CN112446139A - Accelerated test profile optimization method and system, electromechanical product, medium and terminal - Google Patents
Accelerated test profile optimization method and system, electromechanical product, medium and terminal Download PDFInfo
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Abstract
The invention belongs to the technical field of reliability tests, and discloses an accelerated test profile optimization method, a system, an electromechanical product, a medium and a terminal, wherein an accumulated failure model is constructed according to failure distribution and an accelerated model of the electromechanical product; simulating failure data by adopting a Monte Carlo method; estimating model parameters by adopting a maximum likelihood method; the asymptotic variance of the life estimate of the product at normal stress levels is minimized as an optimization criterion. The method is based on three-parameter index-Weibull distribution, uses temperature as the accelerated test stress of the electromechanical product, adopts a stepping mode as the accelerated test stress loading mode of the electromechanical product, and uses the gradual variance minimization of the service life estimation value of the product under the normal stress level as the optimization criterion, thereby solving the problems of long test time and low service life estimation precision of the electromechanical product in the accelerated service life test process.
Description
Technical Field
The invention belongs to the technical field of reliability tests, and particularly relates to an accelerated test profile optimization method, an accelerated test profile optimization system, an electromechanical product, a medium and a terminal.
Background
At present, the basic principle of the accelerated life test is to use higher environmental stress to quickly expose a product to a fault state on the basis of reasonable engineering experience and statistical hypothesis, and further use quickly obtained product fault information to perform reliability evaluation. In the reliability technology, the accelerated life test has the obvious advantage of high test efficiency on products with high reliability and long service life, so that the test time is greatly reduced, the test cost is greatly reduced, and the accelerated life test is widely applied to the fields of new material development, electromechanical equipment development, aerospace research and the like due to the huge advantage.
In order to achieve the goal of accurately evaluating the reliability of a product with high reliability and long service life in a short time, it is necessary to research a more effective accelerated life test profile so as to reduce the test cost (such as test time, cost, etc.) as much as possible while allowing the evaluation accuracy range.
The accelerated life test is generally divided into three basic types, namely a constant stress accelerated life test, a stepping stress accelerated life test and a sequential stress accelerated life test according to different stress applying modes. The method for processing the sequence acceleration test data has high difficulty, relatively few currently obtained research results are obtained, and the method cannot be used in a mature way. Therefore, the constant acceleration test and the step acceleration test are widely applied, but the step acceleration test has lower requirements on test time and sample quantity, has higher test efficiency, and becomes the application trend of the product acceleration test.
On the method for constructing the estimator, the method for estimating the model parameters comprises moment method estimation, least square estimation, Bayesian estimation and maximum likelihood estimation. In general, moment estimators are not unique, and least square estimation is mainly applied to the parameter estimation problem in linear statistical models, so that the method is less applied to accelerated life tests. Bayesian estimation first determines a criterion for goodness and also verifies the goodness of the estimator. The maximum likelihood estimation method is robust and requires a small amount of calculation while maintaining estimation accuracy.
At present, reliability evaluation of electromechanical products is mainly realized by means of an accelerated life test under single-level stress, the accelerated life test time is long, but the accelerated life test under the single-level stress cannot meet the requirements of life and reliability index verification of aerospace electromechanical products due to the fact that the aerospace electromechanical products have the characteristics of long service life and complex environmental stress, and multi-level stress influence factors cannot be combined to be considered.
Through the above analysis, the problems and defects of the prior art are as follows:
(1) in the prior art, an electromechanical product has low test efficiency and poor service life estimation precision in an accelerated service life test process, and the service life estimation precision of the product cannot be ensured within an allowable error range while the accelerated test efficiency is improved.
(2) The existing electromechanical products aiming at the accelerated life test mainly aim at common electromechanical products, and electromechanical products with high reliability and long service life need to control the stress condition of the test so as to ensure the accuracy and the effectiveness of the test.
(3) The existing electromechanical product accelerated life test cannot guide the stress conversion time in the test process, and the influence of the stress conversion time on the test result is not considered.
The difficulty in solving the above problems and defects is: the electromechanical products, especially aerospace electromechanical products, have the characteristics of high reliability and long service life due to the particularity of the application scene, so that the efficiency is low when the electromechanical products are subjected to accelerated life tests, the test efficiency is only improved, and the inaccurate life evaluation is inevitably caused. The requirement for changing stress conditions in the same test is high, and the influences of time factors, environmental factors and product performance threshold values need to be considered. Meanwhile, for aerospace electromechanical products, the effectiveness of each test needs to be ensured in consideration of the production cost and the test cost of the aerospace electromechanical products.
