CN107885928A - Consider the stepstress acceleration Degradation Reliability analysis method of measurement error - Google Patents
Consider the stepstress acceleration Degradation Reliability analysis method of measurement error Download PDFInfo
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Abstract
The present invention proposes the stepstress acceleration Degradation Reliability analysis method for considering measurement error, and its step is as follows：The acceleration degradation experiment data of each sample stepstress are gathered by acceleration degradation experiment, acceleration degradation experiment data are pre-processed；The performance degradation process of product is described based on Generalized Wiener Process, it is determined that accelerating degradation model, establishes the stepstress acceleration degradation model for considering measurement error；The unknown parameter in stepstress acceleration degradation model is estimated using likelihood function；Reliability assessment and life prediction are carried out to product.Measurement error is incorporated into the analysis of stepstress acceleration degraded data by the present invention first, is considered the different time-varying characteristics of average item and variance item, is more met engineering practice；By fully developing the horizontal information between the longitudinal information and various sample under different stress levels between degraded data, information content is increase effectively, improves reliability assessment precision.
Description
Technical Field
The invention relates to the technical field of reliability analysis of acceleration performance degradation tests, in particular to a stepping stress acceleration performance degradation reliability analysis method considering measurement errors.
Background
Due to the limitations of test cost and lead time, the reliability analysis method based on the performance degradation data has higher efficiency compared with the traditional reliability analysis method based on the failure time. However, for some highly reliable and long-life products, only a small amount of performance degradation data is often available in a limited test time, resulting in poor evaluation accuracy. In order to solve the above problems, a reliability analysis method based on a step stress acceleration performance degradation test is an effective approach.
In accelerated performance degradation analysis, the existing methods mostly assume that the performance parameter measurement process is in an ideal state, that is, the uncertainty of the measured performance degradation data is only related to the inherent randomness of product degradation. However, in engineering practice, ideal measurements are often not realistic or too costly. Due to the influence of random factors such as human, measuring instruments and working environments, the measured performance degradation data often contains measurement errors, especially degradation data measured indirectly by the techniques such as sensors and image conversion. If the measurement error in the measurement process is ignored, the random influence factors and the time-varying characteristics of the product degradation process cannot be accurately grasped and described, and the evaluation precision is directly influenced.
Aiming at the problem of reliability analysis of the stepping stress acceleration performance degradation test data of the high-reliability long-life product, the invention provides a method for analyzing the stepping stress acceleration performance degradation reliability by considering the measurement error, so as to solve the problem.
Disclosure of Invention
Aiming at the technical problems that the random influence factors and time-varying characteristics of the product degradation process cannot be accurately grasped and described and the evaluation precision is influenced by neglecting the measurement error of the performance degradation data in the measurement process, the invention provides the stepping stress accelerated performance degradation reliability analysis method considering the measurement error, which effectively increases the information amount, improves the reliability evaluation precision, is convenient and effective and better accords with the actual conditions of engineering.
In order to achieve the purpose, the technical scheme of the invention is realized as follows: a step stress acceleration performance degradation reliability analysis method considering measurement errors comprises the following steps:
the method comprises the following steps: carrying out a stepping stress acceleration performance degradation test on the samples, collecting acceleration performance degradation test data of each sample stepping stress, and preprocessing the acceleration performance degradation test data to remove abnormal values and invalid data;
step two: describing a performance degradation process X (t) of a product based on a generalized Wiener process, determining an accelerated degradation model according to a relation between an accelerated stress level and a degradation rate, and then establishing a step stress accelerated performance degradation model Y (t) considering a measurement error;
step three: estimating unknown parameters of the step stress acceleration performance degradation model Y (t) considering the measurement error in the step two by establishing a likelihood function by utilizing the acceleration performance degradation test data in the step one;
step four: and (4) carrying out reliability evaluation and service life prediction on the product by using the estimated unknown parameters obtained in the third step.
The step of the method for collecting the acceleration performance degradation test data of each sample stepping stress in the step one
The method comprises the following steps:
step 1: m samples are put into the test stand to carry out a step stress acceleration performance degradation test, and each sample is sequentially at an acceleration stress level S1，S2 SLThe test was performed, and S1＜S2＜,...,＜SLL is the number of acceleration stress levels;
step 2: acquisition of accelerated performance degradation test data for individual samples: i th sample at an accelerated stress level SlCorresponding test time ofProcessing the collected performance degradation dataThereby obtaining the step stress acceleration performance degradation test data of the ith sampleWherein, i is 1,2,...,L，ni0＝0；
And step 3: acquiring data of a stepping stress acceleration performance degradation test: acquiring step stress accelerated property degradation test data of m samples as y ═ y1,y2,…,ym)。
The method for determining the accelerated degradation model in the second step comprises the following steps:
step 1: the generalized Wiener process based on the description of the real performance degradation process x (t) of the product is:
X(t)＝a+βΛ+σB(ν)
wherein, X (t) represents the real performance degradation process of the product, Λ ═ Λ (t; η) and ν ═ ν (t; γ) are respectively continuous strict monotone increasing functions related to time t, η and γ are respectively unknown parameter arrays of the functions Λ ═ Λ (t; η) and ν (t; γ), a is the initial performance degradation amount, β is the degradation rate, and a is the unknown parameter array of the function ν ═ ν (t; γ), a is the initial performance degradation amount, and β is the degradation rate N (-) is normally distributed, μ1And σ1Mean and standard deviation, μ, of the initial performance degradation aβAnd σβRespectively the mean value and the standard deviation of the degeneration rate β, sigma is more than or equal to 0 and is a diffusion coefficient, B (v) is a generalized standard Wiener random process, and random terms a, β and B (v) are mutually independent;
step 2: by testing data y for accelerated performance degradation at different stress levels1,y2,…,ymAnalyzing to obtain corresponding degradation rate, and fitting to obtain accelerated degradation model β (S)l)＝bλ(Sl；θ)；
Wherein S islL is the acceleration stress level, L is 1,2,. and L is the acceleration stress level number; b is a transformed degradation rate constant, andμ2and σ2The mean value and the standard deviation of b are respectively; lambda (S)l(ii) a Theta) with respect to the acceleration stress level SlIs a continuous strictly monotonic function of theta being a function lambda (S)l(ii) a θ) of unknown parameters.
λ (S) in the accelerated degradation modell(ii) a θ) determining its form from experience or a selected acceleration model; when the acceleration stress is temperature, an Arrhenius model λ (S) is adoptedl；θ)＝exp(-θ/Sl) (ii) a When the acceleration stress is the electric stress, an inverse power law model is adopted
The method for establishing the stepping stress acceleration performance degradation model Y (t) considering the measurement error in the second step comprises the following steps:
the expression of the acceleration stress level in the whole process of the stepping stress accelerated degradation test is as follows:
wherein, t is the time of the test,indicating the level of stress S from accelerationlIs increased to Sl+1L is an acceleration stress level number;
let lambdal＝λ(Sl(ii) a Theta) andthen, given the initial performance degradation amount a and the transformed degradation rate constant b, the real degradation process x (t) of the step stress accelerated performance degradation test is:
in the formula,
on the basis of determining the specific form of the accelerated degradation model, establishing a stepping stress accelerated performance degradation model considering the measurement error:
wherein Y (t) represents the degradation process of the product performance measured, and the initial performance degradation amount a and the degradation rate β are respectivelyAndε (t) represents a measurement error term and hasσeIs the standard deviation of the error term ε (t), an
Wherein Cov (·) is covariance, and random terms a, B (ν), and ∈ (t) are independent of each other.
