Intelligent diagnosis method for rolling element ... - Semantic Scholar
Computers & Industrial Engineering 60 (2011) 511–518
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Intelligent diagnosis method for rolling element bearing faults using possibility theory and neural network q Huaqing Wang a,⇑, Peng Chen b,⇑ a b
School of Mechanical & Electrical Engineering, Beijing University of Chemical Technology, Chaoyang District, Beisanhuan East Road 15, Beijing 100029, China Department of Environmental Science and Engineering, Faculty of Bioresources Mie University, 1577 Kurimamachiya-cho, Tsu-shi, Mie-ken 514-8507, Japan
a r t i c l e
i n f o
Article history: Received 17 March 2008 Received in revised form 2 March 2010 Accepted 3 December 2010 Available online 13 December 2010 Keywords: Fault diagnosis Neural network Possibility theory Rolling element bearing Centrifugal blower
a b s t r a c t This paper presents an intelligent diagnosis method for a rolling element bearing; the method is constructed on the basis of possibility theory and a fuzzy neural network with frequency-domain features of vibration signals. A sequential diagnosis technique is also proposed through which the fuzzy neural network realized by the partially-linearized neural network (PNN) can sequentially identify fault types. Possibility theory and the Mycin certainty factor are used to process the ambiguous relationship between symptoms and fault types. Non-dimensional symptom parameters are also defined in the frequency domain, which can reflect the characteristics of vibration signals. The PNN can sequentially and automatically distinguish fault types for a rolling bearing with high accuracy, on the basis of the possibilities of the symptom parameters. Practical examples of diagnosis for a bearing used in a centrifugal blower are given to show that bearing faults can be precisely identified by the proposed method. Ó 2010 Elsevier Ltd. All rights reserved.
1. Introduction A rolling element bearing is an important and widely used component of rotating machinery. A fault in a rolling element bearing may cause breakdown of machinery, and serious consequences may arise due to the fault (Lou & Kenneth, 2004; Williams, Ribadeneira, Billington, & Kurfess, 2001; Yang, Mathew, & Ma, 2005, 2006). Therefore, fault diagnosis of rolling element bearings is very important for guaranteeing production efficiency and plant safety. In the field of fault diagnosis, utilization of vibration signals is an effective method for the detection of faults and discrimination of fault types, because the vibration signals carry dynamic information about the machine state (Liu & Ling, 1999; Pusey, 2000; Yang, Stronach, & MacConnell, 2003). Intelligent systems such as neural networks (NNs) have potential applications in pattern recognition and failure diagnosis. Many studies have been carried out to investigate the use of NNs for automatic diagnosis of machinery; most of these methods have been proposed to deal with discrimination of fault types collectively (Alguindigue, Loskiewicz-Buczak, & Uhrig, 1993; Fang, 2006; McCormick & Nandi, 1997; Samanta, Al-Balushi, & Al-Araimi, 2006; Schetinin & Schult, 2006; Wang & Chen, 2009). However, the conventional NNs cannot reflect the possibility of ambiguous diagq
This manuscript was processed by Area Editor Satish Bukkapatnam.
⇑ Corresponding authors. Tel./fax: +81 592319592.
E-mail addresses: [email protected] (H. Wang), [email protected] (P. Chen). 0360-8352/$ - see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.cie.2010.12.004
nosis problems, and will never converge when the first layer symptom parameters have the same values in different states (Christopher, 1995). Furthermore, the number of the faults that are represented in the diagnostic support system is extremely small compared to all the possible faults, and the current knowledge base is therefore very incomplete. The symptoms are also incomplete: many observables are not fully monitored in real time. Knowledge of distinguishing fault is ambiguous in real plant, because the definite relationships between symptom parameters and fault types even in the single fault cannot be easily identified. In addition, the number of fault states to be identified is enormous, and it is very hard to find one symptom parameter or a few symptom parameters that can identify all of those faults simultaneously (Chen & Toyota, 2003). The values of symptom parameters calculated from vibration signals for fault diagnosis are also ambiguous because of the dispersion in the same state. Therefore, it is necessary to solve the ambiguous problem of fault diagnosis and to express uncertainty about the interpretation of the observable. For the above reasons, an intelligent diagnosis method for bearing faults is proposed on the basis of possibility theory and a fuzzy neural network with frequency-domain features of vibration signals. This method is intended to process the ambiguous relationship between the symptom parameters and fault types, and automatically identified faults. A sequential diagnosis approach is also proposed through the partially-linearized neural network (PNN) to identify sequentially the bearing faults. Diagnostic knowledge for the PNN is acquired by possibility theory and the Mycin certainty factor for solving the ambiguous problem of fault
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Fig. 1. The blower system for bearing fault diagnosis, (a) illustration of the blower system, (b) the blower in the field.
diagnosis. In this work, bearing faults are considered which often occur at an early stage, such as the outer race flaw, the inner race flaw, and the roller element flaw. Those faults can be sequentially and automatically distinguished on the basis of the possibility distributions of symptom parameters. Practical examples of fault diagnosis for a rolling bearing used in a centrifugal blower are shown to verify the efficiency of the proposed method. 2. Experimental system for bearing diagnosis In order to verify the efficiency of the methods proposed in this paper, the experimental system of bearing diagnosis is used. Fig. 1 shows the experimental system, including the centrifugal blower (TERAL CLF3), the rolling bearing (NSK 206) and the accelerometers. A 2.2 kW induction motor with three-phases and a maximum revolution of 1420 rpm is employed to drive the blower through two V-belts, and its rotating speed can be varied by a speed controller. NSK 206 bearings were utilized in this study, and specifications of the test bearing, the size of the faults, and other necessary information are listed in Table 1. The faults that most frequently occur in a rolling element bearing at an early stage are the outer race, inner race, and roller element flaws. These faults, shown in Fig. 2, were artificially induced with the use of a wire-cutting machine. As shown in Fig. 3, the two accelerometers are mounted on the bearing housing at the output end of the blower-shaft in order to
measure the vibration signals in the vertical and horizontal directions, respectively. The sampling frequency of the vibration signals is 100 kHz for each channel, and the sampling time is 5 s. The centrifugal blower usually works at a constant speed, and faults often occur at a steady condition; therefore, in the present work, we only considered the steady state. The vibration signals are measured at a constant rotating speed of 800 rpm. A high-pass filter with a 5 kHz cut-off frequency was used to cancel noise in the vibration signals for fault diagnosis. Examples of vibration signals measured in each state after filtering are shown in Fig. 4. The envelope spectrum of the filtered signals could then be obtained and used to calculate the symptom parameters. 3. Symptom parameters and sensitivity evaluation In order to automatically diagnose the machine states by computer, the characteristics of the envelope spectra must be expressed by the symptom parameters (SPs). In this section, we discuss the SPs in the frequency-domain and how to select the better SPs for condition diagnosis. 3.1. Symptom parameters in the frequency-domain In case of the fault diagnosis, the symptom parameters (SPs) usually are used for identifying the conditions of machinery, because they can express the condition information of the machine
Table 1 Bearing information for fault diagnosis. Bearing NSK
Normal (N)
With roller flaw (R)
With inner flaw (I)
With outer flaw (O)
Specification Flaw width Flaw depth Pitch diameter Roller number Roller diameter Pass-frequency
NU206EW – – – – – –
NF206W 0.6 mm 0.3 mm 46.0 mm 13 7.5 mm 79.6 Hz
NU206EW 0.6 mm 0.3 mm 46.5 mm 13 9.0 mm 103.4 Hz
NF206W 0.6 mm 0.3 mm 46.0 mm 13 7.5 mm 72.5 Hz
Fig. 2. Rolling element bearing flaws, (a) outer race flaw (O), (b) inner race flaw (I), (c) roller element flaw (R).
H. Wang, P. Chen / Computers & Industrial Engineering 60 (2011) 511–518
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where N is the number of spectrum lines, fi is the frequency, S(fi) is ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rP N ðf f Þ2 Sðfi Þ i¼1 i , the power spectrum of the enveloped waveform, r ¼ N1 P P and f ¼ Ni¼1 fi Sðfi Þ= Ni¼1 Sðfi Þ. When the rollers pass through the flaw, impulses may appear. The frequency of the impulse waveform is called the pass-frequency of a fault bearing, and can be calculated by the following formulas (Chen & Toyota, 2003).
Sensor1
Sensor2
2
Dfr fR ¼ d
d
!
1 2 cos a D zfr d fo ¼ 1 cos a 2 D zfr d 1 þ cos a fI ¼ 2 D
Fig. 3. The location of sensors.
