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Mechanical Systems and Signal Processing

Mechanical Systems and Signal Processing 40 (2013) 520–544

0888-32http://d

n CorrE-m

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

The automatic selection of an optimal wavelet filter and itsenhancement by the new sparsogram for bearing faultdetectionPart 2 of the two related manuscripts that have a joint title as“Two automatic vibration-based fault diagnostic methodsusing the novel sparsity measurement—Parts 1 and 2”

Peter W. Tse n, Dong WangThe Smart Engineering Asset Management Laboratory (SEAM) and the Croucher Optical Non-destructive Testing and Quality InspectionLaboratory (CNDT), Department of Systems Engineering & Engineering Management, City University of Hong Kong, Tat Chee Avenue,Hong Kong, China

a r t i c l e i n f o

Article history:Received 8 February 2012Received in revised form24 May 2013Accepted 29 May 2013Available online 22 June 2013

Keywords:Rolling element bearingsFault diagnosisMorlet wavelet filterGenetic algorithmSparsity measurementSparsogram

70/$ - see front matter & 2013 Elsevier Ltd.x.doi.org/10.1016/j.ymssp.2013.05.018

esponding author.ail addresses: [email protected] (P.W. Tse

a b s t r a c t

Rolling element bearings are the most important components used in machinery. Bearingfaults, once they have developed, quickly become severe and can result in fatal break-downs. Envelope spectrum analysis is one effective approach to detect early bearing faultsthrough the identification of bearing fault characteristic frequencies (BFCFs). To achievethis, it is necessary to find a band-pass filter to retain a resonant frequency band for theenhancement of weak bearing fault signatures. In Part 1 paper, the wavelet packet filterswith fixed center frequencies and bandwidths used in a sparsogram may not cover awhole bearing resonant frequency band. Besides, a bearing resonant frequency band maybe split into two adjacent imperfect orthogonal frequency bands, which reduce thebearing fault features. Considering the above two reasons, a sparsity measurement basedoptimal wavelet filter is required to be designed for providing more flexible centerfrequency and bandwidth for covering a bearing resonant frequency band. Part 2 paperpresents an automatic selection process for finding the optimal complex Morlet waveletfilter with the help of genetic algorithm that maximizes the sparsity measurement value.Then, the modulus of the wavelet coefficients obtained by the optimal wavelet filter isused to extract the envelope. Finally, a non-linear function is introduced to enhance thevisual inspection ability of BFCFs. The convergence of the optimal filter is fastened by thecenter frequencies and bandwidths of the optimal wavelet packet nodes established bythe new sparsogram. Previous case studies including a simulated bearing fault signal andreal bearing fault signals were used to show that the effectiveness of the optimal waveletfiltering method in detecting bearing faults. Finally, the results obtained from comparisonstudies are presented to verify that the proposed method is superior to the other threepopular methods.

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P.W. Tse, D. Wang / Mechanical Systems and Signal Processing 40 (2013) 520–544 521

1. Introduction

Rolling element bearings are widely used in many machines to support rotating parts and components. Any faults in abearing must be immediately detected to avoid fatal breakdowns, loss of production and even human casualties [1]. Rollingelement bearing faults are mainly caused by localized defects on the outer race, the inner race and the roller elements. Oncea fault is seeded on the surface of a bearing, a series of impacts are generated over short time intervals. These impulsesexcite the resonance frequencies of the bearing housing where the sensor is mounted. The modulating frequency is thebearing fault characteristic frequency, which causes the amplitude modulation phenomenon. Therefore, potential periodicbursts of exponentially decaying sinusoidal vibration occur after one of the components of a bearing proves faulty [2,3].

The basic method for detecting the bearing fault characteristic frequency is the Fourier transform with an envelope analysis.However, the Fourier transform is difficult to directly detect non-stationary transients with weak energy. Therefore, a wavelettransform is considered an effective method for localizing non-stationary transients because the nature of wavelet transform is tocalculate the similarity between a wavelet mother function and the transients to be analyzed [4–7]. However, when a continuouswavelet transform (CWT) is employed to analyze non-stationary transients, too many wavelet scales used in CWT can result inconsiderable redundant information [8]. Besides, the computation of CWT is very time-consuming because the operations withunnecessary scales are involved. On the other hand, a wavelet transform can be also regarded as a series of filtering operations withdifferent wavelet scales. References [9–16] show that a single optimal filtering operation is adequate to extract bearing fault-relatedsignatures. Lin and Qu [9] proposed the wavelet entropy for controlling the shape of the Morlet wavelet for bearing fault diagnosis.Nikolaou and Antoniadis [10] introduced three criteria for the selection of the parameters of a complex shifted Morlet wavelet forbearing fault diagnosis. Qiu et al. [11] considered the Shannon entropy and a periodicity detection method for the selection of theshape and the scale of Morlet wavelet for bearing fault diagnosis. He et al. [12] used differential evolution to optimize a complexMorlet wavelet for extracting envelope signals. Then, a maximum likelihood estimation-based soft-threshold method was appliedto further enhance the signal to noise ratio. Su et al. [13] took the Shannon entropy of the filtered signal as the objective functionand optimized the parameters of the complex Morlet wavelet using genetic algorithms to detect bearing fault characteristicfrequencies. Bozchalooi and Liang [14] selected the shape factor of the complex Morlet wavelet by minimizing the ratio of thegeometric mean to the arithmetic mean and determined the scale using a novel resonance estimation algorithm. Sheen [15]proposed a systematical method to design the parameters of Morlet wavelet to filter out one of the resonance modes of a bearingvibration according to the resonance frequencies of the known bearing vibration modes. Ericsson et al. [16] compared somedifferent vibration analysis with Morlet based wavelet techniques for automatic bearing defect detection and concluded that thewavelet analysis was very suitable to bearing health condition monitoring. From the above results, it is found that the sparsitymeasurements, such as the kurtosis, the smoothness index and Shannon entropy, are some criteria for choosing the parameters ofthe Morlet wavelet.

