CHINESE JOURNAL OF MECHANICAL ENGINEERING
·50·
Vol. 21,aNo. 5,a2008
DOI: 10.3901/CJME.2008.05.050, available online at www.cjmenet.com; www.cjmenet.com.cn
ZHOU Xiong
College of Mechanical Engineering,
Chongqing University,
Chongqing 400044, China
Engineering Training Center,
Chongqing University of Science
and Technology,
Chongqing 401331, China
SEQUENTIAL DIAGNOSIS FOR
A CENTRIFUGAL PUMP BASED
ON FUZZY NEURAL NETWORK*
WANG Huaqing
College of Mechanical and
Electrical Engineering,
Beijing University of Chemical
Technology,
Beijing 100029, China
CHEN Peng
Graduate School of Bioresources,
Mie University,
Tsu 514-8507, Japan
TANG Yike
College of Mechanical Engineering,
Chongqing University,
Chongqing 400044, China
0
Abstract: A sequential diagnosis method is proposed based on a fuzzy neural network realized by
“the partially-linearized neural network (PNN)”, by which the fault types of rotating machinery can be
precisely and effectively distinguished at an early stage on the basis of the possibilities of symptom
parameters. The non-dimensional symptom parameters in time domain are defined for reflecting the
features of time signals measured for the fault diagnosis of rotating machinery. The synthetic
detection index is also proposed to evaluate the sensitivity of non-dimensional symptom parameters
for detecting faults. The practical example of condition diagnosis for detecting and distinguishing
fault states of a centrifugal pump system, such as cavitation, impeller eccentricity which often occur
in a centrifugal pump system, are shown to verify the efficiency of the method proposed in this paper.
Key words: Sequential diagnosis Fuzzy neural network Symptom parameter Centrifugal pump
Rotating machinery
INTRODUCTION∗
In the field of machinery diagnosis, vibration signals are
often used for the detection of mechanical fault and the
discrimination of fault types. Machinery diagnosis depends
largely on the feature analysis of vibration signals measured for
condition diagnosis, so it is important that the feature of the signal
should be sensitively extracted when fault occurs at the state
change of a machine. However, the feature extraction for the fault
diagnosis is difficult since the vibration signals measured at a
point of the machine often contains strong noise [1−3].
In most cases of condition diagnosis for plant rotating
machinery, information for distinguishing faults is ambiguous
because definite relationships between symptoms and fault types
cannot be easily identified [4]. The main reasons can be explained
as follows: ① it is difficult to identify the symptom parameters
for diagnosis by which all fault types can be distinguished
perfectly; ② in an early stage of a fault, the effect of noise in the
signal measured for the diagnosis is so strong that the symptom of
the fault is not evident.
Furthermore, although many method of condition diagnosis
for rotating machinery using neural networks (NN) have been
reported by many studies, they almost dealt with the
discrimination of fault types [5−11]. The problems of the condition
diagnosis using conventional NN are that the NN can not reflect
the possibility grades of the ambiguous diagnosis problems, and it
will never converge when the symptom parameters inputted in the
1st layer have the same values in different states [12].
Based on the above reasons, in order to improve the
efficiency of fault diagnosis and distinguish fault types at an early
stage, this paper proposes a sequential diagnosis method for
rotating machinery using fuzzy neural network by which the state
of machinery can be automatically judged on the basis of the
possibility grades of normal and each abnormal state. Since the
relationship between the values of the symptom parameters and
fault types is ambiguous due to the effect of noise in the time
signals, we use the partially-linearized neural network to solve the
ambiguous problem of the fault diagnosis. Non-dimensional
* This project is supported by Sci-Tech Planning Projects of Chongqing City,
China (No. CSTC2007AA7003). Received January 29, 2008; received in
revised form August 18, 2008; accepted August 20, 2008
symptom parameters (NSP) in time domain are defined to reflect
the features of time signal measured for the fault diagnosis of
rotating machinery. The synthetic detection index (S) is also
proposed to evaluate the sensitivity of NSPs for detecting and
distinguishing faults. In this paper, practical example of fault
diagnosis of a centrifugal pump system will verify that the method
is effective. The method proposed in this paper can also be
applied to other type of rotating machinery.
