Electr EngDOI 10.1007/s00202-016-0441-y

ORIGINAL PAPER

Single point bearing fault diagnosis using simplified frequencymodel

Mohamed Lamine Masmoudi1,2 · Erik Etien1 · Sandrine Moreau1 · Anas Sakout2

Received: 6 October 2015 / Accepted: 17 September 2016© Springer-Verlag Berlin Heidelberg 2016

Abstract The time synchronous averaging (TSA) is a well-known technique used for early detection of bearing failurein electrical machines. This method is very efficient if thecharacteristic default frequency is perfectly known. In thisarticle, a reduced frequency model, derived from ESPRITalgorithm, is used to provide a very accurate estimation ofthe fault frequency. The precision obtained on this frequencyallows to applyTSAalgorithmunder optimal conditions. Theproposed method is tested on simulated and real vibrationsignals for inner and outer ring faults. Finally, a fault indicatoris proposed to discriminate the healthy case from the faultyone.

Keywords Vibration · Diagnosis · Signal processing ·Bearing fault · TSA · ESPRIT · Frequency estimation ·Filtering

1 Introduction

Nowadays, three-phase electrical machines are widely usedin industrial applications. To improve their reliability and

B Mohamed Lamine Masmoudiml_masmoudi@yahoo.com

Erik Etienerik.etien@univ-poitiers.fr

Sandrine Moreausandrine.moreau@univ-poitiers.fr

Anas Sakoutanas.sakout@univ-lr.fr

1 LIAS Laboratory, University of Poitiers, B25, 2 rue PierreBrousse, 86022 Poitiers, France

2 LaSIE Laboratory, University of La Rochelle, La Rochelle,France

availability, an early diagnosis has to be realized. That is why,over the last decade, severalmonitoring techniques have beendeveloped and tested. The different failures which occur inelectrical machines can be classified into stator, rotor or bear-ings faults. According to Electric Power Research Institute(EPRI), bearing faults are the most frequent faults followedby stator and rotor ones [1]. There are several reasons thatcan lead to bearing failure such as mechanical damages,misalignment, wear, lack or loss of lubricant and corrosion.When the smooth surfaces of rolling contact are damaged,high-stress conditions will be imposed on the surface caus-ing a progressive deterioration of the bearing componentsand reducing significantly its lifetime.

Electrical machines can be supervised using several phys-ical quantities measurements such as temperature, magneticflux, mechanical vibrations or electrical currents [2]. Vibra-tion analysis is clearly the preferred approach in industryto detect faulty bearings. After measurement, collected dataare analyzed using signal processing methods classifiedinto three domains: time domain analysis (with differentstatistics-based techniques) [3–7], frequency domain analy-sis (spectral analysis, cepstral analysis, envelope spectrum)[8–12] and time–frequency domain analysis (Wigner–VilleDistribution, Wavelet transform, Short-time Fourier trans-form and Empirical Mode Decomposition) [13–16].

The time synchronous averaging (TSA) [17–19] is wellknown in the field of mechanical fault detection for severaldecades. This approach can be interpreted as a very selectiveband-pass filtering around a frequency defined by the user. Itallows tomake apowerful cleaningof the signal to let the faultfrequencies visible. However, this performance is obtainedonly if the fault frequency is perfectly known. In many cases,the relationships between characteristic bearing frequencieswith geometric considerations cannot determineprecisely thefault frequency and the results provided by TSA can induce

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the user to error. In previous works, high-resolution (HR)technics have been used to obtain this frequency with suffi-cient accuracy. MUltiple SIgnal Classification (MUSIC) andEstimation of Signal Parameters via Rotational InvarianceTechnique (ESPRIT) are the most popular methods [20–22].To reach the desired accuracy, these methods are generallybased on a frequency model with a large number of parame-ters. It leads to several limitations because the search overparameter space is computationally very expensive.

In this paper, a new procedure is proposed. An appropriatepre-filtering greatly reduces the model size in ESPRIT algo-rithm without loss of accuracy. The estimated characteristicfrequency is then used to process vibration signals with TSAprocedure in optimal conditions. In Sect. 2, fault character-istic frequencies expressions are defined. Time synchronousaveraging method is recalled in Sect. 3. In Sect. 4, the enve-lope model is defined and parameters are estimated using theproposed method. Our technique is applied to experimentaldata in Sect. 5 and a fault indicator is presented to discrimi-nate different faulty cases.

