CN109522519B - Dependency evaluation method among multiple performance parameters of ammunition component - Google Patents
Dependency evaluation method among multiple performance parameters of ammunition component Download PDFInfo
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Abstract
The invention provides a method for evaluating the dependency among a plurality of performance parameters of an ammunition component, and provides a reliability model of double-parameter degradation on the basis of Sklar constant force. Further aiming at the model, a two-step method with higher efficiency is provided for estimating the unknown parameters in the model, firstly, modeling analysis is respectively carried out on the degradation processes of the two performance parameters, and the unknown parameters in the edge distribution of the two performance parameters are estimated; then, the edge distribution parameter estimation value is substituted into the overall maximum likelihood function. Through comparison and analysis with measured values stored in a warehouse, the model prediction precision of the parameter 1 calculated by the method reaches 85.92 percent, the model prediction precision of the parameter 2 reaches 88.60 percent, and the model prediction precision of the final double-parameter Copula function value reaches 93.75 percent.
Description
Technical Field
The invention relates to an ammunition component degradation process model.
Background
For products with complex failure mechanisms, there may be multiple performance parameters, and there may be dependencies between the performance parameters. These performance parameters also affect the reliability of the product, and any performance parameter value that meets or exceeds the design specification range will cause a failure. At this time, the influence of degradation of each performance parameter on the reliability of the product must be considered at the same time, and reasonable comprehensive evaluation is performed from multiple angles.
Before introducing Copula functions, it is necessary to explain the clear correlation and dependency differences. There are two words in English that describe the association of random variables: correction and dependency. The former corresponds to correlation and the latter corresponds to dependency.
Correlation is generally only for linear relationships, and when X, Y are uncorrelated, it does not necessarily mean X, Y are independent of each other. Pearson product-moment correlation coefficients are often used in statistics to measure the linear relationship of random variables X, Y:
where ρ is XY For Pearson product-moment correlation coefficients, E (-) and D (-) are the expectation and variance, respectively, of the random variable.
Pearson product-moment correlation coefficients can only be used to measure linear relationships, e.g. random variable X obeys a standard normal distribution, random variable Y = X 2 The scatter diagram of (X, Y) is shown in FIG. 1, it is obvious that X and Y have strong dependence, and the value of Y can be completely determined by X, but the correlation coefficient ρ XY But 0, the pearson product-moment correlation coefficient is said to have a defect in measuring the random variable correlation.
Dependence is an antisense word to Independence (Independence), and Dependence means that random variables X and Y have no Independence in probabilistic representation, and include both linear and nonlinear relationships, which is a more general description. In real life, nonlinear relations exist in large numbers, and therefore, the association relation between random variables is described by using "dependence" here.
The traditional modeling method based on multi-dimensional joint distribution (such as multi-dimensional normal distribution) depends on correlation coefficients, for two-dimensional normal random variables (X, Y), the correlation coefficients of X and Y are 0 and are equivalent to X and Y which are independent, and the characteristic causes that the modeling method is not suitable for occasions with nonlinear dependence. The Copula function-based method can process nonlinear dependent data, and provides a powerful tool for solving the problem.
Definition of two-dimensional Copula function and Sklar's theorem
Copula is originally intended as 'connection', and the modeling idea adopting a Copula function is to decompose joint distribution of random vectors into two parts, namely edge distribution and a Copula function, and connect the edge distribution through the Copula function to serve as joint distribution. This makes the modeling of the joint distribution more flexible and no longer restricted to existing classical joint distributions. The joint distribution of the random vectors (X, Y) is described previously, and the classical joint distribution such as two-dimensional normal distribution is generally adopted, the joint distribution is often obtained through a single distribution family structure, the structure of the distribution family is relatively simple, otherwise, the expression of the joint distribution is too complex. Furthermore, defining a joint distribution is equivalent to implicitly defining an edge distribution, which is often not reasonable. The joint distribution modeling method based on the Copula function can respectively specify the forms of edge distribution and joint structures, so that the model has stronger explanatory power.
The Copula function can be theoretically used for constructing the joint distribution of multi-dimensional random vectors, and only a two-dimensional Copula function is introduced due to some limitations of the Copula function in a high-dimensional situation. The definition and associated theorem of the two-dimensional Copula function is summarized as follows:
definitions 1A two-dimensional Copula function is defined as [0,1]] 2 The function C (u, v) above, which satisfies the following two conditions:
1 holds true for any u, v ∈ [0,1], C (u, 0) = C (0, v) =0, C (u, 1) = u, C (1, v) = v.