The step of estimating the unknown parameters in the step stress acceleration performance degradation model Y (t) considering the measurement error comprises the following steps:
step 1: defining an unknown parameter matrix Θ (μ ═ for a step stress acceleration performance degradation model y (t) considering measurement errors1,μ2,σ1,σ2,θ,σ,η,γ,σe) (ii) a Step stress addingData y of speed performance degradation testiSubject to multivariate normal distribution, yi～MN(μ11i+μ2τi,Σi) And MN (. cndot.) represents a multivariate normal distribution,2 is a covariance matrix of the i-th sample performance degradation data; 1iIs niLColumn vector with all dimension elements 1, 1i＝(1,1,...,1)T；And vij＝ν(tij；γ)；IiIs niLAn identity matrix of dimensions; matrix arrayElement (2) of (1)ijThe expression of (a) is:
in the formula,m is the number of samples put into the step acceleration performance degradation test;
step 2: solving the log-likelihood function of the step stress accelerated performance degradation model Y (t):
the likelihood function of the step stress acceleration performance degradation model is:
the log likelihood function of the step stress acceleration performance degradation model obtained by simultaneously taking logarithms on the two sides of the formula is
In the formula,
and step 3: re-parameterizing the logarithm likelihood function, and estimating unknown parameters in an unknown parameter matrix in a maximized mode through partial derivatives and a multi-dimensional search algorithm:
the parameterized log-likelihood function of the stepping stress acceleration performance degradation model is as follows:
wherein,and
the parameters mu are obtained from the above formula1And mu2First order partial derivatives and parameters thereofFirst order partial derivative of (1) can be obtained
Let the above three equations equal 0 to obtain the parameter μ1、μ2Andrespectively of
In the formula,
will estimate the valueAndthe obtained information is brought into the parameterized log-likelihood function of the step stress acceleration performance degradation modelThe edge log likelihood function of (1):
maximization using a multi-dimensional search algorithmMaximum likelihood estimation can be obtainedAndthen sigma1、σeMaximum likelihood estimates of sum σ are respectivelyAndresulting in a maximum likelihood estimate of the parameter vector theta.
The reliability evaluation method in the fourth step comprises the following steps:
the life T of the product is: t ═ inf { T | x (T) ≧ Df|X(0)＜DfIn which D isfA performance degradation failure threshold for the product;
the conditional probability density distribution function for product life T, given an initial amount of performance degradation a, is:
wherein,and
according to the total probability principle, the probability density distribution function of the service life T is obtained as follows:
wherein,
at the working stress level S0The probability density distribution function for lifetime T is:
wherein,
μβ＝μ2λ(S0(ii) a Theta) and sigmaβ＝σ2λ(S0；θ)；
By means of numerical analysis, the working stress level S is obtained0The life distribution function for the lower product life T is:
thus, the working stress level S is obtained0The reliability function of the product at a given time t:
the method for predicting the service life in the fourth step comprises the following steps:
at the working stress level S0Lower, average life of the productThe approximation is:
wherein, Λ-1(. h) is the inverse of the function Λ (·); solving by numerical analysis:
FT(tR)＝1-R，
the reliable life t with the reliability R can be obtainedRAn estimate of (d).
The invention has the beneficial effects that:
(1) starting from engineering practice, the invention provides a stepping stress acceleration performance degradation reliability analysis method considering the measurement error, and the measurement error is inevitable in practical test, so the method introduces the measurement error into the stepping stress acceleration performance degradation data analysis for the first time, and simultaneously considers different time-varying characteristics of a mean term and a variance term, and has generality. Therefore, the method of the invention is more suitable for the actual situation of engineering.
(2) According to the invention, longitudinal information among degradation data under different stress levels and transverse information among different samples are fully developed and utilized, and the longitudinal information and the transverse information are taken as a whole for statistical inference, so that the information quantity is effectively increased, and the reliability evaluation precision is improved.
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In order to more clearly illustrate the embodiments of the present invention or the technical solutions in the prior art, the drawings used in the description of the embodiments or the prior art will be briefly described below, it is obvious that the drawings in the following description are only some embodiments of the present invention, and for those skilled in the art, other drawings can be obtained according to the drawings without creative efforts.
FIG. 1 is a flow chart of the present invention.
Detailed Description
The technical solutions in the embodiments of the present invention will be clearly and completely described below with reference to the drawings in the embodiments of the present invention, and it is obvious that the described embodiments are only a part of the embodiments of the present invention, and not all of the embodiments. All other embodiments, which can be obtained by a person skilled in the art without inventive effort based on the embodiments of the present invention, are within the scope of the present invention.
As shown in fig. 1, a method for analyzing degradation reliability of acceleration performance of stepping stress considering measurement error includes the following steps:
the method comprises the following steps: and carrying out a stepping stress acceleration performance degradation test on the samples, collecting acceleration performance degradation test data of each sample stepping stress, and preprocessing the acceleration performance degradation test data to remove abnormal values and invalid data.
M samples are put into the step stress acceleration performance degradation test, and m is the number of the samples put into the step stress acceleration performance degradation test. Each sample was sequentially subjected to an acceleration stress level S1，S2 SLThe test was performed, and S1＜S2＜,...,＜SLAnd L is the number of acceleration stress levels.
1. Accelerated performance degradation test data acquisition for individual samples.
yij＝Y(tij) Indicates the ith sample at the jth test time tijMeasured performance degradation parameter1,2, a, m; j ═ 1, 2.., ni1,ni1+1,ni1+2,...,ni2,...,niL. Wherein, when j is ni(l-1)+1,ni(l-1)+2,...,nilWhen y isijFor the ith sample at the ith acceleration stress level SlMeasured Performance parameter, and ni00, L1, 2. In addition, when j is equal to nilWhen t isijIndicating the acceleration stress level S of the ith samplelIs increased to Sl+1The test time of (2).
I th sample at an accelerated stress level SlCorresponding test time ofProcessing the collected performance degradation dataThereby obtaining the step stress acceleration performance degradation test data of the ith sampleWherein n isi0＝0。
2. And acquiring data of the stepping stress accelerated performance degradation test. Acquiring step stress accelerated property degradation test data of m samples as y ═ y1,y2,…,ym)。
3. And (4) preprocessing data. And when abnormal values and other invalid data in the performance degradation data are needed, removing processing is carried out.
Step two: and establishing a real degradation process X (t) of a step stress acceleration performance degradation test according to the acceleration stress level based on the generalized Wiener process, and then establishing a step stress acceleration performance degradation model Y (t) considering the measurement error.
1. An accelerated degradation model is determined.
Through the analysis of the performance degradation data, the fluctuation of the performance degradation curve is mainly caused by two fluctuations, which mainly comprise: random fluctuation on the individual performance degradation curve of the product, namely fluctuation in the product, is mainly caused by the nonuniformity of materials, the inconsistency of manufacturing processes and the like and is expressed as inherent randomness of the product; the random fluctuation of the degradation curves of different product performances, namely the fluctuation among products, is mainly the difference caused by factors such as initial conditions, product shapes, sizes and the like, and is expressed as the randomness among the products. The invention provides a generalized Wiener process for describing a real degradation process of a step stress acceleration performance degradation test of a product, which comprises the following specific steps:
X(t)＝a+βΛ+σB(ν)
wherein, X (t) represents the real performance degradation process of the product, Λ ═ Λ (t; η) and ν ═ ν (t; γ) are respectively continuous strict monotone increasing functions related to time t, η and γ are respectively unknown parameter arrays of the functions Λ ═ Λ (t; η) and ν (t; γ), a is the initial performance degradation amount, β is the drift coefficient, i.e. the degradation rate, and in order to represent the randomness among the products, the parameters a and β are respectively assumed to be random variables, i.e. the degradation rateN (-) is normally distributed, μ1And σ1Mean and standard deviation, μ, of the initial performance degradation aβAnd σβThe mean value and the standard deviation of the degradation rate β are respectively, sigma is more than or equal to 0 and is a diffusion coefficient, B (v) is a generalized standard Wiener random process and is used for describing the volatility and the time variation of the product in an individual, and random items a, β and B (v) are mutually independent.
In analyzing the accelerated performance degradation test data for many products, it is often considered that the accelerated stress level S islAnalyzing the accelerated performance degradation test data under different stress levels to obtain corresponding degradation rates, and fitting to obtain an accelerated degradation model β (S)l)＝bλ(Sl(ii) a Theta), b is the transformed degradation rate constant, and for describing the product-to-product variability, b is assumed to be a random variable andμ2and σ2The mean value and the standard deviation of b are respectively; lambda (S)l(ii) a Theta) with respect to the acceleration stress level SlCan be determined from the selected acceleration model, where θ is a function λ (S)l(ii) a θ) of unknown parameters. The form of the accelerated degradation model may also be determined empirically, for example when the acceleration stress is temperature, an Arrhenius model λ (S) may be usedl；θ)＝exp(-θ/Sl) (ii) a When the acceleration stress is electrical stress, an inverse power law model can be usedAnd the like.
