Jtam-A4.dvi


JOURNAL OF THEORETICAL

AND APPLIED MECHANICS

56, 1, pp. 123-135, Warsaw 2018
DOI: 10.15632/jtam-pl.56.1.123

A NEW METHOD FOR AUTOMATIC DEFECTS DETECTION AND
DIAGNOSIS IN ROLLING ELEMENT BEARINGS USING WALD TEST

Ammar Chiter, Rabah Zegadi, Rais El’Hadi Bekka, Ahmed Felkaoui

Applied Precision Mechanics Laboratory, Institute of Optics and Precision Mechanics, Ferhat Abbas University of Setif 1,

Setif, Algeria; e-mail: chiter.ammar@gmail.com

To detect and to diagnose, the localized defect in rolling bearings, a statistical model based
on the sequential Wald test is established to generate a “hypothetical” signal which takes
the state zero in absence of the defect, and the state one if a peak of resonance causedby the
defect in the bearing is present. The autocorrelation of this signal allows one to reveal the
periodicity of the defect and, consequently, one can establish the diagnosis by comparing the
frequency of the defect with the characteristic frequencies of the bearing. The originality of
this work is the use of theWald test in the signal processing domain. Secondly, this method
permits the detectionwithout considering the level of noise and the number of observations,
it can be used as a support for the Fast Fourier Transform. Finally, the simulated and
experimental signals are used to show the efficiency of thismethod based on theWald test.

Keywords: diagnosis, detection, rolling element bearing, defect, Wald sequential test

1. Introduction

In the industry, a great attention has been given to monitoring conditions and maintenance
for the purpose to improve the quality of production. Edwards et al. (1998) and Tandon and
Choudhury (1999) showed the importance ofmaintenance as the bestway to avoidmaintenance
problems that are often very expensive. And also how the predictive maintenance techniques
have evolved to keep a check of mechanical health by generating information on the machine
condition. In rotatingmachines, the transmission elements: belts, gears andbearingsare ofmajor
interest in industrial maintenance as the operation of a mechanical system heavily depends on
health of these elements. Particularly, the rolling bearing is one of themost critical components
that determine machinery health and its remaining life time in modern production machinery
(Jayaswal et al., 2008). Robust Predictive Health Monitoring tools are needed to guarantee
the healthy state of rolling bearings during the operation. A predictive health monitoring tool
indicates upcoming failures which provide sufficient lead time for maintenance planning, as
showed by El-Thalji and Jantunen (2015), Mann et al. (1995) andRenwick and Babson (1985).

Over the past two decades, several methods have been the subject of studies and develop-
ments. Visibly noticed are revolution methods based on mechanical signal processing, which
are divided into twomain categories, detection and diagnosis, and are based on time-frequency
methods and temporal methods or a combination of both. Thus, many methods are born, the
scalar indicators such as kurtosis, skew, crest factor (Dron et al., 2004; Pachaud et al., 1997),
demodulation and detection of the envelope (Sheen, 2004, 2008), amplitude modulation (Stack
et al., 2004), detection of vibrationmodes (Rizos et al., 1990), de-noising vibratory signals (Bo-
laers et al., 2004), the spectral density analysis (Krejcar and Frischer, 2011), the Fast Fourier
Transform (Lenort, 1995), the statistical model based on hypothesis test asKS-testKolmogorov
and Smirnov (Kar and Mohanty, 2004; Dong et al., 2011; Yang et al., 2005), scalar and vector
statistical time series methods (Kopsaftopoulos and Fassois, 2011), neural networks (Samanta



124 A. Chiter et al.

andAl-Balushi, 2003), wavelets (BendjamaandBoucherit, 2016), blind source separation (Wang
et al., 2014), fuzzy logic (Liu et al., 1996). El-Thalji and Jantunen (2015) andRai andUpadhyay
(2016) reviewed almost all the techniques used in the domain predicting defects.

