Imag

(c4)

(d)

0025

05ndash10

010

Time (s)Real(c5)

ndash10

0

10

Imag

(c5)

(e)

0 02505

ndash50

5

Time (s)Real(c6)

ndash5

0

5

Imag

(c6)

(f )

0 02505

ndash202

Time (s)Real(c7)

ndash2

0

2

Imag

(c7)

(g)

0025

05ndash5

05

Time (s)Real(c8)

ndash5

0

5

Imag

(c8)

(h)

0025

05ndash10

010

Time (s)Real(r)

ndash5

0

5

Imag

(r)

(i)

Figure 9 Decomposition results of the oil lm oscillation signal using the improved BEMD method (N 16 λ 005)

ndash80 0 40 80ndash40Real(c2)

ndash60

ndash30

0

30

60

Imag

(c2)

(a)

ndash80

ndash60

ndash40

ndash20

0

20

40

60

80

Imag

(c3)

ndash40 0 40 80ndash80Real(c3)

(b)

Figure 10 e orbits were made up of c2 (a) and c3 (b) obtained with the improved BEMD method

8 Shock and Vibration

single component fundamental frequency signal and the oillm oscillation signal were not successfully separated Oneof the possible reasons is that in the CLMD algorithm thecomplex signal is only projected onto the x-axis and the y-axis unlike BEMD which projected on multiple directionsIn addition CLMD is a bivariate extension of LMD LMDused a moving average algorithm when tting the signalenvelope which can lter noise to a certain extent esignals other than the noise component c1 in Figure 9 areadded to obtain a ltered oil lm oscillation signal which isthen decomposed by the CLMD method and the rst twodecomposed results are shown in Figure 13 It is seen thatthe single component fundamental frequency signal and

the single component oil lm oscillation signal areseparated

cpf1 and cpf2 consisted of real part signals and imaginarypart signals both of which were composed of the product ofthe envelope signal and the pure frequency modulationfunctione envelope signal is the instantaneous amplitudeof the signal e corresponding instantaneous frequencywas obtained by deriving the inverse function of the cosinepure frequency modulation function e instantaneousamplitude and instantaneous frequency curves are shown inFigure 14 e instantaneous amplitude and frequencyobtained by the CLMD method are smoother than in Fig-ure 12e reason is mainly that the CLMDmethod uses the

ax2ay2

30

40

50

60

70A

mpl

itude

01 02 03 04 050Time (s)

(a)

fx2fy2

70

110

150

Freq

uenc

y (H

z)

01 02 03 04 050Time (s)

(b)

ax3ay3

20

70

120

Am

plitu

de

01 02 03 04 050Time (s)

(c)

fx3fy3

0

50

100

Freq

uenc

y (H

z)01 02 03 04 050

Time (s)

(d)

Figure 11 (a) e instantaneous amplitude of the real part and imaginary part of c2 (b) the instantaneous frequency of the real part andimaginary part of c2 (c) the instantaneous amplitude of the real part and imaginary part of c3 (d) the instantaneous frequency of the real partand imaginary part of c3

0 01 02 03 04 05

2050

800

20

40

60

80

Time (s)ax2

a y2

(a)

3060

90120

0

20

40

60

80

Time (s)ax3

a y3

0 01 02 03 04 05

(b)

Figure 12 e 3D time domain of instantaneous amplitude was made up of (a) ax2 and ay2 and (b) ax3 and ay3

Shock and Vibration 9

moving average ltering algorithm to obtain the signalenvelope curve However this is the result of using theCLMD algorithm after ltering out noise with BEMD If theBEMD algorithm is not used for ltering noise the in-stantaneous amplitude and instantaneous frequency curvesof the single component were not obtained by the CLMDmethod

42 Analysis ofOilWhirl Signal Based on the ImprovedBEMDMethod In the method similar to that presented in Section41 the oil whirl signals of the rotor test rig with a speedparameter of 4320 rpm are collected by two orthogonalsensors as shown in Figure 15 e gure shows the typicalwhirl phenomenon of large circles with embedded smallerones e decomposition results based on the improvedBEMD method (N 16 λ 005) are shown in Figure 16

Only three IMFs appear in Figure 16 and the singlecomponents c2 and c3 and the noise component c1 are suc-cessfully separated from the original signal e HT is appliedto the real and imaginary parts of c2 and c3 respectively andthe instantaneous frequency and instantaneous amplitude areobtained as shown in Figure 17 e three-dimensional timedomain of ax2 and ay2 and of ax3 and ay3 is shown in Figure 18Figure 17 indicates that the frequency of c2 is approximatelytwice the frequency of c3 and that ax2 is larger than ay2 Inaddition ay3 is slightly larger than ax3 but the range of changefor ax3 is greater than the range for ay3 It is inferred that c2 isthe fundamental frequency signal and that c3 is the half-frequency signal in the oil whirl signal

43 Analysis of Looseness and Rotor Rubbing Composite FaultSignal Based on the Improved BEMD Method Loose androtor rubbing composite faults are set on the testequipment shown in Figure 3 in Section 41 Loose fault isset on the nondrive end of the motor and the distancebetween the plastic rod and the shaft is xed near thesensor on the left side of the disk As the rotor speedincreases the vibration increases and the rubbing faultoccurs which is stable at around 1700 rmin e com-posite fault signals are collected by two orthogonal sen-sors as shown in Figure 19 e decomposition resultsbased on the improved BEMD method are shown inFigure 20

Figures 19(c) and 19(d) indicate that the signal com-ponent mainly contain 1X 2X (X 28Hz) and a frequencymodulated signal generated due to time-varying stinessis phenomenon is similar to that described reference [20]Four IMFs appear in Figure 20 and the single components1X 2X signals and the FM signal c2 are successfully separatedfrom the original signal e HT is applied to the real andimaginary parts of c2 c3 and c4 respectively and the in-stantaneous frequency and instantaneous amplitude areobtained as shown in Figure 21 ere are some uctuationsin the frequencies of c2 and c3 but these uctuations aredierent from the random uctuations in the above casesey have obvious regularity and are characteristic of FMsignals e frequency modulation characteristics of c2 aremore obvious than those of c3 In addition by observing theinstantaneous amplitude of c2 it is seen that c2 is still anamplitude modulation signal

minus100

0100

minus100

0

100

Time (s)Real(cpf1)

Imag

(cpf

1)

0 01 02 03 04 05

(a)

Time (s)Real(cpf2) minus100

0100

minus100

0

100

Imag

(cpf

2)

0 01 02 03 04 05

(b)

minus80 0 80minus60

0

60

Real(cpf1)

Imag(cpf1)

(c)

minus100 0 100minus100

0

100

Real(cpf2)

Imag(cpf2)

(d)

Figure 13 e rst two decomposed results of the ltered signal based on the CLMD method

10 Shock and Vibration

0025

05

ndash150

0

150ndash150

0

150

Time (s)Real(z)

Imag

(z)

(a)

ndash150 ndash75 0 75 150ndash150

ndash75

0

75

150

Real(z)

Imag

(z)

(b)

0 72 144 216 2880

40

80

Frequency (Hz)

Am

plitu

de

X 72Y 7321

X 36Y 3433

Real(z)

(c)

Frequency (Hz)0 72 144 216 288

0

40

80

X 36Y 3475

Am

plitu

de

X 72Y 6157

Imag(z)

(d)

Figure 15e oil whirl signal z (a) the 3D time-domain wave of z (b) the 2D plane of z (c) the Fourier spectrum of Real[z] (d) the Fourierspectrum of Imag[z]

30

40

50

60

70

Am

plitu

de

0 01 02 03 04 05Time (s)

ax1ay1

(a)

70

110

150

Freq

uenc

y (H

z)

0 01 02 03 04 05Time (s)

fx1fy1

(b)

0 01 02 03 04 0520

70

120

Time (s)

Am

plitu

de

ax2ay2

(c)

0 01 02 03 04 050

50

100

Time (s)

Freq

uenc

y (H

z)

fx2fy2

(d)

Figure 14 (a) e instantaneous amplitude of the real part and imaginary part of cpf1 (b) the instantaneous frequency of the real part andimaginary part of cpf1 (c) the instantaneous amplitude of the real part and imaginary part of cpf2 (d) the instantaneous frequency of the realpart and imaginary part of cpf2

Shock and Vibration 11

0025

05

ndash5

0

5ndash5

0

5

Time (s)Real(c1)

Imag

(c1)

(a)

0025

05

ndash1000

100ndash100

0

100

Time (s)

Imag

(c2)

Real(c2)

(b)

0025

05

ndash500

50ndash50

0

50

Time (s)

Imag

(c3)

Real(c3)

(c)

0025

05

ndash100

10ndash20

0

20

Time (s)

Imag

(r)

Real(r)

(d)

ndash100 ndash50 0 50 100ndash100

ndash50

0

50

100

Imag

(c2)

Real(c2)

(e)

ndash40 ndash20 0 20 40ndash40

ndash20

0

20

40Im

ag(c

3)

Real(c3)

(f )

Figure 16 e decomposition results of the oil whirl signal based on the improved BEMD method

60

90

120

Am

plitu

de

0 01 02 03 04 05Time (s)

ax2ay2

(a)

50

75

100

Freq

uenc

y (H

z)

0 01 02 03 04 05Time (s)

fx2fy2

(b)

Figure 17 Continued

12 Shock and Vibration

0 01 02 03 04 0520

35

50

Time (s)

Am

plitu

de

ax3ay3

(c)

0 01 02 03 04 0520

35

50

Time (s)

Freq

uenc

y (H

z)

fx3fy3

(d)

Figure 17 e instantaneous amplitude and frequency of c2 and c3 from the oil whirl signal obtained by the HT (a) the instantaneousamplitude of the real part and imaginary part of c2 (b) the instantaneous frequency of the real part and imaginary part of c2(c) the instantaneous amplitude of the real part and imaginary part of c3 (d) the instantaneous frequency of the real part and imaginarypart of c3

0

025

05

40

80

12040

80

120

Time (s)

X 03877Y 9582Z 826

ax2

X 01309Y 9595Z 8452

a y2

(a)

0

025

05

20

35

5020

35

50

Time (s)

X 03955Y 3762Z 3822

ax3

X 008838Y 3608Z 3733

a y3

(b)

Figure 18 e 3D time domain of the instantaneous amplitude of ax2 and ay2 (a) and ax3 and ay3 (b) from the oil whirl signal

0025

05

minus1500

150minus150

0

150

t (s)Real(z)

Imag

(z)

(a)

Imag(z)

Real(z)

minus150

0

150

minus150 0 150

(b)

Figure 19 Continued

Shock and Vibration 13

In order to further verify the correctness of the in-stantaneous amplitude-frequency characteristics of theproposed method the real and imaginary parts of thecomposite fault signal z are analyzed separately using syn-chrosqueezed wavelet transforms (SWT) proposed in ref-erence [21]-e results are shown in Figure 22 It is seen thatthe time-frequency representations of the composite faultsignal z also include the AM-FM signal and the 1X signalwhich proves the correctness of the proposed methodCompared with the SWT method the instantaneousamplitude-frequency characteristics acquired by the HTmethod are relatively straightforward

44 lte Bistable Behavior Analysis of the Fan Rotor Based onBEMD -e bistability of the rotor is a nonlinear behaviorof the rotor-bearing system which is the state in which therotor jumps from one stable state to another forming astep -e bivariate signal of the bistable behavior iscomposed of two signals collected by two displacementsensors from orthogonal locations on the experimentaldevices in literature [22] as shown in Figure 23 Literature[22] shows that the cause of the bistable behavior remainsto be further explored -is paper uses this case to il-lustrate the feasibility of BEMD to analyze nonstationarysignals

0

40

80

Am

plitu

de Real(z)

0 100 200 300 400Frequency (Hz)

(c)

Am

plitu

de

0 100 200 300 400Frequency (Hz)

0

20

40

60

80

Imag(z)

(d)

Figure 19 -e composite fault signal z (a) the 3D time domain wave of z (b) the 2D plane of z (c) the Fourier spectrum of Real[z] (d) theFourier spectrum of Imag[z]

0025

05

minus30

3minus5

0

5

Time (s)Real(c1)

Imag

(c1)

(a)

0025

05

minus200

20minus20

0

20

Time (s)Real(c2 )

Imag

(c2)

(b)

0025

05

minus200

20minus50

0

50

Time (s)Real(c2)

Imag

(c2)

(c)

0025

05

minus1000

100minus100

0

100

Time (s)Real(c3)

Imag

(c3)

(d)

0025

05

minus400

40minus30

0

30

Time (s)Real(r)

Imag

(r)

(e)

Figure 20 -e decomposition results of the composite fault signal based on the improved BEMD method

14 Shock and Vibration

-e x and y signals in the horizontal and vertical di-rections of the left and right bearings respectively from thefan rotors are collected with four displacement sensorsLetting z x+ jy the time and frequency domain plots of z areshown in Figure 24 where the fan rotor speed is 5500 rpm thesampling frequency is 2000Hz and the number of samplingpoints is 1024 -e left and right columns respectively showthe time and frequency domain plots of the vibration signalsfrom the left and right bearings of the fan rotor Bistablebehavior arises in the fan rotor and the amplitudes of the

vibration signals vary significantly in different positions anddirections Further studies are required to explain the causesof this bistability -e present study focuses on extracting thebistable behavioral signal characteristics to verify the feasi-bility of the proposed method

-e decomposition results of the bistable behavioralsignals based on the improved BEMDmethod are shown inFigure 25 c1 c2 c3 and r are separated in order from zusing the improved BEMD method c1 shows a randomarrangement and is considered the high-frequency noise

0

10

20

Am

plitu

de

ax2ay2

0 025 05Time (s)

(a)

0

152

304

Freq

uenc

y (H

z)

0 025 05Time (s)

fy2

fx2

(b)

0

10

20

30

40

Am

plitu

de

ax3ay3

0 025 05Time (s)

(c)

0

56

112

Freq

uenc

y (H

z)

fx3fy3

0 025 05Time (s)

(d)

0 025 050

50

100

Time (s)

Am

plitu

de

ax4ay4

(e)

0 025 050

28

56

Time (s)

Freq

uenc

y (H

z)

fx4fy4

(f )

Figure 21 -e instantaneous amplitude and frequency of c2 c3 and c4 obtained by the HT (a) the instantaneous amplitude of the real partand imaginary part of c2 (b) the instantaneous frequency of the real part and imaginary part of c2 (c) the instantaneous amplitude of the realpart and imaginary part of c3 (d) the instantaneous frequency of the real part and imaginary part of c3 (e) the instantaneous amplitude of thereal part and imaginary part of c4 (f ) the instantaneous frequency of the real part and imaginary part of c4

Shock and Vibration 15

signal c2 is considered to represent the extracted bistablebehavior signals -e HT is applied to the real andimaginary parts of c2 to obtain the instantaneous amplitudeand frequency of c2 from the left and right columns fromFigure 25 as shown in Figure 26 Figure 27 shows thethree-dimensional time domain of ax2 and ay2 from the leftand right columns respectively Figure 26 shows that thevibration signal amplitude on the left side of the fan de-creases from large to small opposite of the behavior ofthe right -e horizontal vibration signal amplitude on theleft side of the fan is larger than that of the vertical di-rection signal opposite of the right -is result validatesthat the vibration signals from different directions orpositions are different when the fan produces bistablebehavior In addition the time of the bistable behavior canbe determined according to the jump point of the am-plitude or frequency

5 Discussion

-e BEMD algorithm decomposes two orthogonal di-rections of vibration signals as a complex signal which is atwo-dimensional digital signal processing method thusensuring that the real and imaginary parts have the samedecomposition scale Similar to EMD the envelope mean iscritical for the decomposition effect of BEMD but the en-velope mean in BEMD is three-dimensional If the numberof projection directions of the complex signal in three-dimensional space is larger the corresponding envelope

signal is also more -us the envelope mean value is moreaccurate and the BEMD decomposition effect is betterIncreasing the number of projection directions can improvemodal aliasing Like EMD BEMD also produces falsecomponents when decomposing signals Generally speakingthe energy of the false components is low and these low-energy false components do not contain fault characteristicinformation and the introduction of the energy thresholdcriterion in the termination condition can increase thedecomposition speed of the BEMD

-e experimental results show that there is a certaindifference in the existence of vibration signal character-istics in different directions when rotating machinery failsIn addition when the number of projection directionsis increased the decomposition speed of BEMD willdecrease

6 Conclusions

We use BEMD and HT to extract the instantaneousamplitude-frequency features of rotor faults A bivariateinstantaneous feature extraction method based on the im-proved BEMD method and the HT is investigated whichextends the fault feature extraction technology to two di-mensions -e BEMD method is suitable to analyze thecomplex multicomponent bivariate signals -e mainsingle-component bivariate signals are separated from themulticomponent bivariate signals of the fan rotor bistabilityfor the oil film oscillation and the oil film vortex using the

0

2

4

6

Time (s)

Freq

uenc

y (H

z)

0 01 02 03 04 052

3

4

5

6

7

8log2 fx

(a)

0

2

4

6

Time (s)

Freq

uenc

y (H

z)

0 01 02 03 04 052

3

4

5

6

7

8log2 fy

(b)

Figure 22 -e results of the composite fault signal z based on SWT the time-frequency representation of (a) the real part of z and (b) theimaginary part of z

Left bearing predstal

Locations of sensors

Fanrotor

Right bearing predstal

Axis

Orthogonal directions

Figure 23 -e schematic diagram of the experimental apparatus

16 Shock and Vibration

Real(z) 00256

0512minus300

0300

minus400

0

400

Time (s)

Imag

(z)

00256

0512minus500

0500

minus500

0

500

Time (s)Real(z)

Imag

(z)

Imag

(z)

minus300 0 300minus500

0

500

Real(z)minus500 0 500

minus500

0

Imag

(z)

Real(z)

500

0 100 200 300 4000

100

200

300

Frequency (Hz)

Am

plitu

de Imag(z)

Frequency (Hz)0 100 200 300 400

0

200

400

Am

plitu

de

Imag(z)

0 100 200 300 4000

70

140

Frequency (Hz)

Am

plitu

de

Real(z)

0 100 200 300 4000

200

400

Frequency (Hz)

Am

plitu

de Real(z)

(a) (b)

Figure 24 -e time and frequency domain plots of the bistable behavior signals

00256

0512

minus800

80minus80

0

80

Time (s)Real(c1 )

Imag

(c1)

Time (s)Real(c1 )

Imag

(c1)

00256

0512

minus400

40minus40

0

40

Time (s)Real(c2) 0

02560512

minus5000

500minus500

0

500

Imag

(c2)

Time (s)Real(c2)

Imag

(c2)

00256

0512

minus5000

500minus500

0

500

Time (s)Real(r) 00256

0512

minus800

80minus80

0

80

Imag

(irc

rm

)

Time (s)Real(r)

Imag

(r)

00256

0512

minus2000

200minus200

0

200

(a) (b)

Figure 25 -e decomposition results of the bistable behavior signals based on the improved BEMD method

Shock and Vibration 17

improved BEMD method For the single-component bi-variate signal the HT is used to obtain the correspondinginstantaneous amplitude and frequency characteristics -eproposed method can examine the detailed information of asingle rotation component

Data Availability

-e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

-e authors declare that they have no conflicts of interest

Authorsrsquo Contributions

All the authors contributed to this work Chuanjin Huangconceived and designed the simulation and experiments anddrafted the manuscript Haijun Song performed the simu-lations and experiments and analyzed the data and

0 0256 05120

50

100

150

Time (s)

Freq

uenc

y (H

z)

fx2

fy2

0 0256 05120

200

400

600

Time (s)

Am

plitu

de

ax2ay2

(a)

fx2

fy2

0 0256 05120

90

180

270

360

Time (s)

Freq

uenc

y (H

z)

0 0256 05120

200

400

600

Time (s)

Am

plitu

de

ax2Data 2

(b)

Figure 26 -e instantaneous amplitude and frequency of c2 from the (a) left and (b) right columns

0

0256

0512

0100

200300

400500

0

100

200

300

400

500

Time (s)

X 04035Y 3509Z 130

X 04235Y 2265Z 3613

X 007Y 1131Z 2191

X 0105Y 3837Z 3312

ax2

a y2

Figure 27 -e three-dimensional time domain of ax2 and ay2

18 Shock and Vibration

Wenping Lei and Yajun Meng performed the experimentsand analyzed the data All the authors contributed to thewriting and discussion of the paper

Acknowledgments

-is research was funded by the Henan Provincial HigherEducation Key Research Project (Grant nos 18A460006 and19A460029) Henan High-Level Innovative Scientific andTechnological Talent Team Construction Project (Grant noC20150034) and Zhengzhou Institute of Technology In-novation Team Project (Grant no CXTD2017K1)

References

[1] R Yan R X Gao and X Chen ldquoWavelets for fault diagnosisof rotary machines a review with applicationsrdquo Signal Pro-cessing vol 96 pp 1ndash15 2014

[2] J Cheng D Yu J Tang and Y Yang ldquoApplication of frequencyfamily separation method based upon EMD and local Hilbertenergy spectrum method to gear fault diagnosisrdquo Mechanismand Machine lteory vol 43 no 6 pp 712ndash723 2008

[3] H Liu and M Han ldquoA fault diagnosis method based on localmean decomposition and multi-scale entropy for rollerbearingsrdquoMechanism andMachinelteory vol 75 pp 67ndash782014

[4] Z Zheng W Jiang Z Wang Y Zhu and K Yang ldquoGear faultdiagnosis method based on local mean decomposition andgeneralized morphological fractal dimensionsrdquo Mechanismand Machine lteory vol 91 pp 151ndash167 2015

[5] W Yang R Court P J Tavner and C J Crabtree ldquoBivariateempirical mode decomposition and its contribution to windturbine condition monitoringrdquo Journal of Sound and Vi-bration vol 330 no 15 pp 3766ndash3782 2011

[6] L Qu X Liu G Peyronne and Y Chen ldquo-e holospectrum anewmethod for rotor surveillance and diagnosisrdquoMechanicalSystems amp Signal Processing vol 3 no 3 pp 255ndash267 1989

[7] F Q Wu and G Meng ldquoCompound rub malfunctions featureextraction based on full-spectrum cascade analysis and SVMrdquoMechanical Systems and Signal Processing vol 20 no 8pp 2007ndash2021 2006

[8] Y Chen Q Gao and Z Guan ldquoSelf-loosening failure analysisof bolt joints under vibration considering the tighteningprocessrdquo Shock and Vibration vol 2017 Article ID 203842115 pages 2017

[9] L Chen J Han W Lei Y Cui and Z Guan ldquoFull-vectorsignal acquisition and information fusion for the fault pre-dictionrdquo International Journal of Rotating Machineryvol 2016 Article ID 5980802 7 pages 2016

[10] C Chen Y Meng and Y Du ldquoApplication of the full vectorspectrum based on EMD in fault diagnosis of bearingsrdquoJournal of Mechanical Strength vol 37 pp 806ndash811 2015

[11] C Huang X Wu and W Cao ldquoLMD-based on full vectorenvelope technique and its application in TRT vibration faultdiagnosisrdquo Electric Power Automation Equipment vol 35pp 168ndash174 2015 in Chinese

[12] X Zhao T H Patel and M J Zuo ldquoMultivariate EMD andfull spectrum based condition monitoring for rotating ma-chineryrdquo Mechanical Systems and Signal Processing vol 27pp 712ndash728 2012

[13] G Rilling P Flandrin P Gonalves and J M Lilly ldquoBivariateempirical mode decompositionrdquo IEEE Signal ProcessingLetters vol 14 no 12 pp 936ndash939 2007

[14] C Park D Looney M M Van Hulle and D P Mandic ldquo-ecomplex local mean decompositionrdquo Neurocomputingvol 74 no 6 pp 867ndash875 2011

[15] N Rehman and D P Mandic ldquoEmpirical mode de-composition for trivariate signalsrdquo IEEE Transactions onSignal Processing vol 58 no 3 pp 1059ndash1068 2010

[16] N Rehman and D P Mandic ldquoMultivariate empirical modedecompositionrdquo Proceedings of the Royal Society A Mathe-matical Physical and Engineering Sciences vol 466 no 2117pp 1291ndash1302 2010

[17] Y Lv R Yuan and G Song ldquoMultivariate empirical modedecomposition and its application to fault diagnosis of rollingbearingrdquo Mechanical Systems and Signal Processing vol 81pp 219ndash234 2016

[18] C Huang Y Meng and W Lei ldquoFull vector envelopetechnique based on complex local mean decomposition andits application in fault feature extraction for rotor systemrdquoJournal of Mechanical Engineering vol 52 no 7 p 69 2016in Chinese

[19] G Rilling P Flandrin and P Goncalves ldquoOn empirical modedecomposition and its algorithmsrdquo in Proceedings of IEEE-EURASIP Workshop on Nonlinear Signal and Image Pro-cessing pp 8ndash11 IEEE Trieste Italy June 2003

[20] L Yang X Chen and S Wang ldquoMechanism of fast time-varying vibration for rotorndashstator contact system with ap-plication to fault diagnosisrdquo Journal of Vibration andAcoustics vol 140 no 1 article 014501 2018

[21] I Daubechies J Lu and H-TWu ldquoSynchrosqueezed wavelettransforms an empirical mode decomposition-like toolrdquoApplied and Computational Harmonic Analysis vol 30 no 2pp 243ndash261 2011

[22] L-S Qu Holospectrum and Holobalancing Technique inMachinery Diagnosis Beijing Science Press Beijing China2007

Shock and Vibration 19

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e HT is applied to the real and imaginary parts of c2and c3 respectively and the instantaneous frequency andinstantaneous amplitude are obtained as shown in Fig-ure 11 where ax2 and ax3 represent the instantaneous

amplitude of the real parts of c2 and c3 respectively ay2and ay3 represent the instantaneous amplitude of theimaginary parts of c2 and c3 respectively fx2 and fx3represent the instantaneous frequency of the real parts of

0025

05

ndash50

5ndash5

0

5

Time (s)Real(c1)

Imag

(c1)

(a)

0025

05

ndash1000

100ndash100

0

100

Time (s)Real(c2 )

Imag

(c2)

(b)

0025

05

ndash1000

100ndash100

0

100

Time (s)Real(c3 )

Imag

(c3)

(c)

0025

05

ndash200

20ndash50

0

50

Time (s)Real(c4 )

Imag

(c4)

Modal aliasing

(d)

0025

05

ndash100

10ndash10

0

10

Time (s)Real(c5 )Im

ag(c

5)

(e)

0025

05

ndash50

5ndash5

0

5

Time (s)Real(c6 )

Imag

(c6)

(f )

0025

05

ndash50

5ndash1

0

1

Time (s)Real(c7)

Imag

(c7)

(g)

0025

05

ndash50

5ndash1

0

1

Time (s)Real(c8)

Imag

(c8)

(h)

0025

05

ndash100

10ndash5

0

5

Time (s)Real(r)

Imag

(r)

(i)

Figure 7 Decomposition results of the oil lm oscillation signal using BEMD (N 4)

ndash80 ndash40 0 40 80ndash60

ndash30

0

30

60

Real(c2)

Imag

(c2)

(a)

ndash100 ndash50 0 50 100ndash100

ndash50

0

50

100

Real(c3)

Imag

(c3)

(b)

ndash80 ndash40 0 40 80ndash60

ndash30

0

30

60

IMFx2

IMF y

2

(c)

ndash100 ndash50 0 50 100ndash100

ndash50

0

50

100

IMFx3

IMF y

3

(d)

Figure 8e orbits made up of (a) the real and imaginary parts of c2 (b) real and imaginary parts of c3 (c) IMFx2 and IMFy2 and (d) IMFx3and IMFy3

Shock and Vibration 7

c2 and c3 respectively and fy2 and fy3 represent the in-stantaneous frequency of the imaginary parts of c2 and c3respectively

In Figure 11 ax3 is much larger than ay3 and their phasesare separated by nearly 180deg which shows that the amplitudeand the phase of the oil lm oscillation signal in dierentdirections can vary fx2 and fy2 and fx3 and fy3 are ap-proximately the same ax2 and ay2 show little change andtheir phases are the same Since BEMD is a bivariate ex-tension of EMD like EMD BEMD also has an endpointeect ere are some uctuations in the instantaneousamplitude and instantaneous frequency due to the end eect

and the edge eect of the Hilbert transform e three-dimensional time domain of ax2 and ay2 and of ax3 and ay3 isshown in Figure 12e amplitude range of c3 is much largerthan that of c2 It is can be inferred that the main componentof oscillation is c3 with the frequency of 52Hz in the oil lmoscillation signal

e decomposition results of the oil lm oscillationsignal are shown in Figure 13 using the CLMD methodproposed in reference [14] e oil lm oscillation signal isdecomposed into four complex product functions cpf1ndashcpf4e noise component is not separated from the oil lmoscillation signal using the CLMD method Moreover the

0025

05

ndash50

5

Time (s)Real(c1 )

ndash5

0

5Im

ag(c

1)

(a)

0025

05

ndash1000

100

Time (s)Real(c2)

ndash100

0

100

Imag

(c2)

(b)

0025

05ndash100

0100

ndash100

0

100

Imag

(c3)

Real(c3 ) Time (s)

(c)

0 02505

ndash100

10

Time (s)Real(c4)

ndash20

0

20

Imag

(c4)

(d)

0025

05ndash10

010

Time (s)Real(c5)

ndash10

0

10

Imag

(c5)

(e)

0 02505

ndash50

5

Time (s)Real(c6)

ndash5

0

5

Imag

(c6)

(f )

0 02505

ndash202

Time (s)Real(c7)

ndash2

0

2

Imag

(c7)

(g)

0025

05ndash5

05

Time (s)Real(c8)

ndash5

0

5

Imag

(c8)

(h)

0025

05ndash10

010

Time (s)Real(r)

ndash5

0

5

Imag

(r)

(i)

Figure 9 Decomposition results of the oil lm oscillation signal using the improved BEMD method (N 16 λ 005)

ndash80 0 40 80ndash40Real(c2)

ndash60

ndash30

0

30

60

Imag

(c2)

(a)

ndash80

ndash60

ndash40

ndash20

0

20

40

60

80

Imag

(c3)

ndash40 0 40 80ndash80Real(c3)

(b)

Figure 10 e orbits were made up of c2 (a) and c3 (b) obtained with the improved BEMD method

