Shock and Vibration 9 (2002) 293–306 293

IOS Press

Wavelet based demodulation of vibration

signals generated by defects in rolling
element bearings
C.T. Yiakopoulos and I.A. Antoniadis
National Technical University of Athens, Department of Mechanical Engineering, Machine Design and Control
Systems Section, P.O. Box 64078, Athens 15710, Greece
E-mail: antogian@central.ntua.gr

Received 11 September 2000

Revised 10 January 2002

Abstract. Vibration signals resulting from roller bearing defects, present a rich content of physical information, the appropriate
analysis of which can lead to the clear identification of the nature of the fault. The envelope detection or demodulation methods
have been established as the dominant analysis methods for this purpose, since they can separate the useful part of the signal
from its redundant contents. The paper proposes a new effective demodulation method, based on the wavelet transform. The
method fully exploits the underlying physical concepts of the modulation mechanism, present in the vibration response of faulty
bearings, using the excellent time-frequency localization properties of the wavelet analysis. The choice of the specific wavelet
family is marginal to their overall effect, while the necessary number of wavelet levels is quite limited. Experimental results and
industrial measurements for three different types of bearing faults confirm the validity of the overall approach.

Keywords: Wavelets, bearing, demodulation, envelope, fault detection

1. Introduction diagnostic potential, since they are based on a more

solid physical background. The corresponding phys-
Bearings are one of the most important and fre- ical mechanism is described in [8]. The general as-
quently encountered components in the vast majority sumption with the enveloping approach is that a mea-
of rotating machines, their carrying capacity and relia- sured signal contains a low-frequency phenomenon that
bility being prominent for the overall machine perfor- acts as the modulator to a high-frequency carrier sig-
mance. Therefore, quite naturally, fault identification
nal. In bearing failure analysis, the low-frequency phe-
of roller bearings has been over the years the subject
nomenon is the impact caused by a small spur or crack;
of extensive research [14]. Vibration analysis has been
the high-frequency carrier is a combination of the natu-
established as the most common and reliable analysis
method. Defects or wear cause impacts at frequencies ral frequencies of the associated rolling element or even
governed by the operating speed of the unit and the of the machine. The goal of the enveloping is first to
geometry of the bearings, which in turn excite various isolate the measured signal in a relatively narrow fre-
machine natural frequencies. Several methods exist to quency band around a specific natural frequency using
exploit this physical effect, based either directly on the a band-pass filter and then demodulate it to produce a
shape of the time domain form of the signal, or on its low-frequency signal, called the “envelope”.
spectral content. Several demodulation methods have been used to
From all those methods, demodulation or envelop- identify faults in rolling element bearings. A “hard-
ing based methods offer a stronger and more reliable ware based” approach, proposed in [3], involves the

