Mechanical Systems and Signal Processing 108 (2018) 58–72

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Mechanical Systems and Signal Processing

journal homepage: www.elsevier .com/locate /ymssp

A parameter-adaptive VMD method based on grasshopperoptimization algorithm to analyze vibration signals fromrotating machinery

https://doi.org/10.1016/j.ymssp.2017.11.0290888-3270/� 2018 Elsevier Ltd. All rights reserved.

⇑ Corresponding author.E-mail address: mqiang@scu.edu.cn (Q. Miao).

Xin Zhang, Qiang Miao ⇑, Heng Zhang, Lei WangSchool of Aeronautics and Astronautics, Sichuan University, Chengdu, Sichuan 610065, PR China

a r t i c l e i n f o

Article history:Received 23 June 2017Received in revised form 9 November 2017Accepted 16 November 2017Available online 22 February 2018

Keywords:Vibration signal analysisFault diagnosisVariational mode decompositionGrasshopper optimization algorithmWeighted kurtosis indexParameter adaptive estimation

a b s t r a c t

The mode number and mode frequency bandwidth control parameter (or quadratic penaltyterm) have significant effects on the decomposition results of the variational mode decom-position (VMD) method. In the conventional VMD method, the values of decompositionparameters are given in advance, which makes it difficult to achieve satisfactory analysisresults. To address this issue, this paper proposes a parameter-adaptive VMD methodbased on grasshopper optimization algorithm (GOA) to analyze vibration signals fromrotating machinery. In this method, the optimal mode number and mode frequency band-width control parameter that match with the analyzed vibration signal can be estimatedadaptively. Firstly, a measurement index termed weighted kurtosis index is constructedby using kurtosis index and correlation coefficient. Then, the VMD parameters are opti-mized by the GOA algorithm using the maximum weighted kurtosis index as optimizationobjective. Finally, fault features can be extracted by analyzing the sensitive mode withmaximum weighted kurtosis index. Two case studies demonstrate that the proposedmethod is effective to analyze machinery vibration signal for fault diagnosis. Moreover,comparisons with the conventional fixed-parameter VMDmethod and the well-known fastkurtogram method highlight the advantages of the proposed method.

� 2018 Elsevier Ltd. All rights reserved.

1. Introduction

Vibration-based time–frequency analysis has been the most commonly and successfully used approach in machineryfault diagnosis, since vibration signal usually contains tremendous information of equipment health conditions [1]. Whenequipment health condition deteriorates during its operation, gear or bearing impacts would increase and impact pulsesof corresponding vibration signal would also increase. Thus, effective vibration signal processing method is significant formachinery prognostics and system health management.

In engineering practices, machinery vibration responses are a superposition of multi-frequency characteristic information[2]. Thus, it is necessary to extract fault features through signal decomposition and filtering for final fault identification.Various methods have been designed for this issue, such as wavelet decomposition [3,4], wavelet packet decomposition[5,6], empirical mode decomposition (EMD) [2,7] and local mean decomposition (LMD) [8,9], etc. However, wavelet

X. Zhang et al. /Mechanical Systems and Signal Processing 108 (2018) 58–72 59

decomposition and wavelet packet decomposition are non-adaptive signal analysis methods because wavelet basis functionis selected in advance [10]. Though EMD and LMD are adaptive signal processing methods, their applications are restricteddue to the existence of mode mixing [8,11]. Noise-assisted techniques such as ensemble EMD and ensemble LMD mitigatethe mode mixing issue to some degree, but computational complexity increases sharply and the added white noises cannotbe effectively eliminated [12–14].

Variational mode decomposition (VMD) is a novel and adaptive signal decomposition method proposed by Dragomiret-skiy and Zosso in 2014 [15]. In contrast to recursive mode decomposition methods (e.g., EMD and LMD), VMD can decom-pose a signal into an ensemble of band-limited intrinsic mode functions, where their center frequencies are estimated on-line and all modes are extracted synchronously. In Dragomiretskiy and Zosso’s experiments [15], VMD outperforms EMDwith regards to tone detection, tone separation, and noise robustness. Wang and Markert [16] presented three applicationsto further demonstrate the performance of VMD. In the field of machinery fault diagnosis, various applications of VMD-basedvibration signal analysis methods have been reported. For instance, Wang et al. [17] first applied VMD to detect multiplesignatures caused by rotor-to-stator rubbing. Yang et al. [18] compared the performance of VMD and EMD for feature extrac-tion in wind turbine condition monitoring. Yang and Jiang [19] proposed a casing fault classification based on VMD, locallinear embedding and support vector machine. Li et al. [20] developed a multi-dimensional variational decomposition forbearing-crack detection. Zhang et al. [21] used VMD to diagnose the rolling bearing fault of multistage centrifugal pump.In [17–21], the VMD decomposition parameters, namely mode number and mode frequency bandwidth control parameter(or quadratic penalty term), are determined based on experience and convenience. This behavior based on previous experi-ence greatly limits the performance of VMD method and may cause inaccurate decomposition results. Consequently, choos-ing appropriate combination of parameters is a critical problem to VMD based signal decomposition. Li et al. [22] proposedan independence-oriented VMD method which finds the most suitable mode number by peak searching and similarity prin-ciple, but this method does not consider the influence of bandwidth control parameter on decomposition results. Shi andYang [23] developed an optimized VMD method for wind turbine condition monitoring. In their method, however, twoparameters are optimized independently. In other words, they neglected the interaction between two parameters, and thusthe algorithmwould easily trap in local optimization. Even though Yan et al. [24] proposed an optimized VMDmethod basedon genetic algorithm which can optimize the two parameters concurrently, the optimization objective function (i.e., envel-ope spectrum entropy value) used in their method only considers the impact properties of the decomposition modes andignores the correlation between the decomposition modes and original signal. This may cause information loss problem.Therefore, the construction of objective function for the VMD parameter optimization would directly influence accuracyand efficiency of final decomposition.

