applied sciences Article
Incipient Fault Feature Extraction of Rolling Bearings Using Autocorrelation Function Impulse Harmonic to Noise Ratio Index Based SVD and Teager Energy Operator Kai Zheng 1 1
2
*
ID
, Tianliang Li 2
ID
, Bin Zhang 1 , Yi Zhang 1 , Jiufei Luo 1 and Xiangyu Zhou 1, *
School of Advanced Manufacturing Engineering, Chongqing University of Posts and Telecommunications, Chongqing 400065, China;
[email protected] (K.Z.);
[email protected] (B.Z.);
[email protected] (Y.Z.);
[email protected] (J.L.) SMRT-NTU Smart Urban Rail Corporate Laboratory, Nanyang Technological University, Singapore 639798, Singapore;
[email protected] Correspondence:
[email protected]; Tel.: +86-187-1628-5005
Received: 2 September 2017; Accepted: 25 October 2017; Published: 30 October 2017
Abstract: The periodic impulse feature is the most typical fault signature of the vibration signal from fault rolling element bearings (REBs). However, it is easily contaminated by noise and interference harmonics. In order to extract the incipient impulse feature from the fault vibration signal, this paper presented an autocorrelation function periodic impulse harmonic to noise ratio (ACFHNR) index based on the SVD-Teager energy operator (TEO) method. Firstly, the Hankel matrix is constructed based on the raw vibration fault signal of rolling bearing, and the SVD method is used to obtain the singular components. Afterwards, the ACFHNR index is employed to measure the abundance of the periodic impulse fault feature for the singular components, and the component with the largest ACFHNR index value is extracted. Moreover, the properties of the ACFHNR index are demonstrated by simulations and the full life cycle of the experiment, showing its superiority over the traditional kurtosis and root mean square (RMS) index for extracting and detecting incipient periodic impulse features. Finally, the Teager energy operator spectrum of the extracted informative signal is gained. The simulation and experimental results indicated that the proposed ACFHNR index based method can effectively detect the incipient fault feature of the rolling bearing, and it shows better performance than the kurtosis and RMS index based methods. Keywords: rolling element bearings (REBs); singular value decomposition (SVD); autocorrelation function impulse harmonic to noise ratio (ACFHNR); teager energy operator (TEO)
1. Introduction Rolling element bearings have been widely used in scenarios such as wind turbines, high-speed railways, and precision machine tools. They generally operate in tough working environments and are easily subject to failures, which may cause machinery to break down and decrease machinery service performance such as manufacturing quality or operation safety [1–4]. When a defect occurs on a rolling bearing surface, the impulses are created in vibration signals [5,6]. As a result, the detection of faults in rolling element bearings is mainly achieved by identifying the frequency of the impulses from the signals [7,8]. For complicated mechanical systems, the rolling bearing often works in complicated environments, and the vibration signals are easily contaminated by environmental noise and other working parts such as the gearbox (misalignment, unbalance, crack on the rotating shaft, looseness, and distortions). Therefore, their early impulse faults often feature weak and low signal to noise ratios (SNR) [5,6]. Appl. Sci. 2017, 7, 1117; doi:10.3390/app7111117
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Many advanced signal decomposition methods have been studied for fault feature extraction in the field of rotary machine diagnosis. The typical multicomponent decomposition methods include short-time Fourier transform [9], wavelet transform [10], Hilbert-Huang transform [11], Empirical Mode Decomposition/Ensemble Empirical Mode Decomposition (EMD/EEMD) [12,13], and Local Mean Decomposition (LMD) [14]. Among a wide range of de-noising and signal enhancing methods available in the literature, noise elimination algorithms based on singular value decomposition (SVD) are considerably faster and easier to implement [5,8]. Also, it is a non-parametric signal analysis tool, which can be implemented without pre-defined basis functions. Therefore, the SVD method has been employed for fault detection in much research [15–19]. In order to extract the impulse fault feature from the vibration signals of the rolling bearing, it is very important to select the effective singular components for reconstructing a new signal when the SVD method is applied. There are many traditional methods such as the median value of singular values, mean value of singular values, and Difference spectrum of singular values (DSSV) based SVD methods [1]. Among them, the well-established traditional median value of singular values based SVD method and the mean value of singular values based SVD method choose the effective singular components with large singular values. It is believed that the larger singular values can contain much more useful fault features when these methods are applied. However, as the literature [1,20] points out, a larger singular value corresponding to singular components may not contain the informative fault feature. The energy value of the informative fault feature may be very weak. A Difference spectrum of singular values (DSSV) is introduced to capture the abrupt change in the singular values, which reflects the boundary between the signal and the noise, based on the principle that the effective informative singular components can be extracted [20,21]. The literature [1] proposed a method to capture the abrupt change based on the correlation coefficient singular value decomposition (CCSVD) method. However, these methods are characterized by capturing the abrupt change and may remove the useful signals and retain the noise signals. As a result, they may not be able to extract the weak fault feature effectively. To solve this problem, one study [5] proposed kurtosis to quantify the impulsive feature of the vibration signal, and the simulation and experiment results indicate that the de-noising method is successful in both the frequency domain and time domain for fault identification. Another study [15] proposed a Periodic Modulation Intensity (PMI) index to select fault features with relatively lower energy levels, and it is robust in the presence of heavy noise and strong random interferences. From the above analysis, it can be seen that it is vital to select suitable singular components for extracting mechanical fault features using the SVD method. The crucial step is to find an index to measure the abundance of fault information for the singular components. Although kurtosis shows high values for extracting the impulsive feature [22], the background noises and rotating frequency-related interference harmonic components in the collected vibration signal would submerge the periodic impulse fault feature. As a result, the kurtosis value may not be able to reveal the useful information. The root-mean-square (RMS) value of the measured signal is applied in industry to measure the vibration intensity and indicate the incipient defect. However, the RMS value has limited applications because it is not sensitive to defect at its early stage, when the defect contributes little energy [23]. Moreover, these two indicators were originally developed in the field of statistics or information theory, thus they mainly focus on the general statistical distribution of a signal but may ignore the specific characteristics of mechanical signals [15]. Motivated by this, an autocorrelation function impulse harmonic to noise ratio index (ACFHNR) is proposed to measure the richness of the periodic impulse fault feature information in this paper. According to the proposed index, the ACFHNR based SVD-Teager energy operator (TEO) method for extracting the incipient fault feature from the vibration signal of the rolling bearing is presented. This paper is organized as follows: Section 2 introduces the theoretical background, and an ACFHNR index is proposed. Based on the proposed index, a novel method for extracting the incipient impulse fault feature of a rolling bearing using the ACFHNR based SVD-TEO is proposed in Section 3. Section 4
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makes a simulation analysis of the proposed method. In Section 5, the proposed method is verified by experimental data, and Section 6 makes several conclusions. 2. Theory Background 2.1. The Fundamental of SVD Denoising Method SVD is an orthogonal decomposition method. For any real matrix A ∈ Rm×n with rank L, it can be decomposed into the product of three matrices [5]: the orthogonal matrix U ∈ Rm×n , the diagonal matrix D = [diag(σ1 , σ2 . . . , σl ), 0], and the orthogonal matrix V ∈ Rm×n , as follows: A = UDVT
(1)
where L = min(m, n), σ1 ≥ σ2 . . . ≥ σl . σi (i = 1, 2 . . . L) are the singular values of the matrix A. There are several steps to denoise the measurement signal with the SVD method. Firstly, the Hankel matrix should be constructed [24–27] as follows: Am =
x (1) x (2) .. . x (m)
x (2) x (3) .. . x ( m + 1)
··· x (n) · · · x ( n + 1) .. .. . . ···
(2)
x( N )
where 1 < n < N, m = N − n + 1. The parameter m is used to determine the number of decomposed components in SVD, which is usually selected less than n [15]. Secondly, the dynamic propriety of the matrix Am can be revealed based on the reconstructed attractor in the reconstructed phase space. As a result, the matrix Am can be expressed as [28]: Am = D + W
(3)
where D is the smooth signal in the reconstructed phase space and W is the noise and interference signal in the reconstructed phase space. To denoise the original signal, it is necessary to find the best approximation matrix for matrix D. According to the singular value decomposition principle, the matrix Am can be decomposed as follows: Am = UEVT
(4)
where E = [diag(σ1 , σ2 . . . , σm ), 0] and L = min(m, n), σ1 ≥ σ2 . . . ≥ σl . U and VT are the left and right singular matrixes, respectively, and they satisfied U = [u1 , u2 . . . , um ], V = [v1 , v2 . . . , vn ]T , where ui is the ith column vector of singular matrix U and vi is the ith column vector of singular matrix V. As a result, Am can be computed as: Am = σ1 u1 v1 + σ2 u2 v2 + . . . + σm um vm
(5)
Finally, according to the singular value decomposition theory and the best approximation theorem of a matrix in the sense of Frobeious norm, it can be obtained that the different kinds of signals are characterized by different singular values [28]. Therefore, by removing the singular values corresponding to the noise components and interference harmonics, we can get the new diagonal matrix E0 = diag σα , σj . . . , σβ , 0 . The new Hankel matrix Am 0 can be computed as: Am 0 = σα uα vα + σj u j v j + . . . + σβ u β v β
(6)
where the matrix Am 0 is supposed to be the best approximation of matrix Am . Compared to the original matrix Am , its noise components and interference harmonics are compressed significantly.
