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Deterministic-random separation in nonstationary

regime

Dany Abboud, Jérôme Antoni, Sophie Sieg-Zieba, Mario Eltabach

To cite this version:

Dany Abboud, Jérôme Antoni, Sophie Sieg-Zieba, Mario Eltabach. Deterministic-random separation

in nonstationary regime. Journal of Sound and Vibration, Elsevier, 2016, Journal of Sound and

Vibration, 362, pp.305-326. �10.1016/j.jsv.2015.09.029�. �hal-01285390�

Deterministic-random separation in nonstationary regime

D. Abboud

a,b,

n

, J. Antoni

a

, S. Sieg-Zieba

b

, M. Eltabach

b

a

Laboratoire Vibrations Acoustique (LVA), Université de Lyon (INSA), F-69621 Villeurbanne Cedex, France

b

Technical Center of Mechanical Industries (CETIM), CS 80067, 60304 Senlis Cedex, France

article info

Article history:

Received 4 November 2014

Received in revised form

15 September 2015

Accepted 18 September 2015

Handling Editor: K. Shin

Available online 31 October 2015

abstract

In rotating machinery vibration analysis, the synchronous average is perhaps the most

widely used technique for extracting periodic components. Periodic components are

typically related to gear vibrations, misalignments, unbalances, blade rotations, recipro-

cating forces, etc. Their separation from other random components is essential in

vibration-based diagnosis in order to discriminate useful information from masking noise.

However, synchronous averaging theoretically requires the machine to operate under

stationary regime (i.e. the related vibration signals are cyclostationary) and is otherwise

jeopardized by the presence of amplitude and phase modulations. A ﬁrst object of this

paper is to investigate the nature of the nonstationarity induced by the response of a

linear time-invariant system subjected to speed varying excitation. For this purpose, the

concept of a cyclo-non-stationary signal is introduced, which extends the class of

cyclostationary signals to speed-varying regimes. Next, a “generalized synchronous

average’’ is designed to extract the deterministic part of a cyclo-non-stationary vibration

signal—i.e. the analog of the periodic part of a cyclostationary signal. Two estimators of

the GSA have been proposed. The ﬁrst one returns the synchronous average of the signal

at predeﬁned discrete operating speeds. A brief statistical study of it is performed, aiming

to provide the user with conﬁdence intervals that reﬂect the "quality" of the estimator

according to the SNR and the estimated speed. The second estimator returns a smoothed

version of the former by enforcing continuity over the speed axis. It helps to reconstruct

the deterministic component by tracking a speciﬁc trajectory dictated by the speed proﬁle

(assumed to be known a priori).The proposed method is validated ﬁrst on synthetic sig-

nals and then on actual industrial signals. The usefulness of the approach is demonstrated

on envelope-based diagnosis of bearings in variable-speed operation.

& 2015 Elsevier Ltd. All rights reserved.

1. Introduction

In most mechanical systems, gears and rolling-element bearings constitute the core elements of the power transmission

chain. Therefore, their diagnosis is critical to ensure the security of the equipment and avoid possible breakdowns in the

Contents lists available at ScienceDirect

journal home page: www.elsevier.com/locate/jsvi

Journal of Sound and Vibration

http://dx.doi.org/10.1016/j.jsv.2015.09.029

0022-460X/& 2015 Elsevier Ltd. All rights reserved.

Abbreviations: CS, cyclostationary; CS1, ﬁrst-order cyclostationary; CS2, second-order cyclostationary; SA, synchronous average; LTI, linear time-

invariant; CNS, cyclo-non-stationary; DRS, deterministic random separation; GSA, generalized synchronous average; dof, degree-of-freedom; KDE, kernel

density estimation; SES, squared envelope spectrum

n

Corresponding author.

E-mail addresses: dany.abboud@insa-lyon.fr, dany.abboud@cetim.fr, d-abboud@live.com (D. Abboud), Jerome.antoni@insa-lyon.fr (J. Antoni),

sophie.sieg-zieba@cetim.fr (S. Sieg-Zieba), mario.eltabach@cetim.fr (M. Eltabach).

Journal of Sound and Vibration 362 (2016) 305–326

system. Interestingly, it was shown that these components—when operating under constant regime—produce symptomatic

cyclostationary (CS) vibrations that can be identiﬁed and characterized by dedicated tools [1]. Speciﬁcally, gears are likely to

generate ﬁrst-order cyclostationary (CS1) vibrations characterized by a periodic mean, whereas rolling-element bearings

generate second-order cyclostationary (CS2) vibrations characterized by a periodic autocovariance function. The former are

deterministic in nature, whilst the latter are random. Their separation is of high practical importance for differential

diagnosis [2–4].

