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Journal ArticleDOI

Deterministic-random separation in nonstationary regime

03 Feb 2016-Journal of Sound and Vibration (Academic Press)-Vol. 362, pp 305-326

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, reciprocating 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 signals and then on actual industrial signals. The usefulness of the approach is demonstrated on envelope-based diagnosis of bearings in variable-speed operation.
Topics: Cyclostationary process (62%), Vibration (52%), Estimator (51%)

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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 rst 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
signali.e. the analog of the periodic part of a cyclostationary signal. Two estimators of
the GSA have been proposed. The rst one returns the synchronous average of the signal
at predened discrete operating speeds. A brief statistical study of it is performed, aiming
to provide the user with condence intervals that reect 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 specic trajectory dictated by the speed prole
(assumed to be known a priori).The proposed method is validated rst 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, rst-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) 305326

system. Interestingly, it was shown that these componentswhen operating under constant regimeproduce symptomatic
cyclostationary (CS) vibrations that can be identied and characterized by dedicated tools [1]. Specically, gears are likely to
generate rst-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 [24].
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
partis 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
difcult to obtain: the operating speed often undergoes some uctuations which are likely to jeopardize the effectiveness of
the SA, even if very low. Since repetitive patterns in rotating machines are intrinsically locked to specic 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 uctuations, signals are subjected to signicant 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 rst 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 uctuation.
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 rst-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-
le
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) 305326306

dependence of the CNS signal, the authors introduced an operatorcoined 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 [1625]. 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 efcient 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 uctuations 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 dened as
m
Y
θ

¼ lim
M-1
1
2M þ1
X
M
m ¼M
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, exible 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 amplication characterized by the system transfer function. In this case, the
transmission path induces maximum amplication 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 amplied/attenuated [11] according to the uctuating
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) 305326 307

the SA is equivalent to applying a narrow-band comb lter [27]; yet, the amplitude distortion and the phase blur results in
energy leakage outside the lobes of the comb 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 dene the GSA, which consists of Fourier series whose complex
exponentials are function of the angle variable,
θ
, and the coefcients 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 coefcients 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) reects 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 coefcients 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 prole. This implies that, beside the operating speed, the Fourier
coefcients 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 coefcients having a speed-dependent joint probability density functioni.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 prole). This model
accounts for changes in the signal structure by introducing a dependency of the Fourier coefcient 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) 305326308

Figures (20)
Citations
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Abstract: : Given a sequence of independent identically distributed random variables with a common probability density function, the problem of the estimation of a probability density function and of determining the mode of a probability function are discussed. Only estimates which are consistent and asymptotically normal are constructed. (Author)

9,261 citations


Journal ArticleDOI
Abstract: This note discusses some aspects of the estimation of the density function of a univariate probability distribution. All estimates of the density function satisfying relatively mild conditions are shown to be biased. The asymptotic mean square error of a particular class of estimates is evaluated.

3,792 citations


Journal ArticleDOI
Robert B. Randall1, Jérôme Antoni2Institutions (2)
TL;DR: This tutorial is intended to guide the reader in the diagnostic analysis of acceleration signals from rolling element bearings, in particular in the presence of strong masking signals from other machine components such as gears.
Abstract: This tutorial is intended to guide the reader in the diagnostic analysis of acceleration signals from rolling element bearings, in particular in the presence of strong masking signals from other machine components such as gears. Rather than being a review of all the current literature on bearing diagnostics, its purpose is to explain the background for a very powerful procedure which is successful in the majority of cases. The latter contention is illustrated by the application to a number of very different case histories, from very low speed to very high speed machines. The specific characteristics of rolling element bearing signals are explained in great detail, in particular the fact that they are not periodic, but stochastic, a fact which allows them to be separated from deterministic signals such as from gears. They can be modelled as cyclostationary for some purposes, but are in fact not strictly cyclostationary (at least for localised defects) so the term pseudo-cyclostationary has been coined. An appendix on cyclostationarity is included. A number of techniques are described for the separation, of which the discrete/random separation (DRS) method is usually most efficient. This sometimes requires the effects of small speed fluctuations to be removed in advance, which can be achieved by order tracking, and so this topic is also amplified in an appendix. Signals from localised faults in bearings are impulsive, at least at the source, so techniques are described to identify the frequency bands in which this impulsivity is most marked, using spectral kurtosis. For very high speed bearings, the impulse responses elicited by the sharp impacts in the bearings may have a comparable length to their separation, and the minimum entropy deconvolution technique may be found useful to remove the smearing effects of the (unknown) transmission path. The final diagnosis is based on “envelope analysis” of the optimally filtered signal, but despite the fact that this technique has been used for 40 years in analogue form, the advantages of more recent digital implementations are explained.

1,485 citations


Journal ArticleDOI
Jérôme Antoni1Institutions (1)
Abstract: This paper is a tutorial on cyclostationarity oriented towards mechanical applications. The approach is voluntarily intuitive and accessible to neophytes. It thrives on 20 examples devoted to illustrating key concepts on actual mechanical signals and demonstrating how cyclostationarity can be taken advantage of in machine diagnostics, identification of mechanical systems and separation of mechanical sources.

436 citations


Journal ArticleDOI
K.R. Fyfe1, E.D.S. Munck1Institutions (1)
Abstract: Vibration analysis of rotating machinery is an important part of industrial predictive maintenance programmes, so that wear and defects in moving parts can be discovered and repaired before the machine breaks down, thus reducing operating and maintenance costs. One method of vibration analysis is known as order tracking. This is a frequency analysis method that uses multiples of the running speed (orders), instead of absolute frequencies (Hz), as the frequency base. Order tracking is useful for machine condition monitoring because it can easily identify speed-related vibrations such as shaft defects and bearing wear. To use order tracking analysis, the vibration signal must be sampled at constant increments of shaft angle. Conventional order tracking data acquisition uses special analog hardware to sample at a rate proportional to the shaft speed. A computed order tracking method samples at a constant rate (i.e. uniform Δt), and then uses software to resample the data at constant angular increments. This study examines which factors and assumptions, inherent in this computed order tracking method, have the greatest effect on its accuracy. Both classical and computed methods were evaluated and compared using a digital simulation. It was found that the method is extremely sensitive to the timing accuracy of the keyphasor pulses and that great improvements in the spectral accuracy were observed when making use of higher-order interpolation functions.

394 citations


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