CN109214094B - Reliability model of multi-degradation process and random impact competition failure system - Google Patents

Reliability model of multi-degradation process and random impact competition failure system Download PDF

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CN109214094B
CN109214094B CN201811068748.XA CN201811068748A CN109214094B CN 109214094 B CN109214094 B CN 109214094B CN 201811068748 A CN201811068748 A CN 201811068748A CN 109214094 B CN109214094 B CN 109214094B
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孙富强
程圆圆
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Beihang University
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Abstract

The invention discloses a novel reliability model of a multi-degradation process and random impact competition failure system. Assuming that random impacts obey a homogeneous poisson process and the probabilities that the impacts are lethal and non-lethal are p (t) and q (t), respectively, wherein a lethal impact, once it occurs, results in immediate system failure; on the basis, the influence of the non-fatal impact on the degradation is considered by correcting the existing non-linear Wiener process degradation model, and a final system reliability function is established by utilizing a time-varying Copula function. Finally, numerical cases are given to illustrate the mathematical model presented herein.

Description

Reliability model of multi-degradation process and random impact competition failure system
Technical Field
The invention belongs to the technical field of reliability analysis of a competitive failure system, and particularly relates to a reliability model of a multi-degradation process and random impact competitive failure system.
Background
Currently, with the development of technologies such as design and manufacturing level and material process, a plurality of systems are more and more complex in structure and function, and often have a plurality of failure mechanisms, which cause a plurality of key performance parameters of the system to degrade simultaneously, and the degradation processes of the performance parameters may affect each other, and certain dependency exists in the degradation data. Meanwhile, the system is continuously influenced by random impact of the external environment in the working process. In general, system failure modes can be classified into the following two types: (1) when the increment of a certain degradation process exceeds a given failure threshold, the system is degraded or soft failed (soft failures). (2) When the system suffers random impact, the system has burst failure (catastrophic failures) or hard failure (hard failures). Therefore, the failure process of the system is usually the result of a dependent competition between multiple performance parameter degradation failures and a random impact induced burst failure.
Due to the complex system failure mechanism, the multiple degradation processes and the random impact are often correlated and mutually influenced. Thus, there are generally two following dependencies in such degradation-impact multi-mode competitive failure systems: (1) dependence between multiple degeneration processes. In practical applications, due to the complex structure and function, a system usually has multiple performance parameter degradations and multiple degradation mechanisms simultaneously, each degradation mechanism may be related to one or more performance parameters, so that degradation processes of the performance parameters are influenced mutually, and certain dependency exists. (2) The dependence between the degradation process and random impact. The mutual influence of the two is mainly reflected in two aspects: on one hand, the degradation can make the system more fragile under the action of random impact, the system fault is accelerated, and the fault is directly caused when the degradation exceeds a threshold value; on the other hand, when a random shock is applied, a phenomenon in which the degradation amount is increased stepwise or the degradation rate is accelerated may occur in the system degradation process. Therefore, for such a complex system with dependent contention failure, reliability evaluation under independent assumption by ignoring the existing dependencies often results in low credibility or even erroneous results.
Therefore, the key for reliability evaluation of a system with multiple degradation processes and random impact competition failure is to respectively establish a degradation model and an impact model and consider the dependence relationship on the basis.
Currently, in the existing degradation-impact competition failure research, an impact model is basically established by using a poisson process to describe different types of impact processes to which a system is subjected. The modeling of the random impact process using the poisson process is mainly based on the following considerations: (1) the poisson process is an important point process, and is reasonable to characterize the single-event effect phenomenon of random impact; (2) the poisson process has a memoryless property, in other words, the impact occurs randomly; (3) the rate of occurrence of the poisson process λ (t) may be of any form, such as being well-chosen to describe the frequency of occurrence of random impacts.