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The averaging value of the anti-diagonals element xi of the matrix Am 0 is calculated, which is regarded as the denoised signal for the original vibration signal [5], as shown in Equation (7): xi =
1 β A β − α + 1 ∑ j=α i− j+1,j
(7)
For traditional SVD denoising methods such as the median value of singular values, the mean value of singular values, and DSSV based SVD denoising, the most energic singular components are reserved. However, for the vibration signal of the rolling elements bearing, the incipient impulse fault feature is often submerged by the noise and interference harmonics, and its energy may be very weak. As a result, the traditional energy based evaluation method of SVD denoising could be invalid for extracting the informative impulse fault feature of a rolling bearing. It is important to find a new method for evaluating the fault information of the singular components. 2.2. Autocorrelation Function Impulse Harmonic to Noise Ratio Index When a defect impacts the rolling surface, cyclic impulses are generated. A variety of methods to decompose the fault signal for extracting the weak periodic impulse feature were proposed such as EEMD [12,13], LMD [14], intrinsic characteristic-scale decomposition (ICD) [29]. However, how to choose the sensitive feature components remains a problem to be solved. Many scholars have proposed a series of indexes such as kurtosis [22], approximate entropy [30], Pearson coefficient [13], or a fusion of these parameters [31] to measure the richness of fault feature information in the decomposed components. However, most of these indicators were originally developed in the field of statistics or information theory, thus they mainly focus on the general statistical distribution of a signal, such as its non-Gaussianity and peakedness, but may ignore the specific characteristics of mechanical signals [15]. Therefore, we proposed a new index, which is called the autocorrelation function periodic impulse harmonic to noise ratio index (ACFHNR). It focuses on the detection of the periodic impulse fault features of the rolling bearing. In the bearing fault signal, the effective feature information is often contaminated by noise. Generally, it is assumed that the noise in the fault signal can be approximated as white noise. The autocorrelation function of white noise reaches the greatest value at the zero point, while it is zero at the nonzero points, which is quite different from the autocorrelation function of the general periodic signal. It can be considered that the autocorrelation function energy is mainly concentrated at the zero point for a white noise signal. Therefore, the autocorrelation function is employed to process the acquired signal. The autocorrelation function reflects the similarity between the signal and itself at different times, which is a statistical method in the time domain, and it can be defined as: R(µ) =
Z
r (t)r (t + µ)dt
(8)
To compute the autocorrelation function of stationary signal s with the length of N, firstly, obtain the envelope signal by Hilbert transform of the measured signal [23,32]: xˆ (t) = H { x (t)} =
1 π
Z ∞ x (τ ) −∞
t−τ
dτ
y(t) = | xˆ (t)|
(9) (10)
Secondly, remove the DC component: s Std(y) =
1 n
∑
n (y i =1 i
− y)
2
(11)
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s(t) =
y(t) − mean(y(t)) std(y)
(12)
According to the theory of stochastic process, the one-dimensional auto-correlation of the generalized stationary signal s with the length of N is defined as follows [33]: R(m) = E s(n + m)s(n)∗ , n = 0, 1, . . . . . . , N − 1, m = 0, ±1, . . . . . . , ±( N − 1)
(13)
where E{.} is the expectation operator and s(n)∗ denotes the conjugation of s(n). As pointed out above, the periodic impulse feature is the most typical sign of a bearing fault. Therefore, we calculated the autocorrelation function of three stimulated signals. Where Signal 1 x (t) is the transient impulse fault sequence of rolling bearings, which can be obtained by Equations (14) and (15). The impulse period is set as T = 0.01 s. Signal 2 N (t) is the noise signal. Signal 3 is the mixed signal of x (t) and N (t), and the SNR is set to −8.1 db. Appl. Sci. 2017, 7, 1117
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+∞
x (t) =
∑
k =−∞
h(t − KT − τk )δj (i ) Ak
ℎ( − − ) ( ) ( ) = ε h(t) = exp − √ 2π f n t sin(2π f n t) t ≥ 0 (2 ) ≥0 ℎ( ) = −1 − 2ε2 2
where
√1 − 2
where
(14) (14)
(15) (15)
( j − 1) π δj (i ) = a sin 2π f h i + ϕ j + c ϕ j =( − 1) 2 ()= 2 + + = 2
The autocorrelation function of three signals can be calculated, as shown in Figure 1a to c, The autocorrelation function of three signals can be calculated, as shown in Figure 1a to c, respectively. From Figure 1a, it 1a, canit be that period of autocorrelation the autocorrelation respectively. From Figure can seen be seen thatthe the impulse impulse period of the function function for the fault sequence is also T ==0.01 which retains the impulse information the original for the fault sequence is also 0.01 s,s,which retains the impulse information of the original of signal. From Figure 1b, 1b, it found thatthat the autocorrelation function of a Gaussian noise signal signal. From Figure it can canbebe found the autocorrelation function ofwhite a Gaussian white noise concentrates its energy at the zero point. In Figure 1c, although the impulse fault feature is submerged signal concentrates its energy at the zero point. In Figure 1c, although the impulse fault feature is by the noise, the impulse period = 0.01 s can be easily observed with the calculation of the submergedautocorrelation by the noise, the impulse period T = 0.01 s can observed calculation function. Moreover, the energy of the noisebeis easily concentrated at thewith zero the point. of the autocorrelation Moreover, ofbethe noise isthrough concentrated at the zero point. Therefore, the function. impulse fault feature andthe theenergy noise can separated the autocorrelation function. Therefore, the impulse fault feature and the noise can be separated through the autocorrelation function.
Figure 1. The autocorrelation functions threekinds kinds of signals, (a) The autocorrelation Figure 1. The autocorrelation functions of of three ofstimulated stimulated signals, (a) The autocorrelation function of period impulse fault signal (b) The autocorrelation function of white noise signal (c) The function of period impulse fault signal; (b) The autocorrelation function of white noise signal; autocorrelation function of signal mixed by period impulse fault signal and white noise signal (c) The autocorrelation function of signal mixed by period impulse fault signal and white noise signal.
Based on the above analysis, the autocorrelation function impulse harmonic to noise ratio index (ACFHNR) is proposed, as shown in Equation (16): ACFHNR = ( where ∑ impulse features.
