F.E. Hernández1, Vicente Atxa2, E. Palomino3, J. Altuna2
1University of Pinar del Río, Marti 270, Pinar del Río, Cuba
2University of Mondragón, Loramendi 4, Mondragón, Spain
3Instituto Superior Politécnico José Antonio Echeverría, Calle 114, No. 11901, Marianao, Cuba.
Abstract-A false indication of failure in rolling element bearings can be reached when cyclostationary processing technique is used for machine condition diagnosis. This problem is due to the fact that the estimated second-order cyclostationary parameters can be altered by first-order cyclostationary signals such as vibrations no related to those produced by defective bearings. The goal of this work is to solve this problem by applying a cumulant-based approach. Four algorithms were implemented. In order to quantify the effectiveness of the algorithm applications, a new function, named Interference Rate Function, is proposed. The appreciable interference immunity in the estimated cumulant-based cyclostationary parameters demonstrated the veracity of the hypothesis.
I. INTRODUCTION
In the group of signal processing techniques applied on machine diagnosis through vibration analysis, the spectral analysis stands out from other techniques. In practical situations, the characteristics of the vibration to analyze (e.g., nonstationary, low signal to noise rate, etc.) does not make the use of this technique to be suitable [1]. In this case, cyclostationary processing technique emerges as one of the most promising procedure used for machine diagnosis.
Second-order cyclostationary analysis has proved to be effective on rolling element bearings diagnosis, as shown in [2], [3], [4] and [5], however, current applications of this technique carry a problem: an indication of failure existence can be achieved while faults in rolling element bearings do not exist. In other words, false alarms can occur. This problem appears when the rotating machine produces certain types of vibrations, for example, vibrations due to unbalances, misalignment, etc. From a different point of view, this problem occurs because of the moment-based approach of the traditional cyclostationary application, which makes the estimated second-order cyclostationary parameter be altered by those vibrations, which are in fact, first-order cyclostationary signals.
This problem can be solved by making the cyclostationary technique approach cumulants. That is why the goal of this work is to apply the cumulant-based cyclostationary processing on vibration analysis in order to resolve the problem of the first-order cyclostationary signals interference.
II. ABOUT THE APPLICATION OF CYCLOSTATIONARY
ANALYSIS ON ROLLING ELEMENT BEARINGS DIAGNOSIS
A signal is cyclostationary of order n in the wide sense if and only if it is possible to find some nth-order nonlinear transformation of the signal that will generate finite-strength additive sine-wave components [6]. Frequencies at which spectral lines appear are called nth-order impure cyclic frequencies α, in opposition to those called nth-order pure cyclic frequencies β, to be explained below.
In general, the application of second-order cyclostationary processing on vibration analysis for diagnosis is based upon the estimation of the spectral parameter known as correlation spectral density (CSD), Sxα( )f 2, which can be calculated in two ways: through the calculation of the autocorrelation cyclic function, and through the calculation of the second-order cyclic periodogram [6, 7]. The application of the cyclostationary theory to rolling element bearings failures starts from the assumption that the vibration generated when a failure exists is second-order cyclostationary, arising then a cyclic frequency equals to the characteristic failure frequency. The characteristic failure frequency is the main parameter used for diagnosing the bearings condition [8]. Although the procedure of failure detecting when applying cyclostationary theory is not clearly exposed in present references, it can be said that the CSD indicates the existence of a failure if it is not zero at the cyclic frequency α equal the characteristic failure frequency, fc. That is, if Sxfc( )f 2≠0 for any f, then a local fault is present in the corresponding component of the rolling element bearings.
In this work, the experimentation is performed by simulating in computer the vibration produced by defective rolling element bearings. The simulation was based on the model described in [2]. Cyclostationary parameters were also calculated in computer. Matlab was the software used for implementing the corresponding mathematical functions and the signals to process.
The absolute value of the CSD, computed at a cyclic frequency α = 300 Hz equals to the characteristic failure frequency of the damaged component of a rolling element bearings, is shown in Fig. 1a. In this case, the vibration is simulated following the characteristics of the model proposed
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in [2]. Fig. 1b shows the absolute value of the CSD computed when two order cyclostationary signals (with two first-order cyclic components at 2100 Hz and 7570 Hz) are added to the signal simulating the vibration produced by defective bearings. It can be observed the effect of such first-order cyclostationary components on the estimation of the CSD.