Among various measurement indexes (e.g., smoothness index [25,26], entropy [27,28], sparsity measurement [29,30], ageneralized dimensionless bearing health indicator [31], correlation coefficient [8,32] and kurtosis index [33–35]) used invibration signal analysis and fault feature extraction, kurtosis index and correlation coefficient are two important indexesand have been widely studied. As a dimensionless index, kurtosis index is extremely sensitive to impact components ofmachinery vibration signal, and meanwhile it is insensitive to working conditions of machine. Thus, it is suitable for impactmeasurement and has been widely used to monitor mechanical damages [33–37]. However, kurtosis index is only dependenton distribution density of impacts. The greater of impact distribution density is, the larger kurtosis index is, and vice versa. Asa result, some impacts with large amplitudes but with disperse distributions may be omitted if the maximum kurtosis indexis used as optimization objective to optimize VMD parameters for fault feature extraction. Correlation coefficient can char-acterize similarity of two signals, but it is susceptible to noises in impact signal detection when it is used as objective func-tion to optimize VMD parameters [37]. Consequently, considering the merits and drawbacks of kurtosis index andcorrelation coefficient, their synthetic measurement index termed weighted kurtosis index is constructed as the objectivefunction for VMD parameter optimization in this paper.

After obtaining the optimization objective function, the optimal parameter combination of mode number and mode fre-quency bandwidth control parameter can be found by many optimization or search algorithms such as particle swarm opti-mization [38], gravitational search algorithm [39], bat algorithm [40], flower pollination algorithm [41], state of mattersearch [42], firefly algorithm [43] and genetic algorithms [44]. In this paper, a novel optimization algorithm named grasshop-per optimization algorithm (GOA) [45] is used to search for the optimal VMD decomposition parameters. GOA is a novelnature-inspired algorithm proposed by Saremi et al. [45] recently, which mimics the swarming behavior of grasshoppersin nature. In [45], the global optimization performance of GOA was demonstrated by solving a set of mathematical testsand some challenging structural design problems.

Based on the above introduction, this paper proposes a parameter-adaptive VMD method based on GOA for vibration sig-nal analysis and machinery fault diagnosis, in which the maximum weighted kurtosis index is defined as the objective func-tion of GOA. The subsequent sections of this paper are organized as follows. The VMD method is briefly introduced inSection 2. The construction of the weighted kurtosis index, the GOA algorithm, and the proposed method are described inSection 3. Two cases of vibration signal analysis in machinery fault diagnosis using the proposed method are conductedin Section 4. Finally, the conclusions are drawn in Section 5.

60 X. Zhang et al. /Mechanical Systems and Signal Processing 108 (2018) 58–72

2. Brief introduction of VMD

VMD is a novel variational method for signal decomposition. It aims to decompose a real valued input signal f into anensemble of sub-signals or modes uk with specific sparsity properties while reproducing the input signal [15]. Each modeis mostly compact around a center frequency wk, and the bandwidth is estimated through the squared L2 norm of the gra-dient. Consequently, the process of VMD can be considered as the construction and solution of a constrained variationalproblem described by Eq. (1).

minfukg;fwkg

Xk

@t ðdðtÞ þ jptÞ � ukðtÞ

� �e�jwkt

��������2

2

( ); subject to

Xk

uk ¼ f ð1Þ

Here, a quadratic penalty term a and Lagrangian multipliers k are used to render the variational problem unconstrained.The quadratic penalty a can guarantee reconstruction accuracy of the signal in the presence of Gaussian noise. The Lagran-gian multipliers k is to enforce constraints strictly. The augmented Lagrangian is described as follows:

Lðfukg; fwkg; kÞ ¼ aXk

@t ðdðtÞ þ jptÞ � ukðtÞ

� �e�jwkt

��������2

2þ f ðtÞ �

Xk

ukðtÞ�����

�����2

2

þ hkðtÞ; f ðtÞ �Xk

ukðtÞi ð2Þ

The saddle point of Eq. (2) corresponding to the solution of Eq. (1) is found using the alternate direction method of mul-tipliers (ADMM). Firstly, the decomposition mode number is specified in advance, and the frequency-domain expression ofmode u1

k , the corresponding center frequency w1k as well as the Largrangian multiplier k1 are initialized. Subsequently, the

modes uk and the center frequencies wk are updated by Eqs (3) and (4), respectively.

