new aplication of real cepstrum

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Mechanical Systems

and

Signal Processing

www.elsevier.com/locate/jnlabr/ymssp

Mechanical Systems and Signal Processing 18 (2004) 10311046

New applications of the real cepstrum to gear signals, including

definition of a robust fault indicator

M. El Badaoui*, F. Guillet, J. Dani"ere

Laboratoire dAnalyse des Signaux et des Processus Industriels (LASPI)-EA-3059, IUT de Roanne 20,

Avenue de Paris., 42334 Roanne, France

Received 11 December 2003; accepted 19 January 2004

Abstract

This article shows the possibilities offered by the use of the power cepstrum for gear system vibratory

diagnosis. A model of signals generated by an accelerometer sensor will be established on which the

theoretical expression for the power cepstrum is partially calculated. This makes it possible to develop an

indicator which is little affected by the signal amplitude, the signal-to-noise ratio, or the position of the

sensor. It is shown that the use of a signal sampled vs. shaft angle enables the cepstrum to preserve its full

resolution and allows the realisation of a synchronous average to isolate each meshing pair. Some

applications of the monitoring procedure are presented.

r 2004 Elsevier Ltd. All rights reserved.

Keywords: Cepstral analysis; Gear box; Angular sampling; Synchronous average; Fault indicator; Diagnosis

1. Introduction

The power cepstrum introduced in 1963 by Bogert [1] was used first for the detection or the

suppression of echoes [24]. It has also been used in vibratory rotating machine diagnosis [58],

because the presence of faults induces in signals some recurrent patterns (echoes). In what follows,we discuss the possibilities offered by the power cepstrum for the vibratory diagnosis of gear

systems. This diagnosis is based on a few properties of the independence of cepstrum peaks. We

use the fact that these properties are almost independent of the nature of the signal in order to

calculate the theoretical expression for the amplitudes and positions of peaks that are exploited.

This expression permits us to define a robust fault indicator. We show that the efficiency of the

cepstrum is maximal when signals are acquired under angular sampling (synchronous cepstral

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*Corresponding author.

E-mail address: [email protected] (M.E. Badaoui).

0888-3270/$ - see front matter r 2004 Elsevier Ltd. All rights reserved.

doi:10.1016/j.ymssp.2004.01.005


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analysis) and that in the case of complex gear systems, the angular sampling permits by

synchronous averaging the isolation of every meshing pair. Finally, we present some cases of

diagnosis using the cepstrum.

2. Modelling of signals received by the sensor

Calculations that we develop have the unique aim of recovering theoretically certain behaviours

of the power cepstrum of an accelerometer signal obtained from a gear system with several stages.

These behaviours have already been observed and have been used in diagnosis [57] we have

found that:

1. The cepstrum includes as many sets of decreasing positive peaks as there are rotating members

in the system. The spacing of these sets corresponds to each rotation period.2. The existence of a comb peak is due exclusively to the presence of a periodicity in the

signal.

3. The amplitude of a comb increases when the energy emitted by the corresponding Member

increases, for example in the case where the member develops a defect.

4. The sum of the first peaks of every comb is constant, therefore when the amplitude of one comb

increases, it is to the detriment of others.

Considering the previous remarks, the structure of signals will be modelled based on the two

following hypotheses:

* The signal emitted by a complex system of gearing is the sum of signals emitted by each of the

rotating members.* The signal emitted by a rotating member is periodic with the rotation period of this

member.

In reality, the second hypothesis is false because a tooth of any wheel is not in contact with the

same matching teeth, in each period of rotation. However, it tends to be true in the case where

local faults are present: for example a defect on one tooth, with perfect homogeneity elsewhere on

the other teeth.

It is precisely this type of defect (generating wide band signals) that can be detected by cepstral

analysis.In the case of only one mesh, this model of the structure is not in contradiction with the model

proposed by Mark [9]. It has the advantage in comparison with part models, to permit the

theoretical calculation of the power cepstrum. This modelling will be justified later by a good

agreement between theory and experiment.

We express the signal emitted by a gear system, including M rotating members, therefore by

st XMk1

pktemak bt; 1

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M.E. Badaoui et al. / Mechanical Systems and Signal Processing 18 (2004) 103110461032


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where pk(t) is the signal emitted during period ak of the rotating organ kand emak is a dirac comb

of Nk impulses of step ak that we will call the multiple echo term expressible by

emak XNk1n0

dnak; 2

b(t) is a noise independent of pk(t).

