CN114169128B - Reliability enhancement test quantitative evaluation method based on Bayes analysis - Google Patents
Reliability enhancement test quantitative evaluation method based on Bayes analysis Download PDFInfo
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Abstract
The invention discloses a reliability strengthening test quantitative evaluation method based on Bayes analysis, which is used for strengthening tests under different conditions, adopts different estimation methods respectively, sets a certain confidence level, evaluates in stages to obtain a comprehensive MTBF value of the reliability strengthening test, overcomes the defect that the sample size required by a classical method is large or faults need to be corrected immediately, solves the problem of small sample reliability strengthening test evaluation, particularly the equipment or whole machine level reliability evaluation, can be popularized to the field of small sample reliability evaluation, saves manpower and material resources, accurately obtains the reliability realization level of a product, and is beneficial to resource allocation decision in product development.
Description
Technical Field
The invention belongs to the technical field of test statistics, and particularly relates to a reliability evaluation technology.
Background
With the continuous development of science and technology, the materials, processes and technology of products are continuously updated in an iterative manner, and the demand of people for products with high reliability is increasing day by day.
Different from the traditional reliability growth test, the reliability strengthening test belongs to an excitation type test. By adopting a high stress level as a test condition, aiming at fully exciting potential defects and weak links existing in the product design, corresponding improvement measures are adopted, the product reliability is accelerated in a short time, the test efficiency is greatly improved, and the cost is reduced.
In the process of the strengthening test, improvement measures are taken on the product, so that the state of the product is changed. The parameters in the acceleration model will also change continuously as the product improves.
At present, classical statistical methods for reliability testing are characterized by: the information upon which the statistical inference is based comes from models and samples, and the distribution of the population is derived from experience or some reasonable assumption. All conclusions of the statistical inference come from probability interpretation, such as maximum likelihood estimation method, and the classical statistical method has a moment method and a maximum likelihood method for point estimation of the parameter theta.
The maximum likelihood method is to construct a likelihood function according to sample distribution, and then calculate a parameter value for making the likelihood function reach a maximum value as a point estimation value of theta, wherein the larger the sample amount is, the more sufficient the sample information is, the more accurate the estimation result is.
When the classical statistical method is used for statistical inference, links such as model selection, statistic construction, sampling distribution determination and the like are needed, and finally a conclusion of certain estimation or hypothesis test on the parameter theta is obtained. Considering that in the selection process of the model, the classical statistical method tries to achieve the purpose of seeking the overall distribution by increasing the sample capacity, and the method requires a large amount of test samples.
Currently, the reliability enhancement test is generally a qualitative test. The existing strengthening test evaluation method generally obtains reliability strengthening test fault data, extrapolates the fault data by adopting a corresponding acceleration model to obtain the fault data under a normal stress level, and then evaluates the reliability of the fault data by adopting an AMSAA model or a classical statistical analysis method to obtain the reliability level of the electronic product to be tested.
The AMSAA model is used for reliability growth evaluation and is suitable for the situation of parent variation, such as an instant correction fault processing mode in a reliability test. Due to the limitation of time, cost and conditions of a test field, the method usually adopts a fault processing mode of delaying correction, namely a matrix is not changed, and the AMSAA growth model is not suitable at the moment.
The information of Bayes statistical analysis is based on model information, sample information and prior information. And deducing and solving posterior distribution of the parameters according to the sample information and the prior information. Byaes statistics recognizes that not only can sample volume be reduced, but in many cases statistical accuracy can be improved using this a priori information.
Compared with a Bayes statistical analysis method, the classical statistical analysis method needs larger sample size, and the higher the test product, particularly the test product, is, the higher the test cost is, the fewer the test prototypes are, and the most of the products of the reliability enhancement test, particularly the level above the equipment level, are small subsamples. The Bayes method is suitable for small samples, and pre-test information is fully utilized, so that manpower and material resources can be saved, and an estimation result superior to a large-sample-size classical statistical analysis method can be obtained.
Disclosure of Invention
The invention provides a reliability strengthening test quantitative evaluation method based on Bayes analysis for solving the problems in the prior art, and adopts the following technical scheme for achieving the purpose.
Carrying out low-temperature stepping stress, high-temperature stepping stress, rapid temperature circulation, vibration stepping stress and comprehensive environmental stress tests, and recording fault data and stress data; dividing test results into failure and non-failure according to whether the product has failure or not; dividing test conditions into step stress and constant stress according to the stress type; and evaluating the average life of the product in the process of having failure step stress, having failure constant stress, having no failure step stress and having no failure constant stress.
Failure step stress: converting failure data, and obtaining average life estimation under normal stress S by adopting a Bayes estimation method based on Gibbs sampling; with constant stress at failure: calculating an acceleration factor, converting the test time of constant stress into the test time of normal stress, obtaining an estimation form of failure rate by adopting a Bayes method, obtaining an estimation value of the failure rate by adopting a Gibbs sampling method, and obtaining average life estimation under Chang Yingli S by adopting a least square method; no failure step stress: calculating an acceleration factor, converting the test time of the stepping stress into the test time of the normal stress, and adopting a failure-free Bayes estimation method; no failure constant stress: and (3) calculating an acceleration factor, converting the test time of the constant stress into the test time of the normal stress, and adopting a failure-free Bayes estimation method.
Setting a certain confidence level according to the test time and the number of failures of the test object under each test condition, generating an MTBF confidence interval of each test condition, calculating and analyzing the usability of each MTBF by adopting a staged quantitative evaluation method, and obtaining a comprehensive MTBF value of the reliability enhancement test through weighted average.
