A method for accelerated mechanism equivalence test and life prediction based on b-s distribution

Abstract

The application provides a kind of based on B-S distribution's accelerated mechanism equivalence test and life prediction method, i.e.a kind of based on B-S distribution's ALT mechanism equivalence test and life prediction method, steps as follows: one: estimate the scale parameter in B-S distribution model;Two: test whether the failure mechanism of product in accelerated life test is equivalent;Three: estimate the shape parameter in B-S distribution model and acceleration model parameter;Four: extrapolate and calculate the product reliable life under normal stress;The application guarantees the accuracy and effectiveness of product reliable life;Strong applicability, high detection precision;Greatly reduce the amount of calculation;The method is scientific, good process, and has broad popularization and application value.

Description

Technical Field

[0001] This invention provides a method for accelerating life testing mechanism equivalence verification and life prediction based on the BS distribution. It relates to a method for accelerating life testing mechanism equivalence verification and life prediction based on the BS distribution. This method is based on an empirical likelihood function and estimation equation for mechanism equivalence verification, and an accelerated model for life prediction. Taking complete failure data obtained from accelerated life testing of products as an example, it transforms the BS distribution into a normal distribution, estimates the scale parameters in the model, establishes an empirical likelihood function and estimation equation, and statistically tests the variance of the data to determine whether the product's failure mechanism has changed under accelerated stress. Based on the assumption of equivalence of the failure mechanism, it estimates the parameters of the product's life distribution model based on the accelerated model, and finally predicts the reliable life of the product under normal stress. This method is applicable to fields such as BS distribution-based accelerated life testing failure mechanism verification and product life assessment. Background Technology

[0002] The Birnbaum-Saunders lifetime distribution is derived from the basic characteristics of the fatigue process. Compared with other common lifetime distributions, such as the log-normal distribution and the Weibull distribution, the BS distribution is more suitable for the study of fatigue failure. It has important application value in the reliability research of mechanical products and electronic products.

[0003] For reliability assessment of high-reliability, long-life products, accelerated life testing (ALT) is a reasonable and feasible method. Effective accelerated life testing requires ensuring that the failure mechanism of the product remains unchanged under different stress levels; otherwise, the extrapolated conclusions cannot truly reflect the product's reliability level under normal stress. Therefore, when conducting reliability assessment and life prediction based on accelerated failure data, verifying whether the product's failure mechanism is the same at each stress level is a crucial step, as this directly affects the effectiveness and accuracy of the extrapolation prediction.

[0004] Currently, relatively mature theoretical systems have been developed for the failure mechanism equivalence test of common lifetime distribution models such as log-normal and Weibull distributions. However, due to the complexity of the BS distribution model, there is a lack of complete methods for its failure mechanism equivalence test. By transforming the BS distribution into a normal distribution and combining the empirical likelihood function with the estimation equation, a new approach is provided for the failure mechanism equivalence test of ALT based on the BS distribution.

[0005] Based on this, the present invention proposes an accelerated life test mechanism equivalence test method based on the BS distribution, and under the condition of equivalent failure mechanism, it provides parameter estimation of the product life distribution model and extrapolation prediction of reliable life under normal working stress based on the accelerated model.

[0006] "BS distribution" refers to the Birnbaum-Saunders lifetime distribution, where the random variable... If it follows a Birnbaum-Saunders distribution, then its cumulative failure function is: In the formula, The cumulative failure function is a standard normal distribution. and These are the shape and scale parameters of the Birnbaum-Saunders distribution, respectively.

[0007] "Log-normal distribution" refers to a random variable that has a log-normal distribution. If it follows a log-normal distribution, then its cumulative failure function is: In the formula, The cumulative failure function is a standard normal distribution. and These are the logarithmic mean and logarithmic standard deviation of the log-normal distribution, respectively.

[0008] The "Weibull distribution" refers to a random variable... If it follows a Weibull distribution, then its cumulative failure function is: In the formula, and These are the shape and scale parameters of the Weibull distribution, respectively.