Typical defects in bearings are localized defects that occur generally in form of tiredness
cracking under cyclic pressure of contact (El-Thalji and Jantunen, 2015; Fajdiga and Sraml,
2009; Glaeser and Shaffer, 1996; Ismail et al., 1990; Tauqir et al., 2000). Thus, the detection of
cracking is frequently based on detection of the attack. During an abnormal operation, a series
of wide band impulses will be generated when the rolling element of the bearing (ball or roller)
(Brie, 2000; Ou et al., 2016) goes above the defect at a frequency determined by the shaft speed,
geometry of the bearing and the site of the defect (Barkov, 1999; Dyer and Stewart, 1978; Feng
et al., 2016; Ma and Li, 1995; Tandon and Choudhury, 1999). The site of the defect depending
on the characteristic frequencies gives the possibility of detecting the presence of the defect and
performing the diagnosis of the defective part.

The difficulty of detection of localized defects (Niu et al., 2015) is related to the bearing
energy whichwill diffuse through awide band of frequency and hence it can be easily immersed
in the noise (Ma andLi, 1995; Van et al., 2016). Thus, under various operating regimes (varying
loads and speeds), manymethods remain inefficient for the prediction (El-Thalji and Jantunen,
2015), because itmayhappen that an excited resonancemodeat thebeginning of the attackmay
not be excited later when the defect has developed (Ma and Li, 1995; Mikhlin and Mytrokhin,
2008). In this paper, and to refer on the sequential analysis developed by Wald in the 1940s
(Schneeweiss, 2005;Wald, 1943, 1945, 1947, 1949;Wald andWolfowitz, 1943, 1948), a composite
hypothesis test is used for the detection and diagnosis of localized defects in rolling bearings. To
this end, it is necessary to be provided with a significant and exact variance without any need
to estimate when the resonances modes occur.

2. Problem position

2.1. Probability Density Function (PDF) of vibrations of rolling bearings

To characterize vibration of rolling bearings, which is supposed to be a stationary stochastic
process, and the PDF can describe the percentage in time when the signal reaches a given
amplitude x. For the given amplitude, the PDF is estimated by

P(x)= lim
∆x→0

Pr{x¬x(t)¬x+∆x}

∆x
= lim
∆x→0

1

∆x

j
∑

i=1

∆ti
T

(2.1)

where T is the total time of observation and ∆ti is the i-th duration while x(t) is inside the
interval [x,x+∆x]. For vibration without a defect, which represents healthy functioning, the
distribution of the amplitude can be considered as aGaussian process. This vibratory signature
will have a well-defined variance σ20 which is different from the variance σ

2
1 of a signal with a

localized fault (Fig. 1) and, consequently, the overall vibration of the bearingwill be constituted
by two alternately periodic parts with different variances (Ma and Li, 1995).

2.2. Sequential Probability Test (SPRT)

Introduce now the sequential probability test (SPRT) of a simple null hypothesis H0 which
indicates the good operating condition and a simple alternative hypothesis H1 which indicates
the presence of a defect, based onN independent observations x1,x2, . . . ,xN having a common
probability density function developed by Wald (1945, 1949) andWeiss (1956).



A new method for automatic defects detection and diagnosis... 125

Fig. 1. Real signal of the rolling bearing with a defect

The hypotheses are

H0 : P(x/H0)=
1

√

(2π)NσN0

exp

(

−
N
∑

i=1

x2i
2σ20

)

H1 : P(x/H1)=
1

√

(2π)NσN1

exp

(

−
N
∑

i=1

x2i
2σ21

)

(2.2)

where x= [x1,x2, . . . ,xN] and σ
2
i are the variances with σ

2
0 <σ

2
1.

For the analysis of any vibratory signal, certainly one of the two variances will be retained
outside the test hypothesisH1 and one will have information whether or not it occurs with one
of the characteristic frequencies of the rolling bearings (inner race, outer race, ball and cage). In
the case of healthy rolling bearings, during a time∆t for the signal x(t), all measurements ofM
observations will have aGaussian distribution given by relation (2.2)1, In the case of a defective
bearing given by (2.2)2 and by varying the number M of observations in the time ∆t, and as
soon asM is sufficiently large, and it is always possible to calculate the estimated variance σ21 of
the acquired vibratory signal with the defect in the rolling bearing. The variances σ20 for healthy
rolling bearings could be calculated by

σ20 =
1

M

M
∑

1

x2i (2.3)

and σ21 = M
−1
∑M
1 x
2
i is considered as an estimated variance of the defect signal. Such an

estimate will lead to a test for probability of both detection or false alarm (Ma and Li, 1995).