In the existing degradation-impact competition failure reliability modeling literature, a simple generalized trajectory model, namely a linear regression model, is mainly used for describing the degradation behavior of the system, and the influence of random impact on the degradation process is considered on the basis. However, the linear regression model assumes that the intrinsic degradation process of the system is deterministic, which is an oversimplification of the actual degradation process. In practice, the degradation process of the system is usually complicated, and has various randomness and uncertainty, and is influenced by environmental factors. Therefore, using a stochastic process to build the degradation model is a more desirable choice because it has a time-dependent structure. However, the current research based on the wiener process degradation model only relates to the situation of linear degradation trajectory. However, in practical engineering applications, due to the complexity of the system structure and the failure structure, the system degradation behavior tends to have nonlinearity. Therefore, when modeling the degradation process of the competitive failure system, the nonlinear degradation behavior of the system should be considered.
Therefore, the method introduces a nonlinear Wiener process and expands the model in the modeling process of the competitive failure system. In addition, most of the existing documents only consider the competitive relationship between a single degradation process and random impact, the reliability modeling problem of a competitive failure system containing a plurality of degradation processes is rarely involved, and the research on the situation that a plurality of dependent degradation processes exist is less.
The invention expects to use a nonlinear Wiener process and a time-varying Copula method to construct a generalized competition failure system reliability model comprehensively considering the dependency between degradation and impact and the dependency between multiple degradation processes.
Disclosure of Invention
Aiming at the defects in the prior art, the invention provides a reliability model of a multi-degradation process and random impact competition failure system.
The model is established by the following method:
(1) random impact model
Assuming that the arrival times of the random impacts obey a homogeneous poisson process { N (t), t >0} with the strength of lambda, and N (t) represents the occurrence times of the random impacts at the time t, the probability of m random impacts can be represented as:
Figure GDA0001826511530000021
the random impacts are classified into lethal impacts and non-lethal impacts, and the probability p (t) of a single random impact being a lethal impact is:
p(t)＝1-exp(-γt) (2)
where γ is a normal number, the probability of this impact being a non-fatal impact is q (t) ═ 1-p (t);
with N1(t) represents the number of fatal impacts occurring at time t, N2(t) represents the number of non-fatal impacts occurring at time t, only if N1When (t) is 0, it can be guaranteed that the system does not have hard failure, and the probability that fatal impact does not occur is as follows:
Figure GDA0001826511530000031
during the [0, t ] time period, the probability that the system is subjected to k times of non-fatal impact actions is as follows:
Figure GDA0001826511530000032
when in [0, t]When no fatal impact occurs in the time period, order N2(t) ═ k; by YjJ-1, 2, …, k denotes the amplitude of each impact, YjIs an independent and uniformly distributed positive random variable, and Y is setjSubject to a normal distribution, i.e.
Figure GDA0001826511530000033
μYAnd σYAre the corresponding mean and standard deviation;
(2) model of the process of degradation
The influence of random impact is not considered, a nonlinear Wiener process is adopted to model the degradation process, and then the degradation model is as follows:
M1:Di(t)＝ν0iΛ(t；θi)+σBiB(t) (5)
in the formula, Di(t) represents the amount of degradation of the i-th degradation process at time t; v. of0iIs a drift coefficient, representing the degradation rate of the degradation process; Λ (t; θ)i) A non-linear characteristic, θ, of non-decreasing time scale function, used to describe the degradation behavioriIs a parameter of the non-linear function; sigmaBiIs the diffusion coefficient; b (-) is standard Brownian Motion (BM);
(3) modified degradation model taking impact effects into account
Further considering the influence of non-lethal impact on the degradation process, the non-lethal impact has two influence mechanisms on the degradation process, the degradation quantity is suddenly changed, the degradation rate is increased, and the model M is subjected to1And (5) correcting:
(a) effect of non-lethal impact on amount of degeneration
Let the impact amplitude YjJ-1, 2, …, k causes the i-th degeneration process to have a step increment of W ij1,2, … n, and WijAnd YjThere is the following relationship between:
Wij＝aiYj (6)
in the formula, aiExpressing the influence of the unit impact amplitude on the i-th degradation process degradation increment, YjObey a normal distribution, then WijAlso obey a normal distribution, i.e.
