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Abstract

Rolling bearing is of great importance in rotating machinery, so the fault diagnosis of rolling bearing is essential to ensure safe operations. The traditional diagnosis approach based on characteristic frequency was shown to be not consistent with experimental data in some cases. Furthermore, two data sets measured under the same circumstance gave different characteristic frequency results, and the harmonic frequency was not linearly proportional to the fundamental frequency. These indicate that existing fault diagnosis is inaccurate and not reliable. This work introduced a new method based on data-driven random fuzzy evidence acquisition and Dempster–Shafer evidence theory, which first compared fault sample data with fuzzy expert system, followed by the determination of random likelihood value and finally obtained diagnosis conclusion based on the data fusion rule. This method was proved to have high accuracy and reliability with a good agreement with experimental data, thus providing a new theoretical approach to fuzzy information processing in complicated numerically controlled equipments.

Keywords

Data-driven random fuzzy set D-S evidence theory data fusion rule

Introduction

As one of the critical parts in rotating machinery, rolling bearing has high probability of failure when exposed in an undesirable working environment. Vibration-based rolling bearing fault diagnosis has always been a hotspot, whose critical research point lies in eliminating overwhelming background signals, which can totally cover fault signal and noise signal and thus makes them undetectable.^1–2 Accurately extracting fault signal is a necessity for a reliable diagnosis. Plenty of significant research has been conducted using following methods:^3–6 (1) ferrographic analysis and ferromagnetic analysis whose acceptance is restricted due to high cost; (2) information processing methods, such as wavelet transform, kernel principal component analysis, neural network, ant colony algorithm, and so on; (3) multi-sensor data fusion method, including Dempster–Shafer (D-S) evidence theory, fuzzy diagnosis, rough set theory, and so on. Until now, the most widely accepted method is multi-sensor data fusion method based on vibration detection. By solving fault diagnosis governing equation, essential parameters are analyzed around characteristic frequency in time-domain, frequency-domain, or time-frequency-domain, and then information containing fault feature is extracted using data processing technology. However, a disagreement was found, as shown in this work, between theoretical calculation based on above method and experimental data, which has not been reasonably explained by now. This work introduces a new method based on data-driven random fuzzy evidence acquisition and D-S evidence theory and a good agreement with experimental data has been accomplished.

Traditional rolling bearing fault diagnosis based on characteristic frequency and its potential issue

Conventional rolling bearing fault diagnosis based on characteristic frequency calculation and experimental verification

When a fault occurs, a significant change in vibration pattern between rolling bearing with bearing pedestal and equipment shell can be observed (Figure 1). This kind of vibration pattern change would generate a periodic signal characterized by certain fault-position-dependent frequencies. Such characteristic frequencies regarding different fault positions are given below

Outer race fault : f_{o} = \frac{z}{2} f_{r} (1 - \frac{d_{1}}{D_{1}} \cos a)

(1)

Inner race fault : f_{i} = \frac{z}{2} f_{r} (1 + \frac{d_{1}}{D_{1}} \cos a)

(2)

Rolling part fault : f_{g} = \frac{D_{1}}{2 d_{1}} f_{r} (1 - \frac{{d_{1}}^{2}}{{D_{1}}^{2}} \cos^{2} a)

(3)

Outer track of rolling bearing : f_{bo} = \frac{f_{r}}{2} (1 - \frac{d_{1}}{D_{1}} \cos a)

(4)

Inner track of rolling bearing : f_{bi} = \frac{f_{r}}{2} (1 + \frac{d_{1}}{D_{1}} \cos a)

(5)

Figure 1.

Setup (top) and schematic (bottom) of the roller-bearing fault diagnosis system.

where $f_{r}$ is the rotation frequency of rolling bearing, z is the roller number in one rolling bearing, $d_{1}$ is the rolling element diameter, $D_{1}$ is the pitch diameter, and a is the pressure angle.

Rolling bearing of 6308 type was chosen in experiments. The rolling bearing properties used were $z = 8$ , $d = 15 mm$ , $D_{1} = 65.5 mm$ , $a = 6 \circ$ , and $f_{r} = 18.6 Hz$ . With the above parameters, $f_{o}$ can be calculated using equation (1), which is equal to 58 Hz. Therefore, corresponding second and third harmonic frequencies are 116 and 174 Hz, respectively. Figure 1 shows two experimental data sets after spectrum transform with outer race fault measured at same operational environment.