ð8Þ ð9Þ ð10Þ
where fR, fo, and fI are the pass-frequency of the roller flaw, the outer race flaw, and the inner race flaw, respectively, z is the number of rolling elements, fr is a rotating frequency, dis a diameter of rolling elements, D is a pitch diameter, andais a contact angle of rolling element. The parameters of the relevant test bearings are shown in Table 1. The pass-frequencies of the roller flaw, the outer race flaw, and the inner race flaw can be obtained of 79.6 Hz, 72.5 Hz, and 103.4 Hz, respectively, by (8)–(10), as shown in Table 1. In this work, we determined five times the maximum pass-frequency (fI is 103.4 Hz) of about 500 Hz as the maximum calculating frequency, and the range of the calculating frequency fi was determined from 0 Hz to 500 Hz. 3.2. Sensitivity evaluation of symptom parameter
Fig. 4. Vibration signals of bearings after filtering (a) Normal state, (b) inner race flaw state, (c) roller race flaw state, (d) outer race flaw state.
indicated by the vibration signal. The SPs are random variables and can be calculated from the vibration signals. Commonly, the SPs can classified into the frequency-domain SPs and the time-domain SPs, and the two types of SPs indicate features of a signal in the frequency-domain and the time-domain, respectively. A large set of symptom parameters has been defined in the pattern recognition field (Fukunaga, 1972). In this study, seven of these parameters in the frequency-domain, which are commonly used for fault diagnosis of plant machinery, are considered (Chen & Toyota, 1998).
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi uPN 2 u i¼1 fi Sðfi Þ P1 ¼ t P N i¼1 Sðfi Þ vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi uPN 4 u fi Sðfi Þ P2 ¼ tPi¼1 N 2 i¼1 fi Sðfi Þ PN 2 i¼1 fi Sðfi Þ P3 ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi PN PN 4 i¼1 Sðfi Þ i¼1 fi Sðfi Þ
r
P4 ¼ f PN ðfi f Þ3 Sðfi Þ P5 ¼ i¼1 3 r N PN 4 i¼1 ðfi f Þ Sðfi Þ P6 ¼ 4 r N PN qffiffiffiffiffiffiffiffiffiffiffiffiffiffi fi f Sðfi Þ i¼1 pffiffiffiffi P7 ¼ rN
For automatic diagnosis, symptom parameters are needed that can sensitively distinguish the fault types. In order to evaluate the sensitivity of an SP for distinguishing two states, such as a normal or an abnormal state, the distinction index (DI) is defined as follows. Suppose that x1and 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 Nðl1 ; r21 Þ and Nðl2 ; r22 Þ, where, l and r are the average and the standard deviation of the SP. The larger jx2 x1j is, the higher the sensitivity of SP for distinguishing the two states. Because z = x2x1 is also normally distributed Nðl2 l1 ; r22 þ r21 Þ, we have the density function about z.
ð1Þ
2 ! z ðl2 l1 Þ 1 f ðzÞ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi exp 2ðr21 þ r22 Þ 2pðr21 þ r22 Þ
ð2Þ
where l2 P l1 (we can obtain the same conclusion when l1 P l2). The probability of x2 < x1 can be calculated by the following formula (Chen, Feng, & Toyota, 2002).
ð3Þ ð4Þ
P0 ¼
Z
ð11Þ
0
f ðzÞdz
ð12Þ
1 2 l1ffiÞ plffiffiffiffiffiffiffiffiffiffi With the substitution u ¼ zð to (11) and (12), the P0 can be obr21 þr22 tained by
Z
DI
2 u exp du 2
ð5Þ
1 P0 ¼ pffiffiffiffiffiffiffi 2p
ð6Þ
where the distinction index (DI) is calculated by
ð7Þ
l2 l1 DI ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r21 þ r22
1
ð13Þ
ð14Þ
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The distinction rate (DR) is defined as
DR ¼ 1 P0
Table 2 The DIs of the SPs for each step.
ð15Þ
It is obvious that the larger the value of the DI, the larger the value of the DR, and therefore the better the SP. Consequently, the DI can be used to evaluate the distinguishing sensitivity of the SP. In this work, the DI should be used for selecting the SP for fault diagnosis. 4. Sequential diagnosis approach based on possibility theory and certainty factor 4.1. Sequential diagnosis approach The number of fault states to be identified is enormous, and it is very hard to find one symptom parameter or a few symptom parameters that can identify all of those faults simultaneously. However, the symptom parameters for identification of two states are quite easy to find. Therefore, a sequential diagnosis method is proposed. In the diagnosis example of a rolling bearing shown in this paper, we must distinguish among four states, namely, the normal state, outer race flaw state, inner race flaw state, and roller element flaw state, sequentially. In those diagnostic stages, we should distinguish only two states in one step. As shown in Fig. 5, the first step of the sequential diagnosis can be used to distinguish the normal state (N) from bearing faults (B) and the unknown state (U) using the corresponding possibility of symptom parameter. The second step can be used to distinguish the inner race flaw (I) from the other bearing faults (the rolling element flaw (R), the outer race flaw (O)) and the unknown state. The last step is used to distinguish the rolling element flaw from the outer race flaw and the unknown state. The DIs of SPs can be calculated by (14), and the values of DIs for each step of the sequential diagnosis are shown in Table 2, respectively. As mentioned in the Section 3.2, the larger the value of the DI, the better the SP will be. Therefore, we used the DI to select the two best symptom parameters (Pi and Pj) for the sequential diagnosis. As shown in Table 2, P1 and P4 are better in the first diagnosis step, because all of the DIs of in each state are large. Similarly, the SPs for other diagnosis steps can be also selected. The other selectred results of the SPs are, P4 and P6 for the second step, and P3 and P7 for the last step, respectively. All of those DIs are larger than 3.45, and therefore all of the distinction rates approach 100%.
SPs
P1
P3
P4
P5
P6
P7
(a) For the first step N:I 5.04 2.64 N:R 5.48 1.92 N:O 6.32 3.40
1.65 1.45 2.97
11.12 6.74 14.32
3.53 3.82 4.00
10.96 1.12 11.26
10.4 4.89 10.67
(b) For the second step I:O 2.82 2.01 I:R 0.89 1.51
4.03 0.50
7.93 3.50
6.24 3.61
6.95 4.62
8.65 3.17
I:O
4.63
2.42
2.73
2.00
3.45
1.96
P2
3.15
monitored in real time. Furthermore, knowledge of distinguishing fault is ambiguous in real plant, because the definite relationships between symptom parameters and fault types even in the single fault cannot be easily identified. The values of symptom parameters calculated from vibration signals for fault diagnosis are also ambiguous because of the dispersion in the same state. Therefore, it is necessary to solve the ambiguous problem of fault diagnosis and to express uncertainty about the interpretation of the observable. Possibility theory is an uncertainty theory devoted to the handling of incomplete information. The basic idea of possibility theory, introduced by Zadeh is to use fuzzy sets not only to represent the gradual aspect of vague concepts such as ‘‘large’’, but also to represent incomplete knowledge, tainted with imprecision and uncertainty. Possibility theory offers a qualitative framework for the modeling of uncertainty and imprecision in reasoning systems. As such, it complements probability theory. However, the probability density function cannot solve uncertainty and incomplete information. In the present work, possibility theory is applied to solving the ambiguous relationship and incomplete information of fault diagnosis. The uncertainty is indicated by a possibility distribution valued on [0, 1]. More details about possibility theory were introduced (Cayrac, Dubois, & Prade, 1996; Dubois & Prade, 2001; Zadeh, 1999; Raufaste, Neves, & Claudette, 2003). The possibility function of the SP can be obtained from its probability density function. When the probability density function of the SP conforms to the normal distribution, it can be changed to a possibility function P(xi) by the following formula (Bendat, 1969).
Pðxi Þ ¼
N X
min fki ;
kk g
ð16Þ
k¼1
ki and kk can be calculated as follows. 4.2. Possibility theory
( ) 1 ðx xÞ2 pffiffiffiffiffiffiffi exp dx; ki ¼ 2r2 xi1 r 2p ( ) Z xk 1 ðx xÞ2 pffiffiffiffiffiffiffi exp kk ¼ dx 2r 2 xk1 r 2p Z
The number of the faults that are represented in the diagnostic support system is extremely small compared to all the possible faults, and the current knowledge base is therefore very incomplete. The symptoms are also incomplete: many observables are not fully
xi
Fig. 5. Flowchart of sequential diagnosis for bearing fault.