3. Sequential test

3.1. The likelihood ratio test with simple choice

The likelihood ratio test (PRT) of the σ21 measurement could then be expressed as follows
(Ma and Li, 1995; Paulson, 1947)

{

if ξ(x)>µ choose H1

if ξ(x)<µ choose H0
(3.1)

where ξ(x) is the likelihood ratio, which is defined by

ξ(x)=
P(x/H1)

P(x/H0)
=
σN1
σN0
exp

(

σ21 −σ
2
0

2σ20σ
2
1

N
∑

i=1

x2i

)

(3.2)



126 A. Chiter et al.

By taking the natural logarithm of the two parts, the test can be simplified into

{

if f(x)>γ choose H1

if f(x)<γ choose H0
(3.3)

where

f(x)=
N
∑

i=1

x2i γ =
2σ20σ

2
1

σ21 −σ
2
0

ln
(σN1
σN0
µ
)

(3.4)

Then probability Pf of the false alarm and the detection probability Pd of the PRT are

Pf =P(f(x)>γ/H0)=P
(

∑

x2i >γ/H0
)

=

∫

∑

x2
i
>γ

P(x/H0) dx

Pd =P(f(x)>γ/H1)=P
(

∑

x2i >γ/H1
)

=

∫

∑

x2
i
>γ

P(x/H1) dx
(3.5)

From equations (2.2) and (3.5)1, the Pf is a decreasing monotone function of the parameter γ.

Integration of equations (3.5) leads to

Pf =exp
(−γ

2σ20

)

Pd =exp
(−γ

2σ21

)

(3.6)

where σ20 is the variance measured during healthy operation, σ
2
1 –variance measured during

unspecified operation and γ – the threshold of the test determined by

γ =−2σ20 lnPf (3.7)

while combining Pd with Pf we will have

Pd =P

σ
2

0

σ
2

1

f
(3.8)

Using equation (3.5)1 to determine the probability of the false alarm Pf which corresponds to
threshold equation (3.7) on the one hand and, on the other, using this same threshold will give
the maximum probability of detection Pd defined by equation (3.5)2 related to the variance σ

2
1

which is an unknown parameter estimated in one duration of the previously signal fixed. It can
be deduced that the uniformlymost powerful test (UMP) exists in the sense ofNeyman-Pearson
criterion which maximizes Pd (3.8) for a given Pf because the optimal probability rate test
(PRT) (3.3) for each σ21 > σ

2
0 could be completely defined apart from the knowledge of the

true variance σ21 of the signal defect. Finally, the UMP test is defined by system (3.3) and is
constructed by equations (3.1), (3.2) with a determined γ by the pre-established false alarm
probability α, where α is the threshold of significance

Pf(γ)=α (3.9)

3.2. Wald sequential test

Contrarily to the classical test (test with a simple choice), one is not obliged tomake a choice
between the two hypothesesH0 andH1, consequently, one deals with another type of test. If the
size of observations is fixed, the construction of the test leads to the sharing of possible values



A new method for automatic defects detection and diagnosis... 127

of the statistical domain in three regions (Wald, 1945; Berger andWald, 1949;Wolfowitz, 1949;
Sobel andWald, 1949)

Ψ(n) =Ψ(x1,x2, . . . ,xn) (3.10)

that is the region of probable values and the region of improbable values (knowing that the basic
hypothesis H0 is true). If a given value of Ψ(x1,x2, . . . ,xn) falls into the region of improbable
values, thebasic hypothesis is rejected.The sequential test, that is, the test based on a sequential
procedure of observation, is built up as follows. For each value of

ν =1,2, . . . ,n,n+1 (3.11)

the domain Γν of possible values of the critical statistics Ψ(x1,x2, . . . ,xn) is divided into three
disjoined regions: ΓH0ν – region of probable values, Γ

H1
ν – region of improbable values and Γ

∗

ν –
region of doubtful values (knowing thatH0 is true)

Γν =Γ
H0
ν ∪Γ

H1
ν ∪Γ

∗

ν (3.12)

where ν = 1,2, . . . with each step ν of the sequential procedure of observation. After having
recorded the observations x1, . . . ,xν, ν =1,2, . . . one makes a decision relying on the following
rulewhichdefines theWald test: ifΨ(x1,x2, . . . ,xn)∈Γ

