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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

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 [

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 [

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 [

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 [

Suppose

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 [

The formula to calculate reliability median rank empirical value is

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 is

The probability density function of two-parameter Weibull distribution is

The reliability function is

Maximum likelihood method [

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

When the maximum likelihood method is used to estimate two-parameter Weibull distribution, the likelihood equation set is obtained as below:

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

The reliability function is

The probability density function of three-parameter Weibull distribution is

The reliability function is

In the process of parameter estimation of three-parameter log-normal distribution, the integral transformation moment method [

The

The exceeding probability weighted moments of observed sample are

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

According to statistic theory, from the reliability empirical vector

Suppose the corresponding discrete failure probability of each failure life data is

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

Let (

Suppose the probability distribution density function with maximum entropy is

The first Lagrangian multiplier

Other

Newton iteration method can be used to solve Lagrangian multiplier vector

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

The value of

The integration variable

Integrate the improved maximum entropy distribution density function

Through comparing the reliability discrete vector obtained from empirical equation (

Suppose that two-parameter Weibull (TWPW) distribution parameter

Verification of empirical model by two-parameter Weibull model.

Suppose that three-parameter Weibull (THPW) distribution parameter

Verification of empirical model by three-parameter Weibull model.

Suppose that two-parameter log-normal (TWPLN) distribution parameter

Verification of empirical model by two-parameter log-normal model.

Suppose that three-parameter log-normal (THPLN) distribution parameter

Verification of empirical model by three-parameter log-normal model.

In the above four simulation examples, it can be seen that the median ranks empirical method can excellently describe two-parameter Weibull distribution, three-parameter Weibull distribution, two-parameter log-normal distribution, and three-parameter log-normal distribution. The reliability empirical values almost completely fall on the known distribution function curve, which accurately describes the distribution law of failure data. In four simulation examples, the standard deviations between empirical vector

Though this empirical formula does not need parameter estimation, is easy to use, and can precisely compute reliability, the estimation results of reliability truth are discrete, fluctuant, and uncertain, making it hard to conduct continuous estimation. Moreover, when failure data repeatedly appear, this formula cannot estimate the failure probability precisely. In order to get more accurate and continuous reliability estimation model, it is necessary to make further study on fitting curve of failure data (researched later).

If using two-parameter log-normal distribution and two-parameter Weibull distribution as reliability model, it is required to verify the accuracy of parameter estimation on two-parameter Weibull distribution and two-parameter log-normal distribution by maximum likelihood method. The improved maximum entropy method does not take function distribution into account, but using simulated data of two-parameter Weibull distribution and two-parameter log-normal distribution to verify the practicability of improved maximum entropy method.

First, generate a group of two-parameter log-normal distributed random numbers by computer, with parameter setting as

Use the reliability empirical equation (

Use maximum likelihood method’s equation (

Use internal mapped method’s equation (

Substitute the estimated parameters into (

Reliability model simulation of two-parameter log-normal distribution and improved maximum entropy distribution.

From Figure

Generate a group of random numbers on two-parameter Weibull distribution by computer, with parameter setting as

Use the reliability empirical equation (

Use maximum likelihood method’s equation (

Use internal mapped method’s equation (

Substitute the estimated parameters into (

Reliability model simulation of two-parameter Weibull distribution and improved maximum entropy distribution.

From Figure

To sum up, from the results of parameter estimation on random numbers of two-parameter log-normal distribution and two-parameter Weibull distribution, it is known that taking maximum likelihood method as parameter estimation method is feasible for these two models with highly accurate estimation results; improved maximum entropy reliability function basically completely coincides with empirical value vector, which proves that this model is suitable for the above two distributions and has small error and high precision.

If using integral transformation moment method, linear moment method, and probability weighted moment method as the parameter estimation method of three-parameter log-normal distribution, it is a must to verify the feasibility of these three-parameter estimation methods to three-parameter log-normal distribution.

Let distribution parameter

Empirical value

Three-parameter log-normal uses integral transformation moment (ITM) method’s equation (

Three-parameter log-normal uses linear moment (LM) method’s equation (

Three-parameter log-normal uses probability weighted moment (PWM) method’s equations (

Improved maximum entropy (ME) uses internal mapped method’s equation (

Substitute the estimated parameters into (

Reliability model simulation of three-parameter log-normal distribution and improved maximum entropy model.

It is observed in Figure

Comparison of three-parameter log-normal distribution and improved maximum entropy distribution.

Estimation method | | | Results |
---|---|---|---|

ITM | 0.2417 | 0.1126 | Valid |

LM | 0.2417 | 0.1063 | Valid |

PWM | 0.2417 | 0.1131 | Valid |

ME | 0.2417 | 0.0916 | Valid |

In reliability function image, it can be found that three estimation methods have good fitting degree. And it is known from

In order to verify the feasibility of

Empirical value

Three-parameter Weibull (THPW) distribution uses

Improved maximum entropy (ME) uses internal mapped method’s equation (

Substitute the estimated parameters into (

Reliability simulation of three-parameter Weibull distribution and improved maximum entropy model.

