Detection of Bearing Faults Using a Novel Adaptive Morphological Update Lifting Wavelet

Chinese Journal of Mechanical Engineering, Nov 2017

The current morphological wavelet technologies utilize a fixed filter or a linear decomposition algorithm, which cannot cope with the sudden changes, such as impulses or edges in a signal effectively. This paper presents a novel signal processing scheme, adaptive morphological update lifting wavelet (AMULW), for rolling element bearing fault detection. In contrast with the widely used morphological wavelet, the filters in AMULW are no longer fixed. Instead, the AMULW adaptively uses a morphological dilation-erosion filter or an average filter as the update lifting filter to modify the approximation signal. Moreover, the nonlinear morphological filter is utilized to substitute the traditional linear filter in AMULW. The effectiveness of the proposed AMULW is evaluated using a simulated vibration signal and experimental vibration signals collected from a bearing test rig. Results show that the proposed method has a superior performance in extracting fault features of defective rolling element bearings.

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Detection of Bearing Faults Using a Novel Adaptive Morphological Update Lifting Wavelet

Detection of Bearing Faults Using a Novel Adaptive Morphological Update Lifting Wavelet Yi-Fan Li 0 1 2 3 4 MingJian Zuo 0 1 2 3 4 Ke Feng 0 1 2 3 4 Yue-Jian Chen 0 1 2 3 4 0 School of Mechatronics Engineering, University of Electronic Science and Technology of China , Chengdu 611731 , China 1 & MingJian Zuo 2 Supported by National Natural Science Foundation of China (51705431, 51375078) and Natural Sciences and Engineering Research Council of Canada , RGPIN-2015-04897 3 Department of Mechanical Engineering, University of Alberta , Edmonton T6G 2G8 , Canada 4 School of Mechanical Engineering, Southwest Jiaotong University , Chengdu 610031 , China The current morphological wavelet technologies utilize a fixed filter or a linear decomposition algorithm, which cannot cope with the sudden changes, such as impulses or edges in a signal effectively. This paper presents a novel signal processing scheme, adaptive morphological update lifting wavelet (AMULW), for rolling element bearing fault detection. In contrast with the widely used morphological wavelet, the filters in AMULW are no longer fixed. Instead, the AMULW adaptively uses a morphological dilation-erosion filter or an average filter as the update lifting filter to modify the approximation signal. Moreover, the nonlinear morphological filter is utilized to substitute the traditional linear filter in AMULW. The effectiveness of the proposed AMULW is evaluated using a simulated vibration signal and experimental vibration signals collected from a bearing test rig. Results show that the proposed method has a superior performance in extracting fault features of defective rolling element bearings. Morphological filter; Lifting wavelet; Adaptive; Rolling element bearing; Fault detection 1 Introduction Rolling element bearings, one of the most important and frequently used components in engineering machinery, play a critical role in system performances [ 1 ]. Effectively detecting the defects of rolling element bearings can provide an assurance for the reliability of machine sets. When a localized fault occurs on the surface of the inner race, outer race or rolling element, the vibration signal would present repetitive peaks which are further modulated by rotational frequencies of machine components. The impulses contain important information about the bearing health status. Therefore, the extraction of cyclic faulty intervals is the essential task in bearing fault detection. Many methods, such as wavelet transform [ 2 ], empirical mode decomposition [ 3 ], higher order spectrum [ 4 ], morphology filter [ 5 ] and order tracking [ 6 ] have been applied