Available online at www.sciencedirect.com
Mathematics and Computers in Simulation 79 (2008) 318–338
Bearing condition monitoring based on shock pulse
method and improved redundant lifting scheme
Li Zhen a,∗ , He Zhengjia b , Zi Yanyang a , Chen Xuefeng a
b
a School of Mechanical Engineering, Xi’an Jiaotong University, 710049 Xi’an, China
State Key Lab for Manufacturing Systems Engineering, Xi’an Jiaotong University, 710049 Xi’an, China
Received 17 January 2007; received in revised form 28 December 2007; accepted 28 December 2007
Available online 17 January 2008
Abstract
Due to the widespread application of rolling element bearings, it is necessary to effectively monitor their health status. The shock
pulse method (SPM) has been widely used as a quantitative method for bearing condition monitoring. However, the shock value
indicating the bearing condition may be mistakenly estimated by direct demodulation in the SPM. To overcome this deficiency, a
new approach based on improved redundant lifting scheme (IRLS) is proposed. The classical redundant lifting scheme is improved
by adding the normalization factors to avoid error propagation of decomposition results, and the IRLS is applied to preprocess the
bearing vibration signals. Then the maximum normalized shock value of detail signals in decomposition results is used as a measure
of the bearing condition. The effectiveness of the proposed method is demonstrated by applying it to both simulated signals and
practical bearing vibration signals under different conditions. The results show that the proposed method is effective for bearing
condition monitoring.
© 2008 IMACS. Published by Elsevier B.V. All rights reserved.
Keywords: Bearing condition monitoring; Redundant lifting scheme; Shock pulse method
1. Introduction
Rolling element bearings play a critical role in industrial applications, and unexpected bearing failures may result
in significant economic losses. Condition monitoring is an effective way to prevent the bearing failures. A detailed
review of vibration and acoustic methods has been introduced by Tandon and Choudhury [25] for bearing condition
monitoring.
The shock pulse method (SPM) has achieved wide acceptance as a quantitative method for detecting the defects
of rolling element bearings. The maximum normalized shock value is used as a measure of bearing condition in the
SPM [1]. It gives a single value indicating the bearing condition, without the need for elaborate data interpretation
as required in some other methods (such as traditional spectral analysis, cepstrum analysis [30], cyclic statistics [10],
empirical mode decomposition (EMD) [3], wavelet transform [15], etc). However, direct demodulation may mistakenly
estimate the shock value in the SPM. To make up this deficiency, wavelet transform based on lifting scheme [7] is used
to preprocess the vibration signals of rolling bearing.
∗
Corresponding author. Tel.: +86 29 82667963; fax: +86 29 82663689.
E-mail address: xjtu lizhen@163.com (L. Zhen).
0378-4754/$32.00 © 2008 IMACS. Published by Elsevier B.V. All rights reserved.
doi:10.1016/j.matcom.2007.12.004
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319
Fig. 1. The forward transform of the lifting scheme.
Wavelet transform is well known for its ability to focus on localized structures in time–frequency domain [11].
Unlike spectral analysis that describes a signal as the sum of sinusoidal functions, wavelet transform decomposes
the signal into wavelet coefficients of various scales in time domain. The advantages are twofold [14]: (1) it is more
effective to extract transient features and (2) it extracts signal features over the entire spectrum, without requiring a
dominant frequency band. Many researchers have investigated the application of wavelet transform to vibration signals
analysis for condition monitoring and fault diagnosis of rolling element bearings [2,9,16,19,21,27]. The lifting scheme
[22,23] is proposed by Sweldens as a new method of wavelet construction. The main feature of the lifting scheme is
that the construction is derived in the time domain. Wavelets with different time–frequency structures are obtained
by the design of prediction operator and update operator. Compared with the classical wavelet transform, the lifting
scheme requires less computation and memory, and can produce integer-to-integer wavelet transform. It is always
perfectly reconstructed no matter how the prediction operator and update operator are designed. The application of
lifting scheme for rotating machinery fault diagnosis has been reported in [5,8]. However, the lifting scheme does
not have time invariant property, (i.e. using the lifting scheme, the decomposition results of a delayed signal are not
the time-shifted version of those of the original signal). This may result in the loss of useful information for bearing
condition monitoring.
