Jtam-A4.dvi


JOURNAL OF THEORETICAL

AND APPLIED MECHANICS

53, 3, pp. 617-628, Warsaw 2015
DOI: 10.15632/jtam-pl.53.3.617

INVERSE METHOD FOR A ONE-STAGE SPUR GEAR DIAGNOSIS

Ali Akrout, Dhouha Tounsi, Mohamed Taktak, Mohamed Slim Abbes,

Mohamed Haddar

Mechanics, Modelling and Production Research Laboratory (LA2MP), National School of Engineers of Sfax,

University of Sfax, Sfax, Tunisia; e-mail: mohamed.taktak@fss.rnu.tn; mohamed.taktak.tn@gmail.com

In this paper, a source separation approach based on the Blind Source Separation (BSS) is
presented. In fact, the IndependentComponentAnalysis (ICA),which is themain technique
of BSS, consists in extracting different source signals from several observed mixtures. This
inverse method is very useful in many fields such as telecommunication, signal processing
and biomedicine. It is also very attractive for diagnosis of mechanical systems such as rota-
ting machines. Generally, dynamic responses of a given mechanical system (displacements,
accelerations and speeds)measured through sensors are used as inputs for the identification
of internal defaults. In this study, the ICA concept is applied to the diagnosis of a one-stage
gearmechanism inwhich two types of defects (the eccentricity error and the localized tooth
defect)are introduced. The finite elementmethod allows determination of the signals corre-
sponding to the acceleration in some locations of the system, and those signalsmay be used
also in the ICA algorithm. Hence, the vibratory signatures of each defect can be identified
by the ICA concept. Thus, a good agreement is obtained by comparing the expected default
signatures to those achieved by the developed inversemethod.

Keywords: Independent Component Analysis (ICA), source identification, gear mechanism,
geometrical defects

1. Introduction

Diagnosis, without disassembling different parts of a givenmachine, represents themain purpose
to be achieved in the defect detection practice. This procedure allows the industry to save
time and money. That is why many approaches found in the literature can be considered as
tools to realize non destructive diagnosis of machines. Usually, any internal default or external
force which excites mechanical systems can be identified by frequency analysis. That allows the
determination of the internal default excitation frequencies or the external excitation forces.
However, the knowledge of the different excitation frequencies existing in the spectrum of the
system vibratory signal is insufficient to determine the type of defaults, its amplitude and its
position in the system. To surpass these limitations, various methods have been developed in
order to solve the source separation problem leading to different algorithms which can be used
formodal analysis of structures and dynamic systems (Antoni et al., 2004; Poncelet et al., 2007;
Kerschen et al., 2007; Abbès et al., 2011; Akrout et al., 2012a,b). One of these techniques is the
Blind Source Separation (BSS) which was presented by many researchers (Antoni and Braum,
2005) and (Antoni, 2005). It consists in separating an independent linear mixture source signal
without any information about the sources by using only the measured vibratory response of
the system. The separation procedure is based on the statistical independency of sources.

In the last few years, an increasing number of works using the ICA concept for detecting
defects inmechanical systems have been found.Many researchers have developed different algo-
rithms based on the ICA technique for the diagnosis of rotating machines (Servière and Fabry,



618 A. Akrout et al.

2004) and (Taktak et al., 2012), bearings (Shalvi andWeinstein, 1990) and structures (Zang and
Friswel, 2004), (Akrout et al., 2012a,b).

In this paper, the ICAmethod is applied to diagnosis a one-stage spur gear system.The spur
gear represents adevice that ensures transmissionofmovement invariousmechanical systems. In
a gear mechanism, there appear frequently many internal defaults which generate disturbances
in the dynamic responses. By using the ICA concept, both the amplitude and the frequency are
used to localize the defaults.

After presentation of the separationmethod including the definition of the ICA in Section 2,
Section 3 presents the separation hypothesis of the ICA concept and the associated different
steps. Section 4 presents the results and the discussion of the application of this technique to a
one-stage spur gear to identify its internal defaults.