ð17Þ
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where r and x are the mean and standard deviation of the SP respectively, and x ¼ x 3r x þ 3r. In the case of sequential diagnosis, the possibility functions and the probability density functions of SP (Pi) used for each step, as examples, are shown in Fig. 6 respectively. In Fig. 6a, N, B, and U are the possibility functions of the normal state, bearing flaw, and the unknown state, respectively; n, o, r, and i are the probability functions of the normal state, outer race flaw, rolling element flaw, and inner race flaw, respectively. In Fig. 6b, I, B, and U are the possibility functions of the inner race flaw, other bearing faults (the rolling element flaw and the outer race flaw), and the unknown state, respectively; i, r, and o are the probability functions of the inner race flaw, the rolling element flaw, and the outer race flaw, respectively. In Fig. 6c, R, O, and U are the possibility functions of the rolling element flaw, the outer race flaw, and the unknown state, respectively; r and o are the probability functions of the rolling element flaw and the outer race flaw, respectively. 4.3. Certainty factor The certainty factor model was introduced by Shortliffe and Buchanan as a method for representation and manipulation of uncertain knowledge in the rule-based expert system MYCIN. The certainty factor CF(h, e) is just a numerical measure between 1 and +1, defined in terms of measures of belief and disbelief, hereh and e mean the hypothesis and evidence, respectively. The certainty factor can handle the problem of a combination of heterogeneous data, and more details for certainty factor have been introduced (Buchanan & Shortliffe, 1984; Lucas, 2001; Binaghi, Luzi, Madella, Pergalani, & Rampint, 1998). For fuzzy inference, the combination functions of the symptom parameters are necessary. Shortliffe and Buchanan have defined a number of combination functions, expressed in terms of certainty factors for the manipulation of certainty factors. The combination function for combining two certainty factors CF(h, e1) and CF(h, e2), due to two different layers of information which have been derived form two co-concluding production rules if ei then h fi, i = 1, 2, is expressed in the following equation (Lucas, 2001; Binaghi et al., 1998).
8 > < w1 þ w2 w2 w1 w þw CFðh; e1 coe2 Þ ¼ 1min 1fjw12j;jw2 jg if > : w1 þ w2 þ w2 w1
if w1 ; w2 > 0 1 < w1 w2 6 0 if
Fig. 6. Examples of possibility functions of Pi for fault diagnosis. (a) For the first step, (b) for the second step, (c) for the third step.
ð18Þ
w1 ; w2 < 0
where CF(h, e1) = w1 and CF(h, e2) = w2. Combination function is commutative and associative in its second argument; so, the order in which production rules are applied has no effect on the final results. In the present work, the combining possibility function of the two selected SPs (Pi and Pj) can be obtained by the certainty factor, and given as follows.
w ¼ wi þ wj wi wj
wN ; wN þ wB þ wU wU w0U ¼ wN þ wB þ wU
w0N ¼
ð19Þ
where w is the combining possibility function, and the wi and wj are possibility function of Pi and Pj, respectively. As mentioned above, the combining possibility functions of SPs in each sequential diagnosis step are obtained as follows. In the first step of the sequential diagnosis, the normalized combination possibility functions of the normal state (wN0 ), bearing fault state (wB0 ), and unknown state (wU0 ) can be obtained through the possibilities of the two selected symptom parameters Pi and Pj (herei = 1, and j = 4), respectively, as follows,
w0B ¼
wB ; wN þ wB þ wU
ð20Þ
where wN = wNi + wNjwNiwNj, wB = wBi + wBjwBi wBj, wU = wUi + wUjwUiwUj, wNi, wBi, wUi (as shown in Fig. 6a) and wNj, wBj, wUj are the possibilities of the normal state, bearing faults, and unknown state obtained by Pi and Pj, respectively. In the second step of the sequential diagnosis, the normalized combination possibility functions of the inner race flaw (wI0 ), other bearing flaws (wOR0 ), and the unknown state (wU0 ) can be obtained through the possibilities of Pi and Pj (here, i = 4, j = 6), respectively, as follows:
wI ; wI þ wOR þ wU wU wU 0 ¼ wI þ wOR þ wU
w0I ¼
wOR 0 ¼
wOR ; wI þ wOR þ wU
ð21Þ
where wI = wIi + wIjwIiwIj, wOR = wORi + wORj wORiwORj, wU = wUi + wUjwUiwUj, wIi, wRi, wUi (as shown in Fig. 6b) and wIj, wORj, wUj are
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H. Wang, P. Chen / Computers & Industrial Engineering 60 (2011) 511–518
the possibilities of inner race flaw, other bearing flaws, and the unknown state obtained by Pi and Pj respectively. Similarly, in the last step of the sequential diagnosis, the normalized combination possibility function of the outer race flaw (wO0 ), rolling element flaw (wR0 ), and the unknown state (wU0 ) can be obtained through the possibilities of symptom parametersPi and Pj (here, i = 3, j = 7), respectively, as follows:
wO ; wO þ wR þ wU wU wU 0 ¼ wO þ wR þ wU
w0O ¼
wR 0 ¼
wR ; wO þ wR þ wU
ð22Þ
where wO = wOi + wOjwOiwOj, wR = wRi + wRj wRiwRj, wU = wUi + wUjwUiwUj, wOi, wRi, wUi (as shown in Fig. 6c), and wOj, wRj, wUj are the possibilities of the inner race flaw, rolling element flaw, and the unknown state obtained by Pi and Pj, respectively. We used a fuzzy neural network to learn the possibilities of each state calculated by (20)–(22), and to identify the bearing faults automatically. The fuzzy neural network can be realized by the PNN described in the later section. 5. Fuzzy neural network for fault diagnosis 5.1. Partially-linearized neural network The main mathematic symbols used in Section 5 are: Nm: the neuron number of the mth layer of an NN, m = 1 to M. ð1;jÞ ð1;jÞ X ð1Þ ¼ fX i g : the pattern input to the 1st layer. Here, X i is the value input to the jth neuron in the input (1st) layer, i = 1 to P, j = 1 to N1. ðM;kÞ X ðMÞ ¼ fX i g : the training (teaching) data for the last layer ðM;kÞ (Mth layer). Here, X i is the output value of the kth neuron in the output (Mth) layer; k = 1 to NM . ð1;jÞ ðM;kÞ X ð1Þ ¼ fX i g and X ðMÞ ¼ fX i g: new data that has not yet been learnt by the NN. ðm;tÞ X i : the value of the tth neuron in the hidden (mth) layer; t = 1 to Nm. W ðmÞ uv : the weight between the uth neuron in the mth layer and the vth neuron in the (m + 1)th layer, m = 1 to M1; u = 1 to Nm; v = 1 to Nm+1. The fuzzy neural network is applied to diagnose the fault types of a rolling bearing by the sequential diagnosis algorithm, and realized with a developed back propagation neural network called as ‘‘the partially-linearized neural network (PNN)’’ (Mitoma, Wang, & Chen, 2008). A back propagation neural network is only used for training the data, and the PNN is used for testing the learned NN. Here, the basic principle of the PNN for the fault diagnosis is described as follows. The neuron number of the mth layer of an NN is Nm. The set ð1;jÞ X ð1Þ ¼ fX i g represents the pattern input to the 1st layer and the ðM;kÞ ðMÞ set X ¼ fX i g is the training data for the last layer (Mth layer). ð1;jÞ Here, i = 1 to P, j = 1 to N1, k = 1 to NM, and, X i : the value input to ðM;kÞ the jth neuron in the input (1st) layer; X i : the output value of the kth neuron in the output (Mth) layer, k = 1 to NM. Even if the NN converges by learning X(1) and X(M), it cannot adequately deal with the ambiguous relationship between the new X(1)⁄ and X(M)⁄, which has not been learnt. In order to predict X(M)⁄ according to the probability distribution of X(1)⁄, partial linear interpolation of the NN is introduced as shown in Fig. 7. In the NN that has converged with the data X(1) and X(M), the following symbols are used.
Fig. 7. The partial linearization of the sigmoid function.
