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Mechanical fault detection using fuzzy index fusion
Tony Boutrosa and Ming Liang
aDepartment of Mechanical Engineering, University of Ottawa, 770 King Edward Avenue, Ottawa, Ont., Canada K1N 6N5
Received 22 June 2006;? revised 26 December 2006;? accepted 3 January 2007.? Available online 23 January 2007.
Abstract
This paper reports a simple, effective and robust fusion approach based on fuzzy logic and Sugeno-style inference engine. Using this method, four condition-monitoring indicators, developed for detection of transient and gradual abnormalities, are fused into one single comprehensive fuzzy fused index (FFI) for reliable machinery health assessment. This approach has been successfully tested and validated in two different applications: tool condition monitoring in milling operations and bearing condition assessment. The FFI differentiates clearly between the normal and abnormal conditions using the same fuzzy rule base. This certainly shows the versatility and robustness of the FFI. As the FFI value always falls between zero and one, it facilitates threshold setting in monitoring conditions of different tools or machinery components. Our experimental study also indicates that the FFI is sensitive to fault severity, capable of differentiating damages caused by an identical fault at different bearing components, but not susceptible to load changes.
Keywords: Condition indicators; Fuzzy fusion; Sugeno inference engine; Tool condition; Bearing condition
Article Outline
Machinery fault detection and machining process monitoring have attracted considerable attention. These tasks have become increasingly difficult due to the complexity of machine structure and operation dynamics. Over the last few decades, many different sensors and condition indicators have been developed in an attempt to achieve more reliable results for different monitoring tasks.
For machinery fault detection, Collacott [1] used the probability density and kurtosis of vibration signalfor bearing defect identification in an early study. It was found that the probability density of the acceleration of a bearing in good condition has a Gaussian distribution, whereas a damaged bearing resulted in a non-Gaussian distribution with dominant tails. Along this line, Dyer and Stewart [2] also used kurtosis for bearing defect detection based on vibration signal. For an undamaged bearing with Gaussian distribution, the kurtosis value was found equal to three. A value greater than three was judged as an indication of impeding failure. However, one disadvantage was noted: the kurtosis value could come down to the level of a normal bearing even when the damage was well advanced. Later, Miyachi and Seki [3] extracted the root-mean-square (r.m.s.) and crest factor from vibration signal to monitor the defects in ball bearings. However, the results were not very successful. Liu and Mengel [4] used the peak amplitude in the frequency domain, peak r.m.s. and the power spectrum as indirect indices for monitoring ball bearing vibration. Heng and Nor [5] reported the application of sound pressure and vibration signals to the detection of bearing faults using a statistical analysis method. The parameters considered in their study included the r.m.s., crest factor and kurtosis. Results obtained through experiments revealed that the statistical parameters were subject to the influence of shaft speed. Recently, Baydar and Ball [6] examined the use of acoustic signal along with vibration signals for monitoring various local faults in a gearbox using the wavelet transform. Two commonly encountered local faults were simulated: tooth breakage and tooth crack. The results suggested that acoustic signals were very effective for the early detection of faults. However, the influence of load variation on the fault detection capability of the acoustic approach was not considered.
For machining process and tool condition monitoring, the task could be more difficult due to the nonlinear process caused by the interaction of the dynamics of material removal, the dynamics of machine tool and machine tool drive [7]. Inasaki [8] developed a monitoring and control system for grinding processes. The system utilized acoustic emission (AE) and power sensors to monitor the grinding process and to construct a control database. Everson and Cheraghi [9] investigated the correlation between the quality of a hole drilled in steel and the AE signal parameters. The AE energy, number of peak amplitudes above a certain threshold and the r.m.s. were used in this investigation. Experimental work was conducted to validate the method. They observed that the AE energy was a good measure but the peak amplitude count as a condition indicator was inefficient in certain cases where signal was short. There is a rich body of literature on tool condition monitoring. Some of the well-cited studies include the use of AE for tool condition assessment [10], [11] and [12], joint use of AE sensor and a force sensor to monitor the tool condition in turning [13], vibration-based tool wear monitoring [14], [15] and [16], force-based tool failure detection [17], [18] and [19], and current-based tool fracture monitoring [20] and [21].
Various condition indicators were used by different researchers. For example, the r.m.s. of the vibration signal in different frequency bands [14], coherence value of vibration signals from two accelerometers [15], r.m.s. and energy of AE signal [9], wavelet coefficient of current or AE signals [12] and [22], and fuzzy transition probability [23] have been used for machining process or tool condition monitoring. The condition indices for machinery monitoring include ratio of a wavelet reference level and its mean calculated from the vibration data [24], normalized harmonics content of the residuals of motor current [25], sound intensity [26], to name just a few.
To synthesize data from different sources, sensor fusion has been well studied [13], [27], [28] and [29]. This paper focuses on the fusion of different indices derived from the same data source collected by a single sensor. Each of the indices has its own merits and shortcomings. The development of an integrated index incorporating all the indices would provide a simple and reliable solution to situations where multiple sensors cannot be conveniently applied.