Further, according to test data, an accumulated damage model or an index damage model is adopted, acceleration factors of high-temperature stepping stress, low-temperature stepping stress, rapid temperature change stress, vibration stepping stress and comprehensive environmental stress are calculated in a folding mode, the high-temperature stepping stress adopts an Arrhenius model and a single stress Eying model to calculate the acceleration factors, the low-temperature stepping stress adopts an index acceleration model to calculate the acceleration factors, the rapid temperature change stress adopts a modified coffee-Manson model and an extension model thereof to calculate the acceleration factors, the vibration stepping stress adopts a Basquin equation and a Miner rule to calculate the acceleration factors, the comprehensive environmental stress divides the test into a rapid temperature change test and a vibration stress test, and the acceleration factors are obtained comprehensively.
In particular, the acceleration factor of the high temperature step stress of the Arrhenius model
Acceleration factor of high-temperature stepping stress of single-stress Eying model
Wherein t is use Is the test time using the stress level, t test Is the test time of the accelerated stress level, T use Is using a stress level, T test Is the acceleration stress level, E a Is activation energy in units of eV, k B The boltzmann constant is used to determine which model to use based on the quality of the results of fitting data between the two models.
The core of the Arrhenius model is activation energy, which is a quantum physics concept and represents an energy barrier to be overcome for microscopically starting recombination between certain particles.
The physical basis of the Arrhenius model is the chemical reaction rate, which describes the failure mechanism of non-mechanical material fatigue in products, which depends on the processes of chemical reaction, corrosion, substance diffusion or migration, etc., and the mathematical form is
Wherein t is L Is a characteristic life or reliability characteristic parameter of the product, and the value is 8.617 multiplied by 10 -5 (eV/K), T is absolute temperature, degrees Celsius +273.15, and C is a constant in units of K.
The single stress Eying model is obtained by deducing from the quantum mechanics principle, the service life characteristic of the product is expressed by the function of absolute temperature, and when the absolute temperature changes in a small rangeWhen the single stress Eying model is similar to the Arrhenius model, the activation energy calculation needs to be obtained, and the mathematical form is
Wherein A, B, C is a constant and the remaining symbols have the same meaning as the Arrhenius model.
In practical engineering applications, the activation energy of the product obtained by testing is not affordable in terms of sample size, cost and cycle time, and generally adopts the recommended standard activation energy, including: the American national defense Manual recommends 0.79eV for electronic products and 0.7 eV-1.0 eV recommended by the JEDEC solid State technology Association standards.
In particular, the acceleration factor of the low-temperature step stress of the exponential acceleration model
b＝8.3×10 -2 Where ξ (T) = a · e b·T The method is a mathematical form of an exponential acceleration model, describes a change rule between the reliability characteristic of a product and low temperature, T is absolute temperature, and a and b are constants.
In particular, the acceleration factor of the fast temperature variation stress of the modified Coffin-Manson model
Wherein N is use 、N test Number of failure cycles, Δ T, for normal and test stresses, respectively test 、ΔT use Temperature transformation ranges of test stress and normal stress respectively, and ftest and fuse are cycle frequencies of the test stress and the normal stress respectively, T max,test 、T max,use The highest cycling temperatures for the test stress and the normal stress, respectively.
The Coffin-Manson model describes the relationship between the plastic strain amplitude and the fatigue life, and the use range is limited to the fatigue life times of less than 10 6 Next, the modified coffee-Manson model N = A · f -α ·ΔT -β ·G(T max ) Where N is the number of cycles to failure, f is the cycle frequency, T is the temperature range of variation, G (T) max ) Is the maximum temperature T of the temperature cycle max The Arrhenius model of (i.e.
The mathematical form of the Basquin equation is NS b Where S is the generalized stress, N is the vibration time or the number of cycles to fatigue, C is a constant, the unit of random vibration is the power spectral density PSD or root mean square value rms, and the unit of sinusoidal vibration is the amplitude.
The mathematical form of Miner rule is L (S) = C -1 ×S -m Describing the lifetime or predicting the determined duration, where S is the stress, describing the impulse-type shock, sinusoidal vibration or random vibration, C is a constant to be determined, C is the time to be determined>0,m is a stress related parameter.
Specifically, vibration stress acceleration is described by a stress-cycle life S-N curve, and an acceleration factor is calculated
Obtaining an S-N curve through a fatigue test of the material, fitting to obtain a parameter m, or inquiring a GJB150A, MIL-STD-810G data handbook to obtain an empirical value of m, wherein the m value of sinusoidal vibration is 6, and the m value of random vibration is 5-8.
In particular, an acceleration factor A of the combined environmental stress F ＝A F_fastT ×A F_var Wherein A is F_fastT Is a rapid temperature variation corresponding to an acceleration factor, A F_var The stress of each test process is integrated by 10 percent of magnitude value lower than the working limit value and is applied to the product, and in a temperature cycle mode, the step length is adopted in the first temperature cycle to be 20 percent of the damage vibration limit, the step length is adopted in the second temperature cycle to be 40 percent of the damage limit, and the like, until the product can not be repaired or the required cycle number is reached.
Further, the Bayes estimation method without failure assumes that the failure distribution follows the exponential distribution, i.e. F (T) = P (T < T) =1-e -λt Wherein t is the working time of the product, F (t) is the accumulated failure rate of the product at the time t, lambda is the failure rate, and is constant under specific stress respectively at different test stresses S i (i =1,2 …, k) test sample, total test time t i0 (i =1,2 …, k), the test time of the same group of test samples of the same test stress is the same, and the number of the corresponding test samples is n i (i =1,2 …, k), no failed sample for each set of tests, and the target stress is S 0 The total time of the equivalent test is t i (i =1,2 …, k), then
Wherein AF (S) j ,S 0 ) Is the test stress S j Relative to the target stress S 0 Acceleration factor of (d), recording test data (t) i ,s i ) Wherein s is i Is t i Number of test samples at time, i.e. s i ＝n i +n i+1 +…+n k Calculating the equivalent test time t based on Bayes method i Corresponding cumulative failure probability estimate
Target stress S is estimated based on least square method 0 Estimated value of failure rate of
Estimate Mean Time Between Failures (MTBF), i.e.
