Mathematical Problems in Engineering

Volume 2018, Article ID 6189527, 21 pages

https://doi.org/10.1155/2018/6189527

## Estimation on Reliability Models of Bearing Failure Data

^{1}Mechatronics Engineering College, Henan University of Science and Technology, Luoyang 471003, China^{2}Collaborative Innovation Center of Machinery Equipment Advanced Manufacturing of Henan Province, Henan University of Science and Technology, Luoyang 471003, China^{3}National Center for Material Service Safety, University of Science and Technology Beijing, Beijing 10083, China^{4}Luoyang Bearing Research Institute Co. Ltd., Luoyang 471039, China

Correspondence should be addressed to Chang Zhen; moc.361@2991cnehz

Received 22 May 2017; Revised 2 February 2018; Accepted 13 February 2018; Published 26 March 2018

Academic Editor: Roman Wendner

Copyright © 2018 Xia Xintao et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The failure data of bearing products is random and discrete and shows evident uncertainty. Is it accurate and reliable to use Weibull distribution to represent the failure model of product? The Weibull distribution, log-normal distribution, and an improved maximum entropy probability distribution were compared and analyzed to find an optimum and precise reliability analysis model. By utilizing computer simulation technology and -s hypothesis testing, the feasibility of three models was verified, and the reliability of different models obtained via practical bearing failure data was compared and analyzed. The research indicates that the reliability model of two-parameter Weibull distribution does not apply to all situations, and sometimes, two-parameter log-normal distribution model is more precise and feasible; compared to three-parameter log-normal distribution model, the three-parameter Weibull distribution manifests better accuracy but still does not apply to all cases, while the novel proposed model of improved maximum entropy probability distribution fits not only all kinds of known distributions but also poor information issues with unknown probability distribution, prior information, or trends, so it is an ideal reliability analysis model with least error at present.

#### 1. Introduction

In machinery products and engineering projects, bearings are the joints and wearing parts in the whole transmission system. Their operational reliability is the basis to establish optimization and improvement strategies and implement failure factor analysis, which directly relates to the operation security of product during service time. In bearing reliability estimation, the selection of failure distribution model is of great importance, because it directly relates to the precision of reliability prediction and has a huge influence on the usability of bearing. If the predicted reliability value is too high and the product performance exceeds a fatigue limit of normal operation, particularly for aerospace, high-speed rail, nuclear reactor, precision meter, and such systems, it will result in major vicious accident or even affect the national security [1, 2]. If the predicted reliability value is too low, the product function cannot be fully exploited, making the product lose its environmental adaption and leading to huge waste of conditional resources. Therefore, in order to guarantee the safe and stable operation of product system, it is essential to implement effective monitoring and diagnosing and precise reliability model estimation for bearings [3–7].

Reliability analysis is aimed at searching for the failure distribution information which can exactly reflect that the failure mechanism of product components accords with the analysis results of failure data. After fitting the fault or failure data into certain distribution form, the reliability estimation and prediction will be carried out, in which the distribution function of product failure time is the basis to study reliability. Since the failure state of bearings may be affected by different operation conditions, such as structural composition, material, load, lubrication, and numerous uncertainty factors, the failure life of actual situation is random, accompanied by multiple failure modes among which each mode can be mutually affected, acted, and dynamically varied [8]. There are some difficulties how to quickly and effectively utilize failure data for precise reliability analysis and model selection of production information. Recently, there is a lack of exact plan to describe their gradual change process during operation, and no perfect theoretical system is formed yet [9]. Traditional reliability estimation theory is established on the basis of a large number of failure data. However, in many cases, the probability distribution of problems met in engineering and experiment are not normal ones. For example, the aircraft bearing, due to its high cost, very few failure data, and extremely high requirements for precision and reliability, has a harsh demand for test equipment. This makes it difficult to implement large sample life test or obtain failure data within limited testing time. So, when the product’s life probability distribution is unknown, and we only have small sample data for reference, it is impossible for us to use existing reliability theory to accurately describe its failure evolving law.