2. And establishing a stepping stress acceleration performance degradation model considering the measurement error.
The expression for the level of the accelerating stress during the entire step stress accelerated degradation test can be described as:
wherein, t is the time of the test,indicating the level of stress S from accelerationlIs increased to Sl+11, 2.
Let lambdal＝λ(Sl(ii) a Theta) andthen, given an initial performance degradation amount a and a transformed degradation rate constant b, the true degradation process x (t) of the step stress accelerated performance degradation test can be described as:
in the formula,
in engineering, performance degradation data are often inevitably influenced by random factors such as human factors, instruments or noises in the measurement process, so that the measured degradation data are not real degradation values, and degradation measurement values often contain measurement errors, particularly accelerated degradation tests. Therefore, it is urgently required to establish a step stress acceleration performance degradation model in consideration of the measurement error. On the basis of determining the specific form of the accelerated degradation model, establishing a stepping stress accelerated performance degradation model considering the measurement error:
wherein Y (t) represents the degradation process of the product performance measured, and the initial performance degradation amount a and the drift coefficient β are respectivelyAndε (t) represents a measurement error term and hasσeIs the standard deviation of the error term epsilon (t) and has covariance
The random terms a, B (v) and epsilon (t) are independent of each other.
Step three: and (3) estimating unknown parameters of the step stress acceleration performance degradation model Y (t) considering the measurement error in the step two by establishing a likelihood function by using the acceleration performance degradation test data in the step one.
1. An unknown parameter matrix of a step stress acceleration performance degradation model Y (t) considering the measurement error is defined.
Defining an unknown parameter matrix of a stepping stress acceleration performance degradation model Y (t) considering measurement errors as theta ═ mu1,μ2,σ1,σ2,θ,σ,η,γ,σe). Defining a matrix for the step stress accelerated property degradation test data collected in the step oneWherein, tauiElement (2) of (1)ijThe expression of (a) is:
in the formula, Λij＝Λ(tij；η)，i＝1,2,...,m，l＝1,2,...,L。
The amount of performance degradation y is known from the basic properties of the modeliFollowing a multivariate normal distribution, there are:
E(yi)＝μ11i+μ2τi
amount of degradation of performance yiThe mean and covariance matrices of (1) are respectively mu11i+μ2τiAndΣifor the ith testThe covariance matrix of sample performance degradation is yi～MN(μ11i+μ2τi,Σi). In the formula 1iIs niLColumn vectors with all dimension elements 1, i.e. 1i＝(1,1,...,1)T；And vij＝ν(tij；γ)；IiIs niLAn identity matrix of dimensions.
2. The log-likelihood function is defined by the likelihood function of the step stress accelerated performance degradation model y (t).
The likelihood function of the step stress acceleration performance degradation model is
The log likelihood function of the step stress acceleration performance degradation model obtained by simultaneously taking logarithms on the two sides of the formula is
In the formula,
3. and re-parameterizing the logarithm likelihood function, and estimating the unknown parameters in the unknown parameter matrix in a maximized mode through partial derivatives and a multi-dimensional search algorithm.
To facilitate parameter estimation, the model is first re-parameterized, such that Andtherefore, the parameterized log-likelihood function of the stepping stress acceleration performance degradation model can be obtained by substituting the re-parameterized parameters into the log-likelihood function
The parameters mu are obtained from the above formula1And mu2First order partial derivative of
Let the above formula equal to 0, and obtain
Obtaining the parameter mu1And mu2Respectively of
In the formula,
solving relevant parameters of log-likelihood function parameterized by stepping stress acceleration performance degradation modelFirst order partial derivative of
Let the above formula equal to 0, the parameters can be obtainedHas a maximum likelihood estimate of
Will estimate the valueAndthe obtained information is brought into the parameterized log-likelihood function of the step stress acceleration performance degradation modelEdge log likelihood function of
Maximization using a multi-dimensional search algorithmMaximum likelihood estimation can be obtainedAndthen sigma1、σeMaximum likelihood estimates of sum σ are respectivelyAndthe maximum likelihood estimate for the resulting parameter vector Θ is:
step four: and (4) carrying out reliability evaluation and service life prediction on the product by using the estimated unknown parameters obtained in the three steps.
1. And (6) reliability evaluation.
Assume a product's performance degradation failure threshold of DfThen, according to the concept of first arrival, the lifetime T of the product is defined as
T＝inf{t|X(t)≥Df|X(0)＜Df}
Given an initial amount of performance degradation a, the conditional probability density distribution function for product life T is known to be
Wherein,and
according to the total probability principle, the probability density distribution function of the service life T can be obtained as
Wherein,considering A, g (t) can be obtained first according to the principle of total probability and thenThus, at the working stress level S0The probability density distribution function of the lifetime T is
Wherein,
μβ＝μ2λ(S0(ii) a Theta) and sigmaβ＝σ2λ(S0；θ)。
Finally, by means of numerical analysis, the level S of the working stress can be obtained0Life distribution function of lower product life T is
Can obtain the working stress level S0The reliability function of the product at a given time t:
2. and (5) predicting the service life. At the working stress level S0Lower, average life of the productCan be approximated as
Wherein, Λ-1(. cndot.) is the inverse of the function Λ (. cndot.). Solving by numerical analysis
FT(tR)＝1-R
The reliable life t with the reliability R can be obtainedRAn estimate of (d).
The above description is only for the purpose of illustrating the preferred embodiments of the present invention and is not to be construed as limiting the invention, and any modifications, equivalents, improvements and the like that fall within the spirit and principle of the present invention are intended to be included therein.
Claims (8)
1. A step stress acceleration performance degradation reliability analysis method considering measurement errors is characterized by comprising the following steps:
the method comprises the following steps: carrying out a stepping stress acceleration performance degradation test on the samples, collecting acceleration performance degradation test data of each sample stepping stress, and preprocessing the acceleration performance degradation test data to remove abnormal values and invalid data;
step two: describing a product degradation process X (t) of the performance of the product changing along with time based on a generalized Wiener process, determining an accelerated degradation model according to the relation between the accelerated stress level and the degradation rate, and then establishing a step stress accelerated performance degradation model Y (t) considering the change of the measurement error along with time;
step three: estimating unknown parameters of the step stress acceleration performance degradation model Y (t) considering the measurement error in the step two by establishing a likelihood function by utilizing the acceleration performance degradation test data in the step one;
step four: and (4) carrying out reliability evaluation on the product by using the estimated unknown parameters obtained in the third step.
2. The method for analyzing reliability of accelerated degradation of stepping stress considering measurement errors according to claim 1, wherein the step one of collecting test data of accelerated degradation of stepping stress of each sample comprises the steps of:
step 1: m samples are put into the test stand to carry out a step stress acceleration performance degradation test, and each sample is sequentially at an acceleration stress level S1，S2，；SLThe test was performed, and S1＜S2＜,...,＜SLL is the number of acceleration stress levels;
step 2: acquisition of accelerated performance degradation test data for individual samples: i th sample at an accelerated stress level SlCorresponding test time ofProcessing the collected performance degradation dataThereby obtaining the step stress acceleration performance degradation test data of the ith sampleWherein, i is 1,2, 1, m, L is 1,2, 1i0＝0；
And step 3: acquiring data of a stepping stress acceleration performance degradation test: acquiring step stress accelerated property degradation test data of m samples as y ═ y1,y2,…,ym)。
3. The method for analyzing reliability of degradation of accelerated performance of step stress considering measurement errors according to claim 1 or 2, wherein the method for determining the accelerated degradation model in the second step is as follows:
step 1: the generalized Wiener process based on the description of the real performance degradation process x (t) of the product is:
X(t)＝a+βΛ+σB(ν)
wherein, X (t) represents the real performance degradation process of the product, Λ ═ Λ (t; η) and ν ═ ν (t; γ) are respectively continuous strict monotone increasing functions related to time t, η and γ are respectively unknown parameter arrays of the functions Λ ═ Λ (t; η) and ν (t; γ), a is the initial performance degradation amount, β is the degradation rate, and a is the unknown parameter array of the function ν ═ ν (t; γ), a is the initial performance degradation amount, and β is the degradation rate N (-) is normally distributed, μ1And σ1Mean and standard deviation, μ, of the initial performance degradation aβAnd σβRespectively the mean value and the standard deviation of the degeneration rate β, sigma is more than or equal to 0 and is a diffusion coefficient, B (v) is a generalized standard Wiener random process, and random terms a, β and B (v) are mutually independent;
step 2: by testing data y for accelerated performance degradation at different stress levels1,y2,…,ymAnalyzing to obtain corresponding degradation rate, and fitting to obtain accelerated degradation model β (S)l)＝bλ(Sl；θ)；
Wherein S islL is the acceleration stress level, L is 1,2,. and L is the acceleration stress level number; b is a transformed degradation rate constant, andμ2and σ2The mean value and the standard deviation of b are respectively; lambda (S)l(ii) a Theta) with respect to the acceleration stress level SlIs a continuous strictly monotonic function of theta being a function lambda (S)l(ii) a θ) of unknown parameters.