Potential problem in traditional fault diagnosis

The experimental data do not agree well with theoretical results, which indicates the following potential problems within this traditional fault diagnosis method:^7,8

The theoretical results fail to agree with the experimental data.

The experimental data are not repeatable.

There is interharmonic error between the second and third harmonics.

Due to the uncertainty of operational environment, such as sensor position, background noise, environment temperature, as well as sensor shifting error and human factor, experimental data are, not surprisingly, random and fuzzy. Therefore, it cannot be accurately predicted by equations (1)–(5). In order to overcome this problem, this work proposed a new method relying on data-driven random fuzzy evidence acquisition and D-S evidence theory.

The method is based on data-driven random fuzzy evidence acquisition and D-S evidence theory. Establishing fault fuzzy system and making data fusion a likelihood measurement value are defined as diagnostic standard to describe the degree to which fault sample and fuzzy database are related. This value reflects the likelihood of fault occurrence, namely, higher value, higher fault probability. Therefore, instead of simply calculating characteristic frequency, this method actually takes the environment factors’ influence into account, which is more realistic and thus improves the diagnosis accuracy.

Fault fuzzy expert system and sample data set matching

Figure 2 shows the signals corresponding to good and faulty conditions along with their frequency spectrum. Based on a large quantity of existing experimental data measured on a variety of rolling bearing fault types, fault fuzzy expert system and fault sample data set were obtained by the following steps.

Figure 2.

The signals corresponding to good (a) and faulty, (b) conditions along with their frequency spectrum.

The establishment of fault fuzzy system

Make observations for n times with time interval $Δ t$ , serving as one data set. Then repeat the above process m times. Assume the first three vibration peaks corresponding to 1X, 2X, and 3X characteristic frequencies, respectively, and any higher order terms are neglected.

Calculate the arithmetic mean value of kth data set

M_{k} = \frac{(x_{k, 1} + x_{k, 2} + x_{k, 3} + \dots + x_{k, n})}{n}

(6)

Calculate standard deviation

σ_{k} = \sqrt{\frac{({(x_{k, 1} - M_{k})}^{2} + {(x_{k, 2} - M_{k})}^{2} + {(x_{k, 3} - M_{k})}^{2} + \dots + {(x_{k, n} - M_{k})}^{2})}{n}}

(7)

Obtain Gaussian membership function

F (x, M_{k}, σ_{k}) = e^{(- \frac{{(x_{k} - M_{k})}^{2}}{2 σ_{k}^{2}})}

(8)

Establish the membership value of fuzzy expert system for fault diagnosis based on m data set

F (x) = {\begin{matrix} e^{(- \frac{{(x - M_{a})}^{2}}{2 σ_{a}^{2}})} x < M_{a} \\ 1 M_{a} \leq x \leq M_{b} \\ e^{(- \frac{{(x - M_{b})}^{2}}{2 σ_{b}^{2}})} x > M_{b} \end{matrix}

(9)

where x is the sensor measurement value, $M_{a} = min_{k = 1, m} (M_{k})$ , and $M_{b} = max_{k = 1, m} (M_{k})$ .

Take the outer race fault as an example. Consider the first three characteristic frequencies; thus, only three Gaussian membership functions are obtained. If inner race fault, outer race fault, rolling element, and rolling bearing track are all studied simultaneously, this leads to 12 Gaussian membership functions. Limited by experiment condition, only nine Gaussian membership functions related to inner race fault, outer race fault, and rolling element fault were studied in this work, which were later proved to be sufficient to validate the accuracy of fuzzy expert system diagnosis.

Membership value of fault sample

The procedure for the determination of membership function of the samples is as follows: first, observations are made for n times with time interval $Δ t$ , which serves as one data set. Three fault types result in three Gaussian membership functions and the Gaussian membership function is obtained as follows

F (x_{d}, M_{d}) = e^{(- \frac{{(x_{d} - M_{d})}^{2}}{2 σ_{d}^{2}})}

(10)

where $\sum_{F}$ is a random data set formed by those whose membership degree is larger than $δ$ in the characteristic parameter space, which means the element numbers vary with the value of $δ$ ; x is likelihood value; $μ_{F} (x)$ is the upper bound for likelihood evaluation; U is observation collection; $δ$ is a constant; and $\sum_{F}$ is a certainty set and can be also interpreted as a cut set in a fuzzy set.