The magnitude of the effect on the CSD produced by the first-order cyclostationary components can be measured through the following factor:
( )
which is called Interference Rate Function (IRF), where “A”
denotes the estimation of the CSD performed when first-order cyclostationary components are added, and “B” denotes the estimation of the CSD when first-order cyclostationary components are not added. A zero value at any frequency f in IRF implies this frequency is not being altered by first-order cyclostationary components. The IRF achieved by substituting the CSD computed and shown in Fig. 1 can be observed in Fig. 2.
0 2000 4000 6000 8000 10000 12000
0
0 2000 4000 6000 8000 10000 12000
0
Fig. 1. Absolute values of the correlation spectral density of a) experimental signal simulating the vibration produced by defective rolling
element bearings, and b) such a simulation signal having added two first-order cyclostationary components at 2100 Hz and 7570 Hz.
0 2000 4000 6000 8000 10000 12000
0
Fig. 2. Interference Rate Function computed via correlation spectral density.
The effect of first-order cyclostationary components on the second-order cyclostationary parameter can be reduced if cumulant-based cyclostationary analysis is applied. This work, in a similar way of moment-based cyclostationary analysis, consists in the calculation of a cyclostationary spectral parameter known as second-order cyclic polyspectrum (CP2),
( )2
Pxβ f . Since interference caused by first-order cyclostationary components is not produced, it is said that the CP2 is a second-order pure cyclostationary parameter and then, arising second-order cyclic frequencies β are considered as pure cyclic frequencies. The CP2 can be also estimated in two ways: through the estimation of the second-order cyclic temporal cumulant [6], and through the convolution of the second-order cyclic periodogram (masked by a special function equal to one everywhere except at those frequencies that arise from impure sine waves, in which case it is equal to zero) with a smoothing window [6].
The procedure of applying the cumulant-based cyclostationary processing on bearing condition monitoring is the same as using the moment-based cyclostationary processing. The difference existing between these two approaches lies in the capability of the cumulant-based cyclostationary processing of providing a more robust estimation of the second-order cyclostationary parameters.
However, it is necessary to take into account of the fact that in practical situations it is not possible to perform a precise estimation of the CP2 as proposed in [6]. This matter is due to the impossibility of carrying out an accurate estimation of the frequencies at which first-order cyclostationary vibrations are produced by real machine in order to form the special window that masks the second-order cyclic periodogram.
III. IMPLEMENTED ALGORITHMS FOR SECOND-ORDER
CYCLIC POLYSPECTRUM ESTIMATION
Firstly, an algorithm, based on the so called “general search problem” and presented by Spooner in [6], is adapted to be expressed by equations in reduced form. The IRF, in function of cumulant-based second-order cyclostationary parameters, can be expressed as:
Frequency f (Hz)
Unit of IRF
Frequency f (Hz)
Unit of Power
α = 300 Hz
Frequency f(Hz)
Unit of Power
α = 300 Hz
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( )
(
22)
( ) ( ) 1
( )
x A
P
x B
P f K f
P f
β β
= β − . (2)
The application of the Spooner algorithm yields the IRF shown in Fig. 3a. In this case, interference reduction can be observed if such a result is compared with that obtained in moment-based approach and shown in Fig. 2.
Another algorithm, founded on the first-order cyclic spectral cumulants estimation, is proposed. This algorithm consists in:
1.-) Compute X f( , )τ =FFTt↔f
{
x t x t( ) ( +τ)}
. 2.-) Threshold detects the bins of X to find{ }
β .3.-) Compute Rxβ( )τ 2= x t x t( ) ( +τ)e−j2πβt . 4.-) Compute X f( , )τ =FFTt↔f
{
x t( +τ)}
. 5.-) Threshold detects the bins of X to find{ }
α1 .6.-) Compute Rxα1( )τ 1= x t( +τ)e−j2πα1t .
7.-) Compute 10 1
10 2
1 1
( ) x (0) x ( )
A R R τ
τ
α α
β
α α β
τ τ
+ =
=
∑
.8.-) Compute Cxβ( )τ 2=Rxβ( )τ 2−Aβ( )τ . 9.-) Compute Pxβ( )f 2=FFTτ↔f
{
Cxβ( )τ 2}
.Fig. 3b shows the IRF achieved as a result of applying this algorithm on second-order cyclostationary parameter estimation. It is clear that the interference is also reduced according to the result reached when applying moment-based cyclostationary parameter estimation as shown in Fig. 2..
