CN110929453A - Copula function failure correlation system-based dynamic fuzzy reliability analysis method - Google Patents
Copula function failure correlation system-based dynamic fuzzy reliability analysis method Download PDFInfo
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Abstract
The invention discloses a Copula function failure-based dynamic fuzzy reliability analysis method for a related system, which comprises the following steps: 1) calculating the dynamic fuzzy reliability of a single part, and substituting the value into a dynamic fuzzy reliability calculation formula of a serial or parallel system based on Copula functions; 2) estimating Copula function parameters, wherein the method comprises the steps of firstly calculating kendall rank correlation coefficients among parts, and then calculating parameter values of each Copula function according to the relationship between the kendall rank correlation coefficients and the Copula function parameters; 3) and obtaining the dynamic fuzzy reliability value of the system through the solved parameter value of the Copula function and the dynamic fuzzy reliability value of the single part.
Description
Technical Field
The invention relates to the analysis of system reliability, in particular to the reliability analysis of a series system and a parallel system, which takes a Copula function as a tool and considers the failure correlation among parts.
Background
Reliability is a quality indicator for a product having a time attribute defined as the ability of the product to perform a specified function under specified conditions within a specified time. The reliability of the product is determined by various factors such as design, manufacture, use, maintenance and the like. Reliability issues were first addressed in the united states because of failures in avionics. During the second war, when the American airplanes are transported to the far east for war, more than 60 percent of the airplanes cannot take off because of the failure of the electron tubes, and the failure rate of the electronic equipment of naval vessels reaches 70 percent. This has also prompted the united states to build reliability advisors for electronic equipment in 1952 and to conduct a thorough reliability study on the design, manufacture, testing, shipping and use of electronic equipment. After a lot of investigation and research, the counselor published the reliability analysis instructive document "military electronic equipment reliability report" in 1957, which is recognized as a foundation document for reliability analysis of electronic equipment, and its basic idea is still widely used.
The reliability of the mechanical system is researched in the united states at first, because a large number of space accidents are caused by mechanical faults in the aerospace aspect, huge manpower and material resource losses are brought to the countries, and the reliability of the mechanical system is researched in the beginning of the last 60 th century by the united states space agency. As the demand for reliability of products is gradually increasing in developed countries such as the united states, the reliability technology is gradually expanding from the field of aerospace military industry to the fields of mechanical equipment, civil engineering and construction, electric power systems, and the like. In the last 60 th century, the reliability technology is applied to the electronic industry in the first place in China, and through years of development and application, the reliability technology is increasingly mature and gradually expands to the fields of aerospace products, military products, mechanical products and the like. The application of mechanical reliability can not only improve the design level and quality of products and reduce the cost of the products, but also solve the problem that the traditional design can not solve, greatly improve the service life of the products, and therefore, the mechanical reliability research has great significance.
Further in the field of reliability, a mechanical structure or component is considered to fail if its generalized strength is less than its threshold of generalized stress. The classical stress-intensity interference model is based on the consideration that the random characteristics of stress and intensity do not change with time, and belongs to the static reliability research. However, in practical engineering, the system is affected by various uncertain factors over time, and the material characteristics, the working environment, the load and the vibration effect of the system change continuously over time, which also causes the generalized stress and the generalized strength of the structure or the part to change dynamically over time, and the randomness and the time variation of the generalized stress and the generalized strength make the dynamic reliability difficult to evaluate correctly. If we continue to use the initial state of the mechanical structure or parts to research the working state of the system in the using process, it is impossible to accurately master the safety and reliability of the system, and even more, the great loss of manpower and financial resources is caused. Meanwhile, in conventional reliability studies, it is generally assumed that structures or parts are independent of each other, but in practical applications, it is sometimes difficult to satisfy independence. Ignoring the correlation between structures or components in the system may result in too large an error in the reliability analysis result, and even an erroneous conclusion may be drawn. Meanwhile, the working conditions are different, and the correlation degrees of the structures or parts are different. It is neither straightforward to use the theory of independence assumptions (considering complete independence between structures or components) nor the theory of weakest links (considering complete linear correlation between structures or components) to compute the reliability of a system. Therefore, the method comprehensively considers the uncertainty of geometry, physics and load, the time-varying property of the strength of the structure or the part and the correlation between the structure or the part in the system, and has important significance in realizing the dynamic and gradual change reliability modeling analysis of the complex mechanical system.
In recent years, Copula functions are widely applied to system static reliability analysis by a plurality of scholars due to the fact that the Copula functions can better handle the correlation problem existing between structures or parts, and a large number of research results are obtained.
Disclosure of Invention
Based on the background technology and the current situation, a Copula function is taken as a tool, failure correlation among parts is considered, and dynamic reliability and dynamic fuzzy reliability of a series system and a parallel system are researched.
