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Extraction of repetitive transients with frequency domain multipoint kurtosis for bearing fault diagnosis To cite this article: Yuhe Liao et al 2018 Meas. Sci. Technol. 29 055012
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Measurement Science and Technology Meas. Sci. Technol. 29 (2018) 055012 (12pp)
https://doi.org/10.1088/1361-6501/aaae99
Extraction of repetitive transients with frequency domain multipoint kurtosis for bearing fault diagnosis Yuhe Liao1,2,3 , Peng Sun1,2, Baoxiang Wang1,2 and Lei Qu1,2 1
Shaanxi Key Laboratory of Mechanical Product Quality Assurance and Diagnostics, Xi’an Jiaotong University, Xi’an, Shaanxi 710049, People’s Republic of China 2 School of Mechanical Engineering, Xi’an Jiaotong University, Xi’an, Shaanxi 710049, People’s Republic of China E-mail:
[email protected] Received 17 December 2017, revised 29 January 2018 Accepted for publication 12 February 2018 Published 10 April 2018 Abstract
The appearance of repetitive transients in a vibration signal is one typical feature of faulty rolling element bearings. However, accurate extraction of these fault-related characteristic components has always been a challenging task, especially when there is interference from large amplitude impulsive noises. A frequency domain multipoint kurtosis (FDMK)-based fault diagnosis method is proposed in this paper. The multipoint kurtosis is redefined in the frequency domain and the computational accuracy is improved. An envelope autocorrelation function is also presented to estimate the fault characteristic frequency, which is used to set the frequency hunting zone of the FDMK. Then, the FDMK, instead of kurtosis, is utilized to generate a fast kurtogram and only the optimal band with maximum FDMK value is selected for envelope analysis. Negative interference from both large amplitude impulsive noise and shaft rotational speed related harmonic components are therefore greatly reduced. The analysis results of simulation and experimental data verify the capability and feasibility of this FDMK-based method Keywords: frequency domain multipoint kurtosis, repetitive transients, rolling element bearing, fault diagnosis (Some figures may appear in colour only in the online journal)
1. Introduction
of the bearing [3]. The high frequency resonance of the entire bearing pedestal can then be excited and sequential repetitive transients are generated. The appearance of repetitive transients in the vibration signal naturally can be seen as a sign of malfunction of REBs. It is feasible to diagnose the bearing faults via vibration analysis [4, 5]. Therefore, whether these repetitive transients can be effectively extracted from the noisy raw vibration signal is the key to the successful fault diagnosis of REBs. Due to the capability of locating transients in different frequency bands, spectral kurtosis (SK) and its derivative techniques have been widely recognized and researched [6]. The fast kurtogram, which is one of the most well known derivations of SK, greatly improves the analysis accuracy and computational efficiency [7]. Furthermore, considering the fact that the analysis results of the kurtogram heavily depend on the frequency
Rolling element bearings (REBs) are critical components in rotating machinery. Due to harsh working environments or conditions, fatigue defects in REBs are not rare in practice. Effective fault diagnosis and condition monitoring of REBs have always been of great significance to ensure the safe and stable operation of the entire equipment. After years of development, different fault diagnosis techniques have been formed [1, 2]. Among all these methods, a vibration analysis-based approach is the most widely used. This is because, according to the fault mechanism analysis, it has already been proved that the fault-related impulses may appear in the vibration signal each time the rolling elements hit the defect on the rolling path 3
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2. Review of relevant topics
resolution, many researchers carried out some targeted studies and fruitful progress was made thereafter [8–13]. However, another noteworthy issue that hinders the application of kurtosis still remains to be resolved. Kurtosis or the kurtogram in itself is unable to distinguish whether the transients being extracted are repetitive or not. The result then could be misleading if there is impulsive noise contained in the vibration signal [3]. This is the reason why this method fails under certain circumstances and the search for solutions to this issue has attracted the attention of many scholars. Based on the signal demodulation analysis, Barszcz et al proposed an alternative method called Protrugram by calculating the kurtosis of the demodulated narrowband envelope spectrum amplitude [14]. This method is feasible and its disadvantage lies mainly in the fact that it is not easy to find the