DOI10.1007/s10010-005-0018-9
Forsch Ingenieurwes(2006)70:90–104
C.T.Yiakopoulos·I.A.Antoniadis
Cyclic bispectrum patterns
of defective rolling element bearing vibration response
Received:1November2005/Published online:13January2006
?Springer-Verlag2005
Abstract The vibration signal from a faulty rolling element bea-ring is not strictly phase locked to the rotational speed of the shaft,due to the variable slip between the bearing elements. For this reason,the classical diagnostic methods,which are ba-sed on Fourier analysis,cannot detect clearly the modulation phenomena and the type of the defect,because the relevant in-formation contained in the second order statistic measures is diminished.The main purpose of this paper is the combina-tion of cyclostationary and higher order statistic analysis me-thods,which can lead to better results,concerning the identi-?cation of the status of the bearing.The paper demonstrates that the cyclic bispectral analysis can lead to a discrete spec-tral structure,enabling the clear detection the type of the bearing fault and the corresponding modulation mechanism,thus signi-?cantly overcoming the corresponding problem of other existing methods.
Das Schwingungsverhalten von zyklischen,bispektralen Strukturen in defekten Lagerelementen Zusammenfassung Das Fehlersignal eines Lagers ist wegen der variablen Versetzung zwischen den Lagerteilen nicht mit seiner Drehgeschwindigkeit in Phase.Daraus folgt,dass die klassische, auf der Fourier-Analyse basierenden Diagnostikmethode nicht in der Lage ist,das Modulationsph?nomen und den Defektstyp ein-deutig zu detektieren.Das Hauptziel dieser Arbeit ist die Kombi-nation der zyklostation?ren Analyse mit einer Statistik h?heren Grades,was zu interessanten Ergebnissen,im Bezug auf den Lagerzustand,führt.Die Studie beweist,dass die zyklische bi-spektrale Analyse in der Lage ist eine diskrete Struktur zu?nden, die den Fehlertyp und den Modulationsmechanismus detektiert, nach dem sie die Probleme der anderen Methoden infolge der Versetzungüberwindet.
C.T.Yiakopoulos·I.A.Antoniadis(u)
National Technical University of Athens,School of Mechanical Engineering,Machine Design and Control Systems Section,
P.O.Box64078,Athens15710,Greece
Email:antogian@central.ntua.gr List of symbols
x(n)Discrete time series
X(k)Discrete Fourier Transform
S xx(k)Power spectrum of a discrete time series
E[]Statistical expectation operator
k,l Discrete frequency variable
B(k,l)Bispectrum
f s Samplin
g frequency
R x(t,τ)Autocorrelation function
αCyclic frequency
m(t)First order moment
T Time period
R a x Fourier coef?cients de?ning the autocorrelation function
S x(α,f)Spectral Correlation Density Function
x(t)Time signal
X(f)Fourier transform of a signal
Bα(k,l)Cyclic Bispectrum
δ(t)Dirac delta function
T d Time period between the impulses
d0Constant amplitude of the impacts
q(t)Distribution function of the load around the rolling element bearing under radial load
q0Maximum load intensity
εLoad distribution factor
θmax Angular extent of the load zone
τLag time
A k Random amplitude of impacts
Q Quality factor
f n Resonance frequency
m Relaxation time
f0Free oscillation frequency
B Amplitude
n(t)Additive background noise
f shaft Shaft rotation speed
BPFO Ball Passing Frequency Outer Race
BPFI Ball Passing Frequency Inner Race
f c Central frequency
91
1Introduction
Vibration-based condition monitoring for rolling element bea-rings has been the subject of extensive research,since bearings constitute one of the most important and frequently encounte-red components of rotating machinery.The vibration response of defective rolling element bearings is typically dominated by mo-dulation effects,characterized by frequencies de?ned by the type of the defect,the shaft rotating speed and the geometry of the bearing[9].This concept has been exploited by a number of tra-ditional condition monitoring methods[14],based on the spec-tral analysis of the vibration signals.These methods assume that the signal statistical properties are stationary and thus,they aim to correlate spectral peaks with prospective defect frequencies.
