The spectral analysis of cyclo-non-stationary signals

Mechanical Systems and Signal Processing ∎ (∎∎∎∎) ∎∎∎–∎∎∎

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The spectral analysis of cyclo-non-stationary signals D. Abboud a,b,n, S. Baudin c,d, J. Antoni a, D. Rémond c, M. Eltabach b, O. Sauvage d a

Vibrations and Acoustic Laboratory (LVA), University of Lyon (INSA), F-69621 Villeurbanne Cedex, France Technical Center of Mechanical industries (CETIM), CS 80067, 60304 Senlis Cedex, France c Laboratory of Contacts and Structural Mechanics (LaMCoS), University of Lyon (INSA), F-69621 Villeurbanne Cedex, France d PSA Peugeot Citroën, Paris, France b

a r t i c l e i n f o

abstract

Article history: Received 3 April 2014 Received in revised form 25 June 2015 Accepted 27 September 2015

Condition monitoring of rotating machines in speed-varying conditions remains a challenging task and an active field of research. Specifically, the produced vibrations belong to a particular class of non-stationary signals called cyclo-non-stationary: although highly non-stationary, they contain hidden periodicities related to the shaft angle; the phenomenon of long term modulations is what makes them different from cyclostationary signals which are encountered under constant speed regimes. In this paper, it is shown that the optimal way of describing cyclo-non-stationary signals is jointly in the time and the angular domains. While the first domain describes the waveform characteristics related to the system dynamics, the second one reveals existing periodicities linked to the system kinematics. Therefore, a specific class of signals – coined angle-time cyclostationary is considered, expressing the angle-time interaction. Accordingly, the related spectral representations, the order-frequency spectral correlation and coherence functions are proposed and their efficiency is demonstrated on two industrial cases. & 2015 Elsevier Ltd. All rights reserved.

Keywords: Cyclo-non-stationarity Angle–time covariance function Order-frequency spectral correlation function Speed-varying conditions Bearing fault detection Gear rattle noise

1. Introduction In the last decade, the cyclostationary theory has played a pivotal role in vibration-based condition monitoring. In particular, it has helped to improve the diagnosis of roller bearings and gears. When it comes to characterizing cyclostationary vibration signals, the spectral correlation is one of the most powerful second-order tool that completely describes the waveform dynamics and the nature of hidden periodicities. Randall et al. [2] were the first who demonstrate its efficiency for rolling element bearing diagnostics. More analytical studies on the same subject together with related estimation issues are found in Refs. [4] and [5]. However, the spectral correlation and its statistical characteristics were all conceived under the assumption of stationary or quasi-stationary regimes wherein the speed profile remains merely constant. In real applications, however, rotating machines are sometimes subjected to large speed variations which jeopardize the stationary regime assumption. Some typical examples are provided by wind turbines or engines during a run-up: in the former case speed variations are endured and in the latter case they are produced on purpose. The corresponding signals are no longer cyclostationary (CS), although they are still exhibiting rhythms produced by some cyclic phenomena. A typical example is the vibration signal produced by a series of (deterministic) impacts locked to some rotating components during a n Corresponding author at: Vibrations and Acoustic Laboratory (LVA), University of Lyon (INSA), F-69621 Villeurbanne Cedex, France. Tel.: þ33 6 95 22 25 73. E-mail addresses: [email protected], [email protected] (D. Abboud).

http://dx.doi.org/10.1016/j.ymssp.2015.09.034 0888-3270/& 2015 Elsevier Ltd. All rights reserved.

Please cite this article as: D. Abboud, et al., The spectral analysis of cyclo-non-stationary signals, Mech. Syst. Signal Process. (2015), http://dx.doi.org/10.1016/j.ymssp.2015.09.034i

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Notations AT AT-CS CS CS1 CS2 SC SCoh OFSC OFSCoh E

angle–time angle–time cyclostationary cyclostationary first-order cyclostationary second-order cyclostationary spectral correlation spectral coherence order-frequency spectral correlation order-frequency spectral coherence Ensemble averaging operator

ℱτ-f ℱt-αt ℱθ-αθ t

θ τ φ f

αt αθ BPOO BPOI

Fourier transform with respect to time lag Fourier transform with respect to time Fourier transform with respect to angle time variable angle variable time-lag variable angle-lag dual spectral frequency (time-lag dual) cyclic frequency (time dual) cyclic order (angle dual) ball pass order on the outer race ball pass order on the inner race

variable speed regime. Obviously, the signal would not be periodic in time due to varying spacing between the impacts. Nevertheless, the spacing of the impacts would be constant in angle – hence defining the notion of “cycle” – but the resulting signal would still not be periodic in angle due to two reasons: first, the response to each impact has constant time characteristics such as resonance frequency and relaxation time, which would become varying in angle; second, the strengths of the impacts are likely to be modulated by the regime. This latter aspect (amplitude modulation caused by speed variations) was investigated in Refs. [8] and [6]. In such cases, the CS framework is generally insufficient to describe and analyze such signals. A similar issue has been encountered in the field of telecommunication [9]. For instance, it has been shown that a signal subjected to a Doppler shifts loses its CS properties due to the relative motion between transmitter and receiver. The novel class of “generalized almost-cyclostationary” processes has been introduced to embody this particular case [10–12]. Despite its relevance in the mentioned field, this generalization is not able to deal with the mechanical signals of interest in this paper. In the last few years, efforts have been directed toward the extension of existing CS tools in nonstationary regimes. In particular, D'Elia et al. [18] were the first to explore the order-frequency approach. Their idea was to replace the frequency– frequency distribution by a frequency-order distribution that jointly describes the time dynamics and the angle periodicities of the signal. They proposed intuitive estimators to extent the SC and cyclic modulation spectrum which were coined as the “α-synchronized spectral correlation density” and “α-synchronized cyclic modulation spectrum”, respectively. Later on, Urbanek et al. [26] proposed an angle–frequency distribution – namely the averaged instantaneous power spectrum – based on a time filtering step followed by an angle averaging operation of the squared output. A similar solution was proposed by Jabłoński et al. [27] who introduced the angular-temporal spectrum to jointly represent the angular and temporal properties of the signal. Other solutions based on the order spectrum of the squared envelope after some preprocessing steps were introduced in Refs. [13–17]. However, all these attempts still lack a formalism with rigorous statistical definitions. The aim of this paper is then to partially fill in this gap by considering an angle/time cyclostationary (AT-CS) approach. This paper is organized as follows: Section 2 is concerned with a brief review of stationary, cyclostationary, and cyclonon-stationary signals as well as the spectral correlation and the spectral coherence. Section 3 introduces the orderfrequency spectral correlation function (OFSC) and the order-frequency spectral coherence function (OFSCoh) from a joint angle–time vision. Next, Section 4 deals with estimation issues by proposing a consistent Welch-based estimator and related statistical thresholds. Eventually, the proposed tools are illustrated in Section 5 on real data from test rigs. A first application is focused with bearing diagnostics and a second one is on the exploitation of these tools to detect gear rattle noise in vibration signals recorded on an automotive gearbox in run-up conditions.