optimal mathematical expression among these possible choices [15]. Antoni put forward a new method named infogram, which replaced the kurtosis with the negentropy of the squared envelope (SE) and the SE spectrum (SES), to detect the impulsive and cyclostationary transients [15]. Li [16] and Wang [17] further made some improvements to infogram. Since the calculation of the negentropy requires that the probability density function of random variables be known in advance, and generally this is not easy in practice, the application of the infogram therefore has been affected to some extent. Xu [18] provided another solution to this problem, which used the envelope harmonic-to-noise ratio instead of kurtosis to deal with the interference problem brought by impulsive noise. In addition, Zhang et al [19] presented an improved method of kurtogram based on correlated kurtosis (CK) [20] to detect the repetitive transients and Gu et al [21] also made some interesting progress by combining wavelet packet transform and frequency domain CK (FDCK). McDonald et al presented the multipoint kurtosis, which is also an important expansion of kurtosis and is suitable for the detection of impulsive and cyclostationary transients [22]. All these studies have done much useful exploration and laid a solid foundation for the follow-up research. This paper tries to make some further progress on this issue. A new parameter, frequency domain multipoint kurtosis (FDMK), is introduced first of all. Our work is partly based on multipoint kurtosis and it is redefined in the frequency domain for the consideration of computational accuracy. Through estimation of the fault characteristic frequency (FCF) with an envelope autocorrelation function [23], the frequency hunting zone can be set for the FDMK. Then, the FDMK, instead of kurtosis, is utilized to generate a fast kurtogram. The SES of the decomposed signal with maximum FDMK is calculated for the fault diagnosis of the REB. This method improves the accuracy of fault diagnosis while ensuring high computational efficiency. This property makes it suitable for this method to be applied in online condition monitoring. The feasibility and effectiveness of this method are tested by experiments. This paper is arranged as follows: a brief review of the relevant theoretical background is given in section 2. In section 3, the proposed method is discussed in detail. Section 4 verifies the effectiveness of this FDMK-based fault diagnosis method with three different experiments. Finally, section 5 gives the discussion and the conclusions of this paper.
For the convenience of further discussion, some relevant topics are briefly reviewed here. 2.1. Spectral kurtosis (SK)
The formal definition of SK was first given by the Wold– Cramer representation, which describes any stochastic nonstationary process Y (t) as the output of a causal, linear and time-varying system [24]: ˆ +∞ Y (t) = e j2πft H (t, f ) dX ( f ), (1) −∞
where dX ( f ) is the orthogonal spectral process of unit variance. H (t, f ) is the time-varying transfer function of the considered system and can be interpreted as the complex envelope of signal Y (t) at frequency f . The SK is then defined as the normalized fourth-order spectral cumulant of signal Y(t) [24]: S4Y ( f ) −2 KY ( f ) = 2 (2) S2Y ( f )
where the second order spectral moment S2nY( f ) is defined as 2n 2n S2nY ( f ) = E |H (t, f ) dX ( f )| = E |H (t, f )| · S2nX (3) and KY ( f ) is the SK of process Y (t) around frequency f . The constant −2 (instead of −3 as in classical kurtosis) here indicates that H (t, f ) is a complex number. Since direct calcul ation of SK is very time consuming, Antoni [7] proposed the fast kurtogram to solve this problem. Essentially, since the fast kurtogram uses 1/3 binary filter banks, the calculation process is much faster than a kurtogram. This property has greatly promoted the industrial application of the fast kurtogram and it is now regarded as a benchmark technique for mechanical fault diagnosis [6]. In view of this, the above mentioned filtering algorithm of the fast kurtogram is also adopted in this paper. The detail of the 1/3 binary filter banks is illustrated in [7] and its decomposition principle is briefly shown in figure 1. The SES of the filtered signal then can be defined as 2 2 (4) SESx = DFT xf [n] , where xf [n] is the filtered signal, which is obtained by the convolution of the original vibration signal x[n] and the filter f [n], as follows: xf [n] = x [n] ⊗ f [n] . (5)
The high efficiency and accuracy of SES in handling the cyclostationary signal has been widely recognized, which makes it the preferred technique in the demodulation process of SK [7]. 2.2. Frequency domain correlated kurtosis (FDCK)
CK is another derivative of kurtosis and is similar to the proposed method in describing the impulsive nature of a 2
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Figure 1. Paving of 1/3 binary filter banks.
