Bearing Condition Monitoring based on the Indicator Generated in Time-frequency Domain

Most condition monitoring systems rely on system-driven generation of indicators or features for early fault detection. However, this strategy requires the prior knowledge on the system kinematics and/or exact structure parameters of monitored system. To address this problem, this paper presents a novel condition monitoring framework where the condition indicator is generated via data-driven method. In this framework, the time-frequency periodogram is extracted from raw vibration signal first. Then, the acquired time-frequency periodogram is mapped by pseudo Perron vector, which is learned from vibration data, to generate the condition indicator. Finally, the bearing can be monitored via analyzing this indicator using gaussian based control chart. Based on experimental results on a publicly-available database, we show the effectiveness of presented framework for early fault detectionin the continuous operation of rolling bearing, indicating its great potentials in real engineering applications.


INTRODUCTION
Bearings are vital mechanical parts of most rotating machines, and frequently used in vast domestic and industrial applications.However, they are also fragile components.According to the report, about 41% of total machine faults are due to bearing (MRWG, 1985) .In this context, accurate and reliable bearing condition monitoring plays a critically important role in the predictive maintenance of rotating machinery, which can not only reduce the maintenance cost but ensure Teng Wang et al.This is an open-access article distributed under the terms of the Creative Commons Attribution 3.0 United States License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
Since it has been widely accepted that the degeneration of bearing can be reflected in the bearing vibration signal, most existing schemes aim to monitor the health of bearing via extracting effective features/indicators from vibration signal using signal processing techniques (Javed, Gouriveau, Zerhouni, & Nectoux, 2014;Dong, Tsui, & Qiang, 2018).These techniques include time-domain, frequency-domain and time-frequency domain approaches.Among them, timefrequency domain approaches have been an extensively studied area in the literature.The most common timefrequency approaches known in the literature are the shorttime Fourier transform (STFT), the empirical mode decomposition (EMD), the continuous wavelet transform (CWT) and the discrete wavelet transform (DWT).A comprehensive review of such approaches can be found in (Zhipeng, Ming, & Fulei, 2013).Comparing with time-domain and frequencydomain approaches, time-frequency domain approaches give an insight of operating condition in both time and frequency domian, where frequency spectrum is continuously extracted over time and then analyzed to inspect the dynamic characteristics of bearing vibration signal.The main advantages of time-frequency domain based method can be boiled down as (1) it is promising in analyzing non-stationary signals (Zhipeng et al., 2013), (2) it is able to inspect the incipient fault information with weak amplitude and short duration (Javed et al., 2014).
To extract condition indicators in time-frequency domain, it is crucial to select early fault related frequency components for monitoring.Take the rolling ball bearing for example, bearing failures are mostly due to the defects in the outer raceway (OR), inner raceway (IR), or ball.Different type of faults will produce unique frequency component (f IR ,f OR ,f ball ) which can be calculated as, According to this, a common strategy to evaluate bearing operating status is monitoring these fault-related frequencies, as conducted in many references (Fernández-Francos, Martínez-Rego, Fontenla-Romero, & Alonso-Betanzos, 2013;Yaguo, Shantao, Liang, & Naipeng, 2017;Gerber, Martin, & Mailhes, 2014;Betta, Liguori, Paolillo, & Pietrosanto, 2002).If an anomaly appears in one of these frequency sub-bands, it is reasonable to believe the corresponding fault occurs.However, this strategy requires the prior knowledge on exact parameters related to the monitored bearing (such as θ,n,f s ,D,d).The acquirement of these parameters is not a trivial work and even impossible in some applications (Combet & Gelman, 2007).To address this problem, this paper presents a novel condition monitoring framework where the condition indicator is generated via data-driven method.Main contributions of this paper are given as follows.
• This paper proposes generating the condition indicator in time-frequency domain by taking into account of the correlations between each pair of frequency components.
• The graph modeling strategy upon time-frequency spectrum is put forward, followed with theoretical interpretation and experimental investigation.The rest of this paper is organized as follows.Section. 2 details the proposed framework.In Section.3, we demonstrate the performance of the proposed framework on a publiclyavailable database, and compare it with a representative method reported in the literature.Finally, Section. 4 concludes this paper and shows our future work.

THE PROPOSED SCHEME FOR BEARING CONDITION MONITORING
This section describes the proposed framework for bearing condition monitoring which contains four steps: timefrequency periodogram extraction, graph modeling, condition indicator generation, fault decision-making.