However,the defective bearing vibration response is not strictly phase-locked to the rotational speed,because of the varia-ble slip motion between the rolling elements of the bearing[1]. Hence,the vibration response of machines with defective rol-ling elements is characterized not only by periodic phenomena, but rather presents a strongly non-linear,non-stationary beha-vior.Since power spectral analysis has the serious drawback of discarding all phase information,it cannot lead to the identi?ca-tion of sets of signals that are phased-coupled.As a result,HOS (Higher Order Statistics)methods[5,11]have been applied with partial success to defective bearing condition monitoring[10].
Parallel,due to the rotation of the machine and the modula-tion effects present in the response,the more general assumption of cyclostationarity[6,7],has been applied to machine condition monitoring[4,8],however also with partial success in the case of defective rolling element bearings[2,13].
The combination of Higher Order Statistics analysis with cyclostationary analysis for machine condition monitoring has been only recently considered and subsequently applied to gear-box fault diagnosis[3,12].
The objective of this paper is to demonstrate that the com-bination of the bispectrum,a third-order statistic powerful tech-nique that can identify non-linear coupling of frequencies,with cyclostationary analysis can increase the diagnostic capability of the existing analysis techniques,leading to more clear diagnostic results.
Section2of the paper presents a brief overview of the basic concepts of Higher Order Statistical methods,cyclostationarity and cyclic bispectral analysis.Section3presents a stochastic mo-del for the dynamic response of a defective bearing.It is shown that even for a small amount of randomness of certain parame-ters like slip,spectral analysis is unable to provide meaningful results.The model of Sect.3is further used in Sect.4to gene-rate signals representing the dynamic behavior of bearings under inner and outer race defects.These signals are further analyzed, using bispectral,cyclostationary and cyclic bispectral analysis methods.It is shown that cyclic bispectral analysis greatly over-comes the problems of the other two methods,exhibiting the nature of the defect in clear and distinct hexagonal frequency pat-terns.Finally,Sect.5veri?es the results in measurements from an industrial installation.2Basic concepts of higher order statistics
and cyclostationarity
A full theoretical treatment for the concepts of this section can be found among others in[5,11]for the Higher Order Statistics Me-
thods,in[6,7]for the cyclostationary analysis and in[3]for the cyclic bispectral analysis.
2.1Higher order statistics measures
Higher Order Statistics measures are extensions to higher orders of
theconventionalsecondorderpowerspectralmeasures[5,11].The second order measures provide satisfactory results if the signal has
a Gaussian probability density function.However,many real-life signals are non-Gaussian.
The most widespread tool in signal processing is the power spec-trum.The power spectrum for a discrete time series x(n)is de?-
ned as follows,using the Discrete Fourier Transform(DFT)X(k) of the signal
S xx(k)=E[X(k)X?(k)],(1) where E[]denotes the statistical expectation operator,k is the discrete frequency variable and the superscripted asterisk deno-
tes complex conjugation.The power spectrum is the distribution of the energy of the signal across the frequency domain.Since it
is a real quantity,it does not contain any phase information.One may consider that the power spectrum is the second moment of
the signal.
Higher Order Statistics measures are formed by the exten-
sion of the de?nitions of second order measures.An alternative way for HOS de?nition is by using polyspectra.Thus,the third
order spectrum can be called bispectrum.Bispectrum decompo-ses the third moment of the signal over the frequency domain in
the same way as the power spectrum decomposes the power of a signal.
The power spectral methods cannot be implemented ef?-ciently on non-linear systems,in which frequency components
couple together,because of the assumption of stationarity upon which they rely.Thus,the limitations of the power spectral me-thods lead to the need for the evaluation of the higher order
properties of the signal.
The bispectrum,likewise the power spectrum,can be de?ned
by the signal DFT as
B(k,l)=E[X(k)X(l)X?(k+1)].(2) We notice that the bispectrum is complex-valued,is de?ned over two spectral frequency axes f1and f2,as presented respectively
by the discrete frequency indices k and l and contains phase information.The power spectrum,governed by the features of
DFT,cannot include information above the Niquist frequency f s/2.Therefore,it is not necessary to compute the values of the bispectrum B(k,l)for all the frequency pairs(k,l),since several
symmetries exist in the plane(k,l).The non-redundant region is
92
called principal domain and de?ned as {k ,l }:0≤k ≤f s /2,
l ≤k ,
2k +1≤f s .