From signals synthesised according to Eq. (1), it is possible to verify that behaviours (3) and (4)

explained earlier are, to a good approximation, independent of the nature of the pk(t) signal and

are determined therefore by their periodicity and their energy. For this purpose we calculated the

power cepstrum of a signal with no noise of size 1024, including periodicities 20 and 63, that is to

say of the form

s p20em20 p63em63: 3

We varied the ratio of the two contributing energies for three different types of signals: p20 and p63diracs, or white noises, an exponential function and a damped sinusoid.

Fig. 1 represents: in (a) the relative height of the cepstrum peak of signal s of period 20 samples

and in (b) the sum of the first peaks of the two combs, all as a function of the square root of the

relative energy that is to sayffiffiffiffiffiffiffiffiffiffiffiffiffiEp20

p=

ffiffiffiffiffiffiffiffiffiffiffiffiffiEp20

p

ffiffiffiffiffiffiffiffiffiffiffiffiffiEp63

p : 4

It can be seen that the character of the cepstrum, used in diagnosis, is relatively independent of the

signal type. It should be noted that the amplitude of the peak in a period of data is practically

proportional to the square root of the energy emitted by the corresponding rotating member

during this period and that the sum of the first peak for each of the two periods is practically

constant and equal to 0.5.

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0

0.2

0.4

0.6

0.8

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

(a)

(b)

Fig. 1. From the power cepstrum no noise of size 1024 and including periodicities 20 and 63 and according to the

square root of the relative energy of the signal p20, in (a) the relative height of the peak to the period 20; in (b) the sum

of peaks to periods 20 and 63. As solid lines: p20 are an exponential and p63 a damped sinusoid. As dotted lines p20

and p63 are diracs. As points : p20 and p63 are two different realisations of white Gaussian noises.

M.E. Badaoui et al. / Mechanical Systems and Signal Processing 18 (2004) 10311046 1033


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3. Calculation of meshing power cepstrum

3.1. The power cepstrum

The power cepstrum of a signal s(t) is defined by [3]

*s FT1ln jSfj 5

with Sf FTst; FT being the Fourier transform.The power cepstrum transforms a convolution product into an addition; if s s1s2 then *s

*s1 *s2: This has been used right from the start for detecting echoes: so if s(t) is a signal containinga pattern and its echo of gain g, with |g|p1 shifted by the quantity a:

st mt gmt a md0 gda mea;g; 6

we will call ea;g single echo term of shift a and of gain g. Its power cepstrum is given by [10]

*ea;g X

nAZ

1n1

2jnjgndna: 7

In the following, we will only present the part t X0. Numerically if the shift a corresponds to a

whole number of samples, *ea;g has the form of alternating peaks, localised to only one sample and

with a spacing of a. The cepstrum possesses an optimal temporal resolution intrinsically because

side effects do not exist for the function ln jSfj and it is real, even and periodic of unit period in

reduced frequency. Fig. 2 represents the power cepstrum of a single echo term of shift a equal to

50 samples. The form of the cepstrum provides an accurate evaluation of a, especially if the signal

m is band limited: indeed if jMfj constant; *m is composed only one sample at the origin. If onthe other hand one supposes m to be an infinitely long sinusoid, then s and *s do not exist. In any

case it is impossible to see an echo in a sum of two sinusoids of infinite size.

The power cepstrum of a finite length signal is of infinite length, consequently, the numerical

calculation of the cepstrum using the discrete Fourier transform will be subject to aliasing.

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50 100 150 200 250 300 350 400 450-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

Sample

Fig. 2. Power cepstrum of single echo with a=50.



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In the case of an echo distributed according to a d distribution, the echo term will be denoted

ed d0 d; and its power cepstrum is given by [10]

*ed X

nAZ

1n1

2jnjdn; 8

where dn means that the d distribution is convolved with itself n times. The gain of the echo

g RN

N dt dt must be less than or equal to unity in modulus. Fig. 3 shows the example of thecepstrum of an echo distributed according to a rectangular window centred on the sample 50 and

calculated numerically.

