NONLINEARITY DETECTION FOR CONDITION MONITORING USING HIGHER-ORDER STATISTICS J W A Fackrelly, S McLaughliny, W B Collisz and P R Whitez, y Dept of Electrical Engineering, University of Edinburgh, Edinburgh EH9 3JL, UK. z Institute of Sound and Vibration Research University of Southampton, Southampton SO17 5NH, UK. Email :
[email protected]
SUMMARY Machine faults are often accompanied by a change in the acoustic and/or vibration signature of the machine involved. To detect such faults non-obtrusively remains the long-standing goal of much work in machine condition monitoring (MCM). The key motivation behind the work described here is the fact that machine faults often result from some sort of nonlinear operation of the machine involved, which may in turn lead to nonlinearities occurring in the machine’s acoustic or vibration signatures. The paper concerns the application of new signal processing techniques to the detection of these nonlinearities. The techniques described are particularly useful in the situation where only a single measurement sensor (e.g. accelerometer) is available.
INTRODUCTION Most traditional signal processing techniques are based on secondorder statistics, such as the power spectrum and autocorrelation function. These measures are useful and in widespread use, since they are easy to implement and have straightforward interpretations as measures of energy. However, these second-order measures are a subset of the “Higher Order Statistics” of the signal, and by using information from the third- and fourth-order measures a fuller picture emerges. In other words, there is information in the signal time series which does not show up in the second-order measures. By looking at higher order measures, and in particular the “polyspectra”, new information about nonlinearities can be obtained. Different polyspectra are sensitive to different types of nonlinear wave coupling. The bispectrum (third-order polyspectrum) is sensitive to quadratic phase coupling (QPC), the trispectrum (fourth-order polyspectrum) is sensitive to cubic phase coupling (CBC) and so on. (It is not usually feasible to obtain reliable estimates of polyspectra of orders higher than four.) There have been several previous applications of HOS to MCM [1, 2, 3, 4, 5], but these have generally concentrated on bispectral analysis only. These papers have concentrated on signals from rotating machines, which have strong periodic components. However, recent evidence [6] has shown that for such signals the polyspectral magnitude can be an ambiguous measure, and that the polyphase (i.e. the phase of the polyspectra) should also be included to detect nonlinearities. However, for signals which do not have these strong periodic components the magnitudes of the polyspectra can still be used to detect nonlinearities. The paper describes recent work on the application of polyspectral techniques to MCM [7, 5]. It begins with a review of two polyspectral measures; the bicoherence and tricoherence, and describes how to estimate them from a finite data record from a single sensor. Two experimental test rigs are then described which illustrate the effectiveness of these measures at detecting machine faults. Finally conclusions are drawn and suggestions for further work are given.
POLYSPECTRA The key tools described in this paper are the bicoherence and tricoherence. These are polyspectral measures at third- and fourth-orders, and represent normalised versions of the bispectrum and trispectrum resepectively. The reason for normalising is related to variance issues outside the scope of this paper (see [8] for a review). If it is assumed that the signal is stationary, then these quantities can be estimated using simple segment-averaging, identical in approach to the Welch periodogram technique for estimating the power spectrum [9]. In fact the power spectrum is needed as part of the normalisation, so this is estimated too. The validity of this assumption of stationarity is difficult to gauge, and some recent work has investigated the bicoherence in non-stationary noise environments [10]. Without the assumption it is practically impossible to estimate polyspectral measures. The measured signal x(n)(n = 0; ::; N ? 1) is divided into K segments, as shown in Figure 1. For clarity it is assumed that the segments do not overlap. Each segment i (i = 0; ::; K ? 1) is first multiplied by a data window (such as a Hamming window), and then its M ?point Discrete Fourier Transform (DFT) Xi (k ) is computed. The DFT is then multiplied by itself in different ways to produce the second-, third- and fourth- order raw polyspectra for that segment. These are then averaged over all K segments to give the un-normalised polyspectra estimates Pˆ (k ); Bˆ (k; l) and Tˆ (k; l; m) (where ˆ is used to denote the estimated quantity);
M
M
M
M
Figure 1: Segment-averaging approach to polyspectral estimation.
