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Mechanical Systems and Signal Processing

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

0888-32

doi:10.1

$ Pat� Cor

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journal homepage: www.elsevier.com/locate/jnlabr/ymssp

An energy operator approach to joint application of amplitude andfrequency-demodulations for bearing fault detection$

Ming Liang �, I. Soltani Bozchalooi

Department of Mechanical Engineering, University of Ottawa, 770 King Edward Avenue, Ottawa, Ontario, Canada, K1N 6N5

a r t i c l e i n f o

Article history:

Received 15 December 2008

Received in revised form

18 May 2009

Accepted 22 December 2009

Keywords:

Energy operator

Bearing fault detection

Frequency demodulation

Amplitude demodulation

High-frequency resonance

70/$ - see front matter & 2010 Elsevier Ltd. A

016/j.ymssp.2009.12.007

ent pending (Serial number 61/119,954).

responding author. Tel.: +1 613 562 5800x626

ail addresses: [email protected] (M. Liang

e cite this article as: M. Liang, I. Soltani Bodulations for bearing fault detection, M

a b s t r a c t

Bearings are among the most frequently used components. Bearing failure could lead to

complete stall of a mechanical system, unpredicted productivity loss for production

facilities or catastrophic consequence for mission-critical equipment. As such, bearing

fault detection and diagnosis is an imperative part of most of preventive maintenance

procedures. This paper presents a parameter independent yet simple to implement fault

detection technique. The Teager energy operator is tailored to extract both the

amplitude and frequency modulations of the vibration signals measured from

mechanical systems. The incorporation of the frequency modulation information into

the proposed bearing fault detection method has eliminated the need for interference

removal steps. As the amplitude demodulation (AD) is also inherent in the energy

operator, the fault frequency can be detected from the spectrum of the energy-

transformed signal. The effectiveness of the proposed method has been validated using

both simulated and experimental data.

& 2010 Elsevier Ltd. All rights reserved.

1. Introduction

The operation of most mechanical systems is based on the transformation of the angular motions to other kinds ofmovement. Bearings play a crucial role in such a process and can be considered as the most commonly used mechanicalcomponents. As their name suggests, they bear much of the load. This together with their harsh operating environmentmakes them one of the most failure-prone machinery elements. It is more so for rolling-element bearings due to theirstructural delicacy.

The failures of rolling-element bearings are often caused by surface faults such as cracks or spalls on roller, inner race orouter race. When these faults come in contact with the mating surfaces during the operation of a bearing, they generatemechanical impulses. These impulses excite a resonance in the entire system including the bearing, the sensor and thestructure where the bearing is mounted. Such a high frequency resonance is damped out quickly due to the structuraldamping characteristics of the system. This process is repeated periodically due to the recurrence of the contact betweenthe fault and the mating surface. As such, the vibrations measured from a faulty bearing can be modeled as a train ofimpulse-excited signatures with damped oscillations (hereafter called fault impulses or simply impulses) (Fig. 1(c)). Thefrequency of repetition of these impulses is related to the geometry of the bearing and the rotational frequency of the shaft,known as fault characteristic frequency [1]. As such, extraction of the frequency components associated with the faultcharacteristic frequency and harmonics would provide us with rich information on bearing conditions.

ll rights reserved.

9; fax: +1 613 562 5177.

), [email protected] (I. Soltani Bozchalooi).

zchalooi, An energy operator approach to joint application of amplitude and frequency-echanical Systems and Signal Processing (2010), doi:10.1016/j.ymssp.2009.12.007

www.elsevier.com/locate/jnlabr/ymssp

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mailto:[email protected]

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Nomenclature

a(t) mathematical model of the vibration of afaulty bearing (acceleration)

A(t) analytic version of a(t)ADM( ) amplitude-demodulated version of a signalAnalytic( ) analytic version of a signalFDM( ) frequency-demodulated version of a signalL amplitude of fault impulser(t) a mixture of fault-generated impulse and

vibration interferencer_ðtÞ transformed version of r(t) or r

_ðtÞ ¼ TrðrðtÞÞ

R(t) analytic version of r(t)s(t) mathematical model of a single fault-gener-

ated impulse (displacement)SIR( ) signal-to interference ratio of a signalSIRADM( )signal-to interference ratio of a signal only

taking into account the transient amplitude-demodulated terms

SIRHF( ) signal-to interference ratio of a signal onlytaking into account the transient high fre-quency terms

Tp time period corresponding to fault character-istic frequency 1/Tp

Tr( ) transformed version of a signal by transform TrTsettling settling time of a damped vibratory systemu(t) unit step functionv(t) vibration interferenceV(t) analytic version of v(t)a amplitude of the vibration interferenceb structural damping characteristicj(t) instantaneous phasetc time constant of an exponentially decaying

vibratory system, equal to the inverse of bti the ith realization of a zero mean, uniformly

distributed random variable, with standarddeviation of 0.01Tp�0.02Tp reflecting the rollerslippage

oIF(t) instantaneous frequencyom the frequency of the interference signalor the excited resonance frequencyC() energy operator

M. Liang, I. Soltani Bozchalooi / Mechanical Systems and Signal Processing ] (]]]]) ]]]–]]]2

However, the vibrations measured from a mechanical system contain not only the fault-related vibrations but also thebackground noise present in any measurement device as well as the vibrations generated by other mechanical componentssuch as gear meshing, shaft imbalance or misalignment. Such noise and interference components would inevitablycomplicate fault detection processes.

0 0.5 1 1.5 20

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Frequency (Hz)

Mag

nitu

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0 0.5 1 1.5 20

0.005

0.01

0.015

Frequency (Hz)

Mag

nitu

de

High Energy Resonance Frequency Band

0 0.5 1 1.5 20

0.005

0.01

0.015

0.02

0.025

0.03

Frequency (Hz)

Mag

nitu

de

0 2000 4000 6000 8000 10000

-2

-1

0

1

2

Sample

Am

plitu

de

x 104 x 104

x 104

Fig. 1. (a) Spectrum of the faulty bearing vibration (displacement) with ringing frequency of 10 000 Hz and fault characteristic frequency of 87.52 Hz,

simulated at 40 KHz sampling rate (no slippage), (b) Spectrum of the acceleration version of the simulated signal (no slippage), (c) a portion of simulated

faulty bearing vibration (acceleration) incorporating roller slippage, (d) spectrum of the signal of part (c).

Please cite this article as: M. Liang, I. Soltani Bozchalooi, An energy operator approach to joint application of amplitude and frequency-demodulations for bearing fault detection, Mechanical Systems and Signal Processing (2010), doi:10.1016/j.ymssp.2009.12.007

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M. Liang, I. Soltani Bozchalooi / Mechanical Systems and Signal Processing ] (]]]]) ]]]–]]] 3

Furthermore, due to roller slippage [2,3], the repetition period of individual impulses may vary slightly. The variation inthe interval between consecutive impulses, represented by independent random values [4], causes smearing of the spectralenergy of the fault signal, making spectral analysis more susceptible to noise and interferences. This is especially vital tothe quality of detection decisions when these faults are at early stages with faint footprints.

To tackle the above difficulties several signal processing techniques have been proposed. In one approach, bandpassfiltering around the excited resonance frequency followed by an amplitude demodulation (AD) step results in a signal withmost of its energy concentrated around the fault characteristic frequency of the bearing and the associated harmonics.Spectral analysis of this signal, i.e., examination of the energy of the frequency components related to the bearing faultcharacteristic frequency and its harmonics, can confirm if a fault exists. This approach known as the high frequencyresonance (HFR) technique is a popular method in the industry [5–11].