ISSN 1070-9622/02/$8.00  2002 – IOS Press. All rights reserved

294 C.T. Yiakopoulos and I.A. Antoniadis / Wavelet based demodulation of vibration signals generated by defects

following steps. The measured signal is passed through used with success in specific case studies for bearing
a band-pass filter to remove all low frequency high- fault detection [5,10,12], as well as for other machine
amplitude signals to keep the dynamic range of the sig- components [1,2,15,16]. The purpose of using the WT
nal within the capabilities of the instruments. The band- in the proposed method, is to obtain the envelope of
pass-filtered signal is passed through a diode, retaining the vibration response of faulty bearings, based on the
only the positive content. The rectified signal is then physical mechanism that generates the modulation ef-
low-pass-filtered to remove the high frequency content. fect and taking into full account its underlying physical
The resulting signal is the low frequency modulation concepts and major conclusions.
with a DC component. This signal is passed through Part 2 of the paper summarizes the basic physical
a capacitor (AC coupled) to produce the demodulated concepts describing the modulation mechanism, inher-
time waveform. ent in faulty bearing response. Part 3 performs a brief
Alternative to “hardware based” approaches, other review of the basic concepts of wavelet analysis, with
demodulation approaches have been also used, based special emphasis on their behavior in the frequency do-
on the Fourier Transform. An approach, based on the main. Part 4 describes the proposed method and anal-
direct use of the FFT, is proposed in [6]. First, an FFT yses the major parameters affecting its performance.
is applied to the N measured, rectangularly windowed Experimental results and industrial measurements for
data points, the lowest [(N/2) + 1] FFT coefficients are three different types of bearing faults are provided in
multiplied by two and the remaining coefficients are part 5, verifying the effectiveness of the method.
set to zero. Then, an inverse FFT is applied to the N
modified FFT coefficients, resulting to an N point pre-
envelope. The squared magnitude for each of the N pre- 2. Modulation of vibration signals generated by
envelope points leads to the final envelope. The advan- roller bearing defects
tage of an FFT-based envelope is that, if the frequency
content of the modulating signal and of the measured Whenever a defect present in one surface of a bearing
modulated signal do not intersect, an exact copy of the strikes another surface, an impact results, exciting the
true envelope can be recovered. resonances of the bearing and of the overall mechan-
A more advanced method is based on the proper ical system. Thus, the pulsation generated by rolling
combination of the FFT with the Hilbert transform [11]. bearing defects, excites vibration at specific defect fre-
The measured signal is passed through a band-pass quencies as well as a high-frequency response in the
filter, in order to isolate a specific high-frequency band, overall machine structure. A well established physi-
that presents in the spectrum relatively high amplitude cal model for the vibration produced by a single point
components, corresponding presumably to a specific defect on the inner race of a rolling element bearing
natural frequency of the machine. This step can be under radial load has been proposed in [8], describing
omitted in many cases with negligible effect. The band- the amplitude modulation of the excitation forces and
pass-filtered signal is then converted into an analytical the corresponding response. The model incorporates
signal. This analytical signal is a complex signal, the the effects of bearing geometry, shaft speed, bearing
real part of which is the band-pass-filtered signal and load distribution, transfer function and the exponential
the imaginary part is the Hilbert transform of the band- decay of vibration.
pass-filtered signal. The magnitude of the analytical Initially, the impact produced by the defect striking
signal corresponds to the envelope of the measured one of the rolling elements is modeled as an ideal im-
signal. pulse, denoted by the Dirac function δ(t). The magni-
This study presents a new demodulating approach, tude of the impulse is depended on the severity of the
based on the Wavelet Transform (WT). The WT has worn and also on the load on the defect at the time of
been established as the most widespread joint time- impact. As the inner race of the bearing rotates, the im-
frequency analysis tool [7,9,13] due among others, to pacts occur periodically at the Ball Passing Frequency
its inherent capability to be realized in real-time in the Inner race frequency (BPFI), which is defined [14] by:
form of a Discrete Wavelet Transform (DWT). The WT

m BD
overcomes the known disability of the Fourier Trans- fBP F I = 1+ cos β fr (1)
form to represent local features of the signal, such as 2 PD
the quite typical impulses, present in the vibration re- where m is the number of rolling elements, BD is
sponse of faulty roller bearings. It has already been the rolling element diameter, P D is the pitch circle
 C.T. Yiakopoulos and I.A. Antoniadis / Wavelet based demodulation of vibration signals generated by defects 295