faulty REB vibration signal. The definition of CK is given by [20] 2 N M m=0 xn−mT n=1 (6) CKM (x) = , M+1 N 2 x n=1 n where N, T and M are the length of x, the period and the shift order, respectively. An improved technique of the fast kurtogram based on CK has also been proposed [19]. Since direct calculation of CK in a time domain is also time consuming, especially when the signal under consideration is of low SNR, it is feasible in theory but may not be practical. FDCK was presented to solve this problem. It is defined by [21] 2 N M m=0 Sk−mT k=1 FDCK (S, T) = , (7) M+1 N 2 S k=1 k
N−L 4 n=1 (tn yn ) Unnormalized MK = 2 , (8) N−L 2 y n=1 n
where n, N and L are the serial number of discrete sequence, the length of the original signal and the filter length, respectively. yn is the deconvolution signal and tn is the target vector. With the normalization factor k, MK is defined as N−L 4 n=1 (tn yn ) MK = k . y, t 2 (9) N−L 2 n=1 yn
When the output y is an integer multiple of the target vector t , the value of MK will be equal to 1. Then we have N−L 2 4 t 1 = k n=1 n 2 , (10) N−L 2 n=1 tn
where T is the FCF of the REB and S is the SES of the decomposed signal. The FDCK-based fault diagnosis method can accurately locate the target frequency and its higher order harmonic components in the frequency domain with high computational efficiency. However, calculation of FDCK requires that the target frequency component should be known accurately in advance, which greatly hinders its application [21].
so
N−L 2 n=1 tn
2
(11) k = N−L . 8 n=1 tn
Put equation (11) into equation (9) and we can get the formal definition of MK as 2 N−L 2 N−L 4 t n=1 n n=1 (tn yn ) MK y, t = N−L . 2 N−L 2 8 n=1 tn n=1 yn (12) The target vector tn , which controls the interested multiple points, is a set of impulse sequences with gradually increasing interval. The components of tn are
2.3. Multipoint kurtosis (MK)
MK was first presented by McDonald in 2016 [22] and our work is also partly inspired by MK. For multiple points controlled by a target vector, equation (8) gives the unnormalized MK, as follows: 3
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T · · · ]N−L T · · · ]N−L
tnR1 = [1 0 · · · 1 0 · · · 0 1 0 tnR2 = [1 0 · · · 0 1 · · · 0 0 1 .. . T tnRR−1 = [1 0 · · · 0 1 · · · 0 0 1 · · · ]N−L T tnRR = [1 0 · · · 0 1 · · · 0 0 1 · · · ]N−L
where Sn and N are the amplitude and the length of the SES, respectively. According to this definition, it can be seen that the frequency interval of the target vectors is an arithmetic progression with its common difference equal to 1. Theoretically, it is required that the frequency hunting zone be set around the FCF for the construction of target vectors. Although the FCF can be precisely calculated with the REB structure parameters, these parameters might not be available either, especially in field applications. In this paper, another approach to the estimation of the FCF with an envelope autocorrelation function is presented. The Hilbert transform of the measured vibration signal x (t) is ˆ 1 +∞ x (τ ) dτ x (t) = H {x (t)} = (18) π −∞ t − τ
(13)
and then we have the target vector tn tn = tnR1 , tnR2 , · · · , tnRR−1 , tnRR N−L×R . (14)
The advantage of MK lies in that, compared with CK, here only a roughly estimated impulse interval is needed. A group of target vectors with equally increasing/decreasing impulse interval, in the form of arithmetic progression, is constructed based on the initially set impulse interval. Repetitive transients can then be correctly extracted through comparison of the similarity between the set of target vectors and the original signal.
