AnImprovedTime-FrequencyAnalysisMethodforInstantaneous...

Research ArticleAn Improved Time-Frequency AnalysisMethod for InstantaneousFrequency Estimation of Rolling Bearing

Zengqiang Ma , Wanying Ruan , Mingyi Chen , and Xiang Li

Department of Electrical and Electronics Engineering, Shijiazhuang Railway University, Shijiazhuang 050043, China

Correspondence should be addressed to Zengqiang Ma; [email protected]

Received 16 May 2018; Revised 16 August 2018; Accepted 23 August 2018; Published 18 September 2018

Academic Editor: Adam Glowacz

Copyright © 2018 ZengqiangMa et al.)is is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Instantaneous frequency estimation of rolling bearing is a key step in order tracking without tachometers, and time-frequencyanalysis method is an effective solution. In this paper, a new method applying the variational mode decomposition (VMD) inassociation with the synchroextracting transform (SET), named VMD-SET, is proposed as an improved time-frequency analysismethod for instantaneous frequency estimation of rolling bearing. )e SET is a new time-frequency analysis method whichbelongs to a postprocessing procedure of the short-time Fourier transform (STFT) and has excellent performance in energyconcentration. Considering nonstationary broadband fault vibration signals of rolling bearing under variable speed conditions,the time-frequency characteristics cannot be obtained accurately by SET alone. )us, VMD-SETmethod is proposed. Firstly, thesignal is decomposed into several intrinsic mode functions (IMFs) with different center frequency by VMD.)en, effective IMFsare selected by mutual information and kurtosis criteria and are reconstructed. Next, the SET method is applied to thereconstructed signal to generate the time-frequency representation with high resolution. Finally, instantaneous frequencytrajectory can be accurately extracted by peak search from the time-frequency representation. )e proposed method is free fromtime-varying sidebands and is robust to noise interference. It is proved by numerical simulated signal analysis and is furthervalidated by lab experimental rolling bearing vibration signal analysis. )e results show this method can estimate the in-stantaneous frequency with high precision without noise interference.

1. Introduction

Rolling bearing is a key part andmost widely used in rotatingmachinery. Its working status directly affects the operationefficiency and service life of mechanical system, therefore, itis very important to diagnose fault of rolling bearing [1–5].In practice, rolling bearing often works under variable speedconditions when the vibration signal shows strong non-stationary characteristics, which increases difficulty of faultdiagnosis. In the field of rolling bearing fault diagnosis undervariable speed [6, 7], order tracking is one of the most ef-fective method [8].)ere are a lot of ways for order tracking.Among them, order tracking based on time-frequencyrepresentation is the most popular method because itdoes not need to install the tachometer, so that it saves spaceand costs, while instantaneous frequency estimation is themost preliminary and vital step.)erefore, how to effectivelyestimate instantaneous frequency from complex vibration

signal with strong noise is a significant issue for further faultdiagnosis of rolling bearing under variable speed conditions.

)ere are many sorts of methods for instantaneousfrequency estimation, and the first class is based on the phasedemodulation method. For example, Coats [9] presented themultistep iterative method to realize phase demodulationand extracted instantaneous frequency information. Fenget al. [10] put forward time-varying demodulation analysismethod by applying ConceFT. Such methods are restrictedbecause the cross terms between the adjacent harmonics willbe generated when the rotating speed varies widely. )esecond class is based on the time-frequency analysis method,to name a few for example, Guo et al. [11] proposed theshort-time Fourier transform (STFT) combined with peaksearch and applied the method to instantaneous frequencyestimation for variable speed motor. But it shows poorproperty under strong noise interference. Zhao et al. [12]devised STFT combined with the Viterbi algorithm to

HindawiShock and VibrationVolume 2018, Article ID 8710190, 18 pageshttps://doi.org/10.1155/2018/8710190

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estimate instantaneous frequency of variable speed vibrationsignal. But the Viterbi algorithm’s complexity results in thismethod’s low computational efficiency, so it is hard to applyto actual working conditions.

To sum up, instantaneous frequency estimation methodbased on the time-frequency analysis is simple in principleand free from speed fluctuation which has broader practicalvalue, so, much attention should be paid to it. For this kindof method, the key to success is time-frequency analysismethod, which should possess high time-frequency reso-lution and good antinoise property. Widely used traditionaltime-frequency analysis methods include short-time Fouriertransform (STFT), wavelet transform (WT), S transform,and Wigner-Ville distribution (WVD), etc. )ese methodshave been applied in many fields and acquired some goodachievements. But, restricted, respectively, by the Heisen-berg uncertainty principle, ineradicable cross-terms, andlarge computational costs, these methods reveal some lim-itations for practical purposes. In recent years, some newtime-frequency analysis methods have been put forward.Among these methods, synchrosqueezed wavelet transforms(SST) attract the most attention, which was proposed byDaubechies et al. [13] in 2011. )is method’s essence is thecombination of wavelet transform and time-frequency reas-signment. Time-frequency aggregation is improved bycompressing wavelet transform coefficient in frequency/scaledirection. However, SST still has many problems, like poorantinoise property and deficiency in processing multicom-ponents signal. )erefore, a lot of scholars proposed variousimproved methods. For example, Feng et al. [14] proposediterative generalized synchrosqueezing transform to addressthe issue of time-frequency blurs of multicomponent andtime-variant frequency signals and applied it to fault diagnosisof wind turbine planetary gearbox under nonstationaryconditions. Wang et al. [15] introduced a matching syn-chrosqueezing transform to process signals composed ofmultiple components with fast varying instantaneous fre-quency that achieved a highly concentrated time-frequencyrepresentation.

Considering these issues, inspired by SST and thetheory of ideal time-frequency analysis, Yu et al. [16]proposed a novel time-frequency analysis method namedSET in 2017, which belongs to a postprocessing procedureof the STFT. Different from the squeezing manner of SST,the main idea of SET is to only retain the time-frequencyinformation of STFT results most related to time-varyingfeatures of the signal and to remove most smeared time-frequency energy, so the time-frequency energy concen-tration can be improved significantly. And Yu successfullyapplied it to analyze instantaneous frequency of bat signaland mechanical vibration signal. Chen et al. [17] dealt withseismic signals making use of SET and realized hydro-carbon detection.