successfully in bearing fault detection and fault diagnosis. Wavelet transform (WT) is one of the most popular signal processing technologies among them. However, the classical WT, both continuous and discrete, are linear [ 7 ]. Because of the fact that a signal often contains information at many scales or resolutions, multi-resolution approaches are indispensable for a thorough understanding of such a signal. Therefore, it is desired to extend WT to nonlinear area. The lifting scheme, proposed by Sweldens [ 8 ], has provided a useful way to design nonlinear wavelets. The flexibility and freedom offered by the lifting scheme have attracted researchers to develop various nonlinear WTs, including morphological ones. Not until 2000, Heijmans and Goutsias [ 9, 10 ] firstly gave the theoretical presentation of a general framework for constructing morphological wavelet (MW). The theoretical foundation of MW is extending the classic wavelet from the linear domain, which is based on convolution, to the nonlinear domain, which is based on morphological operations. In this way, the MW does inherit the multi-dimension and multi-level analysis of wavelet while only involving the purely time domain analysis. As a consequence, a very high computational efficiency is achieved. Following the work of Heijmans and Goutsias [ 9, 10 ], morphological gradient wavelet [ 11, 12 ], morphological undecimated wavelet (MUDW) [ 13–15 ] and morphological undecimated wavelet slices [ 16 ] were proposed and applied for fault detection and fault diagnosis. A major disadvantage of MW and the extended version is that the filter structure is fixed in the whole analysis process, which cannot cope with the sudden impulses and stationary data accurately in one signal at the same time. In many applications, it is desirable to have a filterbank that somehow determines how to shape itself according to the signal being analyzed. Based on this consideration, Piella and Heijmans [ 17 ] proposed an adaptive update lifting wavelet (AULW). The basic idea underlying AULW is to employ different update lifting filters to modify the approximation signal according to the signal local gradient information. However, the essence of AULW is still linear wavelet decomposition. The effectiveness of employing AULW to process the real mechanical vibration signals is actually compromised. A new method of morphological lifting scheme, an adaptive morphological update lifting wavelet (AMULW) is proposed in this study. The aim of AMULW is to address the disadvantages both from the linear wavelet decomposition and from the fixed filter in AULW and MW. In AMULW, the nonlinear morphological dilation-erosion filter, as well as the average filter, is adaptively adopted in the update lifting scheme according to the geometry of the signal. Consequently, the impulsive features would be strengthened and the noise would be suppressed effectively. The rest of this paper is organized as follows: Section 2 briefly introduces the fundamentals of MW; Section 3 proposes the AMULW; the advantage of AMULW over the AULW and MUDW is demonstrated by using a simulated vibration signal in Section 4; Section 5 applies the proposed AMULW technique to experimental signals of rolling element bearings and demonstrates the detection ability of AMULW for an inner race fault bearing and an outer race fault bearing; Conclusions are drawn in Section 6. 