In this context, the synchronous average (SA) is a powerful tool for extracting periodic components [5,6]. Based on prior

knowledge of the desired component, it consists of segmenting the temporal signal into blocks of length equal to the signal

period and averaging them together to extract the periodic waveform. The “residual signal”—which constitutes the random

part—is then obtained by subtracting the estimated periodic part from the original signal. The principle of the SA assumes

the periodic waveform to be stable in time, which in turn requires a constant speed. However, in practice, such a condition is

difﬁcult to obtain: the operating speed often undergoes some ﬂuctuations which are likely to jeopardize the effectiveness of

the SA, even if very low. Since repetitive patterns in rotating machines are intrinsically locked to speciﬁc angular positions, it

thus makes sense to synchronize the SA with respect to angle rather than to time. This problem has found several practical

solutions such as angular sampling or resampling (also known as “computed order tracking” [7]): whereas the former

directly acquires the signal at constant angular intervals, the latter uses a tachometer to resample the time signal at constant

angular increments by interpolation [8,9]. In this case, the cyclostationary property holds in the angle domain and, con-

sequently, the SA is applied on the (resampled) signal. Incidentally, this facilitates the averaging operation since each

angular period then comprises the same (whole) number of angular increments. Nevertheless, the change from a temporal

to an angular variable raises issues about the actual nature of cyclostationarity of rotating machine signals. This was for

instance addressed in Ref. [10] where the authors explored the concept of angle-CS signals and provided the conditions of its

equivalence with the time-CS.

In the case of high speed ﬂuctuations, signals are subjected to signiﬁcant distortions that jeopardize the effectiveness of

the SA. These distortions are basically introduced by (i) variations of the machine power intake and (ii) the effect of linear

time-invariant (LTI) transfers. Whereas the ﬁrst effect essentially results in amplitude modulation, the second one also

induces phase modulation. The latter phenomenon was originally inspected in Ref. [11], before being revisited in Ref. [12]

through a comprehensive theoretical analysis. Non-periodic modulations obviously invalidate the (angle-) CS assumption

and call for a more general description of nonstationary signals. In accordance, “cyclo-non-stationarity” (CNS), a new class

recently introduced in Ref. [13], appears to be a good candidate to describe speed-varying signals: it enfolds highly non-

stationary signals undergoing long-term structural changes in their properties, yet preserving at the same time short-term

cyclic rhythms locked to the angular evolution.

The consideration of CNS signals requires the revision of existing CS tools and in particular of the SA. In order to

accommodate the SA with the phase blur resulting from the transmission path effect, Stander and Heyns [11] proposed an

enhanced version coined “ phase domain averaging”—comprising a phase correction of the cycles before the averaging

operation. The phase information was returned by Hilbert demodulation of a high-energy harmonic (such as a meshing

order) of the resampled acceleration signal. Later, a simpler variant was provided in Ref. [14], coined the

“improved syn-

chronous average”, which consists in resampling the signal with a virtual tachometer signal synthesized via the demodu-

lated phase. This work was inspired from Ref. [9] which introduced a technique to perform angular resampling using the

acceleration signal of a gearbox operating under limited speed ﬂuctuation.

Recently, the same technique was used in Ref. [12] to identify the optimal demodulation band for deterministic/random

separation (DRS) in speed varying conditions. Moreover, the authors in Ref. [15] provided a parametric approach in an

attempt to generalize the SA to the CNS case. Using Hilbert space representation, they decomposed the deterministic

components on to a set of periodic functions multiplied by speed-dependent functions apt to capture long-term evolution

over consecutive cycles. Yet, their method carries the general disadvantages of parametric approaches, namely, the critical

dependence on the basis order. Another attempt was made recently in Ref. [13] where the aim was to remove the deter-

ministic part of the vibrations produced by an internal combustion engine in runup regimes. Assuming a ﬁrst-order Markov

Nomenclature

card

Κ

r

number of cycles belonging to regime r

Κ

r

set of cycle indices belonging to regime r

m

Y

ðθÞ SA of Y

m

Y

ðθ; ωÞ GSA of Y

m

Y

θ

GSA trajectory tracked for a given speed pro-

ﬁle

ω ¼ ωθ

b

m

Y

ðθ; ω

r

Þraw estimator of the GSA of Y at ω

r

b

m

Y

ðθ; ωÞ smoothed estimator of the GSA of Y

P

Y

ðθ; ω

r

Þ mean instantaneous power of Y at ω

r

t time variable

δω speed resolution

Δω speed varying margin

E fg ensemble averaging operator

E jAfg ensemble average operator conditioned to

event A

θ angle variable

θ angular location in the cycle: θϵ 0; Θ

Θ angular period

ω

r

central angular frequency at regime r

D. Abboud et al. / Journal of Sound and Vibration 362 (2016) 305–326306

dependence of the CNS signal, the authors introduced an operator—coined the “cyclic difference”—based on subtracting

each cycle from the previous one in the resampled signal. Despite the good compliance of this method for the particular

addressed application, it suffers from high variability in the more general setting.