Pˆ (k)
=
?1 1 KX
K i=0
Xi(k)Xi(k);
(1)
Bˆ (k; l)
=
Tˆ (k; l; m)
=
?1 1 KX
K i=0
?1 1 KX
K i=0
Xi(k)Xi (l)Xi(k + l);
(2)
Xi(k)Xi (l)Xi(m)Xi(k + l + m):
(3)
Pˆ (k), the second-order polyspectrum, is the familiar power spectrum, and varies with discrete frequency k . Bˆ (k; l), the bispectrum, has two independent frequency axes, and Tˆ (k; l; m) the trispectrum, has three
independent frequency axes.
It is well-known that the power spectrum P (k ) is symmetric around the Nyquist frequency. In a similar way there are many symmetry relations in the (k; l) plane (for the bispectrum) and (k; l; m) space (for the trispectrum). As a result of these it is only necessary to compute the polyspectra in the non-redundant region (also called the “Principal Domain”).
Normalisation The bicoherence and tricoherence are formed as normalised versions of the bispectrum and trispectrum. The definitions are
bˆ (k; l)
=
tˆ(k; l; m)
=
j Bˆ (k; l) j P ; 1 2 ˆ K j Xi(k)Xi(l) j P (k + l) j Tˆ (k; l; m) j P 1 2 Pˆ (k + l + m) j X ( k ) X ( l ) X ( m ) j i i i K
(4)
1 2
1 2
(5)
and it is easy to show that 0 bˆ (k; l) 1 and 0 tˆ(k; l; m) 1.
Phase Coupling Faced with an unkown system, it is possible to identify whether or not nonlinearities make a significant contribution to the output signal by looking for specific relations between the signal phase at different frequencies. Different nonlinearities result in different phase relations. Now the tools of interest here, the bicoherence and tricoherence, detect phase relations which arise from quadratic and cubic nonlinearities respectively. Table 1 shows the types of phase relations the HOS measures are sensitive to.
Filter type Phase relations
Linear none
Nonlinear Quadratic Cubic
f +f = f + f 1
1
Detection tool
2
2
power spectrum bicoherence
f +f +f = f + f + f 1
1
2
3
2
3
tricoherence
Table 1: Outline of choice of signal processing tools for nonlinear signal processing.
Phase Coupling and Signal Symmetry The above discussion has focussed on the interpretation of polyspectra as detectors of particular types of phase coupling, but there is a further interpretation which is also useful [11, 1]. It can be shown that the bispectrum (and hence bicoherence) is closely related to the thirdorder moment, or skewness 3 of the signal, and that the trispectrum (and tricoherence) is closely related to the fourth-order moment, or kurtosis 4 of the signal. Furthermore it can be shown that if a signal has a Gaussian pdf (probability density function) then it has zero polyspectra for all orders higher than two (i.e. its bicoherence, tricoherence are zero). A signal which has zero bicoherence has zero skewness (i.e. 3 = 0 and it has a symmetric pdf) and a signal with zero tricoherence has zero kurtosis (i.e. 4 = 0). (Note that there exists a different kurtosis measure which has 4 = 3 for a Gaussian signal. The choice of kurtosis definition makes no difference to the values, nor to the interpretation, of the estimated polyspectra).
Implementation Scenario The type of MCM detector proposed here is shown schematically in Figure 2. The OK/fault decision is based on a combined feature vector formed from the power spectrum, bicoherence and tricoherence. The power spectrum is sensitive to changes in signal energy, the bicoherence to quadratic nonlinearities (and signal skew) and the tricoherence to cubic nonlinearities (and signal kurtosis). A complete system of this type is beyond the scope of the present work. Instead, it will be demonstrated how the different components of this system (i.e. the three polyspectral measures) respond to two types of simulated machine fault.
x(n)
power spectrum bicoherence
fault decision
tricoherence Figure 2: Schematic representation of MCM system based on polyspectral measures.
EXPERIMENTAL RESULTS To see how well these measures perform as detectors of machine faults, two experiments were carried out. Each experiment was based around a vibration measurement on a vibrating beam. In each case the vibration in the vertical plane was measured using an accelerometer. Power spectra and bicoherences were computed from N = 4096, M = 64 and tricoherences were computed from N = 262144, M = 64. Hamming windows were used throughout, as these give some of the best results [8] for phase coupling detection.