SET is very suitable to analyze time-frequency in-formation of nonstationary signals and can achieve high time-frequency resolution. However, rolling bearing fault vibrationsignal is nonstationary and multicomponent with complexamplitude modulation features, frequency modulation fea-tures, and strong noise interference. SETalone is not sufficient

to accurately extract time-frequency information. )erefore,the authors consider using a preprocessing method to denoiseand decompose multicomponent signal into monocompo-nent signal and then performing SET, in which time-frequency resolution and antinoise property both can beenhanced very well.

Recently, a new adaptive signal decomposition methodnamed variational mode decomposition (VMD) was pro-posed by Dragomiretskiy et al. [18] in 2014, which can selectthe relevant frequency bands of fault signal suppressingnoise interference and decompose the signal into severalmonocomponent signal with high precision. VMD can ef-ficiently overcome the mode mixing and misclassificationproblem of empirical mode decomposition (EMD) andensemble empirical mode decomposition (EEMD) due tononiterative decomposition.)us, it has been widely used insignal decomposition and denoising field, andmany scholarsapplied it to fault diagnosis and other fields in practice[19–21]. VMD can decompose signal into several intrinsicmode functions (IMFs); however, some IMFs are redundantcomponents irrelevant to original signal, and so how toselect efficient components is a key step. Given this, mutualinformation (MI) and kurtosis are introduced to select ef-fective IMFs, which can guarantee that the selected com-ponents contain the useful information to the utmost extentand purify the signal.

In this paper, we propose an improved time-frequencyanalysis method combining VMD with SET to estimateinstantaneous frequency of rolling bearing under variablespeed conditions. First, the signal is decomposed intosome IMFs by VMD. Next, the effective components areselected via MI and kurtosis and reconstructed. )en, SETis carried out for the reconstructed signal. Last, peaksearch is performed on SET time-frequency representa-tion, and instantaneous frequency can be accuratelyextracted.

Hereafter, this paper is structured as follows. In Section 1,we produce the relevant principles including VMD, mutualinformation, kurtosis, SET, and peak search and illustrateconcrete steps of the proposed method. In Section 2, firstly,we compare VMD with EMD, then numerical simulatedsignals are analyzed to prove effectiveness of the proposedmethod and debate antinoise property and estimate accu-racy. In Section 3, the lab experimental vibration signals ofrolling bearing with rising speed and fluctuate speedare employed to further validate the practicability of theproposed method. At last, the conclusions are given inSection 4.

2. Theory

2.1. Variational Mode Decomposition (VMD). )e aim ofVMD is to decompose multicomponent signal into a seriesof band-limited monocomponents with specific sparsityproperties in the bandwidth, and all components arecompact around a center pulsation, moreover, the decom-posed components support reconstruction. It is carried outby working out the following constrained variational opti-mization problem:

2 Shock and Vibration

minuk{ }, ωk{ }

k

zt (δ(t)) +j

πt ∗ uk(t) e−jωkt

��

��

2

2

⎧⎨

⎩

⎫⎬

⎭,

s.t. k

uk(t) � f(t),

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(1)

where uk is the kth component of the original signal, uk denotes a series of modes u1, u2, . . . , uk , ωk is the centerfrequency corresponding to the kth component, ωk de-notes a series of center frequencies corresponding to uk which is represented as ω1,ω2, . . . ,ωk , f(t) is the originalsignal, and δ(t) is the Dirac function.

For this model, first, the analytical signal and its singleside spectrum are obtained via Hilbert transform. )en, it ismultiplied with the exponential factor to modulate the allmodes’ spectrum to the corresponding baseband. Last, theconstraint variational problem is transformed into a non-constrained variational problem by extending the Lagrangefunction and then solved. )e expression is as follows:

L uk , ωk , λ( � αk

zt δ(t) +j

πt ∗ uk e

−jωkt��

��

2

2

+ f(t)− k

uk(t)

��

��

2

2

+ λ(t), f(t) − k

uk(t),

(2)

where α denotes the balancing parameter of the data-fidelityconstraint and λ is the Lagrange multiplier.

)e Lagrange saddle point is acquired using alternatedirection method of multipliers algorithm, which is theoptimal solution of the original variational model. Duringthe solution process, each mode is updated according to thefollowing equation:

un+1k (ω) �

f(ω)−ikû

n

i (ω) + λn(ω)/2

1 + 2α ω−ωnk 2 ,

k ∈ 1, K{ },(3)

where f(ω), ui(ω), λ(ω), un+1k (ω), respectively, denote the

corresponding Fourier transform and n is iterations.)e center frequency is estimated according to the updated

modes’ power spectrum, the center frequency is updated byEquation (4). Further, λ also can be updated. )e updatingprocess ends until the iteration stop condition shown inEquation (5) is satisfied. )en K IMFs can be obtained.

ωn+1k �∞0 ω u

n+1k (ω)

2dω

∞0 u

n+1k (ω)

2dω

, k ∈ 1, K{ }, (4)

k un+1k − u

nk

��22

unk

��22

< ε. (5)

2.2. Mutual Information (MI). MI derived from the entropyof information theory, which is the difference value of tworandom variables uncertainty represents the statisticalcorrelation, and the larger the value, the greater the cor-relation. MI is often used to identify fake components ofEMD, EEMD, and VMD. Some scholars compared MI withcorrelation coefficient, and the result shows that the MI ismore accurate [22]. )e expression of MI is as follows:

MI(X, Y) � H(Y)−H(Y|X), (6)

where MI(X, Y) denotes MI of X and Y, H(Y) denotesthe entropy of Y, and H(Y|X) is the conditional entropy ofY at X.

)e normalized expression is

αi �MIi

max MIi( . (7)

)e threshold value is set for αi, and the correlationbetween the decomposed modes and original signal can bejudged by the threshold value, and if αi is larger than thethreshold value, the corresponding modes are effectivecomponents, otherwise, the corresponding modes are fakeand should be removed.

