# Fault Size Estimation of Bearings Using Multiple Decomposition Techniques with Artificial Neural Network

Authors: Suchi Mishra; Rahul Dubey; Preety D. Swami; Alok Jain
Journal: Scientific Programming
 (2022)

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## Abstract

A running machine generates multi-frequency vibration signals which can be captured by accelerometers. Empirical mode decomposition, wavelet decomposition, and wavelet packet decomposition are the commonly used methods to decompose the multi-frequency signal. Quick fault classification, accurate signal decomposition, and fault size detection are still a problem in machines with rotary components. In the proposed work, fault diameter in rotary part of machine is detected and classified using the machine learning methods. In the first stage, we have employed empirical mode decomposition (EMD) for high-frequency noise removal. Residue signal is obtained by removing first IMF from base signal considering first IMF as a high-frequency noisy signal, followed by wavelet decomposition. Entropy of the wavelet coefficient obtained from 3rd level decomposition of residue signal is calculated which acts as an input parameter to the machine learning techniques to determine the diameter for fault. Three different sets have been taken for inner race, outer race, and ball race correspondingly. The proposed method classifies and detects the fault diameter up to 99.5%. The proposed method can be used for different types of continuous as well as discrete wavelets.


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## Body

## 1. Introduction

The main objective of maintenance department in industries is to keep machineries running. Machine failure makes unwanted downtime across the industries with high maintenance cost. Rolling bearing is considered as a vital component of rotating machines. Several kinds of faults exist in bearings. The faults in the bearing decrease the accuracy and performance of the machine.Bearing fault detection is done using various signal processing techniques. Various available methods to analyse the bearing fault signals are categorized in time-frequency domain [1, 2], time domain [3–5], and frequency domain [6–9]. Image processing techniques have also been explored in works for diagnosis and detection of bearing fault [10, 11].The use of vibration signal in the analysis of bearing faults is also reported in the literature. These vibration signals are recorded using accelerometer which may have noisy background. Kurtosis analysis is also one of the methods applied for the early-stage detection of faults in bearing. In earlier research studies, it is already proved that for kurtosis value more than three, faults are present in the vibration signal [12].Wavelet transform (WT) is applicable for investigation of non-stationary signals. Wavelet transform is used for the decomposition of signal into approximation and detailed coefficient. Application of wavelet transforms in the detection of bearing faults is also reported by Kumar Jha and Swami [13].Various statistical parameters extracted from the wavelet coefficients of vibration signals are utilized for different wavelet functions for fault detection [14].Parey et al applied the combination of artificial Intelligence methods with maximum energy to Shannon entropy extracted from wavelet coefficients for the ball bearing fault classification [15].WT has been extensively used in signal denoising. In [16], Ali Alnuaim et al suggested a procedure for rolling bearing fault diagnosis evolved from wavelet denoising. Wavelet-based function is chosen from impulse response of the bearing system. The authors optimized damping factor and central frequency of wavelet on the basis of kurtosis maximization criteria. Simulated and real rolling vibration signals were both used in [16] to judge the performance of the proposed approach.In [17], reshaping of Morlet wavelet is done by using Shannon entropy. Li and Yin extracted a weak fault signature from rolling bearing from wavelet filter bank [17]. Singular value decomposition is utilized to obtain the periodicity of the signal in the proposed approach in [17].Enhancement of impulsive features and suppression of residual noise were achieved by Rafiee et al. using Morlet wavelet for detection of bearing fault [18]. Jiang et al. optimized the wavelet filter by eliminating the interferential vibration using differential evolution and retrieved the characteristics of fault present in the signal.In [19], Kankar et al. used genetic algorithm to enhance the parameters of Morlet wavelet to maximize the sparsity measurement value. The optimal wavelet filter provides the modulus of the wavelet coefficients which is used to pull out the envelope. Finally, for enhancement of the visual inspection, the ability of bearing fault characteristic frequencies of a non-linear function is utilized.Faulty vibration signals recorded through accelerometer for rolling bearings with respect to different fault condition were preprocessed via Laplace-wavelet transform for feature extraction by Shukla and Dixit [20]. Obtained characteristics from time and frequency domains are utilized as an input to artificial neural network for fault classification. A