factor and n is a constant. As the bearing rotates,

the transmission path and hence the transfer function
between the worn, where the impacts occur, and the
fixed measurement point, vary. This rotation effect
is taken into account by the introduction of a transfer
function r(t), the approximate form of which is shown
graphically in Fig. 1(c). Both q(t) and r(t) are periodic,
with repetition period T r = 1/fr .
The train of impulses w(t), generated by a constant
uniform unit load, is multiplied by the actual load q(t),
experienced by the defect, to give the actual impulses
delivered at the location of the defect. These impulses
are then multiplied by the amplitude of the transfer
function r(t) between the defect and the fixed measure-
ment point. Thus, the excitation on the fixed structure
of the machine is described by a force in the form:
f (t) = w(t)q(t)r(t) (4)
This excitation force, dependent on its location and
spectral content, excites a number of machine natu-
ral frequencies. The total vibration response can be
described by an equation in the form:
x(t) = f (t) ⊗ h(t) = [w(t)q(t)r(t)] ⊗ h(t) (5)
where h(t) is the impulse response function of the entire
Fig. 1. Waveforms involved in the generation of the envelope of machine and ⊗ denotes the convolution operator.
vibrations produced by an inner race defect under radial load [7]: In order to properly identify the fault, only the shape
(a) Impacts produced under a constant uniform unit load, (b) Load of the impulse sequence, as described by the excita-
distribution in the bearing (Stribeck equation), (c) Transfer func-
tion between the worn and the fixed measurement point, (d) Typical tion force pattern f (t) in Eq. (4), is necessary. Thus,
response decay law, (e) Final envelope. the objective of the fault identification procedure, is to
remove from the final response x(t) in Eq. (5) all its
diameter, β is the contact angle and f r is the shaft spectral contents resulting from the structural natural
rotation frequency. frequencies, and isolate its envelope in the following
Thus, the impacts produced by a single point defect form:
under a constant uniform unit load can be modeled as
v(t) = [w(t)q(t)r(t)] ⊗ e(t) (6)
an infinite series of impulses, shown in Fig. 1(a):
 where e(t) represents the typical decay of a resonance
w(t) = w0 δ(t − iTBP F I ) (2) excited by the impacts, assumed to follow an expo-
i
nential decay law with a time constant τ , which is in-
where w0 represents the magnitude of the impulses and dependent of the position at which the impact occurs
TBP F I = 1/fBP F I is the period between the impacts. (Fig. 1(d)):
The load distribution in the bearing is assumed to follow
the Stribeck equation, graphically shown in Fig. 1(b): e(t) = e−t/τ for t > 0
(7)
 n e(t) = 0 elsewhere
1
q(t) = qmax 1 − [1 − cos(2πfr t)] A typical form of the requested envelope is shown in
2ε
Fig. 1(e).
for |Arg(2πfr t)| < ϑmax (3) A signal simulating the vibration response of a bear-
ing with an inner race fault is shown in Fig. 2(a). This
q(t) = 0 elsewhere
signal is generated according to Eq. (5), assuming a ro-
where qmax is the maximum load intensity, θ max is the tation frequency of 36 Hz, a BPFI frequency of 181 Hz
angular extent of the load zone, ε is the load distribution and a single structural resonance at 2000 Hz. Its cor-
296 C.T. Yiakopoulos and I.A. Antoniadis / Wavelet based demodulation of vibration signals generated by defects

Fig. 2. Signal simulating the vibration response of a faulty bearing under an inner race fault: (a) Time waveform, (b) Spectrum (Shaft Speed:
36 Hz, BPFI: 181 Hz, Natural Frequency: 2 KHz, Sampling Frequency: 10 KHz).