unþ1k ðwÞ

f ðwÞPi<kunþ1i ðwÞ �

Pi>ku

ni ðwÞ þ knðwÞ

2

1þ 2aðw�wnkÞ2

ð3Þ

wnþ1k

R10 wjunþ1

k ðwÞj2dwR10 junþ1

k ðwÞj2dwð4Þ

After each updating, the modes and the center frequencies can be obtained. The Largrangian multiplier is also updatedaccording to the following equation.

knþ1ðwÞ knðwÞ þ s f ðwÞ �Xk

unþ1k ðwÞ

!ð5Þ

The above iteration is continued until convergence, namely,

Xk

kunþ1k � un

kk22kun

kk22< e ð6Þ

From the above descriptions, there are four parameters that need to be specified in advance. They are the mode number K ,the mode frequency bandwidth control parameter (or quadratic penalty term) a, the noise-tolerance s and the tolerance ofconvergence criterion e. Compared with the first two parameters, s and e have little influences on the decomposition results,and thus the default values in original VMD algorithm are usually adopted. Since the mode number K is specified withoutany pre-knowledge about signal to be analyzed, it is difficult to guarantee the appropriateness of mode number and thusto guarantee the accuracy and efficiency of signal decomposition. In addition, the mode frequency bandwidth control param-eter a is related to the performance of suppressing noise interference, and it should be carefully selected. Accordingly,searching for optimal parameter combination that matches with signal to be analyzed is the key for VMD method, whichmotivates the following research.

3. Parameter-adaptive VMD method based on GOA

A parameter-adaptive VMD method for vibration signal analysis and fault feature extraction is introduced in this section.Generally speaking, the basic idea of an adaptive method is to search for the optimal parameters by using some search rules,which includes two aspects: (1) the construction of measurement index, and (2) the selection of search methods. These twoaspects are introduced prior to the proposed method.

3.1. Weighted kurtosis index

The measurement index is an important factor in the VMD method, which determines whether the decomposition resultis satisfactory. As described in Section 1, the kurtosis index and correlation coefficient are two important indexes for

X. Zhang et al. /Mechanical Systems and Signal Processing 108 (2018) 58–72 61

vibration signal analysis in machinery fault diagnosis. Kurtosis index is usually used as an impact measurement index tomonitor mechanical damages [36,37]. However, the kurtosis index is only dependent on the distribution density of impacts.As a result, some impacts with large amplitude but with disperse distributions may be omitted if the maximum kurtosisindex is used as optimization objective to optimize VMD parameters. Correlation coefficient can characterize similarity oftwo signals, but it is susceptible to noises in impact signal detection [37]. Consequently, considering the advantages and dis-advantages of these two indexes, their synthetic index termed weighted kurtosis index is constructed to serve as the objec-tive function for the VMD parameter optimization in this work. The weighted kurtosis index KCI is defined as follows.

KCI ¼ KI � jCj ð7Þ

KI ¼1N

PN�1n¼0 x

4ðnÞ1N

PN�1n¼1 x2ðnÞ

� �2 ð8Þ

C ¼ E½ðx� �xÞðy� �yÞ�E½ðx� �xÞ2�E½ðy� �yÞ2�

ð9Þ

where KI is the kurtosis index of signal sequence xðnÞ, N is the signal length, C is the correlation coefficient between signals xand y, and E½�� represents the mathematical expectation. According to Schwartz inequality, the correlation coefficient meetsjCj 6 1. Thus, it can be treated as a weight of kurtosis index and thus KCI is called the weighted kurtosis index.

3.2. GOA

GOA is a nature-inspired algorithm which imitates behavior of grasshopper swarms in nature. The swarming behavior isfound in both nymph and adulthood [45]. There are two main characteristics of the grasshopper swarm behavior. (1) Larvalgrasshoppers move slowly and with small steps, while adults are characterized by long range and abrupt movement. (2)Food source seeking process can be divided into two tendencies: exploration and exploitation. The search agents tend tomove abruptly in exploration, while the behavior becomes local movement during exploitation. The behavior is mathemat-ically modeled as follows.

Xi ¼ Si þ Gi þ Ai ð10Þ

where Xi represents the position of the i-th grasshopper, Si and Gi are social interaction and gravity force on the i-thgrasshopper, respectively, and Ai expresses wind advection. Si, Gi and Ai can be calculated by Eqs. (11), (13) and (14)respectively.