The invention is solved by the following technical scheme:
a dynamic fuzzy reliability analysis method based on a Copula function failure correlation system comprises the following steps:
s1, based on the fuzzy mathematical theory, replacing the strength and stress of a single part with fuzzy strength and fuzzy stress, and then obtaining a part dynamic fuzzy reliability model when the fuzzy strength is not degraded and a part dynamic fuzzy reliability model when the fuzzy strength is degraded under the action of random load by using a horizontal interception theory and a stress-strength interference theory;
s2, based on the dynamic fuzzy reliability of a single part, combining the dynamic fuzzy reliability with a Copula function, substituting the dynamic reliability and parameters of each part, and deducing a dynamic Copula reliability analysis model of a serial-parallel system by using a difference method;
s3, constructing a dynamic fuzzy reliability analysis model of the series-parallel system based on the dynamic fuzzy reliability model of the part with degraded or not degraded fuzzy strength under the random load action constructed in the step S1 and in combination with Copula function analysis in the step S2;
s4, for the dynamic Copula reliability analysis model of the series-parallel system in the step S2, estimating the edge distribution function value of the stress-intensity interference sample of the system and the parameter theta of the Copula function by adopting a nonparametric kernel estimation-maximum likelihood estimation method;
s5, expressing a parameter theta of the Copula function through a kendall rank correlation coefficient, and obtaining the influence of the kendall rank correlation coefficient tau on the dynamic fuzzy reliability of the series and parallel systems, thereby deducing and analyzing the influence of the Copula parameter theta on the dynamic fuzzy reliability of the series and parallel systems;
and S6, calculating the dynamic fuzzy reliability of the system according to the dynamic fuzzy reliability analysis models of the series and parallel systems obtained in the step S3, comparing the calculation result with the result obtained by the Monte Carlo simulation method obtained in advance by the system, and selecting a Copula function model according to the comparison result.
Further, the step S4 specifically includes the following steps:
4a) determining the edge distribution function of the random sample by adopting a nonparametric kernel estimation method, and determining the edge distribution function of the random sample by adopting the nonparametric kernel estimation method; assuming a series system consisting of n parts, firstly simulating T times by adopting a Monte Carlo method, and simulating the strength simulation value sigma of the jth partjiAnd its stress analog value SjiDifference of sigmaji-SjiThe samples are used as the T samples of the jth part and are uniformly recorded as the vector, and the edge probability density function f of the sample of the jth part can be obtainedj(y) and an edge distribution function;
4b) and (4) solving a distribution function of subtracting the stress sample from the strength of each part according to the step 4a), and then calculating a parameter theta of the Copula function by adopting a maximum likelihood method according to the failure sample of each part.
Further, in step S6, fuzzy reliability simulation is performed based on the monte carlo method, and the fuzzy reliability simulation is performed by splitting the fuzzy reliability into two parts, namely, the intercept reliability and the fuzzy reliability, and the simulation steps are as follows:
6b1) firstly, two random numbers lambda are generated by adopting a standard uniform distribution function simulationσiAnd λsiThey respectively represent the interception levels of fuzzy strength and fuzzy stress, and then the probability density function of fuzzy strength interception interval can be determined according to the value of the interception levels
Probability density function of fuzzy stress cut-off interval
6b2) Further, the distribution function of the fuzzy intensity and the probability density function of the fuzzy intensity cut-off interval can be obtained
Extracting the random value sigmajAnd then sigmajSubstituting the fuzzy stress into the distribution function of the fuzzy stress interception interval, and repeating the simulation for M times to obtain the interception reliability
6b3) The process is circularly simulated for N times, so that the fuzzy reliability simulation value P of the part can be obtained when the fuzzy strength of the part is not degradedσ。
Further, in the steps S5 and S6, if the edge distribution of the sample and its corresponding Copula function are known, the kendall rank correlation coefficient τ may be solved.
Further, the value of each Copula function parameter can be derived from the relationship between τ and its parameter when the Copula function is binary Gauss Copula, the relationship between τ and its parameter when the Copula function is binary Gumbel Copula, the relationship between τ and its parameter when the Copula function is binary clayton Copula, the relationship between τ and its parameter when the Copula function is binary Frank Copula, and the relationship between τ and its parameter when the Copula function is binary CClayton Copula.
To select the most suitable Copula function. Let t be t0And aiming at the serial dynamic fuzzy system and the parallel dynamic fuzzy system respectively, and considering the degradation and non-degradation of the fuzzy strength of the parts in the system respectively, selecting different Copula functions according to the model in the step S5, calculating the reliability value of the system in the time t, and simultaneously carrying out fuzzy reliability simulation by using a Monte Carlo method to obtain the reliability value of the system in the time t, thereby selecting the Copula function model with the reliability value closest to the Monte Carlo fuzzy reliability simulation.
Compared with the prior art, the dynamic fuzzy reliability problem of the series-parallel system is solved by applying a horizontal intercept theory, a double-state hypothesis theory, a stress-intensity interference theory and a sequence statistics theory to a timely simultaneous poisson process, fully considering the dynamic fuzzy reliability of the structure or the part when the structure or the part is subjected to a plurality of times of fuzzy load action and the fuzzy intensity of the structure or the part is not degraded, and further deducing a dynamic fuzzy reliability calculation formula based on a Copula function series system and a parallel system by utilizing a Monte Carlo method, namely, the parameter of the Copula function is expressed by a kendall rank correlation coefficient, so that the system reliability under a real condition can be more accurately reflected, and the prediction result of the system reliability is improved.
