CONTRIBUTION TO HIGHER ORDER CYCLIC STATISTICS : APPLICATIONS TO GEAR FAULT DIAGNOSTIC

Amani Raad (*,#), Fan Zhang (**,#), J.Antoni (*,#),Ménad Sidahmed (*,#)

(*) UTC, Laboratoire Heudiasyc, F60205 Compiègne, e-mail: raad@utc.fr (**) CETIM, Senlis, F60304, e-mail: fan.zhang@cetim.fr

(#)LATIM (CNRS/UTC/CETIM) Abstract This paper investigates the potentials of the higher order cyclic statistics to model gear vibration signals. It focuses also on the proposal of new and simple indicators of cyclostationarity and their application to the fault diagnosis in mechanical systems. Indicators of first to fourth order cyclostationarity are introduced and their statistical properties are derived. The repartition of cyclostationarity for different orders can inform about any eventual fault in machines. In order to illustrate the use of these indicators, an industrial application to the spalling diagnosis in a gearbox system is described and discussed. Results demonstrate the effectiveness of these new indicators for a good diagnosis by discriminating between different cyclostationary states. Résumé Ce papier explore les potentiels des statistiques cycliques d’ordre supérieur pour modéliser des signaux vibratoires d’engrenages. De plus, des indicateurs de cyclostationarité globaux et concis sont proposés ainsi que leur application pour faire un diagnostic de défauts dans les systèmes mécaniques. La répartition de la cyclostationarité sur les différents ordres peut informer à propos de la présence éventuelle des défauts. Pour illustrer l’utilisation de ces indicateurs, une application industrielle qui porte sur le diagnostic de défaut d’écaillage dans un engrenage est décrite. Les résultats montrent l’efficacité de ces nouveaux indicateurs pour un bon diagnostic.

1 Introduction

This paper is concerned with the development of signal processing methods to perform an early diagnosis of faults in mechanical systems using vibration signals. Until now, in signal processing, most established methods often rely on a fundamental assumption of stationarity and ergodicity of the involved processes. These notions are appealing because they give the possibility of estimating parameters from a single realization. However, this assumption is a mathematical idealization which, in some case, may be valid only as an approximation to the real situation. Thus, it can exclude many real-life non stationary signals. More particularly, there is a subclass of non stationary signals called cyclostationary signals. These signals are characterized by a periodic variation of their statistical parameters. The theory of estimation of periodically correlated processes, i.e. second order cyclostationary, was first introduced by Hurd (1970) and exploited with success in several domains especially in the diagnosis of gear faults (Capdessus, 1992;

Capdessus et al., 2000; Bouillaut, 2000). For analyzing such signals, several techniques were applied such as spectral analysis, cepstrum (Randall, 1975), time frequency analysis (Wang and McFadden, 1993) and higher order statistics (Nikias and Raghuveer, 1987; Hinish, 1994). For higher orders, the general theory of cyclic statistics has been developed in both the stochastic and fraction of time (FOT) probability frameworks (Gardner, 1994a,b; Giannakis and Dandawate, 1994). Spooner and Napolitano (2001) proposed recently an application of cyclic analysis to the estimation of parameters of modulated signals. Surprisingly, very few applications have been reported in related areas such as mechanical engineering until very recently when it was recognized that cyclostationary processes fit the properties of rotating machinery (Capdessus et al., 2000). However, some precursory work based on higher order cyclostationarity have been concerned with vibration gear signals (Raad et al., 2002). The main objective of this paper is to identify the higher order properties of gear vibration signals by using higher order cyclic statistics. Specifically, the first part of our study is concerned to answer the question: are these signals higher order cyclostationary? and if so, how could they be modeled. It is well known that these signals are cyclostationary at second order because the inherent periodicity in the signal statistics is induced by the periodic movements and rotations of the mechanical components. The cyclostationarity characterizing vibrating signals can inform about the evolution of an eventual fault that would produce repetitive non-linearities and non-stationarities. An important statistical parameter in the study of cyclostationary properties is the nth-order cyclic polyspectrum. However, it is not conceivable to estimate these polyspectrum in the case of on line monitoring because of the high cost of calculation it requires if the order exceeds two. Therefore, concise and global indicators that measure the cyclostationarity for order one to four are proposed in this paper. The idea is based on the work of Prieur and D’urso (1996); Zivanovic and Gardner (1994). However, this paper ameliorates and refines the work of Prieur and D’urso (1996) by placing a clear distinction between first and other order of cyclostationarities. The final objective of the proposed indicators is to characterize the type of fault (spalling, cracks,) as a function of the repartition of the cyclostationarity for different orders. The paper is organized as follows. After presenting the basic principles of cyclostationarity in section 2, we apply these principles on gear vibration signals to extract their properties. The aim of this section is to identify the causes of the higher order cyclostationarity if present. In order to characterize these properties, indicators of cyclostationarity are introduced in section 3. Furthermore, the theory and the statistical properties of the indicators are described in details. To illustrate these methods, an application to industrial vibration signals recorded on a gearbox is provided and discussed. Conclusions are drawn in section 4.