2 for arbitrary u 1 ≤u 2 ,v 1 ≤v 2 ，u 1 ,u 2 ,v 1 ,v 2 ∈[0,1]Is provided with
C(u 2 ,v 2 )-C(u 2 ,v 1 )-C(u 1 ,v 2 )+C(u 1 ,v 1 )≥0
H(x 1 ,x 2 )＝C(F 1 (x 1 ),F 2 (x 2 ))\*MERGEFORMAT(2)
The above is the Sklar theorem proposed by Sklar in 1959. The theorem proves the existence and uniqueness of the Copula function, gives sufficient necessary conditions for the existence of the Copula function, and illustrates that the Copula function is feasible to solve the dependency problem. Sklar's theorem indicates that any two-dimensional joint distribution can be decomposed into a Copula function and two corresponding edge distributions, which makes modeling of the two-dimensional distribution more flexible. The theorem lays a theoretical foundation for the application of the Copula function in the field of multivariate statistical analysis.
Common Copula function
Commonly used Copula functions are an elliptical Copula cluster function, a FGM (Farlie-Gumbel-morgenster) Copula cluster function, and an archimedes Copula cluster function. The Gaussian distribution is one of the elliptical distribution, and although the distribution has no closed representation, the multidimensional joint distribution is simpler to construct, so the method is widely applied to engineering. The biggest disadvantage of the ellipsoids is that the edge distributions are required to be all of the same ellipsoids, and to overcome this difficulty, ellipsoids Copula have been proposed. It has no strict requirement on edge distribution and can belong to different ellipsoids.
The Gaussian Copula function is defined as:
wherein, Σ represents a symmetric positive definite matrix in which the element on the diagonal is 1 and the other elements are ρ; phi ρ Representing a standard multivariate normal distribution cumulative distribution function with a correlation coefficient matrix of sigma; phi -1 (. Cndot.) represents the inverse of a standard normal distribution cumulative distribution function;
i denotes an identity matrix.
Disclosure of Invention
The invention aims to provide a method for evaluating the dependency among a plurality of performance parameters of an ammunition component, and in order to realize the purpose, the scheme is as follows:
the method is characterized in that:
establishing a Copula-based two-parameter degradation model under the premise of assuming that Copula parameters are constant, and providing a corresponding parameter estimation method:
assuming that a certain product has two performance characteristic parameters, the degradation process of each performance characteristic parameter can be described by a degradation track model or a random process model, and is marked as { X 1 (t),X 2 (t)}；
{X 1 (t),X 2 (t) a joint distribution function of
H(X 1 (t),X 2 (t))＝C(F 1 (X 1 (t)),F 2 (X 2 (t))\*MERGEFORMAT(5)
Wherein, F p (X p (t)) is X p (t), p =1,2;
for a product having two performance characteristic parameters, the reliability function is
R(t)＝C(R 1 (t),R 2 (t))\*MERGEFORMAT(6)
Wherein R is p (t) is the reliability function of the p-th performance parameter, p =1,2；
When any one of the performance parameter values reaches or exceeds a threshold value specified by the design, the product is caused to fail; failure threshold is marked as { D 1 ,D 2 }; thus, for a product having two performance characteristic parameters, the reliability is
R(t)＝P((X 1 (t),X 2 (t))∈S)＝P(X 1 (t)＜D 1 ,X 2 (t)＜D 2 )\*MERGEFORMAT(7)
In the formula, S represents a security domain, and the product works normally if and only if all performance parameters are located in the security domain S;
due to P (X) 1 (t)＜D 1 )＝F 1 (D 1 |t)，P(X 2 (t)＜D 2 )＝F 2 (D 2 I t); the formula (7) can be written as
R(t)＝P(X 1 (t)＜D 1 ,X 2 (t)＜D 2 )
＝C(F 1 (D 1 |t),F 2 (D 2 |t))\*MERGEFORMAT(8)
＝C(R 1 (t),R 2 (t))
In the formula F p (X p (t)) is X p (t), p =1,2; r p (t) is a reliability function of the pth performance parameter, p =1,2;
in particular, if the performance characteristic parameters are independent of each other, the reliability function in equation (8) can be simplified to
R(t)＝R 1 (t)R 2 (t)\*MERGEFORMAT(9)
Wherein R is p (t) is a marginal reliability function of the p-th performance characteristic parameter, p =1,2;
for two performance degradation processes { X 1 (t),X 2 (t) having its reliability function at time t of u = R, respectively 1 (t) and v = R 2 (t); then
(P1) for all
C(u,0)＝C(0,v)＝0；
(P2) for all
C(u,1)＝u，C(1,v)＝v；
(P3) for all
max(u+v-1,0)＜C(u,v)＜min(u,v)；
Property (P1) indicates that any one of the performance parameters exceeding the failure threshold will result in a product failure, and property (P2) indicates that if one of the performance parameters does not degrade, the reliability of the product will be determined entirely by the other performance parameter; property (P3) describes the Frechet-Hoeffding interval, R (t) = C (R) 1 (t),R 2 (t)) substituting the property (P3) to obtain a Frechet-Hoeffding interval as follows:
max(R 1 (t)+R 2 (t)-1,0)≤R(t)≤min(R 1 (t),R 2 (t))\*MERGEFORMAT(10)
the property shows that the reliability of the dual-parameter performance degradation failure type product is in the above formula interval;