From Figure

To sum up, it can be known from the simulation results of random numbers obeying three-parameter Weibull distribution, three-parameter log-normal distribution, and improved maximum entropy probability distribution that the reliability estimation methods of the above three models are all feasible with highly accurate estimation results. Improved maximum entropy reliability function basically completely coincides with empirical value vector and is suitable for the above two distribution models with small error and high precision. Meanwhile, it also proves that the distribution of sample data can be ignored in improved maximum entropy application process, and its major feature is that it applies to the poor information issues with unknown probability distribution, trends, or prior information. This is because the improved maximum entropy model does not need parameter estimation or consider any distribution, but objectively processing experimental data. That is to say, there is no ideal model artificially presupposed before data processing, which overcomes the influence produced by subjective factor and parameter estimation error, so that the certainty rule in data change can be directly reflected.

At the same time, median rank empirical model can effectively assess the reliability of product failure data with high precision, so, in the following experimental research section, this empirical value should be taken as a criterion to decide whether the reliability model is good or bad; in two-parameter and three-parameter reliability model verification, multiple parameter estimation methods are feasible and effective with high precision and small error; therefore, they are safe and reliable in the following practical case applications; the novel improved maximum entropy model is suitable for all the above conditions, which has laid a foundation to search for quasi-ideal model in the following sections.

This bearing life reliability research adopted NTN bearing life experimental facility and material samples. The experimental facility is

There are 3 groups of failure data test in total. The failure time recorded by experimental facility is initial data with unit in minute. For easy calculation, failure data is conversed into data with unit in hour and sorted from small to large into a group of vector.

In the first batch of test, the failure data is expressed by

In the second batch of test, the failure data is expressed by

In the third batch of test, the failure data is expressed by

Substitute failure data

Substitute failure data

Substitute failure data

Substitute failure data

Their reliability curves are shown in Figure

Reliability function image of failure data

Substitute failure data

Substitute failure data

Substitute failure data

Substitute failure data

Their reliability curves are shown in Figure

Reliability function image of failure data

Substitute failure data

Substitute failure data

Substitute failure data

Substitute failure data

Their reliability curves are shown in Figure

Reliability function image of failure data

From Figures

Comparative results of three reliability models of failure data.

Distribution model | | Critical value | Standard deviation | | |
---|---|---|---|---|---|

| 0.2024 | 0.2591 | 0.0802 | 1.0295 | 5.9662 |

| 0.2024 | 0.2591 | 0.0899 | 0.6197 | 7.4173 |

| 0.1631 | 0.2591 | 0.0737 | 0.2483 | 5.8670 |

| 0.1547 | 0.2417 | 0.0757 | 1.3854 | 9.0933 |

| 0.1696 | 0.2417 | 0.0685 | 1.0675 | 11.7251 |

| 0.1140 | 0.2417 | 0.0498 | 1.2390 | 9.7120 |

| 0.2400 | 0.2749 | 0.0950 | 1.1562 | 7.0353 |

| 0.2205 | 0.2749 | 0.1008 | 0.5763 | 8.7075 |

| 0.1872 | 0.2749 | 0.0701 | 1.6342 | 4.9040 |

Calculate the standard deviation of reliability function value and empirical points and substitute failure data into empirical equation (

According to (

It can be known from Table

In addition, the life value

Relative errors of each group data under different distributions.

Relative error | | | | | | |
---|---|---|---|---|---|---|

| 314.62% | 149.58% | 11.81% | 13.84% | 29.25% | 64.74% |

| 1.69% | 26.42% | 6.37% | 20.73% | 43.46% | 77.56% |

From Table

In the table, the standard deviations of reliability estimation truth-value vector and reliability empirical value vector for

Empirical value

There-parameter log-normal uses integral transformation moment (ITM) method’s equation (

There-parameter log-normal uses linear moment (LM) method’s equation (

There-parameter log-normal uses probability weighted moment (PWM) method’s equation (

Three-parameter Weibull (THPW) uses

Improved maximum entropy (ME) uses internal mapped method’s equation (

Their reliability images are shown in Figure

Comparison results of failure data

Estimating method | | | Result | Standard deviation |
---|---|---|---|---|

ITM | 0.2127 | 0.2591 | Valid | 0.1116 |

LM | 0.3715 | 0.2591 | Invalid | 0.1486 |

PWM | 0.4615 | 0.2591 | Invalid | 0.2183 |

THPW | 0.1847 | 0.2591 | Valid | 0.0773 |

ME | 0.1631 | 0.2591 | Valid | 0.0737 |

Reliability function image of failure data

Empirical value uses the reliability empirical formula for point estimation on failure data

There-parameter log-normal uses integral transformation moment method for parameter estimation on failure data

There-parameter log-normal uses linear moment method for parameter estimation on failure data

There-parameter log-normal uses probability weighted moment method for parameter estimation on failure data

Three-parameter Weibull uses

Improved maximum entropy uses internal mapped method for probability density estimation on failure data