W ðmÞ uv : the weight between the uth neuron in the mth layer and the vth neuron in the (m+1)th layer, m = 1 to M; u = 1 to Nm; v = 1 to Nm+1. If all these values are memorized by the computer, when new ð1;uÞ ð1;uÞ ð1;uÞ values Xj (1,u)⁄ (X j < Xj < X jþ1 Þ are input into the first layer, the predicted value of the vth neuron (v=1 toNm) in the (m+1)th layer (m = 1 to M 1) can be estimated by
nP ðmþ1;mÞ
Xj
ðmþ1;mÞ
¼ X iþ1
o ðm;uÞ ðmþ1;v Þ ðmþ1;v Þ X j Þ X iþ1 Xi PNm ðmÞ ðm;uÞ ðm;uÞ u¼1 W uv X iþ1 X i
ðm;uÞ Nm ðmÞ u¼1 W uv ðX iþ1
ð23Þ Using the operation above, the sigmoid function is partiallylinearized, as shown in Fig. 8. If a function must be learned, the PNN will learn the points indicated by the symbols shown in Fig. 8. When new data ðs01 , s20 ) are input into the converged PNN, the values depicted by the j symbols corresponding to the data ðs01 ; s02 Þ will quickly be identified as Pe. Thus, the PNN can be used to deal with ambiguous diagnosis problems. As shown in Fig. 8, the new data (s1, s20 ) input into the converged PNN, and which are not learnt by the PNN for recognizing, must satisfy the following condition.
s1ðminÞ < s01 < s1ðmaxÞ and s2ðminÞ < s02 < s2ðmaxÞ
Here, s1(min) and s2(min) are the minimum values, respectively, of s1 and s2, which have been learned by the PNN. Therefore, in this work, the values (Pi and P j ) of symptom parameters input to the PNN for fault diagnosis must satisfy the following condition.
PiðminÞ < Pi < PiðmaxÞ
and P jðminÞ < Pj < PjðmaxÞ
ð25Þ
S2 P a
Pe S2 '
c
b e d
S1 '
ðm;tÞ
: the value of the tth neuron in the hidden (mth) layer, t = 1 Xi to Nm;
ð24Þ
Fig. 8. Interpolation by the PNN.
S1
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(a) For first step
(b) For second step
(c) For third step
Fig. 9. The PNN for the fault classification of a bearing.
Table 3 Examples of training data for the PNN learning. SPs P1
Table 4 Verification results of the learned PNN. State
P4
w0N
(a) For the first step 147.20 0.086 167.46 0.103 199.98 0.154 ... ...
0.67 0.286 ...
(b) For the second step P4 P6 0.645 3.23 0.584 0.221 1.762 1.464 ... ...
wI 1 0.5 0 ...
(c) For the third step P3 P7 0.678 1.99 0.654 4.35 0.654 2.763 ... ...
wR 1 0.25 0.33 ...
0
w0B
w0U
0
0
0
1 0 0.286
...
0
wOR 0 0.25 0.67 ...
0
0.33 0.428 ...
0
wU 0 0.25 0.33 ...
0
wO 0
0.5 0 ...
P1
wU 0 0.5 0.67 ...
P4
(a) In the first step 176.49 0.135 162.75 0.114 204.32 0.995 213.07 1.675 206.93 0.635 207.15 1.52 202.74 0.56 149.35 0.103 P6 P4 (b) In the second step 0.635 0.56 1.675 1.285 1.52 1.24 0.103
2.984 2.79 0.271 1.012 0.27 0.625 4.70
P3 P7 (c) In the third step
5.2. PNN for fault diagnosis and verification Fig. 9 shows the PNN built based on an algorithm of sequential diagnosis proposed in this paper. As an example, for the first step shown in Fig. 9a, the PNN consists of the first layer, one hidden layer, and the last layer. The neurons in the first layer are input as the SPs (P1 and P4). The number of neurons in the hidden layer is seventy determined by trial and error, and the outputs in the last layer are w0N ; w0B , and w0U which are the possibility grades of the normal state, bearing flaws, and unknown state, respectively. The knowledge for training of the PNN is acquired by the possibility theory and the Mycin certainty factor, and it can deal with the vagueness and uncertainty relationships between the symptoms and the fault types. Therefore, the PNN can obtain good convergence when learning the acquired knowledge. Parts of the training data are shown in Table 3. We used the data measured in each state that had not been learned by the PNN to verify the classifiable capability of the PNN. When the value of the symptom parameters is input into the learned PNN, it can quickly and automatically classify those faults according to the possibilities of the corresponding states. The diagnosis results are shown in Table 4. According to the verification results, the possibilities output by the PNN show correct judgments in each state. Therefore, the PNN can precisely distinguish the types of bearing faults on the basis of the possibilities of the symptom parameters.
0.708 0.703 0.675 0.664 0.644
4.068 3.721 2.444 1.799 0.05
wN
0
0.87 0.78 0.011 0.001 0.01 0.002 0.01 0.16 0
wB
0
wU
0.013 0.004 0.86 0.872 0.815 0.875 0.745 0.001 0
wI
wOR
0.858 0.82 0.01 0.014 0.01 0.007 0.19
0.02 0.01 0.94 0.826 0.934 0.871 0.022
0
0
wR
wO
0.021 0.019 0.859 0.84 0.152
0.865 0.789 0.053 0.005 0.000
0
0.12 0.22 0.13 0.127 0.162 0.129 0.232 0.826 wU
0
0.153 0.178 0. 07 0.158 0.09 0.122 0.81 wU
0
0.15 0.231 0.116 0.156 0.85
Judge N N B B B B B U Judge I I O O O O U
or or or or
R R R R
Judge O O R R U
(1) A sequential diagnosis technique was proposed through which the fuzzy neural network realized by the partiallylinearized neural network (PNN) could sequentially distinguish fault types. (2) Knowledge for the PNN was acquired by possibility theory and the Mycin certainty factor, which can represent uncertain diagnostic information. The establishing method of the membership function converting probability distribution function of symptom parameter into possibility function by the possibility theory was proposed, and the combination membership functions of several symptom parameters were obtained by the certainty factor. (3) The non-dimensional symptom parameters were also described in the frequency-domain, and these parameters could reflect the characteristics of the signals measured for fault diagnosis of a rolling bearing. (4) This proposed method had been successfully applied to fault diagnosis of a rolling bearing used in a centrifugal blower, and the faults were sequentially and automatically diagnosed on the basis of the possibilities of the symptom parameters.
6. Conclusions In order to process the ambiguous relationship between the symptom parameters and fault types, and automatically identified faults, an intelligent diagnosis method for bearing faults was proposed on the basis of possibility theory and a fuzzy neural network with frequency-domain features of vibration signals. The main conclusions were obtained as follows.
Acknowledgement This work was supported by the Grants-in-aid for Scientific Research of JSPS (Scientific Research (B) 19360074), National Natural Science Foundation of China under Grant 51075023, and the Fundamental Research Funds for the Central Universities.
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References Alguindigue, I. E., Loskiewicz-Buczak, A., & Uhrig, R. E. (1993). Monitoring and diagnosis of rolling element bearings using artificial neural networks. IEEE Transactions on Industrial Electronics, 40, 209–216. Bendat, J. S. (1969). Probability function for random processes: Prediction of peak, fatigue damage, and catastrophic failure. NASA Report CR-33. Binaghi, E., Luzi, L., Madella, P., Pergalani, F., & Rampint, A. (1998). Slope instability zonation: A comparison between certainty factor and fuzzy dempster-shafer approaches. Natural Hazards, 17, 77–97. Buchanan, B. G., & Shortliffe, E. H. (1984). Rule-based expert systems: The MYCIN experiments of the Stanford Heuristic programming project. Reading, MA: Addison-Wesley. Cayrac, D., Dubois, D., & Prade, H. (1996). Handling uncertainty with possibility theory and fuzzy sets in a satellite fault diagnosis application. IEEE Transactions on Fuzzy Systems, 4(3), 251–269. Chen, P., Feng, F., & Toyota, T. (2002). Sequential diagnosis method for plant machinery by statistical tests and possibility theory. Journal of reliability engineering association of Japan, 24(4), 313–322. Chen, P., & Toyota, T. (1998). Self-Reorganization of feature parameters in frequency domain by genetic programming. Transactions of the Japan Society of Mechanical Engineers Series C, 65(633), 1946–1953. Chen, P., & Toyota, T. (2003). Sequential fuzzy diagnosis for plant machinery. JSME International Journal, Series C, 46(3), 1121–1129. Bishop Christopher, M. I. (1995). Neural networks for pattern recognition. Oxford University Press. Dubois, Didier, & Prade, Henri (2001). Possibility theory, probability theory and multiple-valued logics: A clarification. Annals of Mathematics and Artificial Intelligence, 32, 35–66. Fang, R. M. (2006). Fault diagnosis of induction machine using artificial neural network and support vector machine. Dynamics of Continuous Discrete and Impulsive Systems – Series A-Mathematical Analysis, 13, 658–661. Part 2 Suppl.. Fukunaga, K. (1972). Introduction to statistical pattern recognition. Academic Press. Liu, B., & Ling, S.-F. (1999). On the selection of informative wavelets for machinery diagnosis. Mechanical Systems and Signal Processing, 13, 145–162.