8 Shock and Vibration

single component fundamental frequency signal and the oillm oscillation signal were not successfully separated Oneof the possible reasons is that in the CLMD algorithm thecomplex signal is only projected onto the x-axis and the y-axis unlike BEMD which projected on multiple directionsIn addition CLMD is a bivariate extension of LMD LMDused a moving average algorithm when tting the signalenvelope which can lter noise to a certain extent esignals other than the noise component c1 in Figure 9 areadded to obtain a ltered oil lm oscillation signal which isthen decomposed by the CLMD method and the rst twodecomposed results are shown in Figure 13 It is seen thatthe single component fundamental frequency signal and

the single component oil lm oscillation signal areseparated

cpf1 and cpf2 consisted of real part signals and imaginarypart signals both of which were composed of the product ofthe envelope signal and the pure frequency modulationfunctione envelope signal is the instantaneous amplitudeof the signal e corresponding instantaneous frequencywas obtained by deriving the inverse function of the cosinepure frequency modulation function e instantaneousamplitude and instantaneous frequency curves are shown inFigure 14 e instantaneous amplitude and frequencyobtained by the CLMD method are smoother than in Fig-ure 12e reason is mainly that the CLMDmethod uses the

ax2ay2

30

40

50

60

70A

mpl

itude

01 02 03 04 050Time (s)

(a)

fx2fy2

70

110

150

Freq

uenc

y (H

z)

01 02 03 04 050Time (s)

(b)

ax3ay3

20

70

120

Am

plitu

de

01 02 03 04 050Time (s)

(c)

fx3fy3

0

50

100

Freq

uenc

y (H

z)01 02 03 04 050

Time (s)

(d)

Figure 11 (a) e instantaneous amplitude of the real part and imaginary part of c2 (b) the instantaneous frequency of the real part andimaginary part of c2 (c) the instantaneous amplitude of the real part and imaginary part of c3 (d) the instantaneous frequency of the real partand imaginary part of c3

0 01 02 03 04 05

2050

800

20

40

60

80

Time (s)ax2

a y2

(a)

3060

90120

0

20

40

60

80

Time (s)ax3

a y3

0 01 02 03 04 05

(b)

Figure 12 e 3D time domain of instantaneous amplitude was made up of (a) ax2 and ay2 and (b) ax3 and ay3

Shock and Vibration 9

moving average ltering algorithm to obtain the signalenvelope curve However this is the result of using theCLMD algorithm after ltering out noise with BEMD If theBEMD algorithm is not used for ltering noise the in-stantaneous amplitude and instantaneous frequency curvesof the single component were not obtained by the CLMDmethod

42 Analysis ofOilWhirl Signal Based on the ImprovedBEMDMethod In the method similar to that presented in Section41 the oil whirl signals of the rotor test rig with a speedparameter of 4320 rpm are collected by two orthogonalsensors as shown in Figure 15 e gure shows the typicalwhirl phenomenon of large circles with embedded smallerones e decomposition results based on the improvedBEMD method (N 16 λ 005) are shown in Figure 16

Only three IMFs appear in Figure 16 and the singlecomponents c2 and c3 and the noise component c1 are suc-cessfully separated from the original signal e HT is appliedto the real and imaginary parts of c2 and c3 respectively andthe instantaneous frequency and instantaneous amplitude areobtained as shown in Figure 17 e three-dimensional timedomain of ax2 and ay2 and of ax3 and ay3 is shown in Figure 18Figure 17 indicates that the frequency of c2 is approximatelytwice the frequency of c3 and that ax2 is larger than ay2 Inaddition ay3 is slightly larger than ax3 but the range of changefor ax3 is greater than the range for ay3 It is inferred that c2 isthe fundamental frequency signal and that c3 is the half-frequency signal in the oil whirl signal

43 Analysis of Looseness and Rotor Rubbing Composite FaultSignal Based on the Improved BEMD Method Loose androtor rubbing composite faults are set on the testequipment shown in Figure 3 in Section 41 Loose fault isset on the nondrive end of the motor and the distancebetween the plastic rod and the shaft is xed near thesensor on the left side of the disk As the rotor speedincreases the vibration increases and the rubbing faultoccurs which is stable at around 1700 rmin e com-posite fault signals are collected by two orthogonal sen-sors as shown in Figure 19 e decomposition resultsbased on the improved BEMD method are shown inFigure 20

Figures 19(c) and 19(d) indicate that the signal com-ponent mainly contain 1X 2X (X 28Hz) and a frequencymodulated signal generated due to time-varying stinessis phenomenon is similar to that described reference [20]Four IMFs appear in Figure 20 and the single components1X 2X signals and the FM signal c2 are successfully separatedfrom the original signal e HT is applied to the real andimaginary parts of c2 c3 and c4 respectively and the in-stantaneous frequency and instantaneous amplitude areobtained as shown in Figure 21 ere are some uctuationsin the frequencies of c2 and c3 but these uctuations aredierent from the random uctuations in the above casesey have obvious regularity and are characteristic of FMsignals e frequency modulation characteristics of c2 aremore obvious than those of c3 In addition by observing theinstantaneous amplitude of c2 it is seen that c2 is still anamplitude modulation signal

minus100

0100

minus100

0

100

Time (s)Real(cpf1)

Imag

(cpf

1)

0 01 02 03 04 05

(a)

Time (s)Real(cpf2) minus100

0100

minus100

0

100

Imag

(cpf

2)

0 01 02 03 04 05

(b)

minus80 0 80minus60

0

60

Real(cpf1)

Imag(cpf1)

(c)

minus100 0 100minus100

0

100

Real(cpf2)

Imag(cpf2)

(d)

Figure 13 e rst two decomposed results of the ltered signal based on the CLMD method

10 Shock and Vibration

0025

05

ndash150

0

150ndash150

0

150

Time (s)Real(z)

Imag

(z)

(a)

ndash150 ndash75 0 75 150ndash150

ndash75

0

75

150

Real(z)

Imag

(z)

(b)

0 72 144 216 2880

40

80

Frequency (Hz)

Am

plitu

de

X 72Y 7321

X 36Y 3433

Real(z)

(c)

Frequency (Hz)0 72 144 216 288

0

40

80

X 36Y 3475

Am

plitu

de

X 72Y 6157

Imag(z)

(d)

Figure 15e oil whirl signal z (a) the 3D time-domain wave of z (b) the 2D plane of z (c) the Fourier spectrum of Real[z] (d) the Fourierspectrum of Imag[z]

30

40

50

60

70

Am

plitu

de

0 01 02 03 04 05Time (s)

ax1ay1

(a)

70

110

150

Freq

uenc

y (H

z)

0 01 02 03 04 05Time (s)

fx1fy1

(b)

0 01 02 03 04 0520

70

120

Time (s)

Am

plitu

de

ax2ay2

(c)

0 01 02 03 04 050

50

100

Time (s)

Freq

uenc

y (H

z)

fx2fy2

(d)

Figure 14 (a) e instantaneous amplitude of the real part and imaginary part of cpf1 (b) the instantaneous frequency of the real part andimaginary part of cpf1 (c) the instantaneous amplitude of the real part and imaginary part of cpf2 (d) the instantaneous frequency of the realpart and imaginary part of cpf2

Shock and Vibration 11

0025

05

ndash5

0

5ndash5

0

5

Time (s)Real(c1)

Imag

(c1)

(a)

0025

05

ndash1000

100ndash100

0

100

Time (s)

Imag

(c2)

Real(c2)

(b)

0025

05

ndash500

50ndash50

0

50

Time (s)

Imag

(c3)

Real(c3)

(c)

0025

05

ndash100

10ndash20

0

20

Time (s)

Imag

(r)

Real(r)

(d)

ndash100 ndash50 0 50 100ndash100

ndash50

0

50

100

Imag

(c2)

Real(c2)

(e)

ndash40 ndash20 0 20 40ndash40

ndash20

0

20

40Im

ag(c

3)

Real(c3)

(f )

Figure 16 e decomposition results of the oil whirl signal based on the improved BEMD method

60

90

120

Am

plitu

de

0 01 02 03 04 05Time (s)

ax2ay2

(a)

50

75

100

Freq

uenc

y (H

z)

0 01 02 03 04 05Time (s)

fx2fy2

(b)

Figure 17 Continued

12 Shock and Vibration

0 01 02 03 04 0520

35

50

Time (s)

Am

plitu

de

ax3ay3

(c)

0 01 02 03 04 0520

35

50

Time (s)

Freq

uenc

y (H

z)

fx3fy3

(d)

Figure 17 e instantaneous amplitude and frequency of c2 and c3 from the oil whirl signal obtained by the HT (a) the instantaneousamplitude of the real part and imaginary part of c2 (b) the instantaneous frequency of the real part and imaginary part of c2(c) the instantaneous amplitude of the real part and imaginary part of c3 (d) the instantaneous frequency of the real part and imaginarypart of c3

0

025

05

40

80

12040

80

120

Time (s)

X 03877Y 9582Z 826

ax2

X 01309Y 9595Z 8452

a y2

(a)

0

025

05

20

35

5020

35

50

Time (s)

X 03955Y 3762Z 3822

ax3

X 008838Y 3608Z 3733

a y3

(b)

Figure 18 e 3D time domain of the instantaneous amplitude of ax2 and ay2 (a) and ax3 and ay3 (b) from the oil whirl signal

0025

05

minus1500

150minus150

0

150

t (s)Real(z)

Imag

(z)

(a)

Imag(z)

Real(z)

minus150

0

150

minus150 0 150

(b)

Figure 19 Continued

Shock and Vibration 13

In order to further verify the correctness of the in-stantaneous amplitude-frequency characteristics of theproposed method the real and imaginary parts of thecomposite fault signal z are analyzed separately using syn-chrosqueezed wavelet transforms (SWT) proposed in ref-erence [21]-e results are shown in Figure 22 It is seen thatthe time-frequency representations of the composite faultsignal z also include the AM-FM signal and the 1X signalwhich proves the correctness of the proposed methodCompared with the SWT method the instantaneousamplitude-frequency characteristics acquired by the HTmethod are relatively straightforward

44 lte Bistable Behavior Analysis of the Fan Rotor Based onBEMD -e bistability of the rotor is a nonlinear behaviorof the rotor-bearing system which is the state in which therotor jumps from one stable state to another forming astep -e bivariate signal of the bistable behavior iscomposed of two signals collected by two displacementsensors from orthogonal locations on the experimentaldevices in literature [22] as shown in Figure 23 Literature[22] shows that the cause of the bistable behavior remainsto be further explored -is paper uses this case to il-lustrate the feasibility of BEMD to analyze nonstationarysignals

0

40

80

Am

plitu

de Real(z)

0 100 200 300 400Frequency (Hz)

(c)

Am

plitu

de

0 100 200 300 400Frequency (Hz)

0

20

40

60

80

Imag(z)

(d)

Figure 19 -e composite fault signal z (a) the 3D time domain wave of z (b) the 2D plane of z (c) the Fourier spectrum of Real[z] (d) theFourier spectrum of Imag[z]

0025

05

minus30

3minus5

0

5

Time (s)Real(c1)

Imag

(c1)

(a)

0025

05

minus200

20minus20

0

20

Time (s)Real(c2 )

Imag

(c2)

(b)

0025

05

minus200

20minus50

0

50

Time (s)Real(c2)

Imag

(c2)

(c)

0025

05

minus1000

100minus100

0

100

Time (s)Real(c3)

Imag

(c3)

(d)

0025

05

minus400

40minus30

0

30

Time (s)Real(r)

Imag

(r)

(e)

Figure 20 -e decomposition results of the composite fault signal based on the improved BEMD method

14 Shock and Vibration

-e x and y signals in the horizontal and vertical di-rections of the left and right bearings respectively from thefan rotors are collected with four displacement sensorsLetting z x+ jy the time and frequency domain plots of z areshown in Figure 24 where the fan rotor speed is 5500 rpm thesampling frequency is 2000Hz and the number of samplingpoints is 1024 -e left and right columns respectively showthe time and frequency domain plots of the vibration signalsfrom the left and right bearings of the fan rotor Bistablebehavior arises in the fan rotor and the amplitudes of the

vibration signals vary significantly in different positions anddirections Further studies are required to explain the causesof this bistability -e present study focuses on extracting thebistable behavioral signal characteristics to verify the feasi-bility of the proposed method

-e decomposition results of the bistable behavioralsignals based on the improved BEMDmethod are shown inFigure 25 c1 c2 c3 and r are separated in order from zusing the improved BEMD method c1 shows a randomarrangement and is considered the high-frequency noise

0

10

20

Am

plitu

de

ax2ay2

0 025 05Time (s)

(a)

0

152

304

Freq

uenc

y (H

z)

0 025 05Time (s)

fy2

fx2

(b)

0

10

20

30

40

Am

plitu

de

ax3ay3

0 025 05Time (s)

(c)

0

56

112

Freq

uenc

y (H

z)

fx3fy3

0 025 05Time (s)

(d)

0 025 050

50

100

Time (s)

Am

plitu

de

ax4ay4

(e)

0 025 050

28

56

Time (s)

Freq

uenc

y (H

z)

fx4fy4

(f )

Figure 21 -e instantaneous amplitude and frequency of c2 c3 and c4 obtained by the HT (a) the instantaneous amplitude of the real partand imaginary part of c2 (b) the instantaneous frequency of the real part and imaginary part of c2 (c) the instantaneous amplitude of the realpart and imaginary part of c3 (d) the instantaneous frequency of the real part and imaginary part of c3 (e) the instantaneous amplitude of thereal part and imaginary part of c4 (f ) the instantaneous frequency of the real part and imaginary part of c4

Shock and Vibration 15

signal c2 is considered to represent the extracted bistablebehavior signals -e HT is applied to the real andimaginary parts of c2 to obtain the instantaneous amplitudeand frequency of c2 from the left and right columns fromFigure 25 as shown in Figure 26 Figure 27 shows thethree-dimensional time domain of ax2 and ay2 from the leftand right columns respectively Figure 26 shows that thevibration signal amplitude on the left side of the fan de-creases from large to small opposite of the behavior ofthe right -e horizontal vibration signal amplitude on theleft side of the fan is larger than that of the vertical di-rection signal opposite of the right -is result validatesthat the vibration signals from different directions orpositions are different when the fan produces bistablebehavior In addition the time of the bistable behavior canbe determined according to the jump point of the am-plitude or frequency

5 Discussion

-e BEMD algorithm decomposes two orthogonal di-rections of vibration signals as a complex signal which is atwo-dimensional digital signal processing method thusensuring that the real and imaginary parts have the samedecomposition scale Similar to EMD the envelope mean iscritical for the decomposition effect of BEMD but the en-velope mean in BEMD is three-dimensional If the numberof projection directions of the complex signal in three-dimensional space is larger the corresponding envelope

signal is also more -us the envelope mean value is moreaccurate and the BEMD decomposition effect is betterIncreasing the number of projection directions can improvemodal aliasing Like EMD BEMD also produces falsecomponents when decomposing signals Generally speakingthe energy of the false components is low and these low-energy false components do not contain fault characteristicinformation and the introduction of the energy thresholdcriterion in the termination condition can increase thedecomposition speed of the BEMD

-e experimental results show that there is a certaindifference in the existence of vibration signal character-istics in different directions when rotating machinery failsIn addition when the number of projection directionsis increased the decomposition speed of BEMD willdecrease

6 Conclusions

We use BEMD and HT to extract the instantaneousamplitude-frequency features of rotor faults A bivariateinstantaneous feature extraction method based on the im-proved BEMD method and the HT is investigated whichextends the fault feature extraction technology to two di-mensions -e BEMD method is suitable to analyze thecomplex multicomponent bivariate signals -e mainsingle-component bivariate signals are separated from themulticomponent bivariate signals of the fan rotor bistabilityfor the oil film oscillation and the oil film vortex using the

0

2

4

6

Time (s)

Freq

uenc

y (H

z)

0 01 02 03 04 052

3

4

5

6

7

8log2 fx

(a)

0

2

4

6

Time (s)

Freq

uenc

y (H

z)

0 01 02 03 04 052

3

4

5

6

7

8log2 fy

(b)

Figure 22 -e results of the composite fault signal z based on SWT the time-frequency representation of (a) the real part of z and (b) theimaginary part of z

Left bearing predstal

Locations of sensors

Fanrotor

Right bearing predstal

Axis

Orthogonal directions

Figure 23 -e schematic diagram of the experimental apparatus

16 Shock and Vibration

Real(z) 00256

0512minus300

0300

minus400

0

400

Time (s)

Imag

(z)

00256

0512minus500

0500

minus500

0

500

Time (s)Real(z)

Imag

(z)

Imag

(z)

minus300 0 300minus500

0

500

Real(z)minus500 0 500

minus500

0

Imag

(z)

Real(z)

500

0 100 200 300 4000

100

200

300

Frequency (Hz)

Am

plitu

de Imag(z)

Frequency (Hz)0 100 200 300 400

0

200

400

Am

plitu

de

Imag(z)

0 100 200 300 4000

70

140

Frequency (Hz)

Am

plitu

de

Real(z)

0 100 200 300 4000

200

400

Frequency (Hz)

Am

plitu

de Real(z)

(a) (b)

Figure 24 -e time and frequency domain plots of the bistable behavior signals

00256

0512

minus800

80minus80

0

80

Time (s)Real(c1 )

Imag

(c1)

Time (s)Real(c1 )

Imag

(c1)

00256

0512

minus400

40minus40

0

40

Time (s)Real(c2) 0

02560512

minus5000

500minus500

0

500

Imag

(c2)

Time (s)Real(c2)

Imag

(c2)

00256

0512

minus5000

500minus500

0

500

Time (s)Real(r) 00256

0512

minus800

80minus80

0

80

Imag

(irc

rm

)

Time (s)Real(r)

Imag

(r)

00256

0512

minus2000

200minus200

0

200

(a) (b)

Figure 25 -e decomposition results of the bistable behavior signals based on the improved BEMD method

Shock and Vibration 17

improved BEMD method For the single-component bi-variate signal the HT is used to obtain the correspondinginstantaneous amplitude and frequency characteristics -eproposed method can examine the detailed information of asingle rotation component

Data Availability

-e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

-e authors declare that they have no conflicts of interest

Authorsrsquo Contributions

All the authors contributed to this work Chuanjin Huangconceived and designed the simulation and experiments anddrafted the manuscript Haijun Song performed the simu-lations and experiments and analyzed the data and

0 0256 05120

50

100

150

Time (s)

Freq

uenc

y (H

z)

fx2

fy2

0 0256 05120

200

400

600

Time (s)

Am

plitu

de

ax2ay2

(a)

fx2

fy2

0 0256 05120

90

180

270

360

Time (s)

Freq

uenc

y (H

z)

0 0256 05120

200

400

600

Time (s)

Am

plitu

de

ax2Data 2

(b)

Figure 26 -e instantaneous amplitude and frequency of c2 from the (a) left and (b) right columns

0

0256

0512

0100

200300

400500

0

100

200

300

400

500

Time (s)

X 04035Y 3509Z 130

X 04235Y 2265Z 3613

X 007Y 1131Z 2191

X 0105Y 3837Z 3312

ax2

a y2

Figure 27 -e three-dimensional time domain of ax2 and ay2

18 Shock and Vibration

Wenping Lei and Yajun Meng performed the experimentsand analyzed the data All the authors contributed to thewriting and discussion of the paper

Acknowledgments

-is research was funded by the Henan Provincial HigherEducation Key Research Project (Grant nos 18A460006 and19A460029) Henan High-Level Innovative Scientific andTechnological Talent Team Construction Project (Grant noC20150034) and Zhengzhou Institute of Technology In-novation Team Project (Grant no CXTD2017K1)

References

[1] R Yan R X Gao and X Chen ldquoWavelets for fault diagnosisof rotary machines a review with applicationsrdquo Signal Pro-cessing vol 96 pp 1ndash15 2014

[2] J Cheng D Yu J Tang and Y Yang ldquoApplication of frequencyfamily separation method based upon EMD and local Hilbertenergy spectrum method to gear fault diagnosisrdquo Mechanismand Machine lteory vol 43 no 6 pp 712ndash723 2008

[3] H Liu and M Han ldquoA fault diagnosis method based on localmean decomposition and multi-scale entropy for rollerbearingsrdquoMechanism andMachinelteory vol 75 pp 67ndash782014

[4] Z Zheng W Jiang Z Wang Y Zhu and K Yang ldquoGear faultdiagnosis method based on local mean decomposition andgeneralized morphological fractal dimensionsrdquo Mechanismand Machine lteory vol 91 pp 151ndash167 2015

[5] W Yang R Court P J Tavner and C J Crabtree ldquoBivariateempirical mode decomposition and its contribution to windturbine condition monitoringrdquo Journal of Sound and Vi-bration vol 330 no 15 pp 3766ndash3782 2011

[6] L Qu X Liu G Peyronne and Y Chen ldquo-e holospectrum anewmethod for rotor surveillance and diagnosisrdquoMechanicalSystems amp Signal Processing vol 3 no 3 pp 255ndash267 1989

[7] F Q Wu and G Meng ldquoCompound rub malfunctions featureextraction based on full-spectrum cascade analysis and SVMrdquoMechanical Systems and Signal Processing vol 20 no 8pp 2007ndash2021 2006

[8] Y Chen Q Gao and Z Guan ldquoSelf-loosening failure analysisof bolt joints under vibration considering the tighteningprocessrdquo Shock and Vibration vol 2017 Article ID 203842115 pages 2017

[9] L Chen J Han W Lei Y Cui and Z Guan ldquoFull-vectorsignal acquisition and information fusion for the fault pre-dictionrdquo International Journal of Rotating Machineryvol 2016 Article ID 5980802 7 pages 2016

[10] C Chen Y Meng and Y Du ldquoApplication of the full vectorspectrum based on EMD in fault diagnosis of bearingsrdquoJournal of Mechanical Strength vol 37 pp 806ndash811 2015

[11] C Huang X Wu and W Cao ldquoLMD-based on full vectorenvelope technique and its application in TRT vibration faultdiagnosisrdquo Electric Power Automation Equipment vol 35pp 168ndash174 2015 in Chinese

[12] X Zhao T H Patel and M J Zuo ldquoMultivariate EMD andfull spectrum based condition monitoring for rotating ma-chineryrdquo Mechanical Systems and Signal Processing vol 27pp 712ndash728 2012

[13] G Rilling P Flandrin P Gonalves and J M Lilly ldquoBivariateempirical mode decompositionrdquo IEEE Signal ProcessingLetters vol 14 no 12 pp 936ndash939 2007

[14] C Park D Looney M M Van Hulle and D P Mandic ldquo-ecomplex local mean decompositionrdquo Neurocomputingvol 74 no 6 pp 867ndash875 2011

[15] N Rehman and D P Mandic ldquoEmpirical mode de-composition for trivariate signalsrdquo IEEE Transactions onSignal Processing vol 58 no 3 pp 1059ndash1068 2010

[16] N Rehman and D P Mandic ldquoMultivariate empirical modedecompositionrdquo Proceedings of the Royal Society A Mathe-matical Physical and Engineering Sciences vol 466 no 2117pp 1291ndash1302 2010

[17] Y Lv R Yuan and G Song ldquoMultivariate empirical modedecomposition and its application to fault diagnosis of rollingbearingrdquo Mechanical Systems and Signal Processing vol 81pp 219ndash234 2016

[18] C Huang Y Meng and W Lei ldquoFull vector envelopetechnique based on complex local mean decomposition andits application in fault feature extraction for rotor systemrdquoJournal of Mechanical Engineering vol 52 no 7 p 69 2016in Chinese

[19] G Rilling P Flandrin and P Goncalves ldquoOn empirical modedecomposition and its algorithmsrdquo in Proceedings of IEEE-EURASIP Workshop on Nonlinear Signal and Image Pro-cessing pp 8ndash11 IEEE Trieste Italy June 2003

[20] L Yang X Chen and S Wang ldquoMechanism of fast time-varying vibration for rotorndashstator contact system with ap-plication to fault diagnosisrdquo Journal of Vibration andAcoustics vol 140 no 1 article 014501 2018

[21] I Daubechies J Lu and H-TWu ldquoSynchrosqueezed wavelettransforms an empirical mode decomposition-like toolrdquoApplied and Computational Harmonic Analysis vol 30 no 2pp 243ndash261 2011

[22] L-S Qu Holospectrum and Holobalancing Technique inMachinery Diagnosis Beijing Science Press Beijing China2007

Shock and Vibration 19

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c2 and c3 respectively and fy2 and fy3 represent the in-stantaneous frequency of the imaginary parts of c2 and c3respectively

In Figure 11 ax3 is much larger than ay3 and their phasesare separated by nearly 180deg which shows that the amplitudeand the phase of the oil lm oscillation signal in dierentdirections can vary fx2 and fy2 and fx3 and fy3 are ap-proximately the same ax2 and ay2 show little change andtheir phases are the same Since BEMD is a bivariate ex-tension of EMD like EMD BEMD also has an endpointeect ere are some uctuations in the instantaneousamplitude and instantaneous frequency due to the end eect

and the edge eect of the Hilbert transform e three-dimensional time domain of ax2 and ay2 and of ax3 and ay3 isshown in Figure 12e amplitude range of c3 is much largerthan that of c2 It is can be inferred that the main componentof oscillation is c3 with the frequency of 52Hz in the oil lmoscillation signal

e decomposition results of the oil lm oscillationsignal are shown in Figure 13 using the CLMD methodproposed in reference [14] e oil lm oscillation signal isdecomposed into four complex product functions cpf1ndashcpf4e noise component is not separated from the oil lmoscillation signal using the CLMD method Moreover the

0025

05

ndash50

5

Time (s)Real(c1 )

ndash5

0

5Im

ag(c

1)

(a)

0025

05

ndash1000

100

Time (s)Real(c2)

ndash100

0

100

Imag

(c2)

(b)

0025

05ndash100

0100

ndash100

0

100

Imag

(c3)

Real(c3 ) Time (s)

(c)

0 02505

ndash100

10

Time (s)Real(c4)

ndash20

0

20

Imag

(c4)

(d)

0025

05ndash10

010

Time (s)Real(c5)

ndash10

0

10

Imag

(c5)

(e)

0 02505

ndash50

5

Time (s)Real(c6)

ndash5

0

5

Imag

(c6)

(f )

0 02505

ndash202

Time (s)Real(c7)

ndash2

0

2

Imag

(c7)

(g)

0025

05ndash5

05

Time (s)Real(c8)

ndash5

0

5

Imag

(c8)

(h)

0025

05ndash10

010

Time (s)Real(r)

ndash5

0

5

Imag

(r)

(i)

Figure 9 Decomposition results of the oil lm oscillation signal using the improved BEMD method (N 16 λ 005)

ndash80 0 40 80ndash40Real(c2)

ndash60

ndash30

0

30

60

Imag

(c2)

(a)

ndash80

ndash60

ndash40

ndash20

0

20

40

60

80

Imag

(c3)

ndash40 0 40 80ndash80Real(c3)

(b)

Figure 10 e orbits were made up of c2 (a) and c3 (b) obtained with the improved BEMD method

8 Shock and Vibration

single component fundamental frequency signal and the oillm oscillation signal were not successfully separated Oneof the possible reasons is that in the CLMD algorithm thecomplex signal is only projected onto the x-axis and the y-axis unlike BEMD which projected on multiple directionsIn addition CLMD is a bivariate extension of LMD LMDused a moving average algorithm when tting the signalenvelope which can lter noise to a certain extent esignals other than the noise component c1 in Figure 9 areadded to obtain a ltered oil lm oscillation signal which isthen decomposed by the CLMD method and the rst twodecomposed results are shown in Figure 13 It is seen thatthe single component fundamental frequency signal and

the single component oil lm oscillation signal areseparated

cpf1 and cpf2 consisted of real part signals and imaginarypart signals both of which were composed of the product ofthe envelope signal and the pure frequency modulationfunctione envelope signal is the instantaneous amplitudeof the signal e corresponding instantaneous frequencywas obtained by deriving the inverse function of the cosinepure frequency modulation function e instantaneousamplitude and instantaneous frequency curves are shown inFigure 14 e instantaneous amplitude and frequencyobtained by the CLMD method are smoother than in Fig-ure 12e reason is mainly that the CLMDmethod uses the

ax2ay2

30

40

50

60

70A

mpl

itude

01 02 03 04 050Time (s)

(a)

fx2fy2

70

110

150

Freq

uenc

y (H

z)

01 02 03 04 050Time (s)

(b)

ax3ay3

20

70

120

Am

plitu

de

01 02 03 04 050Time (s)

(c)

fx3fy3

0

50

100

Freq

uenc

y (H

z)01 02 03 04 050

Time (s)

(d)

Figure 11 (a) e instantaneous amplitude of the real part and imaginary part of c2 (b) the instantaneous frequency of the real part andimaginary part of c2 (c) the instantaneous amplitude of the real part and imaginary part of c3 (d) the instantaneous frequency of the real partand imaginary part of c3

0 01 02 03 04 05

2050

800

20

40

60

80

Time (s)ax2

a y2

(a)

3060

90120

0

20

40

60

80

Time (s)ax3

a y3

0 01 02 03 04 05

(b)

Figure 12 e 3D time domain of instantaneous amplitude was made up of (a) ax2 and ay2 and (b) ax3 and ay3

Shock and Vibration 9

moving average ltering algorithm to obtain the signalenvelope curve However this is the result of using theCLMD algorithm after ltering out noise with BEMD If theBEMD algorithm is not used for ltering noise the in-stantaneous amplitude and instantaneous frequency curvesof the single component were not obtained by the CLMDmethod

42 Analysis ofOilWhirl Signal Based on the ImprovedBEMDMethod In the method similar to that presented in Section41 the oil whirl signals of the rotor test rig with a speedparameter of 4320 rpm are collected by two orthogonalsensors as shown in Figure 15 e gure shows the typicalwhirl phenomenon of large circles with embedded smallerones e decomposition results based on the improvedBEMD method (N 16 λ 005) are shown in Figure 16

Only three IMFs appear in Figure 16 and the singlecomponents c2 and c3 and the noise component c1 are suc-cessfully separated from the original signal e HT is appliedto the real and imaginary parts of c2 and c3 respectively andthe instantaneous frequency and instantaneous amplitude areobtained as shown in Figure 17 e three-dimensional timedomain of ax2 and ay2 and of ax3 and ay3 is shown in Figure 18Figure 17 indicates that the frequency of c2 is approximatelytwice the frequency of c3 and that ax2 is larger than ay2 Inaddition ay3 is slightly larger than ax3 but the range of changefor ax3 is greater than the range for ay3 It is inferred that c2 isthe fundamental frequency signal and that c3 is the half-frequency signal in the oil whirl signal