responding spectrum with a sampling rate of 10 KHz shorter windows are chosen, then one will have a higher
is shown in Fig. 2(b). Although just a single struc- time resolution but a coarser frequency resolution. On
tural resonance exists, the spectrum occupies the en- the other hand, if longer windows are chosen, then one
tire frequency band, due to the various modulation ef- will have a higher frequency resolution but a coarser
fects. This clearly indicates the difficulty of bearing time resolution.
fault identification, based just on FFT analysis. To overcome the limitations of the fixed resolution
For other typical bearing faults (e.g. outer race faults, of the STFT in frequency and time domains, a new
ball faults, etc.), the physical modulation mechanism is method, based on wavelets, has been developed [7,9].
essentially the same one to the inner race fault mech- The wavelet transform (WT) is a mathematical tool that
anism, the basic difference being in the shape of the permits the decomposition of a signal in terms of ele-
envelope. mentary contributions, called wavelets. Time-domain
wavelets are simple oscillating amplitude functions of
time, that have large fluctuating amplitudes during a
restricted time period and very low amplitude or zero
3. Wavelet analysis
amplitude outside this time period.
The wavelets are obtained from a single function
The Fourier Transform (FT) is a linear expansion ψ(t) by translation and dilation:
of the signal into sinusoidal waveforms that have in-  
(t) 1 t−τ
finite length in time and that are extremely localized ψ(α, τ ) = √ ψ (8)
in frequency. This results to the total loss of the α α
time-information in the frequency domain. An im- where α is the so-called scaling parameter, τ is the time
provement of the Fourier transform is given by the localization parameter and ψ(t) is called the “mother
Windowed Fourier Transform, called the Short Time wavelet”. The parameters of translation τ ∈ R, and
Fourier Transform (STFT). The STFT is just a series of dilation α > 0, may be continuous or discrete.
FTs, performed on successive portions of a waveform. The WT of a finite energy signal x(t) with the an-
This approach does introduce the opportunity to iden- alyzing wavelet ψ(t) is the convolution of x(t) with a
scaled and conjugated wavelet:
tify time dependent variations in the structure of the  
waveform at various scales, as the window, over which 1 t
W (α, τ ) = x(t) ⊗ √ ψ −
the FT is computed, is moved along the longer wave- α α
form. However, a fixed window size must be used for a  
1 t
given STFT, and in order to obtain good resolution for = x(t) ⊗ √ ψ ∗ (9)
α α
the frequencies that compose the signal, a long window   
is required. The STFT retains the time information 1 ∞ ∗ t−τ
= x(τ )ψ dτ
but has strong time-frequency resolution limitations. If α −∞ α
 C.T. Yiakopoulos and I.A. Antoniadis / Wavelet based demodulation of vibration signals generated by defects 297

Fig. 3. Presentation of a three level wavelet analysis in the frequency domain. The signal is decomposed in a low resolution signal A3 and three
detail functions D1–D3, with corresponding spectra shown.

where ∗ denotes the complex conjugate. Different wavelet shapes are associated with different
Expression Eq. (9) can take the following alternative filter coefficient sequences. Regardless of the wavelet
form: used, the filters that produce the detail functions and the
 ∞
√ low resolution signal in a DWT have a variable band-
W (α, τ ) = α X(f )Ψ∗ (2παf )ei2πf τ df width that depends on the center frequency of those
−∞
√ (10) filters. Figure 3 schematizes the frequency spectrum
= αF −1 {X(f )Ψ∗(2παf )} for each scale in a typical three-level DWT. The figure
shows that each successive detail function in a DWT
where X(f ) and Ψ(f ) are the Fourier transforms of
(D1, D2, D3) has a spectrum with a center frequency
x(t) and ψ(t) respectively, and F −1 denotes the Inverse
at fo,j (level j = 1, 2, 3, . . .) and a bandwidth ∆f o,j
Fourier Transform.
half than that of the previous detail function. Thus, the
Equations (9) and (10) show that the wavelet analysis
frequency resolution improves by a factor of 2 for each
is a time-frequency analysis, or, more properly, a time-
successively larger scale in a DWT and the time reso-
scaled analysis. In particular, Eq. (10) shows that the
lution correspondingly decreases by a factor of 2. Con-
WT acts as also as filter. versely, the time resolution improves by a factor of 2 at
There exist many methods [9] to compute in prac- successively larger scales and the frequency resolution
tice the WT of a waveform. They can be divided in correspondingly decreases by a factor of 2.
two major classes: I) Methods based on the numeri- The center frequency f o,j and bandwidth ∆f o,j of
cal computation of the Continuous Wavelet Transform the jth wavelet’s spectrum become:
(CWT), II) Methods using specially designed filters,
fN fN
that generate a highly efficient Discrete Wavelet Trans- fo,j = + j+1 , j = 1, 2, 3, . . . (11a)
form (DWT), also known as “Multiresolution Decom- 2j 2
position”. fN fN
∆fo,j = j−1
− j , j = 1, 2, 3, . . . (11a)
The CWT is not computationally efficient. The infor- 2 2
mation it displays at closely spaced scales or at closely where fN is half the sampling frequency of the signal
time points is highly correlated and therefore unneces- (Nyquist frequency).
sarily redundant for many applications. The DWT pro- At the wavelet analysis stage shown in Fig. 3, the
vides a non-redundant, highly efficient wavelet repre- DWT analysis has four basic parts, namely a first detail
sentation, that can be implemented with a simple recur- function D1 that captures the high frequencies between
sive filter scheme. The DWT produces only as many 1/2 the Nyquist frequency and the Nyquist frequency, a
coefficients as the number of samples within the origi- second detail function D2 that captures the intermedi-
nal signal, without the loss of any information at all. ate frequencies between 1/4 the Nyquist frequency and
A vibration waveform can be decomposed into its 1/2 the Nyquist frequency, a third detail function D3
DWT coefficients through a simple recursive filter that captures the intermediate frequencies between 1/8
scheme, that consists of a high pass filter and a low pass the Nyquist frequency and 1/4 the Nyquist frequency
filter, whose filter coefficients are uniquely determined and a low resolution signal A3 that captures the low
by the particular wavelet shape used in the analysis. frequencies between 0 and 1/8 the Nyquist frequency.
298 C.T. Yiakopoulos and I.A. Antoniadis / Wavelet based demodulation of vibration signals generated by defects