H0
ν one acceptsH0; ifΨ(x1,x2, . . . ,xn)∈

ΓH1ν one accepts H1 and if Ψ(x1,x2, . . . ,xn) ∈ Γ
∗

ν the problem remains open until the ν-th
observation. For this reason, the region Γ∗ν is called the region of indetermination or the region
of the observation pursuit.
For the establishment of the Wald test of probability, one considers two simple hypotheses

of the following form, seeWald (1945, 1947) andWald andWolfowitz (1948)

H0 The observation is extracted from a density population f(x,θ0)

H1 The observation is extracted from a density population f(x,θ1)
(3.13)

The critical statistics of this test is defined by the relation (Wald, 1945, 1947; Paulson, 1947)

Ψ(ν) = ln
f(x1,θ1) · · ·f(xi,θ1)

f(x1,θ0) · · ·f(xi,θ0)
=
ν
∑

i=1

ln
f(xi,θ1)

f(xi,θ0)
(3.14)

where: f(xi,θ0)=P(x/H0) with θ0 =σ
2
0 and f(xi,θ1)=P(x/H1) with θ1 =σ

2
1.

P(x/H0) and P(x/H1) could be drawn from equation (2.2). So, one establishing the likeli-
hood ratio, the critical statistics would be expressed as follows

Ψ(ν) = ln

{[

exp
(

− 1
2σ2
1

∑ν
i=1x

2
i

)

√

(2π)νσν1

]/[

exp
(

− 1
2σ2
0

∑ν
i=1x

2
i

)

√

(2π)νσν0

]}

(3.15)

After simplification of equation (3.15) and arrangement of the logarithmic term, one gets

Ψ(ν) =
σ21 −σ

2
0

2σ20σ
2
1

ν
∑

i=1

x2i +
ν

2
ln
σ21
σ20

(3.16)

The three regions are defined roughly by relations (3.11), (3.12), (3.13) and (3.16) that define
the completely Wald test (WT) (Aı̈vazian, 1986; Wald, 1947; Weiss, 1956)

ΓH0ν =
{

Ψ : Ψ(ν) ¬ ln
β

1−α

}

ΓH1ν =
{

Ψ : Ψ(ν) ¬ ln
1−β

α

}

Γ∗ν =
{

Ψ : ln
β

1−α
¬Ψ(ν) ¬ ln

1−β

α

}

(3.17)

Wald test (3.17) is more optimal than all tests between hypotheses (3.13) with risks of the first
and second species lower than the respective given values α and β.
Values of α and β (Aı̈vazian, 1986) are: 0.1, 0.05, 0.025, 0.01, 0.005, 0.001, 0.002.



128 A. Chiter et al.

4. Rolling element bearings defects detection

4.1. Detection procedure

The detection procedure is divided into many steps which can be stated as follows:

1 – Take the discrete vibration for M samples, which is larger than the amount of the
characteristic period of the defect.

2 – Select a window of sizeN for the test.

3 – Estimate the variance σ20 by using equation (2.3).

4 – Suggest a choice of α and β.

5 – Position the window at the beginning of recording of the vibration.

6 – Compute Ψ(N) by using equation (3.16).

7 – Define the intervals of the three regions by the terminals a = ln[β/(1 − α)] and
b= ln[(1−β)/α].

8 –Make the test by using equation (3.17).

9 – Generate a hypothetical signal defined by

h(i)=

{

0 if H0 is true (Ψ ¬ a)

1 if H1 is true (Ψ ­ b)

If a¬ Ψ ¬ b, carry on with pursue for data opening another window (here, one does not
make a decision but only increases the size of the window).

10 – After generation of the hypothetical signal, if a defect is present, there will be a data
vector composed of two values 0 and 1. If 1, then appears periodically with a period of the
characteristic frequency of the bearing and is considered defective.

11 –To compare the detected frequencywith themain characteristic frequencies of the rolling
bearings, it would be very easy to locate the defect so the diagnosis could be established
by comparing the multiple of this frequency detected with that of the most well-known
defects.