3.2. Power cepstrum of the multiple echo term

The term ema PN1

n0 dna possesses as Fourier transform:

FTema EMaf 1 e2pifNa

1 e2pifa

FTeNa;1

FTea;1; 9

where eNa;1 and ea;1 are single echo terms of gain 1 and of shifts, respectively, Na and a.

According to definition (5) we will have

*ema *eNa;1 *ea;1 10

that is to say

*ema X

nAZ

1

2jnjdnNa

XnAZ

1

2jnjdna: 11

This cepstrum is composed of a positive comb decreasing by step a and negative comb

decreasing by step Na. To illustrate this result, Fig. 4 represents the cepstrum of a multiple echo

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50 100 150 200 250 300 350 400 450-0.04

-0.03

-0.02

-0.01

0

0.01

0.02

0.03

0.

0.05

04

Sample

Fig. 3. Power cepstrum of a particular distributed echo.



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term extended by zeros up to a considerable length Ns=1024 (for Na=100) in order to minimise

the effect of aliasing of the negative comb.As the size of the Ns signal is actually between (N+1)a and Na the negative comb will be aliased

in positions depending on the value of Ns; consequently, only the positive comb is exploitable,

when its peaks do not occupy positions of aliases. Fig. 5 shows a real case of a multiple echo

cepstrum calculated numerically on a size between (N+1)a and Na with Ns=2048 and a=30.

For the particular case where the size of the comb is equal to the size of the signal ( Ns=Na) emabecomes a comb of diracs of step a, of infinite length, that is to say a cha distribution, that we will

note as pgna (see Fig. 6). As its Fourier transform is a cha function in the frequency domain its

power cepstrum does not exist since it depends on the logarithm of a null quantity on a non-null

support, unless this comb is accompanied by another component, that we will suppose white for

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50 100 150 200 250 300-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

Sample

Fig. 4. Power cepstrum of a multiple echo with a=10.

50 100Sample

150 200 250 300-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

Fig. 5. Power cepstrum of multiple echo with a=30.



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simplicity (impulse or white noise). If one notes as pgna,b the cha added of a component of white

noise b of standard deviation sb; then the calculation of the power cepstrum pgna,b of leads to [11]

*pgna;bt ln Nss

2b g

2d0

a

Nsln

Ns

a

ln Nss2b g2

pgnat; 12

where g 0:577215 is the Euler constant.The amplitude of the comb depends on the period a and consequently on the signal size,

therefore this particular case should be avoided in the context of a diagnosis that is based on the

monitoring the amplitude of cepstrum combs.

3.3. Cepstrum of a multiple mesh signal with no noise

In this case the vibratory signal is modelled by

st XMk1

pktemak: 13

The behaviour of combs of the cepstrum being an independently good approximation of the

signal nature, calculations that follow will be simplified while using instead of pk(t), amplitude

diracs pk, of constant Fourier transform Pk. This supposes that if pk(t) represent white noises,their Fourier transforms are constant as Pk EjPkfj:The signal amplitude only affects the value at the origin of the cepstrum. This value presenting

no interest for the diagnosis, we will suppose, for simplicity, that this amplitude isXMk1

Pk 1: 14

For a signal of size Ns we note that the expression ofs(t) is incomplete; missing are the Mpartial

periods completing the signal of Nkak to Ns. We are going to show that the presence of these

partial periods does not affect the positive part of the combs (used for the diagnosis). Indeed, to

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-1.4

-1.2

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

Amp

Sample

Fig. 6. Power cepstrum of a comb of diracs with the size Ns=500, period a=50.



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simplify, if one looks at the case of only one rotating member, the complete signal is xt p1t

PN1n0 dna p2t

PNn0 dna; where p2(t) corresponds to the signal present in the last partial

period and where p1(t) corresponds to the complement in the whole period. Its Fourier transform

Xf has as its expression Xf 1 P1e2pifNa P2e2pifN1a=1 e2pifa:As in the case of Eq. (9), the denominator is responsible in the cepstrum for the positive comb

decreasing with step a. The numerator is considered as being the Fourier transform of an echo

distributed on two negative samples of heights P1 for the Na abscissa and P2 for the abscissa

(N+1)a. This distribution is responsible in the cepstrum of negative peaks of abscissa different

from na for n oN, and does not therefore interfere with the positive comb.