Figure GDA0001826511530000041
Wherein mui＝aiμY，σi＝aiσY；
When the system is subjected to non-fatal impact, the degradation process has step increment W ij1,2, … n, and the cumulative degradation increment caused by the non-fatal impact on the ith degradation process is recorded as Si(t), expressed as:
Figure GDA0001826511530000042
in the formula, Wi0＝0；
(b) Effect of non-lethal impact on degradation Rate
The system will have accelerated degradation rate after non-fatal impact, and the degradation rate v is setiAnd Si(t) there is a direct proportional relationship between:
Figure GDA0001826511530000043
in the formula, riIs a dependent factor, and the value range is [0, ∞ ];
bringing the degradation rate into the degradation model M1The corrected natural degradation amount of the i-th degradation process is obtained as
Figure GDA0001826511530000044
Comprehensively considering the influence of non-lethal impact on the degeneration quantity and the degeneration rate, the total degeneration quantity X of the i-th degeneration processi(t) from the corrected natural degradation Di(t) cumulative degradation increment S due to random impacti(t) composition, i.e. Xi(t)＝Di(t)+Si(t), the total degradation amount of the i-th degradation process can be expressed as:
Figure GDA0001826511530000045
Xi(t) first crossing its given threshold diIs considered as the degradation failure life T of the i-th degradation process of the systemiObtaining a degradation model M0The corresponding failure and survival probability of the ith degradation process can be known through the FPT distribution;
let the Cumulative Distribution Function (CDF) and Probability Density Function (PDF) of FPT of the i-th degeneration process be respectivelyIs FDi(t) and fDi(t), i ═ 1, 2.., n, then the CDF expression is as follows:
FDi(t)＝P(Xi(t)≥di)＝P(t≥Ti) (11)
further, a conditional probability R is obtained that no degradation failure occurs in the case where the ith degradation process has undergone k times of non-fatal impactsi(t) is:
Figure GDA0001826511530000051
wherein the content of the first and second substances,
Figure GDA0001826511530000052
wherein A isi、Bi、EiAnd GiComprises the following steps:
Ai＝di-ν0iΛ(t；θi)+tν0iΛ′(t；θi)
Bi＝1/ri+ν0iΛ(t；θi)-tν0iΛ′(t；θi)
Figure GDA0001826511530000053
Figure GDA0001826511530000054
(4) system reliability model
The system has n degeneration processes with X degeneration amounti(t), i ═ 1,2, …, n, corresponding failure threshold diWith a time of deterioration of TiThe system can continue to work with two competing risk conditions that must be met: all degradation processes are below their respective failure thresholds, with no fatal impact occurring, and therefore the reliability of the system, r (t), is expressed as:
Figure GDA0001826511530000055
considering that there is a time-varying dependency between multiple degradation processes, a time-varying Copula method is used to obtain a joint distribution function of the multiple degradation processes, and the reliability function of equation (15) can be expressed as:
R(t)＝C(R1(t),R2(t),…,Rn(t)；αt)×P(N1(t)＝0) (16)
in the formula, alphatIs a parameter of a time-varying Copula function, and uses an ARMA (1,10) process to express alphatDynamic evolution equation of
Figure GDA0001826511530000061
Wherein Δ (. cndot.) is secured to αtThe transfer function introduced always in the definition domain, i.e. the logistic transfer function, b0、b1、b2Respectively, are parameters of the dynamic evolution equation.
Compared with the prior art, the invention has the following beneficial effects:
(1) the degradation behavior of a competitive failure system is characterized by using a nonlinear Wiener process, and compared with the conventional generalized trajectory model and a linear Wiener process, the randomness and the nonlinearity in the degradation behavior of the system can be more accurately described;
(2) two influence mechanisms of random impact on the degradation process are comprehensively considered by modifying a Wiener process degradation model: the amount of degradation increases stepwise and the rate of degradation accelerates, characterizing the ability to resist fatal shocks weaker and weaker as the system state gradually decreases, by assuming that the probability of occurrence of fatal shocks increases gradually over time.
(3) A time-varying Copula method is used for representing a dependent structure among a plurality of degradation processes.
Drawings
FIG. 1 is a contention failure mechanism of the system of the present invention;
FIG. 2 is a graph of the reliability function of two degradation processes;
FIG. 3 is a parameter estimation result of a time-varying Copula function;
FIG. 4 is a graph of race fault reliability;
fig. 5 is a system reliability curve.