2. Problem statement 2.1. Stationary, cyclostationary, and cyclo-non-stationary signals The notion of stationary, cyclostationary and cyclo-non-stationary signals is central to this paper; this first subsection provides a brief summary of their definitions with emphasis on the first (mean value) and second orders (covariance functions). Although not general, it often happens that the first two orders are enough to cover many practical purposes in mechanical engineering applications. 2.1.1. Stationary signals Stationary signals are those with constant statistics along the time axis. Formally speaking, a signal xðt Þ is said first-order stationary if its mean (i.e. its first-order moment) does not depend on time t, i.e.
 ð1Þ M 1x ¼ E xðt Þ ; f or all t; Please cite this article as: D. Abboud, et al., The spectral analysis of cyclo-non-stationary signals, Mech. Syst. Signal Process. (2015), http://dx.doi.org/10.1016/j.ymssp.2015.09.034i

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where symbol E denotes the ensemble averaging operator. Similarly, xðt Þ is said second-order stationary if its covariance function (i.e. its “central” second order moment1) C 2x ðt;Þ evaluated at time instants t and t þ depends on the time-lag τ only:
 C 2x ðt;Þ ¼ E ðxðt Þ  M 1x Þðxðt  Þ  M1x Þ ¼ C 2x ðτÞ; f or all t; τ

ð2Þ

*

where denotes complex conjugation. Signals that are stationary on the first and second orders are said weakly stationary. 2.1.2. Cyclostationary signals CS signals are a special case of nonstationary signals that carry hidden periodicities in their structure. A signal is said first-order cyclostationary (CS1) if its first-order statistical moment is a periodic function of period T (also referred to as the signal “cycle”), i.e.
 M 1x ðt Þ ¼ E xðt Þ ¼ M 1x ðt þT Þ; f or all t: ð3Þ Similarly, xðt Þ is said second-order cyclostationary (CS2) if its covariance function is periodic with respect to time:
 C 2x ðt; τÞ ¼ E ðxðt Þ M 1x ðt ÞÞðxðt  τÞ  M 1x ðt  τÞÞ ¼ C 2x ðt þ T; τÞ: ð4Þ Signals that are cyclostationary on the first and second orders are said weakly cyclostationary. Such signals are also referred to as “periodically correlated” signals. CS signals enjoy an insightful representation in terms of a Fourier series X t xðt Þ ¼ c ðt Þej2π iT ; ð5Þ i i where the Fourier coefficients ci ðt Þ are jointly stationary signals and the complex exponentials are parameterized by cyclic frequencies i=T. This representation makes explicit the existence of hidden periodicities in the form of periodic modulations of the random carriers ci ðt Þ. 2.1.3. Cyclo-non-stationary signals CNS signals – which are the topic of this paper – have been recently introduced to extend the cyclostationarity class to cases where a signal still undergoes periodic modulations while at the same time being (strongly) nonstationary on a long term basis [6]. A typical example investigated in Ref. [6] is provided by the noise radiated by an internal combustion engine during a run-up: the noise sources still occur on a periodic basis (actually with respect to the crankangle as will be discussed in Section 2.3), yet they are gradually amplified and their spectral content may progressively change as well. Cyclo-nonstationarity is defined here as a natural extension of the Fourier series representation (5) where the Fourier coefficients are allowed to be jointly nonstationary signals. Contrary to the CS case, this representation is not unique since modulations can be equivalently coded by the complex exponentials or their nonstationary coefficients. As a consequence, cyclo-nonstationarity is a non-property, the reason why it has remained a controversial issue. 2.2. The spectral correlation function (SC) of cyclostationary signals As defined above, the covariance function of a CS signal is periodic with respect to time. Consequently, it accepts a Fourier series2 with non-zero Fourier coefficients at cyclic frequencies i=T, i.e. X i t C 2x ðt; τÞ ¼ C 2x ðτÞej2π iT ð6Þ i

where coefficient C i2x ðτÞ is known as the cyclic covariance function (it is a continuous function of the time-lag τ). This suggests that the same information is more concisely evidenced in the dual Fourier domain. Specifically, the double Fourier transforms relatively to time t and the time-lag τ defines the spectral correlation function (SC)
 ð7Þ S2x ðαt ; f Þ ¼ ℱ t-α C 2x ðt; τÞ t

τ-f

(index t is used to highlight that αt is dual of time in the Fourier transform) which, in the special case of a CS signal, takes the discrete form   X i i ð8Þ S2x ðf Þδ αt  ; S2x ðαt ; f Þ ¼ T i

1 The operation of centering amounts to systematically subtract the mean value from the signal. Note that in the general (nonstationary) case, the
 mean value E xðt Þ is itself a signal (i.e. a function of time). 2 Although similar, the Fourier series (5) and (6) are not identical: the first one has random coefficients which are functions of time t, while the second one has deterministic coefficients which are functions of time-lag.

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n o where Si2x ðf Þ ¼ ℱτ-f C i2x ðτÞ . The SC in Eq. (7) is a frequency–frequency distribution with respect to both the cyclic frequency αt , linked to the cyclic evolution of the waveforms and the spectral frequency f , linked to the waveforms in the signal. While the former reflects the frequency of the modulations, the latter describes the frequency of the carriers. Expression (8) evidences that the SC of a CS signal is discretely distributed along spectral lines parallel to the f-axis and positioned at the cyclic frequencies αt ¼ i=T. This unique signature makes the SC a powerful and robust tool for detecting the presence of and characterizing cyclostationarity in engineering applications [4]. An alternative way of defining the SC is n oo
 1 n S2x ðαt ; f Þ ¼ lim E ℱW xðt Þ ℱW xðt Þe  j2παt t W-1W  1
ð9Þ ¼ lim E X W ðf ÞX W ðf þ αt Þ W-1W
 R þ W=2 where X W ðf Þ ¼ ℱW xðt Þ ¼  W=2 xðt Þe þ j2π tf dt stands for the Fourier transform of signal x(t) over a time interval of finite duration W. This expression provides another interpretation of the SC, which actually justifies its name: it is a measure of the correlation between the frequency components of the signal at f and f þ αt . A related operator is the spectral coherence (SCoh) which has the property of measuring the degree of correlation between two spectral components independently of the signal power spectrum. The SCoh is defined as:

γ 2x ðαt ; f Þ ¼

S2x ðαt ; f Þ

½S2x ðf ÞS2x ðf þ αt Þ1=2

ð10Þ

with S2x ðf Þ the power spectral density of signal x. Just like the regular correlation coefficient, jγ 2x ðαt ; f Þj2 is normalized between 0 and 1: the closer it is to the unity, the stronger the CS component at cyclic frequency αt . Therefore, when a CS signal of cycle 1=αt is embedded in stationary noise, its SCoh jγ 2x ðαt ; f Þj2 can be seen as an indication of the signal-tonoise ratio. 2.3. Time versus angle cyclostationarity The previous definitions are all applied to temporal descriptors of the signals and their dual counterparts expressed as functions of frequencies in Hertz. When considering signals produced by a rotating machine, the true variable of interest is actually the angle of rotation of the machine (i.e. of a reference shaft in the machine) rather than time. With stationary signals, time or angle makes little difference since the signal statistics are constant by definition. However, with CS signals, the two descriptions are not equivalent in general. Strictly speaking, the periodicities impinged by the rotation of the machine should be described according to the angular variable, say θ, and being substituted for time t in all the above definitions. In particular, the Fourier series representation (5) of a CS signal then becomes    X    jiθ ci t θ e ; ð11Þ x t θ ¼ i

  P   or with some abuse of notation, x θ ¼ ci θ ejiθ , where it has been assumed that a full cycle corresponds to 2π radians.    i Strictly speaking, this defines an angle-cyclostationary signal only if ci t θ is stationary in angle. Such an approach has been pursued in several research works [3] where the signals were either directly sampled in angle (by using an encoder signal as the input to the sampling clock of the data acquisition system) or numerically resampled afterwards from time to angle [22]. It is also central to order spectrum analyses and order tracking methods [20,21] which are mainly concerned with the analysis of first-order angle-cyclostationary signals. In the special cases where the machine speed is constant, stationary or cyclostationary,3 it can be shown that an anglecyclostationary signal is also time-cyclostationary [3] – this is the scenario often assumed in the presence of small speed fluctuations. In all other cases, this equivalence does not hold and the choice between a temporal and an angular description may be troublesome as explained hereafter. 2.4. Ambiguities under large speed fluctuations Large speed variations of the rotating machine are typically seen as nonstationarities from the receiver point of view and thus bring to the fore the issue of choosing between a temporal and an angular description. Indeed, whereas angleperiodicity holds for mechanisms resulting from the rotation of mechanical elements whatever the rotational speed, physical phenomena governed by time-dependent dynamical characteristics, such as temporal differential equations, should still be described in time. Unfortunately, the angle-domain covariance function is not able to describe time-dependent phenomena. For instance, an angle-periodic train of Dirac deltas exciting a structural resonance will produce a series of transients with constant ringing in time but periodic positions in angle; thus, an angular representation of the signal will stretch the signal waveform at low speed and squeeze it at high speed [6], destroying the angle-cyclostationary structure of the signal. 3

This latter possibility will be systematically ignored from the subsequent development in the present paper.