Lou, Xinsheng, & Loparo Kenneth, A. (2004). bearing fault diagnosis based on wavelet transform and fuzzy inference. Mechanical Systems and Signal Processing, 18, 1077–1095. Lucas, P. J. F. (2001). Certainty-factor-like structures in Bayesian belief networks. Knowledge-Based systems, 14, 327–335. McCormick, A. C., & Nandi, A. K. (1997). Real-time classification of the rotating shaft loading conditions using artificial neural networks. IEEE Transactions on Neural Networks, 8, 748–756. Mitoma, T., Wang, H., & Chen, P. (2008). Fault diagnosis and condition surveillance for plant rotating machinery using partially-linearized neural network’. Computers & Industrial Engineering, 55(4), 783–794. Pusey, H. C. (2000). Machinery condition monitoring. Journal of Sound and Vibration, 34(5), 6–7. Raufaste, Eric, Neves, Rui da Silva, & Claudette, M. (2003). Testing the descriptive validity of possibility theory in human judgments of uncertainty. Artificial Intelligence, 148, 197–218. Samanta, B., Al-Balushi, K. R., & Al-Araimi, S. A. (2006). Artificial neural networks and genetic algorithm for bearing fault detection. Soft Computing, 10(3), 264–271. Schetinin, V., & Schult, J. (2006). Learning polynomial networks for classification of clinical electroencephalograms. Soft Computing, 10(4), 397–403. Wang, H., & Chen, Peng (2009). Intelligent diagnosis method for a centrifugal pump using features of vibration signals. Neural Computing & Applications, 18, 397–405. Williams, T., Ribadeneira, X., Billington, S., & Kurfess, T. (2001). Rolling element bearing diagnostics in run-to-failure lifetime testing. Mechanical Systems and Signal Processing, 15, 979–993. Yang, Yu, Dejie, Yu, & Junsheng, Cheng (2006). A roller bearing fault diagnosis method based on EMD energy entropy and ANN. Journal of Sound and Vibration, 294, 269–277. Yang, Hongyu, Mathew, Joseph, & Ma, Lin (2005). Fault diagnosis of rolling element bearings using basis pursuit. Mechanical Systems and Signal Processing, 19(2), 341–356. Yang, D. M., Stronach, A. F., & MacConnell, P. (2003). The application of advanced signal processing techniques to induction motor bearing condition diagnosis. Meccanica, 38(2), 297–308. Zadeh, L. A. (1999). Fuzzy sets as basis for a theory of possibility. Fuzzy Sets and Systems, 100(Supplement), 9–34.
Contents lists available at ScienceDirect
Computers & Industrial Engineering journal homepage: www.elsevier.com/locate/caie
Intelligent diagnosis method for rolling element bearing faults using possibility theory and neural network q Huaqing Wang a,⇑, Peng Chen b,⇑ a b
School of Mechanical & Electrical Engineering, Beijing University of Chemical Technology, Chaoyang District, Beisanhuan East Road 15, Beijing 100029, China Department of Environmental Science and Engineering, Faculty of Bioresources Mie University, 1577 Kurimamachiya-cho, Tsu-shi, Mie-ken 514-8507, Japan
a r t i c l e
i n f o
Article history: Received 17 March 2008 Received in revised form 2 March 2010 Accepted 3 December 2010 Available online 13 December 2010 Keywords: Fault diagnosis Neural network Possibility theory Rolling element bearing Centrifugal blower
a b s t r a c t This paper presents an intelligent diagnosis method for a rolling element bearing; the method is constructed on the basis of possibility theory and a fuzzy neural network with frequency-domain features of vibration signals. A sequential diagnosis technique is also proposed through which the fuzzy neural network realized by the partially-linearized neural network (PNN) can sequentially identify fault types. Possibility theory and the Mycin certainty factor are used to process the ambiguous relationship between symptoms and fault types. Non-dimensional symptom parameters are also defined in the frequency domain, which can reflect the characteristics of vibration signals. The PNN can sequentially and automatically distinguish fault types for a rolling bearing with high accuracy, on the basis of the possibilities of the symptom parameters. Practical examples of diagnosis for a bearing used in a centrifugal blower are given to show that bearing faults can be precisely identified by the proposed method. Ó 2010 Elsevier Ltd. All rights reserved.
1. Introduction A rolling element bearing is an important and widely used component of rotating machinery. A fault in a rolling element bearing may cause breakdown of machinery, and serious consequences may arise due to the fault (Lou & Kenneth, 2004; Williams, Ribadeneira, Billington, & Kurfess, 2001; Yang, Mathew, & Ma, 2005, 2006). Therefore, fault diagnosis of rolling element bearings is very important for guaranteeing production efficiency and plant safety. In the field of fault diagnosis, utilization of vibration signals is an effective method for the detection of faults and discrimination of fault types, because the vibration signals carry dynamic information about the machine state (Liu & Ling, 1999; Pusey, 2000; Yang, Stronach, & MacConnell, 2003). Intelligent systems such as neural networks (NNs) have potential applications in pattern recognition and failure diagnosis. Many studies have been carried out to investigate the use of NNs for automatic diagnosis of machinery; most of these methods have been proposed to deal with discrimination of fault types collectively (Alguindigue, Loskiewicz-Buczak, & Uhrig, 1993; Fang, 2006; McCormick & Nandi, 1997; Samanta, Al-Balushi, & Al-Araimi, 2006; Schetinin & Schult, 2006; Wang & Chen, 2009). However, the conventional NNs cannot reflect the possibility of ambiguous diagq
This manuscript was processed by Area Editor Satish Bukkapatnam.
⇑ Corresponding authors. Tel./fax: +81 592319592.
E-mail addresses: [email protected] (H. Wang), [email protected] (P. Chen). 0360-8352/$ - see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.cie.2010.12.004
nosis problems, and will never converge when the first layer symptom parameters have the same values in different states (Christopher, 1995). Furthermore, the number of the faults that are represented in the diagnostic support system is extremely small compared to all the possible faults, and the current knowledge base is therefore very incomplete. The symptoms are also incomplete: many observables are not fully monitored in real time. Knowledge of distinguishing fault is ambiguous in real plant, because the definite relationships between symptom parameters and fault types even in the single fault cannot be easily identified. In addition, the number of fault states to be identified is enormous, and it is very hard to find one symptom parameter or a few symptom parameters that can identify all of those faults simultaneously (Chen & Toyota, 2003). The values of symptom parameters calculated from vibration signals for fault diagnosis are also ambiguous because of the dispersion in the same state. Therefore, it is necessary to solve the ambiguous problem of fault diagnosis and to express uncertainty about the interpretation of the observable. For the above reasons, an intelligent diagnosis method for bearing faults is proposed on the basis of possibility theory and a fuzzy neural network with frequency-domain features of vibration signals. This method is intended to process the ambiguous relationship between the symptom parameters and fault types, and automatically identified faults. A sequential diagnosis approach is also proposed through the partially-linearized neural network (PNN) to identify sequentially the bearing faults. Diagnostic knowledge for the PNN is acquired by possibility theory and the Mycin certainty factor for solving the ambiguous problem of fault
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Fig. 1. The blower system for bearing fault diagnosis, (a) illustration of the blower system, (b) the blower in the field.