4. The method of claim 3, wherein λ (S) in the accelerated degradation model is a step stress accelerated degradation reliability analysis method considering measurement errorl(ii) a θ) determining its form from experience or a selected acceleration model; when the acceleration stress is temperature, an Arrhenius model λ (S) is adoptedl；θ)＝exp(-θ/Sl) (ii) a When the acceleration stress is the electric stress, an inverse power law model is adopted
5. The method for analyzing reliability of step-by-step stress acceleration performance degradation considering measurement errors according to claim 3, wherein the method for establishing the step-by-step stress acceleration performance degradation model y (t) considering measurement errors in the second step is as follows:
the expression of the acceleration stress level S in the whole process of the stepping stress accelerated degradation test is as follows:
<mrow> <mi>S</mi> <mo>=</mo> <mfenced open = "{" close = ""> <mtable> <mtr> <mtd> <msub> <mi>S</mi> <mn>1</mn> </msub> </mtd> <mtd> <mrow> <mn>0</mn> <mo>&le;</mo> <mi>t</mi> <mo>&le;</mo> <msub> <mi>t</mi> <msub> <mi>n</mi> <mn>1</mn> </msub> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <msub> <mi>S</mi> <mn>2</mn> </msub> </mtd> <mtd> <mrow> <msub> <mi>t</mi> <msub> <mi>n</mi> <mn>1</mn> </msub> </msub> <mo><</mo> <mi>t</mi> <mo>&le;</mo> <msub> <mi>t</mi> <msub> <mi>n</mi> <mn>2</mn> </msub> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mtable> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> </mtable> </mtd> <mtd> <mtable> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> </mtable> </mtd> </mtr> <mtr> <mtd> <msub> <mi>S</mi> <mi>L</mi> </msub> </mtd> <mtd> <mrow> <msub> <mi>t</mi> <msub> <mi>n</mi> <mrow> <mi>L</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> </msub> <mo><</mo> <mi>t</mi> <mo>&le;</mo> <msub> <mi>t</mi> <mi>n</mi> </msub> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>,</mo> </mrow>
wherein, t is the time of the test,indicating the level of stress S from accelerationlIs increased to Sl+1L is an acceleration stress level number;
let lambdal＝λ(Sl(ii) a Theta) andthen, given the initial performance degradation amount a and the transformed degradation rate constant b, the real degradation process x (t) of the step stress accelerated performance degradation test is:
<mrow> <mi>X</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>=</mo> <mfenced open = "{" close = ""> <mtable> <mtr> <mtd> <mrow> <mi>a</mi> <mo>+</mo> <msub> <mi>b&lambda;</mi> <mn>1</mn> </msub> <mi>&Lambda;</mi> <mo>+</mo> <mi>&sigma;</mi> <mi>B</mi> <mrow> <mo>(</mo> <mi>v</mi> <mo>)</mo> </mrow> </mrow> </mtd> <mtd> <mrow> <mn>0</mn> <mo>&le;</mo> <mi>t</mi> <mo>&le;</mo> <msub> <mi>t</mi> <msub> <mi>n</mi> <mn>1</mn> </msub> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mi>a</mi> <mo>+</mo> <mi>b</mi> <mrow> <mo>&lsqb;</mo> <mrow> <msub> <mi>&lambda;</mi> <mn>2</mn> </msub> <mrow> <mo>(</mo> <mrow> <mi>&Lambda;</mi> <mo>-</mo> <msub> <mi>&Lambda;</mi> <msub> <mi>n</mi> <mn>1</mn> </msub> </msub> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <msub> <mi>&lambda;</mi> <mn>1</mn> </msub> <msub> <mi>&Lambda;</mi> <msub> <mi>n</mi> <mn>1</mn> </msub> </msub> </mrow> <mo>&rsqb;</mo> </mrow> <mo>+</mo> <mi>&sigma;</mi> <mi>B</mi> <mrow> <mo>(</mo> <mi>v</mi> <mo>)</mo> </mrow> </mrow> </mtd> <mtd> <mrow> <msub> <mi>t</mi> <msub> <mi>n</mi> <mn>1</mn> </msub> </msub> <mo><</mo> <mi>t</mi> <mo>&le;</mo> <msub> <mi>t</mi> <msub> <mi>n</mi> <mn>2</mn> </msub> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mtable> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> </mtable> </mtd> <mtd> <mtable> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> </mtable> </mtd> </mtr> <mtr> <mtd> <mrow> <mi>a</mi> <mo>+</mo> <mi>b</mi> <mrow> <mo>&lsqb;</mo> <mrow> <msub> <mi>&lambda;</mi> <mi>L</mi> </msub> <mrow> <mo>(</mo> <mrow> <mi>&Lambda;</mi> <mo>-</mo> <msub> <mi>&Lambda;</mi> <msub> <mi>n</mi> <mrow> <mi>L</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> </msub> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <munderover> <mi>&Sigma;</mi> <mrow> <mi>l</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow> <mi>L</mi> <mo>-</mo> <mn>1</mn> </mrow> </munderover> <msub> <mi>&lambda;</mi> <mi>l</mi> </msub> <msub> <mi>&Delta;&Lambda;</mi> <msub> <mi>n</mi> <mi>l</mi> </msub> </msub> </mrow> <mo>&rsqb;</mo> </mrow> <mo>+</mo> <mi>&sigma;</mi> <mi>B</mi> <mrow> <mo>(</mo> <mi>v</mi> <mo>)</mo> </mrow> </mrow> </mtd> <mtd> <mrow> <msub> <mi>t</mi> <msub> <mi>n</mi> <mrow> <mi>L</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> </msub> <mo><</mo> <mi>t</mi> <mo>&le;</mo> <msub> <mi>t</mi> <mi>n</mi> </msub> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>,</mo> </mrow>
in the formula,
on the basis of determining the specific form of the accelerated degradation model, establishing a stepping stress accelerated performance degradation model Y (t) considering measurement errors as follows:
<mrow> <mi>Y</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>=</mo> <mfenced open = "{" close = ""> <mtable> <mtr> <mtd> <mrow> <mi>a</mi> <mo>+</mo> <msub> <mi>b&lambda;</mi> <mn>1</mn> </msub> <mi>&Lambda;</mi> <mo>+</mo> <mi>&sigma;</mi> <mi>B</mi> <mrow> <mo>(</mo> <mi>v</mi> <mo>)</mo> </mrow> <mo>+</mo> <mi>&epsiv;</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </mtd> <mtd> <mrow> <mn>0</mn> <mo>&le;</mo> <mi>t</mi> <mo>&le;</mo> <msub> <mi>t</mi> <msub> <mi>n</mi> <mn>1</mn> </msub> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mi>a</mi> <mo>+</mo> <mi>b</mi> <mrow> <mo>&lsqb;</mo> <mrow> <msub> <mi>&lambda;</mi> <mn>2</mn> </msub> <mrow> <mo>(</mo> <mrow> <mi>&Lambda;</mi> <mo>-</mo> <msub> <mi>&Lambda;</mi> <msub> <mi>n</mi> <mn>1</mn> </msub> </msub> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <msub> <mi>&lambda;</mi> <mn>1</mn> </msub> <msub> <mi>&Lambda;</mi> <msub> <mi>n</mi> <mn>1</mn> </msub> </msub> </mrow> <mo>&rsqb;</mo> </mrow> <mo>+</mo> <mi>&sigma;</mi> <mi>B</mi> <mrow> <mo>(</mo> <mi>v</mi> <mo>)</mo> </mrow> <mo>+</mo> <mi>&epsiv;</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </mtd> <mtd> <mrow> <msub> <mi>t</mi> <msub> <mi>n</mi> <mn>1</mn> </msub> </msub> <mo><</mo> <mi>t</mi> <mo>&le;</mo> <msub> <mi>t</mi> <msub> <mi>n</mi> <mn>2</mn> </msub> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mtable> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> </mtable> </mtd> <mtd> <mtable> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> </mtable> </mtd> </mtr> <mtr> <mtd> <mrow> <mi>a</mi> <mo>+</mo> <mi>b</mi> <mrow> <mo>&lsqb;</mo> <mrow> <msub> <mi>&lambda;</mi> <mi>L</mi> </msub> <mrow> <mo>(</mo> <mrow> <mi>&Lambda;</mi> <mo>-</mo> <msub> <mi>&Lambda;</mi> <msub> <mi>n</mi> <mrow> <mi>L</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> </msub> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <munderover> <mi>&Sigma;</mi> <mrow> <mi>l</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow> <mi>L</mi> <mo>-</mo> <mn>1</mn> </mrow> </munderover> <msub> <mi>&lambda;</mi> <mi>l</mi> </msub> <msub> <mi>&Delta;&Lambda;</mi> <msub> <mi>n</mi> <mi>l</mi> </msub> </msub> </mrow> <mo>&rsqb;</mo> </mrow> <mo>+</mo> <mi>&sigma;</mi> <mi>B</mi> <mrow> <mo>(</mo> <mi>v</mi> <mo>)</mo> </mrow> <mo>+</mo> <mi>&epsiv;</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </mtd> <mtd> <mrow> <msub> <mi>t</mi> <msub> <mi>n</mi> <mrow> <mi>L</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> </msub> <mo><</mo> <mi>t</mi> <mo>&le;</mo> <msub> <mi>t</mi> <mi>n</mi> </msub> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>,</mo> </mrow>
wherein Y (t) represents the degradation process of the product performance measured, and the initial performance degradation amount a and the degradation rate β are respectivelyAndε (t) represents a measurement error term and hasσeIs the standard deviation of the error term ε (t), an
<mrow> <mi>C</mi> <mi>o</mi> <mi>v</mi> <mo>{</mo> <mi>&epsiv;</mi> <mrow> <mo>(</mo> <msub> <mi>t</mi> <mi>i</mi> </msub> <mo>)</mo> </mrow> <mo>,</mo> <mi>&epsiv;</mi> <mrow> <mo>(</mo> <msub> <mi>t</mi> <mi>j</mi> </msub> <mo>)</mo> </mrow> <mo>}</mo> <mo>=</mo> <mfenced open = "{" close = ""> <mtable> <mtr> <mtd> <msubsup> <mi>&sigma;</mi> <mi>e</mi> <mn>2</mn> </msubsup> </mtd> <mtd> <mrow> <mi>i</mi> <mo>=</mo> <mi>j</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mrow> <mi>i</mi> <mo>&NotEqual;</mo> <mi>j</mi> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>,</mo> </mrow>
Wherein Cov (·) is covariance, and random terms a, B (ν), and ∈ (t) are independent of each other.
6. The method for analyzing reliability of degradation of stepping stress acceleration performance considering measurement errors according to claim 1 or 5, wherein the step of estimating the unknown parameters in the model Y (t) of degradation of stepping stress acceleration performance considering measurement errors comprises the following steps:
step 1: defining an unknown parameter matrix Θ (μ ═ for a step stress acceleration performance degradation model y (t) considering measurement errors1,μ2,σ1,σ2,θ,σ,η,γ,σe) (ii) a Step stress acceleration performance degradation test data yiSubject to multivariate normal distribution, yi～MN(μ11i+μ2τi,Σi) And MN (. cndot.) represents a multivariate normal distribution,is a covariance matrix of the i-th sample performance degradation data; 1iIs niLColumn vector with all dimension elements 1, 1i＝(1,1,...,1)T；And vij＝ν(tij；γ)；IiIs niLAn identity matrix of dimensions; matrix arrayElement (2) of (1)ijThe expression of (a) is:
<mrow> <msub> <mi>&tau;</mi> <mrow> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>=</mo> <mfenced open = "{" close = ""> <mtable> <mtr> <mtd> <mrow> <msub> <mi>&lambda;</mi> <mn>1</mn> </msub> <msub> <mi>&Lambda;</mi> <mrow> <mi>i</mi> <mi>j</mi> </mrow> </msub> </mrow> </mtd> <mtd> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>...</mn> <mo>,</mo> <mi>m</mi> <mo>;</mo> <mi>j</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>...</mn> <mo>,</mo> <msub> <mi>n</mi> <mrow> <mi>i</mi> <mn>1</mn> </mrow> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>&lambda;</mi> <mn>2</mn> </msub> <mrow> <mo>(</mo> <mrow> <msub> <mi>&Lambda;</mi> <mrow> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>-</mo> <msub> <mi>&Lambda;</mi> <mrow> <msub> <mi>in</mi> <mn>1</mn> </msub> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <msub> <mi>&lambda;</mi> <mn>1</mn> </msub> <msub> <mi>&Lambda;</mi> <mrow> <msub> <mi>in</mi> <mn>1</mn> </msub> </mrow> </msub> </mrow> </mtd> <mtd> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>...</mn> <mo>,</mo> <mi>m</mi> <mo>;</mo> <mi>j</mi> <mo>=</mo> <msub> <mi>n</mi> <mrow> <mi>i</mi> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mn>1</mn> <mo>,</mo> <msub> <mi>n</mi> <mrow> <mi>i</mi> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mn>2</mn> <mo>,</mo> <mn>...</mn> <mo>,</mo> <msub> <mi>n</mi> <mrow> <mi>i</mi> <mn>2</mn> </mrow> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mtable> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> </mtable> </mtd> <mtd> <mtable> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> </mtable> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>&lambda;</mi> <mi>L</mi> </msub> <mrow> <mo>(</mo> <mrow> <msub> <mi>&Lambda;</mi> <mrow> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>-</mo> <msub> <mi>&Lambda;</mi> <msub> <mi>n</mi> <mrow> <mi>i</mi> <mrow> <mo>(</mo> <mrow> <mi>L</mi> <mo>-</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> </mrow> </msub> </msub> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <munderover> <mi>&Sigma;</mi> <mrow> <mi>l</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow> <mi>L</mi> <mo>-</mo> <mn>1</mn> </mrow> </munderover> <msub> <mi>&lambda;</mi> <mi>l</mi> </msub> <msub> <mi>&Delta;&Lambda;</mi> <mrow> <msub> <mi>in</mi> <mi>l</mi> </msub> </mrow> </msub> </mrow> </mtd> <mtd> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>...</mn> <mo>,</mo> <mi>m</mi> <mo>;</mo> <mi>j</mi> <mo>=</mo> <msub> <mi>n</mi> <mrow> <mi>i</mi> <mrow> <mo>(</mo> <mrow> <mi>L</mi> <mo>-</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> </mrow> </msub> <mo>+</mo> <mn>1</mn> <mo>,</mo> <msub> <mi>n</mi> <mrow> <mi>i</mi> <mrow> <mo>(</mo> <mrow> <mi>L</mi> <mo>-</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> </mrow> </msub> <mo>+</mo> <mn>2</mn> <mo>,</mo> <mn>...</mn> <mo>,</mo> <msub> <mi>n</mi> <mrow> <mi>i</mi> <mi>L</mi> </mrow> </msub> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>,</mo> </mrow>