In particular, with p i Represents an equivalent test time of t i Cumulative probability of failure, i.e. p i ＝f(t i ) From (1-p) i ) 2 Estimating a priori density as a kernel of density, i.e.
Wherein
Is a specific constant, and is calculated according to the general calculation mode of empirical distribution function of reliability test data processing
By
Computing
Namely that
According to likelihood function corresponding to non-failure test result
P is calculated by Bayes formula i The posterior distribution density of (i.e.)
Taking into account the square loss to obtain
An estimated value of, i.e.
In particular, by functions
Describing cumulative failure probability estimates
Relation to failure rate λ, where ∈ i Is cumulative probability of failureEstimated value
The generated error is that the least square method is adopted to solve the square loss corresponding to the failure rate
Let dQ (λ)/d λ =0, calculate the least squares estimate of the parameter λ, i.e.
Further, the Bayes estimation method with failure step stress assumes that failure distribution follows exponential distribution, i.e., that
Wherein the step stress is S i T is the working time of the product, lambda i Is the step stress S of the product i The failure rate at time t, being a fixed constant,
is the product is under step stress S i The cumulative probability of failure at time t,
at step stress S for the product i The instantaneous probability of failure at time t, i.e. the failure probability density function,
is the product under stress S i Reliability at time t, working time of failed sample is
Wherein
The total number of samples is n, and the test pause time is tau according to the characteristics of the timing truncation i Time τ i The number of failed samples of (2) is r i From the memoryless of the exponential distribution, a likelihood function of the step stress test is constructed, i.e.
Order to
Wherein R is i Is a step stress S 1 To S i Total number of failures of R = R k Is the total number of failures, τ, of the step test i ' is the cumulative working time of all samples, simplifying the likelihood function, i.e.
Set the product at step stress S 1 The failure rate of is λ 1 I.e. by
Wherein θ = λ 2 /λ 1 Is the step stress S of the product 2 Lower phase ratio in step stress S 1 Acceleration coefficient of (e), theta>0, by function
Description of the invention
According to the calculation mode of the stepping stress type and the acceleration factor, an Arrhenius model, an exponential acceleration model and an S-N curve equation are respectively selected to describe acceleration models of high-temperature stepping, low-temperature stepping and vibration stepping, and functions are simplified
Namely, it is
Let λ = λ 1 、
To obtain
Selecting a prior distribution of theta
1≤k 1 ＜θ＜k 2 Suppose the product is under step stress S 1 The prior distribution of the failure rates lambda follows the Gamma distribution, the density function
Wherein 0 < lambda < ∞, alpha>0,β>0, alpha and beta are hyper-parameters, the prior distribution takes the uniform distribution on (0,1) and (0,c), respectively, c>0 is a constant, the probability densities are pi (beta) =1 (0 < alpha < 1) and pi (beta) =1/c (0 < beta < c), respectively, and if alpha and beta are independent, the multilayer prior probability density of lambda is constant
Obtaining a probability density function of a joint posterior distribution of parameters (lambda, theta, alpha, beta) from a prior distribution of the likelihood functions
Wherein
Describing relevant test information, and sequentially integrating the formula to obtain the marginal probability density of each parameter and the probability density of lambda full-condition posterior distribution
Probability density of full condition posterior distribution of theta
The probability density pi (alpha-theta, lambda, alpha, tau) oc-beta of the alpha full-condition posterior distribution α λ α The probability density pi (beta | theta, lambda, alpha, tau) · beta of the full-condition posterior distribution of/gamma (alpha), beta α e -λβ Calculating the parameter mean value of the combined posterior distribution by adopting a Gibbs sampling method, generating random numbers of the full-condition posterior distribution of theta and alpha by adopting a round-robin sampling method, and adopting Gamma distribution Gamma (alpha + r, beta + T) 2 ) And Γ (α +1, λ) generate random numbers with a full condition posterior distribution of λ and β, setting a starting point (λ) (0) ,θ (0) ,α (0) ,β (0) ) Posterior distribution pi from full condition (lambda | theta) (n-1) ,α (n-1) ,β (n-1) τ) to produce λ (n) Posterior distribution pi from full condition (theta | lambda) (n) ,α (n-1) ,β (n-1) τ) producing θ (n) Posterior distribution pi (alpha | lambda) from full condition (n) ,θ (n) ,β (n-1) τ) production of α (n) Posterior distribution pi (beta | lambda) from full condition (n) ,θ (n) ,α (n) τ) production of β (n) Then (λ) (n) ,θ (n) ,α (n) ,β (n) N =1,2, …, M, M +1, …, M) are Gibbs iteration samples for parameters (λ, θ, α, β), where M is the sample volume discarded before Gibbs iteration samples reach steady state, M>m is the total sample volume, resulting in multi-layered Bayes parameter estimation of λ, θ, α, β
Obtaining an acceleration equation of a stepping stress test model by parameter estimation, and calculating target stress S 0 Average MTBF of (a).
Further, bayes estimation method with failure constant stress, S i Is constant stress, the working time of the failed sample is
n i Is a constant stress S i According to the nature of the timing truncation, τ i Is a constant stress S i Test pause time of r i Is at τ i Number of previous failed samples, order
Likelihood function of the test data
Wherein H = { (λ) 1 ,λ 2 ,…,λ k ):0＜λ 1 ≤λ 2 ≤…≤λ k Instruction of
Wherein T is i Is a constant stress S i The accumulated working time of all samples of, simplifying the likelihood function, i.e.
