synchronous analysis for diagnostics of rolling bearings

International Journal of Rotating Machinery 2005:2, 128–134c© 2005 Z. Gosiewski and A. Romaniuk

Synchronous Analysis for Diagnostics of RollingBearings in the Turbojet Engine

Zdzisław GosiewskiDepartment of Mechanical Engineering, Technical University of Bialystok, ul. Wiejska 45C, 15-351 Bialystok, PolandEmail: [email protected]

Andrzej Romaniuk

ul. Sw. Bonifacego 74 m. 160, 02-936 Warszawa, PolandEmail: [email protected]

Received 12 October 2003

Designed in the Aviation Institute, the K-15 turbojet engine has got rolling bearings, which answers with frequency 5.87 × ωr

the unbalance excitation. The signal with such frequency indicates a fault of the outer race of the rolling bearing. A set of thedigital synchronous filters was used for the K-15 vibration spectrum analysis. A procedure of filtration was performed by thecomputer software. The sychronous summation of the measured signals was carried out before the spectrum analysis. Two caseswere considered: the engine with a small force due to unbalance (a small angular velocity of the rotor), and the engine with a bigunbalance force (high angular velocity). In the first case, the outer race frequency was not observed, despite the existence of thevibration amplitude (caused by unknown disturbances) with such frequency before the synchronous summation. In the secondcase, the outer race frequency after synchronous summation has enlarged amplitude while other spectrum components in itsvicinity have been damped. It underlines the usefulness of the synchronous analysis in the vibration diagnostics of the rotatingmachinery.

Keywords and phrases: diagnostics, nonstationery vibrations, synchronous summation, synchronous filtration, turbojet engine,rolling bearings.

1. INTRODUCTION

Proper diagnostics of a rolling bearing condition is of greatimportance in rotating machinery maintenance. This pro-cedure is based on measurement of vibration level. Gener-ally, the level of vibration generated by the rolling bearing isvery low not only in proportion to other sources of vibra-tion but to noise as well. For this reason, the methods of sep-aration of bearing-generated signal must be implemented.The well-known and widely described method is the high-frequency resonance technique which utilitizes a resonanceof the bearing-support structure [7]. The support structureplays a role of a mechanical filter. This enables us not onlyto separate the bearing signal in the frequency domain but toperform amplification as well [9].

The main disadvantage of this method is that in the high-frequency region, the frequency components from variousresonances are mixed and as the bandpass frequency of sucha mechanical filter is strictly defined, measurement can be

This is an open access article distributed under the Creative CommonsAttribution License, which permits unrestricted use, distribution, andreproduction in any medium, provided the original work is properly cited.

done for the strictly defined rotating frequency. In the paper,the alternative approach is proposed. We measure the rollingbearing vibration signal directly and simultaneously utilizethe information about forces acting in the bearing planes. Toimprove the signal-to-noise coefficient, the synchronous fil-tration and the synchronous summation are proposed.

2. THE CAUSES OF VIBRATION GENERATION

2.1. Equations of motion for the rigid shaftwith flexible foundation

In the rotating machinery, most of the external excitationsare synchronous to the constant factor with rotor angularvelocity ωr . So more, such excitation as the rotor unbal-ance causes sub- or multisynchronous vibrations of rotatingmachinery’s components, for example, of rolling bearings.So, for proper diagnostics of the bearings, the knowledge offorces acting in the bearing planes is strongly recommended.

The procedure will be explained on the simplified exam-ple of a massless shaft with a rigid disk (see Figure 1). Theexample is rather loosely connected with the K-15 engine butgives a good insight into the proposed procedure.

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Synchronous Analysis for Diagnostics of Rolling Bearings 129

The bearingplane (a)

The bearingplane (b)

αx

a b

β

y

ωz

Figure 1: Model of the rigid rotor with the flexible foundation.