2.3. Kurtosis. Kurtosis is a dimensionless parameter de-scribing the waveform peak, which is sensitive to the impulsecharacteristics of signal. For the discrete variable x, thenormalized fourth order central moment is called kurtosis,which is defined as follows:

K �E(x− μ)4

σ4, (8)

where K is kurtosis value, E(x− μ)4 denotes the fourth ordermathematic expectation, μ is the mean value, and σ is thestandard deviation.

When the rolling bearing is under normal workingcondition, the amplitude probability density of the vibrationsignal is close to the normal distribution, when the kurtosisvalue is about 3, which is a stationary or weak stationaryprocess. However, when there is damage impulse due to therolling bearing elements pitting or cracks, the amplitudeprobability density will deviate from the normal distribu-tion, and the kurtosis value increases, that is, the moreimpulsive the signal is, the larger the kurtosis value becomes.)us, the mode with larger kurtosis contains more abundantfault information [23].

2.4. Synchroextracting Transform (SET). SET is a novel time-frequency analysis method, and it is a postprocessing pro-cedure of the STFT, which is a more energy concentratedtime-frequency representation than classical time-frequencyanalysis methods and can effectively describe time-frequency characteristics.

For a multicomponent signal s(t), which can be seen asthe sum of n nonstationary modes, its expression is asfollows:

Shock and Vibration 3

s(t) � n

k�1sk(t) �

n

k�1Ak(t)e

iφk(t),

φk+1′ (t)−φk′(t)> 2Δ,

(9)

where sk, Ak,φk, respectively, denote the kth mode, thecorresponding instantaneous amplitude, and instantaneousphase; φk′ is the first-order derivative of φk and denotesinstantaneous frequency; and Δ is the frequency support ofwindow function.

SET is based on STFT, and the STFT representationGe(t,ω) of original signal s(t) is shown as the followingform:

Ge(t,ω) ≈ n

k�1Ak(t)g ω−φk′(t)( e

iφk(t), (10)

where g(·) denotes the Fourier transform of the windowfunction g, g ∈ L2(R).

According to (10), we can calculate instantaneous fre-quency by

φ′(t,ω) � n

k�1φk′(t,ω) � −i

ztGe(t,ω)Ge(t,ω)

. (11)

In order to enhance the time-frequency resolution, Yudesigned an operator to only retain the time-frequencyinformation most related to time-frequency characteristicsof the target signal from STFT representation, which canremove the irrelevant interference and smeared time-frequency energy. )e SET is formulated as

Te(t,ω) � Ge(t,ω)δ ω−φ′(t,ω)( , (12)

where δ(ω−φ′(t,ω)) � 1, ω � φ′(t,ω),

0, else,which is

named the synchroextracting operator (SEO).According to (10) and (12), the following expression can

be deduced:

Te(t,ω)ω−

n

k�1φk′(t)�0� Ge(t,ω)

ω−n

k�1φk′(t)�0

≈ n

k�1Ak(t)g(0)e

iφk(t).(13)

In this way, we can obtain a sharper time-frequencyrepresentation than STFT and extract instantaneous fre-quency with a highly precise degree.

2.5. Peak Search. After obtaining the time-frequency rep-resentation, we need to extract the instantaneous frequencycurve from it. For a time-frequency analysis method withhigh precision and high time-frequency aggregation, thepeak search algorithm can extract the instantaneous fre-quency from the time-frequency diagram accurately, and thepeak search principle is simple and the efficiency is high.)erefore, this paper uses the peak search method to extractthe instantaneous frequency curve. )e steps to extractinstantaneous frequency curve from the time-frequencydiagram by peak search are as follows:

(1) Time-frequency representation is obtained.

(2) Select the starting point of search. In the time fre-quency diagram, a point is selected as the startingpoint in the region where the peak value of thetracking component is prominent. After selecting thestarting point, the peak value of the time frequencydiagram is searched according to the followingequation:

IFE n1, k1( � argmaxk0−p≤ k≤ k0+p

SPEC n1, k( ,

IFE ni, ki( � argmaxn�ni±1

ki±1−p≤k≤ki±1+p

SPEC(n, k){ },

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(14)

where ni � n1 ± 1, n1 ± 2, . . . , ni ∈ (0, M− 1); ki ∈ (0, N− 1);M denotes the number of time lines in the time-frequencygrid, N denotes the number of frequency lines in the time-frequency grid, IFE is the peak search function, argmax is theparameter when the objective function takes the maximumvalue, and SPEC is the corresponding time-frequency rep-resentation. (n1, k1) is the first instantaneous frequencycoordinate obtained from peak search with (n1, k0) as thestarting point. p denotes the range of peak search; (ni, ki) isthe instantaneous frequency coordinate corresponding toeach time after peak search.

(3) Instantaneous frequency curve fitting. )e leastsquares fit is performed on the discrete in-stantaneous frequency obtained above. According tothe trend of instantaneous frequency change of eachpoint, the number of polynomials is selected. Nor-mally, the rotation speed will not be abrupt, so wecan choose low-order polynomial fitting. Take thesecond-order polynomial as an example. )e fittingformula is as follows:

f(t) � at2

+ bt + c, (15)

where t is time; f(t) denotes instantaneous frequency fittingfunction; and a, b, and c are undetermined coefficients.

)e squared error is as follows:

F(a, b, c) � M

i�1

f ti( − f ni( 2. (16)

According to these restrictive conditions: zF/za � 0,zF/zb � 0, zF/zc � 0, a, b, and c can be determined.

2.6. VMD-SET Analysis Procedure for Rolling Bearing FaultSignal. Considering rolling bearing fault signal showsstrong nonstationary under variable speed condition andis complex multicomponent signal contaminated bystrong noise, it is difficult to accurately estimate in-stantaneous frequency for time-frequency analysis aloneeven with high time-frequency resolution. )us, it isnecessary to preprocess signal to denoise and decomposethe original. VMD has a solid theoretical foundation,which decomposes a multicomponent signal into a set of


quasiorthogonal IMFs with different center frequency innonrecursively way and is suitable to process the rollingbearing fault vibration signal. However, not all IMFs are valid,so we select the effective IMFs by MI and kurtosis, which notonly removes the noise interference but also obtains themonocomponent signal containing the most useful in-formation. After that, we perform SET and can eliminate themost-smeared time-frequency energy and get clear time-frequency representation.