Because an ultrasonic echo signal is similar to a vibration transient signal, the Morlet wavelet is also used to extract weakultrasonic echo signatures in ultrasonic non-destructive testing. The major difference is that only the bandwidth of Morletwavelet is required to be optimized for enhancing the ultrasonic time of flight resolution [17,18], because the echoes have acenter frequency similar to the emitted impulse. Considering the sparsity measurement used in non-destructive testing, Lianget al. [19] revised the blind deconvolution algorithm based on the proposed nonlinear function to improve the time resolution ofthe ultrasonic signal. They [20] also used a sparse solution to enhance the estimated accuracy of the time of flight of ultrasonicechoes. Chen et al. [21] employed the ensemble empirical mode decomposition method to analyze magnetic flux leakage signalsand chose the most useful intrinsic mode function based on sparsity value. Therefore, the sparsity measurement used in non-destructive testing may be a potential method for the optimization of complex Morlet wavelet used in vibration analysis. Part 2paper features an optimal method that maximizes the sparsity measurement used in non-destructive testing to choose optimalparameters for the complex Morlet wavelet through the help of a genetic algorithm that enhances the accuracy of bearing faultdiagnoses. Moreover, a wavelet filtering operation with an optimal center frequency and bandwidth is required only once toextract bearing fault related signatures. By combining the optimal wavelet filtering with an envelope spectrum analysis (with orwithout non-linear transform), an intelligent process has been realized that automatically selects the parameters of a Morletwavelet filter for detecting bearing fault characteristic frequencies. It should be noted that the previous research on theparameter selections of the optimal Morlet wavelet filter did not illustrate how to properly set the initial center frequencies andbandwidths for speeding up the convergence of the optimal Morlet wavelet filter [9–14,17,18]. The new sparsogram reported inPart 1 paper is capable of providing the proper initial center frequencies and bandwidths for the use of genetic algorithm for theoptimization of the complex Morlet wavelet filter.

The organization of Part 2 paper is given as follows. Section 2 introduces the newmethod by which the complex Morlet waveletparameters are optimized through a genetic algorithm and the maximum sparsity measurement value. Section 3 presents asimulated bearing fault signal and real bearing fault signals collected from an experimental motor to investigate and validate thenew method. Some comparisons with other popular methods are done in Section 4. Finally, Section 5 concludes the paper.

2. The maximum sparsity measurement for the optimization of the complex Morlet wavelet filter by genetic algorithmand its enhancement by the new sparsogram

The method proposed in Part 1 paper is a fast algorithm that gives an approximate analysis. However, one of the inherentdeficiencies of binary wavelet packet filters is the energy leakage problem because any wavelet packet filter banks are not

P.W. Tse, D. Wang / Mechanical Systems and Signal Processing 40 (2013) 520–544522

perfect to orthogonally divide a frequency band. In other words, the overlapping frequency band between the imperfectorthogonal wavelet filters always exist no matter whatever a wavelet mother function is used [22]. If a bearing resonantfrequency band is located in one of the overlapping frequency bands generated by binary wavelet packet filter banks, thebearing resonant frequency band must be split into two adjacent imperfect orthogonal frequency bands. Besides, thefrequency localization of binary wavelet packet filters becomes poor as the wavelet packet decomposition depth increases[23]. Consequently, the bearing fault signatures are reduced. In order to illustrate the inherent deficiency of binary waveletpacket filters, a simulated bearing fault signal with one resonant frequency was analyzed at first using the new sparsogram.The simulated bearing fault signal was built based on equation (14) in Part 1 paper. Here, in order to simplify the analysisprocess, only one resonant frequency, f3, (equal to 3375 Hz) was considered. The remaining parameters of the simulatedfault signal are the same as those used in equation (14) in Part 1 paper. The simulated bearing fault signal with one resonantfrequency band is mathematically expressed as follow:

yðkÞ ¼∑re−α�ðk−r�Fs=f m−τr Þ=Fs � sin ð2πf 3 � ðk−r � Fs=f m−τrÞ=FsÞ: ð1Þ

A normally distributed random heavy noise signal with a mean of 0 and a variance of 0.65 was added to Eq. (1) to corruptthe simulated signal. The simulated bearing fault signal corrupted by heavy noise is plotted in Fig. 1(a) and its correspondingfrequency spectrum is plotted in Fig. 1(b). According to the frequency support division provided by binary wavelet packettransform, as introduced in Section 2.1 of Part 1 paper, the resonant frequency band of Eq. (1) is coincidently located in oneof the overlapping frequency bands of wavelet packet filter banks. It means that the resonant frequency 3375 Hz issimultaneously distributed into two imperfect orthogonal frequency bands, namely the frequency band (3000 Hz to3375 Hz) of wavelet packet node (4, 8) and the frequency band (3375 Hz to 3750 Hz) of wavelet packet node (4, 9). Thesparsogram was used to extract the bearing fault signatures. The paving of the sparsogram is plotted in Fig. 2(a), in whichthe optimal wavelet packet node (4, 9) is selected for further envelope analysis. In Fig. 2(b), the frequency spectrum of thewavelet packet filter related to wavelet packet node (4, 9) shows that the frequency localization of the wavelet packet filteris not good because it contains an undesired frequency band range from 2200 Hz to 2600 Hz. The frequency spectrum of thesignal filtered by the wavelet packet filter is plotted in Fig. 2(c), where the energy leakage can be clearly seen at thefrequency range from 2200 Hz to 2600 Hz. Besides, the frequency band of the wavelet packet filter related to wavelet packetnode (4, 9) only covers the frequency range from 3275 Hz to 3750 Hz, which is the part of the simulated resonant frequencyband as shown in Fig. 1(b). According to the above analyses, the simulated bearing fault signatures are reduced by the binarywavelet packet filter related to the optimal wavelet packet node. The envelope spectrum of the signal extracted fromwavelet packet node (4, 9) is enhanced by a non-linear transform that is introduced at the end of this section, and the finalresult is depicted in Fig. 2(d). Although the results obtained by the sparsogram are effective in detecting the simulated faultsignatures, the visual inspection ability is reduced because the simulated resonant frequency band is only partly retainedand the undesired frequency band corrupted by noise still exists.

This section presents an automatic selection process for finding the optimal Morlet wavelet filter. It is capable ofextracting one of the bearing resonant frequency bands due to its flexibility in selecting a proper center frequency andbandwidth. The optimal parameters of the complex Morlet wavelet can be selected by a genetic algorithm through thefinding of the maximal sparsity measurement value. Once the optimal complex Morlet wavelet has been obtained, then it

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Fig. 1. A simulated bearing fault signal with one resonant frequency band corrupted by heavy noise: (a) the temporal signal; (b) the frequency spectrum ofthe signal.

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Fig. 2. The results obtained by applying the sparsogram to the simulated signal with one resonant frequency band corrupted by heavy noise: (a) the pavingof sparsogram; (b) the frequency spectrum of the wavelet packet filter related to wavelet packet node (4, 9); (c) the frequency spectrum of the signalextracted from wavelet packet node (4, 9); (d) the envelope spectrum of the filtered signal enhanced by a non-linear transform.


can be used to perform single filtering operation. The modulus of the filtered signal has the ability for amplitudedemodulation. The concerned bearing fault characteristic frequency can be detected by envelope spectrum analysis with orwithout a non-linear transform. Moreover, the new sparsogram as stated in Part 1 paper can provide proper initial centerfrequencies and bandwidths for the use of a genetic algorithm. Hence, the convergence process becomes much faster.A flowchart for illustrating the new method is shown in Fig. 3. The details of the proposed method are presented in thefollowing paragraphs.