We come back to the case of the multiple meshing signal without noise modelled by Eq. (13), its

Fourier transform is

Sf XM

k1

1 e2pifNkak

1 e2pifak

Pkf; 15

we have then

Sf

PMk1Pk1 e

2pifNkakQ

jak1 e2pifajQM

j11 e2pifaj

; 16

the numerator is 1 P

j cje2pifdj an elementary calculation on this polynomial shows that

Pj cj

1; that is to say the Fourier transform of d0 P

j cjddj d0 d; therefore of a term of echodistributed according to dof gain 1, then Sf FTed=Q

Mj1 FTeaj;1; the power cepstrum is

therefore

*s XMk1

XnAZ

1

2jnjdnak

" #

XnAZ

1

2jnjdn: 17

The first term of this expression shows that there exists in *s; M positive combs decreasing bystep ak. We focus now on M combs decreasing by the step characteristic ak of every rotating

member. Consequently we examine the expression of abscissas nak, that is to say the only ones

capable of altering the previous combs, that is to say not to keep d, the terms of abscissa ak.

The FT of these terms is the numerator, noted Num, of Eq. (16) in which one eliminates the

term e2pifNkak:

Num X

k

PkYnak

1 e2pifan

" #?

X

k

Pk 1 Xnak

e2pifan

!" #?

1 X

k

PkXnak

e2pifan

" #?

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1

Xk XnakPne

2pifak

" #?

1 X

k

1 Pke2pifak ?;

that is to say

Sf 1

PMk11 Pke

2pifak ?QMj11 e

2pifaj18

whence the cepstrum of the signal will be

*s XM

k1 XnA

Z

1

2jnjdnak X

nA

Z

1 Pkn

2jnjdnak

" #? : 19

The part clarified by this result is the exploitable part *se of *s; because the other terms occupypositions of linear combinations ofak, that correspond to greater delays than the size of the signal

in general and which generate aliased peaks. This exploitable part *se is given by

*se XMk1

XnAZ

1 1 Pkn

2jnjdnak

" #: 20

One recovers M positive combs decreasing by the step characteristic ak of every member in

rotation.

To illustrate these results, we calculate the power cepstrum of a signal simulated on the basis of

Eq. (13), with M=2. One replaces Pk by the expected value of the pk(t) FT modulus, thenconsidering condition (20), we write

Pk sk

ffiffiffiffiffiak

pPM

i1 siffiffiffiffi

aip : 21

One notes that Eq. (20) replicates the exploitable part of the cepstrum well, as testified by Fig. 7

that exhibits the first 10,000 points of the power cepstrum calculated on a signal containing the

two periodicities 512 and 1613, p1 and p2 being a standard deviation of unity. The O indicates

the theoretical value of cepstrum peaks according to Eq. (20).

The first peak of every comb has for amplitude Ak Pk=2 this amplitude is proportional to the

square root of the energy emitted during a period ak by the rotating organ k. One notices that thesum of the first M peaks defined by sfp

PMk1 Ak has a value of 0.5. This remarkable result is

used to make the diagnosis: when a rotating organ in a period develops a defect, the resulting

energy increases, therefore the cepstral peak of abscissa aiwill increase to the detriment of abscissa

akai peaks.

3.4. Cepstrum of a multiple meshing signal with noise

The exact calculation of the power cepstrum of a noisy signal is complex and not completely

resolved [12]. We sketch here the calculation of the power cepstrum of a linear combination of

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multiple echoes in the presence of noise. This calculation has for objective to explain the evolution

of the sum of the first peaks sfp according to the signal-to-noise ratio.

The signal is expressed by Eq. (1) where b(t) is an independent noise of pk.

Taking Eq. (18) its Fourier transform has for expression.

Sf 1

PMk11 Pke

2pifak ?