Other algorithm is proposed by Napolitano and Spooner in [9]. This algorithm is based upon the median, and the positive outcome of its application on pure cyclostationary parameter estimation and interference reduction is shown in Fig. 3c.
0 2000 4000 6000 8000 10000 12000
0 2 4 6 8 10 12 14 16 18
a)
0 2000 4000 6000 8000 10000 12000
0 5 10 15 20 25 30 35
b)
0 2000 4000 6000 8000 10000 12000
0 2 4 6 8 10 12 14 16
c)
0 2000 4000 6000 8000 10000 12000
0 2 4 6 8 10 12 14 16
d)
Fig. 3. Interference Rate Function computed via second-order cyclic polyspectrum, applying a) Spooner’s algorithm, b) algorithm based upon first-order cyclostationary components, c) median-based algorithm, and d) median-based algorithm discarding the two most deviated components.
Unit of IRF
Frequency f(Hz) Frequency f (Hz)
Frequency f (Hz) Frequency f(Hz)
Unit of IRF Unit of IRF Unit of IRF
APPLICATION OF CUMULANT-BASED CYCLOSTATIONARY PROCESSING 143
A low magnitude of interference caused by first-order cyclostationary components is shown in Fig. 3c. Even more, if some cyclostationary components inside the window that performs the median procedure are discarded, for example, two components (components that lie in positions very separated from the median value in the windows), the results achieved are enhanced, as shown in Fig. 3d.
IV. CONCLUSIONS
The cumulant-based cyclostationary processing, applied on diagnosis of rolling element bearings, allows to reduce the alteration produced by first-order cyclostationary components in the estimated second-order spectral parameter. Then false alarms occurrence is reduced too.
The best results were achieved by applying median-based algorithms in order to estimate the cumulant-based cyclostationary parameters.
The definition of a new function, named IRF, allowed to quantify the effect produced by first-order cyclostationary signals on the estimated second-order cyclostationary parameter, and then, to compare the results obtained by the application of different algorithms.
REFERENCES
[1] F.E. Hernández, “Aplicación del procesamiento cicloestacionario de vibraciones, avanzado y de segundo orden, a la detección de fallos locales en cojinetes de
rodamientos,” PhD dissertation presented at the Mechanic Department, Instituto Superior Politécnico José Antonio Echeverría, La Habana, Cuba, january 2006.
[2] R.B. Randall, J. Antoni, and S. Chobsaard, “The relationship between spectral correlation and envelope analysis in the diagnostics of bearing faults and other cyclostationary machine signals,” Mechanical Systems and Signal Processing, vol. 15, pp. 945-962, 2001.
[3] I. Antoniadis, and G. Glossiotis, “Cyclostationary analysis of rolling-element bearing vibration signals,” Journal of Sound and Vibration, vol. 248, pp. 829-845, 2001.
[4] A. McCormick, and A.K. Nandi, “Cyclostationarity in rotating machine vibrations,” Mechanical Systems and Signal Processing, vol. 12, pp. 225-242, 1998.
[5] J. Antoni, and R.B. Randall, “Differential diagnosis of gear and bearing faults,” ASME Journal of Sound and Vibration, vol. 124, pp. 165-171, 2002.
[6] C. Spooner, “Higher-Order Statistics for Nonlinear Processing of Cyclostationary Signals,” in Cyclostationarity in Communications and Signal Processing, Ed. William Gardner, IEEE Press, 1994.
[7] A.V. Dandawate, “Exploiting cyclostationary higher-order statistics in signal processing,” dissertation presented at the Engineering School of Applied Sciences, University of Virginia, 1993.
[8] R.B. Randall, “State of the art in monitoring rotating machinery,” Annals of the International Conference on Noise and Vibration Engineering, Belgium, 2002.
[9] A. Napolitano, and C. Spooner, “Median-Based Cyclic Polyspectrum Estimation,” IEEE Transactions on Signal Processing, vol. 48, pp. 1462-1466, 2000.
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