Si ¼XN

j¼1;j–i

sðdijÞdij ¼XN

j¼1;j–i

sðjxi � xjjÞ xi � xjdij

ð11Þ

sðrÞ ¼ fe�r=l � e�r ð12Þ

where dij ¼ jxi � xjj (with the specified initial interval of ½1;4� in GOA algorithm) is the distance between the i-th and j-th

grasshopper, sð�Þ is used to define the social forces (attraction and repulsion) of grasshoppers, dij ¼ ðxi � xjÞ=dij representsa unit vector from the i-th to the j-th grasshopper, f and l are the intensities of attraction and attractive length scale, respec-tively. The details of how the parameters f and l affect grasshopper social interaction can be seen in [45]. Generally, f ¼ 1:5and l ¼ 0:5 are chosen.

Gi ¼ �geg ð13Þ

where g is the gravitational constant, and eg indicates a unity vector towards the center of earth.

Ai ¼ uew ð14Þ

where u represents a constant drift, and ew is a unity vector in the wind direction.

The following equation can be obtained by substituting S, G and A into Eq. (10).

Xi ¼XN

j¼1;j–i

sðjxj � xijÞ xj � xidij

� geg þ uew ð15Þ

where the function sð�Þ is illustrated in Eq. (12), and the parameter N represents the number of grasshoppers.It should be noted that Eq. (15) will not be utilized in the swarm simulation and optimization algorithm because it pre-

vents the algorithm from exploring and exploiting the search space around a solution [45]. This equation is used to simulatethe interactions between grasshoppers in a swarm. In the GOA algorithm, optimization problems are solved using a modified

62 X. Zhang et al. /Mechanical Systems and Signal Processing 108 (2018) 58–72

equation which is presented without considering the gravity (no G part) and assumption that the wind direction (A part) isalways towards the optimization target.

Xdi ¼ c

XNj¼1;j–i

cubd � lbd

2s xdj � xdi��� ���� � xj � xi

dij

!þ Td ð16Þ

where ubd and lbd are the upper and lower bounds in Dth dimension, respectively, Td represents the optimization target (thebest solution obtained so far), and the parameter c is a decreasing coefficient which is calculated by following equation.

c ¼ c�max� lc�max� c�min

Lð17Þ

where l and L indicate the current iteration and total number of iterations respectively, c max and c min represent the max-imum and minimum values of parameter c .

It should be noted that the next positon of a grasshopper is determined by its current position, the target position and thepositon of other grasshoppers, which is indicated in Eq. (16). The first half of Eq. (16) considers the positions of other

grasshoppers and implements the interactions of grasshoppers in nature. The second half (i.e., Td) simulates the tendenciesof grasshoppers to move towards food source. Parameter c simulates the deceleration of grasshoppers approaching foodsource and eventually eating food. The varying mechanism of parameter c can guarantee the GOA algorithm does not con-verge towards the target too quickly so as to avoid being trapped in local optimization, and realizes a fast convergence rate inthe last steps of optimization [45]. In addition, since the target is not known at the beginning of a real optimization problem,the fittest grasshopper (the one with the best objective value) is defined as the target in the GOA algorithm [45].

The above mathematical model forms the theoretical foundation of GOA algorithm. The optimization process is as fol-lows. Firstly, generate a number of random solutions. Update the positions of all search agents according to Eq. (16). Subse-quently, update the positon of the best target found so far in each iteration. Furthermore, calculate the parameter c using Eq.(17) and normalize the initial distances between grasshoppers in each iteration. Update the positon until meet the end cri-terion. Finally, return to the best target which is considered as the best global optimum solution. The pseudo code of thisalgorithm is illustrated in Fig. 1 [45].

3.3. Proposed method

The proposed parameter-adaptive VMD method realizes parameter optimization by the GOA algorithm using the maxi-mum weighted kurtosis index as objective function, which can be described as the form of Eq. (18). Since the original GOA isan optimization algorithm to find the minimum value, the opposite of weighted kurtosis index is served as the objectivefunction in the proposed method.

fitness ¼ minc¼ðK;aÞ

f�KCIigs:t:K ¼ 2;3; . . . ;7a 2 ½1000; 10;000�

8>><>>: ð18Þ

where fitness represents the fitness or objective function, KCIi (i ¼ 1;2; . . . ;K) is the weighted kurtosis index of the VMDdecomposition mode ui, and c ¼ ðK;aÞ is the parameter group of VMD to be optimized. In this study, K takes an integerin the interval of ½2;7�, and a is assigned in the interval of ½1000; 10;000�. The details of this method are summarized asfollows:

Initialize the swarm Xi(i=1,2,…,n)Initialize c_max, c_min, and maximum number of iterationsCalculate the fitness of each search agentT=the best search agentwhile (l<Max number of iterations)

Update c using Eq. (10) for each search agent

Normalize the distances between grasshoppers in [1,4]Update the position of the current search agent by Eq . (11)

Bring the current search agent back if it goes outside boundaries end forUpdate T, if there is a better solutionl=l+1

end whileReturn T

Fig. 1. Pseudo code of GOA algorithm.