It is also worth mentioning that in our previous work (Taktak et al., 2012), the observed
signals (accelerations) are determined on two nodes of a discretized crankcase flexible plate.
However, in this article, the vibratory responses are calculated in the pinion and the gear centre,
respectively.

2. Inverse method for dynamic problem: the Independent Component Analysis
(ICA)

2.1. ICA definition

The ICA method was defined by Comon (1990) as a statistical technique that aims at
decomposing a random signal X (measured signal) into a linear combination of independent
signals (source signals). Then, the ICA problem is formulated as follows

X=AS (2.1)

whereX is a column vector ofM output observations, which represents an instantaneous linear
mixture of source signals S defined by a column vector of N sources. A is a M ×N mixing
matrix. The equation of the ICA concept is described as follows (Zang et al., 2004; Abbès et al.,
2011; Akrout et al., 2012)

Xj(t)=
N∑

i=1

ajiSi(t) j=1, . . . ,M (2.2)

2.2. ICA indeterminacies

In spite of its advantages in separating source signals, two ambiguities are inherent to the
ICAmethod: it aims at identification of both source signals and themixingmatrix with certain
indeterminacies that include arbitrary scaling and permutation. The mixture model can be
expressed as follows (Antoni, 2005; Tong et al., 1991)

X=AP−1PS (2.3)

whereP is the permutation matrix.

In the same way, the observations can be rewritten as follows

X=AD−1DS (2.4)

D is a diagonal matrix. So any scalar multiplier to one of the source Si could be avoided by
dividing the corresponding column of the matrixA.



Inverse method for a one-stage spur gear diagnosis 619

These two indeterminacies imply that it is impossible to obtain a unique determination of
the original sources. So the separatingmatrixB is estimated by the productBAwhich is equal
to the diagonal matrixD up to the permutation matrixP

BA=DP (2.5)

In order to reduce the shape indeterminacy in the model and save the waveform of the original
signals, a constraint of unit variance is imposed on the sources S to eliminate the scale factor
(Antoni, 2005; Tong et al., 1991)

E[SS+] = I (2.6)

where (+) denotes the conjugate, and I is theN×N identity matrix.

The ICA concept assumes that the sources Si must be statically independent uncorrelated
and have a non-Gaussian distribution. Then, to perform the estimated source signals task and
themixingmatrixA, the ICA requires some separation assumptions which are presented in the
following section.

2.3. Separation assumptions

2.3.1. Statistical independency of the sources

The separation can be achieved only when all the components of the separating system
are mutually and statically independent. The components of a random vector X are mutually
independent if and only if the probability density of the vector X is equal to the marginal
probability density (Akrout et al., 2012a,b; Hyvärinen and Oja, 2000)

Px(x)=
N∏

i=1

Pxi(xi) (2.7)

Many formulations are used to assure the statistical independence of the sources such as the
mutual information which is based on the Kulback divergence (Comon, 1990)and the likelihood
estimation (Geata and Lacoume, 1990).

2.3.2. Uncorrelated sources

The statistical independence of the sources imposes that the sources must be uncorrelated.
So, two variables Y1 and Y2 are uncorrelated if and only if their covariance is equal to zero
(Akrout et al., 2012a,b; Comon, 1990)

E{Y1Y2}−E{Y1}E{Y2}=0 (2.8)

2.4. Non-Gaussianity distribution of the sources

Another very useful hypothesis in the source separation is the non-Gaussianity of the so-
urces (Akrout et al., 2012a,b; Comon, 1990). This criterion is especially essential in the case
of stationary white sources since the non-Gaussianity of the sources is needed to achieve the
separation.

The Gaussianity of a signal is defined as the difference between the distributions of the
signal and the Gaussian signal with the same power. Instantaneous linear mixture sources can
be separated by maximizing the non-Gaussianity of the output signal, which is obtained by a
linear combination of the observations.