The significance of solving the problems and the defects is as follows: meanwhile, the test efficiency and the service life estimation precision of the accelerated service life test are ensured, the time cost can be ensured while the reliability index method of the electromechanical product is obtained, and the method has guiding significance for the improvement and design of the product. The stress condition is changed in the same test, the test efficiency of the accelerated life test can be effectively improved, the stress conversion time and the stress level are simultaneously used as design variables for testing, and the accuracy, the stability and the rapidity of the test can be effectively improved. The method takes an electromechanical product as a test object, simulates failure data through a Monte Carlo method, estimates model parameters through a maximum likelihood method, and performs optimization design of an accelerated life test section on the product by taking the gradual variance minimization of a life estimation value as an optimization criterion.
Disclosure of Invention
Aiming at the problems in the prior art, the invention provides an accelerated test profile optimization method, an accelerated test profile optimization system, an electromechanical product, a medium and a terminal. In particular to an accelerated test profile optimization design method based on three-parameter index-Weibull distribution.
The invention is realized in such a way that an accelerated test profile optimization method comprises the following steps:
1) building an accumulated failure model according to the failure distribution and the acceleration model of the electromechanical product;
2) simulating failure data by adopting a Monte Carlo method;
3) estimating model parameters by adopting a maximum likelihood method;
4) the asymptotic variance of the estimated lifetime is minimized as an optimization criterion.
Further, the step 1) comprises the following steps:
1-1) selecting three-parameter index-Weibull distribution which can well fit various types of test data and is widely applied to the reliability field as failure distribution of electromechanical products.
Using a three-parameter index-Weibull distribution as the failure distribution of the electromechanical product, wherein the distribution function of the three-parameter index-Weibull distribution is as follows:
in the formula: η is a scale parameter (η >0), m is a first shape parameter (β >0), and α is a second shape parameter (α > 0).
The probability density function of the three-parameter index-weibull distribution is:
1-2) the temperature with the greatest effect on promoting the failure mechanism was selected as the acceleration stress for the acceleration test.
Since the acceleration stress considered is only temperature, using the arrhenius acceleration model, it can be expressed as:
in the formula: η is a scale parameter (product characteristic lifetime); t is ambient temperature (. degree. C.); eaActivation energy (eV); k is Boltzmann's constant, and has a value of 8.617 × 10-5eV/K; a is a coefficient to be determined.
1-3) constructing an accumulative failure model by using a Nelson accumulative failure theory.
Nelson cumulative failure theory (i.e., the CE model) assumes that the remaining life of a product depends only on the portion that has accumulated failure at the time and the stress level at the time, regardless of the cumulative mode. This assumption is made by Nelson on the basis of physical principles. If the life distribution of the product is F (t), the assumed mathematical meaning is: a certain product at stress level SiThe lower working time is tiThen t isiCumulative probability of failure of inner product is Fi(ti) Equivalent to at stress level SjThe lower working time is tjCumulative probability of failure F of hour productj(tj) I.e. Fi(ti)＝Fj(tj). From this assumption, lifetime data at different stress levels can be translated.
In order to keep the failure mechanism of the product consistent under different stress levels, the shape parameters of the distribution function of the product under different stress levels are assumed to be kept consistentIs constant, i.e. the shape parameter m of the two-parameter weibull distribution remains constant. According to Nelson cumulative failure theory, from F1(t1)＝F2(τ1) Obtaining:
the above formula is simplified to obtain:
in the formula: tau is1Expressed at stress level S1Lower test t1Time conversion to stress level S2Cumulative equivalent test time of (c).
Further, the monte carlo simulation failure data in the step 2) comprises:
1-1) determining the distribution rule of failure model parameters and acceleration model parameters of the electromechanical product by using engineering experience and referring to manuals and documents;
1-2) assigning the failure model parameters of the electromechanical product according to the distribution rule of the failure model parameters and the acceleration model parameters to obtain an accumulated failure model (namely a CE model);
1-3) generating n random numbers uniformly distributed between (0,1), and then sequencing to obtain a cumulative failure probability sequence p of the electromechanical product;
1-4) generating a failure time sequence t of the electromechanical product by using an inverse function method according to the cumulative failure probability sequence p and the CE model.
Further, in the step 3), a maximum likelihood estimation method is selected for parameter estimation of the model.
Since the accelerated test generally has the situations of test truncation, uncertain failure modes and the like, the burst-type failure mode data is usually incomplete data. The Maximum Likelihood Estimation (MLE) is not only suitable for complete data, but also suitable for incomplete data, so that the MLE is mainly selected to carry out statistical analysis on the accelerated test burst type failure mode data.
Supposing a parameter theta distribution with probability density function of f (x; theta) to make the observed result be x1,x2,…,xn. For any continuous random variable, the probability of x in an interval around x, which is dx, is f (x; θ) dx. If x1,x2,…,xnIndependently, then the likelihood function is:
taking the maximum value of the likelihood function indicates that the parameter estimate at that time is more likely to result in the generation of observed data. Due to dx1,dx2,…,dxnThe product of (a) does not affect the result of the parameter estimation when the likelihood function takes the maximum value, so the likelihood function is often expressed as:
determining the optimal estimate of the parameter using the likelihood function may be accomplished by deriving a likelihood function of the observed value and obtaining a logarithmic form thereof, the logarithmic form being:
the logarithmic form is then given a partial derivative equal to 0, resulting in:
the maximum value of the likelihood function obtained by solving this system of equations is the optimal estimate of the parameter.