Detailed Description
In order to make the objects, technical solutions and advantages of the present invention more apparent, the present invention is further described in detail below with reference to examples, and it should be understood that the specific examples described herein are only for the purpose of explaining the present invention and are not intended to limit the present invention.
During the whole life cycle, the system is subjected to a multi-phase competition-dependent failure process consisting of a plurality of degradation processes and random impacts. The system undergoes two dependent failure processes: soft failures caused by degradation and hard failures caused by impact, whichever occurs first, result in system failure. The above failure processes are interdependent in that they have certain common factors, such as the same environmental/operational stresses, usage history, material quality, and system maintenance. Therefore, the dependency between multiple failure processes should be considered in the competitive failure model.
In order to establish a reliability model for a competitive failure system of multiple degradation processes and random impacts, the following assumptions are made:
a) the system has n degradation processes, and the accumulated degradation amount of any one process exceeds the critical failure threshold value of the process, so that the system fails. All degradation processes can be described using a non-linear wiener process.
b) The random shock experienced by the system can be classified as a non-lethal shock (Y as shown in FIG. 1)1,Y2,Y3) And fatal impact (Y shown in FIG. 1)4) Two types are provided. Wherein, the probability that a single random impact is a fatal impact is p (t), and once the single random impact occurs, the system can be caused to be failed instantaneously; the probability of a non-fatal impact is q (t) -1-p (t). There are two ways that non-lethal impact can affect the degradation process: the amount of degeneration is abrupt and the rate of degeneration is increased as shown in fig. 1. Assuming that p (t) is an increasing function over time,the characteristic that the impact resistance of the system is gradually reduced along with the performance degradation is shown.
c) There are two competing failure mechanisms for the system: hard failure due to fatal impact (t shown in FIG. 1)4Hard failure from time-to-time fatal impact); and fatal impact does not occur, the cumulative degradation of a degradation process in the system exceeds its critical failure threshold (degradation process X shown in FIG. 1)1Over d1The case(s).
d) The method is characterized in that the dependence exists among a plurality of degradation processes of the system, and the dependence is described by adopting a time-varying Copula method.
The following is the process of establishing a reliability model of a multi-degradation process and random impact competition failure system:
(1) random impact model
Assuming that the arrival times of the random impacts obey a homogeneous poisson process { N (t), t >0} with the strength of lambda, and N (t) represents the occurrence times of the random impacts at the time t, the probability of m random impacts can be represented as:
Figure GDA0001826511530000071
the random impacts experienced by the system include both lethal and non-lethal impacts. Let the probability p (t) that a single random impact is a fatal impact be:
p(t)＝1-exp(-γt) (19)
where γ is a normal number, the probability of this impact being a non-fatal impact is q (t) ═ 1-p (t). Lethal and non-lethal impacts are subjected to a non-homogeneous poisson process with intensities λ p (t) and λ q (t), respectively.
Next, N is used1(t) represents the number of fatal impacts occurring at time t, N2(t) represents the number of times of occurrence of non-fatal impact at time t, and N (t) is N1(t)+N2(t), and N1(t) and N2(t) are independent of each other. Only when N is present1When the value (t) is 0, the system can be ensured not to generate hard failure, and the probability that fatal impact does not occur is as follows:
Figure GDA0001826511530000072
meanwhile, in the [0, t ] time period, the probability that the system is subjected to k times of non-fatal impact actions is as follows:
Figure GDA0001826511530000081
if at [0, t]In the time period, no fatal impact occurs, and N is2(t) ═ k; by YjJ-1, 2, …, k denotes the amplitude of each impact, YjAre independent and equally distributed positive random variables. Is provided with YjSubject to a normal distribution, i.e.
Figure GDA0001826511530000082
μYAnd σYAre the corresponding mean and standard deviation.