2 Bearing characteristic frequencies

Let us consider the case where the inner ring of a ball bearingis damaged by a failure mechanismmentioned above.When-ever one of the ball rolls on the defect, an impulsive forceoccurs and causes the vibration of the bearing. The bearingresponse (inner, outer rings) occurs at its natural frequencyand it is quickly attenuated with time due to depreciation ofthe impulse. The fundamental frequency of vibration signalis directly linked to the ball passage on the defect.

The objective is to detect the bearing defect’s fundamentalfrequency. It can be predicted from the bearing characteristicgeometry and the rotation speed of themechanical shaft [23].

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

fo(Hz) = frNb

2

(

1 − Db

DpcosΦ

)

fi (Hz) = frNb

2

(

1 + Db

DpcosΦ

)

fc(Hz) = fr1

2

(

1 − Db

DpcosΦ

)

fb(Hz) = frDp

Db

(

1 − D2b

D2pcos2Φ

)

(1)

where

– fb the characteristic frequency of a ball,– fi the characteristic frequency of the inner ring,– fo the characteristic frequency of the outer ring,– fc the characteristic frequency of the cage,– fr the rotation frequency of the mechanical shaft,– Nb the number of balls,

Dp

Db

Φ

Fig. 1 Schematic diagram of a ball bearing

– Db the ball diameter (see Fig. 1),– Dp the pitch diameter (see Fig. 1),– Φ the contact angle (see Fig. 1).

We consider in this work that the bearings are radial, i.e.,with a contact angleΦ = 0◦.Moreover, these frequencies arecalculated with the consideration that the contacts ball/ringare perfectly punctual and that the balls are rolling with-out sliding. In real bearing, the balls slide at the same timeas they roll on the slopes. To take into account this slidingphenomenon, a multiplicative sliding factor of frequency isintroduced, defined as the ratio between the rolling distanceand the slidingdistance. In practice, this factor varies between0.96 and 0.98 for a healthy bearing and it is neglected in thepresent case [24].

3 Time synchronous averaging

3.1 Envelope analysis

The envelope analysis [25] is used to monitor the high-frequency response of the mechanical system with periodicshocks such as gear or bearing defects. A pulse is generatedeach time a component gets into contact with a defected sur-face of the bearing. This pulse has an extremely short durationcompared to the interval between pulses. The pulse energy ofthe fault is distributed at a very low level over a wide range offrequencies. This wide energy distributionmakes the bearingdefects difficult to detect by conventional spectrum analysis,specially with the presence of external vibrations producedby other machine components. Fortunately, the impact oftenexcites a resonance in the system at a much higher frequencythan the vibrations generated by the other components. Thisenergydistribution is generally concentrated in a narrowband

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0 1000 2000 3000 4000 5000 60000

0.002

0.004

0.006

0.008

0.01

0.012

0.014Spectrum - healthy bearing

Mag

nitude(dB)

0 1000 2000 3000 4000 5000 60000

0.01

0.02

0.03

0.04

0.05

0.06Spectrum - Inner ring fault

Frequency (Hz)

Mag

nitude(dB)

Resonance frequency

Fig. 2 The spectrum for two signals, without fault (top) and with aninner ring fault (bottom)

near bearing resonance which can be easily detected. Figure2 shows two spectra calculated from vibration signals whichcan be found in the database of the web site “Data BearingCenter” [26]. The first spectrum (top) refers to a healthy bear-ing. The second signal (bottom) carries an inner ring fault.Examining this figure closely, the resonance frequency frescan be identified visually around 3000 Hz.

The high-frequency technique focuses on the narrow bandcontaining only the vibrations’ fault frequencies to define thetype of bearing failure. This technique consists in treating thisenergy with an envelope detector. First of all, data are fil-tered with a band-pass filter around the resonance frequencyfres. The center frequency of the band-pass filter should beselected to coincidewith the resonance frequency of the stud-ied spectrum. The bandwidth of the filter should be at leasttwice the highest characteristic frequency of the defect, thiswill ensure that the filter will pass the carrier frequency and atleast one pair of modulation sidebands. In practice, the band-width should be slightly larger to accommodate the first twopairs of modulation sidebands around the carrier frequency.As a second step, the envelope of the signal is recovered byamplitude demodulation technique. The Hilbert transform isused to perform this treatment [27]. This demodulation isequivalent to translating the spectrum of the filtered signal,initially centered around the resonant frequency to 0 Hz.