Time-frequency periodogram extraction
Derived from the short-time Fourier transform(STFT), timefrequency periodogram estimates the power spectral density of bearing vibration signal.Let us denote the signal as This transform is discrete in both time and frequency domains where M is time-skip and ∆f is frequency-skip, Y (m, k) is the output at time mM and frequency k∆f , the operator ( * ) denotes conjugation.The proper window ω can be selected according to the characteristics of processed signals by considering the amplitude resolution, frequency resolution, time resolution, etc.Note that the Hanning window with a size of 1*2000 is used in this paper.
The short-time periodogram is then calculated by, Similarly, P (m, k) is the output at time index 0 ≤ m ≤ M and frequency index 1 ≤ k ≤ K.
Since it has been well-recognized that the bearing early fault can reflect in the energy distribution over frequency domain, there is a clear motivation to investigate the short-time periodogram of bearing vibration signal, where the power spectral density is continuously extracted over time.However, if the prior knowledge of monitored bearing is not acquirable, how can we extract the fault-related frequency component?This paper proposes mapping the short-time periodogram using pseudo Perron vector which is calculated from graph model.

Graph modeling strategy
A typical graph G consists of a set of nodes and a set of edges, i.e., G = {V, L}.The motivations of employing graph modeling strategy upon time-frequency periodogram are boiled down as follows, • Graph describes/represents the time-frequency periodogram from a global view, so that both the time information and frequency information are taken into account.
• By introducing edges, the correlation between each pair of frequency component can be described, which is the main innovative point of this paper.Concretely, this paper proposes extracting condition indicator via considering the correlations between each pair of frequency components.
a) Regard each frequency sample k = 1, 2, ..., K as a node, and connect each pair of two nodes i and j as an edge L i,j ; b) To reflect the correlation between each pair of nodes, calculate the weight e i,j and sign it to edge L i,j ; c) Algebraically express the graph G as an adjacent matrix α, i.e., α = {e i,j }, which is symmetric and square.
In this paper, the weight of edge is calculated via Eq.( 6), where Cov(•, •) is the covariance.ℵ is mean removal operation.Intuitively, weight e ij describes the covariance between each pair of frequency components.

Condition indicator generation
The condition indicator in this paper is generated by mapping the time-frequency periodogram using pseudo Perron vector.
To illustrate this point, let us start with a quick review of graph spectrum.
In the other hand, according to Perron-Frobenius theorem (Berman & Plemmons, 1994), any nonnegative matrix has a nonnegative principal eigenvalue called the Perron root and any other eigenvalue is strictly smaller than the Perron root in absolute value.The corresponding eigenvector of Perron root is so-called Perron vector which is widely used in many graph based applications such as web page rank, population statis-tics.It is worth mentioning that the Perron-frobenius theorem only applies to nonegative matrix.However, as the elements in α = {e i,j } can be negative value in this paper, the V 1 can not be regard as Perron vector.In this regard, we name V 1 as pseudo Perron vector.
Since pseudo Perron vector V 1 is with a size of 1 × K and time-frequency periodogram P (k, m) is with a size of K ×m, the condition index C 1 (m) is acquired by mapping P (k, m) where m is time index.

Fault decision-making
In order to use the condition indicator C 1 (m) for early fault detection, we first make an assumption upon C 1 (m) as follows, Lemma 1: When bearing operates in normal condition, C 1 (m) is stable.While, the occurrence of fault will cause dramatic fluctuation of C 1 (m).
Note that the experimental validation and theoretical interpretation of Lemma 1 are given in Experiment.3.2 and Appendix.B respectively.
Lemma 1 enables the early fault can be detected using gaussian based control chart.Since condition indicator C 1 (m) is stable when bearing operates normally, we can defined a confidence interval based on a simple assumption that C 1 (m) follows gaussian distribution (Teng, Guoliang, Jie, & Peng, 2018).Any early fault will induce C 1 (m) to exceed this confidence interval.The formulation of fault decision-making is given as follows, H 0 : C 1 (t) ∈ A, i.e., normal operation.
where A = [µ t−1 − nσ t−1 , µ t−1 + nσ t−1 ] is the confidence interval, µ t−1 is the mean value and σ t−1 is the standard deviation computed from past data, i.e., {C 1 (1), C 1 (2), ..., C 1 (t − 1)}.n is the parameter that controls precision and recall.A suitable value of n can be confirmed empirically or by a prior estimation.In the following experiments, n is fixed as 5 empirically.

EXPERIMENT
In this section, we aim to demonstrate the effectiveness and superiority of the proposed framework.Experiment I and Experiment II are conducted on a publicly-available database to answer the following two questions separately: • Can the generated indictor reflect the bearing operating condition effectively.
• Does the presented condition monitoring strategy outperforms the representative methods in this area.

Experiment materials
The bearing vibration data used for testing are provided by Center on Intelligent Maintenance Systems (IMS), University of Cincinnati (Hai, Lee, Jing, & Gang, 2006).As shown in Fig. 4, four bearings were installed on one shaft which was driven by an AC motor and coupled by rub belts.Two accelerometers were mounted on each of bearing to collect the vibration signals in two perpendicular spatial axes.Totally, three times of run-to-failure experiments (Experiment 1∼3) were conducted on this platform.It is worth mentioning that the vibration data snapshots were collected with a interval of 10 minutes and we changed the interval into 300 minutes to decrease the computational load.In this study, as exhibited in Appendix.A, all six run-to-failure signals collected from these experiments are tested and their descriptions are given in Table .1.