Data matching and fusion

The process is as follows. First, match fault sample data with fuzzy expert system in terms of Gaussian membership functions, identify fault feature, and determine fuzzy likelihood. Then, normalize matching value and calculate basic probability assignment (BPA)⁹ from data fusion of fuzzy evidence data system. BPA here means fault detect evidence. The determination of BPA, which is the main and first step in D-S theory, is still an open puzzle. The result data are used to build a normal distribution–based model for each attribute of the data. Then, a nested structure BPA function can be constructed, using the relationship between the test data and the normal distribution model. Figure 3 shows likelihood value as a function of frequency for outer race fault sample and fuzzy rule base. Take the 1X frequency of outer race as a reference; fault sample membership curve is obtained with parameters $n = 40$ , $m = 5$ .

Figure 3.

Outer race fault membership curve around 1X characteristic frequency compared with fuzzy rule base.

Random sets and fuzzy sets

Assume $(Ω, F, P)$ is a probability space and $(U, B_{U})$ is a measurable space, where U is observation collection and $B_{U}$ is corresponding Borel domain. Define fuzzy set-valued mapping.

$X : Ω \to P (U)$ , where P(U) is power set consisting of every subset in U. If for $\forall A \in B_{U}, X (A) = {ω \in Ω : X (ω) \land A \neq \emptyset} \in F$ , $χ$ is then considered strongly measurable.

Assume $δ$ is a number which is randomly distributed between [0, 1], and define

\sum_{F} \overset{def}{=} \sum_{δ} (μ_{F}) = {χ \in u | δ \leq μ_{F} (χ)}

(11)

where $\sum_{F}$ is a random data set formed by those whose membership degree is larger than $δ$ in the characteristic parameter space, which means the element numbers vary with the value of $δ$ . When $δ$ is a constant, $\sum_{F}$ is a certainty set and can be also interpreted as a cut set in a fuzzy set.

Similarly, the random set for fault sample can be defined as

θ \overset{def}{=} \sum_{δ} (μ_{0}) = {x \in u | δ \leq μ_{0} (χ)}

(12)

If fuzzy fault feature parameters $θ$ and $\sum_{F}$ have a good agreement with each other, this indicates the occurrence of fault in sample. Such matching is a probabilistic likelihood phenomenon; therefore, $θ$ and $\sum_{F}$ are randomly varied, and in some cases, they are orthogonal to each other, which is called conflict situation and also considered as inspection pending mode.

Failing to predict the occurrence of fault, however, does not necessarily indicate the orthogonality of the data set. In some cases, there exists similarity between fault sample and fuzzy database. Therefore, it is fair to define a likelihood measurement value, which describes the degree to which fault sample and fuzzy database are related. This value reflects the likelihood of fault occurrence, namely, higher value, higher fault probability. Correspondingly, we can get

\begin{matrix} ρ (θ | F) = ⪻ (θ \cap \sum_{F} \neq \emptyset) = p (δ) \leq (μ_{F} (χ) \land μ_{0} (χ)) \\ = sup_{χ} min (μ_{F} (χ) \land μ_{0} (χ)) \end{matrix}

(13)

Equation (13) shows a minimum filter operation between $μ_{F}$ and $μ_{0} (χ)$ for every single value $χ$ and then picks up the $χ$ corresponding to maximum value in the previous filter operation as likelihood value, which can also be obtained by determining y values of three intersection points in Figure 2 between the membership curve of fault sample and three membership curves from fuzzy database around 1X characteristic frequency. Then, a fuzzy vector can be obtained, $μ_{1} = (μ_{11}, μ_{12}, μ_{13})$ ; similarly, vectors $μ_{2} = (μ_{21}, μ_{22}, μ_{23})$ and $μ_{3} = (μ_{31}, μ_{32}, μ_{33})$ can be obtained around 2X and 3X characteristic frequencies, respectively.