The principal object of this paper is to provide a novel operator, coined the “generalized synchronous average” (GSA),

which happens to be an extension of the classical SA to the CNS case, with the aim of enhancing DRS in variable regime. It is

assumed that the rotational speed is precisely known; its estimation is beyond the scope of this paper and may be found in

references such as [16–25]. This paper is organized as follows: Section 2 states the problem with a particular attention to

formulating the transmission path effect from a CNS view. Section 3 introduces the GSA operator, two dedicated estimators

of it and a statistical study in order to evaluate the estimator quality according to the SNR and the estimated speed. Section 4

validates the effectiveness of the GSA on numerical signals and Section 5 on actual industrial signals. Eventually, in light of

the obtained results, the paper is sealed with a general conclusion in Section 6.

2. Problem statement

This section starts with a brief review of the SA. Then, the transmission path effect is qualitatively addressed with a link

made with CNS.

2.1. Synchronous average: a brief review

The synchronous average (SA)—widely termed as the time synchronous average [26]—is an efﬁcient technique for the

extraction of periodic waveforms from a noisy signal. It consists of averaging periodic sections of the signal —known as cycles

—assuming a priori knowledge of the period. Under the assumption of cycloergodicity [10], the SA is perhaps the best

candidate for estimating the mean function of a CS signals. In practice, it is however jeopardized when applied in the time-

domain due to existing speed ﬂuctuations or if the signal period is not a multiple of the sampling period. In such cases,

angular resampling can be a simple preprocessing solution consisting of expressing the signal in terms of the angular

variable

θ

of the machine instead of time t. Precisely, let Y θ

be an angle-CS signal of cycle

Θ

(for instance

Θ

¼2

π

); its SA is

then deﬁned as

m

Y

θ

¼ lim

M-1

1

2M þ1

X

M

m ¼M

Y θþmΘ

(1)

where 2M þ 1ðÞstands for the number of averaged cycles and

θA 0; Θ

for the angular location in the cycle (i.e.

θ ¼ θ kΘ

with k the greatest integer smaller than θ=Θ). The SA is widely used in vibration analysis of rotating machinery to separate

deterministic waveforms (such as produced by imbalances, misalignments, anisotropic rotors, ﬂexible coupling, gear

meshing and other phenomena) from other competing but random sources.

2.2. Transmission path effect

A primary goal in machine diagnostics is to infer the cause of abnormal vibrations. In most cases, accelerometers are

remoted from the sources due to non-intrusion constraints and there exists a transmission path effect characterized by the

transfer function of the structure. In cases where the sensor is intentionally used outside its nominal bandwidth (e.g. as a

shock sensor), its transfer function should be considered as well. Consequently, the measured signal is to be interpreted as

the output of an LTI system. This system is characterized by ordinary differential equations in time, independently of the

excitation nature. As a result, the response of the system to a complex exponential temporal waveform is kept unchanged

except for a phase shift and an amplitude ampliﬁcation characterized by the system transfer function. In this case, the

transmission path induces maximum ampliﬁcation and phase shift when the frequency content of the excitation coincides

with a resonance of the transfer function. This also holds true in general for time-CS signals: a LTI system excited by a time-

CS signal returns a time-CS output. However, the situation is not the same for angle-CS signals. In the case of large speed

variations, the corresponding response is instantaneously delayed and ampliﬁed/attenuated [11] according to the ﬂuctuating

amplitude and frequency content of the input signal. From a temporal view, the system response is the convolution of a

nonstationary excitation with a LTI transfer function: it is therefore nonstationary as well. Conversely, from an angular point

of view the order content of the excitation remains constant, yet the transfer function becomes angle-varying; therefore, the

response of the system to an angle-CS excitation is CNS in general, and angle-CS in the particular case of periodic, stationary

or cyclostationary operating speed [10] (see Fig. 1 ).

The present paper is mainly concerned with angle-CS1 excitation and the related transmission path effect on gear

vibrations which jeopardizes the (traditional) SA. The issue can be explained from different but equivalent points of view.