in the formula, Λij＝Λ(tij；η)，m is the number of samples put into the step acceleration performance degradation test;
step 2: solving the log-likelihood function of the step stress accelerated performance degradation model Y (t):
the likelihood function of the step stress acceleration performance degradation model is:
<mrow> <mi>L</mi> <mrow> <mo>(</mo> <mi>&Theta;</mi> <mo>)</mo> </mrow> <mo>=</mo> <munderover> <mo>&Pi;</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>m</mi> </munderover> <mo>{</mo> <msup> <mrow> <mo>(</mo> <mn>2</mn> <mi>&pi;</mi> <mo>)</mo> </mrow> <mrow> <mo>-</mo> <mfrac> <msub> <mi>n</mi> <mrow> <mi>i</mi> <mi>L</mi> </mrow> </msub> <mn>2</mn> </mfrac> </mrow> </msup> <mo>|</mo> <msub> <mi>&Sigma;</mi> <mi>i</mi> </msub> <msup> <mo>|</mo> <mrow> <mo>-</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </msup> <mi>exp</mi> <mo>&lsqb;</mo> <mo>-</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <msup> <mrow> <mo>(</mo> <msub> <mi>y</mi> <mi>i</mi> </msub> <mo>-</mo> <msub> <mi>&mu;</mi> <mn>1</mn> </msub> <msub> <mn>1</mn> <mi>i</mi> </msub> <mo>-</mo> <msub> <mi>&mu;</mi> <mn>2</mn> </msub> <msub> <mi>&tau;</mi> <mi>i</mi> </msub> <mo>)</mo> </mrow> <mi>T</mi> </msup> <msubsup> <mi>&Sigma;</mi> <mi>i</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msubsup> <mrow> <mo>(</mo> <msub> <mi>y</mi> <mi>i</mi> </msub> <mo>-</mo> <msub> <mi>&mu;</mi> <mn>1</mn> </msub> <msub> <mn>1</mn> <mi>i</mi> </msub> <mo>-</mo> <msub> <mi>&mu;</mi> <mn>2</mn> </msub> <msub> <mi>&tau;</mi> <mi>i</mi> </msub> <mo>)</mo> </mrow> <mo>&rsqb;</mo> <mo>}</mo> <mo>,</mo> </mrow>
the log likelihood function of the step stress acceleration performance degradation model obtained by simultaneously taking logarithms on the two sides of the formula is
<mrow> <mi>l</mi> <mrow> <mo>(</mo> <mi>&Theta;</mi> <mo>)</mo> </mrow> <mo>=</mo> <mo>-</mo> <mfrac> <mi>N</mi> <mn>2</mn> </mfrac> <mi>l</mi> <mi>n</mi> <mrow> <mo>(</mo> <mn>2</mn> <mi>&pi;</mi> <mo>)</mo> </mrow> <mo>-</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <munderover> <mo>&Sigma;</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>m</mi> </munderover> <mi>l</mi> <mi>n</mi> <mo>|</mo> <msub> <mi>&Sigma;</mi> <mi>i</mi> </msub> <mo>|</mo> <mo>-</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <munderover> <mo>&Sigma;</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>m</mi> </munderover> <msup> <mrow> <mo>(</mo> <msub> <mi>y</mi> <mi>i</mi> </msub> <mo>-</mo> <msub> <mi>&mu;</mi> <mn>1</mn> </msub> <msub> <mn>1</mn> <mi>i</mi> </msub> <mo>-</mo> <msub> <mi>&mu;</mi> <mn>2</mn> </msub> <msub> <mi>&tau;</mi> <mi>i</mi> </msub> <mo>)</mo> </mrow> <mi>T</mi> </msup> <msubsup> <mi>&Sigma;</mi> <mi>i</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msubsup> <mrow> <mo>(</mo> <msub> <mi>y</mi> <mi>i</mi> </msub> <mo>-</mo> <msub> <mi>&mu;</mi> <mn>1</mn> </msub> <msub> <mn>1</mn> <mi>i</mi> </msub> <mo>-</mo> <msub> <mi>&mu;</mi> <mn>2</mn> </msub> <msub> <mi>&tau;</mi> <mi>i</mi> </msub> <mo>)</mo> </mrow> </mrow>
In the formula,
and step 3: re-parameterizing the logarithm likelihood function, and estimating unknown parameters in an unknown parameter matrix in a maximized mode through partial derivatives and a multi-dimensional search algorithm:
the parameterized log-likelihood function of the stepping stress acceleration performance degradation model is as follows:
<mrow> <mi>l</mi> <mrow> <mo>(</mo> <mi>&Theta;</mi> <mo>)</mo> </mrow> <mo>=</mo> <mo>-</mo> <mfrac> <mi>N</mi> <mn>2</mn> </mfrac> <mo>&lsqb;</mo> <mi>l</mi> <mi>n</mi> <mrow> <mo>(</mo> <mn>2</mn> <mi>&pi;</mi> <mo>)</mo> </mrow> <mo>+</mo> <mi>ln</mi> <mo>(</mo> <msubsup> <mi>&sigma;</mi> <mn>2</mn> <mn>2</mn> </msubsup> <mo>)</mo> <mo>&rsqb;</mo> <mo>-</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <munderover> <mo>&Sigma;</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>m</mi> </munderover> <mi>l</mi> <mi>n</mi> <mo>|</mo> <msub> <mi>&Psi;</mi> <mi>i</mi> </msub> <mo>|</mo> <mo>-</mo> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <msubsup> <mi>&sigma;</mi> <mn>2</mn> <mn>2</mn> </msubsup> </mrow> </mfrac> <munderover> <mo>&Sigma;</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>m</mi> </munderover> <msup> <mrow> <mo>(</mo> <msub> <mi>y</mi> <mi>i</mi> </msub> <mo>-</mo> <msub> <mi>&mu;</mi> <mn>1</mn> </msub> <msub> <mn>1</mn> <mi>i</mi> </msub> <mo>-</mo> <msub> <mi>&mu;</mi> <mn>2</mn> </msub> <msub> <mi>&tau;</mi> <mi>i</mi> </msub> <mo>)</mo> </mrow> <mi>T</mi> </msup> <msubsup> <mi>&Psi;</mi> <mi>i</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msubsup> <mrow> <mo>(</mo> <msub> <mi>y</mi> <mi>i</mi> </msub> <mo>-</mo> <msub> <mi>&mu;</mi> <mn>1</mn> </msub> <msub> <mn>1</mn> <mi>i</mi> </msub> <mo>-</mo> <msub> <mi>&mu;</mi> <mn>2</mn> </msub> <msub> <mi>&tau;</mi> <mi>i</mi> </msub> <mo>)</mo> </mrow> <mo>,</mo> </mrow>
wherein,and
the parameters mu are obtained from the above formula1And mu2First order partial derivatives and parameters thereofFirst order partial derivative of (1) can be obtained