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By way of an example, consider the angle-periodically modulated signal   xðt Þ ¼ p θðt Þ εðt Þ ð12Þ     where p θ ¼ p θ þ2π is angular periodic and εðt Þ is zero-mean stationary random noise. Introducing the Fourier series   P p θ ¼ pi ejiθ , it is readily seen that signal x(t) has the Fourier series representation i X pi εðt Þejiθðt Þ ð13Þ xðt Þ ¼ i

with time-stationary random Fourier coefficients ci ðt Þ ¼ pi εðt Þ. This is generally neither a time nor an anglecyclostationary signal:

 when the signal is viewed in time the complex exponentials ejiθðtÞ are not periodic and,     when viewed in angle, the random Fourier coefficients ci t θ are no longer stationary (although they are with respect to t) except for the specific cases when θ is linearly related to t, that is when the rotational speed θ_ ¼ dθ=dt is constant, or when θ is itself a stationary process. This is actually an example of a CNS signal as defined in Section 2.1. Let us now investigate the consequence on the signal statistics. Since M 1x ðt Þ ¼ 0, second-order statistics only are of concerned. Assuming from now on that θðt Þ is a deterministic function, the time-domain covariance function of signal x(t) is     C 2x ðt; τÞ ¼ p θðt Þ p θðt  τÞ C ε ðτÞ ð14Þ with C ε ðτÞ the covariance function of εðτÞ. Except for the very specific case when the rotational speed θ_ is constant,      p θðt Þ is generally not a periodic function of time t (i.e. there is no T a 0 such that p θðt Þ ¼ p θðt þ T Þ ); therefore C 2x ðt; τÞ is not periodic either and the signal not time-cyclostationary. Consequently, in the “temporal” SC, spectral lines parallel to the spectral frequency axis will spread all over the cyclic frequency axis, and the modulation information will be lost. Alternatively, the angle description of the covariance function is                 C 2x θ; φ ¼ Efx t θ xðt θ  φ Þ g ¼ p θ p θ  φ C ε t θ t θ  φ ð15Þ          which is not periodic in angle since C ε t θ  t θ  φ a C ε ðt θ þ2π  t θ þ 2π  φ in general (except again for the very specific case when the rotational speed θ_ is constant). Consequently, the “angular” SC will be distorted along the spectral order axis (i.e. the dual of the angle-lag φ). Therefore, this specific signal is neither time-cyclostationary nor angle-cyclostationary; it is actually cyclo-non-stationary. To make this example more tactile, let us consider the simple signal composed of a constant-magnitude chirp (Fig. 1(a)),   p θ , modulating a colored noise, εðt Þ, of limited frequency-band [0.2 Hz, 0.3 Hz] (Fig. 1(b)). The obtained signal as well as its spectrogram are respectively reported in Fig. 1(c) and (d). The signal length is set to 40,000 points and the sampling frequency is equal to unity. Fig. 2(a) displays the instantaneous frequency profile of the modulating chirp, and Fig. 2(b) the “temporal” SC of the signal; it is clear that the angle-periodicity of the modulation cannot be detected, yet the spectral 

Fig. 1. Synthetic example of a CNS signal made of a constant-magnitude chirp modulating a colored noise. (a) Constant-magnitude chirp, (b) Welch spectrum of the colored noise, (c) synthetic CNS signal, and (d) spectrogram of the signal.

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Frequency-Frequency Spectral Correlation

Instantaneous frequency Spectral frequency [Hz]

Magnitude [Hz]

0.02

0.015

0.01

0.005

0.45

0.3

0.4 0.35

0.25

0.3

0.2

0.25 0.2

0.15

0.15

0.1

0.1 0.05

0.05 0

0 0

0.5

1

1.5

2

2.5

3

3.5

0.5

1

Time [s]

0.4

2

0.35 0.3

1.5

0.25 0.2

1

0.15 0.1

0.5

Spectral frequency [Hz]

Spectral order [shaft order X]

0.45

3

0.3

2.5

3

0.8

0.25 0.6

0.2 0.15

0.4

0.1

0 2

1

0.35

0

Cyclic order [shaft order X]

1.2

0.4

0.05 1.5

2.5

0.45

0.05 1

2

Order-Frequency Spectral Correlation

Order-OrderSpectralCorrelation X

0.5

1.5

Cyclic frequency [Hz]

0.2 0.5

1

1.5

2

2.5

3

Cyclic order [shaft order X]

Fig. 2. (a) Instantaneous frequency of the modulating function, (b) “temporal” spectral correlation of the time signal, (c) “angular” spectral correlation of the angular-resampled signal, and (d) order-frequency spectral correlation of the signal.

content of the colored noise is partly identified. Alternatively, Fig. 2(c) displays the “angular” SC of the simulated signal (orders on the vertical axis are divided by the averaged instantaneous frequency to get values close to frequencies). Clearly, the frequency content of the colored noise is completely lost, while the cyclic content of the chirp is reasonably returned by a spectral line at the first shaft cyclic order. In conclusion, performing the SC either in the time or the angle domain will not return complete information about the signal in the general scenario with large speed fluctuations: kinematic information related to angle-periodic rotations of the machine is lost in the temporal representation, whereas structural information related to time-dependent phenomena is lost in the angular representation. Thus, the choice between a time and an angle description appears to be a controversial issue in the CNS scenario.

3. Theoretical solutions 3.1. The joint angle–time vision As previously shown, neither the time nor the angle-domain covariances are able to describe the periodic correlation property under nonstationary regime. Ideally, an angular vision should be used for characterizing periodic modulations and a temporal vision for the carrier waveform. Hence, the optimal solution for describing such signals is to consider angle and time jointly instead of independently. This is embodied by the following Fourier series representation, X ci ðt ÞejiθðtÞ ; ð16Þ xðt Þ ¼ i

where the complex exponentials ejiθ are explicitly expressed in angle and the Fourier coefficients ci ðt Þ in time. Whether ci ðt Þ is a stationary or nonstationary signal leads to two different but important properties that are investigated in the next subsection. 3.2. Angle–time cyclostationary and cyclo-non-stationary signals A particular case unveiled by the Fourier representation (16) is when ci ðt Þ is time-stationary and the correlation time is short compared to the cycle duration. Such signals have recently been shown to be angle–time cyclostationary (AT-CS), a Please cite this article as: D. Abboud, et al., The spectral analysis of cyclo-non-stationary signals, Mech. Syst. Signal Process. (2015), http://dx.doi.org/10.1016/j.ymssp.2015.09.034i