43 Analysis of Looseness and Rotor Rubbing Composite FaultSignal Based on the Improved BEMD Method Loose androtor rubbing composite faults are set on the testequipment shown in Figure 3 in Section 41 Loose fault isset on the nondrive end of the motor and the distancebetween the plastic rod and the shaft is xed near thesensor on the left side of the disk As the rotor speedincreases the vibration increases and the rubbing faultoccurs which is stable at around 1700 rmin e com-posite fault signals are collected by two orthogonal sen-sors as shown in Figure 19 e decomposition resultsbased on the improved BEMD method are shown inFigure 20

Figures 19(c) and 19(d) indicate that the signal com-ponent mainly contain 1X 2X (X 28Hz) and a frequencymodulated signal generated due to time-varying stinessis phenomenon is similar to that described reference [20]Four IMFs appear in Figure 20 and the single components1X 2X signals and the FM signal c2 are successfully separatedfrom the original signal e HT is applied to the real andimaginary parts of c2 c3 and c4 respectively and the in-stantaneous frequency and instantaneous amplitude areobtained as shown in Figure 21 ere are some uctuationsin the frequencies of c2 and c3 but these uctuations aredierent from the random uctuations in the above casesey have obvious regularity and are characteristic of FMsignals e frequency modulation characteristics of c2 aremore obvious than those of c3 In addition by observing theinstantaneous amplitude of c2 it is seen that c2 is still anamplitude modulation signal

minus100

0100

minus100

0

100

Time (s)Real(cpf1)

Imag

(cpf

1)

0 01 02 03 04 05

(a)

Time (s)Real(cpf2) minus100

0100

minus100

0

100

Imag

(cpf

2)

0 01 02 03 04 05

(b)

minus80 0 80minus60

0

60

Real(cpf1)

Imag(cpf1)

(c)

minus100 0 100minus100

0

100

Real(cpf2)

Imag(cpf2)

(d)

Figure 13 e rst two decomposed results of the ltered signal based on the CLMD method

10 Shock and Vibration

0025

05

ndash150

0

150ndash150

0

150

Time (s)Real(z)

Imag

(z)

(a)

ndash150 ndash75 0 75 150ndash150

ndash75

0

75

150

Real(z)

Imag

(z)

(b)

0 72 144 216 2880

40

80

Frequency (Hz)

Am

plitu

de

X 72Y 7321

X 36Y 3433

Real(z)

(c)

Frequency (Hz)0 72 144 216 288

0

40

80

X 36Y 3475

Am

plitu

de

X 72Y 6157

Imag(z)

(d)

Figure 15e oil whirl signal z (a) the 3D time-domain wave of z (b) the 2D plane of z (c) the Fourier spectrum of Real[z] (d) the Fourierspectrum of Imag[z]

30

40

50

60

70

Am

plitu

de

0 01 02 03 04 05Time (s)

ax1ay1

(a)

70

110

150

Freq

uenc

y (H

z)

0 01 02 03 04 05Time (s)

fx1fy1

(b)

0 01 02 03 04 0520

70

120

Time (s)

Am

plitu

de

ax2ay2

(c)

0 01 02 03 04 050

50

100

Time (s)

Freq

uenc

y (H

z)

fx2fy2

(d)

Figure 14 (a) e instantaneous amplitude of the real part and imaginary part of cpf1 (b) the instantaneous frequency of the real part andimaginary part of cpf1 (c) the instantaneous amplitude of the real part and imaginary part of cpf2 (d) the instantaneous frequency of the realpart and imaginary part of cpf2

Shock and Vibration 11

0025

05

ndash5

0

5ndash5

0

5

Time (s)Real(c1)

Imag

(c1)

(a)

0025

05

ndash1000

100ndash100

0

100

Time (s)

Imag

(c2)

Real(c2)

(b)

0025

05

ndash500

50ndash50

0

50

Time (s)

Imag

(c3)

Real(c3)

(c)

0025

05

ndash100

10ndash20

0

20

Time (s)

Imag

(r)

Real(r)

(d)

ndash100 ndash50 0 50 100ndash100

ndash50

0

50

100

Imag

(c2)

Real(c2)

(e)

ndash40 ndash20 0 20 40ndash40

ndash20

0

20

40Im

ag(c

3)

Real(c3)

(f )

Figure 16 e decomposition results of the oil whirl signal based on the improved BEMD method

60

90

120

Am

plitu

de

0 01 02 03 04 05Time (s)

ax2ay2

(a)

50

75

100

Freq

uenc

y (H

z)

0 01 02 03 04 05Time (s)

fx2fy2

(b)

Figure 17 Continued

12 Shock and Vibration

0 01 02 03 04 0520

35

50

Time (s)

Am

plitu

de

ax3ay3

(c)

0 01 02 03 04 0520

35

50

Time (s)

Freq

uenc

y (H

z)

fx3fy3

(d)

Figure 17 e instantaneous amplitude and frequency of c2 and c3 from the oil whirl signal obtained by the HT (a) the instantaneousamplitude of the real part and imaginary part of c2 (b) the instantaneous frequency of the real part and imaginary part of c2(c) the instantaneous amplitude of the real part and imaginary part of c3 (d) the instantaneous frequency of the real part and imaginarypart of c3

0

025

05

40

80

12040

80

120

Time (s)

X 03877Y 9582Z 826

ax2

X 01309Y 9595Z 8452

a y2

(a)

0

025

05

20

35

5020

35

50

Time (s)

X 03955Y 3762Z 3822

ax3

X 008838Y 3608Z 3733

a y3

(b)

Figure 18 e 3D time domain of the instantaneous amplitude of ax2 and ay2 (a) and ax3 and ay3 (b) from the oil whirl signal

0025

05

minus1500

150minus150

0

150

t (s)Real(z)

Imag

(z)

(a)

Imag(z)

Real(z)

minus150

0

150

minus150 0 150

(b)

Figure 19 Continued

Shock and Vibration 13

In order to further verify the correctness of the in-stantaneous amplitude-frequency characteristics of theproposed method the real and imaginary parts of thecomposite fault signal z are analyzed separately using syn-chrosqueezed wavelet transforms (SWT) proposed in ref-erence [21]-e results are shown in Figure 22 It is seen thatthe time-frequency representations of the composite faultsignal z also include the AM-FM signal and the 1X signalwhich proves the correctness of the proposed methodCompared with the SWT method the instantaneousamplitude-frequency characteristics acquired by the HTmethod are relatively straightforward

44 lte Bistable Behavior Analysis of the Fan Rotor Based onBEMD -e bistability of the rotor is a nonlinear behaviorof the rotor-bearing system which is the state in which therotor jumps from one stable state to another forming astep -e bivariate signal of the bistable behavior iscomposed of two signals collected by two displacementsensors from orthogonal locations on the experimentaldevices in literature [22] as shown in Figure 23 Literature[22] shows that the cause of the bistable behavior remainsto be further explored -is paper uses this case to il-lustrate the feasibility of BEMD to analyze nonstationarysignals

0

40

80

Am

plitu

de Real(z)

0 100 200 300 400Frequency (Hz)

(c)

Am

plitu

de

0 100 200 300 400Frequency (Hz)

0

20

40

60

80

Imag(z)

(d)

Figure 19 -e composite fault signal z (a) the 3D time domain wave of z (b) the 2D plane of z (c) the Fourier spectrum of Real[z] (d) theFourier spectrum of Imag[z]

0025

05

minus30

3minus5

0

5

Time (s)Real(c1)

Imag

(c1)

(a)

0025

05

minus200

20minus20

0

20

Time (s)Real(c2 )

Imag

(c2)

(b)

0025

05

minus200

20minus50

0

50

Time (s)Real(c2)

Imag

(c2)

(c)

0025

05

minus1000

100minus100

0

100

Time (s)Real(c3)

Imag

(c3)

(d)

0025

05

minus400

40minus30

0

30

Time (s)Real(r)

Imag

(r)

(e)

Figure 20 -e decomposition results of the composite fault signal based on the improved BEMD method

14 Shock and Vibration

-e x and y signals in the horizontal and vertical di-rections of the left and right bearings respectively from thefan rotors are collected with four displacement sensorsLetting z x+ jy the time and frequency domain plots of z areshown in Figure 24 where the fan rotor speed is 5500 rpm thesampling frequency is 2000Hz and the number of samplingpoints is 1024 -e left and right columns respectively showthe time and frequency domain plots of the vibration signalsfrom the left and right bearings of the fan rotor Bistablebehavior arises in the fan rotor and the amplitudes of the

vibration signals vary significantly in different positions anddirections Further studies are required to explain the causesof this bistability -e present study focuses on extracting thebistable behavioral signal characteristics to verify the feasi-bility of the proposed method

-e decomposition results of the bistable behavioralsignals based on the improved BEMDmethod are shown inFigure 25 c1 c2 c3 and r are separated in order from zusing the improved BEMD method c1 shows a randomarrangement and is considered the high-frequency noise

0

10

20

Am

plitu

de

ax2ay2

0 025 05Time (s)

(a)

0

152

304

Freq

uenc

y (H

z)

0 025 05Time (s)

fy2

fx2

(b)

0

10

20

30

40

Am

plitu

de

ax3ay3

0 025 05Time (s)

(c)

0

56

112

Freq

uenc

y (H

z)

fx3fy3

0 025 05Time (s)

(d)

0 025 050

50

100

Time (s)

Am

plitu

de

ax4ay4

(e)

0 025 050

28

56

Time (s)

Freq

uenc

y (H

z)

fx4fy4

(f )

Figure 21 -e instantaneous amplitude and frequency of c2 c3 and c4 obtained by the HT (a) the instantaneous amplitude of the real partand imaginary part of c2 (b) the instantaneous frequency of the real part and imaginary part of c2 (c) the instantaneous amplitude of the realpart and imaginary part of c3 (d) the instantaneous frequency of the real part and imaginary part of c3 (e) the instantaneous amplitude of thereal part and imaginary part of c4 (f ) the instantaneous frequency of the real part and imaginary part of c4

Shock and Vibration 15

signal c2 is considered to represent the extracted bistablebehavior signals -e HT is applied to the real andimaginary parts of c2 to obtain the instantaneous amplitudeand frequency of c2 from the left and right columns fromFigure 25 as shown in Figure 26 Figure 27 shows thethree-dimensional time domain of ax2 and ay2 from the leftand right columns respectively Figure 26 shows that thevibration signal amplitude on the left side of the fan de-creases from large to small opposite of the behavior ofthe right -e horizontal vibration signal amplitude on theleft side of the fan is larger than that of the vertical di-rection signal opposite of the right -is result validatesthat the vibration signals from different directions orpositions are different when the fan produces bistablebehavior In addition the time of the bistable behavior canbe determined according to the jump point of the am-plitude or frequency

5 Discussion

-e BEMD algorithm decomposes two orthogonal di-rections of vibration signals as a complex signal which is atwo-dimensional digital signal processing method thusensuring that the real and imaginary parts have the samedecomposition scale Similar to EMD the envelope mean iscritical for the decomposition effect of BEMD but the en-velope mean in BEMD is three-dimensional If the numberof projection directions of the complex signal in three-dimensional space is larger the corresponding envelope

signal is also more -us the envelope mean value is moreaccurate and the BEMD decomposition effect is betterIncreasing the number of projection directions can improvemodal aliasing Like EMD BEMD also produces falsecomponents when decomposing signals Generally speakingthe energy of the false components is low and these low-energy false components do not contain fault characteristicinformation and the introduction of the energy thresholdcriterion in the termination condition can increase thedecomposition speed of the BEMD

-e experimental results show that there is a certaindifference in the existence of vibration signal character-istics in different directions when rotating machinery failsIn addition when the number of projection directionsis increased the decomposition speed of BEMD willdecrease

6 Conclusions

We use BEMD and HT to extract the instantaneousamplitude-frequency features of rotor faults A bivariateinstantaneous feature extraction method based on the im-proved BEMD method and the HT is investigated whichextends the fault feature extraction technology to two di-mensions -e BEMD method is suitable to analyze thecomplex multicomponent bivariate signals -e mainsingle-component bivariate signals are separated from themulticomponent bivariate signals of the fan rotor bistabilityfor the oil film oscillation and the oil film vortex using the

0

2

4

6

Time (s)

Freq

uenc

y (H

z)

0 01 02 03 04 052

3

4

5

6

7

8log2 fx

(a)

0

2

4

6

Time (s)

Freq

uenc

y (H

z)

0 01 02 03 04 052

3

4

5

6

7

8log2 fy

(b)

Figure 22 -e results of the composite fault signal z based on SWT the time-frequency representation of (a) the real part of z and (b) theimaginary part of z

Left bearing predstal

Locations of sensors

Fanrotor

Right bearing predstal

Axis

Orthogonal directions

Figure 23 -e schematic diagram of the experimental apparatus

16 Shock and Vibration

Real(z) 00256

0512minus300

0300

minus400

0

400

Time (s)

Imag

(z)

00256

0512minus500

0500

minus500

0

500

Time (s)Real(z)

Imag

(z)

Imag

(z)

minus300 0 300minus500

0

500

Real(z)minus500 0 500

minus500

0

Imag

(z)

Real(z)

500

0 100 200 300 4000

100

200

300

Frequency (Hz)

Am

plitu

de Imag(z)

Frequency (Hz)0 100 200 300 400

0

200

400

Am

plitu

de

Imag(z)

0 100 200 300 4000

70

140

Frequency (Hz)

Am

plitu

de

Real(z)

0 100 200 300 4000

200

400

Frequency (Hz)

Am

plitu

de Real(z)

(a) (b)

Figure 24 -e time and frequency domain plots of the bistable behavior signals

00256

0512

minus800

80minus80

0

80

Time (s)Real(c1 )

Imag

(c1)

Time (s)Real(c1 )

Imag

(c1)

00256

0512

minus400

40minus40

0

40

Time (s)Real(c2) 0

02560512

minus5000

500minus500

0

500

Imag

(c2)

Time (s)Real(c2)

Imag

(c2)

00256

0512

minus5000

500minus500

0

500

Time (s)Real(r) 00256

0512

minus800

80minus80

0

80

Imag

(irc

rm

)

Time (s)Real(r)

Imag

(r)

00256

0512

minus2000

200minus200

0

200

(a) (b)

Figure 25 -e decomposition results of the bistable behavior signals based on the improved BEMD method

Shock and Vibration 17

improved BEMD method For the single-component bi-variate signal the HT is used to obtain the correspondinginstantaneous amplitude and frequency characteristics -eproposed method can examine the detailed information of asingle rotation component

Data Availability

-e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

-e authors declare that they have no conflicts of interest

Authorsrsquo Contributions

All the authors contributed to this work Chuanjin Huangconceived and designed the simulation and experiments anddrafted the manuscript Haijun Song performed the simu-lations and experiments and analyzed the data and

0 0256 05120

50

100

150

Time (s)

Freq

uenc

y (H

z)

fx2

fy2

0 0256 05120

200

400

600

Time (s)

Am

plitu

de

ax2ay2

(a)

fx2

fy2

0 0256 05120

90

180

270

360

Time (s)

Freq

uenc

y (H

z)

0 0256 05120

200

400

600

Time (s)

Am

plitu

de

ax2Data 2

(b)

Figure 26 -e instantaneous amplitude and frequency of c2 from the (a) left and (b) right columns

0

0256

0512

0100

200300

400500

0

100

200

300

400

500

Time (s)

X 04035Y 3509Z 130

X 04235Y 2265Z 3613

X 007Y 1131Z 2191

X 0105Y 3837Z 3312

ax2

a y2

Figure 27 -e three-dimensional time domain of ax2 and ay2

18 Shock and Vibration

Wenping Lei and Yajun Meng performed the experimentsand analyzed the data All the authors contributed to thewriting and discussion of the paper

Acknowledgments

-is research was funded by the Henan Provincial HigherEducation Key Research Project (Grant nos 18A460006 and19A460029) Henan High-Level Innovative Scientific andTechnological Talent Team Construction Project (Grant noC20150034) and Zhengzhou Institute of Technology In-novation Team Project (Grant no CXTD2017K1)

References

[1] R Yan R X Gao and X Chen ldquoWavelets for fault diagnosisof rotary machines a review with applicationsrdquo Signal Pro-cessing vol 96 pp 1ndash15 2014

[2] J Cheng D Yu J Tang and Y Yang ldquoApplication of frequencyfamily separation method based upon EMD and local Hilbertenergy spectrum method to gear fault diagnosisrdquo Mechanismand Machine lteory vol 43 no 6 pp 712ndash723 2008

[3] H Liu and M Han ldquoA fault diagnosis method based on localmean decomposition and multi-scale entropy for rollerbearingsrdquoMechanism andMachinelteory vol 75 pp 67ndash782014

[4] Z Zheng W Jiang Z Wang Y Zhu and K Yang ldquoGear faultdiagnosis method based on local mean decomposition andgeneralized morphological fractal dimensionsrdquo Mechanismand Machine lteory vol 91 pp 151ndash167 2015

[5] W Yang R Court P J Tavner and C J Crabtree ldquoBivariateempirical mode decomposition and its contribution to windturbine condition monitoringrdquo Journal of Sound and Vi-bration vol 330 no 15 pp 3766ndash3782 2011

[6] L Qu X Liu G Peyronne and Y Chen ldquo-e holospectrum anewmethod for rotor surveillance and diagnosisrdquoMechanicalSystems amp Signal Processing vol 3 no 3 pp 255ndash267 1989

[7] F Q Wu and G Meng ldquoCompound rub malfunctions featureextraction based on full-spectrum cascade analysis and SVMrdquoMechanical Systems and Signal Processing vol 20 no 8pp 2007ndash2021 2006

[8] Y Chen Q Gao and Z Guan ldquoSelf-loosening failure analysisof bolt joints under vibration considering the tighteningprocessrdquo Shock and Vibration vol 2017 Article ID 203842115 pages 2017

[9] L Chen J Han W Lei Y Cui and Z Guan ldquoFull-vectorsignal acquisition and information fusion for the fault pre-dictionrdquo International Journal of Rotating Machineryvol 2016 Article ID 5980802 7 pages 2016

[10] C Chen Y Meng and Y Du ldquoApplication of the full vectorspectrum based on EMD in fault diagnosis of bearingsrdquoJournal of Mechanical Strength vol 37 pp 806ndash811 2015

[11] C Huang X Wu and W Cao ldquoLMD-based on full vectorenvelope technique and its application in TRT vibration faultdiagnosisrdquo Electric Power Automation Equipment vol 35pp 168ndash174 2015 in Chinese

[12] X Zhao T H Patel and M J Zuo ldquoMultivariate EMD andfull spectrum based condition monitoring for rotating ma-chineryrdquo Mechanical Systems and Signal Processing vol 27pp 712ndash728 2012

[13] G Rilling P Flandrin P Gonalves and J M Lilly ldquoBivariateempirical mode decompositionrdquo IEEE Signal ProcessingLetters vol 14 no 12 pp 936ndash939 2007

[14] C Park D Looney M M Van Hulle and D P Mandic ldquo-ecomplex local mean decompositionrdquo Neurocomputingvol 74 no 6 pp 867ndash875 2011

[15] N Rehman and D P Mandic ldquoEmpirical mode de-composition for trivariate signalsrdquo IEEE Transactions onSignal Processing vol 58 no 3 pp 1059ndash1068 2010

[16] N Rehman and D P Mandic ldquoMultivariate empirical modedecompositionrdquo Proceedings of the Royal Society A Mathe-matical Physical and Engineering Sciences vol 466 no 2117pp 1291ndash1302 2010

[17] Y Lv R Yuan and G Song ldquoMultivariate empirical modedecomposition and its application to fault diagnosis of rollingbearingrdquo Mechanical Systems and Signal Processing vol 81pp 219ndash234 2016

[18] C Huang Y Meng and W Lei ldquoFull vector envelopetechnique based on complex local mean decomposition andits application in fault feature extraction for rotor systemrdquoJournal of Mechanical Engineering vol 52 no 7 p 69 2016in Chinese

[19] G Rilling P Flandrin and P Goncalves ldquoOn empirical modedecomposition and its algorithmsrdquo in Proceedings of IEEE-EURASIP Workshop on Nonlinear Signal and Image Pro-cessing pp 8ndash11 IEEE Trieste Italy June 2003

[20] L Yang X Chen and S Wang ldquoMechanism of fast time-varying vibration for rotorndashstator contact system with ap-plication to fault diagnosisrdquo Journal of Vibration andAcoustics vol 140 no 1 article 014501 2018

[21] I Daubechies J Lu and H-TWu ldquoSynchrosqueezed wavelettransforms an empirical mode decomposition-like toolrdquoApplied and Computational Harmonic Analysis vol 30 no 2pp 243ndash261 2011

[22] L-S Qu Holospectrum and Holobalancing Technique inMachinery Diagnosis Beijing Science Press Beijing China2007

Shock and Vibration 19

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single component fundamental frequency signal and the oillm oscillation signal were not successfully separated Oneof the possible reasons is that in the CLMD algorithm thecomplex signal is only projected onto the x-axis and the y-axis unlike BEMD which projected on multiple directionsIn addition CLMD is a bivariate extension of LMD LMDused a moving average algorithm when tting the signalenvelope which can lter noise to a certain extent esignals other than the noise component c1 in Figure 9 areadded to obtain a ltered oil lm oscillation signal which isthen decomposed by the CLMD method and the rst twodecomposed results are shown in Figure 13 It is seen thatthe single component fundamental frequency signal and

the single component oil lm oscillation signal areseparated

cpf1 and cpf2 consisted of real part signals and imaginarypart signals both of which were composed of the product ofthe envelope signal and the pure frequency modulationfunctione envelope signal is the instantaneous amplitudeof the signal e corresponding instantaneous frequencywas obtained by deriving the inverse function of the cosinepure frequency modulation function e instantaneousamplitude and instantaneous frequency curves are shown inFigure 14 e instantaneous amplitude and frequencyobtained by the CLMD method are smoother than in Fig-ure 12e reason is mainly that the CLMDmethod uses the

ax2ay2

30

40

50

60

70A

mpl

itude

01 02 03 04 050Time (s)

(a)

fx2fy2

70

110

150

Freq

uenc

y (H

z)

01 02 03 04 050Time (s)

(b)

ax3ay3

20

70

120

Am

plitu

de

01 02 03 04 050Time (s)

(c)

fx3fy3

0

50

100

Freq

uenc

y (H

z)01 02 03 04 050

Time (s)

(d)

Figure 11 (a) e instantaneous amplitude of the real part and imaginary part of c2 (b) the instantaneous frequency of the real part andimaginary part of c2 (c) the instantaneous amplitude of the real part and imaginary part of c3 (d) the instantaneous frequency of the real partand imaginary part of c3

0 01 02 03 04 05

2050

800

20

40

60

80

Time (s)ax2

a y2

(a)

3060

90120

0

20

40

60

80

Time (s)ax3

a y3

0 01 02 03 04 05

(b)

Figure 12 e 3D time domain of instantaneous amplitude was made up of (a) ax2 and ay2 and (b) ax3 and ay3

Shock and Vibration 9

moving average ltering algorithm to obtain the signalenvelope curve However this is the result of using theCLMD algorithm after ltering out noise with BEMD If theBEMD algorithm is not used for ltering noise the in-stantaneous amplitude and instantaneous frequency curvesof the single component were not obtained by the CLMDmethod

42 Analysis ofOilWhirl Signal Based on the ImprovedBEMDMethod In the method similar to that presented in Section41 the oil whirl signals of the rotor test rig with a speedparameter of 4320 rpm are collected by two orthogonalsensors as shown in Figure 15 e gure shows the typicalwhirl phenomenon of large circles with embedded smallerones e decomposition results based on the improvedBEMD method (N 16 λ 005) are shown in Figure 16

Only three IMFs appear in Figure 16 and the singlecomponents c2 and c3 and the noise component c1 are suc-cessfully separated from the original signal e HT is appliedto the real and imaginary parts of c2 and c3 respectively andthe instantaneous frequency and instantaneous amplitude areobtained as shown in Figure 17 e three-dimensional timedomain of ax2 and ay2 and of ax3 and ay3 is shown in Figure 18Figure 17 indicates that the frequency of c2 is approximatelytwice the frequency of c3 and that ax2 is larger than ay2 Inaddition ay3 is slightly larger than ax3 but the range of changefor ax3 is greater than the range for ay3 It is inferred that c2 isthe fundamental frequency signal and that c3 is the half-frequency signal in the oil whirl signal

43 Analysis of Looseness and Rotor Rubbing Composite FaultSignal Based on the Improved BEMD Method Loose androtor rubbing composite faults are set on the testequipment shown in Figure 3 in Section 41 Loose fault isset on the nondrive end of the motor and the distancebetween the plastic rod and the shaft is xed near thesensor on the left side of the disk As the rotor speedincreases the vibration increases and the rubbing faultoccurs which is stable at around 1700 rmin e com-posite fault signals are collected by two orthogonal sen-sors as shown in Figure 19 e decomposition resultsbased on the improved BEMD method are shown inFigure 20

Figures 19(c) and 19(d) indicate that the signal com-ponent mainly contain 1X 2X (X 28Hz) and a frequencymodulated signal generated due to time-varying stinessis phenomenon is similar to that described reference [20]Four IMFs appear in Figure 20 and the single components1X 2X signals and the FM signal c2 are successfully separatedfrom the original signal e HT is applied to the real andimaginary parts of c2 c3 and c4 respectively and the in-stantaneous frequency and instantaneous amplitude areobtained as shown in Figure 21 ere are some uctuationsin the frequencies of c2 and c3 but these uctuations aredierent from the random uctuations in the above casesey have obvious regularity and are characteristic of FMsignals e frequency modulation characteristics of c2 aremore obvious than those of c3 In addition by observing theinstantaneous amplitude of c2 it is seen that c2 is still anamplitude modulation signal

minus100

0100

minus100

0

100

Time (s)Real(cpf1)

Imag

(cpf

1)

0 01 02 03 04 05

(a)

Time (s)Real(cpf2) minus100

0100

minus100

0

100

Imag

(cpf

2)

0 01 02 03 04 05

(b)

minus80 0 80minus60

0

60

Real(cpf1)

Imag(cpf1)

(c)

minus100 0 100minus100

0

100

Real(cpf2)

Imag(cpf2)

(d)

Figure 13 e rst two decomposed results of the ltered signal based on the CLMD method

10 Shock and Vibration

0025

05

ndash150

0

150ndash150

0

150

Time (s)Real(z)

Imag

(z)

(a)

ndash150 ndash75 0 75 150ndash150

ndash75

0

75

150

Real(z)

Imag

(z)

(b)

0 72 144 216 2880

40

80

Frequency (Hz)

Am

plitu

de

X 72Y 7321

X 36Y 3433

Real(z)

(c)

Frequency (Hz)0 72 144 216 288

0

40

80

X 36Y 3475

Am

plitu

de

X 72Y 6157

Imag(z)

(d)

Figure 15e oil whirl signal z (a) the 3D time-domain wave of z (b) the 2D plane of z (c) the Fourier spectrum of Real[z] (d) the Fourierspectrum of Imag[z]

30

40

50

60

70

Am

plitu

de

0 01 02 03 04 05Time (s)

ax1ay1

(a)

70

110

150

Freq

uenc

y (H

z)

0 01 02 03 04 05Time (s)

fx1fy1

(b)

0 01 02 03 04 0520

70

120

Time (s)

Am

plitu

de

ax2ay2

(c)

0 01 02 03 04 050

50

100

Time (s)

Freq

uenc

y (H

z)

fx2fy2

(d)

Figure 14 (a) e instantaneous amplitude of the real part and imaginary part of cpf1 (b) the instantaneous frequency of the real part andimaginary part of cpf1 (c) the instantaneous amplitude of the real part and imaginary part of cpf2 (d) the instantaneous frequency of the realpart and imaginary part of cpf2

Shock and Vibration 11

0025

05

ndash5

0

5ndash5

0

5

Time (s)Real(c1)

Imag

(c1)

(a)

0025

05

ndash1000

100ndash100

0

100

Time (s)

Imag

(c2)

Real(c2)

(b)

0025

05

ndash500

50ndash50

0

50

Time (s)

Imag

(c3)

Real(c3)

(c)

0025

05

ndash100

10ndash20

0

20

Time (s)

Imag

(r)

Real(r)

(d)

ndash100 ndash50 0 50 100ndash100

ndash50

0

50

100

Imag

(c2)

Real(c2)

(e)

ndash40 ndash20 0 20 40ndash40

ndash20

0

20

40Im

ag(c

3)

Real(c3)

(f )

Figure 16 e decomposition results of the oil whirl signal based on the improved BEMD method

60

90

120

Am

plitu

de

0 01 02 03 04 05Time (s)

ax2ay2

(a)

50

75

100

Freq

uenc

y (H

z)

0 01 02 03 04 05Time (s)

fx2fy2

(b)

Figure 17 Continued

12 Shock and Vibration

0 01 02 03 04 0520

35

50

Time (s)

Am

plitu

de

ax3ay3

(c)

0 01 02 03 04 0520

35

50

Time (s)

Freq

uenc

y (H

z)

fx3fy3

(d)

Figure 17 e instantaneous amplitude and frequency of c2 and c3 from the oil whirl signal obtained by the HT (a) the instantaneousamplitude of the real part and imaginary part of c2 (b) the instantaneous frequency of the real part and imaginary part of c2(c) the instantaneous amplitude of the real part and imaginary part of c3 (d) the instantaneous frequency of the real part and imaginarypart of c3

0

025

05

40

80

12040

80

120

Time (s)

X 03877Y 9582Z 826

ax2

X 01309Y 9595Z 8452

a y2

(a)

0

025

05

20

35

5020

35

50

Time (s)

X 03955Y 3762Z 3822

ax3

X 008838Y 3608Z 3733

a y3

(b)

Figure 18 e 3D time domain of the instantaneous amplitude of ax2 and ay2 (a) and ax3 and ay3 (b) from the oil whirl signal

0025

05

minus1500

150minus150

0

150

t (s)Real(z)

Imag

(z)

(a)

Imag(z)

Real(z)

minus150

0

150

minus150 0 150

(b)

Figure 19 Continued

Shock and Vibration 13

In order to further verify the correctness of the in-stantaneous amplitude-frequency characteristics of theproposed method the real and imaginary parts of thecomposite fault signal z are analyzed separately using syn-chrosqueezed wavelet transforms (SWT) proposed in ref-erence [21]-e results are shown in Figure 22 It is seen thatthe time-frequency representations of the composite faultsignal z also include the AM-FM signal and the 1X signalwhich proves the correctness of the proposed methodCompared with the SWT method the instantaneousamplitude-frequency characteristics acquired by the HTmethod are relatively straightforward