It has the following characteristics:
1) the method includes the steps that when the strength of structural parts is not degraded along with time and the strength of the parts is degraded along with time, a dynamic reliability model of a series-parallel system related to part failure is constructed by combining a Copula function;
2) the nonparametric kernel estimation method and the maximum likelihood estimation method are combined to accurately estimate the parameters of different Copula function models;
3) and when the non-time degradation of the fuzzy strength of the part and the time degradation of the fuzzy strength of the part are respectively considered, a dynamic fuzzy reliability model of the series and parallel connection system related to the part failure is constructed by combining the Copula function.
Drawings
FIG. 1 is a flow diagram of a specific embodiment of the present invention;
FIG. 2 is a graph of three commonly used truncated distributions;
FIG. 3 is a diagram of a series system;
fig. 4 is a diagram of a parallel system.
Detailed Description
The following describes the embodiment of the present invention with reference to the flow chart shown in fig. 1.
The invention discloses a Copula function failure correlation system-based dynamic fuzzy reliability analysis method, which comprises the following steps:
1. motion blur reliability calculation for individual parts
1.1 two-state hypothesis fuzzy reliability calculation formula
Based on fuzzy mathematical theory, assume fuzzy strength
And fuzzy stress
Are respectively a membership function of
And
if the blurring strength is greater than the blurring stress
The structure is safe if the blurring strength is less than the blurring stress
The structure fails. Given truncation level λσ，λsAll belong to a closed interval of 0 to 1, and the intensity intercept interval of
The stress cutoff interval is
Is provided with
And
when their probability density functions are respectively
And
according to the horizontal cutting theory and the stress-intensity interference theory, the expression of the reliability and the failure probability can be obtained as follows:
then at the truncation level lambdaσAnd λsAs integral variables, in the intervals [0,1] respectively]To pair
And
the approximate solutions of the fuzzy reliability and the fuzzy failure probability obtained by performing double integration are respectively as follows:
1.2 part dynamic fuzzy reliability model when fuzzy strength is not degraded
The structure or component being subjected to fuzzy loads m times during operation
The action is equivalent to that m samples are extracted from the fuzzy load total samples, which are abbreviated as
A structure or component is considered reliable if it does not fail when subjected to a fuzzy load multiple times. Thus, when it is assumed that the blurring strength of the structure or component does not occur with timeIn the process of degradation, if the structure or the part does not fail under the maximum fuzzy load, the structure or the part does not fail under the fuzzy load of m times, namely the fuzzy reliability of the structure or the part under the fuzzy load of m times can be equivalent to the fuzzy reliability of the structure or the part under the maximum fuzzy load of m times:
in the formula:
indicating the blurring strength of the structure or component.
Suppose that
Representing the stress at maximum creep load of a structure or component,
expressing the corresponding membership function, and calculating the lambda of the membership function according to the horizontal intercept theorymaxThe horizontal cut set is:
when in use
Then, according to the horizontal intercept theory, several commonly used intercept distributions are utilized to obtain the (6) middle position
Has an inner probability density function of
That is to sayAs shown in fig. 2, the expression is one of the formulas (7) to (9):
1) uniform distribution:
2) linear distribution:
3) truncated normal distribution:
wherein:
C0the standard coefficients are normally distributed for the introduced truncation.
Further, the distribution function can be obtained as
The probability density function of the equivalent fuzzy stress to which the structure or component is subjected when subjected to the fuzzy load m times can be expressed as:
the same assumption
Indicating the blurring strength of the structure or component,
expressing the corresponding membership function, and obtaining the lambda of the membership function according to the horizontal cut-set theoryσThe horizontal cut set is:
according to
Can also be obtained in
Has an inner probability density function of
From the above two-state hypothesis theory one can obtain the difference in λσ、λsThe reliability and failure probability at the level of the intercept are respectively:
then the level of truncation is lambdaσAnd λsRespectively as integral variables in the closed interval [0,1]]The above two-fold integral is respectively carried out on the formula (12) and the formula (13), and the calculation formula of the fuzzy reliability and the fuzzy failure probability of the structure or the part under the action of the fuzzy load for m times can be obtained as follows:
and further deducing a fuzzy reliability formula when the strength of the structure or the part is not degraded under the action of m times of fuzzy load.
The poisson process is generally used for describing events which independently occur in disjoint time and interval, and the cumulative action times of random loads on structures or parts are consistent with a poisson process mathematical model. The number of occurrences of the fuzzy load in a fixed time can therefore be described using a poisson distribution process, with a particular expression shown in equation (16).
