Rolling bearing random truncation data reliability evaluation method

Abstract

The invention relates to a random truncation data reliability evaluation method, in particular to a rolling bearing random truncation data reliability evaluation method. The method comprises the following steps: acquiring random truncation test data, establishing a life distribution model through two-parameter Weibull distribution, converting the Weibull distribution into extreme value distribution, calculating parameter estimators by using a least square method principle, further performing reliability evaluation, calculating a reliable life single-side confidence lower limit and a reliability single-side confidence lower limit, and calculating a reliability life single-side confidence lower limit and a reliability degree single-side confidence lower limit. And finally, verifying the accuracy by adopting a Monte-Carlo method. According to the method, incomplete information in the service life data of the rolling bearing is fully developed and utilized, so that the information amount is effectively increased, and the reliability evaluation precision is improved.

Description

Technical Field

[0001] The invention relates to a reliability evaluation method for randomly truncated data, in particular to a reliability evaluation method for randomly truncated data of a rolling bearing. Background Art

[0002] During reliability life tests of rolling bearings, some bearings may be forced to withdraw from the test due to various force majeure factors, limited by test conditions or other factors. This results in interrupted data and incomplete data. Incomplete data is a common problem in reliability engineering, and randomly truncated data is a difficult problem for statistical analysis in engineering. To improve the accuracy of reliability assessment of rolling bearings, it is of great significance to conduct reliability assessment research using randomly truncated data.

[0003] At present, the maximum likelihood method is mainly used to perform statistical analysis on incomplete data, but it is impossible to perform high-confidence reliability assessment and life prediction, and the research on random truncated data processing methods in the field of reliability engineering is relatively rare. In order to solve the problem of reliability assessment of rolling bearing randomly truncated test data, the present invention proposes a rolling bearing randomly truncated data reliability assessment method to solve the above problem. Summary of the Invention

[0004] In order to solve the above technical problems, the present invention proposes a method for reliability assessment of randomly truncated data of rolling bearings, which can achieve high-confidence reliability assessment of randomly truncated data.

[0005] The technical solution adopted by the present invention is: a reliability evaluation method for randomly truncated data of rolling bearings, which mainly includes the following steps:

[0006] Step S1: Collect random tail test data of rolling bearings, put n samples into random tail test, and generate n=n q +m q The randomly truncated life data of rolling bearings, where n q is the number of failed specimens, m q is the number of terminated samples;

[0007] Step S2: establishing a rolling bearing life distribution model through a two-parameter Weibull distribution;

[0008] Step S3: performing required parameter estimation calculations for the life distribution model, converting the Weibull distribution into an extreme value distribution, converting the data collected in step S1 into standard extreme value distribution randomly truncated data, and using the mean and covariance of the order statistics of the standard extreme value distribution randomly truncated data to calculate the required parameter estimates through the principle of least squares, thereby obtaining parameter estimates of the Weibull distribution;

[0009] Step S4: Reliability assessment: Calculate the one-sided confidence lower limit of the reliable life and the one-sided confidence lower limit of the reliability based on the parameter estimation calculation results;

[0010] Step S5: using the Monte-Carlo method to generate randomly truncated data that obey the Weibull distribution, repeating the simulation process, comparing the parameter estimation results of the method of the present invention with those of the traditional method, and verifying the accuracy of the method of the present invention.

[0011] As a preferred solution, in step S1, the rolling bearing randomly truncated life data is sorted according to the following formula:

[0012]

[0013] Where, t i (i=1,2,…,n q is the failure data of the i-th specimen in the randomly censored data, is the i-th interrupt data, n j represents the total number of failure data of the first j blocks (i.e., blocks 1 to j), m j Represents the total number of sample termination data of the first j blocks, j = 1, 2, ..., q.

[0014] As a preferred solution, in step S2, the two-parameter Weibull distribution core formula is:

[0015]

[0016] Where α is the shape parameter, β is the scale parameter, t is the life failure time, and F(·) is the life distribution function.