44 lte Bistable Behavior Analysis of the Fan Rotor Based onBEMD -e bistability of the rotor is a nonlinear behaviorof the rotor-bearing system which is the state in which therotor jumps from one stable state to another forming astep -e bivariate signal of the bistable behavior iscomposed of two signals collected by two displacementsensors from orthogonal locations on the experimentaldevices in literature [22] as shown in Figure 23 Literature[22] shows that the cause of the bistable behavior remainsto be further explored -is paper uses this case to il-lustrate the feasibility of BEMD to analyze nonstationarysignals

0

40

80

Am

plitu

de Real(z)

0 100 200 300 400Frequency (Hz)

(c)

Am

plitu

de

0 100 200 300 400Frequency (Hz)

0

20

40

60

80

Imag(z)

(d)

Figure 19 -e composite fault signal z (a) the 3D time domain wave of z (b) the 2D plane of z (c) the Fourier spectrum of Real[z] (d) theFourier spectrum of Imag[z]

0025

05

minus30

3minus5

0

5

Time (s)Real(c1)

Imag

(c1)

(a)

0025

05

minus200

20minus20

0

20

Time (s)Real(c2 )

Imag

(c2)

(b)

0025

05

minus200

20minus50

0

50

Time (s)Real(c2)

Imag

(c2)

(c)

0025

05

minus1000

100minus100

0

100

Time (s)Real(c3)

Imag

(c3)

(d)

0025

05

minus400

40minus30

0

30

Time (s)Real(r)

Imag

(r)

(e)

Figure 20 -e decomposition results of the composite fault signal based on the improved BEMD method

14 Shock and Vibration

-e x and y signals in the horizontal and vertical di-rections of the left and right bearings respectively from thefan rotors are collected with four displacement sensorsLetting z x+ jy the time and frequency domain plots of z areshown in Figure 24 where the fan rotor speed is 5500 rpm thesampling frequency is 2000Hz and the number of samplingpoints is 1024 -e left and right columns respectively showthe time and frequency domain plots of the vibration signalsfrom the left and right bearings of the fan rotor Bistablebehavior arises in the fan rotor and the amplitudes of the

vibration signals vary significantly in different positions anddirections Further studies are required to explain the causesof this bistability -e present study focuses on extracting thebistable behavioral signal characteristics to verify the feasi-bility of the proposed method

-e decomposition results of the bistable behavioralsignals based on the improved BEMDmethod are shown inFigure 25 c1 c2 c3 and r are separated in order from zusing the improved BEMD method c1 shows a randomarrangement and is considered the high-frequency noise

0

10

20

Am

plitu

de

ax2ay2

0 025 05Time (s)

(a)

0

152

304

Freq

uenc

y (H

z)

0 025 05Time (s)

fy2

fx2

(b)

0

10

20

30

40

Am

plitu

de

ax3ay3

0 025 05Time (s)

(c)

0

56

112

Freq

uenc

y (H

z)

fx3fy3

0 025 05Time (s)

(d)

0 025 050

50

100

Time (s)

Am

plitu

de

ax4ay4

(e)

0 025 050

28

56

Time (s)

Freq

uenc

y (H

z)

fx4fy4

(f )

Figure 21 -e instantaneous amplitude and frequency of c2 c3 and c4 obtained by the HT (a) the instantaneous amplitude of the real partand imaginary part of c2 (b) the instantaneous frequency of the real part and imaginary part of c2 (c) the instantaneous amplitude of the realpart and imaginary part of c3 (d) the instantaneous frequency of the real part and imaginary part of c3 (e) the instantaneous amplitude of thereal part and imaginary part of c4 (f ) the instantaneous frequency of the real part and imaginary part of c4

Shock and Vibration 15

signal c2 is considered to represent the extracted bistablebehavior signals -e HT is applied to the real andimaginary parts of c2 to obtain the instantaneous amplitudeand frequency of c2 from the left and right columns fromFigure 25 as shown in Figure 26 Figure 27 shows thethree-dimensional time domain of ax2 and ay2 from the leftand right columns respectively Figure 26 shows that thevibration signal amplitude on the left side of the fan de-creases from large to small opposite of the behavior ofthe right -e horizontal vibration signal amplitude on theleft side of the fan is larger than that of the vertical di-rection signal opposite of the right -is result validatesthat the vibration signals from different directions orpositions are different when the fan produces bistablebehavior In addition the time of the bistable behavior canbe determined according to the jump point of the am-plitude or frequency

5 Discussion

-e BEMD algorithm decomposes two orthogonal di-rections of vibration signals as a complex signal which is atwo-dimensional digital signal processing method thusensuring that the real and imaginary parts have the samedecomposition scale Similar to EMD the envelope mean iscritical for the decomposition effect of BEMD but the en-velope mean in BEMD is three-dimensional If the numberof projection directions of the complex signal in three-dimensional space is larger the corresponding envelope

signal is also more -us the envelope mean value is moreaccurate and the BEMD decomposition effect is betterIncreasing the number of projection directions can improvemodal aliasing Like EMD BEMD also produces falsecomponents when decomposing signals Generally speakingthe energy of the false components is low and these low-energy false components do not contain fault characteristicinformation and the introduction of the energy thresholdcriterion in the termination condition can increase thedecomposition speed of the BEMD

-e experimental results show that there is a certaindifference in the existence of vibration signal character-istics in different directions when rotating machinery failsIn addition when the number of projection directionsis increased the decomposition speed of BEMD willdecrease

6 Conclusions

We use BEMD and HT to extract the instantaneousamplitude-frequency features of rotor faults A bivariateinstantaneous feature extraction method based on the im-proved BEMD method and the HT is investigated whichextends the fault feature extraction technology to two di-mensions -e BEMD method is suitable to analyze thecomplex multicomponent bivariate signals -e mainsingle-component bivariate signals are separated from themulticomponent bivariate signals of the fan rotor bistabilityfor the oil film oscillation and the oil film vortex using the

0

2

4

6

Time (s)

Freq

uenc

y (H

z)

0 01 02 03 04 052

3

4

5

6

7

8log2 fx

(a)

0

2

4

6

Time (s)

Freq

uenc

y (H

z)

0 01 02 03 04 052

3

4

5

6

7

8log2 fy

(b)

Figure 22 -e results of the composite fault signal z based on SWT the time-frequency representation of (a) the real part of z and (b) theimaginary part of z

Left bearing predstal

Locations of sensors

Fanrotor

Right bearing predstal

Axis

Orthogonal directions

Figure 23 -e schematic diagram of the experimental apparatus

16 Shock and Vibration

Real(z) 00256

0512minus300

0300

minus400

0

400

Time (s)

Imag

(z)

00256

0512minus500

0500

minus500

0

500

Time (s)Real(z)

Imag

(z)

Imag

(z)

minus300 0 300minus500

0

500

Real(z)minus500 0 500

minus500

0

Imag

(z)

Real(z)

500

0 100 200 300 4000

100

200

300

Frequency (Hz)

Am

plitu

de Imag(z)

Frequency (Hz)0 100 200 300 400

0

200

400

Am

plitu

de

Imag(z)

0 100 200 300 4000

70

140

Frequency (Hz)

Am

plitu

de

Real(z)

0 100 200 300 4000

200

400

Frequency (Hz)

Am

plitu

de Real(z)

(a) (b)

Figure 24 -e time and frequency domain plots of the bistable behavior signals

00256

0512

minus800

80minus80

0

80

Time (s)Real(c1 )

Imag

(c1)

Time (s)Real(c1 )

Imag

(c1)

00256

0512

minus400

40minus40

0

40

Time (s)Real(c2) 0

02560512

minus5000

500minus500

0

500

Imag

(c2)

Time (s)Real(c2)

Imag

(c2)

00256

0512

minus5000

500minus500

0

500

Time (s)Real(r) 00256

0512

minus800

80minus80

0

80

Imag

(irc

rm

)

Time (s)Real(r)

Imag

(r)

00256

0512

minus2000

200minus200

0

200

(a) (b)

Figure 25 -e decomposition results of the bistable behavior signals based on the improved BEMD method

Shock and Vibration 17

improved BEMD method For the single-component bi-variate signal the HT is used to obtain the correspondinginstantaneous amplitude and frequency characteristics -eproposed method can examine the detailed information of asingle rotation component

Data Availability

-e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

-e authors declare that they have no conflicts of interest

Authorsrsquo Contributions

All the authors contributed to this work Chuanjin Huangconceived and designed the simulation and experiments anddrafted the manuscript Haijun Song performed the simu-lations and experiments and analyzed the data and

0 0256 05120

50

100

150

Time (s)

Freq

uenc

y (H

z)

fx2

fy2

0 0256 05120

200

400

600

Time (s)

Am

plitu

de

ax2ay2

(a)

fx2

fy2

0 0256 05120

90

180

270

360

Time (s)

Freq

uenc

y (H

z)

0 0256 05120

200

400

600

Time (s)

Am

plitu

de

ax2Data 2

(b)

Figure 26 -e instantaneous amplitude and frequency of c2 from the (a) left and (b) right columns

0

0256

0512

0100

200300

400500

0

100

200

300

400

500

Time (s)

X 04035Y 3509Z 130

X 04235Y 2265Z 3613

X 007Y 1131Z 2191

X 0105Y 3837Z 3312

ax2

a y2

Figure 27 -e three-dimensional time domain of ax2 and ay2

18 Shock and Vibration

Wenping Lei and Yajun Meng performed the experimentsand analyzed the data All the authors contributed to thewriting and discussion of the paper

Acknowledgments

-is research was funded by the Henan Provincial HigherEducation Key Research Project (Grant nos 18A460006 and19A460029) Henan High-Level Innovative Scientific andTechnological Talent Team Construction Project (Grant noC20150034) and Zhengzhou Institute of Technology In-novation Team Project (Grant no CXTD2017K1)

References

[1] R Yan R X Gao and X Chen ldquoWavelets for fault diagnosisof rotary machines a review with applicationsrdquo Signal Pro-cessing vol 96 pp 1ndash15 2014

[2] J Cheng D Yu J Tang and Y Yang ldquoApplication of frequencyfamily separation method based upon EMD and local Hilbertenergy spectrum method to gear fault diagnosisrdquo Mechanismand Machine lteory vol 43 no 6 pp 712ndash723 2008

[3] H Liu and M Han ldquoA fault diagnosis method based on localmean decomposition and multi-scale entropy for rollerbearingsrdquoMechanism andMachinelteory vol 75 pp 67ndash782014

[4] Z Zheng W Jiang Z Wang Y Zhu and K Yang ldquoGear faultdiagnosis method based on local mean decomposition andgeneralized morphological fractal dimensionsrdquo Mechanismand Machine lteory vol 91 pp 151ndash167 2015

[5] W Yang R Court P J Tavner and C J Crabtree ldquoBivariateempirical mode decomposition and its contribution to windturbine condition monitoringrdquo Journal of Sound and Vi-bration vol 330 no 15 pp 3766ndash3782 2011

[6] L Qu X Liu G Peyronne and Y Chen ldquo-e holospectrum anewmethod for rotor surveillance and diagnosisrdquoMechanicalSystems amp Signal Processing vol 3 no 3 pp 255ndash267 1989

[7] F Q Wu and G Meng ldquoCompound rub malfunctions featureextraction based on full-spectrum cascade analysis and SVMrdquoMechanical Systems and Signal Processing vol 20 no 8pp 2007ndash2021 2006

[8] Y Chen Q Gao and Z Guan ldquoSelf-loosening failure analysisof bolt joints under vibration considering the tighteningprocessrdquo Shock and Vibration vol 2017 Article ID 203842115 pages 2017

[9] L Chen J Han W Lei Y Cui and Z Guan ldquoFull-vectorsignal acquisition and information fusion for the fault pre-dictionrdquo International Journal of Rotating Machineryvol 2016 Article ID 5980802 7 pages 2016

[10] C Chen Y Meng and Y Du ldquoApplication of the full vectorspectrum based on EMD in fault diagnosis of bearingsrdquoJournal of Mechanical Strength vol 37 pp 806ndash811 2015

[11] C Huang X Wu and W Cao ldquoLMD-based on full vectorenvelope technique and its application in TRT vibration faultdiagnosisrdquo Electric Power Automation Equipment vol 35pp 168ndash174 2015 in Chinese

[12] X Zhao T H Patel and M J Zuo ldquoMultivariate EMD andfull spectrum based condition monitoring for rotating ma-chineryrdquo Mechanical Systems and Signal Processing vol 27pp 712ndash728 2012

[13] G Rilling P Flandrin P Gonalves and J M Lilly ldquoBivariateempirical mode decompositionrdquo IEEE Signal ProcessingLetters vol 14 no 12 pp 936ndash939 2007

[14] C Park D Looney M M Van Hulle and D P Mandic ldquo-ecomplex local mean decompositionrdquo Neurocomputingvol 74 no 6 pp 867ndash875 2011

[15] N Rehman and D P Mandic ldquoEmpirical mode de-composition for trivariate signalsrdquo IEEE Transactions onSignal Processing vol 58 no 3 pp 1059ndash1068 2010

[16] N Rehman and D P Mandic ldquoMultivariate empirical modedecompositionrdquo Proceedings of the Royal Society A Mathe-matical Physical and Engineering Sciences vol 466 no 2117pp 1291ndash1302 2010

[17] Y Lv R Yuan and G Song ldquoMultivariate empirical modedecomposition and its application to fault diagnosis of rollingbearingrdquo Mechanical Systems and Signal Processing vol 81pp 219ndash234 2016

[18] C Huang Y Meng and W Lei ldquoFull vector envelopetechnique based on complex local mean decomposition andits application in fault feature extraction for rotor systemrdquoJournal of Mechanical Engineering vol 52 no 7 p 69 2016in Chinese

[19] G Rilling P Flandrin and P Goncalves ldquoOn empirical modedecomposition and its algorithmsrdquo in Proceedings of IEEE-EURASIP Workshop on Nonlinear Signal and Image Pro-cessing pp 8ndash11 IEEE Trieste Italy June 2003

[20] L Yang X Chen and S Wang ldquoMechanism of fast time-varying vibration for rotorndashstator contact system with ap-plication to fault diagnosisrdquo Journal of Vibration andAcoustics vol 140 no 1 article 014501 2018

[21] I Daubechies J Lu and H-TWu ldquoSynchrosqueezed wavelettransforms an empirical mode decomposition-like toolrdquoApplied and Computational Harmonic Analysis vol 30 no 2pp 243ndash261 2011

[22] L-S Qu Holospectrum and Holobalancing Technique inMachinery Diagnosis Beijing Science Press Beijing China2007

Shock and Vibration 19

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moving average ltering algorithm to obtain the signalenvelope curve However this is the result of using theCLMD algorithm after ltering out noise with BEMD If theBEMD algorithm is not used for ltering noise the in-stantaneous amplitude and instantaneous frequency curvesof the single component were not obtained by the CLMDmethod

42 Analysis ofOilWhirl Signal Based on the ImprovedBEMDMethod In the method similar to that presented in Section41 the oil whirl signals of the rotor test rig with a speedparameter of 4320 rpm are collected by two orthogonalsensors as shown in Figure 15 e gure shows the typicalwhirl phenomenon of large circles with embedded smallerones e decomposition results based on the improvedBEMD method (N 16 λ 005) are shown in Figure 16

Only three IMFs appear in Figure 16 and the singlecomponents c2 and c3 and the noise component c1 are suc-cessfully separated from the original signal e HT is appliedto the real and imaginary parts of c2 and c3 respectively andthe instantaneous frequency and instantaneous amplitude areobtained as shown in Figure 17 e three-dimensional timedomain of ax2 and ay2 and of ax3 and ay3 is shown in Figure 18Figure 17 indicates that the frequency of c2 is approximatelytwice the frequency of c3 and that ax2 is larger than ay2 Inaddition ay3 is slightly larger than ax3 but the range of changefor ax3 is greater than the range for ay3 It is inferred that c2 isthe fundamental frequency signal and that c3 is the half-frequency signal in the oil whirl signal

43 Analysis of Looseness and Rotor Rubbing Composite FaultSignal Based on the Improved BEMD Method Loose androtor rubbing composite faults are set on the testequipment shown in Figure 3 in Section 41 Loose fault isset on the nondrive end of the motor and the distancebetween the plastic rod and the shaft is xed near thesensor on the left side of the disk As the rotor speedincreases the vibration increases and the rubbing faultoccurs which is stable at around 1700 rmin e com-posite fault signals are collected by two orthogonal sen-sors as shown in Figure 19 e decomposition resultsbased on the improved BEMD method are shown inFigure 20

Figures 19(c) and 19(d) indicate that the signal com-ponent mainly contain 1X 2X (X 28Hz) and a frequencymodulated signal generated due to time-varying stinessis phenomenon is similar to that described reference [20]Four IMFs appear in Figure 20 and the single components1X 2X signals and the FM signal c2 are successfully separatedfrom the original signal e HT is applied to the real andimaginary parts of c2 c3 and c4 respectively and the in-stantaneous frequency and instantaneous amplitude areobtained as shown in Figure 21 ere are some uctuationsin the frequencies of c2 and c3 but these uctuations aredierent from the random uctuations in the above casesey have obvious regularity and are characteristic of FMsignals e frequency modulation characteristics of c2 aremore obvious than those of c3 In addition by observing theinstantaneous amplitude of c2 it is seen that c2 is still anamplitude modulation signal

minus100

0100

minus100

0

100

Time (s)Real(cpf1)

Imag

(cpf

1)

0 01 02 03 04 05

(a)

Time (s)Real(cpf2) minus100

0100

minus100

0

100

Imag

(cpf

2)

0 01 02 03 04 05

(b)

minus80 0 80minus60

0

60

Real(cpf1)

Imag(cpf1)

(c)

minus100 0 100minus100

0

100

Real(cpf2)

Imag(cpf2)

(d)

Figure 13 e rst two decomposed results of the ltered signal based on the CLMD method

10 Shock and Vibration

0025

05

ndash150

0

150ndash150

0

150

Time (s)Real(z)

Imag

(z)

(a)

ndash150 ndash75 0 75 150ndash150

ndash75

0

75

150

Real(z)

Imag

(z)

(b)

0 72 144 216 2880

40

80

Frequency (Hz)

Am

plitu

de

X 72Y 7321

X 36Y 3433

Real(z)

(c)

Frequency (Hz)0 72 144 216 288

0

40

80

X 36Y 3475

Am

plitu

de

X 72Y 6157

Imag(z)

(d)

Figure 15e oil whirl signal z (a) the 3D time-domain wave of z (b) the 2D plane of z (c) the Fourier spectrum of Real[z] (d) the Fourierspectrum of Imag[z]

30

40

50

60

70

Am

plitu

de

0 01 02 03 04 05Time (s)

ax1ay1

(a)

70

110

150

Freq

uenc

y (H

z)

0 01 02 03 04 05Time (s)

fx1fy1

(b)

0 01 02 03 04 0520

70

120

Time (s)

Am

plitu

de

ax2ay2

(c)

0 01 02 03 04 050

50

100

Time (s)

Freq

uenc

y (H

z)

fx2fy2

(d)

Figure 14 (a) e instantaneous amplitude of the real part and imaginary part of cpf1 (b) the instantaneous frequency of the real part andimaginary part of cpf1 (c) the instantaneous amplitude of the real part and imaginary part of cpf2 (d) the instantaneous frequency of the realpart and imaginary part of cpf2

Shock and Vibration 11

0025

05

ndash5

0

5ndash5

0

5

Time (s)Real(c1)

Imag

(c1)

(a)

0025

05

ndash1000

100ndash100

0

100

Time (s)

Imag

(c2)

Real(c2)

(b)

0025

05

ndash500

50ndash50

0

50

Time (s)

Imag

(c3)

Real(c3)

(c)

0025

05

ndash100

10ndash20

0

20

Time (s)

Imag

(r)

Real(r)

(d)

ndash100 ndash50 0 50 100ndash100

ndash50

0

50

100

Imag

(c2)

Real(c2)

(e)

ndash40 ndash20 0 20 40ndash40

ndash20

0

20

40Im

ag(c

3)

Real(c3)

(f )

Figure 16 e decomposition results of the oil whirl signal based on the improved BEMD method

60

90

120

Am

plitu

de

0 01 02 03 04 05Time (s)

ax2ay2

(a)

50

75

100

Freq

uenc

y (H

z)

0 01 02 03 04 05Time (s)

fx2fy2

(b)

Figure 17 Continued

12 Shock and Vibration

0 01 02 03 04 0520

35

50

Time (s)

Am

plitu

de

ax3ay3

(c)

0 01 02 03 04 0520

35

50

Time (s)

Freq

uenc

y (H

z)

fx3fy3

(d)

Figure 17 e instantaneous amplitude and frequency of c2 and c3 from the oil whirl signal obtained by the HT (a) the instantaneousamplitude of the real part and imaginary part of c2 (b) the instantaneous frequency of the real part and imaginary part of c2(c) the instantaneous amplitude of the real part and imaginary part of c3 (d) the instantaneous frequency of the real part and imaginarypart of c3

0

025

05

40

80

12040

80

120

Time (s)

X 03877Y 9582Z 826

ax2

X 01309Y 9595Z 8452

a y2

(a)

0

025

05

20

35

5020

35

50

Time (s)

X 03955Y 3762Z 3822

ax3

X 008838Y 3608Z 3733

a y3

(b)

Figure 18 e 3D time domain of the instantaneous amplitude of ax2 and ay2 (a) and ax3 and ay3 (b) from the oil whirl signal

0025

05

minus1500

150minus150

0

150

t (s)Real(z)

Imag

(z)

(a)

Imag(z)

Real(z)

minus150

0

150

minus150 0 150

(b)

Figure 19 Continued

Shock and Vibration 13

In order to further verify the correctness of the in-stantaneous amplitude-frequency characteristics of theproposed method the real and imaginary parts of thecomposite fault signal z are analyzed separately using syn-chrosqueezed wavelet transforms (SWT) proposed in ref-erence [21]-e results are shown in Figure 22 It is seen thatthe time-frequency representations of the composite faultsignal z also include the AM-FM signal and the 1X signalwhich proves the correctness of the proposed methodCompared with the SWT method the instantaneousamplitude-frequency characteristics acquired by the HTmethod are relatively straightforward

44 lte Bistable Behavior Analysis of the Fan Rotor Based onBEMD -e bistability of the rotor is a nonlinear behaviorof the rotor-bearing system which is the state in which therotor jumps from one stable state to another forming astep -e bivariate signal of the bistable behavior iscomposed of two signals collected by two displacementsensors from orthogonal locations on the experimentaldevices in literature [22] as shown in Figure 23 Literature[22] shows that the cause of the bistable behavior remainsto be further explored -is paper uses this case to il-lustrate the feasibility of BEMD to analyze nonstationarysignals

0

40

80

Am

plitu

de Real(z)

0 100 200 300 400Frequency (Hz)

(c)

Am

plitu

de

0 100 200 300 400Frequency (Hz)

0

20

40

60

80

Imag(z)

(d)

Figure 19 -e composite fault signal z (a) the 3D time domain wave of z (b) the 2D plane of z (c) the Fourier spectrum of Real[z] (d) theFourier spectrum of Imag[z]

0025

05

minus30

3minus5

0

5

Time (s)Real(c1)

Imag

(c1)

(a)

0025

05

minus200

20minus20

0

20

Time (s)Real(c2 )

Imag

(c2)

(b)

0025

05

minus200

20minus50

0

50

Time (s)Real(c2)

Imag

(c2)

(c)

0025

05

minus1000

100minus100

0

100

Time (s)Real(c3)

Imag

(c3)

(d)

0025

05

minus400

40minus30

0

30

Time (s)Real(r)

Imag

(r)

(e)

Figure 20 -e decomposition results of the composite fault signal based on the improved BEMD method

14 Shock and Vibration

-e x and y signals in the horizontal and vertical di-rections of the left and right bearings respectively from thefan rotors are collected with four displacement sensorsLetting z x+ jy the time and frequency domain plots of z areshown in Figure 24 where the fan rotor speed is 5500 rpm thesampling frequency is 2000Hz and the number of samplingpoints is 1024 -e left and right columns respectively showthe time and frequency domain plots of the vibration signalsfrom the left and right bearings of the fan rotor Bistablebehavior arises in the fan rotor and the amplitudes of the

vibration signals vary significantly in different positions anddirections Further studies are required to explain the causesof this bistability -e present study focuses on extracting thebistable behavioral signal characteristics to verify the feasi-bility of the proposed method

-e decomposition results of the bistable behavioralsignals based on the improved BEMDmethod are shown inFigure 25 c1 c2 c3 and r are separated in order from zusing the improved BEMD method c1 shows a randomarrangement and is considered the high-frequency noise

0

10

20

Am

plitu

de

ax2ay2

0 025 05Time (s)

(a)

0

152

304

Freq

uenc

y (H

z)

0 025 05Time (s)

fy2

fx2

(b)

0

10

20

30

40

Am

plitu

de

ax3ay3

0 025 05Time (s)

(c)

0

56

112

Freq

uenc

y (H

z)

fx3fy3

0 025 05Time (s)

(d)

0 025 050

50

100

Time (s)

Am

plitu

de

ax4ay4

(e)

0 025 050

28

56

Time (s)

Freq

uenc

y (H

z)

fx4fy4

(f )

Figure 21 -e instantaneous amplitude and frequency of c2 c3 and c4 obtained by the HT (a) the instantaneous amplitude of the real partand imaginary part of c2 (b) the instantaneous frequency of the real part and imaginary part of c2 (c) the instantaneous amplitude of the realpart and imaginary part of c3 (d) the instantaneous frequency of the real part and imaginary part of c3 (e) the instantaneous amplitude of thereal part and imaginary part of c4 (f ) the instantaneous frequency of the real part and imaginary part of c4

Shock and Vibration 15

signal c2 is considered to represent the extracted bistablebehavior signals -e HT is applied to the real andimaginary parts of c2 to obtain the instantaneous amplitudeand frequency of c2 from the left and right columns fromFigure 25 as shown in Figure 26 Figure 27 shows thethree-dimensional time domain of ax2 and ay2 from the leftand right columns respectively Figure 26 shows that thevibration signal amplitude on the left side of the fan de-creases from large to small opposite of the behavior ofthe right -e horizontal vibration signal amplitude on theleft side of the fan is larger than that of the vertical di-rection signal opposite of the right -is result validatesthat the vibration signals from different directions orpositions are different when the fan produces bistablebehavior In addition the time of the bistable behavior canbe determined according to the jump point of the am-plitude or frequency

5 Discussion

-e BEMD algorithm decomposes two orthogonal di-rections of vibration signals as a complex signal which is atwo-dimensional digital signal processing method thusensuring that the real and imaginary parts have the samedecomposition scale Similar to EMD the envelope mean iscritical for the decomposition effect of BEMD but the en-velope mean in BEMD is three-dimensional If the numberof projection directions of the complex signal in three-dimensional space is larger the corresponding envelope

signal is also more -us the envelope mean value is moreaccurate and the BEMD decomposition effect is betterIncreasing the number of projection directions can improvemodal aliasing Like EMD BEMD also produces falsecomponents when decomposing signals Generally speakingthe energy of the false components is low and these low-energy false components do not contain fault characteristicinformation and the introduction of the energy thresholdcriterion in the termination condition can increase thedecomposition speed of the BEMD

-e experimental results show that there is a certaindifference in the existence of vibration signal character-istics in different directions when rotating machinery failsIn addition when the number of projection directionsis increased the decomposition speed of BEMD willdecrease

6 Conclusions

We use BEMD and HT to extract the instantaneousamplitude-frequency features of rotor faults A bivariateinstantaneous feature extraction method based on the im-proved BEMD method and the HT is investigated whichextends the fault feature extraction technology to two di-mensions -e BEMD method is suitable to analyze thecomplex multicomponent bivariate signals -e mainsingle-component bivariate signals are separated from themulticomponent bivariate signals of the fan rotor bistabilityfor the oil film oscillation and the oil film vortex using the

0

2

4

6

Time (s)

Freq

uenc

y (H

z)

0 01 02 03 04 052

3

4

5

6

7

8log2 fx

(a)

0

2

4

6

Time (s)

Freq

uenc

y (H

z)

0 01 02 03 04 052

3

4

5

6

7

8log2 fy

(b)

Figure 22 -e results of the composite fault signal z based on SWT the time-frequency representation of (a) the real part of z and (b) theimaginary part of z

Left bearing predstal

Locations of sensors

Fanrotor

Right bearing predstal

Axis

Orthogonal directions

Figure 23 -e schematic diagram of the experimental apparatus

16 Shock and Vibration

Real(z) 00256

0512minus300

0300

minus400

0

400

Time (s)

Imag

(z)

00256

0512minus500

0500

minus500

0

500

Time (s)Real(z)

Imag

(z)

Imag

(z)

minus300 0 300minus500

0

500

Real(z)minus500 0 500

minus500

0

Imag

(z)

Real(z)

500

0 100 200 300 4000

100

200

300

Frequency (Hz)

Am

plitu

de Imag(z)

Frequency (Hz)0 100 200 300 400

0

200

400

Am

plitu

de

Imag(z)

0 100 200 300 4000

70

140

Frequency (Hz)

Am

plitu

de

Real(z)

0 100 200 300 4000

200

400

Frequency (Hz)

Am

plitu

de Real(z)

(a) (b)

Figure 24 -e time and frequency domain plots of the bistable behavior signals

00256

0512

minus800

80minus80

0

80

Time (s)Real(c1 )

Imag

(c1)

Time (s)Real(c1 )

Imag

(c1)

00256

0512

minus400

40minus40

0

40

Time (s)Real(c2) 0

02560512

minus5000

500minus500

0

500

Imag

(c2)

Time (s)Real(c2)

Imag

(c2)

00256

0512

minus5000

500minus500

0

500

Time (s)Real(r) 00256

0512

minus800

80minus80

0

80

Imag

(irc

rm

)

Time (s)Real(r)

Imag

(r)

00256

0512

minus2000

200minus200

0

200

(a) (b)

Figure 25 -e decomposition results of the bistable behavior signals based on the improved BEMD method

Shock and Vibration 17

improved BEMD method For the single-component bi-variate signal the HT is used to obtain the correspondinginstantaneous amplitude and frequency characteristics -eproposed method can examine the detailed information of asingle rotation component

Data Availability

-e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

-e authors declare that they have no conflicts of interest

Authorsrsquo Contributions

All the authors contributed to this work Chuanjin Huangconceived and designed the simulation and experiments anddrafted the manuscript Haijun Song performed the simu-lations and experiments and analyzed the data and

0 0256 05120

50

100

150

Time (s)

Freq

uenc

y (H

z)

fx2

fy2

0 0256 05120

200

400

600

Time (s)

Am

plitu

de

ax2ay2

(a)

fx2

fy2

0 0256 05120

90

180

270

360

Time (s)

Freq

uenc

y (H

z)

