Statistical condition monitoring based on vibration signals - QUT ePrints

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Zhang, Sheng and Mathew, Joseph and Ma, Lin and Sun, Yong and Mathew, Avin D. (2004) Statistical condition monitoring based on vibration signals. In Vyas, N. S. and Rao, J. S. and Mathew, Joseph and Ma, Lin and Raghuram, V., Eds. Proceedings VETOMAC-3 & ACSIM-2004, pages pp. 1238-1243, New Delhi, India.

© Copyright 2004 (please consult author)

Statistical condition monitoring based on vibration signals Sheng Zhang, Joseph Mathew, Lin Ma, Yong Sun and Avin Mathew CRC for Integrated Engineering Asset Management, School of Mechanical, Manufacturing and Medical Engineering, Queensland University of Technology, Brisbane, QLD 4001, Australia Designing control limits for condition monitoring is an important aspect of setting maintenance schedules and has been virtually ignored by researchers to date. This paper proposes a novel statistical process control tool, the Weighted Loss function CUSUM (WLC) chart, for the detection of condition variation. The control limit was designed using baseline condition data, where the process was fitted by an autoregressive model and the residuals were used as the chart statistic. The condition variation is reflected by the changes of mean and variance of the statistic’s distribution against baseline condition, which can be detected by a single WLC chart. The approach was evaluated using a case study which showed that the chart can detect faulty conditions as well as their severity. The proposed approach has the advantage of requiring healthy baseline data only for the design of condition classifiers. It is applicable in numerous practical situations where data from faulty conditions are unavailable. 1. Introduction Condition monitoring involves fault diagnosis and condition prognosis. Few studies have focused the design of condition control limits – an important topic given that control limits initiate maintenance activities and trigger diagnostic tasks in automatic fault detection. Ideally, control limits should be established such that false alarms are not triggered under baseline conditions, whilst faults can be detected without delay. In practice, the ‘red-line’ is widely used as the control limit which comes from standards or application experience. Such control limits have drawbacks in that false alarms and miss alarms are difficult to control. Some work has been directed to the effort of determining the control limit dynamically [1-2] with limited success. The selection of parameters sensitive to condition is critical for condition monitoring. Based on the parameters, it is desirable to design mathematically derived control limits. The ideas from statistical process control, which aims to detect the mean shift and/or variance shift based on assumptions of the statistical distribution, have been utilised by condition monitoring applications in recent years [3-4]. The control limit design for control charts provides a mathematically controllable approach to balance the Type I error (the probability of fault triggering under healthy conditions) and Type II error (the probability of a healthy condition being indicated when faults exist). A suitable statistic is required for control chart operation. The statistic is usually assumed to follow a certain probability distribution. The identically and independently distributed (iid) normal distribution is commonly adopted. Some time-domain features of vibration signals under healthy conditions follow normal distributions [5]. However, these features are usually auto-correlated and do not meet the requirement of iid. At the same time, features in the frequency-domain and time-frequency domain are not used for statistical condition monitoring purposes. The violation of the assumed iid normal distribution leads to false alarms [6]. Several approaches have been investigated to derive statistics satisfying the requirement. One straightforward method is to average the auto-correlated signals. The resultant data approximate an iid normal distribution according to the central limit theorem. Jun and Suh [7] applied time-domain averaged vibration signals to the Shewhart X , EWMA (exponentially weighted moving average) and adaptive EWMA control charts for tool breakage detection. Kullaa [8] used the changes of modal parameters as statistics and applied them to univariate and multivariate X , CUSUM (cumulative sum) and EWMA charts for structural condition monitoring. Both authors only considered mean shifts of the statistical distribution. Another way to derive the required statistics is to fit the auto-correlated process using a time series model [9]. The residuals between the fitted signals and observed signals generally follow an iid normal distribution and can be used in control chart operation. Fugate et al [10] applied the residuals to the joint X and S charts to monitor a concrete bridge column for both mean and variance shifts. As the condition is commonly indicated by multi-parameters, Baydar et al [11] presented a multivariate statistical analysis methodology for gearbox monitoring, where principle component analysis was used to reduce the feature dimension. The Q and T2 statistics were adopted as condition indicators. In this paper, an autoregressive model introduced in Section 2 is used to approximate the vibration time series signals from healthy conditions. The residuals follow the iid normal distribution. The deterioration of condition results

in changes of the residual distribution, reflected by both mean shift and variance shift. Section 3 presents a novel Weighted Loss function CUSUM (WLC) chart capable of sensing both types of shifts. A case study in Section 4 shows that the control chart is sensitive to the detection of faulty conditions and fault severity. Section 5 presents the conclusion. 2. Residual Model for Condition Monitoring Vibration signals are commonly used in deciphering equipment condition. Autoregressive models have been used in the past to approximate the underlying condition behaviour. Denote x(t ) as the observed vibration signal, then a pordered autoregressive model AR(p) is given by p

x (t )  a 0 

 a x (t  k )   k

(1)