However, the major challenge in the application of the HFR technique is the proper selection of the center frequencyand bandwidth of the bandpass filter. Many researches have focused on the development of efficient and robust methodsof estimating the proper center frequency and optimum bandwidth of the bandpass filter. Spectral kurtosis and kurtogram,proposed by Antoni and Randall for machinery fault detection and diagnosis are examples of such efforts [12,13]. Also, Qiuet al. [14,15] used the minimal Shannon entropy to select the proper bandwidth and a periodicity detection method todetermine center frequency. In another research [16] Lin et al. selected the parameters of Morlet wavelet based on thekurtosis maximization criterion. Bozchalooi and Liang proposed a smoothness index guided approach to find the best scaleand shape-factor of Gabor wavelet corresponding to the center frequency and bandwidth of a Gaussian filter [17]. In [18],the same authors developed a resonance frequency estimation algorithm based on the variations of the shaft rotationalfrequency. The estimated resonance was used as the center frequency of a wavelet-based filtering method.

In addition to the above, Self Adaptive Noise Cancellation (SANC) technique [2,19–21] was successfully used to removethe vibration interferences generated by other sources such as gear meshing, shaft imbalance, etc. A variant of SANC,Discrete Random Separation (DRS), was also proposed to separate the faulty bearing impulsive vibrations from other signalcomponents. DRS could achieve results similar to those obtained by SANC, but with considerably less computational effort[20]. For such techniques to perform successfully, parameters such as filter length and delay duration should be selectedproperly. Cyclostationarity characteristic of the bearing vibrations due to random slippage of the rollers was also efficientlyused to detect the existence of bearing faults in the presence of strong quasi-periodic interferences generated by othervibration sources [3,4].

This paper presents a joint amplitude and frequency demodulation approach to bearing fault detection. The informationcontained in the amplitude and frequency modulations are simultaneously derived by a simple mathematicaltransformation step. The energy operator used for the frequency and amplitude demodulation of the speech signals[22–27] is adopted for this purpose. The advantages of this method include:

(a)

Plde

It is parameter free and thus relieves the burden involved in selecting and re-calibrating parameters.
(b) It incorporates both the amplitude and frequency modulation information thereby potentially leading to more reliable
detection results.
(c) It eliminates the enveloping step due to the inclusion of the amplitude demodulation. (d) It boosts the signal-to-interference ratio and thus could enhance detection performance. (e) It does not rely on prior knowledge about the excited resonance of the monitored system. (f) It requires a simple transformation and is easy to implement. (g) It provides excellent time resolution and hence is responsive to fault onset in high speed rotating machinery. (h) The parameter-free nature makes the method more versatile.
The above advantages have been demonstrated by applying the proposed method to experimental data measured fromfaulty bearings. The applicability of the proposed technique is based on the same assumptions as those applied to the HFRtechnique [5], i.e.: (1) the frequency of the fault-excited resonance is higher than the frequency of the vibrationinterferences [5], and (2) a reasonable level of periodicity can be attributed to the fault-generated impulses so that thedetection of the periodic signal component associated with the fault characteristic frequencies or harmonics can be reliablytreated as a fault feature. It should be pointed out that the proposed method is directly based on the energy operatorinstead of direct frequency and amplitude-demodulations.

The paper hereafter is organized as follows: Section 2 explains the roles of amplitude demodulation (AD) and frequencydemodulation (FD) in bearing fault detection. Section 3 introduces the energy operator and illustrates its application incondition monitoring. The experimental results are presented in Section 4. Section 5 concludes the paper.

2. Amplitude/frequency demodulation

In the following subsection, the behaviour of AD in the absence of any interference removal, bandpass filtering or de-noising scheme is investigated. The effect of background noise and vibration interferences on AD will be investigatedseparately.

ease cite this article as: M. Liang, I. Soltani Bozchalooi, An energy operator approach to joint application of amplitude and frequency-modulations for bearing fault detection, Mechanical Systems and Signal Processing (2010), doi:10.1016/j.ymssp.2009.12.007

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2.1. Amplitude demodulation

In the simplest form, the vibrations of a faulty bearing in response to an induced fault impulse may be modeled as theimpulse response of a single degree of freedom mass-spring-damper [21]:

sðtÞ ¼ Ae�bt sinorðtÞuðtÞ ð1aÞ

Hence, the vibrations (displacement) of a faulty bearing with fault characteristic frequency of 1/Tp can be modeled as:

xðtÞ ¼XM

m ¼ �M

smðt�mTp�Xm

i ¼ �M

tiÞ ¼XM

m ¼ �M

Ame�bðt�mTp�Pm

i ¼ �MtiÞ sinorðt�mTp�

Xmi ¼ �M

tiÞuðt�mTp�Xm

i ¼ �M

tiÞ ð1bÞ

where Am is the amplitude of the mth fault impulse, u(t) a unit step function, Tp the time period corresponding to the faultcharacteristic frequency, b the structural damping characteristic, or the excited resonance frequency, ti represents theeffect of random slippage of the rollers and is the ith realization of a zero mean, uniformly distributed random variable,with standard deviation of 0.01Tp�0.02Tp and x(t) is the vibration signal containing 2M+1 fault generated impulses, s(t).

As such vibrations are often measured using an accelerometer, the measured vibration signal can be expressed in aformat of acceleration, i.e.,

s00ðtÞ ¼d2sðtÞ

dt2¼ L cosðortþyÞe�btuðtÞ ð1cÞ

where L¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiA2ðb2

�o2r Þ

2þ4A2b2o2

r

qand y¼ p� tanððAðb2

�o2r ÞÞ=ð�2AborÞÞ.

Eqs. (1b) and (1c) lead to

aðtÞ ¼XM

m ¼ �M

Lme�bðt�mTp�Pm

i ¼ �MtiÞ cos

�orðt�mTp�

Xmi ¼ �M

tiÞþym

�uðt�mTp�

Xm

i ¼ �M

tiÞ ð1dÞ

According to the above, the introduced phase parameter y accounts for the difference between the displacement andacceleration models. This difference can be better noted in frequency domain where [21,28]:

sðoÞ ¼ A

2

1

bþ jðo�orÞþ

1

bþ jðoþorÞ

� �

and,

s00 ðoÞ ¼ �Ao2

2

1

bþ jðo�orÞþ

1

bþ jðoþorÞ

� �ð2Þ

where sðoÞ and s00 ðoÞ are the Fourier transforms of s(t) and s00(t), respectively.Figs. 1(a) and (b) illustrate the spectrums of simulated faulty bearing vibrations in terms of displacement and

acceleration respectively, without consideration of amplitude modulations (e.g. for inner race fault) or slippage. Due to theimpulsive nature of fault vibration signals, the frequency content of the vibrations measured from faulty bearings spreadsover a wide frequency band as shown in these figures. As a result, the fault characteristic frequency and its harmonics canbe easily masked by noise and interferences.

It is well known that the resonance excitation phenomenon forms a high-energy frequency band. This is also observedin Figs. 1(a) and (b) where the high energy frequency band concentrates around the excited resonance frequency or

(10 KHz in this example). As can be seen from these figures, the power spectral density of the signal at low frequencies,especially near the fault characteristic frequency, is considerably lower than that in the resonance frequency region. Thisphenomenon is more pronounced in the spectrum of the acceleration signal which can be expected from the multiplyingfactor o2 present in Eq. (2). As a result, the fault feature represented by the characteristic frequency and first fewharmonics is more vulnerable to noise and interferences compared to the spectral content of the resonance frequencyband.

In addition, roller slippage leads to random deviation of the time interval between the consecutive fault impulses fromthe theoretical fault characteristic period and causes the spectrum of the fault signal to smear. This phenomenon has amore severe effect on the higher frequency regions [21]. Fig. 1(c) illustrates simulated acceleration signal with the effect ofrandom variations incorporated into the fault repetition period (slippage with standard deviation of 0.01Tp). The spectrumof this simulated signal is shown in Fig. 1(d). As one can see from this figure, due to the smearing effect, the spectral densitylevel has been reduced substantially, hence undermining the effectiveness of spectral analysis in the presence ofbackground noise. In addition, the harmonics of the fault characteristic frequency in the high frequency region shown inthe close-up view of Fig. 1(b) can no longer be observed in this spectrum (close-up view of Fig. 1(d)). This phenomenonfurther complicates fault detection based on spectral analysis.