The time invariant property is particularly important in signal processing such as features detection and noise reduction [17]. The redundant lifting scheme (RLS) [4] possesses time invariant property and overcomes the disadvantage
of lifting scheme by getting rid of the split step and zero padding of prediction operator and update operator. The
approximation and detail signals at all levels are the same length as the original signal in the redundant lifting scheme.
However, the RLS may lead to error propagation of decomposition results.
In the previous paper, the authors have demonstrated the high potential of the RLS to detect bearing fault [29]. In
this paper, we bring this idea and improve the algorithm of RLS. A new method based on SPM and IRLS is proposed to
quantitatively indicate the bearing condition. To make up the deficiency of classical RLS, the cause of error propagation
using classical redundant lifting scheme is analyzed. The RLS is improved by adding the normalization factors. To
overcome the misestimate of the shock value with direct demodulation in the SPM, IRLS is applied to preprocess
the bearing vibration signals. And the maximum normalized shock value of detail signals is used as a measure of the
bearing condition. We find that the proposed method is more effective for bearing condition monitoring.
The structure of the paper is organized as follows. In Section 2, the theories of lifting scheme and redundant lifting
scheme are reviewed briefly. The deficiency of redundant lifting scheme is discussed. In Section 3, the algorithm of IRLS
is introduced. In Section 4, the SPM, Hilbert transform and the proposed method are explained. Section 5 describes
the disadvantage of SPM based on direct demodulation, and demonstrates the performance of the proposed method by
simulation experiment. In Section 6, the proposed method is used for bearing condition monitoring. Conclusions are
given in Section 7.
Fig. 2. The scaling function and wavelet function with N = 6, Ñ = 6.
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Fig. 3. The detail signal of original signal using the lifting scheme decomposition.
Fig. 4. The detail signal of delayed signal using the lifting scheme decomposition.
Fig. 5. The forward transform of redundant lifting scheme.
Fig. 6. The distortion occurs in decomposition results using redundant lifting scheme.
L. Zhen et al. / Mathematics and Computers in Simulation 79 (2008) 318–338
321
Fig. 7. The inverse lifting scheme with d(1) = 0.
Fig. 8. The inverse lifting scheme with s(1) = 0.
Fig. 9. The forward and inverse transform of IRLS.
2. The lifting scheme and redundant lifting scheme
2.1. The lifting scheme
The lifting scheme is a powerful tool to construct biorthogonal wavelets in the spatial domain [22,23]. The
implementation of lifting scheme is simple, fast and efficient. The lifting scheme consists in three main steps.
Fig. 10. The decomposition results of signals shown in Fig. 6 using IRLS.
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Fig. 11. The flow chart of the proposed method for bearing condition monitoring.
(0)
In the split step, the original signal x = (xi )i␧z is split into the even samples s(0) = (si )i ∈ Z , and the odd samples
(0)
(0)
d = (d i )i ∈ Z ,
si (0) = x2i ,
d i (0) = x2i+1
(1)
In the prediction step, we apply an operator P on s(0) to predict d(0) . The prediction error d(1) is regarded as the
detail signal of x,
d (1) = d (0) − P(s(0) )
(2)
where P = [p(1), p(2), . . ., p(N)] is the prediction operator, and N is the number of prediction coefficients.
In the update step, an update of even samples s(0) is accomplished by using an update operator U to detail signal
(1)
d and adding the result to s(0) , the update sequence s(1) can be regarded as the approximation signal of x,
s(1) = s(0) + U(d (1) )
(3)
where U = [u(1), u(2), . . . , u(Ñ)] is the update operator, and Ñ is the number of update coefficients.
Let s(1) be the input signal for lifting scheme, the detail and approximation signals at the lower resolution level
can be obtained. The inverse lifting scheme can be performed by reversing the prediction and update operators and
changing each ‘+’ into ‘−’ and vice versa. Fig. 1 shows the forward transform of the lifting scheme.
The prediction operator and update operator can be designed by the interpolation subdivision method introduced
in [24]. The scaling function and wavelet function with N = 6, Ñ = 6 are shown in Fig. 2. They are symmetrical and
compactly supported. The shape of the wavelet function is very similar to an impulse. Therefore, it is desirable to
extract the transient components from vibration signals.
L. Zhen et al. / Mathematics and Computers in Simulation 79 (2008) 318–338
323
Fig. 12. The periodic impulse signal with a period of 0.0167 s and its Hilbert spectrum.