X. Zhang et al. /Mechanical Systems and Signal Processing 108 (2018) 58–72 63

(1) Input the machinery vibration signal xðtÞ. Set the ranges of the VMD parameters to be optimized, and initialize theGOA parameters including the maximum iteration number L and search agents n.

(2) Decompose the vibration signal using VMD, and calculate the fitness of all modes. Save the minimum fitness for eachiteration of GOA.

(3) Determine whether the termination condition is achieved, i.e., whether l P L. If yes, end the iteration. Else, let l ¼ lþ 1,and continue the iteration.

(4) Obtain and save the optimal parameters and the minimum fitness.(5) Decompose the original signal using the VMD method with the optimized parameters, and calculate the KCI of all

decomposition modes.(6) Select the sensitive mode that contains the main fault information, and the mode with the maximum KCI is defined as

the sensitive mode in this study.(7) The selected mode is further analyzed by Hilbert envelope spectrum to extract fault features for final fault diagnosis.

The flowchart of the proposed method is illustrated in Fig. 2.

4. Case validations

In this section, two case studies on vibration signal analysis and fault diagnosis are conducted to verify the effectivenessand feasibility of the proposed method. Furthermore, the comparisons with other methods are carried out to evaluate theadvantages of the proposed method.

4.1. Case 1: gear vibration signal analysis and fault diagnosis

4.1.1. Data acquisitionIn this case, the gear fault vibration signal was collected from the Spectra Quest’s Machinery Fault Simulator (MFS) shown

in Fig. 3(a), which includes a 1 HP motor, a variable speed controller, two rolling element bearings, and a data acquisitionsystem, etc. The tested single-stage bevel gearbox (type: Hub City M2) was driven by the motor through a 3/400 diametersshaft and two double groove ‘‘V” belt drive with a reduction ratio of 2.527. Specification of the gear transmission systemis listed in Table 1. The worn tooth defect (see Fig. 3(b)) was planted in small gear by wire cut electrical discharge machining.Vibration signal was collected by an accelerometer mounted on the top of gearbox housing shell. The sampling frequencywas 10 kHz, and the signal length was 8192. The experiment was conducted under a constant motor speed of 5400 rpm.

Vibration signal x(t)

Signal decomposition using VMD

Calculate fitness of each mode ui

Set VMD parameter ranges and initialize GOA algorithm

l L?

No

l=l+1

Yes

Save the optimal parameters

Signal decomposition using VMD with the optimal parameters, and

calculate the KCI of all modes

Select the sensitive mode with the maximum KCI

Fault feature extraction by time-frequency analysis

Fault diagnosis

Fig. 2. Flowchart of the parameter-adaptive VMD method for machinery vibration signal analysis.

Signal conditioner

Tachometer Rolling bearing

Test bevel gearbox

Manual speed controller

Accelerometer

Tested gear with worn tooth

(a) (b)

Data acquisition system

Fig. 3. (a) Experimental rig of MFS and (b) The tested gear with worn tooth.

Table 1Specification of the gear transmission system.

Bevel gear parameter Value

Transmission ratio 1.5:1Number of teeth (z1=z2) 27/18Pressure angle 20�Large gear pitch diameter 1.687500

Small gear pitch diameter 1.12500

Large gear contact angle 56�190

Small gear contact angle 33�410

64 X. Zhang et al. /Mechanical Systems and Signal Processing 108 (2018) 58–72

Therefore, the fault characteristic frequency f s of the tested gear (small gear) and the meshing frequency f m can be calculatedby following equations:

f s ¼5400

2:527� 60¼ 35:62 Hz ð19Þ

f m ¼ f s � z2 ¼ 641:2 Hz ð20Þ

4.1.2. Gear fault diagnosisThe time-domain waveform and Hilbert envelope spectrum of the original vibration signal are shown in Fig. 4. As shown

in this figure, it is difficult to find periodic impact components in the time-domain waveform. Although the small gear faultcharacteristic frequency can be observed, a large number of interference frequency components are distributed in low-frequency band. These interferences are generally caused by noises and other rotating components, which affect the faultdiagnosis accuracy.

0 100 200 300 400 500 600 7000

5

x 10-3

Frequency (Hz)

Am

plitu

de (g

)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8-0.5

0

0.5

Am

plitu

de (g

)

Time (s)

fs

(a)

(b)

Fig. 4. Original gear vibration signal: (a) Time-domain waveform and (b) Hilbert envelope spectrum.

0 2 4 6 8 10

-100

-100.2

X: 4Y: -1.51

Objective space

Iteration

Best

scor

e ob

tain

ed so

far

Fig. 5. Case 1: GOA convergence curve for VMD parameter optimization.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8-0.1

0

0.1

Time (s)

u2

-0.2

0

0.2

u1

1 20

1

2

KC

I i

1.51

0.587

Fig. 6. The modes and the corresponding KCI obtained by the proposed VMD method (K = 2, a = 9000).