2 Cyclostationarity 2.1 Definition

In general, a cyclostationary process is a stochastic process that exhibits some hidden periodicities in its structure. Formally, a stochastic process ( )tx is said to be

strict-sense nth order cyclostationary with cycle T if its joint probability density function ( )1 1, , ; ,n np x x t tK Kx is periodic in t with period T , i.e. if ( ) ( )1 1 1 1, , ; , , , ; ,n n n np x x t t p x x t T t T= + +x xK K K K (1) In general, in rotating machines, the statistics of their vibration signals are periodic because of different cyclic mechanical phenomena. For example, the cyclic modifications in the geometry of the machine, the cyclic changes in torques, rotations of the anisotropic components, etc.. produce periodic modulations in the statistics of vibration signals. Depending on the structure of the physical machines and phenomena which take place there, the signals resulting can exhibit various types of cyclostationnarity. 2.2 Pure and impure cyclostionarity There are two possible ways to define cyclostationarity based respectively on the use of moments or cumulants.

The first approach deals with the periodicity of the nth-order moment, while the second deals with the periodicity of the nth-order cumulant. The use of nth-order cumulants is more advantageous than nth-order moments because it is often the case that an nth-order moment is impure in the sense that it is made up in part or wholly of products of lower order moments (Gardner, 1994a).

To purify the nth-order impure moments, all the impure terms provided from lower orders must be extracted. This is exactly what is realized by using cumulants by opposition to moments. For n=2 and 3, the purification is easily achieved by extracting the first–order moment (i.e. its periodic part) to the signal. For higher orders, Leonov’s type formula must be used (Gardner, 1994a). 2.3 Application to gear signals

2.3.1 Causes producing first order cyclostationarity of gear signals

Gear transmission is a complex mechanical process. In operational mode, the system might get deformed, vibrates and generates noise. To have a solid understanding of this phenomenon, it is essential to characterize the gear behavior. Studies for more than 60 years have worked on the mechanical aspects modeled by weight-spring models (Gregory,1963), (Velex,1988). In this case the gears are likely to be rigid cylinders related by the gear stiffness which represents the elastic link between teeth.

Shock between the teeth is traditionally one of the proposed theories that produces vibrations inside gears (MARK,1978), (Harris,1958). In this model the source of vibration is due to the wheel variation position with respect to the unloaded gear. This deviation is called the static transmission error, which is composed of the sum of the teeth deformation and the deviation of the tooth profile. Another model presented by (Stuckey 1981), related the vibration to the modulation phenomena. The transmission error between the healthy and the faulty system is represented by the apparition of some new modulations between phenomena.

These errors can be divided into two categories:

1. Form errors The profile errors are representative of the variations existing between the

theoretical profile of teeth and the real profile. These errors can be generated during manufacture or during operation by wear and deterioration of the profiles. They can be repetitive from one tooth to another one (or on a whole number of teeth) and are then associated to the emergence of the peak at the meshing frequency and its harmonics.

The surface quality depends on the quality of the teeth cutting. The errors related to the kinematics of the cutting machine can generate undulations on the cut profile, which results in the apparition of lines called in the literature phantom lines because they cannot be connected to any characteristic frequency of the system. In the automobile field, for example, these frequencies appear particularly harmful because they are on a level comparable with the level of the fundamental line of gear meshing. These defects affecting surface can also be caused by a bad lubrication or by an oxidation disadvantaged by the reheating of the lubricant.

2. Position errors

The eccentricity defect traduces the non concentricity between the primitive cylinder axis of the gear and the rotation axis on which the gear is dependent. It introduces an amplitude modulation harmonic at the rotational frequency, which is traduced by the apparition of side bands around the gear mesh frequency and of its harmonics. These side bands are composed of several spaced lines of rotation frequencies of each wheel.