(2) Model of the process of degradation
In the existing competitive failure modeling analysis, most researches describe the degradation process of the system by using a linear model X (t) ═ ω + β t, and the simple linear model is not suitable due to the randomness and nonlinearity of the degradation process of the system. The invention models the degradation process by using a nonlinear Wiener process. On one hand, the Wiener process has independent incremental Markov characteristics, and can more reasonably reflect the randomness of system degradation; on the other hand, the nonlinearity of the degradation behavior can be more accurately described through a nonlinear Wiener process, and the limitation that only a strict monotonous degradation phenomenon can be described is not provided.
When the influence of random impact is not considered, a nonlinear Wiener process is adopted to model the degradation process, and the degradation model is as follows:
M1:Di(t)＝ν0iΛ(t；θi)+σBiB(t) (22)
in the formula, Di(t) represents the amount of degradation of the i-th degradation process at time t (for simplicity, assume the initial degradation value is 0 or make the initial degradation value by transformation)Is 0); v. of0iIs a drift coefficient, representing the degradation rate of the degradation process; Λ (t; θ)i) A non-linear characteristic, θ, of non-decreasing time scale function, used to describe the degradation behavioriIs a parameter of the non-linear function; sigmaBiIs the diffusion coefficient; b (-) is the standard Brownian Motion (BM).
(3) Modified degradation model taking impact effects into account
When there is no non-fatal impact, the system follows M1The described track is degenerated, however, in engineering practice, the system is in a complex working environment and is constantly influenced by random impact in an external environment, and non-fatal impact has two influence mechanisms on the degeneration process: the amount of degeneration is abrupt and the rate of degeneration is increased. The invention passes through the pair model M1And correcting, comprehensively considering the two influences, and representing the dependency between the degradation and the impact.
(a) Effect of non-lethal impact on amount of degeneration
The impact of a non-fatal impact on the amount of degradation is primarily due to the impact amplitude, but the same impact amplitude is different for the amount of damage per degradation process. Let the impact amplitude YjJ-1, 2, …, k causes the i-th degeneration process to have a step increment of WijI is 1,2, … n. Suppose WijAnd YjThere is a linear relationship between:
Wij＝aiYj (23)
in the formula, aiExpressing the influence of the unit impact amplitude on the degradation increment of the ith degradation process; when a isiWhen 1, the amplitude representing the impact applies an equivalent value to the i-th degeneration process; when a isiAt 0, the amplitude representing the impact does not increment the i-th degeneration process. Y isjSubject to the assumption of a normal distribution, then WijAlso obey a normal distribution, i.e.
Figure GDA0001826511530000095
Wherein mui＝aiμY，σi＝aiσY。
When the system is affected by abnormalityAfter the action of the catastrophic impact, the degradation process takes place in a step increment W ij1,2, … n, and the cumulative degradation increment caused by the non-fatal impact on the ith degradation process is recorded as Si(t), then Si(t) can be represented by a composite poisson process:
Figure GDA0001826511530000091
in the formula, Wi0＝0。
(b) Effect of non-lethal impact on degradation Rate
Non-lethal impact may also alter the rate of degradation of the system while affecting the amount of degradation. Aiming at the phenomenon that the degradation rate is accelerated after the system is subjected to non-lethal impact, the invention assumes the degradation rate viAnd Si(t) there is a direct proportional relationship between the two, taking into account the dependencies that exist between them:
Figure GDA0001826511530000092
in the formula, riIs a dependent factor, has a value range of [0, ∞), and is obviously vi/v0i>1, which characterizes the degradation rate acceleration. If riA value of 0 indicates that a non-lethal impact does not affect the rate of degradation.
Bringing the degradation rate into the degradation model M1The corrected natural degradation amount of the i-th degradation process is obtained as
Figure GDA0001826511530000093
The total degradation X of the i-th degradation process after considering the above two effects of non-fatal impacti(t) from the corrected natural degradation Di(t) cumulative degradation increment S due to random impacti(t) composition, i.e. Xi(t)＝Di(t)+Si(t), then the ith degenerationThe total degradation of the process can be expressed as:
Figure GDA0001826511530000094
Xi(t) first crossing its given threshold diIs considered as the degradation failure life T of the i-th degradation process of the systemi. By finding a degradation model M0The corresponding failure and survival probability of the i-th degeneration process can be known.