the method for estimating the model parameters comprises the following steps:
the first step is as follows: solving edge distribution parameters
1) For the covariate model in the nonlinear drift wiener process, the parameters to be estimated and the connection function μ z In connection with, when the connection function is μ z ＝ae -b/z When the parameter to be solved is beta p ＝[a p ,b p ,σ p ]P =1,2; degradation increment x pkij Has a probability density function of
The log-likelihood function is
In the formula
Is the observed degradation increment, Δ (t) kij )＝t kij -t ki,j-1 Is the interval between two detections, tau kij ＝τ(t kij )-τ(t ki,j-1 ) Is the interval time after the conversion,
a probability density function that is a standard normal distribution;
2) For the covariate model in the inverse Gaussian process, the parameter to be estimated is beta p ＝[a p ,b p ,σ p ]P =1,2, wherein a p ,b p Is the connection function mu z The parameter (1) of (1); degradation increment x pkij Has a probability density function of
The log-likelihood function is
In the formula
Is the observed increment of degradation, τ kij ＝τ(t kij )-τ(t ki,j-1 ) Is the detection interval time after the conversion,
is about a p ,b p A function of (a);
the second step is that: solving Copula function parameters;
the first step has already solved two edgesParameters of the distribution
By substituting this into equation (32), a likelihood function for parameter α can be obtained:
in the formula
And
respectively a cumulative distribution function and a probability density function of the p-th edge distribution,
is a density function of C (·; α), x pkij Is the observed increment of degradation, p =1,2;
after omitting the constant term, equation (37) can be simplified as:
the estimated parameter value of the parameter α may be expressed as:
wherein Θ is the parameter space;
because the likelihood function only comprises one to two parameters, the maximum likelihood estimation value can be obtained by an optimization algorithm such as Newton-Raphson and the like.
Drawings
FIG. 1 shows a random variable X following a standard normal distribution with a random variable Y = X 2 (X, Y), wherein the correlation coefficient is 0, but X and Y are not independent;
FIG. 2 is an equivalent series model;
FIG. 3 is a parameter 1 degradation trace at 70 ℃;
FIG. 4 is a parameter 2 degradation trace at 70 ℃;
FIG. 5 is a parameter 1 degradation trace at 85 ℃;
FIG. 6 is a parameter 2 degradation trace at 85 ℃;
FIG. 7 is a parameter 1 degradation trace at 100 ℃;
FIG. 8 is a plot of the parameter 2 degradation trace at 100 deg.C
FIG. 9 is the mean value of the parameter 1 degradation data (70 ℃) under a power function fit;
FIG. 10 is the mean value of the parameter 1 degradation data (85 ℃ C.) under a power function fit;
FIG. 11 is the mean value of the parameter 1 degradation data (100 ℃ C.) under a power function fit.
Detailed Description
The invention is further described with reference to the following drawings and specific embodiments.
1.1 reliability model
The invention adopts Copula function to describe the dependency between two performance parameters. Firstly, a single-parameter degradation model based on a non-stationary incremental process is established, then a Copula-based double-parameter degradation model is established on the premise that Copula parameters are constant, and a corresponding parameter estimation method is provided.
For a product with two key performance parameters, the present invention makes the following assumptions:
the degradation behavior of both performance parameters can be modeled with a non-stationary incremental process established below.
There is a dependency between two performance parameters, whose dependency structure can be described by Copula function.
Any performance parameter value that meets or exceeds the design tolerance will result in product failure.
The parameters of the Copula function do not change with time and stress.
Assuming a certain product has two performance characteristic parameters, the degradation process of each performance characteristic parameter can be processed by a degradation track model or random processEquation model description, denoted as { X } 1 (t),X 2 (t) }. According to theorem 1, { X 1 (t),X 2 (t) } a joint distribution function of
H(X 1 (t),X 2 (t))＝C(F 1 (X 1 (t)),F 2 (X 2 (t))\*MERGEFORMAT(5)
Wherein, F p (X p (t)) is X p (t), p =1,2.
R(t)＝C(R 1 (t),R 2 (t))\*MERGEFORMAT(6)
Wherein R is p (t) is the reliability function of the p-th performance parameter, p =1,2.
For a product with two performance parameters, the reliability diagram can be equivalent to the tandem model shown in fig. 2.
When any one of the performance parameter values meets or exceeds a design-specified threshold, product failure will result. Failure threshold is noted as { D 1 ,D 2 }. Thus, for a product having two performance characteristic parameters, the reliability is
R(t)＝P((X 1 (t),X 2 (t))∈S)＝P(X 1 (t)＜D 1 ,X 2 (t)＜D 2 )\*MERGEFORMAT(7)
Where S represents a security domain, the product operates normally if and only if all performance parameters are within the security domain S.