Assuming the product is at constant stress S i Failure rate of i Obeys the Gamma distribution Ga (alpha) i ,β i ) In which α is i ,β i Is known to be a defined hyper-parameter, dependent on the respective stress S i Independent of each other, to obtain lambda i ,…,λ k Combined prior density function of
From Bayes' theorem to λ 1 ,λ 2 ,…,λ k Combined a posteriori density function of
Calculating lambda i Bayes posterior mean of
Wherein (lambda) 1 ,λ 2 ,…,λ k ) Belongs to H, if i is less than j, 0 is less than lambda i ≤λ j 、 E(λ i |D)≤E(λ j |D)、
Obtaining lambda by adopting a Gibbs sampling method i Full condition posterior distribution of
Truncated Gamma distribution of obedient intervals, i.e.
Wherein G is i Is a Gamma distribution Ga (. Alpha.) i +r i ,β i +T i ) The distribution function of (a) is,
is the inverse of it, U is a random sample of uniformly distributed U (0,1), if there is no information prior, then α is s ＝β s =0, and (lambda) is obtained through Gibbs sampling iteration in t steps 1 ,λ 2 ,…,λ k ) M-fold Gibbs sample
Then
Obtaining the target stress S according to the calculation mode of the acceleration factor 0 Equivalent failure rate estimate of
And failure rate estimation
And
further, assuming that the average failure-free interval T of the products obeys exponential distribution, randomly extracting n products, and performing a timing end-cutting life test under a stress condition, wherein the stopping time is T 0 At t 0 If the front r products fail, the tail sample t is cut off regularly 1 ≤t 2 ≤…≤ t r ≤t 0 R < n, likelihood function of sample
t 1 ≤t 2 ≤…≤t r ≤t 0 Wherein
Is the total test time, order
Then
Its jacobian determinant
To obtain omega 1 ,…,ω r Combined density function of
According to omega 1 ,…,ω r Independently distributed to obtain common distribution Exp (lambda) = Ga (1, lambda), and obtained according to additivity of independent Gamma variable
The 1-alpha approximate confidence interval for MTBF is
The tail probability alpha/2 is merged into the 1-alpha/2 of the other side to obtain the lower confidence limit of the 1-alpha on one side
Specifically, the estimated value of the 1-alpha approximate confidence interval is reserved, the total number of the estimated value is k, and if the engineering experience information is less, the comprehensive MTBF is calculated by adopting an averaging method, namely
If the engineering experience information is more, a Delphi-AHP model is adopted to calculate the comprehensive MTBF, namely
The invention has the beneficial effects that: the method overcomes the defect that a sample size is large based on a classical method or an AMSAA model needs to adopt an instant correction fault processing mode, solves the problem of small sample reliability strengthening tests, particularly equipment or whole machine level reliability evaluation, can be popularized to the field of small sample reliability evaluation, saves manpower and material resources, accurately obtains the reliability realization level of products and is beneficial to resource allocation decision in product development by adopting a Bayes estimation method based on Gibbs sampling aiming at four combination modes of low temperature stepping, high temperature stepping, rapid temperature circulation and vibration stepping in the reliability strengthening tests and decomposing comprehensive environmental stress into stepping stress, constant stress and test fault and no fault.
Drawings
Fig. 1 is an evaluation flowchart, fig. 2 is a low-temperature step stress test diagram, fig. 3 is a high-temperature step stress test diagram, fig. 4 is a rapid temperature change cycle test diagram, fig. 5 is a vibration step stress test diagram, fig. 6 is a comprehensive environmental stress test diagram, fig. 7 is a low-temperature step stress simulation diagram, and fig. 8 is a vibration step stress simulation diagram.
Detailed Description
The technical scheme of the invention is specifically explained in the following by combining the attached drawings.
The evaluation process is as shown in fig. 1, and includes firstly judging whether failure data exists or not, then judging whether the failure data is stepping stress or constant stress, then applying different models, setting a confidence interval, and evaluating a comprehensive MTBF value.
The low-temperature stepping stress test is shown in figure 2, starting from 0 ℃, taking a step every time when the temperature is reduced by 5 ℃ and ending at-45 ℃, wherein the temperature is kept for 5h at 0 ℃, then the temperature is kept for 2h at each stage, the temperature change rate between adjacent temperature steps is 10 ℃/min, the fault occurs for 1 time at-45 ℃, and the recurrence still exists at-40 ℃.
As shown in figure 3, the high-temperature stepping stress test starts from 60 ℃, is one step when the temperature is reduced by 5 ℃, and ends when the temperature is reduced by 70 ℃, wherein the temperature is preserved for 5h at 60 ℃, the temperature is preserved for 2h at each stage, the temperature change rate between adjacent temperature steps is 10 ℃/min, and no fault exists.
The rapid temperature change cycle test is shown in FIG. 4, the low temperature section is used at-40 ℃, the high temperature section is used at 70 ℃, the heat preservation time of each temperature section is 4h, the temperature change rate of the rapid temperature change is 30 ℃/min, and no fault exists.
As shown in fig. 5, the vibration stepping stress test starts from 18grms, each 2grms is a stepping step, and the highest step reaches 24grms, and 3 groups of vibration stepping tests are continuously developed in three directions of an X axis (axial direction), a Y axis (vertical direction) and a Z axis (lateral direction), each vibration stress step is kept for 3min, the vibration stress lifting speed is quickly ignored, and two faults occur in total, namely, when the 2 nd group steps by 24grms, and when the 3 rd group steps by 18 grms.