4.2. Test plan

Based on the detection procedure described in Section 4.1, a test plan can be established
which is shown by the procedure diagram shown in Fig. 2. So that the experiment is valid,
one chooses N as a small fraction of the characteristic period of the defect, that is to say one
fifth (Ma and Li, 1995). By examining step 10 in the detection procedure in Section 4.1 (to
show if there is periodicity), one uses autocorrelation of the signal, a peak in the autocorrelation
function reveals the periodicity of the signal, and the value of the time of this peak will give
the period of the defect Td. Consequently, one can determine the frequency of the defect fd, and
comparing it with the characteristic frequency fc, one can establish the diagnosis.

5. Validation of the model by simulated and experimental signals

5.1. Validation of the model by simulated signals

5.1.1. Generation of the simulated signals

To simulate the defect, a bearing of the type NJ2204ECP has been used. The shaft speed is
n=1500rpm, the characteristic frequencies are determined by the relations fromAppendixA1,



A new method for automatic defects detection and diagnosis... 129

Fig. 2. Test plan

where the frequency of the cage is: fcage = 0.39fr (9.74Hz), the frequency of the outer race:
for = 0.39Zfr (87.68Hz), the frequency of the inner race fir = 0.61Zfr (137.32Hz), and the
frequency of the ball: fre = 4.754fr (118.85Hz); where: fr = 25Hz, Z is the number of balls.
For NJ2204ECP: Z = 9, d = 7.5mm, D = 34mm, α = 0. The reference signal (Fig. 3a) is
taken as a sinusoid of frequency 25Hz, amplitude equal to unit and a null phase. The simulated
defect signal (Fig. 3b) is considered as the sum of a sinusoid of frequency 25Hz, amplitude
equal to unit and the null phase, a sinusoid of frequency 87Hz of amplitude 10 times the unit
(representing a defect of frequency 87Hz, which corresponds to the frequency of the outer race,
as one can use the function pulstran available in Matlab which generates a series of impulses),
and aGaussianwhite noise centeredwith variance equal to 1 generated by the function “randn”
available in Matlab with a signal noise ratio SNR= 20dB. The thresholds of significances are
fixed at α=0.05 and β=0.002.

Fig. 3. (a) Reference signal, (b) defect signal

5.1.2. Interpretation

One can say that periodicity of a hypothetical signal (h-signal, Fig. 4a) appeared in the
function of autocorrelation (Fig. 4b) reveals the existence of a defect. Todetermine its frequency,
one carries out Fourier fast transform (FFT) of the hypothetical signal, which reveals visually
the frequency of the defect (87Hz) which corresponds indeed to the characteristic frequency of
the outer race (Fig. 4c). Consequently, one can affirm that the plan suggested for detection and
diagnosis of the defect in the bearing has succeeded and tomade diagnosis of the defective part.
During healthy running, the hypothetical signal will be zero, the autocorrelation of the h-signal
will not reveal any periodicity, and the FFTwill confirm the absence of the defect.



130 A. Chiter et al.

Fig. 4. (a) Hypothetical signal, (b) autocorrelation of hypothetical signal, (c) FFT of hypothetical signal

5.2. Validation of the model by experimental signals

5.2.1. Generation of the experimental signals

The test stand consists of a reinforced concrete frame, isolated from the ground by shock-
absorbing studs. Two rows of shafts each having diameter of 60mm and length of 680mm are
mounted in anopen loopandfixed to the chassis by four rollingbearingswith anaverage stiffness
of 3 ·107daN/m as shown in Fig. 5.

Fig. 5. Architecture of the teststand (RB – roller bearing, BB – ball bearing)

Thebearings in thevicinity of the test gear pair have ball bearings of the type 6012,while the
outer bearings are roller bearings of the type NU1013. The shaft lines are connected in rotation
by test gears. The applied speed and torque are measured by an electronic device composed of
a motor and a brake.

The dynamic behavior of the system can be studied usingmeasurements of the acceleration,
transmission error and noise. The accelerations are measured using piezoelectric accelerometers
ENDEVCO 224C whose resonance frequency is 32kHz. The accelerometers are mounted by
gluing small duralumin pellets onto the accelerometers which are screwed. The tests are carried
out on a spurgearwithhelical teeth.Thegear ratio is 36/38withmodulusm=2.Thegeometric
characteristics of the ball and roller bearing are given in Table 1.