0 0256 05120

200

400

600

Time (s)

Am

plitu

de

ax2Data 2

(b)

Figure 26 -e instantaneous amplitude and frequency of c2 from the (a) left and (b) right columns

0

0256

0512

0100

200300

400500

0

100

200

300

400

500

Time (s)

X 04035Y 3509Z 130

X 04235Y 2265Z 3613

X 007Y 1131Z 2191

X 0105Y 3837Z 3312

ax2

a y2

Figure 27 -e three-dimensional time domain of ax2 and ay2

18 Shock and Vibration

Wenping Lei and Yajun Meng performed the experimentsand analyzed the data All the authors contributed to thewriting and discussion of the paper

Acknowledgments

-is research was funded by the Henan Provincial HigherEducation Key Research Project (Grant nos 18A460006 and19A460029) Henan High-Level Innovative Scientific andTechnological Talent Team Construction Project (Grant noC20150034) and Zhengzhou Institute of Technology In-novation Team Project (Grant no CXTD2017K1)

References

[1] R Yan R X Gao and X Chen ldquoWavelets for fault diagnosisof rotary machines a review with applicationsrdquo Signal Pro-cessing vol 96 pp 1ndash15 2014

[2] J Cheng D Yu J Tang and Y Yang ldquoApplication of frequencyfamily separation method based upon EMD and local Hilbertenergy spectrum method to gear fault diagnosisrdquo Mechanismand Machine lteory vol 43 no 6 pp 712ndash723 2008

[3] H Liu and M Han ldquoA fault diagnosis method based on localmean decomposition and multi-scale entropy for rollerbearingsrdquoMechanism andMachinelteory vol 75 pp 67ndash782014

[4] Z Zheng W Jiang Z Wang Y Zhu and K Yang ldquoGear faultdiagnosis method based on local mean decomposition andgeneralized morphological fractal dimensionsrdquo Mechanismand Machine lteory vol 91 pp 151ndash167 2015

[5] W Yang R Court P J Tavner and C J Crabtree ldquoBivariateempirical mode decomposition and its contribution to windturbine condition monitoringrdquo Journal of Sound and Vi-bration vol 330 no 15 pp 3766ndash3782 2011

[6] L Qu X Liu G Peyronne and Y Chen ldquo-e holospectrum anewmethod for rotor surveillance and diagnosisrdquoMechanicalSystems amp Signal Processing vol 3 no 3 pp 255ndash267 1989

[7] F Q Wu and G Meng ldquoCompound rub malfunctions featureextraction based on full-spectrum cascade analysis and SVMrdquoMechanical Systems and Signal Processing vol 20 no 8pp 2007ndash2021 2006

[8] Y Chen Q Gao and Z Guan ldquoSelf-loosening failure analysisof bolt joints under vibration considering the tighteningprocessrdquo Shock and Vibration vol 2017 Article ID 203842115 pages 2017

[9] L Chen J Han W Lei Y Cui and Z Guan ldquoFull-vectorsignal acquisition and information fusion for the fault pre-dictionrdquo International Journal of Rotating Machineryvol 2016 Article ID 5980802 7 pages 2016

[10] C Chen Y Meng and Y Du ldquoApplication of the full vectorspectrum based on EMD in fault diagnosis of bearingsrdquoJournal of Mechanical Strength vol 37 pp 806ndash811 2015

[11] C Huang X Wu and W Cao ldquoLMD-based on full vectorenvelope technique and its application in TRT vibration faultdiagnosisrdquo Electric Power Automation Equipment vol 35pp 168ndash174 2015 in Chinese

[12] X Zhao T H Patel and M J Zuo ldquoMultivariate EMD andfull spectrum based condition monitoring for rotating ma-chineryrdquo Mechanical Systems and Signal Processing vol 27pp 712ndash728 2012

[13] G Rilling P Flandrin P Gonalves and J M Lilly ldquoBivariateempirical mode decompositionrdquo IEEE Signal ProcessingLetters vol 14 no 12 pp 936ndash939 2007

[14] C Park D Looney M M Van Hulle and D P Mandic ldquo-ecomplex local mean decompositionrdquo Neurocomputingvol 74 no 6 pp 867ndash875 2011

[15] N Rehman and D P Mandic ldquoEmpirical mode de-composition for trivariate signalsrdquo IEEE Transactions onSignal Processing vol 58 no 3 pp 1059ndash1068 2010

[16] N Rehman and D P Mandic ldquoMultivariate empirical modedecompositionrdquo Proceedings of the Royal Society A Mathe-matical Physical and Engineering Sciences vol 466 no 2117pp 1291ndash1302 2010

[17] Y Lv R Yuan and G Song ldquoMultivariate empirical modedecomposition and its application to fault diagnosis of rollingbearingrdquo Mechanical Systems and Signal Processing vol 81pp 219ndash234 2016

[18] C Huang Y Meng and W Lei ldquoFull vector envelopetechnique based on complex local mean decomposition andits application in fault feature extraction for rotor systemrdquoJournal of Mechanical Engineering vol 52 no 7 p 69 2016in Chinese

[19] G Rilling P Flandrin and P Goncalves ldquoOn empirical modedecomposition and its algorithmsrdquo in Proceedings of IEEE-EURASIP Workshop on Nonlinear Signal and Image Pro-cessing pp 8ndash11 IEEE Trieste Italy June 2003

[20] L Yang X Chen and S Wang ldquoMechanism of fast time-varying vibration for rotorndashstator contact system with ap-plication to fault diagnosisrdquo Journal of Vibration andAcoustics vol 140 no 1 article 014501 2018

[21] I Daubechies J Lu and H-TWu ldquoSynchrosqueezed wavelettransforms an empirical mode decomposition-like toolrdquoApplied and Computational Harmonic Analysis vol 30 no 2pp 243ndash261 2011

[22] L-S Qu Holospectrum and Holobalancing Technique inMachinery Diagnosis Beijing Science Press Beijing China2007

Shock and Vibration 19

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Submit your manuscripts atwwwhindawicom

0025

05

ndash150

0

150ndash150

0

150

Time (s)Real(z)

Imag

(z)

(a)

ndash150 ndash75 0 75 150ndash150

ndash75

0

75

150

Real(z)

Imag

(z)

(b)

0 72 144 216 2880

40

80

Frequency (Hz)

Am

plitu

de

X 72Y 7321

X 36Y 3433

Real(z)

(c)

Frequency (Hz)0 72 144 216 288

0

40

80

X 36Y 3475

Am

plitu

de

X 72Y 6157

Imag(z)

(d)

Figure 15e oil whirl signal z (a) the 3D time-domain wave of z (b) the 2D plane of z (c) the Fourier spectrum of Real[z] (d) the Fourierspectrum of Imag[z]

30

40

50

60

70

Am

plitu

de

0 01 02 03 04 05Time (s)

ax1ay1

(a)

70

110

150

Freq

uenc

y (H

z)

0 01 02 03 04 05Time (s)

fx1fy1

(b)

0 01 02 03 04 0520

70

120

Time (s)

Am

plitu

de

ax2ay2

(c)

0 01 02 03 04 050

50

100

Time (s)

Freq

uenc

y (H

z)

fx2fy2

(d)

Figure 14 (a) e instantaneous amplitude of the real part and imaginary part of cpf1 (b) the instantaneous frequency of the real part andimaginary part of cpf1 (c) the instantaneous amplitude of the real part and imaginary part of cpf2 (d) the instantaneous frequency of the realpart and imaginary part of cpf2

Shock and Vibration 11

0025

05

ndash5

0

5ndash5

0

5

Time (s)Real(c1)

Imag

(c1)

(a)

0025

05

ndash1000

100ndash100

0

100

Time (s)

Imag

(c2)

Real(c2)

(b)

0025

05

ndash500

50ndash50

0

50

Time (s)

Imag

(c3)

Real(c3)

(c)

0025

05

ndash100

10ndash20

0

20

Time (s)

Imag

(r)

Real(r)

(d)

ndash100 ndash50 0 50 100ndash100

ndash50

0

50

100

Imag

(c2)

Real(c2)

(e)

ndash40 ndash20 0 20 40ndash40

ndash20

0

20

40Im

ag(c

3)

Real(c3)

(f )

Figure 16 e decomposition results of the oil whirl signal based on the improved BEMD method

60

90

120

Am

plitu

de

0 01 02 03 04 05Time (s)

ax2ay2

(a)

50

75

100

Freq

uenc

y (H

z)

0 01 02 03 04 05Time (s)

fx2fy2

(b)

Figure 17 Continued

12 Shock and Vibration

0 01 02 03 04 0520

35

50

Time (s)

Am

plitu

de

ax3ay3

(c)

0 01 02 03 04 0520

35

50

Time (s)

Freq

uenc

y (H

z)

fx3fy3

(d)

Figure 17 e instantaneous amplitude and frequency of c2 and c3 from the oil whirl signal obtained by the HT (a) the instantaneousamplitude of the real part and imaginary part of c2 (b) the instantaneous frequency of the real part and imaginary part of c2(c) the instantaneous amplitude of the real part and imaginary part of c3 (d) the instantaneous frequency of the real part and imaginarypart of c3

0

025

05

40

80

12040

80

120

Time (s)

X 03877Y 9582Z 826

ax2

X 01309Y 9595Z 8452

a y2

(a)

0

025

05

20

35

5020

35

50

Time (s)

X 03955Y 3762Z 3822

ax3

X 008838Y 3608Z 3733

a y3

(b)

Figure 18 e 3D time domain of the instantaneous amplitude of ax2 and ay2 (a) and ax3 and ay3 (b) from the oil whirl signal

0025

05

minus1500

150minus150

0

150

t (s)Real(z)

Imag

(z)

(a)

Imag(z)

Real(z)

minus150

0

150

minus150 0 150

(b)

Figure 19 Continued

Shock and Vibration 13

In order to further verify the correctness of the in-stantaneous amplitude-frequency characteristics of theproposed method the real and imaginary parts of thecomposite fault signal z are analyzed separately using syn-chrosqueezed wavelet transforms (SWT) proposed in ref-erence [21]-e results are shown in Figure 22 It is seen thatthe time-frequency representations of the composite faultsignal z also include the AM-FM signal and the 1X signalwhich proves the correctness of the proposed methodCompared with the SWT method the instantaneousamplitude-frequency characteristics acquired by the HTmethod are relatively straightforward

44 lte Bistable Behavior Analysis of the Fan Rotor Based onBEMD -e bistability of the rotor is a nonlinear behaviorof the rotor-bearing system which is the state in which therotor jumps from one stable state to another forming astep -e bivariate signal of the bistable behavior iscomposed of two signals collected by two displacementsensors from orthogonal locations on the experimentaldevices in literature [22] as shown in Figure 23 Literature[22] shows that the cause of the bistable behavior remainsto be further explored -is paper uses this case to il-lustrate the feasibility of BEMD to analyze nonstationarysignals

0

40

80

Am

plitu

de Real(z)

0 100 200 300 400Frequency (Hz)

(c)

Am

plitu

de

0 100 200 300 400Frequency (Hz)

0

20

40

60

80

Imag(z)

(d)

Figure 19 -e composite fault signal z (a) the 3D time domain wave of z (b) the 2D plane of z (c) the Fourier spectrum of Real[z] (d) theFourier spectrum of Imag[z]

0025

05

minus30

3minus5

0

5

Time (s)Real(c1)

Imag

(c1)

(a)

0025

05

minus200

20minus20

0

20

Time (s)Real(c2 )

Imag

(c2)

(b)

0025

05

minus200

20minus50

0

50

Time (s)Real(c2)

Imag

(c2)

(c)

0025

05

minus1000

100minus100

0

100

Time (s)Real(c3)

Imag

(c3)

(d)

0025

05

minus400

40minus30

0

30

Time (s)Real(r)

Imag

(r)

(e)

Figure 20 -e decomposition results of the composite fault signal based on the improved BEMD method

14 Shock and Vibration

-e x and y signals in the horizontal and vertical di-rections of the left and right bearings respectively from thefan rotors are collected with four displacement sensorsLetting z x+ jy the time and frequency domain plots of z areshown in Figure 24 where the fan rotor speed is 5500 rpm thesampling frequency is 2000Hz and the number of samplingpoints is 1024 -e left and right columns respectively showthe time and frequency domain plots of the vibration signalsfrom the left and right bearings of the fan rotor Bistablebehavior arises in the fan rotor and the amplitudes of the

vibration signals vary significantly in different positions anddirections Further studies are required to explain the causesof this bistability -e present study focuses on extracting thebistable behavioral signal characteristics to verify the feasi-bility of the proposed method

-e decomposition results of the bistable behavioralsignals based on the improved BEMDmethod are shown inFigure 25 c1 c2 c3 and r are separated in order from zusing the improved BEMD method c1 shows a randomarrangement and is considered the high-frequency noise

0

10

20

Am

plitu

de

ax2ay2

0 025 05Time (s)

(a)

0

152

304

Freq

uenc

y (H

z)

0 025 05Time (s)

fy2

fx2

(b)

0

10

20

30

40

Am

plitu

de

ax3ay3

0 025 05Time (s)

(c)

0

56

112

Freq

uenc

y (H

z)

fx3fy3

0 025 05Time (s)

(d)

0 025 050

50

100

Time (s)

Am

plitu

de

ax4ay4

(e)

0 025 050

28

56

Time (s)

Freq

uenc

y (H

z)

fx4fy4

(f )

Figure 21 -e instantaneous amplitude and frequency of c2 c3 and c4 obtained by the HT (a) the instantaneous amplitude of the real partand imaginary part of c2 (b) the instantaneous frequency of the real part and imaginary part of c2 (c) the instantaneous amplitude of the realpart and imaginary part of c3 (d) the instantaneous frequency of the real part and imaginary part of c3 (e) the instantaneous amplitude of thereal part and imaginary part of c4 (f ) the instantaneous frequency of the real part and imaginary part of c4

Shock and Vibration 15

signal c2 is considered to represent the extracted bistablebehavior signals -e HT is applied to the real andimaginary parts of c2 to obtain the instantaneous amplitudeand frequency of c2 from the left and right columns fromFigure 25 as shown in Figure 26 Figure 27 shows thethree-dimensional time domain of ax2 and ay2 from the leftand right columns respectively Figure 26 shows that thevibration signal amplitude on the left side of the fan de-creases from large to small opposite of the behavior ofthe right -e horizontal vibration signal amplitude on theleft side of the fan is larger than that of the vertical di-rection signal opposite of the right -is result validatesthat the vibration signals from different directions orpositions are different when the fan produces bistablebehavior In addition the time of the bistable behavior canbe determined according to the jump point of the am-plitude or frequency

5 Discussion

-e BEMD algorithm decomposes two orthogonal di-rections of vibration signals as a complex signal which is atwo-dimensional digital signal processing method thusensuring that the real and imaginary parts have the samedecomposition scale Similar to EMD the envelope mean iscritical for the decomposition effect of BEMD but the en-velope mean in BEMD is three-dimensional If the numberof projection directions of the complex signal in three-dimensional space is larger the corresponding envelope

signal is also more -us the envelope mean value is moreaccurate and the BEMD decomposition effect is betterIncreasing the number of projection directions can improvemodal aliasing Like EMD BEMD also produces falsecomponents when decomposing signals Generally speakingthe energy of the false components is low and these low-energy false components do not contain fault characteristicinformation and the introduction of the energy thresholdcriterion in the termination condition can increase thedecomposition speed of the BEMD

-e experimental results show that there is a certaindifference in the existence of vibration signal character-istics in different directions when rotating machinery failsIn addition when the number of projection directionsis increased the decomposition speed of BEMD willdecrease

6 Conclusions

We use BEMD and HT to extract the instantaneousamplitude-frequency features of rotor faults A bivariateinstantaneous feature extraction method based on the im-proved BEMD method and the HT is investigated whichextends the fault feature extraction technology to two di-mensions -e BEMD method is suitable to analyze thecomplex multicomponent bivariate signals -e mainsingle-component bivariate signals are separated from themulticomponent bivariate signals of the fan rotor bistabilityfor the oil film oscillation and the oil film vortex using the

0

2

4

6

Time (s)

Freq

uenc

y (H

z)

0 01 02 03 04 052

3

4

5

6

7

8log2 fx

(a)

0

2

4

6

Time (s)

Freq

uenc

y (H

z)

0 01 02 03 04 052

3

4

5

6

7

8log2 fy

(b)

Figure 22 -e results of the composite fault signal z based on SWT the time-frequency representation of (a) the real part of z and (b) theimaginary part of z

Left bearing predstal

Locations of sensors

Fanrotor

Right bearing predstal

Axis

Orthogonal directions

Figure 23 -e schematic diagram of the experimental apparatus

16 Shock and Vibration

Real(z) 00256

0512minus300

0300

minus400

0

400

Time (s)

Imag

(z)

00256

0512minus500

0500

minus500

0

500

Time (s)Real(z)

Imag

(z)

Imag

(z)

minus300 0 300minus500

0

500

Real(z)minus500 0 500

minus500

0

Imag

(z)

Real(z)

500

0 100 200 300 4000

100

200

300

Frequency (Hz)

Am

plitu

de Imag(z)

Frequency (Hz)0 100 200 300 400

0

200

400

Am

plitu

de

Imag(z)

0 100 200 300 4000

70

140

Frequency (Hz)

Am

plitu

de

Real(z)

0 100 200 300 4000

200

400

Frequency (Hz)

Am

plitu

de Real(z)

(a) (b)

Figure 24 -e time and frequency domain plots of the bistable behavior signals

00256

0512

minus800

80minus80

0

80

Time (s)Real(c1 )

Imag

(c1)

Time (s)Real(c1 )

Imag

(c1)

00256

0512

minus400

40minus40

0

40

Time (s)Real(c2) 0

02560512

minus5000

500minus500

0

500

Imag

(c2)

Time (s)Real(c2)

Imag

(c2)

00256

0512

minus5000

500minus500

0

500

Time (s)Real(r) 00256

0512

minus800

80minus80

0

80

Imag

(irc

rm

)

Time (s)Real(r)

Imag

(r)

00256

0512

minus2000

200minus200

0

200

(a) (b)

Figure 25 -e decomposition results of the bistable behavior signals based on the improved BEMD method

Shock and Vibration 17

improved BEMD method For the single-component bi-variate signal the HT is used to obtain the correspondinginstantaneous amplitude and frequency characteristics -eproposed method can examine the detailed information of asingle rotation component

Data Availability

-e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

-e authors declare that they have no conflicts of interest

Authorsrsquo Contributions

All the authors contributed to this work Chuanjin Huangconceived and designed the simulation and experiments anddrafted the manuscript Haijun Song performed the simu-lations and experiments and analyzed the data and

0 0256 05120

50

100

150

Time (s)

Freq

uenc

y (H

z)

fx2

fy2

0 0256 05120

200

400

600

Time (s)

Am

plitu

de

ax2ay2

(a)

fx2

fy2

0 0256 05120

90

180

270

360

Time (s)

Freq

uenc

y (H

z)

0 0256 05120

200

400

600

Time (s)

Am

plitu

de

ax2Data 2

(b)

Figure 26 -e instantaneous amplitude and frequency of c2 from the (a) left and (b) right columns

0

0256

0512

0100

200300

400500

0

100

200

300

400

500

Time (s)

X 04035Y 3509Z 130

X 04235Y 2265Z 3613

X 007Y 1131Z 2191

X 0105Y 3837Z 3312

ax2

a y2

Figure 27 -e three-dimensional time domain of ax2 and ay2

18 Shock and Vibration

Wenping Lei and Yajun Meng performed the experimentsand analyzed the data All the authors contributed to thewriting and discussion of the paper

Acknowledgments

-is research was funded by the Henan Provincial HigherEducation Key Research Project (Grant nos 18A460006 and19A460029) Henan High-Level Innovative Scientific andTechnological Talent Team Construction Project (Grant noC20150034) and Zhengzhou Institute of Technology In-novation Team Project (Grant no CXTD2017K1)

References

[1] R Yan R X Gao and X Chen ldquoWavelets for fault diagnosisof rotary machines a review with applicationsrdquo Signal Pro-cessing vol 96 pp 1ndash15 2014

[2] J Cheng D Yu J Tang and Y Yang ldquoApplication of frequencyfamily separation method based upon EMD and local Hilbertenergy spectrum method to gear fault diagnosisrdquo Mechanismand Machine lteory vol 43 no 6 pp 712ndash723 2008

[3] H Liu and M Han ldquoA fault diagnosis method based on localmean decomposition and multi-scale entropy for rollerbearingsrdquoMechanism andMachinelteory vol 75 pp 67ndash782014

[4] Z Zheng W Jiang Z Wang Y Zhu and K Yang ldquoGear faultdiagnosis method based on local mean decomposition andgeneralized morphological fractal dimensionsrdquo Mechanismand Machine lteory vol 91 pp 151ndash167 2015

[5] W Yang R Court P J Tavner and C J Crabtree ldquoBivariateempirical mode decomposition and its contribution to windturbine condition monitoringrdquo Journal of Sound and Vi-bration vol 330 no 15 pp 3766ndash3782 2011

[6] L Qu X Liu G Peyronne and Y Chen ldquo-e holospectrum anewmethod for rotor surveillance and diagnosisrdquoMechanicalSystems amp Signal Processing vol 3 no 3 pp 255ndash267 1989

[7] F Q Wu and G Meng ldquoCompound rub malfunctions featureextraction based on full-spectrum cascade analysis and SVMrdquoMechanical Systems and Signal Processing vol 20 no 8pp 2007ndash2021 2006

[8] Y Chen Q Gao and Z Guan ldquoSelf-loosening failure analysisof bolt joints under vibration considering the tighteningprocessrdquo Shock and Vibration vol 2017 Article ID 203842115 pages 2017

[9] L Chen J Han W Lei Y Cui and Z Guan ldquoFull-vectorsignal acquisition and information fusion for the fault pre-dictionrdquo International Journal of Rotating Machineryvol 2016 Article ID 5980802 7 pages 2016

[10] C Chen Y Meng and Y Du ldquoApplication of the full vectorspectrum based on EMD in fault diagnosis of bearingsrdquoJournal of Mechanical Strength vol 37 pp 806ndash811 2015

[11] C Huang X Wu and W Cao ldquoLMD-based on full vectorenvelope technique and its application in TRT vibration faultdiagnosisrdquo Electric Power Automation Equipment vol 35pp 168ndash174 2015 in Chinese

[12] X Zhao T H Patel and M J Zuo ldquoMultivariate EMD andfull spectrum based condition monitoring for rotating ma-chineryrdquo Mechanical Systems and Signal Processing vol 27pp 712ndash728 2012

[13] G Rilling P Flandrin P Gonalves and J M Lilly ldquoBivariateempirical mode decompositionrdquo IEEE Signal ProcessingLetters vol 14 no 12 pp 936ndash939 2007

[14] C Park D Looney M M Van Hulle and D P Mandic ldquo-ecomplex local mean decompositionrdquo Neurocomputingvol 74 no 6 pp 867ndash875 2011

[15] N Rehman and D P Mandic ldquoEmpirical mode de-composition for trivariate signalsrdquo IEEE Transactions onSignal Processing vol 58 no 3 pp 1059ndash1068 2010

[16] N Rehman and D P Mandic ldquoMultivariate empirical modedecompositionrdquo Proceedings of the Royal Society A Mathe-matical Physical and Engineering Sciences vol 466 no 2117pp 1291ndash1302 2010

[17] Y Lv R Yuan and G Song ldquoMultivariate empirical modedecomposition and its application to fault diagnosis of rollingbearingrdquo Mechanical Systems and Signal Processing vol 81pp 219ndash234 2016

[18] C Huang Y Meng and W Lei ldquoFull vector envelopetechnique based on complex local mean decomposition andits application in fault feature extraction for rotor systemrdquoJournal of Mechanical Engineering vol 52 no 7 p 69 2016in Chinese

[19] G Rilling P Flandrin and P Goncalves ldquoOn empirical modedecomposition and its algorithmsrdquo in Proceedings of IEEE-EURASIP Workshop on Nonlinear Signal and Image Pro-cessing pp 8ndash11 IEEE Trieste Italy June 2003

[20] L Yang X Chen and S Wang ldquoMechanism of fast time-varying vibration for rotorndashstator contact system with ap-plication to fault diagnosisrdquo Journal of Vibration andAcoustics vol 140 no 1 article 014501 2018

[21] I Daubechies J Lu and H-TWu ldquoSynchrosqueezed wavelettransforms an empirical mode decomposition-like toolrdquoApplied and Computational Harmonic Analysis vol 30 no 2pp 243ndash261 2011

[22] L-S Qu Holospectrum and Holobalancing Technique inMachinery Diagnosis Beijing Science Press Beijing China2007

Shock and Vibration 19

International Journal of

AerospaceEngineeringHindawiwwwhindawicom Volume 2018

RoboticsJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Active and Passive Electronic Components

VLSI Design

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Shock and Vibration

Hindawiwwwhindawicom Volume 2018

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawiwwwhindawicom

Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Control Scienceand Engineering

Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Journal ofEngineeringVolume 2018

SensorsJournal of

Hindawiwwwhindawicom Volume 2018

International Journal of

RotatingMachinery

Hindawiwwwhindawicom Volume 2018

Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Navigation and Observation

International Journal of

Hindawi

wwwhindawicom Volume 2018

Advances in

Multimedia

Submit your manuscripts atwwwhindawicom

0025

05

ndash5

0

5ndash5

0

5

Time (s)Real(c1)

Imag

(c1)

(a)

0025

05

ndash1000

100ndash100

0

100

Time (s)

Imag

(c2)

Real(c2)

(b)

0025

05

ndash500

50ndash50

0

50

Time (s)

Imag

(c3)

Real(c3)

(c)

0025

05

ndash100

10ndash20

0

20

Time (s)

Imag

(r)

Real(r)

(d)

ndash100 ndash50 0 50 100ndash100

ndash50

0

50

100

Imag

(c2)

Real(c2)

(e)

ndash40 ndash20 0 20 40ndash40

ndash20

0

20

40Im

ag(c

3)

Real(c3)

(f )

Figure 16 e decomposition results of the oil whirl signal based on the improved BEMD method

60

90

120

Am

plitu

de

0 01 02 03 04 05Time (s)

ax2ay2

(a)

50

75

100

Freq

uenc

y (H

z)

0 01 02 03 04 05Time (s)

fx2fy2

(b)

Figure 17 Continued

12 Shock and Vibration

0 01 02 03 04 0520

35

50

Time (s)

Am

plitu

de

ax3ay3

(c)

0 01 02 03 04 0520

35

50

Time (s)

Freq

uenc

y (H

z)

fx3fy3

(d)

Figure 17 e instantaneous amplitude and frequency of c2 and c3 from the oil whirl signal obtained by the HT (a) the instantaneousamplitude of the real part and imaginary part of c2 (b) the instantaneous frequency of the real part and imaginary part of c2(c) the instantaneous amplitude of the real part and imaginary part of c3 (d) the instantaneous frequency of the real part and imaginarypart of c3

0

025

05

40

80

12040

80

120

Time (s)

X 03877Y 9582Z 826

ax2

X 01309Y 9595Z 8452

a y2

(a)

0

025

05

20

35

5020

35

50

Time (s)

X 03955Y 3762Z 3822

ax3

X 008838Y 3608Z 3733

a y3

(b)

Figure 18 e 3D time domain of the instantaneous amplitude of ax2 and ay2 (a) and ax3 and ay3 (b) from the oil whirl signal

0025

05

minus1500

150minus150

0

150

t (s)Real(z)

Imag

(z)

(a)

Imag(z)

Real(z)

minus150

0

150

minus150 0 150

(b)

Figure 19 Continued

Shock and Vibration 13

In order to further verify the correctness of the in-stantaneous amplitude-frequency characteristics of theproposed method the real and imaginary parts of thecomposite fault signal z are analyzed separately using syn-chrosqueezed wavelet transforms (SWT) proposed in ref-erence [21]-e results are shown in Figure 22 It is seen thatthe time-frequency representations of the composite faultsignal z also include the AM-FM signal and the 1X signalwhich proves the correctness of the proposed methodCompared with the SWT method the instantaneousamplitude-frequency characteristics acquired by the HTmethod are relatively straightforward

44 lte Bistable Behavior Analysis of the Fan Rotor Based onBEMD -e bistability of the rotor is a nonlinear behaviorof the rotor-bearing system which is the state in which therotor jumps from one stable state to another forming astep -e bivariate signal of the bistable behavior iscomposed of two signals collected by two displacementsensors from orthogonal locations on the experimentaldevices in literature [22] as shown in Figure 23 Literature[22] shows that the cause of the bistable behavior remainsto be further explored -is paper uses this case to il-lustrate the feasibility of BEMD to analyze nonstationarysignals

0

40

80

Am

plitu

de Real(z)

0 100 200 300 400Frequency (Hz)

(c)

Am

plitu

de

0 100 200 300 400Frequency (Hz)

0

20

40

60

80

Imag(z)

(d)

Figure 19 -e composite fault signal z (a) the 3D time domain wave of z (b) the 2D plane of z (c) the Fourier spectrum of Real[z] (d) theFourier spectrum of Imag[z]

0025

05

minus30

3minus5

0

5

Time (s)Real(c1)

Imag

(c1)

(a)

0025

05

minus200

20minus20

0

20

Time (s)Real(c2 )

Imag

(c2)

(b)

0025

05

minus200

20minus50

0

50

Time (s)Real(c2)

Imag

(c2)

(c)

0025

05

minus1000

100minus100

0

100

Time (s)Real(c3)

Imag

(c3)

(d)

0025

05

minus400

40minus30

0

30

Time (s)Real(r)

Imag

(r)

(e)

Figure 20 -e decomposition results of the composite fault signal based on the improved BEMD method

14 Shock and Vibration

-e x and y signals in the horizontal and vertical di-rections of the left and right bearings respectively from thefan rotors are collected with four displacement sensorsLetting z x+ jy the time and frequency domain plots of z areshown in Figure 24 where the fan rotor speed is 5500 rpm thesampling frequency is 2000Hz and the number of samplingpoints is 1024 -e left and right columns respectively showthe time and frequency domain plots of the vibration signalsfrom the left and right bearings of the fan rotor Bistablebehavior arises in the fan rotor and the amplitudes of the

vibration signals vary significantly in different positions anddirections Further studies are required to explain the causesof this bistability -e present study focuses on extracting thebistable behavioral signal characteristics to verify the feasi-bility of the proposed method

-e decomposition results of the bistable behavioralsignals based on the improved BEMDmethod are shown inFigure 25 c1 c2 c3 and r are separated in order from zusing the improved BEMD method c1 shows a randomarrangement and is considered the high-frequency noise