and the envelope signal E [x (t)] is E[x(t)] = x2 (t) + x 2 (t). (19)
3. FDMK-based method
Removing the DC component, we have
It can be seen that MK has the ability to extract repetitive transients. In the proposed method, the kurtosis index in the fast kurtogram is replaced by MK. The details and several key points of this method will be discussed below. According to the original definition of MK, the target vectors always start from an impulse, as shown in equation (13). This, in general, is surely not the case in practice. In fact, the first impulse collected seldom happens to be located at the sampling start time. Direct calculation of MK in the time domain then is not feasible. Considering the fact that the FCF and its higher order harmonics are separately distributed in the frequency spectrum at a certain interval, it would be more practical to calculate MK in the frequency domain. Moreover, removing the first impulse in the target vectors at the sampling start time can ensure the impulse sequence of one target vector matches exactly with that of the concerned frequency components. Therefore, the target vectors tn are redefined in the frequency domain as
E [x (t)] = E [x (t)] − mean {E [x (t)]} . (20)
The autocorrelation of E [x (t)] is ˆ rE[x(t)] (τ ) = E[x (t)]E[x (t + τ )]dt. (21) Finding the serial number Nτmax that corresponds to the maximum of equation (21) and the FCF can be estimated by Fs FCF = , (22) Nτmax
where τ is the delay time and Fs is the sampling frequency. Take the FCF as the center frequency and the frequency hunting zone then can be set. Generally, the larger the hunting zone is, the lower the computational efficiency will be. A smaller hunting zone may tend to bring greater risk that the possible hunting target could be excluded. In consideration of that, here the frequency hunting zone is set around the FCF with a suitable zone width of 10 Hz (i.e. [FCF-5 Hz, FCF + 5 Hz]) according to previous exper imental analysis results. With the FCF being estimated and the frequency hunting zone being set, the FDMK then can be calculated using equation (17). The detailed fault diagnosis process is as follows. Firstly, the FCF of the original vibration signal is estimated and the frequency hunting zone is set. Secondly, 1/3 binary filter banks are applied to decompose the signal into a certain number of sub-signals located in different frequency bands. Then, the FDMKs of each decomposed sub-signal are calculated. The one with the maximum FDMK value is selected and filtered out. Finally, the SES of the filtered signal is calculated for the fault diagnosis of the REB. Figure 2 gives the flow chart of this method. A simulation analysis is used to test the performance of this method. The vibration signal of an REB with outer-race fault is simulated as
T
tnRR = [0 0 · · · 1 0 · · · 0 1 0 · · · ]N T tnR2 = [0 0 · · · 0 1 · · · 0 0 1 · · · ]N .. (15) . T tnRR−1 = [0 0 · · · 1 0 · · · 0 1 0 · · · ]N T tnR1 = [0 0 · · · 0 1 · · · 0 0 1 · · · ]N ,
and we have
tn = tnR1 , tnR2 , · · · , tnRR−1 , tnRR N×R . (16) Note the difference between equations (13) and (15). The impulses at the sampling start time in the redefined target vector are removed here in equation (15). Based on that, the FDMK is given by 2 N N 2 4 n=1 tn n=1 (tn Sn ) FDMK = N , 2 N 8 2 n=1 tn S n=1 n (17) 4
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Figure 2. Flow chart of the FDMK-based method.