(3)
The principal domain forms a triangle,which in turn is divided into two new triangles,called inner and outer triangles.
Since the bispectrum is related to the skewness of a signal,it can be used to track asymmetric non-linearities.The bispec-trum is zero when the signal is not skewed.The amplitude of the bispectrum at the bifrequency (k,l)measures the amount of coupling between the spectral components at the frequencies k,l and k+l.Such a coupling means that there exists a quadratic non-linearity in the signal.2.2Cyclostationary analysis
Most established statistical signal processing methods consider the signal statistical properties as stationary.In contrast to them,cyclostationary analysis assumes periodically time-varying sta-tistics [6,7].A signal is termed as nth-order cyclostationary,if its statistical moments up to nth order are periodically time-dependent.The fundamental frequency αof such a periodicity is called the “cyclic frequency”of the signal .
The ?rst order cyclostationarity is related just to the ?rst or-der moment of a signal m (t )=E [x (t )]=m (t +T ).
(4)
First order cyclostationarity is practically used in vibration ana-lysis through the implementation of the rotation synchronous averaging technique.
The second-order cyclostationarity indicates a periodically time-varying autocorrelation function.The autocorrelation func-tion of a signal x(t)can be de?ned by the formula R x (t ,τ)=E [x (t +τ/2)x (t ?τ/2)].
(5)
Since the above function is periodic,it can be expanded into a Fourier series
R x (t ,t ?τ)=
α
R α
x (τ)e j2π(t ?τ/2),(6)
where R αχare the Fourier coef?cients de?ning the cyclic autocor-relation function and α=1/T is the cycle frequency.The cycle autocorrelation function indicates the amount of energy in the signal due to the cyclostationary components at frequency α.In the same way that the power spectral density (PSD)func-tion of a signal can be computed by the Fourier transform of its stationary autocorrelation function,the Spectral Correlation Density Function (SCDF)can be calculated by the Fourier Trans-form of the cyclic autocorrelation function,with respect to the time shift τ.
The SCDF can be also expressed as S x (α,f )=E [X (f ?α/2)X ?(f +α/2)],
(7)
where X(f)is the Fourier transform of x(t).According to Eq.7,
the SCDF measures the correlation between frequency com-ponents centered on a frequency f and separated by a frequency
shift equal to α/2.Moreover,the Degree of Cyclostationarity function DCS of the signal is de?ned as
DCS α
= f S αf (f ) 2
f S 0f (f ) 2.
(8)The Degree of Cyclostationarity Function resembles the functio-nality and simplicity of the PSD of the envelope of the signal.2.3Cyclic bispectrum
The objective of this paper is focused on certain bispectral fea-tures of the signals resulting from defective bearings,which are able to reveal the nature of the defect.Cyclic bispectral analysis combines the cyclostationary and the higher-order attributes of the defective bearing vibration signal.Cyclostationary analysis can identify the correlation of the several modulation mecha-nisms present in the signal through the cyclic frequency that governs the periodicity of the statistical moments of the signal.On the other hand,bispectrum can detect sets of frequency com-ponents that are phase-coupled.In other words,the proposed algorithm is based on the computation of the bispectrum with a simple modi?cation to introduce the notion of the cyclic fre-quency [12].Thus,the cyclic bispectrum of a discrete signal x(n)is de?ned as
B α(k ,l )=E [X (k )X (l )X ?(k +l ?α)].
(9)
The presentation of the cyclic bispectrum in Eq.9requires
three different frequency axes:A)The spectral axes f 1and f 2introduced by the bispectral analysis as presented respectively by the indices k and l (s.Eq.2),and B)The cyclic frequency axis α.Since the presentation of the results in three different axes is practically almost impossible,the actual cyclic bispectral analysis involves the computation of Eq.9just for a set of speci?c values for the cyclic frequency α.Thus,contour plots are practically generated,similar to those of a traditional bispectral analysis.However,each one of them corresponds to a speci?c cyclic frequency α.