As will be explained in the following section, amplitude demodulation when applied to the full-band signal-noisemixture (i.e. no bandpass filtering) can play a significant role in the extraction of the fault feature namely faultcharacteristic frequency and the associated harmonics in such circumstances.


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2.1.1. Amplitude demodulation directly applied to signal-background noise mixture

The analytic version of the signal model of Eq. (1d) can be written as:

AnalyticðaðtÞÞ ¼ AðtÞ �XM

m ¼ �M


i ¼ �MtiÞe

j

�or ðt�mTp�

Pm

i ¼ �MtiÞþym

�uðt�mTp�

Xmi ¼ �M

tiÞ ð3Þ

The approximation sign is due to the wideband nature of the above signal model and also due to the deviation of theHilbert’s transform of the signal from its quadrature component caused by the leakage of the positive spectral componentsinto the negative spectral region [29,30]. Such deviation will result in an amplitude demodulation which does not perfectlymatch the actual envelope, i.e., the exponentially decaying term Lme�bðt�mTp�

Pm

i ¼ �MtiÞ in the above signal model. However,

one should note that in the context of machinery fault detection, the purpose of AD is to extract a low-frequencycomponent that reveals the periodic fault characteristic instead of finding the true envelope of the fault impulse. Therefore,such deviations should not cause a problem.

The amplitude-demodulated signal can then be written as:

ADMðaðtÞÞ �XM

m ¼ �M


i ¼ �MtiÞuðt�mTp�

Xm

i ¼ �M

tiÞ ð4Þ

where ADM stands for ‘‘Amplitude Demodulated’’. Most of the energy of this signal is concentrated around the low-frequency region. As such, the spectral content of the fault characteristic frequency and the associated harmonics willstand out of the background noise. In addition, random roller slippage has a considerably diminished effect in the lowfrequency region [21]. This fact further indicates the importance of the amplitude demodulation step.

Fig. 2(a) shows the spectrum of Fig. 1(d) with added simulated white Gaussian noise. The SNR of the resulting mixture is�15 db. According to Fig. 2(a), the resulting spectrum is completely covered by noise frequency content, making it verydifficult to draw any conclusions regarding the health state of the bearing. In addition, the wide-band nature of theresulting signal is evident from this figure. The above signal is amplitude demodulated by finding the magnitude of theanalytical version of the signal–noise mixture as is, without any bandpass filtering. Fig. 2(b) illustrates the envelopespectrum of this noise–signal mixture. As one can see, following the amplitude demodulation, spectral analysis shouldsuffice to extract the fault characteristic frequency and the associated harmonics.

This example indicates that the envelope analysis applied on the full-band signal can perform reasonably well in thepresence of intense white Gaussian background noise. Accordingly, one may conclude that such noises do not criticallyaffect the detection performance. In addition, one should note that for extremely low SNR signals where the AD step isunable to bring about sufficient improvement for fault detection, background noise removal techniques such as waveletthresholding can be applied [31,32].

However, in practice in addition to the faulty bearing vibration signals and background noise, the measured signalcontains vibration interferences generated by other mechanical components (e.g. gear meshing, shaft imbalance, shaftmisalignment, etc). In the following, the effect of such interferences on the effectiveness of the amplitude demodulationstep is evaluated.

2.1.2. Effect of low-frequency high-amplitude interfering signals on the AD step

The intensity of such interfering vibrations is usually substantial compared to weak vibration signatures of faultybearings. This is especially true when the measuring device is not installed in the near vicinity of the faulty bearing due totechnical or operational limitations.

0 0.5 1 1.5 20

0.002

0.004

0.006

0.008

0.01

Frequency (Hz)

Mag

nitu

de

200 400 600 800 10000

0.005

0.01

0.015

0.02

0.025

Frequency (Hz)

Mag

nitu

de

87.74 Hz 175.5 Hz

263.2 Hz 350.9 Hz

438.7 Hz

526.4 Hz

x 104

Fig. 2. (a) Spectrum of the mixture of the signal presented in Fig. 1(c) and simulated white Gaussian noise (SNR:�15db), and (b) spectrum of the

amplitude-demodulated version of the signal associated with Fig. 2(a), amplitude demodulation is applied on the full-band signal–noise mixture (no

bandpass filtering).


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To better understand the effect of such interferences on the AD result, consider a single fault-generated impulse mixedwith an added interference as simple as a low frequency, large amplitude single harmonic, given below:

rðtÞ ¼ s00ðtÞþvðtÞ ¼ Le�bt cosðortþyÞþacosðomtÞ tZ0 ð5Þ

where a 5L and orbom. Here abL emphasizes the unique bearing fault detection environment where the fault signal issubstantially weaker than interference signal components. The assumption of orbom is relevant to many industrialapplications as the excited resonance frequency or is often much higher than the frequency of the interference vibrationom [5].

Then one can write:

AnalyticðrðtÞÞ ¼ RðtÞ � Le�btþ jðor tþyÞ þaejðomtÞ tZ0 ð6Þ

The amplitude-demodulated version of the above equation is given by:

ADMðrðtÞÞ �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðLe�bt cosðortþyÞþa cosðomtÞÞ2þðLe�bt sinðortþyÞþa sinðomtÞÞ2

q�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiL2e�2btþa2þ2Lae�bt cosððor�omÞtþyÞ

p� aþLe�bt cosððor�omÞtþyÞ

tZ0

According to the above, though the additive interference is successfully amplitude demodulated, the bearing fault impulseis almost intact, only affected with a small shift in the ringing frequency. In other words, the AD step mainly affects thedominant signal component(s), a cosomt, in this case. As such, it can be stated that the effectiveness of the amplitudedemodulation step relies on the relative strength of the fault-generated impulses with respect to the interfering signalcomponents. One should note that the sole purpose of the above example is to illustrate the effect of interferences on theenveloping step, though the interfering signals encountered in practice could be far more complex than a single harmonic.

The above observation will lead us to devising a new approach to bearing fault detection. Before presenting the newmethod, it is necessary to define an index, signal-to-interference ratio (SIR), to quantify the relative strength of impulsivefault signals-to-vibration interferences. The details are given in the next subsection.

2.2. Signal-to-interference ratio

To quantify the level of vibration interferences for a vibration signal mixture r(t)=a(t)+v(t) (a(t) is the faulty bearingvibration and v(t) is the vibration interference), one can define the SIR as follows:

SIRðrÞ ¼ð1=TÞ

R T0 a2ðtÞdt

ð1=TÞR T

0 v2ðtÞdt¼

Pa

Pvð7Þ

where T is the signal length, Pa the power of fault impulses and Pv the power of the vibration interference. One can furtherdefine the SIR of a transformed signal r

_ðtÞ ¼ TrðrðtÞÞ ¼ aT ðtÞþvT ðtÞ as:

SIRðr_Þ ¼ð1=TÞ

R T0 a2

T ðtÞdt

ð1=TÞR T

0 v2T ðtÞdt

ð8Þ

where aT and vT respectively represent the transient associated with the fault impulse and the interfering signalcomponent of the transformed signal and are not necessarily equal to Tr(a(t)) and Tr(v(t)). Such a generalized form ofdefinition removes the linearity constraint from the transformation Tr( ).

We aim to find a transformation Tr( ) that amplifies the relative strength of the fault signal components of impulsive,transient nature with respect to that of the interferences so that in the presence of a fault signal component the following holds:

SIRðr_Þ4SIRðrÞ; or

SIRðr_Þ

SIRðrÞ41:

In the following, it is shown that a transformation Tr( ) that converts the measured signal to the frequency-demodulated version of the same signal is one such transformation.