Due to P (X) 1 (t)＜D 1 )＝F 1 (D 1 |t)，P(X 2 (t)＜D 2 )＝F 2 (D 2 I t). The formula (7) can be written as
R(t)＝P(X 1 (t)＜D 1 ,X 2 (t)＜D 2 )
＝C(F 1 (D 1 |t),F 2 (D 2 |t))
＝C(R 1 (t),R 2 (t))
\*MERGEFORMAT(8)
In the formula F p (X p (t)) is X p (t), p =1,2.R is p (t) is the reliability function of the p-th performance parameter, p =1,2.
In particular, if the performance characteristic parameters are independent of each other, the reliability function in equation (8) can be simplified to
R(t)＝R 1 (t)R 2 (t)\*MERGEFORMAT(9)
Wherein R is p (t) is the marginal reliability function of the p-th performance characteristic parameter, p =1,2.
Property 1Copula function has very good statistical properties, for two performance degradation processes { X } 1 (t),X 2 (t) having its reliability function at time t as u = R, respectively 1 (t) and v = R 2 (t) of (d). Then (P1) for all
C(u,0)＝C(0,v)＝0；
(P2) for all
C(u,1)＝u，C(1,v)＝v；
(P3) for all
max(u+v-1,0)＜C(u,v)＜min(u,v)。
Property (P1) indicates that any one performance parameter exceeding the failure threshold will result in a product failure, and property (P2) indicates that if one of the performance parameters does not degrade, the reliability of the product will be determined entirely by the other performance parameter. Property (P3) describes the Frechet-Hoeffding interval, R (t) = C (R) 1 (t),R 2 (t)) substituting the property (P3) to obtain a Frechet-Hoeffding interval as follows:
max(R 1 (t)+R 2 (t)-1,0)≤R(t)≤min(R 1 (t),R 2 (t))\*MERGEFORMAT(10)
the property shows that the reliability of the dual-parameter performance degradation failure type product is in the above formula interval.
Two-parameter regression model parameter estimation
Parameter estimation method under 1.1 rated stress
Assuming a total of n samples are input, each sample is at
The moment is detected, the p-th performance parameter of the i-th sample is at t ij The performance measurement value at that time is recorded as
i＝1,2,...,n,j＝0,1,2,...,m i P =1,2; wherein n is the number of samples, m i Is the number of detections for the ith sample. The degradation data may be represented in the form:
let α be the parameter in the Copula function, β p For the model parameters of the p-th performance degradation process,
for the p-th performance parameter of the i-th sample in the time interval t i,j-1 ,t ij ]An increment of degradation within, wherein
The joint probability density function of the degradation increments is
In the formula F p (·；β p ) And f p (·；β p ) Respectively the cumulative distribution function and the probability density function of the p-th edge distribution,
is a function of the density of C (.;. Alpha.).
From equation (12) and the independent incremental property, a likelihood function that yields experimental data of
The corresponding log-likelihood function is
Equation (14) includes a plurality of unknown parameters, β being the simplest model for example 1 ,β 2 Each containing three parameters and alpha containing one to two parameters. It is not easy, even feasible, to directly use the maximum likelihood estimation method to solve the model parameters. The observation shows that the parameter to be estimated consists of two parts: the first part is the parameter β = [ β ] in the edge distribution 1 ,β 2 ]The second part is the parameter α in the Copula function. The two-step estimation method of the invention comprises the steps of firstly fitting each edge distribution in sequence, and solving the parameter beta = [ beta ] in the edge distribution 1 ,β 2 ](ii) a Then selecting a proper Copula structure, and obtaining the edge distribution parameter estimation value obtained in the first step
The likelihood function shown in the formula (14) is substituted, and the estimated value of the Copula parameter alpha is obtained by using a maximum likelihood estimation method. Because each step only contains a small number of parameters, the calculation difficulty is obviously reduced, and the solving efficiency is also improved. Xu, joe and wang-fang, etc. have studied the effectiveness and progressive normality of this two-step estimation, and compared with the whole maximum likelihood estimation method, point out that this method has higher efficiency, the edge distribution parameter estimated value and Copula parameter estimated value are very close to the real parameter.
The first step is as follows: and solving the edge distribution parameters.
Separately estimating parameters of two edge distributions
1) For a simple model X (t) = sigma in a nonlinear drift wiener process p B(t)+μ p τ p (t) for β p ＝[μ p ,σ p ,γ p ]P =1,2, wherein γ p Is tau p Unknown parameters in (t), e.g.
Increment of degradation x pij Has a probability density function of
The log-likelihood function is
In the formula
Is the observed increase in degradation, τ ij ＝τ(t ij )-τ(t i,j-1 ) Is the two detection interval after conversion, Δ t ij ＝t ij -t i,j-1 Is the time interval between two detections,
is a probability density function of a standard normal distribution.