Awavelet transform can be explained by two individual aspects, namely the inner product operation and the convolutionoperation. The inner product operation is conducted with the translated and dilated mother wavelet functions.The convolution operation is conducted with the time reversal and dilated mother wavelet functions. These two operationscan be expressed in [24]

Wf ðu; sÞ ¼Z þ∞

−∞f ðtÞ 1ffiffi

sp γ

t−us

� �dt ¼ ⟨f ðtÞ;ψu;sðtÞ⟩¼ f ðtÞ n ψ ′

sðtÞ; ð2Þ

ψu;sðtÞ ¼1ffiffis

p ψt−us

� �; ð3Þ

Perform the optimal complex Morletwavelet filtering

Load an original vibration signal

Obtain the optimal Morlet waveletparameters by genetic algorithm

Extract the envelope by the modulus ofsignal filtered by optimal filtering

Identify bearing fault characteristicfrequency

Start

Estimate fitness

Initialize genetic algorithm

Perform genetic operations: selection,crossover and mutation

End Optimal parameters for the complexMolert wavelet

Yes

Estimate fitness and update populationwith new individuals

No

Perform spectrum analysis with orwithout a non-linear transform

Generate a sparsogram

Exceed maximum number ofgeneration ?

Fig. 3. The flowchart of the proposed method.

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Fig. 4. The frequency spectra: (a) the complex Morlet wavelet with the different bandwidths; (b) the complex Morlet wavelet with the different centerfrequencies.


ψ s0 ðtÞ ¼ 1ffiffi

sp γ

−ts

� �; ð4Þ

where s is the scale parameter and u is the translation parameter. ⟨; ⟩ is the inner product operation and n is the convolutionoperation. The γðt−u=sÞ is the complex conjugate of the signal ψðt−u=sÞ. The convolution of two signals can be fast calculatedby taking the inverse Fourier transform of the product of the Fourier transform of the two signals. In other words, if the scaleof a wavelet transform is fixed, it can be regarded as a filtering process. Further, Eq. (2) can be interpreted as

Wf ðu; sÞ ¼ f ðuÞ n ψ s0 ðuÞ ¼ F−1r ðFðf Þ � ffiffi

sp

ϕs0 ðsf ÞÞ; ð5Þ

where Fr−1 is the inverse Fourier transform. Fðf Þ and ϕs

0 ðf Þ are the Fourier transform of f ðuÞ and ψ ′sðuÞ.In our research, the complex Morlet wavelet was considered because of its flexible center frequency and bandwidth

[9–16]

ψðtÞ ¼ sffiffiffiπ

p e−s2t2 ej2πf ct ; ð6Þ

where f c and s are the center frequency and the bandwidth, respectively. Its Fourier transform is taken and given as

ϕðf Þ ¼ eð−π2=s2Þ�ðf−f cÞ2 ; ð7Þ

its corresponding frequency band is limited in the band ½f c−ðs=2Þ; f c þ ðs=2Þ�. The wavelet function must satisfy theadmission condition:

ϕð0Þ ¼Z þ∞

−∞ψðtÞdt ¼ 0: ð8Þ

Considering Eqs. (7) and (8), it is deduced:

ϕð0Þ ¼ e−f2c π

2=s2 : ð9Þ

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Fig. 5. The results obtained by the developed optimal Morlet filtering method for the simulated signal with one resonant frequency band corrupted byheavy noise: (a) the convergence of the optimal complex Morlet wavelet filter; (b) the frequency spectrum of the optimal complex Morlet wavelet filter; (c)the frequency spectrum of the signal obtained by the optimal complex Morlet wavelet filtering; (d) the envelope spectrum of the filtered signal with a non-linear transform.


Reference [12] indicates that if ðf c=sÞ41:3, then ϕð0Þ≈0. The frequency spectrum of Eq. (7) with the fixed centerfrequency is plotted in Fig. 4(a). Here, f c is 3000 Hz and bandwidths s are 500 Hz, 800 Hz and 1000 Hz. The frequencyspectrum of Eq. (7) with the fixed band is plotted in Fig. 4(b). Here, s is 400 Hz and frequencies f c are 1500 Hz, 2000 Hz and2500 Hz.

For bearing fault diagnosis, it is necessary to properly select a suitable center frequency and bandwidth for the complexMorlet wavelet filter so that the selected frequency band will contain enough fault signatures in the resonance frequencyband. To select the optimal center frequency and bandwidth automatically and intelligently, a genetic algorithm wasemployed to maximize the objective function, which is the sparsity measurement in this paper. As suggested by Holland[25], a genetic algorithm is an effective and simple tool for generating optimal solutions for the optimization and searchproblems [26–28]. The other advantages of genetic algorithm are given as follow. Compared with the traditional search for a

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Fig. 6. The results obtained by the developed optimal Morlet wavelet filtering method for the simulated bearing fault signal (two resonant frequencybands) mixed with heavy noise: (a) the convergence of the optimal complex Morlet wavelet filter; (b) the frequency spectrum of the optimal complexMorlet wavelet filter; (c) the frequency spectrum of the signal obtained by the optimal filtering.

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Fig. 7. The frequency spectra obtained by the method proposed in Part 2 paper for the simulated signal mixed with heavy noise: (a) the envelope spectrumof the filtered signal without a non-linear transform; (b) the envelope spectrum of the filtered signal with a non-linear transform.


global optimal solution, genetic algorithm searches the global optimal value by a parallel form, which avoids finding a localoptimal solution. Genetic algorithm uses more reasonable probability based rules rather than deterministic rules for findingthe global optimal solution. Genetic algorithm can be used to solve an optimal problem when an analytical solution isdifficult to be found. The process of using a genetic algorithm to optimize the parameter selection for a complex Morletwavelet filter with the maximum sparsity measurement value is described as follows.

Step 1. Initialization. It is necessary to encode the center frequency and the bandwidth in binary, as strings of 0 and 1.The lengths of binary code for each center frequency and bandwidth are 13 and 12, respectively. Binary code lengths maybe changed according to the user's requirements [27]. Encoding the center frequency and bandwidth forms achromosome. The population size of the chromosomes was set at 50. The initial values of the chromosomes could berandomly generated in the initialization process. Zhang and Randall [29] suggested that the initial center frequency and

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Fig. 8. The results obtained by the developed optimal Morlet filtering method for outer race fault signal: (a) the convergence of the complex Morletwavelet filter; (b) the frequency spectrum of the optimal complex Morlet wavelet filter; (c) the frequency spectrum of the signal obtained by the optimalfiltering.

