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Fault diagnosis method based on integration of RSSD and wavelet transform to rolling bearing
Accepted Manuscript Fault diagnosis method based on integration of RSSD and wavelet transform to rolling bearing Chen Baojia, Shen Baojia, Chen Fafa, Tian Hongliang, Xiao Wenrong, Fajun Zhang, Chunhua Zhao PII: DOI: Reference:
S0263-2241(18)30639-0 https://doi.org/10.1016/j.measurement.2018.07.043 MEASUR 5727
To appear in:
Measurement
Received Date: Revised Date: Accepted Date:
4 March 2018 14 July 2018 16 July 2018
Please cite this article as: C. Baojia, S. Baojia, C. Fafa, T. Hongliang, X. Wenrong, F. Zhang, C. Zhao, Fault diagnosis method based on integration of RSSD and wavelet transform to rolling bearing, Measurement (2018), doi: https:// doi.org/10.1016/j.measurement.2018.07.043
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Fault diagnosis method based on integration of RSSD and wavelet transform to rolling bearing Chen Baojiaa,b, Shen Baojiaa, Chen Fafaa*, Tian Honglianga, Xiao Wenronga,b, Fajun Zhanga,b, Chunhua Zhaoa,b a
Hubei Key Laboratory of Hydroelectric Machinery Design & Maintenance, China Three Gorges University, Yichang 443002, Hubei, China;
b
College of Mechanical and Power Engineering, China Three Gorges University, Yichang 443002,Hubei,China
Abstract: To solve the problem of early fault diagnosis of rolling bearing under strong background noise, a fault diagnosis method based on integration of Resonance-based Sparse Signal Decomposition (RSSD) and Wavelet Transform (WT) is proposed in this paper. The RSSD method is combined with quality factor optimization using genetic algorithm and sub-band reconstruction. Firstly, the early fault vibration signal of the rolling bearing is decomposed by RSSD. The kurtosis value of the low resonance component is taken as the objective function to optimize the combination of high and low quality factors with genetic algorithm. Then, the master sub-band is selected out to reconstruct the low resonance component based on the principle of energy dominant distribution. It can reduce the noise interference and enhance the impulse characteristic of the fault signal. Finally, characteristics of local optimization and multi-resolution of wavelet analysis considered, the multi-scale wavelet decomposition is applied to the reconstructed low resonance component to extract the fault features of the bearing failure deeply. The effectiveness and application value of the method are proved by two different diagnosis cases of rolling bearing faults. By comparisons, the fault feature extraction ability of the proposed method is prior to WPD method and similar or prior to EMD method for different bearing fault signals. Key words: quality factor; sub-band reconstruction; rolling bearing; resonance-based sparse signal decomposition; wavelet transform
1 Introduction The vibration characteristics of the rolling bearing faults are mainly determined by the rotation of the shaft, the bearing structure, the damage form, the transmission path between the bearing and the system etc [1]. When incipient failure of the rolling bearing emerges, the impulse force caused by the damage will induce the high
*
Corresponding author.
E-mail address: [email protected] (F. Chen).
frequency natural vibration of the bearing system, which is also modulated by the periodic impact force [2,3]. The vibration signal mainly contains the harmonic component, the fault impulse component relating to the revolutions and the background noise. With the development of the faults, the dynamic vibration signals of the rolling bearing often demonstrate nonlinearity and non-stationarity [4]. Some components may have the similar center frequency and overlapping frequency bands. Therefore, the key issue for rolling bearing faults diagnosis is to isolate and highlight the fault impacting ingredients from the original vibration signals to reduce the interference of harmonics and noise. The application prerequisite of the Fourier Transform (FT) is that the signals can be expressed as a linear superposition of the stationary sine waves with different frequencies. It lacks an accurate description of the local time-frequency variation presentation of the signal. In recent years, the nonlinear and nonstationary signal processing has been a research hotspot, especially in the field of mechanical fault diagnosis. In order to extract fault features from the transient weak signal under the strong background noise, Yan [5] proposed a new wind turbine gearbox fault diagnosis method based on continuous wavelet transform (WT) and stationary subspace analysis. Zheng [6] proposed an improved Empirical Mode Decomposition (EMD) fault diagnosis method, which contained Generalized Empirical Mode Decomposition (GEMD) and Improved Direct Quadrature (IDQ) demodulation. Several different Intrinsic Mode Functions (IMF) are obtained by defining different mean curves and in each rank the best IMF is selected. The improved empirical AM-FM decomposition and IDQ are used to demodulate the component. Wang and Chen [7,8] analyzed the filter characteristics and noise effects of the Local Mean Decomposition (LMD) method respectively and gave out the SNR (signal to noise ratio) threshold of LMD for signal waveform and instantaneous frequency extraction. Then LMD was applied to the fault diagnosis of rotor robbing and locomotive bogie. A new adaptive stochastic resonance method was developed by Xiang [9] using an artificial bee colony algorithm and stochastic resonance theory. In order to obtain the maximum stochastic resonance output SNR, the structural parameters of the system have been adaptively optimized by an artificial bee colony algorithm to realize the adaptive adjustment of the bistable system parameters. The above signal processing methods decompose the signal into multiple frequency bands by analytical expressions or self-adaptive methods. Whereafter the feature extraction methods are adopted on the specific frequency band to get some fault characteristic quantities so as to implement diagnosis procedure. If the center frequency of the harmonic components, fault impulse components and the background noise superpose together, the multi-components of the signals can’t be separated out by general frequency band decomposition or linear filtering methods. Whereas the signals can be decomposed based on their resonance properties (different quality
factor Q), rather than on frequency or scale, as provided by FT and WTs. In 2011, Selesnick [10] proposed a new signal processing method, namely, resonance-based sparse signal decomposition (RSSD). The RSSD method can be used to separate a complex signals into a high-resonance component consisting of multiple simultaneous sustained oscillations and a low-resonance component consisting of non-oscillatory transients of unspecified shape and duration [11]. In [12,13], RSSD was successfully introduced to the fault diagnosis of bearing and gearbox. In order to detect multiple faults in gearboxes, Zhang [14] proposed a novel method called resonance-based sparse signal decomposition with comb filter (RSSD-CF) and the stepwise optimization strategy was applied for parameter optimization of the RSSD method.
In most existing RSSD-based fault diagnosis methods, the analyzing procedure is only adopted on the decomposed low-resonance component. It is thought that the low-resonance components are composed of transient shock components. Due to the noise interference, the components with different resonance properties are considered to have great coupling effect. It is difficult to observe the characteristic frequency of the fault rolling bearing when it is not dominant, especially for incipient failure. In addition, the information mining capability of the RSSD method for transient impulse components is inadequate for the reason that it does not have multi-scale band analysis capability similar to WT, EMD or LMD. On the basis of the above analysis, this paper presents a fault diagnosis method based on the integration of RSSD and WT. The quality factor optimization with genetic algorithm (GA) and sub-band reconstruction are introduced to improve the RSSD method. The kurtosis value of the low resonance component is taken as the objective function to optimize the combination of high and low quality factors with GA. The master sub-band is selected to reconstruct the low resonance component based on the principle of energy dominant distribution, which can reduce the noise interference and enhance the impulse characteristic of the fault signal. Characteristics of local optimization and multiresolution of wavelet analysis considered, the multi-scale wavelet decomposition is applied to the reconstructed low resonance component to extract the fault features of the bearing failure deeply. This paper is organized as follows: Section 2 introduces the RSSD method briefly. Section 3 focuses on the details of quality factor optimization and sub-band reconstruction. Section 4 discusses the diagnostic process, which integrates the RSSD method with the wavelet analysis technique. Section 5 deals with the experimental verification of the proposed method and gives out the comparative analysis results with Wavelet Packet Decomposition (WPD) and EMD. The conclusion of this study is formulated in the last section.
2 Resonance-based Sparse Signal Decomposition The RSSD method is actually a sparse representation jointly using the high and the low Tunable Q-Factor
Wavelet Transform (TQWT). The key issue of RSSD is to construct the tunable wavelet. In the early studies, Selesnick[11] explored Rational-Dilation Wavelet Transform (RADWT), which can attain a range of redundancies and Q-factors making the transform more flexible than the dyadic wavelet transform. The tunable range of Q-factor can be extended by adjusting the sampling size q to improve frequency resolution. In contrast, the TQWT is simpler conceptually, which can be more efficiently implemented using radix-2 Fast Fourier Transformation (FFT), and its parameters are more easily related to the Q-factor of the transform.
2.1 Tunable Q-Factor Wavelet Transform The TQWT is fully discrete, has the perfect reconstruction property, is modestly overcomplete, is developed in terms of iterated two-channel filter banks, and implemented using the Discrete Fourier Transform (DFT) based on the resonant nature of the signal [12]. The tunable-Q wavelet transform is based on the multirate filter bank illustrated in Fig. 1.