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property that enlarges the class of CS signals [7]. Some models of AT-CS processes are given in Ref. [7]. Not only are AT-CS signals of practical interest (the situation is likely to occur in many applications in accordance to the discussion in Sections 2.3 and 2.4), but also of theoretical interest since they pave the way as how to proceed with CNS signals. As a matter of fact, the next subsection investigates the covariance function that results from such a model. Note that the terminology “Time–angle periodically correlated signal” has been introduced in Ref. [7]. Periodically correlated signals are strictly equivalent to second-order cyclostationary signals. Since the latter terminology seems to prevail in mechanics, it will be systematically used in this work. The adjective “second-order” will omitted for the sake of simplicity. 3.3. Angle–time covariance function The requirement of a joint angle–time vision is not that simple when it comes to the covariance function, because the latter involves two time instants – or angle positions – that must conform to the same units. One solution introduced in Ref.     [7] is to consider two time instants, t θ and t θ  τ, locked on a given angle position θ and spaced apart by a given time-lag τ. Thus, the angle–time (AT) covariance function is defined as  
       C 2x θ; τ ¼ E x t θ x t θ  τ : ð17Þ Loosely speaking, this function describes the cyclic evolution of the modulation by means of the angle variable and the waveform characteristic of the carrier by means of the time-lag variable. For AT-CS signals, the AT covariance function is angle-periodic with respect to variable θ and thus accepts Fourier series with non-zero Fourier coefficients at cyclic orders i   X i C 2x θ; τ ¼ C 2x ðτÞejiθ ; ð18Þ i

where the coefficient C i2x ðτÞ stands for the cyclic covariance function. A simple example of an AT-CS signal is provided by the signal of Section 2.4 when the carrier εðt Þ is white. Thus, its AT covariance function reads          C 2x θ; τ ¼ Efx t θ xðt θ  τÞ g ¼ pðθÞ2 σ 2ε δðτÞ ¼ C 2x θ þ 2π ; τ ð19Þ which is obviously angle-periodic and thus accepts a Fourier series in the form of Eq. (18). In general, CNS signals will not produce a periodic AT covariance function. Yet, it is asserted that in many cases of practical interest they can be approximated as locally angle-periodic, that is there is range of limited speed fluctuations     where C 2x θ; τ  C 2x θ þ 2π ; τ . Coming back to the previous example and allowing εðt Þ to be colored,           C 2x θ; τ ¼ p θ p θ t θ  τ  φ C ε ðτÞ  pðθÞ2 C ε ðτÞ ð20Þ provided the temporal extent of C ε ðτÞ – i.e. the correlation length of εðt Þ – is short as compared to the time interval   R θþπ T θ ¼ θ  π dθ corresponding to an angular period. 3.4. Order-frequency spectral correlation function (OFSC) Similarly to the SC, the OFSC is defined as the double Fourier transform of the AT-covariance function:  
  S 2x αθ ; f ¼ ℱ θ-α C 2x θ; τ

τ-f

θ

ð21Þ

where αθ stands for “order”, a quantity without unit that counts the number of events occurring per rotation of the reference shaft. It is emphasized that the first Fourier transform maps angle (in radian) to order (without dimension) while the second one maps time (in seconds) to frequency (in Hertz). After some calculus (see Appendix A), the above definition can be expressed with respect to temporal Fourier transforms as: n n oo  
 1 S 2x αθ ; f ¼ lim E ℱW xðt Þ :ℱW xðt Þe  jαθ θðt Þ θ_ ðt Þ ; ð22Þ W-1ΦðW Þ R where θ_ ðt Þ ¼ ddtθ stands for the instantaneous angular speed, ΦðW Þ ¼ W θ̇dt is the angular sector spanned during the time interval W, and ℱW f:g is defined as in Eq. (9). The latter expression bears obvious similarity with Eq. (9) except that the tÞ _ second Fourier transform involves a multiplication of thensignal by e  jαθ θðo θðt Þ instead of e  jαt t ; this will not correspond to a   frequency shift in general so that, contrary to Eq. (9), ℱW xðt Þe  jαθ θðtÞ θ_ ðt Þ aX W f þ αθ . Therefore, the interpretation of the OFSC is between two transformed versions of the signal, yet not necessarily between spectral components. Specifically, the OFSC is equivalent to the (cross-) spectral density   S 2x αθ ; f ¼ Sxxαθ ðf Þ; ð23Þ of signals x(t) and xαθ ðt Þ ¼ xðt Þe  jαθ θðtÞ θð̇ t ÞW Φ (by convention, Sxy ðf Þ denotes the cross-spectral density of signals x and y). Eventually, it is noteworthy that the presence of θð̇ t Þ in the transform guarantees orthogonality of the Fourier basis when expressed with respect to angle θ instead of time t and consequently conserves the signal energy. Please cite this article as: D. Abboud, et al., The spectral analysis of cyclo-non-stationary signals, Mech. Syst. Signal Process. (2015), http://dx.doi.org/10.1016/j.ymssp.2015.09.034i

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For AT-CS signals, the OFSC takes the particular form   X   S 2x αθ ; f ¼ S 2x ði; f Þδ αθ  i ;

ð24Þ

i

which clearly indicates the presence of spectral lines S 2x ði; f Þ parallel to the spectral frequency order at the cyclic orders i that carries the cyclic information of the waveform. Similarly to the CS case, the order-frequency spectral coherence function (OFSCoh) is defined as     S2x αθ ; f Γ2x αθ ; f ¼ ð25Þ ½S2x ð0; f ÞS2xα ð0; f Þ1=2 θ

or, equivalently, is expressed as 



Γ2x αθ ; f ¼

Sxxαθ ðf Þ

ð26Þ

½S2x ðf ÞS2xα ðf Þ1=2 θ

after making use of Eq. (23). The next section addresses how to estimate these quantities. An illustrative representation of the proposed approach is provided in Fig. 3 wherein an AT-CS process is shown to be the upshot between a rotating mechanism (kinematics) and a time-domain stochastic process. The failure of the classical approach stems from the non-linearity of the angle–time relationship: the periodicity is destroyed for the time-domain covariance function which results in a misleading representation of the SC whereas the frequency of the carriers is lost in the angle-domain covariance function. On the contrary, the AT-covariance enjoys the peculiarity of complying with this nonlinearity and results in a more expressive representation of angle and time components, thus providing a symptomatic bispectral representation embodied in the OFSC.

4. Practical solutions: estimation issues One difficulty with the spectral descriptors of non-stationary signals is the non-availability of generally consistent estimators. Cyclostationarity is one exception and it happens that AT-cyclostationarity, as defined in this paper, is another one to some extent. A robust estimator of the OFSC is proposed hereafter, which has the advantage to operate entirely in the time-domain without the need to resample the signal in the angle domain – which is often a source of errors if not done carefully [24].

Fig. 3. An illustrative representation of the proposed approach.

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4.1. Discussion on a previous work As previously mentioned, D’Elia et al. [18] were the first who inspected the need of an order-frequency distribution for a prominent description of cyclic components in rolling element bearing signals when operating under nonstationary speed conditions. Indeed, they expanded the Welch estimator of the SC by adding two resampling operations: the first was applied to perform the order shifting operation, while the second was used to back up to the time domain before performing the Fourier transform. Notwithstanding the practical efficiency of the proposed algorithm, it was fully intuitive needing to be supported by rigorous theoretical proofs. In the following, authors aim at filling in this gap by building a consistent Welchbased estimator of the OFSC starting from the theoretical background of the quantities introduced in (17) and (22) as well as the related confidence intervals. 4.2. A Welch-based estimator Similarly to the SC, the estimation of the OFSC follows the classical lines of “stationary” spectral analysis enjoying the particularity provided in (22). In particular, the Welch estimator4 – which consists of substituting the ensemble average operator by an average over weighted blocks – is probably the most practical because of its implementation easiness, enjoying a low computational cost using short FFT’s of fixed size [25].
L  1 Subsequently, the OFSC estimator of a finite-length record x½n n ¼ 0 sampled with period Δ will be denoted by 
N w  1 ðLÞ  S^ 2x αθ ; f . Let w½n n ¼ 0 be a window of Nw points and let ws ½n ¼ w½n  sR be its shifted version by R samples. The increment R is set between 1 and Nw to provide the opportunity of possible overlaps. By replacing the ensemble averaging operator by a finite average over segments and by omitting the limit to infinity of the integration interval, one finds (see the proof in Appendix B),  ðLÞ  S^ 2x αθ ; f ¼