The comprehensive environmental stress test is shown in fig. 6, the low temperature section is used at-40 ℃, the high temperature section is used at 70 ℃, the heat preservation time of each temperature section is 4h, the temperature change rate among temperature steps is 15 ℃/min, and the random vibration stress of 19.2grms (24 grms multiplied by 0.8) is continuously applied in the whole test process without failure.
According to the recorded results of the reliability strengthening tests, the strengthening tests in different stages are oriented, the reliability is quantitatively evaluated in stages, faults exist in low-temperature stepping and vibration stepping, the Bayes evaluation method under stepping stress is selected as the stepping stress test, and the Bayes evaluation method without faults and failure data in high-temperature stepping, rapid temperature change and comprehensive stress tests is selected.
1) Quantitative evaluation of low temperature step stress test
And (3) according to the low-temperature stepping stress and fault records, arranging to obtain low-temperature stepping strengthening test data, wherein the specific process is as follows:
quantitative estimation of MTBF (mean time between failure) for solving low-temperature stepping stress test by adopting Gibbs sampling algorithmBefore Gibbs sampling, the values of independent parameters r and T except lambda, theta, alpha and beta can be determined according to fault data 1 、T 2 The calculation method of (2):
And (3) the low-temperature stepping total test stage k =10, and the total test fault number meets the following conditions:
Function corresponding to low-temperature stepping test
Thus:
therefore, T can be obtained 1 Comprises the following steps:
By
Therefore T 2 Can be simplified into a function of an unknown parameter theta to obtain
Changes with changes in θ values during Gibbs sampling:
at the same time, before Gibbs sampling, it is necessary to set the ranges of parameters λ, θ, α, β, λ ∈ [0, ∞ ]]， α∈[0,1]，β∈[0,500]And the parameter theta may be set based on empirical stress levels,
thus setting
On the basis, gibbs sampling simulation with the single total step size of 2000 is carried out, the change of the mean value of the hyper-parameters along with the sampling times in the single Gibbs sampling process is shown in figure 7, and it can be seen that the parameter values are gradually converged along with the increase of the iteration times.
Although theoretically, the steady-state solution of the parameters can be finally obtained from the single Gibbs sampling result, the parameter convergence process still has certain randomness, the convergence speed possibly has different, in order to ensure the stability of the result, 1000 repeated experiments are carried out in total, and only the 1000 times after each sampling are reserved as the steady state to participate in statistics, because S in the example is adopted 0 ＝S 1 =0 ℃, so the average MTBF of the product at the target stress is estimated as:
MTBF low temperature ＝362.9h
2) Quantitative evaluation of high temperature step stress test
And (3) according to the high-temperature stepping stress, arranging to obtain high-temperature stepping strengthening test data:
a Bayes evaluation method without failure data is adopted to solve the quantitative estimated value of MTBF in the high-temperature stepping stress test, and the specific process is as follows:
Conversion of temperature units to K, taking into account an acceleration model under high temperature stress
The acceleration factors in the high-temperature stage are obtained as follows:
AF(S 1 ,S 0 )＝1，AF(S 2 ,S 0 )＝1.4330，AF(S 3 ,S 0 )＝2.0322
the test time for extrapolating the observation points under normal stress is:
t 1 ＝2，t 2 ＝4.8661，t 3 ＝8.9305
according to s i ＝n i +n i+1 +…+n k The following can be calculated:
s 1 ＝3，s 2 ＝2，s 3 ＝1
3) Quantitative evaluation of rapid temperature change stress test
According to the provided rapid temperature change stress, 5 rapid temperature change circulation observation periods are contained in total, and the rapid temperature change strengthening test data are obtained by arranging:
a Bayes evaluation method without failure data is adopted to solve the quantitative estimation value of the MTBF of the rapid temperature change stress test, and the specific process is as follows:
Considering acceleration under rapid temperature-dependent stressModel, converting temperature unit into K, determining AF (S) i ,S 0 ) The calculation method of (2) can obtain the acceleration factors of the rapid temperature change stage as follows:
AF(S i ,S 0 )＝3.8132i＝1,2,3,4,5
the test time for extrapolating the observation points under normal stress is:
t 1 ＝30.97，t 2 ＝61.94，t 3 ＝92.92，t 2 ＝123.88，t 3 ＝154.86
at the same time, according to s i ＝n i +n i+1 +…+n k The following can be calculated:
s 1 ＝5，s 2 ＝4，s 3 ＝3，s 4 ＝2，s 5 ＝1
4) Quantitative evaluation of vibration step stress test
And according to the provided vibration stepping stress and fault records, arranging to obtain vibration stepping strengthening test data:
according to a Gibbs sampling algorithm of a Bayes evaluation method under step stress, a quantitative estimation value of MTBF of a vibration step stress test is solved, and irrelevant parameters r and T except lambda, theta, alpha and beta are determined according to fault data before Gibbs sampling is carried out 1 、T 2 Specifically, the calculation method of (1):
Vibration stepping total test stage k =12, and the total number of test faults satisfies:
Function corresponding to vibration step test
Thus:
since the time unit is converted into h, T can be obtained 1 ：
Obtaining R i And τ i ', simplification
The value of (c) is also a function of the unknown parameter θ, and varies with the variation of θ during Gibbs sampling.