[0009] "Accelerated Life Testing (ALT) technology" refers to a testing technique that places products in harsh stress environments, such as high pressure, high temperature, high humidity, and high salt spray, to induce rapid product failure under high stress conditions, thereby obtaining sufficient product reliability information while saving testing costs and time. Summary of the Invention

[0010] (1) The purpose of this invention is to provide a method for verifying the equivalence of accelerated life mechanism and predicting life based on the BS life distribution model, that is, a method for verifying the equivalence of failure mechanism in accelerated life test; under the condition of equivalence of failure mechanism, the parameters of the product's life distribution model are estimated based on the accelerated model, and finally the reliable life of the product under normal stress is predicted. (2) Technical solution: The following basic settings are required for this invention: Setting 1 for different stress levels Accelerated product lifespan All follow a Black-Scholes distribution, with their probability density function and cumulative failure function as follows: ① Probability density function: (1) In the formula: is the probability density function of the standard normal distribution; ② Cumulative failure function: (2) In the formula: The cumulative failure function is a standard normal distribution. and stress levels The scale and shape parameters of the lower BS distribution; Setting 2 for different stress levels If the product's failure mechanism remains unchanged, then the shape parameter of the BS distribution... Keep it at a constant value; Based on the above assumptions, this invention provides a method for equivalence testing and lifetime prediction of acceleration mechanisms based on the BS distribution, specifically a method for equivalence testing and lifetime prediction of ALT mechanisms based on the BS distribution, which is implemented through the following steps: Step 1: Estimate the scaling parameters in the BS distribution model; Accelerated life testing has a total of Group stress level, i.e. , record the The full-life failure data obtained from observations at the group stress level are as follows: (3) In the formula: For the first Number of failure samples under the group stress level; right Make the following transformation: (4) In the formula: For the first Scale parameters of the BS distribution under group stress levels; definition To transform invalid data, it must follow a normal distribution with a mean of 0, i.e.: (5) In the formula: For the first Shape parameters of the BS distribution under group stress level; For data that failed to convert. ,have: (6) In the formula: ; The scale parameter can then be obtained. The estimated value is: (7) In the formula: ; Step 2: Verify whether the failure mechanisms of the products are the same in accelerated life testing; I. Calculate the empirical log-likelihood ratio of adjacent stresses; The first The shape parameter of the BS distribution under the group stress level is denoted as: , No. The shape parameter of the BS distribution under the group stress level is denoted as: The null hypothesis is: (8) In the formula: ; The alternative hypothesis is: (9) The scaling parameters of the BS distribution model are obtained from step one. Then, according to equation (4), the conversion failure data can be obtained as follows: (10) For the Group and No. For each set of failure data, an estimation equation is constructed: (11) (12) In the formula: For the first Transition failure data under group stress levels; For the first Transition failure data under group stress levels; make and Let be the empirical likelihood probability value, then the empirical log-likelihood ratio is: (13) In the formula: , ,and: (14) (15) remember , ,but: (16) In the formula: ; II. Calculate the test statistic; After obtaining the empirical log-likelihood ratio between adjacent stresses, let: (17) when At that time, there were: (18) In the formula: , , , It is a gamma function; The test statistic for the equivalence of failure mechanisms is defined as follows: (19) In the formula: Indicates the number of stress levels; For a given confidence level ,like If the failure mechanism remains unchanged under each accelerated stress level, then it is assumed that the failure mechanism has not changed; otherwise, the failure mechanism has changed, and the location of the change in the failure mechanism is determined. for: (20) In the formula: For the standard maximum distribution Quantiles, or standard maximum distributions, refer to the distributions where random variables... If it follows a standard maximum distribution, then its cumulative failure function is: ; Step 3: Estimate the shape parameters and acceleration model parameters in the BS distribution model; I. Estimate the shape parameters of the BS distribution model; For the accelerated life test data that passed the test in step two, the estimated value of the shape parameter in the BS distribution model can be obtained according to equation (6): (twenty one) In the formula: The conversion failure data obtained from equation (5) This represents the total number of failure samples in accelerated life testing. II. Estimate the parameters of the acceleration model; The accelerated model for the BS distribution is as follows: (twenty two) In the formula: The model parameters to be estimated are... It is a function of accelerating stress; After determining the acceleration model, the estimated values ​​of the parameters in the acceleration model are calculated using the least squares estimation method; Step 4: Extrapolate and calculate the product's reliable lifespan under normal stress; For a given reliability Normal working stress conditions Substitute the data into the acceleration model and extrapolate to calculate the reliable life of the product under normal stress. ; The reliable lifetime of the BS distribution is: (twenty three) In the formula: The scale parameters of the BS distribution under normal stress are derived from the acceleration model. For standard normal distribution quantiles; In step three, the "least squares estimation method" refers to: When the dependent variable With independent variable When there is a linear relationship, we have: (twenty four) in, Considering interference in the experiment or measurement, and The linear relationship is not strict, therefore: (25) In the formula: It follows a certain distribution with a mean of 0; Assuming a total of In the second trial, the dependent variable is in the third trial. Group independent variables The observed values ​​are as follows , All independent variables Construct a matrix ,all Constructing vectors Due to the influence of random interference factors, and The relationship is not strictly linear. Represents error; set up It is a parameter The estimated value, if satisfy: (26) Then it is called It is a parameter The least squares estimate; where, This represents the sum of squared errors between the observed and predicted values. Through the above steps, this invention achieves the equivalence test of the failure mechanism of the BS distribution model based on the empirical likelihood function and the estimation equation, ensuring the accuracy and effectiveness of extrapolation prediction of the reliable life of products under normal working stress. The failure mechanism equivalence test method of this invention has strong applicability to the BS distribution model and high detection accuracy. The parameter estimation method of this invention greatly reduces the amount of calculation while ensuring accuracy by transforming the BS distribution into a normal distribution, eliminating the need to solve transcendental equations, and providing explicit expressions for the parameter estimation results, making it highly operable. The method described in this invention is scientific, has good manufacturability, and has broad application value. Attached Figure Description