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Fig. 9. The frequency spectra obtained by the developed optimal Morlet filtering method for outer race fault signal: (a) the envelope spectrum of thefiltered signal without a non-linear transform; (b) the envelope spectrum of the filtered signal with a non-linear transform.

FigMoopt

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bandwidth provided by the fast Kurtogram could be used to fasten the convergence process of the genetic algorithm.Based on the similar idea, the frequency band of the optimal wavelet packet node provided by the new sparsogram iscapable of providing a proper initial center frequency and bandwidth for the use of a genetic algorithm, because theproper initial values help the genetic algorithm to search for the best solution within a narrow solution zone. It isnecessary to define a termination condition. The maximum number of generations 50 was artificially set to repeat 50times in the natural evolution process. To speed up the convergence process again, some constraints were taken intoaccount [10,12,13]. First, the upper cut-off frequency of the complex Morlet wavelet filter should be lower than half of thesampling frequency Fs, namely Nyquist frequency. Second, to reduce the interruption from the lower shaft rotatingfrequency, misalignment, unbalance, etc., the lower cut-off frequency of the complex Morlet wavelet filter should beseveral times larger than the rotating shaft frequency. The lower and upper center frequencies are the values of Fs � 0:1and Fs � 0:4. Third, to effectively extract the faulty impacts, the bandwidth of the complex Morlet wavelet should beseveral times larger than the largest bearing fault characteristic frequency to cover the resonant frequency band.

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. 10. The results obtained by the developed optimal Morlet filtering method for the inner race fault signal: (a) the convergence of the proposed complexrlet wavelet filter; (b) the frequency spectrum of the optimal complex Morlet wavelet filter; (c) the frequency spectrum of the signal obtained by theimal filtering.

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. 11. The frequency spectra obtained by the method proposed in Part 2 paper for inner race fault signal: (a) the envelope spectrum of the filtered signalhout a non-linear transform; (b) the envelope spectrum of the filtered signal with a non-linear transform.

Figwavfilte

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Step 2. Fitness estimation. The fitness function scores how good a candidate solution in each generation should be.The candidate solution is also called an individual. The objective function value based linear ranking was used tocalculate the fitness of each candidate solution. To construct the objective function, a sparsity measurement wasemployed. Although L0 norm has a great ability to measure the sparsity of a signal, it is likely to be affected by noisebecause the noise makes the signal become non-zero. Therefore, S is the approximate sparsity measurement, which isdefined as follow according to References [19–21]:

S¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi∑L

t ¼ 1ðzðtÞÞ2q∑L

t ¼ 1jzðtÞj¼ ∥zðtÞ∥2

∥zðtÞ∥1; ð10Þ

where ∥zðtÞ∥2 and ∥zðtÞ∥1 are the norms of L2 and L1, respectively. L is the length of the signal. zðtÞ is the modulus of thewavelet coefficients given by the complex Morlet wavelet filter, namely the envelope signal. In other words, zðtÞ is thedemodulated signal. The candidate solution that has the largest sparsity value also has the highest fitness.

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. 12. The results obtained by the developed optimal Morlet filtering method for bearing ball fault signal: (a) the convergence of the complex Morletelet filter; (b) the frequency spectrum of the optimal complex Morlet wavelet filter; (c) the frequency spectrum of the signal obtained by the optimalring.

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. 13. The frequency spectra obtained by the developed optimal Morlet filtering method for the ball fault signal: (a) the envelope spectrum of the filteredal without a non-linear transform; (b) the envelope spectrum of the filtered signal with a non-linear transform.

Ficosp

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Step 3. Performing genetic operations: selection, crossover and mutation. In the process of each successive generation, aproportion of chromosomes with higher fitness were kept. Here, the ratio was set to 0.1. The rest of the chromosomeswere randomly selected again by the stochastic universal selection method to breed a new selected generation. Based onStep 2, the fitter solution candidates were selected. The new selected chromosomes were used for reproduction. A newgeneration of population was generated by crossover and mutation from those selected chromosomes and a newcandidate solution was created by sharing a pair of candidate solutions selected in the last generation. It should be notedthat the reproduction process was probabilistic. Single point crossover was used and the crossover probability value wasset at 0.7. To maintain genetic diversity from one generation to the next, a mutation probability was used to change therandom bit in a chromosome. In our research, the mutation probability was chosen as 0.002.Step 4. Iteration. Step 2 was repeated on new individuals to calculate individual fitness and replace the least-fitpopulation with new individuals.Step 5. Termination. As mentioned in Step 1, the maximum number of generations was defined as the terminationcondition. Steps 3 and 4 were repeated until the termination condition was satisfied.

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g. 14. The results obtained by the developed optimal Morlet filtering method for bearing multi-fault signal (the outer race and inner race faults): (a) thenvergence of the optimal complex Morlet wavelet filter; (b) the frequency spectrum of the optimal complex Morlet wavelet filter; (c) the frequencyectrum of signal obtained by the optimal filtering.

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. 15. The frequency spectra obtained by the developed optimal Morlet filtering method for the multi-fault (outer race and inner race faults) signal: (a)envelope spectrum of the filtered signal without a non-linear transform; (b) the envelope spectrum of the filtered signal with a non-linear transform.


optimal filter is the envelope signal zoptðtÞ. Bearing fault characteristic frequencies can be used to judge fault types through

After completing the selection of the optimal center frequency and bandwidth, the optimal complex Morlet wavelet filterwas used to enhance the weak bearing fault vibration signal. The modulus of the wavelet coefficients obtained by the

frequency spectrum analysis of the envelope signal. It is usually known that bearing fault characteristic frequenciesdominate the frequency spectrum after amplitude demodulation [30], thus it is interesting to enhance its visual inspectionability by removing unrelated noise in frequency spectrum. To achieve the desired fault frequency, a non-linear functionused in ultrasonic signal processing [19] was employed to depress noise. Assume the envelope signal has a zero mean, whichcan be realized by subtracting the mean of the envelope signal from the envelope signal. The non-linear function is

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Fig. 16. The results obtained by the developed optimal Morlet filtering method without initial values provided by the sparsogram for the simulated signal(two resonant frequency bands) mixed with heavy noise: (a) the convergence of the optimal complex Morlet wavelet filter; (b) the frequency spectrum ofthe optimal complex Morlet wavelet filter; (c) the frequency spectrum of the signal obtained by the optimal filtering.

