CN110096820A - Method for predicting residual life of k/n (G) system when unit life obeys Weibull distribution - Google Patents
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Abstract
The invention provides a residual life prediction method of a k/N (G) system when the unit life obeys Weibull distribution, which comprises the steps of firstly randomly extracting samples of N k/N (G) system composition units for life test, obtaining the failure time of each sample, and estimating the failure probability of the failure time of each sample; calculating point estimates of life distribution parameters of k/n (G) system composition units based on a horizontal error function; predicting the remaining life of the k/n (G) system according to the point estimation of the life distribution parameters of the k/n (G) system composition units. The invention well solves the problem of predicting the residual service life of a k/n (G) system when the unit service life obeys Weibull distribution through the steps, and has simple steps, clear result and easy operation.
Description
Technical field
The invention belongs to predicting residual useful life fields, and in particular to a kind of unit institute for Weibull Distributed Units
The method for predicting residual useful life of k/n (G) system of composition.
Background technique
Reliability refers to ability of the product under the defined conditions with completion predetermined function in the defined time [with reference to text
It offers: the Changsha Guo Bo, Wu little Yue systems reliability analysis [M]: publishing house, the National University of Defense technology, 2002:5-6.].Reliability
Evaluation problem is the important content of reliability consideration, and the index of assessment reliability includes reliability, service life and remaining life etc.,
Middle remaining life indicates product remaining normal working hours after current time.
Reliability is the build-in attribute of product, is the important indicator for measuring product quality quality, therefore the reliability of product
Problem is extremely important, especially in defence and military field.Through improving the reliability of product, k/ frequently with redundancy in engineering
N (G) system is a kind of wherein common mode.K/n (G) system, which refers to, forms a system by n same unit, when at least
When k unit works normally, k/n (G) system can be worked normally.
The key of prediction product remaining life is obeyed using the sample data estimation life of product of life of product test
Parameter in distribution then predicts remaining life in conjunction with the service life distribution of product.But for k/n (G) system,
Due to being difficult to be collected into the sample data of k/n (G) lifetime of system, so can not Direct Modeling and analysis k/n (G) system lifetim
Distribution, then can not the remaining life directly to k/n (G) system predict.But then, it is collected into k/n (G) system group
It is to be relatively easy at the service life sample data of unit.It then can be first with the service life sample number of k/n (G) system component units
According to come model and analyze k/n (G) system component units service life distribution, in conjunction with the pass of k/n (G) system and its component units
System, analysis k/n (G) system lifetim distribution, and the remaining life of k/n (G) system is predicted.
In existing research, the reliable of k/n (G) system is analyzed when being distributed according to the service life of k/n (G) system component units
Property index when, mostly use exponential distribution describe the service life of component units distribution [bibliography: Li Xiaofei's belief system refers to
Target approximate confidence limit analysis [D] Wenzhou University, 2014；Shi little Lin is stepped up under II type truncated sample that k/N (G) system can
By confidence interval [J] the firepower and command and control of property index, 2012,37 (4): 30-33.].The probability density letter of exponential distribution
Number is
Wherein t > 0 is the service life of component units, and θ > 0 is the distribution parameter of exponential distribution.But existing theory shows product
Service life obey Weibull distribution [bibliography: Ling Dan Weibull distribution model and its application in Mechanical Reliability mostly
Research [D] University of Electronic Science and Technology, 2011.], and the probability density function of Weibull distribution is
Wherein t > 0 is the service life of product, and m > 0 and η > 0 are respectively the form parameter and scale parameter of Weibull distribution.
It being apparent from from formula (2), the exponential distribution in formula (1) is a kind of special shape of Weibull distribution, i.e., as form parameter m=1, prestige
Boolean's distribution will become exponential distribution.When it is assumed that k/n (G) system component units Weibull Distributed Units when, it is how right
The reliability index of k/n (G) system is analyzed, and existing research is also deficienter, especially in this case how to k/n
(G) remaining life of system is predicted, does not there is related public technology also at present.
Summary of the invention
For in the prior art when cell life obey Weibull distribution when k/n (G) system predicting residual useful life technology
The problem of shortage, the predicting residual useful life side of k/n (G) system when present invention provides a kind of cell life obedience Weibull distribution
Method.
The technical scheme is that
The method for predicting residual useful life of k/n (G) system, includes the following steps: when cell life obeys Weibull distribution
(1) sample for randomly selecting N number of k/n (G) system component units carries out life test, when obtaining the failure of each sample
Between, wherein N >=2；
(2) estimate the failure probability of the out-of-service time of each sample；
(3) point estimation of k/n (G) system component units service life distribution parameter is calculated based on horizontal error function；
(4) the remaining longevity of k/n (G) system is predicted according to the point estimation of k/n (G) system component units service life distribution parameter
Life.
In (1) of the invention, the working condition of each sample is observed in life test, if a certain moment starts certain a sample not
It can work on, then the moment is the out-of-service time of the sample.The sample of the sample obtained by life test in the present invention
Notebook data, that is, sample out-of-service time.