diagnosis. In this work, bearing faults are considered which often occur at an early stage, such as the outer race flaw, the inner race flaw, and the roller element flaw. Those faults can be sequentially and automatically distinguished on the basis of the possibility distributions of symptom parameters. Practical examples of fault diagnosis for a rolling bearing used in a centrifugal blower are shown to verify the efficiency of the proposed method. 2. Experimental system for bearing diagnosis In order to verify the efficiency of the methods proposed in this paper, the experimental system of bearing diagnosis is used. Fig. 1 shows the experimental system, including the centrifugal blower (TERAL CLF3), the rolling bearing (NSK 206) and the accelerometers. A 2.2 kW induction motor with three-phases and a maximum revolution of 1420 rpm is employed to drive the blower through two V-belts, and its rotating speed can be varied by a speed controller. NSK 206 bearings were utilized in this study, and specifications of the test bearing, the size of the faults, and other necessary information are listed in Table 1. The faults that most frequently occur in a rolling element bearing at an early stage are the outer race, inner race, and roller element flaws. These faults, shown in Fig. 2, were artificially induced with the use of a wire-cutting machine. As shown in Fig. 3, the two accelerometers are mounted on the bearing housing at the output end of the blower-shaft in order to
measure the vibration signals in the vertical and horizontal directions, respectively. The sampling frequency of the vibration signals is 100 kHz for each channel, and the sampling time is 5 s. The centrifugal blower usually works at a constant speed, and faults often occur at a steady condition; therefore, in the present work, we only considered the steady state. The vibration signals are measured at a constant rotating speed of 800 rpm. A high-pass filter with a 5 kHz cut-off frequency was used to cancel noise in the vibration signals for fault diagnosis. Examples of vibration signals measured in each state after filtering are shown in Fig. 4. The envelope spectrum of the filtered signals could then be obtained and used to calculate the symptom parameters. 3. Symptom parameters and sensitivity evaluation In order to automatically diagnose the machine states by computer, the characteristics of the envelope spectra must be expressed by the symptom parameters (SPs). In this section, we discuss the SPs in the frequency-domain and how to select the better SPs for condition diagnosis. 3.1. Symptom parameters in the frequency-domain In case of the fault diagnosis, the symptom parameters (SPs) usually are used for identifying the conditions of machinery, because they can express the condition information of the machine
Table 1 Bearing information for fault diagnosis. Bearing NSK
Normal (N)
With roller flaw (R)
With inner flaw (I)
With outer flaw (O)
Specification Flaw width Flaw depth Pitch diameter Roller number Roller diameter Pass-frequency
NU206EW – – – – – –
NF206W 0.6 mm 0.3 mm 46.0 mm 13 7.5 mm 79.6 Hz
NU206EW 0.6 mm 0.3 mm 46.5 mm 13 9.0 mm 103.4 Hz
NF206W 0.6 mm 0.3 mm 46.0 mm 13 7.5 mm 72.5 Hz
Fig. 2. Rolling element bearing flaws, (a) outer race flaw (O), (b) inner race flaw (I), (c) roller element flaw (R).
H. Wang, P. Chen / Computers & Industrial Engineering 60 (2011) 511–518
513
where N is the number of spectrum lines, fi is the frequency, S(fi) is ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rP N ðf f Þ2 Sðfi Þ i¼1 i , the power spectrum of the enveloped waveform, r ¼ N1 P P and f ¼ Ni¼1 fi Sðfi Þ= Ni¼1 Sðfi Þ. When the rollers pass through the flaw, impulses may appear. The frequency of the impulse waveform is called the pass-frequency of a fault bearing, and can be calculated by the following formulas (Chen & Toyota, 2003).
Sensor1
Sensor2
2
Dfr fR ¼ d
d
!
1 2 cos a D zfr d fo ¼ 1 cos a 2 D zfr d 1 þ cos a fI ¼ 2 D
Fig. 3. The location of sensors.
ð8Þ ð9Þ ð10Þ
where fR, fo, and fI are the pass-frequency of the roller flaw, the outer race flaw, and the inner race flaw, respectively, z is the number of rolling elements, fr is a rotating frequency, dis a diameter of rolling elements, D is a pitch diameter, andais a contact angle of rolling element. The parameters of the relevant test bearings are shown in Table 1. The pass-frequencies of the roller flaw, the outer race flaw, and the inner race flaw can be obtained of 79.6 Hz, 72.5 Hz, and 103.4 Hz, respectively, by (8)–(10), as shown in Table 1. In this work, we determined five times the maximum pass-frequency (fI is 103.4 Hz) of about 500 Hz as the maximum calculating frequency, and the range of the calculating frequency fi was determined from 0 Hz to 500 Hz. 3.2. Sensitivity evaluation of symptom parameter
Fig. 4. Vibration signals of bearings after filtering (a) Normal state, (b) inner race flaw state, (c) roller race flaw state, (d) outer race flaw state.
indicated by the vibration signal. The SPs are random variables and can be calculated from the vibration signals. Commonly, the SPs can classified into the frequency-domain SPs and the time-domain SPs, and the two types of SPs indicate features of a signal in the frequency-domain and the time-domain, respectively. A large set of symptom parameters has been defined in the pattern recognition field (Fukunaga, 1972). In this study, seven of these parameters in the frequency-domain, which are commonly used for fault diagnosis of plant machinery, are considered (Chen & Toyota, 1998).
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi uPN 2 u i¼1 fi Sðfi Þ P1 ¼ t P N i¼1 Sðfi Þ vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi uPN 4 u fi Sðfi Þ P2 ¼ tPi¼1 N 2 i¼1 fi Sðfi Þ PN 2 i¼1 fi Sðfi Þ P3 ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi PN PN 4 i¼1 Sðfi Þ i¼1 fi Sðfi Þ
r
P4 ¼ f PN ðfi f Þ3 Sðfi Þ P5 ¼ i¼1 3 r N PN 4 i¼1 ðfi f Þ Sðfi Þ P6 ¼ 4 r N PN qffiffiffiffiffiffiffiffiffiffiffiffiffiffi fi f Sðfi Þ i¼1 pffiffiffiffi P7 ¼ rN
For automatic diagnosis, symptom parameters are needed that can sensitively distinguish the fault types. In order to evaluate the sensitivity of an SP for distinguishing two states, such as a normal or an abnormal state, the distinction index (DI) is defined as follows. Suppose that x1and 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 Nðl1 ; r21 Þ and Nðl2 ; r22 Þ, where, l and r are the average and the standard deviation of the SP. The larger jx2 x1j is, the higher the sensitivity of SP for distinguishing the two states. Because z = x2x1 is also normally distributed Nðl2 l1 ; r22 þ r21 Þ, we have the density function about z.
ð1Þ
2 ! z ðl2 l1 Þ 1 f ðzÞ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi exp 2ðr21 þ r22 Þ 2pðr21 þ r22 Þ
ð2Þ
where l2 P l1 (we can obtain the same conclusion when l1 P l2). The probability of x2 < x1 can be calculated by the following formula (Chen, Feng, & Toyota, 2002).
ð3Þ ð4Þ
P0 ¼
Z
ð11Þ
0
f ðzÞdz
ð12Þ
1 2 l1ffiÞ plffiffiffiffiffiffiffiffiffiffi With the substitution u ¼ zð to (11) and (12), the P0 can be obr21 þr22 tained by
Z
DI
2 u exp du 2
ð5Þ
1 P0 ¼ pffiffiffiffiffiffiffi 2p
ð6Þ
where the distinction index (DI) is calculated by
ð7Þ
l2 l1 DI ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r21 þ r22
1
ð13Þ
ð14Þ
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The distinction rate (DR) is defined as
DR ¼ 1 P0
Table 2 The DIs of the SPs for each step.
ð15Þ
It is obvious that the larger the value of the DI, the larger the value of the DR, and therefore the better the SP. Consequently, the DI can be used to evaluate the distinguishing sensitivity of the SP. In this work, the DI should be used for selecting the SP for fault diagnosis. 4. Sequential diagnosis approach based on possibility theory and certainty factor 4.1. Sequential diagnosis approach The number of fault states to be identified is enormous, and it is very hard to find one symptom parameter or a few symptom parameters that can identify all of those faults simultaneously. However, the symptom parameters for identification of two states are quite easy to find. Therefore, a sequential diagnosis method is proposed. In the diagnosis example of a rolling bearing shown in this paper, we must distinguish among four states, namely, the normal state, outer race flaw state, inner race flaw state, and roller element flaw state, sequentially. In those diagnostic stages, we should distinguish only two states in one step. As shown in Fig. 5, the first step of the sequential diagnosis can be used to distinguish the normal state (N) from bearing faults (B) and the unknown state (U) using the corresponding possibility of symptom parameter. The second step can be used to distinguish the inner race flaw (I) from the other bearing faults (the rolling element flaw (R), the outer race flaw (O)) and the unknown state. The last step is used to distinguish the rolling element flaw from the outer race flaw and the unknown state. The DIs of SPs can be calculated by (14), and the values of DIs for each step of the sequential diagnosis are shown in Table 2, respectively. As mentioned in the Section 3.2, the larger the value of the DI, the better the SP will be. Therefore, we used the DI to select the two best symptom parameters (Pi and Pj) for the sequential diagnosis. As shown in Table 2, P1 and P4 are better in the first diagnosis step, because all of the DIs of in each state are large. Similarly, the SPs for other diagnosis steps can be also selected. The other selectred results of the SPs are, P4 and P6 for the second step, and P3 and P7 for the last step, respectively. All of those DIs are larger than 3.45, and therefore all of the distinction rates approach 100%.