We can determine the forces by solving differential equa-tions of the rotor motion [3, 5]:

mx +(bxa + bxb

)x +

(−abxa + bxb)β +

(kxa + kxb

)x

+(−akxa + akxb

)β = meω2

r cosωrt,(1)

my +(bya + byb

)y +

(−abya + bbyb)α +

(kya + kyb

)y

+(−akya + bkyb

)α = meω2

r sinωrt,(2)

Iβ + Ipωrα +(−abxa + bbxb

)x +

(a2bxa + b2bxb

)β

+(−akxa + bkxb

)x +

(a2kxa + b2kxb

)β

= (Ip − I)τω2

r cos(ωrt + βτ

),

(3)

Iα− Ipωrβ + (−abya + bbyb) y + (a2bya + b2byb)α

+ (−akya + bkyb)y + (a2kya + b2kyb)α

= (Ip − I)τω2r cos(ωrt + βτ),

(4)

where bxa, bxb, bya, byb; kxa, kxb, kya, kyb are the dampingand stiffness coefficients, for the left a and right b supportin x and y axes, respectively, I , Ip are the axis and the polarmoments of inertia, respectively, m is the mass, e is the masseccentricity, τ is the the angular misalignment of the princi-pal axis of moment of the rotor with respect to the centerlineof the shaft.

Equations (1)–(4) can be expressed in the matrix form

Mz + Gz + Bz + Kz = V, (5)

where: M is the mass matrix, G is the gyroscopic matrix, Bis the damping matrix, K is the stiffness matrix, V is theunbalance vector, z is the rotor’s displacement vector, andz = [x β y −α]T .

The rotor motion can also be expressed in terms of thecoordinates of the rotor center at the bearing planes, that is,wb = [xa xb ya yb]T , or by the coordinates of the rotor cen-ter at measurement planes, that is, wm = [xc xd yc yd]T . Foralgebraic simplicity, it will be assumed that the bearing and

measurement planes coincide. The coordinates can be trans-formed one into another according to the following formula:

w = T1z, (6)

T1 =[

Tb 00 Tb

], Tb =

[1 −a1 b

]. (7)

According to virtual work rule, we have VTδz = PTδw,where P are unbalance forces in the bearing planes, andδw = T1δz. Finally, we have

V = TT1 P. (8)

Introducing (6) and (8) into matrix equation (5), we will ob-tain the motion equations in the coordinates connected withmeasurement/bearing planes

MT−1w + GT−1w + BT−1w + KT−1w = TT1 P. (9)

Since the measurements are done in the coordinates w, wecan determine the unbalance forces P acting in the measure-ment/bearing planes for given angular velocity by solvingthese differential equations. From the opposite side, when wehave unbalance forces, we can calculate (from (9)) displace-ments for different angular velocities. In the further consid-erations, we use a signal denoted by w(t) as a contact energygenerated by proper component of force P.

2.2. Model of the bearing vibration signal

Su and Lin [9] proposed the following model of the bearingvibration signal:

xm(t) =∫ t

−∞d(τ)w(τ)am(τ)e−αm(t−τ) cosωm(t − τ)dτ, (10)

Xm(ω) = [D(ω)∗W(ω)∗ Am(ω)]Hm(ω), (11)

where Xm(ω) is a frequency characteristic of the mth (high)mode of vibration, D(ω) is the Fourier transform of d(t),d(t) is the impulse train (e.g., generated by defected bearing),W(ω) is the Fourier transform of w(t), w(t) is the weightingfunction describing the contact energy (e.g., caused by iner-tia forces generated by rotor unbalance),Am(ω) is the Fouriertransform of am(t), am(t) reflects the structural characteris-tics of the transmission path between the input and outputat resonance of mth mode, Hm(ω) is the Fourier transformof e−αmt cosωmt, it is the description of bandpass mechanicalfilter connected with resonance of mth mode, and αm is thedamping factor of the bearing-housing ensemble, it can beconsidered as a quality of mechanical filter.

We analyze the periodic properties of the functions w(t)and am(t) as proposed by McFadden and Smith [7]. A mag-nitude of the contact energy w(t) is mainly influenced by theloadings associated with the misalignment or the dynamicunbalance of the shaft, the axial loading, the radial load-ing, the preload, and the manufacturing imperfections. InTable 1, the periodic properties of different loads w(t) areshown.

130 International Journal of Rotating Machinery

Table 1: Periodic characteristics of various loadings and the transmission paths for different defects.

Causes of periodicitiesPeriodicities for different defects

Outer race Inner race Roller

Stationary loading Without periodicity1f1

1fcage

Loading due to shaft unbalance1f1

Without periodicity1

( f1 − fcage)

Loadings due to roller errors1fcage

1( f1 − fcage)

Without periodicity

Transmission path Without periodicity1f1

1fcage

,2froll

From (10) and (11), we can deduce that the energy of thesignal components with the defect characteristic frequencydepends on the weighting function describing the contact en-ergyw(t) and the structural characteristics am(t) of the trans-mission path between the input and the output. The energyof the signal components depends on stability of character-istic frequencies of the contact energy and the transmissionpath between the input and the output. The synchronous fil-tration, due to changing of the rotating frequency, eliminatesnonstability of the characteristic frequencies.

