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AR pole trajectory in condition monitoring studies

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AR POLE TRAJECTORY IN CONDITION MONITORINGSTUDIES
Suguna Thanagasundram
1
, Kalu Ram Gurung
1
, Y.Feng*
1
and Fernando SoaresSchlindwein
1
1
University of Leicester, Department of Engineering, University Road,Leicester LE1 7RH, UK st89@le.ac.uk
Abstract
In this paper a novel approach is proposed for vibration based fault detection studies by thetracking of pole movements in the complex z domain. Vibration signals obtained from the ball bearings from a High Vacuum (HV) and Low Vacuum (LV) ends of a dry vacuum pumprun in normal and fault conditions are modeled as time variant AR (Autoregressive) series.The positions of the poles which are the roots of the AR coefficient polynomial vary for every frame of vibration data. It is a known fact that as defects such as spalls and cracks startto appear on the ball bearings, the amplitude of the vibrations of characteristic defectfrequencies increase. Faults can be predicted by movement of poles in the complex plane asthe pole positions are expected to move closer to the unit circle as the severity of the defectincreases. The area of the region swept by the migratory poles loci and their distances fromthe unit circle can be useful fault indicators. From the position of the poles inside the unitcircle, classification and quantification of the main spectral peak of defect frequencies can beeasily performed, leading to the possibility of having frame to frame monitoring of spectral parameters of interest. The AR pole positions also allow an easier quantitative estimation of the spectral parameters. The pole representation facilitates the easier understanding of thespectral characteristics of the process because of the one-to-one correspondence between the poles and the AR spectral peaks. This method has interesting potential applications incondition monitoring and diagnostic applications. The description of the movement of the poles is shown to be particularly important for the study of harmonic components of thesignal. This analysis has been validated with actual data obtained from the pump and initialresults obtained are very promising.
Eds.: J. Eberhardsteiner, H.A. Mang, H. Waubke
Suguna Thanagasundram, Kalu Ram Gurung, Y.Feng and Fernando Soares Schlindwein
INTRODUCTION
Vibration analysts often rely on PSDs (Power Spectral Densities) of vibration data tomonitor the health of moving parts of machinery. Common failures such as bearingfaults and gear problems can be detected by trending major frequency componentsand their amplitudes. Most of the frequency domain methods used in industry todayare based on the FFT (Fast Fourier Transform) technique. The major shortcoming of the spectral estimation technique is that a large number of frequency componentshave to be monitored due to the complexity of the system. A standard approach inevaluating an instantaneous frequency implies the computation of the whole spectrumfirst and then estimation of the amplitude of a particular frequency of interest. For instance, if ball bearing defect frequencies [1] such as BPFO (Ball Pass Frequency of Outer Race), BPFI (Ball Pass Frequency of Inner Race), BSF (Ball Spin Frequency)and FTF (Fundamental Train Frequency, also known as Cage Frequency) are to bedetected, FFT spectra are computed for frame sizes of concern and then the spectraare filtered to monitor the presence of the fault frequencies. Such a process can befirstly be time consuming as whole frames of data have to be estimated. Secondly, itcan be power intensive as it involves processing time proportional tocomputations where is the sample size. Another concern of the FFT technique isthat a large enough sample size has to be used for the spectral estimation for reasonable resolution capabilities as the resolution of the FFT tool is inversely proportional to the frame size utilized. This might not be appropriate in real timeapplications.
N N
2
log
N
An interesting alternative is to evaluate directly the frequencies of interest. Inthis case only those frequencies have to estimated instead of the whole spectrum. This provides a reduction in computing time and effort facilitating real time estimation. Inthis study, characteristic bearing defect frequencies are extracted from the polefrequencies of a parametric AR model [2]. Using the AR estimation method, it becomes unnecessary for the calculation of the whole spectrum. Instead justevaluation of the pole frequencies of interest from the derived AR parameters wouldsuffice as AR modelling allows spectral decomposition. This often just involves thecalculation of the AR coefficients and the variance of the input vibration signal. Smallorder AR models can efficiently estimate the pole frequencies which correspond tothe poles of the bearing defect frequencies. The AR technique also only requires afraction of samples as that required by the FFT method for the same resolution. Whencompared to the traditional FFT method, the resolution of the AR technique is higher due to its implicit extrapolated autocorrelation sequence. This means that smaller sample sizes can be used for PSD estimation. Power and frequency of each bearingdefect spectral component can be extracted from the position and residual of each pole. The time varying behaviour of the spectral components can also be studied bytracking the movement of the AR poles. This investigation reports a study of thedetection of an inner race bearing fault of a dry vacuum pump through the mapping of AR poles from its vibration signal. This computational method seems to be veryattractive for condition monitoring applications as the method provides a more
ICSV13, July 2-6, 2006, Vienna, Austria
immediate comprehension of the spectral process characteristics when expressed interms of poles and AR spectral components.
THEORY BEHIND AR MODELLING
An AR process of model order can be described by Eq.(1) where are the AR parameters, is white noise with zero mean and variance and
t
is the discrete-time index [2]. The same equation expressed in the z-transform domain is stated asEq.(2).
p
k
a
][
t e
2
σ
(1)][][][
1
t ek t xat x
k pk
+−−=
∑
=
(
2
)
][)(][
1
z E za z X z X
k k pk
+−=
−=
∑
If is defined as AR polynomial of the model transfer function relating the inputto output, Eq.(2) can be rewritten as Eq.(3) where and are the z-transforms of and respectively and is defined by Eq.(4).