Fig. 4. Indicative application of the WT in the frequency domain: (a) Original waveform of the signal, (b) Fourier transform of the signal, (c)
Fourier transform of the Morlet wavelet, (d) Application of the wavelet transform operation of Eq. (10) in the frequency domain, (e) Result of the
wavelet transform in the time domain.
2 2
Several wavelet families have been developed [7] to (t) = cj e−σj t ei2πfo,j t (12)
define the exact shape of the wavelet ψ(t). For demon-
where cj is a positive parameter, σ j determines the
stration purposes of the basic concepts of wavelet anal-
width of the wavelet and hence the width of the fre-
ysis, the complex-valued Morlet wavelet is used as a
quency band and f 0,j is the center frequency of the
typical example. The Morlet wavelet is defined in the
band. The parameter c j is equal to:
time domain as a sinusoidal wave multiplied by a Gaus- √
sian function: cj = σj / π (13)
 C.T. Yiakopoulos and I.A. Antoniadis / Wavelet based demodulation of vibration signals generated by defects 299

The Fourier transform of the Morlet’s wavelet is

given by:
2
− π2 (f −fo,j )2
σ
Ψ(f ) = e j (14)
where Ψ(f ) = Ψ∗ (f ), since Ψ(f ) is real.
This wavelet has a Gaussian shape in the frequency
domain, where the center frequency f o,j is given by
Eq. (11a) and the corresponding frequency range for
level j is:


σj σj
fN fN
fo,j − , fo,j + = , (15)
2 2 2j 2j−1
The parameter σ j becomes:
fN
σj = (16)
2j
The procedure for the calculation of the wavelet
transform in the frequency domain using Eq. (10) is
demonstrated in Fig. 4. The Morlet wavelet is applied.
Figure 4(a) indicates the original signal and Fig. 4(b)
the real part of its Fourier Transform. Figure 4(c) indi-
cates the Fourier Transform Ψ ∗ (2παf ) of the Morlet
wavelet, generated according to Eq. (15) for a given
center frequency f o,j (level j) and Nyquist frequency
fN . Figure 4(d) indicates the product X(f )Ψ ∗ (2παf ),
representing the band-pass filtering of the time-domain
signal. Finally, Fig. 4(e) represents the inverse Fourier
transform of the filtered signal. This waveform is
the wavelet transform of the input signal for a given
level j.
Although the calculation of the Wavelet Transform in
practice is performed directly in the time domain using
the DWT transform, the different presentation shown
in Fig. 4 has been chosen to clearly illustrate the effect
of the wavelet transform in the frequency domain. Fig. 5. Schematic presentation of the proposed demodulation
method.