For an actual rolling bearing vibration signal, the pro-posed method can be generated following the procedurelisted below:

(1) Decompose the original signal into a number ofIMFs by VMD. )e VMD parameters are set as thedefault value. α � 2000, τ � 0.

(2) Calculate the MI between each mode and theoriginal signal and each IMF’s kurtosis value, andthe components are removed that the MI is lessthan the threshold value and the kurtosis value isless than 3, and the other IMFs are selected. In thispaper, the MI threshold value is determined as 0.1.)rough a large number of experimental dataanalysis, the results show that it is the most ap-propriate to set the threshold value as 0.1, when thesignal can retain the most useful information andeliminate noise effectively.

(3) Add the selected IMFs to get the reconstructed signal,then apply SET to the reconstructed signal. Wheninstantaneous frequency features can be shown clearlyin the SET time-frequency representation.

(4) Extract instantaneous frequency curve via peaksearch based on SET time-frequency spectrum.

)e VMD-SET analysis flowchart for instantaneousfrequency estimation of rolling bearing is shown in Figure 1.

)e characteristics and advantages of the proposedmethod include (1) the application of VMD, which canrealize the signal decomposing and denoising; (2) VMD’seffective components are selected by combing MI withkurtosis, in which the acquired components can guaranteeaccuracy and completeness to the utmost extent; and (3) theproposedmethod possesses strong noise resistance, excellenttime-frequency resolution, and high estimation precision ofinstantaneous frequency.

3. Simulated Signal Analysis and Comparison

Rolling bearing vibration signal under variable speed iscomplex multicomponent signal. In different workingconditions, the signal can be demodulated by differentfrequency, and the estimation accuracy of instantaneousfrequency will be greatly affected due to strong backgroundnoise. )us, in this paper, we design two kinds ofdemodulated signal (linear frequency modulation and si-nusoidal frequency modulation multicomponent signals)to simulate rolling bearing vibration signal under twoworking conditions, which can demonstrate the method’seffectiveness more convincingly.

3.1. Comparison of Signal Decomposition Effects betweenVMD and EMD. First of all, let’s discuss why we choose topreprocess signals with VMD. In order to illustrate therationality of VMD used in this article, we compare VMDwith EMD to test its noise immunity and signal de-composition accuracy. A multicomponent harmonic signalis constructed, and the center frequencies of three com-ponents are 2Hz, 24Hz, and 288Hz, respectively, as shownin (17), where η represents Gaussian additive noise witha mean of 0 and a variance of 0.1.

f(t) � cos(2 × 2πt) +14cos(24 × 2πt)

+116

cos(288 × 2πt) + η.

(17)

)e signal is decomposed by VMD and EMD, re-spectively, and the waveforms and spectrums of the ob-tained components are as shown in Figure 2. It can be seenthat the three components of the signal can be completelyseparated by VMD, and the center frequency of eachcomponent is consistent with the true value. )e de-composition accuracy is very high, and the noise can beeliminated very well, and there is no modal aliasing. Five

Start

Rolling bearing vibration signal

VMD

Reconstructed signal

Mutualinformation Kurtosis

SET

Peak search

Instantaneous frequency curve

End

Time-frequencyrepresentation

Figure 1: VMD-SET analysis flowchart for instantaneous fre-quency estimation of rolling bearing.


modes are obtained by EMD, in which only the centerfrequencies of the two modes correspond to the true valuesof 24Hz and 288Hz.)e other modes do not correspond toan accurate center frequency, and we can see that the modalaliasing is severe.)e waveforms have severe distortion, thesignal decomposition accuracy is very low, and the abilityto eliminate noise is weak.

In order to be a suitable preprocessing method for SET,good noise immunity and high-precision decomposition

capability are necessary. From the above analysis, it can beseen that VMD has advantages over EMD. So, we chooseVMD as a preprocessing method for SET.

3.2. Linear Frequency Modulation (LFM) Signal Validating.)e simulated multicomponent vibration signal of linefrequency modulation is established, and the instantaneousangular frequency is

Am

plitu

de

Time (s) Frequency (Hz)

0 0.5 1–0.5

0

0.5

0 50 100

Am

plitu

de

0

0.2

0.4

24Hz

0 0.5 1–2

0

2

0 50 1000

1

2

2Hz

0 0.5 1–0.2

0

0.2

0 200 400 6000

0.05

0.1

288Hz

(a)

0 0.5 1–0.5

0

0.5

0 0.5 1–0.2

0

0.2

0 0.5 1

Am

plitu

de

–0.1

0

0.1

0 0.5 1–0.5

0

0.5

Time (s)0 0.5 1

–0.2

0

0.2

0 200 400 6000

0.05

0.1

0 200 400 6000

0.01

0.02

0 200 400 600

Am

plitu

de

0

0.01

0.02

0 200 400 6000

0.2

0.4

Frequency (Hz)0 200 400 600

0

0.01

0.02

140Hz

7Hz

24Hz

288Hz

66Hz

(b)

Figure 2: Signal decomposition comparison: (a) waveforms and spectrums of VMD; (b) waveforms and spectrums of EMD.


ω(t) � 2.5πt + 30π. (18)

So, the instantaneous frequency is

f(t) � 1.25t + 15. (19)

)e simulated signal is

x(t) � sin t

0ω(τ)dτ + 0.8 sin 0.66

t

0ω(τ)dτ

+ 0.7 sin 0.5t

0ω(τ)dτ + η(t),

(20)

where η(t) denotes Gaussian white noise, 0≤ t≤ 20s.)e signal’s sampling frequency is 100Hz, and we added

strong noise to the signal to get SNR (signal-to-noise ratio)of −12 dB, which can make the advantage of the proposedmethod in antinoise more prominent.)e signal’s waveformis shown in Figure 3. )e VMD result is shown in Figure 4(IMF1 to IMF3 from top to bottom), from which we can seethree components of original signal with different frequencyare separated completely (the number of VMD’s mode is setas 3 according to the signal). )en, calculate each compo-nent’s kurtosis value and the MI between each componentand the original signal; the results are displayed in Table 1. Itis observed that three components are effective according tothe above principle. )e SET time-frequency representationof the reconstructed signal is shown in Figure 5, from whichthe frequency curves containing base frequency, 0.66 and 0.5times frequency are clearly seen without noise interference,which can accurately describe the time-frequency in-formation of the signal.