3Stochastic nature of defective rolling element bearings vibration response
A large number of models have been used to describe the dyna-mic behavior of rolling element bearings under different types of defects.According to the traditional approach [9],the repetitive impacts produced by a localized bearing defect can be described by a train of Dirac delta functions δ(t)of the following form
d (t )=????
???q (t )d 0N k =0δ(t ?kT d ),inner race defect d 0N k =0
δ(t ?kT d ),outer race defect
,(10)where T d is the period between the impulses,de?ned as the reci-procal of the characteristic defect frequency of a rolling element
93 bearing,d0is the amplitude of the impulse force characterizing
the severity of the defect and q(t)is the distribution function of
the load around the rolling element bearing under radial load,
approximated typically by the well-known Stribeck equation[9]
q(t)=
q0[1?(1/2ε)(1?cosθ)]n,for|θ|<θmax
0,elsewhere
(11)
However,signi?cant variations of the dynamic response of a defective rolling element can exist,when certain parameters of the above equation are assumed to present rather a stochastic than a deterministic behavior[1].For example,the amplitude of the impacts,excluding the load distribution around the circumference of the bearing,depend upon the variation in the dynamic stiffness of the assembly,the waviness of the rolling elements and races,and the existence of off-sized balls in the ball complement.Therefore, the series of the impacts should be rather considered as randomly modulated in amplitude.
Parallel,rolling element bearings experience an amount of slip motion.As a consequence,the impulses never occur exactly at the same position from one cycle to another.
Therefore,a more realistic model for the impact train of a defective rolling element bearing should be of the following form[1]
d(t)=?
???
???
q(t)
N
k=0
A kδ(t?kT d?τk),inner race defect
N
k=0
A kδ(t?kT d?τk),outer race defect
,(12)
where A k are random variables for the amplitude of the kth impulse force with a mean value of d0and with a probability den-sity function typically assumed to be normal(Gaussian),andτk is a random variable for the time lag between two impacts due to the presence of slip,which is typically assumed to be of zero mean and of a normal(Gaussian)probability density function.
The impulse train of Eq.12generates impulsive excitation forces which excite resonances in the bearing and in the machine structure.Assuming for simplicity that the excited structure is a linear MDOF system,the structural response to each impulse can be written as
s(t)=
M
i=1
B i e
?t
m i cos(2πf0i t)(13)
and
m i=
Q i
πf ni
,f0i=f ni
1?
1
4Q2i
and Q i>
1
2
,(14)
where i=1,M are the excited modes,and for each mode i,m i is the relaxation time,f ni is the resonance frequency,and Q i is the quality factor.
Therefore,the dynamic response x(t)resulting from an indu-ced defect in a bearing can be expressed by
d(t)=
?
???
???
???
???
???
???
???
???
???
???
N
k=0
q(t)A kδ(t?kT d?τk)?
M
i=1
B i e
?t
m i cos(2πf0i t)
+n(t)
for inner race defect
N
k=0
A kδ(t?kT d?τk)?
M
i=1
B i e
?t
m i cos(2πf0i t)
+n(t)
for outer race defect
,
(15) where the symbol?denotes convolution and n(t)is an additive background noise.
In order to exhibit the effect of the stochastic nature of certain critical parameters like slip,on the vibration spectra,a typical example is considered.The simulated signal generated by Eq.15, corresponds to the typical response of a bearing with an outer race defect.Theshaftrotationspeedf shaft is19.35Hz.Thecharacteristic bearing defect frequency BPFO(Ball Passing Frequency Outer race)is equal to3.75times the shaft rotation speed,leading to an estimation of the BPFO around72.47Hz.The excited natural frequency of the system f n1is assumed to be equal to3975Hz. The signal consists of16,384samples and the sampling rate is equal to40KHz.The quality factor Q1is equal to12.Finally,the slip percentage is assumed to vary from0up to1.44%.
Figure1illustrates the spectra of the simulated signals.In perfectagreementtotheexpectedresultsofthemodelde?nedin[9], a set of harmonics of the BPFO frequency of72.47Hz appears for zero slip.Their amplitudes are maximum in the frequency band around the resonance frequency of3975Hz.This frequency does not appear in the spectrum,since it does not necessarily coincide to a harmonic of the BPFO frequency.