Where n (t) is the total number of times the random load acts during the period of (0, t), and satisfies the following conditions:
(1) when N (0) is 0, the number of occurrences of the load is 0, that is, N (0) is 0;
(2) the probability of the occurrence of the load is not related to each other in each time period, namely 0 < t is arbitrarily selected1＜···＜tm，N(t1),N(t2)-N(t1),···,N(tm)-N(tm-1) Are not related to each other;
(3) the initial time does not influence the number of times of load application, and the number of times of load application is only related to the length of a time period, namely for any s, t is more than or equal to 0, m is more than 0, and P [ N (s + t) -N(s) -m ] ═ P [ N (t) -m ];
(4) for any t > 0 and sufficiently small Δ t > 0, there is P [ N (t + Δ t) -N (t) ═ 1]＝λ1Δ t + o (Δ t) and P [ N (t + Δ t) -N (t) > 2]＝ο(Δt)；
Therefore, by using the total probability formula, the Poisson distribution (16) and the fuzzy reliability formula (14), the fuzzy reliability formula when the strength of the structure or the part is not degraded under the action of m times of fuzzy loads can be obtained as follows:
the Taylor expansion using an exponential function can reduce the above equation to:
in the formula, t represents a time variable,
as a horizontal truncation of the membership function
Distribution function of internal blurring stress, wherein
And
respectively representing the upper and lower limits, λ, of the horizontal cut-sets of the membership functions1The parameters of the simultaneous poisson process are shown,
is a horizontal truncated set
Distribution function of internal load.
1.3 part dynamic fuzzy reliability model in fuzzy strength degradation
Assuming the fuzzy strength of a structure or component at a certain time τ
The fuzzy strength of the structure or the part at a certain moment t can be expressed by a function of the initial fuzzy strength and time, and the probability of the fuzzy load acting on the part in a time period (tau, tau + delta tau) is known to be lambda in combination with the Poisson random process1(t) Δ t. Thus, the reliability of the motion blur of the structure or component at that moment
Can be expressed as:
in the formula
And representing the dynamic fuzzy reliability of the structure or the part at the time t, and the specific expression is as follows. Setting the fuzzy strength of the structure or part and the corresponding horizontal cut set of the fuzzy stress at t moment
And
then the probability density functions of the fuzzy strength and the fuzzy stress are obtained according to the level intercept theory described above as
And
further can be obtained by combining the theory of stress-intensity interference and dual-state hypothesis
Comprises the following steps:
this is obtained by shifting the term for equation (19):
dividing both sides of the equation of equation (21) by Δ τ simultaneously yields:
further, the method can be obtained as follows:
solving the differential equation for equation (23) yields:
according to the known condition t being 0,
in this case, the differential constant can be obtained
Therefore, the dynamic fuzzy reliability formula of the structure or the part under the action of multiple fuzzy loads when the strength is degraded can be obtained as follows:
2. system reliability calculation based on Copula function
2.1 reliability calculation of series systems
The static series system composed of n parts is shown in fig. 3, wherein the numbers 1-n in the figure represent the 1 st to n-th parts, the reliability of the system can be calculated by integrating the joint probability density function of the parts, and the expression is as follows:
in the formula, Zi＝σi-Si(σiIndicates the strength of the part, SiIndicating the stress to which the part is subjected), ZiRepresenting the function of the ith part, fZ(. represents (Z)1,Z2,···,Zn) The joint probability density function of (a).
However, the difficulty of calculating the reliability of the system by using the formula (26) is that the joint probability density function of the system is difficult to determine during calculation, and the difficulty of calculating by using the formula (27) is that if the number of system parts exceeds a certain number, the multiple integral calculation amount is huge, and if the correlation between the parts is not considered, the parts are assumed to be independent,
wherein R is1·R2·…·R3The reliability of each constituent element of the bit system.
If the components are considered to be in a complete correlation relationship, the reliability of the series system can be equivalent to the reliability of the minimum-reliability component in the series system, because the series system cannot normally work according to the correlation relationship as long as the series system cannot normally work, so the reliability calculation formula of the series system in the complete correlation is as follows:
Rs＝min(P(Z1＞0),P(Z2＞0),···,P(Zn＞0))＝min(R1,R2,···,Rn) (28)
however, in practical engineering applications, the components cannot be completely independent from each other or completely related to each other, and the actual correlation relationship between the components is always between the two components. Therefore, the reliability of the series system can be analyzed more accurately only by accurately processing the actual correlation between the parts.
Setting a series system of n units with a joint distribution function of FZ(z1,z2,···,zn)，Zi＝σi-Si(σiIndicates the strength of the part, SiIndicating the stress to which the part is subjected), ZiIndicating the function of the ith part, Fi(. cndot.) represents the distribution function of the functional function of the ith part, then there is an n-dimensional Copula function according to Sklar's theorem such that: fZ(z1,z2,···,zn)＝C(F1(z1),F2(z2),···,Fn(zn) Because F isi(zi) Is continuous, so C (F)1(z1),F2(z2),···,Fn(zn) Is unique, where C (-) represents an n-dimensional Copula function. The reliability of the series system is then:
in the formula Fi(0) Representing probability of failure, and Δ representing differential sign, e.g.
Representing a double difference; while
Representing double differences and so on.