The spectrum of the demodulated signal contains frequen-cies defined by the harmonics of the characteristic frequencyfd of the fault surrounded by multiple modulation sidebandsof the shaft frequency fr . A classical model is given in [17]to describe the spectral content of fault signals:

f = m. fd + n · fr (2)

a

b

Fig. 3 The envelope spectrum for two signals with a inner ring faultand b outer ring fault

where

– m and n are relative integers,– fr is the shaft frequency,– fd is the fault frequency ( fi for an inner ring fault or fofor an outer ring fault).

Figure 3 shows the spectrum of the envelope signal fortwo distinct defects obtained from the Bearing Data Cen-ter [26]. Figure 3a illustrates the envelope signal for innerbearing fault. We can notice easily the appearance of theinner fault frequency and its harmonics (the theoretical innerfault frequency calculated using the characteristic geometryand the rotation speed corresponds to 159.93 Hz). Figure3brepresents the envelope signal for outer bearing fault. Thefrequency fault in this case corresponds to 106.4 Hz. Figure4 presents a zoomed capture of the envelope signal for innerring fault to put the emphasis on the harmonics of the innerring characteristic frequency n · fi = n · 159.93 Hz modu-lated by the shaft frequencies fr = 29.5Hz.A simplemethodto detect a bearing fault consists in isolating the defect fre-quency fd and its harmonics. It can be performed using theTime Synchronous Averaging technique (TSA) recalled inthe following section.

3.2 Time synchronous averaging (TSA)

3.2.1 Principle of TSA

It has been demonstrated in [17] that the synchronous aver-aging y(t) of a time signal x(t) using a trigger signal withthe frequency ft is equivalent to the convolution:

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0 50 100 150 200 2500

0.02

0.04

0.06

0.08

0.1

0.12

0.14

Mag

nitude(dB)

Frequency(Hz)

Envelope spectrum - Inner ring fault

fi = 159, 5Hz

2.fr = 59, 14Hz

fi − 2.fr = 100Hz

fi − fr = 130Hz

fi + fr = 189Hz

fi + 2.fr = 218, 6Hz

fr = 29, 48Hz

Fig. 4 Envelope spectrum of a signal with inner ring defect (zoom)

y(t) = x(t) ∗ c(t) (3)

where c(t) is a train of M pulses of amplitude 1/M , spacedwith an interval of Tt = 1/ ft given by [18]:

c(t) = 1

M

M−1∑

i=0

δ(t + i · Tt ) (4)

In the frequency domain, y(t) defined in (3) is equivalentto multiplying the Fourier transform of the signal x(t) by thethe Fourier transform of c(t), as it is shown below:

Y ( f ) = X ( f ) · C( f ) (5)

where C( f ) is the Fourier transform of c(t), which can beconsidered as a comb filter function:

C( f ) = 1

M

sin(πMTt f )

sin(πTt f )(6)

Increasing the number ofM pulses reduces the comb teeth,and reduces the amplitude of the sidelobes between teeth. Fora large number of M , only exact frequency multiples of thetrigger frequency ft are passed through. Thus, the synchro-nous averaging can be interpreted in the frequency domainas a selective filtering that occurs at an integer multiple of thetrigger frequency ft . It also can be seen as a complete removalof all frequencies, except those which are synchronized withthe frequency ft . The selectivity of this filter increases withthe number of pulses M .

Applying the TSA on the envelope signal by synchroniz-ing the trigger with the fault frequency fd , reduces the model(2) to:

f = m · fd (7)

This filtering eliminates the multiple components of therotational frequency fr , combinations of the frequencies frand fd and all spectral components independent of the fault.As an example, to detect the presence of an inner ring fault,the frequency of the trigger should be set to the fault fre-quency of the inner ring ft = fi . The TSA will select fi andits harmonics. In the next section, a simulated signal is builtto study the sensibility of TSA with trigger frequency.