[0017] As a preferred solution, in step S3, in order to facilitate parameter estimation, a variable transformation y=lnt is introduced to transform the Weibull distribution function formula in step S2 into an extreme value distribution, whose distribution function is:

[0018]

[0019] Where σ and μ are the scale parameter and location parameter of the extreme value distribution, respectively, which are obtained by arranging and transforming the distribution function of the Weibull distribution:

[0020]

[0021] At the same time, the data collected in step S1 is converted into:

[0022]

[0023] The data collected in step S1 is converted into standard extreme value distribution randomly truncated data, The standard extreme value distribution function is obtained:

[0024] F(y*)=1-exp[-exp(y*)] (6)

[0025] where F(·) is the lifetime distribution function, and F(y*) does not contain any unknown parameters.

[0026] As a preferred solution, in step S3, the data collected in S1 is transformed into a set of randomly truncated data from the standard extreme value distribution F(y*), and the mean and covariance of the order statistics of the randomly truncated data of the standard extreme value distribution are respectively:

[0027]

[0028]

[0029]

[0030] Where E(·) is the mean, Var(·) is the variance, Cov(·) is the covariance, M(y i +r) is the mean of the i+rth order statistic, is the second-order origin moment of the i+rth order statistic, is the second-order mixed moment of the i1+r1th order statistic and the i2+r1+r2th order statistic, and the transformation is:

[0031]

[0032] Among them, h (j+1)kr If i=n j +k failed data, that is, the kth (1≤k≤nth) of the j+1th (0≤j≤q-1) failed data j+1 -n j ) failure data y i If we continue to test the interrupted data before, there are only r interrupted data in y i The previous failure event A (j+1)kr The probability of occurrence, It is Failure data and The covariance of the failure data; It's an event Probability of occurrence;

[0033] Transforming formula (10), we can get It is in the event In case of incident The conditional probability of occurrence is equivalent to computing the random censored data: Events Probability of occurrence.

[0034] As a preferred solution, in step S3, in formulas (7) to (9), M(y i +r) and It can be regarded as the mean and second-order origin moment of the i+rth order statistic in n complete failure data. It is equivalent to the second-order mixed moment of the i1+r1th order statistics and the i2+r1+r2th order statistics in n complete failure data, because:

[0035] so

[0036] y i =μ+σu i +ε i i=1,2,…,n q (13)

[0037] In the formula And there is

[0038] E(ε i )=0 (14)

[0039]

[0040] set up as well as According to the Gauss-Markov theorem, the residual sum of squares Q is

[0041]

[0042] Taking the derivative with respect to μ and σ,

[0043]

[0044] The estimates of parameters μ and σ can be obtained:

[0045]

[0046]

[0047] In the formula

[0048]

[0049]

[0050]

[0051] According to formula (4), formula (18), and formula (19), the two parameter estimates of the Weibull distribution are:

[0052]

[0053] Where α is the shape parameter, β is the scale parameter, μ is the location parameter of the extreme value distribution, and σ is the scale parameter of the extreme value distribution.

[0054] As a preferred solution, in step S4, the randomly truncated data in formula (1) is divided into failed data and terminated data, where n q Failure data and nn q Abort data For the Weibull distribution in formula (2), let

[0055] x=t α (twenty four)

[0056] Where x is a random variable that obeys exponential distribution, for the randomly censored data t1, t2, ..., t n ,make

[0057]

[0058] Where x k (k=1,2,…,n) is a set of randomly truncated data of the random variable x. For the Weibull distributed random variable t, the confidence level is γ and the reliability is R. The one-sided lower confidence limit of the reliable life for

[0059]

[0060] The reliability R(t) confidence level of its given life t is the reliability one-sided confidence lower limit R l (t) is

[0061]

[0062] in The degrees of freedom are 2n q +2 χ 2 The lower gamma quantile of the distribution.

[0063] The beneficial effects of the present invention are:

[0064] First, based on practical engineering practice, this paper proposes a reliability assessment method for rolling bearings using randomly censored data. Randomly censored data are common in reliability engineering. This paper proposes an optimal linear unbiased estimation method for rolling bearing life distribution parameters using randomly censored data, and derives one-sided confidence limits for rolling bearing reliability and reliable life. This method offers higher assessment accuracy and facilitates engineering applications.