The step error characterizes the defect of the angular localization of one tooth compared to its theoretical position. It is generally considered as a random type fault with non localized frequencies.

The misalignment can be defined by two angles: the inclination angle corresponding to an angular variation in the plan consisted of the two axes of gear rotation and the deviation angle in a plan perpendicular to the precedent and parallel to the axes. The deviation angle induced a side error in the teeth localization.

These form of periodic waves with probably random additive stationary noise, are first order cyclostationary, because these errors induce a change in the “centering” of the synchronous mean. To generate this type of signals, we propose a model of signal. This model is extremely simplistic but rather generic. It includes the general case of vibration signals in rotating machinery. These gear signals can be modeled in the form of exponential modulated in amplitude and phase as follows

( ) ( ) e ij

ii

ϖ θθ θ= ∑x a (2)

where θ is an angular variable relating to the axis rotation, the ϖi’ are the "angular velocities" relating to the mesh frequency and its harmonics and the ai(θ) are complex and deterministic functions.

2.3.2 Causes producing higher order cyclostationarity of gear signals

In the previous paragraph, we highlighted the causes generating first order cyclostationnarity of gear signals. This cyclostationnarity is present because of the inherent periodicities produced inside. In this paragraph, we will explore other causes leading to a cyclostationnarity at different orders. Those can be divided into two categories:

1. Macroscopic/microscopic point of view

There is a difference in the mechanical structures between the macroscopic

and microscopic scale. From the macroscopic point of view, the vibrations appear periodic (stationary waves). However, the modeling of the contact phenomena in the gears with respect to the microscopic scale induces transitory phenomena at very high frequency exceeding several tens of KHz, on the contact surface between several teeth at the meshing time, which are rather modeled as stationary processes (constraint waves). They are generated mainly by two categories of defects: distributed defects on all the teeth and those localized on some individual teeth. Among the distributed ones, we can find wear and micropitting. Spalling represents an example of localized defects.

These stochastic processes are at least second order cyclostationary. To model these processes, let us use the same generic model introduced by (2). However, it is supposed now that ai(θ).are stochastic processes. To study the cyclostationarity, let us consider the nth-order cumulant of ( )θx given by :

( )( )

1 2 1 2

1 2 1 2

1

2

1

1 2 1 2,...,

2 1 1

1

( ), ( ),..., ( ) ( ), ( ),..., ( )

exp ( ... )

exp ( ... )

n n

n n

n

n

n

i i i n i i i ni i

i n i n

i n i

Cum X X X Cum

j

j

ε εε ε ε εθ θ θ θ θ θ

ε ϖ τ ε ϖ τ

θ ε ϖ ε ϖ−

=

× + +

× + +

∑ a a a

(3)

where we use εi =1 or * whether the natural or the conjugate form of x(θ) is considered and τi = θi - θ. Since the ai(θ) are jointly stationary, their cumulants

1

1 1( ),..., ( )n

n ni iCum εε θ θ τ − + a a are functions of the angle differences τ1,…,τn-1 only and thus

the right hand side of Eq. (3) is a function of θ in the last exponential only. This proves that the nth-order cumulant function of x(θ) is in general periodic in θ (with non-zero period) if at least one nth-order joint cumulant of the ai(θ) is non-zero. The simplicity but generality of this conclusion is worth mentioning.

The model (2) can thus be used as well in the case of first order and higher order cyclostationary when the complex modulations ai(θ) are jointly stationary and mutually dependent stochastic processes.

2. Speed Variations The last paragraph showed the cyclostationary character of the gear vibration

processes when the signal is sampled into angular domain. This character is also valid in temporal for a rotating machine at a constant speed Ω so that the length of the cycle is constant. However, there are always small fluctuations of the speed making T a random variable with zero mean. In this paragraph, we study the effect of the speed variations on a time sampled signal.

Let us consider again the model introduced by (2). Since all the cinematic variables in the gears are periodic in rotation angles, the signals are thus intrinsically cyclostationary in angle rather than in time. Consequently, a justified choice is to acquire the signals compared to an angular variable rather than a temporal variable so that the property of the cyclostationnarity can be preserved. Two solutions are generally adopted:

1. The signal is directly acquired under angular sampling, for a fixed number N of pulses by cycle, the signal is sampled with a constant angular step of 2 / Nπ . This procedure has the advantage in real time operation and can be used to follow very fast variations in the speed.