ement, is ‘filtered’ and drawn away of the measured

4. Wavelet based demodulation signal. Figure 5 illustrates schematically the proposed
approach.
The efficiency of the WT can be fully exploited in The vibration, measured by an accelerometer mounted
the demodulation of vibration signals, resulting from on the casing of the machine near the bearing, is first
bearing defects. The development of the proposed en- squared to obtain the absolute value of the modulated
veloping approach is based on the fact that the mea- signal. The squaring procedure offers a number of ad-
sured signal, as described in part 2, contains a low- vantages. First, a waveform is produced, possessing
frequency component, which acts as the modulator to a only positive values, as does the final form of the ex-
high-frequency carrier signal. The goal of the envelop- pected envelope. Then, squaring escalates the differ-
ing approach is to isolate the low-frequency informa- ences of the peak variations, in order that the peaks are
tion of the measured signal that contains the percus- more discrete in the signal. Finally, as shown [4], the
sive frequencies caused by the bearing defect. Thus, squaring procedure, is able to transfer the most impor-
the high-frequency carrier signal, which contains the tant frequency content of the signal to lower frequency
natural frequencies of the associated race or rolling el- bands.
300 C.T. Yiakopoulos and I.A. Antoniadis / Wavelet based demodulation of vibration signals generated by defects

Fig. 6. Low resolution approximations A1, A2, A3 of of the signal of Fig. 2, using the Daubechies wavelet family of order 2.

Fig. 7. Details and spectrum of the low resolution approximation A2 of Fig. 2.

Then, using an N level wavelet transform, the rec- techniques like the FFT transform, in order to derive
tified signal is decomposed into its approximation and other specific signal features.
detail waveforms. The approximation waveform A N , The selection of the proper decomposition-level N is
which contains the low-frequency components of the critical for the method. It depends on the extent of the
signal, is the requested envelope. Optionally, it can low-frequency region, where the characteristic bearing
be further processed using alternative signal processing frequencies are expected to appear and on the sampling
 C.T. Yiakopoulos and I.A. Antoniadis / Wavelet based demodulation of vibration signals generated by defects 301

Fig. 8. Alternative signal simulating the vibration response of a faulty bearing under an inner race fault: (a) Time waveform, (b) Spectrum (Shaft
Speed: 24.5 Hz, BPFI: 245 Hz, Natural Frequency: 2 KHz, Sampling Frequency: 20 KHz).

Fig. 9. Low resolution approximations A2, A3, A4 of of the signal of Fig. 12, using the Daubechies wavelet family of order 2.

rate. The first factor is known by the geometry of the exposed. The proper decomposition level N is then
bearings that are monitored and on the shaft speed. subsequently selected in such a way, that the frequency
Characteristic calculation formula, similar in form to content of the approximation waveform A N completely
Eq. (1), exist in the literature [14]. The selection of the covers the low frequency region, without intersecting
sampling rate determines the total frequency bandwidth with the frequency band dominated by the structural
of the monitored signal. This bandwidth should be resonances.
selected as high as necessary, in order to include a The procedure for the selection of the proper decom-
number of structural natural frequencies, exited by the position level is shown in two characteristic examples.
characteristic impulses of the bearing defect. Thus, the In the first example, the signal of Fig. 2 is used. Wavelet
measured signal includes all the relevant information decomposition is performed at three different levels N
necessary for allowing the fault features to be properly = 1, 2 and 3. The resulting approximations A1, A2 and
302 C.T. Yiakopoulos and I.A. Antoniadis / Wavelet based demodulation of vibration signals generated by defects

Fig. 10. Details and spectrum of the low resolution approximation A3 of Fig. 8.