1
CENTRIFUGAL PUMP SYSTEM FOR FAULT
DIAGNOSIS
The centrifugal pump system for the condition diagnosis is
shown in Fig. 1.
(b) Equipment of centrifugal pump on field
Fig. 1
Centrifugal pump system for the condition diagnosis
CHINESE JOURNAL OF MECHANICAL ENGINEERING
The states to be diagnosed for the centrifugal pump system
are normal state, cavitation, impeller eccentricity. Cavitation
phenomenon is one of the sources of instability in a centrifugal
pump. Cavitation can cause more undesirable effects, such as
deterioration of the hydraulic performance (drop in head-capacity
and efficiency curves), damage of the pump by pitting and erosion
and structure vibration and resulting noise [13]. To prevent the
occurrence of the cavitation, we have to detect it at early stage of
cavitation phenomenon from the vibration signal of a pump
system.
There were six accelerometers used to measure vibration
signals for the detection of faults. The locations of the sensors are
shown in Fig. 2. Two sensors were put at the pump inlet, another
two at the pump outlet and one sensor at the motor and pump
housing respectively. The sampling frequency of signal measurement is 50 kHz, and the sampling time is 10 s. The vibration
signals in each state shown in Fig. 3 are measured at a constant
speed (2 500 r/min) and constant flow rate (13.5 m3/h) of water.
Fig. 2
·51·
For the purpose of making the signals comparable regardless
of differences in magnitude, the filtered signals of each state are
normalized by using the following formula before calculating the
symptom parameters
xi =
xi′ − μ ′x
σ′
(1)
where μ ′ and σ ′ are the mean and standard deviation of the
filtered signals series xi′(i = 1, 2," , N ) , respectively. xi is the
ith element of the signal series after normalization.
The location of the sensors
Fig. 4
Signals after filtration
Using the normalized signals, the ten NSPs in the time
domain are calculated as follows, respectively
N
p1 =
∑ ( xi − μ )3
i =1
( N − 1)σ 3
(2)
N
p2 =
∑ ( xi − μ ) 4
i =1
( N − 1)σ 4
p3 =
σp
μp
(3)
(4)
M
Fig. 3
Original signals measured in each state
p4 =
Low-pass filter with 800 Hz cut-off frequency is used for
canceling the noise. The vibration signals measured in each state
after filtration are shown in Fig. 4.
SYMPTOM PARAMETERS FOR FAULT
DIAGNOSIS
Non-dimensional symptom parameter
Many symptom parameters have been defined in the pattern
recognition field [14]. Here, ten of non-dimensional symptom
parameters (NSP) in time domain, commonly used for the fault
diagnosis of plant machinery, are considered.
i =1
( M − 1)σ p 3
(5)
M
p5 =
2
∑ ( x pi − μ p )3
∑ ( x pi − μ p )4
i =1
( M − 1)σ p 4
p6 =
2.1
σv
μv
(6)
(7)
K
p7 =
∑ ( xvi − μv )3
i =1
( K − 1)σ v 3
(8)
YZHOU Xiong, et al: Sequential diagnosis for a centrifugal pump based on fuzzy neural networky
·52·
the synthetic detection index (S) is defined as follows
K
p8 =
∑ ( xvi − μv )4
i =1
( K − 1)σ v 4
p9 =
M
Z
K
p10 =
Z
(9)
M
S = ∑ DI i
(17)
i =1
(10)
(11)
3
SEQUENTIAL DIAGNOSIS METHOD
For the purpose of distinguishing faults effectively, a
sequential diagnosis method is proposed (Fig. 5). The inference of
the sequential diagnosis is as follows.
where, μ and σ are the mean and standard deviation of
normalized signal series xi , respectively, x pi (i=1, 2, " , M) and
xvi (i=1, 2, " , K) are the peak value and valley value of xi,
respectively, μ p and σ p are the mean and standard deviation
of peak values x pi , respectively, μv and σ v are the mean and
standard deviation of the valley values xvi , respectively, and
Z is the number of xi passing zero.