Let the Cumulative Distribution Function (CDF) and Probability Density Function (PDF) of FPT of the i-th degeneration process be FDi(t) and fDi(t), i ═ 1, 2.., n, then the CDF expression is as follows:
FDi(t)＝P(Xi(t)≥di)＝P(t≥Ti) (28)
further, a conditional probability R is obtained that no degradation failure occurs in the case where the ith degradation process has undergone k times of non-fatal impactsi(t) is:
Figure GDA0001826511530000101
wherein the content of the first and second substances,
Figure GDA0001826511530000102
wherein A isi、Bi、EiAnd GiComprises the following steps:
Ai＝di-ν0iΛ(t；θi)+tν0iΛ′(t；θi)
Bi＝1/ri+ν0iΛ(t；θi)-tν0iΛ′(t；θi)
Figure GDA0001826511530000103
Figure GDA0001826511530000104
(4) system reliability model
The system has n degeneration processes with X degeneration amounti(t), i ═ 1,2, …, n, corresponding failure threshold diWith a time of deterioration of Ti. As can be seen from the definition of soft failure, the system fails immediately as soon as any one of the n degradation processes exceeds its corresponding failure threshold. In addition, when a fatal impact occurs, the system can experience hard failure. The system can still continue to work and therefore must satisfy two competing risk conditions: 1) all degradation processes are below their corresponding failure thresholds; 2) no fatal impact occurs.
Thus, the reliability of the system, r (t), can be expressed as:
Figure GDA0001826511530000111
if the degradation processes are independent of each other, the system reliability of equation (15) can be expressed as:
Figure GDA0001826511530000112
considering that the dependency exists among the multiple degradation processes, the Copula method is used for obtaining the joint distribution function of the multiple degradation processes. In addition, in order to reflect that the dependency relationship between the degradation processes may change with time, the time-varying Copula function is used to perform the calculation of the multivariate joint distribution, and then the reliability function of the system is:
R(t)＝C(R1(t),R2(t),…,Rn(t)；αt)×P(N1(t)＝0) (34)
in the formula, alphatIs a parameter of the time-varying Copula function, is a quantity varying with time, and is usually expressed by a dynamic evolution equation. The most widely used ARMA (1,10) procedure is used to express alphatDynamic evolution equation of
Figure GDA0001826511530000113
Wherein Δ (. cndot.) is secured to αtThe transformation function introduced always within the defined domain, the logistic transformation function. Table 1 shows several common time-varying Copula functional forms and their logistic transfer functions; b0、b1、b2Respectively, are parameters of the dynamic evolution equation.
TABLE 1 common time-varying Copula function
Figure GDA0001826511530000114
The time-varying Copula functions are of various types, an AIC criterion and a BIC criterion are selected as test standards of goodness of fit, and the smaller the AIC and BIC values are, the higher the fitting degree of the model is.
The competition failure system model of the invention is adopted to carry out example analysis on a competition failure system containing two degradation processes and an impact process. Let λ be 3 × 10 as the generation intensity of random impact-3The probability of fatal impact is p (t) 1-exp (-gamma t), gamma is 0.0002, the amplitude of random impact follows normal distribution, and the average value is muY0.6, variance σY0.01. The first degradation process has a respective amount of degradation of X1(t) corresponding to a soft fail threshold of d1The degradation rate is a drift function of 12
Figure GDA0001826511530000121
Wherein v01＝1×10-3，r1＝0.05，W1j＝a1Yj，a10.75 as a function of time scale
Figure GDA0001826511530000124
θ11.3, diffusion coefficient σB10.2; the second degradation process has a respective amount of degradation of X2(t) corresponding to a soft fail threshold of d2The degradation rate is a drift function of 7
Figure GDA0001826511530000122
Wherein v02＝3×10-4，r2＝0.02，W2j＝a2Yj，a20.1, time scale function of
Figure GDA0001826511530000123
θ21.3, diffusion coefficient σB20.1; through the calculation formula (12) of the system reliability, the conditional probability curves of the two degradation processes without degradation failure can be obtained as shown in fig. 2.