As noted above, several complementary indices can be developed from the same signal. For instance, one fault index may be suitable for capturing transient incidents and another could be sensitive to a trend of gradual changes. Obviously, using only one of the two can be misleading if both abrupt faults and gradual deterioration are important concerns. Additionally, the enormous on-line information from different monitoring indices requires a tremendous amount of efforts and time to process, comprehend and analyze if each index is considered separately. For similar reasons, Goebel [30] proposed a system for the fusion of diagnostic information. This fusion method deals mainly with conflict resolution and fault coverage discrepancies. A hierarchal weight manipulation approach was employed to refine the output.
Another important aspect is the difficulty in threshold setting when many different indices are used, each with its own threshold setting scheme. This is further complicated by the application-dependent nature of many indices. For example, the threshold values of the widely used r.m.s. of vibration signal could differ substantially in detecting a bearing ball defect and a bearing outer race anomaly. The development of a single synthesized, dimensionless and normalized index would lead to a welcoming relief from the threshold setting chore.
For the above reasons, a fusion approach is proposed in this paper based on fuzzy logic and the Sugeno style inference engine. The method turns the effects of several condition indicators into a single comprehensive fuzzy fused index (FFI) for quick fault monitoring. As the values of FFI are bounded by 0 and 1, threshold setting can be simplified accordingly. The details of the condition indices, fusion process and experimental work are described in the following sections.
To improve the detection efficiency of transient (i.e. short-duration) and gradually developed anomalies, the monitoring data were grouped into “mini-groups” and “sub-groups”. Each “mini-group” contains K samples and each “sub-group” is composed of J “mini-groups”. The size of the “mini-group” defines the search resolution for transient anomalies whereas the “sub-group” size represents the minimum required duration for decision-making. Four indicators are derived based on the power, standard deviation and correlation factor formulas. Denoting k as the index for data samples (k=1, 2?,…,?K), j for “mini-groups” (j=1,2?,…,?J) and i for “sub-groups” (i=1,2?,…,?I), the four indices are derived as follows.
The power condition indicator (PCI) reflects the power fluctuation magnitude of each mini-group within a sub-group. It represents the normalized deviation of each mini-group power from the average level with respect to the variation range. For each mini-group, the PCI is obtained as
(1)
where Pij is the power of mini-group j in sub-group i, defines the mean “mini-group” power within the same “sub-group”, Pi,max and Pi,min represent the maximum and minimum “mini-group” powers within “sub-group” i. They are given as follows:
(2)
(3)
where VLHP,ij(k) represents the data sample k in the jth mini-group of ith sub-group obtained after bandpass filtering.
Obviously, the maximum value of the PCI, i.e., MPCI, can capture the abrupt changes in the signal and hence is a good indicator of suddenly developed events. It is calculated for every sub-group as follows:
(4)
Though standard deviation is a direct measure of dispersion of the signal and has been used directly for condition monitoring, a normalized variation of standard deviations would be a less situation-dependent indicator of transient events. This indicator is defined as
(5)
where σij is the standard deviation of mini-group power, σi,max and σi,min the maximum and minimum “mini-group” standard deviation within “sub-group” i, and the mean “min-group” standard deviation with respect to sub-group i. These parameters are calculated as follows:
(6)
(7)
with
(8)
Similar to MPCI, the maximum value of SDCI would provide more clear sign of transient incidents which is calculated as
(9)
It should also be noted that an indicator stemming from standard deviation cannot be applied alone since it just represents the percent of deviation in a signal with respect to its average level. For this reason, we suggest that MSDCI be applied along with MPCI, an indicator of signal strength.
The power correlation factor (PCF) defines the variation of the instantaneous power mean, calculated at sub-group level, with respect to the reference power level computed at initial state that is usually assumed “normal”. It can be expressed as
(10)
where SSVp,i is the sum of the squared deviations with respect to the varying mean and SSRp,i is the sum of the squared errors of power relative to a reference mean power, , i.e., the average power of the first Jb mini-groups, all for sub-group i. They are obtained as follows:
(11)
(12)
where is the power mean over a window of J mini-groups. and are written as
(13a)
(13b)
If the state of the system under monitoring does not change with time, SSVp,i will be equal to SSRp,i. As the system starts to deteriorate, the average power of the mini-groups in a sub-group starts to increase accordingly (i.e., the sound pressure level increases in a milling operation as tool wear size increases or vibration level steps up in the bearing in the presence of a fault). Therefore, SSVp,i, the sum of the squared deviations of the mini-groups power with respect to the mean power (i.e., ) will be less than SSRp,i, the squared deviations with respect to the reference level (i.e., ) showing as such an indication about degradation. Consequently, the PCF, which was originally equal to one, starts to decrease towards zero.