∑
( (0)
)
(16)
) is the energy value of the autocorrelation function ( ) in the th period of (0) is the energy value of the autocorrelation function ( ) at the zero point.
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Based on the above analysis, the autocorrelation function impulse harmonic to noise ratio index (ACFHNR) is proposed, as shown in Equation (16): ACFHNR =
∑ik=1 Ren (kNT ) Ren (0)
(16)
where ∑ik=1 Ren (kNT ) is the energy value of the autocorrelation function R(µ) in the kth period of impulse features. Ren (0) is the energy value of the autocorrelation function R(µ) at the zero point. It is recommended that k = 3. Obviously, the larger the ACFHNR index, the more periodic impulse Appl. Sci. information 2017, 7, 1117 is contained in the vibration signal. The algorithm of the ACFHNR index is shown 6 of 19 feature in Table 1. feature information is contained in the vibration signal. The algorithm of the ACFHNR index is shown in Table 1. algorithm of the autocorrelation function impulse harmonic to noise ratio Table 1. The (ACFHNR) index. Table 1. The algorithm of the autocorrelation function impulse harmonic to noise ratio (ACFHNR) index. Input: Signal x(t), Prior fault frequency f b , Sampling frequency f s Input: Signal x(t), Prior fault frequency
, Sampling frequency
Procedure:Procedure:
1. x(t) Theissignal x(t) isto subjected Hilbert transform 1. The signal subjected Hilberttotransform ) and the 2. theCompute thethe (Interval Interval )* 2. Compute Z (t) and point N =point (1/ f b )*= f(1/ s 3. Compute Autocorrelation function R (µ) ( ) 3. theCompute the Autocorrelation function k 4. Set Ren4. = 0, Compute en ( NT ) = ∑i(=1 R ) en Set = 0,RCompute ( i )← , Seti + i ←1 + 1 = (∑kNT ), Set 5. Compute R 0 ( ) en 5. Compute (0) ( (0)/ Output: ACFHNR Output: ACFHNR = Ren ( NT=)/Ren )
(0)
The vibration fault signal of a rolling bearing is sensitive to the kurtosis value. Therefore, we The vibration fault signal of a rolling bearing is sensitive to the kurtosis value. Therefore, we made made a comparative analysis of the proposed ACFHNR index and the kurtosis. The kurtosis can be a comparative analysis of the proposed ACFHNR index and the kurtosis. The kurtosis can be calculated calculated by Equation (17): by Equation (17): x̅ 4 11 N x ((i ))− − (17) Kurt(i() )== ∑ (17) N i =1 σt where x is the mean value of the signal x (i ) (and σt is the standard deviation of x (i ). The tested signal ) and where ̅ is the mean value of the signal is the standard deviation of ( ). The tested can be obtained by mixing the transient impulse fault sequence x(t) and the noise signal n(t). signal can be obtained by mixing the transient impulse fault sequence x(t) and the noise signal n(t).
= x ((t)) + + n((t)) S((t)) =
(18) (18)
Based on the mixed mixed signal signal S((t)),, we weset setthe theSNR SNRto to−−14.5 db to to 00 db db and and calculated calculated the the kurtosis kurtosis Based on the 14.5 db value and ACFHNR index. The simulation result is shown in Figure 2. value and ACFHNR index. The simulation result is shown in Figure 2.
Figure ACFHNR index index and and the the kurtosis. kurtosis. Figure 2. 2. The The comparative comparative simulation simulation of of the the proposed proposed ACFHNR
From Figure 2, it can be seen that, when the input SNR of the stimulated signal changed from to 0 db to −14.5 db, both the kurtosis value and the ACFHNR index gradually decreased, which indicated that both the kurtosis factor and the ACFHNR index can distinguish signals with different SNR of the input signal. However, when the SNR of the input signal is less than −5.5 dB and larger than −14.5 dB, the ACFHNR index has better resolution than the kurtosis, indicating that the proposed ACFHNR index can be used for identifying the incipient periodic impulse feature of the
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From Figure 2, it can be seen that, when the input SNR of the stimulated signal changed from to 0 db to −14.5 dB, both the kurtosis value and the ACFHNR index gradually decreased, which indicated that both the kurtosis factor and the ACFHNR index can distinguish signals with different SNR of the input signal. However, when the SNR of the input signal is less than −5.5 db and larger than −14.5 dB, the ACFHNR index has better resolution than the kurtosis, indicating that the proposed ACFHNR index can be used for identifying the incipient periodic impulse feature of the rolling bearings. 2.3. Teager Energy Operator The Teager energy operator is a nonlinear operator proposed by Teager and Kaiser. It has high temporal resolution and can adaptively detect an instantaneous change of signal [34–36]. As a result, it is suitable for extracting the impulse fault features of rolling bearings. Moreover, as the Teager energy signal contains the total energy of the signal, including the information of amplitude modulation (AM) and frequency modulation (FM) caused by impulse fault feature, the Teager energy operator can improve the signal to noise ratio (SNR) and make the impulse fault feature extracted from a rolling bearing more reliable [34]. It is also capable of increasing the signal-to-noise ratio of the spectrum, sharpening the spectral peaks that reveal the presence of faults. The Teager energy operator for the continued signal v(t) can be expressed as: . 2 .. ϕc [v(t)] = v(t) − v(t)v(t)
(19)
Moreover, the Teager energy operator for the discrete signal x (n) can be expressed as: ϕ[ x (n)] = [ x (n)]2 − x (n − 1) x (n + 1)
(20)
Therefore, considering that the Teager energy operator has good adaptability to detect the transient components of a signal, it is employed to detect the fault features of rolling bearings in this paper. 3. The Proposed Method Under the motivation of the above analysis, a method combined the ACFHNR index based on the SVD and Teager energy operator (TEO) demodulation technology is proposed for the early fault diagnosis of rolling bearings. The framework of the proposed method is shown in Figure 3, and its implementation can be described as follows: Firstly, select the appropriate delay step to construct Hankel matrix. When the computational cost and calculation accuracy are taken into account, there is a trade-off between computational efficiency and decomposition performance when deciding an appropriate choice of m [15]. In this paper, the m is set as 15 ≤ m ≤ 30, and the singular value decomposition is performed for the constructed Hankel matrix. Secondly, for the original vibration signal, the theoretical fault characteristic frequency can be calculated according to the geometry size and defect frequencies multiple of running speeds. Therefore, the prior period of impulses T can be computed through Equation T = f s/ f , where f is the fault characteristic frequency of the rolling element bearings, while fs is the sampling frequency. With the prior period of impulses T, the ACFHNR index for each singular component can be calculated. Thirdly, the singular component with the largest ACFHNR index is employed to construct the new Hankel matrix. The average of the inverse diagonal elements of the reconstructed Hankel matrix is calculated, which is the denoising signal of the raw vibration signal. Finally, we implemented the Teager energy operator envelope analysis to the reconstructed signal for early fault diagnosis. The fault type can be decided on the basis of the extracted characteristic frequency information.
calculated. Thirdly, the singular component with the largest ACFHNR index is employed to construct the new Hankel matrix. The average of the inverse diagonal elements of the reconstructed Hankel matrix is calculated, which is the denoising signal of the raw vibration signal. Finally, we implemented the Teager energy operator envelope analysis to the reconstructed Appl. Sci. signal 2017, 7,for 1117early fault diagnosis. The fault type can be decided on the basis of the extracted characteristic frequency information.