H 0 ( w)
LPS a
v0 (n)
LPS
1
a
H 0* ( w)
y0 ( n )
x ( n)
y ( n)
H1 ( w)
HPS b
HPS
1
v1 (n)
b
H1* ( w)
y1 (n)
Fig. 1. Analysis and synthesis filter banks for the tunable-Q wavelet transform. *
In Fig. 1, x(n) is the original signal, H i = 0,1 ( w) , Hi =0,1 (w) are the frequency response functions of the analysis and synthesis filters respectively. LPS and HPS represent low-pass scaling and high-pass scaling respectively. vi = 0,1 (n) are the sub-band signals obtained after decomposition. yi = 0,1 ( n ) are the synthetic signals. The scaling parameters satisfy
2 ì ïb = Q + 1 ï b ï ía = 1 r ï ïs.t. 0 < a , b £ 1 ï î
(1)
a + b >1
where ݎrepresents the redundancy rate. Once the filter bank parameters ߙ and ߚ are determined, the
values of the Q-factor and ݎcan be calculated. The essence of the multilayer decomposition process is that
the signal passes through sequentially from a high frequency band-pass filter to a low frequency band-pass filter. It is a successive iteration procedure. The RSSD decomposes the signal into components with different resonance properties. Suppose that the observed signal x can be represented as
x = x1 + x2 + n
(2)
where ݔ1 and ݔ2 are components with different oscillation behavior, and n is the noise. To decompose the
analysis signal, the Morphological Component Analysis (MCA) is applied in the RSSD method [10,16]. The goal of MCA is to estimate/determine ݔ1 and ݔ2 individually. Assuming that the signals ݔ1 and ݔ2 can be sparsely represented in base functions (or frames) ܵ1 and ܵ2 respectively, (obtained by TQWT), they can be estimated by
minimizing the objective function
J˄W1 ,W2 ) = x - S1W1 - S 2W2
2 2
+ l1 W1 1 + l2 W2
1
(3)
where ܹ1 and ܹ2 denote the transform coefficients of the signals ݔ1 and ݔ2. ܵ1 and ܵ2 represent the inverse
TQWT with the high and the low Q-factors respectively. ߣ1 and ߣ2 are the regularization parameters of the high
and low resonance components. The sparse signal decomposition method based on compound Q-factor bases uses the Split Augmented Lagrangian Searching Algorithm (SALSA). It minimizes the target function , by iteratively
updating the transform coefficients ܹ1 and ܹ2. Eventually, it effectively separates the high resonance component
and low resonance component and extracts the transient impulse component. After ܹ1 and ܹ2 are obtained, the estimated high and low resonance components (solved by matching) are shown as follows
xˆ1 = S1W1 ,
xˆ2 = S2W2
(4)
2.2 Selection and Influence of TQWT Parameters It is important to adaptively select the parameters of RSSD for purpose of feature extraction of the analysis signal because those parameters have a great effect on the decomposition results. 7he Q-factor of a wavelet function can be adaptively designed and selected according to the oscillations of the analyzed signal. The high Q-factor base is used to match the high resonance component (the random vibration and strong noise of the rolling bearing itself), while the low Q-factor base is used to match the low resonance component (fault impulse component). Different Q-factor represents different time-frequency resolution of the waveform function and overlap of the adjacent two sub-bands. So the choices of Q-factor values greatly affect the correlation (degree of coupling) of the high and low resonance components. As is shown in Eq. (5), the literature [10] gives the correlation analysis expression of the quality factors Q1 and Q2.
rmax (Q1 , Q2 ) =
Q2 + 1/ 2 £1 Q1 + 1/ 2
(5)
It is assumed that Q1> Q2. If Q2 = 1 , increasing Q1 can reduce the correlation and improve the resolution. The larger the difference between Q1 and Q2 is, the lower the correlation is. However, the quality factor is too high,
and its wavelet base function may not match the oscillation characteristic of high resonance signal, producing the singular signal and reducing the accuracy and reliability of RSSD. It is necessary to choose the reasonable quality factors Q1 and Q2 to obtain a better decomposition effect. The redundancy factor r is generally between 3.0 and 4.0 in practice. With r ³ 3 , the pass-band of the level-
frequency response will not have a ‘flat top’ (wherein the frequency response is equal to a constant over a sub-interval of its pass-band) as discussed in [17-19]. The decomposition level L determines the number of elements constructed by the wavelet basis function, which affects the center frequency and the bandwidth of each layer sub-band. With the increase of L, the center frequency and bandwidth of each sub-band are getting smaller. The relationship between L, Q and r is shown in Eq. (6).
L=
log( N / 4(Q + 1)) log(Q + 1 / (Q + 1 - 2 / r )
(6)
N represents the signal length. In practical application, the choice of parameter L needs to be considered to ensure that the decomposed sub-band frequency covers the resonance frequency band of each component.
3 Q-Factor Optimization and Sub-band Reconstruction 3.1 Q-Factor Optimization Based on Signal Kurtosis Index In traditional RSSD method, the Q-factors are usually assigned integer as Q2 = 1 and Q2 = 3 ~ 9 . The randomness of the choice determines the quality of decomposition. If the signal that carries the fault information is decomposed into components with specific resonance attribute, a wavelet-based function library that matches the fault signal oscillation characteristics needs to be constructed. Therefore, how to adaptively select quality factors is the key problem in the RSSD method. It's important to note that the Q-factor in the actual application could be non-integer. The kurtosis index K v can be used to describe the degree of the signal deviates from the normal distribution, which is calculated based on Eq. (7).
1 Kv =
N
Nå i =1
( xi - u x ) 4
é1 N 2ù ê N å ( xi - u x ) ú i = 1 ë û
2
(7)
where ux is the mean of the signal and N is the signal length. When the mechanical equipment is working under normal condition, the vibration amplitudes obey normal distribution, K v » 3 . As the failure degree increases, the absolute value of K v gets larger. Therefore, the kurtosis of the decomposed low resonance component is taken as the optimization objective function in this study to select the proper Q-factors. As widely
used in some literatures [20–22], GA is applied in this paper to optimize Q1 and Q2. The changing precision of the Q-factors is 0.01 and the decimal code is adopted. The main parameters of GA are set as follows: l
population size 50,
l
number of generations 200,
l
genetic rate 0.1,
l
crossover rate 0.7,
l
mutation rate 0.02.
The specific steps of GA can be referred to the literature [23]. According to the combination of quality factors obtained by GA, the kurtosis of the low resonance component in theory is the largest, and the fault information is the most. It can be deduced theoretically that the combination of Q-factors obtained by GA optimization corresponds with the low resonance component with the largest kurtosis. It contains more fault information, especially for impulse component.
3.2 Sub-band Reconstruction In the strong background noise working environment, the high and low resonance components of the rolling bearing fault signal decomposed by RSSD have a high coupling degree due to noise interference. It may weaken the impact component in the low resonance component. In order to highlight the rolling bearing failure characteristics, it is necessary to reconstruct the sub-bands selectively of the low resonance components. In [24], the low-resonance component was further decomposed into a set of sub-signals by the TQWT and the proper signal was reconstructed from some selected sub-signals combined with kurtosis analysis to extract the fault characteristics of the rolling bearing. However, the reconstruction method cumulated sub-band components only from high frequency to low frequency according to the filtering characteristic of the Q-factors. The relationships between the sub-band frequency and their proportions in the signal energy are not in consideration. It is inevitable that the noise on some sub-bands should be introduced to the reconstructed signal. In this paper, the sub-band proportion to total energy-based reconstruction method is considered. The detail process is listed as follows: (1) Adopt TQWT on the original signal to obtain the high and low resonance components. Then calculate the energy Ej of L+1 sub-band signals and its percentage ej in the total energy respectively. The calculation equations are shown as
Nj ì 2 ï E j = å W ji i ïï í Ej ´ 100% J +1 ïe j = E ï å j ïî j =1
(8)
where Wji represents the ith wavelet coefficient lies in the jth level and Nj is the length of the jth level wavelet coefficient. (2) Select out the dominant sub-band components, whose energy percentage ej is greater than 1.5 times sub-band average energy, as shown in Eq. (9).
e j ³ 1.5 / ( L + 1) ´ %
(9)
(3) Superimpose the sub-band components together linearly to reconstruct a new low resonance component.
4 Diagnosis Process The RSSD method decomposes the signal according to the resonance properties of the different components rather than divide the signal based on the frequency band. WT is able to analyze localization and multiresolution precisely, which can expand the signal in multi-scale. If the RSSD method is incorporated with the WT, they will complement each other in two aspects of signal resonance attribute and local time-frequency characteristic analysis. The new proposed method is suitable for information deep mining and incipient faults diagnosing to the rolling bearing. The specific diagnostic process is shown in Fig. 2 and the detail is introduced as follows: (1) Through the experiment, the original vibration signals of the rolling bearing fault are obtained. The signals are decomposed by the combination of the quality factor obtained by GA optimization according to the kurtosis principle of the low resonance component described in section 3.1. (2) Sub-band selection and reconstruction are applied to the low resonance components according to the method described in Section 3.2. (3) Wavelet multi-scale decomposition is used to analyze the new reconstructed low resonance component. Finally, the Hilbert demodulation method is used to extract the signal envelope spectrum and obtain the bearing fault characteristics.
Fig. 2. Diagnosis flow chart
5 Experimental Case Study In order to demonstrate the performance of proposed method, this section presents two application examples for the detection of localized defects from the real vibration signals acquired through the rolling bearing fault experiment. In addition, the effectiveness of Hilbert demodulation and regular RSSD method are compared in this section.