SX 1 n o
 1 _ ðnÞe  jαθ θðnÞ DTFT w ð n Þx ð n Þ DTFT w ð n Þx ð n Þ θ s s ΦSjjwjj2 s ¼ 0

ð27Þ

where S stands for the greatest integer smaller than or equal to L RNw þ 1, DTFT is the discrete time Fourier transform, and jjwjj2 stands for the window energy. Once again, the previous equation can be equivalently written as ðLÞ ðLÞ S^ xx ðν; f Þ ¼ S^ xxα ðf Þ;

ð28Þ

θ

ðLÞ S^ xy ðf Þ

the estimator of the cross-power spectrum of x and y over L samples. The same with xαθ ðnÞ ¼ xðnÞe  jαθ θðnÞ θ_ ðnÞW Φ and applies to the OFSCoh estimator defined by  ðLÞ  S^ 2x αθ ; f b ðLÞ α ; f  ¼ Γ ; ð29Þ θ 2x ðLÞ ðLÞ ½S^ 2x ð0; f ÞS^ 2xα ð0; f Þ1=2 θ

or, equivalently, b ðLÞ α ; f  ¼ Γ θ 2x

ðLÞ S^ xxα ðf Þ θ

ðLÞ ðLÞ ½S^ 2x ðf ÞS^ 2xα ðf Þ1=2

ð30Þ

θ

after substituting expression (28) into (29). Similarly to the SC where Δαt  L1Δ, the cyclic order resolution is proportional to the inverse of the angular sector spanned R during the time interval W, i.e. Δ 2απθ  Φð1W Þ. Noting that ΦðW Þ ¼ W θð̇ t Þdt, the following condition is obtained for the cyclic order resolution, 2π 1 2π o o Δαθ  R ; W Ωmax W Ωmin W θ ð̇ t Þdt

ð31Þ

with Ωmin and Ωmax the minimum and the maximum angular speeds, respectively, expressed in [rad/s]. Accordingly, care should be taken to respect the inequality (31) while choosing the cyclic order increment. Readers will find the related Matlab routine in Appendix C. Coming back to the numerical example of Section 2.4, Fig. 2(d) shows the OFSC calculated by the proposed estimator. Interestingly, the angle-periodic modulation imposed by the chirp and the frequency content carried by the colored noise are both conserved in the distribution. The cyclic content is accurately pictured by the spectral line parallel to the spectralfrequency axis at the first-shaft order, while the frequency content is correctly identified in the expected spectral band. Though the signal is strictly speaking CNS due to the correlated nature of the noise, it still enjoys the AT-CS property to a very good approximation; this is explained by the fact that the noise correlation length (inversely proportional to the noise spectral bandwidth 0.2 Hz) is much smaller than the smallest period (inversely proportional to the maximal instantaneous frequency 0.02 Hz), making the signal conform to the approximation provided in (20). 4

Also known as the averaged cyclic periodogram.

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Fig. 4. Test rig located at CETIM.

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4.3. Estimator variance and threshold Proceeding with the calculation reported in Ref. [5], the variance of the OFSC is expressed as n ðLÞ  X o X Var S^ 2x αθ ; f  : S 2x ð0; f Þ:S 2xα ð0; f Þ ¼ : S2x ðf Þ:S2xα ðf Þ θ

where X

P

¼

θ

ð32Þ

is the variance reduction factor, SX 1 s¼0

Rw ðsRÞ2

S  jsj ; S2

ð33Þ

with Rw ½n the autocorrelation function of the data-window ws ðnÞ. An estimate of the variance is thus simply obtained by estimating the quantities S2x ðf Þ and S2xα ðf Þ. In turn, this makes possible to find a statistical threshold to assess the presence θ of angle-periodic components. Usually, two hypotheses are proposed. The first one, H0, denies the presence of an AT-CS component at the cyclic order αθ , whereas the second one, H1, affirms its presence. Accordingly, it can be proved that H0 is Please cite this article as: D. Abboud, et al., The spectral analysis of cyclo-non-stationary signals, Mech. Syst. Signal Process. (2015), http://dx.doi.org/10.1016/j.ymssp.2015.09.034i

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Fig. 6. The estimated SC (magnitude) of the accelerometer signals Acc1 (a) and Acc2 (d); the cyclic frequency resolution is Δα ¼ f rot *0.02 ¼0.514 Hz. The estimated OFSC (magnitude) of the accelerometer signals Acc1 (b) and Acc2 (e) with corresponding close-ups (c) and (f); the cyclic order resolution is Δαθ ¼ 0:01. For all figures: Hanning windows, overlap¼ 66,6%, L¼ 256000 points, N w ¼ 256, Δf ¼ 100 Hz. The signals Acc1 and Acc2 are those of Fig. 2.

Table 1 Characteristics of the rolling element bearing (Deep groove ball 6820). Pitch diameter [mm]

Rolling element diameter [mm]

Number of rolling elements

Contact angle

Outer, housing, diameter [mm]

Inner, bore, diameter [mm]

Ball pass orderouter race (BPOO)

Ball pass orderinner race (BPOI)

25

6

8

0.00

35

15

3.04

4.96

rejected if vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n u  o d ^ ðLÞ  ðLÞ   u  ^  tVar S 2x αθ ; f : χ 21  p;2 ; S 2x αθ ; f  Z 2

ð34Þ ðLÞ

^ ðα ; f Þ has a where χ 21  p;2 stands for the percentile of the chi-square law with two degrees of freedom. Interestingly, Γ θ 2x constant threshold due to its implicit self-whitening operation resulting from the spectral-normalization. In this case H0 is rejected if ðLÞ   Γ 2x αθ ; f j2 Z jc

P 2

χ 21  p;2 :

ð35Þ

Expression (35) makes it clear that the OFSCoh is more convenient to handle than the OFSC due to its fixed threshold, albeit they both carry the same information.

5. Application part This section has for objective to prove the efficiency of the OFSC and OFSCoh algorithms in two very different industrial applications. The first one aims at diagnosing rolling element bearing and the second one concerns the detection of gear rattle noise in automotive vehicles. Please cite this article as: D. Abboud, et al., The spectral analysis of cyclo-non-stationary signals, Mech. Syst. Signal Process. (2015), http://dx.doi.org/10.1016/j.ymssp.2015.09.034i

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5.1. Diagnostics of rolling element bearing under speed-varying regimes In precursory works, it has been recognized that vibrations produced by faulty rolling-element bearing are essentially random in their nature [2,1]. Indeed, the fault signature appears as cyclically correlated components locked to the rotational shaft speed via the fault characteristic order. Consequently, in stationary speed conditions, the diagnostic information is completely preserved in the SC constituting a symptomatic pattern of spectral lines as pointed out in Section 2.2. When operating under speed-varying conditions, the nature of the bearing signal turns to become cyclo-non-stationary. This is principally imposed by the angle-periodic occurrence of the fault which are captured by the complex exponentials of Eq. (16). Yet, the AT dualism appears as an inner property in the signal because of the interaction between time and angle-dependent phenomena. Precisely, a faulty bearing signal can be viewed as a series of cyclic impacts locked to the crankangle and exciting structural resonances. Clearly, the positions of the impact excitations are described by the shaft angle while the resonance responses are governed by differential equations that impose time-invariant resonance frequencies and relaxation times. Despite the presence of this dualism, two reasons prevent the signal from being AT-CS, namely (i) the correlated nature of the carrier which is actually the system impulse response, and (ii) the long-term modulation. The first reason could be overridden once recognizing that relaxation times of mechanical systems are usually short compared to cycle durations, whereas the second restricts the signal length so that the long-term energy evolution remains limited. In these conditions, bearing signals can be approximated to be AT-CS, thus enjoying a periodic AT-covariance function standing for a symptomatic property of the fault signature in the order-frequency plane. After a brief description of the test rig, the proposed estimators with their relative thresholds will be evaluated respectively in (i) stationary and (ii) non-stationary speed-varying conditions. Please cite this article as: D. Abboud, et al., The spectral analysis of cyclo-non-stationary signals, Mech. Syst. Signal Process. (2015), http://dx.doi.org/10.1016/j.ymssp.2015.09.034i