feedforward multi-layer perceptron neural network, which consists of three layers, was used for the categorization in [20].Lempel–Ziv complexity along with CWT was utilized to characterize the bearing faults in [21]. CWT was utilized to find best location of faults and to remove interferences of noise and irrelevant signal components. The envelope and high-frequency carrier signal are drawn from wavelet coefficient followed by Lempel–Ziv complexity value calculation for all.The hybrid CWT-SVM method was proposed by Qiu et al. for bearing fault detection in induction motor in [22].The CWT with autocorrelation improvement was given by He et al. in [23] to allow minor defect detection of bearings. The coefficients of CWT are calculated after reducing dimensions of vibration signal. The ratio of energy to Shannon entropy was applied as a parameter to select the base wavelet followed by extraction of the statistical feature from wavelet coefficient. In the last stage, soft computing techniques were applied to do the classification of bearing faults [24].The detection of bearing fault using morphological wavelet slicing was proposed by Tiwari and Shukla in [25]. Li et al. proposed a multi-scale autocorrelation approach for the analysis of the bearing fault. Fourier transform of extracted temporal components is evaluated to view morphological slices in frequency domain. Its three-dimensional representation was reduced by the presence of noise. Hence, autocorrelation function was used to produce a multi-scale autocorrelation spectrogram to maximize the autocorrelation values of all frequencies.For non-linear and non-stationary signals, Peter and Wang [26] suggested an EMD technique that overcomes disadvantages of wavelet transform. Al Raheem et al. [27] used the footprint of Hilbert plot as a metric to characterize distinct ball bearing faults. The suggested approach is estimated to be 99.98% efficient [27].The polar diagram in [28] is used to improve the periodic transient signals caused by bearing defects. In the method proposed in [28], signal is represented in time scale domain using CWT, and then it maps wavelet coefficient into polar diagram. Transient characteristics would thus be synchronously improved at a definite area if mapping time period equals that of transient vibrations.Stalin et al. proposed adaptive wavelet to improve impulsive features generated due to bearing [29]. The proposed scheme enhanced the impulsive components and simultaneously depressed the noise.Application of artificial neural network for the classification of rolling bearing faults can be extensively found in the literature [30–33]. In [30], Konar and Chattopadhyay used PCA along with SVD for bearing fault detection. Li et al. applied the EMD method followed by singular value decomposition. Elman neural network is used in later stage for classification of bearing faults. Classification of ball bearing faults is also done using various artificial neural network methods in [32, 33].The detection of fault size is still unexplored in the literature. In the proposed work, EMD is used on bearing signal to eliminate high-frequency noisy component. First, IMF is measured to be a high-frequency signal; therefore, it is removed from the bearing signal to get the residue [34]. The three-level wavelet decomposition of the residue signal is done. The entropy of the third level wavelet coefficient has been calculated which is further utilized as an input parameter for feedforward neural network to categorize bearing signal into respective classes [35].This paper is organized as follows. In Section2, EMD, WT, and features based on entropy are discussed. In Section 3, the proposed method is discussed. In Section 4, results and discussion are given, followed by conclusion and future scope in Section 5.


## 2. Theory

### 2.1. Empirical Mode Decomposition

Peter and Wang presented the decomposition of a signal through the EMD method in [26]. The EMD method decomposed a signal into an IMF from high-frequency component to low-frequency component. There are two necessary conditions for the component to be IMF, written below:(i)
Number of zero crossing and number of extrema in the each signal in the whole dataset essentially either equal or differ at most by one.(ii)
At any point, the mean value of envelope explained by local maxima and local minima is zero.The EMD algorithm is summarized below.Signalx (t) is subjected to calculate every probable extrema. Lower and upper envelopes Xmint and Xmaxt, respectively, are obtained through interpolation among minima and corresponding maxima [36]. The mean value at of the envelope is determined using equation (1), which is further deducted from signal xt to get ht using equation (2).(1)at=Xmint+Xmaxt2,(2)ht=xt−at.It is checked whetherht is an IMF. If it is an IMF, the residual signal rt is drawn by repeating the above steps using equation (3), or else replace xt with ht, and do entire process again.(3)rt=xt−ht.IMF1t,IMF2t,IMF3t,…,IMFNt are different frequency components of xt from high-frequency component to low-frequency component. Residual component of a signal x is used to display its central tendency (t). By combining all IMFs and residual r, the original signal x (t) may be reconstructed (t).