2.3. Frequency demodulation (FD)

2.3.1. The notion of frequency demodulation

In the conventional bearing fault detection techniques that rely on the detection of the fault characteristic frequenciessuch as the high frequency resonance technique, the amplitude modulations of the faulty bearing vibrations are retained(Eq. (4)) for spectral analysis whereas the frequency modulation information is completely discarded. Since fault detectionis very complex, all sources of information should be fully exploited to obtain reliable results. As such, the disregard of theinformation provided by the frequency modulations may not be an apposite strategy. This new point of view is elaboratedin the following.


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2.3.2. The effect of FD on the SIR

Consider the analytic signal of Eq. (3) presented here for convenience:

AnalyticðaðtÞÞ ¼ AðtÞ �XM

m ¼ �M


i ¼ �MtiÞejðor ðt�mTp�

Pm

i ¼ �MtiÞþymÞuðt�mTp�

Xm

i ¼ �M

tiÞ

Disregarding the discontinuity at certain transition points, t¼mTpþPm

i ¼ �M ti, where the instantaneous frequencyexperiences abrupt changes, the frequency-demodulated form of the signal can be written as:

FDMðaðtÞÞ �XM

m ¼ �M

or

�uðt�mTp�

Xm

i ¼ �M

tiÞ�uðt�mTp�Xm

i ¼ �M

ti�TsettlingÞ

�ð9Þ

where FDM stands for ‘‘Frequency Demodulated’’, and Tsettling is the settling time of the fault-excited resonance associatedwith the structural damping of the system.

It is reasonable to assume that TsettlingoTp. This inequality reflects the assumption that a ringing period excited by thefault impact damps out to a very small value before a new impulse is generated [33].

With this assumption Eq. (9) represents a pulse train with a period corresponding to the fault characteristic frequency.Therefore, the spectral content of the pulse train given in Eq. (9) would provide the necessary information for faultdetection. To illustrate, the simulated faulty bearing signal of Fig. 1(c) is frequency demodulated. The FDM result isobtained using Hilbert’s transform (applied on the full-band signal). The phase of the resulting analytic signal is thenunwrapped (Fig. 3(a)) and differenced to obtain the frequency-demodulated result shown in Fig. 3(b). As one can see fromthe close-up view of a single frequency-demodulated impulse (Fig. 3b), the resulting instantaneous frequencies deviatefrom the theoretical pulse train (oscillations around 10 KHz mean frequency and also negative frequencies at the two ends)due to the full-band application of frequency demodulation. The spectral content of the frequency-demodulated result(Fig. 3(c)) illustrates several harmonics of the fault characteristic frequency.

1000 2000 3000 4000 5000 6000-2

-1

0

1

2x 104

Sample

Freq

uenc

y (H

z)

1000 2000 3000 4000 5000 60000

500

1000

1500

2000

Sample

Unw

rapp

ed P

hase

(rad

)

0 500 1000 1500 20000

500

1000

1500

Frequency (Hz)

Mag

nitu

de

87.74 Hz 175.5 Hz

263.2 Hz

350.9 Hz

438.7 Hz

Fig. 3. (a) Unwrapped phase of the analytical version of the signal of Fig. 1(c), (b) Frequency-demodulated version of the signal of Fig. 1(c), and (c),

Spectral content of the frequency-demodulated version of the signal of Fig. 1(c).


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The above example illustrated that the periodic changes in the frequency content of the signal can reveal the periodicnature of the fault signal. This observation can be utilized to extract the transient high frequency fault impulses from amixture of impulses and vibration interferences.

As explained in [29], instantaneous frequency is equal to the first moment of the Wigner–Ville distribution with respectto frequency, i.e.:

oIF ðtÞ ¼

R1�1

f WDðt; f ÞdfR1�1

WDðt; f Þdf

The above equation has an implication to the extraction of high frequency transients mixed with interferences.According to this equation, at the appearance of a high frequency transient in the signal, the instantaneous frequency oIF(t)will experience a drift. Denoting the Wigner–Ville distribution of the additive transient by WDT(t,f), the amount of drift willdepend on the integral

R1�1

f WDTðt; f Þdf . In other words, the drift in the instantaneous frequency depends on both thestrength of the transient reflected by WDT(t,f) and the frequency values, f, associated with WDT(t,f). The higher thetransient frequency, the more effective the identification of the transient. For bearing fault detection, the fault inducedringing is of high frequency and thus leads to pronounced instantaneous frequency, i.e., fault footprint.

The following elaborates on the effect of vibration interferences on the FD operation again by considering the analyticsignal mixture model of Eq. (6). To determine the instantaneous frequency, we denote the instantaneous phase of theanalytic form (Eq. (6)) by j(t) which is given by

jðtÞ ¼ tan�1 Le�bt sinðortþyÞþa sinomt

Le�bt cosðortþyÞþa cosomt

� �Then, the instantaneous frequency oIF(t) of the signal mixture can be written as:

oIF ðtÞ ¼djdt¼

d

dttan�1 Le�bt sinðortþyÞþa sinomt

Le�bt cosðortþyÞþa cosomt

� �� Using the equality ðd=dxÞðtan�1ðgðxÞÞÞ ¼ ðg0ðxÞ=ð1þg2ðxÞÞÞ, one obtains

oIF ðtÞ ¼Lae�bt

�bsinððom�orÞt�yÞþðorþomÞcosððom�orÞt�yÞ

�þL2ore�2btþa2om

L2e�2btþa2þ2Lae�bt cosððom�orÞt�yÞ

¼Lae�bt

�bsinððom�orÞt�yÞþðor�omÞcosððom�orÞt�yÞ

�þL2ore�2btþa2om

a2 L2

a2 e�2btþ2 La e�bt cosððom�orÞt�yÞþ1

� ��

L

abe�bt sinððom�orÞt�yÞþL

a ðor�omÞe�bt cosððom�orÞt�yÞþom ð10Þ

In the last step, we neglected the first two terms inside the bracket of the denominator, and the quadratic termL2ore

�2bt in the numerator based on the assumption abL.The transient signal component of the transformed signal associated with fault impulse is given as

L

abe�bt sinððom�orÞt�yÞþ

L

aðorþomÞe

�bt cosððom�orÞt�yÞ

and the interfering component is represented by om. In the following, the signal-to-interference ratio of the transformedsignal oIF(t) is obtained based on Eq. (8). In doing so, as before it is assumed that individual impulses damp outcompletely prior to the appearance of the next impulse. Then, considering the periodic nature of the bearing signal onecan write:

SIRðoIF Þ ¼ð1=TpÞ

R Tp

0 ½ðL=aÞbe�bt sin�ðom�orÞt�y

�þðL=aÞðorþomÞe�bt cos

�ðom�orÞt�y

��2 dt

o2m

�ð1=TpÞ

R10 ½ðL=aÞbe�bt sin

�ðom�orÞt�y

�þðL=aÞðorþomÞe�bt cos

�ðom�orÞt�y

��2 dt

o2m

and according to Eq. (7),

SIRðrÞ ¼ð1=TpÞ

R Tp

0 L2e�2bt cos2ðortþyÞdt

ð1=TÞR T

0 a2 cos2ðomtÞdt�ð1=TpÞ

R10 L2e�2bt cos2ðortþyÞdt

ð1=TÞR T

0 a2 cos2ðomtÞdt

Following some manipulations one obtains:

SIRðoIF Þ �b

2o2mðb

2þo2

r Þ�

L2b2

2a2Tpcos2yþ

L2o2r

2a2Tpcos2y�

L2bor

a2Tpsin2y

" #

�or

2o2mðb

2þo2

r Þ�

L2b2

2a2Tpsin2yþ

L2o2r

2a2Tpsin2yþ

L2bor

a2Tpcos2y

" #þ

L2ðb2þo2

r Þ

4o2mba2Tp

ð11Þ


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and

SIRðrÞ �L2

2Tpba2þ

L2ðb cos2y�or sin2yÞ2Tpa2ðb2

þo2r Þ

ð12Þ

A comprehensible comparison between the SIR of the original and frequency-demodulated signal, given by Eqs. (11)and (12) is not straightforward. As such, in the following the relationship between the excited resonance or and thestructural damping b is examined. The purpose of such analysis is to make a tangible comparison between the SIR of thetransformed signal (Eq. (11)) and the SIR of the original signal-interference mixture (Eq. (12)).