y (t) =
i
Ai s (t − Ti − δi ) +
j
In this case, the FCF, Fo, is 80 Hz. The sampling frequency is 10 kHz and the data length is 10 240. There are two harmonic components contained in the simulation signal and their amplitudes, frequencies and phases are 0.1, 0.08, 30 Hz, 60 Hz, π/6 and −π/3, respectively. The parameters of the amplitude modulation signal, i.e. fr, fm and A, are 30 Hz, 800 Hz and 0.6. The SNR of the simulation signal with additive Gaussian noise is −12 dB. Other parameters related to the system impulse response are listed in Table 1. Another two methods, the fast kurtogram and the FDCKbased method, are also applied here for comparison. The analysis result of the fast kurtogram is first shown in figure 4. Two prominent areas marked by dashed yellow frames in figure 4(a) are exactly the resonance bands of the two interference impulses. However, the resonance band of the faultrelated repetitive transients, which should be located in the dashed red frame in figure 4(a), is not detected. Figure 4(b) gives the waveform of the results of the fast kurtogram and its envelope spectrum is shown in figure 4(c). The FCF comp onent is completely submerged in the background noise. Almost no useful information is observed. Obviously, the diagnosis process has been seriously interfered with by impulsive noises. Figure 5 gives the results of the FDCK-based method. It also fails to extract the repetitive transients, as shown in figure 5(a). The resonance band detected deviates obviously. Only the rotating frequency component and its higher order harmonic counterparts can be seen in figure 5(c). However, the concerned FCF component is still not found. Finally, the FDMK-based method is applied and the results are shown in figure 6. Compared with the above mentioned fast kurtogram and the FDCK-based method, the FDMK-based method solves the interference problem effectively. The resonance band of repetitive transients is successfully identified,
Bj s (t − Tj )
Ck sin (2πfk t + ϕk ) + x (t) + n (t) . k (23) Besides the fault-related repetitive transients, here we considered four different kinds of interferences to simulate a bearing vibration signal in complex conditions. Therefore, this signal consists of five parts, as shown on the right-hand side of equation (23). The first one is the fault-related impulsive component, i.e. the repetitive transients. Ai is the ampl itude, Ti is the nominal time interval between two adjacent impulses, δi is the random jitter generated by rolling element slip, which usually accounts for 1–2% of Ti . The second part is the interference term related to nonrepetitive impulsive noise, where Bj and Tj are the amplitude and the time of the jth impulse. The impulse expression s(t) can be further expanded as +
s (t) = e−αt sin (2πfr t + ϕ) , (24)
where α , fr and ϕ are the coefficient of resonance damping, the structure resonance frequency and the phase, respectively. The third part represents the shaft rotation related harmonic components, where Ck, fk and ϕk are the amplitude, frequency and phase, respectively. The fourth part is an amplitude modulated signal, which simulates the possible interference from adjacent components and can be shown as x (t) = (1 + cos (2πfr t)) ∗ A cos(2πfm t), (25)
where fr is the modulation signal frequency, fm is the carrier signal frequency and A is the amplitude coefficient. Finally, the fifth part, n(t), represents the background white noise. The simulated signal and its components are given in figure 3. 5
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Figure 3. The simulation signal of a REB with outer race defect. (a) Impulsive fault response; (b) interference impulses; (c) harmonic components generated by shaft rotation; (d) AM signal; (e) white noise; (f) the combined signal.
theoretical value of the FCF, denoted by F0, is 102.9 Hz. The motor power is 3 horsepower. The maximum width of the defect is about 0.014 inches. It was said that the analysis of this data is not easy [25] and we choose it to test the capability of this method. Figure 7 gives the waveform and its spectrum. Several impulses with large amplitude can be seen in figure 7(a) and there are three distinct resonance bands in figure 7(b). This kind of signal generally indicates the existence of a bearing fault. However, no more information can be observed in figure 7. The data needs to be further analyzed. The FDMK-based method is applied and the analysis result is given in figure 8. The resonance band of fault-related repetitive transients is clearly identified, as shown in figure 8(a). In addition, the nonrepetitive impulses are also removed in the filtered signal, see figure 8(b). The concerned FCF, F0, and its second order harmonic component are both highlighted in the envelope spectrum of the filtered signal, see figure 8(c). The outer race fault has been successfully diagnosed. In the second experiment a bearing with a ball defect is considered. The data comes from [26] and the detailed information can be found in [11]. The sampling frequency is 80 kHz and data length is 80 000. The FCF of this fault, denoted by Fb, is 64 Hz. Figure 9 gives the time domain waveform and its frequency spectrum. There are several nonrepetitive interference impulses in the time domain waveform. One narrowband resonance region, jointly excited by both the fault-related repetitive transients
Table 1. Parameter values of the simulation signal.