### 2.2. Wavelet Transform (WT)

The wavelet is a progressive research filed for the researcher in the engineering over last 20 years. Some of the researchers have contributed substantially to the use of the wavelet transform in bearing fault identification [37]. Wavelet transform is a mathematical tool that changes signals from time domain to time-frequency domain. Wavelet transform is broadly classified as CWT, DWT, and WPT [38]. Wavelet transform passes one-dimensional signal via filter bank consisting of LPF and HPF. The signal component passing through the low-pass filter is recognized as approximation coefficient (CA1) while the component extracted from the high-pass filter is recognized as detailed coefficient (CD1). This is one-level wavelet decomposition, and achieved coefficients are known as first-level wavelet coefficients [39]. Further, the approximation and detailed coefficient is passed again through LPF and HPF to get second-order wavelet coefficients. In the proposed work, three-level wavelet decomposition has been done in order to get third-level wavelet coefficients [40].The structural outline for the wavelet decomposition step is represented in Figure1.Figure 1
Two-dimensional wavelet decomposition.


### 2.3. Features Based on Entropies

Entropy is used to measure uncertainty in the signal [41]. The unpredictability of time series is estimated by entropy.The power spectrum of the signal was utilized by spectral entropy to determine the regularity of time series. Normalized Shannon entropy was used to evaluate the spectral entropy [24, 34]. Spectral entropy is the parameter to quantify spectral complexity of time series. Power spectral density (PSD) is defined as the power distribution of signal based on frequencies existing in signal. If the power level of each frequency component is represented by Pf, the ratio of the power level consistent to every frequency to total power ∑Pf is called normalization of the power (npf) and can be written as(4)npf=Pf∑Pf.Entropy is calculated by multiplication of power level according to every frequency and inverse logarithm of equal power level. Finally, spectral entropy (ShEn1) is calculated by [41](5)ShEn1=∑Pflog1Pf.


## 2.1. Empirical Mode Decomposition

Peter and Wang presented the decomposition of a signal through the EMD method in [26]. The EMD method decomposed a signal into an IMF from high-frequency component to low-frequency component. There are two necessary conditions for the component to be IMF, written below:(i)
Number of zero crossing and number of extrema in the each signal in the whole dataset essentially either equal or differ at most by one.(ii)
At any point, the mean value of envelope explained by local maxima and local minima is zero.The EMD algorithm is summarized below.Signalx (t) is subjected to calculate every probable extrema. Lower and upper envelopes Xmint and Xmaxt, respectively, are obtained through interpolation among minima and corresponding maxima [36]. The mean value at of the envelope is determined using equation (1), which is further deducted from signal xt to get ht using equation (2).(1)at=Xmint+Xmaxt2,(2)ht=xt−at.It is checked whetherht is an IMF. If it is an IMF, the residual signal rt is drawn by repeating the above steps using equation (3), or else replace xt with ht, and do entire process again.(3)rt=xt−ht.IMF1t,IMF2t,IMF3t,…,IMFNt are different frequency components of xt from high-frequency component to low-frequency component. Residual component of a signal x is used to display its central tendency (t). By combining all IMFs and residual r, the original signal x (t) may be reconstructed (t).


## 2.2. Wavelet Transform (WT)

The wavelet is a progressive research filed for the researcher in the engineering over last 20 years. Some of the researchers have contributed substantially to the use of the wavelet transform in bearing fault identification [37]. Wavelet transform is a mathematical tool that changes signals from time domain to time-frequency domain. Wavelet transform is broadly classified as CWT, DWT, and WPT [38]. Wavelet transform passes one-dimensional signal via filter bank consisting of LPF and HPF. The signal component passing through the low-pass filter is recognized as approximation coefficient (CA1) while the component extracted from the high-pass filter is recognized as detailed coefficient (CD1). This is one-level wavelet decomposition, and achieved coefficients are known as first-level wavelet coefficients [39]. Further, the approximation and detailed coefficient is passed again through LPF and HPF to get second-order wavelet coefficients. In the proposed work, three-level wavelet decomposition has been done in order to get third-level wavelet coefficients [40].The structural outline for the wavelet decomposition step is represented in Figure1.Figure 1
Two-dimensional wavelet decomposition.