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Where k is the time delay,  is the unobserved Gaussian white noise with zero mean and standard deviation  . Several methods are available to derive the model coefficients, ak , k  0,...p , such as the Yule-Walker method which is solved by the Levinson-Durbin recursion [9]. It is essential to select an appropriate order of p for the fitting of the underlying model, because a smaller p under-fits the process while a larger p over-fits the process. Minimising the Akaike Information Criterion (AIC) given below is an acceptable approach for the choice of the model order.

AIC(k )  log( k2 )  2k

(2)

Where the first item decreases with an increasing order k, and the second item restrains the increase of order. A well fitted AR model reflects the underlying condition behaviour and is capable of forecasting the condition progress. Furthermore, the residual e(t ) between the predicted signal xˆ (t ) and observed signal x(t ) will follow an iid normal distribution, where e(t )  x(t )  xˆ (t )

 x (t )  a 0 

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The properties of the residual distribution can be tested via its estimated probability density function [12]. When a change of the condition occurs, e.g. from healthy to faulty, the AR model derived from the baseline healthy condition does not fit the changed condition. Using the AR model to approximate the changed condition will result in a changed residual distribution, which is reflected by a mean shift, a variance shift, or more often by both shifts simultaneously. Applying a control chart to the residuals for detection of mean and variance shifts provides a novel approach to detect the condition variation. 3. WLC Control Chart Control charts are a powerful tool in statistical process control, providing an important role in product and process industries for quality assurance and process stabilisation [13]. Different control charts have been developed to detect process variation, e.g. mean shift and/or variance shift. For mean shift detection, the Shewhat X chart is prevalent due to its ease of operation. The EWMA and CUSUM charts are commonly used for relatively small shift detection. The S chart and R chart are suitable for variance shift detection. The joint use of X and S (or R) charts can detect mean and variance shifts concurrently. A recently developed chart, the Weighted Loss function CUSUM (WLC) chart, aims at detecting both types of shifts in one chart and has demonstrated success [14-15]. The WLC chart is based on the CUSUM schema and uses the Weighted Loss function (WL) as the statistic

WL  s 2  (1   )( x   0 ) 2

(5)

Where, x and s are the sample mean and sample standard deviation, 0 is the in-control (healthy condition) mean, and  (0 ≤  ≤ 1) is the weighting factor. The statistic to be updated and plotted in CUSUM manner for the WLC chart is At.

A0  0 , At  max(0, At 1  WLt  k A )

(6)

Where, kA is the reference parameter. The WLC chart is capable of detecting both mean shift and variance shift and outperforms the joint X &S charts in terms of both ease of chart operation and chart performance. To design a WLC chart is to decide the control limit HA, the reference parameter kA and the weighting factor  at the predefined shift point, given the sample size n and allowable Type I error. The designed chart is optimal for predefined shift detection and thus is deemed to be optimal for all other shifts while maintaining the designated Type I error. A detailed procedure for the WLC chart design can be found in [14-15] where the charting process is described using the Markov chain method. 4. Case Study Misalignment is a relatively common fault in rotating machines. A marginal level of misalignment does not usually result in a significant variation of vibration amplitude. It however will deteriorate long term machine performance. For example, misalignment generates a force which is applied to the supported bearing. When the force increases, the life expectancy of the bearing decreases by the cube of the force increment [16]. Therefore, developing a condition indicator for shaft misalignment detection is a handy tool. The test rig in Figure 1 was employed to simulate the offset misalignment scenario. The shaft with a wheel is supported by two rolling ball bearings, and driven by a motor through a pair of flexible couplings. A screw mechanism is designed to adjust the left bearing horizontally and introduce misalignment. Screw for misalignment Left adjustment bearing