2.2. The redundant lifting scheme
The time invariance [11] of an operator L means that if the input f(t) is delayed by τ, fτ (t) = f(t − τ), then the output
is also delayed by τ:
g(t) = Lf (t) ⇒ g(t − τ) = Lfτ (t)
(4)
In the lifting scheme, the time invariant property is not ensured because there exists the split step and the length of
approximation signal and detail signal decreases.
Suppose the original signal is
x = {. . . , x2j−4 , x2j−3 , x2j−2 , x2j−1 , x2j , x2j+1 , x2j+2 , x2j+3 , x2j+4 , . . .}
The prediction operator is P = {p1 , p2 }, and the update operator is U = {u1 , u2 }. The decomposition procedure of
original signal is given in Fig. 3. The detail signal d of original signal can be calculated by
d[j] = x2j+1 − (p1 x2j + p2 x2j+2 )
(5)
when delaying the original signal by one time step, the detail signal d of the delayed signal can be computed by
d[j] = x2j − (p1 x2j−1 + p2 x2j+1 ).
Fig. 13. The periodic impulse signal with a period of 0.0125 s and its Hilbert spectrum.
(6)
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Fig. 14. The simulated signal with two periodic impulse signals and its Hilbert spectrum.
The decomposition procedure of the delayed signal is shown in Fig. 4. Obviously, the detail signal d of delayed
signal is not equal to the detail signal d of original signal, so the time invariant property is not ensured in the lifting
scheme.
In the redundant lifting scheme, the split step is discarded. The redundant prediction operator P(l) and the redundant
update operator U(l) are computed by padding the prediction operator P and update operator U with zeros at the
corresponding level l. The redundant lifting scheme possesses time invariant property and well keeps the signal
information. The forward transform of redundant lifting scheme is shown in Fig. 5. The decomposition results of an
approximation signal ṡ(l) at level l with redundant lifting scheme are expressed by following equations,
ḋ
(l+1)
= ṡ(l) − P (l) ṡ(l)
ṡ(l+1) = ṡ(l) + U (l) ḋ
where ḋ
(l+1)
(7)
(l+1)
(8)
and ṡ(l+1) are detail signal and approximation signal at level l + 1.
Fig. 15. The decomposition results of simulation signal using IRLS.
L. Zhen et al. / Mathematics and Computers in Simulation 79 (2008) 318–338
325
Fig. 16. The Hilbert spectrums of detail signals r(1) and r(2) .
2.3. The deficiency of redundant lifting scheme
Since the redundant lifting scheme does not control the error propagation, the decomposition results using redundant lifting scheme always bring distortion. The prediction operator and update operator with N = 6, Ñ = 6 are used
to demonstrate the deficiency of redundant lifting scheme. The inspected signal is a practical vibration signal of
rolling bearing with heavy outer-race defects. The waveform contains periodic impulses. The range of original
signal is [−0.8, 0.8]. The decomposition results are shown in Fig. 6. From Fig. 6, the range of the detail sig(1)
nal (ḋ ) is [−1.5, 1.5], which far exceeds the range of original signal. Moreover, the energy of the signal is
given by the classical Euclidian norm. The total energy of the original signal is 98.26, but the total energy of the
decomposition results is 360.64. It shows that the distortion occurs in decomposition results using redundant lifting
scheme.
Fig. 17. The decomposition results of simulation signal using redundant lifting scheme.
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Fig. 18. The Hilbert spectrums of detail signals ḋ
(1)
(2)
and ḋ .
3. The improved redundant lifting scheme (IRLS)
3.1. The cause of the error propagation in redundant lifting scheme
The error propagation can be controlled by inverse transform in the lifting scheme [7,20]. We analyze the cause
of the error propagation in the redundant lifting scheme by comparing the decomposition results of redundant lifting
scheme with the reconstruction results of lifting scheme. The signal is performed 1-level decomposition by using
lifting scheme then reconstructed by oneself. The total energy of reconstruction results is approximately equal to that
of original signal. From the forward transform of lifting scheme, the detail signal d(1) and the approximation signal
s(1) can be expressed as follows:
(1)
d i = x2i+1 −
N
pr x2i−N+2r
r=1
Fig. 19. The decomposition results of the simulation signal using the Jiang’s method.
(9)
L. Zhen et al. / Mathematics and Computers in Simulation 79 (2008) 318–338
Fig. 20. The Hilbert spectrums of detail signals d̂
(1)
si = x2i +
Ñ
(1)
uj d i−Ñ/2+j−1
(1)
327
(2)
and d̂ .