In order to highlight the advantage and necessity of theproposed method, we provided the time-frequency repre-sentation of SETalone for comparison, as Figure 6. It is seenthat the representation is seriously influenced by noise, andthe instantaneous frequency curve with base frequency isweakly displayed, and the other frequency components al-most disappear. At last, the instantaneous frequency isextracted from SET time-frequency representation andVMD-SET time-frequency representation, respectively, bypeak search. )e comparison result is displayed in Figure 7,as you can see, the instantaneous frequency estimated by theproposed method is mainly in accordance with the realinstantaneous frequency, while the instantaneous frequencyestimated based on SET differs greatly from the true result.

In order to illustrate quantitatively the estimation ac-curacy of the two methods, we calculated the percentagevalue of the estimation error using Equation (21). By cal-culation, the estimation error of the proposed method is2.83%, and the error of SET is 46.82%.

ξ �

��

Nn�1(f(n)− f(n))

2

��

Nn�1

f2(n)

, (21)

where ξ denotes the percentage value of the estimated error,f(n) denotes estimated frequency, f(n) denotes real fre-quency. and N is the corresponding point.

3.3. Sinusoidal Frequency Modulation (SFM) SignalValidating. )e simulated multicomponent vibration signalof sinusoidal frequency modulation is established, and theinstantaneous angular frequency is

ω(t) � 2π(7.5− 2.5 cos(t)). (22)

So, the instantaneous frequency is

f(t) � 7.5− 2.5 cos(t). (23)

)e simulated signal is

x(t) � sin t

0ω(τ)dτ + 0.6 sin 2

t

0ω(τ)dτ

+ 0.4 sin 4t

0ω(τ)dτ + η(t),

(24)

where η(t) denotes Gaussian white noise, 0≤ t≤ 20s.)e signal’s sampling frequency is also 100Hz. We still

added strong noise to the signal, and the SNR is equal to−12 dB. )e signal’s waveform is shown in Figure 8, andthe VMD result is shown in Figure 9 (IMF1 to IMF3 fromtop to bottom), from which we can see three components oforiginal signal with different frequency are separatedcompletely (the number of VMD’s mode is set as 3 accordingto the signal). )en calculate each component’s kurtosisvalue and the MI between each component and the originalsignal; the results are displayed in Table 2. It is observed thatthree components are effective according to the aboveprinciple, so the reconstructed signal is obtained by addingthree IMFs. )e SET time-frequency representation of thereconstructed signal is shown in Figure 10, from which threemodulated frequency curves are clearly displayed withoutnoise interference, which can accurately describe the time-frequency information of the signal. However, the time-frequency representation of SET alone is displayed in Fig-ure 11. It is seen that the representation is seriously influ-enced by noise, and the instantaneous frequency curve issubmerged in the strong noise hard to identify. At last, theinstantaneous frequency is extracted from SET time-frequency representation and VMD-SET time-frequency

Time (s)0 5 10 15 20

Am

plitu

de

–15

–10

–5

0

5

10

Figure 3: Waveform of LFM signal.


representation, respectively, by peak search. )e comparisonresult is displayed in Figure 12, as you can see, the in-stantaneous frequency estimated by VMD-SET is nearlyidentical with the real instantaneous frequency, while theinstantaneous frequency estimated based on SET deviatesgreatly from the true result. )e reason is that the SNR is solow that noise energy is larger than the signal energy, and thenthe estimation error is nonnegligible. However, the proposed

method adopts VMD to preprocess the signal, which cangreatly reduce noise and effectively decompose the signal. Bycalculation according to Equation (21), the estimation error ofthe proposedmethod is 1.58%, and the error of SET is 45.97%.

3.4. Antinoise Property and Estimation Precision Analysis.In this paper, we further analyze the antinoise property andthe precision of instantaneous frequency estimation for theproposed method. To be more persuasive, the authors madea lot of tests and addedwhite noise to the LFM signal and SFMsignal tomake SNR vary from −20 dB to 10 dB.We performedVMD-SET time-frequency analysis on LFM signal and SFMsignal under different SNR and calculated instantaneousfrequency estimation error under different SNR.

In order to save space, we give VMD-SET time-frequency representation for LFM signal and SFM signalunder four groups of SNR. )e representation for LFMsignal is shown in Figure 13, and the representation for SFMis shown in Figure 14. It is seen that the time-frequencyresolution remains high under low SNR, and each in-stantaneous frequency curve is clearly displayed, whichmanifests that the proposed method is possessed of goodantinoise property.

In order to validate the performance of the proposedmethod strictly, we adopt the quantified indicator to illus-trate the effectiveness. )e more energy-concentrated time-frequency representation manifests the better the ability ofthe time-frequency location and the better the character-ization of time-varying feature. )e Rényi entropy canevaluate the energy concentration of time-frequency rep-resentation, so we adopt it to validate the performance ofVMD-SET and SET. )e smaller the Rényi entropy is, thehigher the time-frequency resolution, and its definitionequation is shown in the following equation:

–5

0

5

0

0.5

0

0.5

1

Time (s)0 10 20

–5

0

5

Am

plitu

de

–10

0

10


Am

plitu

de

0

0.5

Figure 4: Waveform and spectrum of IMFs by VMD for LFM signal.

Table 1: )e kurtosis values and mutual information (MI) of IMFsfor LFM signal.

IMF IMF1 IMF2 IMF3Kurtosis 4.78 3.57 4.13MI 0.856 0.769 0.985

Freq

uenc

y (H

z)

5

10

15

20

25

30

35

40

45

Time (s)5 10 15 20

0

0.2

0.4

0.6

0.8

1

1.2

1.4

Figure 5: Time-frequency representation of LFM signal based onVMD-SET.


