QUT Digital Repository: http;;//eprints.qut.edu.au

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 statistics 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 )(tx as the observed vibration signal, then a p-

ordered autoregressive model AR(p) is given by

p

k

k ktxaatx

1

0 )()( (1)

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, pkak ,...0, , 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.

kkAIC k 2)log()(2 (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 )(te between the predicted signal )( tx and observed signal )(tx will follow an iid

normal distribution, where

)()()( txtxte (3)

p

k

k ktxaatx

1

0 )()(

(4)

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

2

02 ))(1( xsWL (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.

00 A , ),0max( 1 Attt kWLAA (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.

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.

Shaft Left

bearing

Screw for misalignment

adjustment

380 mm

Right

bearing

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Figure 2. Vibration signals and residuals

a: misalignment = 0mm; b: misalignment = 0.5mm;

c: misalignment = 1.5mm

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

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 ( 0183.0,0018.0 11 ). The

design parameters are deduced as HA= 0.00038, kA = 0.00012, and 3.0 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

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.

6. References

[1] DeCoste D. (2000) Learning envelopes for fault detection and state summarization. IEEE Aerospace Conference, 337-344.

[2] Majanne Y., Lautala P. & Lappalainen R. (1995) Condition monitoring and diagnosis in a solid fuel gasification process. Control Engineering Practice, 3(7), 1017-1021.

[3] Wang W. (2000) A model to determine the optimal critical level and the monitoring intervals in condition-based maintenance. International Journal of Production Research, 38(6), 1425-1436.

[4] Cassady R., Bowden R. O., Liew L. & Pohl E. A. (2000) Combining preventive maintenance and statistical process control: a preliminary investigation. IIE Transactions, 32, 471-478.

[5] Toyota T., Niho T. & Chen P. (2000) Condition monitoring and diagnosis of rotating machinery by Gram-Charlier expansion of vibration signal. Fourth International Conference on Knowledge-based Intelligent Engineering

Systems & Allied Technologies, Brighton, UK, 541-544.

[6] Montgomery D. C. (2001) Introduction to Statistical Quality Control. John Wiley & Sons, New York. [7] Jun C. K. & Suh S. H.(1999) Statistical tool breakage detection schemes based on vibration signals in NC milling.

International Journal of Machine Tools & Manufacture. 39, 1733-1746.

[8] Kullaa J. (2003) Damage detection of the Z24 bridge using control charts. Mechanical Systems and Signal Processing, 17(1), 163-170.

[9] Wang W. Y. & Wong A. K. (2002) Autoregressive model-based gear fault diagnosis, Journal of vibration and acoustics. Transactions of ASME, 124, 1-8.

[10] Fugate M. L., Sohn H. & Farrar C. R. (2001) Vibration-based damage detection using statistical process control. Mechanical Systems and Signal Processing, 15(4), 707-721.

[11] Baydar N., Chen Q. & Ball A. (2002) Detection of incipient tooth defect in helical gears using multivariate statistics. Mechanical Systems and Signal Processing, 15(2), 303-321.

[12] Martinez W. L. & Martinez A. R. (2002) Computational Statistics Handbook with MATLAB. Chapman & Hall/CRC.

[13] Douglas C. M. (1997) Introduction to Statistical Quality Control, Third Edition. John Wiley & Sons Inc., New York.

[14] Wu, Z. and Tian, Y. (2002) Adjusted-loss-function CUSUM Chart for monitoring the process mean and variance. Working paper, School of Mechanical and Production Engineering, Nanyang Technological University,

Singapore.

[15] Zhang S. and Wu Z. (2003) Monitoring the process mean and variance by the WLC scheme with variable sampling intervals. Under review by IEE.

[16] Teutsch R. and Sauer B. (2004) An alternative slicing technique to consider pressure concentrations in non-Hertzian line contacts. Journal of Tribology, 126(3), 436-442.

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Figure 6. WLC control chart under three conditions

a: misalignment = 0mm; b: misalignment = 0.5mm;

c: misalignment = 1.5mm