However,the most important feature of Fig.1is that even for a small amount of slip,the spectral components characterizing the defect are no longer visible,and that the excited frequency band contains redundant information.The reason is that all information of the second-order properties(energy,variance)is constrained by the degree and the occurrence rate of the slip phenomenon.Thus, the additive noise and randomness,which are generated by the bearing slip,“encage the energy of the deterministic part”of the defect andof the modulation components[1]leading toa reduction of the SNR and to a distortion of the data attributes.Due to this fact,the usage of an advanced statistical procedure is necessary.
4Characteristic patterns of defective rolling element bearing response
In order to check the effectiveness of the advanced statistical analysis tools under consideration,i.e.bispectrum,second order cyclostationary analysis and cyclic bispectral analysis,a set of simulated signals is?rst considered,generated by the stochastic model of(15).
94
Fig.1a–e.Spectra of signals si-mulating the vibration response of a bearing under an outer race defect with slip variable up to a 0%,b 0.18%,c 0.54%,d 1.08%and e 1.44%
The ?rst signal considered corresponds to the typical re-sponse of a bearing with an outer race defect.The ball pas-sing frequency of the outer race BPFO is chosen equal to 71.4Hz and the excited structural natural frequencies f n 1and f n 2are chosen equal to 2859Hz and 3682Hz,respectively.Both the quality factors Q 1and Q 2are chosen equal to 11.The sampling frequency of the simulated signal is 40KHz,its length is equal to 8,192samples and the shaft rotation speed f shaft is assumed to be 18.3Hz.Finally,it is assu-med,that the rolling elements of the bearing have a deviation around the defect frequency (slip percentage)from 0up to 1.07%.
Figure 2presents the stationary signal without slip,together with the corresponding spectra.Figure 3presents the signal with a variable slip percentage up to 1.07%.Similar to most of the spectra of Fig.1,the spectrum in Fig.3b fails to provide a clear and valuable tool for detecting and characterizing the type of the defect.Observing also the waveform in Fig.3a,contrary to the waveform in Fig.2a,it is noted that the impacts,which form a useful “signature”of the modulation type and pulse shapes,are “slinked”due to the slip effect.As a consequence,the energy of the stochastic part is signi?cantly higher than that of the determi-nistic part [1].
Then,cyclostationary analysis is performed.Figure 4illu-strates a part of the contour plot of the produced SCDF for the simulated signal.All the points of the SCDF plot lie in skew lines.The distinct points form a series of complex rhombic struc-tures which is supposed to be related to the fault type and the modulation mechanism due to the defective bearings [2].The spacing of these points must be equal to the BPFO frequency and its harmonics in the horizontal direction and the double of these in the vertical direction.According to these notes,this characteri-stic pattern will provide evidence of the correlation of the natural frequencies with the defective frequencies.Hence,if these condi-
95
Fig.2a,b.Simulated response of a bearing under an ou-ter race defect with zero slip a Waveform and b
Spectrum
Fig.3a,b.Simulated response of a bearing under an outer race defect with slip variable up to 1.07%a Waveform and b Spectrum
tions stand,this correlation veri?es the speci?c modulation effect present in the signal [2].
The useful information from a diagnostic viewpoint is con-centrated around the excited regions due the dynamic behavior of rolling element bearings under different types of defects.Thus,our diagnostic research is focused in the frequency band bet-ween 2and 4KHz in the horizontal direction,because it is the excited region around the natural frequencies of the system.It is observed in the Fig.4that the dominant distinct points shape characteristic rhombic structures.Contrary to our expectations,the vertical and horizontal coordinates of the points and the di-mensions of the diagonal axes of the rhombs do not correspond to the bearing defect frequency or its harmonics.Thus,the pat-tern appearing in the Fig.4is indeterminate and the modulation effect generated by the bearing defect frequency BPFO cannot be easily detected.It is clear from the SCDF plot of Fig.4that the degree of the inherent correlation between the spectral components associated with the defect frequency and the modulations attenuates,because of the loss of information resulting from the random components.This loss is due to the fact that the slip of the rolling elements yields a randomness that ampli?es the role of the background noise in the signal.Thus,the linkbetweenthe signal components withthe same cyclic frequency is misquoted,leading the whole number of the related cyclic correlations to attenuate.