5.1 Description of the Experiment 1¾DC drive motor 2¾Pulley 3¾Bearing pedestal 4¾Support housing 5¾Coupling 6¾DC load motor 7¾Vibration acceleration sensors 8¾tachometer
Fig. 3. Rolling bearing failure testing platform
The structure of the rolling bearing fault testing platform and its picture are presented in Fig. 3. The experiment system is composed of a DC drive motor, pulley, bearing pedestal, support housing, coupling, a DC load motor and other accessories. In the experiment, two CA-YD-106 piezoelectric acceleration sensors positioned on the bearing pedestal (one vertical Y and one horizontal X) were used to collect the vibration signals. One eddy current transducer installed near the couping was used as tachometer to detect the shaft rotational speed. The testing system includes an INV306U-5160 type intelligent signal acquisition instrument and a DLF–4 type multi-channel charge amplifier. Experiments were performed on four 6038 different ball bearings. One was normal and the other three ball bearings had an artificially added local flake fault on individual outer race (located in the middle of the race, about 7 mm2, 0.2 mm deep), inner race (located in the middle of the race, about 3.6 mm2, 0.1 mm deep) and rolling element (spot damage). The structure parameters of the rolling bearing are shown in Table 1. Table 1. The structure parameters of the rolling bearing
Bearing
outer race
inner race
pitch
number
ball
tapered
of balls
diameter
contact angle
8
15 mm
0°
width model
diameter
diameter
diameter
6308
90 mm
40 mm
65 mm
23 mm
The shaft rotational speed was kept constantly at 1620RPM, so that the rotational frequency f r maintained 27Hz. The data sampling frequency was 20kHz and the sampling length was 8192 points. It is well known that the fundamental fault frequency of a defective ball bearing can be calculated based on ball bearing geometry. The ball pass frequency on inner race (BPFO) f i , the ball pass frequency on outer race (BPFO) f o , and the ball spin frequency BSF f b are respectively given by Eq. (10), (11) and (12) according to the fundamental equations of
motions proposed by Patel et al. [25].
fo =
Nb 8 15 d cos a (1 ) f r = ´ (1 - ) ´ 27 = 83Hz 2 2 65 D
(10)
fi =
Nb 8 15 d cos a (1 + ) f r = ´ (1 + ) ´ 27 = 133Hz 2 2 65 D
(11)
fb =
D d 65 15 (1 - ( ) 2 cos 2 a ) f r = ´ (1 - ( ) 2 ) ´ 27 = 55.4Hz 2d D 2 ´15 65
(12)
Where, D is pitch diameter, d is ball diameter, N b is number of balls and a is tapered contact angle. The fault characteristic of the outer race is more obvious for short transmission path and it is relatively easier to diagnose than other faulty type. So in this paper, the diagnosis process is only adopted in the faults of inner race and rolling element.
5.2 Inner Race Fault Diagnosis The time domain waveform, frequency spectrum and its Hilbert envelope spectrum of the bearing vibration
signal with inner race fault are shown in Fig. 4 (a), (b) and (c). There are some impacting ingredients with apparent periodicity can be observed from its time domain waveform. The frequency spectrum is very complex and the useful fault information is hidden in the strong noisy background. Due to the effect of modulation, the signal frequency energy mainly concentrated in the vicinity of 4000 Hz. Hilbert transform (HT) is applied on the vibration signal for demodulation analysis. There are three obvious spectrum peaks can be seen in Fig. 4 (c): 159 Hz, 237 Hz and 502 Hz, which correspond with the values of f i + f r , 4 fi - f r and 2 fi - f r . However, the
inner race fault frequency f i can’t be extracted directly only through Hilbert demodulation analysis. With the influence of environmental noise, signal energy decay and the modulation of the rotational frequency, the signal fault feature is getting relatively weak and submerged in the background noise.
A/mv
a
t/s
A/mv
b
f/Hz
A/mv
c +] +]
+]
f/Hz
Fig. 4. The inner race fault vibration signal: (a) time domain waveform (b) frequency spectrum (c) envelope spectrum In order to verify the effectiveness of the proposed method, the inner race fault vibration signal of the rolling bearing is first decomposed by RSSD. According to the signal processing process shown in Fig.2, the kurtosis maximum of the low resonance component is taken as the objective function and the quality factor is optimized with GA. Simultaneously, the sampling frequency and the sub-band coverage basic theory considered, the quality
factor and other parameters are selected as Q1=3.75ˈQ2=1.06ˈr1=r2=3.5ˈL1=35ˈL2=15. The high resonance component decomposed by RSSD of the vibration signal is shown in Fig. 5 (a), which mainly contains the harmonic components. The low resonance component is shown in Fig. 5 (b). It can be observed that the transient impact component in the signal is relatively obvious.
A/mv
D
t/s
A/mv
E
t/s
Fig. 5. Inner race fault vibration signal RSSD result: (a) high resonance component (b) low resonance component The sub-band energy distribution of the low resonance component can be calculated with Eq. (8) and shown in Fig. 6. It can be seen that the energy of sub-band 2, 5, 6, 7, 16 is relatively dominant and satisfies the sub-band selection condition specified in Eq. (9). It means that these sub-bands contain the main fault impact information, thus they are selected to reconstruct the low resonant signal. SUBBAND ENERGY (% OF TOTAL)
DISTRIBUTION OF SIGNAL ENERGY 15
10
5
0
0
2
4
6
8 SUBBAND
10
12
14
16
Fig. 6. Sub-band energy distribution for low resonance components
A/mv
D
t/s
A/mv
E
f/Hz A/mv
F
f/Hz G
fi-fr
fr
fi fi+fr
A/mv
2fr
3fr
f/Hz Fig. 7. Low resonance component master sub-band reconstruction signal: (a) time domain waveform (b) frequency spectrum (c) the frequency spectrum of the first layer detail after WT (d) the envelope spectrum of the first layer detail after WT The sub-band reconstruction signal of the low resonance component is shown in Fig. 7(a), in which the impact characteristics are more obvious. The FFT transform is directly applied on the master sub-band reconstruction signal and its frequency spectrum is shown in Fig. 7(b). The frequency spectrum is very complex and the fault information can’t be acquired from it. In order to further extract the local time-frequency feature of the reconstructed signal, WT is introduced. Daubechies family of wavelet packets seems to resemble the bearing vibration signal most and db 10 is adopted as the mother wavelet. The reconstructed signal is decomposed with db10 wavelet into 3 layers. Fig. 7 (c) and (d) show the frequency spectrum and the envelope spectrum of the first layer detail signal respectively. They are very similar, which means that they have the similar analysis results. Because the envelope spectrum amplitude is larger and has clear demodulation meaning, the detailed analysis is
implemented on Fig. 7 (d). From it, rotational frequency accompanied by its harmonics ( f r , 2 f r and 3 f r ), the
inner race fault characteristic frequency f i and its modulation frequency ( f i ± f r ) can be seen clearly. According to the rolling bearing vibration theory, these frequency components coincide with the characteristics of the inner race fault. The weak fault feature is successfully extracted out by the proposed method. For comparison, the low resonance component without parameters optimization, sub-band reconstruction and WT are also analyzed. Fig. 8 is the envelope spectrum of the low resonance component obtained by regular RSSD. The inner race fault characteristic frequency f i and its modulation frequency ( f i - f r , 2 fi - f r ) can be observed, but the amplitude is not dominant. The fault feature recognition effect is poorer than the proposed
A/mv
method.
fr
fi-fr fi
2fi-fr
f/Hz Fig. 8. The envelope spectrum of the low resonance component obtained by regular RSSD method WT and EMD are two most frequently used time-frequency signal processing methods in the area of fault diagnosis of bearing up to now. The comparative analysis results of WT and EMD methods are also given out. WPD is employed to decompose the inner race fault vibration signal into 3 layers and the result is shown in Fig. 9. As before, db10 is adopted as the mother wavelet. According to WPT theory, there are total 8 frequency band 1
components ( P3
P38 ) from low to high frequency will be obtained, where superscript represents the frequency
band number and the subscript represents decomposition layer number. The frequency spectrum analysis 1 corresponding to P3
P38 is shown in Fig. 9(b). The inner race fault characteristic frequency components are
mainly distributed in the first frequency band and its details locating in the range of 0-1200Hz is shown in Fig. 10. Similar to Fig. 4(c), there are three obvious spectrum peaks (159 Hz, 237 Hz and 502 Hz) corresponding to the values of f i + f r , 4 fi - f r and 2 fi - f r can be clearly observed. However, the inner race fault frequency
f i can’t be extracted. The fault feature recognition effect of WT is poorer than the proposed method.
P P
P
A/mv
P31 P32 P33 P34
A/mv
5 3 6 3 7 3
P38
5 0 -5 5 0 -5 10 0 -10 10 0 -10 5 0 -5 5 0 -5 5 0 -5 5 0 -5
0
0.1
0.2
0.3
0.4
0.2 0.1 0 0.1 0.05 0 0.4 0.2 0 0.4 0.2 0 0.2 0.1 0 0.1 0.05 0 0.1 0.05 0 0.1 0.05 0
0
2000
4000
6000
8000
10000
f/Hz
t/s (a)
(b)
Fig. 9. The WPD result of the Inner race fault vibration signal: (a) time domain waveform (b) frequency spectrum 0.2
+]
A/mv
0.15
+]
+]
0.1 0.05 0
0
200
400
600
800
1000
1200
f/Hz Fig. 10. The detailed frequency spectrum of the first WPD frequency band The EMD method is also applied to analyze the above vibration acceleration signal. As shown in Fig. 11, there are a total of 11 IMFs components and 1 residue obtained. Considering the apparent periodical impulses existing in IMF1, the HT is applied to extract the modulation frequency and the Hilbert spectrum is shown in Fig. 12. By comparison with Fig 7(d), it can be found that the fault feature extraction ability is similar to the proposed method. Except rotational frequency f r and its harmonics ( 2 f r , 3 f r ), the inner race fault characteristic frequency f i and its modulation frequency ( f i ± f r ) also can be found in Fig. 12. With the increase of frequency, the harmonicity of the Hilbert spectrum decreases.