X. Zhang et al. /Mechanical Systems and Signal Processing 108 (2018) 58–72 65

The above vibration signal is processed by the proposed method. Here, the GOA parameters are set as follows: the searchagents n ¼ 30 and the maximum iteration number L ¼ 10. The GOA convergence curve for finding the minimum �KCI ofVMD decomposition modes is shown in Fig. 5. The optimal parameter combination of mode number K and mode frequencybandwidth control parameter a, as well as the maximum KCI found by GOA are 2, 9000 and 1.51, respectively. Decomposi-tion result of the original signal using the VMD method with the optimal parameters is presented in Fig. 6. As mentioned inSection 3.1, KCI is also used as the measurement index to select the sensitive mode, which is calculated and shown in Fig. 6.The first mode u1 with the maximumweighted kurtosis index (KCI1 ¼ 1:51) is selected as the sensitive mode for fault featureextraction. It should be noted that besides the selected mode, others are also analyzed using Hilbert envelope spectrum so as

0

0.01

0.02

Am

plitu

de (g

)

0 100 200 300 400 500 600 7000

1

2

3x 10

-3

Frequency (Hz)

0

2

4

6

x 10-3

fs 2fs3fs4fs 5fs 6fs

Original signal

u1

u2

Fig. 7. The Hilbert envelope spectrums of the original signal and the modes obtained by the proposed VMD method. Here, the first mode u1 with themaximum kurtosis index (KCI1 = 1.51) is the sensitive mode.

66 X. Zhang et al. /Mechanical Systems and Signal Processing 108 (2018) 58–72

to validate the effectiveness of KCI for sensitive mode selection. Fig. 7 displays the Hilbert envelope spectrums of all themodes obtained by the proposed method. Meanwhile, the frequency spectrum of the original vibration signal is also plottedin this figure, so that the advantages of the proposed method can be clearly seen.

From the Hilbert envelope spectrums, the fault features are centralized in the selected mode u1. The small gear fault char-acteristic frequency f s as well as its harmonics (2f s;3f s, 4f s, 5f s and 6f s) can be clearly observed in the spectrum of mode u1. Incontrast, no obvious fault features can be identified from the spectrum of mode u2. These analysis results demonstrate thatthe proposed method is effective to analyze vibration signal for fault feature extraction, and the definition of sensitive modegiven in Section 3.3 is feasible.

4.1.3. Comparisons with other methodsTo further evaluate the advantages of the proposed method, two methods (i.e., the conventional VMD with fixed param-

eters specified in advance and the well-known fast kurtogram method [33]) are used to analyze the same signal for faultfeature extraction.

In the conventional VMD method, the mode number and mode frequency bandwidth control parameter are assignedempirically. In this case, the mode number is specified as K ¼ 4 and the mode frequency bandwidth control parameteradopts the default value in the original VMD algorithm, namely a ¼ 2000. Fig. 8 shows the decomposition modes obtainedby the fixed-parameter VMD method, and Fig. 9 presents the corresponding Hilbert envelope spectrums of the four modes.Though the frequency associated with the small gear fault can be identified, the amplitude of fault characteristic frequency f sis much smaller than that in Fig. 7. Meanwhile, the harmonic frequencies cannot be observed clearly.

-0.2

0

0.2

u1

-0.10

0.1

u2

-0.1

0

0.1

u3

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8-0.1

0

0.1

u4

Time (s)

Fig. 8. The modes obtained by the conventional fixed-parameter VMD method (K = 4, a = 2000).

0

0.005

0.01

u1

fs 2fs

0246

x 10-3

u2

0

1

2

3 x 10-3

u3

0 100 200 300 400 500 600 7000

1

2

3 x 10-3

u4

Frequency (Hz)

Fig. 9. Case 1: The Hilbert envelope spectrums of the modes obtained by the conventional fixed-parameter VMD method.

0 100 200 300 400 500 600 7000

2

4

6x 10

-5

Frequency (Hz)

Am

plitu

de (g

)

0 0.2 0.4 0.6 0.80

0.05

0.1

Am

plitu

de (g

)

Time (s)

(a) (c)

(b)

fs

1/3fm

2/3fmfs

Frequency (Hz)

Leve

l kK

max=1.2 @ level 3, Bw= 1250Hz, f

c=3125Hz

0 2000 4000 6000 8000 10000

01

1.62

2.6

33.6

44.6

50

0.2

0.4

0.6

0.8

1

1.2

Fig. 10. The results obtained by fast kurtogram for Case 1: (a) Kurtogram of the original signal. (b) The filtered signal and (c) its squared envelope spectrum.

Channel #1 Accelerometer

Channel #2 Accelerometer

Channel #3 Accelerometer

Speedometer

(a) (b)

Tested bearing

Fig. 11. (a) Experimental rig and (b) The tested bearing with inner race defect.

Table 2Specification of the tested bearing.

Test bearing (HRB6304) Value

Pitch diameter (mm) 36Contact angle (�) 0Ball diameter (mm) 9.6Number of balls 7Inner radius (mm) 13.2Outer radius (mm) 22.8

Table 3Fault characteristic frequencies of the tested bearing.