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Fig. 17. The frequency spectra obtained by the developed optimal Morlet filtering method without initial values provided by sparsogram for the simulatedsignal (two resonant frequency bands) mixed with heavy noise: (a) the envelope spectrum of the filtered signal without a non-linear transform; (b) theenvelope spectrum of the filtered signal with a non-linear transform.


performed on the envelope signal zoptðf Þ:

Zoptðf Þ ¼zoptðf Þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

∑Fs=2−1g ¼ 0;g≠f ðzoptðgÞÞ2=∑

Fs=2−1g ¼ 0;g≠f jzoptðgÞj

q −zoptðf Þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

∑Fs=2−1g ¼ 0 ðzoptðgÞÞ2=∑Fs=2−1

g ¼ 0 jzoptðgÞjq

0B@

1CA ð11Þ

where zoptðf Þ is the frequency spectrum of zoptðtÞ. The reasons that Eq. (11) is able to depress noise are given as follows.Considering the second term on the right hand side of Eq. (11), it is a constant value at a frequency point. It can be regardedas a threshold at that frequency point. It means that the value of Eq. (11) at that frequency point is the first term on the righthand side of Eq. (11) subtracts the threshold at that frequency point. If the value of zoptðf Þ at that frequency point is large, thefirst term on the right hand side of Eq. (11) becomes large. Consequently, the difference of the first term and the second termon the right hand side of Eq. (11) becomes larger. Therefore, by using Eq. (11), only the major components in envelopespectrum can be retained.

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Fig. 18. The results obtained by the developed optimal Morlet filtering method without initial values provided by the sparsogram for bearing outer racefault signal: (a) the convergence of the optimal complex Morlet wavelet filter; (b) the frequency spectrum of the optimal complex Morlet wavelet filter; (c)the frequency spectrum of the signal obtained by the optimal filtering.

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Fig. 19. The frequency spectra obtained by the developed optimal Morlet filtering method without initial values provided by sparsogram for bearing outer race faultsignal: (a) the envelope spectrum of the filtered signal without a non-linear transform; (b) the envelope spectrum of the filtered signal with a non-linear transform.


For a comparison, the optimal complex Morlet wavelet filter based method proposed in this paper was applied to process thesame simulated signal with one resonant frequency band produced by Eq. (1). The results are shown in Fig. 5. The convergenceprocess is plotted in Fig. 5(a). In this figure, the optimal center frequency and bandwidth are 3351 Hz and 1010 Hz, respectively. Thefrequency support range of the optimal Morlet wavelet filter starts from 2846 Hz to 3856 Hz, which can properly cover thesimulated resonant frequency band as shown in Fig. 1(b). The frequency band of the signal filtered by the optimal Morlet wavelet isplotted in Fig. 5(c). The envelope spectrum of the filtered signal with the non-linear transform is given in Fig. 5(d), where it can beseen that the simulated fault characteristic frequency and its harmonics obtained by the optimal Morlet filtering are clearer thanthose as shown in Fig. 2(d). From the above analyses, it is concluded that the optimal Morlet wavelet filtering has better ability toextract a proper resonant frequency band for bearing fault diagnosis.

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Fig. 20. The results obtained by the developed optimal Morlet filtering method without initial values provided by the sparsogram for the bearing innerrace fault signal: (a) the convergence of the optimal complex Morlet wavelet filter; (b) the frequency spectrum of the optimal complex Morlet waveletfilter; (c) the frequency spectrum of the signal obtained by the optimal filtering.

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Fig. 21. The frequency spectra obtained by the developed optimal Morlet filtering method without initial values provided by the sparsogram for bearinginner race fault signal: (a) the envelope spectrum of the filtered signal without a non-linear transform; (b) the envelope spectrum of the filtered signal witha non-linear transform.


3. Case studies for validating the proposed method

3.1. A simulated bearing fault signal with two resonant frequency bands

In simulation, the same simulated bearing fault signal (two resonant frequency bands) mixed with heavy noise used inSection 3.1 of Part 1 paper was firstly employed to validate the effectiveness of the optimal complex Morlet wavelet filteringmethod proposed in Part 2 paper.

In order to speed up the convergence of the optimal complex Morlet wavelet filter, two pairs of initial center frequenciesand bandwidths for finding the optimal complex Morlet wavelet filter were set by considering the wavelet packet nodes(4, 4) and (4, 11) provided by the sparsogram in Section 3.1 of Part 1 paper. These nodes indicated the approximate locationsof the resonant frequency bands detected by the sparsogram. Recalling that the frequency band of a specific wavelet packet

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Fig. 22. The results obtained by the developed optimal Morlet filtering method without initial values provided by the sparsogram for bearing ball faultsignal: (a) the convergence of the optimal complex Morlet wavelet filter; (b) the frequency spectrum of the optimal complex Morlet wavelet filter; (c) thefrequency spectrum of the signal obtained by the optimal filtering.

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Fig. 23. The frequency spectra obtained by the developed optimal Morlet filtering method without initial values provided by the sparsogram for bearingball fault signal: (a) the envelope spectrum of the filtered signal without a non-linear transform; (b) the envelope spectrum of the filtered signal with anon-linear transform.


node (j, p) is located in the frequency range ½p� Fs=2Jþ1; ðpþ 1Þ � Fs=2

Jþ1�, the center frequency and bandwidth for thespecific wavelet packet node are ½ð2pþ 1Þ � Fs=2

Jþ2; Fs=2Jþ1�. Therefore, the frequency bands of wavelet packet nodes (4, 4)

and (4, 11) were ranged from 1500 Hz to 1875 Hz and from 4125 Hz to 4500 Hz, respectively. The center frequencies andbandwidths of wavelet packet nodes (4, 4) and (4, 11) were calculated as (1687.5 Hz, 375 Hz) and (4312.5 Hz, 375 Hz),respectively. The convergence process of the proposed optimal filter is plotted in Fig. 6(a). The frequency spectrum of theoptimal complex Morlet wavelet filter is shown in Fig. 6(b), where it is found that the optimal center frequency andbandwidth are 1716 Hz and 1195 Hz. The frequency spectrum of the signal obtained by the optimal complex Morlet waveletfiltering is given in Fig. 6(c). The result obtained by the optimal filtering illustrates that the optimal complex Morlet waveletfilter can keep the most useful fault signatures for bearing fault diagnosis.

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Fig. 24. The results obtained by the developed optimal Morlet filtering method without initial values provided by the sparsogram for bearing multi-faultsignal: (a) the convergence of the optimal complex Morlet wavelet filter; (b) the frequency spectrum of the optimal complex Morlet wavelet filter; (c) thefrequency spectrum of the signal obtained by the optimal filtering.

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Fig. 25. The frequency spectra obtained by the developed optimal Morlet filtering method without initial values provided by the sparsogram for bearingmulti-fault signal: (a) the envelope spectrum of the filtered signal without a non-linear transform; (b) the envelope spectrum of the filtered signal with anon-linear transform.


The envelope spectra, with and without a non-linear transform, of the signal filtered by the optimal complex Morletwavelet filtering are displayed in Fig. 7(a) and (b). The envelope spectrum with a non-linear transform clearly has bettervisual inspection. In other words, the non-linear transform has the ability to keep the major components, namely theamplitudes of the modulating frequency and its harmonics.