IMF1
IMF2 IMF3 IMF4 IMF5 IMF6 IMF7
RES
IMF11
IMF10
IMF9
IMF8
A/mv
10 0 -10 5 0 -5 5 0 -5 2 0 -2 2 0 -2 1 0 -1 0.5 0 -0.5 0.5 0 -0.5 0.2 0 -0.2 0.2 0 -0.2 0.1 0 -0.1 -0.005 -0.01 -0.015
0
0.1
0.2
0.3
0.4
t/s Fig. 11 The EMD result of the inner race fault vibration signal 0.8
fi
fr
0.8
0.6
0.6
0.4
fi-fr
A/mv
3fr fi+fr
0.2
0.4
0
0.2 0
2fr
0
200
400
600
0
50
100
800
150
1000
200
250
300
350
1200
f/Hz Fig. 12 The Hilbert spectrum of IMF1
5.3 Rolling Element Fault Diagnosis Fig. 13 is the rolling element fault vibration signal of the rolling bearing. Some non-periodic impulse components exist in the time domain waveform. Similar to the inner race vibration signal, the Hilbert demodulation analysis is adopted on the rolling element fault vibration signal and its envelope spectrum is shown
400
in Fig. 13(b). There are some obvious spectrum peaks located on the frequency of 181Hz, 195Hz, 327Hz and 535Hz. However, the rolling element fault characteristic frequency f b and its harmonics can’t be observed. Due
to the influence of the complicated transmission path, the useful fault information of the signal is hidden in the strong noisy background.
A/mv
a
t/s b
A/mv
535Hz 181Hz
195Hz 327Hz
f/Hz
Fig. 13. The rolling element fault vibration signal and its envelope spectrum: (a) time domain waveform (b) envelope spectrum Subsequently, the proposed method is also applied to analyze the rolling element fault vibration signal. The GA is first used to optimize the quality factor and other parameters of RSSD in order to maximize the kurtosis maximum of the low resonance component. The obtained parameters of RSSD are listed as Q1=4.92ˈQ2=1.13ˈ r1=r2=3.5ˈL1=40ˈL2=15. The high and low resonant components of the signal after decomposition are shown in Fig. 14 (a) and (b).
A/mv
a
t/s
A/mv
b
t/s
Fig. 14. The RSSD result rolling of element fault vibration signal: (a) High resonance component (b) Low resonance component Then the sub-band energy distribution of the low resonance component is calculated with Eq. (8) and shown in Fig. 15. The energy of the sub-bands 1, 2, and 16 is relatively dominant and satisfies the sub-band selection condition. These sub-bands are selected as master sub-bands to reconstruct the low resonance component. Fig. 16(a) is the time domain waveform of the reconstructed low resonance component. It can be observed that the fault impact component is greatly highlighted. Similarly, the reconstructed signal is decomposed with db10 wavelet into 3 layers. The envelope spectrum of the first-order detail is presented in Fig. 16(b) and the second, fourth and sixth harmonic ( 2 f b , 4 f b and 6 f b ) of the fault characteristic frequency can be apparently seen. The
SUBBAND ENERGY (% OF TOTAL)
extracted frequency components coincide with the fault type of the rolling bearing.
DISTRIBUTION OF SIGNAL ENERGY 50 40 30 20 10 0
0
2
4
6
8 10 SUBBAND
12
14
Fig. 15. Sub-band energy distribution for low resonance components
16
A/mv
a
t/s
A/mv
b 2fb 68Hz
149Hz
4fb 6fb
f/Hz Fig. 16. Low resonance component master sub-band reconstruction signal: (a) Time domain waveform (b) the envelope spectrum of the first layer detail after WT For comparison, the envelope spectrum analysis is also applied to the low resonance component obtained by regular RSSD and the result is shown in Fig. 17. Some obvious spectrum peaks located on the frequency of 181Hz, 195Hz, 327Hz and 535Hz can be seen. Among them, except that the frequency 195 Hz and 327 Hz correspond with 2 f b + f r and 6 f b respectively, the rest frequency has nothing to do with the fault characteristic frequency.
A/mv
The fault feature recognition effect is poorer than the proposed method.
535Hz 327Hz 181Hz 195Hz
f/Hz Fig. 17. The envelope spectrum of the low resonance component obtained by regular RSSD method Similar with the previous analysis to the inner race fault vibration signal, WT and EMD are also employed to process the rolling element fault vibration signals. The time domain waveforms and their corresponding frequency 1
spectrums of the 8 frequency band components ( P3
P38 ) obtained by WPD method are shown in Fig. 18. Db10
wavelet is also taken as the mother wavelet and the decompose layer number is 3. The detailed spectrum locating in the range of 0-1200Hz of the first frequency band component is shown in Fig. 19. It has similar analysis result
to Fig. 13(b), there are four obvious spectrum peaks located on the frequency of 181Hz, 195Hz, 327Hz and 535Hz. However, the rolling element fault characteristic frequency f b and its harmonics can’t be observed. It means that
10 0 -10 10 0 -10 20 0 -20 20 0 -20 5 0 -5 5 0 -5 5 0 -5 10 0 -10
1 0.5 0 0.4 0.2 0 0.5
A/mv
P34 P38
P37
P36
P35
A/mv
P33
P32
P31
the fault feature recognition effect of WT is poorer than the proposed method.
0
0.1
0.2
0.3
0.4
0 1 0.5 0 0.2 0.1 0 0.2 0.1 0 0.1 0.05 0 0.2 0.1 0
0
2000
4000
6000
8000
10000
f/Hz
t/s
Fig. 18. The WPD result of the rolling element fault vibration signal (a) time domain waveform (b) frequency spectrum
A/mv
0.8 0.6
535Hz
0.4
181Hz 195Hz 327Hz
0.2 0
0
200
400
600
800
1000
1200
f/Hz Fig. 19. The detailed frequency spectrum of the first WPD frequency band The EMD result of the rolling element fault vibration signal is shown in Fig. 20, which includes 11 IMFs components and 1 residue. The HT is also applied on IMF1 to extract the modulation frequency and the Hilbert spectrum is shown in Fig. 21. By comparison with Fig 16(b), it can be found that the fault feature extraction ability is similar to the proposed method. Except 149Hz, the second and the fourth harmonic ( 2 f b , 4 f b ) of the rolling element fault characteristic frequency can be seen, which coincide with the fault type of the rolling bearing. But the amplitudes are not dominant and smaller than that of 16(b). It means that the fault feature recognition effect of EMD method is poorer than the proposed method.
IMF1
IMF2 IMF3 IMF4 IMF5 IMF6
A/mv
IMF7 IMF8 IMF9 IMF10
RES
IMF11
20 0 -20 10 0 -10 10 0 -10 5 0 -5 2 0 -2 2 0 -2 0.5 0 -0.5 0.5 0 -0.5 0.2 0 -0.2 0.1 0 -0.1 0.2 0 -0.2 1.6 1.4 0
0.1
0.2
0.3
0.4
t/s Fig. 20 The EMD result of the rolling element fault vibration signal 1.5
A/mv
149Hz
2fb
1
4fb
0.5
0
0
100
200
300
400
500
600
f/Hz Fig. 21 The Hilbert spectrum of IMF1
6 Conclusion The present study proposes a new kind of fault diagnosis method integrating RSSD with WT. The method includes parameters optimization, RSSD, master sub band selection and reconstruction, WT and Hilbert demodulation analysis. Through two fault diagnosis case study of rolling bearing, the conclusion can be
drawn as follows. (1) During the incipient failure of rolling bearings, the characteristic of the fault signal is relatively weak and vulnerable to environmental noise and signal energy decay. If the signal is merely analyzed by RSSD, the impact fault features are not evident and can’t be extracted. The RSSD method combined with parameter optimization and sub-band reconstruction can effectively highlight the weak impact fault characteristics. (2) The RSSD method can decompose complex signals into a high resonance component containing continuous oscillating components and a low resonance component containing transient impacting ingredients according to the signal resonance properties. The WT signal processing method has excellent local time-frequency multiresolution resolution ability, which can decompose the signal in a multiscale way from the frequency domain perspective. The fault information can be deeply mined from two aspects of signal resonance attribute and local time-frequency characteristic analysis based on the fusion of RSSD and WT technique. (3) The effectiveness of the method is proved by two diagnosis cases to early different type rolling bearing faults. The results show that the proposed method is better than regular RSSD method in the weak fault feature highlighting. The fault feature recognition ability of the proposed method is prior to WPD method and similar or prior to EMD method for different bearing fault signals.