QM

j11 e2pifaj

Bf: 22

So b(t) is white and of expected value of the B FT modulus constant a similar calculation forshows that the exploitable part of the cepstrum is given by

*se XMk1

XnAZ

1 1 Pk B=1 Bn

2jnjdnak

" #: 23

The noise has the effect of decreasing the amplitude of combs and in particular the first peaks of

each them [11,13], that have for amplitude Ak Pk=21 B and their sum is equal to

sfp 0:5

1 B: 24

This thus depends on the signal-to-noise ratio, so if b(t) is white Gaussian, of standard deviation

sb; and estimating B as the Pk one will take

Bsb

ffiffiffiffiffiffiNs

pPM

i1 skffiffiffiffiffi

akp : 25

Fig. 8 represents the evolution of the sum of the first cepstral peaks of a signal constructed

according to Eq. (1), whose periods akare equal to 512 and 1613. Simulations are made for signals

pk(t) that are Gaussian white noises (the signal) to which was added an independent Gaussian

white noise (the noise). On the abscissa, we have the ratio of the noise to the standard deviation of

the signal.

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-0.1

-0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Fig. 7. Power cepstrum of a signal made according to Eq. (1) without noise. The O represents the theoretical height

of the cepstrum peaks (Eq. (20)).



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We note that actually the diminution of the sfp cannot be due only to the noise of measurement,

but also to the presence in the acquired signal of anything independent of the meshing, for

example rolling noise.

3.5. Utilisation of the power cepstrum for diagnosis

Numerous simulations have allowed us to observe that the given theoretical expressions

replicate the previously calculated cepstrum sufficiently well when M=2, but less well as Mincreases. However the main properties of the cepstrum as revealed by our calculations are still

valid: the amplitude of the first peaks is proportional to the square root of energies developed

during a period and their sum depends only on the signal-to-noise ratio.

Numerically, cepstrum peaks are concentrated in one or two samples at most provided that the

successive periods of each rotating member are rigorously constant. In the real case if periods are

distributed, it is advisable to replace the amplitudes represented in Eq. (23), by the areas of peaks

in the experimental cepstrum. This fact can also be justified by Eq. (8) that shows that the first

peak of the cepstrum replicates the distribution of a single echo, which by Eq. (23) can be

interpreted as the positive part of the cepstrum of a multiple echo of distributed period. If there is

widening of a peak, there will be a reduction of its height in the same proportion, that is to say arisk of disappearance in the noise. Consequently, to preserve the optimum resolution of the

cepstrum we recommend its utilisation on signals acquired by sampling at equal angular intervals

using a shaft encoder placed on the input shaft of the gear system. In this way, the angular periods

are constant and independent of possible variations of rotation speed. The angular sampling also

gives the important advantage of being able to isolate, in a complex gear system, each mesh, by

synchronous averaging: the signal is divided in to sections whose size corresponds to the angle

developed by number of tooth meshes equal to the lowest common multiple (lcm) of the tooth

numbers of the two wheels participating in the meshing. The averaging of these sections is

constructive for the involvement of each of these wheels and destructive for the others and for any

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0 0.5 1 1.5 2 2.5 3 3.5 40

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Fig. 8. Sum of the first peaks of a signal containing two periodicities 512 and 1613 as a function of noise to signal

(standard deviation) ratio.



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noise. The monitoring of complex systems consists in calculating the corresponding cepstrum for

every meshing pair isolated as described above. It is then possible to define the temporal evolution

of the relative difference of normalised areas of the first peaks associated with each of the two

wheels:

dt Apt=Ap0 Art=Ar0

Apt=Ap0 Art=Ar0; 26

Ap and Ar are, respectively, the areas of the first peaks of combs associated with the pinion and the

wheel of each meshing pair.

Supposing that the gear system originally does not have any defect. d(t) will tend to a value of

1 when the wheel presents a defect, and to +1 when it is associated with the pinion. We note

that d(t) can remain near zero if the two develop a defect simultaneously. This unlikely event can

be revealed by following the sum of areas st Apt Art: In this case, the signal-to-noise ratioincreases and s(t) will tend toward 0.5. In the case of a system with several meshes this ambiguity

can be removed easily by examining the adjacent mesh. In general, the indicator d(t) will reveal a

defect if it passes a threshold fixed according to the standard deviation of its estimates, this

standard deviation (difficult to foresee theoretically) can be estimated experimentally during a

period of working with the healthy gear.

In summary, d(t) is independent of the signal amplitude, of the position of the sensor, and the

signal-to-noise ratio; only its standard deviation depends on this last parameter.

4. Applications

4.1. Gear system with two trains of meshing

Results presented in this paragraph have the sole aim of showing a power cepstrum calculated

from experimental signals, as well as the separation of meshing in a complex system. At the time

of this experiment no defect has been experienced.