(10)
j=1
when d(l) = 0, the inverse transform of the lifting scheme is shown in Fig. 7. The reconstructed signal ŝ(1) of approx(1)
(1)
(1)
imation signal s(1) is calculated. The even samples ŝ(1)
e = {ŝ2i } and the odd samples ŝo = {ŝ2i+1 } are given by the
following equations:
(1)
(1)
ŝ2i = si
(1)
ŝ2i+1
=
N
(11)
(1)
pr si+r−N/2
(12)
r=1
Fig. 21. The vibration signals of bearing under different conditions: (a) healthy bearing, (b) weak outer-race defect and (c) heavy outer-race defect.
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Fig. 22. The decomposition results using IRLS of the vibration signal of healthy bearing.
Fig. 23. The Hilbert spectrum of detail signal r(1) in Fig. 22.
Fig. 24. The decomposition results using IRLS of the vibration signal with weak outer-race defect.
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329
Fig. 25. The Hilbert spectrum of detail signal r(1) in Fig. 24.
Fig. 26. The decomposition results using IRLS of the vibration signal with heavy outer-race defect.
(1)
when s(1) = 0, the inverse transform of the lifting scheme is shown in Fig. 8. The even samples d̂ e =
(1)
(1)
(1)
{d̂ 2i } and the odd samples d̂ o = {d̂ 2i+1 } of the reconstructed signal d̂
equations:
(1)
Ñ
(1)
(1)
d̂ 2i = − uj d i−Ñ/2+j−1
are expressed by the following
(13)
j=1
Fig. 27. The Hilbert spectrum of detail signal r(1) in Fig. 26.
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Fig. 28. Maximum normalized values of bearing under different conditions using the proposed method.
(1)
(1)
d̂ 2i+1 = d̂ i +
N
(1)
pr d̂ 2i−N+2r
(14)
r=1
(1)
(1)
From the forward transform of redundant lifting scheme, the even samples ḋ e = {ḋ 2i } and the odd samples
(1)
(1)
= {ḋ 2i+1 } of the detail signal ḋ are described as,
(1)
ḋ o
(1)
ḋ 2i = x2i −
N
(15)
pr x2i−N+2r−1
r=1
(1)
ḋ 2i+1 = x2i+1 −
N
pr x2i−N+2r
(16)
r=1
(1)
(1)
(1)
(1)
The even samples ṡe = {ṡ2i } and the odd samples ṡo = {ṡ2i+1 } of the approximation signal ṡ(1) are expressed as
follows:
(1)
ṡ2i = x2i +
Ñ
(1)
uj ḋ 2i−Ñ+2j−1
(17)
j=1
(1)
ṡ2i+1 = x2i+1 +
Ñ
(1)
(18)
uj ḋ 2i−Ñ+2j
j=1
From Eqs. (9)–(11), we obtain
Ñ
N
(1)
ŝ2i = x2i +
uj x2i−Ñ+2j−1 −
pr x2i−Ñ+2j−2−N+2r
j=1
(19)
r=1
Considering Eqs. (16) and (17), we have
(1)
ṡ2i = x2i +
Ñ
j=1
uj (x2i−Ñ+2j−1 −
N
pr x2i−Ñ+2j−2−N+2r )
(20)
r=1
(1)
(1)
From Eqs. (19) and (20), we can see that the even samples ṡe = {ṡ2i } of the approximation signal ṡ(1) are equal
(1)
(1)
(1)
(1)
to the even samples ŝ(1)
e = {ŝ2i } of reconstructed signal ŝ . Moreover, compared with the odd samples ŝo = {ŝ2i+1 }
L. Zhen et al. / Mathematics and Computers in Simulation 79 (2008) 318–338
(1)
331
(1)
of reconstructed signal ŝ(1) , the odd samples ṡo = {ṡ2i+1 } of the approximation signal ṡ(1) is calculated by different
interpolation method. The approximation signal ṡ(1) cannot bring the error propagation. Therefore, the error propagation
(1)
of decomposition results using redundant lifting scheme is caused by the distortion of the detail signal ḋ .