Frequency column Value (Hz)

Rotational frequency f r 30.67Inner race fault frequency f i 4:43� f r ¼ 135:8Outer race fault frequency f o 2:566� f r ¼ 78:7Ball fault frequency f b 1:742� f r ¼ 53:4Cage fault frequency f c 0:367� f r ¼ 11:3

X. Zhang et al. /Mechanical Systems and Signal Processing 108 (2018) 58–72 67

0 100 200 300 400 500 600 7000

0.02

0.04

Frequency (Hz)A

mpl

itude

(g)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8-0.5

0

0.5

Am

plitu

de (g

)

Time (s)(a)

(b)

Fig. 12. Original bearing vibration signal: (a) Time-domain waveform and (b) Hilbert envelope spectrum.

2 4 6 8 10

-100.551

-100.553 X: 2

Y: -3.578

Objective space

Iteration

Best

scor

e ob

tain

ed so

far

Fig. 13. Case 2: GOA convergence curve for VMD parameter optimization.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8-0.05

0

0.05

u6

Time (s)

1 2 3 4 5 60

5

KC

I i

-0.050

0.05

u2

-0.2

0

0.2

u1

-0.020

0.02

u3

-0.02

0

0.02

u4

-0.05

0

0.05

u5

3.074 2.7191.37 1.346 1.616

3.578

Fig. 14. The modes and the corresponding KCI obtained by the proposed VMD method (K = 6, a = 1000).

68 X. Zhang et al. /Mechanical Systems and Signal Processing 108 (2018) 58–72

0

0.02

0.04

0 100 200 300 400 500 600 7000

2

4x 10

-3

Frequency (Hz)

0

1

2

3x 10

-30

0.5

1

1.5x 10

-30

1

2x 10

-3

Am

plitu

de (g

) 0

2

4x 10

-30

0.02

0.04u1

u2

u3

u4

u5

u6fr fi

2fifr

Original signal

Fig. 15. The Hilbert envelope spectrums of the original signal and the modes obtained by the proposed. VMD method. Here, the sixth mode u6 with themaximum kurtosis index (KCI6 = 3.578) is the sensitive mode.

X. Zhang et al. /Mechanical Systems and Signal Processing 108 (2018) 58–72 69

Fig. 10 shows the analysis results using the fast kurtogram method. From the squared envelope spectrum of the filteredsignal, the small gear fault characteristic frequency f s as well as the meshing frequency components 1=3f m and 2=3f m can beobserved. However, a great deal of interference components around the low-frequency band make the spectrum peak at f s aswell as the sidebands at meshing frequency components indistinct.

The proposed method demonstrates its advantages through these comparison results. It achieves much better effect forgear vibration signal analysis and fault feature extraction than the conventional fixed-parameter VMDmethod. Furthermore,it performs much better than the fast kurtogram method in gear fault feature extraction. To exclude the possibility that theabove results are obtained because of a particular signal, another case study on bearing vibration signal analysis would beconducted in the next subsection.

4.2. Case 2: bearing vibration signal analysis and fault diagnosis

4.2.1. Data acquisitionIn this subsection, the proposed method is applied to analyze the bearing fault vibration signal collected from an aero

engine rotor-bearing fault simulator [46]. The experimental system consists of a test rig, three miniature accelerometersof type 4508 from Brüel & Kjær Sound & Vibration Measurement A/S, and an NI 9234 signal acquisition module, etc. Theexperimental rig and tested bearing with inner race fault are shown in Fig. 11. The tested bearing type is HRB6304, whosespecification is illustrated in Table 2. A crack with a width of 0.6 mm planted by electrical discharge machining was used tosimulate the impacts generated by the damage of bearing outer raceway. The vibration signal was collected at a rotatingspeed of 1840 rpm. The sampling frequency and data points were 10 kHz and 8192 respectively. According to the bearingspecification and rotating speed, the fault characteristic frequencies of the tested bearing are calculated and shown inTable 3.

-0.020

0.02

u2

-0.2

0

0.2

u10 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

-0.04-0.02

00.020.04

u3

Time (s)

Fig. 16. The modes obtained by the conventional fixed-parameter VMD method (K = 3, a = 2000).

0 100 200 300 400 500 600 7000

1

2

3x 10

-3

Frequency (Hz)

0

0.5

1

1.5x 10

-3

Am

plitu

de (g

)

0

0.01

0.02

fr

fi

fr

u1

u2

u3

Fig. 17. Case 2: The Hilbert envelope spectrums of the modes obtained by the conventional fixed-parameter VMD method.

70 X. Zhang et al. /Mechanical Systems and Signal Processing 108 (2018) 58–72

4.2.2. Bearing inner race fault diagnosisFig. 12 shows the time-domain waveform and Hilbert envelope spectrum of the original bearing fault vibration signal.

Due to the strong interferences of noises and other signals generated by rotating components, no useful information canbe obtained for fault diagnosis.