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Figure 3. The framework of the proposed method. Figure 3. The framework of the proposed method. Appl. Sci. 2017, 7, 1117
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4. Simulation Analysis 4. Simulation Analysis
4.1. The Stimulated Signal
4.1.verify The Stimulated Signal To the validity of the proposed method, the simulated fault signals of the rolling bearing are analyzed in thisthe section. simulated of the faults of ball bearings can bebearing calculated To verify validityThe of the proposedsignal method, the early simulated fault signals of the rolling are analyzed this(21) section. simulated of the early faults of ball bearings can be calculated by Equations (14),in(15), andThe (22), where I signal t is the interference signal, x t is the transient impulse () () ( ) ( ) by Equations (14), (15), (21) and (22), where is the interference signal, is the transient fault signal, and n(t) is the noise signal. impulse fault signal, and
( ) is the noise signal.
S(t)( = ) =x (t() )++I ((t)) + + n( t)) I (t)( =) =Ah ∑
n sin(2π f hi)) i =1 sin(2
(21) (21) (22) (22)
The simulation result of the fault signal is shown in Figure 4; Figure 4a is the transient periodic The simulation result of the fault signal is shown in Figure 4; Figure 4a is the transient periodic impulse fault signal, Figure 4b is interference Figure4c4cisisthe the noise signal. impulse impulse fault signal, Figure 4bthe is the interferencesignal, signal, and and Figure noise signal. TheThe impulse periodperiod is setistoset0.01 S, and the resonance frequency is 2000 HZ. Moreover, 6000 sampling points to 0.01 S, and the resonance frequency is 2000 HZ. Moreover, 6000 sampling points of of the signals are obtained. The sampling setatat1212 KHZ. signal-to-noise ratio (SNR) the signals are obtained. The samplingfrequency frequency isisset KHZ. TheThe signal-to-noise ratio (SNR) is set− to13.2 −13.2dB, dB,and andthe the signal-to-interference signal-to-interference signal ratio (SIR) is setistoset −4.5 In dB. order showto theshow is set to signal ratio (SIR) to db. −4.5 Intoorder signalclearly, clearly,only only3000 3000 sample sample points 4. 4. The mixed signal andand its envelope the signal pointsare areplotted plottedininFigure Figure The mixed signal its envelope spectrum are plotted in Figure 5. It can be seen that the informative features in the spectrum of the spectrum are plotted in Figure 5. It can be seen that the informative features in the spectrum of the original signal cannot be identified. Therefore, the proposed method is performed on the simulated original signal cannot be identified. Therefore, the proposed method is performed on the simulated signal to detect the incipient fault features. signal to detect the incipient fault features.
The stimulated signals of rolling element bearings. (a) The transient impulse fault FigureFigure 4. The4.stimulated fault fault signals of rolling element bearings. (a) The transient impulse fault signal; signal; (b) The interference signal; (c) The noise signal. (b) The interference signal; (c) The noise signal.
Figure 4. 1117 The stimulated fault signals of rolling element bearings. (a) The transient impulse fault Appl. Sci. 2017, 7, 9 of 19 signal; (b) The interference signal; (c) The noise signal.
Figure5.5.(a) (a)The Thestimulated stimulatedfault faultvibration vibrationsignal; signal;(b) (b)The Theenvelope envelopespectrum spectrumofof(a). (a). Figure
4.2.The ThePerformance PerformanceAnalysis Analysisofofthe theProposed ProposedMethod Method 4.2. Appl. Sci. 1117 9 of 19 In 2017, order7, to analyze the decomposition performance of SVD method, we studied the relationship
In order to analyze the decomposition performance of SVD method, we studied the relationship between thesingular singularvalues valuesof ofthe theSVD SVDmethod methodand anddifferent differentkinds kindsof ofsignals signalsfirst. first.Firstly, Firstly,the theHankel Hankel between the matrix is constructed when applying the SVD denoise method, and there is a high degree of matrix Am is constructed when applying the SVD denoise method, and there is a high degree of correlation between the adjacent rows [28]. As a result, the matrix must be a morbid matrix, and correlation between the adjacent rows [28]. As a result, the matrix Am must be a morbid matrix, and its its rank is far less than . Therefore, for an SVD denoising approach, the useful information should rank r is far less than l. Therefore, for an SVD denoising approach, the useful information should be be determined by some singular components. determined by some singular components. In order to separate the effective periodic impulse component, the interference component, and In order to separate the effective periodic impulse component, the interference component, and the the noise component of the fault signal, it is important to verify whether there were dislocations noise component of the fault signal, it is important to verify whether there were dislocations filtering filtering the characteristics of the singular value distributions of the different signals. Hence, the the characteristics of the singular value distributions of the different signals. Hence, the periodic periodic impulse fault signal, the interference signal, and the white noise signal are set as reference impulse fault signal, the interference signal, and the white noise signal are set as reference signals to signals to build three groups of mixed signals. Mixed signal 1 is comprised of the interference signal build three groups of mixed signals. Mixed signal 1 is comprised of the interference signal and the and the noise signal, and mixed signal 2 is composed of the periodic impulse fault signal and the noise signal, and mixed signal 2 is composed of the periodic impulse fault signal and the noise signal, noise signal, while mixed signal 3 is comprised of the periodic impulse fault signal, the interference while mixed signal 3 is comprised of the periodic impulse fault signal, the interference signal, and the signal, and the noise signal. The three reference signals and the three mixed signals are analyzed noise signal. The three reference signals and the three mixed signals are analyzed based on the SVD based on the SVD denoise method. The delay step is set as m = 20. The singular values of these six denoise method. The delay step is set as m = 20. The singular values of these six signals are obtained, signals are obtained, as shown in Figures 6 and 7. as shown in Figures 6 and 7. Interestingly, it can be found that the singular values of mixed signals are added by the reference Interestingly, it can be found that the singular values of mixed signals are added by the reference signals, as shown in Figures 6 and 7. For example, the singular values of mixed signal 1 are added by signals, as shown in Figures 6 and 7. For example, the singular values of mixed signal 1 are added by the the interference signal and the white noise signal. This showed that different singular values interference signal and the white noise signal. This showed that different singular values correspond correspond to different kinds of signals. Therefore, the sensitive impulse feature signal can be to different kinds of signals. Therefore, the sensitive impulse feature signal can be reconstructed with reconstructed with the appropriate singular components, selected based on certain principles of the the appropriate singular components, selected based on certain principles of the SVD method, thereby SVD method, thereby realizing the fault features extraction of the rolling bearings. realizing the fault features extraction of the rolling bearings.
Figure Figure6.6. The The singular singular components componentsof ofthese thesesix sixsignals. signals.
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Figure 6. The singular components of these six signals.
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Figure Figure7.7.The Thesingular singularcomponents. components. (a) (a) The The interference interference signal signal and and mixed mixed signal signal 1;1;(b) (b)The Theimpulse impulse signal signaland andmixed mixedsignal signal2.2.
Inorder orderto toverify verifythe the performance performanceof of proposed proposedACFHNR-SVD-TEO ACFHNR-SVD-TEOmethod, method,the thesimulated simulatedfault fault In vibration signal of the rolling bearing is analyzed with the feature extraction method. The delay step vibration signal the rolling bearing is analyzed with the feature extraction method. The delay m ismset as m For comparison index value value step is set as =m 20. = 20. For comparisonpurposes, purposes,the thekurtosis kurtosisvalue valueand and the the ACFHNR index corresponding to to the the 20 20singular singular components components of of SVD SVD are arecomputed, computed, as asshown shown in inFigure Figure8.8. Figure Figure88 corresponding exhibitedthat thatthe thekurtosis kurtosisand andthe theACFHNR ACFHNRvalues valuesof ofthe thesecond secondand andthe the third thirdsingular singularcomponents components exhibited Appl. Sci. 2017, 7, 1117 10 of 19 are larger larger than than the other and thethe third singular components are are other values, values,which whichtells tellsus usthat thatthe thesecond second and third singular components the the effective signal features. However, apparently, it can be found that that the ACFHNR indexindex is very are effective signal features. However, apparently, it can be found the ACFHNR is sensitive to the fault components, as the ACFHNR indexes corresponding to the second and the third very sensitive to the fault components, as the ACFHNR indexes corresponding to the second and the singular components are much larger than the others, while the kurtosis did not have such good third singular components are much larger than the others, while the kurtosis did not have such good performance. As a consequence, the ACFHNR index has much better resolution than the kurtosis for performance. As a consequence, the ACFHNR index has much better resolution than the kurtosis for extracting fault features. extracting fault features.