Type of defect:To simulate the scaling on thebearings, a notchof 1.7mmanddepthof 0.088mm
is made using a fine grinder as shown in Fig. 6. The roller bearing is removable without “NU
type” destruction or specialized tooling.



A new method for automatic defects detection and diagnosis... 131

Table 1.Geometric characteristics of the ball and roller bearing

Geometric characteristics Ball bearing Roller bearing

Middle diameter to center of balls D [mm] 77.7 80.55

Diameter of ball d [mm] 9 7

Number of balls Z 14 21

Angle of contact α 0 0

Fig. 6. Defective inner race geometry of the roller bearing

Monitoring conditions: The applied load is equal to 12daNmand 4300rpm speed test. The cha-
racteristic frequencies of the ball bearing and the roller bearing are calculated by the geometrical
formulas given in Appendix A1.

Table 2.Characteristic frequencies of the ball and roller bearings

Bearing type Fr [Hz] Fcage [Hz] For [Hz] Fir [Hz] Fer [Hz]

6012 71.67 31.68 443.56 559.77 627.02

NU1013 71.67 32.72 687.11 817.89 830.91

Fr – rotating frequency, Fcage – frequency of the cage,
For – frequency of the outer race, Fir – frequency of the inner race,
Fer – frequency of the ball or roller

Experimental signals: The acquired reference signal and the acquired signal defect are shown in
Fig. 7a and 7b.

5.2.2. Interpretation

The detection and diagnostic plan applied to the experimental signals shown in Fig. 7a and
Fig. 7b is able to detect the fault frequency applied to the bearing inner ring shown in Fig. 8a.
It shows the presence of state “1” of the hypothetical signal and Fig. 8b shows a frequency of
814Hz very close to the fault frequency which is equal to 817.89Hz. It indicates that the plan
has reacted well in establishing a correct diagnosis.

6. Diagnosis plan

To establish a good diagnosis of defects, it is necessary to know a significant number of defects.
Thus, by comparing the frequency detected by theWald test presented before with the charac-
teristic frequencies we can locate the defect. By comparing the defect frequency with the main
defects of the rolling bearings (Barkov, 1999), we can establish the diagnosis by using the fre-
quency of modulation presented in the work of Barkov (1999). For the plan suggested by Fig. 2
it is possible to establish the diagnosis of the bearing defective part and its nature.



132 A. Chiter et al.

Fig. 7. (a) Experimental reference signal, (b) experimental defect signal

Fig. 8. (a) Experimental hypothetical signal, (b) FFT of the experimental hypothetical signal

7. Conclusion

The detection and diagnosis plan based on theWald test is described. This plan can be applied
tomeasurements of the bearings vibration signals with andwithout defects under various loads
and speeds.The effectiveness of the suggested detection plan is illustrated inFig. 4 for the simu-
lated signal and in Fig. 8 for the experimental signal. The plan works very well with vibratory
signals of wide bands. Finally, the plan is very promising for automatic detection and diagnostic
applications.

Acknowledgment

I address all my gratitude to the Dynamics and Control of Structure team of the Contacts and

StructuresMechanics Laboratory (LaMCoS), research unit (UMR5259) of INSALyonFrance andCNRS

(INSIS, Institute of Engineering Sciences and Systems) for providing me with the experimental signals

realized on their own test stands and, particularly, Associate ProfessorMahfoud Jarir, also the director

of LaMCoS laboratory.

Appendix 1

Characteristic frequencies of the bearing (Barkov, 1999):
— frequency of the cage

fcage =
fr
2

(

1−
d

D
cosα

)

— frequency of the outer race

for =Z
fr
2

(

1−
d

D
cosα

)



A new method for automatic defects detection and diagnosis... 133

— frequency of the inner race

fir =Z
fr
2

(

1+
d

D
cosα

)

— frequency of the ball

fre = fr
d

D

[

1+
(D

d

)2
cosα

]

whereα is the angle of contact, d [mm] – diameter of the ball,D [mm] –middle distance to the
center of balls, Z – number of balls, fr [Hz] – rotating frequency (fr = n/60), n [rpm] – shaft
speed.

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Manuscript received November 13, 2016; accepted for print July 28, 2017