<mrow> <mfrac> <mrow> <mo>&part;</mo> <mi>l</mi> <mrow> <mo>(</mo> <mi>&Theta;</mi> <mo>)</mo> </mrow> </mrow> <mrow> <mo>&part;</mo> <msub> <mi>&mu;</mi> <mn>1</mn> </msub> </mrow> </mfrac> <mo>=</mo> <mfrac> <mn>1</mn> <msubsup> <mi>&sigma;</mi> <mn>2</mn> <mn>2</mn> </msubsup> </mfrac> <munderover> <mo>&Sigma;</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>m</mi> </munderover> <msubsup> <mn>1</mn> <mi>i</mi> <mi>T</mi> </msubsup> <msubsup> <mi>&Psi;</mi> <mi>i</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msubsup> <mrow> <mo>(</mo> <msub> <mi>y</mi> <mi>i</mi> </msub> <mo>-</mo> <msub> <mi>&mu;</mi> <mn>1</mn> </msub> <msub> <mn>1</mn> <mi>i</mi> </msub> <mo>-</mo> <msub> <mi>&mu;</mi> <mn>2</mn> </msub> <msub> <mi>&tau;</mi> <mi>i</mi> </msub> <mo>)</mo> </mrow> </mrow>
<mrow> <mfrac> <mrow> <mo>&part;</mo> <mi>l</mi> <mrow> <mo>(</mo> <mi>&Theta;</mi> <mo>)</mo> </mrow> </mrow> <mrow> <mo>&part;</mo> <msub> <mi>&mu;</mi> <mn>2</mn> </msub> </mrow> </mfrac> <mo>=</mo> <mfrac> <mn>1</mn> <msubsup> <mi>&sigma;</mi> <mn>2</mn> <mn>2</mn> </msubsup> </mfrac> <munderover> <mo>&Sigma;</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>m</mi> </munderover> <msubsup> <mi>&tau;</mi> <mi>i</mi> <mi>T</mi> </msubsup> <msubsup> <mi>&Psi;</mi> <mi>i</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msubsup> <mrow> <mo>(</mo> <msub> <mi>y</mi> <mi>i</mi> </msub> <mo>-</mo> <msub> <mi>&mu;</mi> <mn>1</mn> </msub> <msub> <mn>1</mn> <mi>i</mi> </msub> <mo>-</mo> <msub> <mi>&mu;</mi> <mn>2</mn> </msub> <msub> <mi>&tau;</mi> <mi>i</mi> </msub> <mo>)</mo> </mrow> </mrow>
<mrow> <mfrac> <mrow> <mo>&part;</mo> <mi>l</mi> <mrow> <mo>(</mo> <mi>&Theta;</mi> <mo>)</mo> </mrow> </mrow> <mrow> <mo>&part;</mo> <msubsup> <mi>&sigma;</mi> <mn>2</mn> <mn>2</mn> </msubsup> </mrow> </mfrac> <mo>=</mo> <mo>-</mo> <mfrac> <mi>N</mi> <mrow> <mn>2</mn> <msubsup> <mi>&sigma;</mi> <mn>2</mn> <mn>2</mn> </msubsup> </mrow> </mfrac> <mo>+</mo> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <msubsup> <mi>&sigma;</mi> <mn>2</mn> <mn>4</mn> </msubsup> </mrow> </mfrac> <munderover> <mo>&Sigma;</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>m</mi> </munderover> <msup> <mrow> <mo>(</mo> <msub> <mi>y</mi> <mi>i</mi> </msub> <mo>-</mo> <msub> <mi>&mu;</mi> <mn>1</mn> </msub> <msub> <mn>1</mn> <mi>i</mi> </msub> <mo>-</mo> <msub> <mi>&mu;</mi> <mn>2</mn> </msub> <msub> <mi>&tau;</mi> <mi>i</mi> </msub> <mo>)</mo> </mrow> <mi>T</mi> </msup> <msubsup> <mi>&Psi;</mi> <mi>i</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msubsup> <mrow> <mo>(</mo> <msub> <mi>y</mi> <mi>i</mi> </msub> <mo>-</mo> <msub> <mi>&mu;</mi> <mn>1</mn> </msub> <msub> <mn>1</mn> <mi>i</mi> </msub> <mo>-</mo> <msub> <mi>&mu;</mi> <mn>2</mn> </msub> <msub> <mi>&tau;</mi> <mi>i</mi> </msub> <mo>)</mo> </mrow> </mrow>
Let the above three equations equal 0 to obtain the parameter μ1、μ2Andrespectively of
<mrow> <msub> <mover> <mi>&mu;</mi> <mo>^</mo> </mover> <mn>1</mn> </msub> <mo>=</mo> <mfrac> <mrow> <msub> <mi>C</mi> <mn>2</mn> </msub> <msub> <mi>B</mi> <mn>1</mn> </msub> <mo>-</mo> <msub> <mi>C</mi> <mn>1</mn> </msub> <msub> <mi>B</mi> <mn>2</mn> </msub> </mrow> <mrow> <msub> <mi>B</mi> <mn>1</mn> </msub> <msub> <mi>A</mi> <mn>2</mn> </msub> <mo>-</mo> <msub> <mi>A</mi> <mn>1</mn> </msub> <msub> <mi>B</mi> <mn>2</mn> </msub> </mrow> </mfrac> </mrow>
<mrow> <msub> <mover> <mi>&mu;</mi> <mo>^</mo> </mover> <mn>2</mn> </msub> <mo>=</mo> <mfrac> <mrow> <msub> <mi>C</mi> <mn>1</mn> </msub> <msub> <mi>A</mi> <mn>2</mn> </msub> <mo>-</mo> <msub> <mi>A</mi> <mn>1</mn> </msub> <msub> <mi>C</mi> <mn>2</mn> </msub> </mrow> <mrow> <msub> <mi>B</mi> <mn>1</mn> </msub> <msub> <mi>A</mi> <mn>2</mn> </msub> <mo>-</mo> <msub> <mi>A</mi> <mn>1</mn> </msub> <msub> <mi>B</mi> <mn>2</mn> </msub> </mrow> </mfrac> </mrow>
<mrow> <msubsup> <mover> <mi>&sigma;</mi> <mo>^</mo> </mover> <mn>2</mn> <mn>2</mn> </msubsup> <mo>=</mo> <mfrac> <mn>1</mn> <mi>N</mi> </mfrac> <munderover> <mo>&Sigma;</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>m</mi> </munderover> <msup> <mrow> <mo>(</mo> <msub> <mi>y</mi> <mi>i</mi> </msub> <mo>-</mo> <msub> <mover> <mi>&mu;</mi> <mo>^</mo> </mover> <mn>1</mn> </msub> <msub> <mn>1</mn> <mi>i</mi> </msub> <mo>-</mo> <msub> <mover> <mi>&mu;</mi> <mo>^</mo> </mover> <mn>2</mn> </msub> <msub> <mi>&tau;</mi> <mi>i</mi> </msub> <mo>)</mo> </mrow> <mi>T</mi> </msup> <msubsup> <mi>&Psi;</mi> <mi>i</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msubsup> <mrow> <mo>(</mo> <msub> <mi>y</mi> <mi>i</mi> </msub> <mo>-</mo> <msub> <mover> <mi>&mu;</mi> <mo>^</mo> </mover> <mn>1</mn> </msub> <msub> <mn>1</mn> <mi>i</mi> </msub> <mo>-</mo> <msub> <mover> <mi>&mu;</mi> <mo>^</mo> </mover> <mn>2</mn> </msub> <msub> <mi>&tau;</mi> <mi>i</mi> </msub> <mo>)</mo> </mrow> </mrow>
In the formula,
will estimate the valueAndthe obtained information is brought into the parameterized log-likelihood function of the step stress acceleration performance degradation modelThe edge log likelihood function of (1):
<mrow> <mi>l</mi> <mrow> <mo>(</mo> <mover> <mi>&Theta;</mi> <mo>~</mo> </mover> <mo>)</mo> </mrow> <mo>=</mo> <mo>-</mo> <mfrac> <mi>N</mi> <mn>2</mn> </mfrac> <mo>&lsqb;</mo> <mi>l</mi> <mi>n</mi> <mrow> <mo>(</mo> <mn>2</mn> <mi>&pi;</mi> <mo>)</mo> </mrow> <mo>+</mo> <mn>1</mn> <mo>&rsqb;</mo> <mo>-</mo> <mfrac> <mi>N</mi> <mn>2</mn> </mfrac> <mi>l</mi> <mi>n</mi> <mrow> <mo>(</mo> <msubsup> <mover> <mi>&sigma;</mi> <mo>^</mo> </mover> <mn>2</mn> <mn>2</mn> </msubsup> <mo>)</mo> </mrow> <mo>-</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <munderover> <mo>&Sigma;</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>m</mi> </munderover> <mi>l</mi> <mi>n</mi> <mo>|</mo> <msub> <mi>&Psi;</mi> <mi>i</mi> </msub> <mo>|</mo> <mo>,</mo> </mrow>
maximization using a multi-dimensional search algorithmMaximum likelihood estimation can be obtainedAndthen sigma1、σeMaximum likelihood estimates of sum σ are respectivelyAndresulting in a maximum likelihood estimate of the parameter vector theta.