Figure 8. 8. The The relationship relationship between between the the singular singular value value and and (a) (a) the the kurtosis kurtosis or or (b) (b) the the ACFHNR ACFHNR index. Figure index.
From the result of of thethe SVD method is shown in Figure 9. Ten the analysis analysisabove, above,the thedecomposition decomposition result SVD method is shown in Figure 9. singular components are plotted, as shown in Figure 9a. Moreover, the singular components and Ten singular components are plotted, as shown in Figure 9a. Moreover, the singular components their Teager energy operator spectra are obtained. Figure 9b,c correspond to the second and the third singular components, respectively. One can easily see that the fault frequency and its harmonics are clearly detected detected in inthe thesecond secondand andthe thethird thirdsingular singularcomponents. components. Based above analysis, it can Based onon thethe above analysis, it can be be the ACFHNR very sensitive to the fault information, whichbetter has much better seenseen the ACFHNR indexindex is veryissensitive to the fault information, which has much performance performance than for the extracting kurtosis forthe extracting the weak periodic impulse fault feature. Moreover, the than the kurtosis weak periodic impulse fault feature. Moreover, the proposed proposed ACFHNR-SVD-TEO to beateffective at the extracting thefault incipient fault ACFHNR-SVD-TEO method is method proved is to proved be effective extracting incipient features of features of rolling bearings. rolling bearings.
their Teager energy operator spectra are obtained. Figure 9b,c correspond to the second and the third singular components, respectively. One can easily see that the fault frequency and its harmonics are clearly detected in the second and the third singular components. Based on the above analysis, it can be seen the ACFHNR index is very sensitive to the fault information, which has much better performance than the kurtosis for extracting the weak periodic impulse fault feature. Moreover, the Appl. Sci. proposed 2017, 7, 1117 ACFHNR-SVD-TEO method is proved to be effective at extracting the incipient fault 11 of 19 features of rolling bearings.
Figure 9. The singular components, (a) Teager their Teager energy operator spectrum; second Figure 9. The singular components, (a) their energy operator spectrum; (b) (b) TheThe second spectrum; spectrum; and (c) The third spectrum. and (c) The third spectrum. Appl. Sci. 2017, 7, 1117
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Additionally, for comparison purposes, other well-established methods such as the median value comparison other well-established methods as the of singularAdditionally, values basedforSVD method purposes, and the Difference spectrum based SVDsuch method aremedian employed value the of singular values based SVD method Difference spectrum based SVD method are to process stimulated vibration signal, andand thethe results are shown in Figure 10a,b, respectively. employed to process the stimulated vibration signal, and the results are shown in Figure 10a,b, It can be seen that the median value of the singular values based on SVD cannot identify the fault respectively. It can be seen that the median value of the singular values based on SVD cannot identify features, as shown in Figure 10a. Although the Difference spectrum based SVD method can detect the fault features, as shown in Figure 10a. Although the Difference spectrum based SVD method can the fault frequency f b and its second harmonic 2 f b , the fault features are not obvious, as shown in detect the fault frequency and its second harmonic 2 , the fault features are not obvious, as Figureshown 10b. The Kurtogram of the stimulated fault vibration signal is shown in Figure 11a, from which in Figure 10b. The Kurtogram of the stimulated fault vibration signal is shown in Figure 11a, the optimal carrier is determined 2000 HZ, and the center frequency 1000 HZ.isTherefore, from which thefrequency optimal carrier frequencyas is determined as 2000 HZ, and the centerisfrequency 1000 the optimal frequency band isfrequency 1000 HZband to 3000 HZ.HZ Astoa3000 result, filter is designed HZ. Therefore, the optimal is 1000 HZ.the As aband-pass result, the band-pass filter is to extract the potential effective fault features vibration The envelop of the designed to extract the potential effectivefrom faultthe features fromsignal. the vibration signal. spectrum The envelop spectrum of presented the filtered signal is presented in Figure 11b. Itthat can the be found the fault3frequency filtered signal is in Figure 11b. It can be found fault that frequency f b and 4 f b3can be and However, 4 can be the detected. However,are thealso faultnot features are also not obvious. Therefore, traditional well-SVD detected. fault features obvious. Therefore, traditional well-established established SVD denoising and the fast Kurtogram can obtain little diagnostic information. denoising and the fast Kurtogram can obtain little diagnostic information.
FigureFigure 10. The feature signals and (a)The The median value of singular 10. extracted The extracted feature signals andtheir theirenvelop envelop spectrum. spectrum. (a) median value of singular values based SVD method; (b) The Difference spectrum singular values based SVD method. values based SVD method; (b) The Difference spectrum singular values based SVD method.
Figure 10. The extracted feature signals and their envelop spectrum. (a) The median value of singular Appl. Sci. 2017, 7, 1117 values based SVD method; (b) The Difference spectrum singular values based SVD method.
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Figure 11. Optimal (a) Optimal resonanceband bandobtained obtained by envelope spectrum. Figure 11. (a) resonance by Kurtogram; Kurtogram;(b)(b) envelope spectrum.
5. Experiment Results Appl. Sci. 2017, 7, 1117
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In this section, the proposed method is verified by the experimental data. The fault data for the 5. Experiment Results bearing roller comes from the bearing data center at the Laboratory of Case Western Reserve University. The test bed consists Horsepower (HP) motors, torque transducer, dynamometer, In this section,of thetwo proposed method is verified by thea experimental data. Theand faultadata for the as shown in Figure 12. The test bearings (SKF6205, Svenska Kullager-Fabriken group. Switzerland), bearing roller comes from the bearing data center at the Laboratory of Case Western Reserve University. The end test and bed fan consists of two Horsepower (HP)motor motors, a torque a including the drive end bearings, support the rotor shaft. transducer, Three typesand of Single shown Figure inner 12. Therace test fault bearings (SKF6205, Svenska group.to the pointdynamometer, faults (outer as race faultin(ORF), (IRF), and ball fault Kullager-Fabriken (BF) were introduced Switzerland), the drive machining end and fan[37]. end All bearings, support the motor rotor shaft. Three test bearings usingincluding electro-discharge bearing fault data are annotated with bearing types of Single point faults (outer race fault (ORF), inner race fault (IRF), and ball fault (BF) were geometric, operating condition, and fault information. In the experiment, the vibration signals of introduced to the test bearings using electro-discharge machining [37]. All bearing fault data are the rolling bearing were collected by three accelerometers, which were mounted to the housing with annotated with bearing geometric, operating condition, and fault information. In the experiment, the magnetic bases at the three, six, and 12 o’clock positions. The fault frequency of the ball fault can be vibration signals of the rolling bearing were collected by three accelerometers, which were mounted computed follows: to the as housing with magnetic bases at the three, six, and 12 o’clock ! positions. The fault frequency of d 2 the ball fault can be computed as follows:D 2 f bp f r = 1 − ( ) cos α f r (23) 2d D =
1−( )
(23)
2 where f r is the rotating frequency, D is the pitch diameter, d is the balling diameter, and α is the contact is the rotating frequency, D is the pitch d is multiple the ballingofdiameter, the fault where angle. According to the geometry size and defect diameter, frequencies runningand speed,isthe contact angle. According to the geometry size and defect frequencies multiple of running speed, the frequency of the ball fault is obtained, which is about f bp f r = 137 Hz. fault frequency of the ball fault is obtained, which is about = 137 Hz. In this section, the vibration signals of the drive end bearing are analyzed. To verify the In this section, the vibration signals of the drive end bearing are analyzed. To verify the effectiveness of the proposed method, ball fault 1, with fault diameters of seven miles, is analyzed. effectiveness of the proposed method, ball fault 1, with fault diameters of seven miles, is analyzed. The rotation speed of ball fault 1 is 1772 Thesampling sampling frequency is set tokHz 12 kHz in the The rotation speed of ball fault 1 is 1772r/min. r/min. The frequency is set to 12 in the experiment. The original signal and its envelope spectrum are plotted in Figure 13. From Figure experiment. The original signal and its envelope spectrum are plotted in Figure 13. From Figure 13, 13, it canit be thatthat fault frequencies by noise noiseand andirrelevant irrelevant harmonic components. canfound be found fault frequenciesare areflooded flooded by harmonic components. Therefore, the proposed ACFHNR-SVD-TEO method is employed for incipient fault feature Therefore, the proposed ACFHNR-SVD-TEO method is employed for incipient fault feature extraction extraction of the fault signal. of the fault signal.