The settling time Tsettling of the transient impulsive component is defined as the time interval from the start of theresonance excitation up to the point where the oscillations remain within 0.67% of the initial amplitude (L). For a vibratorysystem with an exponentially decaying factor of e�bt, this time duration is 5 times (e�5=0.0067) of the time constanttc ¼ ð1=bÞ or Tsettling=(5/b).

The frequency equivalent of the settling time is therefore (1/Tsettling)=(b/5). In other words, for bEor, the fault-generated transient impulsive component of the measured signal would contain approximately five full periods ofoscillations as shown in Fig. 4(a). For any structural damping larger than the excited resonance, there will be fewereffective oscillations. For b4or no visible oscillations would be detected. However, for the real fault-generated impulsesmeasured from a faulty bearing, a condition similar to Fig. 4(a) is rare and strong oscillations (e.g., in Fig. 4(b)) can beobserved in practical situations. This observation suggests that the assumption bror is valid in the majority ofapplications. This assumption is a quantified version of the same assumption inherent in many other fault detectiontechniques that a resonance region associated with the ringing of the fault impulse exists in the frequency domain. As thedamping increases, the resonance region diminishes.

For (b4or) the fault signal approximately represents an impulse train. An impulse train in time domain translates to animpulse train in the frequency domain where no resonance frequency band is observable. As such, through the aboveassumption no significant restriction is added to the problem at hand. By assuming that bror, which leads to at least fiveoscillations (Fig. 4(a)), it is ensured that a reasonable level of ringing is induced by the fault impulse.

With the above assumption, one can evaluate Eqs. (11) and (12) for the limiting cases (1) bEor and (2) b5or. For the

first case bEor, from Eqs. (11) and (12) one can write SIRðoIF Þ �b � or

ðor=omÞ2½L2=4Tpa2or �, SIRðrÞ �

b � or

ð3L2=4Tpora2Þ, and

hence

SIRðoIF Þ �or

om

� �2 SIRðrÞ

3ð13Þ

Similarly, for the second case b5or, one obtains:

SIRðoIF Þ �b5or or

om

� �2 L2

4Tpa2b

� �and SIRðrÞ �

b5or L2

2Tpa2b:

Therefore,

SIRðoIF Þ �or

om

� �2 SIRðrÞ

2ð14Þ

As a result, the upper and lower bounds of SIR(oIF) can be obtained from Eqs. (13) and (14), i.e.,

or

om

� �2 SIRðrÞ

3oSIRðoIF Þr

or

om

� �2 SIRðrÞ

2

0 500 1000 1500-3

-2

-1

0

1

2

3

Sample

Am

plitu

de

0 0.2 0.4 0.6 0.8 1

-0.5

0

0.5

Time (s)

Am

plitu

de

x10-3

Fig. 4. (a) A single simulated fault impulse with structural damping equal to the excited resonance frequency (i.e., bEor), (b) an example of real fault

generated impulses measured from a faulty bearing with outer race fault (in this case, boor) [34].


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As ðor=omÞ2ðSIRðrÞ=3Þ should be greater than SIR(r) for the frequency demodulation to be effective, a conservative

criterion for the frequency-demodulation step to be able to increase the SIR of the signal-interference mixture would be:

or 41:7om

As can be observed in Eq. (10), such an increase in SIR is due to the exponentially decaying oscillating component

L

abe�bt sinððom�orÞt�yÞþ

L

aðorþomÞe

�bt cosððom�orÞt�yÞ:

This clearly suggests that after the FD transform a more prominent high-energy resonance region would emerge around(or�om) or approximately around the same excited resonance frequency as that of the original signal, which couldsubstantially facilitate the detection of weak fault signals when followed by an additional amplitude demodulation step.

2.3.3. Simulation result

In this subsection the SIR improving effect of FD is illustrated using simulations. For this purpose, simulated faultybearing vibration of Fig. 1(c) is mixed with several harmonics of the gear meshing frequency of 250 Hz, and harmonics ofthe shaft rotational frequency of 25 Hz. The resulting signal mixture and the corresponding frequency domainrepresentation are illustrated in Figs. 5(a) and (b), respectively.

According to Fig. 5(b), the spectrum of the simulated signal is dominated on the low frequency region by the simulatedinterfering vibration frequency components.

The signal mixture of Fig. 5(a) is then frequency-demodulated using Hilbert’s transform. The phase information of theresulting analytic signal is then unwrapped. The difference between adjacent unwrapped phase valuesoIF ðnDtÞ ¼ ððjððnþ1ÞDtÞ�jðnDtÞÞ=ðDtÞÞ, assuming Dt=1, is used as an estimate of the frequency-demodulated data pointsoIF ðtÞ ¼ ððdjðtÞ=ðdtÞÞ. The frequency-demodulated result is illustrated in Fig. 6(a). As one can see the transient impulses areextracted from the mixture. The overshoots and glitches shown in the FD result are observed at the zero-crossing points.Such zero-crossing results in errors when calculating the phase of the analytical signal which is then further amplifiedthrough differencing and leads to the illustrated glitches.

Fig. 6(b) illustrates the amplitude-demodulated version of the FD result (solid line), superposed on the FD result (dottedline). Amplitude demodulation is applied on the full-band FD result. Due to the wideband nature of the FD signal, theamplitude-demodulated result deviates from the actual expected envelope. This deviation can be observed from the close-up view of the amplitude-demodulated version of a single transient. The ripples on the AD result are due to the full-bandapplication of the amplitude demodulation. However, as explained earlier, in the context of machinery fault detection suchdeviations do not cause significant concerns.

Following envelope analysis of the frequency-demodulated signal, fault characteristic frequency and many of theassociated harmonics can be easily detected. Fig. 7(a) illustrates the resulting spectrum. A portion of the resulting envelopespectrum is shown in Fig. 7(b). The fault characteristic frequency and its harmonics are shown in this figure.

To summarize the above results, one can state that the transient impulsive faulty bearing signal components of thesignal mixture may be amplified with respect to the interferences through frequency demodulation. In other words, FD canincrease the SIR of the signal mixture and hence strengthen the effect of AD. Hence, following the FD step the energy of thesignal can be moved towards the low frequency region where fault characteristic frequency and its harmonics reside byenveloping, i.e., amplitude demodulating the frequency-demodulated signal.

0 1000 2000 3000 4000 5000-10

-5

0

5

10

Sample

Am

plitu

de

0 0.5 1 1.5 2-150

-100

-50

0

50

Frequency (Hz)

Mag

nitu

de (d

B)

Shaft Rotational Frequency and Harmonics

Gear Meshing

High energy resonance

frequency band

x 104

Fig. 5. (a) Simulated signal of Fig. 1(c) mixed with large amplitude vibration interferences, and (b) the corresponding frequency domain representation.


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0 0.5 1 1.5 20

100

200

300

400

500

Frequency (Hz)

Mag

nitu

de

0 200 400 600

50100150200250300350

Frequency (Hz)

Mag

nitu

de

87.74 Hz

175.5 Hz 263.2 Hz

350.9 Hz 438.7 Hz

526.4 Hz

x 104

Fig. 7. (a) Spectrum of the amplitude-demodulated version of the FD result, shown in Fig. 6(b) and (b), spectrum of part (a) on the range 0–600 Hz.