3. THE ISOLATION OF THE ROTOR AND THE ROLLINGBEARINGS SIGNAL COMPONENTS

3.1. Synchronous averaging

The application of the correlation function Rxx(τ) is usefulwhen we do not know the characteristic feature of a sig-nal under investigation. To eliminate or reduce randomnessof the digital signal x(k), we can average x(k) by its inputautocovariance sequence Rxx(k) and its Fourier spectrumH(Ω)—by input autopower spectrum Sxx(Ω), where Ω is thenormalized frequency. In this case, output autocovarianceRyy(k) and output autopower spectrum Syy(Ω) result fromestimates Rxx(k) and Sxx(Ω) of the input signal, respectively[1],

Ryy(k) = h(k)∗ h(−k)∗ Rxx(k),

Syy(Ω) = H(Ω)H∗(Ω)Sxx(Ω),(12)

where h(k) is the system impulse response.The correlation function gives us information about the

existence of periodical components, but we are unable to re-store these components in a time domain.

By means of synchronous averaging, we can eliminate allnonperiodic signals [2]:

y(t) = 1mp

mp∑i=0

[ϕd(t) + ni(t)

]

= ϕd(t) +1mp

mp∑i=0

ni(t) ∼= ϕd(t),

(13)

where ϕd(t) is the periodic signal, ni(t) are the nondetermin-istic components, mp is the number of periods.

This property is of great importance because it allows usto select one-by-one deterministic signals from the measuredtime sequence.

The synchronous averaging operation is similar to a syn-chronous filtration. When the number m of periods in-creases, the averaging signal approaches the periodic signal.

We denote by Tss a total time of average and introducea frequency of the useful signal

fss = 1Tss

. (14)

In some applications, the disadvantage of this procedure isthat not only the fundamental component fss of signal is de-tected but also the components with frequencies

fss, 2 fss, 3 fss, . . . . . . ,n fss, . . . , (15)

where n is the integral positive number.For these reasons, synchronous averaging can be applied

with full satisfaction only as a preprocessing procedure priorto the synchronous filtration.

The advantage of the synchronous averaging is thata phenomenon of sidebands does not appear. The phe-nomenon of sidebands is dangerous especially in the case ofa signal, which contains components with large levels in pro-portion to the components of our interest.

3.2. Synchronous filtration

A way of vibration signal measurement, which is synchro-nized with rotor normalized angular velocity Ωr , was pre-sented in [4] for purposes of the automatic balancing sys-tem. Nowadays, a development of the computerized calcu-lation methods enables the practical application of the syn-chronized digital filtration.

A structure scheme of the synchronized vibration mea-surement path is shown in Figure 2. [8].

In a standard application, the digital filter realizes oper-ations on a data set in which samples are evenly distributedalong the time axis, but to obtain phase compatibility, we areforced to scale the localization of samples in time. We can ex-plain this on an example of the impulse response matrix ofthe bandpass filter [6]:

h(n) =(

11nπ

)sin(nΩ1

)cos(nΩc

), (16)


x(t)

Transducer

Chargeamplifier

Antyaliasingfilter

A/Ctransducer

PCcomputer

ωrTacho

impulsatorFrequency multiplier Software

Figure 2: Instrumentation setup of the synchronous vibration mea-surement.

where Ωc = fc/2π fd, fc is the center frequency of bandpassfilter, fc = 2πΩ, Ω is a normalized (on the unit circle) fre-quency, fd is the sampling frequency, Ω1 is the frequencydefining the band of the filter. Taking into account frequencycorrection coefficient f / fc = k f , (16) has the form

h(n) =(

1nπ

)sin(nk fΩ1

)cos(nk fΩc

). (17)

We have to know the coefficient k f to implement the al-gorithm, which allows us to change frequency of the filterunder investigation. This classic realization of the trackingfilter is inefficient. We are obliged to continuously measurefrequency and after each measurement to change the coeffi-cients of the filter. A delay between frequency measurementand changing the coefficients is the source of error. To elim-inate such error, we have introduced the synchronous filtra-tion.

3.3. Differences between the synchronousfiltration and Fourier transform

We can calculate a frequency spectrum (amplitude andphase) of the signal by using of the synchronous digital filter(SDF). Determination of each frequency line (sample) of thespectrum is realized in the time domain by the single band-pass filter.