The formula (29) is a static reliability calculation formula based on the Copula function cascade system, and on the basis, a dynamic Copula reliability analysis model calculation formula of the cascade system can be derived as follows:
wherein t represents a time variable, Δ is a differential sign, and Xj(t) represents the function of the jth part at the moment t, and the dynamic reliability of the part is Rj(t) dynamic failure probability of Fj(t), then: fj(t)＝1-Rj(t), wherein j ═ 1,2, ·, n.
When two parts are connected in series, the dynamic reliability of the series system can be derived from equation (30):
when three parts are connected in series, the dynamic reliability of the series system can be derived from equation (30):
2.2 reliability calculation of parallel systems
A parallel system with n units is shown in FIG. 4, where the numbers 1-n represent the 1 st to n th parts, and the joint distribution function is FZ(z1,z2,···,zn)zi＝σi-Si(σiIndicates the strength of the part, SiRepresenting the stress to which the part is subjected), ziIndicating the function of the ith part, Fi(. cndot.) represents the distribution function of the functional function of the ith part, then there is an n-dimensional Copula function according to Sklar's theorem such that: fZ(z1,z2,···,zn)＝C(F1((z1)),F2(z2),···,Fn(zn) Because F isi(zi) Is continuous, so C (F)1(z1),F2(z2),···,Fn(zn) Is unique, where C (-) represents an n-dimensional Copula function. The reliability of the parallel system is:
the formula (33) is a static reliability calculation formula based on the Copula function parallel system, and on the basis, a Copula reliability analysis model calculation formula of the parallel system can be deduced as follows:
in the formula Xj(t) represents the function of the jth part at the moment t, and the dynamic reliability of the part is Rj(t) dynamic failure probability of Fj(t), then: fj(t)＝1-Rj(t), wherein j ═ 1,2, ·, n.
3. Fuzzy reliability calculation based on Copula function
3.1 Serial System dynamic fuzzy reliability calculation
According to the Copula theory and the formulas (29) and (30) of the dynamic reliability of the series system, and the reliability or the failure probability in the Copula theory is converted into the dynamic fuzzy reliability values of the structure or the part when the strength is not degraded and the strength is degraded which are respectively calculated according to the formulas (18) and (25), then the dynamic fuzzy reliability model based on the Copula function series system can be similarly deduced as follows:
in the above formula
Represents the motion blur reliability value, C (u) of the nth part1,u2,···,un) Is a Copula function, wherein
When two parts are connected in series, the motion blur reliability of the series system can be derived from equation (35):
3.2 parallel system dynamic fuzzy reliability calculation
According to the Copula theory and the dynamic reliability formulas (33) and (34) of the parallel system, the reliability or the failure probability is converted into the dynamic fuzzy reliability values of the structure or the part when the strength is not degraded and the strength is degraded, which are respectively calculated according to the formulas (18) and (25), and then the dynamic fuzzy reliability model based on the Copula function parallel system can be similarly deduced as follows:
in the above formula
Representing the motion blur reliability value of the nth part.
When two parts are connected in series, the dynamic fuzzy reliability of the parallel system can be deduced by the formula (37):
4. estimation of Copula function parameters
For different situations, people adopt different methods to estimate parameters of the Copula function, and the common methods mainly include a maximum likelihood estimation method, a distribution estimation method and a semi-parameter estimation method, but the estimation methods all need to assume or estimate a sample distribution rule in advance to estimate the parameter values of the Copula function, and the estimation accuracy cannot be guaranteed. In order to solve the defects, the method combines nonparametric kernel estimation and a maximum likelihood method to estimate the parameters of the Copula function, and the specific process is as follows:
1) firstly, an nonparametric kernel estimation method is adopted to determine the edge distribution function of a random sample. Assuming a series system consisting of n parts, firstly simulating T times by adopting a Monte Carlo method, and simulating the strength simulation value sigma of the jth partjiAnd its stress analog value SjiDifference of sigmaji-SjiAs samples and unifying the T samples of the jth part as vector Yj＝(Yj1,Yj2,…,Yji,…,YjT) (j ═ 1,2, …, n; i 1,2, …, T), and then vector Y of samplesjSubstituting equation (39) can estimate the density function of the edge distribution function of the random sample to obtain the jth part sample YjEdge distribution density function f at variable yjThe estimated formula of (-) is:
in the formula kj(. h) represents a kernel function for the jth part, the kernel function satisfying the following condition:
hjfor window width, the optimal window width may be:
wherein
The estimated value of the standard deviation of the sample is expressed by the formula
The mean represents a function of the median number,
subtracting the stress sample vector Y for the jth part strengthjThe median of (3).
When k isj(. cndot.) is normally distributed:
the edge distribution function for the sample of the jth part can be written as:
where φ (-) represents the probability density function of a standard normal distribution, and φ (-) represents the distribution function of a standard normal distribution.