Further, said step 4) selects the progressive variance minimization of the estimated lifetime as an optimization criterion.
When an accelerated life test is performed on a product with the characteristics of high reliability and long service life, the gradual variance of the P-order fractional life MLE estimated value of the product under normal stress is usually selected as the reliability life characteristic quantity of the product. In order to improve the estimation precision of the characteristic quantity, the accelerated life test scheme is optimized by taking the minimum progressive variance of the MLE estimated value of the P-order fractional life of the product under normal stress as a criterion.
Generally, the maximum likelihood estimator has progressive unbiasedness and progressive normality, so that under a normal stress level, the progressive variance of the product P-order quantile life is obtained as follows:
AV(xP)＝H·Σ·HT＝H·F-1·HT
in the formula: x is the number ofPThe method is characterized in that the method is a P-order quantile life of an electromechanical product under a normal stress level, H is a first-order partial derivative of a P-order quantile life expression to parameters needing to be optimized, and sigma is a variance-covariance matrix of a model maximum likelihood estimator. The variance-covariance matrix Σ is generally difficult to obtain, and is reciprocal to the information matrix F according to the MLE estimation theory, that is, Σ F-1Therefore, the covariance matrix is usually obtained by inverting the information matrix
Another object of the present invention is to provide an accelerated test profile optimization system for implementing the accelerated test profile optimization method, the accelerated test profile optimization system including:
the accumulated failure model building module is used for building an accumulated failure model according to the failure distribution and the acceleration model of the electromechanical product;
the parameter simulation and estimation module is used for simulating failure data by adopting a Monte Carlo method and estimating model parameters by adopting a maximum likelihood method;
and the optimization criterion acquisition module is used for minimizing the gradual variance of the service life estimation value of the electromechanical product under the normal stress level as an optimization criterion.
Another object of the present invention is to provide an electromechanical product designed by implementing the accelerated test profile optimization method.
It is a further object of the invention to provide a computer arrangement comprising a memory and a processor, the memory storing a computer program which, when executed by the processor, causes the processor to perform the accelerated test profile optimization method.
It is another object of the present invention to provide a computer-readable storage medium storing a computer program which, when executed by a processor, causes the processor to perform the steps of: building an accumulated failure model according to the failure distribution and the acceleration model of the electromechanical product;
simulating failure data by adopting a Monte Carlo method;
estimating model parameters by adopting a maximum likelihood method;
minimizing the progressive variance of the life estimates of the electromechanical product at normal stress levels as an optimization criterion.
Another object of the present invention is to provide an electromechanical product information data processing terminal, which is configured to implement any one of the above accelerated test profile optimization methods.
By combining all the technical schemes, the invention has the advantages and positive effects that:
the acceleration profile optimization method provided by the invention can determine the acceleration stress level factor and the stress conversion time factor of the three-step acceleration test of the electromechanical product, achieves the purpose of optimizing the test scheme, can realize the reliability test and evaluation with high efficiency and high precision, and can save the test time and test samples.
The accumulated failure model constructed according to the failure distribution and acceleration model can well describe the failure process of the electromechanical product, and can objectively reflect the failure mechanism of the product under the influence of environmental factors.
The invention utilizes the Monte Carlo method to simulate failure data, can avoid complex modeling process, and can make the simulation data better close to the actual situation of the research object.
The method adopts the maximum likelihood method to estimate the model parameters, can effectively reduce the accumulated error caused by distribution estimation, improves the calculation accuracy, and can well solve the problem of information loss.
The invention selects the minimization of the gradual variance of the life estimation value as an optimization criterion, and can improve the estimation precision of the life as much as possible under the condition of meeting the requirement of reducing the number of test samples.
The invention adopts the step acceleration test to replace the constant acceleration test, can effectively reduce the number of test samples and reduce the test time, and further improves the efficiency of the acceleration test.
According to the failure distribution method, the three-parameter index-Weibull distribution is selected to replace the index distribution to serve as the failure distribution of the electromechanical product, so that the actual distribution condition of the electromechanical product can be better approached, and meanwhile, failure data can be better fitted.
The invention selects the temperature as the acceleration stress of the acceleration test, can well reflect the physical state and energy information of the electromechanical product, and can accelerate the failure speed of the product.
The invention selects the problems of the Nelson cumulative failure theory model in the aspect of statistical analysis, and can well reflect the relation between each stress level and the service life characteristic of the electromechanical product.
Compared with the test scheme before improvement, the test based on the invention reduces the precision of life estimation, improves the test efficiency, saves the test cost and greatly reduces the test samples.
Drawings
In order to more clearly illustrate the technical solutions of the embodiments of the present application, the drawings needed to be used in the embodiments of the present application will be briefly described below, and it is obvious that the drawings described below are only some embodiments of the present application, and it is obvious for those skilled in the art that other drawings can be obtained from the drawings without creative efforts.