SPs
P1
P3
P4
P5
P6
P7
(a) For the first step N:I 5.04 2.64 N:R 5.48 1.92 N:O 6.32 3.40
1.65 1.45 2.97
11.12 6.74 14.32
3.53 3.82 4.00
10.96 1.12 11.26
10.4 4.89 10.67
(b) For the second step I:O 2.82 2.01 I:R 0.89 1.51
4.03 0.50
7.93 3.50
6.24 3.61
6.95 4.62
8.65 3.17
I:O
4.63
2.42
2.73
2.00
3.45
1.96
P2
3.15
monitored in real time. Furthermore, knowledge of distinguishing fault is ambiguous in real plant, because the definite relationships between symptom parameters and fault types even in the single fault cannot be easily identified. The values of symptom parameters calculated from vibration signals for fault diagnosis are also ambiguous because of the dispersion in the same state. Therefore, it is necessary to solve the ambiguous problem of fault diagnosis and to express uncertainty about the interpretation of the observable. Possibility theory is an uncertainty theory devoted to the handling of incomplete information. The basic idea of possibility theory, introduced by Zadeh is to use fuzzy sets not only to represent the gradual aspect of vague concepts such as ‘‘large’’, but also to represent incomplete knowledge, tainted with imprecision and uncertainty. Possibility theory offers a qualitative framework for the modeling of uncertainty and imprecision in reasoning systems. As such, it complements probability theory. However, the probability density function cannot solve uncertainty and incomplete information. In the present work, possibility theory is applied to solving the ambiguous relationship and incomplete information of fault diagnosis. The uncertainty is indicated by a possibility distribution valued on [0, 1]. More details about possibility theory were introduced (Cayrac, Dubois, & Prade, 1996; Dubois & Prade, 2001; Zadeh, 1999; Raufaste, Neves, & Claudette, 2003). The possibility function of the SP can be obtained from its probability density function. When the probability density function of the SP conforms to the normal distribution, it can be changed to a possibility function P(xi) by the following formula (Bendat, 1969).
Pðxi Þ ¼
N X
min fki ;
kk g
ð16Þ
k¼1
ki and kk can be calculated as follows. 4.2. Possibility theory
( ) 1 ðx xÞ2 pffiffiffiffiffiffiffi exp dx; ki ¼ 2r2 xi1 r 2p ( ) Z xk 1 ðx xÞ2 pffiffiffiffiffiffiffi exp kk ¼ dx 2r 2 xk1 r 2p Z
The number of the faults that are represented in the diagnostic support system is extremely small compared to all the possible faults, and the current knowledge base is therefore very incomplete. The symptoms are also incomplete: many observables are not fully
xi
Fig. 5. Flowchart of sequential diagnosis for bearing fault.
ð17Þ
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where r and x are the mean and standard deviation of the SP respectively, and x ¼ x 3r x þ 3r. In the case of sequential diagnosis, the possibility functions and the probability density functions of SP (Pi) used for each step, as examples, are shown in Fig. 6 respectively. In Fig. 6a, N, B, and U are the possibility functions of the normal state, bearing flaw, and the unknown state, respectively; n, o, r, and i are the probability functions of the normal state, outer race flaw, rolling element flaw, and inner race flaw, respectively. In Fig. 6b, I, B, and U are the possibility functions of the inner race flaw, other bearing faults (the rolling element flaw and the outer race flaw), and the unknown state, respectively; i, r, and o are the probability functions of the inner race flaw, the rolling element flaw, and the outer race flaw, respectively. In Fig. 6c, R, O, and U are the possibility functions of the rolling element flaw, the outer race flaw, and the unknown state, respectively; r and o are the probability functions of the rolling element flaw and the outer race flaw, respectively. 4.3. Certainty factor The certainty factor model was introduced by Shortliffe and Buchanan as a method for representation and manipulation of uncertain knowledge in the rule-based expert system MYCIN. The certainty factor CF(h, e) is just a numerical measure between 1 and +1, defined in terms of measures of belief and disbelief, hereh and e mean the hypothesis and evidence, respectively. The certainty factor can handle the problem of a combination of heterogeneous data, and more details for certainty factor have been introduced (Buchanan & Shortliffe, 1984; Lucas, 2001; Binaghi, Luzi, Madella, Pergalani, & Rampint, 1998). For fuzzy inference, the combination functions of the symptom parameters are necessary. Shortliffe and Buchanan have defined a number of combination functions, expressed in terms of certainty factors for the manipulation of certainty factors. The combination function for combining two certainty factors CF(h, e1) and CF(h, e2), due to two different layers of information which have been derived form two co-concluding production rules if ei then h fi, i = 1, 2, is expressed in the following equation (Lucas, 2001; Binaghi et al., 1998).
8 > < w1 þ w2 w2 w1 w þw CFðh; e1 coe2 Þ ¼ 1min 1fjw12j;jw2 jg if > : w1 þ w2 þ w2 w1
if w1 ; w2 > 0 1 < w1 w2 6 0 if
Fig. 6. Examples of possibility functions of Pi for fault diagnosis. (a) For the first step, (b) for the second step, (c) for the third step.
ð18Þ
w1 ; w2 < 0
where CF(h, e1) = w1 and CF(h, e2) = w2. Combination function is commutative and associative in its second argument; so, the order in which production rules are applied has no effect on the final results. In the present work, the combining possibility function of the two selected SPs (Pi and Pj) can be obtained by the certainty factor, and given as follows.
w ¼ wi þ wj wi wj
wN ; wN þ wB þ wU wU w0U ¼ wN þ wB þ wU
w0N ¼
ð19Þ
where w is the combining possibility function, and the wi and wj are possibility function of Pi and Pj, respectively. As mentioned above, the combining possibility functions of SPs in each sequential diagnosis step are obtained as follows. In the first step of the sequential diagnosis, the normalized combination possibility functions of the normal state (wN0 ), bearing fault state (wB0 ), and unknown state (wU0 ) can be obtained through the possibilities of the two selected symptom parameters Pi and Pj (herei = 1, and j = 4), respectively, as follows,
w0B ¼
wB ; wN þ wB þ wU
ð20Þ
where wN = wNi + wNjwNiwNj, wB = wBi + wBjwBi wBj, wU = wUi + wUjwUiwUj, wNi, wBi, wUi (as shown in Fig. 6a) and wNj, wBj, wUj are the possibilities of the normal state, bearing faults, and unknown state obtained by Pi and Pj, respectively. In the second step of the sequential diagnosis, the normalized combination possibility functions of the inner race flaw (wI0 ), other bearing flaws (wOR0 ), and the unknown state (wU0 ) can be obtained through the possibilities of Pi and Pj (here, i = 4, j = 6), respectively, as follows:
wI ; wI þ wOR þ wU wU wU 0 ¼ wI þ wOR þ wU
w0I ¼
wOR 0 ¼
wOR ; wI þ wOR þ wU
ð21Þ
where wI = wIi + wIjwIiwIj, wOR = wORi + wORj wORiwORj, wU = wUi + wUjwUiwUj, wIi, wRi, wUi (as shown in Fig. 6b) and wIj, wORj, wUj are
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the possibilities of inner race flaw, other bearing flaws, and the unknown state obtained by Pi and Pj respectively. Similarly, in the last step of the sequential diagnosis, the normalized combination possibility function of the outer race flaw (wO0 ), rolling element flaw (wR0 ), and the unknown state (wU0 ) can be obtained through the possibilities of symptom parametersPi and Pj (here, i = 3, j = 7), respectively, as follows:
wO ; wO þ wR þ wU wU wU 0 ¼ wO þ wR þ wU
w0O ¼
wR 0 ¼
wR ; wO þ wR þ wU
ð22Þ
where wO = wOi + wOjwOiwOj, wR = wRi + wRj wRiwRj, wU = wUi + wUjwUiwUj, wOi, wRi, wUi (as shown in Fig. 6c), and wOj, wRj, wUj are the possibilities of the inner race flaw, rolling element flaw, and the unknown state obtained by Pi and Pj, respectively. We used a fuzzy neural network to learn the possibilities of each state calculated by (20)–(22), and to identify the bearing faults automatically. The fuzzy neural network can be realized by the PNN described in the later section. 5. Fuzzy neural network for fault diagnosis 5.1. Partially-linearized neural network The main mathematic symbols used in Section 5 are: Nm: the neuron number of the mth layer of an NN, m = 1 to M. ð1;jÞ ð1;jÞ X ð1Þ ¼ fX i g : the pattern input to the 1st layer. Here, X i is the value input to the jth neuron in the input (1st) layer, i = 1 to P, j = 1 to N1. ðM;kÞ X ðMÞ ¼ fX i g : the training (teaching) data for the last layer ðM;kÞ (Mth layer). Here, X i is the output value of the kth neuron in the output (Mth) layer; k = 1 to NM . ð1;jÞ ðM;kÞ X ð1Þ ¼ fX i g and X ðMÞ ¼ fX i g: new data that has not yet been learnt by the NN. ðm;tÞ X i : the value of the tth neuron in the hidden (mth) layer; t = 1 to Nm. W ðmÞ uv : the weight between the uth neuron in the mth layer and the vth neuron in the (m + 1)th layer, m = 1 to M1; u = 1 to Nm; v = 1 to Nm+1. The fuzzy neural network is applied to diagnose the fault types of a rolling bearing by the sequential diagnosis algorithm, and realized with a developed back propagation neural network called as ‘‘the partially-linearized neural network (PNN)’’ (Mitoma, Wang, & Chen, 2008). A back propagation neural network is only used for training the data, and the PNN is used for testing the learned NN. Here, the basic principle of the PNN for the fault diagnosis is described as follows. The neuron number of the mth layer of an NN is Nm. The set ð1;jÞ X ð1Þ ¼ fX i g represents the pattern input to the 1st layer and the ðM;kÞ ðMÞ set X ¼ fX i g is the training data for the last layer (Mth layer). ð1;jÞ Here, i = 1 to P, j = 1 to N1, k = 1 to NM, and, X i : the value input to ðM;kÞ the jth neuron in the input (1st) layer; X i : the output value of the kth neuron in the output (Mth) layer, k = 1 to NM. Even if the NN converges by learning X(1) and X(M), it cannot adequately deal with the ambiguous relationship between the new X(1)⁄ and X(M)⁄, which has not been learnt. In order to predict X(M)⁄ according to the probability distribution of X(1)⁄, partial linear interpolation of the NN is introduced as shown in Fig. 7. In the NN that has converged with the data X(1) and X(M), the following symbols are used.