Similar to PCF, the standard deviation correlation factor (SDCF) is developed to study the variation of the standard deviation with respect to a selected reference level (e.g., a level computed for normal state). PCF concentrates on the energy content of the signal, whereas the aim of SDCF is to analyze the fluctuations of the signal due to anomalies. As the system under monitoring degrades gradually, the fluctuations in the signal become more frequent and repeatable (e.g. fluctuations due to rubbing in case of tool wear or impacting due to bearing faults). This implies an increase in the dynamic average of the standard deviation and consequently a reduction in the correlation factor. The SDCF is obtained by
(14)
where
(15)
(16)
where is the mean value of the standard deviation over a window of J mini-groups and is the reference mean of the standard deviation of the first Jb mini-groups in sub-group i. They are written as
(17a)
(17b)
Obviously, the use of any one of them alone could mislead the fault detection decision but the simultaneous application of them in parallel would cause confusion and slow down the detection decision. In addition, the four indices also require different thresholds. This situation, along with the complementary nature of the four indices, motivates us to develop fuzzy fusion approach (Fig. 1) to take advantage of all the indices. To be consistent in the detection scheme with respect to the interval [0, 1] (i.e., the perfect normal condition corresponds to 0 and the worst abnormal condition corresponds to 1, not otherwise), we use the complements of PCF and SDCF, denoted as CPCF and CSDCF, respectively. The four monitoring indices, MPCI, MSDCI, CPCF and CSDCF, as shown in Fig. 1 are defined with respect to each “sub-group” and are all normalized values (i.e. values varying between 0 and 1). An index value close to 0 indicates a normal condition whereas a value approaching 1 illustrates an abnormal case. As illustrated above, to improve the fault detection efficiency and reduce unnecessary effort in selecting threshold for each monitoring index, the four monitoring indicators are fused into one single fuzzy output as described in the following sections.
To associate crisp inputs with fuzzy sets, membership functions have to be defined. For computational efficiency, the triangular membership functions are used for inputs and spike membership functions are employed for outputs. In this study, four fuzzy sets are proposed for the inputs and three others for the outputs (Fig. 2). The input fuzzy sets are selected as follows: ZV, zero value; SV, small value; MV, medium value; BV, big value. On the other hand, the output fuzzy sets are defined in this order: NR, normal range; MR, middle range; and AR, abnormal range.
One of the most important components of this fuzzy fusion module is the detection rule base. The fuzzy rules, which determine the output norm based on the inputs, represent the knowledge gained through learning and experience of domain experts. Each of these rules is written as an IF (ranges of the four condition indicators)—THEN (consequence, i.e., machine condition). As shown in Figs. 2 and 3, there are four input fuzzy sets associated with four condition indicators. This leads to a total of 256 rules (44). For illustration purpose, only part of the rules is presented in Table 2. The complete list of the rules is available in [31]. These rules are used for both machining tool and bearing condition monitoring.
Each set of condition indicators generally activates several fuzzy rules. For example, consider Fig. 3. If MPCI and MSDCI fall respectively between MV and BV, and SV and MV, CPCF triggers ZV and SV and CSDCF activates SV and MV, eight points in total will be triggered at different heights (h). This leads to 16 combinations of fuzzy sets. Referring to Table 2, the To explain briefly the meaning of these rules, we consider for example rule (166). It can be translated as: if the MPCI and the maximum SDCI (MSDCI) fall within a medium range (MV) and the complements of the power and standard-deviation correlation factors have SV, the machine condition will be considered to be in the AR. All the rules activated by a set of inputs, i.e., MPCI, MSDCI, CPCF, and CSDCF, should be taken into consideration and hence jointly contribute to the strength of the final output, i.e., FFI. The FFI will be defined by the fuzzy inference engine as illustrated in the next section.
The role of the fuzzy inference engine is to perform the fuzzy operations necessary for the determination of the FFI. During this phase, the fused index is computed based on the fuzzy inputs and the activated rules. To reduce the computing time for on-line applications, the Sugeno-type inference engine is selected in this study. This type of engines provides a very quick response comparing with the well-known Mandani-type inference engine. The difference between the two techniques is mainly due to the selection of the output membership function. With the Sugeno method, the output membership is presented by a spike (Fig. 2b) instead of a complete triangle used by the Mandani inference engine. In addition to its computational efficiency, the Sugeno method has other attractive characteristics such as guaranteed continuity of the output surface and appealing effectiveness to work with optimization and adaptive techniques [32].
The output, i.e. FFI, is simply the weighted (with respect to the spike heights truncated according to the inference rule) average of the spike locations [33], i.e.:
(18)
where Lr is the location of the output spike associated with rule r, Hr is the minimum height (or membership) associated with rule r and R is the total number of activated rules.
It should be noted that Eq. (18) is a general expression of FFI and its application is not limited to the fusion of the four fault indices mentioned above. Additional fault indices can be easily incorporated if necessary.
References
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