Figure 12. The bearing fault test bedfrom fromthe thebearing bearing data the Laboratory of Case Western Figure 12. The bearing fault test bed datacenter centeratat the Laboratory of Case Western Reserve University Reserve University.
Figure 12. The bearing fault test bed from the bearing data center at the Laboratory of Case Western Appl. Sci. 2017, 7, 1117 Reserve University
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Appl. Figure Sci. 2017,13. 7, 1117 13 of 19 The envelope spectrum of ball fault 1.(a)1.(a) TheThe rawraw faultfault signal ; (b) ;The Figure Theoriginal originalsignal signaland andthe the envelope spectrum of ball fault signal (b) Appl. Sci. 2017,13. 7, 1117 13 of 19
envelope spectrum of theofraw signal. The envelope spectrum the fault raw fault signal .
Firstly, the delay step is set as m = 15. The kurtosis value and the ACFHNR index value Firstly, the delay step is set as m = 15. The kurtosis value and the ACFHNR index value corresponding to these 15 singular components of SVD are calculated, as shown in Figure 14. Figure Firstly, thetodelay is set as m = 15. The kurtosis value and ACFHNR index value corresponding these step 15 singular components of SVD are calculated, as the shown in Figure 14. Figure 14 exhibited that the ACFHNR value of the first singular component is larger than the ACRHNR corresponding to these 15 singular components of SVD are calculated, as shown in Figure 14. Figure 14 14 exhibited that the ACFHNR value of the first singular component is larger than the ACRHNR values of the other components, which tells us that the first component is the effective signal feature. exhibited thatother the ACFHNR value of thetells firstussingular is larger than the ACRHNR values values of the components, which that thecomponent first component is the effective signal feature. Therefore, the first singular component is processed, and its Teager energy operator spectrum is of the other tells usis that the firstand component the effective signal feature. Therefore, thecomponents, first singularwhich component processed, its Teagerisenergy operator spectrum is obtained, as shown in Figure 15. Fault frequencies such as the + , + 2 and 2 ,, Therefore, the first singular component is processed, and its Teager energy operator spectrum obtained, as shown in Figure 15. Fault frequencies such as the + , + 2 and 2 is can be clearly identified, demonstrating that the proposed method,,can successfully detect the incipient obtained, as shown in Figure 15. Fault frequencies such asmethod the f bp can f r and 2 f bp f r can be clearly identified, demonstrating that the proposed incipient f r,, fsuccessfully bp f r + f r , f bpdetect f r + 2 the fault features of rolling element bearings. Also, it shows that the first singular component is the effective can be clearly identified, demonstrating that the proposed method can successfully detect the incipient fault features of rolling element bearings. Also, it shows that the first singular component is the effective component. However, the kurtosis values corresponding to the third and fourth singular components fault features of rolling Also, it shows thatthird the and first fourth singular component is the component. However, theelement kurtosisbearings. values corresponding to the singular components are much larger than the other kurtosis values, as shown in Figure 14, indicating that the kurtosis values effective However, the kurtosis values corresponding to the third and singular are much component. larger than the other kurtosis values, as shown in Figure 14, indicating that thefourth kurtosis values may provide the wrong information for extracting the sensitive fault information. components than thefor other kurtosis as shown in Figure 14, indicating that the may provide are the much wronglarger information extracting thevalues, sensitive fault information. kurtosis values may provide the wrong information for extracting the sensitive fault information.
Figure 14. 14. The The comparison comparison of of the the proposed proposed ACFHNR ACFHNR index index and and the the kurtosis. kurtosis. Figure Figure 14. The comparison of the proposed ACFHNR index and the kurtosis.
Figure (a)The Theextracted extractedfault faultfeature feature signal and Teager energy operator spectrum of Figure 15. 15. (a) signal and thethe (b)(b) Teager energy operator spectrum of ball Figure 15. (a) The extracted fault feature signal and the (b) Teager energy operator spectrum of ball ball faultfault 1. 1. fault 1.
To further evaluate the performance of the proposed method, the fault data for a bearing roller To further evaluate the performance of the proposed method, the fault data for a bearing roller from the Intelligent Maintenance System, University of Cincinnati, are explored. In the test bed, two from the Intelligent Maintenance System, University of Cincinnati, are explored. In the test bed, two PCB 353B33 High Sensitivity Quartz ICP (Integrated Circuits Piezoelectric) accelerometers (PCB PCB 353B33 High Sensitivity Quartz ICP (Integrated Circuits Piezoelectric) accelerometers (PCB Piezotronics Inc., Depew, NY, USA) are mounted for data acquisition (one vertical and one Piezotronics Inc., Depew, NY, USA) are mounted for data acquisition (one vertical and one horizontal) for each bearing, and a radial load of 2721.6 kg is applied to the bearings. Meanwhile, the horizontal) for each bearing, and a radial load of 2721.6 kg is applied to the bearings. Meanwhile, the
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To further evaluate the performance of the proposed method, the fault data for a bearing roller from the Intelligent Maintenance System, University of Cincinnati, are explored. In the test bed, two PCB 353B33 High Sensitivity Quartz ICP (Integrated Circuits Piezoelectric) accelerometers (PCB Piezotronics Inc., Depew, NY, USA) are mounted for data acquisition (one vertical and one horizontal) for each bearing, and a radial load of 2721.6 kg is applied to the bearings. Meanwhile, the vibration signals are measured at an interval of every 10 min. The sampling frequency is set to 20 KHz, and the data length is 20,480 points [38]. The shaft rotating speed of the motor is 2000 rpm. Four bearings, named Bearings 1 to 4, are installed on one shaft, as shown in Figure 16. At the end of the experiment, an outer race fault is discovered in test bearing 1. Appl.accelerated Sci. 2017, 7, 1117 14 of 19
Figure 16. 16.The Thebearing bearing fault bed the from the Intelligent Maintenance System, University of Figure fault test test bed from Intelligent Maintenance System, University of Cincinnati, Cincinnati, (a) The physical (b) map. The sketch map. (a) The physical map; (b) Themap; sketch
The fault frequency of the outer race fault can be computed as follows: The fault frequency of the outer race fault can be computed as follows: 1 (24) =1 1 − d 2 f bp f o = Z 1 − cos α f r (24) 2 D is the rotating frequency, D is the pitch diameter, d is the balling diameter, Z is the number where of the balls, and α is the contact angle. Equation (23), the faultZ frequency of the where f r is the rotating frequency, D is theAlso, pitchaccording diameter, to d is the balling diameter, is the number of out racer of the bearing can be computed, which is about 236 HZ [29]. Based on this prior fault the balls, and α is the contact angle. Also, according to Equation (23), the fault frequency of the out frequency, ACFHNR value which of all the fault vibration signals in on thethis whole life of the testing racer of the the bearing can beindex computed, is about 236 HZ [29]. Based prior fault frequency, are ACFHNR calculated,index as shown comparison the life kurtosis RMS Mean the valuein ofFigure all the 17a. faultFor vibration signalspurposes, in the whole of theand testing are(Root calculated, Square) values are computed. The results are shown in Figures 18a and 19a, in which the sub graphs as shown in Figure 17a. For comparison purposes, the kurtosis and RMS (Root Mean Square) values in the figures are local are enlargement of dashed are computed. Thethe results shown in Figures 18aline andboxes. 