2) And (4) solving a distribution function of subtracting the stress sample from the strength of each part according to the previous step, and then calculating a parameter theta of the Copula function by adopting a maximum likelihood method according to the failure sample of each part. Assuming the strength minus stress of the jth part is less than zero, it is designated as a failed sample yjk＝σji-Sji＜0，(j＝1,2,…,n；k＝1,2,…,l；l<T) wherein k represents the number of failures of the part; all failure samples of the jth part are recorded as vectorsZj＝(yj1,yj2,…,yjk,…,yjl)，(j＝1,2,…,n；k＝1,2,…,l；l<T); will ZjSubstituting formula (43) to obtain the edge distribution function value vector of the jth part failure sample as (F)j(yj1),Fj(yj2),…,Fj(yjk),…,Fj(yjl))，U＝[uj1,uj2,…,ujk,…,ujl]Namely, the function value of the edge distribution of each part failure sample of the series system is put into a matrix U,
and (j ═ 1,2, …, n; (k ═ 1,2, …, l; (l < T)), u ═ 1,2, …, l; (l < T)jk TIs ujkTransposing; each column U of UjkRespectively substituted into the probability density function c (u) of the n-dimensional Copula function1k,u2k,…,ujk,…,unk) In (2), a likelihood function is available:
probability density function f of samplej(. cndot.) can be obtained from formula (39). Further taking the logarithm of both sides of the equation above can obtain:
according to mathematical knowledge, the above formula is further derived, and then the derived function is made equal to zero, so that the value of the Copula function parameter theta when the above formula reaches the maximum value can be obtained, and the process can be simplified and expressed as follows:
θ＝arg max lnLθ(46)
5. copula function parameter representation and kendall rank solving method
Let { (x)1,y1),(x2,y2),···,(xN,yN) A sample is composed of N sets of observations of a random sample vector (X, Y), where X, Y denote successive random variables. By using
The number of combinations of r samples out of N samples is shown, and it is apparent that the samples are included in total
Term by observed value (x)i,yi) And (x)j,yj) I ≠ j, j ≠ 1,2, ·, N, and two sets of observations (x) in each combinationi,yi) And (x)j,yj) I ≠ j is either identical or not. Will be provided with
The combination of items being divided into two parts, i.e.
Where c represents the number of two sets of observations that are consistent and d represents the number of two sets of observations that are inconsistent, then one can define:
is a sample { (x)1,y1),(x2,y2),···,(xN,yN) The kendall rank correlation coefficient of.
Suppose (x)1,y1) And (x)2,y2) For random vectors which are independent of each other and have the same distribution
τ≡P[(x1-x2)(y1-y2)＞0]-P[(x1-x2)(y1-y2)＜0](48)
Is the kendall rank correlation coefficient.
Assuming that the edge distributions of the random variables X and Y are f (X), g (Y), respectively, and the Copula function corresponding to the edge distributions is C (u, v), and u ═ f (X), v ═ g (Y), u, v ∈ [0,1], then the kendall rank correlation coefficient is obtained by double integration of the Copula function C (u, v):
according to the kendall rank correlation theory, if the edge distribution of the sample and the corresponding Copula function thereof are known, the kendall rank correlation coefficient τ can be solved through a formula (49), so that the parameters of the Copula function in the sample can also be solved through solving the kendall rank correlation coefficient of the sample. The Copula function parameter used in the method has the following relation with the kendall rank correlation coefficient:
when the Copula function is binary Gauss Copula, τ has the following relationship with its parameters:
when the Copula function is binary Gumbel Copula, τ has the following relationship with its parameters:
when the Copula function is binary Clayton Copula, τ has the following relationship with its parameters:
when the Copula function is binary Frank Copula, τ has the following relationship with its parameters:
when the Copula function is a binary CClayton Copula, the expression of the binary distribution function is:
C(u1,u2；θ)＝u1+u2-1+[(1-u1)-θ+(1-u2)-θ-1]-1/θ，θ∈(0,∞) (54)
τ is related to its parameters by:
the values of the Copula function parameters can be deduced back from equation (49) or directly from equations (50), (53) and (55). I.e. assuming a period of time t ═ t0And aiming at the serial dynamic fuzzy system and the parallel dynamic fuzzy system respectively, and considering the degradation and non-degradation of the fuzzy strength of the parts in the system respectively, selecting different Copula functions according to the model in the step S3, calculating the reliability value of the system in the time t, and simultaneously carrying out fuzzy reliability simulation by using a Monte Carlo method to obtain the reliability value of the system in the time t, thereby selecting the Copula function model with the reliability value closest to the Monte Carlo fuzzy reliability simulation.
Firstly, we use the formula (62) to calculate each part
The method is characterized by sequentially putting the samples into matrixes, wherein the simulation times are N times, so that each matrix has N sample values, the number of the samples is totally two, the logarithm of harmonic and harmonic dissonance in the two matrixes is compared, and the kendall rank correlation coefficient of the two parts can be solved by programming according to a kendall rank solving formula (49).
6. Monte Carlo simulation of fuzzy reliability
According to equation (17), the simulation is performed by splitting into the following two parts:
1) simulation of truncation reliability
According to the horizontal intercept theory, if the membership function of the fuzzy strength and the fuzzy stress is known, the corresponding intercept intervals can be obtained, and then the probability density functions can be respectively obtained according to the intercept intervals
And
and finally, combining a stress-intensity interference theory to obtain an expression of reliability and failure probability. The clipping reliability will be described below
The specific calculation process of (1).