Then, the proposed method is applied to analyze the above bearing vibration signal. The maximum iteration number andsearch agents of GOA algorithm are set as the same as those in Case 1. The GOA convergence curve for finding the minimum�KCI of VMD decomposition modes is shown in Fig. 13. The maximum KCI obtained by GOA is 3.578 and the correspondingoptimal parameters are K ¼ 6 and a ¼ 1000. The modes decomposed by the VMD method using the optimal parameters arepresented in Fig. 14. The KCI of each mode is also calculated and shown in Fig. 14. The mode u6 with the maximum KCI isselected as the sensitive mode to extract fault features. Other modes are also analyzed to further evaluate the effectivenessof KCI for sensitive mode selection. The Hilbert envelope spectrums of all the six modes are displayed in Fig. 15. As shown inthis figure, the equipment running status information is centralized in the selected mode, which indicates the effectivenessand feasibility of the rule for sensitive mode selection.

As shown in Fig. 15, the rotational frequency f r , and characteristic frequency f i associated with bearing inner race fault aswell as its second-order harmonic 2f i can be identified obviously. Moreover, the sidebands of rotational frequency appearson both sides of f i, which shows that the inner race fault characteristic frequency f i is modulated by the rotation frequency f r .It can be concluded that the bearing inner race has serious local defect, which is consistent with the experimental scheme.

4.2.3. Comparisons with other methodsAs described in Section 4.1.3, the conventional fixed-parameter VMD method and the fast kurtogram method are chosen

for comparison study. For the conventional VMD method, K ¼ 3 and a ¼ 2000 are adopted. The decomposition modes areshown in Fig. 16, and the corresponding Hilbert envelope spectrums are presented in Fig. 17. As shown in Fig. 17, the rota-tional frequency f r and its harmonics �f r can be observed, but the fault characteristic frequency f i is almost buried by otherinterference frequency components. That is to say, it cannot provide accurate and reliable information for fault diagnosis.

Fig. 18 shows the analysis results of the fast kurtogrammethod. As shown in Fig. 18(c), the rotational frequency f r and thefault characteristic frequency f i as well as the sidebands can be identified. However, the signal energy had been greatly

0 50 100 150 200 250 3000

0.5

1x 10

-5

Frequency (Hz)

Am

plitu

de (g

)

0 0.05 0.1 0.15 0.2 0.25 0.3 0.350

0.01

0.02

0.03

Am

plitu

de (g

)

Time (s)

fi

(a) (c)

(b)

fr

Frequency (Hz)

Leve

l kK

max=2.2 @ level 4, Bw= 312.5Hz, f

c=4843.75Hz

0 1000 2000 3000 4000 5000

0

1

1.6

2

2.6

3

3.6

4

0.5

1

1.5

2

fr

Fig. 18. The results obtained by fast kurtogram for Case 2: (a) Kurtogram of the original signal. (b) The filtered signal and (c) its squared envelope spectrum.

X. Zhang et al. /Mechanical Systems and Signal Processing 108 (2018) 58–72 71

weakened by filtering, resulting in the frequency amplitudes far less than those in Fig. 15. Therefore, if the signal itself isrelatively weak, this method may not be able to successfully extract the fault features.

In summary, this case study further verifies the effectiveness of the proposed method to analyze vibration signal for faultfeature extraction. Meanwhile, the comparisons with the conventional fixed-parameter VMDmethod and the fast kurtogrammethod highlight the advantages of the proposed method.

5. Conclusions

This paper proposes a parameter-adaptive VMD method based on GOA to analyze vibration signal from rotating machin-ery. The proposed method overcomes the VMD parameter selection problem, which can adaptively obtain the optimalparameter combination of the mode number and mode frequency bandwidth control parameter that matches with signalto be analyzed. Firstly, the weighted kurtosis index constructed using the kurtosis index and the correlation coefficient con-siders not only the impact properties of decomposition modes but also the correlation between the modes and original sig-nal. Thus, this index can avoid the omission of important information as much as possible. Then, the optimal VMDdecomposition parameters are found by a novel and efficient optimization algorithm (i.e., GOA) using the maximumweighted kurtosis index as optimization objective. Finally, the decomposition mode with the maximum weighted kurtosisindex is selected as sensitive mode to extract fault features. Two case studies on machinery vibration signal analysis andfault diagnosis demonstrate the effectiveness and feasibility of the proposed method. In addition, the comparisons withthe conventional fixed-parameter VMD method and the well-known fast kurtogram method verify the advantages of theproposed method in vibration analysis and fault feature extraction. Therefore, it can be concluded that this study has somepotential values for machinery fault diagnosis.

Acknowledgements

This research was supported by the National Natural Science Foundation of China (Nos. 51675355 & 51275554). In addi-tion, we would like to thank Professor Guo Chen at Nanjing University of Aeronautics and Astronautics for providing thebearing experimental data to validate the proposed method. The authors would also like to thank the editor and anonymousreviewers for their constructive comments in improving this paper.

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