2. If the angular acquisition is not possible or even inadequate in some applications, the signal is acquired under temporal sampling jointly with a tachometer signal, giving the information of the law speed of the machine. This procedure requires a re-sampling of the signal, based on more or less elaborate techniques according to the type of interpolation chosen.

First of all, it is compulsory to assume that the relationship between the generic angular variable θ and time t is bijective, which is most likely if the machine has enough inertia so that there is no backlash. Therefore,

-( ) ( ) ( )

tt t t t u duθ

∞= Ω⋅ + = Ω⋅ + ∫ϕ ν (4)

where Ω is the nominal angular speed of the machine and ν(t) is the (random) speed fluctuation around Ω (for θ(t) to be bijective, it must hold that | ( ) |t < Ων for any t). Note that from the time point of view θ(t) is a random function of the independent variable t, whereas t(θ) is a random function of the independent variable θ from the angle domain

By inserting this relationship into Eq. (2), the time version of the signal x(θ) becomes : ( ) ( ) e ij t

ii

t t ω= ∑X A (5)

where ( ) [ ( )]t tθ=X x , ( )( ) ( ( )) ij t

i it t t e ϖ= Ω +A a ϕϕ and i iω ϖ= Ω . To check if X(t) is first order cyclostationary or not, one has to compute its first-order moment (expected value).

• First Order Cyclostationnarity

The first order moment of ( )tX can be written as follows : ( ) ( )( ) ( ) ( ) e ei ij t j t

ii

m t E E t t t ϖ ω= Ω +∑X a ϕϕ ϕ ϕ (6)

with E⋅|ϕ(t) and Eϕ⋅ the expected value conditioned to ϕ(t) and the expected value with respect to ϕ(t) respectively. Taking into account the stationarity of ai(Ωt+ϕ(t)),

( ) ( ) ( )iiE t t t mΩ + = aa ϕ ϕ which is independent of time, we obtain

( )( ) e e ( )ei i i

i i

j t j t j t

i im t m E m tϖ ω ω= =∑ ∑X a A

ϕϕ (7)

with ( ) e ( ; )i

i i

jm t m p t dϖ φ φ φ= ∫A a ϕ where ( ; )p tφϕ is the probability density function of ϕ(t). Therefore the first-order moment mX(t) is time periodic if and only if the speed fluctuation has periodic statistics such that ( ; ) ( ; )p t p t Tφ φ= +ϕ ϕ .

• Second Order Cyclostationnarity The second order cumulant ( autocovariance function) of ( )tX is: * *

2 1 2 1 2 1 2( , ) ( ) ( ) ( ) ( )c t t E t t m t m t= −X X XX X (8) which , upon (3), becomes :

( ) ( ) 2 1

1

( ) ( )* *2 1 2 1 1 2 2 1 2 1 2

,( )

( , ) ( ) ( ) ( ), ( ) e ( ) ( )

e e

j i

i j

j j i

j t j ti j

i jj jt

c t t E E t t t t t t m t m tϖ ϖ

τω ω ω

−

−

= Ω + Ω + −

×

∑X A Aa a ϕ ϕϕ ϕ ϕ ϕ ϕ

(9) with 2 1t tτ = − .

Taking into account the stationarity of ( ( )i t tΩ +a ϕ ,

( ) ( ) *1 1 2 2 1 2( ) ( ) ( ), ( )i jE t t t t t tΩ + Ω +a aϕ ϕ ϕ ϕ = ( )2 1( ) ( )

i jK t tτΩ + −a a ϕ ϕ , we obtain

( ) 2 1 1

1

( ) ( ) ( )2 1 2 2 1

,( )

1 2,

( , ) ( ) ( ) e e e

( , ) e e

j i j j i

i j

j j i

i j

j t j t j jt

i jj jt

i j

c t t E C t t

c t t

ϖ ϖ τω ω ω

τω ω ω

τ − −

−

= Ω + −

=

∑∑

X a a

A A

ϕ ϕϕ ϕ ϕ

(10)

with 2 1

1 2 2 1 2 1 2 1 2 1 2( , ) ( ) e ( , ; , )j i

i j i j

j jc t t c p t t d dϖ φ ϖ φτ φ φ φ φ φ φ−= Ω + −∫A A a a ϕ and 2 1 2 1 2( , ; , )p t tϕ φ φ the

joint probability density of ϕ(t1) and ϕ(t2). Therefore, the second order cumulant 2 1 2( , )Xc t t is periodic in time if and only if the joint probability density 2 1 2 1 2( , ; , )p t tφ φϕ is

periodic. • The third order cumulant of X(t) is :