First, according to the previous discussion, the system response is no longer angle-CS1, which invalidates the working

assumption of the SA [12]. Alternatively, from an angular point of view, the averaging process over multiple shaft rotations

usually results in an energy loss principally caused by the induced phase blur. Finally, from the order domain point of view,

D. Abboud et al. / Journal of Sound and Vibration 362 (2016) 305–326 307

the SA is equivalent to applying a narrow-band comb ﬁlter [27]; yet, the amplitude distortion and the phase blur results in

energy leakage outside the lobes of the comb ﬁ lter.

In summary, when rotating machines undergo large speed variations, the resulting vibration signals lose their angle-CS

properties; thus, invalidates the application of traditional CS tools such as the SA. The aim of the next section is to char-

acterize such signals which preserve repetitive patterns related to the cyclic behavior of the excitation, yet with angle

(or time) varying properties.

2.3. Speed dependence

This section introduces the basic model that will serve to deﬁne the GSA, which consists of Fourier series whose complex

exponentials are function of the angle variable,

θ

, and the coefﬁcients are only dependent on the speed, ω.

To avoid confusions between angle and time domains, the accent mark “tilde” will refer to the temporal representation of

the signal. Building on the discussion of Section 2.2, a CNS process can be seen as the response

~

Yt

ðÞ

¼ Yt

θ

of a LTI system,

say

~

htðÞ, to an angle-CS excitation

~

XtðÞ¼X

θ tðÞ

¼

X

α

c

α

X

θ tðÞ

e

j2

πα

θ

tðÞ

Θ

; (2)

where

Θ stands for the angular period of X θ tðÞ

, c

α

X

θ

are mutually (angle-) stationary Fourier coefﬁcients and

θ t

ðÞ

¼

R

t

0

ω t

ðÞ

dt is the angular position of the reference. In short

~

YtðÞ¼

~

htðÞ

~

XtðÞ; (3)

where stands for the convolution product. In the particular case of constant operating speed (i.e.

θ t

ðÞ

¼ ω

0

t) the response

takes the particular form

Y

θ

¼

X

α

c

α

Y

θ; ω

0

e

j2

πα

θ

tðÞ

Θ

¼

X

α

c

α

X

θ

A

αω

0

=2

e

jk

Φαω

0

=2

ðÞ

e

j2

πα

θ

Θ

; αA Z; (4)

where AfðÞe

jk

Φ

fðÞ

¼ HfðÞis the system frequency response function. Actually, Eq. (4) reﬂects the pivotal role of the speed not

only on the amplitude, but also on the phase of the system response.

In the case of speed varying excitation, the response would turn into a convolution between an angle-CS excitation and

an LTI system, which is less straightforward than Eq. (4) because the input signal would no longer be time-CS in general. One

alternative would be to express the convolution directly in the angular domain, i.e. symbolically

Y

θ

¼ h

θ

Y

θ

; (5)

where h

θ

¼

~

ht

θ

would then become angle-varying owing to the nonlinearity of the angle-time relationship. In this

case, the Fourier coefﬁcients of the response at a given instant are principally dependent on the operating speed at that

instant, as well as past and future instances of the speed proﬁle. This implies that, beside the operating speed, the Fourier

coefﬁcients of the response are also dependent on its higher derivatives at the evaluated instant, i.e.

~

YtðÞ¼

X

α

c

α

Y

θ tðÞ; ω tðÞ;

_

ω tðÞ;

€

ω tðÞ;

:::

ω tðÞ…

e

j2

πα

θ

tðÞ

Θ

: (6)

Under mild conditions, the dependence on higher derivative orders can be neglected and the variations of both

amplitude and phase are consequently modeled with an explicit dependence on the speed only:

~

YtðÞ¼

X

α

c

α

Y

θ tðÞ; ω tðÞ

e

j2

πα

θ

tðÞ

Θ

; (7)

where c

α

Y

θ; ω

stands for the complex Fourier coefﬁcients having a speed-dependent joint probability density function—i.e.

mutually stationary stochastic process for a constant speed

ω. The angular representation of a CNS signal can be deduced as

Y

θ

¼

X

α

c

α

Y

θ;

~

ωθ

e

j2

πα

θ

Θ

; (8)

with

~

ωθ

¼

ω t θ

. Note that Eq. (7) is perfectly valid for runup regimes (where higher-order derivatives of the speed are

nil) and approximately valid in the case of modest accelerations (i.e. smooth variability of the speed proﬁle). This model

accounts for changes in the signal structure by introducing a dependency of the Fourier coefﬁcient on speed, while the cyclic

LTI

Time-CS

(stationary speed case)

Time-CS

LTI

Angle-CS

(varying speed case)

Angle-CS if the speed is periodic,

stationary or CS

CNS elsewhere

Fig. 1. Transmission path effect on cyclostationary excitations.

D. Abboud et al. / Journal of Sound and Vibration 362 (2016) 305–326308