A3 respectively, are shown in Fig. 6. Approximation BPFI frequency and the shaft rotation frequency char-
A1 covers the range between 0 Hz and 2500 Hz, half acterizing the fault, clearly appear again in the spec-
the Nyquist frequency of 5000 Hz. Since the structural trum. Compared to the analysis of the first example,
resonance of 2000 Hz is also present in this range, the although the structural resonance and the characteris-
level A1 contains additional high frequency compo- tic bearing fault frequencies are in the same region,
nents, which render the demodulation procedure inef- an additional analysis level was necessary, since the
ficient. Approximations A2 and A3 cover respectively sampling frequency was doubled.
the ranges 0–1250 Hz and 0–625 Hz. Both isolate well The selection of the specific wavelet family to be
the low frequency region of the characteristic bearing used has a marginal effect on the method, since the
fault frequencies from the structural resonance region primary effect of the wavelet analysis with respect to
and as shown in Fig. 6, both are practically equivalent the proposed method is to isolate the low frequency
as envelope estimators. Thus, for computational pur- component of the signal, preserving its specific local
poses, only analysis up to level N = 2 is necessary in features in time. However, the Daubechies wavelet of
this case. A corresponding detail view of the waveform order 2 presents a slightly better behavior, indicating a
A2 and its Fourier analysis is shown in Fig. 7. The better high frequency component isolation.
BPFI frequency of 181 Hz and its modulation by the
shaft rotation frequency of 36 Hz, both characterizing
the fault, clearly appear now in the spectrum. 5. Experimental results
The signal used in the second example and its cor-
responding spectrum are shown in Fig. 8. Wavelet de- Three characteristic cases are presented, each one
composition is performed at three different levels N = been typical of a different type of bearing fault. Case A
2, 3 and 4 and the resulting approximations A2, A3 presents an inner race fault, case B presents an outer
and A4 respectively, are shown in Fig. 9. Since ap- race fault and case C a rolling element fault. The
proximation A2 covers also in this case the range 0– measurements in cases A and C where conducted on
2500 Hz, where the structural resonance of 2000 Hz is a machinery fault simulator manufactured by Spec-
also present, it is inappropriate as envelope estimator. taQuest, in order to study signatures of common ma-
Both approximations A3 and A4, covering respectively chinery faults. The measurements in Case B where
the ranges 0–1250 Hz and 0–625 Hz, isolate well the conducted on a fan motor at the industrial installations
low frequency region from the structural resonance re- of Aluminium of Greece S.A.
gion. Thus, analysis only up to level N = 3 is neces- In all cases, the measuring device is based on a Pen-
sary in this case. A corresponding detail view of the tium II/266 MHz portable computer, equipped with a
waveform A3 and its spectrum is shown in Fig. 10. The PCMCIA DAQCard-1200 data acquisition card from
 C.T. Yiakopoulos and I.A. Antoniadis / Wavelet based demodulation of vibration signals generated by defects 303

Fig. 11. Mesurements and analysis results of the bearing of Case A, representing an inner race fault: (a) Measured signal, (b) Spectrum of the
measured signal, (c) Envelope predicted by the proposed approach, (d) Spectrum of the envelope.

National Instruments. This is an 8 channel software- in the signal and a number of modulation indicating
configurable 12-bit data acquisition card, with a total side-bands are observable in the spectrum, the source
sampling rate capacity of 100 KHz. A B&K type 8325 and the nature of the fault cannot be identified with-
accelerometer is used, with a sensitivity of 97.3 mV/g out further processing. Figure 11(c) indicates the en-
and a dynamic range of 1 Hz to 10 KHz. The code of the velope produced by the demodulation procedure pro-
algorithm that is used in the data acquisition procedure posed in this paper. A three level wavelet analysis is
and signal analysis has been developed under the Lab- performed and the approximation function A3 is re-
VIEW programming environment of National Instru- tained, corresponding to the range 0–1024 Hz. The
ments. The wavelet transform of the measured signal envelope clearly follows with great accuracy the shape
is accomplished at the MATLAB wavelet toolbox. and the local features of the spikes of the original sig-
The bearing examined in Case A consists of 8 balls, nal, without containing its high frequency components,
has a ball diameter equal to 0.2813 inches, a pitch di- clearly demonstrating the value of the WT. For a more
ameter equal to 1.1228 inches and a contact angle equal clear identification of the nature of the fault, Fig. 11(d)
to 0 deg. The rotor speed is 36.52 Hz and the sam- presents the spectrum of the envelope. The spectrum of
pling frequency is 16384 KHz. Figure 11(a) illustrates the demodulated signal in Fig. 11(d), compared to the
the measured signal and Fig. 11(b) the corresponding spectrum of the original signal in Fig. 11(b), presents
spectrum. Although a “spiky” behavior is observable a far more clear structure, revealing peaks to the rotor
304 C.T. Yiakopoulos and I.A. Antoniadis / Wavelet based demodulation of vibration signals generated by defects

