CONDITION MONITORING OF ROTATING MACHINERY USING CYCLIC AUTOREGRESSIVE MODELS Andrew C MCCORMICK and Asoke K NANDI Signal Processing Division, Department of Electronic and Electrical Engineering University of Strathclyde, 204 George Street, Glasgow G1 1XW U.K. Tel: +44-141-548-2663 ? Fax: +44-141-552-2478 ? E-mail
[email protected] ABSTRACT So far the modeling of vibrations for fault diagnosis in machinery has been achieved using time invariant autoregressive models which assume the signal to be stationary. The periodic nature of rotating machinery however gives rise to vibrations which are cyclostationary. This paper applies a periodically time varying autoregressive lter to better model such signals. Experimental results indicate that a model based approach can be applied to fault diagnosis where vibration data from faulty machines are available and to fault detection where only vibration data from normal operation are available. 1 INTRODUCTION
The use of vibration data in the monitoring of machine condition involves the use of signal processing to extract condition dependent features from the signal. From these features, the condition can be classi ed using discriminant analysis techniques ranging from thresholding to arti cial neural networks [6, 5]. By using autoregressive techniques to model the vibrations [2], a change in machine condition can be detected by a change in the di erence between the predicted signal and the measured signal. As a machine's condition deteriorates through wear, the model will not t the signal so well and consequently the error will increase. By modeling signals for normal and fault conditions, it is possible to classify the machine's condition using a simple best t algorithm. While the vibrations of rotating machinery can be assumed to be stationary, the periodic nature of such machinery gives rise to signals which are more appropriately considered as cyclostationary [4]. This can be exploited by extending linear time-invariant autoregressive models of the signal into periodic time-variant models which better represent the signal. In this paper the extension of linear AR models to linear periodic AR models is investigated, and applied to the modeling of vibration time series collected from a small rotating machine for the purpose of fault detection and diagnosis. A small Bently-Nevada machine set consisting of a shaft with a ywheel attached was used to generate the vibration data. It was possible to con gure the machine to pro-
duce several fault conditions. The vibrations were measured using accelerometers attached to bearing blocks and recorded using a DSP card attached to a PC and analyzed using MATLAB. In section 2 the cyclostationary nature of rotating machine vibrations is discussed. In section 3 the extension of linear autoregressive modeling to incorporate periodic components and its application to fault detection is shown. Experimental results based on real data are presented in section 4 and conclusions are given in section 5.
2 CYCLOSTATIONARY VIBRATIONS
A process x(t) is said to be wide sense cyclostationary if its autocorrelation rxx (t; ) = E fx(t + 2 )x(t ? 2 )g varies periodically with time: rxx(t; ) = rxx (t + kT; );
k = 0; 1; 2:::
(1)
where t represents time, represents a time-shift and T is the fundamental period of some underlying periodic
physical process. The rotation of machinery provides a periodicity in vibration signals which can interact in a non-linear manner with random external events to produce signals which are cyclostationary; for example amplitude modulation exists in gear-box vibrations [7]. The degree of second-order cyclostationary components, or spectrally correlated components can be seen in the Spectral Correlation Density Function (SCDF) which is the Fourier transform of the time varying autocorrelation with respect to time t and time shift : (f ) = Sxx
Z1 Z1Efx(t + )x(t ? )ge? +
+
?1 ?1
2
2
j 2( t+f ) dtd
(2) If the process is strictly stationary then the SCDF will be zero everywhere except along the = 0 line where the SCDF will be the power spectrum of the signal. With rotating machinery it is easy to measure a once per revolution pulse and therefore samples occurring at the same position in each revolution can be easily correlated. Since the process is not ergodic, a simple time average is not sucient as an expectation operation however estimates of the autocorrelation separated by one revolution in time
can be considered as members of an ensemble and averaged to produce a better estimate. From this estimate of the autocorrelation the SCDF can be calculated. The SCDF of the vibrations of the machine in its normal condition is shown in gure 1. Clearly there exist components which do not lie on the = 0 line indicating that the signal is cyclostationary. 6
x 10 2 1.5
The time varying coecients can be expanded into a Fourier series to give a periodic autoregressive model with cyclic components existing at harmonics of the rotation frequency: y(n) =
X a y(n ? k) (6) X X a cos( 2 n )y(n ? k) + T X X b sin( 2 n )y(n ? k) + (n) + N
k=1
k;0)
(
N M
k=1 =1 N M
1 0.5 0
k=1 =1
10 5 0 −5 f
−10
−10
−15
−20
−5
0
5
10
15
20
alpha
Fig. 1: SCDF of Machine Vibrations
3 AUTOREGRESSIVE MODELING FAULT DETECTION
FOR
If a stationary time series can be modeled as an Nth-order linear time-invariant autoregressive process: y(n) =
X a y(n ? k) + (n) N
k=1
k
(3)
then if the model can be identi ed, it can be used as a one-step-ahead predictor
X y^(n) = a y(n ? k) N
k=1
k
(4)
with an error which depends on the statistics of (n). If the signal is a stationary linear autoregressive process then an AR model will provide the minimum mean square error in prediction. There exist two approaches to exploiting this error: 1) if it is only possible to obtain vibrations from the machine working in the normal condition then if the variance of the prediction error increases signi cantly beyond the variance when the model was designed, this could indicate a fault; and 2) if vibration data is available for the machine in faulty conditions then it is possible to model normal and fault conditions. The condition can be assigned to that of the model which predicts the signal most accurately. If the time series is non-stationary, a time-invariant model will not provide the optimal predictor. If the time series is cyclostationary, a periodically time varying model [8] may provide a better predictor; for example
X y(n) = h (n(mod T))y(n ? k) + (n) N
k=1
k
(5)
k; )
(
k; )
(
T
The SCDF gives some indication of which M frequencies to include in the model. However the model order, N , is not so easily determined. The approach used to determine both the model order N and number of cyclic sections M was to start with a linear AR model of the signal. The Akaike Information Criterion [1] was used choose the model order N and the coecients were determined from the time-invariant part of the autocorrelation using the Yule-Walker equations. Cyclic sections of the same order were then added for each harmonic frequency T , one at a time, with the coecients being determined using a least mean square optimization algorithm. Cyclic sections were added until the additional sections no longer reduced the residual. This approach results in a model which is an improvement on the AR model, however it may not be the optimal periodic AR model.