3.2.2 Simulation of a reference signal

In presence of fault, the envelope signal will contain frequen-cies defined by the model (2). This model can be reduced to(7) only if the TSA procedure is performed with the exactdefault frequency. To quantify the sensibility of TSA toany error on frequency estimation, a simulation approachis proposed. A signal built with 105 frequencies (from 0to 1267 Hz), corresponding to model (2) and defined by0 ≤ m ≤ 7 and 0 ≤ n ≤ 5 is considered. The purposeof simulating a reference signal is to have a signal with thesame characteristics as the real one (containing just the faultfrequencies and its harmonics). So the fault frequency is wellknown in this case. On other hand, the simulation approachallows to add noise with a well-known signal to noise ratio(SNR) value to study the ESPRIT sensibility to noisy envi-ronment. From the Bearing Data Center, a real vibrationsignal corresponding to an inner ring fault ( fd = fi ) isused to manually estimate frequencies corresponding to themodel (2). Magnitudes and phases of the inner ring fault areshown in Fig. 5. Let us note that the frequencies estimatedmanually are not accurate enough to be used for synchro-nous averaging. However, the obtained envelope signal is agood approximation of a real envelope signal and its charac-teristics (frequencies, magnitudes and phases) are perfectlyknown and can be used to study the TSA performances. Now,a reference signal (noted xiref(t) in the following) is sim-ulated by adding 105 cosines corresponding to each value(frequencies, magnitudes and phases) determined by manualprocedure. Figure 6 compares the simulated signal (bottom)with the real signal (top). In the same way, a reference signalfor an outer ring fault is simulated from the experimental data(noted xoref(t)).

3.2.3 Application of TSA

Table 1 shows the difference between the theoretical fi.theoand the real fault frequency fi.real of the inner ring. Thetheoretical frequencies are calculated using the bearing char-acteristic geometry and the rotation speed of the mechanicalshaft. The real fault frequencies are determined by man-ual procedure. It can be seen in this table that the errorincreases with the harmonics. In this work, the number ofpulses M = 50 is chosen so that the pulse width is greater

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0 50 100 150 200 250−80

−60

−40

−20

0

Frequency (Hz)

Mag

nitude(dB)

150 160 170 180 190 200 210 220 230 240 250−3

−2

−1

0

1

2

3

Frequency (Hz)

Phase(R

adians)

Fig. 5 Zoom of envelope spectrum of a signal with inner ring fault.Markers o show frequencies, magnitudes and phases estimated by man-ual procedure

2.8 2.81 2.82 2.83 2.84 2.85 2.86 2.87 2.88 2.89 2.90

0.5

1

1.5

Realsign

al

Time (s)

2.8 2.81 2.82 2.83 2.84 2.85 2.86 2.87 2.88 2.89 2.90

0.5

1

1.5

Sim

ulatedsign

al

Time (s)

Fig. 6 Vibration signals spectrumwith inner ring fault, real signal (top)and simulated signal (bottom)

than the double of the expected maximum deviation (in ourcase it has been chosen greater than 2 × 2Hz = 4Hz).

In the following, TSA is applied to the two simulated refer-ence signals xiref(t) and xoref(t). The used trigger frequencyft is the true inner ring characteristic frequency. The resultsare shown in Fig. 7 where the inner ring fault can be clearlyidentified. The characteristic frequency fi and its harmonicshave been well isolated by TSA filtering. As expected, inthe case of an outer ring fault, the chosen trigger ft = fi isnot adapted for this case and the characteristic frequenciesare eliminated by the TSA procedure, except the frequenciescorresponding coincidentally to the inner ring frequency andits harmonics (3 fo = 2 fi ∼= 318Hz, 6 fo = 4 fi ∼= 636Hz,9 fo = 6 fi ∼= 954Hz). At this stage, it is improbable to

Table 1 Deviation between theoretical and estimated frequencies of asignal with inner ring defect

fi 2 fi 3 fi 4 fi 5 fi

fi.theo (Hz) 159.9 319.86 479.79 639.7 799.6

fi.real (Hz) 159.55 319.15 478.64 638.12 797.6

|δ f | 0.35 0.71 1.15 1.58 2

Err (%) 0.21 0.22 0.24 0.24 0.25

0 500 1000 15000

0.02

0.04

0.06

0.08

Mag

nitude(dB)

ft = fi

Frequency (Hz)

0 500 1000 15000

0.05

0.1

0.15

Mag

nitude(dB)

a Inner ring fault: Time Synchronous Averaging with

b Outer ring fault: Time Synchronous Averaging with ft = fi

Frequency (Hz)

fi

2.fi 3.fi6.fi

8.fi

4.fi

5.fi

7.fi

Fig. 7 The average spectrum for the reference signals xiref (t), xoref (t)with ft = fi . a For inner ring fault and b for outer ring fault

conclude that the inner ring is damaged. This exceptionalequality can be explained by the fact that the ratio between thepitch Diameter and the ball Diameter is equal to 5 (accord-ing to the geometric characteristics of the bearing 6205-2RSJEM SKF: Nb = 9, Dball = 7.94mm, Dpitch = 39mm,Φ = 0◦). The Eq. (1) defined in Sect. 2 becomes:

⎧⎪⎪⎨

⎪⎪⎩

fo(Hz) = frNb

2

(4

5

)

fi (Hz) = frNb

2

(6

5

) (8)

According to this equation, the frequencies 3 fo, 6 fo, 9 foare respectively equal to 2 fi , 4 fi , 6 fi for

DpDb

= 5. In theSect. 5.1, a procedure has been established to discriminateinner bearing fault from other cases.