Symmetric Fault In the first experiment (see Figure 3) two measurements were taken of a vibrating aluminium beam, first in a normal condition (called “OK”) and then with the fixing screw loosened so that the beam rattled (called “faulty”). This fault is likely to influence the vibration in a similar way at the top and the bottom of the vibration cycle of the beam, and so is called a symmetric fault. Figure 3 shows that the shape of the power spectrum hardly changes when the fault occurs. The bicoherence (Figure 4) does not change a great deal either, confirming that the fault is of a symmetric nature (i.e. it does not introduce significant skew into the signal). However, the tricoherence changes a great deal, as shown in Figure 5. The OK condition has a maximum tricoherence of about 0.03, but the faulty condition has a maximum tricoherence of about 0.62. Thus the tricoherence appears to be sensitive to faults of a symmetric nature.
Asymmetric fault In the second experiment (Figure 6) two measurements were taken of the vibration of a steel beam. First the normal vibration was measured (called “OK”) and then a magnet was brought into proximity with the lower side of the beam to cause the vibration to become skewed (called “faulty”). This is an asymmetric fault since it will affect the bottom of the vibration cycle more than it will affect the top. Figure 6 shows that the introduction of the magnet causes a large change in the power spectrum, but the power spectrum does not really provide much information about the nature of the interference. As discussed earlier the bicoherence is very sensitive to signal skewness, and this is illustrated in Figure 7, with a maximum of 0:1 in the “OK” condition and a maximum of 0:9 in the “faulty” condition. The tricoherence is primarily sensitive to the signal kurtosis, and although there is a change in the tricoherence when the fault occurs, the level of tricoherence is 0:15, which is very small compared to the bicoherence level. Thus from the HOS evidence it is possible to say that in this case the interference is predominantly asymmetric, manifesting itself in a change in the bicoherence.
CONCLUSIONS The bicoherence and tricoherence have been investigated as tools for fault detection in two MCM experiments. These results indicate that by using both measures, extra information about the origin of the machine fault can be obtained. These measures can be computed from a single sensor measurement, and estimated in a way similar to the power spectrum. At the moment the information held in the polyspectral plots has to be interpreted by the experimenter, but in future it might be possible to implement an automated fault-detection system based on these measures.
ACKNOWLDEGEMENTS The authors would like to acknowledge the support of EPSRC and The Royal Society.
accelerometer
shaker
OK faulty
-40 dB
adjustable nut
beam
-60 -80 -100 0
0.1
0.2
0.3
0.4
0.5
f1
Figure 3: First Experiment. Set up (left) and Power Spectra (right). 0.5
0.5
0
0.5
0 f1
0.5 0
f2
0
0.5
0 f1
0.5 0
f2
Figure 4: First Experiment. Bicoherence magnitude for “OK” signal (left) and “faulty” signal (right). Scale : 0 bˆ (k; l) 0:5
Figure 5: First Experiment. Tricoherence magnitude for “OK” signal (left) and “faulty” signal (right). The size of shaded balls is proportional to the tricoherence magnitude at that trifrequency. Scale : 0 tˆ(k; l; m) 0:026 (left) and 0 tˆ(k; l; m) 0:629 (right)
beam
accelerometer -40
magnets shaker
dB
OK faulty -60
-80 0
0.1
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0.3
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Figure 6: Second experiment : Set up (left) and power spectra (right). 1.0 0.5 0
0.5
0 f1
0.5 0
f2
1.0 0.5 0
0.5
0 f1
0.5 0
f2
Figure 7: Second experiment : Bicoherence magnitude for “OK” signal (left) and “faulty” signal (right). Scale : 0 b(k; l) 1
Figure 8: Second experiment : Tricoherence magnitude for “OK” signal (left) and “faulty” signal (right). The size of shaded balls is proportional to the tricoherence magnitude at that trifrequency. Scale : 0 tˆ(k; l; m) 0:011 (left) and 0 tˆ(k; l; m) 0:149 (right)
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