2 Morphological Wavelet Consider a family Vj, Vj?1 and Wj?1 of signal spaces. There are two analysis operators wj" and xj" which together decompose a signal in the direction of increasing j. The signal analysis operator wj" maps a signal from Vj to Vj?1 and the detail analysis operator xj" maps a signal from Vj to Wj?1. wj"ðwj#ðxÞ þ xj#ðyÞÞ ¼ x xj"ðwj#ðxÞ þ xj#ðyÞÞ ¼ y x 2 Vjþ1; y 2 Wjþ1 x 2 Vjþ1; y 2 Wjþ1 ð1Þ where x is the approximation signal and y is the detail signal. The signal and detail analysis operators correspond to a low pass and a high pass filter, respectively [ 10 ]. On the other hand, a synthesis operator ? proceeds in the direction of decreasing j. For the purpose of yielding a complete signal representation, the analysis mapping (wj", xj"): Vj ? Vj?1 9 Wj?1 and synthesis mapping (wj# þ xj#): Vj?1 9 Wj?1 ? Vj should be inverse of each other, which means the following condition should be satisfied: wj#wj"ðxÞ þ xj#xj"ðxÞ ¼ x x 2 Vj This condition is called the perfect reconstruction [ 17 ]. A raw signal x0 can be decomposed with the following recursive analysis scheme: x0 ! fx1; y1g ! fx2; y2; y1g ! ! fxn; yn; yn 1; yn 2; ; y2; y1g And x0 also can be exactly reconstructed from xn, yn, yn1, yn-2, …, y2, y1 through the recursive synthesis scheme shown in Eq. (4). xj ¼ wj#ðxjþ1Þ þ xj#ðyjþ1Þ j ¼ n 1; n 2; ; 0 In MW, the analysis operator wj" and the synthesis operator wj# are morphological operators. Some MWs have been established, such as morphological Haar wavelet [ 18 ]. ð2Þ ð3Þ ð4Þ 3 Adaptive Morphological Update Lifting Wavelet 3.1 Principles of AMULW The schematic of the proposed AMULW in one stage is illustrated in Figure 2. In AMULW, the update lifting is applied to modify the approximation signal, while the detail signal remains unchanged in the decomposition process, namely y01 ¼ y1. The AMULW comprises three main steps. First, the raw signal x0 is split into two parts, producing an approximation signal x1 and a detail signal y1. This partition can be fulfilled by some special wavelet transform. The simplest one is to directly split x0 into odd and even samples [ 19 ]: x1 ¼ w"ðx0ÞðnÞ ¼ x0ð2nÞ y1 ¼ x"ðx0ÞðnÞ ¼ x0ð2n þ 1Þ Then, a two-valued decision map D is used to control the choice of the update filter, which is expressed as follows: DðnÞ ¼ ( 1 0 gðnÞ [ T T gðnÞ where T is the threshold value; g(n) denotes the local gradient information of adjacent three samples in a signal, which is defined as: gðnÞ ¼ jx1ðnÞ y1ðn 1Þj þ jy1ðnÞ x1ðnÞj Subsequently, the approximation signal x1 is updated using the information hidden in the detail signal y1, yielding a new approximation signal x01. If D(n) = 1, the morphological dilation-erosion filter is used as the update operator U1: U1ðnÞ ¼ y1ðn 1Þ ^ y1ðnÞ y1ðn 1Þ _ y1ðnÞ x01ðnÞ ¼ x1ðnÞ þ U1ðnÞ ¼ x1ðnÞþ y1ðn 1Þ ^ y1ðnÞ y1ðn 1Þ _ y1ðnÞ where _ represents morphological dilation operation and ^ represents morphological erosion operation. If D(n) = 0, the average filter is utilized as the update operator U0: U0ðnÞ ¼ y1ðn 1Þ þ y1ðnÞ ð11Þ ð12Þ ð13Þ ð14Þ ð15Þ ð16Þ 1 x01ðnÞ ¼ 3 ðx1ðnÞ þ U0ðnÞÞ The above analysis process is limited to the scope of one stage decomposition. A raw signal x0 can be decomposed into multiple levels as a multiple stage decomposition involves: On account of the presence of the decision map D, the update lifting operator U can be adaptively selected to modify the approximation signal. The morphological dilation-erosion filter is adopted to strengthen the impulsive features when there is an impulse in a signal, while the average filter is employed to smooth a signal at the time when the signal amplitudes vary weakly. 3.2 The Selection of the Threshold T The function of the threshold T is to distinguish the stationary data and the impulses in a signal. If T is too large, some useful impulsive information might be smoothed by using the average filter; on the contrary, if T is too small, some tiny fluctuations might be treated as impulses. In the present paper, T is defined as: T ¼ k maxðgðnÞÞ k 2 ½0; 1 In order to reduce the adverse effects of the threshold T, we perform the AMULW multiple times with k increasing from 0 to 1, with a step size of 0.1. As a result, 11 analysis results would be obtained. Then, characteristic frequency intensity coefficient (CFIC) [ 20 ] of the above 11 signals are computed. The CFIC can be expressed as: N P Aifc CFIC ¼ i¼M1 P Afj j¼1 where Aifc is the amplitude of the ith harmonic of the