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Fig. 9. Estimated SC (magnitude) of the accelerometer signals Acc1 (a) and Acc2 (e); the cyclic frequency resolution is Δα¼ 0.01*f rot ¼ 0:254 Hz. The OFSC (magnitude) of accelerometer signals Acc1 (b) and Acc2 (f) with corresponding close-ups (c) and (g). The estimated OFSCoh (magnitude) of accelerometer signals Acc1 (d) and Acc2 (h). The cyclic order resolution is set to be Δαθ ¼ 0:02. For all figures: Hanning windows, overlap¼66,6%, L ¼ 256,000 points, Nw ¼ 256, Δf¼ 100 Hz.

5.1.1. Test rig specifications Experimental tests have been conducted on the test rig of Fig. 4 that is located at CETIM.5 The test rig comprises an asynchronous motor supplied by a variable-speed drive to control the motor speed, followed by a spur gear with 18 teeth and ratio of 1. Two identical rolling element bearings are installed after the spur gear: the first (B1) – the healthy bearing – is closer to the gears than the faulty one (B2) that is branched to an alternator by means of a belt simulating a fixed load. The bearing characteristics are given in Table 1. An optical keyphasor of ‘Brawn’ type is fixed close to the motor output to measure the rotational shaft position (see Fig. 4). In addition, two accelerometers Acc1 and Acc2 are mounted on bearings B1 and B2, respectively, in the Z-direction to measure the vibrations. The sampling frequency is set to 25.6 kHz. 5.1.2. Stationary-speed varying conditions As a preliminary step, it is useful to assess the functionality of the OFSC in quasi-stationary regimes in which cases it should return similar results than the traditional SC. The first experiment is conducted under (quasi-) constant speed – 1546 rpm with  0.5% fluctuations – imposed by the electric motor (see Fig. 5(a)). Fig. 5(b) illustrates a 4 s record of bearing vibration signals captured by Acc1 and Acc2, while Fig. 5(c) displays the corresponding Welch-spectra. In the following, the angle-domain synchronous average is subtracted from the signals in order to eliminate the deterministic components produced by the gear system [19]. Afterward, the OFSC estimator proposed in Section 4.2 together with the classical SC are applied to the residual part of signals, and the obtained results are reported in Fig. 6. It is seen that the SC and the OFSC return virtually the same distributions for both accelerometer signals. The outer-race fault symptom is prominent in Acc2 around the spectral-frequency 7 kHz and at the cyclic orders (resp. frequencies) corresponding to the bearing outer-race signature m  BPOOþn with m ¼ 71, 72…; n¼ 71 [2] — (see Fig. 6(d)–(f)). For Acc1, 5

Technical center for mechanical industries, Senlis, France.

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Fig. 11. The estimated OFSCoh (squared-magnitude) evaluated at the cyclic orders BPOO and BPOI of Acc1 (resp. a, b), and Acc2 (resp. c, d) with its 1% level of significance (dashed line).

Please cite this article as: D. Abboud, et al., The spectral analysis of cyclo-non-stationary signals, Mech. Syst. Signal Process. (2015), http://dx.doi.org/10.1016/j.ymssp.2015.09.034i

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Fig. 12. Illustration of gear rattle noise: angular speed of the leading gear (black) and of the unloaded gear (red) and relative displacement. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Fig. 13. (a) Gearbox on the LaMCoS test bench and (b) accelerometer signal for 6° misalignment (arbitrary units).

the signature also appears along a narrow spectral frequency band located at 2 kHz (Fig. 6(a)–(c)), which is better evidenced in Fig. 7(a). Add to this the appearance of CS2 at the shaft order together with its harmonics. This is believed to be due to the fact that gears were exposed to excessive wear. In order to assess the presence of outer and inner faults, the OFSC magnitude is reported in Fig. 7 at critical cyclic orders, namely the BPOO and BPOI, together with its 1% level of significance. Interestingly, the threshold is prominently exceeded at the BPOO for Acc1 and Acc2, thus indicating the presence of an outer-race fault; on the other hand the threshold is not exceeded at the BPOI, thus indicating the healthiness of the inner-race component. The use of the OFSCoh and the SCoh returns even more meaningful representation which is not shown here due to lack of space, yet its illustration is postponed to the next section in the case of large-speed variation.

5.1.3. Nonstationary speed-varying conditions In the second test, a run-up speed profile (from 1200 to 1900 rpm) has been applied to the electric motor (see Fig. 8(a)) over 15 s duration signals captured by Acc1 and Acc2 presented in Fig. 8(b). Subsequently, the SC and OFSC are estimated on the residuals of both signals Acc1 and Acc2 (i.e. after removal of deterministic components), and the results reported in Fig. 9. As expected, the classical SC fails in detecting modulations related to the angle as evidenced by the absence of spectral lines (Fig. 9(a) and (e)): the signal information are completely lost. Actually, the cyclic frequency components spread all over the cyclic frequency axis because of the speed variation. On the other hand, the spectral lines already revealed in the quasiconstant case (see Fig. 6(a) and (d)) are preserved with the proposed OFSC at the cyclic orders of the outer-race signature (Fig. 9(b) and (f)), thus indicating the presence of the bearing outer-race fault. The OFSC preserves the signal information because of its capability to accommodate speed variations, and consequently is a powerful tool for rolling element bearing fault detection in non-stationary conditions. Remarkably, the hidden cyclic components in the weak-power spectral frequency bands are better revealed in the coherence because of its inherent power normalization. This is shown in Fig. 9(d) and (h) where the spectral lines stretch over a wider spectral frequency band, providing a clearer fault signature for both Acc1 (Fig. 9(d)) and Acc2 (Fig. 9(h)). In short, the strength of the cyclic component is no more dependent on the power spectrum density of the signal; whence the authors recommend the use of the OFSCoh rather than the OFSC. Please cite this article as: D. Abboud, et al., The spectral analysis of cyclo-non-stationary signals, Mech. Syst. Signal Process. (2015), http://dx.doi.org/10.1016/j.ymssp.2015.09.034i

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Fig. 14. (a) Accelerometer signal (arbitrary units). (b) Input speed (red dotted line) composed of the superposition of a linear average speed (black) and a fluctuating speed (dimensionless). (c) Zoom on the input speed. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Fig. 15. Spectral coherence estimated with SCoh algorithm (a) and OFSCoh algorithm (b). Hanning windows, Overlap 66,6%, N w ¼ 256, Δf ¼ 50 Hz, Δαθ ¼ 0:0085 evt=tr.