## 2.3. Features Based on Entropies

Entropy is used to measure uncertainty in the signal [41]. The unpredictability of time series is estimated by entropy.The power spectrum of the signal was utilized by spectral entropy to determine the regularity of time series. Normalized Shannon entropy was used to evaluate the spectral entropy [24, 34]. Spectral entropy is the parameter to quantify spectral complexity of time series. Power spectral density (PSD) is defined as the power distribution of signal based on frequencies existing in signal. If the power level of each frequency component is represented by Pf, the ratio of the power level consistent to every frequency to total power ∑Pf is called normalization of the power (npf) and can be written as(4)npf=Pf∑Pf.Entropy is calculated by multiplication of power level according to every frequency and inverse logarithm of equal power level. Finally, spectral entropy (ShEn1) is calculated by [41](5)ShEn1=∑Pflog1Pf.


## 3. Proposed Method

This paper presents a novel scheme according to various entropy measures to classify the ball bearing fault. The data are randomly sampled to get different data size samples. The EMD has been applied to samples for the removal of high-frequency noisy component [42]. The high-frequency component has been deducted from raw signal to eliminate effectual noise. The complex vibration signal has been analysed in time-frequency domain for better accuracy of classification [43]. That raw signal is split into 23 subsignals, i.e., 8 scales with 3 levels of decomposition. The entropy of wavelet coefficients has been calculated which is utilized as feature for ball bearing fault’s classification.


### 3.1. Proposed Algorithm

Letr⟶xt, where xt is sample comprising of population of vibrating data.Sample space is arbitrarily picked from collection of population, and EMD is performed. EMD divides the signal into IMFs, which are monotonic frequency components.(6)ht=EMDr,where ht is a vector consisting of an IMF obtained using the EMD of signal space r.The first IMF, i.e., the highest frequency component, is deducted from raw signal to obtain residue using the following equation:(7)res=r−ht1,where res is the residue obtained after removing first IMF [ht1], from signal space r.Wavelet coefficients are obtained by transforming the residue (res) into the DWT up to third level of decomposition using equation (8). Equation (8) represents the mathematical model to calculate the DWT of the residue res.(8)dwtres=1√2j∫resψ∗t−k2j2jdt.The entropy of the obtained wavelet coefficient (dwtres) and maximum energy to Shannon entropy ratio are then evaluated as(9)En=−∑jPdwtreslogPdwtres,ShEn1=Pdwtreslog1Pdwtres.Calculated entropy for different set is utilized as an input parameter to neural network for classification.The structural outline of the proposed technique is presented in Figure2.Figure 2
Structural outline of the proposed work.


## 3.1. Proposed Algorithm

Letr⟶xt, where xt is sample comprising of population of vibrating data.Sample space is arbitrarily picked from collection of population, and EMD is performed. EMD divides the signal into IMFs, which are monotonic frequency components.(6)ht=EMDr,where ht is a vector consisting of an IMF obtained using the EMD of signal space r.The first IMF, i.e., the highest frequency component, is deducted from raw signal to obtain residue using the following equation:(7)res=r−ht1,where res is the residue obtained after removing first IMF [ht1], from signal space r.Wavelet coefficients are obtained by transforming the residue (res) into the DWT up to third level of decomposition using equation (8). Equation (8) represents the mathematical model to calculate the DWT of the residue res.(8)dwtres=1√2j∫resψ∗t−k2j2jdt.The entropy of the obtained wavelet coefficient (dwtres) and maximum energy to Shannon entropy ratio are then evaluated as(9)En=−∑jPdwtreslogPdwtres,ShEn1=Pdwtreslog1Pdwtres.Calculated entropy for different set is utilized as an input parameter to neural network for classification.The structural outline of the proposed technique is presented in Figure2.Figure 2
Structural outline of the proposed work.