Their reliability images are shown in Figure

Comparison results of failure data

Estimating method | | | Result | Standard deviation |
---|---|---|---|---|

ITM | 0.2172 | 0.2417 | Valid | 0.0813 |

LM | 0.3204 | 0.2417 | Invalid | 0.1160 |

PWM | 0.3627 | 0.2417 | Invalid | 0.1541 |

THPW | 0.1795 | 0.2417 | Valid | 0.0651 |

ME | 0.1140 | 0.2417 | Valid | 0.0498 |

Reliability function image of failure data

Empirical value uses the reliability empirical formula for point estimation on failure data

There-parameter log-normal uses integral transformation moment method for parameter estimation on failure data

There-parameter log-normal uses linear moment method for parameter estimation on failure data

There-parameter log-normal uses probability weighted moment method for parameter estimation on failure data

Three-parameter Weibull uses

Improved maximum entropy uses internal mapped method for probability density estimation on failure data

Their reliability images are shown in Figure

Comparison results of failure data

Estimating method | | | Result | Standard deviation |
---|---|---|---|---|

ITM | 0.2304 | 0.2749 | Valid | 0.1207 |

LM | 0.4397 | 0.2749 | Invalid | 0.1727 |

PWM | 0.5652 | 0.2749 | Invalid | 0.2533 |

THPW | 0.1907 | 0.2749 | Valid | 0.0831 |

ME | 0.1872 | 0.2749 | Valid | 0.0702 |

Reliability function image of failure data

Three failure data groups were estimated by three-parameter log-normal distribution, three-parameter Weibull distribution, and improved maximum entropy probability distribution. Their reliability images are shown in Figures

In Tables

The

Improved maximum entropy reliability curves participate in fitting in three examples, and the fitting curves almost coincide with empirical values. Its standard deviation is the minimum in three groups of life test compared to previous two models, declaring its highest fitting degree. It is observed in Figures

Through calculation in test examples, it can be obtained that the reliability estimation method of improved maximum entropy probability distribution is suitable for all situations, and it is observed that improved maximum entropy reliability model estimation approach has a perfect effect on reliability estimation on bearing failure data. In reliability model estimation on three failure data groups, the two-parameter log-normal distribution of

Compare the standard deviations of three groups of experimental feasible reliability models.

Group number | 2-parameter std. deviation | 3-parameter std. deviation | ME std. deviation |
---|---|---|---|

| 0.0802 (TWPLN) | 0.0773 (THPW) | 0.0737 |

| 0.0685 (TWPW) | N/A | 0.0498 |

| 0.0950 (TWPLN) | 0.0831 (THPW) | 0.0702 |

Compare the relative life errors of three groups of experimental feasible models.

Group number | 2-parameter | 2-parameter | 3-parameter | 3-parameter |
---|---|---|---|---|

| 314.62% | 1.69% | 189.37% | 9.32% |

| 13.84% | 20.73% | N/A | N/A |

| 29.25% | 43.46% | 43.91% | 39.78% |

The results display that, in comparison with three reliability models, the standard deviation between estimated truth-value of improved maximum entropy reliability model and empirical value vector is the smallest. Taking the improved maximum entropy model as datum, other two models possess a larger relative life error under failure probability of 10% and 50%, with maximum relative error of 314.58%, which means the log-normal distribution and Weibull distribution possess low precision and large error in reliability prediction of bearing failure data. Once again, this proves that the novel improved maximum entropy probability distribution being taken as reliability model estimation method possesses the best effect and lowest error in estimating bearing failure life data.

In order to verify that the novel proposed model of maximum entropy distribution can be applied to poor information problem with small sample and unknown probability distribution, another

Initial data series are

With the help of improved maximum entropy method, the reliability estimation results of this bearing failure data group are shown in Figure

Improved maximum entropy reliability function image of small sample data

Depending on reliability empirical value vector

To sum up, in simulation test, three reliability models display good fitting. In other words, in theoretical state, log-normal distribution, Weibull distribution, and improved maximum entropy model can be applied in reliability analysis on product performance failure problems. But in experiment part, we made comparative analysis of three reliability estimation methods according to actual bearing failure data, with the results showing that both standard deviation and relative error are the smallest between reliability empirical value vector

Though the reliability estimation of bearing performance failure data can be realized by the above models, their prediction precision is far different. In practical application, we cannot rush to a conclusion by reliability obtained from single model.

The standard deviations of two-parameter log-normal distribution for

For a novel improved maximum entropy model compared to Weibull distribution and log-normal distribution, the relative life error and standard deviation of its truth-value vector and empirical value vector are the smallest, in which, the maximum relative error of log-normal distribution reaches 314.58%, and Weibull distribution reaches 189.37%.

Whether there are really other distribution models (quasi-ideal distribution model) to calculate bearing failure performance reliability or not, at least, the improved maximum entropy probability distribution is a precise simulation to such quasi-ideal distribution model.

The novel proposed model of maximum entropy probability distribution does not take parameter distribution into account and allows poor information issues with unknown probability distribution, unknown prior information, or trends. This provides important theoretical reference to many uncertain information and poor information issues in engineering and even aerospace field and remedies the deficiency of classic statistics.

The authors declare that they have no conflicts of interest.

This project is supported by the Natural Science Foundation of Henan Province of China (Grant no. 162300410065).