Bispectrum is an alternative method that can be typically used.Thebispectrumappearsasanindicatorofnon-lineartransformations that are related to the signal skewness.Despite the inadequate second-order features,useful third-order information could be preserved in the signal,governed by non-linear phenomena.
The resulting bispectral analysis of the signal under slip is illustrated in Fig.5.Unfortunately,this method fails also to pro-
96
Fig.4.Contour representation of the second order cyclostationary analysis of the signal of Fig.
3a
Fig.5.Contour representation of the bispectrum of the signal of Fig.3a
duce valuable results.The contour plot in Fig.5cannot present a useful pattern that could exhibit the type of the defect.
The reason is that the success of the third-order statistics (bispectrum)depends upon the degree of nonlinearity and non-Gaussianity present in the received data.Contrary,a signal which is assumed to be Gaussian disappears at third cumulant orders.The randomness generated by the slip in the signal under consi-deration,ampli?es the Gaussianity of the signal and reduces the phase relation and the third-order statistical behavior.Thus,the non-Gaussian nature of the signal tends to be minimal,with the increase of the extent and the occurrence rate of the slip motion.Additionally,the fact that the bispectrum is estimated for the en-tire frequency domain results to the reduction of the effectiveness of the method.
The last step of the analysis involves the application of the cyclic bispectrum method.According to the analysis in Sect.2.3,the actual cyclic bispectral analysis involves the computation of Eq.9for a set of speci?c values for the cyclic frequency α.Ac-cording to Eq.15,the most suitable set of cyclic frequencies includes the defect frequency BPFO and the excited natural fre-quencies.Although the BPFO frequency can be determined by the type of the bearing and the rotational speed,the excited natu-ral frequencies cannot be practically estimated,since they do not appear in the spectrum.
97
Fig.6.Contour representation of the cyclic bispec-trum of the signal of Fig.3a for a cyclic frequency αequal to the BPFO frequency of 71.4
Hz
Fig.7.Contour representation of the cyclic bispec-trum of the signal of Fig.3a for a cyclic frequency αequal to the central frequency fc of 2891Hz
Thus,for the practical application of the cyclic bispectrum,two values for the cyclic frequency αare selected:A)The defect frequency BPFO,estimated to 71.4Hz and B)The frequency f c that corresponds to the largest peak in the spectrum (2891Hz),assumed to be close to a resonance frequency.
Figure 6presents the frequency pattern for the ?rst selection of the cyclic frequency.It is noted that the pattern appearing is dominated by the presence of a number of distinct points.How-ever,these points still do not present a regular structure and thus,they cannot provide an indication of the defect.
The reason is that the bearing defect frequency BPFO is not rigidly linked and strictly phased-locked to the shaft frequency,due to the extent of the slip motion.Thus,the BPFO is not strongly related with phase-couplings and correlations between the other spectral components,due to cyclostationarity.
Figure 7presents the pattern appearing for the second se-lection of the cyclic frequency.The contour plot is governed by the clear presence of a number of distinct points that form a large cross and a hexagon around the central frequency f c of 2891Hz.An enlarged presentation of this area is further shown in Fig.8.Two characteristic hexagons clearly appear now around the central frequency pair (f c ,f c ),in the area between 2400and 3400Hz in the f 1-axis and the f 2-axis of the spectral frequencies.