620 A. Akrout et al.

The non-Gaussianity can be evaluated by the kurtosis or the fourth order cumulate. The
Kurtosis of a variable y is expressed as follows (Comon, 1990)

Kurt(y)=E{y4}−3(E{y2})2 (2.9)

Anothermeasure of non-Gaussianity which can be used is the negentropy theorywhich is based
on the theoretical quantity of the differential entropy (Wiggins, 1978).The differential entropy
of a variable y with a density f(y) can be defined as follows (Hyvärinen and Oja, 2000)

H(y)=−

∫
f(y)log(f(y)) dy (2.10)

So, the negentropy which is equal to zero for the Gaussian variable can be written as follows
(Hyvärinen andOja, 2000)

J(y)=H(ygauss)−H(y) (2.11)

where ygauss is a randomvariablewhich has the same covariancematrix as y. So, the negentropy
is always non negative and it is equal to zero in the case of Gaussian variable.

3. ICA steps

3.1. Pre-processing for ICA

Before applying the ICA algorithm and to guarantee the uncorrolatedness of the sources, it
is useful to do some pre-processing. These pre-processingsmake the ICA estimation easy. These
pre-processings are presented in the following sections.

3.1.1. Centring

The first pre-treatment is the centring. Each observed vector X must be centred, i.e., sub-
tracts its mean vector m=E{X}; so,Xwill be the zero mean variable which implies that {S}
is the zero mean as well (Hyvärinen and Oja, 2000).

3.1.2. Whitening

The second pre-treatment used in ICA is whitening of the observationsX. This means that
before applying the ICA algorithm, we transform the vector X to a white vector X̃ which has
uncorrelated components, and the variance of its components is equal to unity and (Thirion-
-Moreau, Moreau, 2000)

X̃=WX (3.1)

W is the whiteningmatrix, it is determined by the eigenvalues decomposition of the covariance
matrix which is defined by

Rx(0)=E{XX
+}=AA+ =UDUT (3.2)

where U is the eigenvector orthogonal matrix of E{XXT} and D is the diagonal matrix of its
eigenvalues.

So,we can write

E{(WX)(WX)+}=WW+E{XX+}= I (3.3)



Inverse method for a one-stage spur gear diagnosis 621

Then, the whiteningmatrix can be defined as follows

W=D−1/2UT (3.4)

This transformation allows definition of a newmatrix Ã

X̃=D−1/2UTAS= ÃS (3.5)

à is an orthogonal mixingmatrix, then we can write

E{X̃X̃T}= ÃE{SST}ÃT = ÃÃT = I (3.6)

So, the whitening of the observations contributes to reduction of the number of parameters to
be estimated. Instead of estimating n2 parameters, only the elements of the new orthogonal
matrix à are considered. We noticed that an orthogonal matrix has n(n− 1)/2 degrees of
freedom (Hyvärinen and Oja, 2000).

3.2. ICA algorithm

3.2.1. Kurtosis maximization

In the literature, many contrast functions have been proposed. Among them the Kurtosis
defined by the normalized fourth order marginal cumulate is the commonly used statistical
quantity in the ICA concept (Comon, 1994). The kurtosis was firstly used in blind separation
of seismic signals and digital communication (Shalvi andWeinstein, 1990; Wiggins, 1978).
The separation of sources using ICA is based on the determination of each column of the

separating matrix W. Then it determines the source related to this column. The extracting
vector of the mixing matrix can be defined as follows (Akrout et al., 2012a,b; Zarzoso and
Comon, 2010)

Y=WHX (3.7)

where Y is the estimated source and W is the separating matrix, (·)H denotes the conjugate
transpose operator. W must maximize the Kurtosis in order to guarantee the non-Gaussianity
distribution. This Kurtosis is defined as follows (Akrout et al., 2012a,b; Zarzoso and Comon,
2010)

K(ω)=
E{|y|4}−2E2{|y|2}−|E{y2}|2

E2{|y|2}
(3.8)

In order to maximize this function, an exact line search of the absolute Kurtosis contrast is
performed, which is defined as (Zarzoso and Comon, 2010)

µopt =arg
(
K(w+µg)

)
(3.9)

where g is the search direction defined as the gradient g=∇w
(
K(ω)

)
. The exact line search is

determined after determination of roots of a fourth degrees polynomial, so the roots leading to
the absolute maximum of the contrast along the search direction are chosen.
After the determination of the column of the separating matrix, a deflation approach is

considered in order to extract the source corresponding to the determined column.