Time (s)0 5 10 15 20

Am

plitu

de

–15

–10

–5

0

5

10

15

Figure 8: Waveform of SFM signal.

Time (s)0 5 10 15 20

Freq

uenc

y (H

z)

15

20

25

30

35

40

SETTrue resultVMD-SET

Figure 7: Comparison between estimated instantaneous frequency and true result of LFM signal.

Freq

uenc

y (H

z)

5

10

15

20

25

30

35

40

45

Time (s)5 10 15 20

0

0.5

1

1.5

2

2.5

3

Figure 6: Time-frequency representation of LFM signal based on SET.


R(q) �log ni�1p

qi(

1− q, (25)

where R denotes Rényi entropy and q≥ 0, q≠ 1, (p1,p2, . . . , pn) denotes the probability distribution of arbitrarydiscrete variables.

Under different SNR, the Rényi entropies of SET andVMD-SET time-frequency representations are respectivelyshown in Figures 15 and 16. It is seen that the Rényi entropy

increases with the reduction of SNR, which indicates thatnoise makes influence on time-frequency resolution. )eRényi entropy of SET changes sharply, which proves that ithas poor antinoise property. While due to preprocessing byVMD in the proposed method, the noise interference isremoved greatly. We can see the Rényi entropy of VMD-SETkeeps minimum in different SNR and changes not obviously,which validates that the proposed method possesses the bestability to improve the time-frequency resolution withoutnoise interference.

Moreover, we calculate the estimation error of LFMsignal and SFM signal under different SNR based onVMD-SET. It is listed in Figure 17. As you can see, theestimation error is kept within 3% when the SNR is above−20 dB, and the precision is sufficient to meet the accuracyrequirement.

–5

0

5

0

0.5

1

0

0.2

0.4

Time (s)0 10 20

–5

0

5A

mpl

itude

–5

0

5


Am

plitu

de

0

0.2

0.4

Figure 9: Waveform and spectrum of IMFs by VMD for SFM signal.

Table 2: )e kurtosis values and mutual information (MI) of IMFsfor SFM signal.

IMF IMF1 IMF2 IMF3Kurtosis 3.28 4.87 5.12MI 0.723 0.859 0.967

Freq

uenc

y (H

z)

5

10

15

20

25

30

35

40

45

Time (s)5 10 15 20

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Figure 10: Time-frequency representation of SFM signal based onVMD-SET.

Freq

uenc

y (H

z)

5

10

15

20

25

30

35

40

45

Time (s)5 10 15 20

0

0.5

1

1.5

2

2.5

Figure 11: Time-frequency representation of SFM signal based onSET.


4. Lab Experimental Validating

In this section, we apply the proposed method to the realrolling bearing vibration signal to further validate thepracticability and effectiveness. )e real data are from themechanical fault diagnosis laboratory in ShijiazhuangRailway University, and the experimental setup is displayedin Figure 18. Figure 18(a) is the simulation experimentplatform. Figure 18(b) is the rolling bearing for testing and

fault outer race, which is NU205EM type. Figure 18(c) is theCA-YD-188 acceleration sensor. )e experimental in-stallation is shown in Figure 18(d), where, 1 denotes the CA-YD-188 acceleration sensor and 2 denotes the ICP lasertachometer.

)e sampling frequency of vibration signal is 25600Hz,and the sampling frequency of the laser tachometer is 1 kHz.And the true rotational frequency is obtained by five-pointformula according to the signal from laser tachometer, which

Time (s)

Freq

uenc

y (H

z)

10

20

30

40

5 10 15 20

(a)

Time (s)

Freq

uenc

y (H

z)

10

20

30

40

5 10 15 20

(b)

Time (s)

Freq

uenc

y (H

z)

10

20

30

40

5 10 15 20

(c)

Freq

uenc

y (H

z)

Time (s)5 10 15 20

10

20

30

40

(d)

Figure 13: VMD-SET time-frequency representation of LFM signal under different SNR: (a) 10 dB, (b) 0 dB, (c) −10 dB, and (d) −20 dB.

Time (s)0 5 10 15 20

Freq

uenc

y (H

z)

4

5

6

7

8

9

10

11

True resultSETVMD-SET

Figure 12: Comparison between estimated instantaneous frequency and true result of SFM signal.


is used to compare to estimated frequency. In order to amplydemonstrate the effectiveness of the proposed method, wemake analysis under two conditions, respectively, the con-dition with rising speed and complex fluctuated speed.

4.1. Frequency Estimation Validating with Rising Speed.)e vibration signal waveform is shown in Figure 19, whichis rising speed condition with strong noise. )e rotational

frequency varies from 11.4Hz to 24.6Hz. Firstly, the signal isprocessed by VMD (the mode is set as 6 according to test),and the result is displayed in Figure 20 (IMF1 to IMF6 fromtop to bottom). )en, calculate the corresponding MI andkurtosis shown in Table 3, so we can see that the kurtosisand MI of IMF1 to IMF3 are eligible, so these componentsare effective. However, two indicators of IMF4 and IMF5are all not qualified, and for IMF6 the kurtosis is acceptablebut the MI is incompetent, so it is still invalid. )us, the

Time (s)

Freq

uenc

y (H

z)10

20

30

40

5 10 15 20

(a)

Time (s)

Freq

uenc

y (H

z)

10

20

30

40

5 10 15 20

(b)

Time (s)

Freq

uenc

y (H

z)

10

20

30

40

5 10 15 20

(c)

Time (s)

Freq

uenc

y (H

z)

10

20

30

40

5 10 15 20

(d)

Figure 14: VMD-SET time-frequency representation of SFM signal under different SNR: (a) 10 dB, (b) 0 dB, (c) −10 dB, and (d) −20 dB.

SNR (dB)–20 –15 –10 –5 0 5 10

Rény

i ent

ropy

7

8

9

10

11

12

13

SETVMD-SET

Figure 15: )e Rényi entropies of the time-frequency representation based on SET and VMD-SET under different SNR (LFM).


5SNR (dB)

–20 –15 –10 –5 0 10

Rény

i ent

ropy

4

5

6

7

8

9

10

VMD-SETSET

Figure 16: )e Rényi entropies of the time-frequency representation based on SET and VMD-SET under different SNR (SFM).