Now, let us consider the case where the theoretical charac-teristic frequency is not determined precisely. Indeed, severalparameters in (1) can be poorly calculated or measured suchas the rotation frequency ( fr ). In these cases, the triggerfrequency is different from the real characteristic. Figure 8shows the results obtained with ft = 0, 97 fi (error of 3 %).It can be seen that a correct fault identification cannot beobtained. The use of time synchronous averaging with the-

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0 500 1000 15000

0.005

0.01

0.015

0.02

0.025

0.03

0.035Magnitude(dB)

Frequency (Hz)

Fig. 8 The spectrumof the reference signals xiref (t)with ft = 0, 97· fifor inner ring fault

oretical characteristic frequency does not guarantee that theresult is exploitable. The fault frequency must be estimatedwith a high precision to extract the faulty cases from healthyones. In the following, a high-resolutionmethod is performedto determine precisely this frequency.

4 Identification of the characteristic frequencyusing a high-resolution method

4.1 ESPRIT method: general framework

The acronym ESPRIT stands for Estimation of SignalParameters viaRotational InvarianceTechnique. ESPRIT isan approach based on subspace techniques which are knownto be very powerful estimation techniques [28]. The purposeof using ESPRIT is to estimate the parameters of the signalmodeled in (9) and particularly, to estimate the frequencymodulations induced by the bearing fault. This techniqueis based on a decomposition of the signal x(t) into twoeigenspaces: the signal generated by the damped sinusoidalcomponents s(t) andnoise subspace xk(t) forming its orthog-onal complementary subspace.

x(t) = s(t) + xk(t)

=K−1∑

k=0

ake−δk t ei(ωk t+ϕk) + xk(t)

=K−1∑

k=0

αk ztk + xk(t) (9)

The complex parameters αk = akeiϕk , zk = e−δk eiωk are,respectively, the amplitude and the pole of the kth component

of the signal s(t). The parameter ak represents the amplitudeat the origin, ϕk the phase at the origin, δk the damping andωk the pulsation. In general case, these four parameters haveto be estimated for the K components of the signal. In thiswork, the pulsation ωk = 2π fk implies that the frequencyfk = ωk/2π has to be determined with high precision.As a first step, the Hankel matrix is calculated from the

discrete signal (9) [28,29].

X (t) =

⎡

⎢⎢⎢⎣

xt−l+1 xt−l+2 · · · xtxt−l+2 xt+1

......

xt−l+n xt−l+n+1 · · · xt+n−1

⎤

⎥⎥⎥⎦

(10)

where the integers l and n are defined as N = n + l − 1samples of the signal (in the application of this algorithmn = l = Nt ). Applying the singular value decomposition(SVD) on the Hankel matrix, a diagonal matrix S of the samedimension as X , and unitary matricesU and V are calculatedso that X = U · S · V T .

ESPRIT algorithm is based on a strong property, rotationalinvariance of space signal. Define the two matrices U↑, U↓of dimension K − 1 containing, respectively, the K − 1 lastrows and the K − 1 first rows of the matrix U , where K isthe length of the eigenvalues vector {zk} of the model.

After determining the twomatricesU↑,U↓, the calculationof the spectral matrix is given by (11), where (†) is referredto the pseudoinverse of the matrix.

φ(t) = (U↓)† ·U↑ (11)

In particular, the eigenvalues of the spectral matrix φ(t)are the eigenvalues {zk}k=0,1,...,K−1 of the model. Once thecomplex poles are determined, the damping δk and the fre-quency fk can be calculated by:

δk = − ln(|zk |) (12)

fk = 1

2πarg(zk) (13)

As a second step, the Vandermonde matrix V n of dimen-sion K × n is calculated using the complex poles determinedbefore as follows:

V n =

⎡

⎢⎢⎢⎣

1 1 · · · 1z1 z2 · · · zK...

