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Mechanical Systems and Signal Processing (2001) 15(5), 873}885

doi:10.1006/mssp.2001.1413, available online at http://www.idealibrary.com on

USE OF THE MOVING CEPSTRUM INTEGRAL

TO DETECT AND LOCALISE TOOTH SPALLS

IN GEARS

M. ELBADAOUI, J. ANTONI, F. GUILLET ANDJ. DANIEERE

Laboratoire d+Analyse des Signaux et des Processus Industriels (LASPI)~EA-3059, IUT deRoanne, 20, Avenue de Paris, 42 334 Roanne, France. E-mail: badaoui@univ-st-etienne.fr

AND

P. VELEX

Laboratoire de Me&canique des Contacts, UMR CNRS 5514, INSA de Lyon, BaLt. 113,

20 Avenue Albert Einstein, 69 621 Villeurbanne Cedex, France

The objective of this paper is to propose a new indicator for the vibratory diagnosis ofgear systems. This indicator is deduced from the power cepstrum of the accelerometer signal.A model aimed at simulating the contributions of local tooth defects such as spalls to thegear dynamic behaviour is set-up. The pinion and the gear of a pair are modelled as two rigidcylinders with all six degrees of freedom connected by a series of springs which represent gearbody and gear tooth compliances on the base plane. It permits us to foresee the shape ofthe excitation induced by the presence of spalls. From an analytical analysis of the equa-tions of motion, a detection technique based upon the acceleration power cepstrum

is proposed. The identi"

cation of the spalls is provided by the fact that the power cepstrumof the excitation that it generates is strictly negative, in contrast to that of a normalexcitation. A tool of detection and localisation, using this property, has been de"ned. It is"rst tested on acceleration signals simulated by numeric integration of the model, then onreal signals.

2001 Academic Press

1. INTRODUCTION

Rotating machine diagnosis is becoming more important because it contributes to safety in

critical applications such as aeronautics but, more generally, because it reduces equipment

downtime and maintenance costs. There are two main types of diagnostic techniques in usetoday, i.e. vibration monitoring and debris monitoring techniques, which have both had

limited success in detecting gear failures. Focusing on vibration analysis, some methods use

synchronously averaged signals in order to reduce random noise and contributions unre-

lated to the particular gear of interest. The current tendency is to"nd techniques capable of

detecting the occurrence of faults at an early stage. The simplest time domain analyses are

based on statistical indicators such as kurtosis[1, 2]whose amplitude can be related to the

state of the gear. More sophisticated techniques can be employed, including amplitude and

phase demodulation by using Hilbert transforms[3], time}frequency transforms based on

the Wigner}Ville decomposition [4], wavelet transforms [5}7], etc. Due to the lack of

sensitivity of these methods for early detection, alternate methods relying on stress wavepropagation (acoustic emission) have been considered[8, 9]but they are strongly dependent

on the propagation path and the interfaces between the wave source and the sensors.

Among the numerous available signal processing techniques used in vibration monitor-

ing, it has been demonstrated in early papers [10}13] that the power cepstrum of the

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acceleration signal is e$cient for detecting localised tooth faults such as spalls. The

detection principle is based on the increase of the energy level associated with the rotational

period of the defective wheel and it has been shown that the proposed defect identi"er is

independent of the sensor position, the signal amplitudes and the signal-to-noise ratio

[14}16]. However, results were limited to the de"nition and use of a global defect indicator

with no information on the defect position. The present paper presents some complement-ary theoretical developments and an extension towards defect localisation. The organisa-

tion of the paper is as follows. First, a mechanical model of a pinion}gear pair is used to

characterise some properties of the dynamic response of a geared system with a tooth defect.

Next, it is demonstrated that the additional excitation generated by a tooth fault can be

e!ectively extracted from acceleration signals by virtue of the power cepstrum properties.

Finally, some applications to simulated and experimental signals illustrate the bene"ts of

the methodology.

2. DYNAMIC MODEL AND STRUCTURE OF THE DYNAMIC RESPONSE WITHLOCALISED TOOTH DEFECTS

In order to qualitatively analyse the in#uence of tooth spalls on gear dynamic response,

a simpli"ed version of the model of Velex and Maatar[17]is used and extended to account

for localised tooth faults. A pinion and a gear of a pair are modelled as two rigid cylinders

connected by a set of lumped sti!nesses which accounts for contact, tooth and gear body

de#ections (Fig. 1). The elemental sti!nesses are associated with all potential points of

contact on the base plane as well as elemental normal deviationse(, t) in order to simulateactual tooth#ank geometries. According to rigid body kinematics, the corresponding lines

of contact are translated at constant speed and all relevant parameters are recalculated at

each time step of the meshing process.Quasi-analytical indications on the response structure can be obtained if the following

simpli"cations are introduced:

(a) except for localised tooth defects, the pinion and the gear are geometrically perfect;

(b) the mesh sti!ness per unit of contact length of a pair k

is supposed to be constant

along the face width and the path of contact;

(c) the contributions of bending angles are averaged over one mesh period so that

a constant structure vector < [i.e. a vector connecting mesh de#ections to the

system degrees of freedom, seeequation (4)below] can be used.

In these conditions, the equations of motion read as

[M]x(#[C]xR#[K]x#[K(t,x)]x"F

#F(e(H(t))#G(t,x) (1)

and the analytical expressions of the two quantities which largely control the geared system

dynamic response to tooth defects are

[K(t, x)]"k

H((, t)) d