The inverse digital Fourier transform (IDFT) algorithmmultiplies samples in the frequency domain (which repre-sents frequency spectrum of the input signal we want to re-store in the time domain) by IDFT coefficients (which repre-sents frequency spectrum of each filter). Because multiplica-tion in the frequency domain corresponds to convolution inthe time domain as an output of the IDTF, we get a signal inthe time domain.

We should decrease or increase the number of IDFT co-efficients to equal them to the number of input samples.Increasing of the coefficients number allows us to use thegreater number of samples in the time domain and leads tothe following advantages.

(a) Diminution of a discontinuity impact between thefirst and the last samples in the time domain. The impulse δ,which represents this discontinuity, is represented in the fre-quency domain by evenly distributed spectrum (a whitenoise). The ratio of the white noise to signal diminishes withthe increase of the number of the input samples.

(b) Detection of low-frequency modulation is possiblefor the higher resolution in the frequency domain.

The output of the bandpass filter is a sinusoid withchanging amplitude. The amplitude change depends on dy-namic properties of the input signal and the band of the fil-ter. As far as DFT is concerned, we get the set of sinusoidalcomponents with the constant amplitude and phase shift forevery normalized frequency.

DFT analyzes a set of samples taken during the time pe-riod given a priori. Shifting the set by one sample in a positivetime direction, we can obtain the all-pass filter. We obtain arequired transfer characteristic of the desired synchronousfilter by the calculation of chosen frequency components.Next, we have the filter output by summation of the com-ponents in the time domain, while the characteristic of thedigital filter is encoded in its impulse response (filter coeffi-cients).

For a given frequency spectrum, IDFT restores a signal inthe time domain, SDF calculates the output signal by a con-volution of the input signal with an impulse response of thedesired filter. DFT calculates the spectrum components so wecan treat it as the set of the bandpass filters. We obtain eachfrequency component by the multiplication of the transformcoefficients by the input signal samples.

3.4. The comparison of recursive andnonrecursive filters

Sidebands

Sidebands are connected only with the nonrecursive filtersdue to an impulse response truncation. The impulse re-sponse truncation is realized by the multiplication of theinput signal by a window function. Because the multiplica-tion in the time domain is equivalent to the convolution inthe frequency domain, the sidebands occur. Due to the side-bands occurrence, the output signal contains not only theuseful components but useless components as well. We canput the useless components into two categories: synchronousand nonsynchronous ones.

The synchronous summation procedure eliminates thenonsynchronous components from the input signal. Elimi-nation of the synchronous components, which frequenciesare the multiple of measured signal, can be obtained by de-signing of special filter algorithm, so these components donot occur in the sidebands. DFT with the inherent rectangu-lar window is an example.

Phase linearity

The recursive filters introduce the nonlinearity to the phasecharacteristics.

Memory

The nonrecursive filters need a finite volume of the proces-sor memory which is a positive feature. They memorize onlythe number of the impulse samples equal to the number ofimpulse response coefficients in the filter algorithm. For thisreason, a casual distortion signal whips out, after using in thecalculation procedure, the number of impulse samples equal


to the number of impulse response coefficients. The recur-sive filter due to the recursive character of its algorithm theo-retically remembers unlimited number of samples. Using therecursive filter as the synchronous filter, it is not possible toobtain full synchronism, because it is impossible to restrictthe number of the input signal samples, which we take tocalculate the output sample, to the number equal to impulseresponse coefficients.

Initial condition

Using the recursive filter, we always have to remove at thebeginning some number of output samples as they are unre-liable. The number of removed samples depends on the errorwe can approve.

The efficiency of the filter algorithm

For the continuous work with known initial conditions orafter transient period, the recursive filters are faster becausethey realize only a few operations of the multiplication. Forinterrupted work, their efficiency is similar to the efficiencyof nonrecursive filters. For long transient period, their timeefficiency may be even worse.

3.5. Fast algorithms

In the case of the synchronous filters, it is possible to de-sign a faster algorithm due to the feature that all componentsare multiple of the first component. Such convolution fea-tures like association and commutation allow us to reducethe number of operations. This procedure is possible onlyfor the nonrecursive filters since their principle of operationis based on the convolution.