SNR (dB)–20 –15 –10 –5 0 5 10

Erro

r (%

)

0

0.5

1

1.5

2

2.5

3

3.5

4

LFM signalSFM signal

Figure 17: Instantaneous frequency estimation errors based on the proposed method under different SNR.

(a)

Figure 18: Continued.


reconstructed signal is obtained generated by adding IMF1,IMF2, and IMF3. )e VMD-SET time-frequency repre-sentation of reconstructed signal is shown in Figure 21, andit is seen that the time-frequency energy of base rotational

frequency is the most high, which can be regarded as thetarget of peak search. At last, according to the representa-tion, the instantaneous rotational frequency is extracted viapeak search, and the result is shown in Figure 22. Comparing

Outer race fault

(b)

(c)

1

2

(d)

Figure 18: Experimental setup: (a) QPZZ-II fault simulation platform, (b) test rolling bearing, (c) acceleration sensor, and (d) experimentalinstallation.

Time (s)0 2 4 6 8 10

Am

plitu

de (m

·s–2 )

–4

–3

–2

–1

0

1

2

3

4

Figure 19: Waveform of rolling bearing vibration.


the true result, we can see they are almost coincident, and theerror is 1.46% by calculating, which is a desired result.

4.2. Frequency Estimation Validating with Fluctuated Speed.)e vibration signal waveform is shown in Figure 23, which isfluctuated speed condition with strong noise. )e rotational

frequency changes between 6Hz and 20Hz back and forth, sothe extraction of instantaneous rotational frequency is verydifficult. Firstly, the signal is processed by VMD (the mode isset as 6 according to test), and the result is shown in Figure 24(IMF1 to IMF6 from top to bottom). )en, calculate thecorresponding MI and kurtosis shown in Table 4, so we cansee that the kurtosis and MI of IMF1 to IMF3 are all eligible,so these components are effective. And the other componentsare all unqualified, and should be removed. )us, thereconstructed signal is obtained generated by adding IMF1,IMF2, and IMF3. )e VMD-SET time-frequency represen-tation of reconstructed signal is shown in Figure 25, and it is

0.2 10505

0210

20

Am

plitu

de ×

10–

3 (m

·s–2 )

100

2010

010

50

0 5000 10000Frequency (Hz)

15000

0–0.20.1

0–0.1

0.1A

mpl

itude

(m·s–

2 )

0–0.1

10

–10.5

0–0.5

0.50

–0.50 0.05

Time (s)0.1

Figure 20: Waveform and spectrum of IMFs by VMD for rolling.

Table 3: )e kurtosis values and mutual information (MI) of IMFsfor real signal with rising speed.

IMF IMF1 IMF2 IMF3 IMF4 IMF5 IMF6Kurtosis 3.78 4.13 4.87 2.84 2.65 3.02MI 0.856 0.923 0.985 0.076 0.069 0.085

Freq

uenc

y (H

z)

0

20

40

60

80

100

Time (s)2 4 6 8

×105

1

2

3

4

5

6

Figure 21: Time-frequency representation of rolling bearing vi-bration signal with rising speed based on VMD-SET.

2 4 6 8Time (s)

0 10

Freq

uenc

y (H

z)

10

15

20

25

Estimated resultTrue Result

Figure 22: Comparison between estimated instantaneous rota-tional frequency and true result of rolling bearing vibration signalwith rising speed.


seen that the base rotational frequency component is the mostconspicuous, and its time-frequency energy concentration isthe best. At last, according to the representation, the in-stantaneous rotational frequency is extracted via peak search,and the result is shown in Figure 26. Comparing the trueresult, we can see they are almost identical, and the error is0.98% by calculating, which is a very ideal result.

5. Conclusion

In this paper, we propose an improved time-frequency analysismethod named VMD-SET for instantaneous frequency esti-mation of rolling bearing.)e proposedmethod uses VMD andmutual information and kurtosis indicators to select the effectivecomponents and then performs SET on the effective compo-nents, which not only retains the merits of VMD and SET, butalso improves the signal decomposition accuracy and enhancesthe antinoise ability and time-frequency resolution. Comparedwith the SET time-frequency analysis alone, the advantages ofthe proposedmethod are obvious. Two simulation signalmodelswith different SNR and two groups of real signals separatelyunder rising speed and fluctuated speed conditions have beenused to prove the effectiveness of the proposed method, and the

Time (s)0 5 10 15 20 25 30 35 40

Am

plitu

de (m

·s–2 )

–4

–3

–2

–1

0

1

2

3

4

Figure 23: Waveform of rolling bearing vibration.

–0.050

0.05

024

–0.050

0.05

00.5

1

–0.010

0.01

00.5

1

Am

plitu

de ×

10–

3 (m

·s–2 )

–0.10

0.1

Am

plitu

de (m

·s–2 )

024

–0.10

0.1

024

Time (s)0 0.05 0.1

–0.20

0.2

Frequency (Hz)0 5000 10000 150000

24

Figure 24: Waveform and spectrum of IMFs by VMD for rolling.

Table 4: )e kurtosis values and mutual information (MI) of IMFsfor real signal with fluctuated speed.

IMF IMF1 IMF2 IMF3 IMF4 IMF5 IMF6Kurtosis 4.58 4.93 4.12 2.93 2.26 2.69MI 0.905 0.986 0.725 0.086 0.048 0.067


results show that the proposed method can be successfullyapplied to the instantaneous frequency estimation of rollingbearing vibration signal under different complex workingconditions, and the estimation accuracy is enough tomeet actualrequirement, which is an instantaneous frequency estimationmethod with practical application value.

Conflicts of Interest

)e authors declare that they have no conflicts of interest.

Acknowledgments

)is work was supported by National Natural ScienceFoundation of China (Grant nos. U1534202, 11372199,11572206, and 51405313).

References

[1] B. Ju, H. Zhang, Y. Liu et al., “A feature extraction methodusing improved multi-scale entropy for rolling bearing faultdiagnosis,” Entropy, vol. 20, no. 4, p. 212, 2018.