fault characteristic frequency, N is the number of the harmonics of the fault characteristic frequency, Afj is the amplitude of the frequencies analyzed, and M is the number of all frequency components. The CFIC represents the portion of the fault frequency amplitude to the overall frequency amplitude. A larger value of CFIC implies a better fault detection performance. Other statistic criterion, such as kurtosis [ 21 ], smoothness index [ 22 ], crest factor [ 23 ], peak energy [ 24 ] and fusion criterion of kurtosis, smoothness index and crest factor [ 25 ], can also be used as a guide to select the fault relevant information. In this study, CFIC is employed. The signal with the largest CFIC value is picked out from the whole 11 candidates and only this signal will be utilized in the fault detection. 3.3 Perfect Reconstruction Assume that a, b and c are lifting wavelet coefficients: x01ðnÞ ¼ ax1ðnÞ þ by1ðn 1Þ þ cy1ðnÞ Ref. [ 17 ] has proved in theory that in order to fulfill perfect reconstruction, it requires: a þ b þ c ¼ M where M is a constant and usually set as 1. In Eq. (11), a ¼ b ¼ c ¼ 1=3, which meets the requirement of Eq. (18). For Eq. (9), we discuss the influence of the value of y1(n-1) and y1(n) on perfect reconstruction. In summary, there are three situations: Case 1: y1(n - 1) [ y1(n), at this moment Eq. (9) can be simplified as x01ðnÞ ¼ x1ðnÞ þ y1ðnÞ y1ðn 1Þ Case 2: y1(n - 1) \ y1(n), at this moment Eq. (9) can be simplified as x01ðnÞ ¼ x1ðnÞ þ y1ðn 1Þ y1ðnÞ Case 3: y1(n - 1) = y1(n), at this moment Eq. (9) can be simplified as ð17Þ ð18Þ ð19Þ ð20Þ ð21Þ x01ðnÞ ¼ x1ðnÞ Regardless the value of y1(n - 1) and y1(n), the situations in Eqs. (19)–(21) accords with the requirement of Eq. (18). Therefore, the proposed AMULW fits the perfect reconstruction condition. ð22Þ ð23Þ ð24Þ ð25Þ sðtÞ ¼ s1ðtÞ þ s2ðtÞ þ s3ðtÞ where s1(t) is the periodic impulse response model of a defective bearing [ 21 ]: s1ðtÞ ¼ X Aiuðt i iT siÞ þ nðtÞ where i is the number of impulses; Ai is the amplitude modulator; T stands for the fault impacts repetitive period, which gives the reciprocal of fault characteristic frequency fc; si represents the minor and random time fluctuation around T; n(t) denotes the stationary noise; and u( ) is fault impulse function: uðtÞ ¼ e at sinð2pf0t þ uÞ where a is the decay parameter, f0 the fault excited resonance frequency and u the original phase. For an inner race fault, Ai in Eq. (23) can be simplified as [ 27 ] Ai ¼ A0 þ A00 cosð2pfAðiT þ siÞ þ uAÞ where A0 and A00 are the mean and maximum of the amplitude modulator, respectively; fA equals to the shaft frequency; iT þ si denotes the special time point at the ith impulse; and uA is related to the location of the sensor. The parameters for the simulation of vibration signal s1(t) are shown in Table 1. In Eq. (22), S2(t) = 0.2sin(2p10t) ? 0.4 cos(2p25t) simulates two interference frequencies, s3(t) represents a Gaussian white noise. The signal-to-noise ratio of s(t) is 0 dB. Figure 3 presents the simulated signal s(t) and its corresponding fast Fourier transform (FFT) spectrum. As it can be seen that the fault related characteristic frequency 16 Hz along with its harmonics are totally overwhelmed by the interference signals (10 and 25 Hz) and the noise. The approximation signal obtained by AMULW is presented in Figure 4(a). This approximation signal obtained by AMULW is at level 3 of the decomposition process. Wavelet decomposition algorithm is the downsampling procedure that the length of approximation signal and detail signal would halve as the decomposition level increasing. Suppose that the length of raw signal is L, the data number of approximation signal and detail signal A vibration signal s(t) of rolling element bearing under an inner race fault condition is simulated [ 26 ] to evaluate the effective of the proposed AMULW. The sampling frequency fs of signal s(t) is 4096 Hz and the samples are 