At present, the researches of bearing failure data are mostly about Weibull distribution, log-normal distribution, gamma distribution, and binomial distribution reliability analysis methods. In particular, the Weibull distribution and log-normal distribution are widely used in reliability theoretical analysis. Though this has achieved certain results, they show large error and low precision in the process of product reliability estimation [10–15]. The accuracy of reliability models increasingly got the attention of scholars and experts. Rodriguez-Picon et al. [16] considered a gamma process to marginally model the degradation of a performance characteristic through two degradation test phases performed sequentially and obtained a robust model to get reliability estimates considering the effect of two serial degradation tests. Reuben et al. [17, 18] proposed a reliability evaluation method by using Weibull equation and made reliability estimation on product failure data of gearbox bearing and ceramic material, respectively, where results showed very small discrepancy between its fitting curve and the points of failure data. Through the reliability modeling disposing of failure process of large-scale and complicated machinery equipment, Pulcini [19] declared that its failure strength is not so monotonous, and on this basis, he proposed the reliability analysis model of nonhomogeneous Poisson process. To solve the real-time online reliability problems, Hong and Meeker [20] proposed an intelligent reliability estimation method based on dynamic state information changes, which brought much convenience to timely judge the dynamic running state of workpiece. Khaleghei and Makis [21] proposed a new competing risk model to calculate the conditional mean residual life and conditional reliability function of a system subject to two dependent failure modes, namely, degradation failure and catastrophic failure. In order to ensure that the classifier can correctly inspect the system failure information, Hwang and Lee [22] presented a new approach to overcome class imbalance problem and human factor influence by using classification technique, thus speedily and effectively implementing system reliability estimation. Zhang et al. [23] applied ANSYS/PDS module to make simulated analysis on the reliability of agricultural machinery chassis drive axle housing and probed into the influence of random variables such as geometric dimension, load, and material strength on drive axle housing. By exploring discrete random variable and analyzing the expectation interval and information capacity of certain entropy, Aviyente et al. [24, 25] successfully solved the time frequency distribution problem and interval forecast problem of entropy. Xia [26] proposed a grey bootstrap method based on poor information theory, which conducted a reliability analysis of zero-failure data when the probability distribution information is known or unknown in life test, thus providing a strong theoretical reference to the reliability of poor information of zero-failure data.

Based on this, lots of topics and literature sources are mentioned associated with bearing capacity, distribution types, reliability, lifetime, and so on, because there is a very close relationship between them. Firstly, bearing capacity, lubrication condition, rotational speed, and other working conditions are important determinants affecting the bearing lifetime, and the set of the same batch bearing lifetime makes up a number of failure data under the above working conditions. Secondly, the distribution types of a number of failure data can be obtained according to statistical theory, and then its probability density function can be acquired easily. As we all know, the probability density function is the hub of data analysis and solution, and then, according to the probability density function and the given integral interval, the failure probability of bearings can be obtained during their service. Finally, using the unit one to subtract the failure probability, the reliability of bearing failure data is acquired. Therefore, these topics on bearing capacity, distribution types, reliability, and lifetime have a very close coherent interlocking, and all of them have an evidently direct or indirect relationship with the calculation of the reliability of bearing failure data.

This article used the failure data obtained from simulated test and bearing life failure test and made comparative analysis via log-normal distribution, Weibull distribution, and improved maximum entropy distribution, so as to select the optimum and precise reliability analysis model. First, the reliability empirical value calculated by Johnson [27] method was taken as standard. In specific analysis, two-parameter log-normal distribution was compared with two-parameter Weibull distribution, and three-parameter log-normal distribution was compared with three-parameter Weibull distribution, while in parameter estimation process of three-parameter log-normal distribution, the integral transformation moment method, linear moment method, and probability weighted moment method were used for comparative analysis, respectively. The research indicates that Weibull distribution is not applicable to all bearing failure conditions, and sometimes, the log-normal distribution has a smaller standard deviation and lower relative life error in reliability analysis. Then, the reliability estimation method for improved maximum entropy probability distribution was put forward to make reliability analysis on failure data, and this novel method has a high fitting degree and can be applied to all failure cases; when comparatively analyzing with Weibull distribution model and log-normal distribution model, both of the standard deviation and relative life error between its reliability truth-value vector and empirical value vector are minimum.