[0065] Secondly, the method of the present invention effectively increases the amount of information and improves the accuracy of reliability assessment by fully developing and utilizing the incomplete information in the rolling bearing life data. BRIEF DESCRIPTION OF THE DRAWINGS

[0066] In order to more clearly illustrate the embodiments of the present invention or the technical solutions in the prior art, the following briefly introduces the drawings required for use in the embodiments or the description of the prior art. Obviously, the drawings described below are only some embodiments of the invention. For ordinary technicians in this field, other drawings can be obtained based on these drawings without paying any creative work.

[0067] Figure 1 is a flow chart of the method of the present invention;

[0068] Figure 2 is the relative error of the shape parameters of the bearing life distribution;

[0069] Figure 3 is the relative error of the bearing life distribution scale parameter;

[0070] Figure 4 is the relative error of the bearing's reliable life;

[0071] Figure 5 is the relative error of bearing reliability. DETAILED DESCRIPTION

[0072] The present invention is described in detail below by way of exemplary embodiments, but it should be understood that elements, structures, and features of one embodiment may be beneficially combined in other embodiments without further description.

[0073] It should be noted that, unless otherwise defined, the technical or scientific terms used herein shall have the ordinary meaning understood by persons having ordinary skills in the field to which the invention belongs. The words "one," "an," "the," and similar expressions used in the patent application specification and claims of the present invention do not express quantitative limitations, but rather indicate the presence of at least one; words such as "include" or "comprise" indicate that the elements or objects preceding "include" or "comprise" include the elements or objects listed after "include" or "comprise" and their equivalents, but do not exclude other elements or objects with the same function.

[0074] In order to more clearly describe the specific embodiments of the present invention, Figure 1 -Attached Figure 5 Describe this embodiment:

[0075] Step 1: Collect random censored test data of rolling bearings. The specific implementation method is as follows:

[0076] This example analyzes the random truncated life test data of NU206E cylindrical roller bearings. The bearing life follows a Weibull distribution. The results are shown in Table 1: Table 1 NU206E cylindrical roller bearings Randomly truncated lifespan data

[0077] n (n=15 in this embodiment) samples are put into random truncation test. All samples are tested from time zero to obtain random truncation life data of n rolling bearings. q (This embodiment n q =10) failure data and m q (This embodiment m q =5) stop data:

[0078]

[0079] Step 2: Establish a rolling bearing life distribution model. The specific implementation is as follows:

[0080] This embodiment adopts the rolling bearing life to obey the two-parameter Weibull distribution, whose distribution function is:

[0081]

[0082] Where α is the shape parameter; β is the scale parameter; and t is the life failure time.

[0083] Step 3: Estimation of life distribution model parameters. The specific implementation process is as follows:

[0084] Let y = lnt, transform equation (2) into extreme value distribution, and its distribution function is

[0085]

[0086] Where σ and μ are the scale parameter and location parameter of the extreme value distribution, respectively, and

[0087]

[0088] make Then we can get the standard extreme value distribution function:

[0089] F(y*)=1-exp[-exp(y*)] (5)

[0090] Then the mean and covariance of the order statistics of the randomly truncated data of the standard extreme value distribution are:

[0091]

[0092]

[0093]

[0094] In formula (6) and formula (7):

[0095]

[0096] Among them, h (j+1)kr If i=n j +k failed data, that is, the kth (1≤k≤nth) of the j+1th (0≤j≤q-1) failed data j+1 -n j ) failure data y i If we continue to test the interrupted data before, there are only r interrupted data in y i The previous failure event A (j+1)kr Probability of occurrence.

[0097] In formula (8), It is Failure data and The covariance of the failure data; It's an event The probability of occurrence is obtained by formula (9); It is in the event In case of incident The conditional probability of occurrence is equivalent to computing the random censored data:

[0098] Events The probability of occurrence is still obtained by formula (9).

[0099] In formulas (6) to (8), M(y i +r) and It can be regarded as the mean and second-order origin moment of the i+rth order statistic in n complete failure data. It is equivalent to the second-order mixed moment of the i1+r1th order statistics and the i2+r1+r2th order statistics in n complete failure data. They can be calculated by formula or obtained by looking up the table.