Fig. 1 is a flowchart of an accelerated test profile optimization method based on a three-parameter index-weibull distribution according to an embodiment of the present invention.
Fig. 2 is a flowchart of an accelerated test profile optimization method based on a three-parameter index-weibull distribution according to an embodiment of the present invention.
FIG. 3 is a cross-section of a three-step accelerated test design for an electromechanical product provided in an embodiment of the present invention;
fig. 4 is a schematic process diagram for building a cumulative failure model according to an embodiment of the present invention.
Detailed Description
The three-parameter index-index Weibull distribution accelerated test profile optimization method provided by the invention can be applied to a multi-step accelerated life test and can be further applied to the optimization design of a multi-step stress level (namely the stress level number is more than 3) test scheme. In engineering practice, electromechanical products are under the action of various stresses (such as temperature, humidity, vibration) and the like, so that the method can be further applied to the optimized design based on the accelerated life test of the humidity stress, the vibration stress and the like.
In order to make the objects, technical solutions and advantages of the present invention more apparent, the present invention is further described in detail with reference to the following embodiments. It should be understood that the specific embodiments described herein are merely illustrative of the invention and are not intended to limit the invention.
Aiming at the problems in the prior art, the invention provides an accelerated test profile optimization method based on three-parameter index-Weibull distribution, and the invention is described in detail below with reference to the accompanying drawings.
As shown in fig. 1, the method for optimizing an accelerated test profile based on a three-parameter exponential-weibull distribution according to an embodiment of the present invention includes:
and S101, constructing an accumulative failure model according to the failure distribution and acceleration model of the electromechanical product.
And S102, simulating failure data by adopting a Monte Carlo method.
And S103, estimating model parameters by adopting a maximum likelihood method.
And S104, minimizing the gradual variance of the estimated service life as an optimization criterion.
Step S101 includes:
1-1) selecting a three-parameter index-Weibull distribution which can well fit various types of test data as failure distribution of the electromechanical product; 1-2) selecting the temperature with the maximum promotion effect on the failure mechanism as the acceleration stress of an acceleration test; 1-3) constructing an accumulative failure model by using a Nelson accumulative failure theory.
Using a three-parameter index-Weibull distribution as the failure distribution of the electromechanical product, wherein the distribution function of the three-parameter index-Weibull distribution is as follows:
where η is a scale parameter (η >0), m is a first shape parameter (β >0), and α is a second shape parameter (α > 0).
The probability density function of the three-parameter index-weibull distribution is:
since the acceleration stress considered is only temperature, using the arrhenius acceleration model, it can be expressed as:
in the formula: η is a scale parameter (product characteristic lifetime); t is ambient temperature (. degree. C.); eaActivation energy (eV); k is Boltzmann's constant, and has a value of 8.617 × 10-5eV/K; a is a coefficient to be determined.
Nelson cumulative failure theory (i.e., the CE model) assumes that the remaining life of a product depends only on the portion that has accumulated failure at the time and the stress level at the time, regardless of the cumulative mode. This assumption is made by Nelson on the basis of physical principles. If the life distribution of the product is F (t), the assumed mathematical meaning is: a certain product at stress level SiThe lower working time is tiThen t isiCumulative probability of failure of inner product is Fi(ti) Equivalent to at stress level SjThe lower working time is tjCumulative probability of failure F of hour productj(tj) I.e. Fi(ti)＝Fj(tj). From this assumption, lifetime data at different stress levels can be translated.
In order to keep the failure mechanism of the product consistent under different stress levels, the shape parameter of the distribution function of the product under different stress levels is assumed to be unchanged, namely the shape parameter m of the two-parameter Weibull distribution is kept unchanged. According to Nelson cumulative failure theory, from F1(t1)＝F2(τ1) Obtaining:
the above formula is simplified to obtain:
in the formula: tau is1Expressed at stress level S1Lower test t1Time conversion to stress level S2Cumulative equivalent test time of (c).
Step S102 includes: 1-1) determining the distribution rule of failure model parameters and acceleration model parameters of the electromechanical product by using engineering experience and referring to manuals and documents; 1-2) assigning the failure model parameters of the electromechanical product according to the distribution rule of the failure model parameters and the acceleration model parameters to obtain an accumulated failure model (namely a CE model); 1-3) generating n random numbers uniformly distributed between (0,1), and then sequencing to obtain a cumulative failure probability sequence p of the electromechanical product; 1-4) generating a failure time sequence t of the electromechanical product by using an inverse function method according to the cumulative failure probability sequence p and the CE model.
In step S103, a maximum likelihood estimation method is selected for parameter estimation of the model.
Since the accelerated test generally has the situations of test truncation, uncertain failure modes and the like, the burst-type failure mode data is usually incomplete data. The Maximum Likelihood Estimation (MLE) is not only suitable for complete data, but also suitable for incomplete data, so that the MLE is mainly selected to carry out statistical analysis on the accelerated test burst type failure mode data.