And fitting the joint degradation distribution by adopting four time-varying Copula, including a time-varying Normal Copula function, a time-varying Gumbel Copula function, a time-varying Clayton Copula and a time-varying Frank Copula function. Fig. 3 shows the parameter evolution process curves of the four time-varying Copula functions, where the solid line is the parameter evolution process curve of the time-varying Copula function, and the dotted line is the parameter evolution process curve of the static Copula function. It can be seen from the figure that the parameter evolution process curve of the time-varying Copula function has obvious changes, so that when the dependency relationship between the degradation processes is considered, the condition that the dependency relationship may change should not be ignored, sometimes the effect of fitting the joint distribution of the time-varying Copula function is better, and the time-varying Copula function can reflect the change condition of the dependent structure measure along with time.
As shown in fig. 4, it can be seen that, assuming that there is a great difference between the independent and dependent competitive failure reliability curves in the degradation processes, the reliability model obtained without considering the dependency may underestimate the health status of the system, which may result in making an erroneous decision and bringing economic loss, and therefore it is very necessary to reasonably consider the dependent structure in the degradation processes.
Table 2 gives the fitting results of the time-varying copula function, and it can be seen that the ranking results of AIC and BIC are consistent. Best used to fit the joint degradation profile is a time-varying Gumbel Copula function with AIC of-1253.013 and BIC of-1243.058; next is the time-varying Normal Copula function with an AIC of-1191.207 and a BIC of-1181.252.
TABLE 2 fitting Effect of time-varying Copula function
Figure GDA0001826511530000131
The joint distribution function of the time-varying Gumbel Copula function fitting degradation process is selected according to the sorting result of Table 2, and the reliability of the competitive failure model of the system is further obtained as shown in FIG. 5, which is obviously higher than the reliability based on the independence assumption condition. It is shown that when there is a dependency between the degradation processes, it needs to be fully considered in the degradation modeling.

Claims (1)

1. A method for establishing a reliability model of a multi-degradation process and random impact competition failure system is characterized by comprising the following steps:
(1) random impact model
Assuming that the arrival times of the random impacts obey a homogeneous poisson process { N (t), t >0} with the strength of lambda, and N (t) represents the occurrence times of the random impacts at the time t, the probability of m random impacts can be represented as:
Figure FDA0002665824640000011
the random impacts are classified into lethal impacts and non-lethal impacts, and the probability p (t) of a single random impact being a lethal impact is:
p(t)＝1-exp(-γt) (2)
where γ is a normal number, the probability of this impact being a non-fatal impact is q (t) ═ 1-p (t);
with N1(t) represents the number of fatal impacts occurring at time t, N2(t) represents the number of non-fatal impacts occurring at time t, only ifN1When (t) is 0, it can be guaranteed that the system does not have hard failure, and the probability that fatal impact does not occur is as follows:
Figure FDA0002665824640000012
during the [0, t ] time period, the probability that the system is subjected to k times of non-fatal impact actions is as follows:
Figure FDA0002665824640000013
when in [0, t]When no fatal impact occurs in the time period, order N2(t) ═ k; by YjJ-1, 2, …, k denotes the amplitude of each impact, YjIs an independent and uniformly distributed positive random variable, and Y is setjSubject to a normal distribution, i.e.
Figure FDA0002665824640000014
μYAnd σYAre the corresponding mean and standard deviation;
(2) model of the process of degradation
The influence of random impact is not considered, a nonlinear Wiener process is adopted to model the degradation process, and then the degradation model is as follows:
M1:Di(t)＝ν0iΛ(t；θi)+σBiB(t) (5)
in the formula, Di(t) represents the amount of degradation of the i-th degradation process at time t; v. of0iIs a drift coefficient, representing the degradation rate of the degradation process; Λ (t; θ)i) A non-linear characteristic, θ, of non-decreasing time scale function, used to describe the degradation behavioriIs a parameter of the non-linear function; sigmaBiIs the diffusion coefficient; b (-) is standard Brownian motion;
(3) modified degradation model taking impact effects into account
Further consider the effect of non-lethal impact on the degradation process, which has two typesMechanism of influence, mutation of degeneration amount and increase of degeneration rate, on model M1And (5) correcting:
(a) effect of non-lethal impact on amount of degeneration
Let the impact amplitude YjJ-1, 2, …, k causes the i-th degeneration process to have a step increment of Wij1,2, … n, and WijAnd YjThere is the following relationship between:
Wij＝aiYj (6)
in the formula, aiExpressing the influence of the unit impact amplitude on the i-th degradation process degradation increment, YjObey a normal distribution, then WijAlso obey a normal distribution, i.e.