2) Model for random effect in nonlinear drift wiener process
In other words, beta p ＝[ω p ,δ p ,σ p ,γ p ]，p＝1,2，γ p Is tau p Unknown parameters in (t)Number, e.g.
Degradation increment x pij Has a probability density function of
The log-likelihood function is
In the formula
Is the observed increment of degradation, τ ij ＝τ(t ij )-τ(t i,j-1 ) Is the interval time of two detections after conversion, Δ t ij ＝t ij -t i,j-1 Is the time interval between two detections.
3) For the simple model IG (μ τ (t), σ τ) in the inverse Gaussian process 2 (t)) in respect of beta p ＝[μ p ,σ p ,γ p ]，p＝1,2，γ p Is tau p Unknown parameters in (t), e.g.
Degradation increment x pij Has a probability density function of
The log-likelihood function is
In the formula
Is the observed increment of degradation, τ ij ＝τ(t ij )-τ(t i,j-1 ) Is the converted detection interval time.
4) Model of stochastic effects in inverse Gaussian IG (μ τ (t), σ τ 2 (t)),μ -1 ～TN(ω,δ -2 ) In other words, beta p ＝(σ p ,γ p ,ω p ,δ p )，p＝1,2，γ p Is tau p Unknown parameters in (t), e.g.
Degradation increment x pi Has a joint probability density function of
The log-likelihood function is
In the formula
Is the observed increment of degradation, τ ij ＝τ(t ij )-τ(t i,j-1 )，
The second step is that: and solving Copula function parameters.
The first step has already determined the parameters of the two edge distributions
Substituting this into equation (14) yields a likelihood function for the parameter α:
in the formula
And
respectively a cumulative distribution function and a probability density function of the p-th performance parameter,
is a density function of C (·; α), x pij Is the observed increase in degradation, p =1,2.
The second half of equation (23) is independent of parameter α, does not affect the final result, and can be discarded. The new likelihood function is:
the maximum likelihood estimate of the parameter α can be expressed as:
where Θ is the parameter space.
For the single-parameter Copula function, the likelihood function only contains one parameter α, and the maximum likelihood estimation value can be easily obtained. Taking Frank Copula as an example, the Copula density function is:
the Copula density function is substituted into the formula (25) to obtain:
among other single-parameter Copula functions
Can also be solved by a similar method.
For a two-parameter Copula function, the likelihood function contains two parameters α = [ α = 1 ,α 2 ]. Taking Joe-Clayton Copula as an example, the Copula density function is:
substituting Copula density function in formula (28) into formula (25) yields:
parameters in other two-parameter Copula functions
Can also be solved by a similar method.
Equation (29) includes only two parameters, and the optimal solution can be obtained by a common optimization algorithm, such as a Newton-Raphson algorithm.
2.2 parameter estimation method under constant acceleration stress
The method of estimating the parameters under nominal stress is discussed above, and on this basis, the method of estimating the parameters under constant acceleration stress is discussed. Assuming that a product has two performance characteristic parameters, the degradation process of each performance characteristic parameter can be described by a degradation track model or a random process model. Assumed stress level z k Lower throw-in n k Samples of total number of samples
l is the total number of stresses. Sample at t kij The moment is detected, the p characteristic parameter of the ith sample under the k stress is t kij The performance measurement value at that time is recorded as
k＝1,2,...,l,i＝1,2,...,n k ,j＝0,1,2,...,m ki ,p＝1,2。n k Is the number of samples at the k stress level, m ki Is the number of detections for the ith sample at the kth stress level. The degradation data may be represented in the form shown in table 1:
TABLE 1 Dual parameter accelerated degradation test data for products
Let α be the parameter in the Copula function, β p For the model parameters of the p-th performance degradation process,
for performance parameters in the time interval t ki,j-1 ,t kij ]An internally observed increment of degradation, wherein
The joint probability density function of the degradation increments is
In the formula F p (·；β p ) And f p (·；β p ) Respectively a cumulative distribution function and a probability density function of the p-th edge distribution,
is a density function of C (·; α).
From the independent incremental properties, a likelihood function of experimental data is obtained as
The corresponding log-likelihood function is
Model parameters are also estimated using a two-step estimation method.
The first step is as follows: and solving the edge distribution parameters.
3) For the covariate model in the nonlinear drift wiener process, the parameters to be estimated and the connection function μ z In connection with, when the connection function is μ z ＝ae -b/z When the parameter to be solved is beta p ＝[a p ,b p ,σ p ]P =1,2. Increment of degradation x pkij Has a probability density function of
The log-likelihood function is
In the formula
Is the observed degradation increment, Δ (t) kij )＝t kij -t ki,j-1 Is the time interval between two detections, tau kij ＝τ(t kij )-τ(t ki,j-1 ) Is the interval time after the conversion,
is a probability density function of a standard normal distribution.