98
Fig.8.Zoom of the cyclic bispectrum contour plot of Fig.7in the area between 2400and 3400
Hz
Fig.9a,b.Simulated response of a bearing under an in-ner race defect with zero slip a Waveform and b Spectrum
It should be noted that the dimensions of the hexagon are clo-sely associated with the sums and the differences between the frequency f c ,the bearing defect frequency BPFO and its third harmonic 3xBPFO.This hexagon structure forms a frequency pattern which clearly indicates the type of the bearing defect.This clear structure is due to the fact that the cyclic bispec-trum approach can better extract the periodically time-varying third-order statistics for a cyclic frequency αclose to the natural frequency of the system.This frequency is more rigidly correla-ted with each one of the other spectral lines.Thus,the capability of the bispectrum in exhibiting non-linearities of the signal is better exploited into this speci?c frequency domain,rather than in the entire frequency band,due the strength of the cyclostatio-nary nature of this central frequency f c .The next signal considered corresponds to the typical re-sponse of a bearing with an inner race defect.The shaft ro-tation speed f shaft is 17.5Hz.The characteristic bearing defect frequency BPFI (Ball Passing Frequency Inner Race)is equal to 9.3times the shaft rotation speed,leading to an estimation of the BPFI around 163Hz.The excited natural frequencies of the system f n 1and f n 2are assumed to be equal to 1036Hz and 1783Hz,respectively.The signal consists of 2,048samples and the sampling rate is equal to 5KHz.The quality factors Q 1and Q 2are respectively equal to 9and 17.Finally,the slip percentage is considered to be variable up to 16%.Although this value is not realistic,it is considered as a worst case scenario.
Figure 9presents the waveform and the spectrum of the si-gnal under zero slip.Spectral lines characterizing the bearing
99
Fig.10a,b.Simulated response of a bearing under an inner race defect with slip variable up to 16%a Waveform and b
Spectrum
Fig.11.Contour representation of the cyclic bispectrum of the signal of Fig.10a for a cyclic frequency αequal to the central frequency fc of 1801Hz
defect and the modulations dominate in the spectrum.Figure 10presents the waveform and the spectrum of the stationary si-gnal under slip.The period and the slip between the oscillating bursts appear in the waveform of Fig.10a.The spectral analysis of Fig.10b fails to exhibit the defect.
Then,the cyclic bispectrum is implemented with a value for the cyclic frequency αchosen equal to f c =1801Hz,which is the largest spectral line of the spectrum,close to the natural fre-quency of 1783Hz.
The resulting plot is illustrated in Fig.11.The frequency do-main is governed by the clear presence of distinct points that form a large cross and a hexagon around the central frequency pair of (f c ,f c ).
A more clear view of the hexagon is presented in Fig.12.A hexagon,similar in structure to the one of Fig.8,is formed by the sums and the differences of the central frequency f c and the bearing defect frequency BPFI,as shown by the peaks,which are connected with dash-lines.The appearance of this hexagon clearly characterizes the modulation of the central frequency by the bearing defect frequency BPFI.
Furthermore,the frequency pair (f c ,f c )and the other points of the hexagon are surrounded by another family of peaks with
100
Fig.12.Zoom of the cyclic bispectrum contour plot of Fig.11in the area between 1,500and 2100
Hz
Fig.13a,b.Waveforms of signals measured on the bearing of a Power Plant fan in a 17/12/2002and b 03/09/2003
low amplitude,that form smaller hexagons.The coordinates of these points are sums and differences of the central frequency f c ,the bearing defect frequency BPFI and the rotational speed f shaft .The existence of these hexagons indicates a second modulation of the central frequency f c and the defect frequency BPFI by the shaft rotational speed f shaft of 17.5Hz.
Thus,the cyclic bispectrum analysis is again able to present a clear and discrete frequency pattern,from which the type of the defect and the modulation mechanisms can be deduced.
5An industrial application
The measurements were conducted on a fan at the industrial in-stallations of a thermoelectric power station of PPC in Lavrion,Attica.The shaft rotation speed during the measurements was around 1236rpm (20.6Hz).The monitored bearing is of an SKF type 22244C3,with a BPFO frequency equal to 6.49times the shaft rotation speed,leading to a theoretical estimation of the
101
Fig.14a,b.Spectra of the si-gnals of Fig.13a Signal of 17/12/2002and b Signal of
03/09/2003
Fig.15a,b.Probability Densities of the signals of Fig.13a Signal of 17/12/2002and b Signal of 03/09/2003
expected BPFO frequency around 134Hz.Two signals were re-corded almost nine months appart.Each signal is 16384samples long and recorded with a sampling rate of 20KHz.