Acknowledgments This program is jointly supported by the projects of National Nature Science Foundation of China (51775307, 51605255), Natural Science Foundation of Hubei Province of China (2018CFB671) and Hubei Key Laboratory of Hydroelectric Machinery Design & Maintenance (No.2016KJX15 & No.2016KJX02).
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S0263-2241(18)30639-0 https://doi.org/10.1016/j.measurement.2018.07.043 MEASUR 5727
To appear in:
Measurement
Received Date: Revised Date: Accepted Date:
4 March 2018 14 July 2018 16 July 2018
Please cite this article as: C. Baojia, S. Baojia, C. Fafa, T. Hongliang, X. Wenrong, F. Zhang, C. Zhao, Fault diagnosis method based on integration of RSSD and wavelet transform to rolling bearing, Measurement (2018), doi: https:// doi.org/10.1016/j.measurement.2018.07.043
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Fault diagnosis method based on integration of RSSD and wavelet transform to rolling bearing Chen Baojiaa,b, Shen Baojiaa, Chen Fafaa*, Tian Honglianga, Xiao Wenronga,b, Fajun Zhanga,b, Chunhua Zhaoa,b a
Hubei Key Laboratory of Hydroelectric Machinery Design & Maintenance, China Three Gorges University, Yichang 443002, Hubei, China;
b
College of Mechanical and Power Engineering, China Three Gorges University, Yichang 443002,Hubei,China
Abstract: To solve the problem of early fault diagnosis of rolling bearing under strong background noise, a fault diagnosis method based on integration of Resonance-based Sparse Signal Decomposition (RSSD) and Wavelet Transform (WT) is proposed in this paper. The RSSD method is combined with quality factor optimization using genetic algorithm and sub-band reconstruction. Firstly, the early fault vibration signal of the rolling bearing is decomposed by RSSD. The kurtosis value of the low resonance component is taken as the objective function to optimize the combination of high and low quality factors with genetic algorithm. Then, the master sub-band is selected out to reconstruct the low resonance component based on the principle of energy dominant distribution. It can reduce the noise interference and enhance the impulse characteristic of the fault signal. Finally, characteristics of local optimization and multi-resolution of wavelet analysis considered, the multi-scale wavelet decomposition is applied to the reconstructed low resonance component to extract the fault features of the bearing failure deeply. The effectiveness and application value of the method are proved by two different diagnosis cases of rolling bearing faults. By comparisons, the fault feature extraction ability of the proposed method is prior to WPD method and similar or prior to EMD method for different bearing fault signals. Key words: quality factor; sub-band reconstruction; rolling bearing; resonance-based sparse signal decomposition; wavelet transform
1 Introduction The vibration characteristics of the rolling bearing faults are mainly determined by the rotation of the shaft, the bearing structure, the damage form, the transmission path between the bearing and the system etc [1]. When incipient failure of the rolling bearing emerges, the impulse force caused by the damage will induce the high
*
Corresponding author.
E-mail address: [email protected] (F. Chen).
frequency natural vibration of the bearing system, which is also modulated by the periodic impact force [2,3]. The vibration signal mainly contains the harmonic component, the fault impulse component relating to the revolutions and the background noise. With the development of the faults, the dynamic vibration signals of the rolling bearing often demonstrate nonlinearity and non-stationarity [4]. Some components may have the similar center frequency and overlapping frequency bands. Therefore, the key issue for rolling bearing faults diagnosis is to isolate and highlight the fault impacting ingredients from the original vibration signals to reduce the interference of harmonics and noise. The application prerequisite of the Fourier Transform (FT) is that the signals can be expressed as a linear superposition of the stationary sine waves with different frequencies. It lacks an accurate description of the local time-frequency variation presentation of the signal. In recent years, the nonlinear and nonstationary signal processing has been a research hotspot, especially in the field of mechanical fault diagnosis. In order to extract fault features from the transient weak signal under the strong background noise, Yan [5] proposed a new wind turbine gearbox fault diagnosis method based on continuous wavelet transform (WT) and stationary subspace analysis. Zheng [6] proposed an improved Empirical Mode Decomposition (EMD) fault diagnosis method, which contained Generalized Empirical Mode Decomposition (GEMD) and Improved Direct Quadrature (IDQ) demodulation. Several different Intrinsic Mode Functions (IMF) are obtained by defining different mean curves and in each rank the best IMF is selected. The improved empirical AM-FM decomposition and IDQ are used to demodulate the component. Wang and Chen [7,8] analyzed the filter characteristics and noise effects of the Local Mean Decomposition (LMD) method respectively and gave out the SNR (signal to noise ratio) threshold of LMD for signal waveform and instantaneous frequency extraction. Then LMD was applied to the fault diagnosis of rotor robbing and locomotive bogie. A new adaptive stochastic resonance method was developed by Xiang [9] using an artificial bee colony algorithm and stochastic resonance theory. In order to obtain the maximum stochastic resonance output SNR, the structural parameters of the system have been adaptively optimized by an artificial bee colony algorithm to realize the adaptive adjustment of the bistable system parameters. The above signal processing methods decompose the signal into multiple frequency bands by analytical expressions or self-adaptive methods. Whereafter the feature extraction methods are adopted on the specific frequency band to get some fault characteristic quantities so as to implement diagnosis procedure. If the center frequency of the harmonic components, fault impulse components and the background noise superpose together, the multi-components of the signals can’t be separated out by general frequency band decomposition or linear filtering methods. Whereas the signals can be decomposed based on their resonance properties (different quality
factor Q), rather than on frequency or scale, as provided by FT and WTs. In 2011, Selesnick [10] proposed a new signal processing method, namely, resonance-based sparse signal decomposition (RSSD). The RSSD method can be used to separate a complex signals into a high-resonance component consisting of multiple simultaneous sustained oscillations and a low-resonance component consisting of non-oscillatory transients of unspecified shape and duration [11]. In [12,13], RSSD was successfully introduced to the fault diagnosis of bearing and gearbox. In order to detect multiple faults in gearboxes, Zhang [14] proposed a novel method called resonance-based sparse signal decomposition with comb filter (RSSD-CF) and the stepwise optimization strategy was applied for parameter optimization of the RSSD method.
In most existing RSSD-based fault diagnosis methods, the analyzing procedure is only adopted on the decomposed low-resonance component. It is thought that the low-resonance components are composed of transient shock components. Due to the noise interference, the components with different resonance properties are considered to have great coupling effect. It is difficult to observe the characteristic frequency of the fault rolling bearing when it is not dominant, especially for incipient failure. In addition, the information mining capability of the RSSD method for transient impulse components is inadequate for the reason that it does not have multi-scale band analysis capability similar to WT, EMD or LMD. On the basis of the above analysis, this paper presents a fault diagnosis method based on the integration of RSSD and WT. The quality factor optimization with genetic algorithm (GA) and sub-band reconstruction are introduced to improve the RSSD method. The kurtosis value of the low resonance component is taken as the objective function to optimize the combination of high and low quality factors with GA. The master sub-band is selected to reconstruct the low resonance component based on the principle of energy dominant distribution, which can reduce the noise interference and enhance the impulse characteristic of the fault signal. Characteristics of local optimization and multiresolution of wavelet analysis considered, the multi-scale wavelet decomposition is applied to the reconstructed low resonance component to extract the fault features of the bearing failure deeply. This paper is organized as follows: Section 2 introduces the RSSD method briefly. Section 3 focuses on the details of quality factor optimization and sub-band reconstruction. Section 4 discusses the diagnostic process, which integrates the RSSD method with the wavelet analysis technique. Section 5 deals with the experimental verification of the proposed method and gives out the comparative analysis results with Wavelet Packet Decomposition (WPD) and EMD. The conclusion of this study is formulated in the last section.
2 Resonance-based Sparse Signal Decomposition The RSSD method is actually a sparse representation jointly using the high and the low Tunable Q-Factor
Wavelet Transform (TQWT). The key issue of RSSD is to construct the tunable wavelet. In the early studies, Selesnick[11] explored Rational-Dilation Wavelet Transform (RADWT), which can attain a range of redundancies and Q-factors making the transform more flexible than the dyadic wavelet transform. The tunable range of Q-factor can be extended by adjusting the sampling size q to improve frequency resolution. In contrast, the TQWT is simpler conceptually, which can be more efficiently implemented using radix-2 Fast Fourier Transformation (FFT), and its parameters are more easily related to the Q-factor of the transform.
2.1 Tunable Q-Factor Wavelet Transform The TQWT is fully discrete, has the perfect reconstruction property, is modestly overcomplete, is developed in terms of iterated two-channel filter banks, and implemented using the Discrete Fourier Transform (DFT) based on the resonant nature of the signal [12]. The tunable-Q wavelet transform is based on the multirate filter bank illustrated in Fig. 1.