[0100] Then we can find the estimates of parameters μ and σ:

[0101]

[0102]

[0103] Where,

[0104]

[0105]

[0106]

[0107] According to formula (4), formula (11), and formula (12), the two parameter estimates of Weibull distribution are:

[0108]

[0109] Step 4: Reliability assessment;

[0110] For a Weibull distributed random variable t, the one-sided confidence lower limit of the reliable life with a confidence level of γ and a reliability of R is

[0111]

[0112] The one-sided confidence lower limit of the reliability R(t) with a confidence level of γ for a given life t is

[0113]

[0114] in The degrees of freedom are 2n q +2 χ 2 The lower gamma quantile of the distribution.

[0115] Step 5: Use Monte-Carlo method to simulate and verify the accuracy of the model;

[0116] The true values ​​of the shape parameter and scale parameter of the bearing life distribution are set to 2 and 300 respectively. The Monte-Carlo method is used to generate randomly truncated data that obeys the Weibull distribution and has a sample size of n. The simulation process is repeated 1000 times. The relevant parameters of the simulation model are shown in Table 2. For different sample sizes, the method of the present invention and the nonlinear least squares method are used for parameter estimation. The calculated results are shown in Table 3, and the relative errors between the estimated values ​​and the true values ​​of the shape parameter and scale parameter are shown in the attached table. Figure 2 and attached Figure 3 shown. Table 2 Model parameter selection Table 3 Weibull Distribution parameter estimation results

[0117] For different sample sizes, the parameter estimates calculated by the method in this paper and the nonlinear least squares method are respectively calculated using formula (17) and formula (18) to obtain the reliable life t with a confidence level of γ = 0.7 and a reliability of R = 0.9. R The one-sided confidence lower limit and the one-sided confidence lower limit of the reliability R(t) with a confidence level of γ = 0.7 and a given life of t = 50h are shown in Table 4 below. The corresponding reliable life and reliability results are compared with the reliable life and reliability under the true value condition. The relative errors are as follows: Figure 4 and Figure 5 shown. Table 4 Calculation results of reliable life and reliability

[0118] By the attached Figure 2 and attached Figure 3 It can be seen that compared with the nonlinear least squares method, the relative error between the parameter estimation value and the true value obtained by the method of the present invention is smaller, and the observation of the attached Figure 4 and attached Figure 5 It can be seen that the one-sided lower confidence limits of the reliable life and the one-sided lower confidence limits of the reliability obtained by the method of the present invention are closer to the true values. In general, the method of the present invention is more accurate than the nonlinear least squares method, which shows the accuracy of the method of the present invention.

[0119] It should be noted that although the present invention has been described with reference to the above embodiments, the present invention may also have other various embodiments. Without departing from the spirit and scope of the present invention, it is obvious that those skilled in the art may make various corresponding changes and modifications to the present invention, and such changes and modifications shall fall within the scope of protection of the appended claims and their equivalents.

Claims

1. A method for reliability assessment of rolling bearing randomly truncated data, characterized in that: The steps include: Step S1: Collect random tail test data of rolling bearings, put n samples into random tail test, and generate n=n q +m q The randomly truncated life data of rolling bearings, where n q is the number of failed specimens, m q is the number of terminated samples; Step S2: establishing a rolling bearing life distribution model through a two-parameter Weibull distribution; Step S3: performing required parameter estimation calculations for the life distribution model, converting the Weibull distribution into an extreme value distribution, converting the data collected in step S1 into standard extreme value distribution randomly truncated data, and using the mean and covariance of the order statistics of the standard extreme value distribution randomly truncated data to calculate the required parameter estimates through the principle of least squares, thereby obtaining parameter estimates of the Weibull distribution; Step S4: Reliability assessment: Calculate the one-sided confidence lower limit of the reliable life and the one-sided confidence lower limit of the reliability based on the parameter estimation calculation results to obtain the reliability data of the rolling bearing randomly truncated data; Step S5: Generate randomly truncated data that obey the Weibull distribution using the Monte-Carlo method, repeat the simulation process, and verify the accuracy of the calculated data.