The signals are shown in Fig.13.A series of impacts are present in the waveforms,providing an initial indication of a pos-sible defect.The time spacing between the impulses of the ?rst signal is almost constant and equal to 0.007sec [Fig.13a].This time interval corresponds to the theoretical estimation of the ex-pected BPFO frequency.Two frequency bands are shown to be excited in the spectrum of the ?rst signal,each one of them in-cluding characteristic,distinct and equidistant spectral lines with a frequency spacing around 142Hz,which corresponds to the actual BPFO.Contrary,only one excited frequency band appears in the spectrum of the second signal [Fig.14b],coinciding with the se-cond frequency band of the ?rst signal.However,in this case,the band does not contain any spectral peaks,able to represent the type of the defect.This is due to the extent and the frequent occurrence of the slip motion,which can be observed in the wa-veform of the second signal [Fig.13b].
It is also interesting to note that the kurtosis of the se-cond signal is decreased and the probability density function of the second signal has been transformed from leptokurtic to mesokurtic (Fig.15).Thus,the probability density function of the second signal approaches the normal (Gaussian)distribu-tion,whose kurtosis is equal to zero,providing evidence about
102
Fig.16a,b.Second Order Cyclostationary Analysis of the signal of Fig.13b:a Entire frequency band,b Zoom in the area between3200and3700Hz of the f spectral frequency
axis
Fig.17a,b.Bispectrum of the signal of the Fig.13b:a Entire frequency band,b Zoom in the area between3200and3800Hz in the f1-and f2-spectral frequency axes
the trend of the slip motion to increase the Gaussianity of the
signal.
Next,cyclostationary and bispectral analysis are performed
in the second signal of Fig.13b,with corresponding results pre-
sented in the Figs.16and17.
The contour representation of the cyclostationary approach
in Fig.16is characterized by a semi-rhombic structure around
the excited?eld[Fig.16b].Unfortunately,the coordinates of
the points that form the rhombus,and the distances between
them cannot provide any meaningful information about the de-
fect type.
The bispectrum plot in Fig.17is dominated by a quadran-
gular structure in the area between3400and3600Hz in the f1-
and the f2-spectral frequency axes.However,the length of the
103 Fig.18a,b.Contour representation of the cyclic bispectrum of the signal of Fig.13b for a cyclic frequencyαequal to the central frequency fc of3450Hz: a Entire frequency band,b Zoom in the area between3200and3700Hz in the f1-and f2-spectral frequency axes
sides of the square and the coordinates of its points are not re-
lated to the frequencies characterizing the type of the bearing defect.
Finally,cyclic bispectral analysis is performed for a cy-clic frequencyαequal to3450Hz,the highest spectral line of
Fig.14b.Figure18a presents the pattern produced.The contour plot is dominated by the presence of a number of distinct points
that form hexagons around the central frequency f c.For better resolution,this part of the picture is enlarged in Fig.18b.Two
characteristic hexagons around the central frequency pair(f c,f c) can be clearly observed.The distances of the hexagons are equal
to the sums and the differences between the central frequency f c, the bearing defect frequency BPFO and its subharmonic multi-ple exactly at(1/2)xBPFO.The presence of this second hexagon, which is associated with the subharmonic of the BPFO,requi-res particular attention,since it provides a further indication of the complex non-linear mechanisms present in the vibration re-sponse of defective bearings.
6Conclusions
Contrary to spectral analysis,second order cyclostationary ana-lysis or bispectral analysis,the cyclic bispectral analysis results to a clear frequency pattern,able to characterize the type of the defect even for large amounts of randomness and Gaussianity of the signal,which are a result of the slip motion between the rolling elements of the bearing.This pattern appears for a cycle frequencyαequal to the largest spectral line f c,closest to a re-sonance frequency.In this way,the cyclic bispectrum analysis is focused to the extraction of the periodically time-varying third-order statistics just for this single frequency and not for the entire frequency band.
Since this frequency is rigidly connected to the other spectral components of the signal,the characteristic frequency pattern consists of characteristic hexagons around the central frequency pair(f c,f c).The dimensions of the hexagon are closely associa-ted with the sums and the differences between the frequency f c, the harmonics or subharmonics of the bearing defect frequency, as well as the shaft rotational speed for certain types of defect. Acknowledgement This work is co-funded by the European Social Fund (75%)and National Resources(25%)–Operational Program for Educatio-nal and V ocational Training II(EPEAEK II)and particularly the Program PYTHAGORAS.
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