H 0 ( w)
LPS a
v0 (n)
LPS
1
a
H 0* ( w)
y0 ( n )
x ( n)
y ( n)
H1 ( w)
HPS b
HPS
1
v1 (n)
b
H1* ( w)
y1 (n)
Fig. 1. Analysis and synthesis filter banks for the tunable-Q wavelet transform. *
In Fig. 1, x(n) is the original signal, H i = 0,1 ( w) , Hi =0,1 (w) are the frequency response functions of the analysis and synthesis filters respectively. LPS and HPS represent low-pass scaling and high-pass scaling respectively. vi = 0,1 (n) are the sub-band signals obtained after decomposition. yi = 0,1 ( n ) are the synthetic signals. The scaling parameters satisfy
2 ì ïb = Q + 1 ï b ï ía = 1 r ï ïs.t. 0 < a , b £ 1 ï î
(1)
a + b >1
where ݎrepresents the redundancy rate. Once the filter bank parameters ߙ and ߚ are determined, the
values of the Q-factor and ݎcan be calculated. The essence of the multilayer decomposition process is that
the signal passes through sequentially from a high frequency band-pass filter to a low frequency band-pass filter. It is a successive iteration procedure. The RSSD decomposes the signal into components with different resonance properties. Suppose that the observed signal x can be represented as
x = x1 + x2 + n
(2)
where ݔ1 and ݔ2 are components with different oscillation behavior, and n is the noise. To decompose the
analysis signal, the Morphological Component Analysis (MCA) is applied in the RSSD method [10,16]. The goal of MCA is to estimate/determine ݔ1 and ݔ2 individually. Assuming that the signals ݔ1 and ݔ2 can be sparsely represented in base functions (or frames) ܵ1 and ܵ2 respectively, (obtained by TQWT), they can be estimated by
minimizing the objective function
J˄W1 ,W2 ) = x - S1W1 - S 2W2
2 2
+ l1 W1 1 + l2 W2
1
(3)
where ܹ1 and ܹ2 denote the transform coefficients of the signals ݔ1 and ݔ2. ܵ1 and ܵ2 represent the inverse
TQWT with the high and the low Q-factors respectively. ߣ1 and ߣ2 are the regularization parameters of the high
and low resonance components. The sparse signal decomposition method based on compound Q-factor bases uses the Split Augmented Lagrangian Searching Algorithm (SALSA). It minimizes the target function , by iteratively
updating the transform coefficients ܹ1 and ܹ2. Eventually, it effectively separates the high resonance component
and low resonance component and extracts the transient impulse component. After ܹ1 and ܹ2 are obtained, the estimated high and low resonance components (solved by matching) are shown as follows
xˆ1 = S1W1 ,
xˆ2 = S2W2
(4)
2.2 Selection and Influence of TQWT Parameters It is important to adaptively select the parameters of RSSD for purpose of feature extraction of the analysis signal because those parameters have a great effect on the decomposition results. 7he Q-factor of a wavelet function can be adaptively designed and selected according to the oscillations of the analyzed signal. The high Q-factor base is used to match the high resonance component (the random vibration and strong noise of the rolling bearing itself), while the low Q-factor base is used to match the low resonance component (fault impulse component). Different Q-factor represents different time-frequency resolution of the waveform function and overlap of the adjacent two sub-bands. So the choices of Q-factor values greatly affect the correlation (degree of coupling) of the high and low resonance components. As is shown in Eq. (5), the literature [10] gives the correlation analysis expression of the quality factors Q1 and Q2.
rmax (Q1 , Q2 ) =
Q2 + 1/ 2 £1 Q1 + 1/ 2
(5)
It is assumed that Q1> Q2. If Q2 = 1 , increasing Q1 can reduce the correlation and improve the resolution. The larger the difference between Q1 and Q2 is, the lower the correlation is. However, the quality factor is too high,
and its wavelet base function may not match the oscillation characteristic of high resonance signal, producing the singular signal and reducing the accuracy and reliability of RSSD. It is necessary to choose the reasonable quality factors Q1 and Q2 to obtain a better decomposition effect. The redundancy factor r is generally between 3.0 and 4.0 in practice. With r ³ 3 , the pass-band of the level-
frequency response will not have a ‘flat top’ (wherein the frequency response is equal to a constant over a sub-interval of its pass-band) as discussed in [17-19]. The decomposition level L determines the number of elements constructed by the wavelet basis function, which affects the center frequency and the bandwidth of each layer sub-band. With the increase of L, the center frequency and bandwidth of each sub-band are getting smaller. The relationship between L, Q and r is shown in Eq. (6).
L=
log( N / 4(Q + 1)) log(Q + 1 / (Q + 1 - 2 / r )
(6)
N represents the signal length. In practical application, the choice of parameter L needs to be considered to ensure that the decomposed sub-band frequency covers the resonance frequency band of each component.
3 Q-Factor Optimization and Sub-band Reconstruction 3.1 Q-Factor Optimization Based on Signal Kurtosis Index In traditional RSSD method, the Q-factors are usually assigned integer as Q2 = 1 and Q2 = 3 ~ 9 . The randomness of the choice determines the quality of decomposition. If the signal that carries the fault information is decomposed into components with specific resonance attribute, a wavelet-based function library that matches the fault signal oscillation characteristics needs to be constructed. Therefore, how to adaptively select quality factors is the key problem in the RSSD method. It's important to note that the Q-factor in the actual application could be non-integer. The kurtosis index K v can be used to describe the degree of the signal deviates from the normal distribution, which is calculated based on Eq. (7).
1 Kv =
N
Nå i =1
( xi - u x ) 4
é1 N 2ù ê N å ( xi - u x ) ú i = 1 ë û
2
(7)
where ux is the mean of the signal and N is the signal length. When the mechanical equipment is working under normal condition, the vibration amplitudes obey normal distribution, K v » 3 . As the failure degree increases, the absolute value of K v gets larger. Therefore, the kurtosis of the decomposed low resonance component is taken as the optimization objective function in this study to select the proper Q-factors. As widely
used in some literatures [20–22], GA is applied in this paper to optimize Q1 and Q2. The changing precision of the Q-factors is 0.01 and the decimal code is adopted. The main parameters of GA are set as follows: l
population size 50,
l
number of generations 200,
l
genetic rate 0.1,
l
crossover rate 0.7,
l
mutation rate 0.02.
The specific steps of GA can be referred to the literature [23]. According to the combination of quality factors obtained by GA, the kurtosis of the low resonance component in theory is the largest, and the fault information is the most. It can be deduced theoretically that the combination of Q-factors obtained by GA optimization corresponds with the low resonance component with the largest kurtosis. It contains more fault information, especially for impulse component.
3.2 Sub-band Reconstruction In the strong background noise working environment, the high and low resonance components of the rolling bearing fault signal decomposed by RSSD have a high coupling degree due to noise interference. It may weaken the impact component in the low resonance component. In order to highlight the rolling bearing failure characteristics, it is necessary to reconstruct the sub-bands selectively of the low resonance components. In [24], the low-resonance component was further decomposed into a set of sub-signals by the TQWT and the proper signal was reconstructed from some selected sub-signals combined with kurtosis analysis to extract the fault characteristics of the rolling bearing. However, the reconstruction method cumulated sub-band components only from high frequency to low frequency according to the filtering characteristic of the Q-factors. The relationships between the sub-band frequency and their proportions in the signal energy are not in consideration. It is inevitable that the noise on some sub-bands should be introduced to the reconstructed signal. In this paper, the sub-band proportion to total energy-based reconstruction method is considered. The detail process is listed as follows: (1) Adopt TQWT on the original signal to obtain the high and low resonance components. Then calculate the energy Ej of L+1 sub-band signals and its percentage ej in the total energy respectively. The calculation equations are shown as
Nj ì 2 ï E j = å W ji i ïï í Ej ´ 100% J +1 ïe j = E ï å j ïî j =1
(8)
where Wji represents the ith wavelet coefficient lies in the jth level and Nj is the length of the jth level wavelet coefficient. (2) Select out the dominant sub-band components, whose energy percentage ej is greater than 1.5 times sub-band average energy, as shown in Eq. (9).
e j ³ 1.5 / ( L + 1) ´ %
(9)
(3) Superimpose the sub-band components together linearly to reconstruct a new low resonance component.
4 Diagnosis Process The RSSD method decomposes the signal according to the resonance properties of the different components rather than divide the signal based on the frequency band. WT is able to analyze localization and multiresolution precisely, which can expand the signal in multi-scale. If the RSSD method is incorporated with the WT, they will complement each other in two aspects of signal resonance attribute and local time-frequency characteristic analysis. The new proposed method is suitable for information deep mining and incipient faults diagnosing to the rolling bearing. The specific diagnostic process is shown in Fig. 2 and the detail is introduced as follows: (1) Through the experiment, the original vibration signals of the rolling bearing fault are obtained. The signals are decomposed by the combination of the quality factor obtained by GA optimization according to the kurtosis principle of the low resonance component described in section 3.1. (2) Sub-band selection and reconstruction are applied to the low resonance components according to the method described in Section 3.2. (3) Wavelet multi-scale decomposition is used to analyze the new reconstructed low resonance component. Finally, the Hilbert demodulation method is used to extract the signal envelope spectrum and obtain the bearing fault characteristics.
Fig. 2. Diagnosis flow chart
5 Experimental Case Study In order to demonstrate the performance of proposed method, this section presents two application examples for the detection of localized defects from the real vibration signals acquired through the rolling bearing fault experiment. In addition, the effectiveness of Hilbert demodulation and regular RSSD method are compared in this section.