* *1 2 3 1 1 2 2 3 3

*1 2 3

*1 2 3 2 1 3 3 1 2

*1 2 3

( , , ) ( ) ( ))( ( ) ( ))( ( ) ( )

( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

2 ( ) ( ) ( )

c t t t E t m t t m t t m t

E t t t

E t E t t E t E t t E t E t t

E t E t E t

= − − −

=

− − −

+

3X X X X

* *

(X X XX X XX X X X X X X X XX X X

(11) which, upon the use of (3), gives :

( ) ( ) ( )

3 2 1 1 12

*1 2 3 1 1 2 2 3 3 1 2 3

, ,( ) ( ) ( ) ( )*

1 2 3

( , , ) ( ) ( ) ( ) ( ), ( ), ( )

e 2 ( ) ( ) ( ) e e ek j i j k j ik

i j k

i j ki j kj t j t j t j jtj

c t t t E E t t t t t t t t t

m t m t m tϖ ϖ ϖ τ ω ω ω ωτ ω+ − + −

= Ω + Ω + Ω +

× + ×

∑3X

A A A

a a aϕ

ϕ ϕ ϕ

ϕ ϕ ϕ ϕ ϕ ϕ(12)

with 1 2 1-t tτ = and 2 3 2-t tτ = Taking into account the stationarity of ( ( )i t tΩ +a ϕ ,

( ) ( ) ( ) *1 1 2 2 3 3 1 2 3( ) ( ) ( ) ( ), ( ) , ( )i j kE t t t t t t t t tΩ + Ω + Ω +a a aϕ ϕ ϕ ϕ ϕ ϕ =

( )3 2 1( ) ( ) ( )i jc t t tτΩ + + −

ka a a ϕ ϕ ϕ , we obtain

( ) 3 2 1 1 12

1 12

( ) ( ) ( ) ( )1 2 3 3 2 1

, ,( )

1 2 3, ,

( , , ) ( ) ( ) ( ) e e e e

( , , ) e e e

k j i j k j ik

i j k

j k j ik

i j k

j t j t j t j jtj

i j kj jtj

i j k

c t t t E c t t t

c t t t

ϖ ϖ ϖ τ ω ω ω ωτ ω

τ ω ω ω ωτ ω

τ + − + −

+ −

= Ω + + −

=

∑∑

3X a a a

A A A

ϕ ϕ ϕϕ ϕ ϕ ϕ

(13) with 3 2 1

1 2 3 3 2 1 2 1 2 3 1 2 3 1 2 3( , , ) ( ) e ( , , ; , , )k j i

i j K i j k

j j jc t t t c p t t t d d dϖ φ ϖ φ ϖ φτ φ φ φ φ φ φ φ φ φ+ −= Ω + + −∫A A A a a a ϕ

et 2 1 2 3 1 2 3( , , ; , , )p t t tϕ φ φ φ the joint probability density of 1( )tϕ , 2( )tϕ et 3( )tϕ . Therefore,

the third order cumulant 3 1 2 3( , , )Xc t t t is periodic in time if and only if the joint probability density 1 2 3 1 2 3( , , ; , , )p t t tφ φ φ3ϕ is periodic. For higher orders, the demonstration follows the same steps elaborated for orders 1 till 3. In conclusion, we can say The speed variations induce cyclostationarity at all orders in gear signals if and only if these variations are cyclostationary processes at all orders.

2.3.3 Other specificities of gear signals

1. Impure/Pure Cyclostationarity

With gear signals, it is essential to make the distinction between pure and impure cyclostationnarity. Indeed, it is significant to accept that the pure cyclostationary signals only at order 1 (a mixture of sinusoidal signals with additive stationary random components) must be analyzed with traditional analysis such as Fourier and does not require the cyclostationary analysis.

Gear signals are impure cyclostationary. Unfortunately, this last point was not entirely recognized in some preceding work concerning the gear vibration signals. In order to study these signals at different orders, it is of primary importance to extract only the pure cyclostationary part of the considered order. Precursory work treats this distinction such as (Capdessus, 2000), (Lejeune,1997), (Raad,2002).