Supposing a parameter theta distribution with probability density function of f (x; theta) to make the observed result be x1,x2,…,xn. For any continuous random variable, the probability of x in an interval around x, which is dx, is f (x; θ) dx. If x1,x2,…,xnIndependently, then the likelihood function is:
taking the maximum value of the likelihood function indicates that the parameter estimate at that time is more likely to result in the generation of observed data. Due to dx1,dx2,…,dxnThe product of (a) does not affect the result of the parameter estimation when the likelihood function takes the maximum value, so the likelihood function is often expressed as:
determining the optimal estimate of the parameter using the likelihood function may be accomplished by deriving a likelihood function of the observed value and obtaining a logarithmic form thereof, the logarithmic form being:
the logarithmic form is then given a partial derivative equal to 0, resulting in:
the maximum value of the likelihood function obtained by solving this system of equations is the optimal estimate of the parameter.
In step S104, the progressive variance minimization of the estimated lifetime is selected as an optimization criterion.
When an accelerated life test is performed on a product with the characteristics of high reliability and long service life, the gradual variance of the P-order fractional life MLE estimated value of the product under normal stress is usually selected as the reliability life characteristic quantity of the product. In order to improve the estimation precision of the characteristic quantity, the accelerated life test scheme is optimized by taking the minimum progressive variance of the MLE estimated value of the P-order fractional life of the product under normal stress as a criterion.
Generally, the maximum likelihood estimator has progressive unbiasedness and progressive normality, so that under a normal stress level, the progressive variance of the product P-order quantile life is obtained as follows:
AV(xP)＝H·Σ·HT＝H·F-1·HT
in the formula: x is the number ofPThe method is characterized in that the method is a P-order quantile life of an electromechanical product under a normal stress level, H is a first-order partial derivative of a P-order quantile life expression to parameters needing to be optimized, and sigma is a variance-covariance matrix of a model maximum likelihood estimator. The variance-covariance matrix Σ is generally difficult to obtain, and is reciprocal to the information matrix F according to the MLE estimation theory, that is, Σ F-1Therefore, the covariance matrix is usually obtained by inverting the information matrix.
The invention is further described with reference to specific examples.
Examples
As shown in fig. 2, the present invention is specifically implemented by the following processes:
1) building an accumulated failure model according to the failure distribution and acceleration model of the electromechanical product:
1 determining accelerated test Profile and design variables
For the convenience of analysis, the invention optimally designs the three-step acceleration test profile. As shown in FIG. 3, SmIn order to not change the maximum acceleration stress of a failure mechanism, the maximum acceleration stress can be determined according to a small amount of test or simulation; t is tcThe tail-cutting time can be determined according to engineering experience or a few background tests; s1And S2To an accelerating stress level; t is t1And t2The stress conversion time; s0The normal working stress level of the product.
Due to more parameters, analysis is inconvenient. In order to simplify the calculation and make the relation model between the value of each parameter and the result of the optimization model more general, the model needs to be standardized. Order:
in summary, there are 4 design variables for the three-step acceleration test, which are the acceleration stress level factors u1And u2And a stress conversion time factor v1And v2. The corresponding constraint conditions are:
0＜u1＜u2＜1；0＜v1＜v2＜1
determining failure distribution
The service life models of common products mainly comprise exponential distribution, normal distribution and Weibull distribution, wherein the Weibull distribution model is widely applied, and the Weibull distribution model has changeable function forms and can well fit various types of data. However, the electromechanical device product has a complex structure, the failure rate function of the electromechanical device product is often a non-monotonic function, the failure rate functions of exponential distribution, normal distribution and Weibull distribution are monotonic functions, and the failure rate functions are not in accordance with the actual situation, so a second shape parameter is introduced into the two-parameter Weibull distribution function to form a three-parameter exponential-Weibull distribution function, so that the failure rate function of the electromechanical device product is a non-monotonic function, and the cumulative failure distribution function of the electromechanical device product is:
where η is a scale parameter (η >0), m is a first shape parameter (β >0), and α is a second shape parameter (α > 0).
The probability density function of the three-parameter index-weibull distribution is:
in order to keep the failure mechanism of the product consistent under different stress levels, it is assumed that the shape parameters of the distribution function, i.e. the shape parameters β and α of the three-parameter exponential-weibull distribution, remain unchanged under different stress levels.
Determining acceleration model
The essence of the accelerated life test is: the key to extrapolating the life characteristics of a product at normal stress levels when the life characteristics of the product are at high stress levels is to establish a mathematical relationship between the life characteristics and the stress levels, which is referred to as an acceleration model or acceleration equation. Common single stress acceleration models include an Arrhenius model, an inverse power law model, an Eying model, and a generalized Eying model. The temperature with the largest promotion effect on the failure mechanism is selected as the acceleration stress of the acceleration test, so that an Arrhenius acceleration model is adopted and can be expressed as follows:
in the formula: η is a scale parameter (product characteristic lifetime); t is ambient temperature (. degree. C.); eaActivation energy (eV); k is Boltzmann's constant, and has a value of 8.617 × 10-5eV/K; a is a coefficient to be determined.