3.2. The algorithm of the IRLS
The IRLS is proposed to control the error propagation by adding the normalization factors. The forward and inverse
transforms of IRLS are shown in Fig. 9. The al+1 and bl+1 are the normalization factors at level l + 1. The decomposition
results of the approximation signal ṡ(l) at level l with IRLS are expressed as follows:
(l+1)
= ṡ(l) − P (l) ṡ(l)
(l+1)
= ṡ
ḋ
ṡ
r
(l+1)
(21)
(l) (l+1)
+ U ḋ
(l)
= bl+1 ḋ
(22)
(l+1)
(23)
c(l+1) = al+1 ṡ(l+1)
(24)
(l+1)
and ṡ(l+1) are computed by the classical redundant lifting scheme. r (l+1) and c(l+1) are the detail signal
where ḋ
and approximation signal at level l + 1 using IRLS.
Suppose the total energy of the approximation signal ṡ(l) is described as
Esl =
n
(l) 2
[ṡi ] .
(25)
i=1
The total energies of ṡ(l+1) and ḋ
Esl+1 =
n
(l+1)
are given by Esl+1 and Edl+1 , respectively
(l+1) 2
[ṡi
] ,
(26)
i=1
Edl+1 =
n
(l+1) 2
[ḋ i
]
(27)
i=1
The total energies of c(l+1) and r (l+1) are expressed by Ecl+1 and Erl+1 , respectively
Ecl+1 =
n
(l+1) 2
[ci
]
(28)
i=1
Erl+1 =
n
(l+1) 2
[ri
]
(29)
i=1
If the error propagation is controlled, the total energy of original signal should be approximately equal to that of
decomposition results, i.e. Esl = Ecl+1 + Erl+1 . Furthermore, the approximation signal ṡ(l+1) cannot bring the error
propagation in decomposition results using redundant lifting scheme. The normalization factors are calculated by the
following equations:
al+1 = 1
bl+1 =
(30)
Esl − Esl+1
Edl+1
(31)
The inspected signal shown in Fig. 6 is performed 1-level decomposition by using IRLS. The results are given
in Fig. 10. Compared with Fig. 6, the total energy of detail signal r(1) and approximation signal c(1) is 98.26, which
is equal to that of the original signal. The range of the detail signal varies from [−1.5, 1.5] to [−0.8, 0.8], and the
distortion of the decomposition results using redundant lifting scheme is eliminated by the normalization factors.
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4. Hilbert transform and SPM and the proposed method
4.1. Hilbert transform
As is well known, for a signal x(t), the Hilbert transform is defined by
1 ∞ x(t)
H[x(t)] =
dτ
π −∞ t − τ
Demodulation is accomplished by forming a complex-valued signal A[x(t)], which is defined as
√
A[x(t)] = x(t) + iH[x(t)] = a(t) eiφ(t) , i = −1
(32)
(33)
The complex time-domain signal can be converted from the real/imaginary format to the magnitude/phase format,
a(t) = x2 (t) + H 2 [x(t)],
(34)
φ(t) = arctan
H[x(t)]
x(t)
(35)
The envelope represents an estimate of the modulation phenomenon in the signal. In addition, it removes carrier
signals and decreases the influence of irrelevant information for bearing fault detection.
4.2. The shock pulse method (SPM)
The shock pulse method (SPM) [31] has been developed by SPM Instrument AB in the early 1970s in Sweden. It
relies on the fact that when a ball or roller contacts with a damaged area of raceway or with debris in the bearing, an
impulse of vibration is generated. The maximum normalized shock value (dB) gives an indication of bearing condition.
It is defined as,
dB = 20 log
2000 SV
ND0.6
(36)
where N denotes rotating speed of the bearing, and D is the inner diameter. SV is the shock value, which should be
calculated by demodulation of bearing vibration signals. With dB value, the bearing running state can be estimated as
[31]:
⎫
(1) 0 ≤ dB < 20 healthy bearing ⎪
⎬
(2) 20 ≤ dB < 35 weak fault
(37)
⎪
⎭
(1) 35 ≤ dB < 60 heavy fault
The bounds of three condition zones were empirically established by testing bearings under variable operating
condition in the early 1970s. For over 35 years, based on three condition zones, the shock pulse method (SPM)
has been very successfully used to obtain a reliable diagnosis of the operating condition of rolling element bearings
[1,12,13,18,26]. Additionally, the bounds have been applied for the design of shock pulse meters, which have achieved
wide acceptance in industries.