Last, the OFSC and OFSCoh are evaluated at the BPOO and BPOI together with their 1% levels of significance (Figs. 10 and 11, respectively). The obtained results are virtually similar to those obtained under stationary speed conditions, which reflects the robustness of the proposed approach to speed changes: thresholds are exceeded when evaluated at BPOO for Acc1 and Acc2 (resp. Fig. 10(a) and (c)), but not at the BPOI (see Fig. 10(b) and (d)), thus clearly indicating the presence of an outer-race fault. Again, the cyclic components are more conspicuous for OFSCoh than for OFSC. This is particularly true for the cyclic component located at 2 kHz in Acc2 (see Figs.10(c) and 11 (c)). 5.2. Detection of gear rattle noise The second application of the OFSCoh concerns the detection of gearbox rattle noise. This is a typical AT-CS phenomenon which characterizes the operation of gears. In this application all variables will be dimensionless for confidential reasons. Gear rattle noise due to impacts between teeth of unloaded gears is a current problem for car manufacturers [23]. These impacts are principally caused by the engine acyclism transmitted to the primary shaft of the gearbox after being partially filtered by the clutch. In a four-cylinder, four-stroke combustion engine there are 2 explosions per revolution of the crankshaft. The acyclism being then principally composed of the second harmonic of the rotation speed (denoted H 2 ), the periodic rattle is composed of 4 impacts per revolution of the primary shaft: one impact on the leading flank and one on the led flank of the gear per period H 2 (Fig. 12). The appearance and the level of rattle noise depend on the operating conditions, particularly on the rotation speed. In order to scan a large panel of operating conditions, gearboxes are often tested in runup conditions. It is then very interesting to use the OFSCoh to detect the appearance of the impacts. 5.2.1. Presentation of the test bench The test bench used in this study is located at the LaMCoS (Contact and Structural Mechanics Laboratory), University of Lyon, France. It is composed of an automotive gearbox driven by an electrical motor which is speed controlled. In order to simulate engine acyclism, a Cardan joint is placed between the electrical motor and the input of the gearbox. This device introduces a cyclic torsional excitation twice per rotation. A load is applied to the gearbox by an electrical generator which is torque controlled. The gearbox shown in Fig. 13(a) is equipped with different transducers (accelerometers and microphones). The third gear is engaged and a speed ramp with a variation of 2430 rpm is applied during 73 s. An angular encoder Please cite this article as: D. Abboud, et al., The spectral analysis of cyclo-non-stationary signals, Mech. Syst. Signal Process. (2015), http://dx.doi.org/10.1016/j.ymssp.2015.09.034i

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17

Fig. 16. : Squared OFSCoh estimated for each signal portion at the order 4 evt/rev for misalignments 0° (a), 4°(b), 6°(c), 8° (d). Hanning windows, Overlap 66,6%, N w ¼ 256, Δf ¼ 50 Hz.

with a resolution of 60 pulses per revolution is positioned on the bench between the electrical motor and the input of the gearbox. One accelerometer signal is considered with a sample frequency of 25.6 kHz (Fig. 13(b)) and 4 misalignments of the Cardan joint are tested (0°, 4°, 6° and 8°). Those misalignments induce different levels of acyclism. For example, at 2000 rpm, the acyclism of H2 measured at the input of the gearbox is around 250 rad/s² for 4° misalignment, 600 rad/s² for 6° misalignment and 1250 rad/s² for 8° misalignment. 5.2.2. Detection with the order/frequency spectral coherence (OFSCoh) 5.2.2.1. Presentation of the method. The objective in this application is to detect the speed at which the rattle noise appears. Being composed of impacts this noise is characterized by a wide frequency band in Hertz (it is a sparse signal in time domain) and by the angle-periodicity of 4 impacts per revolution of the primary shaft for the periodic case. An orderfrequency tool is then particularly well suited to the detection of rattle noise. The proposed method consists of dividing the accelerometer signal into constant angular records. Consequently, all the portions of the signal have the same number of primary shaft revolutions. As illustrated in Fig. 14(a) the long-term amplitude modulation is negligible and each successive portion of the signal can then be considered AT-CS. In order to deal only with second order AT-CS, the deterministic part is previously removed with synchronous averaging in the angle domain. For each signal portion the OFSCoh is then estimated for the order 4 evt/rev in order to construct a map “speed in rev/min vs. frequency in Hz” (for confidential reason, the presented maps will be here expressed in “signal portions vs. frequency in Hz”). This is possible only if the energy is exactly localized at the order 4 evt/rev. The next subsection will demonstrate the advantage of using the AT-CS algorithm even for a limited speed variation. 5.2.2.2. Preliminary study. Fig. 15 presents the squared spectral coherence estimated with the SCoh and OFSCoh estimators for orders around 4 evt/rev and for a signal portion with a speed variation of 3.2% of the mean value in 2.5 s (Fig. 14(a)). An acyclism is generated with the Cardan joint: the input speed is then composed of the superposition of a linear average speed and a fluctuating speed (Fig. 14(b)). In Fig. 15(a) it is observed that with the CS assumption the energy is dispersed while with the OFSCoh estimator it is exactly localized on the order 4 evt/rev (Fig. 15(b)). This example demonstrates the benefit of using the OFSCoh even when the speed variation is small. 5.2.2.3. Application to rattle detection. To effectively visualize the evolution of the order 4 evt/rev according to the speed, the analysis is here focused only on this order and not on a wider range. A comparison of results for the 4 positions of the Cardan joint is presented. The signals are divided into 30 portions. For each portion the OFSCoh is estimated with Eq. (30) for αθ ¼ 4 and presented on a map “signal portion vs. frequency” (Fig. 16). The signal portion information is directly linked to a mean input speed value. Fig. 16(a) corresponds to a test without acyclism (0° misalignment). This map highlights very weak Please cite this article as: D. Abboud, et al., The spectral analysis of cyclo-non-stationary signals, Mech. Syst. Signal Process. (2015), http://dx.doi.org/10.1016/j.ymssp.2015.09.034i

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18

amplitudes of the order 4 evts/rev. There are only few lines which evolve proportionally to speed and can be attributed to the 4th harmonic of the order 1 evt/rev caused by the eccentricity of the driving gear. From the 4° misalignment (Fig. 16(b)– (d)) the maps highlight wideband areas where the amplitude is higher. These areas depend on the misalignment, in other words on the amplitude of the acyclism, which is characteristic of rattle noise. For example for the 8° misalignment a wideband area is detected from the 7th portion while it is not detected at this speed for the 4° misalignment and with a smaller intensity with the 6° misalignment. For these wideband areas the order 4 evt/rev is then greater than in the remainder of the signal and they thus correspond to the occurrence of the impacts – or at least more intense impacts than elsewhere – generating the audible rattle noise. The interest of such maps is to permit the visualization of the bandwidth of a given order according to the speed. It enables to keep an angle/time duality to characterize the phenomena. In particular for the order 4 evt/rev, the bandwidth enables the dissociation between the components produced by impacts (appearing on a large frequency band) and those produced by the second harmonic of the acyclism (localized in frequency).

6. Conclusion A particular difficulty when analyzing mechanical signals produced by systems undergoing large speed variations is to enlarge the cyclostationary class to a wider one coined “cyclo-non-stationarity”. It was inspected that, in certain mechanical applications, cyclo-non-stationarity evidences a marked interaction between time and angle-dependent components. This interaction is materialized by the angle-periodicity of the correlation measure of two versions of the signals shifted by a constant time-lag, giving birth to a novel class of processes coined ‘angle–time cyclostationarity’. The double Fourier transform of the angle–time covariance function defines an order-frequency spectral distribution of the energy (the OFSC) that jointly localizes angle-periodicity of the modulation and the spectral property of the carrier in Hertz. A first objective of the paper was to set up the foundations of ‘angle–time cyclostationarity’ and its related tools. Although the idea has been partially discussed in a precedent paper, the actual originality is to provide (i) a rigorous definition of the OFSC together with its normalized form (i.e. OFSCoh), (ii) a Welch-based estimator, and (iii) the related statistics. Another objective of this paper is to demonstrate the effectiveness of the proposed theory in the field of rotating machines operating under nonstationary conditions. For this purpose, two very different applications have been considered. The first one is concerned with the diagnosis of rolling element bearing whereas the second one aims at detecting gear rattle noise. Interesting results have been obtained in both cases. In addition, the OFSCoh shows some benefits over the OFSC due to its inherent power normalization that leads to a more accurate representation of the SNR on the one hand and a constant statistical threshold on the other hand. However, the proposed approach assumes that time-dependent components are independent of the operating speed, which may be acceptable for modest speed variations. In practice, from the authors' experience, these components may undergo structural changes as the speed varies more widely, causing substantial changes in the frequency content. Another point not addressed here is the long-term modulation effect which has the disadvantage of destroying the angle–time cyclostationary property of the signal; thus its compensation constitutes an emerging field of investigation. Last, the authors believe that the proposed approach is the first step towards the consideration of more complicated phenomena for systems operating under speed-varying conditions: cyclo-non-stationarity constitutes a raising field of investigation.