#### 2. Mathematical Model

##### 2.1. Classical Reliability Empirical Value

Suppose is a failure data series group of research object, and each failure data is unequal and nonredundant, which is denoted aswhere is the vector composed of such failure data group, is the th failure data of this data series, is the serial number of th failure data, and is the number of failure data.

In case that the probability distribution or distribution parameter of failure data is unknown, the reliability of life failure data of research object can be nonparametric estimated using Johnson’s median rank empirical value formula. The reliability empirical formula [28, 29] of such method can be expressed by vector aswhere refers to reliability empirical value vector.

The formula to calculate reliability median rank empirical value iswhere is the th failure data and is the number of failure data.

##### 2.2. Two-Parameter Log-Normal Distribution and Weibull Distribution

Both distributions are common reliability models in engineering applications, especially the Weibull distribution which is widely used in analyzing bearing failure data and has achieved good research results.

The probability density function of two-parameter log-normal distribution is

The reliability function iswhere is the random variable of life, is the proportional parameter, is the shape parameter, , , and .

The probability density function of two-parameter Weibull distribution is

The reliability function iswhere is the random variable of life, is the proportional parameter, is the shape parameter, , , and .

###### 2.2.1. Parameter Estimation

Maximum likelihood method [30, 31] is widely used in the parameter estimation of all kinds of reliability models, which is one of the frequently used parameter estimation methods. For two-parameter log-normal distribution and two-parameter Weibull distribution, the maximum likelihood method is used for parameter estimation of these two models, respectively.

When the maximum likelihood method is used to estimate two-parameter log-normal distribution, the likelihood equation set is obtained as below:where is the likelihood function of log-normal distribution.

When the maximum likelihood method is used to estimate two-parameter Weibull distribution, the likelihood equation set is obtained as below:where is the likelihood function of Weibull distribution. Using the iterative method to solve the equation set, we can acquire the estimated values of two parameters in two-parameter Weibull distribution.

##### 2.3. Three-Parameter Log-Normal Distribution and Weibull Distribution

The probability density function of three-parameter log-normal distribution is

The reliability function iswhere is the random variable of life, is the parameter of log-normal distribution: is the proportional parameter and is the shape parameter, and is the location parameter. ,, and .

The probability density function of three-parameter Weibull distribution is

The reliability function iswhere* t *is the random variable of life,* μ *is the proportional parameter,

*is the shape parameter, and*

*σ**is the location parameter. , , and .*

*τ*###### 2.3.1. The Parameter Estimation of Three-Parameter Log-Normal Distribution

In the process of parameter estimation of three-parameter log-normal distribution, the integral transformation moment method [32], linear moment method [33], and probability weighted moment method [34] were used for comparative analysis, respectively. These parameter estimation methods are mature and widely used in many fields, and the details are as follows.

*(**1) Integral Transformation Moment Method*. In the following formulas, is the location parameter, which can be determined by mean value , coefficient of variation , and coefficient of skew , that is,wherewith where stands for standard normal distribution.

*(2) L-Moment Method*where , , , , , , and are constants. refers to standard normal distribution, and linear moments , , and are determined by life data of given sample.

*(3) Probability Weighted Moment Method*where and are function of , both of which cannot be expressed by explicit formulation; and are coefficient of variation and coefficient of skew of variable , respectively. , , and are the probability weighted moments [35] of zero-order, first-order, and two-order, respectively.withwhere and are transition variables to solve function value for and .