...

zn−11 zn−1

2 · · · zn−1K

⎤

⎥⎥⎥⎦

(14)

where

– K is the length of the eigenvalues vector {zk} of themodel.

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The complex amplitude αk is calculated by multiplying thepseudoinverse of the matrix V n by the x(t) as it is shown in(15). This leads to the determination of the amplitude ak andthe phase ϕk at the origin.

αk = (V n)† · x(t) (15)

ak = |αk | (16)

ϕk = arg(αk) (17)

This method will be applied to the envelope of the vibra-tion signals to determine all the parameters of the model (9)and especially to estimate the fault frequencies. This methodwill be implemented in MATLAB environment and appliedto different vibrations signals to identify the defects of theouter or the inner rings.

4.2 Algorithm simplification by pre-filtering

High-resolution methods are very efficient for signals con-taining a large number of spectral components. The richer infrequency components the signal is, the higher the numberof model frequencies characterizing the signal. It leads tolarge matrices (the Hankel matrix Nt × Nt will have a largedimension) and long computation times. In our case, only thefault characteristic frequency is needed with high precision.Consequently, ESPRIT algorithm can be simplified and thenumber of model frequencies considerably reduced (reduc-ing the length of the eigenvalues vector K ). However, thesignal must be pre-filtered to limit the range of the analysisaround the characteristic frequency.

To find optimal values for parameters Nt (order of theHankelmatrix) and K (length of the eigenvalues vector {zk}),a white noise xk(t) is added to the reference signal:

xrefk(t) = xref(t) + xk(t), (18)

where xref(t) represents the signal built in Sect. 3.2.2. Themagnitude of xk(t) is tuned to obtain a desired signal to noiseratio defined by:

SNR(dB) = 10log

[RMS(xref)

RMS(xk)

]2

, (19)

with RMS(x) defined as the root mean square of the signalx(t).

xrefk(t) is now filtered around the desired characteristicfrequency ( fi ). Results are shown in Table 2, respectively,for K = 3 and 2 and Nt = 30 and 3. Estimations of the char-acteristic frequency are performed on the one hand for thereference signal (inner ring fault) without band-pass filtering.On the other hand, this estimation is made with a pre-filteredsignal. The relative error is given as:

Table 2 Characteristic frequency estimation: optimal parameters foran inner ring defect ( fd = fi )

SNR (dB)

−1 0 1 10

Witouth K 3 3 3 3

filtering Nt 30 30 30 30

Err(%) 0, 98 0, 93 0, 86 0, 8

Calculation time = 1, 74s

With K 2 2 2 2

pre- Nt 3 3 3 3

filtering Err(10−3%) 8 4, 6 0, 85 0, 5

Calculation time = 0, 6s

Err = freal − festfreal

× 100 (20)

where

– freal is the real frequency.– fest is the estimated one.

It can be seen that the pre-filtering around fi plays animportant role in the precision of the estimated frequency:it reduces the frequency components around fi and makesits identification easier by choosing the highest amplitude ofthose components. It reduces also the matrix dimensions cal-culated in the ESPRIT algorithm and consequently reducesthe computation time. It can be seen also in Table 2 the impactof the noise on the proposed technique where the error isdecreasing (decreasing from 8 × 10−3 to 5 × 10−4 %) withthe increase of the SNR. On the simulated signal, the char-acteristic frequency is obtained with a very good accuracyfor K = 2 and Nt = 3 (an error about 10−4 %), whichindicates the poor influence of the noise on the proposedtechnique.

4.3 Influence of the contact angle Φ

The proposed technique is applied on inner ring fault signalsby introducing an error on the contact angle Φ to prove itseffectiveness regardless of the error introduced during themeasurement.

It can be noted in Table 3 that the calculated characteris-tic frequency fi th varies for each contact angle value unlike

Table 3 Influence of contact angleΦ on the estimation of characteristicfrequency

The calculated fi th (Hz) The estimated fi (Hz)

Φ = 0◦ 159.92 159.54

Φ = 10◦ 159.52 159.54

Φ = 20◦ 158.30 159.54

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Fig. 9 The proposed approach for single bearing fault detection

the estimated frequency fi . This means that the proposedtechnique is totally independent of the contact angle value.In the next section, this methodology will be used for faultdetection on real data.