4 EXPERIMENTAL RESULTS
The vibration time series were recorded using a small motor test set which could simulate two faults: unbalance and rubbing. The unbalance was introduced by attaching a 1g weight to the ywheel; the rubbing was introduced by screwing a threaded brass rod through a threaded hole in a bearing block until it rubbed against the shaft. These faults could be introduced independently allowing four machine conditions to be simulated: NN (no faults); NR (rub fault); WN (weight attached); WR (weight and rub faults applied). Time series showing six revolutions for each of these conditions are plotted in gure 2. To test the use of periodic AR models for condition monitoring, time series of 5000 samples representing over 30 revolutions of the machine for each condition were used to determine models for each of the four conditions. Table 1 shows the model order selected, the number of frequency components needed to be added, the mean magnitude error for the AR model and for the periodic AR model. Clearly the periodic models produce smaller errors than their AR counterparts and consequently are better predictors of the signal. The models for the rub conditions are of a lower order than the non-rub conditions and this may be the cause of their higher average prediction error.
V/mV NN V/mV NR V/mV WN V/mV WR
2000 0 −2000 0 2000
100
200
300
400
500
600
700
800
900
100
200
300
400
500
600
700
800
900
100
200
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0 −2000 0 2000 0 −2000 0 2000 0 −2000 0
Fig. 2: Vibration Time Series for Machine Conditions Table 1: Periodic Signal Models
Condition NN NR WN WR Model Order 67 27 39 16 Frequency components 10 5 5 5 AR error 11.24 14.59 8.83 35.77 Periodic AR error 6.43 8.95 5.25 16.91 To test the usefulness of these models in predicting the machine's performance, an additional time series of 15000 samples recorded for each condition was input into all four models and the average prediction error calculated. The mean magnitude error in predicting each time series using each model shown in table 2. Table 2: Prediction Errors of Test Time Series
Time Series NN NR WN WR
NN 6.79 52.3 21.8 48.4
Model NR WN 9.71 21.7 13.8 35.9 17.9 6.84 24.8 43.6
WR 28.6 62.7 19.6 18.9
By reading across the columns, the smallest prediction error for each time series can be used to diagnose the condition. In these cases it is clear that the condition can be classi ed correctly. Using a model to detect a change from the known condition to some new condition does not however appear to work so convincingly. By reading each column in isolation, the prediction error for each time series can be compared with the prediction error for the training time series. A signi cant change should indicate that the machine is in a di erent condition, however the de nition of what is a signi cant change is entirely subjective. Using this approach to decide whether the test time series are due to di erent conditions for both the NN and WN models, the error appears to give a good indication when the machine being in a di erent condition. The prediction error for data from the same condition is only
slightly larger than that of the training data, but in all the other cases, the prediction error has signi cantly increased to well over three times that of the training data. However if this approach were used with the models for the NR and WR conditions, a failure to detect some of the di erent conditions would result. This is likely to be because of the poorer average prediction error obtained for the training time series. It is also possible that the extra uncertainty introduced by the rubbing makes these time series less easy to predict and consequently the models are more tolerant of changes in condition. The model orders for these two conditions are signi cantly smaller than those for the other two conditions and this may be the cause of the higher prediction error.
5 CONCLUSIONS
It has been shown that when autoregressively modeling the vibrations of rotating machinery the cyclostationarity of the signals can be exploited to improve the models by adding periodicaly varying sections. A method for deriving periodic-AR models by extending an existing AR model has been used in modeling vibrations for fault detection. Using these models classi cation of machine condition using models for normal and fault conditions has been demonstrated. The application of this model based approach to situations where it is only possible to obtain data for the system under normal conditions has been discussed. The method used to chose the model orders may not be the best for periodic models as it is based upon the stationary statistics of the vibrations and is only one of many possible model order selection criteria [3]. Therefore further investigation of model order selection based upon periodic features is desirable.
6 ACKNOWLEDGMENT
The authors wish to acknowledge the support of the EPSRC and DRA Winfrith.
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[5] A. C. McCormick and A. K. Nandi. Classi cation of rotating machine condition using arti cial neural networks. Proceedings of The Institution of Mechanical Engineers: Part C. To be published in 1997. [6] A. C. McCormick and A. K. Nandi. Real time classi cation of rotating shaft loading conditions using arti cial neural networks. IEEE Transactions on Neural Networks. To be published in 1997. [7] P. D. McFadden and J. D. Smith. A signal processing technique for detecting local defects in a gear from the signal average of the vibration. Proceedings of the Institution of Mechanical Engineers: Part C, 199:287{ 292, 1985. [8] M. Pagano. Periodic and multiple autoregression. Annals of Statistics, 6:1310{1317, 1978.