0

10

20

Am

plitu

de

ax2ay2

0 025 05Time (s)

(a)

0

152

304

Freq

uenc

y (H

z)

0 025 05Time (s)

fy2

fx2

(b)

0

10

20

30

40

Am

plitu

de

ax3ay3

0 025 05Time (s)

(c)

0

56

112

Freq

uenc

y (H

z)

fx3fy3

0 025 05Time (s)

(d)

0 025 050

50

100

Time (s)

Am

plitu

de

ax4ay4

(e)

0 025 050

28

56

Time (s)

Freq

uenc

y (H

z)

fx4fy4

(f )

Figure 21 -e instantaneous amplitude and frequency of c2 c3 and c4 obtained by the HT (a) the instantaneous amplitude of the real partand imaginary part of c2 (b) the instantaneous frequency of the real part and imaginary part of c2 (c) the instantaneous amplitude of the realpart and imaginary part of c3 (d) the instantaneous frequency of the real part and imaginary part of c3 (e) the instantaneous amplitude of thereal part and imaginary part of c4 (f ) the instantaneous frequency of the real part and imaginary part of c4

Shock and Vibration 15

signal c2 is considered to represent the extracted bistablebehavior signals -e HT is applied to the real andimaginary parts of c2 to obtain the instantaneous amplitudeand frequency of c2 from the left and right columns fromFigure 25 as shown in Figure 26 Figure 27 shows thethree-dimensional time domain of ax2 and ay2 from the leftand right columns respectively Figure 26 shows that thevibration signal amplitude on the left side of the fan de-creases from large to small opposite of the behavior ofthe right -e horizontal vibration signal amplitude on theleft side of the fan is larger than that of the vertical di-rection signal opposite of the right -is result validatesthat the vibration signals from different directions orpositions are different when the fan produces bistablebehavior In addition the time of the bistable behavior canbe determined according to the jump point of the am-plitude or frequency

5 Discussion

-e BEMD algorithm decomposes two orthogonal di-rections of vibration signals as a complex signal which is atwo-dimensional digital signal processing method thusensuring that the real and imaginary parts have the samedecomposition scale Similar to EMD the envelope mean iscritical for the decomposition effect of BEMD but the en-velope mean in BEMD is three-dimensional If the numberof projection directions of the complex signal in three-dimensional space is larger the corresponding envelope

signal is also more -us the envelope mean value is moreaccurate and the BEMD decomposition effect is betterIncreasing the number of projection directions can improvemodal aliasing Like EMD BEMD also produces falsecomponents when decomposing signals Generally speakingthe energy of the false components is low and these low-energy false components do not contain fault characteristicinformation and the introduction of the energy thresholdcriterion in the termination condition can increase thedecomposition speed of the BEMD

-e experimental results show that there is a certaindifference in the existence of vibration signal character-istics in different directions when rotating machinery failsIn addition when the number of projection directionsis increased the decomposition speed of BEMD willdecrease

6 Conclusions

We use BEMD and HT to extract the instantaneousamplitude-frequency features of rotor faults A bivariateinstantaneous feature extraction method based on the im-proved BEMD method and the HT is investigated whichextends the fault feature extraction technology to two di-mensions -e BEMD method is suitable to analyze thecomplex multicomponent bivariate signals -e mainsingle-component bivariate signals are separated from themulticomponent bivariate signals of the fan rotor bistabilityfor the oil film oscillation and the oil film vortex using the

0

2

4

6

Time (s)

Freq

uenc

y (H

z)

0 01 02 03 04 052

3

4

5

6

7

8log2 fx

(a)

0

2

4

6

Time (s)

Freq

uenc

y (H

z)

0 01 02 03 04 052

3

4

5

6

7

8log2 fy

(b)

Figure 22 -e results of the composite fault signal z based on SWT the time-frequency representation of (a) the real part of z and (b) theimaginary part of z

Left bearing predstal

Locations of sensors

Fanrotor

Right bearing predstal

Axis

Orthogonal directions

Figure 23 -e schematic diagram of the experimental apparatus

16 Shock and Vibration

Real(z) 00256

0512minus300

0300

minus400

0

400

Time (s)

Imag

(z)

00256

0512minus500

0500

minus500

0

500

Time (s)Real(z)

Imag

(z)

Imag

(z)

minus300 0 300minus500

0

500

Real(z)minus500 0 500

minus500

0

Imag

(z)

Real(z)

500

0 100 200 300 4000

100

200

300

Frequency (Hz)

Am

plitu

de Imag(z)

Frequency (Hz)0 100 200 300 400

0

200

400

Am

plitu

de

Imag(z)

0 100 200 300 4000

70

140

Frequency (Hz)

Am

plitu

de

Real(z)

0 100 200 300 4000

200

400

Frequency (Hz)

Am

plitu

de Real(z)

(a) (b)

Figure 24 -e time and frequency domain plots of the bistable behavior signals

00256

0512

minus800

80minus80

0

80

Time (s)Real(c1 )

Imag

(c1)

Time (s)Real(c1 )

Imag

(c1)

00256

0512

minus400

40minus40

0

40

Time (s)Real(c2) 0

02560512

minus5000

500minus500

0

500

Imag

(c2)

Time (s)Real(c2)

Imag

(c2)

00256

0512

minus5000

500minus500

0

500

Time (s)Real(r) 00256

0512

minus800

80minus80

0

80

Imag

(irc

rm

)

Time (s)Real(r)

Imag

(r)

00256

0512

minus2000

200minus200

0

200

(a) (b)

Figure 25 -e decomposition results of the bistable behavior signals based on the improved BEMD method

Shock and Vibration 17

improved BEMD method For the single-component bi-variate signal the HT is used to obtain the correspondinginstantaneous amplitude and frequency characteristics -eproposed method can examine the detailed information of asingle rotation component

Data Availability

-e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

-e authors declare that they have no conflicts of interest

Authorsrsquo Contributions

All the authors contributed to this work Chuanjin Huangconceived and designed the simulation and experiments anddrafted the manuscript Haijun Song performed the simu-lations and experiments and analyzed the data and

0 0256 05120

50

100

150

Time (s)

Freq

uenc

y (H

z)

fx2

fy2

0 0256 05120

200

400

600

Time (s)

Am

plitu

de

ax2ay2

(a)

fx2

fy2

0 0256 05120

90

180

270

360

Time (s)

Freq

uenc

y (H

z)

0 0256 05120

200

400

600

Time (s)

Am

plitu

de

ax2Data 2

(b)

Figure 26 -e instantaneous amplitude and frequency of c2 from the (a) left and (b) right columns

0

0256

0512

0100

200300

400500

0

100

200

300

400

500

Time (s)

X 04035Y 3509Z 130

X 04235Y 2265Z 3613

X 007Y 1131Z 2191

X 0105Y 3837Z 3312

ax2

a y2

Figure 27 -e three-dimensional time domain of ax2 and ay2

18 Shock and Vibration

Wenping Lei and Yajun Meng performed the experimentsand analyzed the data All the authors contributed to thewriting and discussion of the paper

Acknowledgments

-is research was funded by the Henan Provincial HigherEducation Key Research Project (Grant nos 18A460006 and19A460029) Henan High-Level Innovative Scientific andTechnological Talent Team Construction Project (Grant noC20150034) and Zhengzhou Institute of Technology In-novation Team Project (Grant no CXTD2017K1)

References

[1] R Yan R X Gao and X Chen ldquoWavelets for fault diagnosisof rotary machines a review with applicationsrdquo Signal Pro-cessing vol 96 pp 1ndash15 2014

[2] J Cheng D Yu J Tang and Y Yang ldquoApplication of frequencyfamily separation method based upon EMD and local Hilbertenergy spectrum method to gear fault diagnosisrdquo Mechanismand Machine lteory vol 43 no 6 pp 712ndash723 2008

[3] H Liu and M Han ldquoA fault diagnosis method based on localmean decomposition and multi-scale entropy for rollerbearingsrdquoMechanism andMachinelteory vol 75 pp 67ndash782014

[4] Z Zheng W Jiang Z Wang Y Zhu and K Yang ldquoGear faultdiagnosis method based on local mean decomposition andgeneralized morphological fractal dimensionsrdquo Mechanismand Machine lteory vol 91 pp 151ndash167 2015

[5] W Yang R Court P J Tavner and C J Crabtree ldquoBivariateempirical mode decomposition and its contribution to windturbine condition monitoringrdquo Journal of Sound and Vi-bration vol 330 no 15 pp 3766ndash3782 2011

[6] L Qu X Liu G Peyronne and Y Chen ldquo-e holospectrum anewmethod for rotor surveillance and diagnosisrdquoMechanicalSystems amp Signal Processing vol 3 no 3 pp 255ndash267 1989

[7] F Q Wu and G Meng ldquoCompound rub malfunctions featureextraction based on full-spectrum cascade analysis and SVMrdquoMechanical Systems and Signal Processing vol 20 no 8pp 2007ndash2021 2006

[8] Y Chen Q Gao and Z Guan ldquoSelf-loosening failure analysisof bolt joints under vibration considering the tighteningprocessrdquo Shock and Vibration vol 2017 Article ID 203842115 pages 2017

[9] L Chen J Han W Lei Y Cui and Z Guan ldquoFull-vectorsignal acquisition and information fusion for the fault pre-dictionrdquo International Journal of Rotating Machineryvol 2016 Article ID 5980802 7 pages 2016

[10] C Chen Y Meng and Y Du ldquoApplication of the full vectorspectrum based on EMD in fault diagnosis of bearingsrdquoJournal of Mechanical Strength vol 37 pp 806ndash811 2015

[11] C Huang X Wu and W Cao ldquoLMD-based on full vectorenvelope technique and its application in TRT vibration faultdiagnosisrdquo Electric Power Automation Equipment vol 35pp 168ndash174 2015 in Chinese

[12] X Zhao T H Patel and M J Zuo ldquoMultivariate EMD andfull spectrum based condition monitoring for rotating ma-chineryrdquo Mechanical Systems and Signal Processing vol 27pp 712ndash728 2012

[13] G Rilling P Flandrin P Gonalves and J M Lilly ldquoBivariateempirical mode decompositionrdquo IEEE Signal ProcessingLetters vol 14 no 12 pp 936ndash939 2007

[14] C Park D Looney M M Van Hulle and D P Mandic ldquo-ecomplex local mean decompositionrdquo Neurocomputingvol 74 no 6 pp 867ndash875 2011

[15] N Rehman and D P Mandic ldquoEmpirical mode de-composition for trivariate signalsrdquo IEEE Transactions onSignal Processing vol 58 no 3 pp 1059ndash1068 2010

[16] N Rehman and D P Mandic ldquoMultivariate empirical modedecompositionrdquo Proceedings of the Royal Society A Mathe-matical Physical and Engineering Sciences vol 466 no 2117pp 1291ndash1302 2010

[17] Y Lv R Yuan and G Song ldquoMultivariate empirical modedecomposition and its application to fault diagnosis of rollingbearingrdquo Mechanical Systems and Signal Processing vol 81pp 219ndash234 2016

[18] C Huang Y Meng and W Lei ldquoFull vector envelopetechnique based on complex local mean decomposition andits application in fault feature extraction for rotor systemrdquoJournal of Mechanical Engineering vol 52 no 7 p 69 2016in Chinese

[19] G Rilling P Flandrin and P Goncalves ldquoOn empirical modedecomposition and its algorithmsrdquo in Proceedings of IEEE-EURASIP Workshop on Nonlinear Signal and Image Pro-cessing pp 8ndash11 IEEE Trieste Italy June 2003

[20] L Yang X Chen and S Wang ldquoMechanism of fast time-varying vibration for rotorndashstator contact system with ap-plication to fault diagnosisrdquo Journal of Vibration andAcoustics vol 140 no 1 article 014501 2018

[21] I Daubechies J Lu and H-TWu ldquoSynchrosqueezed wavelettransforms an empirical mode decomposition-like toolrdquoApplied and Computational Harmonic Analysis vol 30 no 2pp 243ndash261 2011

[22] L-S Qu Holospectrum and Holobalancing Technique inMachinery Diagnosis Beijing Science Press Beijing China2007

Shock and Vibration 19

International Journal of

AerospaceEngineeringHindawiwwwhindawicom Volume 2018

RoboticsJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Active and Passive Electronic Components

VLSI Design

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Shock and Vibration

Hindawiwwwhindawicom Volume 2018

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawiwwwhindawicom

Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Control Scienceand Engineering

Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Journal ofEngineeringVolume 2018

SensorsJournal of

Hindawiwwwhindawicom Volume 2018

International Journal of

RotatingMachinery

Hindawiwwwhindawicom Volume 2018

Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Navigation and Observation

International Journal of

Hindawi

wwwhindawicom Volume 2018

Advances in

Multimedia

Submit your manuscripts atwwwhindawicom

0 01 02 03 04 0520

35

50

Time (s)

Am

plitu

de

ax3ay3

(c)

0 01 02 03 04 0520

35

50

Time (s)

Freq

uenc

y (H

z)

fx3fy3

(d)

Figure 17 e instantaneous amplitude and frequency of c2 and c3 from the oil whirl signal obtained by the HT (a) the instantaneousamplitude of the real part and imaginary part of c2 (b) the instantaneous frequency of the real part and imaginary part of c2(c) the instantaneous amplitude of the real part and imaginary part of c3 (d) the instantaneous frequency of the real part and imaginarypart of c3

0

025

05

40

80

12040

80

120

Time (s)

X 03877Y 9582Z 826

ax2

X 01309Y 9595Z 8452

a y2

(a)

0

025

05

20

35

5020

35

50

Time (s)

X 03955Y 3762Z 3822

ax3

X 008838Y 3608Z 3733

a y3

(b)

Figure 18 e 3D time domain of the instantaneous amplitude of ax2 and ay2 (a) and ax3 and ay3 (b) from the oil whirl signal

0025

05

minus1500

150minus150

0

150

t (s)Real(z)

Imag

(z)

(a)

Imag(z)

Real(z)

minus150

0

150

minus150 0 150

(b)

Figure 19 Continued

Shock and Vibration 13

In order to further verify the correctness of the in-stantaneous amplitude-frequency characteristics of theproposed method the real and imaginary parts of thecomposite fault signal z are analyzed separately using syn-chrosqueezed wavelet transforms (SWT) proposed in ref-erence [21]-e results are shown in Figure 22 It is seen thatthe time-frequency representations of the composite faultsignal z also include the AM-FM signal and the 1X signalwhich proves the correctness of the proposed methodCompared with the SWT method the instantaneousamplitude-frequency characteristics acquired by the HTmethod are relatively straightforward

44 lte Bistable Behavior Analysis of the Fan Rotor Based onBEMD -e bistability of the rotor is a nonlinear behaviorof the rotor-bearing system which is the state in which therotor jumps from one stable state to another forming astep -e bivariate signal of the bistable behavior iscomposed of two signals collected by two displacementsensors from orthogonal locations on the experimentaldevices in literature [22] as shown in Figure 23 Literature[22] shows that the cause of the bistable behavior remainsto be further explored -is paper uses this case to il-lustrate the feasibility of BEMD to analyze nonstationarysignals

0

40

80

Am

plitu

de Real(z)

0 100 200 300 400Frequency (Hz)

(c)

Am

plitu

de

0 100 200 300 400Frequency (Hz)

0

20

40

60

80

Imag(z)

(d)

Figure 19 -e composite fault signal z (a) the 3D time domain wave of z (b) the 2D plane of z (c) the Fourier spectrum of Real[z] (d) theFourier spectrum of Imag[z]

0025

05

minus30

3minus5

0

5

Time (s)Real(c1)

Imag

(c1)

(a)

0025

05

minus200

20minus20

0

20

Time (s)Real(c2 )

Imag

(c2)

(b)

0025

05

minus200

20minus50

0

50

Time (s)Real(c2)

Imag

(c2)

(c)

0025

05

minus1000

100minus100

0

100

Time (s)Real(c3)

Imag

(c3)

(d)

0025

05

minus400

40minus30

0

30

Time (s)Real(r)

Imag

(r)

(e)

Figure 20 -e decomposition results of the composite fault signal based on the improved BEMD method

14 Shock and Vibration

-e x and y signals in the horizontal and vertical di-rections of the left and right bearings respectively from thefan rotors are collected with four displacement sensorsLetting z x+ jy the time and frequency domain plots of z areshown in Figure 24 where the fan rotor speed is 5500 rpm thesampling frequency is 2000Hz and the number of samplingpoints is 1024 -e left and right columns respectively showthe time and frequency domain plots of the vibration signalsfrom the left and right bearings of the fan rotor Bistablebehavior arises in the fan rotor and the amplitudes of the

vibration signals vary significantly in different positions anddirections Further studies are required to explain the causesof this bistability -e present study focuses on extracting thebistable behavioral signal characteristics to verify the feasi-bility of the proposed method

-e decomposition results of the bistable behavioralsignals based on the improved BEMDmethod are shown inFigure 25 c1 c2 c3 and r are separated in order from zusing the improved BEMD method c1 shows a randomarrangement and is considered the high-frequency noise

0

10

20

Am

plitu

de

ax2ay2

0 025 05Time (s)

(a)

0

152

304

Freq

uenc

y (H

z)

0 025 05Time (s)

fy2

fx2

(b)

0

10

20

30

40

Am

plitu

de

ax3ay3

0 025 05Time (s)

(c)

0

56

112

Freq

uenc

y (H

z)

fx3fy3

0 025 05Time (s)

(d)

0 025 050

50

100

Time (s)

Am

plitu

de

ax4ay4

(e)

0 025 050

28

56

Time (s)

Freq

uenc

y (H

z)

fx4fy4

(f )

Figure 21 -e instantaneous amplitude and frequency of c2 c3 and c4 obtained by the HT (a) the instantaneous amplitude of the real partand imaginary part of c2 (b) the instantaneous frequency of the real part and imaginary part of c2 (c) the instantaneous amplitude of the realpart and imaginary part of c3 (d) the instantaneous frequency of the real part and imaginary part of c3 (e) the instantaneous amplitude of thereal part and imaginary part of c4 (f ) the instantaneous frequency of the real part and imaginary part of c4

Shock and Vibration 15

signal c2 is considered to represent the extracted bistablebehavior signals -e HT is applied to the real andimaginary parts of c2 to obtain the instantaneous amplitudeand frequency of c2 from the left and right columns fromFigure 25 as shown in Figure 26 Figure 27 shows thethree-dimensional time domain of ax2 and ay2 from the leftand right columns respectively Figure 26 shows that thevibration signal amplitude on the left side of the fan de-creases from large to small opposite of the behavior ofthe right -e horizontal vibration signal amplitude on theleft side of the fan is larger than that of the vertical di-rection signal opposite of the right -is result validatesthat the vibration signals from different directions orpositions are different when the fan produces bistablebehavior In addition the time of the bistable behavior canbe determined according to the jump point of the am-plitude or frequency

5 Discussion

-e BEMD algorithm decomposes two orthogonal di-rections of vibration signals as a complex signal which is atwo-dimensional digital signal processing method thusensuring that the real and imaginary parts have the samedecomposition scale Similar to EMD the envelope mean iscritical for the decomposition effect of BEMD but the en-velope mean in BEMD is three-dimensional If the numberof projection directions of the complex signal in three-dimensional space is larger the corresponding envelope

signal is also more -us the envelope mean value is moreaccurate and the BEMD decomposition effect is betterIncreasing the number of projection directions can improvemodal aliasing Like EMD BEMD also produces falsecomponents when decomposing signals Generally speakingthe energy of the false components is low and these low-energy false components do not contain fault characteristicinformation and the introduction of the energy thresholdcriterion in the termination condition can increase thedecomposition speed of the BEMD

-e experimental results show that there is a certaindifference in the existence of vibration signal character-istics in different directions when rotating machinery failsIn addition when the number of projection directionsis increased the decomposition speed of BEMD willdecrease

6 Conclusions

We use BEMD and HT to extract the instantaneousamplitude-frequency features of rotor faults A bivariateinstantaneous feature extraction method based on the im-proved BEMD method and the HT is investigated whichextends the fault feature extraction technology to two di-mensions -e BEMD method is suitable to analyze thecomplex multicomponent bivariate signals -e mainsingle-component bivariate signals are separated from themulticomponent bivariate signals of the fan rotor bistabilityfor the oil film oscillation and the oil film vortex using the

0

2

4

6

Time (s)

Freq

uenc

y (H

z)

0 01 02 03 04 052

3

4

5

6

7

8log2 fx

(a)

0

2

4

6

Time (s)

Freq

uenc

y (H

z)

0 01 02 03 04 052

3

4

5

6

7

8log2 fy

(b)

Figure 22 -e results of the composite fault signal z based on SWT the time-frequency representation of (a) the real part of z and (b) theimaginary part of z

Left bearing predstal

Locations of sensors

Fanrotor

Right bearing predstal

Axis

Orthogonal directions

Figure 23 -e schematic diagram of the experimental apparatus

16 Shock and Vibration

Real(z) 00256

0512minus300

0300

minus400

0

400

Time (s)

Imag

(z)

00256

0512minus500

0500

minus500

0

500

Time (s)Real(z)

Imag

(z)

Imag

(z)

minus300 0 300minus500

0

500

Real(z)minus500 0 500

minus500

0

Imag

(z)

Real(z)

500

0 100 200 300 4000

100

200

300

Frequency (Hz)

Am

plitu

de Imag(z)

Frequency (Hz)0 100 200 300 400

0

200

400

Am

plitu

de

Imag(z)

0 100 200 300 4000

70

140

Frequency (Hz)

Am

plitu

de

Real(z)

0 100 200 300 4000

200

400

Frequency (Hz)

Am

plitu

de Real(z)

(a) (b)

Figure 24 -e time and frequency domain plots of the bistable behavior signals

00256

0512

minus800

80minus80

0

80

Time (s)Real(c1 )

Imag

(c1)

Time (s)Real(c1 )

Imag

(c1)

00256

0512

minus400

40minus40

0

40

Time (s)Real(c2) 0

02560512

minus5000

500minus500

0

500

Imag

(c2)

Time (s)Real(c2)

Imag

(c2)

00256

0512

minus5000

500minus500

0

500

Time (s)Real(r) 00256

0512

minus800

80minus80

0

80

Imag

(irc

rm

)

Time (s)Real(r)

Imag

(r)

00256

0512

minus2000

200minus200

0

200

(a) (b)

Figure 25 -e decomposition results of the bistable behavior signals based on the improved BEMD method

Shock and Vibration 17

improved BEMD method For the single-component bi-variate signal the HT is used to obtain the correspondinginstantaneous amplitude and frequency characteristics -eproposed method can examine the detailed information of asingle rotation component

Data Availability

-e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

-e authors declare that they have no conflicts of interest

Authorsrsquo Contributions

All the authors contributed to this work Chuanjin Huangconceived and designed the simulation and experiments anddrafted the manuscript Haijun Song performed the simu-lations and experiments and analyzed the data and

0 0256 05120

50

100

150

Time (s)

Freq

uenc

y (H

z)

fx2

fy2

0 0256 05120

200

400

600

Time (s)

Am

plitu

de

ax2ay2

(a)

fx2

fy2

0 0256 05120

90

180

270

360

Time (s)

Freq

uenc

y (H

z)

0 0256 05120

200

400

600

Time (s)

Am

plitu

de

ax2Data 2

(b)

Figure 26 -e instantaneous amplitude and frequency of c2 from the (a) left and (b) right columns

0

0256

0512

0100

200300

400500

0

100

200

300

400

500

Time (s)

X 04035Y 3509Z 130

X 04235Y 2265Z 3613

X 007Y 1131Z 2191

X 0105Y 3837Z 3312

ax2

a y2

Figure 27 -e three-dimensional time domain of ax2 and ay2

18 Shock and Vibration

Wenping Lei and Yajun Meng performed the experimentsand analyzed the data All the authors contributed to thewriting and discussion of the paper

Acknowledgments

-is research was funded by the Henan Provincial HigherEducation Key Research Project (Grant nos 18A460006 and19A460029) Henan High-Level Innovative Scientific andTechnological Talent Team Construction Project (Grant noC20150034) and Zhengzhou Institute of Technology In-novation Team Project (Grant no CXTD2017K1)

References

[1] R Yan R X Gao and X Chen ldquoWavelets for fault diagnosisof rotary machines a review with applicationsrdquo Signal Pro-cessing vol 96 pp 1ndash15 2014

[2] J Cheng D Yu J Tang and Y Yang ldquoApplication of frequencyfamily separation method based upon EMD and local Hilbertenergy spectrum method to gear fault diagnosisrdquo Mechanismand Machine lteory vol 43 no 6 pp 712ndash723 2008

[3] H Liu and M Han ldquoA fault diagnosis method based on localmean decomposition and multi-scale entropy for rollerbearingsrdquoMechanism andMachinelteory vol 75 pp 67ndash782014

[4] Z Zheng W Jiang Z Wang Y Zhu and K Yang ldquoGear faultdiagnosis method based on local mean decomposition andgeneralized morphological fractal dimensionsrdquo Mechanismand Machine lteory vol 91 pp 151ndash167 2015

[5] W Yang R Court P J Tavner and C J Crabtree ldquoBivariateempirical mode decomposition and its contribution to windturbine condition monitoringrdquo Journal of Sound and Vi-bration vol 330 no 15 pp 3766ndash3782 2011

[6] L Qu X Liu G Peyronne and Y Chen ldquo-e holospectrum anewmethod for rotor surveillance and diagnosisrdquoMechanicalSystems amp Signal Processing vol 3 no 3 pp 255ndash267 1989

[7] F Q Wu and G Meng ldquoCompound rub malfunctions featureextraction based on full-spectrum cascade analysis and SVMrdquoMechanical Systems and Signal Processing vol 20 no 8pp 2007ndash2021 2006

[8] Y Chen Q Gao and Z Guan ldquoSelf-loosening failure analysisof bolt joints under vibration considering the tighteningprocessrdquo Shock and Vibration vol 2017 Article ID 203842115 pages 2017

[9] L Chen J Han W Lei Y Cui and Z Guan ldquoFull-vectorsignal acquisition and information fusion for the fault pre-dictionrdquo International Journal of Rotating Machineryvol 2016 Article ID 5980802 7 pages 2016

[10] C Chen Y Meng and Y Du ldquoApplication of the full vectorspectrum based on EMD in fault diagnosis of bearingsrdquoJournal of Mechanical Strength vol 37 pp 806ndash811 2015

[11] C Huang X Wu and W Cao ldquoLMD-based on full vectorenvelope technique and its application in TRT vibration faultdiagnosisrdquo Electric Power Automation Equipment vol 35pp 168ndash174 2015 in Chinese

[12] X Zhao T H Patel and M J Zuo ldquoMultivariate EMD andfull spectrum based condition monitoring for rotating ma-chineryrdquo Mechanical Systems and Signal Processing vol 27pp 712ndash728 2012

[13] G Rilling P Flandrin P Gonalves and J M Lilly ldquoBivariateempirical mode decompositionrdquo IEEE Signal ProcessingLetters vol 14 no 12 pp 936ndash939 2007

[14] C Park D Looney M M Van Hulle and D P Mandic ldquo-ecomplex local mean decompositionrdquo Neurocomputingvol 74 no 6 pp 867ndash875 2011

[15] N Rehman and D P Mandic ldquoEmpirical mode de-composition for trivariate signalsrdquo IEEE Transactions onSignal Processing vol 58 no 3 pp 1059ndash1068 2010

[16] N Rehman and D P Mandic ldquoMultivariate empirical modedecompositionrdquo Proceedings of the Royal Society A Mathe-matical Physical and Engineering Sciences vol 466 no 2117pp 1291ndash1302 2010

[17] Y Lv R Yuan and G Song ldquoMultivariate empirical modedecomposition and its application to fault diagnosis of rollingbearingrdquo Mechanical Systems and Signal Processing vol 81pp 219ndash234 2016

[18] C Huang Y Meng and W Lei ldquoFull vector envelopetechnique based on complex local mean decomposition andits application in fault feature extraction for rotor systemrdquoJournal of Mechanical Engineering vol 52 no 7 p 69 2016in Chinese

[19] G Rilling P Flandrin and P Goncalves ldquoOn empirical modedecomposition and its algorithmsrdquo in Proceedings of IEEE-EURASIP Workshop on Nonlinear Signal and Image Pro-cessing pp 8ndash11 IEEE Trieste Italy June 2003

[20] L Yang X Chen and S Wang ldquoMechanism of fast time-varying vibration for rotorndashstator contact system with ap-plication to fault diagnosisrdquo Journal of Vibration andAcoustics vol 140 no 1 article 014501 2018

[21] I Daubechies J Lu and H-TWu ldquoSynchrosqueezed wavelettransforms an empirical mode decomposition-like toolrdquoApplied and Computational Harmonic Analysis vol 30 no 2pp 243ndash261 2011

[22] L-S Qu Holospectrum and Holobalancing Technique inMachinery Diagnosis Beijing Science Press Beijing China2007

Shock and Vibration 19

International Journal of

AerospaceEngineeringHindawiwwwhindawicom Volume 2018

RoboticsJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Active and Passive Electronic Components

VLSI Design

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Shock and Vibration

Hindawiwwwhindawicom Volume 2018

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawiwwwhindawicom

Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Control Scienceand Engineering

Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Journal ofEngineeringVolume 2018

SensorsJournal of

Hindawiwwwhindawicom Volume 2018

International Journal of

RotatingMachinery

Hindawiwwwhindawicom Volume 2018

Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Navigation and Observation

International Journal of

Hindawi

wwwhindawicom Volume 2018

Advances in

Multimedia

Submit your manuscripts atwwwhindawicom

In order to further verify the correctness of the in-stantaneous amplitude-frequency characteristics of theproposed method the real and imaginary parts of thecomposite fault signal z are analyzed separately using syn-chrosqueezed wavelet transforms (SWT) proposed in ref-erence [21]-e results are shown in Figure 22 It is seen thatthe time-frequency representations of the composite faultsignal z also include the AM-FM signal and the 1X signalwhich proves the correctness of the proposed methodCompared with the SWT method the instantaneousamplitude-frequency characteristics acquired by the HTmethod are relatively straightforward

44 lte Bistable Behavior Analysis of the Fan Rotor Based onBEMD -e bistability of the rotor is a nonlinear behaviorof the rotor-bearing system which is the state in which therotor jumps from one stable state to another forming astep -e bivariate signal of the bistable behavior iscomposed of two signals collected by two displacementsensors from orthogonal locations on the experimentaldevices in literature [22] as shown in Figure 23 Literature[22] shows that the cause of the bistable behavior remainsto be further explored -is paper uses this case to il-lustrate the feasibility of BEMD to analyze nonstationarysignals