[0011] Figure 1 This is a flowchart of the method described in this invention. Detailed Implementation

[0012] The present invention will now be described in further detail with reference to the embodiments.

[0013] A certain mechanical bearing has a stress cycle of 30MPa when working under normal environmental conditions. A total of 48 samples were taken and divided into 4 groups of 12 samples each. Accelerated life test was carried out on them according to the accelerated stress shown in Table 1. Table 1. Stress Levels in Accelerated Life Tests of a Certain Mechanical Bearing

[0014] The collected complete failure data are shown in Table 2: Table 2 Accelerated life test data of mechanical bearings

[0015] The unit for failure life data is hours; This invention provides a method for equivalence testing and lifetime prediction of acceleration mechanisms based on the Black-Scholes distribution, specifically a method for equivalence testing and lifetime prediction of ALT mechanisms based on the Black-Scholes distribution. Figure 1 As shown, this can be achieved through the following steps: Step 1: Estimate the scaling parameters in the BS distribution model; The scale parameters in the BS distribution model under various acceleration stress levels are calculated according to equation (7), and the calculation results are shown in Table 3. Table 3. Estimated values ​​of scale parameters of the BS distribution model under different stress levels.

[0016] Step 2: Verify whether the failure mechanisms of the products are the same in accelerated life testing; I. Calculate the empirical log-likelihood ratio of adjacent stresses; According to equation (4), the conversion failure data corresponding to the accelerated life data can be obtained. The calculation results are shown in Table 4. Table 4 Mechanical bearing conversion failure data

[0017] The empirical log-likelihood ratio between adjacent accelerating stresses in each group is calculated according to equation (16). The calculation results are shown in Table 5. Table 5. Empirical log-likelihood ratios between adjacent accelerating stresses

[0018] II. Calculate the test statistic; Take the confidence level as According to equation (19), we can obtain: Therefore, it can be determined that the failure mechanism of this batch of mechanical bearings did not change under the four sets of accelerated stress levels. Step 3: Estimate the shape parameters and acceleration model parameters in the BS distribution model; I. Estimate the shape parameters of the BS distribution model; According to equation (21), the estimated values ​​of the shape parameter in the BS distribution model are: ; II. Estimate the parameters of the acceleration model; For fatigue testing, the characteristic lifespan of the product With stress cycle Conforming to the inverse power law model, according to equation (22), the acceleration model can be obtained as follows:

[0019] In the formula: These are the parameters to be estimated in the model. For the first Stress cycle levels in accelerated group tests; We can obtain the following from the least squares estimation method: ; Step 4: Extrapolate and calculate the product's reliable lifespan under normal stress; Based on the parameter estimation results in step three, given a reliability Compared with normal working stress level Then, according to equation (22), we can obtain: ; According to equation (23), the reliable lifespan of the product is... ; Therefore, based on the calculations and evaluations mentioned above, the product's 95% reliable lifespan under normal stress is 888.99 hours; In summary, this invention presents a method for accelerating failure mechanism equivalence testing and life prediction based on the BS distribution. It is a failure mechanism equivalence testing method based on empirical likelihood functions and estimation equations, and a life prediction method based on an accelerated model. The specific steps of this method are: 1. Estimating the scale parameters in the BS distribution model; 2. Verifying whether the failure mechanisms of the products are equivalent in accelerated life testing; 3. Estimating the shape parameters in the BS distribution model and the accelerated model parameters; 4. Extrapolating and calculating the reliable life of the product under conventional stress. This invention is applicable to fields such as accelerating failure mechanism equivalence testing and product life assessment using the BS distribution, and has high practicality and strong operability.