2. Poly-/Quasi-Cyclostationarity

The definition of cyclostationarity relies on the existence of a unique and finite

cycle with respect to which the machinery is periodic. However, in practice, a common cycle could prove to be extremely long and practically impossible to observe. It is typically the case of the complex gearboxes, where the least common cycle is a multiple of the cycles of each gears in the system and can be long. Consequently, gear signals are called poly-cyclostationary.

If the design of the gearbox does not permit the existence of a common cycle, for example when some rotating elements are not joined together, or, if only a part of this cycle is taken into account, then gear signals are known as quasi-cyclostationary.

3 Indicators of cyclostationarity 3.1 General theory

Let the first order moment, second, third and fourth order cumulants of the

signal )(tx be denoted respectively as 1 ( )xM t , 2 ( , )xC t τ , ),,( 213x ττtC and ),,,( 3214x τττtC . The cyclic moment 1xM α is defined as the Fourier coefficients of the first order moment and similarly the cyclic cumulants ( )nxcα τ as the Fourier coefficients of the nth-order cumulants with respect to t. The latter are conveniently taken at lags zero for

measuring a degree of cyclostationarity because they summarize all the cyclic spectral information according to the following projection theorem:

2 2

3 3 1 2 1 2

4 4 1 2 3 1 2 3

(0) ( )

(0,0) ( , )

(0,0,0) ( , , )

x x

x x

x x

c S d

c S d d

c S d d d

α α

α α

α α

ν ν

ν ν ν ν

ν ν ν ν ν ν

=

=

=

∫∫

∫

(14)

where ( )nxS

α ν is the nth-order polyspectrum of ( )x t , i.e. the Fourier transform of ( )nxcα τ .

The problem of defining a measure of the degree of second-order cyclostationary processes was first addressed in (Gardner, 1994a,b; Zivanovic and Gardner, 1994). In (Zivanovic and Gardner,1994), the authors proposed a degree of cyclostationarity for each cyclic frequency α defined by: 2 0 2

2 2| ( ) | | ( ) |x xDCS c d c dα α τ τ τ τ= ∫ ∫ (15)

In this paper, simplified indicators of cyclostationarity from order one to four are proposed and defined as follows: 0 2

20

ˆ ˆˆ (0) | |nnx x nxI c Pα

α

−

≠

= ∑ (16)

where 1 1

ˆ ˆx xP Mα α= and ˆ ˆ ( )nx nxP cα α= 0 for n=2,3,4.

These indicators are motivated by the facts that: 1. they are monotonic and increasing functions of the degree of nth-order

cyclostationarity, 2. they are theoretically zero if the process is stationary, 3. they are normalized by the energy of the signal to be without dimension, 4. they generalize the well-known standardized cumulants, i.e. the classical

skewness and kurtosis by giving them a ‘cyclic’ counterpart. 3.2 Estimation techniques

Consistent estimators, for discrete signal x(m), are defined as below (Gardner, 1994a):

12

20

13

30

14 -

4 2x 2x0

1ˆ (0) ( )exp( 2 m / )

1ˆ (0,0) ( )exp( 2 m / )

1ˆ (0,0,0) ( )exp( 2 m / ) -3 c (0)c (0)

NN

x cmNN

x cmNN

x cm

c x m j NN

c x m j NN

c x m j NN

α

α

α α β β

β

π α

π α

π α

−→∞

=−→∞

=−→∞

=

= −

= −

= −

∑

∑

∑ ∑

(17)

where 1ˆ( ) ( ) ( )c xx m x m M m= − is the centered signal obtained after extracting its

synchronous average 1ˆ ( )xM m .

3.3 Statistical properties of the indicators

The objective of this paragraph is to evaluate the statistical properties (bias and variance) of the indicators from order one to three to be able to put a threshold defining the degree of cyclostationarity at different orders. The properties of 4xI will not be presented here. The problem can be stated in a hypotheses-testing framework defined as below:

0

1

: stationary signal: cyclostationary signal

HH

(18)

• Bias of the indicators 2

02

1ˆ | | (0)

mnx nxn

x

E I E Pc

α

α

≈ ∑ (19)

This approximation is valid because )0(ˆ)0( 0

202 xx cc ff with high probability under the

null hypothesis. One can obtain at the end:

0 22

1ˆ ( )ˆ (0) n

mnx n x

x

E I RN c α

α≈ ∑ (20)

where 2 nxR is the moment power spectrum of ( )nx m .