0 1000 2000 3000 4000 5000-2

-1

0

1

2

3x 104

Sample

Freq

uenc

y (H

z)

0 1000 2000 3000 4000 5000-2

-1

0

1

2x 104

Sample

Freq

uenc

y (H

z)

Over-shoot

Glitch

Ripples caused by the full-band application of AD

Fig. 6. (a) The frequency-demodulated version (solid line) of the mixture shown in Fig. 5(a) (superposed on the scaled mixture (dotted line). Please note

that the superposition is only for comparison purpose, and the FD result and the original time signal reflect different aspects, i.e., frequency and

amplitude, respectively), and (b) amplitude-demodulated result applied on the full-band frequency-demodulated signal shown in part (a) (superposed on

FD result).


Motivated by this outcome, we propose a single transformation that incorporates both amplitude and frequencymodulations of the measured signal into the diagnosis algorithm. The concept of energy operator used in speech signalprocessing is adopted for this purpose. This is elaborated in the following section.

3. Energy operator

3.1. Adopting energy operator for fault detection

The energy operator C() applied on a continuous signal g(t) is defined as [24]:

CðgðtÞÞ ¼dgðtÞ

dt

� �2

�gðtÞd2gðtÞ

dt2:

The derived version of this operator has been used to separate the frequency and amplitude modulations of speechsignals. It is logic to conjecture that the extraction of certain information (e.g., the transient impulsive fault signatures)relying on the combined amplitude and frequency demodulations could be carried out directly by the operator in itsoriginal form. Now consider an arbitrary signal g(t)

gðtÞ ¼ f ðtÞ cosðjðtÞÞ; ð15Þ

where f(t) is the time varying amplitude and oIF ðtÞ ¼ ðdjðtÞ=dtÞ is the time varying instantaneous frequency of signal g(t).


dx.doi.org/10.1016/j.ymssp.2009.12.007

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It can be shown [23,25] that for the continuous signal g(t) of form Eq. (15):

CðgðtÞÞ � f 2ðtÞdjðtÞ

dt

� �2

¼ f 2ðtÞo2IF ðtÞ: ð16Þ

The discrete time equivalent of the energy operator is given as [23]:

CdðgðnÞÞ ¼ g2ðnÞ�gðn�1Þgðnþ1Þ: ð17Þ

Similarly it is shown [23] that:

CdðgðnÞÞ ¼ a2ðnÞ sin2ðOðnÞÞ ð18Þ

where g(n)= f(n) cos(j(n)) and O(n)=j(n)�j(n�1).According to Eqs. (16) and (18) the energy operator extracts both the amplitude-modulation (AM) and frequency-

modulation (FM) information of the signal. Although the energy operator has mainly been used to separate the amplitudeand frequency modulations of a given speech signal [22–27], the separation of such information is not necessary in thecontext of machinery fault diagnosis.

The information of interest for machinery fault detection is the transient nature of the fault impulses resulting fromboth amplitude and frequency modulations as explained in the previous sections. This makes the energy operator avaluable means of accentuating such transient characteristics relative to the stationary components of the signal such asgear meshing and shaft imbalance vibrations.

Another very attractive feature of this operator is its simplicity in application and excellent time resolution because it,as evidenced in Eq. (17), performs on only three consecutive samples of the measured signal. Fig. 8 illustrates the flowchartof the proposed algorithm.

3.2. Teager Energy Operator (TEO) as a means to incorporate amplitude demodulation and SIR enhancement

As before, to assess the effect of energy transformation applied to a mixture of fault impulses and vibrationinterferences, consider the model of Eq. (5), repeated here for convenience:

rðtÞ ¼ s00ðtÞþvðtÞ ¼ Le�bt cosðortþyÞþa cosðomtÞ tZ0

The energy transformed version of this mixture can be written as:

CðrðtÞÞ ¼drðtÞ

dt

� �2

�rðtÞd2rðtÞ

dt2¼ L2o2

r e�2btzfflfflfflfflfflfflffl}|fflfflfflfflfflfflffl{ADM

þLahðtÞe�btzfflfflfflfflfflfflffl}|fflfflfflfflfflfflffl{HF

þa2o2m tZ0 ð19Þ

where

hðtÞ ¼ ½2bom cosðortþyÞsinomtþ2orom sinðortþyÞ sinomtþo2m cosðortþyÞ cosomt

�b2cosðortþyÞ cosomt�2bor cosomt sinðortþyÞþo2r cosðortþyÞ cosomt�

As shown above, the energy transformed signal-interference mixture consists of three terms. The first two terms aretransient components (amplitude-demodulated term denoted by ADM and high frequency component denoted by HF) ofthe signal evidenced by the exponentially decaying multiplying factors (e�bt ; e�2bt) and the third term a2om

2 representsthe interfering signal component. As illustrated in Eq. (19), the first term L2o2

r e�2bt is the squared envelope of the faultimpulse, scaled by a factor of or

2. In other words, energy transformation can successfully amplitude demodulate the fault-generated impulse despite the presence of the interfering component, which as illustrated earlier is difficult to achievethrough a regular amplitude-demodulation step. This observation suggests that the spectral analysis can be directlyapplied to the energy transformed signal-interference mixture.

In addition to the above, the appearance of the scaling factor or2 suggests that the amplitude-demodulation effect of

energy operator may also be accompanied by SIR improvement for a better detection performance. To better illustrate thisproperty of the energy transformation we derive the signal-to-interference ratio of the transformed signal, only taking intoaccount the amplitude-demodulated term (ADM term). In this analysis the exponentially decaying oscillating term (secondterm of Eq. (19) or HF term) is disregarded because it will not contribute to fault detection due to its high frequencycontent. Thus

SIRADMðCðrÞÞ ¼L4o4

r

4bTpa4o4m

where the subscript ADM emphasizes the fact that only the amplitude-demodulated term is included in the abovederivation. Comparing this result with the SIR of the original signal mixture model of Eq. (5) one can write:

or

om

� �4 L2

3a2

� �SIRðrÞrSIRADMðCðrÞÞr

or

om

� �4 L2

2a2

� �SIRðrÞ ð20Þ

where the lower and upper bounds are obtained for the limiting cases bEor and b5or, respectively.


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Acquire a block of vibration data r (t)

Transform data r (t) using the energy operator r (t) = Ψ(r (t))^

In the FFT of the transformed signal r (t), look ^

for the fault characteristic frequency and its harmonics for fault detection

Fault detected?

n = n + 1

n = 1

Alert the User

n > N?

Yes

No

No

Yes

r (t) = r (t)^

Fig. 8. Flowchart of the proposed algorithm.


The implication of the above equation may be better explained using a numerical example. According to Eq. (20) for afault impulse immersed in vibration interferences with an amplitude 10 times that of the fault impulse, i.e., a=10L, and anexcited resonance frequency that is five times of the interference frequency, i.e., or=5om, energy transformation canincrease the relative strength of amplitude-demodulated transient with respect to vibration interference by 2–3 times. Itshould be emphasized that the above SIR improvement is obtained by considering only the amplitude-demodulated term.One should note that even for cases where SIRADM(C(r)) is not considerably increased with respect to the original signalSIR, the fault detection performance may still be improved as the fault impulses are successfully amplitude demodulated.As shown above, the level of this improvement is dependent on: (1) the signal-to-interference ratio of the original mixturereflected by (L2/a2), and (2) the ratio of the frequency of the excited resonance to that of the interference, represented byðo2

r =o4mÞ in Eq. (20).

3.2.1. Repetitive application of the energy operator

A closer examination of Eq. (20) indicates that if the right-hand side is small due to a very low SIR(r) and a low ratio ofthe frequency of the excited resonance to the frequency of the interference, the strength of the amplitude-demodulatedterm, SIRADM(C(r)), becomes insignificant. In these circumstances, fault characteristic frequency and the associatedharmonics might not be visible in the spectrum of the energy-transformed signal and hence the fault cannot be detected.This may be resolved by repeatedly applying the energy operator. The rationale is explained in the following.