][
z H
][
z X
][
z E
][
t x
][
t e
][
z H
][)(][
z E z H z X
=
(
3
)
∑
=−
+=
pk k k
za z H
1
11)((4)The poles, , are obtained by finding the roots of the
AR
coefficient polynomial in the denominator of . An AR model's transfer function contains poles in the denominator plus only trivial zeroes in the numerator at z=0, so it isreferred to as an "all-pole" model. Since the coefficients of are real, the rootsmust be real or complex conjugate pairs. The number of poles in z plane equals to theAR model order.
k
z
][
z H
)(
z H
∏
=−
+⋅=+++=
pk k p
z z z z z z z z z
z H
0110
)1(1)().........)((
1)((5)There are four main methods for the estimation of the coefficients: Yule-Walker, Burg, Least Squares Forward method and Least Squares Forward Backwardmethod [2]. The PSD of the output signal,, is related to the PSD of the inputsignal which is noise. can be obtained with the variance once thecoefficients are known.
k
a
)(
f P
x
)(
f P
x
2
σ
k
a
Suguna Thanagasundram, Kalu Ram Gurung, Y.Feng and Fernando Soares Schlindwein
212
2
1)(
t f j
e z pk k k x
zat f P
∆
==−
∑
+∆=
π
σ
(
6
)
Each pair of complex conjugate poles in Eq.(5) has a one to one relation withthe AR spectrum, , in the z domain. A
th
)(
f P
x
p
order AR model with poles willhave peak frequencies where
p
m
2
pm
=
when is even and
p
2)1(
+=
pm
whenis odd. Not all poles give rise to peaks in the AR spectrum. Only the poles whichare close to the unit circle give rise to sharp peaks in the AR frequency spectrum (seeFigure 1). The other poles are equally distributed around the unit circle to create anequiripple ‘flat’ PSD estimation. The closer the poles are to the unit circle, the bigger are the amplitude of the peaks. For stability, all poles must lie within the z-plane unitcircle, thus the magnitude of each must be less than unity. The advantage of the poletracking method is that the symmetry of the z plane representation can beconveniently exploited and we can disregard the poles in the negative imaginary plane, halving the number of poles to be tracked.
p
Figure 1: Monitoring the peaks obtained from the AR model. In this case, AR model order was 8 and sampling frequency was 2000 Hz. Notice that it is only the pole nearest to the unit circle that gives rise to a sharp peak in the AR spectrum.
Each pole has a phase
k
z
k
φ
and a magnitude (distance from the srcin). Byknowing the pole position inside the unit circle, the central frequency of each peak can be obtained from the phase
k
r
k
f
k
φ
if the sampling frequency is known (Eq.(7)).The position of the more significant poles vary with the every frame of datadepending on whether they were obtained from no-fault or faulty conditions. The areamapped out by these critical migratory poles is given by Eq.(8).
s
f
( ) ( )( )
π φ π
2ReImtan2
1
sk k k k
f z z f
×=⋅⋅=
−
(7)
( )
k k
r Area
φ
∆⋅∆⋅=
2
21(8)
ICSV13, July 2-6, 2006, Vienna, Austria
RELATING BEARING CHARACTERISTIC DEFECT FREQUENCIESWITH POLE POSITIONS
The angles of the pole locations of the characteristic defect frequencies for the ball bearings [1] are fixed as they are determined by the application and the geometricalshape of the ball bearings used but the distances of the pole locations from the unitcircle are determined by the levels of vibration they cause. It is known that as defectssuch as spalls and cracks start to appear on the ball bearings the amplitude of thevibrations increase (seeFigure 2). We can determine what is the appropriate alarmlevel of vibration from standards such as the IS0 10816 [3] and ISO 7919 [4] andtranslate these to relative amplitudes and alarm levels and hence corresponding pole positions for the characteristic ball bearing defect frequencies.
Figure 2: Typical movement of pole as defect becomes more severe (from 1 to 3) and amplitude of vibration of characteristic defect frequency begins to increase.
The Barden bearing specifications for the BOC Edwards IGX dry vacuum pump thatwas used as the testbed in this experimentation are: number of balls = 9, pitchdiameter = 46.2 mm, ball diameter = 9.5mm and contact angle = 24.97 degrees. For acase of the pump rotating at a speed of 100
Hz
, the theoretical ball bearing defectfrequencies and their relative pole phase angles
k
φ
for a sampling rate of 2000
H
zare shown inFigure 3.In reality, a bearing with an inner race fault has BPFIoccurring at slightly less than 534.10
Hz
as it was observed that the rotating speed of the pump’s rotor shaft, on which the bearing case was directly connected to, wasoften less than the set speed of the pump due to rotor slip. It varied with externalrunning conditions like the ultimate pressure of the inlet of the pump. The runningspeed of the pump had to be determined very accurately for the diagnostics scheme,as this was the frequency which was used in the calculations of the bearing defectfrequencies. This effect was taken to consideration when translating the bearingdefect frequencies into pole positions.
s
f
xxx
Characteristic Defect Frequency11
3
Amplitude of frequency peak increases as defect becomesmore severe
322
0.50.5
11
Movement of pole as amplitudeof frequency peak grows000.5100
2
FrequencySampling
f
(Hz)
CharacteristicDefect Frequency

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