Fig. 7. The partial linearization of the sigmoid function.
W ðmÞ uv : the weight between the uth neuron in the mth layer and the vth neuron in the (m+1)th layer, m = 1 to M; u = 1 to Nm; v = 1 to Nm+1. If all these values are memorized by the computer, when new ð1;uÞ ð1;uÞ ð1;uÞ values Xj (1,u)⁄ (X j < Xj < X jþ1 Þ are input into the first layer, the predicted value of the vth neuron (v=1 toNm) in the (m+1)th layer (m = 1 to M 1) can be estimated by
nP ðmþ1;mÞ
Xj
ðmþ1;mÞ
¼ X iþ1
o ðm;uÞ ðmþ1;v Þ ðmþ1;v Þ X j Þ X iþ1 Xi PNm ðmÞ ðm;uÞ ðm;uÞ u¼1 W uv X iþ1 X i
ðm;uÞ Nm ðmÞ u¼1 W uv ðX iþ1
ð23Þ Using the operation above, the sigmoid function is partiallylinearized, as shown in Fig. 8. If a function must be learned, the PNN will learn the points indicated by the symbols shown in Fig. 8. When new data ðs01 , s20 ) are input into the converged PNN, the values depicted by the j symbols corresponding to the data ðs01 ; s02 Þ will quickly be identified as Pe. Thus, the PNN can be used to deal with ambiguous diagnosis problems. As shown in Fig. 8, the new data (s1, s20 ) input into the converged PNN, and which are not learnt by the PNN for recognizing, must satisfy the following condition.
s1ðminÞ < s01 < s1ðmaxÞ and s2ðminÞ < s02 < s2ðmaxÞ
Here, s1(min) and s2(min) are the minimum values, respectively, of s1 and s2, which have been learned by the PNN. Therefore, in this work, the values (Pi and P j ) of symptom parameters input to the PNN for fault diagnosis must satisfy the following condition.
PiðminÞ < Pi < PiðmaxÞ
and P jðminÞ < Pj < PjðmaxÞ
ð25Þ
S2 P a
Pe S2 '
c
b e d
S1 '
ðm;tÞ
: the value of the tth neuron in the hidden (mth) layer, t = 1 Xi to Nm;
ð24Þ
Fig. 8. Interpolation by the PNN.
S1
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(a) For first step
(b) For second step
(c) For third step
Fig. 9. The PNN for the fault classification of a bearing.
Table 3 Examples of training data for the PNN learning. SPs P1
Table 4 Verification results of the learned PNN. State
P4
w0N
(a) For the first step 147.20 0.086 167.46 0.103 199.98 0.154 ... ...
0.67 0.286 ...
(b) For the second step P4 P6 0.645 3.23 0.584 0.221 1.762 1.464 ... ...
wI 1 0.5 0 ...
(c) For the third step P3 P7 0.678 1.99 0.654 4.35 0.654 2.763 ... ...
wR 1 0.25 0.33 ...
0
w0B
w0U
0
0
0
1 0 0.286
...
0
wOR 0 0.25 0.67 ...
0
0.33 0.428 ...
0
wU 0 0.25 0.33 ...
0
wO 0
0.5 0 ...
P1
wU 0 0.5 0.67 ...
P4
(a) In the first step 176.49 0.135 162.75 0.114 204.32 0.995 213.07 1.675 206.93 0.635 207.15 1.52 202.74 0.56 149.35 0.103 P6 P4 (b) In the second step 0.635 0.56 1.675 1.285 1.52 1.24 0.103
2.984 2.79 0.271 1.012 0.27 0.625 4.70
P3 P7 (c) In the third step
5.2. PNN for fault diagnosis and verification Fig. 9 shows the PNN built based on an algorithm of sequential diagnosis proposed in this paper. As an example, for the first step shown in Fig. 9a, the PNN consists of the first layer, one hidden layer, and the last layer. The neurons in the first layer are input as the SPs (P1 and P4). The number of neurons in the hidden layer is seventy determined by trial and error, and the outputs in the last layer are w0N ; w0B , and w0U which are the possibility grades of the normal state, bearing flaws, and unknown state, respectively. The knowledge for training of the PNN is acquired by the possibility theory and the Mycin certainty factor, and it can deal with the vagueness and uncertainty relationships between the symptoms and the fault types. Therefore, the PNN can obtain good convergence when learning the acquired knowledge. Parts of the training data are shown in Table 3. We used the data measured in each state that had not been learned by the PNN to verify the classifiable capability of the PNN. When the value of the symptom parameters is input into the learned PNN, it can quickly and automatically classify those faults according to the possibilities of the corresponding states. The diagnosis results are shown in Table 4. According to the verification results, the possibilities output by the PNN show correct judgments in each state. Therefore, the PNN can precisely distinguish the types of bearing faults on the basis of the possibilities of the symptom parameters.
0.708 0.703 0.675 0.664 0.644
4.068 3.721 2.444 1.799 0.05
wN
0
0.87 0.78 0.011 0.001 0.01 0.002 0.01 0.16 0
wB
0
wU
0.013 0.004 0.86 0.872 0.815 0.875 0.745 0.001 0
wI
wOR
0.858 0.82 0.01 0.014 0.01 0.007 0.19
0.02 0.01 0.94 0.826 0.934 0.871 0.022
0
0
wR
wO
0.021 0.019 0.859 0.84 0.152
0.865 0.789 0.053 0.005 0.000
0
0.12 0.22 0.13 0.127 0.162 0.129 0.232 0.826 wU
0
0.153 0.178 0. 07 0.158 0.09 0.122 0.81 wU
0
0.15 0.231 0.116 0.156 0.85
Judge N N B B B B B U Judge I I O O O O U
or or or or
R R R R
Judge O O R R U
(1) A sequential diagnosis technique was proposed through which the fuzzy neural network realized by the partiallylinearized neural network (PNN) could sequentially distinguish fault types. (2) Knowledge for the PNN was acquired by possibility theory and the Mycin certainty factor, which can represent uncertain diagnostic information. The establishing method of the membership function converting probability distribution function of symptom parameter into possibility function by the possibility theory was proposed, and the combination membership functions of several symptom parameters were obtained by the certainty factor. (3) The non-dimensional symptom parameters were also described in the frequency-domain, and these parameters could reflect the characteristics of the signals measured for fault diagnosis of a rolling bearing. (4) This proposed method had been successfully applied to fault diagnosis of a rolling bearing used in a centrifugal blower, and the faults were sequentially and automatically diagnosed on the basis of the possibilities of the symptom parameters.
6. Conclusions In order to process the ambiguous relationship between the symptom parameters and fault types, and automatically identified faults, an intelligent diagnosis method for bearing faults was proposed on the basis of possibility theory and a fuzzy neural network with frequency-domain features of vibration signals. The main conclusions were obtained as follows.
Acknowledgement This work was supported by the Grants-in-aid for Scientific Research of JSPS (Scientific Research (B) 19360074), National Natural Science Foundation of China under Grant 51075023, and the Fundamental Research Funds for the Central Universities.
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