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By deriving the signal-to-interference ratio of the transformed result, this time by only taking into account the highfrequency component denoted by HF in Eq. (19), it can be shown that:

or

om

� �4 SIRðrÞ

3rSIRHFðCðrÞÞr

or

om

� �4 SIRðrÞ

2ð21Þ

where as before, the lower and upper bounds are obtained for the limiting cases bEor and b5or, respectively.According to Eq. (21), SIRHFðCðrÞÞis always larger than SIR(r) for or 41:3om, regardless of the original SIR of the signal-

interference mixture. Larger SIRHFðCðrÞÞ in the transformed signal is tantamount to a larger (L2/a2) in Eq. (20). As a result, itcan be inferred that by re-applying the energy transformation on the transformed signal, the intensity of the amplitude-demodulated term can be further increased, i.e.,

SIRADMðCðCðrÞÞÞ4SIRADM

�CðrÞ

�ð22Þ

This result can be generalized to N consecutive energy transformations.However, one should note that high-frequency noise amplification is an unwanted by-product of the TEO application.

Therefore, the number of such iterations (N in the flowchart of Fig. 8) has to be limited, which obviously depends on theintensity of the high-frequency noise in the raw signal.

3.3. Simulation results

In this section the proposed method is evaluated on simulated mixture of faulty bearing signal, background noise andvibration interferences. Here, to further mimic the practical situations, the simulated fault impulses are amplitudemodulated with the shaft rotational frequency, representing the cases where faults are located on the inner race of thebearing. As before, random slippage of the rollers is considered to have a standard deviation of 0.01Tp. The resultingsimulated bearing signal is shown in Fig. 9(a). This signal is then mixed with white Gaussian noise resulting in a mixturewith a SNR of �15 db. The noisy signal is further blended with vibration interferences consisting of several harmonics ofgear meshing frequency (250 Hz) and shaft rotational frequency (25 Hz). The resulting mixture is plotted in Fig. 9(b).Fig. 9(c) illustrates the spectrum of the mixture. As shown in this figure, the existence of fault cannot be concluded. For this

0 2000 4000 6000 8000 10000-10

-5

0

5

10

Sample

Am

plitu

de

0 2000 4000 6000 8000 10000-2

-1

0

1

2

Sample

Am

plitu

de

0 0.5 1 1.5 2

-80

-60

-40

-20

0

20

Frequency (Hz)

Mag

nitu

de (d

B)

Shaft Rotational Frequency and Harmonics

Gear Meshing

x 104

Fig. 9. (a) Simulated fault impulses, (b) Simulated fault impulses mixed with vibration interferences and white noise, and (c) Spectrum of the signal

mixture.


dx.doi.org/10.1016/j.ymssp.2009.12.007

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0.5 1 1.5 20

0.02

0.04

0.06

0.08

Frequency (Hz)

Mag

nitu

de

0.5 1 1.5 20

0.005

0.01

0.015

0.02

0.025

0.03

Frequency (Hz)

Mag

nitu

de

Fault Characteristic Frequency and Harmonics

Fault Characteristic Frequency and Harmonics

x 104 x 104

Fig. 10. (a) Spectrum of the energy transformed version of the signal presented in Fig. 9(b) and (b), envelope spectrum of the bandpass filtered version of

the signal shown in Fig. 9(b).


reason, the signal is then transformed using the proposed energy operator. The spectrum of the energy-transformed signalis illustrated in Fig. 10(a). As one can see, several harmonics of fault characteristic frequency can now be clearly detectedfrom this spectrum.

To compare the results with those of the conventional HFR technique, a bandpass filter is designed using the Parks–McClellan algorithm [35] with the center frequency tuned to the simulated resonance (10 KHz), i.e., the ringing frequencyof the fault impulses and a bandwidth of 2 KHz. The signal of Fig. 9(b) is then bandpass filtered. The spectrum of theamplitude-demodulated version of the bandpass filtered result is shown in Fig. 10(b). The fault characteristic frequenciescan be detected as shown in this figure.

A comparison between Figs. 10(a) and (b) shows that the performance of the proposed method is on a par with that ofthe conventional HFR technique. One may specially appreciate the effectiveness of the proposed technique in view that theabove comparison is made assuming that proper bandpass filter parameters are available, which is not the case in reality.

It should be emphasized that the result presented in Fig. 10(a) is obtained by directly applying the energy operator tothe simulated signal mixture of Fig. 9(b) without any prior de-noising process to remove intrinsic background noise. Theremoval of such noises prior to the application of energy operator would improve the outcome of the proposed method butit may nevertheless be unnecessary as the fault signature can already be detected.

In the next section, the proposed method is evaluated using experimentally measured faulty bearing data.

4. Experimental evaluation

4.1. Application to fault detection for bearing inner race

A SpectraQuest Machinery Fault Simulator (MFK-PK5M) as shown in Fig. 11 is used in this experiment. A well-balancedmass rotor (200 thick, 400 in diameter and 11.1 lbs) is mounted on a 5/800 steel shaft and supported by two normal bearings oftype ER10 K (inside, outside, pitch and ball diameters are respectively 0.625000, 1.850000, 1.319000 and 0.312500) with eightrolling elements (balls). The simulator is powered by a 3-hp AC motor which is controlled by a Hitachi drive (SJ200-022NFU). The AC motor operates on a bearing with pre-seeded inner race fault of unknown dimensions. The AC motor’sfaulty bearing is of type NSK-6203 (inside, outside, pitch and ball diameters are respectively 0.669300, 1.574800, 1.14200 and0.26600) with eight rolling elements (balls). To introduce additional vibration interferences, two unbalanced rotors (0.500

thick, 600 in diameter and 1.42 lb each, with two 0.011 lb unbalancing weights mounted to the outer perimeter of each disc)are also installed on the 5/800 steel shaft. In addition, a gearbox is connected to the driving shaft via a belt connection. Anaccelerometer (Montronix VS100-100) with 100 mV/g sensitivity and 1–12 kHz sensitivity range is used to collect thevibration signal. The signal is fed to an NI AT-MIO-16DE-10 DAQ card and then collected through LabVIEW. The signalprocessing is done using MATLAB on a Pentiums 4/2.52 GHz PC.

The shaft speed is set at 1428 RPM (23.8 Hz), leading to bearing fault characteristic frequency of 117 Hz (=4.932fr,specified in the simulator user’s manual). The rotational speed of the main shaft is reduced by a factor of 2.56 due to thelarger pulley on the driving shaft of the gearbox (see Fig. 11). The driven gear has 18 teeth. Hence the meshing frequency ofthe gearbox is 18�main shaft rotational frequency/2.56 (=7.03�main shaft rotational frequency). For this test, themeshing frequency is 167.3 Hz (=7.03�23.8 Hz). The accelerometer is installed on the simulator base at a location awayfrom the AC motor (Fig. 11). The vibration signal is sampled at 20,000 samples/sec.


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Fig. 12. (a) A portion of the measured signal, and (b) spectrum of the measured signal.

Normal Bearing

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Gearbox

Belt

Sensor

Coupling

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bearing

AC Drive

Normal Bearing

Unbalancing rotors

Fig. 11. Experimental setup #1.


Fig. 12(a) displays a portion of the measured data. As one can see from this figure no signal component featuringimpulsive characteristics can be observed from the time domain signal. The spectrum of the measured vibration signal isshown in Fig. 12(b). As one can see from this figure, the frequency components associated with the interfering vibrations atgear meshing, 167.3 Hz and associated harmonics as well as the harmonics of the shaft rotational frequency (23.8 Hz)dominate the spectrum of the measured signal. As a result, the fault characteristic frequency and its harmonics cannot bedetected.