4096. The signal s(t) is made up of three parts: 2000 2500 Sampling points N 500 1000 1500 3000 3500 4000 would be L=2n after n decompositions. Therefore, only 512 samples are shown in Figure 4(a). The parameter k is set at 0.2 because at this time the analysis result has the maximum CFIC value. As shown in Figure 4(b), the impulsive characteristic frequency of 16, 32, 48, 64 and 80 Hz are identified clearly and the interferences of 10 and 25 Hz are totally restrained. The simulated signal shown in Figure 3(a) is analyzed by AULW [ 17 ]. Figure 5 displays the analysis results in the time domain and its FFT spectrum. Although some impulsive characteristic frequencies are detected, 10 Hz and 25 Hz as well as other interferences are still present. Comparing with Figure 4(b), the AULW is less effective than the proposed method to extract impulsive features and restrain interferences. The MUDW [ 14 ] procedure is applied to the simulated signal with three levels of decomposition. The approximation signal at level 3 and its frequency spectrum are shown in Figure 6. MUDW is a non-decimation decomposition method and it omits down-sampling in the forward transform. Therefore, the approximation signal and the detail signal are of the same length as the raw signal whatever level they are at. Unfortunately, the spectral line 25 Hz in Figure 6(b) is visually obvious, which comes 200 300 Sampling points N 400 500 1000 2000 Sampling points N 3000 4000 from the interference signal s2(t). Hence, in this case, AMULW is more effective in extracting the features. 5 Experimented Lab Signal Analysis In this section, the performance of the proposed AMULW is graphically evaluated by using vibration signals of rolling element bearings. The experiment was conducted at Engineering Reliability, Prognostic and Health Management Laboratory, University of Electronic Science and Technology of China. The experimental set-up consists of a shaft, two bearings and a bevel gearbox. It is driven by a 2.24 kW three phase electrical motor controlled by a motor speed controller. The bearing system is under consideration for fault diagnosis. An accelerometer (with sensitivity of 100 mV/g and frequency range 0–10 kHz) and a shaft encoder are used for capturing the vibration signals and rotating speed signals simultaneously. The whole set-up arrangement is displayed in Figure 7. 5.1 Analysis of the Vibration Signal of Bearing with an Inner Race Fault For the rotor with a rotating speed at 1800 r/min, vibration signal of the defective bearing with an inner race fault and its FFT spectrum are illustrated in Figure 8, sampled at a frequency of 20480 Hz. The theoretical rotational frequency fr of the shaft is 30 Hz and ball pass frequency in inner race (BPFI) is 163 Hz. But the BPFI cannot be identified in Figure 8(b). The AMULW method is applied to the rolling element bearing vibration signal presented in Figure 8(a). Figure 9(a) shows the approximation signal at the third level of AMULW decomposition. The parameter k is determined as 0.2 according to multiple trials as well. Figure 9(b) plots the frequency spectrum where the defective feature frequencies of the bearing can be identified. The BPFI together with its second-order and third-order harmonics, 2BPFI and 3BPFI, side frequencies BPFI - 2fr, 1.5 -2s 1.0 a/m0.5 ion 0.0 t ra-0.5 e l ce-1.0 cA-1.5 0 5000 10000 Sampling points N 15000 20000 100 200 300 Frequency f /Hz 400 500 Figure 8 Vibration signal of inner race fault bearing: (a) timedomain waveform, (b) frequency spectrum 0 500 1000 1500 Sampling points N 2000 2fr BPFI-2fr BPFI BPFI+2fr 2BPFI-2fr 2BPFI 2BPFI+2fr 3BPFI-2fr 2500 3BPFI 100 200 300 Frequency f /Hz 400 500 BPFI ? 2fr, 2BPFI - 2fr, 2BPFI ? 