5 Experimental results

The MATLAB environment is used to implement the pro-posed technique. According to previous sections, the fol-lowing procedure is proposed to detect single point bearingfailure using vibration signals (Fig. 9):

1. Band-pass filtering of the signal around the resonancefrequency fres;

2. Calculation of the Hilbert transform and the envelopesignal;

3. Band-pass filtering of the envelope signal around the the-oretical characteristic frequency fd ;

4. Characteristic frequency estimation using pre-filteringand simplified ESPRIT algorithm;

5. Time synchronous averaging of the envelope signal withthe estimated characteristic frequency;

6. Decision making by calculating a fault indicator.

Table 4 shows results for experimental data from the Bear-ing Data Center [26]. Two bearings with inner ring fault aretested. The diameter of the artificial pitmade on the inner ringvaries from 0.007 to 0.021 in. In the following, this diameteris called “fault diameter”. According to Sect. 4.2, the follow-

Table 4 Inner ring fault: characteristic frequency estimation using pre-filtering and ESPRIT algorithm

fi th = 159.93Hz

Fault diameter (in.) 0.007 0.021

Estimated frequency (Hz) 159.54 159.79

Table 5 Outer ring fault: characteristic frequency estimation using pre-filtering and ESPRIT algorithm

foth = 105.87Hz

Fault diameter (in.) 0.007 0.021

Estimated frequency (Hz) 106.29 106.03

ing parameters are chosen for the ESPRIT algorithm K = 2and Nt = 3 as optimal values to achieve enough accuracyas well as suitable computation time. The rotation frequencyof the mechanical shaft is fr = 29.53Hz. According to (1),the theoretical characteristic frequency of the inner ring isfi th = 159.93Hz.The same tests are realized for an outer ring fault. The

results are given in Table 5. In this case, the theoretical char-acteristic frequency of the outer ring is foth = 105.87Hz.

5.1 Results of TSA application

Tables 4 and 5 show that the simplified ESPRIT algo-rithm gives a good estimation of the characteristic fre-quencies. These frequencies can now be used to performthe TSA filtering with the required accuracy. Figures 10and 11 show the results of TSA applied to faulty bearingswith frequencies shown in Tables 4 and 5. Characteris-tic frequencies and their harmonics are correctly isolatedand all external components are filtered which leads to aclean signal containing just information about the bearingfault.

The proposed technique has also proved its effectivenessin function of different operating points when the motor loadwas increased from 0 to 3 hp and the rotation speed wasslightly reduced from 1797 to 1730 rpm. Figure 12 showsthe results of applying TSA technique on inner bearing faultsignals with different motor load. According to this figure,the characteristic frequency and its multiples are perfectlyisolated.

Finally, a bearing fault indicator is proposed to distinguishthe faulty case from the healthy one by calculating the sumof the maximum amplitude of the signal Y ( f ) defined in (5)for f = n · fd as it is shown in (21). This indicator is appliedto signals with inner ring fault, outer ring fault and healthysignals for different motor loads.

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0.05

0.1

0.15

0.2

0.25

Mag

nitude(dB)

, 007 inch

0 500 1000 15000

0.05

0.1

0.15

0.2

0.25

Mag

nitude(dB)

Frequency (Hz)

a Fault diameter 0

b Fault diameter 0, 021 inch

fi

2.fi3.fi 4.fi

5.fi 6.fi7.fi 8.fi

Fig. 10 The average spectrum for an inner ring fault with two differentfault diameters a 0.007 and b 0.021 in. (M = 50)

0 500 1000 15000

0.1

0.2

0.3

0.4

Mag

nitude(dB)

, 007 inch

0 500 1000 15000

0.1

0.2

0.3

0.4

Mag

nitude(dB)

Frequency (Hz)

a Fault diameter 0

b Fault diameter 0, 021 inch

2.fo

3.fo 4.fo

fo

5.fo

6.fo

8.fo10.fo9.fo

7.fo

Fig. 11 The average spectrum for an outer ring fault with two differentfault diameters a 0.007 and b 0.021 in. (M = 50)

Ind =10∑

n=1

Y (n · fd) (21)

where

– fd represents one of the characteristic frequencies ( fi orfo).

– n represents the multiples of the characteristic frequency.

Figure 13 shows the evolution of the proposed indicatorfor several cases with fd = fi . A threshold (indth = 0.2)is chosen as twice the maximum amplitude of the indicatorvalue when applying the TSA on the outer ring fault signal.