Taking logarithm of two sides of the formula respectively to obtain:
and then ordering:
the acceleration model can then be converted into a log-linear relationship:
in the formula, gamma1And gamma2In order to determine the coefficient to be determined,
is a function related to temperature stress.
Fourthly, building an accumulative failure model
And constructing an accumulative failure model by using a Nelson accumulative failure theory. Nelson cumulative failure theory (i.e., the CE model) assumes that the remaining life of a product depends only on the portion that has accumulated failure at the time and the stress level at the time, regardless of the cumulative mode. This assumption is made by Nelson on the basis of physical principles. If the life distribution of the product is F (t), the assumed mathematical meaning is: a certain product at stress level SiThe lower working time is tiThen t isiCumulative probability of failure of inner product is Fi(ti) Equivalent to at stress level SjThe lower working time is tjCumulative probability of failure F of hour productj(tj) I.e. Fi(ti)＝Fj(tj). Based on this assumption, lifetime data at different stress levels can be translated, as shown in FIG. 4.
In order to keep the failure mechanism of the product consistent under different stress levels, the shape parameter of the distribution function of the product under different stress levels is assumed to be unchanged, namely the shape parameter m of the two-parameter Weibull distribution is kept unchanged. According to Nelson cumulative failure theory, from F1(t1)＝F2(τ1) Obtaining:
the above formula is simplified to obtain:
in the same way, from F2(t2-t1+τ1)＝F3(τ2) Obtaining:
in the formula: tau is1Expressed at stress level S1Lower test t1Time conversion to stress level S2Cumulative equivalent test time of2Expressed at stress level S2Lower test (t)2-t1+τ1) Time conversion to stress level SmCumulative equivalent test time of (c).
Thus, the cumulative distribution function of the test piece failure time X is:
the corresponding probability density function is:
in the formula:
2) simulation of failure data using a Monte Carlo method
Firstly, the distribution rule of failure model parameters and acceleration model parameters of the electromechanical product is determined by using engineering experience and referring to manuals and documents.
The accelerated stress is selected as the temperature stress, the level number of the stepping stress is 3, the normal temperature stress is 20 ℃, the maximum temperature stress is 90 ℃, and the test tail-ending time is 9 h. According to engineering experience and a thorough test, solving the true value of the model parameters as follows:
α＝2.587，β＝0.715，γ1＝9.993，γ2＝4.826
assigning the failure model parameters of the electromechanical product according to the distribution rule of the failure model parameters and the acceleration model parameters to obtain an accumulated failure model (namely, a CE model).
And thirdly, generating n random numbers which are uniformly distributed between (0,1), and then sequencing to obtain the cumulative failure probability sequence p of the electromechanical product.
And generating the failure time sequence of the electromechanical product by using an inverse function method according to the cumulative failure probability sequence p and the CE model.
3) Estimating model parameters using maximum likelihood method
Since the accelerated test generally has the situations of test truncation, uncertain failure modes and the like, the burst-type failure mode data is usually incomplete data. The Maximum Likelihood Estimation (MLE) is not only suitable for complete data, but also suitable for incomplete data, so that the MLE is mainly selected to carry out statistical analysis on the accelerated test burst type failure mode data.
Suppose that the ith sample is at time xiFailure, 3 indicator functions are defined:
the log-likelihood function for the ith sample is then:
Li＝I3{I2[I1lnf1(xi)+(1-I1)lnf2(xi)]+(1-I2)lnf3(xi)}+(1-I3)lnF3(tc)
when the number of samples put into the test is n, then the log-likelihood function for all samples is:
solving the system of first order partial derivatives of the log-likelihood function L:
because the equation set is a transcendental equation set and the expression of each equation is long through derivation calculation, a specific closed-form solution cannot be solved, and an intelligent optimization algorithm is adopted for solving.
4) The asymptotic variance of the estimated lifetime is minimized as an optimization criterion.
(ii) optimization criterion
When an accelerated life test is performed on a product with the characteristics of high reliability and long service life, the gradual variance of the P-order fractional life MLE estimated value of the product under normal stress is usually selected as the reliability life characteristic quantity of the product. In order to improve the estimation accuracy of the feature quantity, reduce the number of samples, and shorten the test time, it is necessary to optimize an accelerated life test scheme. The accelerated life test scheme is optimized by taking the minimum progressive variance of the MLE estimated value of the P-order fractional life of the product under normal stress as a criterion.