At a known probability density function
And
thereafter, for convenience in describing this process, equation (56) is rewritten to the following generalized form:
further make according to equation (58)
The above equation can be simplified to:
thus equation (58) can be reduced to:
equation (61) is equivalent to solving the sample mean of the integral, and can be simplified as:
in the formula, σjIs a function of probability density
The random sample values that are extracted are,
the distribution function M representing the fuzzy stress intercept interval represents the simulation times.
2) Simulation of fuzzy reliability
The same equation (57) can be reduced approximately to the following form:
in which N denotes the number of N times lambda of the cyclic simulation of step 5a)σAnd λsRespectively, the fuzzy strength and the truncation level of the fuzzy stress.
The specific simulation steps are as follows: firstly, two random numbers lambda are generated by adopting a standard uniform distribution function simulationσiAnd λsiThey respectively represent the interception levels of fuzzy strength and fuzzy stress, and then the probability density function of fuzzy strength interception interval can be determined according to the value of the interception levels
Probability density function of fuzzy stress cut-off interval
Further, their distribution function can be found and then the probability density function can be calculated according to the formula (62)
Extracting a random number sigmajAnd then sigmajSubstituting the fuzzy stress into the distribution function of the fuzzy stress interception interval, and repeating the simulation for M times to obtain the interception reliability
Finally, according to a formula (63), circularly simulating the process for N times to obtain a fuzzy reliability simulation value P of the part when the fuzzy strength of the part is not degradedσ. The simulation method of the fuzzy reliability of the part when the fuzzy strength of the part is degraded is similar to the process, and the simulation value of the fuzzy reliability of the part when the fuzzy strength of the part is degraded can be obtained only by substituting the reliability value into the formula (25).
The present invention is not limited to the above-mentioned embodiments, and based on the technical solutions disclosed in the present invention, those skilled in the art can make some substitutions and modifications to some technical features without creative efforts according to the disclosed technical contents, and these substitutions and modifications are all within the protection scope of the present invention.
Claims (10)
1. A dynamic fuzzy reliability analysis method based on a Copula function failure correlation system is characterized by comprising the following steps:
s1, based on the fuzzy mathematical theory, replacing the strength and stress of a single part with fuzzy strength and fuzzy stress, and then obtaining a part dynamic fuzzy reliability model when the fuzzy strength is not degraded and a part dynamic fuzzy reliability model when the fuzzy strength is degraded under the action of random load by using a horizontal interception theory and a stress-strength interference theory;
s2, based on the dynamic fuzzy reliability of a single part, combining the dynamic fuzzy reliability with a Copula function, substituting the dynamic reliability and parameters of each part, and deducing a dynamic Copula reliability analysis model of a serial-parallel system by using a difference method;
s3, constructing a dynamic fuzzy reliability analysis model of the series-parallel system based on the dynamic fuzzy reliability model of the part with degraded or not degraded fuzzy strength under the random load action constructed in the step S1 and in combination with Copula function analysis in the step S2;
s4, for the dynamic Copula reliability analysis model of the series-parallel system in the step S2, estimating the edge distribution function value of the stress-intensity interference sample of the system and the parameter theta of the Copula function by adopting a nonparametric kernel estimation-maximum likelihood estimation method;
s5, expressing a parameter theta of the Copula function through a kendall rank correlation coefficient, and obtaining the influence of the kendall rank correlation coefficient tau on the dynamic fuzzy reliability of the series and parallel systems, thereby deducing and analyzing the influence of the Copula parameter theta on the dynamic fuzzy reliability of the series and parallel systems;
and S6, calculating the dynamic fuzzy reliability of the system according to the dynamic fuzzy reliability analysis models of the series and parallel systems obtained in the step S3, comparing the calculation result with the result obtained by the Monte Carlo simulation method obtained in advance by the system, and selecting a Copula function model according to the comparison result.
2. The method according to claim 1, wherein in step S1, the reliability model of the dynamic fuzzy of the part is determined when the fuzzy strength is not degraded
The following were used:
in the formula, t represents a time variable,
as a horizontal truncation of the membership function
Distribution function of internal blurring stress, wherein
And
respectively representing the upper and lower limits, λ, of the horizontal cut-sets of the membership functions1The parameters of the simultaneous poisson process are shown,
is a horizontal truncated set
A distribution function of internal loads;
part dynamic fuzzy reliability model during fuzzy strength degradation
In the formula, τ represents a time variable,
and representing the dynamic fuzzy reliability of the structure or the part at the time t.
3. The method according to claim 1, wherein in step S2, the dynamic Copula reliability analysis model R of the tandem systemS(t) the calculation formula is:
wherein t represents a time variable, Δ is a differential sign, and Xj(t) is the function of the jth part at time t, C (-) is the Copula function of the system, Ri(t) dynamic reliability of the part, Fi(t) is the dynamic failure probability of the part, where i ═ 1,2, …, n.