Data fusion rule

After normalizing $μ_{1}, μ_{2}, μ_{3}$ and finishing D-S evidence data fusion, a diagnosis vector can be obtained. A combination with BPA value is a necessity for accurate diagnosis, which requires certain criterion. This work assumes the following criteria based on previous experience:¹⁰

Maximum BPA should exist and must be larger than 0.5 for fault to occur.

Uncertainty m(⊙) should be below some certain threshold which is defined as 0.2 here.

The BPA criterion for fault occurrence should have at least 0.2 difference value with other BPA.

Results and discussion

Validation of fault diagnosis method

All the previous work first extracted feature information related to the calculated characteristic frequencies, followed by the time-domain, frequency-domain, as well as time-frequency-domain information processing based on rough set, wavelet packet theory, and so on.^11–12 Fault diagnosis method therefore can be obtained by information fusion technology. Based on the above analysis, this work proposes a new data-driven fuzzy fault diagnosis method,^13–16 which is also applicable to the driven data set. In order to validate this method, we conducted experiments with parameters $n = 40$ , $m = 5$ . A total of 200 data sets were obtained and summarized in Table 2. Table 2 clearly proves the high accuracy of this diagnosis method. The inconsistency and inaccuracy were most likely caused by the measurement error during the experiments due to environment uncertainty. On the other hand, although our fault fuzzy expert system has shown high diagnosis accuracy with sufficient database by obtaining high diagnosis accuracy, there is always a necessity to enrich it with more experimental data and thus reduce the inconsistency and inaccuracy in the diagnosis process.

The application capability of diagnosis methods

All the previous work was conducted under same operational environment, namely, the same experiment table, same background temperature, and so on. A total of n data sets were measured and then separated into two groups, one containing n − m data sets and the other one consisting of the remaining m data sets. These two groups were for concluding diagnosis method and validation purpose, respectively. Due to same operational environment, there is high consistency and similarity among n data sets, which make validation of the method less convincing and less applicable to other fault type or even the same fault type but measured under different environment. The high accuracy arising from the data consistency has always been controversial in this field. Fuzzy expert system with 200 data sets from rolling bearing A is established and then this fuzzy expert system is used to test 40 data sets from rolling bearing B with outer-bearing fault following the diagnosis method introduced in this article. The important parameters of fuzzy expert system of rolling bearing A and fault sample rolling bearing B are listed in Table 1.

Table 1.

Diagnosis methods verification by inverse analysis based on 200 data sets.

Fault type	Accurate case	Misjudge case	Uncertainty case
Outer race	199	0	1
Rolling element	198	1	1
Inner race	198	0	2

Figure 3 shows data fusion of membership curves from fuzzy expert system and fault sample at 1X, 2X, and 3X frequencies. The likelihood value with respect to these three fault types is then obtained from Figure 4. The values are concluded in Table 2.

Figure 4.

Data fusion of membership curves from fuzzy expert system and fault sample at (a) 1X, (b) 2X, and (c) 3X frequencies.

Table 2.

Parameters in fuzzy expert system of rolling bearing A and fault sample rolling bearing B.

Fault type	Outer race fault (rolling bearing A)				Rolling element Fault (rolling bearing A)				Inner race fault (rolling bearing A)				Fault sample (rolling bearing B)
Fault type	M_a	M_b	a	b	M_a	M_b	a	b	M_a	M_b	a	b	M	c
1X frequency	57.90	58.45	19.063	17.102	71.3	71.75	9.905	7.987	86.8	86.85	17.642	19.92	58.714	10.904
2X frequency	114.6	116.65	34.252	21.713	142.15	142.9	26.41	32.87	174.6	175.8	36.989	33.326	117.571	17.619
3X frequency	172.85	174.85	31.936	29.654	212.7	214.1	33.693	33.252	265.05	265.7	35.721	30.436	177.143	20.809