4.3. The proposed method
The SPM combined with demodulation method is a useful method of estimating bearing running state. The maximum
normalized shock value (dB) is used as a measure of bearing condition. However, the direct demodulation may
mistakenly estimate the shock value indicating the bearing condition. The IRLS is considered to preprocess the
vibration signal of rolling bearing before demodulation. Fig. 11 shows the flow chart of the proposed method for
bearing condition monitoring.
L. Zhen et al. / Mathematics and Computers in Simulation 79 (2008) 318–338
333
Table 1
The geometric parameters of the tested bearing
Ball diameter d (mm)
Pitch diameter E (mm)
Contact angle α (◦ )
Number of rolling elements z
68
450
0
17
5. The analysis of simulation experiment
To verify the feasibility of the proposed method for bearing condition monitoring, we first use a simulated signal.
The simulated signal is composed of two periodic impulse signals. One is described as 0.1 e−2π×16t sin(2π × 2000t)
with a period of 0.0167 s, and the other is expressed as 0.1 e−2π×25t sin(2π × 5000t) with a period of 0.0125 s.
The periodic impulse signal with a period of 0.0167 s and its Hilbert spectrum are shown in Fig. 12. The peak value
of 60 Hz (1/0.0167 s) is 0.0147. The other with a period of 0.0125 s and its Hilbert spectrum are shown in Fig. 13. The
peak value of 80 Hz (1/0.0125 s) is 0.0105.
The simulated signal is shown in Fig. 14a. It is directly demodulated by the Hilbert transform. The Hilbert spectrum
is shown in Fig. 14b. Theoretically, the demodulation spectrum should have two peaks at 60 Hz and 80 Hz, and their
values are equal to 0.0147 and 0.0105. But from Fig. 14b, the peak values of 60 Hz and 80 Hz are 0.0096 and 0.0064,
respectively. They are all less than real values. It indicates that the direct demodulation may mistakenly estimate useful
information.
To extract accurately the useful information from the simulated signal, the simulated signal is performed 4-level
decomposition by using IRLS. The results are shown in Fig. 15. The two periodic impulse signals are at the different
levels. The Hilbert spectrums of detail signals r(1) and r(2) are shown in Fig. 16. The peak value of 60 Hz (1/0.0167 s)
is 0.0145 in Fig. 16a, and the peak value of 80 Hz (1/0.0125 s) is 0.0102 in Fig. 16b. The relative errors of the peak
values at 60 Hz and 80 Hz with their real values are 1.361% and 2.857%, respectively. They are approximately equal
to the real values. Therefore, the IRLS is very effective as a preprocessor for feature extraction.
To demonstrate the effectiveness of the IRLS, the same signal is analyzed by using redundant lifting scheme for
comparison. Fig. 17 shows decomposition results using redundant lifting scheme. Since the error propagation is not
(1)
(2)
controlled, it brings the distortion of detail signals. The Hilbert spectrums of detail signals ḋ and ḋ are shown in
Fig. 18. The peak value of 60 Hz (1/0.0167 s) is 0.0289 in Fig. 18a, and the peak value of 80 Hz (1/0.0125 s) is 0.0191
in Fig. 18b. They far exceed the real values.
The wavelet transform based on lifting scheme, which was introduced in [7], is also used to preprocess the simulated
signal. The decomposition results are shown in Fig. 19. The peak value of 60 Hz is 0.0132 in Fig. 20a, and the peak
value of 80 Hz is 0.0094 in Fig. 20b. Compared with direct demodulation, the Jiang’s method is effective. However,
the relative errors of the peak values at 60 Hz and 80 Hz with their real values are 10.20% and 10.38%, respectively.
They are more than the relative errors of decomposition results using IRLS.
6. The analysis of practical bearing vibration signals
In the present work, the propose method is used to evaluate the bearings under different conditions. The geometric
parameters of the roller bearing are listed in Table 1. The vibration signals are picked up at a constant speed of 390 rpm.
Based on the geometric parameters and the rotating speed of the bearing, the characteristic frequency of the outer-race
defects can be calculated by the following formula,
fo =
1
f
2
1−
d
cos α z
E
(38)
Using the above formula, the characteristic frequency of outer-race defect is calculated to be at 46.88 Hz.
Fig. 21 shows the bearing vibration signals under the condition of health, weak outer-race defect and heavy outerrace defect. Although the evident impulses occur in the vibration signal with heavy outer-race defect, it is hardly
possible to evaluate the bearing condition only through such signals.