3.2.2. Deflation approach

The deflation approach allows escaping from a multiple inputs/multiple outputs system
into a system with a single input/multiple outputs. The deflation approach is used to extract
successively the sources. This ensures that each source is chosen oncewithmultiplying the exact
factor. After the determination of the column vector of the separationmatrixW bymaximizing
the kurtosis contrast, the deflation approach is used in order to determine the estimated source.



622 A. Akrout et al.

4. Diagnosis of a one-stage gear mechanism

The purpose of this part is to provide an illustration of the capability of the ICA algorithm
to separate and extract signals from a given mechanical system which is a one-stage spur gear
mechanism. The ICA concept is applied to this system and the source identification results are
presented and discussed.

4.1. System presentation

The studied system is a one-stage spur gear mechanism. It can be divided into two blocks
as presented in Fig. 1a. The first block is composed by the input shaft (1) and a pinion. The
second block is constituted by the gear and the output shaft (2). Both the pinion and the gear
are supposed to be rigid. The shafts, which are assumed to be flexible, are discretized by two
node beam finite elements (Fig. 1b) with 6 degrees of freedom per node.

Fig. 1. Studied case: (a) model of a one-stage spur gear, (b) discretized single spur gear

The shafts are supported by bearings each modelled by two linear springs. The gear mesh
stiffness is modelled by a linear spring acting on the line of action of the meshing teeth.
The expression of the displacement on the line of action is expressed as (Chaari et al., 2008)

δ(t) = (z1−z2)sinγ+(y1−y2)cosγ+θxprbp+θxgrbg (4.1)

where zi and yi (i = 1,2) are respectively the centre displacements of the pinion and gear.
θxp and θxg are angular rotations of the pinion and the gear centre, respectively. rbp and rbg are
the basis radius of the pinion and the gear, respectively. γ is the pressure angle.
The equation of motion of the system is defined as (Chaari et al., 2008)

Mq̈+Cq̇+K(t)q=Fext (4.2)

where M, C and K(t) are respectively the global mass, damping and stiffness matrices of the
whole system. q is the vector of degrees of freedom,Fext is the external applied forces vector.
The mass matrix M is a diagonal matrix which can be written as follows (Chaari et al.,

2008)

M=




M1 0 0 0
0 Mp 0 0
0 0 Mg 0
0 0 0 M2




(4.3)

Mi (i=1,2) is themassmatrix of the ith shaft;Mp andMg are respectively themassmatrices
of the pinion and the gear (Chaari et al., 2008).
K(t) includes the shafts stiffness, bearing stiffness and the time varying gear mesh stiffness

matrix.The stiffness matrix of the shafts is determined by the FE model presented in Fig. 1b.



Inverse method for a one-stage spur gear diagnosis 623

Then, the matrix K(t) can be divided into a mean stiffness matrix K and a fluctuating one
KGS(t) (Chaari et al., 2008)

K(t)=K+KGS(t)

K=




Kshaft′1′ KC1 0 0
KTC1 Kbearing′1′ 0 0
0 0 Kbearing′2′ KC2
0 0 KTC2 Kshaft′2′




(4.4)

Kshaft′i′ and Kbearing′i′ (i= 1,2) are the stiffness matrices of the ith shaft. KC1 and KC2 are
the coupling matrices between the bearings and shafts. The bearing stiffness matrices contain
the translation stiffness along with the z and y directions (Chaari et al., 2008) (Ky1 = Kz1 =
Ky2 =Kz2 =10

8N/m).
KGS(t) is the fluctuation stiffness matrix defined as follows (Chaari et al., 2008)