There are two effective and reliable methods for validation.¹⁷ One is replacing with an unknown fault type rolling bearing and then following the procedure as introduced above to conclude diagnosis result.^18,19 If a good agreement with the actual sample fault type can be obtained, the feasibility of the proposed method is thus validated. Although this method is highly accurate, detecting the actual fault type of this arbitrary rolling bearing experimentally requires extra work, which makes it less operable. The other one is to replace with a known fault and then following the same procedure as above. This method clearly avoids the fault detecting problem in the first method. So the following work is based on the second method, which first establishes fuzzy expert system with 200 data sets from rolling bearing A and then uses this fuzzy expert system to analyze another 40 data sets from rolling bearing B with known outer-bearing fault. If the fault diagnosis result is outer-bearing fault, the feasibility of the proposed method can thus be validated. The important parameters on fuzzy expert system of rolling bearing A and fault sample rolling bearing B are listed in Table 3.

Table 3.

Likelihood value of three fault types.

Fault type	Likelihood value of outer race fault	Likelihood value of rolling element fault	Likelihood value of inner race fault	Uncertainty degree
1X frequency	0.9838	0.8421	0.6999	0.0162
2X frequency	0.9514	0.5567	0.2854	0.0486
3X frequency	0.9871	0.7734	0.4365	0.0129
D-S fusion	0.9838	0.8421	0.6999	0.0162

D-S fusion: Dempster–Shafer fusion.

After normalizing data from Table 4 and applying D-S evidence data fusion, BPA values for the three fault types are obtained, which are listed in Table 4. From Table 4, the diagnosis results indicate that rolling bear B had outer race fault which matched experiments very well, thus demonstrating high accuracy of this diagnosis method based on data-driven random fuzzy evidence acquisition and D-S evidence theory.

Table 4.

BPA values for the three fault types.

Fault type	Outer race BPA	Rolling element BPA	Inner race BPA	Uncertainty degree	Diagnosis result
1X frequency	0.5164	0.3022	0.1549	0.0265	Outer race fault
2X frequency	0.3870	0.3312	0.2753	0.0065	Uncertain
3X frequency	0.4466	0.3499	0.1975	0.0060	Uncertain
D-S data fusion	0.6726	0.235	0.0456	0.0468	Outer race fault

BPA: Basic Probability Assignment; D-S data fusion: Dempster–Shafer data fusion.

Table 5 shows the comprehensive evaluation of the three fault types. According to this table, it is clearly shown that the fault diagnosis method based on data-driven random fuzzy evidence acquisition and D-S evidence theory has very high diagnosis accuracy, which provides a promising diagnosis method in equipment safety monitoring.

Table 5.

Comprehensive evaluation regarding the three fault types.

	Accuracy (%)	Specificity (%)	Sensitivity (%)
Outer race fault	95.3	94.1	98.7
Inner race fault	94.7	96.5	94.6
Rolling element fault	97.8	96.9	95.7

Conclusion

As an increasing requirement in the safety of large computer numerically controlled (CNC) equipment, fault diagnosis, condition monitoring, and fault prediction have drawn more and more attention recently. This work introduced a data-driven fuzzy random evidence acquisition method combined with D-S evidence data fusion, which shows good agreement with experimental data and proves to be accurate and reliable. Therefore, this work provides an effective approach for condition monitoring and fault diagnosis.

Footnotes

Academic Editor: Duc T Pham

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the National Natural Science Foundation of China (Grant No. 51075220), A Project of Shandong Province Higher Educational Science and Technology Program (Grant No. J13LB11), Specialized Research Fund for the Doctoral Program of Higher Education (Grant No. 20123721110001), and Science & Technology Basic Research Program of Qingdao (Grant No. 12-1-4-4-(3)-JCH).

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Rolling bearing fault diagnosis method based on data-driven random fuzzy evidence acquisition and Dempster–Shafer evidence theory

Abstract

Keywords

Introduction

Traditional rolling bearing fault diagnosis based on characteristic frequency and its potential issue

Conventional rolling bearing fault diagnosis based on characteristic frequency calculation and experimental verification

Potential problem in traditional fault diagnosis

Fault fuzzy expert system and sample data set matching

The establishment of fault fuzzy system

Membership value of fault sample

Data matching and fusion

Random sets and fuzzy sets

Data fusion rule

Results and discussion

Validation of fault diagnosis method

The application capability of diagnosis methods

Conclusion

Footnotes

Declaration of Conflicting Interests

Funding

References