4. THE SELECTION METHODS OF THE SIGNALPROCESSING ALGORITHM

When frequency of the signal under investigation is known,the synchronous filtration gives much better results than cor-relation function or the power spectrum density functionSxx. In this case, our task is to separate deterministic com-ponents with known periodicity while all other componentsshould be removed. The determination of the rotor unbal-ance is a typical practical application.

We can also use the synchronous filtration for rollingbearing diagnostics. In this case, the frequency of the charac-teristic signal for each part of the bearing is known. Becausein this case phase relations are of no importance, we can ap-ply stochastic methods as well. The usage of the nonrecur-sive filter with a rectangular window combined with the syn-chronous filtration allows us to remove components whosefrequency is the multiple of the band center frequency. Thiskind of the nonrecursive filter is characterized by the high-level sideband, so the nonharmonic components are pre-sented in the output signal. We can damp the nonharmoniccomponents in two ways: choosing the window parametersto achieve the low level of the sidebands or applying two fil-ters in a cascade; one with the rectangular window and nextwith, for example, the Hamming window.

SDF

CPPL-3

Figure 3: The K-15 turbojet engine with the acceleration trans-ducer CPPL-3.

Application of two connected filters means that we getthe impulse response matrix h(n) whose number of coeffi-cients is two times greater than number of coefficients in ma-trices h1(n) and h2(n). Better solution is to design the filterwith the Hamming window, which is able to damp the un-useful harmonics, because in a calculation procedure, it usestwo times less coefficients than in the case of two filters in thecascade, so it runs faster, and introduces less phase shift.

A number of impulse response coefficients used as filtercoefficients determine the filter frequency resolution. Insteadof applying the input signal, the number of samples equal tothe filter coefficients number, we can use a reduced numberof input samples, by repeating the beginning portion of sam-ples. For example, if the number of samples is reduced fourtimes, we should repeat it four times. This procedure reducestime necessary to measure samples but increases distortionconnected with discontinuity of nonsynchronic signals.

Before filtration, we should carry out the synchronoussummation on the input signal. The synchronous summa-tion can be executed on the output signal of the trackingfilter, in order to enlarge amplitude of synchronous com-ponents. The synchronous filter allows us to realize anotheraveraging operation in the time domain. We can calculatethe output signal using the number of input samples beingthe multiple of the filter coefficients.

The other method is the ensemble average of frequencyspectrum.

5. SYNCHRONOUS FILTRATION APPLICATION

During the test on the engine test stand, a vibration acceler-ation signal in the radial direction of the K-15 turbojet en-gine with the single-suction impeller was recorded. The lo-cation of the acceleration transducer type CPPL-3 is shownin Figure 3. The aim of the transducer is to measure the over-all vibration. The complete procedure with implementationof equations of motion presented in Section 2.1 was not yetperformed, the usage of synchronous digital filtration withimplementation of four transducers will enable us in the fu-ture to calculate the forces acting in the bearings plane.

Figure 4 shows the ensemble average of the frequencyspectrum of the vibration acceleration signal. The funda-mental synchronous frequency component f1 is clearly seen.The multiple of the fundamental frequency and the other


3

3

2

2

1

1

0

Vib

rati

onac

cele

rati

ong

0 1 2 3 4 5 6 7 8

f / f1

Figure 4: The ensemble average of the frequency spectrum of thevibration acceleration signal radial components, 14.5 samples perrevolution.

Before synchronous summation0.20

0.180.16

0.14

0.120.10

0.080.06

0.040.02

0.00

Acc

eler

atio

ng

5 5.5 6 6.5 7

f / f1

5.87

(a)

Before synchronous summation0.03

0.03

0.02

0.02

0.01

0.01

0.00

Acc

eler

atio

ng

5 5.5 6 6.5 7

f / f1

5.87

(b)

Figure 5: Frequency spectrum. Rotational speed (a) 6857 rev/m,(b) 14500 rev/m. Sampling frequency is 58 samples/rev.

frequencies synchronous with the fundamental component,for example, the 5.87× f1 component generated by the outer-race irregularity of the frontal rolling bearing, are also seen.

The characteristic outer-race frequency of the QBJ55SP5roller bearing was determined using the manufacturer infor-mation.