• Variance of the indicators 2 2

2ˆ ˆ ˆ| | | |m m mnx nx xVar I E I E I= − (21)

At first,

2 20 22

1ˆ ˆ ˆ| | | | (0)

mnx nx nxn

x

E I E P Pc

α β

α β

≈ ∑∑ (22)

Assuming that the estimates nxP

α are asymptotically gaussian and using (Dandawate and Giannakis,1994), one can obtain :

2 2 22 0 2

2

1ˆ| | ( | ( ) | | ( ) |N (0)

| ( ) | )

n n

n

nx n nx nxnx

nx

E I R Rc

R

α β

α α βα β

α β

α β

β

−

+

= + −

+

∑ ∑∑∑∑

(23)

Finally, the expression of the variance is given as:

22 0 2

2ˆ | ( ) |N (0) nnx n nx

nx

Var I Rc

α β

α β

β+= ∑∑ (24)

3.4 Application to diagnostics

The objective of this paragraph is to propose an application of the indicators to

make a diagnostic of any mechanical system composed of two (or more) classes 1ϖ and 2ϖ , each characterized by a set of specific cyclic frequencies. The idea is to monitor the evolution of the degrees of cyclostationarity over each class in order to detect any abnormal changes in the condition of the system and also to discriminate between the classes. The protocol is resumed in tab.1. where CSn characterizes the nth-order cyclostationarity:

Table 1. Monitoring protocol

Set 1 Set 2 CS1 1 1 1 2( )xI ϖ ϖ ϖ− ∩ 1 2 1 2( )xI ϖ ϖ ϖ− ∩ CS2 2 1 1 2( )xI ϖ ϖ ϖ− ∩ 2 2 1 2( )xI ϖ ϖ ϖ− ∩ CS3 3 1 1 2( )xI ϖ ϖ ϖ− ∩ 3 2 1 2( )xI ϖ ϖ ϖ− ∩ CS4 4 1 1 2( )xI ϖ ϖ ϖ− ∩ 4 2 1 2( )xI ϖ ϖ ϖ− ∩

For example, consider 1 1 / ck Tϖ = and 2 2 / ck Tϖ = two sets of harmonically related cyclic frequencies –as would happen in rotating machinery- of finite cardinal 2K+1 (number of harmonics in the frequency band of interest) where icT is the fundamental cyclic period in class n. Suppose further that 1 2 1 1/ cN k Tϖ ϖ∩ = . The expressions (21) and (24) for 1;2n = then simplify as:

2 1 10

12

21 12 0 2

1

(0) ( 1)ˆ (0)2M ( 1)ˆ ( )

(0) n

nxnx n

x

nx n nxpnx

C M NE I NNcNVar I C p

N N c

−≈

−≈ ∑

(25)

where 1M is the number of samples corresponding to cT1 .

3.5 Application to the diagnosis of a gearbox.

The system under examination is a power circulating gear-testing machine. It is composed of two single-stage gear units mounted back to back. Both units contain a pair of spur gears. The first pair under study has an equal number of 20 teeth and the

second has 40/40 teeth. The test gear, i.e. the 20 teeth gear, has a rotation speed of 1000 rpm.

The objective of this paragraph is to characterize the cyclostationarity relative to each gear wheel of rotating frequencies

1 2and r rf f . α must be within a specific set of

values that are different for wheel 1 and wheel 2. For wheel 1, the set of possible cyclic frequencies is equal to: 1 1 1 1 11 1, 2 , , 0, , ,r r r r rf f nf N f nN fα ∈ −K K (26) where 1N is the tooth number of the wheel.

The interpretation of (26) can be formulated as follows: α can not be equal to the meshing frequency or any of its harmonics. Idem for wheel 2. Unfortunately, for this experiment, the rotating frequencies for the two wheels are equal: ==

21 rr ff 16.67 Hz. Therefore, no possible distinction between the two wheels can be obtained. The meshing frequency mf is equal to 330 Hz.

Reported tests concern gradual development of spalling on individual teeth. The experiment was carried out for two weeks. On the last day, the spalling was in an advanced stage close to the breakage of two teeth. Every day, the test bench was stopped after recording the vibration signals in order to correlate the gear teeth state with the measured signals.

The data recording was set-up with the following parameters: the data were collected with an accelerometer of sampling rate 80 kHz and the length of records was 160000 (2s): 256 signals were acquired during the measurement campaign, and angular re-sampled by using a top reference and interpolation techniques. The synchronization is the same by respect to any of the two wheels.