4. The method according to claim 3, wherein in step S2, Copula reliability analysis model R of parallel systemP(t) the calculation formula is:
wherein t represents a time variable, Δ is a differential sign, and Xj(t) represents the function of the jth part at time t, Rj(t) dynamic reliability of the part, Fj(t) is the dynamic failure probability, where j ═ 1,2, …, n.
5. The method according to claim 1, wherein in step S3, a dynamic fuzzy reliability analysis model of the cascade system is constructed
Comprises the following steps:
in the formula (I), the compound is shown in the specification,
representing the motion blur reliability value of the nth part, t representing a time variable, C (u)1,u2,…,un) Is a Copula function, wherein
6. The method according to claim 1, wherein in step S3, a dynamic fuzzy reliability analysis model of the parallel system is constructed
Comprises the following steps:
where C (·, …,) is a Copula function,
representing the motion blur reliability value of the nth part.
7. The method according to claim 1, wherein the step S4 comprises the following steps:
4a) determining the edge distribution function of the random sample by adopting a nonparametric kernel estimation method, and determining the edge distribution function of the random sample by adopting the nonparametric kernel estimation method; assuming a series system consisting of n parts, firstly simulating T times by adopting a Monte Carlo method, and simulating the strength simulation value sigma of the jth partjiAnd its stress analog value SjiDifference of difference Yji＝σji-SjiAs samples and unifying the T samples of the jth part as vector Yj＝(Yj1,Yj2,…,Yji,…,YjT) J is 1,2, …, n; i is 1,2, …, T, and the edge probability density function f of the sample of the jth part can be obtainedj(y) and an edge distribution function Fj(y)：
Wherein y represents the difference between the strength and the stress, kj(. h) represents a kernel function of the jth part, satisfies
hjFor the window width, the optimum window width may be selected as
Wherein the content of the first and second substances,
represents an estimate of the standard deviation of the sample, phi (-) is a standard normal distribution function;
4b) solving a distribution function of subtracting the stress sample from the strength of each part according to the step 4a), and then calculating a parameter theta of a Copula function by adopting a maximum likelihood method according to the failure sample of each part, wherein the log likelihood function is as follows:
in the formula, cθk(u1k,u2k,…,unk) As Copula function C (u)1,u2,…,un) Partial derivatives of (d);
the derivation of equation (45) finds the log-likelihood function lnLθThe maximum value point can obtain the value of the Copula function parameter theta:
θ＝arg max(lnLθ) (46)。
8. the method according to claim 1, wherein in step S6, the fuzzy reliability simulation is performed based on the monte carlo method, and the fuzzy reliability simulation is performed by splitting the fuzzy reliability into two parts, i.e. the truncated reliability and the fuzzy reliability, and the result is as follows:
6a) reliability of truncation
The simulation of (2):
in the formula, σjProbability density function being the fuzzy intensity intercept interval
The random sample values that are extracted are,
representing a distribution function of a fuzzy stress interception interval, wherein M represents simulation times;
6b) fuzzy reliability simulation:
in which N denotes the number of N times lambda of the cyclic simulation of step 6a)σAnd λsRespectively representing the fuzzy strength and the truncation level of the fuzzy stress;
the specific simulation steps are as follows:
6b1) firstly, two random numbers lambda are generated by adopting a standard uniform distribution function simulationσiAnd λsiThey respectively represent the interception levels of fuzzy strength and fuzzy stress, and then the probability density function of fuzzy strength interception interval can be determined according to the value of the interception levels
Probability density function of fuzzy stress cut-off interval
6b2) Further, the distribution function of the two can be obtained, and then the probability density function of the fuzzy intensity intercept interval is calculated according to the formula (53)
Extracting the random value sigmajAnd then sigmajSubstituting the fuzzy stress into the distribution function of the fuzzy stress interception interval, and repeating the simulation for M times to obtain the interception reliability
6b3) Finally, according to a formula (54), the process is circularly simulated for N times, and the fuzzy reliability simulation value P of the part when the fuzzy strength of the part is not degraded can be obtainedσ。
9. The method according to claim 1, wherein in steps S5 and S6, if the edge distribution of the samples and their corresponding Copula functions are known, the kendall rank correlation coefficient τ can be solved by formula (57):
in the formula, C (u)1,u2,…,un) Is a Copula function, where ui∈[0,1],i＝1,2,…,n。
10. The method of claim 9, wherein the parameter θ of the Copula function is related to the kendall rank correlation coefficient τ as follows:
when the Copula function is binary Gauss Copula, τ has the following relationship with its parameters:
when the Copula function is binary Gumbel Copula, τ has the following relationship with its parameters:
when the Copula function is binary Clayton Copula, τ has the following relationship with its parameters:
when the Copula function is binary Frank Copula, τ has the following relationship with its parameters:
wherein t represents a time variable;
when the Copula function is a binary CClayton Copula, the relationship between τ and its parameters is:
according to the formulas (49) - (55), the values of the Copula function parameters can be reversely deduced and substituted into the dynamic fuzzy reliability model of the serial-parallel system in step S5, and the obtained result is compared with the sampling result obtained in claim 8, so as to select the most suitable Copula function.
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