Energy operator is applied to the measured signal for two iterations (N=2, in the flowchart of Fig. 8). Fig. 14demonstrates the frequency domain representation of the resulting transformed signal. The first three harmonics (at 117,234 and 351 Hz) of the fault characteristic frequency can be clearly detected from this result.

To compare the results obtained based on the proposed algorithm of Fig. 8 with those of the conventional HFRtechnique, a bandpass filter is designed using the Parks–McClellan method [35]. To locate the resonance frequency bandthe vibration signal is measured when the faulty AC motor is replaced with a healthy one. A portion of the signal measuredfrom the healthy motor and the associated spectrum are illustrated in Figs. 13(a) and 13(b), respectively. By comparing thespectrum of the faulty and healthy signals as shown in Figs. 13(c) and (d) a region (1550–1900 Hz) of high-SNR associatedwith the excited resonance frequency can be identified. The parameters of the filter are adjusted to keep the high SNRresonance frequency band (shown in Fig. 13(c)) and remove the noise and interferences. Fig. 15(a) illustrates the bandpassfiltered signal. The associated envelope spectrum is plotted in Fig. 15(b). Though the fault characteristic frequency can bedetected from both the spectrum obtained based on HFR method (Fig. 15(b)) and that of the energy transformed result(Fig. 14), only two harmonics of the fault frequency can be observed in Fig. 15(b) instead of three in Fig. 14. One mayfurther appreciate the performance of the proposed method by noting that in the comparison step it is assumed thatprecise knowledge regarding the filter parameters is available which is not always true in practice. In the following, theproposed method is further evaluated on the experimental data measured from a bearing with outer race fault.


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Fig. 13. (a) A portion of the signal measured from the same setup (experiment #1) when faulty AC motor is replaced with a healthy one, (b) spectrum of

the signal of part (a), (c) the first 2 KHz of the spectrum of Fig. 12(b), and (d) the first 2 KHz of the spectrum of Fig. 13(b).

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Fig. 14. Spectrum of the resulting signal after two iterations of energy operator applied to the original signal.



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Faulty BearingNormal Bearing

Loads

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AC Drive

Fig. 16. Experimental setup #2.


4.2. Application to fault detection for bearing outer race

The same simulator as that used in the previous experiment is employed in this test. The AC motor with faulty bearingis replaced with a healthy motor. Then, a bearing with an outer race fault (it was created by the manufacturer and thedimensions of the fault are unknown) and the same geometrical characteristics as the normal bearings used in previousexperiment is installed on the right side of the shaft (Fig. 16). The unbalancing rotors were replaced with an additionalbalanced mass rotor with the same dimensions as that used in the previous experiment. Similar to the previousexperiment, the gearbox is driven by the shaft via a belt connection. The accelerometer is installed in the immediateproximity of the gearbox and away from the faulty bearing. The vibration signal is sampled at 20,000 samples/sec. Theshaft speed is set at 1422 RPM (23.7 Hz) leading to bearing fault characteristic frequency of 72 Hz (=3.052fr, specified in thesimulator user’s manual). The experimental setup is shown in Fig. 16.

Part of the raw data is shown in Fig. 17(a). As before, the fault impulses cannot be observed from the time signal. Thespectrum of the measured signal is plotted in Fig. 17(b). According to this figure, the spectrum of the measured signal isdominated by frequency components associated with the gearbox meshing frequency (166.6 Hz=7.03�23.7 Hz), shaftrotational frequency (23.7 Hz) and the associated harmonics. As such, the fault characteristic frequency (72 Hz) and itsharmonics cannot be detected from such spectral information. The raw signal of Fig. 17(a) is then transformed once (N=1,in flowchart of Fig. 8) using the energy operator. Fig. 19 presents the spectrum of the transformed signal. As shown in thisfigure, fault characteristic frequency as well as five of its harmonics at 144, 216, 288, 360 and 432 Hz can be clearlydetected from the result.


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Fig. 17. (a) A portion of the measured signal, and (b) spectrum of the measured signal.

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Shaft rotational frequency

& harmonics Gear meshing & harmonics

Fig. 18. (a) A portion of the signal measured from the same setup (experiment #2) when faulty bearing is replaced with a healthy one, (b) spectrum of the

signal of part (a), (c) the first 2 KHz of the spectrum of Fig. 17(b), and (d), the first 2 KHz of the spectrum of Fig. 18(b).


As proved in Section 3, the increase in the SIRADM depends on the ratio between the frequency of excited resonance andthe frequency of interference. As such, higher resonance frequency would clearly lead to better results. In theseexperiments (experiments #1 and #2), the excited resonance frequencies (approximately 1750 Hz for experiment #1 and


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1000 Hz for experiment #2) are not considerably higher than the frequency of interfering vibrations. Nevertheless, despitesuch an unfavourable circumstance, the transformation is still able to extract the fault-related vibration componentssuccessfully. However, the energy transformation was unable to amplify the ADM-fault impulses to a degree such that theycompletely dominate the interfering signal components. As such, the interfering frequency components are still visible inthe spectrum of the transformed signals (Figs. 14 and 19). Yet, these frequency components do not cause any problem forfault detection as the fault characteristic frequency and harmonics have clearly stood out.

Similar to the previous test, to compare the result with that of the HFR technique, a bandpass filter is designed using theParks–McClellan method [35] to cover the high-SNR frequency band of the measured signal. Similar to the previousexperiment, the resonance frequency band is located by comparing the faulty signal spectrum with that of the signalmeasured from the setup with healthy bearings. A portion of the signal measured from the setup using healthy bearings isshown in Fig. 18(a). This time signal is very similar to the signal of Fig. 17(a). The spectrum of the signal is illustrated inFig. 18(b). By comparing the spectrum of the faulty and healthy signals as shown in Fig. 18(c) and (d), the resonancefrequency band is found as 850–1150 Hz. Fig. 20(a) illustrates the resulting bandpass filtered signal. The fault-generatedimpulses can be easily observed in this figure. Fig. 20(b) displays the envelope spectrum of the bandpass-filtered signal.The effectiveness of the proposed fault detection approach can be evidenced by comparing Figs. 19 and 20(b). Similar to theprevious experiment, one would appreciate the proposed method by noting that such precisely designed bandpass filtersare rarely available in practice, particularly for online monitoring applications.

According to the experimental results, a simple energy transformation requiring no prior knowledge about themechanical system and incorporating no analytical parameters can outperform the conventional methods which are basedon several computationally expensive and practically challenging steps. The effectiveness in fault detection as well as itssimplicity in application and computation would make the proposed method much more appealing to the industry thanthe conventional techniques. A provisional US patent has been filed based on the proposed invention.

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Fig. 20. (a) Bandpass filtered signal, and (b) envelope spectrum of the bandpass filtered signal.


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5. Conclusion

In this paper a new approach is proposed to detect bearing faults. The algorithm involves a simple transformation and aspectral analysis step. The measured signal is first transformed using the energy operator. This transformation stepincorporates both the frequency and amplitude modulations caused by the fault impulses and increases the relativestrength of the amplitude-demodulated impulses with respect to vibration interferences. This leads to more effectivebearing fault detection. Another advantage of the proposed method is that, unlike in other bearing fault-detectionmethods, no parameters need to be selected and no prior knowledge about the structure where the bearing is mounted orthe bearing itself (other than the fault characteristic frequencies) is required. The proposed method has been applied onthe signals measured from bearings with inner and outer race faults. The results have shown that the proposed method caneffectively reveal the fault characteristic frequency and the associated harmonics in the presence of intense vibrationinterferences.

Acknowledgments

This study was supported by Natural Science and Engineering Research Council of Canada, which is greatly appreciated.The authors would also like to thank the two anonymous reviewers for their detailed comments and suggestions.

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http://www.eecs.cwru.edu/laboratory/bearing

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