3.2. Experimental bearing fault signals analyzed by the proposed method

To simulate the localized bearing faults that develop on the surface of different bearing components, some realexperimental data were produced as described in Section 3.2 of Part 1 paper. The experiment platform and the inspectedbearing are shown in Fig. 11(a) and (b) of Part 1 paper, respectively. An outer race fault, an inner race fault, a rolling elementfault and multiple faults (outer race and inner race faults) were separately introduced to the inspected bearing, as previouslydescribed in Part 1 paper. The sampling frequency was equal to 80 kHz. Each sampled fault signal with 16,000 samples wasused for analysis. The bearing outer race fault characteristic frequency, the bearing inner race fault characteristic frequencyand the bearing ball spinning frequency were calculated as 136 Hz, 192 Hz and 64 Hz, respectively, as reported in Part1 paper.

The bearing outer race fault signal, the bearing inner race fault signal and the rolling element fault signal are plotted inFig. 12(a), (b) and (c) of Part 1 paper, respectively. The temporal waveform of an extra bearing vibration signal from a bearingthat suffered the outer race and the inner race defects is plotted in Fig. 13(a) of Part 1 paper. The proposed method wasapplied to each of the abovementioned faulty signal types. In Section 3.2 of Part 1 paper, for processing the outer race faultsignal by using the sparsogram, the results showed that wavelet packet node (4, 3) had the largest sparsity value. Thefrequency band of wavelet packet node (4, 3) was ranged from 7500 Hz to 10,000 Hz. Therefore, from this node, a pair of

Table 1The comparisons of the developed optimal Morlet filtering method with and without the initial values provided by the sparsogram.

Initial valuesprovided by

Convergence times Optimal center frequencies and bandwidths

Signal 1(s)

Signal 2(s)

Signal 3(s)

Signal 4(s)

Signal 5(s)

Signal 1(Hz)

Signal 2(Hz)

Signal 3(Hz)

Signal 4(Hz)

Signal 5(Hz)

Sparsogram 3.29 1.10 1.53 2.45 2.26 (1716, 1195) (8073, 4640) (16,005, 5740) (12,829, 2006) (22,395, 9916)Random number 6.22 3.92 1.97 5.56 3.50 (1664, 705) (8108, 4594) (16,017, 5790) (12,843, 2129) (22,400, 9940)

Notes: Signal 1 is the simulated signal (two resonant frequency bands) corrupted by heavy noise. Signal 2 is the bearing outer race fault signal. Signal 3 isthe inner race fault signal. Signal 4 is the ball fault signal. Signal 5 is the multi-fault signal.

5 10 15 20 25 30 35 40 45 504.235

4.24

4.245

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0.5

1

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500

1000

1500

Fig. 26. The results obtained by the kurtosis based optimal Morlet filtering method for simulated signal mixed (two resonant frequency bands) with heavynoise: (a) the convergence of the optimal complex Morlet wavelet filter; (b) the frequency spectrum of the optimal complex Morlet wavelet filter; (c) thefrequency spectrum of the signal obtained by the optimal filtering.


initial value for the center frequency and the bandwidth was determined as (8750 Hz, 2500 Hz). The convergence of theproposed method is plotted in Fig. 8(a). The optimal center frequency and bandwidth of the optimal complex Morletwavelet are automatically determined as 8073 Hz and 4640 Hz. After the automatic parameter selection process, thefrequency spectrum of the optimal filter is shown in Fig. 8(b). The frequency spectrum of the signal obtained by the optimalcomplex Morlet wavelet filtering is plotted in Fig. 8(c).

The envelope spectra of the filtered signal, with and without a non-linear transform, are given in Fig. 9(a) and (b). Theresults indicate that the proposed method successfully extracts the outer race fault characteristic frequency and itsharmonics. The results obtained by the proposed method without a non-linear transform show more harmonics, while the

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Fig. 27. The frequency spectra obtained by the kurtosis based optimal Morlet filtering method for simulated signal mixed (two resonant frequency bands)with heavy noise: (a) the envelope spectrum of the filtered signal without a non-linear transform; (b) the envelope spectrum of the filtered signal with anon-linear transform.

5 10 15 20 25 30 35 40 45 500.826

0.828

0.83

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Fig. 28. The results obtained by the smoothness index based optimal Morlet filtering for simulated signal mixed (two resonant frequency bands) withheavy noise: (a) the convergence of the complex Morlet wavelet filter; (b) the frequency spectrum of the optimal complex Morlet wavelet; (c) thefrequency spectrum of the signal obtained by the optimal filtering.


results obtained by the proposed method with a non-linear transform have a cleaner spectrum for depressing the noise.Therefore, the proposed method with a non-linear transform can generate a better visual inspection for bearing faultdiagnosis.

For processing the inner race fault signal by using the sparsogram, the results showed that wavelet packet node (4, 2) andnode (4, 6) had the largest and the second largest values from sparsity measurement. According to the frequency supportdecomposition of the binary wavelet packet filters, the frequency bands of wavelet packet nodes (4, 2) and (4, 6) wereranged from 5000 Hz to 7500 Hz and from 15,000 Hz to 17,500 Hz, respectively. Based on the results, the initial values of thecenter frequencies and bandwidths for the optimization of the method proposed in this paper could be determined as

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Fig. 29. The frequency spectra obtained by the smoothness index based optimal Morlet filtering method for simulated signal (two resonant frequencybands) mixed with heavy noise: (a) the envelope spectrum of the filtered signal without a non-linear transform; (b) the envelope spectrum of the filteredsignal with a non-linear transform.

5 10 15 20 25 30 35 40 45 509.928

9.93

9.932

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1

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1500

Fig. 30. The results obtained by the Shannon entropy based optimal Morlet filtering for simulated signal (two resonant frequency bands) mixed with heavynoise: (a) the convergence of the complex Morlet wavelet filter; (b) the frequency spectrum of optimal complex Morlet wavelet; (c) the frequency spectrumof the signal obtained by the optimal filtering.


(6250 Hz, 2500 Hz) and (16,250 Hz, 2500 Hz). The convergence of the optimal complex Morlet wavelet filter for processingthe inner race faulty signal is plotted in Fig. 10(a). The optimal center frequency and bandwidth finally selected by thegenetic algorithmwere 16,005 Hz and 5740 Hz respectively. The frequency spectrum of the optimal complex Morlet waveletfilter and that of the signal filtered by the optimal complex Morlet wavelet filter are shown in Fig. 10(b) and (c).The envelope spectrum of the filtered signal indicates the existence of the inner race fault characteristic frequency in Fig. 11(a). With the aid of a non-linear transform, the result in Fig. 11(b) clearly shows the inner race fault characteristic frequency.Note that heavy noise in the envelope spectrum has been depressed by the non-linear transform.