Fig. 12. Mesurements and analysis results of the bearing of Case B, representing an outer race fault: (a) Measured signal, (b) Spectrum of the
measured signal, (c) Envelope predicted by the proposed approach, (d) Spectrum of the envelope.

speed fs , the characteristic bearing frequency BPFI (= of the analysis are shown in Fig. 13, using again a 3
181.1 Hz) and its second harmonic 2*BPFI. Therefore, level WT. The spectrum of the original signal presents
the nature of the fault can be clearly identified. a broad band spectrum, indicating a rather strong mod-
The bearing examined in Case B is of type 6324MC3 ulation. Only the spectrum of the envelope allows the
manufactured by SKF. The rotor speed is about identification of the fault, revealing peaks at the rotor
1,500 rpm. The sensor is mounted near the bearing speed fs , the characteristic Ball Spin Frequency BSF
at the horizontal direction. The sampling rate used is (BSF = 1.871 × Rotor speed) and its second harmonic.
20 KHz and the number of samples is 32,768. The
same type of analysis as in Case A is performed, using
a 3 level wavelet analysis. The results are presented in 6. Conclusion
Fig. 12. The low-frequency information transferred to
the spectrum of the demodulated signal reveals peaks The excellent time-frequency localization capabili-
to the rotor speed fs, the characteristic Ball Passing Fre- ties of the wavelet transform, enhanced by the squaring
quency Outer race BPFO (= 78.12 Hz) and its second preprocessing phase of the signal, are able to exhibit the
and third harmonic, confirming also in this Case the underlying physical modulation mechanism, present in
validity of the approach. the vibration response of faulty bearings, leading to a
The last Case C presented, was accomplished with new effective demodulation procedure. Key element
the same type of bearing as in Case A, exhibiting now a for the effectiveness of this demodulation procedure is
fault on the rolling elements. The rotor speed is about first that the choice of the specific wavelet family to
1,450 rpm and the values of the rest of the measure- perform the analysis has a marginal effect. Also, for
ment parameters are the same as in Case A. The results the typical defect frequencies, resonance regions, and
 C.T. Yiakopoulos and I.A. Antoniadis / Wavelet based demodulation of vibration signals generated by defects 305

Fig. 13. Mesurements and analysis results of the bearing of Case C, representing a ball fault: (a) Measured signal, (b) Spectrum of the measured
signal, (c) Envelope predicted by the proposed approach, (d) Spectrum of the envelope.

sampling frequencies encountered in faulty bearing re- Mechanical Systems and Signal Processing 14(5) (2000), 763–
sponse, the number of the necessary wavelet levels can 788.
[5] J.C. Li and J. Ma, Wavelet decomposition of vibrations for
be quite limited in practice, a typical value being three detection of bearing-localized defects, NDT&E International
levels of approximation. The above facts render the 30(3) (1997), 143–149.
overall procedure quite simple conceptually and fast [6] D.E. Lyon, Thoughts on 1996 CMVA presenters’ questions:
computationally, taking into account the efficiency of ‘Peakness’ methods for bearing fault diagnosis, Vibrations
14(3) (1998), 5–11.
the DWT. The experimental results clearly confirm the [7] S. Mallat, A wavelet tour of signal processing, Academic
effectiveness of the proposed method. Press, CA, 1998.
[8] P.D. McFadden and J.D. Smith, Model for the Vibration pro-
duced by a single point defect in a rolling element bearing,
Journal of Sound and Vibration 96(1) (1984), 69–82.
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[3] R.L. Eshleman, Comments on rolling element bearing analy- tory surveillance methods and diagnostic techniques, Senlis,
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[13] W.J. Staszewski, Wavelet based compression and feature se- [15] W.J. Wang and P.D. McFadden, Application of wavelets to
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