7. The method for analyzing reliability of degradation of stepping stress acceleration performance considering measurement error according to claim 6, wherein the reliability evaluation in the fourth step is as follows:
the life T of the product is: t ═ inf { T | x (T) ≧ Df|X(0)＜DfIn which D isfA performance degradation failure threshold for the product;
the conditional probability density distribution function for product life T, given an initial amount of performance degradation a, is:
<mrow> <msub> <mi>f</mi> <mi>T</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>|</mo> <mi>a</mi> <mo>)</mo> </mrow> <mo>&cong;</mo> <mfrac> <mn>1</mn> <mi>A</mi> </mfrac> <mi>g</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>|</mo> <mi>a</mi> <mo>)</mo> </mrow> <mo>;</mo> </mrow>
wherein,and
<mrow> <mi>g</mi> <mrow> <mo>(</mo> <mrow> <mi>t</mi> <mo>|</mo> <mi>a</mi> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mfrac> <mn>1</mn> <mrow> <mi>&tau;</mi> <msqrt> <mrow> <mn>2</mn> <mi>&pi;</mi> <mrow> <mo>&lsqb;</mo> <mrow> <msup> <mi>&sigma;</mi> <mn>2</mn> </msup> <mi>&tau;</mi> <mo>+</mo> <msubsup> <mi>&sigma;</mi> <mi>&beta;</mi> <mn>2</mn> </msubsup> <msup> <mi>&Lambda;</mi> <mn>2</mn> </msup> </mrow> <mo>&rsqb;</mo> </mrow> </mrow> </msqrt> </mrow> </mfrac> <mfrac> <mrow> <mi>d</mi> <mi>&tau;</mi> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> <mi>exp</mi> <mrow> <mo>{</mo> <mrow> <mo>-</mo> <mfrac> <msup> <mrow> <mo>&lsqb;</mo> <mrow> <msub> <mi>D</mi> <mi>f</mi> </msub> <mo>-</mo> <msub> <mi>&mu;</mi> <mi>&beta;</mi> </msub> <mi>&Lambda;</mi> </mrow> <mo>&rsqb;</mo> </mrow> <mn>2</mn> </msup> <mrow> <mn>2</mn> <mrow> <mo>&lsqb;</mo> <mrow> <msup> <mi>&sigma;</mi> <mn>2</mn> </msup> <mi>&tau;</mi> <mo>+</mo> <msubsup> <mi>&sigma;</mi> <mi>&beta;</mi> <mn>2</mn> </msubsup> <msup> <mi>&Lambda;</mi> <mn>2</mn> </msup> </mrow> <mo>&rsqb;</mo> </mrow> </mrow> </mfrac> </mrow> <mo>}</mo> </mrow> <mrow> <mo>{</mo> <mrow> <msub> <mi>D</mi> <mi>f</mi> </msub> <mo>-</mo> <mrow> <mo>&lsqb;</mo> <mrow> <mi>&Lambda;</mi> <mo>-</mo> <mi>h</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mi>&tau;</mi> </mrow> <mo>&rsqb;</mo> </mrow> <mfrac> <mrow> <msub> <mi>D</mi> <mi>f</mi> </msub> <msubsup> <mi>&Lambda;&sigma;</mi> <mi>&beta;</mi> <mn>2</mn> </msubsup> <mo>+</mo> <msub> <mi>&mu;</mi> <mi>&beta;</mi> </msub> <msup> <mi>&sigma;</mi> <mn>2</mn> </msup> <mi>&tau;</mi> </mrow> <mrow> <msup> <mi>&sigma;</mi> <mn>2</mn> </msup> <mi>&tau;</mi> <mo>+</mo> <msubsup> <mi>&sigma;</mi> <mi>&beta;</mi> <mn>2</mn> </msubsup> <msup> <mi>&Lambda;</mi> <mn>2</mn> </msup> </mrow> </mfrac> </mrow> <mo>}</mo> </mrow> <mo>;</mo> </mrow>
according to the total probability principle, the probability density distribution function of the service life T is obtained as follows:
<mrow> <msub> <mi>f</mi> <mi>T</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>=</mo> <msubsup> <mo>&Integral;</mo> <mrow> <mo>-</mo> <mi>&infin;</mi> </mrow> <mrow> <mo>+</mo> <mi>&infin;</mi> </mrow> </msubsup> <msub> <mi>f</mi> <mi>T</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>|</mo> <mi>a</mi> <mo>)</mo> </mrow> <mi>f</mi> <mrow> <mo>(</mo> <mi>a</mi> <mo>)</mo> </mrow> <mi>d</mi> <mi>a</mi> </mrow>
wherein,
at the working stress level S0The probability density distribution function for lifetime T is:
<mrow> <msub> <mi>f</mi> <mi>T</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>=</mo> <mfrac> <mn>1</mn> <msup> <mi>A</mi> <mo>&prime;</mo> </msup> </mfrac> <mi>g</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow>
wherein, μβ＝μ2λ(S0(ii) a Theta) and sigmaβ＝σ2λ(S0；θ)；
By means of numerical analysis, the working stress level S is obtained0The life distribution function for the lower product life T is:
<mrow> <msub> <mi>F</mi> <mi>T</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>=</mo> <mi>P</mi> <mo>{</mo> <mi>T</mi> <mo>&le;</mo> <mi>t</mi> <mo>}</mo> <mo>=</mo> <msubsup> <mo>&Integral;</mo> <mn>0</mn> <mi>t</mi> </msubsup> <msub> <mi>f</mi> <mi>T</mi> </msub> <mrow> <mo>(</mo> <mi>u</mi> <mo>)</mo> </mrow> <mi>d</mi> <mi>u</mi> <mo>;</mo> </mrow>
thus, the working stress level S is obtained0The reliability function of the product at a given time t:
<mrow> <mi>R</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>=</mo> <mn>1</mn> <mo>-</mo> <msub> <mi>F</mi> <mi>T</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>=</mo> <mn>1</mn> <mo>-</mo> <msubsup> <mo>&Integral;</mo> <mn>0</mn> <mi>t</mi> </msubsup> <msub> <mi>f</mi> <mi>T</mi> </msub> <mrow> <mo>(</mo> <mi>u</mi> <mo>)</mo> </mrow> <mi>d</mi> <mi>u</mi> <mo>.</mo> </mrow>
8. the method for analyzing reliability of degradation of stepping stress acceleration performance considering measurement error as claimed in claim 7, wherein the method for predicting life in the fourth step is:
at the working stress level S0Lower, average life of the productThe approximation is:
<mrow> <mover> <mi>t</mi> <mo>&OverBar;</mo> </mover> <mo>=</mo> <msup> <mi>&Lambda;</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mrow> <mo>(</mo> <mfrac> <mrow> <msub> <mi>D</mi> <mi>f</mi> </msub> <mo>-</mo> <msub> <mi>&mu;</mi> <mn>1</mn> </msub> </mrow> <mrow> <msub> <mi>&mu;</mi> <mn>2</mn> </msub> <mi>&lambda;</mi> <mrow> <mo>(</mo> <msub> <mi>S</mi> <mn>0</mn> </msub> <mo>;</mo> <mi>&theta;</mi> <mo>)</mo> </mrow> </mrow> </mfrac> <mo>)</mo> </mrow> <mo>,</mo> </mrow>
wherein, Λ-1(. h) is the inverse of the function Λ (·); solving by numerical analysis:
FT(tR)＝1-R，
the reliable life t with the reliability R can be obtainedRAn estimate of (d).
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| CN113094923B (en) * | 2021-04-22 | 2022-04-12 | 哈尔滨工业大学 | Multi-parameter related degradation product reliability evaluation method considering multi-source uncertainty |
| CN113094923A (en) * | 2021-04-22 | 2021-07-09 | 哈尔滨工业大学 | Multi-parameter related degradation product reliability evaluation method considering multi-source uncertainty |
| CN113486455A (en) * | 2021-04-29 | 2021-10-08 | 河南科技大学 | Self-lubricating joint bearing accelerated degradation reliability assessment and service life prediction method |
| CN114626248A (en) * | 2022-03-30 | 2022-06-14 | 北京航空航天大学 | Spiral spring reliability evaluation method based on multi-stress accelerated degradation data |
| CN114626248B (en) * | 2022-03-30 | 2023-04-18 | 北京航空航天大学 | Spiral spring reliability evaluation method based on multi-stress accelerated degradation data |
| CN115828648A (en) * | 2023-02-15 | 2023-03-21 | 中国科学技术大学 | Method and device for predicting service life of electronic packaging interconnection structure after fire |
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Application publication date: 20180406 Assignee: Luoyang Tejia Bearing Co.,Ltd. Assignor: HENAN University OF SCIENCE AND TECHNOLOGY Contract record no.: X2021980005473 Denomination of invention: Reliability analysis method of step stress accelerated performance degradation considering measurement error Granted publication date: 20201030 License type: Common License Record date: 20210629 |