Figure FDA0002665824640000021
Wherein mui＝aiμY，σi＝aiσY；
When the system is subjected to non-fatal impact, the degradation process has step increment Wij1,2, … n, and the cumulative degradation increment caused by the non-fatal impact on the ith degradation process is recorded as Si(t), expressed as:
Figure FDA0002665824640000022
in the formula, Wi0＝0；
(b) Effect of non-lethal impact on degradation Rate
The system will have accelerated degradation rate after non-fatal impact, and the degradation rate v is setiAnd Si(t) there is a direct proportional relationship between:
Figure FDA0002665824640000023
in the formula, riIs a dependent factor, and the value range is [0, ∞ ];
bringing the degradation rate into the degradation model M1The corrected natural degradation amount of the i-th degradation process is obtained as
Figure FDA0002665824640000024
Comprehensively considering the influence of non-lethal impact on the degeneration quantity and the degeneration rate, the total degeneration quantity X of the i-th degeneration processi(t) from the corrected natural degradation Di(t) cumulative degradation increment S due to random impacti(t) composition, i.e. Xi(t)＝Di(t)+Si(t), the total degradation amount of the i-th degradation process can be expressed as:
Figure FDA0002665824640000031
Xi(t) first crossing its given threshold diThe Time First Passage Time, FPT for short, is considered as the degradation failure life T of the i-th degradation process of the systemiObtaining a degradation model M0The corresponding failure and survival probability of the ith degradation process can be known through the FPT distribution;
let the cumulative distribution function and probability density function of FPT of i-th degeneration process be FDi(t) and fDi(t), i ═ 1, 2.., n, then the CDF expression is as follows:
FDi(t)＝P(Xi(t)≥di)＝P(t≥Ti) (11)
further, a conditional probability R is obtained that no degradation failure occurs in the case where the ith degradation process has undergone k times of non-fatal impactsi(t) is:
Figure FDA0002665824640000032
in the formula (I), the compound is shown in the specification,
Figure FDA0002665824640000033
wherein A isi、Bi、EiAnd GiComprises the following steps:
Ai＝di-ν0iΛ(t；θi)+tν0iΛ′(t；θi)
Bi＝1/ri+ν0iΛ(t；θi)-tν0iΛ′(t；θi)
Figure FDA0002665824640000034
Figure FDA0002665824640000035
(4) system reliability model
The system has n degeneration processes with X degeneration amounti(t), i ═ 1,2, …, n, corresponding failure threshold diWith a time of deterioration of TiThe system can continue to work with two competing risk conditions that must be met: all degradation processes are below their respective failure thresholds, with no fatal impact occurring, and therefore the reliability of the system, r (t), is expressed as:
R(t)＝P(X1(t)<d1,X2(t)<d2,K,Xn(t)<dn)·P(N1(t)＝0)
＝P(R1(t),R2(t),...,Rn(t))·P(N1(t)＝0) (15)
considering that there is a time-varying dependency between multiple degradation processes, a time-varying Copula method is used to obtain a joint distribution function of the multiple degradation processes, and the reliability function of equation (15) can be expressed as:
R(t)＝C(R1(t),R2(t),...,Rn(t)；αt)×P(N1(t)＝0) (16)
in the formula, alphatIs a parameter of a time-varying Copula function, and uses an ARMA (1,10) process to express alphatDynamic evolution equation of
Figure FDA0002665824640000041
Wherein Δ (. cndot.) is secured to αtThe transfer function introduced always in the definition domain, i.e. the logistic transfer function, b0、b1、b2Respectively, a parameter of the dynamic evolution equationt-1Is a parameter of the time-varying Copula function at the time of t-1;
the degradation process is abrasion loss, crack size or corrosion loss, and the random impact is vibration impact, temperature impact, overstress or overload.
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