The recordings were made on a healthy two stage gear system with helical teeth (Fig. 9) for

which the input shaft rotation frequency is 25 Hz.

The accelerometer signals were acquired by angular sampling of 512 samples per rotation of the

motor shaft. To analyse the vibratory state of each gearing train, the power cepstrum wascalculated from the same signal, after synchronous averaging according to the fundamental period

of that train (least common multiplelcm).

Fig. 10 represents the power cepstrum of the vibratory signal synchronous by averaged

according to the lcm of the number of teeth on the pinion and the wheel of the first mesh. We

show the information relative to this mesh only. One can see the two combs of peaks associated

respectively with the pinion P1 and the wheel R1 of period 512 and 1613 samples.

Fig. 11 represents the signal power cepstrum average according to the lcm of the second mesh.

There we can see the comb associated with the pinion R3 of period 1613 samples and the first peak

of the comb associate with the wheel R4 of period 8736 samples.

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4.2. Gear system with one meshing train, case of signals with low noises

The recordings were carried out at CETIM on a gear system with a train of gearing with a ratio

of 20/21 functioning continuously until its destruction. The test was of length 12 days with a daily

mechanical appraisal; measurements were collected every 24 h. These signals have already beenused on several occasions to demonstrate diagnostic procedures [6, 14]. Fig. 12 shows the

evolution of cepstrum peaks (A1 and A2), their sum and indicator d(t) as a function of the

acquisition day.

It can be seen that the cepstrum peak of the wheel that develops the defect (A1) increases to

the detriment of the other. The sum of the first peaks staying near 0.5 indicates that signals

have low noise. The evolution of the indicator d(t) indicates clearly that from the eighth day,

the pinion is developing a defect. This result agrees with the report of the appraisal made on

this gear system, that indicates, on the eighth day, the appearance of spalling beginning on the

teeth 15/16.

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-0.1

-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08

R1

R2

0.1

Fig. 10. Power cepstrum calculated according the first gear periodicity.

R1, 20 teeth

R2, 63 teeth

R3, 12 teeth

R4, 65 teeth

Fig. 9. Gear system with two meshing trains.



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4.3. Gear system with one meshing train, case of signal noisy signal

The accelerometer signals were acquired by the survey and research group of EDF Chatou on a

test rig. These signals are existing and were exploited by Fontanive in 1992 [7].

The ratio is 56/15. The wheel rotation frequency is 12.5 Hz, and its meshing frequency is 700 Hz.

Signals were acquired with a sampling frequency of 6400 Hz.

During the fatigue test, fifteen measurements were acquired. From the tenth measurement, the

contact surface of the wheel presents a growing spalling that led to the cracking of a tooth at the

time of the fifteenth appraisal. The pinion remained healthy.

Fig. 13 shows the evolution of cepstrum peaks, their sum and indicator d(t) according to

acquisition index.

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-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Sum

day

A1

A2

d

Fig. 12. Evolution of cepstrum peaks A1 and A2, the sum and the indicator d, for the CETIM gears.

1000 2000 3000 4000 5000 6000 7000 8000 9000 10000

-0.1

-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08

0.1

R3 R4

Fig. 11. Power cepstrum calculated according the second gear periodicity.



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Between the first and the ninth acquisition, there is not an obvious defect and the sum of the

first peaks is practically constant and near 0.15. This small value indicates that the signals are

noisy, or include a contribution independent of the meshing. One notes that at the tenth

acquisition, a defect appears on the wheel with 56 teeth, the signal of meshing increases in relation

to the noise (assumed to be of constant variance) and consequently the sum of the first peaks

comes closer to 0.5.

These results correspond to those of Ref. [7], also obtained from the power cepstrum. We bring

an additional theoretical justification of these results in addition.

5. Discussion

The work that we have presented confirms that the power cepstrum proves to be an efficient

tool for the monitoring of gears. We contribute, in relation to the previous works on this topic, a

theoretical justification of the shape of the power cepstrum of the meshing signal. The acquired

results are in agreement with observations, and permit us to define a very robust detector of faults,

since it is independent of the signal amplitude, of the signal-to-noise ratio and the position of the

sensor.This present detector has maximal efficiency when signals are acquired using angular sampling

by means of a shaft encoder. In this case the resolution of the cepstrum is maximal and it is

possible, from the same signal, to isolate each of meshing stages by an appropriate synchronous

average. In this context only one sensor need be used if it is placed judiciously to hear all

meshes.