5.1 Description of the Experiment 1¾DC drive motor 2¾Pulley 3¾Bearing pedestal 4¾Support housing 5¾Coupling 6¾DC load motor 7¾Vibration acceleration sensors 8¾tachometer
Fig. 3. Rolling bearing failure testing platform
The structure of the rolling bearing fault testing platform and its picture are presented in Fig. 3. The experiment system is composed of a DC drive motor, pulley, bearing pedestal, support housing, coupling, a DC load motor and other accessories. In the experiment, two CA-YD-106 piezoelectric acceleration sensors positioned on the bearing pedestal (one vertical Y and one horizontal X) were used to collect the vibration signals. One eddy current transducer installed near the couping was used as tachometer to detect the shaft rotational speed. The testing system includes an INV306U-5160 type intelligent signal acquisition instrument and a DLF–4 type multi-channel charge amplifier. Experiments were performed on four 6038 different ball bearings. One was normal and the other three ball bearings had an artificially added local flake fault on individual outer race (located in the middle of the race, about 7 mm2, 0.2 mm deep), inner race (located in the middle of the race, about 3.6 mm2, 0.1 mm deep) and rolling element (spot damage). The structure parameters of the rolling bearing are shown in Table 1. Table 1. The structure parameters of the rolling bearing
Bearing
outer race
inner race
pitch
number
ball
tapered
of balls
diameter
contact angle
8
15 mm
0°
width model
diameter
diameter
diameter
6308
90 mm
40 mm
65 mm
23 mm
The shaft rotational speed was kept constantly at 1620RPM, so that the rotational frequency f r maintained 27Hz. The data sampling frequency was 20kHz and the sampling length was 8192 points. It is well known that the fundamental fault frequency of a defective ball bearing can be calculated based on ball bearing geometry. The ball pass frequency on inner race (BPFO) f i , the ball pass frequency on outer race (BPFO) f o , and the ball spin frequency BSF f b are respectively given by Eq. (10), (11) and (12) according to the fundamental equations of
motions proposed by Patel et al. [25].
fo =
Nb 8 15 d cos a (1 ) f r = ´ (1 - ) ´ 27 = 83Hz 2 2 65 D
(10)
fi =
Nb 8 15 d cos a (1 + ) f r = ´ (1 + ) ´ 27 = 133Hz 2 2 65 D
(11)
fb =
D d 65 15 (1 - ( ) 2 cos 2 a ) f r = ´ (1 - ( ) 2 ) ´ 27 = 55.4Hz 2d D 2 ´15 65
(12)
Where, D is pitch diameter, d is ball diameter, N b is number of balls and a is tapered contact angle. The fault characteristic of the outer race is more obvious for short transmission path and it is relatively easier to diagnose than other faulty type. So in this paper, the diagnosis process is only adopted in the faults of inner race and rolling element.
5.2 Inner Race Fault Diagnosis The time domain waveform, frequency spectrum and its Hilbert envelope spectrum of the bearing vibration
signal with inner race fault are shown in Fig. 4 (a), (b) and (c). There are some impacting ingredients with apparent periodicity can be observed from its time domain waveform. The frequency spectrum is very complex and the useful fault information is hidden in the strong noisy background. Due to the effect of modulation, the signal frequency energy mainly concentrated in the vicinity of 4000 Hz. Hilbert transform (HT) is applied on the vibration signal for demodulation analysis. There are three obvious spectrum peaks can be seen in Fig. 4 (c): 159 Hz, 237 Hz and 502 Hz, which correspond with the values of f i + f r , 4 fi - f r and 2 fi - f r . However, the
inner race fault frequency f i can’t be extracted directly only through Hilbert demodulation analysis. With the influence of environmental noise, signal energy decay and the modulation of the rotational frequency, the signal fault feature is getting relatively weak and submerged in the background noise.
A/mv
a
t/s
A/mv
b
f/Hz
A/mv
c +] +]
+]
f/Hz
Fig. 4. The inner race fault vibration signal: (a) time domain waveform (b) frequency spectrum (c) envelope spectrum In order to verify the effectiveness of the proposed method, the inner race fault vibration signal of the rolling bearing is first decomposed by RSSD. According to the signal processing process shown in Fig.2, the kurtosis maximum of the low resonance component is taken as the objective function and the quality factor is optimized with GA. Simultaneously, the sampling frequency and the sub-band coverage basic theory considered, the quality
factor and other parameters are selected as Q1=3.75ˈQ2=1.06ˈr1=r2=3.5ˈL1=35ˈL2=15. The high resonance component decomposed by RSSD of the vibration signal is shown in Fig. 5 (a), which mainly contains the harmonic components. The low resonance component is shown in Fig. 5 (b). It can be observed that the transient impact component in the signal is relatively obvious.
A/mv
D
t/s
A/mv
E
t/s
Fig. 5. Inner race fault vibration signal RSSD result: (a) high resonance component (b) low resonance component The sub-band energy distribution of the low resonance component can be calculated with Eq. (8) and shown in Fig. 6. It can be seen that the energy of sub-band 2, 5, 6, 7, 16 is relatively dominant and satisfies the sub-band selection condition specified in Eq. (9). It means that these sub-bands contain the main fault impact information, thus they are selected to reconstruct the low resonant signal. SUBBAND ENERGY (% OF TOTAL)
DISTRIBUTION OF SIGNAL ENERGY 15
10
5
0
0
2
4
6
8 SUBBAND
10
12
14
16
Fig. 6. Sub-band energy distribution for low resonance components
A/mv
D
t/s
A/mv
E
f/Hz A/mv
F
f/Hz G
fi-fr
fr
fi fi+fr
A/mv
2fr
3fr
f/Hz Fig. 7. Low resonance component master sub-band reconstruction signal: (a) time domain waveform (b) frequency spectrum (c) the frequency spectrum of the first layer detail after WT (d) the envelope spectrum of the first layer detail after WT The sub-band reconstruction signal of the low resonance component is shown in Fig. 7(a), in which the impact characteristics are more obvious. The FFT transform is directly applied on the master sub-band reconstruction signal and its frequency spectrum is shown in Fig. 7(b). The frequency spectrum is very complex and the fault information can’t be acquired from it. In order to further extract the local time-frequency feature of the reconstructed signal, WT is introduced. Daubechies family of wavelet packets seems to resemble the bearing vibration signal most and db 10 is adopted as the mother wavelet. The reconstructed signal is decomposed with db10 wavelet into 3 layers. Fig. 7 (c) and (d) show the frequency spectrum and the envelope spectrum of the first layer detail signal respectively. They are very similar, which means that they have the similar analysis results. Because the envelope spectrum amplitude is larger and has clear demodulation meaning, the detailed analysis is
implemented on Fig. 7 (d). From it, rotational frequency accompanied by its harmonics ( f r , 2 f r and 3 f r ), the
inner race fault characteristic frequency f i and its modulation frequency ( f i ± f r ) can be seen clearly. According to the rolling bearing vibration theory, these frequency components coincide with the characteristics of the inner race fault. The weak fault feature is successfully extracted out by the proposed method. For comparison, the low resonance component without parameters optimization, sub-band reconstruction and WT are also analyzed. Fig. 8 is the envelope spectrum of the low resonance component obtained by regular RSSD. The inner race fault characteristic frequency f i and its modulation frequency ( f i - f r , 2 fi - f r ) can be observed, but the amplitude is not dominant. The fault feature recognition effect is poorer than the proposed
A/mv
method.
fr
fi-fr fi
2fi-fr
f/Hz Fig. 8. The envelope spectrum of the low resonance component obtained by regular RSSD method WT and EMD are two most frequently used time-frequency signal processing methods in the area of fault diagnosis of bearing up to now. The comparative analysis results of WT and EMD methods are also given out. WPD is employed to decompose the inner race fault vibration signal into 3 layers and the result is shown in Fig. 9. As before, db10 is adopted as the mother wavelet. According to WPT theory, there are total 8 frequency band 1
components ( P3
P38 ) from low to high frequency will be obtained, where superscript represents the frequency
band number and the subscript represents decomposition layer number. The frequency spectrum analysis 1 corresponding to P3
P38 is shown in Fig. 9(b). The inner race fault characteristic frequency components are
mainly distributed in the first frequency band and its details locating in the range of 0-1200Hz is shown in Fig. 10. Similar to Fig. 4(c), there are three obvious spectrum peaks (159 Hz, 237 Hz and 502 Hz) corresponding to the values of f i + f r , 4 fi - f r and 2 fi - f r can be clearly observed. However, the inner race fault frequency
f i can’t be extracted. The fault feature recognition effect of WT is poorer than the proposed method.
P P
P
A/mv
P31 P32 P33 P34
A/mv
5 3 6 3 7 3
P38
5 0 -5 5 0 -5 10 0 -10 10 0 -10 5 0 -5 5 0 -5 5 0 -5 5 0 -5
0
0.1
0.2
0.3
0.4
0.2 0.1 0 0.1 0.05 0 0.4 0.2 0 0.4 0.2 0 0.2 0.1 0 0.1 0.05 0 0.1 0.05 0 0.1 0.05 0
0
2000
4000
6000
8000
10000
f/Hz
t/s (a)
(b)
Fig. 9. The WPD result of the Inner race fault vibration signal: (a) time domain waveform (b) frequency spectrum 0.2
+]
A/mv
0.15
+]
+]
0.1 0.05 0
0
200
400
600
800
1000
1200
f/Hz Fig. 10. The detailed frequency spectrum of the first WPD frequency band The EMD method is also applied to analyze the above vibration acceleration signal. As shown in Fig. 11, there are a total of 11 IMFs components and 1 residue obtained. Considering the apparent periodical impulses existing in IMF1, the HT is applied to extract the modulation frequency and the Hilbert spectrum is shown in Fig. 12. By comparison with Fig 7(d), it can be found that the fault feature extraction ability is similar to the proposed method. Except rotational frequency f r and its harmonics ( 2 f r , 3 f r ), the inner race fault characteristic frequency f i and its modulation frequency ( f i ± f r ) also can be found in Fig. 12. With the increase of frequency, the harmonicity of the Hilbert spectrum decreases.