KGS(t)=




0 0 0 0 0 0 0 0 0 0 0 0
0 s4 s5 s6 0 0 0 −s4 −s5 s8 0 0
0 s5 s3 s7 0 0 0 −s5 s3 s9 0 0
0 s6 s7 s10 0 0 0 −s6 s7 s12 0 0
0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0
0 −s4 −s5 −s6 0 0 0 s4 s5 −s8 0 0
0 −s5 −s3 −s7 0 0 0 s5 s3 −s9 0 0
0 s8 s9 s12 0 0 0 −s8 −s9 s11 0 0
0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0




Ke(t) (4.5)

whereKe(t) is the time varying gear mesh stiffness.
The terms si (i=1, . . . ,12) are given in Table 1.

Table 1.Coefficients si

s1 sinγ s7 rbp sinγ

s2 cosγ s8 rbg cosγ

s3 sin
2γ s9 rbg sinγ

s4 cos
2γ s10 r

2
bp

s5 sinγ cosγ s11 r
2
bg

s6 rbp cosγ s12 rbgrbp

The damping matrix of the system is determined from a combination between the mass
matrix of the system and the mean stiffness matrix

C=αM+βK (4.6)

where α and β are two constants which influence the stability of the system. The properties of
the studied system are presented in Tables 2 and 3.

4.2. Geometrical and teeth defect models

In order to disturb the dynamic response of the studied system, some defects have been
introduced into the spur gear. The vibration sources (internal and external excitations) of the
dynamic transmission are multiple. However, the internal sources have considerable effects on



624 A. Akrout et al.

Table 2.Parameters of the spur gear (Hidaka et al., 1979)

Pinion Gear

Teeth numbers 20 40

Mass [kg] 0.6659 2.663

Inertia Moment [kgm2] 2.996 ·10−4 0.0048

Base diameter circle [m] 0.0564 0.1128

Rotation speed [rpm] 1500 750

Torque [Nm] 10 −20

Module [mm] 3

Pressure angle [deg] 20

Gear mesh frequency [Hz] 500

Table 3.Parameters of the input and output shafts (Hidaka et al., 1979)

Shaft characteristics value

Material steel

Young’s modulus [MPa] 2 ·105

Length [m] 0.45

Section [m2] 3.84 ·10−3

Poisson’s coefficient 0.29

Density [kg/m3] 7858

the dynamic response of the system.As examples, of internal excitation sources, we canmention
assembly defects which are characterized by an excitation frequency which corresponds to the
rotation frequency of the shafts as well as geometric deviation which is characterized by an
excitation frequency corresponding to the gear mesh rotation frequency.
The geometric deviation comesmainly frommanufacturing defects; they are duplicated typi-

cally at thegearmesh frequency.Wecandistinguishtheprofileerrorwhichpresents thedifference
between the actual profile and the theoretical profile of the teeth. According to Munro (1967),
the profile error which is duplicated on each tooth, induces a transmission error marked at the
gear mesh frequency and its harmonics.
The eccentricity error represents the difference between the real and theoretical location of

the axes of the gear polar moment of inertia. Munro (1967) shows that the eccentricity of the
wheel is characterized by the presence of amain peak at the rotation frequency in the spectrum
of the static transmission.
Furthermore, teeth defects, such as spalling, breakage and crack, can affect considerably the

dynamic behaviour of the gear transmission. In that case, several dynamic models have been
developed in order to achieve some diagnosis solutions (Munro, 1967; Chaari et al., 2009).
The eccentricity error and the localized tooth defect have the following expressions (Hidaka

et al., 1979; Wajag and Kahraman, 2002)

Fdef1 =Ke(t)ec sin(2πfrt) Fdef2 =Ke(t)ec sin(2πfent) (4.7)

where fr is the rotation frequency of the shaft, ec is the excitation amplitude, fen is the gear
mesh frequency.Therefore, the equation ofmotion of the spurgear transmission canbe expressed
as follows

M, q̈+Cq̇+K(t)q=F−Fdef1−Fdef2 (4.8)

Equation (4.10) is solved using the Newmarkmethod. Then, vibratory responses of the system
are determined and used to reconstruct the forces related to different system failures.