4) For the covariate model in the inverse Gaussian process, the parameter to be estimated is beta p ＝[a p ,b p ,σ p ]P =1,2, wherein a p ,b p Is a connection function mu z The parameter (1). Degradation increment x pkij Has a probability density function of
The log-likelihood function is
In the formula
Is the observed increase in degradation, τ kij ＝τ(t kij )-τ(t ki,j-1 ) Is the detection interval time after the conversion,
is about a p ,b p Is measured as a function of (c).
The second step is that: and solving Copula function parameters.
The first step has already determined the parameters of the two edge distributions
By substituting it into equation (32), a likelihood function for the parameter α can be obtained:
in the formula
And
respectively a cumulative distribution function and a probability density function of the p-th edge distribution,
is a density function of C (.;. Alpha.), x pkij Is the observed increase in degradation, p =1,2.
After omitting the constant term, equation (37) can be simplified as:
the estimated parameter value of the parameter α may be expressed as:
where Θ is the parameter space.
Since the likelihood function only comprises one to two parameters, the maximum likelihood estimation value can be easily obtained by an optimization algorithm such as Newton-Raphson and the like.
2 ammunition Assembly reliability evaluation
2.1 accelerated degradation test data
Aiming at the ammunition assembly, a constant stress temperature accelerated degradation test is carried out, 70 ℃, 85 ℃ and 100 ℃ are selected as 3 accelerated stress levels, and 5 samples are respectively arranged at each temperature. For each sample, two performance degradation parameters, parameter 2 and parameter 1, were measured periodically, 9, 12, and 9 times at three stresses. The complete performance degradation parameter measurement data is shown in tables 2-4 and FIGS. 3-8.
TABLE 2 degradation data of Performance parameters at 70 deg.C
TABLE 3 degradation data of Performance parameters at 85 deg.C
TABLE 4 degradation data of Performance parameters at 100 deg.C
According to the parameter estimation method of the double-parameter model and the two-step method, firstly, the two performance parameters of the parameter 2 and the parameter 1 are subjected to translation processing, namely, the initial drift amount is translated to 0, then, the translated parameters are respectively modeled, then, the dependent relationship of the two parameters is modeled based on the Copula function, and parameter estimation and reliability estimation are carried out.
2.2 parameter 1
And (3) aiming at the single performance parameter of the parameter 1, adopting a common degradation track model to perform modeling analysis:
Y(t)＝λ 1 γ 1 Λ 1 (t)+ε 1 \*MERGEFORMAT(40)
wherein, Λ 1 (t) is a non-linear function, γ 1 Is a parameter of a covariate such that,
is an error term.
Since the reliability tests of ammunition components are carried out under temperature-accelerated stress, the commonly used arrhenius model is used to model the covariate parameters, i.e.
The formula (40) becomes
By
It can be seen that, at any time t,
let its probability density function be f Y (y)：
Wherein the content of the first and second substances,
is a function of the probability density of a standard normal distribution,
is Λ 1 Unknown parameters in (t).
It was further found that, under the same temperature stress, E [ Y (t)]∝Λ 1 (t), therefore by regression analysis of the mean of all 5 sample degradation data under a single stress, a preliminary determination can be made 1 (t) basic form. Here, three forms of a linear function, an exponential function and a power function are used as alternative models for analysis, and as a result, as shown in table 5 and fig. 9 to 11, it is found that the fitting effect of the power function is good, so that the power function is selected here
TABLE 5 correlation coefficient R of mean values of parameter 1 degradation data under different fitting functions 2
At the determination of Λ 1 After the basic form of (t), the unknown parameters in the degradation model can be estimated by using a maximum likelihood estimation method, wherein the maximum likelihood function is as follows:
wherein (t) i,j,k ,y i,j,k ) The time point and the parameter 1 quantity of the ith sample at the jth measurement under the kth stress, and K is the total stressNumber, n k Number of samples under k-th stress, m i The number of measurements for the ith sample.
The maximum value of the maximum likelihood function is solved by using intelligent optimization methods such as a genetic algorithm and the like, and the estimation values of all parameters are obtained as shown in the following table:
TABLE 6 PARAMETER 1 degeneration model parameter estimation
From this, the ammunition parameters
Wherein epsilon 1 ～N(0,0.2207 2 )。
Thus, the mean value of parameter 1 predicted by the model to be 10 years of storage at 25.7 ℃ is
Where, -1.2333 is the average of parameter 1 for all samples at the start of the experiment.
And the actual measurement values of the parameter 1 of the mechanical ammunition assembly stored in a two-shot warehouse (25.7 degrees) are recorded as-0.7 and-1.1 respectively by actually measuring the parameter 1, namely the average value of the actual measurement of the parameter 1 is
Thus, for parameter 1, the prediction accuracy of the model
2.3 parameter 2
For the performance degradation parameter of parameter 2, a similar degradation model is used for processing, i.e. X (t) = λ 2 γ 2 Λ 2 (t)+ε 2 Wherein
In the form of a non-linear function,
is a covariate that is related to the acceleration stress,
is an error term.