Concerning wavelet packet nodes (4, 5) and (4, 3) obtained by the sparsogram in the case of the ball fault signal in Part 1paper, the frequency bands of wavelet packet nodes (4, 5) and (4, 3) were ranged from 7500 Hz to 10,000 Hz and from

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Fig. 31. The frequency spectra obtained by the Shannon entropy based optimal Morlet filtering method for simulated signal (two resonant frequencybands) mixed with heavy noise: (a) the envelope spectrum of the filtered signal without a non-linear transform; (b) the envelope spectrum of the filteredsignal with a non-linear transform.

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Fig. 32. The results obtained by the kurtosis based optimal Morlet filtering method for bearing ball fault signal: (a) the convergence of the optimal complexMorlet wavelet filter; (b) the frequency spectrum of the optimal complex Morlet wavelet filter; (c) the frequency spectrum of the signal obtained byoptimal filtering.


12,500 Hz to 15,000 Hz, respectively. Therefore, two pairs of initial center frequencies and bandwidths were found to be(8750 Hz, 2500 Hz) and (13,750 Hz, 2500 Hz). The convergence process of the proposed optimal filter is shown in Fig. 12(a)for the real ball fault signal. The optimal center frequency of 12,830 Hz and the bandwidth of 1984 Hz were finally selectedby the genetic algorithm. The frequency spectrum of the optimal complex Morlet wavelet is plotted in Fig. 12(b).The frequency spectrum of the signal obtained by the optimal complex Morlet wavelet filtering is given in Fig. 12(c).The envelope spectrum of the filtered signal indicates that the bearing with a ball fault is successfully detected in Fig. 13(a).To enhance the visual inspection, the result using a non-linear transform in Fig. 13(b) illustrates that the non-lineartransform not only keeps the bearing ball spinning frequency and its harmonics but also suppresses a significant amountof noise.

For processing the bearing multi-fault signal, the optimal wavelet packet node (4, 9) was selected by the sparsogram inPart 1 paper. The frequency range of wavelet packet node (4, 9) was ranged from 22,500 Hz to 25,000 Hz. A pair of initialcenter frequency and bandwidth for the use of genetic algorithm was set to (23,750 Hz, 2500 Hz). The convergence processof the optimal wavelet filter is plotted in Fig. 14(a). At last, the optimal center frequency and bandwidth are 22,395 Hz and

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Fig. 33. The frequency spectra obtained by the kurtosis based optimal Morlet filtering method for bearing ball fault signal: (a) the envelope spectrum of thefiltered signal a without non-linear transform; (b) the envelope spectrum of the filtered signal with a non-linear transform.

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Fig. 34. The results obtained by the smoothness index based optimal Morlet filtering method for bearing ball fault signal: (a) the convergence of theoptimal complex Morlet wavelet filter; (b) the frequency spectrum of the optimal complex Morlet wavelet filter; (c) the frequency spectrum of the signalobtained by the optimal filtering.


9916 Hz. The frequency spectrum of the optimal wavelet filter is shown in Fig. 14(b), while the signal obtained by theoptimal wavelet filtering is plotted in Fig. 14(c). The envelope spectrum of the signal filtered by the optimal wavelet is shownin Fig. 15(a), where the outer race fault characteristic frequency, inner race fault characteristic frequency and theirharmonics are found to demonstrate the existence of the outer race and inner race faults. After a non-linear transform wasapplied to depress the noise existing in Fig. 15(a), bearing outer race fault characteristic frequency and the first harmonic ofthe bearing inner race fault characteristic frequency are retained in Fig. 15(b).

In this paper, all computations were conducted using MATLAB installed on a desktop with 3.1 GHz CPU and 4 GB (3.24 GBusable) RAM. The convergence times of the proposed optimal Morlet wavelet filter enhanced by the sparsogram for thesimulated signal mixed with heavy noise, the bearing outer race fault signal, the bearing inner race fault signal, the bearingball fault signal and the bearing multi-fault signal are 3.29 s, 1.10 s, 1.53 s,2.45 s and 2.26 s, respectively.

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Fig. 35. The frequency spectra obtained by the smoothness index based optimal Morlet filtering for bearing ball fault signal: (a) the envelope spectrum ofthe filtered signal without a non-linear transform; (b) the envelope spectrum of the filtered signal with a non-linear transform.

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Fig. 36. The results obtained by the Shannon entropy based optimal Morlet filtering method for bearing ball fault signal: (a) the convergence of the optimalcomplex Morlet wavelet filter; (b) the frequency spectrum of the optimal complex Morlet wavelet filter; (c) the frequency spectrum of the signal obtainedby the optimal filtering.


4. Comparison studies

4.1. A comparison study for the convergence of the optimal complex Morlet wavelet filter without using initial center frequenciesand bandwidths provided by the sparsogram

Section 3 demonstrates that the initial guessing of the center frequencies and bandwidths provided by the sparsogramcan speed up the convergence process of the optimal complex Morlet wavelet filter. Nonetheless, it is necessary to conduct acomparison study to investigate the performance of the optimal complex Morlet wavelet filter without using the properinitial guessing values. In Section 4.1, the initial center frequencies and bandwidths for the use of genetic algorithm wererandomly generated. For extensive comparisons, the same bearing fault signals, including the simulated signal with tworesonant frequency bands and those caused by the outer race fault, the inner race fault, the ball fault and the multiple faults,were investigated again. Without setting the proper initial values for the center frequencies and bandwidths, theconvergence of the proposed optimal Morlet wavelet filter becomes slower as shown in Figs. 16(a), 18(a), 20(a), 22(a)and 24(a). Specifically, the convergence times for processing the simulated signal mixed with heavy noise, the bearing outerrace fault signal, the bearing inner race fault signal, the ball fault signal and the multi-fault signal are 6.22 s, 3.92 s, 1.97 s,5.56 s and 3.5 s, respectively. Compared with the convergence times with proper initial guessing values, the convergencetimes required here are largely increased. The optimal center frequencies and bandwidths for the frequency spectra of thesimulated signal with two resonant frequency bands, the outer race fault signal, the inner race fault signal, the ball faultsignal and the multi-fault signal are displayed in Figs. 16(b), 18(b), 20(b), 22(b) and 24(b) respectively. The frequency spectraof the signals extracted from the optimal filtering bands for the cases of having two resonant frequency bands, the outer racefault, the inner race fault, the ball fault and the multiple faults are plotted in Figs. 16(c), 18(c), 20(c), 22(c) and 24(c)respectively.