###### 2.3.2. The Parameter Estimation of Three-Parameter Weibull Distribution

The -order exceeding probability weighted moment [36] equation of Weibull distribution iswhere is gamma function; for convenience, is valued as , and we can obtain , , and , and the three parameters of three-parameter Weibull distribution are

The exceeding probability weighted moments of observed sample are

##### 2.4. Improved Maximum Entropy Reliability Model

Improved maximum entropy method can make an optimal estimation with minimum subjective bias on unknown probability distribution. Firstly, according to reliability empirical formula, the reliability empirical vector can be obtained for failure data. Secondly, using the empirical value of vector to adversely deduce a frequency vector for discrete failure, a statistical histogram is acquired, which is convenient to be used to calculate Lagrangian multipliers, and it is different for the traditional maximum entropy to use amount of sample data to solve the Lagrangian multipliers. Then, based on an internal mapped method, probability density function for the improved maximum entropy can be obtained. Finally, reliability function for estimate true value is then acquired by integrating the function of .

###### 2.4.1. Discrete Failure Frequency Vector

According to statistic theory, from the reliability empirical vector in (2), we can get the discrete cumulative failure probability vector :where is the reliability of the th failure data of initial data in (1) and is the number of initial data.

Suppose the corresponding discrete failure probability of each failure life data is . For the first data, that is, when , let its failure probability be . So, from the second data, that is, when *, *the corresponding failure probability of each failure life can be obtained by cumulatively subtracting the elements in vector successively as follows: , .

So the discrete failure frequency vector of its failure life data is

Let (30) correspond to statistic histogram, in which the abscissa is discrete failure life data , and ordinate is the frequency , that corresponds to class mid-value of each group. Normally, the histogram can be expanded to group, that is, , and let , , . Here, the processing of histogram is beneficial for utilizing Newton’s method to solve Lagrangian multipliers in maximum entropy probability density function in the following.

###### 2.4.2. Maximum Entropy Probability Distribution Density Function

Suppose the probability distribution density function with maximum entropy iswhere is the random variable of life; is origin moment order, generally let , and commonly ; is the th Lagrangian multiplier, , totally .

The first Lagrangian multiplier is

Other Lagrangian multipliers shall satisfy

Newton iteration method can be used to solve Lagrangian multiplier vector .

###### 2.4.3. Improved Maximum Entropy Probability Distribution Numerical Solution

There are some difficulties to obtain the solution procedure of probability distribution by improved maximum entropy method. To achieve a quick numerical solution with good convergence, this article adopted the internal mapped Newton iteration method. First, the failure data series were mapped onto dimensionless interval , . Then, the mapping data were sorted from small to large into* Q*-2 groups, and histogram was drawn, to obtain mid-value and frequency of each group. Later, the histogram is extended into* Q *groups; namely, , .

The value of order origin moment changes to

The integration variable turns into mapped variable , and integrating range is mapped onto . The improved maximum entropy probability distribution density function changes towhere and are interval mapping parameters.

Integrate the improved maximum entropy distribution density function in interval . The obtained cumulative failure probability function isTherefore, the improved maximum entropy reliability estimation truth function can be expressed as

#### 3. Computer Simulation Verification

##### 3.1. Reliability Median Ranks Empirical Model Verification

Through comparing the reliability discrete vector obtained from empirical equation (3) with the reliability of simulated failure data of known Weibull distribution and log-normal distribution, we verified the feasibility of median ranks empirical model of failure data.

*Example 1 (two-parameter Weibull distribution simulation example). *Suppose that two-parameter Weibull (TWPW) distribution parameter . Let the value range of reliability simulated vector be 0.95~0.05 with interval of −0.05. According to the inverse function of two-parameter Weibull reliability, we got 19 simulation data , which is failure data. Then according to reliability median ranks empirical model in (3), empirical point can be figured out to compose empirical vector , and the results are shown in Figure 1.