At the same time, before Gibbs sampling, it is necessary to set the ranges of the parameters λ, θ, α, β, and empirically set λ ∈ [0, ∞]，α∈[0,1]，β∈[0,500]And the parameter theta may be set based on empirical stress levels,
thus setting
The Gibbs sampling mode which is the same as the low-temperature stepping mode is adopted, the change of the mean value of the hyper-parameters along with the sampling times in the single Gibbs sampling process is shown in figure 8, in order to ensure the stability of the result, 1000 repeated experiments are also carried out in total, only 1000 times after each sampling is kept as the stable state to participate in statistics, and because S in the embodiment is used as the stable state to participate in statistics 0 ＝S 1 =18grms, so the average MTBF of the product at the target stress is estimated as:
MTBF vibration ＝0.53h
5) Quantitative evaluation of comprehensive environmental stress test
According to the provided comprehensive environmental stress, the observation period comprising 5 comprehensive environmental stress cycles is obtained in a finishing way, and the comprehensive environmental stress strengthening test data are obtained:
the quantitative evaluation method without failure data is adopted to solve the quantitative estimated value of MTBF of the comprehensive environmental stress test, and the specific process is as follows:
AF is determined by converting the temperature unit into K in consideration of an acceleration model under rapid temperature change stress (S) i ,S 0 ) The calculation method of (2) can obtain the acceleration factors of the comprehensive environmental stress stage as follows:
AF(S i ,S 0 )＝5.0720 i＝1,2,3,4,5
the test time for observation points under normal stress can be extrapolated as:
t 1 ＝41.82，t 2 ＝83.63，t 3 ＝125.45，t 2 ＝167.26，t 3 ＝209.08
according to s i ＝n i +n i+1 +…+n k The following can be calculated:
s 1 ＝5，s 2 ＝4，s 3 ＝3，s 4 ＝2，s 5 ＝1
6) Comprehensive evaluation of MTBF calculation results of each stage
In summary, using the confidence level of α =0.1 to calculate the MTBF confidence interval for each phase, the MTBF calculation for each phase can be obtained:
| test phase | MTBF (h) under Normal stress | Confidence interval |
| Low temperature stepping | 362.9 | [62.13,1379.91] |
| High temperature stepping | 25.35 | [4.65,4542.3] |
| Quick temperature change | 636.88 | [80.63,78766.42] |
| Vibration stepping | 0.82 | [0.50,4.79] |
| Combined stress | 859.86 | [108.85,106344.27] |
Analysis shows that all the evaluation results are within the confidence interval under the current confidence level, and the comprehensive MTBF of the product strengthening test can be obtained:
the above-described embodiments are not intended to limit the present invention, and any modifications, equivalents, improvements, etc. made within the spirit and principle of the present invention are included in the scope of the present invention.
Claims (10)
1. A reliability strengthening test quantitative evaluation method based on Bayes analysis is characterized by comprising the following steps: carrying out low-temperature stepping stress, high-temperature stepping stress, rapid temperature circulation, vibration stepping stress and comprehensive environmental stress tests, and recording fault data and stress data; dividing the test result into failure and non-failure according to whether the product has failure or not; dividing test conditions into step stress and constant stress according to the stress type; evaluating the average life of the product under the conditions of failure step stress, failure constant stress, no failure step stress and no failure constant stress;
for step stress with failure: converting failure data, and obtaining average life estimation under normal stress S by adopting a Bayes estimation method based on Gibbs sampling; for constant stress with failure: calculating an acceleration factor, converting the test time of constant stress into the test time of normal stress, obtaining an estimation form of failure rate by adopting a Bayes method, obtaining an estimation value of failure rate by adopting a Gibbs sampling method, and obtaining average life estimation under Chang Yingli S by adopting a least square method; for no failure step stress: calculating an acceleration factor, converting the test time of the stepping stress into the test time of the normal stress, and adopting a Bayes estimation method without failure; for constant stress without failure: calculating an acceleration factor, converting the test time of the constant stress into the test time of the normal stress, and adopting a failure-free Bayes estimation method;
setting a certain confidence level according to the test time and the number of failures of the test object under each test condition, generating an MTBF confidence interval of each test condition, calculating and analyzing the availability of each MTBF by adopting a staged quantitative evaluation method, and obtaining a comprehensive MTBF value of the reliability enhancement test through weighted average.
2. The Bayes analysis based reliability enhancement test quantitative evaluation method according to claim 1, wherein said calculating an acceleration factor comprises: according to test data, converting acceleration factors of high-temperature stepping stress, low-temperature stepping stress, rapid temperature change stress, vibration stepping stress and comprehensive environmental stress by adopting an accumulated damage model or an exponential damage model; the method comprises the steps of calculating an acceleration factor by adopting an Arrhenius model and a single stress Eying model for high-temperature stepping stress, calculating an acceleration factor by adopting an exponential acceleration model for low-temperature stepping stress, calculating an acceleration factor by adopting a modified coefficient-Manson model and an extension model thereof for rapid temperature change stress, calculating an acceleration factor by adopting a Basquin equation and a Miner rule for vibration stepping stress, splitting a test into a rapid temperature change test and a vibration stress test by synthesizing environmental stress, analyzing and calculating, and synthesizing to obtain the acceleration factor.