## 4. Results and Discussion

The data were obtained from Case Western Reserve University’s Bearing Data Centre, United States. Arrangement of the machine is shown in Figure3.Figure 3
Test stand.The specification and arrangement details are available in [35]. The test bearings used for the setup are of SKF and NTN make. The bearing information is provided in Table 1.Table 1
Bearing specifications.
BearingInner diameter (inch)Outer diameter (inch)Ball diameter (inch)Drive end bearing0.98432.04720.3126Fan end bearing0.66931.57480.2656In the proposed algorithm, EMD is employed for removal of high-frequency noisy component. EMD decomposed a signal into IMF which varies from high frequency to low frequency. Initial IMF (IMF1) is deducted from original signal (r) to get residue signal (res). Discrete wavelet transform of residue (res) is calculated up to 3rd level of decomposition. The entropy of the wavelet coefficient obtained after the 3rd level decomposition is calculated which is further utilized as a parameter to classify the ball bearing fault diameter into different output classes. The dataset has been split into subsets by choosing arbitrary samples. The proposed methodology has been tested for various CWT and DWT. The feature sets (entropy values) obtained from 3rd level wavelet coefficients are shown in Table 2 with respect to various discrete wavelets.Table 2
Result of the proposed algorithm at different decomposition levels of DWTs.
S.No.WaveletFault typeFault diameterLevel 1 entropyLevel 2 entropyLevel 3 entropy1DaubechiesInner race0.0074.994024.966484.61392Outer race4.392043.843013.196693Ball race4.686355.157625.584554Inner race0.0145.445145.895796.258435Outer race4.617995.197925.783216Ball race4.517114.87395.131637Inner race0.0214.679984.081593.420838Outer race4.497684.191763.814569Ball race4.97635.538786.095071HaarInner race0.0074.963654.980414.598732Outer race4.456683.941933.262673Ball race4.619415.091475.54824Inner race0.0145.455925.969776.264685Outer race4.568915.116795.713526Ball race4.471734.812144.980677Inner race0.0214.7334.190143.540178Outer race4.488654.140263.684529Ball race4.889525.415895.928381Sym4Inner race0.0074.94874.921794.494122Outer race4.36563.833433.143843Ball race4.612945.095495.495364Inner race0.0145.448335.952356.206375Outer race4.641245.218615.79346Ball race4.476534.797584.984087Inner race0.0214.683244.096733.447038Outer race4.490064.195833.704239Ball race4.945635.502946.023541Coif1Inner race0.0074.963414.972984.667522Outer race4.396773.896733.301273Ball race4.656555.117175.536614Inner race0.0145.421365.9486.28545Outer race4.621125.176045.838126Ball race4.440074.778095.027327Inner race0.0214.668714.10033.444368Outer race4.500124.257863.715989Ball race4.950245.515636.0299Feedforward neural network is utilized with 25 neurons in hidden layer. In the proposed work, feedforward neural network has been used. Training of model has been done using the scaled conjugate gradient-based technique. Moller [36] developed the SCG algorithm, that is according to conjugate directions but does not do a line scan at every iteration like previous conjugate gradient algorithms [44]. Increasing computational cost of system [37]. SCG was created to eliminate the time-consuming process of searching for lines. The SCG method's step size is calculated by quadratic approximation of error function, making it extra robust and independent of user-described parameter. The proposed algorithm has been tested over well-established dataset available online at [35]. The entire dataset has been separated into training and testing sets. For training, 70% of the dataset has been used and the remaining 30% is used for testing and validation. The accuracy of suggested algorithm is achieved after 10-fold cross validation. To validate the performance, cross entropy is used [45]. Three different sets for inner, outer, and ball race faults have been utilized as input parameters to neural network. The network has to classify input set into various fault diameter classes. The diameters of faults are 0.007 inch, 0.014 inch, and 0.021 inch. Figure 4 shows the neural network used to classify the input vector for the proposed method [46].Figure 4