2.2
Sensitivity evaluation of symptom parameter
The quality of a symptom parameter (SP), which will be used
for distinguishing two states, such as normal or abnormal state, is
derived in the following way. Supposing that x1 and x2 are the SP
values calculated from the signals measured in state 1 and state 2,
respectively, and they conform respectively to the normal
distributions N ( μ1 ,σ 12 ) and N ( μ2 ,σ 2 2 ) . Where, μ and σ
are the average and the standard deviation of the SP. The larger the
value of x1 − x2 is, the higher the sensitivity of distinguishing
the two states by the SP is. Because Z = x2 − x1 is conform to the
normal distribution N ( μ 2 − μ1 ,σ 2 2 + σ 12 ) , we have the density
function about Z
f ( z) =
[15−16]
⎛ [ z − ( μ2 − μ1 )]2 ⎞
⎟
exp ⎜
⎜ 2(σ 12 + σ 2 2 ) ⎟
2π(σ 12 + σ 2 2 )
⎝
⎠
1
(12)
where, μ 2 ≥ μ1 (we can obtain the same conclusion when
μ 2 ≥ μ1 ). The probability of x2 < x1 can be calculated by the
following formula
0
P0 = ∫−∞ f (z )dz
where 1 − P0
substitution
(13)
is called “distinction rate (DR)”. With the
μ=
z − ( μ2 − μ1 )
σ 12 + σ 2 2
(14)
From Eqs. (14) and (15), the P0 can be obtained by
P0 =
1
2π
− DI
∫−∞
⎛ u2 ⎞
exp ⎜ − ⎟ du
⎝ 2⎠
(15)
Fig. 5
Flow chart of sequential diagnosis for a pump
In the first step, if the possibility grades of normal state (N)
and abnormal states (A) are gN and gA, and gN>gA, then the state is
judged as “normal state (N)”, else proceed to the next step.
In the second step, if the possibility grades of cavitations (C)
and other abnormal faults (A) are gC and gA, and gC>gA, then the
state is judged as “cavitations (C)”, else proceed to the next step.
In the third step, if the possibility grades of impeller
eccentricity (E) and another fault are gD and gA, respectively, and
gD>gA, then the state is judged as “impeller eccentricity (E)”, else
it will be judged as “unknown abnormal state (UA)”.
In this paper, two best NSP pi and p j , are selected by Eq.
(17) for each state of the sequential diagnosis, respectively. The
selection results of the NSP are as follows.
p9 and p10 : for the first step to distinguish normal state
from abnormal states;
p1 and p2 : for the second step to distinguish cavitation
from other abnormal states;
p1 and p2 : for the third step to distinguish impeller
eccentricity from unknown states.
It shows the DI of the NSP ( pi and p j ) for each step to
distinguish the two kinds of state, respectively (Tables 1, 2). Since
all of those DI are larger than 2.4, all of the detection rates (DR)
are larger than 99%.
Table 1 DI of NSP for the first step
NSP
N:C
N:E
P9 p9
8.3
7.9
P10
3.5
2.4
Table 2 DI of NSP for the second step
and third step
NSP
C:E
p1
2.9
p2
3.0
The training data for the learning of fuzzy neural network
can be obtained by the values of NSP calculated by the vibration
signal measured in each state. If the pi and pj are selected for
distinguishing state k from another state, and their mean value and
standard deviation are pik , p jk and σik , σ jk , respectively, the
(16)
training data for distinguishing state k from another state are
calculated as follows.
If pik − 2σ ik < pi < pik + 2σ ik and p jk − 2σ jk < p j < p jk +
It is obvious that the larger the value of DR is, the larger the
value of DR (DR =1−P0) will be, and therefore, the better the SP
will be. So DI can be used as the index of the quality to evaluate
the distinguishing sensitivity of SP.
If the number of symptom parameters used for diagnosis is M,
else the state is judged as another state.
According to the method stated above, these show the
training data for distinguishing normal state, cavitation, impeller
eccentricity unknown states, respectively (Figs. 6−8). These
training data will be used for the learning of the fuzzy neural
network.
where, distinction index DI is calculated by
DI =
μ2 − μ1
σ 12 + σ 2 2
2σ jk , then the state is judged as state k with 100% possibility,
CHINESE JOURNAL OF MECHANICAL ENGINEERING
·53·
the symbols are used as follows.
X ( m,t ) is the value of the tth neuron in the hidden (the mth)
layer; t = 1 , 2, " , N m .