Experiment I: Performance on early fault detection
In this Section, the performance of proposed framework is validated.As mentioned in Fig. 2, in the learning phase, the pseudo Perron vector is learned first.Then this vector is used for mapping time-frequency periodogram to acquire the condition indicator in the testing phase.It is worth mentioning that the more similar the learning signal and testing signal are, the better performance will be achieved.Since signal 1 and 2, signal 3 and 4, signal 5 and 6 are with the same type of fault (seen Table.tor is highly related with the degradation process of tested bearing.The tendency of C 1 (m) is consistent with the assumption made in Lemma 1, namely when bearing operates normally, C 1 (m) shows a stable tendency and the early fault can cause a dramatic fluctuation which can be detected by Eq. ( 8).In summary, all of early faults were successfully detected without any false detections.

Experiment II: Comparison with representative works
As mentioned in Section. 1, the main challenge of condition monitoring from time-frequency domain is extracting faultrelated frequency components.To cope with this issue, existing studies (Gerber et al., 2014;Fernández-Francos et al., 2013;Betta et al., 2002) mainly relies on the selection of frequency sub-bands/components which are highly related with the potential faults.Generally, the selection strategy can be classified into following two categories, • Prior knowledge based selection: This strategy requires the knowledge on the system kinematics and exact structure parameters of monitored system.For example, using the parameters of bearing structure and rotating speed, the fault-related frequency f IR ,f OR ,f ball can be calculated.Then, the sub-bands are determined as (Yaguo et al., 2017).
• Experience based selection: In this strategy, the decision about the size of the sub-band is made empirically.Generally, two requirements have to be met for sub-band selection (Fernández-Francos et al., 2013): (a) it should not be sensitive to noise and (b) it should be able to accurately concentrate the diagnosis in the band where the fault is significant.Intuitively, if the size of the sub-band is too narrow, the method will be very sensitive to noise.On the other hand, too wide sub-bands would not allow us to accurately localise the exact band where the fault is evident.Since this paper focus on monitoring the condition without the prior knowledge, we only compare the presented work with the methods in the second category.Concretely, the method reported in (Fernández-Francos et al., 2013) is taken as a competitor , in which the size of sub-band is determined as 200Hz and then the average energies of each band are calculated to form a sub-band energy pattern.Finally, the early fault is detected using one-class SVM upon this energy pattern.We implemented this method on the same database used in this paper.Moreover, apart from using the sub-bands with the size of 200Hz, we also report the detection performance using 50Hz, 100Hz sub-bands.The precision indicates the probability that a detection is a true positive; recall is the probability that a true positive can be detected.
It can be seen from Table .2 that even though sub-band based methods performed well on Recall, these achieved an extremely low performance on P recision i.e., 0.2% with 50Hz sub-bands, 0.3% with 100Hz sub-bands, 0.4% with 200Hz sub-bands.The main reason behind this lies in the existence of noises over frequency domain (Carden & Fanning, 2004).these noises are unrelated with early faults but induce the false detections in the monitoring phase.In comparison, our method achieved a perfect performance in terms of P recision and Recall.It is worth mentioning that our method dramatically outperforms sub-band based method on P recision, which demonstrates that mapping time-frequency periodogram via pseudo Perron vector can effectively distinguish the fault-related frequency information from irregularly fluctuated noises (see Appendix.B for details).

CONCLUSION
In this paper, we have presented a novel framework for bearing condition monitoring based on the condition indicator generated in time-frequency domain.In this framework, the time-frequency periodogram is extracted from the raw vibration signal first.Then, the acquired time-frequency periodogram is mapped by pseudo Perron vector to generate the condition indicator.Finally, the bearing can be monitored via gaussian based control chart.In the experiment, we show the effectiveness of proposed framework, which outperforms sub-band selection based method in this area.
Meanwhile, this paper shows the potential of mapping timefrequency spectrum for early fault prognostic.Alone with this line, this work can be expanded to early fault diagnosis, which is the future work of this study.

Figure 2 .
Figure 2. The flowchart of proposed framework.

Figure 3 .
Figure 3. Illustrative example of graph modeling: (a) Timefrequency periodogram that is assumed to contain 12 frequency components.(b) Modeled graph.(c) Adjacent matrix α of the graph.

Figure 4 .
Figure 4. Bearing test platform for fault detection experiment.
1), in this experiment, signals with the same fault type are taken as training signals mutually.For example, signal 1 was used as learning signal to test signal 2, and vice versa.The condition monitoring results are shown in Fig. 5. From Fig. 5, we can see that the extracted condition indica-
k) output of short-time periodogram e ij edge weight α adjacent matrix for the representation of a graph V 1 pseudo Perron vector C 1 (m) condition index

Table 1 .
The description of all six run-to-failure signals in IMS database.

Table 2 .
The detection result of bearing failure