0

40

80

Am

plitu

de Real(z)

0 100 200 300 400Frequency (Hz)

(c)

Am

plitu

de

0 100 200 300 400Frequency (Hz)

0

20

40

60

80

Imag(z)

(d)

Figure 19 -e composite fault signal z (a) the 3D time domain wave of z (b) the 2D plane of z (c) the Fourier spectrum of Real[z] (d) theFourier spectrum of Imag[z]

0025

05

minus30

3minus5

0

5

Time (s)Real(c1)

Imag

(c1)

(a)

0025

05

minus200

20minus20

0

20

Time (s)Real(c2 )

Imag

(c2)

(b)

0025

05

minus200

20minus50

0

50

Time (s)Real(c2)

Imag

(c2)

(c)

0025

05

minus1000

100minus100

0

100

Time (s)Real(c3)

Imag

(c3)

(d)

0025

05

minus400

40minus30

0

30

Time (s)Real(r)

Imag

(r)

(e)

Figure 20 -e decomposition results of the composite fault signal based on the improved BEMD method

14 Shock and Vibration

-e x and y signals in the horizontal and vertical di-rections of the left and right bearings respectively from thefan rotors are collected with four displacement sensorsLetting z x+ jy the time and frequency domain plots of z areshown in Figure 24 where the fan rotor speed is 5500 rpm thesampling frequency is 2000Hz and the number of samplingpoints is 1024 -e left and right columns respectively showthe time and frequency domain plots of the vibration signalsfrom the left and right bearings of the fan rotor Bistablebehavior arises in the fan rotor and the amplitudes of the

vibration signals vary significantly in different positions anddirections Further studies are required to explain the causesof this bistability -e present study focuses on extracting thebistable behavioral signal characteristics to verify the feasi-bility of the proposed method

-e decomposition results of the bistable behavioralsignals based on the improved BEMDmethod are shown inFigure 25 c1 c2 c3 and r are separated in order from zusing the improved BEMD method c1 shows a randomarrangement and is considered the high-frequency noise

0

10

20

Am

plitu

de

ax2ay2

0 025 05Time (s)

(a)

0

152

304

Freq

uenc

y (H

z)

0 025 05Time (s)

fy2

fx2

(b)

0

10

20

30

40

Am

plitu

de

ax3ay3

0 025 05Time (s)

(c)

0

56

112

Freq

uenc

y (H

z)

fx3fy3

0 025 05Time (s)

(d)

0 025 050

50

100

Time (s)

Am

plitu

de

ax4ay4

(e)

0 025 050

28

56

Time (s)

Freq

uenc

y (H

z)

fx4fy4

(f )

Figure 21 -e instantaneous amplitude and frequency of c2 c3 and c4 obtained by the HT (a) the instantaneous amplitude of the real partand imaginary part of c2 (b) the instantaneous frequency of the real part and imaginary part of c2 (c) the instantaneous amplitude of the realpart and imaginary part of c3 (d) the instantaneous frequency of the real part and imaginary part of c3 (e) the instantaneous amplitude of thereal part and imaginary part of c4 (f ) the instantaneous frequency of the real part and imaginary part of c4

Shock and Vibration 15

signal c2 is considered to represent the extracted bistablebehavior signals -e HT is applied to the real andimaginary parts of c2 to obtain the instantaneous amplitudeand frequency of c2 from the left and right columns fromFigure 25 as shown in Figure 26 Figure 27 shows thethree-dimensional time domain of ax2 and ay2 from the leftand right columns respectively Figure 26 shows that thevibration signal amplitude on the left side of the fan de-creases from large to small opposite of the behavior ofthe right -e horizontal vibration signal amplitude on theleft side of the fan is larger than that of the vertical di-rection signal opposite of the right -is result validatesthat the vibration signals from different directions orpositions are different when the fan produces bistablebehavior In addition the time of the bistable behavior canbe determined according to the jump point of the am-plitude or frequency

5 Discussion

-e BEMD algorithm decomposes two orthogonal di-rections of vibration signals as a complex signal which is atwo-dimensional digital signal processing method thusensuring that the real and imaginary parts have the samedecomposition scale Similar to EMD the envelope mean iscritical for the decomposition effect of BEMD but the en-velope mean in BEMD is three-dimensional If the numberof projection directions of the complex signal in three-dimensional space is larger the corresponding envelope

signal is also more -us the envelope mean value is moreaccurate and the BEMD decomposition effect is betterIncreasing the number of projection directions can improvemodal aliasing Like EMD BEMD also produces falsecomponents when decomposing signals Generally speakingthe energy of the false components is low and these low-energy false components do not contain fault characteristicinformation and the introduction of the energy thresholdcriterion in the termination condition can increase thedecomposition speed of the BEMD

-e experimental results show that there is a certaindifference in the existence of vibration signal character-istics in different directions when rotating machinery failsIn addition when the number of projection directionsis increased the decomposition speed of BEMD willdecrease

6 Conclusions

We use BEMD and HT to extract the instantaneousamplitude-frequency features of rotor faults A bivariateinstantaneous feature extraction method based on the im-proved BEMD method and the HT is investigated whichextends the fault feature extraction technology to two di-mensions -e BEMD method is suitable to analyze thecomplex multicomponent bivariate signals -e mainsingle-component bivariate signals are separated from themulticomponent bivariate signals of the fan rotor bistabilityfor the oil film oscillation and the oil film vortex using the

0

2

4

6

Time (s)

Freq

uenc

y (H

z)

0 01 02 03 04 052

3

4

5

6

7

8log2 fx

(a)

0

2

4

6

Time (s)

Freq

uenc

y (H

z)

0 01 02 03 04 052

3

4

5

6

7

8log2 fy

(b)

Figure 22 -e results of the composite fault signal z based on SWT the time-frequency representation of (a) the real part of z and (b) theimaginary part of z

Left bearing predstal

Locations of sensors

Fanrotor

Right bearing predstal

Axis

Orthogonal directions

Figure 23 -e schematic diagram of the experimental apparatus

16 Shock and Vibration

Real(z) 00256

0512minus300

0300

minus400

0

400

Time (s)

Imag

(z)

00256

0512minus500

0500

minus500

0

500

Time (s)Real(z)

Imag

(z)

Imag

(z)

minus300 0 300minus500

0

500

Real(z)minus500 0 500

minus500

0

Imag

(z)

Real(z)

500

0 100 200 300 4000

100

200

300

Frequency (Hz)

Am

plitu

de Imag(z)

Frequency (Hz)0 100 200 300 400

0

200

400

Am

plitu

de

Imag(z)

0 100 200 300 4000

70

140

Frequency (Hz)

Am

plitu

de

Real(z)

0 100 200 300 4000

200

400

Frequency (Hz)

Am

plitu

de Real(z)

(a) (b)

Figure 24 -e time and frequency domain plots of the bistable behavior signals

00256

0512

minus800

80minus80

0

80

Time (s)Real(c1 )

Imag

(c1)

Time (s)Real(c1 )

Imag

(c1)

00256

0512

minus400

40minus40

0

40

Time (s)Real(c2) 0

02560512

minus5000

500minus500

0

500

Imag

(c2)

Time (s)Real(c2)

Imag

(c2)

00256

0512

minus5000

500minus500

0

500

Time (s)Real(r) 00256

0512

minus800

80minus80

0

80

Imag

(irc

rm

)

Time (s)Real(r)

Imag

(r)

00256

0512

minus2000

200minus200

0

200

(a) (b)

Figure 25 -e decomposition results of the bistable behavior signals based on the improved BEMD method

Shock and Vibration 17

improved BEMD method For the single-component bi-variate signal the HT is used to obtain the correspondinginstantaneous amplitude and frequency characteristics -eproposed method can examine the detailed information of asingle rotation component

Data Availability

-e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

-e authors declare that they have no conflicts of interest

Authorsrsquo Contributions

All the authors contributed to this work Chuanjin Huangconceived and designed the simulation and experiments anddrafted the manuscript Haijun Song performed the simu-lations and experiments and analyzed the data and

0 0256 05120

50

100

150

Time (s)

Freq

uenc

y (H

z)

fx2

fy2

0 0256 05120

200

400

600

Time (s)

Am

plitu

de

ax2ay2

(a)

fx2

fy2

0 0256 05120

90

180

270

360

Time (s)

Freq

uenc

y (H

z)

0 0256 05120

200

400

600

Time (s)

Am

plitu

de

ax2Data 2

(b)

Figure 26 -e instantaneous amplitude and frequency of c2 from the (a) left and (b) right columns

0

0256

0512

0100

200300

400500

0

100

200

300

400

500

Time (s)

X 04035Y 3509Z 130

X 04235Y 2265Z 3613

X 007Y 1131Z 2191

X 0105Y 3837Z 3312

ax2

a y2

Figure 27 -e three-dimensional time domain of ax2 and ay2

18 Shock and Vibration

Wenping Lei and Yajun Meng performed the experimentsand analyzed the data All the authors contributed to thewriting and discussion of the paper

Acknowledgments

-is research was funded by the Henan Provincial HigherEducation Key Research Project (Grant nos 18A460006 and19A460029) Henan High-Level Innovative Scientific andTechnological Talent Team Construction Project (Grant noC20150034) and Zhengzhou Institute of Technology In-novation Team Project (Grant no CXTD2017K1)

References

[1] R Yan R X Gao and X Chen ldquoWavelets for fault diagnosisof rotary machines a review with applicationsrdquo Signal Pro-cessing vol 96 pp 1ndash15 2014

[2] J Cheng D Yu J Tang and Y Yang ldquoApplication of frequencyfamily separation method based upon EMD and local Hilbertenergy spectrum method to gear fault diagnosisrdquo Mechanismand Machine lteory vol 43 no 6 pp 712ndash723 2008

[3] H Liu and M Han ldquoA fault diagnosis method based on localmean decomposition and multi-scale entropy for rollerbearingsrdquoMechanism andMachinelteory vol 75 pp 67ndash782014

[4] Z Zheng W Jiang Z Wang Y Zhu and K Yang ldquoGear faultdiagnosis method based on local mean decomposition andgeneralized morphological fractal dimensionsrdquo Mechanismand Machine lteory vol 91 pp 151ndash167 2015

[5] W Yang R Court P J Tavner and C J Crabtree ldquoBivariateempirical mode decomposition and its contribution to windturbine condition monitoringrdquo Journal of Sound and Vi-bration vol 330 no 15 pp 3766ndash3782 2011

[6] L Qu X Liu G Peyronne and Y Chen ldquo-e holospectrum anewmethod for rotor surveillance and diagnosisrdquoMechanicalSystems amp Signal Processing vol 3 no 3 pp 255ndash267 1989

[7] F Q Wu and G Meng ldquoCompound rub malfunctions featureextraction based on full-spectrum cascade analysis and SVMrdquoMechanical Systems and Signal Processing vol 20 no 8pp 2007ndash2021 2006

[8] Y Chen Q Gao and Z Guan ldquoSelf-loosening failure analysisof bolt joints under vibration considering the tighteningprocessrdquo Shock and Vibration vol 2017 Article ID 203842115 pages 2017

[9] L Chen J Han W Lei Y Cui and Z Guan ldquoFull-vectorsignal acquisition and information fusion for the fault pre-dictionrdquo International Journal of Rotating Machineryvol 2016 Article ID 5980802 7 pages 2016

[10] C Chen Y Meng and Y Du ldquoApplication of the full vectorspectrum based on EMD in fault diagnosis of bearingsrdquoJournal of Mechanical Strength vol 37 pp 806ndash811 2015

[11] C Huang X Wu and W Cao ldquoLMD-based on full vectorenvelope technique and its application in TRT vibration faultdiagnosisrdquo Electric Power Automation Equipment vol 35pp 168ndash174 2015 in Chinese

[12] X Zhao T H Patel and M J Zuo ldquoMultivariate EMD andfull spectrum based condition monitoring for rotating ma-chineryrdquo Mechanical Systems and Signal Processing vol 27pp 712ndash728 2012

[13] G Rilling P Flandrin P Gonalves and J M Lilly ldquoBivariateempirical mode decompositionrdquo IEEE Signal ProcessingLetters vol 14 no 12 pp 936ndash939 2007

[14] C Park D Looney M M Van Hulle and D P Mandic ldquo-ecomplex local mean decompositionrdquo Neurocomputingvol 74 no 6 pp 867ndash875 2011

[15] N Rehman and D P Mandic ldquoEmpirical mode de-composition for trivariate signalsrdquo IEEE Transactions onSignal Processing vol 58 no 3 pp 1059ndash1068 2010

[16] N Rehman and D P Mandic ldquoMultivariate empirical modedecompositionrdquo Proceedings of the Royal Society A Mathe-matical Physical and Engineering Sciences vol 466 no 2117pp 1291ndash1302 2010

[17] Y Lv R Yuan and G Song ldquoMultivariate empirical modedecomposition and its application to fault diagnosis of rollingbearingrdquo Mechanical Systems and Signal Processing vol 81pp 219ndash234 2016

[18] C Huang Y Meng and W Lei ldquoFull vector envelopetechnique based on complex local mean decomposition andits application in fault feature extraction for rotor systemrdquoJournal of Mechanical Engineering vol 52 no 7 p 69 2016in Chinese

[19] G Rilling P Flandrin and P Goncalves ldquoOn empirical modedecomposition and its algorithmsrdquo in Proceedings of IEEE-EURASIP Workshop on Nonlinear Signal and Image Pro-cessing pp 8ndash11 IEEE Trieste Italy June 2003

[20] L Yang X Chen and S Wang ldquoMechanism of fast time-varying vibration for rotorndashstator contact system with ap-plication to fault diagnosisrdquo Journal of Vibration andAcoustics vol 140 no 1 article 014501 2018

[21] I Daubechies J Lu and H-TWu ldquoSynchrosqueezed wavelettransforms an empirical mode decomposition-like toolrdquoApplied and Computational Harmonic Analysis vol 30 no 2pp 243ndash261 2011

[22] L-S Qu Holospectrum and Holobalancing Technique inMachinery Diagnosis Beijing Science Press Beijing China2007

Shock and Vibration 19

International Journal of

AerospaceEngineeringHindawiwwwhindawicom Volume 2018

RoboticsJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Active and Passive Electronic Components

VLSI Design

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Shock and Vibration

Hindawiwwwhindawicom Volume 2018

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawiwwwhindawicom

Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Control Scienceand Engineering

Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Journal ofEngineeringVolume 2018

SensorsJournal of

Hindawiwwwhindawicom Volume 2018

International Journal of

RotatingMachinery

Hindawiwwwhindawicom Volume 2018

Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Navigation and Observation

International Journal of

Hindawi

wwwhindawicom Volume 2018

Advances in

Multimedia

Submit your manuscripts atwwwhindawicom

-e x and y signals in the horizontal and vertical di-rections of the left and right bearings respectively from thefan rotors are collected with four displacement sensorsLetting z x+ jy the time and frequency domain plots of z areshown in Figure 24 where the fan rotor speed is 5500 rpm thesampling frequency is 2000Hz and the number of samplingpoints is 1024 -e left and right columns respectively showthe time and frequency domain plots of the vibration signalsfrom the left and right bearings of the fan rotor Bistablebehavior arises in the fan rotor and the amplitudes of the

vibration signals vary significantly in different positions anddirections Further studies are required to explain the causesof this bistability -e present study focuses on extracting thebistable behavioral signal characteristics to verify the feasi-bility of the proposed method

-e decomposition results of the bistable behavioralsignals based on the improved BEMDmethod are shown inFigure 25 c1 c2 c3 and r are separated in order from zusing the improved BEMD method c1 shows a randomarrangement and is considered the high-frequency noise

0

10

20

Am

plitu

de

ax2ay2

0 025 05Time (s)

(a)

0

152

304

Freq

uenc

y (H

z)

0 025 05Time (s)

fy2

fx2

(b)

0

10

20

30

40

Am

plitu

de

ax3ay3

0 025 05Time (s)

(c)

0

56

112

Freq

uenc

y (H

z)

fx3fy3

0 025 05Time (s)

(d)

0 025 050

50

100

Time (s)

Am

plitu

de

ax4ay4

(e)

0 025 050

28

56

Time (s)

Freq

uenc

y (H

z)

fx4fy4

(f )

Figure 21 -e instantaneous amplitude and frequency of c2 c3 and c4 obtained by the HT (a) the instantaneous amplitude of the real partand imaginary part of c2 (b) the instantaneous frequency of the real part and imaginary part of c2 (c) the instantaneous amplitude of the realpart and imaginary part of c3 (d) the instantaneous frequency of the real part and imaginary part of c3 (e) the instantaneous amplitude of thereal part and imaginary part of c4 (f ) the instantaneous frequency of the real part and imaginary part of c4

Shock and Vibration 15

signal c2 is considered to represent the extracted bistablebehavior signals -e HT is applied to the real andimaginary parts of c2 to obtain the instantaneous amplitudeand frequency of c2 from the left and right columns fromFigure 25 as shown in Figure 26 Figure 27 shows thethree-dimensional time domain of ax2 and ay2 from the leftand right columns respectively Figure 26 shows that thevibration signal amplitude on the left side of the fan de-creases from large to small opposite of the behavior ofthe right -e horizontal vibration signal amplitude on theleft side of the fan is larger than that of the vertical di-rection signal opposite of the right -is result validatesthat the vibration signals from different directions orpositions are different when the fan produces bistablebehavior In addition the time of the bistable behavior canbe determined according to the jump point of the am-plitude or frequency

5 Discussion

-e BEMD algorithm decomposes two orthogonal di-rections of vibration signals as a complex signal which is atwo-dimensional digital signal processing method thusensuring that the real and imaginary parts have the samedecomposition scale Similar to EMD the envelope mean iscritical for the decomposition effect of BEMD but the en-velope mean in BEMD is three-dimensional If the numberof projection directions of the complex signal in three-dimensional space is larger the corresponding envelope

signal is also more -us the envelope mean value is moreaccurate and the BEMD decomposition effect is betterIncreasing the number of projection directions can improvemodal aliasing Like EMD BEMD also produces falsecomponents when decomposing signals Generally speakingthe energy of the false components is low and these low-energy false components do not contain fault characteristicinformation and the introduction of the energy thresholdcriterion in the termination condition can increase thedecomposition speed of the BEMD

-e experimental results show that there is a certaindifference in the existence of vibration signal character-istics in different directions when rotating machinery failsIn addition when the number of projection directionsis increased the decomposition speed of BEMD willdecrease

6 Conclusions

We use BEMD and HT to extract the instantaneousamplitude-frequency features of rotor faults A bivariateinstantaneous feature extraction method based on the im-proved BEMD method and the HT is investigated whichextends the fault feature extraction technology to two di-mensions -e BEMD method is suitable to analyze thecomplex multicomponent bivariate signals -e mainsingle-component bivariate signals are separated from themulticomponent bivariate signals of the fan rotor bistabilityfor the oil film oscillation and the oil film vortex using the

0

2

4

6

Time (s)

Freq

uenc

y (H

z)

0 01 02 03 04 052

3

4

5

6

7

8log2 fx

(a)

0

2

4

6

Time (s)

Freq

uenc

y (H

z)

0 01 02 03 04 052

3

4

5

6

7

8log2 fy

(b)

Figure 22 -e results of the composite fault signal z based on SWT the time-frequency representation of (a) the real part of z and (b) theimaginary part of z

Left bearing predstal

Locations of sensors

Fanrotor

Right bearing predstal

Axis

Orthogonal directions

Figure 23 -e schematic diagram of the experimental apparatus

16 Shock and Vibration

Real(z) 00256

0512minus300

0300

minus400

0

400

Time (s)

Imag

(z)

00256

0512minus500

0500

minus500

0

500

Time (s)Real(z)

Imag

(z)

Imag

(z)

minus300 0 300minus500

0

500

Real(z)minus500 0 500

minus500

0

Imag

(z)

Real(z)

500

0 100 200 300 4000

100

200

300

Frequency (Hz)

Am

plitu

de Imag(z)

Frequency (Hz)0 100 200 300 400

0

200

400

Am

plitu

de

Imag(z)

0 100 200 300 4000

70

140

Frequency (Hz)

Am

plitu

de

Real(z)

0 100 200 300 4000

200

400

Frequency (Hz)

Am

plitu

de Real(z)

(a) (b)

Figure 24 -e time and frequency domain plots of the bistable behavior signals

00256

0512

minus800

80minus80

0

80

Time (s)Real(c1 )

Imag

(c1)

Time (s)Real(c1 )

Imag

(c1)

00256

0512

minus400

40minus40

0

40

Time (s)Real(c2) 0

02560512

minus5000

500minus500

0

500

Imag

(c2)

Time (s)Real(c2)

Imag

(c2)

00256

0512

minus5000

500minus500

0

500

Time (s)Real(r) 00256

0512

minus800

80minus80

0

80

Imag

(irc

rm

)

Time (s)Real(r)

Imag

(r)

00256

0512

minus2000

200minus200

0

200

(a) (b)

Figure 25 -e decomposition results of the bistable behavior signals based on the improved BEMD method

Shock and Vibration 17

improved BEMD method For the single-component bi-variate signal the HT is used to obtain the correspondinginstantaneous amplitude and frequency characteristics -eproposed method can examine the detailed information of asingle rotation component

Data Availability

-e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

-e authors declare that they have no conflicts of interest

Authorsrsquo Contributions

All the authors contributed to this work Chuanjin Huangconceived and designed the simulation and experiments anddrafted the manuscript Haijun Song performed the simu-lations and experiments and analyzed the data and

0 0256 05120

50

100

150

Time (s)

Freq

uenc

y (H

z)

fx2

fy2

0 0256 05120

200

400

600

Time (s)

Am

plitu

de

ax2ay2

(a)

fx2

fy2

0 0256 05120

90

180

270

360

Time (s)

Freq

uenc

y (H

z)

0 0256 05120

200

400

600

Time (s)

Am

plitu

de

ax2Data 2

(b)

Figure 26 -e instantaneous amplitude and frequency of c2 from the (a) left and (b) right columns

0

0256

0512

0100

200300

400500

0

100

200

300

400

500

Time (s)

X 04035Y 3509Z 130

X 04235Y 2265Z 3613

X 007Y 1131Z 2191

X 0105Y 3837Z 3312

ax2

a y2

Figure 27 -e three-dimensional time domain of ax2 and ay2

18 Shock and Vibration

Wenping Lei and Yajun Meng performed the experimentsand analyzed the data All the authors contributed to thewriting and discussion of the paper

Acknowledgments

-is research was funded by the Henan Provincial HigherEducation Key Research Project (Grant nos 18A460006 and19A460029) Henan High-Level Innovative Scientific andTechnological Talent Team Construction Project (Grant noC20150034) and Zhengzhou Institute of Technology In-novation Team Project (Grant no CXTD2017K1)

References

[1] R Yan R X Gao and X Chen ldquoWavelets for fault diagnosisof rotary machines a review with applicationsrdquo Signal Pro-cessing vol 96 pp 1ndash15 2014

[2] J Cheng D Yu J Tang and Y Yang ldquoApplication of frequencyfamily separation method based upon EMD and local Hilbertenergy spectrum method to gear fault diagnosisrdquo Mechanismand Machine lteory vol 43 no 6 pp 712ndash723 2008

[3] H Liu and M Han ldquoA fault diagnosis method based on localmean decomposition and multi-scale entropy for rollerbearingsrdquoMechanism andMachinelteory vol 75 pp 67ndash782014

[4] Z Zheng W Jiang Z Wang Y Zhu and K Yang ldquoGear faultdiagnosis method based on local mean decomposition andgeneralized morphological fractal dimensionsrdquo Mechanismand Machine lteory vol 91 pp 151ndash167 2015

[5] W Yang R Court P J Tavner and C J Crabtree ldquoBivariateempirical mode decomposition and its contribution to windturbine condition monitoringrdquo Journal of Sound and Vi-bration vol 330 no 15 pp 3766ndash3782 2011

[6] L Qu X Liu G Peyronne and Y Chen ldquo-e holospectrum anewmethod for rotor surveillance and diagnosisrdquoMechanicalSystems amp Signal Processing vol 3 no 3 pp 255ndash267 1989

[7] F Q Wu and G Meng ldquoCompound rub malfunctions featureextraction based on full-spectrum cascade analysis and SVMrdquoMechanical Systems and Signal Processing vol 20 no 8pp 2007ndash2021 2006

[8] Y Chen Q Gao and Z Guan ldquoSelf-loosening failure analysisof bolt joints under vibration considering the tighteningprocessrdquo Shock and Vibration vol 2017 Article ID 203842115 pages 2017

[9] L Chen J Han W Lei Y Cui and Z Guan ldquoFull-vectorsignal acquisition and information fusion for the fault pre-dictionrdquo International Journal of Rotating Machineryvol 2016 Article ID 5980802 7 pages 2016

[10] C Chen Y Meng and Y Du ldquoApplication of the full vectorspectrum based on EMD in fault diagnosis of bearingsrdquoJournal of Mechanical Strength vol 37 pp 806ndash811 2015

[11] C Huang X Wu and W Cao ldquoLMD-based on full vectorenvelope technique and its application in TRT vibration faultdiagnosisrdquo Electric Power Automation Equipment vol 35pp 168ndash174 2015 in Chinese

[12] X Zhao T H Patel and M J Zuo ldquoMultivariate EMD andfull spectrum based condition monitoring for rotating ma-chineryrdquo Mechanical Systems and Signal Processing vol 27pp 712ndash728 2012

[13] G Rilling P Flandrin P Gonalves and J M Lilly ldquoBivariateempirical mode decompositionrdquo IEEE Signal ProcessingLetters vol 14 no 12 pp 936ndash939 2007

[14] C Park D Looney M M Van Hulle and D P Mandic ldquo-ecomplex local mean decompositionrdquo Neurocomputingvol 74 no 6 pp 867ndash875 2011

[15] N Rehman and D P Mandic ldquoEmpirical mode de-composition for trivariate signalsrdquo IEEE Transactions onSignal Processing vol 58 no 3 pp 1059ndash1068 2010

[16] N Rehman and D P Mandic ldquoMultivariate empirical modedecompositionrdquo Proceedings of the Royal Society A Mathe-matical Physical and Engineering Sciences vol 466 no 2117pp 1291ndash1302 2010

[17] Y Lv R Yuan and G Song ldquoMultivariate empirical modedecomposition and its application to fault diagnosis of rollingbearingrdquo Mechanical Systems and Signal Processing vol 81pp 219ndash234 2016

[18] C Huang Y Meng and W Lei ldquoFull vector envelopetechnique based on complex local mean decomposition andits application in fault feature extraction for rotor systemrdquoJournal of Mechanical Engineering vol 52 no 7 p 69 2016in Chinese

[19] G Rilling P Flandrin and P Goncalves ldquoOn empirical modedecomposition and its algorithmsrdquo in Proceedings of IEEE-EURASIP Workshop on Nonlinear Signal and Image Pro-cessing pp 8ndash11 IEEE Trieste Italy June 2003

[20] L Yang X Chen and S Wang ldquoMechanism of fast time-varying vibration for rotorndashstator contact system with ap-plication to fault diagnosisrdquo Journal of Vibration andAcoustics vol 140 no 1 article 014501 2018

[21] I Daubechies J Lu and H-TWu ldquoSynchrosqueezed wavelettransforms an empirical mode decomposition-like toolrdquoApplied and Computational Harmonic Analysis vol 30 no 2pp 243ndash261 2011

[22] L-S Qu Holospectrum and Holobalancing Technique inMachinery Diagnosis Beijing Science Press Beijing China2007

Shock and Vibration 19

International Journal of

AerospaceEngineeringHindawiwwwhindawicom Volume 2018

RoboticsJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Active and Passive Electronic Components

VLSI Design

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Shock and Vibration

Hindawiwwwhindawicom Volume 2018

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawiwwwhindawicom

Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Control Scienceand Engineering

Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Journal ofEngineeringVolume 2018

SensorsJournal of

Hindawiwwwhindawicom Volume 2018

International Journal of

RotatingMachinery

Hindawiwwwhindawicom Volume 2018

Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Navigation and Observation

International Journal of

Hindawi

wwwhindawicom Volume 2018

Advances in

Multimedia

Submit your manuscripts atwwwhindawicom

signal c2 is considered to represent the extracted bistablebehavior signals -e HT is applied to the real andimaginary parts of c2 to obtain the instantaneous amplitudeand frequency of c2 from the left and right columns fromFigure 25 as shown in Figure 26 Figure 27 shows thethree-dimensional time domain of ax2 and ay2 from the leftand right columns respectively Figure 26 shows that thevibration signal amplitude on the left side of the fan de-creases from large to small opposite of the behavior ofthe right -e horizontal vibration signal amplitude on theleft side of the fan is larger than that of the vertical di-rection signal opposite of the right -is result validatesthat the vibration signals from different directions orpositions are different when the fan produces bistablebehavior In addition the time of the bistable behavior canbe determined according to the jump point of the am-plitude or frequency

5 Discussion

-e BEMD algorithm decomposes two orthogonal di-rections of vibration signals as a complex signal which is atwo-dimensional digital signal processing method thusensuring that the real and imaginary parts have the samedecomposition scale Similar to EMD the envelope mean iscritical for the decomposition effect of BEMD but the en-velope mean in BEMD is three-dimensional If the numberof projection directions of the complex signal in three-dimensional space is larger the corresponding envelope

signal is also more -us the envelope mean value is moreaccurate and the BEMD decomposition effect is betterIncreasing the number of projection directions can improvemodal aliasing Like EMD BEMD also produces falsecomponents when decomposing signals Generally speakingthe energy of the false components is low and these low-energy false components do not contain fault characteristicinformation and the introduction of the energy thresholdcriterion in the termination condition can increase thedecomposition speed of the BEMD

-e experimental results show that there is a certaindifference in the existence of vibration signal character-istics in different directions when rotating machinery failsIn addition when the number of projection directionsis increased the decomposition speed of BEMD willdecrease

6 Conclusions

We use BEMD and HT to extract the instantaneousamplitude-frequency features of rotor faults A bivariateinstantaneous feature extraction method based on the im-proved BEMD method and the HT is investigated whichextends the fault feature extraction technology to two di-mensions -e BEMD method is suitable to analyze thecomplex multicomponent bivariate signals -e mainsingle-component bivariate signals are separated from themulticomponent bivariate signals of the fan rotor bistabilityfor the oil film oscillation and the oil film vortex using the

0

2

4

6

Time (s)

Freq

uenc

y (H

z)

0 01 02 03 04 052

3

4

5

6

7

8log2 fx

(a)

0

2

4

6

Time (s)

Freq

uenc

y (H

z)

0 01 02 03 04 052

3

4

5

6

7

8log2 fy

(b)