The tool that we have just defined is not well adapted to an absolute and prompt diagnosis, but

is more applicable to the monitoring of gearboxes in operation. It can be of service however as a

manufacturing quality control system if the indicative parameter d is initialised from a healthy

gear system.

ARTICLE IN PRESS

2 4 6 8 10 12 14-0.2

0

0.2

0.4

0.6

0.8d

A2

A1

sum

day

Fig. 13. Evolution of cepstrum peaks A1 and A2, the sum and the indicator d, for the EDF gears.



16/16

Our work is continuing currently, on the one hand to improve the theoretical expression of the

meshing signal power cepstrum, while using some more realistic models, and on the other hand to

improve the signal-to-noise ratio of the experimental cepstrum.

Acknowledgements

The authors gratefully acknowledge the support of CETIM (Centre des Etudes Techniques des

Industries M!ecaniques de Senlis), EDF Chatou and G.D.R.-P.R.C. ISIS unit!e G-720 which

provided the experimental results.

References

[1] B.P. Bogert, M.J.R. Healy, J.W. Tukey, The frequency analysis of time series of echoes: cepstrum pseudo-autocovariance cross-cepstrum and saphe cracking, in: M. Rosemblah (Ed.), Proceedings of the Symposium on

Time Series Analysis, Wiley, New York, 1963, pp. 209243.

[2] D.G. Childers, D.P. Skinner, R.C. Kemerait, The cepstrum: a guide to processing, Proceedings of the IEEE 65 (10)

(1977) 14281443.

[3] A.V. Oppenheim, R.W. Schaffer, Discrete-Time Signal Processing, Prentice-Hall, Englewood Cliffs, NJ, 1989.

[4] J.T. Kim, R.H. Lyon, Cepstral analysis as a tool for robust processing deverberation and detection of transients,

Mechanical Systems and Signal Processing (1992) 115.

[5] R.B. Randall, Cepstrum analysis and gearbox fault diagnosis, Br.uel and Kjaer Technical Bulletin, Edition 2, 1980,

pp. 119.

[6] C. Capdessus, M. Sidahmed, Analyse des vibrations dun engrenage: cepstre, corr!elation spectre, Traitement du

Signal 8 (5) (1992) 365372.

[7] C. Fontanive, P. Prieur, Surveillance et diagnostic des engrenages, Progr"

es r!

ecents des m!

ethodes de surveillanceacoustiques et vibratoires, Senlis, 2729 October 1992, pp. 639649.

[8] M. El Badaoui, V. Cahouet, F. Guillet, J. Dani"ere, P. Velex, Modelling and detection of localized tooth defects in

geared systems, Proceedings 1999 ASME Design Engineering Technical Conferences, Las Vegas, Nevada,

September 1215, 1999.

[9] W.D. Mark, Analysis of the vibratory excitation of gear systems: basic theory, Journal of the Acoustical Society of

America (1978) 14091430.

[10] T. Fournel, J. Daniere, M. Moine, J. Pigeon, M. Courbon, J.P. Schon, Utilisation du cepstre d!energie pour la

v!elocim!etrie par images de particules, Traitement du Signal 9 (3) (1992) 267271.

[11] M. El Badaoui, F. Guillet, J. Daniere, Surveillance des syst"emes complexes "a engrenages par lanalyse cepstrale

synchrone, Revue Traitement du Signal (TS) 16 (5) (1999) 371381.

[12] J.C. Hassab, R. Boucher, A probabilistic analysis of time delay extraction by the cepstrum in stationary gaussman

noise, IEEE Transactions on Information Theory IT 22 (4) (1976).

[13] M. El Badaoui, Contribution au diagnostic vibratoire des r!educteurs complexes "a engrenages par lanalyse

cepstrale, Th"ese de Doctorat, Universit!e Jean-Monnet de St.-Etienne, Juillet, 1999.

[14] K. Drouiche, M. Sidhamed, Y. Grenier, D!etection de d!efauts dengrenages par analyse vibratoire, Traitement du

Signal 8 (5) (1992) 331.

ARTICLE IN PRESS


new aplication of real cepstrum

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