Acknowledgment This work was supported by the CETIM and it was supported by PSA Peugeot Citroën within the framework of the OpenLab Vibro-Acoustic-Tribology@Lyon. It was performed within the framework of the Labex CeLyA of University of Lyon, operated by the French National Research Agency (ANR-10-LABX-0060/ANR-11-IDEX-0007).

Appendix A: Proof of Eq. (22) By substituting expression (17) in (21), one obtains   1 S xx αθ ; f ¼ lim

φ-1φ

Z θ1 þ φ Z θ ¼ θ1

þ1

τ ¼ 1


       E x t θ x t θ  τ e  j2πτf e  jαθ θ dτdθ;

ðA1Þ

where the integral is defined over an angular window of length ϕ and the limit is taken to infinity to ensure convergence of the integration. Let W be the time interval corresponding to the angular span ϕ. The key point is to carefully rewrite the

 W interval of integration over τ as t  W where W grows up to infinity after applying the variable change θ ¼ θðt Þ. 2;tþ 2 Please cite this article as: D. Abboud, et al., The spectral analysis of cyclo-non-stationary signals, Mech. Syst. Signal Process. (2015), http://dx.doi.org/10.1016/j.ymssp.2015.09.034i

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Thereby,   S 2x αθ ; f ¼ lim

1 W-1 Φ ðW Þ

Z

W=2 t ¼  W=2

Z

tþW 2

τ ¼ t  W2


 E xðt Þxðt  τÞ e  j2πτf e  jαθ θðtÞ θ_ ðt Þdτdt;

with θ_ ðt Þ ¼ ddtθ. Another change of variable u ¼ t  τ leads to Z W=2 Z þ W=2  
 1 E xðt ÞxðuÞ e  j2πτf e þ j2π uf e  jαθ θðtÞ θ_ ðt Þdudt: S 2x αθ ; f ¼ lim W-1 Φ ðW Þ t ¼  W=2 u ¼  W=2

ðA2Þ

ðA3Þ

Note that the statistical expectation consists of an integral weighted by the probability density function, thus it can be extracted and the integrals rearranged as follows according to Fubini’s theorem, (Z ) Z þ W=2 þ W=2 1 þ j2π uf  j2π tf  jαθ θðt Þ _ E xðuÞe du xðt Þ:e e θðt Þdt ; ðA4Þ S 2x ðν; f Þ ¼ lim W-1 Φ ðW Þ u ¼  W=2 t ¼  W=2 this being the developed form of (22). In the particular case of constant speed θ_ ðt Þ ¼ Ω0 where ΦðW Þ ¼ Ω0 W, expression (A4) boils down to     Ω0 S 2x αθ ; f ¼ S2x αt ¼ αθ ;f ; ðA5Þ 2π which is the spectral correlation.

Appendix B: Construction of the Welch-based estimator Starting from equation (A4), by omitting the limit to infinity of the integration interval, and by replacing the ensemble average operator by a finite average over segments, one obtains  ðLÞ  S^ 2x αθ ; f ¼

1   S  1 N  1

N  1 w w   P P P ΦSw2 Δ ws ðnÞx ðnÞej2π nf Δ Δ ws ðnÞxðnÞe  jαθ θðnÞ e  j2π nf Δ θ_ ðnÞ ; s¼0

n¼0

ðB1Þ

n¼0

with Δ the sampling period. Next, the above expression can be expressed by means of the discrete time Fourier transform (DTFT) as  ðLÞ  S^ 2x αθ ; f ¼

1

SX 1

ΦSjjwjj2 s ¼ 0

n o
 DTFT ws ðnÞxðnÞ DTFT ws ðnÞxðnÞθ_ ðnÞe  jαθ θðnÞ ;

ðB2Þ

which in turn, may be computed by the fast Fourier transform (FFT).

Appendix C: Matlab routine of the OFSC estimator with its related variance reduction factor

function Spec ¼OFSC(y,x,alpha,theta_prime,theta,nfft,Noverlap,Window) % Spec ¼ OFSC (y,x,alpha,theta_prime,theta,nfft,Noverlap,Window) % Welch’s estimate of the (cross) Order-Frequency Spectral Correlation % function (OFSC) of signals y and x at cyclic order alpha: % x and y are divided into K overlapping blocks (Noverlap taps), % each of which is detrended, windowed and zero-padded to length % nfft. Input arguments nfft, Noverlap, and Window are as in % function PSD or PWELCH of Matlab. Denoting by Nwind the window % length, it is recommended to use nfft ¼2*NWind and Noverlap¼ % 2/3*Nwind (hanning window) or Noverlap¼ 1/2*Nwind (halfsine % window). % alpha (scalar) is the desired cyclic order [without unit] % theta (vector) is the shaft angular position [rad] % theta_prime (vector) is the instantaneous angular speed [rad/s] % Note: use analytic signal to avoid correlation between þ and – % frequencies % ——— % Outputs % ————— % Spec is a structure organized as follows: % Spec.S¼ the OFSC vector % Spec.f¼ vector of frequencies % Spec.K¼ number of blocks % Spec.Varduc ¼Variance Reduction factor

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% ———————————————————————— % Ref 1: J. Antoni, "Cyclic SPECTRAL Analysis in Practice", Mechanical % Systems and Signal Processing, Volume 21, Issue 2, February 2007, % Pages 597-630. % Ref 2: The present paper. % ———————————————————————— if length(Window)¼ ¼ 1 Window¼ hanning(Window); end Window¼Window(:); n ¼length(x); % Number of data points nwind¼ length(Window); % length of window if nwind o ¼Noverlap,error(’Window length must be 4 Noverlap’);end y ¼y(:); x ¼x(:); theta¼ theta(:); theta_prime¼theta_prime(:); K ¼fix((n-Noverlap)/(nwind-Noverlap)); % Number of windows index¼ 1:nwind; f ¼(0:nfft-1)/nfft; CPS¼0; y ¼y.*exp(-1i*alpha*theta).* theta_prime; for i ¼1:K xw ¼Window.*x(index); yw ¼Window.*y(index); Yw1¼ fft(yw,nfft); % Yw(f þ a/2) or Yw(f) Xw2¼ fft(xw,nfft); % Xw(f-a/2) or Xw(f-a) CPS¼ Yw1.*conj(Xw2) þ CPS; index¼ index þ (nwind - Noverlap); end PHI¼theta(end)-theta(1); % Angular window length % normalize KMU¼K*norm(Window)\widehat2* PHI; % Normalizing scale factor¼ ¼ 4 asymptotically unbiased CPS¼CPS/KMU; % variance reduction factor Window¼Window(:); Delta¼ nwind - Noverlap; R2w¼xcorr(Window); k ¼nwind þDelta:Delta:min(2*nwind-1,nwind þDelta*(K-1)); if length(k) 41 Var_Reduc¼R2w(nwind)\widehat2/K þ 2/K*(1-(1:length(k))/K)*(R2w(k).\widehat2); else Var_Reduc¼R2w(nwind)\widehat2/K; end % set up output parameters Spec.S¼CPS; Spec.f¼f; Spec.K¼K; Spec.Varduc¼Var_Reduc;

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