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Fig. 37. The frequency spectra obtained by the Shannon entropy based optimal Morlet filtering method for bearing ball fault signal: (a) the envelopespectrum of the filtered signal without a non-linear transform; (b) the envelope spectrum of the filtered signal with a non-linear transform.

Table 2The comparisons of the proposed complex Morlet wavelet filtering method, the kurtosis based Morlet wavelet filtering method, the smoothness indexbased Morlet wavelet filtering method and the Shannon entropy based Morlet wavelet filtering method.

Metric for Morlet wavelet filtering Effective in detecting bearing faults? Visual inspection ability Optimal center frequency and bandwidth

Case 1 Case 2 Case 1 Case 2 Case 1 (Hz) Case 2 (Hz)

Sparsity Yes Yes High High (1716, 1195) (12,830, 1984)Kurtosis Yes Yes High High (1541, 1975) (13,637, 11,058)Smoothness index Yes Yes High High (1552, 1408) (12,867, 1101)Shannon entropy Yes Yes High High (1832, 835) (12,914, 1420)

Notes: Case 1 is about the simulated signal corrupted by heavy noise. Case 2 is about the real laboratorial ball fault signal. The initial center frequencies andbandwidths are provided by the sparsogram.


Even though the proposed optimal Morlet wavelet filtering method without having the proper initial center frequenciesand bandwidths provided by the sparsogram may take much longer calculation times, the results shown in Figs. 17, 19, 21,23 and 25 demonstrate that the proposed optimal filtering method is still effective in distinguishing different bearing faultsignatures. The envelop spectra of the filtered signal without the application of the non-linear transform are shown inFigs. 17(a), 19(a), 21(a), 23(a) and 25(a) for the simulated signal, the outer race fault signal, the inner race fault signal, the ballfault signal and the multiple fault signal, respectively. Whilst, the envelope spectra of the filtered signal with the applicationof the non-linear transform are shown in Figs. 17(b), 19(b), 21(b), 23(b) and 25(b). Note that with the help from the non-linear transform, many of the faulty signals become easier to be observed. The comparisons of the convergence times, theselected optimal center frequency and its bandwidths for different fault signals with and without the employment of theproper initial values are summarized in Table 1. From the results, it can be concluded that the initial guessing valuesprovided by the sparsogram are able to enhance the performance of the proposed method.

4.2. A comparison study with three popular methods

Three popular metrics introduced in Section 4.2 of Part 1 paper were used to replace the sparsity measurement functionused in the proposed optimal Morlet wavelet filtering method for selecting the optimal parameters. The metrics are thekurtosis [12], the smoothness index [14] and Shannon entropy [13]. In the analyses, two signals including the simulatedsignal (two resonant frequency bands) mixed with heavy noise and the bearing ball fault signal were investigated. Here, forfair comparisons and speeding up the convergence process, the same proper initial center frequencies and bandwidthsprovided by the sparsogram in Sections 3.1 and 3.2 of this paper were used in this section.

For the simulated signal, the results obtained by the kurtosis based optimal wavelet filtering are given in Figs. 26 and 27.In Fig. 26(c), one simulated resonant frequency band at a lower frequency band is retained. Besides, the bandwidth obtainedby the maximum kurtosis method covers a wide frequency band. In Fig. 27, it is seen that the kurtosis based optimal Morletwavelet filtering method is effective in detecting the simulated fault signatures. In Figs. 28(c) and 30(c), the results obtainedfrom the smoothness index and Shannon entropy based optimal Morlet filtering methods demonstrate that both methodsare useful in the selection of proper center frequencies and bandwidths for wavelet filtering. They can also recover thesimulated fault signatures from heavy noise as shown in Figs. 29 and 31. From these results, it is concluded that the sparsitymeasurements, such as kurtosis, the smoothness index and Shannon entropy, can provide similar results as long as theoptimal Morlet wavelet filter have been employed.

On the other hand, in Part 1 paper, the kurtosis, the smoothness index and the Shannon entropy were used to quantifythe envelope of the wavelet packet coefficients obtained by binary wavelet packet transform. However, in Table 1 of Part 1paper, they failed to detect any fault signatures hidden in heavy noise. Therefore, the performance of the optimal Morletwavelet filter is better than that of binary wavelet packet transform for the extraction of simulated fault signaturescorrupted by heavy noise.

In order to further verify our method, we used the real bearing ball fault signal for further analyses. The results shown inFigs. 32 to 37 demonstrate that the optimal Morlet wavelet filtering is very effective in extracting the bearing fault features.Besides, it is seen that even though different methods generate different center frequencies and bandwidths for the optimalMorlet wavelet filter, their abilities in displaying the detecting bearing fault characteristic frequencies were roughly similar.The performance comparisons of the proposed Morlet wavelet filtering method, the kurtosis based Morlet filtering method,the smoothness index based Morlet filtering method and the Shannon entropy based Morlet filtering method is summarizedin Table 2. It is concluded that the important step is to obtain the optimal parameters of the optimal complex Morlet waveletfilter through the help of genetic algorithm. It is worth to note here again that, the sparsogram, as reported in Part 1 paper,provides a faster bearing fault diagnosis method. The optimal Morlet wavelet filter, as reported Part 2 paper, is able toperform reasonably fast inspection on the collected vibration signals that the impacts are the fault features.

5. Conclusion

This paper presented an intelligent bearing fault diagnosis method with a joint algorithm based on the complex Morletwavelet filter and genetic algorithm for maximizing the sparsity measurement value. The proposed method was validatedby the simulated bearing and real bearing fault signals. The results show that the proposed method can effectively identifythe bearing fault characteristic frequency and its harmonics. In addition, the results illustrate that the proposed methodwithout a non-linear transform reveals more harmonics of the bearing fault characteristic frequency while keeping noisethat might confuse the maintenance staff if he is diagnosing the bearing health status from the signal frequency spectrum.On the other hand, the proposed method with a non-linear transform results in a cleaner frequency spectrum. Therefore, theproposed method is easier to identify the bearing fault characteristic frequency and its harmonics. The sparsitymeasurement has been proven capable of extracting vital features for revealing various types of bearing faults. With thehelp of genetic algorithm, the entire bearing fault detection process can be fully automatic and its accuracy can besignificantly enhanced. Besides, the optimal wavelet packet node established by the new sparsogram is able to speed up theconvergence process during the optimization of the complex Morlet wavelet filter. Finally, it is concluded that the optimalMorlet wavelet filtering based methods are better than the binary wavelet packet transform based methods in the


extraction of the bearing fault features, especially when the bearing fault signatures have been overwhelmed byheavy noise.

Acknowledgments

The work described in this paper was fully supported by a Grant from the Research Grants Council of the Hong KongSpecial Administrative Region, China (Project No. CityU 122011) and a Grant from City University of Hong Kong (Project No.7008187).

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