Figure 22 -e results of the composite fault signal z based on SWT the time-frequency representation of (a) the real part of z and (b) theimaginary part of z

Left bearing predstal

Locations of sensors

Fanrotor

Right bearing predstal

Axis

Orthogonal directions

Figure 23 -e schematic diagram of the experimental apparatus

16 Shock and Vibration

Real(z) 00256

0512minus300

0300

minus400

0

400

Time (s)

Imag

(z)

00256

0512minus500

0500

minus500

0

500

Time (s)Real(z)

Imag

(z)

Imag

(z)

minus300 0 300minus500

0

500

Real(z)minus500 0 500

minus500

0

Imag

(z)

Real(z)

500

0 100 200 300 4000

100

200

300

Frequency (Hz)

Am

plitu

de Imag(z)

Frequency (Hz)0 100 200 300 400

0

200

400

Am

plitu

de

Imag(z)

0 100 200 300 4000

70

140

Frequency (Hz)

Am

plitu

de

Real(z)

0 100 200 300 4000

200

400

Frequency (Hz)

Am

plitu

de Real(z)

(a) (b)

Figure 24 -e time and frequency domain plots of the bistable behavior signals

00256

0512

minus800

80minus80

0

80

Time (s)Real(c1 )

Imag

(c1)

Time (s)Real(c1 )

Imag

(c1)

00256

0512

minus400

40minus40

0

40

Time (s)Real(c2) 0

02560512

minus5000

500minus500

0

500

Imag

(c2)

Time (s)Real(c2)

Imag

(c2)

00256

0512

minus5000

500minus500

0

500

Time (s)Real(r) 00256

0512

minus800

80minus80

0

80

Imag

(irc

rm

)

Time (s)Real(r)

Imag

(r)

00256

0512

minus2000

200minus200

0

200

(a) (b)

Figure 25 -e decomposition results of the bistable behavior signals based on the improved BEMD method

Shock and Vibration 17

improved BEMD method For the single-component bi-variate signal the HT is used to obtain the correspondinginstantaneous amplitude and frequency characteristics -eproposed method can examine the detailed information of asingle rotation component

Data Availability

-e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

-e authors declare that they have no conflicts of interest

Authorsrsquo Contributions

All the authors contributed to this work Chuanjin Huangconceived and designed the simulation and experiments anddrafted the manuscript Haijun Song performed the simu-lations and experiments and analyzed the data and

0 0256 05120

50

100

150

Time (s)

Freq

uenc

y (H

z)

fx2

fy2

0 0256 05120

200

400

600

Time (s)

Am

plitu

de

ax2ay2

(a)

fx2

fy2

0 0256 05120

90

180

270

360

Time (s)

Freq

uenc

y (H

z)

0 0256 05120

200

400

600

Time (s)

Am

plitu

de

ax2Data 2

(b)

Figure 26 -e instantaneous amplitude and frequency of c2 from the (a) left and (b) right columns

0

0256

0512

0100

200300

400500

0

100

200

300

400

500

Time (s)

X 04035Y 3509Z 130

X 04235Y 2265Z 3613

X 007Y 1131Z 2191

X 0105Y 3837Z 3312

ax2

a y2

Figure 27 -e three-dimensional time domain of ax2 and ay2

18 Shock and Vibration

Wenping Lei and Yajun Meng performed the experimentsand analyzed the data All the authors contributed to thewriting and discussion of the paper

Acknowledgments

-is research was funded by the Henan Provincial HigherEducation Key Research Project (Grant nos 18A460006 and19A460029) Henan High-Level Innovative Scientific andTechnological Talent Team Construction Project (Grant noC20150034) and Zhengzhou Institute of Technology In-novation Team Project (Grant no CXTD2017K1)

References

[1] R Yan R X Gao and X Chen ldquoWavelets for fault diagnosisof rotary machines a review with applicationsrdquo Signal Pro-cessing vol 96 pp 1ndash15 2014

[2] J Cheng D Yu J Tang and Y Yang ldquoApplication of frequencyfamily separation method based upon EMD and local Hilbertenergy spectrum method to gear fault diagnosisrdquo Mechanismand Machine lteory vol 43 no 6 pp 712ndash723 2008

[3] H Liu and M Han ldquoA fault diagnosis method based on localmean decomposition and multi-scale entropy for rollerbearingsrdquoMechanism andMachinelteory vol 75 pp 67ndash782014

[4] Z Zheng W Jiang Z Wang Y Zhu and K Yang ldquoGear faultdiagnosis method based on local mean decomposition andgeneralized morphological fractal dimensionsrdquo Mechanismand Machine lteory vol 91 pp 151ndash167 2015

[5] W Yang R Court P J Tavner and C J Crabtree ldquoBivariateempirical mode decomposition and its contribution to windturbine condition monitoringrdquo Journal of Sound and Vi-bration vol 330 no 15 pp 3766ndash3782 2011

[6] L Qu X Liu G Peyronne and Y Chen ldquo-e holospectrum anewmethod for rotor surveillance and diagnosisrdquoMechanicalSystems amp Signal Processing vol 3 no 3 pp 255ndash267 1989

[7] F Q Wu and G Meng ldquoCompound rub malfunctions featureextraction based on full-spectrum cascade analysis and SVMrdquoMechanical Systems and Signal Processing vol 20 no 8pp 2007ndash2021 2006

[8] Y Chen Q Gao and Z Guan ldquoSelf-loosening failure analysisof bolt joints under vibration considering the tighteningprocessrdquo Shock and Vibration vol 2017 Article ID 203842115 pages 2017

[9] L Chen J Han W Lei Y Cui and Z Guan ldquoFull-vectorsignal acquisition and information fusion for the fault pre-dictionrdquo International Journal of Rotating Machineryvol 2016 Article ID 5980802 7 pages 2016

[10] C Chen Y Meng and Y Du ldquoApplication of the full vectorspectrum based on EMD in fault diagnosis of bearingsrdquoJournal of Mechanical Strength vol 37 pp 806ndash811 2015

[11] C Huang X Wu and W Cao ldquoLMD-based on full vectorenvelope technique and its application in TRT vibration faultdiagnosisrdquo Electric Power Automation Equipment vol 35pp 168ndash174 2015 in Chinese

[12] X Zhao T H Patel and M J Zuo ldquoMultivariate EMD andfull spectrum based condition monitoring for rotating ma-chineryrdquo Mechanical Systems and Signal Processing vol 27pp 712ndash728 2012

[13] G Rilling P Flandrin P Gonalves and J M Lilly ldquoBivariateempirical mode decompositionrdquo IEEE Signal ProcessingLetters vol 14 no 12 pp 936ndash939 2007

[14] C Park D Looney M M Van Hulle and D P Mandic ldquo-ecomplex local mean decompositionrdquo Neurocomputingvol 74 no 6 pp 867ndash875 2011

[15] N Rehman and D P Mandic ldquoEmpirical mode de-composition for trivariate signalsrdquo IEEE Transactions onSignal Processing vol 58 no 3 pp 1059ndash1068 2010

[16] N Rehman and D P Mandic ldquoMultivariate empirical modedecompositionrdquo Proceedings of the Royal Society A Mathe-matical Physical and Engineering Sciences vol 466 no 2117pp 1291ndash1302 2010

[17] Y Lv R Yuan and G Song ldquoMultivariate empirical modedecomposition and its application to fault diagnosis of rollingbearingrdquo Mechanical Systems and Signal Processing vol 81pp 219ndash234 2016

[18] C Huang Y Meng and W Lei ldquoFull vector envelopetechnique based on complex local mean decomposition andits application in fault feature extraction for rotor systemrdquoJournal of Mechanical Engineering vol 52 no 7 p 69 2016in Chinese

[19] G Rilling P Flandrin and P Goncalves ldquoOn empirical modedecomposition and its algorithmsrdquo in Proceedings of IEEE-EURASIP Workshop on Nonlinear Signal and Image Pro-cessing pp 8ndash11 IEEE Trieste Italy June 2003

[20] L Yang X Chen and S Wang ldquoMechanism of fast time-varying vibration for rotorndashstator contact system with ap-plication to fault diagnosisrdquo Journal of Vibration andAcoustics vol 140 no 1 article 014501 2018

[21] I Daubechies J Lu and H-TWu ldquoSynchrosqueezed wavelettransforms an empirical mode decomposition-like toolrdquoApplied and Computational Harmonic Analysis vol 30 no 2pp 243ndash261 2011

[22] L-S Qu Holospectrum and Holobalancing Technique inMachinery Diagnosis Beijing Science Press Beijing China2007

Shock and Vibration 19

International Journal of

AerospaceEngineeringHindawiwwwhindawicom Volume 2018

RoboticsJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Active and Passive Electronic Components

VLSI Design

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Shock and Vibration

Hindawiwwwhindawicom Volume 2018

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawiwwwhindawicom

Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Control Scienceand Engineering

Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Journal ofEngineeringVolume 2018

SensorsJournal of

Hindawiwwwhindawicom Volume 2018

International Journal of

RotatingMachinery

Hindawiwwwhindawicom Volume 2018

Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Navigation and Observation

International Journal of

Hindawi

wwwhindawicom Volume 2018

Advances in

Multimedia

Submit your manuscripts atwwwhindawicom

Real(z) 00256

0512minus300

0300

minus400

0

400

Time (s)

Imag

(z)

00256

0512minus500

0500

minus500

0

500

Time (s)Real(z)

Imag

(z)

Imag

(z)

minus300 0 300minus500

0

500

Real(z)minus500 0 500

minus500

0

Imag

(z)

Real(z)

500

0 100 200 300 4000

100

200

300

Frequency (Hz)

Am

plitu

de Imag(z)

Frequency (Hz)0 100 200 300 400

0

200

400

Am

plitu

de

Imag(z)

0 100 200 300 4000

70

140

Frequency (Hz)

Am

plitu

de

Real(z)

0 100 200 300 4000

200

400

Frequency (Hz)

Am

plitu

de Real(z)

(a) (b)

Figure 24 -e time and frequency domain plots of the bistable behavior signals

00256

0512

minus800

80minus80

0

80

Time (s)Real(c1 )

Imag

(c1)

Time (s)Real(c1 )

Imag

(c1)

00256

0512

minus400

40minus40

0

40

Time (s)Real(c2) 0

02560512

minus5000

500minus500

0

500

Imag

(c2)

Time (s)Real(c2)

Imag

(c2)

00256

0512

minus5000

500minus500

0

500

Time (s)Real(r) 00256

0512

minus800

80minus80

0

80

Imag

(irc

rm

)

Time (s)Real(r)

Imag

(r)

00256

0512

minus2000

200minus200

0

200

(a) (b)

Figure 25 -e decomposition results of the bistable behavior signals based on the improved BEMD method

Shock and Vibration 17

improved BEMD method For the single-component bi-variate signal the HT is used to obtain the correspondinginstantaneous amplitude and frequency characteristics -eproposed method can examine the detailed information of asingle rotation component

Data Availability

-e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

-e authors declare that they have no conflicts of interest

Authorsrsquo Contributions

All the authors contributed to this work Chuanjin Huangconceived and designed the simulation and experiments anddrafted the manuscript Haijun Song performed the simu-lations and experiments and analyzed the data and

0 0256 05120

50

100

150

Time (s)

Freq

uenc

y (H

z)

fx2

fy2

0 0256 05120

200

400

600

Time (s)

Am

plitu

de

ax2ay2

(a)

fx2

fy2

0 0256 05120

90

180

270

360

Time (s)

Freq

uenc

y (H

z)

0 0256 05120

200

400

600

Time (s)

Am

plitu

de

ax2Data 2

(b)

Figure 26 -e instantaneous amplitude and frequency of c2 from the (a) left and (b) right columns

0

0256

0512

0100

200300

400500

0

100

200

300

400

500

Time (s)

X 04035Y 3509Z 130

X 04235Y 2265Z 3613

X 007Y 1131Z 2191

X 0105Y 3837Z 3312

ax2

a y2

Figure 27 -e three-dimensional time domain of ax2 and ay2

18 Shock and Vibration

Wenping Lei and Yajun Meng performed the experimentsand analyzed the data All the authors contributed to thewriting and discussion of the paper

Acknowledgments

-is research was funded by the Henan Provincial HigherEducation Key Research Project (Grant nos 18A460006 and19A460029) Henan High-Level Innovative Scientific andTechnological Talent Team Construction Project (Grant noC20150034) and Zhengzhou Institute of Technology In-novation Team Project (Grant no CXTD2017K1)

References

[1] R Yan R X Gao and X Chen ldquoWavelets for fault diagnosisof rotary machines a review with applicationsrdquo Signal Pro-cessing vol 96 pp 1ndash15 2014

[2] J Cheng D Yu J Tang and Y Yang ldquoApplication of frequencyfamily separation method based upon EMD and local Hilbertenergy spectrum method to gear fault diagnosisrdquo Mechanismand Machine lteory vol 43 no 6 pp 712ndash723 2008

[3] H Liu and M Han ldquoA fault diagnosis method based on localmean decomposition and multi-scale entropy for rollerbearingsrdquoMechanism andMachinelteory vol 75 pp 67ndash782014

[4] Z Zheng W Jiang Z Wang Y Zhu and K Yang ldquoGear faultdiagnosis method based on local mean decomposition andgeneralized morphological fractal dimensionsrdquo Mechanismand Machine lteory vol 91 pp 151ndash167 2015

[5] W Yang R Court P J Tavner and C J Crabtree ldquoBivariateempirical mode decomposition and its contribution to windturbine condition monitoringrdquo Journal of Sound and Vi-bration vol 330 no 15 pp 3766ndash3782 2011

[6] L Qu X Liu G Peyronne and Y Chen ldquo-e holospectrum anewmethod for rotor surveillance and diagnosisrdquoMechanicalSystems amp Signal Processing vol 3 no 3 pp 255ndash267 1989

[7] F Q Wu and G Meng ldquoCompound rub malfunctions featureextraction based on full-spectrum cascade analysis and SVMrdquoMechanical Systems and Signal Processing vol 20 no 8pp 2007ndash2021 2006

[8] Y Chen Q Gao and Z Guan ldquoSelf-loosening failure analysisof bolt joints under vibration considering the tighteningprocessrdquo Shock and Vibration vol 2017 Article ID 203842115 pages 2017

[9] L Chen J Han W Lei Y Cui and Z Guan ldquoFull-vectorsignal acquisition and information fusion for the fault pre-dictionrdquo International Journal of Rotating Machineryvol 2016 Article ID 5980802 7 pages 2016

[10] C Chen Y Meng and Y Du ldquoApplication of the full vectorspectrum based on EMD in fault diagnosis of bearingsrdquoJournal of Mechanical Strength vol 37 pp 806ndash811 2015

[11] C Huang X Wu and W Cao ldquoLMD-based on full vectorenvelope technique and its application in TRT vibration faultdiagnosisrdquo Electric Power Automation Equipment vol 35pp 168ndash174 2015 in Chinese

[12] X Zhao T H Patel and M J Zuo ldquoMultivariate EMD andfull spectrum based condition monitoring for rotating ma-chineryrdquo Mechanical Systems and Signal Processing vol 27pp 712ndash728 2012

[13] G Rilling P Flandrin P Gonalves and J M Lilly ldquoBivariateempirical mode decompositionrdquo IEEE Signal ProcessingLetters vol 14 no 12 pp 936ndash939 2007

[14] C Park D Looney M M Van Hulle and D P Mandic ldquo-ecomplex local mean decompositionrdquo Neurocomputingvol 74 no 6 pp 867ndash875 2011

[15] N Rehman and D P Mandic ldquoEmpirical mode de-composition for trivariate signalsrdquo IEEE Transactions onSignal Processing vol 58 no 3 pp 1059ndash1068 2010

[16] N Rehman and D P Mandic ldquoMultivariate empirical modedecompositionrdquo Proceedings of the Royal Society A Mathe-matical Physical and Engineering Sciences vol 466 no 2117pp 1291ndash1302 2010

[17] Y Lv R Yuan and G Song ldquoMultivariate empirical modedecomposition and its application to fault diagnosis of rollingbearingrdquo Mechanical Systems and Signal Processing vol 81pp 219ndash234 2016

[18] C Huang Y Meng and W Lei ldquoFull vector envelopetechnique based on complex local mean decomposition andits application in fault feature extraction for rotor systemrdquoJournal of Mechanical Engineering vol 52 no 7 p 69 2016in Chinese

[19] G Rilling P Flandrin and P Goncalves ldquoOn empirical modedecomposition and its algorithmsrdquo in Proceedings of IEEE-EURASIP Workshop on Nonlinear Signal and Image Pro-cessing pp 8ndash11 IEEE Trieste Italy June 2003

[20] L Yang X Chen and S Wang ldquoMechanism of fast time-varying vibration for rotorndashstator contact system with ap-plication to fault diagnosisrdquo Journal of Vibration andAcoustics vol 140 no 1 article 014501 2018

[21] I Daubechies J Lu and H-TWu ldquoSynchrosqueezed wavelettransforms an empirical mode decomposition-like toolrdquoApplied and Computational Harmonic Analysis vol 30 no 2pp 243ndash261 2011

[22] L-S Qu Holospectrum and Holobalancing Technique inMachinery Diagnosis Beijing Science Press Beijing China2007

Shock and Vibration 19

International Journal of

AerospaceEngineeringHindawiwwwhindawicom Volume 2018

RoboticsJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Active and Passive Electronic Components

VLSI Design

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Shock and Vibration

Hindawiwwwhindawicom Volume 2018

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawiwwwhindawicom

Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Control Scienceand Engineering

Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Journal ofEngineeringVolume 2018

SensorsJournal of

Hindawiwwwhindawicom Volume 2018

International Journal of

RotatingMachinery

Hindawiwwwhindawicom Volume 2018

Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Navigation and Observation

International Journal of

Hindawi

wwwhindawicom Volume 2018

Advances in

Multimedia

Submit your manuscripts atwwwhindawicom

improved BEMD method For the single-component bi-variate signal the HT is used to obtain the correspondinginstantaneous amplitude and frequency characteristics -eproposed method can examine the detailed information of asingle rotation component

Data Availability

-e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

-e authors declare that they have no conflicts of interest

Authorsrsquo Contributions

All the authors contributed to this work Chuanjin Huangconceived and designed the simulation and experiments anddrafted the manuscript Haijun Song performed the simu-lations and experiments and analyzed the data and

0 0256 05120

50

100

150

Time (s)

Freq

uenc

y (H

z)

fx2

fy2

0 0256 05120

200

400

600

Time (s)

Am

plitu

de

ax2ay2

(a)

fx2

fy2

0 0256 05120

90

180

270

360

Time (s)

Freq

uenc

y (H

z)

0 0256 05120

200

400

600

Time (s)

Am

plitu

de

ax2Data 2

(b)

Figure 26 -e instantaneous amplitude and frequency of c2 from the (a) left and (b) right columns

0

0256

0512

0100

200300

400500

0

100

200

300

400

500

Time (s)

X 04035Y 3509Z 130

X 04235Y 2265Z 3613

X 007Y 1131Z 2191

X 0105Y 3837Z 3312

ax2

a y2

Figure 27 -e three-dimensional time domain of ax2 and ay2

18 Shock and Vibration

Wenping Lei and Yajun Meng performed the experimentsand analyzed the data All the authors contributed to thewriting and discussion of the paper

Acknowledgments

-is research was funded by the Henan Provincial HigherEducation Key Research Project (Grant nos 18A460006 and19A460029) Henan High-Level Innovative Scientific andTechnological Talent Team Construction Project (Grant noC20150034) and Zhengzhou Institute of Technology In-novation Team Project (Grant no CXTD2017K1)

References

[1] R Yan R X Gao and X Chen ldquoWavelets for fault diagnosisof rotary machines a review with applicationsrdquo Signal Pro-cessing vol 96 pp 1ndash15 2014

[2] J Cheng D Yu J Tang and Y Yang ldquoApplication of frequencyfamily separation method based upon EMD and local Hilbertenergy spectrum method to gear fault diagnosisrdquo Mechanismand Machine lteory vol 43 no 6 pp 712ndash723 2008

[3] H Liu and M Han ldquoA fault diagnosis method based on localmean decomposition and multi-scale entropy for rollerbearingsrdquoMechanism andMachinelteory vol 75 pp 67ndash782014

[4] Z Zheng W Jiang Z Wang Y Zhu and K Yang ldquoGear faultdiagnosis method based on local mean decomposition andgeneralized morphological fractal dimensionsrdquo Mechanismand Machine lteory vol 91 pp 151ndash167 2015

[5] W Yang R Court P J Tavner and C J Crabtree ldquoBivariateempirical mode decomposition and its contribution to windturbine condition monitoringrdquo Journal of Sound and Vi-bration vol 330 no 15 pp 3766ndash3782 2011

[6] L Qu X Liu G Peyronne and Y Chen ldquo-e holospectrum anewmethod for rotor surveillance and diagnosisrdquoMechanicalSystems amp Signal Processing vol 3 no 3 pp 255ndash267 1989

[7] F Q Wu and G Meng ldquoCompound rub malfunctions featureextraction based on full-spectrum cascade analysis and SVMrdquoMechanical Systems and Signal Processing vol 20 no 8pp 2007ndash2021 2006

[8] Y Chen Q Gao and Z Guan ldquoSelf-loosening failure analysisof bolt joints under vibration considering the tighteningprocessrdquo Shock and Vibration vol 2017 Article ID 203842115 pages 2017

[9] L Chen J Han W Lei Y Cui and Z Guan ldquoFull-vectorsignal acquisition and information fusion for the fault pre-dictionrdquo International Journal of Rotating Machineryvol 2016 Article ID 5980802 7 pages 2016

[10] C Chen Y Meng and Y Du ldquoApplication of the full vectorspectrum based on EMD in fault diagnosis of bearingsrdquoJournal of Mechanical Strength vol 37 pp 806ndash811 2015

[11] C Huang X Wu and W Cao ldquoLMD-based on full vectorenvelope technique and its application in TRT vibration faultdiagnosisrdquo Electric Power Automation Equipment vol 35pp 168ndash174 2015 in Chinese

[12] X Zhao T H Patel and M J Zuo ldquoMultivariate EMD andfull spectrum based condition monitoring for rotating ma-chineryrdquo Mechanical Systems and Signal Processing vol 27pp 712ndash728 2012

[13] G Rilling P Flandrin P Gonalves and J M Lilly ldquoBivariateempirical mode decompositionrdquo IEEE Signal ProcessingLetters vol 14 no 12 pp 936ndash939 2007

[14] C Park D Looney M M Van Hulle and D P Mandic ldquo-ecomplex local mean decompositionrdquo Neurocomputingvol 74 no 6 pp 867ndash875 2011

[15] N Rehman and D P Mandic ldquoEmpirical mode de-composition for trivariate signalsrdquo IEEE Transactions onSignal Processing vol 58 no 3 pp 1059ndash1068 2010

[16] N Rehman and D P Mandic ldquoMultivariate empirical modedecompositionrdquo Proceedings of the Royal Society A Mathe-matical Physical and Engineering Sciences vol 466 no 2117pp 1291ndash1302 2010

[17] Y Lv R Yuan and G Song ldquoMultivariate empirical modedecomposition and its application to fault diagnosis of rollingbearingrdquo Mechanical Systems and Signal Processing vol 81pp 219ndash234 2016

[18] C Huang Y Meng and W Lei ldquoFull vector envelopetechnique based on complex local mean decomposition andits application in fault feature extraction for rotor systemrdquoJournal of Mechanical Engineering vol 52 no 7 p 69 2016in Chinese

[19] G Rilling P Flandrin and P Goncalves ldquoOn empirical modedecomposition and its algorithmsrdquo in Proceedings of IEEE-EURASIP Workshop on Nonlinear Signal and Image Pro-cessing pp 8ndash11 IEEE Trieste Italy June 2003

[20] L Yang X Chen and S Wang ldquoMechanism of fast time-varying vibration for rotorndashstator contact system with ap-plication to fault diagnosisrdquo Journal of Vibration andAcoustics vol 140 no 1 article 014501 2018

[21] I Daubechies J Lu and H-TWu ldquoSynchrosqueezed wavelettransforms an empirical mode decomposition-like toolrdquoApplied and Computational Harmonic Analysis vol 30 no 2pp 243ndash261 2011

[22] L-S Qu Holospectrum and Holobalancing Technique inMachinery Diagnosis Beijing Science Press Beijing China2007

Shock and Vibration 19

International Journal of

AerospaceEngineeringHindawiwwwhindawicom Volume 2018

RoboticsJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Active and Passive Electronic Components

VLSI Design

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Shock and Vibration

Hindawiwwwhindawicom Volume 2018

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawiwwwhindawicom

Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Control Scienceand Engineering

Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Journal ofEngineeringVolume 2018

SensorsJournal of

Hindawiwwwhindawicom Volume 2018

International Journal of

RotatingMachinery

Hindawiwwwhindawicom Volume 2018

Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Navigation and Observation

International Journal of

Hindawi

wwwhindawicom Volume 2018

Advances in

Multimedia

Submit your manuscripts atwwwhindawicom

Wenping Lei and Yajun Meng performed the experimentsand analyzed the data All the authors contributed to thewriting and discussion of the paper

Acknowledgments

-is research was funded by the Henan Provincial HigherEducation Key Research Project (Grant nos 18A460006 and19A460029) Henan High-Level Innovative Scientific andTechnological Talent Team Construction Project (Grant noC20150034) and Zhengzhou Institute of Technology In-novation Team Project (Grant no CXTD2017K1)

References

[1] R Yan R X Gao and X Chen ldquoWavelets for fault diagnosisof rotary machines a review with applicationsrdquo Signal Pro-cessing vol 96 pp 1ndash15 2014

[2] J Cheng D Yu J Tang and Y Yang ldquoApplication of frequencyfamily separation method based upon EMD and local Hilbertenergy spectrum method to gear fault diagnosisrdquo Mechanismand Machine lteory vol 43 no 6 pp 712ndash723 2008

[3] H Liu and M Han ldquoA fault diagnosis method based on localmean decomposition and multi-scale entropy for rollerbearingsrdquoMechanism andMachinelteory vol 75 pp 67ndash782014

[4] Z Zheng W Jiang Z Wang Y Zhu and K Yang ldquoGear faultdiagnosis method based on local mean decomposition andgeneralized morphological fractal dimensionsrdquo Mechanismand Machine lteory vol 91 pp 151ndash167 2015

[5] W Yang R Court P J Tavner and C J Crabtree ldquoBivariateempirical mode decomposition and its contribution to windturbine condition monitoringrdquo Journal of Sound and Vi-bration vol 330 no 15 pp 3766ndash3782 2011

[6] L Qu X Liu G Peyronne and Y Chen ldquo-e holospectrum anewmethod for rotor surveillance and diagnosisrdquoMechanicalSystems amp Signal Processing vol 3 no 3 pp 255ndash267 1989

[7] F Q Wu and G Meng ldquoCompound rub malfunctions featureextraction based on full-spectrum cascade analysis and SVMrdquoMechanical Systems and Signal Processing vol 20 no 8pp 2007ndash2021 2006

[8] Y Chen Q Gao and Z Guan ldquoSelf-loosening failure analysisof bolt joints under vibration considering the tighteningprocessrdquo Shock and Vibration vol 2017 Article ID 203842115 pages 2017

[9] L Chen J Han W Lei Y Cui and Z Guan ldquoFull-vectorsignal acquisition and information fusion for the fault pre-dictionrdquo International Journal of Rotating Machineryvol 2016 Article ID 5980802 7 pages 2016

[10] C Chen Y Meng and Y Du ldquoApplication of the full vectorspectrum based on EMD in fault diagnosis of bearingsrdquoJournal of Mechanical Strength vol 37 pp 806ndash811 2015

[11] C Huang X Wu and W Cao ldquoLMD-based on full vectorenvelope technique and its application in TRT vibration faultdiagnosisrdquo Electric Power Automation Equipment vol 35pp 168ndash174 2015 in Chinese

[12] X Zhao T H Patel and M J Zuo ldquoMultivariate EMD andfull spectrum based condition monitoring for rotating ma-chineryrdquo Mechanical Systems and Signal Processing vol 27pp 712ndash728 2012

[13] G Rilling P Flandrin P Gonalves and J M Lilly ldquoBivariateempirical mode decompositionrdquo IEEE Signal ProcessingLetters vol 14 no 12 pp 936ndash939 2007

[14] C Park D Looney M M Van Hulle and D P Mandic ldquo-ecomplex local mean decompositionrdquo Neurocomputingvol 74 no 6 pp 867ndash875 2011

[15] N Rehman and D P Mandic ldquoEmpirical mode de-composition for trivariate signalsrdquo IEEE Transactions onSignal Processing vol 58 no 3 pp 1059ndash1068 2010

[16] N Rehman and D P Mandic ldquoMultivariate empirical modedecompositionrdquo Proceedings of the Royal Society A Mathe-matical Physical and Engineering Sciences vol 466 no 2117pp 1291ndash1302 2010

[17] Y Lv R Yuan and G Song ldquoMultivariate empirical modedecomposition and its application to fault diagnosis of rollingbearingrdquo Mechanical Systems and Signal Processing vol 81pp 219ndash234 2016

[18] C Huang Y Meng and W Lei ldquoFull vector envelopetechnique based on complex local mean decomposition andits application in fault feature extraction for rotor systemrdquoJournal of Mechanical Engineering vol 52 no 7 p 69 2016in Chinese

[19] G Rilling P Flandrin and P Goncalves ldquoOn empirical modedecomposition and its algorithmsrdquo in Proceedings of IEEE-EURASIP Workshop on Nonlinear Signal and Image Pro-cessing pp 8ndash11 IEEE Trieste Italy June 2003

[20] L Yang X Chen and S Wang ldquoMechanism of fast time-varying vibration for rotorndashstator contact system with ap-plication to fault diagnosisrdquo Journal of Vibration andAcoustics vol 140 no 1 article 014501 2018

[21] I Daubechies J Lu and H-TWu ldquoSynchrosqueezed wavelettransforms an empirical mode decomposition-like toolrdquoApplied and Computational Harmonic Analysis vol 30 no 2pp 243ndash261 2011

[22] L-S Qu Holospectrum and Holobalancing Technique inMachinery Diagnosis Beijing Science Press Beijing China2007

Shock and Vibration 19

International Journal of

AerospaceEngineeringHindawiwwwhindawicom Volume 2018

RoboticsJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Active and Passive Electronic Components

VLSI Design

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Shock and Vibration

Hindawiwwwhindawicom Volume 2018

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawiwwwhindawicom

Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Control Scienceand Engineering

Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Journal ofEngineeringVolume 2018

SensorsJournal of

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