[2] K. Li, L. Su, J. Wu et al., “A rolling bearing fault diagnosismethod based on variational mode decomposition and animproved kernel extreme learningmachine,”Applied Sciences,vol. 7, no. 10, p. 1004, 2017.

[3] L. Ou, D. Yu, and H. Yang, “A new rolling bearingfault diagnosis method based on GFT impulse componentextraction,”Mechanical Systems and Signal Processing, vol. 81,pp. 162–182, 2016.

[4] A. Glowacz, W. Glowacz, Z. Glowacz, and J. Kozik, “Earlyfault diagnosis of bearing and stator faults of the single-phaseinduction motor using acoustic signals,” Measurement,vol. 113, pp. 1–9, 2018.

[5] Y. Li, M. Xu, H. Zhao, and W. Huang, “Hierarchical fuzzyentropy and improved support vector machine based binarytree approach for rolling bearing fault diagnosis,”Mechanismand Machine Deory, vol. 98, pp. 114–132, 2016.

[6] Y. Ren, W. Li, Z. Zhu, Z. Tong, and G. Zhou, “A new faultfeature for rolling bearing fault diagnosis under varying speedconditions,”Advances inMechanical Engineering, vol. 9, no. 6,pp. 1–11, 2017.

[7] S. Lu, J. Guo, Q. He, F. Liu, Y. Liu, and J. Zhao, “A novelcontactless angular resampling method for motor bearingfault diagnosis under variable speed,” IEEE Transactions onInstrumentation and Measurement, vol. 65, no. 11,pp. 2538–2550, 2016.

[8] A. Brandt, T. Lagö, K. Ahlin et al., “Main principles andlimitations of current order tracking methods,” Sound andVibration, vol. 39, no. 3, pp. 19–22, 2016.

[9] M. D. Coats and R. B. Randall, “Order-tracking with andwithout a tacho signal for gear fault diagnostics,” in Pro-ceedings of Acoustics, Fremantle, Australia, 2012.

[10] Z. P. Feng, X. W. Chen, and T. Y. Wang, “Time-varyingdemodulation analysis for rolling bearing fault diagnosisunder variable speed conditions,” Journal of Sound Vibration,vol. 400, pp. 71–85, 2017.

[11] Y. Guo, S. R. Qin, B. P. Tang et al., “Order tracking of rotatingmachinery based on instantaneous frequency estimation,”Chinese Journal of Mechanical Engineering, vol. 39, no. 3,pp. 32–36, 2003, in Chinese.

[12] X. P. Zhao, X. L. Zhao, R. T. Hou et al., “A new method forinstantaneous frequency estimation of run-up or run-downvibration signal for rotating machinery,” Journal of Me-chanical Engineering, vol. 47, no. 7, pp. 103–108, 2011, inChinese.

[13] I. Daubechies, F. J. Lu, and H. T. Wu, “Synchrosqueezedwavelet transforms: an empirical mode decomposition-liketool,” Applied and Computational Harmonic Analysis, vol. 30,no. 2, pp. 243–261, 2011.

[14] Z. P. Feng, X. W. Chen, and M. Liang, “Iterative generalizedsynchrosqueezing transform for fault diagnosis of windturbine planetary gearbox under non-stationary conditions,”Mechanical Systems and Signal Processing, vol. 52-53, no. 1,pp. 360–375, 2015.

[15] S. B. Wang, X. F. Chen, I. W. Selesnick, Y. Guo, C. Tong, andX. Zhang, “Matching synchrosqueezing transform: a usefultool for characterizing signals with fast varying instantaneousfrequency and application to machine fault diagnosis,” Me-chanical Systems and Signal Processing, vol. 100, pp. 242–288,2018.

Time (s)0 5 10 15 20 25 30 35 40

Freq

uenc

y (H

z)

6

8

10

12

14

16

18

20

22

Estimated resultTrue result

Figure 26: Comparison between estimated instantaneous rota-tional frequency and true result of rolling bearing vibration signalwith fluctuated speed.

Freq

uenc

y (H

z)

0

10

20

30

40

50

Time (s)5 10 15 20 25 30 35

×105

2

4

6

8

10

12

14

16

Figure 25: Time-frequency representation of rolling bearing vi-bration signal with fluctuated speed based on VMD-SET.


[16] G. Yu, M. Yu, and C. Xu, “Synchroextracting transform,”IEEE Transactions on Industrial Electronics, vol. 64, no. 10,pp. 8042–8054, 2017.

[17] H. Chen, J. Kang, Y. Chen, D. Xu, and Y. Hu, “An improvedtime-frequency analysis method for hydrocarbon detectionbased on EWTand SET,” Energies, vol. 10, no. 8, p. 1090, 2017.

[18] K. Dragomiretskiy and D. Zosso, “Variational mode de-composition,” IEEE Transactions on Signal Processing, vol. 62,no. 3, pp. 531–544, 2014.

[19] Z. Q. Ma, Y. C. Li, Z. Liu et al., “Rolling bearing’s fault featureextraction based on variational mode decomposition andteager energy operator,” Journal of Vibration and Shock,vol. 35, no. 13, pp. 134–139, 2016, in Chinese.

[20] Y. C. Li, Research on Fault Diagnosis for Rolling Bearing Basedon Variational Mode Decomposition, Shijiazhuang TiedaoUniversity, Shijiazhuang, China, 2016, in Chinese.

[21] W. Liu, S. Cao, Z. Wang et al., “Spectral decomposition forhydrocarbon detection based on VMD and Teager-Kaiserenergy,” IEEE Geoscience and Remote Sensing Letters,vol. 14, no. 4, pp. 539–543, 2017.

[22] C. L. Liu, Y. J. Wu, and C. G. Zhen, “Rolling bearing faultdiagnosis based on variational mode decomposition and fuzzyc means clustering,” Proceedings of the CSEE, vol. 35, no. 13,pp. 3358–3365, 2015, in Chinese.

[23] Z. Q. Ma, X. Y. Liu, J. J. Zhang et al., “Application of VMD-ICA combined method in fault diagnosis of rolling bearings,”Journal of Vibration and Shock, vol. 36, no. 13, pp. 201–207,2017, in Chinese.


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