19a, in which the sub graphs in the figures are In the run-to-failure experiment, it is assumed that the bearing worked at the stationary stage the local enlargement of dashed line boxes. fromIn group 10 to group 500. The discrete probability distribution the ACFHNR, kurtosis,stage and the run-to-failure experiment, it is assumed that the bearingfor worked at the stationary RMS group index 10 at to thegroup stationary stage is Gaussian distribution, as for presented in Figure 20. The from 500. The discrete probability distribution the ACFHNR, kurtosis, andalarm RMS thresholds for each parameter are set as the mean value plus four times the stand deviation [23]. Then index at the stationary stage is Gaussian distribution, as presented in Figure 20. The alarm thresholds the each threshold of the three canvalue be obtained, as shown in Table Also, the [23]. alarmThen time the for for parameter are setparameters as the mean plus four times the stand2.deviation RMS, kurtosis theparameters ACFHNR index derived.asThe alarm 88 alarm h 40 min, that threshold of theand three can beare obtained, shown in time Tableof2.RMS Also,isthe timewhile for RMS, of kurtosis is 107 h 50 min, and the ACFHNR warn time is 88 h 40 min, as shown in Figures 17a, 18a, kurtosis and the ACFHNR index are derived. The alarm time of RMS is 88 h 40 min, while that of and 19a.isMeanwhile, compared with the envelope harmonic-to-noise (EHNR) index proposed kurtosis 107 h 50 min, and the ACFHNR warn time is 88 h 40 min, as ratio shown in Figures 17a, 18a and in the literature [23], the ACFHNR canharmonic-to-noise detect faults 1 h and min earlier, the alarm 19a. Meanwhile, compared with the index envelope ratio50(EHNR) indexwhere proposed in the time of the EHNR index is 90 h 30 min [23]. literature [23], the ACFHNR index can detect faults 1 h and 50 min earlier, where the alarm time of the Moreover, wehproposed the concept of sensitivity to quantify the three indexes, which can be EHNR index is 90 30 min [23]. calculated by Equation (25), where ∆d is the of the three indexes in thewhich time ∆can . The Moreover, we proposed the concept of difference sensitivityvalue to quantify the three indexes, be sensitivity can be deemed the gradient of the fitted curve in the incipient fault stage. calculated by Equation (25), where ∆d is the difference value of the three indexes in the time ∆t. The sensitivity can be deemed the gradient of the fitted ∆ curve in the incipient fault stage. = ∆
(25)
The sensitivity values in the incipient stages of the three indexes are calculated. For the ACFHNR and RMS indexes, the data from group 532 to group 700 are calculated, as shown in Figures 17b and 18b. For the kurtosis index, the data from group 647 to group 700 are calculated, as shown in Figure 19b. The sensitivity values for the three indexes are presented in Table 3, and it can be found that the sensitivity value of ACFHNR is much larger than that of kurtosis or RMS.
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S=
∆d ∆t
(25)
The sensitivity values in the incipient stages of the three indexes are calculated. For the ACFHNR and RMS indexes, the data from group 532 to group 700 are calculated, as shown in Figures 17b and 18b. For the kurtosis index, the data from group 647 to group 700 are calculated, as shown in Figure 19b. The sensitivity values for the three indexes are presented in Table 3, and it can be found Appl.the Sci. 2017, 7, 1117 value of ACFHNR is much larger than that of kurtosis or RMS. 15 of 19 that sensitivity Appl. Sci. Sci. 2017, 2017, 7, 7, 1117 1117 Appl.
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Figure 17. The ACFHNR value for the cycle bearing (b) The 1; sensitivity of ACFHNR. Figure 17. (a)(a) The ACFHNR value forfull thelife full lifeofcycle of 1; bearing (b) The value sensitivity value Figure 17. (a) (a) The The ACFHNR ACFHNRvalue value for for the the full full life life cycle cycle of of bearing bearing 1; 1; (b) (b) The The sensitivity sensitivityvalue value of of ACFHNR. ACFHNR. Figure 17. of ACFHNR.
Figure 18. (a) The kurtosis value for the full life cycle of bearing 1; (b) The sensitivity value of kurtosis. Figure18. 18.(a) (a)The Thekurtosis kurtosisvalue valuefor forthe thefull fulllife lifecycle cycleof ofbearing bearing1; 1;(b) (b)The Thesensitivity sensitivityvalue valueof ofkurtosis. kurtosis. Figure 18. kurtosis value for the full cycle bearing 1; sensitivity value of kurtosis. Figure
Figure 19. (a) The RMS value for the full life cycle of bearing 1; (b) The sensitivity value of RMS. Figure19. 19.(a) (a)The TheRMS RMSvalue valuefor forthe thefull fulllife lifecycle cycleof ofbearing bearing1; 1;(b) (b)The Thesensitivity sensitivityvalue valueof ofRMS. RMS. Figure 19. (a) The RMS value for the full life cycle of bearing 1; (b) The sensitivity value of RMS. Figure
Appl. Sci.Figure 2017, 7,19. 1117 (a) The RMS value for the full life cycle of bearing 1; (b) The sensitivity value of RMS. 16 of 19
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Table 2. The threshold values of the three indexes.
The index ACFHNRfor parameters Kurtosis RMS Figure Figure 20. 20. The The discrete discrete probability probability distribution distribution for parameters at at the the stationary stationary stage: stage: (a) (a) ACFHNR; ACFHNR; Threshold value 0.15615 3.76849 0.0812 (b) root-mean-square (RMS); (c) kurtosis. (b) root-mean-square (RMS); (c) kurtosis. 2. The value threshold values the three indexes. Table 3. TheTable sensitivity of the three of indexes in the incipient fault stage.
The index The Index Sensitivity
Threshold value
ACFHNR ACFHNR 0.0261
0.15615
Kurtosis Kurtosis 0.0033 3.76849
RMSRMS 1.55714 × 10−4
0.0812
From Figures 17–20, at least two conclusions can be drawn: (1) Both the RMS and the ACFHNR index are able Table to detect faults earliervalue thanofthe factor, can detect the fault information 3. The sensitivity thekurtosis three indexes in as thethey incipient fault stage. at group 532 (at a time of about 88 h 40 min) while the kurtosis factor detected the fault information at group 647 (atIndex time of about 107ACFHNR h 50 min); (2) Compared with the kurtosis and The Kurtosis RMSRMS indexes, the sensitivity value of the ACFHNR index is the largest in the incipient fault stage, where the sensitivity Sensitivity 0.0261 0.0033 1.55714 × 10−4 value of ACFHNR index is 0.0261, while the sensitivity values of kurtosis and the RMS are 0.0033 and 1.55714 × 10−4, respectively. According to the above conclusions, it can be found that the ACFHNR index From Figures 17–20, at least two conclusions can be drawn: (1) Both the RMS and the ACFHNR is superior to the traditional RMS and kurtosis indexes for detecting the early faults of rolling bearings. index are able to detect faults earlier than the kurtosis factor, as they can detect the fault information Moreover, the early fault vibration signal of group 533 is analyzed. The sample lengths are set at group 532 (at a time of about 88 h 40 min) while the kurtosis factor detected the fault information as 10,000. The original vibration signal and its corresponding envelope demodulation spectrum are at group 647 (at time of about 107 h 50 min); (2) Compared with the kurtosis and RMS indexes, shown in Figure 21. Theouter race fault frequency = 236 HZ can be detected, which further the sensitivity value of the ACFHNR index is the largest in the incipient fault stage, where the indicates the effectiveness of the ACFHNR index for detecting the early weak fault vibration signal sensitivity value of ACFHNR index is 0.0261, while the sensitivity values of kurtosis and the RMS are directly. To further prove the usefulness of the proposed ACFHNR-SVD-TEO method, the vibration 0.0033 and 1.55714 × 10−4 , respectively. According to the above conclusions, it can be found that the signal of group 533 is processed. ACFHNR index is superior to the traditional RMS and kurtosis indexes for detecting the early faults of The delay step is set as m = 15. Also, both the kurtosis values and the ACFHNR index values rolling bearings. corresponding to these 15 singular components are calculated, as shown in Figure 22. It can be seen Moreover, the early fault vibration signal of group 533 is analyzed. The sample lengths are set that the ACFHNR value of the first singular component is the largest, which means that the first as 10,000. The original vibration signal and its corresponding envelope demodulation spectrum are component is the effective signal feature. Thus the signal of the first singular component is processed shown in Figure 21. Theouter race fault frequency f bp f o = 236 HZ can be detected, which further using the proposed method, as shown in Figure 23. The fault frequencies , 2 , 3 , and indicates the effectiveness of the ACFHNR index for detecting the early weak fault vibration signal 4 can be clearly detected. Once again, this processed result proves the usefulness of the directly. To further prove the usefulness of the proposed ACFHNR-SVD-TEO method, the vibration proposed method to detect the early faults of the rolling element bearings. signal of group 533 is processed.