3. The Bayes analysis based reliability enhancement test quantitative evaluation method according to claim 2, wherein the Arrhenius model calculates an acceleration factor of high temperature step stress
Calculated by the single stress Eying modelAcceleration factor of high temperature step stress
Wherein t is use Is the test time using the stress level, t test Is the test time of the acceleration stress level, T use Is using a stress level, T test Is the acceleration stress level, E a Is activation energy in units of eV, k B The model is a boltzmann constant, and which model is used is determined according to the quality of a result of fitting data of the two models;
the acceleration factor of the low-temperature stepping stress calculated by the exponential acceleration model
b＝8.3×10 -2 Where ξ (T) = a · e b·T The method is a mathematical form of an exponential acceleration model, describes a change rule between the reliability characteristic and the low temperature of a product, T is absolute temperature, and a and b are constants;
acceleration factor of rapid temperature variation stress calculated by the modified coffee-Manson model
Wherein N is use 、N test Number of failure cycles, Δ T, for normal and test stresses, respectively test 、ΔT use Temperature transformation ranges of the test stress and the normal stress, respectively, f test 、f use The cycle frequencies of the test stress and the normal stress, T, respectively max,test 、T max,use The maximum cycling temperatures for the test stress and the normal stress, respectively;
acceleration factor of vibration stepping stress calculated by the Basquin equation and Miner rule
By stress-cycle life S-N curvesThe line describes the fatigue test of the material, a parameter m is obtained by fitting, or an empirical value of m is obtained by inquiring a GJB150A, MIL-STD-810G data handbook, the value of m of sinusoidal vibration is 6, and the value of m of random vibration is 5-8;
acceleration factor A of the synthetic environmental stress F ＝A F_fastT ×A F_var Wherein A is F_fastT Is a rapid temperature variation corresponding to an acceleration factor, A F_var The stress of each test process is integrated by 10 percent of magnitude value lower than the working limit value and is applied to the product, and in a temperature cycle mode, the step length is adopted in the first temperature cycle to be 20 percent of the damage vibration limit, the step length is adopted in the second temperature cycle to be 40 percent of the damage limit, and the like, until the product can not be repaired or the required cycle number is reached.
4. The Bayes analysis based reliability enhancement test quantitative evaluation method of claim 3, wherein the failure-free Bayes estimation method comprises: assuming that the failure distribution follows an exponential distribution, i.e., F (T) = P (T < T) =1-e -λt Wherein t is the working time of the product, F (t) is the accumulated failure rate of the product at the time t, lambda is the failure rate, and is constant under specific stress respectively at different test stress S i (i =1,2 …, k) test samples for a total test time
The test time of the same group of test samples with the same test stress is the same, and the number of the corresponding test samples is n i (i =1,2 …, k), none of the tests in each set failed, and the target stress was S 0 The total time of the equivalent test is t i (i =1,2 …, k), then
Wherein AF (S) j ,S 0 ) Is the test stress S j Relative to the eyeStandard stress S 0 Acceleration factor of (d), recording test data (t) i ,s i ) Wherein s is i Is t i Number of test samples at time, i.e. s i ＝n i +n i+1 +…+n k Calculating the equivalent test time t based on Bayes method i Corresponding cumulative failure probability estimate
Target stress S is estimated based on least square method 0 Estimate of failure rate of
Estimate Mean Time Between Failures (MTBF), i.e.
5. The Bayes analysis based reliability enhancement test quantitative evaluation method according to claim 4, wherein the Bayes analysis based reliability enhancement test quantitative evaluation method calculates an equivalent test time t i Corresponding cumulative failure probability estimate
The method comprises the following steps: by p i Represents an equivalent test time of t i Cumulative probability of failure, i.e. p i ＝f(t i ) From (1-p) i ) 2 Estimating the prior density as a kernel of the density, i.e.
Wherein
Is a general meter of empirical distribution function processed according to reliability test dataComputing means, order
By
Calculating out
Namely that
According to likelihood function corresponding to non-failure test result
Calculation of p using Bayes' formula i The posterior distribution density of (i.e.)
Taking into account the square loss to obtain
Is estimated, i.e.
6. The Bayes analysis-based reliability augmentation test quantitative evaluation method according to claim 5, wherein the target stress S is estimated based on a least square method 0 Estimate of failure rate of
The method comprises the following steps: using functions
Describing cumulative failure probability estimates
Relation to failure rate λ, where ∈ i Is an accumulated failure probability estimate
The generated error is that the least square method is adopted to solve the square loss corresponding to the failure rate
Let dQ (λ)/d λ =0, a least-squares estimate of the parameter λ is calculated, i.e.
7. The Bayes analysis based reliability enhancement test quantitative evaluation method according to claim 3, wherein the step stress with failure estimation method comprises: assuming that the failure distribution follows an exponential distribution, i.e.
Wherein the step stress is S i T is the working time of the product, lambda i Is the step stress S of the product i The failure rate at time t, being a fixed constant,
is the product is under step stress S i The cumulative probability of failure at time t,
at step stress S for the product i The instantaneous probability of failure, i.e. the probability of failure density function,
is the product under stress S i Reliability at time t, working time of failed sample is
Wherein
The total number of samples is n, and the test pause time is tau according to the characteristics of the timing truncation i Time τ i The number of failed samples of (2) is r i From the memoryless of the exponential distribution, a likelihood function of the step stress test is constructed, i.e.
And
order to
Wherein R is i Is a step stress S 1 To S i Total number of failures of R = R k Is the total number of failures, τ, of the step test i ' is the cumulative working time of all samples, simplifying the likelihood function, i.e.