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Fig. 29. Maximum normalized values of bearing under different conditions using redundant lifting scheme decomposition.
The vibration signals are performed 4-level decomposition by using IRLS. The decomposition results of the vibration
signal of healthy bearing are shown in Fig. 22. The amplitude of characteristic frequency is very small in the Hilbert
spectrum shown in Fig. 23. The decomposition results of the vibration signal with weak outer-race defect are shown
in Fig. 24. The detail signal r(1) contains the predominant impulses, and its Hilbert spectrum is shown in Fig. 25. The
characteristic frequency of 46.88 Hz is exacted clearly. Fig. 26 illustrates the decomposition results of the vibration
signal with heavy outer-race defect. The Hilbert spectrum of detail signal at the first level is shown in Fig. 27. From
Fig. 27, the amplitude of the characteristic frequency (46.88 Hz) is far more than that in Fig. 25.
Based on the IRLS decomposition and Hilbert spectrum, the maximum normalized shock values (dB) of bearings
under different conditions are given in Fig. 28. The dB value of healthy bearing is 16.5, indicating good bearing
condition. The dB value of bearing with weak outer-race defect is 31.6, indicating caution zone. The maximum
normalized value of bearing with heavy outer-race defect is 49.8, indicating the damaged bearing condition.
For comparison, the redundant lifting scheme is used to preprocess the same vibration signals. The maximum
normalized values of bearings are shown in Fig. 29. From Fig. 29, the dB value of bearing with weak outer-race
Fig. 30. The EMD decomposed results of the vibration signal of healthy bearing.
L. Zhen et al. / Mathematics and Computers in Simulation 79 (2008) 318–338
335
Fig. 31. The Hilbert spectrum of IMF1 in Fig. 30.
Fig. 32. The EMD decomposed results of the vibration signal with weak outer-race defect.
defect is 37.3 greater than 35. It indicates that the bearing condition with weak defect is mistakenly estimated as heavy
defect.
The Hilbert–Huang transform is also applied to analyze the bearing vibration signals under different conditions.
The vibration signals are decomposed into some intrinsic mode functions (IMFs) by empirical mode decomposition
(EMD) [6]. The decomposition results using EMD are given in Figs. 30–34, respectively. Fig. 31 shows the Hilbert
spectrum of IMF1 in Fig. 30. Fig. 33 shows the Hilbert spectrum of IMF1 in the EMD decomposed results of the
vibration signal with weak outer-race defect.
Fig. 33. The Hilbert spectrum of IMF1 in Fig. 32.
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L. Zhen et al. / Mathematics and Computers in Simulation 79 (2008) 318–338
Fig. 34. The EMD decomposed results of the vibration signal with heavy outer-race defect.
Fig. 35. The Hilbert spectrum of IMF1 in Fig. 34.
Fig. 36. The decomposed results using STFT of the vibration signal with weak outer-race defect.
L. Zhen et al. / Mathematics and Computers in Simulation 79 (2008) 318–338
337
Fig. 37. The Hilbert spectrum at the frequency range (3200 Hz and 4000 Hz) in Fig. 36.
For the heavy outer-race defect, the IMF1 includes the most dominant fault information. The Hilbert spectrum of
IMF1 shown in Fig. 35 can reveal clearly the characteristic frequency of outer-race defect. Compared with the results
shown in Fig. 25, the fault components caused by weak outer-race defect are difficult to identify using visual inspection
in Figs. 32 and 33. The short time Fourier transform (STFT) [28] and envelope detector are used to analyze the signal
with weak outer-race defect. The results are given in Figs. 36 and 37. The fault features are not extracted effectively
from Figs. 36 and 37. Therefore, the proposed method is more effective to evaluate the bearing condition.
7. Conclusion
The method of combining IRLS and SPM has been proposed for bearing condition monitoring. Based on the analysis
of the cause of decomposition distortion using the redundant lifting scheme, the classical redundant lifting scheme has
been improved by adding the normalization factors. With the IRLS and SPM, the bearing condition can be accurately
estimated. The proposed method is verified by simulated signals and bearing vibration signals. The results demonstrate
that the proposed method is more effective for bearing condition monitoring.
Acknowledgements
This work was supported by the key project of National Natural Science Foundation of China (No. 50335030),
National High-tech R&D Program of China (863 Program) (2006AA04Z430), National Basic Research Program of
China (No. 2005CB724100).
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