At any one of the time points t,
let its probability density function be f X (x)：
And similarly, estimating unknown parameters in the degradation model by using a maximum likelihood estimation method, wherein the maximum likelihood function is as follows:
wherein (t) i,j,k ,x i,j,k ) Is the time point of the ith sample at the jth measurement under the kth stress and the quantity of the parameter 2, wherein K is the total stress number, n k Number of samples under k-th stress, m i The number of measurements for the ith sample.
The maximum value of the maximum likelihood function is solved by using intelligent optimization methods such as a genetic algorithm and the like, and the estimation values of all parameters are obtained as shown in the following table:
TABLE 7 parameter 2 degradation model parameter estimation
| Parameter(s) | λ 2 | σ 2 | η 2 | β 2 |
| Estimated value | 7.6911 | 0.1808 | 0.2259 | 0.0075 |
From this, the parameters
Wherein epsilon 2 ～N(0,0.1808 2 ). Thus, the mean value of parameter 2, predicted by the model to be stored for 10 years at 25.7 °, is
Where-0.4 is the average of parameter 2 for all samples at the start of the test.
And actually measuring the quantity of the parameter 2 stored in the two storerooms (at 25.7 ℃), and recording the actual measurement values of-0.5 and-0.4 respectively when the parameter 2 is stored for 10 years, namely the actual measurement average value of the parameter 2 is
Thus, for parameter 1, the prediction accuracy of the model
2.4 two parameter modeling
When the correlation between the parameter 2 and the parameter 1 is considered, the Gaussian Copula function is used for carrying out two-parameter modeling, and the basic form is
Wherein, Σ represents a symmetric positive definite matrix in which an element on a diagonal is 1 and other elements are ρ; phi ρ Representing a standard multivariate normal distribution cumulative distribution function with a correlation coefficient matrix of sigma; phi (phi) of -1 (. Cndot.) represents an inverse of a standard normal distribution cumulative distribution function;
i denotes an identity matrix.
For the two-dimensional Gaussian Copula under the condition of double parameters, the functional form can be written as
Wherein the content of the first and second substances,
representing a standard binary normal distribution probability density function with a correlation coefficient p.
Based on the Gaussian Copula function and the parameter 1 and parameter 2 degradation models, the total likelihood function can be obtained as
Wherein F Y (x ijk ),F X (y ijk ) The cumulative distribution functions of parameter 1 and parameter 2, respectively. Taking logarithm to the likelihood function to obtain
From the two-step method proposed above, the unknown parameters in the two edge distributions of parameter 1 and parameter 2 can be estimated first, and the results are shown in table 6 and table 7 above; these parameter estimates are substituted (56) to obtain a new likelihood function as follows.
Solving the maximum value of the likelihood function by an intelligent optimization method such as a genetic algorithm and the like can further estimate the unknown parameter rho =0.0089 in the Gaussian Copula function.
Therefore, for the dependent two-parameter degradation process, the average value of the Copula function of the model prediction stored for 10 years at 25.7 ℃ is
The measured values of the parameters 1 and 2 of the ammunition assembly are (-0.5, -0.7) and (-0.4, -1.1) respectively under the condition of warehouse storage, and the measured average value of the Copula function is
Thus, for this two-parameter degradation process, the prediction accuracy of the model
Aiming at a double-parameter degradation process with a dependency relationship, the invention firstly provides a method for evaluating the dependency among a plurality of performance parameters of an ammunition component, and provides a reliability model of double-parameter degradation on the basis of Sklar constant force. Further aiming at the model, a two-step method with higher efficiency is provided for estimating the unknown parameters in the model, firstly, modeling analysis is respectively carried out on the degradation processes of the two performance parameters, and the unknown parameters in the edge distribution of the two performance parameters are estimated; and then substituting the edge distribution parameter estimation value into the whole maximum likelihood function to estimate the unknown parameters in the Copula function.
Aiming at the actual degradation process of the ammunition component, two key degradation parameters, namely a parameter 1 and a parameter 2, are considered, a proposed two-parameter degradation model is used for modeling analysis, and the edge distribution parameters of two performance degradation quantities and unknown parameters in a Gaussian Copula function are respectively obtained by utilizing a genetic algorithm and a maximum likelihood estimation method. Through comparison and analysis with measured values stored in a warehouse, the model prediction precision of the parameter 1 calculated by calculation reaches 85.92%, the model prediction precision of the parameter 2 calculated by calculation reaches 88.60%, and the model prediction precision of the final double-parameter Copula function value reaches 93.75%.