Set the product at step stress S 1 The failure rate of is λ 1 I.e. by
Wherein θ = λ 2 /λ 1 Is the step stress S of the product 2 Lower phase ratio in step stress S 1 Acceleration factor of theta > 0, using a function
Description of the invention
According to the calculation mode of the stepping stress type and the acceleration factor, an Arrhenius model, an exponential acceleration model and an S-N curve equation are respectively selected to describe acceleration models of high-temperature stepping, low-temperature stepping and vibration stepping, and functions are simplified
Namely, it is
Let λ = λ 1 、
To obtain
Selecting a prior distribution of theta
Suppose the product is under step stress S 1 The prior distribution of the failure rates lambda follows the Gamma distribution, the density function
Where 0 < λ < ∞, α > 0, β > 0, α and β are hyper-parameters, the prior distributions are uniformly distributed over (0,1) and (0,c), respectively, c > 0 is a constant, the probability densities are π (α) =1 (0 < α < 1) and π (β) =1/c (0 < β < c), respectively, if α and β areBeta is independent, then lambda's multi-layer prior probability density
Obtaining a probability density function of a joint posterior distribution of parameters (lambda, theta, alpha, beta) from a prior distribution of the likelihood functions
Wherein
Describing relevant test information, and sequentially integrating the formula to obtain the marginal probability density of each parameter and the probability density of lambda full-condition posterior distribution
Probability density of full condition posterior distribution of theta
The probability density pi (alpha-theta, lambda, beta, tau) oc-beta of the alpha full-condition posterior distribution α λ α The probability density pi (beta | theta, lambda, alpha, tau) · beta of the full-condition posterior distribution of/gamma (alpha), beta α e -λβ Calculating the parameter mean value of the combined posterior distribution by adopting a Gibbs sampling method, generating random numbers of the full-condition posterior distribution of theta and alpha by adopting a round-robin sampling method, and adopting Gamma distribution Gamma (alpha + r, beta + T) 2 ) And Γ (α +1, λ) generates random numbers with a full condition posterior distribution of λ and β, setting a starting point (λ) (0) ,θ (0) ,α (0) ,β (0) ) Posterior distribution pi from full condition (lambda | theta) (n-1) ,α (n-1) ,β (n-1) τ) to produce λ (n) Posterior distribution from full Conditionπ(θ∣λ (n) ,α (n-1) ,β (n-1) τ) producing θ (n) Posterior distribution pi (alpha | lambda) from full condition (n) ,θ (n) ,β (n-1) τ) production of α (n) Posterior distribution pi (beta | lambda) from full condition (n) ,θ (n) ,α (n) τ) production of β (n) Then (λ) (n) ,θ (n) ,α (n) ,β (n) N =1,2, …, M, M +1, …, M) becomes a Gibbs iteration sample for the parameter (λ, θ, α, β), where M is the sample volume discarded before Gibbs iteration samples reach steady state, M > M is the total sample volume, resulting in a multi-layered Bayes parameter estimate for λ, θ, α, β
Obtaining an acceleration equation of a stepping stress test model by parameter estimation, and calculating a target stress S 0 Average MTBF of (a).
8. The Bayes analysis based reliability enhancement test quantitative evaluation method according to claim 1, wherein the estimation method of the constant stress with failure comprises: let S i Is constant stress, the working time of the failed sample is
n i Is a constant stress S i According to the nature of the timing truncation, τ i Is a constant stress S i Test pause time of r i Is at τ i Number of previous failed samples, order
Likelihood function of test data
Wherein H = { (λ) 1 ,λ 2 ,…,λ k ):0＜λ 1 ≤λ 2 ≤…≤λ k Instruction of
Wherein T is i Is a constant stress S i The accumulated working time of all samples of, simplifying the likelihood function, i.e.
Assuming the product is at constant stress S i Failure rate of i Obeys the Gamma distribution Ga (alpha) i ,β i ) In which α is i ,β i Is known to be a defined hyper-parameter, dependent on the respective stress S i Independent of each other, to obtain lambda i ,…,λ k Combined prior density function of
From Bayes' theorem to λ 1 ,λ 2 ,…,λ k Combined a posteriori density function of
Calculating lambda i Bayes posterior mean of
Wherein (lambda) 1 ,λ 2 ,…,λ k ) E is H, if i is less than j, 0 is less than lambda i ≤λ j 、E(λ i |D)≤E(λ j |D)、
Obtaining lambda by adopting a Gibbs sampling method i Full condition posterior distribution of
Truncated Gamma distribution of obedient intervals, i.e.
Wherein G i Is a Gamma distribution Ga (. Alpha.) i +r i ,β i The distribution function of + Ti, gi-1 is its inverse, U is a random sample of uniformly distributed U (0,1), if no information is a priori, then α s ＝β s =0, and (lambda) is obtained through Gibbs sampling iteration in t steps 1 ,λ 2 ,…,λ k ) M weight of Gibbs sample
Then
According to the calculation mode of the acceleration factor, obtaining the target stress S 0 Equivalent failure rate estimate of
And failure rate estimation
And
9. the Bayes analysis based reliability enhancement test quantitative evaluation method according to claim 1, wherein said generating MTBF confidence intervals for each test condition comprises: assuming that the average fault-free interval T of the products obeys exponential distribution, randomly extracting n products, and carrying out a timing end-cutting service life test under the stress condition, wherein the stopping time is T 0 At t, at 0 If the front r products fail, the tail sample t is cut off regularly 1 ≤t 2 ≤…≤t r ≤t 0 R < n, likelihood function of sample
t 1 ≤t 2 ≤…≤t r ≤t 0 Wherein
Is the total test time, order
Then
Its jacobian determinant
To obtain omega 1 ,…,ω r Combined density function of
According to omega 1 ,…,ω r Independently distributed to obtain common distribution Exp (lambda) = Ga (1, lambda), and obtained according to additivity of independent Gamma variable
The 1-alpha approximate confidence interval for MTBF is
The tail probability alpha/2 is merged into the 1-alpha/2 of the other side to obtain the lower confidence limit of the 1-alpha on one side
10. The Bayes analysis based reliability enhancement test quantitative evaluation method according to claim 9,the method is characterized in that the weighted average is used for obtaining the comprehensive MTBF value of the reliability enhancement test, and the method comprises the following steps: keeping the estimated value of the 1-alpha approximate confidence interval, totaling k, and if the engineering experience information is less, calculating the comprehensive MTBF by adopting an average method, namely
If the engineering experience information is more, calculating the comprehensive MTBF by adopting a Delphi-AHP model, namely
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