2fr, 3BPFI - 2fr, and the second harmonic of modulation frequency 2fr are all detected. These frequency components clearly indicate an inner race fault on the bearing. In Figure 10, the analysis results of AULW are presented, from which it can be found that the 4fr and 3BPFI could be clearly detected in the Fourier spectrum, while the BPFI and its side frequencies are not particularly evident. Therefore, the accuracy of using AULW to diagnose this inner race fault bearing is actually poor. Figure 11 demonstrates the analysis results of MUDW. In Figure 11(b), some characteristic frequencies are 3fr 100 4fr 2fr 100 10000 Sampling points N 5000 15000 20000 detected, such as BPFI, 2BPFI, 3BPFI, BPFI ? 2fr, BPFI - 2fr, etc. These frequency components indicate an inner race fault on the bearing. The effectiveness of employing MUDW to detect this bearing fault is much better than AULW. However, these fault related features shown in Figure 11(b) is not as apparent as Figure 9(b). 5.2 Analysis of the Vibration Signal of Bearing with Outer Race Fault The vibration signal of a bearing with a fault in outer race was collected at the sampling frequency of 20480 Hz. The shaft rotation frequency fr and ball pass 10000 Sampling points N 5000 15000 20000 100 200 300 Frequency f /Hz 400 500 Figure 12 Vibration signal of outer race fault bearing: (a) timedomain waveform, (b) frequency spectrum 500 BPFO 100 1000 1500 Sampling points N 2BPFO 3BPFO 200 300 Frequency f /Hz 2000 2500 400 500 frequency in outer race (BPFO) are 30 Hz and 107 Hz, respectively. Figure 12 shows the time waveform and the FFT spectrum of this signal. The fault frequency BPFO cannot be determined, as shown in the FFT spectrum of Figure 12(b). The AMULW analysis results are shown in Figure 13. The approximation signal at the third level of AMULW decomposition and its FFT spectrum, are presented in Figure 13(a) and (b) respectively. The parameter k is selected at 0.2. The BPFO and its second harmonic 2BPFO and third harmonic 3BPFO can be clearly identified in Figure 13(b). There is a good match between the estimated 3fr 100 10000 Sampling points N 15000 20000 fr 2fr 3fr BPFO+2fr 100 200 300 Frequency f /Hz 400 500 features and the actual fault features associated with the rolling element bearing with the outer race defect. Figures 14 and 15 describe the analysis results of AULW and MUDW, respectively. The frequency spectrum in Figures 14(b) and 15(b) are not suitable for indicating a bearing with an outer race fault. Therefore, both of the two methods fail to identify this fault. The ability of AULW and MUDW for detecting the rolling element bearing with outer race fault is inferior to AMULW. 6 Conclusions In this paper, an AMULW with perfect reconstruction is developed to detect rolling element bearing faults. Compared with Fourier transforms using the same filter and wavelets being translation and dilation of one given function, lifting scheme adapts local data irregularities in the transform process. In the proposed AMULW, two filters are adaptively employed. The nonlinear morphological dilation-erosion filter is effective to extract impulses while the average filter is suitable for removing noise. Therefore, AMULW can reasonably process a non-stationary mechanical vibration signal comprising of impulses and interferences. The experimental evaluation results have shown that the proposed AMULW is capable of extracting the impulsive features of the bearing vibration signals. 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Yue-Jian Chen born in 1991, is a PhD candidate majoring in Mechanical Engineering at the University of Alberta, Canada. He received his Master of Science degree from Nanjing University of Science and Technology , China, in 2015 . His main research interests include signal processing, condition monitoring, and fault diagnosis . E-mail : .


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Yi-Fan Li, MingJian Zuo, Ke Feng, Yue-Jian Chen. Detection of Bearing Faults Using a Novel Adaptive Morphological Update Lifting Wavelet, Chinese Journal of Mechanical Engineering, 2017, 1-9, DOI: 10.1007/s10033-017-0186-1