The frequency analysis by means of the synchronous fil-ter set was carried out. In order to enlarge the amplitude forthe characteristic frequency 5.87 × f1 of the rolling bearing,the synchronous averaging was performed prior to the fre-

After synchronous summation0.20

0.180.16

0.14

0.12

0.10

0.080.06

0.040.02

0.00

Acc

eler

atio

ng

5 5.5 6 6.5 7

f / f1

5.87

(a)

After synchronous summation0.03

0.025

0.02

0.015

0.01

0.005

0

Acc

eler

atio

ng

5 5.2 5.4 5.6 5.8 6 6.2 6.4 6.6 6.8 7

f / f1

5.145.87

6.6

(b)

Figure 6: Frequency spectrum after synchronous averaging. Rota-tional speed (a) 6857 rev/m, (b) 14500 rev/m. Sampling frequencyis 58 samples/rev.

quency analysis. The two cases were taken into account: theengine with small unbalance forces caused by relatively smallrotor angular velocity, the engine with big unbalance forcescaused by high angular velocity. For small unbalance forces(see left part of Figures 5 and 6), the fundamental compo-nent f1 is hardly seen. In the case of the engine with smallunbalance forces, after the synchronous summation, the 5,87multiple of the rotating frequency was not seen, despite thepresence of such a component before the operation of thesynchronous summation.

In the case of the engine’s vibration signal where the fun-damental is dominant (see right part of Figures 5 and 6), af-ter the synchronous summation of the input signal, the 5,87multiple of rotating frequency has got the enlarged ampli-tude and the nonsynchronic components in its vicinity aredamped.

6. PRACTICAL IMPORTANCE

The high-frequency modulation technique is widely con-sidered in rolling bearing diagnostics. The main reason for


doing this is the necessity of amplification of weak sig-nal generated by bearings. The disadvantage of this tech-nique is that due to modulation property, it is impossibleto use this technique when rotating frequency of the mea-sured object is changing because the modulation frequencyis determined by the bearing-disk support parameters andwe cannot change it. The other disadvantage is that it is diffi-cult to know the amplitude of the measured signal to estimatethe dimension of a defect. We cannot estimate the shape ofthe defect by measuring the vibration signal using the high-frequency resonance technique.

In our approach, the direct low-frequency signal firsttime proposed by [10] is used. Having both the informationof the force which is applied to the bearing and the bearingresponse, we can estimate the kind and dimension of the de-fect. The experimental results depend on proper extracting ofthe signal of interest from distortions so application of syn-chronous digital filtration improves them. Due to phase lin-earity of nonrecursive filters, it is possible to realize addition-subtraction operations in the time domain on a measuredsignal components prior to the digital signal processing.

REFERENCES

[1] J. V. Candy, Signal Processing: the Modern Approach, McGraw-Hill, New York, NY, USA, 1988, pp. 83–87.

[2] C. Cempel, Podstawy wibroakustycznej diagnostyki maszyn,Wydawn. Nauk. Techn., Warszawa, Polska, 1982, pp. 74–77.

[3] Z. Gosiewski and A. Muszynska, Dynamika maszynwirnikowych, Technical University of Koszalin, Koszalin,Poland, 1992, pp. 38–52.

[4] Z. Gosiewski, Aktywne regulowanie poziomu drganmaszyn wirnikowych ze sztywnym wirnikiem o zmien-nym niewywazenie, Ph.D. thesis, IPPT PAN, Warszawa,Poland, 1981, pp. 45–50.

[5] M. Lalanne and G. Ferraris, Rotordynamics Prediction in En-gineering, John Wiley & Sons, New York, NY, USA, 1998, pp.31–33.

[6] P. A. Lynn and W. Fuerst, Introductory Digital Signal Processingwith Computer Applications, John Wiley & Sons, New York,NY, USA, 1994, pp. 138–144.

[7] P. D. McFadden and J. D. Smith, “Model for the vibration pro-duced by a single point defect in a rolling element bearing,”Journal of Sound and Vibration, vol. 96, no. 1, pp. 69–82, 1984.

[8] A. Romaniuk, “Filtr sledzacy w wibroakustycznej diagnos-tyce wirnika i łozysk tocznych,” Zeszyty Naukowe PolitechnikiRzeszowskiej, Nr 186, Mechanika zeszyt 56, Awionika, tom 1,pp. 201–208, 2001.

[9] Y.-T. Su and S.-J. Lin, “On initial fault detection of a taperedroller bearing: frequency domain analysis,” Journal of Soundand Vibration, vol. 155, no. 1, pp. 75–84, 1992.

[10] C. S. Sunnersjo, “Rolling bearing vibrations—The effects ofgeometrical imperfections and wear,” Journal of Sound and Vi-bration, vol. 98, no. 4, pp. 455–474, 1985.

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