0 500 1000 15000

0.05

0.1

0.15

0.2

Mag

nitude(dB) Motor load 1 hp

0 500 1000 15000

0.05

0.1

0.15

0.2

Mag

nitude(dB)

Motor load 2 hp

0 500 1000 15000

0.05

0.1

0.15

0.2

Mag

nitude(dB)

Frequency (Hz)

Motor load 3 hp

2.fi3.fi

5.fi4.fi

fi

6.fi7.fi 8.fi

Fig. 12 The average spectrum for inner bearing fault with a 1 hpmotorload, b 2 hp motor load and c 3 hp motor load

1 2 3 4 50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8Mag

nitude(dB)

No loadMotor load = 1hpMotor load = 2hpMotor load = 3hp

a. Healthy cases

b. Inner ring fault(fault diameter0,007)

c. Inner ring fault(fault diameter0,021)

d. Outer ring fault(fault diameter0,007)

e. Outer ring fault(fault diameter0,021)

Fig. 13 Evolution of the indicator Ind for a healthy signal, b signalwith 0.007 in. inner ring fault diameter, c signal with 0.021 in. innerring fault diameter, d signal with 0.007 in. outer ring fault diameter ande signal with 0.021 in. outer ring fault diameter

This figure proves that the proposed indicator can clearlydistinguish the inner ring fault from others cases.

Figure 14 shows the evolution of the proposed indica-tor for several cases with fd = fo. This figure illustratesthe effectiveness of the proposed indicator which can clearlydiscriminate the outer ring fault from others cases.

Figure 9 resumes the proposed technique. It allows detect-ing single bearing fault. To characterize the fault (defectedcomponent, fault severity) two steps are required. The firstone is the estimation of the real fault frequency which ismostly different from the theoretical one using ESPRIT algo-rithm. The second step is vibration signals filtering from all

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1 2 3 4 50

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

Mag

nitude(dB)

No loadMotor load = 1hpMotor load = 2hpMotor load = 3hp

e. Outer ring fault(fault diameter 0,021)

c. Inner ring fault(fault diameter 0,021)

d. Outer ring fault(fault diameter 0,007)

b. Inner ring fault(fault diameter 0,007)

a. Healthy cases

Fig. 14 Evolution of the indicator I nd for a healthy signal, b signalwith 0.007 in. inner ring fault diameter, c signal with 0.021 in. innerring fault diameter, d signal with 0.007 in. outer ring fault diameter ande signal with 0.021 in. outer ring fault diameter

spectral components except those which coincide with realfault frequencies and their harmonics (respectively, fi andfo). Once the TSA is applied, an indicator Ind is calculatedfor the two fault frequencies. The final step of this procedureis decision making by evaluating the calculated indicator. Ifthe indicator exceeded a certain threshold, the bearing willbe considered defected, otherwise it is healthy.

6 Conclusion

This paper deals with the improvement of time synchronousaveraging using a simplified ESPRIT algorithm applied tosimulated and real vibration signals. The simplification hasbeen obtained by a pre-filtering of the envelope signal aroundthe fault frequency. ESPRIT algorithm has been also con-siderably simplified and the number of model frequenciesreduced to the minimum by choosing optimal values for thetwo parameters Nt and K to achieve enough accuracy aswell as fast calculation time. Finally, the resulting algorithmis very fast and provides very accurate fault frequencies. Thisaccuracy permits to use these frequencies in a TSA filteringwith good performances. This algorithm is applied to vibra-tion signals with inner and outer ring faults and different faultdiameters. The obtained results confirm that the combinationbetween TSA, ESPRIT and the pre-filtering achieves betterperformances than the only application of TSA to vibra-tion signals. The proposed algorithm shows its efficiency fordetection of the inner and the outer ring fault for differentfault diameters. The proposed procedure is able to distin-guish inner from outer faults by setting the trigger frequencyft equal to the fault characteristic frequencies. If the obtained

frequencies are not a train of this characteristic frequencyand its harmonics, then the bearing is considered healthy.The proposed indicator is characterized by its simplicity andeffectiveness. The results show its capability of distinguish-ing the inner ring bearing fault from the outer ring bearingfault and the healthy bearing for different fault diameters andmotor loads.

The proposed approach was successfully tested for sin-gle point bearing faults with various fault cases and loadconditions. Further investigations are required to study thegeneralized roughness faults (multi-point bearing faults).

Acknowledgments This work has been realized through the collab-oration between LaSIE laboratory, University of La Rochelle, Franceand LIAS laboratory, University of Poitiers, France, with the financialsupport of FEDER program no: 33288-2010.

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