Amplitude Resonance damping coefficient Resonance frequency
Repetitive First interference impulse impulse
Second interference impulse
1.8 420
2 120
3.8 260
0.8 KHz
3.7 KHz
1.8 KHz
which is marked by the dashed red frame in figure 6(a). The FCF and its higher order harmonic components are also very clear in figure 6(c). 4. Experimental verification The vibration signals of REBs with seeded defects are analyzed with the proposed method in this section. Here we considered three different kinds of faults, which are located in the outer race, rolling element and inner race, respectively. The REB data of the first experiment comes from the Case Western Reserve University (CWRU) Bearing Data Center Website. The document serial number of the selected data is OR014@6_3 (X200_DE_time of 12k Drive End Bearing Fault Data). The test speed is 1723 r min−1. The seeded fault is located in load bearing zone of the outer race and the 6
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Figure 4. The results of the fast kurtogram. (a) Fast kurtogram; (b) the filtered signal; (c) the envelope spectrum of (b).
Figure 5. The results of the FDCK-based method. (a) FDCK; (b) the filtered signal; (c) the envelope spectrum of (b).
Figure 6. The results of the FDMK-based method. (a) FDMK-based method; (b) the filtered signal; (c) the envelope spectrum of (b).
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Figure 7. The waveform and spectrum of the selected data. (a) Waveform; (b) frequency spectrum.
Figure 8. The result of the FDMK-based method. (a) FDMK; (b) the filtered signal; (c) the envelope spectrum of (b).
Figure 9. The waveform (a) and the frequency spectrum (b) of the bearing vibration signal.
12(b). However, FCF and its higher order harmonic comp onents can only be clearly identified in the envelope spectrum of the signal filtered by FDMK, as shown in figure 12(c). In comparison, identification of these components with the fast kurtogram and the FDCK-based method fails. The FCF and its harmonic components are almost submerged by background noise, see figures 10(c) and 11(c). The last experiment was carried out on the ball bearing fault simulation test rig of our own laboratory. The test rig, together with the sensor arrangement, is shown in figure 13. The schematic diagram of the test rig is given in figure 14. The function of the main spindle box is to isolate motor vibration. The test bearing can be loaded both radially and axially (the test bearing is a single-row deep groove ball bearing, so no axial load is actually applied in this experiment). The radial static load is provided by a heavy weight through a simple pulley device and horizontally applied (along
and the interference impulse, appears in the frequency spectrum, as shown in figure 9(b). Accurate fault diagnosis cannot be achieved with only the waveform and frequency spectrum. For the convenience of our discussion, here we applied all the above three methods to analyze the data. Figures 10–12 give the analysis results of the fast kurtogram, the FDCK- and the FDMK-based method, respectively. The resonance bands are identified in all the three methods, see the area surrounded by the dashed frame in figures 10(a), 11(a) and 12(a). Although the center frequencies of each resonance band are slightly different, they all cover approximately the same frequency band. This is because the resonance band excited by impulsive noises overlaps with that of the concerned repetitive transients. This certainly increases the difficulty of identification and extraction of the FCF. Nonrepetitive transients with large amplitude appear in all the time domain waveforms of the filtered signal, see figures 10(b), 11(b) and 8
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Figure 10. The result of the fast kurtogram. (a) Fast kurtogram; (b) the filtered signal; (c) the envelope spectrum of (b).
Figure 11. The results of the FDCK-based method. (a) FDCK; (b) the filtered signal; (c) the envelope spectrum of (b).
Figure 12. The results of the FDMK-based method. (a) FDMK; (b) the filtered signal; (c) the envelope spectrum of (b).
the Z axis) to the test bearing. The size of the load can be adjusted by changing the weight. The side view on the righthand side of figure 14 gives the detail of this axial loading device. In order to shorten the transmission path of the vibration signal as much as possible, the accelerometer is mounted
in the load bearing zone on the shaft sleeve, which is rigidly attached to the bearing outer ring. Here a PCB three-direction accelerometer (type 356A17) is used. The sensor sensitivities in the three directions, as shown in the enlarged view on the right-hand side of figure 13, are 502 mV g−1 (X direction), 9
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Figure 13. The test rig and sensor arrangement. Accelerometer Coupling Motor
A-View
Axial loading device Main Spindle Box
Belt
Bolt Test bearing
Pulley
A-View
Keyphasor
Radial loading device
Accelerometer Weight
Weight
Figure 14. The test rig schematic diagram and detail of the loading device.