Fig. 6
Training data for distinguishing normal state from abnormal state
Wuv( m ) is the weight between the uth neuron in the mth layer
and the vth neuron in the (m+1)th layer; m=1, 2, " , M; u=1, 2, " ,
Nm; v=1, 2, " , Nm+1.
If all of these values are remembered by computer, when new
u )∗
u)
u )∗
u)
values X (1,
( X (1,
< X (1,
< X (1,
j
j
j
j +1 ) are inputted to the first
layer, the predicted value of the vth neuron ( v = 1 , 2, " , N m ) in
the (m+1)th layer ( m = 1, 2, " , M−1) will estimated by
Nm
X (j m +1, v ) = X i(+m1+1, v ) −
Fig. 7
{∑ Wuv( m ) ( X i(+m1,u ) − X (j m ,u ) )}( X i(+m1+1, v ) − X (j m +1, v ) )
v =0
Training data for distinguishing cavitation from another fault
Nm
∑ Wuv( m) ( X i(+m1,u ) − X i( m,u ) )
v =0
(18)
In the above way, the sigmoid function is partially linearized
(Fig. 9). If a function must be learned, the PNN will learn the
points indicated by symbol ● shown (Fig. 10). When new data
( S1′ , S′2 ) are input into the converged PNN, the values indicted by
symbol ■ corresponding to the data ( S1′ , S ′2 ) will be quickly
Fig. 8
4
Training data for distinguishing eccentricity from unknown faults
identified as Pe. Thus, the PNN can be used for dealing with
ambiguous diagnosis problems.
FUZZY NEURAL NETWORK FOR THE
DIAGNOSIS AND VERIFICATION
4.1
Partially-linearized neural network
The partially-linearized neural network (PNN) can solve
ambiguous diagnosis problems, and can distinguish fault types on
the basis of the probability distributions of the machine conditions
when the diagnosing is being done[4]. In present work, the PNN is
implemented for fault diagnosis, and the basic principle of the
PNN is described as follows.
The neuron number of the mth layer of an NN is Nm. The set
X (1) = { X i(1, j ) } expresses the pattern inputted to the 1st layer and
the set X ( M ) = { X i( M , k ) } is the training data for the last layer (the
Mth layer). Here, i=1, 2, " , P, j=1, 2, " , N1, k=1, 2, " , NM.
X i(1, j ) is the value inputted to the jth neuron in the input (the
1st) layer.
X i( M , k ) is the output value of the kth neuron in the output (the
Mth) layer.
Even if the NN converge by learning X (1) and X ( M ) , it
cannot adequately deal with the ambiguous relationship between
new X (1)∗ and X ( M )∗ , which have not been learned. In order to
predict X i ( M )∗ according to the probability distribution of X (1)∗ ,
partially linear interpolation of the NN is introduced (Fig. 9), we
called it “partially-linearized neural network (PNN)”.
Fig. 10
Interpolation by the PNN
The new data ( S1′ , S′2 ) inputted into the converged PNN,
which are not learned by the PNN for recognition, must satisfy the
following condition
S1(min) < S1′ < S1(max) and S 2(min) < S 2′ < S 2(max)
(19)
where S1(min) , S 2(min) and S1(max) , S 2(max) are the minimum and
maximum value of S1 and S 2 , which have been learned by the
PNN.
Therefore, in this paper the verify values of NSP ( pi* and
p*j ) input to the PNN for distinguishing the state k, must satisfy
the following condition
pik (min) < pi* < pik (max) and p jk (min) < p*j < p jk (max)
(20)
where pik (min) , p jk (min) and pik (max) , p jk (max) are the minimum
and maximum values of pi and p j , respectively.
4.2
Fig. 9
Partial linearization of the sigmoid function
In the NN which has converged by the data X (1) and X ( M ) ,
Diagnosis and verification
A back propagation neural network is only used for training
the data, and the PNN is used for testing the learned NN. Fig. 11
shows the PNN built for the fault diagnosis of a centrifugal pump
system on the basis algorithm of sequential diagnosis (Fig. 5). The
training data for the PNN learning are shown (Figs. 6−8). The
PNN are quickly convergent when the training data are being
learned. We used the data measured in each known state, which
have not been learned by the PNN to verify the diagnostic
capability of the PNN. Examples of fault diagnosis by the learnt
PNN are shown in Tables 3−5). In the cases of the verifications,
the data input to the learned PNN have not been trained, and the
YZHOU Xiong, et al: Sequential diagnosis for a centrifugal pump based on fuzzy neural networky
·54·
PNN correctly and quickly judged the states (N: normal state, C:
cavitation, E: impeller eccentricity, A: abnormal faults, UA:
unknown abnormal states), which are expressed by the
possibilities gN, gC, gE, gA and gUA.