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Figure 1. Test rig for experiments The data acquisition system includes an ENDEVCO 256HX-10 piezoelectric accelerometer, a PCB 482A20 ICP signal conditioner, a KROHN-HITE 3202 Filter, a Daq-308 I/O card, and a laptop with DaqEZ Pro data acquisition application software package. The accelerometer was mounted on the housing of the right bearing. The acceleration signals were acquired when the shaft was adjusted to three positions, misalignment = 0mm (healthy condition), misalignment = 0.5mm (slightly faulty condition), and misalignment = 1.5mm (severely faulty condition). The rotating speed was 960rpm and the sampling frequency was 5000Hz with a cut off low pass frequency of 1000Hz. Figures 2 (a), (b) and (c) display three signals over five revolutions by the blue line. The amplitude of vibration signals increases with increasing misalignment and becomes especially significant for large levels of misalignment. The signal acquired when the shaft was well aligned is deemed to represent the healthy machine condition. The data of the signal generally follow a normal distribution. However, they are auto-correlated. The AR method was used to model the baseline healthy condition and obtain the residuals with the expected property of iid. Taking a signal with data length of 1560 (five revolutions), an AR(1) model was searched with minimum AIC. The residuals are displayed using black line in Figure 2 together with the acquired signals. The probability density function of the residuals was approximated by the averaged shifted histogram (ASH) method [12] and illustrated in Figure 3 (a). The property of normal distribution was tested and displayed in Figure 3(b). The dashed line is a linear fit to join the first and third quartiles of the ordered residuals. A linear plot explains that the residuals can be modelled by a normal distribution. The auto-correlation of the residuals was calculated and displayed in Figure 3 (c). It is highly auto-correlated only at lag = 0. Therefore, the residuals derived from the AR(1) model under baseline healthy condition follow the iid normal distribution and can be used as a statistic for WLC control charts.

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Figure 3. Signal test for misalignment = 0mm a: estimated pdf; b: normal distribution test c: autocorrelation test

The AR(1) model was applied to the signals under misalignment conditions and the residuals were produced. For the two cases of misalignments, Figures 2 (b) and (c) display the acquired signals and residuals. Figures 4-5 display similar information against Figure 3. The residual distribution has changed in terms of both mean and variance for the two cases of misalignments. For example, under healthy conditions,  0 = 0.0007 and  0 = 0.0130. With misalignment = 0.5mm, 1 =0.0018 and  1 =0.0183; and with misalignment = 1.5mm,  2 =0.0015 and  2 =0.0277. Figures 4-5 (b) and (c) show that the residuals do not follow the iid normal distribution and indicate a change of conditions.

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Figure 4. Signal test for misalignment = 0.5mm a: estimated pdf; b: normal distribution test c: autocorrelation test

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Figure 5. Signal test for misalignment = 1.5mm a: estimated pdf; b: normal distribution test c: autocorrelation test

The residuals under healthy conditions were used to design the WLC control chart with an allowable Type I error of 0.001, and sample size n = 4. The chart was designed optimally for detection of shift ( 1  0.0018,  1  0.0183 ). The design parameters are deduced as HA= 0.00038, kA = 0.00012, and   0.3 for the WLC chart. The residuals are presented on the WLC control chart. Figures 6 (a), (b) and (c) display the three conditions. There are no false alarm points beyond the control limit for healthy conditions despite the designed 0.001 Type I error. For the two cases of misalignment, a significant number of points lie beyond the control limit, indicating that the condition is faulty. It was noted that 14 points (out-of-control points) lay beyond the control limit for slight misalignment and 89 points were beyond the control limit for significant misalignment. The number of out-of-control points therefore is a direct indicator of the severity of faults. 5. Conclusion

6. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16]

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The Weighted Loss function CUSUM (WLC) control chart can detect both mean and variance shifts and provides a favourable approach for control limit design. It was successfully used for condition monitoring with shaft misalignment being the case study. The residuals between the fitted vibration signals using autoregressive models and the observed vibration signals were a suitable statistic for WLC charts. It was shown that the chart can distinguish between healthy and faulty conditions, and indicate fault severity. The control charts are an unsupervised means to identify the conditions since their design requires only data from healthy conditions, which are applicable in situations where faulty condition data are lacking.

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Figure 6. WLC control chart under three conditions a: misalignment = 0mm; b: misalignment = 0.5mm; c: misalignment = 1.5mm

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