Figure 15. The time domain waveform (a) and the frequency spectrum (b) of the faulty bearing vibration signal.
interference was suspected to be the possible cause, it cannot be confirmed yet at this stage without further information. The presence of four resonance bands in figure 15(b) indicates that the bearing is under faulty conditions. Accurate diagnosis then is made. Figure 16 gives the analysis results of the proposed method. Fault-related characteristic information is clearly identified with FDMK. It seems that the effect of impulsive interference on the analytical result of FDMK is removed. This FDMK-based method successfully identifies the second and third resonance bands in figure 15(b). No apparent impulsive interference comp onents can be found in the filtered signal, see figure 16(b), and the envelope spectrum clearly highlights the FCF and its higher order harmonic components, as shown in figure 16(c). Moreover, the envelope spectrum also shows the modulation sidebands, which are centered in the FCF and its higher order harmonics.
Table 2. The structural parameters of the test bearing.
Diameter of ball
Number of ball
Diameter of inner raceway
Pitch diameter
15 mm
8
40 mm
65.5 mm
Contact angle 0°
496 mV g−1 (Y direction) and 471 mV g−1 (Z direction), respectively. The test speed is 1800 rev min−1. Signal sampling frequency is 15 000 Hz and the data length is 150 000. The detailed structural parameters of the test bearing, with seeded inner race defect, are listed in table 2. Theoretically calculated FCF, denoted by Fi, is 147.5 Hz in this case. Only the vibration signal collected from radial direction, i.e. the Z direction, is considered here. Figure 15 gives its time domain waveform and frequency spectrum. There are two unexpected abnormal interruptions in the signal, as shown in figure 15(a). Although electromagnetic 10
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Figure 16. The results of the FDMK-based method. (a) FDMK; (b) waveform of the filtered signal; (c) the envelope spectrum of (b).
5. Discussion and conclusion
use the theoretically calculated FCFs of all possible bearing faults instead to set the frequency hunting zones. The proposed method then needs to be implemented repeatedly for each FCF until the bearing fault is identified. Obviously, the amount of computation in this process is greatly increased, but the high computational efficiency of this method still ensures that these searching processes can be done very quickly. We are still working on the improvement of this method. A more robust FCF estimation approach is under development. In addition, this paper verifies the validity of the proposed method in the diagnostic problem of REBs under single fault condition. Its effectiveness for the compound faults diagnostic problem is also pending further investigation.
One typical feature of faulty REB vibration is that there are repetitive transients in the signal. How to effectively extract or identify these characteristic components has been seen as one feasible solution for the accurate fault diagnosis of REBs. In view of that, research on kurtosis-related topics has been a hotspot in recent years. Up to now, the efforts concerning SK-related REB fault diagnosis methods have focused mainly on two topics. One tries to build a decomposition algorithm with higher accuracy and the other intends to develop a more powerful index for the effective detection of repetitive transients under noisy conditions. However, whether the index adopted is capable of locating the optimal frequency band that contains the fault-related characteristic information is the key to an accurate diagnosis. In view of that, this paper therefore made some attempts in this area. A diagnostic approach, which is based on a newly developed index, FDMK, is proposed. Analysis shows that, since FDMK is calculated with the envelope spectra of the filtered signal, this approach can accurately locate the optimal frequency band with high computational efficiency. The interference of nonrepetitive impulses and rotating speed related harmonic components can be effectively removed, which ensures the accurate extraction of the fault-related repetitive transients. The results of simulation and experiments verify that the bearing fault diagnostic ability is greatly improved with this FDMK-based method. In addition, since it adopts the same computation approach as the fast kurtogram, this method maintains high computational efficiency at the same time and ensures that the frequency band searching process, which could be very time consuming otherwise, is accomplished efficiently. These properties show that this method can not only be used for off-line diagnosis, but also matches the requirements of online analysis. One thing to note is that, in some complicated cases (e.g. signal with extremely low SNR), the estimated FCF may be seriously distorted and direct use of this method could be misleading. A feasible solution at present to this problem is to
Acknowledgments This work is supported by the National Natural Science Foundation of China (Grant No. 51575424), and the National Science and Technology Major Project (Grant No. 2014ZX04001191), which are highly appreciated by the authors. ORCID iDs Yuhe Liao
https://orcid.org/0000-0001-7948-9934
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