Figure 21.21. (a)The original vibration group 533; (b)(b) The envelope demodulation spectrum ofof Figure (a)The original vibrationsignal signalofof group 533; The envelope demodulation spectrum group 533. group 533.
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The delay step is set as m = 15. Also, both the kurtosis values and the ACFHNR index values corresponding to these 15 singular components are calculated, as shown in Figure 22. It can be seen that the ACFHNR value of the first singular component is the largest, which means that the first component is the effective signal feature. Thus the signal of the first singular component is processed using the proposed method, as shown in Figure 23. The fault frequencies f bp f o , 2 f bp f o , 3 f bp f o , and 4 f bp f o can be clearly Once again, this processed result proves the usefulness of the proposed method Appl. Sci.detected. 2017, 7, 1117 17 ofto 19 Appl. Sci. 2017, 7, 1117 17 of 19 detect the early faults of the rolling element bearings.
Figure22. 22.The The comparison of the proposed ACFHNR index and the kurtosis. Figure The comparison of the proposed ACFHNR index and the kurtosis.
Figure fault feature signal of of group 533; (b)533; The (b) teager Figure23. 23.(a)The (a)Theextracted extracted fault feature signal of of of group group Theenergy teageroperator energyspectrum operator Figure 23. (a)The extracted fault feature signal of 533; (b) The teager energy operator of group 533. spectrumof ofgroup group533. 533. spectrum
6. Conclusions 6.Conclusions Conclusions 6. The incipient periodic impulse feature of the vibration signal from fault rolling element bearings Theincipient incipientperiodic periodicimpulse impulsefeature featureof ofthe thevibration vibrationsignal signalfrom fromfault faultrolling rollingelement elementbearings bearings The isis often contaminated by noise and interference harmonics. In this paper, aanovel early fault detection often contaminated by noise and interference harmonics. In this paper, novel early fault detection is often contaminated by noise and interference harmonics. In this paper, a novel early fault detection approach based on ACFHNR-SVD ACFHNR-SVDcombined combinedwith withTeager Teagerenergy energyoperator operator is presented extracting approachbased presented forfor extracting the approach on ACFHNR-SVD combined with Teager energy operator isispresented for extracting the the incipient periodic impulse feature. The main contributions of this paper can be summarized incipientperiodic periodicimpulse impulsefeature. feature.The Themain maincontributions contributionsof ofthis thispaper papercan canbe besummarized summarizedas asfollows: follows: incipient as follows: (1) Compared Compared with with the the kurtosis kurtosis index, index, the the sensitive sensitive singular singular components components obtained obtained by by the the SVD SVD (1) method can canwith be clearly identified with the proposed ACFHNR index. Moreover, the incipient (1) method Compared the kurtosis index, the sensitive singular components obtained by the SVD be clearly identified with the proposed ACFHNR index. Moreover, the incipient fault features of the vibration vibration signal from the rolling rolling element bearings can be be detected by the method can be clearly identifiedsignal with the proposed ACFHNR index. Moreover, the incipient fault fault features of the from the element bearings can detected by the proposed ACFHNR-SVD method combined with Teager Teager energy operator method. They are are features ofACFHNR-SVD the vibration signal fromcombined the rolling with element bearings can operator be detected by the proposed proposed method energy method. They proved by the simulation and experimental results. ACFHNR-SVD method combined with Teager energy operator method. They are proved by the proved by the simulation and experimental results. (2) TheACFHNR ACFHNR index ableto toresults. detectfaults faultsearlier earlierthan thanthe thekurtosis kurtosisindex. index.Moreover, Moreover,compared compared simulation andindex experimental (2) The isisable detect withACFHNR kurtosis and RMS indexes, the sensitivity value of the ACFHNR index is the largest inthe the (2) with The index is able to detect faults earlier than the kurtosis index. Moreover, compared kurtosis and RMS indexes, the sensitivity value of the ACFHNR index is the largest in incipient fault stage, demonstrating its superiority over the traditional RMS and kurtosis indexes with kurtosis RMS indexes, theits sensitivity value thetraditional ACFHNRRMS indexand is the largest in the incipient fault and stage, demonstrating superiority overofthe kurtosis indexes for detecting the early fault features of the vibration signal. This is proved by the full life cycle incipient fault stage, demonstrating its superiority over the traditional RMS and kurtosis indexes for detecting the early fault features of the vibration signal. This is proved by the full life cycle of the bearing degradation data. for detecting the early fault features of the vibration signal. This is proved by the full life cycle of of the bearing degradation data. the bearing degradation data. In this this preliminary study, the ACFHNR ACFHNR index index and and the the proposed proposed ACFHNR-SVD-TEO ACFHNR-SVD-TEO method method In preliminary study, the are tested and demonstrated to be effective in bearing fault detection using the simulation signal and are tested and demonstrated to be effective in bearing fault detection using the simulation signal and two sets sets of of experimental experimental data. data. In In our our future future work, work, the the applicability applicability of of the the ACFHNR ACFHNR index index will will be be two considered for other methods such as ensemble empirical mode decomposition and local mean considered for other methods such as ensemble empirical mode decomposition and local mean decomposition. decomposition.
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In this preliminary study, the ACFHNR index and the proposed ACFHNR-SVD-TEO method are tested and demonstrated to be effective in bearing fault detection using the simulation signal and two sets of experimental data. In our future work, the applicability of the ACFHNR index will be considered for other methods such as ensemble empirical mode decomposition and local mean decomposition. Acknowledgments: This research was supported by the Chongqing Research Program of Basic Research and Frontier Technology (Grant No. cstc2017jcyjAX0151), the Science and Technology Research Program of the Chongqing Municipal Education Commission (Grant No. KJ1704109), and the National Natural Science Foundation of China (Grant No. 51775076). The authors would like to thank the Case Western University Bearing Data Center and Intelligent Maintenance System, University of Cincinnati, for providing the experimental data. Author Contributions: Kai Zheng wrote the manuscript and analyzed the data. Tianliang Li and Xiangyu Zhou analyzed and interpreted the data. After the manuscript was finished, Bin Zhang, Yi Zhang and Jiufei Luo reviewed the manuscript. Conflicts of Interest: The authors declare no conflict of interest.
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