Neural network design used for the proposed method.Tables2 and 3 show the result of the proposed algorithm with discrete wavelets and continuous wavelets, respectively. The results obtained using CWT are depicted in Table 3. The outcomes of Tables 2 and 3 state that variation between the values of entropy with respect to fault diameter at various wavelets can be applied as a criterion for analysis of ball bearing fault diameter [47]. The confusion matrix and ROC for two sets are depicted in Figures 5–7. Figures 5 and 6 demonstrate confusion matrix and ROC for set obtained through entropy of inner race fault for the classification of fault diameter in bearings [48], and Figure 7 shows the confusion matrix for set obtained from entropy of outer race fault for classification [49] of fault diameter in the bearings. It can be concluded from Table 2 that for any kind of wavelet, when fault diameter changes from 0.007 to 0.014, in case of inner race, the entropy increases in the range of 30–40%, while for fault [50] diameter change from 0.014 to 0.021, the entropy [51] was found to be decreased in the range of 40–50%.Table 3
Result of the proposed algorithm at levels 1, 2, and 3 for various continuous wavelet types.
S.No.WaveletFault diameterFault typeLevel 1 entropyLevel 2 entropyLevel 3 entropy1meyr0.007Inner race4.620444.569863.931292Outer race4.645713.894113.386383Ball race4.032864.490544.5429640.014Inner race4.259684.619584.228945Outer race3.772344.294084.461336Ball race4.145354.54874.3335170.021Inner race4.776444.030043.087918Outer race4.748213.991343.871679Ball race4.026794.46184.2431gaus10.007Inner race4.653054.674214.36272Outer race4.626214.61544.799623Ball race4.076474.074694.2023540.014Inner race4.292134.416894.303215Outer race3.861093.848884.41366Ball race4.145744.183514.3416470.021Inner race4.834514.773624.477318Outer race4.763854.564614.131119Ball race4.094554.137984.466491morl0.007Inner race4.616264.545683.842582Outer race4.585463.782413.330313Ball race4.023014.433364.2520340.014Inner race4.019124.47644.521515Outer race3.77884.257784.440576Ball race4.120734.568794.1732870.021Inner race4.782854.014783.204648Outer race4.737453.980083.307799Ball race4.006254.502744.292131mexh0.007Inner race4.657864.324524.227732Outer race4.637034.441814.232453Ball race4.095613.854893.701340.014Inner race4.292334.137974.23465Outer race3.838453.657363.792286Ball race4.159314.027974.1880270.021Inner race4.811954.47224.112258Outer race4.784424.472254.10399Ball race4.089373.895224.12463Figure 5
Confusion matrix for inner race fault set.Figure 6
ROC for inner race fault set.Figure 7
Confusion matrix for outer race fault set.In case of outer race, when fault [52] diameter changes from 0.007 to 0.014, entropy increases by the value found in the range of 70–80%, while for 0.014 to 0.021 diameter change, entropy value decreases in the range of 30–40%. In case of ball race, when fault diameter changes from 0.007 to 0.014, entropy decreases by the value found in the range of 5–10%, while for 0.014 to 0.021 diameter change, entropy value increases in the range of 15–20%.


## 5. Conclusion

This paper aims at determining the fault size and their correct classification. The authors present a new approach for diagnosis of bearing fault size by applying multiple decomposition techniques like EMD and wavelet decomposition. The EMD approach is utilized to divide original signal into monotonic frequency components. High-frequency components are deducted from the raw signal to get the residue whose wavelet decomposition is performed later. Therefore, EMD is utilized for the removal of high-frequency noisy component. The entropy associated with the third level wavelet coefficients of residue is calculated which is used to classify the ball bearing fault diameter into respective classes. Three fault diameters 0.007″, 0.014″, and 0.021″ were considered. The results of the proposed technique as reflected in the confusion matrix prove that the procedure used to classify the fault diameter size is more efficient than the empirical methods. With the proposed method, the achieved classification accuracy is about 100%. It can be concluded that when fault diameter increases from 0.007 to 0.014, inner race and outer race have increment in entropy value in particular range, while ball race has decrement in the value of entropy, whereas when fault diameter increases from 0.014 to 0.021, inner and outer race entropy has a decrement in specific range, while ball race entropy has been increased. Hence, on the basis of variation in entropy value for different kinds of fault, fault type as well as fault size can be diagnosed using the proposed algorithm.



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