Fig. 11 PNN for the fault diagnosis of a centrifugal pump system
Table 3
P9
1.244
1.288
Verification results for the first step
P10
1.17
1.18
gN
0.985
0.978
gA
0.005
0.007
Judge
N
N
#
#
#
#
#
1.454
1.339
1.309
1.226
0.044
0.054
0.95
0.94
A
A
#
#
#
#
#
Table 4
P1
0.033
0.012
Verification results for the second step
P2
2.175
2.185
gC
0.988
0.986
gA
0.0003
0.001
Judge
C
C
#
#
#
#
#
0.146
0.182
2.584
2.804
0.032
0.003
0.943
0.96
A
A
#
#
#
#
#
Table 5
P1
0.176
0.024
Verification results for the third step
P2
2.777
2.618
gE
0.998
0.672
gUA
0.002
0.274
Judge
E
E
#
#
#
#
#
0.339
0.317
2.739
2.756
0.034
0.131
0.938
0.93
UA
UA
#
#
#
#
#
According to the test results, the probability grades output by
the PNN show the correct judgment in each state. Therefore, the
PNN can precisely distinguish the type of pump system fault on
the basis of the possibility distributions of symptom parameters.
New experimental data are used (Tables 3−5).
5
CONCLUSIONS
(1) For the purpose of improving the efficiency of the
condition diagnosis for plant rotating machinery and
distinguishing fault types at an early stage, a sequential diagnosis
method for a pump using fuzzy neural network by which the state
of machinery can be automatically judged on the basis of the
possibility grades of the normal and each abnormal state is
proposed.
(2) Since the relationship between the values of the symptom
parameters and fault types is ambiguous owing to the effect of
noise in the time signals, PNN as a fuzzy neural network and the
possibility grade were applied to solve the ambiguous problem of
the condition diagnosis. NSP in time domain were defined, which
can reflect the characteristics of time signal measured for the fault
diagnosis of rotating machinery. The synthetic detection index
was also proposed to evaluate the sensitivity of NSP for detecting
and distinguishing faults.
(3) The practical example of diagnosis of a centrifugal pump
system for detecting fault states, such as cavitation and impeller
eccentricity which often occured in pump, were shown to verify
the efficiency of the method proposed. The method proposed can
also be applied to other type of rotating machinery.
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Biographical notes
ZHOU Xiong is currently a vice professor in Engineering Training Center,
Chongqing University of Science and Technology, China. He received his
master’s degree from College of Mechanical Engineering, Chongqing
University, China, in 1998. He is also a PhD candidate in College of
Mechanical Engineering, Chongqing University, China. His research interests
include fault diagnosis.
Tel: +86-23-65022027; E-mail: cq_mecc@yahoo.com.cn
WANG Huaqing received his BS degree and MS degree from School of
Mechanical and Electrical Engineering, Beijing University of Chemical
Technology, China, in 1995 and 2002, respectively. He is currently a teacher in
Beijing University of Chemical Technology and a doctoral candidate in Mie
University, Japan. His research interest includes fault diagnosis of plant
machinery and signal processing.
Tel: +86-10-64446043; E-mail: wanghq_buct@hotmail.com
CHEN Peng graduated from the doctoral course of the Kyushu University,
Japan in 1990, and is currently a professor in Department of Environmental
Science and Technology, Mie University, Japan. His research interest includes
condition diagnosis of plant machinery, information and signal processing.
Tel: +8159-2319592; E-mail: chen@bio.mie-u.ac.jp
TANG Yike was born in 1949. He is currently a professor and a doctoral
advisor in College of Mechanical Engineering, Chongqing University, China.
His research interests include mechanical vibration and fault diagnosis.
Tel: +86-23-65022027; E-mail: yktang@cqu.edu.cn