Lifetime progressive variance
The variance-covariance matrix of the model parameter estimates is:
the common covariance matrix is difficult to obtain, and the covariance matrix sigma andthe information matrix F is reciprocal, i.e. has ∑ F-1Therefore, the covariance matrix is usually obtained by inverting the information matrix. The information matrix can be obtained by the mathematical expectation of the log-likelihood function of n samples to the negative second order partial derivative matrix of each model parameter, and the method comprises the following steps:
wherein:
the scale parameter estimation value under the normal stress level can be obtained according to the acceleration model as follows:
the cumulative distribution function at normal stress levels is thus obtained as:
therefore, the P-order quantile life MLE estimated value of the product is as follows:
typically, the maximum likelihood estimator has progressive unbiased and progressive normality, so that it can be found at normal stress level S0The progressive variance of the log life of the P-th quantiles of the following products is:
Var(lnxP)＝H·V·HT＝H·F-1·HT
wherein:
problem optimization
By the above derivation, the design variable u alone can be obtained1、u2、v1And v2The associated progressive variance expression. The final attribute is the following nonlinear optimization problem:
Find u1,u2,v1,v2
min Var(lnxP)
s.t.0＜u1＜u2＜1
0＜v1＜v2＜1
through derivation calculation, elements in the information matrix F are too complex, and a specific definite integral expression cannot be solved, so that a numerical integration method is adopted for approximate solution. Considering that on one hand, the known integration interval is needed when numerical integration is used, on the other hand, a plurality of variable substitutions are involved in the whole derivation process, and meanwhile, constraint conditions exist, a nonlinear programming algorithm is used for carrying out optimization solution. While the classical nonlinear programming algorithm is mostly solved by adopting a gradient descent method, which needs to use an explicit function expression and is obviously not applicable, the method of random Monte-Carlo simulation is decided to be used for optimal solution under the comprehensive consideration, and the optimal variable u corresponding to the minimum value of the progressive variance is solved1 *、u2 *、v1 *And v2 *。
The invention is further described below in connection with simulation experiments.
Now the product is at normal stress level S0(T0＝20℃、V05Grms) of log median life estimates
Minimum as optimization criterion to accelerate stressHorizontal factor u1、u2And a stress conversion time factor v1、v2To design variables, a three-step accelerated test protocol was optimized.
The result of performing 1000 times of random optimization solution by adopting a Monte-Carlo simulation method is as follows:
u1 *＝0.377、u2 *＝0.727、v1 *＝0.102、v2 *＝0.213
the corresponding progressive variance minimum at this time is 1.511.
If the accelerated test scheme of the traditional uniform design is adopted:
u1 0＝1/3、u2 0＝2/3、v1 0＝1/3、v2 0＝2/3
the asymptotic variance obtained at this time was 2947.895.
Obviously, the progressive variance of the optimized life estimation value is smaller than that of the traditional uniform design, which shows that the optimized accelerated test scheme has higher estimation precision than that of the traditional uniform design.
The invention discloses an application software system of an accelerated test profile optimization method of an electromechanical product, which can calculate a model parameter, an accelerated stress level factor, a stress conversion time factor and a corresponding progressive variance minimum value of a three-parameter index-Weibull distribution by setting a normal temperature, a maximum temperature and a test tail-ending time.
In the reliability life test of an electromechanical product, it is difficult to determine a test stress to be applied in the test and an acting time thereof. The software system is mainly applied to the design of an accelerated life test of an electromechanical product, improves the reliability of the electromechanical product, enables the electromechanical product to be more suitable for the harsh and complex service environment, can determine parameters in the test, can improve the test efficiency and reduce the test cost. The method can help further analyze the action mechanism of the electromechanical product, and further provide technical support for development and use guarantee of the electromechanical product.
It should be noted that the embodiments of the present invention can be realized by hardware, software, or a combination of software and hardware. The hardware portion may be implemented using dedicated logic; the software portions may be stored in a memory and executed by a suitable instruction execution system, such as a microprocessor or specially designed hardware. Those skilled in the art will appreciate that the apparatus and methods described above may be implemented using computer executable instructions and/or embodied in processor control code, such code being provided on a carrier medium such as a disk, CD-or DVD-ROM, programmable memory such as read only memory (firmware), or a data carrier such as an optical or electronic signal carrier, for example. The apparatus and its modules of the present invention may be implemented by hardware circuits such as very large scale integrated circuits or gate arrays, semiconductors such as logic chips, transistors, or programmable hardware devices such as field programmable gate arrays, programmable logic devices, etc., or by software executed by various types of processors, or by a combination of hardware circuits and software, e.g., firmware.
The above description is only for the purpose of illustrating the present invention and the appended claims are not to be construed as limiting the scope of the invention, which is intended to cover all modifications, equivalents and improvements that are within the spirit and scope of the invention as defined by the appended claims.
Claims (10)
1. An accelerated test profile optimization method, characterized by comprising:
building an accumulated failure model according to the failure distribution and the acceleration model of the electromechanical product;
simulating failure data by adopting a Monte Carlo method;
estimating model parameters by adopting a maximum likelihood method;
minimizing the progressive variance of the life estimates of the electromechanical product at normal stress levels as an optimization criterion.