Figure 1 shows a spectrum of a vibration signal. It can be seen from the inspection of this figure that the spectrum is divided into a discrete and a continuous part. The continuous part obviously appears after 16 kHz.

Figure 2 presents the four indicators nxI when all signals are low-pass filtered (cutting frequency equal to 16 kHz). The synchronous mean clearly increases at the end of the campaign, this being a classical result conforming to these obtained in (Capdessus,1992). For the other three indicators, a little increase of cyclostationarity can be observed from the state 232 which corresponds to the last day of acquisition. Another remark can be deduced that the amplitude of the indicators increases with the orders. In order to evaluate the proprieties of these indicators, we calculated the bias and the variance. The thresholds corresponding to 3 , et 3ix ix ix ix ixE V E E V+ − , i from 1 to 4, are exposed in the same figure. Several important conclusions can be deduced. When the signals are low-pass filtered, the first order indicator presents cyclostationarity increasing along the evolution of the measurements. Idem for order 2. However, the third and fourth order indicator do not exhibit any cyclostationarity, the threshold is always higher than the values of the indicators.

Figure.3 shows the evolution of the four indicators when signals are high-pass filtered. As it was predicted from inspection of the spectrum, the amplitude of the synchronous mean decreases by comparison with Fig.2. In the contrary, 2 xI , 3xI and

4xI increase in a spectacular way. This result is very clear on the 174th state, which corresponds to the 8th day, i.e. 3 days before the end of measurements. To be able to confirm this result, we calculate also the thresholds corresponding to

3 , et 3ix ix ix ix ixE V E E V+ − , i from 2 to 4. Intentionally, we did not put a threshold for the first order indicator because we already know that this indicator will not show any cyclostationarity after the high-pass filtering. This increase in amplitudes confirms that higher order cyclostationarity do appear for high frequencies after gear mesh harmonics and grows bigger when the default of spalling is evolving. Furthermore, the cyclostationarity is visible for all orders, which emphasizes that gear vibration signals are cyclostationary for all orders. A test of stationarity called reverse arrangements test was performed on these indicators. Results show that they are non-stationary at 99%. In addition, to evaluate the properties of these indicators, we have computed the bias and the variance of these indicators conforming to equations (25). We found that the null hypothesis 0H was rejected at a level of significance of 0.13%.

0 0.5 1 1.5 2 2.5 3 3.5 4

x 104

-50

-40

-30

-20

-10

0

10

20

30

Fréquence

Figure 1 : DSP of a gear vibration signal

3

3

ix ix

ix

ix ix

E VE

E V

+

−

Figure 2 : Evolution of cyclostationary indicators with different states of the gearbox system

when all signals are low pass filtered and threshold at 3σ

3

3

ix ix

ix

ix ix

E VE

E V

+

−

Figure 3 : Evolution of cyclostationary indicators with different states of the gearbox system when all signals are high pass filtered and threshold at 3σ .

4 Conclusion

This paper is focused on the analysis of vibration signals using the higher order cyclostationary techniques. To introduce these non familiar notions, we reviewed the basic definition of cyclostationarity with an emphasize on the “pure” and “impure” approaches based respectively on cumulants and moments. We proposed a generic model of gear signals, based on the knowledge of the mechanical phenomena. This model enabled us to demonstrate the cyclostationarity of these signals at several orders. More precisely, we have shown that :

- the geometric or position errors induce a periodic part in gear signals and consist the base of first order cyclostationarity,

- the contact phenomena induce transitory phenomena at high frequencies and also higher order cyclostationarities,

- the speed variations contribute to the higher order cyclostationarity if and only if these variations are cyclostationary processes at these orders,

- vibration signals are impure poly-cyclostationary at all orders. This paper deals also with some new indicators of cyclostationarity to monitor and follow up the state of mechanical systems. The main idea in the development of these indicators was to exploit cyclostationarity at different orders without using nth-order polyspectra which are inappropriate for real-time computation. The proposed indicators are expressed in terms of cumulants and are normalized. They generalize the degrees of cyclostationarity introduced by Zivanovic and Gardner (1994). The statistical properties of these indicators were derived and could be used in designing a suitable threshold in hypotheses testing. They are therefore useful for several applications, and in particular for diagnostics. Such an application was detailed in this paper, where the repartition of cyclostationarity of over different orders was used to monitor a mechanical system. Results are promising and encourage further use of the proposed indicators of cyclostationarity for diagnosis objective.

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