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.
Notes
Supported by National Natural Science Foundation of China (51705431, 51375078) and Natural Sciences and Engineering Research Council of Canada (RGPIN-2015-04897).
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.
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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.
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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
Figure 1 illustrates a one stage uncoupled MW decomposition scheme [10].
×
Consider a family V_{
j
}, V_{
j+1} and W_{
j+1} of signal spaces. There are two analysis operators \(\psi_{j}^{ \uparrow }\) and \(\omega_{j}^{ \uparrow }\) which together decompose a signal in the direction of increasing j. The signal analysis operator \(\psi_{j}^{ \uparrow }\) maps a signal from V_{
j
} to V_{
j+1} and the detail analysis operator \(\omega_{j}^{ \uparrow }\) maps a signal from V_{
j
} to W_{
j+1}.
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 (\(\psi_{j}^{ \uparrow }\), \(\omega_{j}^{ \uparrow }\)): V_{
j
} → V_{
j+1} × W_{
j+1} and synthesis mapping (\(\psi_{j}^{ \downarrow } + \omega_{j}^{ \downarrow }\)): V_{
j+1} × W_{
j+1} → V_{
j
} should be inverse of each other, which means the following condition should be satisfied:
And x_{0} also can be exactly reconstructed from x_{
n
}, y_{
n
}, y_{
n-1}, y_{
n-2}, …, y_{2}, y_{1} through the recursive synthesis scheme shown in Eq. (4).
In MW, the analysis operator \(\psi_{j}^{ \uparrow }\) and the synthesis operator \(\psi_{j}^{ \downarrow }\) are morphological operators. Some MWs have been established, such as morphological Haar wavelet [18].
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 \(y^{\prime}_{1} = y_{1}\).
×
The AMULW comprises three main steps. First, the raw signal x_{0} is split into two parts, producing an approximation signal x_{1} and a detail signal y_{1}. This partition can be fulfilled by some special wavelet transform. The simplest one is to directly split x_{0} into odd and even samples [19]:
Subsequently, the approximation signal x_{1} is updated using the information hidden in the detail signal y_{1}, yielding a new approximation signal \(x^{\prime}_{1}\). If D(n) = 1, the morphological dilation-erosion filter is used as the update operator U_{1}:
Correspondingly, x_{1} can be easily reconstructed from \(x^{\prime}_{1}\) and \(y^{\prime}_{1}\) as Eq. (12) or (13). Once x_{1} is obtained, the synthesis signal x_{0} is also easy to get.
The above analysis process is limited to the scope of one stage decomposition. A raw signal x_{0} 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:
$$T = k \cdot \hbox{max} (g(n))\quad \quad k \in [0,1]$$
(15)
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:
where \(A_{{if_{c} }}\) is the amplitude of the ith harmonic of the fault characteristic frequency, N is the number of the harmonics of the fault characteristic frequency, \(A_{{f_{j} }}\) 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 α, β and γ are lifting wavelet coefficients:
Ref. [17] has proved in theory that in order to fulfill perfect reconstruction, it requires:
$$\alpha + \beta + \gamma = M$$
(18)
where M is a constant and usually set as 1.
In Eq. (11), \(\alpha = \beta = \gamma = {1 \mathord{\left/ {\vphantom {1 3}} \right. \kern-0pt} 3}\), which meets the requirement of Eq. (18). For Eq. (9), we discuss the influence of the value of y_{1}(n−1) and y_{1}(n) on perfect reconstruction. In summary, there are three situations:
Case 1: y_{1}(n − 1) > y_{1}(n), at this moment Eq. (9) can be simplified as
Case 3: y_{1}(n − 1) = y_{1}(n), at this moment Eq. (9) can be simplified as
$$x^{\prime}_{1} (n) = x_{1} (n)$$
(21)
Regardless the value of y_{1}(n − 1) and y_{1}(n), the situations in Eqs. (19)–(21) accords with the requirement of Eq. (18). Therefore, the proposed AMULW fits the perfect reconstruction condition.
4 Simulated Signal Analysis
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 f_{
s
} of signal s(t) is 4096 Hz and the samples are 4096. The signal s(t) is made up of three parts:
$$s(t) = s_{1} (t) + s_{2} (t) + s_{3} (t)$$
(22)
where s_{1}(t) is the periodic impulse response model of a defective bearing [21]:
where i is the number of impulses; A_{
i
} is the amplitude modulator; T stands for the fault impacts repetitive period, which gives the reciprocal of fault characteristic frequency f_{
c
}; \(\tau_{i}\) represents the minor and random time fluctuation around T; n(t) denotes the stationary noise; and u(\(\cdot\)) is fault impulse function:
where A_{0} and \(A^{\prime}_{0}\) are the mean and maximum of the amplitude modulator, respectively; f_{
A
} equals to the shaft frequency; \(iT + \tau_{i}\) denotes the special time point at the ith impulse; and \(\varphi_{A}\) is related to the location of the sensor.
The parameters for the simulation of vibration signal s_{1}(t) are shown in Table 1.
Table 1
Parameters used in the simulation of signal s_{1}(t)
i
A
T
\(\tau_{i}\)
n(t)
16
[3, 4]
1/16
0
0
a
f_{0}
ϕ
ϕ_{
A
}
100
600
0
0
In Eq. (22), S_{2}(t) = 0.2sin(2π10t) + 0.4 cos(2π25t) simulates two interference frequencies, s_{3}(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 down-sampling 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 would be \({\raise0.7ex\hbox{$L$} \!\mathord{\left/ {\vphantom {L {2^{n} }}}\right.\kern-0pt} \!\lower0.7ex\hbox{${2^{n} }$}}\) 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 from the interference signal s_{2}(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 f_{
r
} 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 − 2f_{
r
}, BPFI + 2f_{
r
}, 2BPFI − 2f_{
r
}, 2BPFI + 2f_{
r
}, 3BPFI − 2f_{
r
}, and the second harmonic of modulation frequency 2f_{
r
} 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 4f_{
r
} 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 detected, such as BPFI, 2BPFI, 3BPFI, BPFI + 2f_{
r
}, BPFI − 2f_{
r
}, 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 f_{
r
} and ball pass 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 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. It outperforms AULW in detection both of an inner race fault and an outer race fault of a rolling element bearing and it outperforms MUDW in detection of an outer race fault of a rolling element bearing.
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