2. The accelerated test profile optimization method of claim 1, wherein the constructing the cumulative failure model based on the failure distribution and the acceleration model of the electromechanical product comprises:
1) selecting three-parameter index-Weibull distribution as failure distribution of electromechanical products: the distribution function of the three-parameter exponential-weibull distribution is:
in the formula: eta is a scale parameter, eta is greater than 0, m is a first shape parameter, beta is greater than 0, alpha is a second shape parameter, and alpha is greater than 0;
the probability density function of the three-parameter index-Weibull distribution is:
2) the temperature with the greatest effect on promoting the failure mechanism was selected as the acceleration stress of the acceleration test: using an arrhenius acceleration model, expressed as:
in the formula: eta is a scale parameter; t is the ambient temperature; eaTo activate energy; k is Boltzmann's constant, and has a value of 8.617 × 10-5eV/K; a is a coefficient to be determined;
3) and (3) constructing an accumulative failure model by using a Nelson accumulative failure theory:
according to Nelson cumulative failure theory, F1(t1)＝F2(τ1) Obtaining:
simplifying to obtain:
in the formula: tau is1Expressed at stress level S1Lower test t1Time conversion to stress level S2Cumulative equivalent test time of (c).
3. The method of claim 1, wherein simulating failure data using a monte carlo method comprises:
determining the distribution rule of failure model parameters and acceleration model parameters of the electromechanical product;
assigning the failure model parameters of the electromechanical product according to the distribution rule of the failure model parameters and the acceleration model parameters to obtain a cumulative failure model (CE model);
generating n random numbers uniformly distributed between (0,1), and sequencing to obtain a cumulative failure probability sequence p of the electromechanical product;
and generating a failure time sequence t of the electromechanical product by using an inverse function method according to the cumulative failure probability sequence p and the CE model.
4. An accelerated test profile optimization method according to claim 1, wherein said estimating model parameters using maximum likelihood comprises:
the probability density function is f (x; theta) parameter theta distribution, and the observation result is x1,x2,…,xn(ii) a For any continuous random variable, the probability of x in an interval around x, which is dx, is f (x; θ) dx. If x1,x2,…,xnIndependently, the likelihood function is:
taking the maximum value of the likelihood function to show that the parameter estimation value at the moment causes the generation of observation data; dx (x)1,dx2,…,dxnThe product of (a) does not affect the result of the parameter estimation when the likelihood function takes the maximum value, and the likelihood function is expressed as:
determining the optimal estimation of the parameters by using a likelihood function, and calculating the partial derivative by an expression and making the partial derivative equal to 0 by deducing the likelihood function of the observed value and obtaining a logarithm form; the maximum value of the likelihood function obtained by solving this system of equations is the optimal estimate of the parameter.
5. An accelerated test profile optimization method according to claim 1, wherein said selecting a progressive variance minimization of estimated lifetime as an optimization criterion comprises:
under the normal stress level, the progressive variance of the P-order quantile life of the electromechanical product is as follows:
AV(xP)＝H·Σ·HT＝H·F-1·HT
in the formula: x is the number ofPThe method comprises the steps of obtaining a P-order quantile life of an electromechanical product under a normal stress level, obtaining a first-order partial derivative of a P-order quantile life expression to parameters needing to be optimized, and obtaining a variance-covariance matrix of a model maximum likelihood estimator; according to the MLE estimation theory, the variance-covariance matrix Σ is reciprocal to the information matrix F, where Σ ═ F-1And solving the covariance matrix by inverting the information matrix.
6. An accelerated test profile optimization system for implementing the accelerated test profile optimization method according to any one of claims 1 to 5, the accelerated test profile optimization system comprising:
the accumulated failure model building module is used for building an accumulated failure model according to the failure distribution and the acceleration model of the electromechanical product;
the parameter simulation and estimation module is used for simulating failure data by adopting a Monte Carlo method and estimating model parameters by adopting a maximum likelihood method;
and the optimization criterion acquisition module is used for minimizing the gradual variance of the service life estimation value of the electromechanical product under the normal stress level as an optimization criterion.
7. An electromechanical product, which is designed by implementing the method for optimizing an accelerated test profile according to any one of claims 1 to 5.
8. A computer device comprising a memory and a processor, the memory storing a computer program that, when executed by the processor, causes the processor to perform the accelerated test profile optimization method of any one of claims 1 to 5.
9. A computer-readable storage medium storing a computer program which, when executed by a processor, causes the processor to perform the steps of: building an accumulated failure model according to the failure distribution and the acceleration model of the electromechanical product;
simulating failure data by adopting a Monte Carlo method;
estimating model parameters by adopting a maximum likelihood method;
minimizing the progressive variance of the life estimates of the electromechanical product at normal stress levels as an optimization criterion.
10. An electromechanical product information data processing terminal, characterized in that the electromechanical product information data processing terminal is used for implementing the accelerated test profile optimization method of any one of claims 1 to 5.
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