Inverse method for a one-stage spur gear diagnosis 625

4.3. Numerical simulation

As mentioned above, the first step in the simulation process is the introduction of both
eccentricity and tooth defaults. The waveforms of the two internal defects introduced in the
gear transmission are presented in Figs. 2 and 3. The first one (Fig. 2) corresponds to the
temporary evolution of the eccentricity error. The second one (Fig. 3) corresponds to the time
evolution of the tooth defect.

Fig. 2. Time-evolution of the eccentricity error

Fig. 3. Time-evolution of the localized tooth defect

In order to extract the original sources, the observed signals (accelerations) are determined at
both the centre of the pinion and the centre of the gear and by taking into account the presence
of the tooth failure and the eccentricity error. The rotational speed of the pinion is equal to
N = 1500tr/min. The period is Te = 0.002s which corresponds to the gear mesh frequency
fe =500Hz.
Then, these calculated dynamic reponses are used as inputs in the ICA separation algorithm

to estimate the waveforms of the real internal defects in the spur gear as well as the excitation
frequency and, finally, to identify the type of defect.
After determination of the observed signals, the ICA algorithm is applied to determine the

estimated sources which correspond to the internal defects of the spur gear. The presence of the
rotation frequency of the shaft in the spectrum of the estimated sources indicates clearly the
position defect with the corresponding waveform. So, the eccentricity error can be identified.
If only the gear mesh frequency and its harmonics are observed, we can stipulate that there is
a tooth failure with the waveform and we can distinguish between the different tooth failures
which may exist.
Figure 4 presents the evolution of the estimated source extracted by the ICAmethod from

the observed signals versus time superimposed with the original source, which presents the
eccentricity error. The estimated source has almost the same waveform and amplitude as the
original source. The two sources have the same excitation frequency.



626 A. Akrout et al.

Fig. 4. Original and estimated time-evolution of the eccentricity defect

In Fig. 5, the spectrum of the estimated source corresponding to the eccentricity error is
presented.We notice the presence of a peak at the rotation frequency of the shaft which is equal
to 25Hz and other peaks at the gear mesh frequency and its harmonics that correspond to the
excitation frequency of the eccentricity error.

Fig. 5. Spectrum of the estimated signal of the eccentricity defect

Figure 6 shows the time-evolution of both original and estimated source corresponding to
the tooth defect. So, one can clearly see the efficiency of the developed inverse method for
reconstruction of the introduced internal defects. The amplitudes and the waveforms of the
original and the estimated excitation source are almost identical.

Fig. 6. Original and estimated time-evolution of the localized tooth defect

In Fig. 7, the spectrum of the second estimated source is presented. Here, we noticed the
presence of three peaks corresponding to the gearmesh frequency and its harmonics. Hence, the
excitation frequency of the tooth defect can be identified (fe =500Hz).



Inverse method for a one-stage spur gear diagnosis 627

Fig. 7. Spectrum of the estimated signal of the second defect

From all the results presented above, we conclude that the estimated sources reconstructed
by the developed inverse method(ICA algorithm) have the same waveforms and excitation fre-
quencies as the internal defects considered present in the system. So, the ICAmethod gives good
results for dynamic diagnosis of a one-stage spur gear and canbe employed for other complicated
mechanical systems.

5. Conclusion

In this paper, the ICAmethod has been presented and applied to a mechanical system defined
bya one-stage spur gear in order to identify the eventual internal defects. Two different types of
internal defects have been introduced in the equation ofmotion: the eccentricity error and tooth
defect. Then, by using the dynamic response signals as inputs in the ICA algorithm, we have
succeeded to recover the estimated sources which correspond to real internal sources introduced
in the gear system. So, the usefulness of the proposedmethod is demonstrated.

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Manuscript received April 8, 2014; accepted for print January 30, 2015