Claims (1)
1. A method for evaluating a dependency between a plurality of performance parameters of an ammunition assembly, comprising:
establishing a Copula-based two-parameter degradation model under the premise of assuming that Copula parameters are constant, and providing a corresponding parameter estimation method:
assuming that a certain product has two performance characteristic parameters, the degradation process of each performance characteristic parameter can be described by a degradation track model or a random process model, and is marked as { X 1 (t),X 2 (t)}；{X 1 (t),X 2 (t) a joint distribution function of
H(X 1 (t),X 2 (t))＝C(F 1 (X 1 (t)),F 2 (X 2 (t))
Wherein, F p (X p (t)) is X p (t), p =1,2;
for a product having two performance characteristic parameters, the reliability function is
R(t)＝C(R 1 (t),R 2 (t))
Wherein R is p (t) is a reliability function of the pth performance parameter, p =1,2;
for the two-dimensional Gaussian Copula under the condition of double parameters, the functional form is written as
Wherein the content of the first and second substances,
representing a standard binary normal distribution probability density function with a correlation coefficient of rho;
when any one of the performance parameter values reaches or exceeds a threshold value specified by the design, the product is caused to fail; the failure threshold is noted as { D 1 ,D 2 }; thus, for a product having two performance characteristic parameters, the reliability is
R(t)＝P((X 1 (t),X 2 (t))∈S)＝P(X 1 (t)＜D 1 ,X 2 (t)＜D 2 )
In the formula, S represents a security domain, and the product works normally if and only if all performance parameters are located in the security domain S; due to P (X) 1 (t)＜D 1 )＝F 1 (D 1 |t)，P(X 2 (t)＜D 2 )＝F 2 (D 2 I t); written as R (t) = P (X) 1 (t)＜D 1 ,X 2 (t)＜D 2 )
＝C(F 1 (D 1 |t),F 2 (D 2 |t))
＝C(R 1 (t),R 2 (t))
In the formula F P (X P (t)) is X P (t), p =1,2; r is P (t) is a reliability function of the pth performance parameter, p =1,2;
if the performance characteristic parameters are independent of each other, the reliability function in the formula is simplified to
R(t)＝R 1 (t)R 2 (t)
Wherein R is p (t) is a marginal reliability function of the p-th performance characteristic parameter, p =1,2;
for two performance degradation processes { X 1 (t)，X 2 (t) having its reliability function at time t of u = R, respectively 1 (t) and v = R 2 (t); then
(Pl) for all
C(n，0)＝C(0，v)＝0；
(P2) for all
C(n，1)＝u，C(1，v)＝v；
(P3) for all
max(u+v，-1，0)＜C(u，v)＜min(u，v，)；
Property (P1) indicates that any one of the performance parameters exceeding the failure threshold will result in a product failure, and property (P2) indicates that if one of the performance parameters does not degrade, the reliability of the product will be determined entirely by the other performance parameter; property (P3) describes the Frechet-Hoeffding interval, R (t) = C (R) 1 (t)，R 2 (t)) substituting the property (P3) to obtain a Frechet-Hoeffding interval as follows:
max(R 1 (t)+R 2 (t)-1，0)≤R(t)≤min(R 1 (t)，R 2 (t))
the property shows that the reliability of the dual-parameter performance degradation failure type product is in the above formula interval;
the method for estimating the model parameters comprises the following steps:
the first step is as follows: solving edge distribution parameters
1) For the covariate model in the nonlinear drift wiener process, the parameters to be estimated and the connection function μ z In connection with, when the connection function is μ z ＝ae -b/z When the parameter to be solved is beta p ＝[a p ，b p ，σ p ]P =1,2; degradation increment x pkij Has a probability density function of
The log-likelihood function is
In the formula
Is the observed degradation increment, Δ (tk) ij )＝tk ij -tk i，j-1 Is the time interval between two detections, τ k ij ＝τ(tk ij )-τ(tk i，j-1 ) Is the interval time after the conversion,
a probability density function that is a standard normal distribution;
2) For the covariate model in the inverse Gaussian process, the parameter to be estimated is beta p ＝[a p ，b p ，σ p ]，p＝1，2，
Wherein a is p ,b p Is the connection function mu z The parameter (1) of (1); degradation increment x pkij Has a probability density function of
The log-likelihood function is
In the formula
Is the observed increment of degradation, τ kij ＝τ(t kij )-τ(t ki,j-1 ) Is the detection interval time after the conversion,
is about a p ,b p A function of (a);
the second step is that: solving Copula function parameters;
the first step has already determined the parameters of the two edge distributions
A likelihood function is obtained for the parameter α:
in the formula
And
respectively a cumulative distribution function and a probability density function of the p-th edge distribution,
is a density function of C (.;. Alpha.), x pkij Is the observed increment of degradation, p =1,2;
after the constant term is omitted, the method is simplified as follows:
the estimated parameter value of the parameter α is expressed as:
wherein Θ is the parameter space;
since the likelihood function only comprises one to two parameters, the maximum likelihood estimated value is obtained by a Newton-Raphson optimization algorithm.
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