Claims

1. A method for acceleration mechanism equivalence test and lifetime prediction based on the Black-Scholes distribution, requiring the following settings: Setting 1: For different stress levels Accelerated product lifespan All follow a Black-Scholes distribution, with their probability density function and cumulative failure function as follows: ① Probability density function: （1） In the formula: The probability density function is the standard normal distribution. ② Cumulative failure function: （2） In the formula: The cumulative failure function is a standard normal distribution. and stress levels The scale and shape parameters of the lower BS distribution; Setting 2: For different stress levels If the product's failure mechanism remains unchanged, then the shape parameter of the BS distribution... Keep it at a constant value; Its features are, The specific steps are as follows: Step 1: Estimate the scaling parameters in the BS distribution model; Accelerated life testing has a total of Group stress level, i.e. , record The full-life failure data obtained from observations at the group stress level are as follows: （3） In the formula: For the first Number of failure samples under the group stress level; right Make the following transformation: （4） In the formula: For the first Scale parameters of the BS distribution under group stress levels; definition To transform invalid data, it must follow a normal distribution with a mean of 0, i.e.: （5） In the formula: For the first Shape parameters of the BS distribution under group stress level; For data that failed to convert. ,have: （6） In the formula: ; The scale parameter is thus obtained. The estimated value is: （7） In the formula: ; Step 2: Verify whether the failure mechanisms of the products are the same in accelerated life testing; 2.1 Calculate the empirical log-likelihood ratio of adjacent stresses; The first The shape parameter of the BS distribution under the group stress level is denoted as: , No. The shape parameter of the BS distribution under the group stress level is denoted as: The null hypothesis is: （8） In the formula: ; The alternative hypothesis is: （9） The scaling parameters of the BS distribution model are obtained from step one. Then, according to equation (4), the conversion failure data is obtained as follows: （10） For the Group and No. For each set of failure data, an estimation equation is constructed: （11） （12） In the formula: For the first Transition failure data under group stress levels; For the first Transition failure data under group stress levels; make and Let be the empirical likelihood probability value, then the empirical log-likelihood ratio is: （13） In the formula: , ,and: （14） （15） remember , ,but: （16） In the formula: ; 2.2 Calculate the test statistic; After obtaining the empirical log-likelihood ratio between adjacent stresses, let: （17） when At that time, there were: （18） In the formula: , , , It is a gamma function; The test statistic for the equivalence of failure mechanisms is defined as follows: （19） In the formula: Indicates the number of stress levels; For a given confidence level ,like If so, it is considered that the failure mechanism of the product has not changed under each group of accelerated stress levels; Otherwise, the failure mechanism has changed, and the location where the failure mechanism has changed is... for: （20） In the formula: For the standard maximum distribution Quantiles, or "standard maximum distributions," refer to the distributions where a random variable... If it follows a standard maximum distribution, then its cumulative failure function is: ; Step 3: Estimate the shape parameters and acceleration model parameters in the BS distribution model; 3.1 Estimate the shape parameters of the BS distribution model; For the accelerated life test data that passed the test in step two, the estimated value of the shape parameter in the BS distribution model is obtained according to equation (5): （21） In the formula: The conversion failure data obtained from equation (4) This represents the total number of failure samples in accelerated life testing. 3.2 Estimate the parameters of the acceleration model; The accelerated model for the BS distribution is as follows: （22） In the formula: The model parameters to be estimated are... It is a function of accelerating stress; After determining the acceleration model, the estimated values ​​of the parameters in the acceleration model are calculated using the least squares estimation method; Step 4: Extrapolate and calculate the product's reliable lifespan under normal stress; For a given reliability Normal working stress conditions Substitute the data into the acceleration model and extrapolate to calculate the reliable life of the product under normal stress. ; The reliable lifetime of the BS distribution is: （23） In the formula: The scale parameters of the BS distribution under normal stress are derived from the acceleration model. For standard normal distribution quantiles; In step three, the "least squares estimation method" refers to: When the dependent variable With independent variable When there is a linear relationship, we have: （24） in, Considering interference in the experiment or measurement, and The linear relationship is not strict, therefore: （25） In the formula: It follows a distribution with a mean of 0; Let's assume a total of In the second trial, the dependent variable is in the third trial. Group independent variables The observed values ​​are as follows , All independent variables Construct a matrix ,all Constructing vectors Due to the influence of random interference factors, and The relationship is not strictly linear. Represents error; set up It is a parameter The estimated value, if satisfy: （26） Then it is called It is a parameter The least squares estimate; where, This represents the sum of squared errors between the observed and predicted values.

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