2. A rolling bearing random truncated data reliability assessment method according to claim 1, characterized in that: In step S1, the rolling bearing randomly truncated life data is sorted according to the following formula: Where, t i (i=1,2,…,n q ) is the failure data of the i-th specimen in the randomly censored data, is the i-th interrupt data, n j represents the total number of failure data of the first j blocks (i.e., blocks 1 to j), m j Represents the total number of sample termination data of the first j blocks, j = 1, 2, ..., q.

3. The method for reliability assessment of rolling bearing randomly truncated data according to claim 1, characterized in that: In step S2, the two-parameter Weibull distribution core formula is: Where α is the shape parameter, β is the scale parameter, t is the life failure time, and F(·) is the life distribution function.

4. The method for reliability assessment of rolling bearing randomly truncated data according to claim 1, characterized in that: In step S3, in order to facilitate parameter estimation, the variable transformation y=lnt is introduced to transform the Weibull distribution function formula in step S2 into an extreme value distribution, whose distribution function is Where σ and μ are the scale parameter and location parameter of the extreme value distribution, respectively, which are obtained by arranging and transforming the distribution function of the Weibull distribution: At the same time, the data collected in step S1 is converted into: The data collected in step S1 is converted into standard extreme value distribution randomly truncated data, The standard extreme value distribution function is obtained: F(y*)=1-exp[-exp(y*)] (6) where F(·) is the lifetime distribution function, and F(y*) does not contain any unknown parameters.

5. The method for reliability assessment of rolling bearing randomly truncated data according to claim 4, characterized in that: In step S3, the data collected in S1 is transformed into a set of randomly truncated data from the standard extreme value distribution F(y*). The mean and covariance of the order statistics of the randomly truncated data of the standard extreme value distribution are: Where E(·) is the mean, Var(·) is the variance, Cov(·) is the covariance, M(y i +r) is the mean of the i+rth order statistic, is the second-order origin moment of the i+rth order statistic, is the second-order mixed moment of the i1+r1th order statistic and the i2+r1+r2th order statistic, and the transformation is: Among them, h (j+1)kr If i=n j +k failed data, that is, the kth (1≤k≤nth) of the j+1th (0≤j≤q-1) failed data j+1 -n j ) failure data y i If we continue to test the interrupted data before, there are only r interrupted data in y i The previous failure event A (j+1)kr The probability of occurrence, It is Failure data and The covariance of the failure data; It's an event Probability of occurrence; Transform formula (10) to obtain It is in the event In case of incident The conditional probability of occurrence is equivalent to computing the random censored data: Events Probability of occurrence.

6. The method for reliability assessment of rolling bearing randomly truncated data according to claim 5, characterized in that: In step S3, in formulas (7) to (9), M(y i +r) and is regarded as the mean and second-order origin moment of the i+rth order statistic in n complete failure data, It is equivalent to the second-order mixed moment of the i1+r1th order statistics and the i2+r1+r2th order statistics in n complete failure data, because: so In the formula And there is set up as well as Then the residual sum of squares Q is The estimates of the parameters μ and σ can be obtained by taking the derivative of μ and σ: In the formula According to formula (4), formula (17), and formula (18), the two parameter estimates of Weibull distribution are: Where α is the shape parameter, β is the scale parameter, μ is the location parameter of the extreme value distribution, and σ is the scale parameter of the extreme value distribution.

7. The method for reliability assessment of rolling bearing randomly truncated data according to claim 1, characterized in that: In step S4, the randomly truncated data in formula (1) are divided into failure data and termination data, where n q Failure data t1, t2, ..., and nn q Abort data …,t n , for the Weibull distribution in formula (2), let x=t α (23) Where x is a random variable that obeys exponential distribution, for the randomly censored data t1, t2, ..., t n ,make Where x k (k=1,2,…,n) is a set of randomly truncated data of the random variable x. For the Weibull distributed random variable t, the confidence level is γ and the reliability is R. The one-sided lower confidence limit of the reliable life for The reliability R(t) confidence level of its given life t is the reliability one-sided confidence lower limit R l (t) is in The degrees of freedom are 2n q +2 χ 2 The lower gamma quantile of the distribution.

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