IMF1
IMF2 IMF3 IMF4 IMF5 IMF6 IMF7
RES
IMF11
IMF10
IMF9
IMF8
A/mv
10 0 -10 5 0 -5 5 0 -5 2 0 -2 2 0 -2 1 0 -1 0.5 0 -0.5 0.5 0 -0.5 0.2 0 -0.2 0.2 0 -0.2 0.1 0 -0.1 -0.005 -0.01 -0.015
0
0.1
0.2
0.3
0.4
t/s Fig. 11 The EMD result of the inner race fault vibration signal 0.8
fi
fr
0.8
0.6
0.6
0.4
fi-fr
A/mv
3fr fi+fr
0.2
0.4
0
0.2 0
2fr
0
200
400
600
0
50
100
800
150
1000
200
250
300
350
1200
f/Hz Fig. 12 The Hilbert spectrum of IMF1
5.3 Rolling Element Fault Diagnosis Fig. 13 is the rolling element fault vibration signal of the rolling bearing. Some non-periodic impulse components exist in the time domain waveform. Similar to the inner race vibration signal, the Hilbert demodulation analysis is adopted on the rolling element fault vibration signal and its envelope spectrum is shown
400
in Fig. 13(b). There are some obvious spectrum peaks located on the frequency of 181Hz, 195Hz, 327Hz and 535Hz. However, the rolling element fault characteristic frequency f b and its harmonics can’t be observed. Due
to the influence of the complicated transmission path, the useful fault information of the signal is hidden in the strong noisy background.
A/mv
a
t/s b
A/mv
535Hz 181Hz
195Hz 327Hz
f/Hz
Fig. 13. The rolling element fault vibration signal and its envelope spectrum: (a) time domain waveform (b) envelope spectrum Subsequently, the proposed method is also applied to analyze the rolling element fault vibration signal. The GA is first used to optimize the quality factor and other parameters of RSSD in order to maximize the kurtosis maximum of the low resonance component. The obtained parameters of RSSD are listed as Q1=4.92ˈQ2=1.13ˈ r1=r2=3.5ˈL1=40ˈL2=15. The high and low resonant components of the signal after decomposition are shown in Fig. 14 (a) and (b).
A/mv
a
t/s
A/mv
b
t/s
Fig. 14. The RSSD result rolling of element fault vibration signal: (a) High resonance component (b) Low resonance component Then the sub-band energy distribution of the low resonance component is calculated with Eq. (8) and shown in Fig. 15. The energy of the sub-bands 1, 2, and 16 is relatively dominant and satisfies the sub-band selection condition. These sub-bands are selected as master sub-bands to reconstruct the low resonance component. Fig. 16(a) is the time domain waveform of the reconstructed low resonance component. It can be observed that the fault impact component is greatly highlighted. Similarly, the reconstructed signal is decomposed with db10 wavelet into 3 layers. The envelope spectrum of the first-order detail is presented in Fig. 16(b) and the second, fourth and sixth harmonic ( 2 f b , 4 f b and 6 f b ) of the fault characteristic frequency can be apparently seen. The
SUBBAND ENERGY (% OF TOTAL)
extracted frequency components coincide with the fault type of the rolling bearing.
DISTRIBUTION OF SIGNAL ENERGY 50 40 30 20 10 0
0
2
4
6
8 10 SUBBAND
12
14
Fig. 15. Sub-band energy distribution for low resonance components
16
A/mv
a
t/s
A/mv
b 2fb 68Hz
149Hz
4fb 6fb
f/Hz Fig. 16. Low resonance component master sub-band reconstruction signal: (a) Time domain waveform (b) the envelope spectrum of the first layer detail after WT For comparison, the envelope spectrum analysis is also applied to the low resonance component obtained by regular RSSD and the result is shown in Fig. 17. Some obvious spectrum peaks located on the frequency of 181Hz, 195Hz, 327Hz and 535Hz can be seen. Among them, except that the frequency 195 Hz and 327 Hz correspond with 2 f b + f r and 6 f b respectively, the rest frequency has nothing to do with the fault characteristic frequency.
A/mv
The fault feature recognition effect is poorer than the proposed method.
535Hz 327Hz 181Hz 195Hz
f/Hz Fig. 17. The envelope spectrum of the low resonance component obtained by regular RSSD method Similar with the previous analysis to the inner race fault vibration signal, WT and EMD are also employed to process the rolling element fault vibration signals. The time domain waveforms and their corresponding frequency 1
spectrums of the 8 frequency band components ( P3
P38 ) obtained by WPD method are shown in Fig. 18. Db10
wavelet is also taken as the mother wavelet and the decompose layer number is 3. The detailed spectrum locating in the range of 0-1200Hz of the first frequency band component is shown in Fig. 19. It has similar analysis result
to Fig. 13(b), there are four obvious spectrum peaks located on the frequency of 181Hz, 195Hz, 327Hz and 535Hz. However, the rolling element fault characteristic frequency f b and its harmonics can’t be observed. It means that
10 0 -10 10 0 -10 20 0 -20 20 0 -20 5 0 -5 5 0 -5 5 0 -5 10 0 -10
1 0.5 0 0.4 0.2 0 0.5
A/mv
P34 P38
P37
P36
P35
A/mv
P33
P32
P31
the fault feature recognition effect of WT is poorer than the proposed method.
0
0.1
0.2
0.3
0.4
0 1 0.5 0 0.2 0.1 0 0.2 0.1 0 0.1 0.05 0 0.2 0.1 0
0
2000
4000
6000
8000
10000
f/Hz
t/s
Fig. 18. The WPD result of the rolling element fault vibration signal (a) time domain waveform (b) frequency spectrum
A/mv
0.8 0.6
535Hz
0.4
181Hz 195Hz 327Hz
0.2 0
0
200
400
600
800
1000
1200
f/Hz Fig. 19. The detailed frequency spectrum of the first WPD frequency band The EMD result of the rolling element fault vibration signal is shown in Fig. 20, which includes 11 IMFs components and 1 residue. The HT is also applied on IMF1 to extract the modulation frequency and the Hilbert spectrum is shown in Fig. 21. By comparison with Fig 16(b), it can be found that the fault feature extraction ability is similar to the proposed method. Except 149Hz, the second and the fourth harmonic ( 2 f b , 4 f b ) of the rolling element fault characteristic frequency can be seen, which coincide with the fault type of the rolling bearing. But the amplitudes are not dominant and smaller than that of 16(b). It means that the fault feature recognition effect of EMD method is poorer than the proposed method.
IMF1
IMF2 IMF3 IMF4 IMF5 IMF6
A/mv
IMF7 IMF8 IMF9 IMF10
RES
IMF11
20 0 -20 10 0 -10 10 0 -10 5 0 -5 2 0 -2 2 0 -2 0.5 0 -0.5 0.5 0 -0.5 0.2 0 -0.2 0.1 0 -0.1 0.2 0 -0.2 1.6 1.4 0
0.1
0.2
0.3
0.4
t/s Fig. 20 The EMD result of the rolling element fault vibration signal 1.5
A/mv
149Hz
2fb
1
4fb
0.5
0
0
100
200
300
400
500
600
f/Hz Fig. 21 The Hilbert spectrum of IMF1
6 Conclusion The present study proposes a new kind of fault diagnosis method integrating RSSD with WT. The method includes parameters optimization, RSSD, master sub band selection and reconstruction, WT and Hilbert demodulation analysis. Through two fault diagnosis case study of rolling bearing, the conclusion can be
drawn as follows. (1) During the incipient failure of rolling bearings, the characteristic of the fault signal is relatively weak and vulnerable to environmental noise and signal energy decay. If the signal is merely analyzed by RSSD, the impact fault features are not evident and can’t be extracted. The RSSD method combined with parameter optimization and sub-band reconstruction can effectively highlight the weak impact fault characteristics. (2) The RSSD method can decompose complex signals into a high resonance component containing continuous oscillating components and a low resonance component containing transient impacting ingredients according to the signal resonance properties. The WT signal processing method has excellent local time-frequency multiresolution resolution ability, which can decompose the signal in a multiscale way from the frequency domain perspective. The fault information can be deeply mined from two aspects of signal resonance attribute and local time-frequency characteristic analysis based on the fusion of RSSD and WT technique. (3) The effectiveness of the method is proved by two diagnosis cases to early different type rolling bearing faults. The results show that the proposed method is better than regular RSSD method in the weak fault feature highlighting. The fault feature recognition ability of the proposed method is prior to WPD method and similar or prior to EMD method for different bearing fault signals.
Acknowledgments This program is jointly supported by the projects of National Nature Science Foundation of China (51775307, 51605255), Natural Science Foundation of Hubei Province of China (2018CFB671) and Hubei Key Laboratory of Hydroelectric Machinery Design & Maintenance (No.2016KJX15 & No.2016KJX02).
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