In (2) of the invention, remember that the sample data of the N number of sample obtained through life test is t1,…,tNIf t1≤…≤
tN, i is referred to as sample data tiOrder, according to formula (3) sample estimates data tiFailure probability:
The implementation method of (3) of the invention is as follows:
(3.1) distribution function of Weibull distribution is linearized:
The probability density function of Weibull distribution is
Wherein t > 0 is the service life of product, and m > 0 and η > 0 are respectively the form parameter and scale parameter of Weibull distribution；
According to the probability density function of Weibull distribution in formula (2), the distribution function that can obtain Weibull distribution is
Wherein t > 0 is the service life of product, and m > 0 and η > 0 are respectively the form parameter and scale parameter of Weibull distribution.
Logarithm operation is taken to the distribution function of Weibull distribution twice, formula (4) can be converted to
Ln (- ln [1-F (t)])=mlnt-mln η (5)
(3.2) horizontal error function of the point estimation about Weibull distribution parameters is established:
Thought based on fitting of distribution constructs horizontal error function
WhereinIt is the sample data t acquired by formula (3)iFailure
Probability, xi=lnti, t1≤…≤tNFor the out-of-service time of sample, m and η are the form parameter and scale parameter of Weibull distribution.
(3.3) point estimation of Weibull distribution parameters is calculated:
According to the necessity condition of function minimum [bibliography: Gan Yingai, rich equal operational research (the 3rd edition) Beijing in field:
Publishing house, Tsinghua University, 2005.] it is found that error function S is inclined about the single order of m and η when error function S minimum in formula (6)
Derivative is 0, i.e.,
It can be obtained by abbreviation
To acquire the point estimation of k/n (G) system component units service life distribution parameter m and ηWith
In step (4) of the present invention, the predicting residual useful life value after k/n (G) system τ moment is
WhereinFor incomplete gamma functions.Formula (14) is according to k/n (G) system group
At the remaining life of point estimation prediction k/n (G) system of cell life distribution parameter, wherein the derivation process of formula (14) is such as
Under:
The service life for remembering n component units in k/n (G) system is T1,…,Tn, and be T by ascending order arrangement postscript(1)≤…≤
T(n).Then according to k/n (G) system and the relationship of component units it is found that k/n (G) system lifetim TsFor T(n-k+1), i.e., n composition
Big rear (n-k+1) a service life is discharged to from small in cell life.According to the concept [bibliography: cogongrass poem pine, soup of order statistic
Silver-colored, Beijing Wang Lingling reliability statistics [M]: Higher Education Publishing House, 2008:44-45.] it is found that T(n-k+1)Probability it is close
Spending function is
The probability density function f (t) and distribution function F (t) of Weibull distribution, can obtain k/n in substitution formula (2) and formula (4)
(G) probability density function of lifetime of system is
To the f in formula (9)s(t) abbreviation is carried out, can be obtained
According to the definition of remaining life, it is known that the probability density function of the remaining life L after k/n (G) system τ moment is
Then remaining life L after k/n (G) system τ moment is contemplated to be
The probability density function f of k/n (G) lifetime of system in substitution formula (10)s(t) and abbreviation formula (12) it, can obtain
WhereinFor incomplete gamma functions.
The final distribution parameter point estimation being distributed using k/n (G) the system unit service life acquired in step (2)With?
Predicting residual useful life value after to k/n (G) system τ moment is
Advantageous effects of the invention:
As described above, current invention assumes that the Weibull Distributed Units of k/n (G) system unit, first according to unit sample
The lifetime data of product acquires the distribution parameter point estimation of Weibull distribution, closes further according to the structure of k/n (G) system unit and system
System, provides the probability density function of k/n (G) lifetime of system, then derives the remaining life expectation of k/n (G) system, final to utilize
The distribution parameter point estimation of Weibull distribution acquires the predicting residual useful life value of k/n (G) system.The present invention is through the above steps
The predicting residual useful life problem of k/n (G) system when cell life obeys Weibull distribution is well solved, and step is simple,
As a result clear, it is easily operated.
Specific embodiment
In order to which the purposes, technical schemes and advantages of the disclosed invention are more clearly understood with reference to specific embodiments
And with reference to the accompanying drawings, the present invention is described in more detail.It should be noted that not described in attached drawing or specification description
Content and part English are abbreviated as content known to those of ordinary skill in technical field.It is given in the present embodiment
Some special parameters are only as demonstration, and the value can change accordingly to suitably be worth in different real-time modes.
Momenttum wheel is the critical component of satellite attitude control system, is substantially electronic product, therefore single momenttum wheel
Weibull Distributed Units [bibliography: Liu Qiang, Huang Xiuping, Zhou Jinglun, golden light, momenttum wheel of the Sun Quan based on the physics of failure
Bayesian reliability assesses [J] aviation journal, 2009,30 (8)].In order to guarantee the reliability of momenttum wheel work, in satellite knot
K/n (G) system, specially 3/4 (G) system, i.e. n=4, k=3 are designed in structure for it.This example is directed to momenttum wheel 3/4 (G)
System, and using the out-of-service time data of one group of 9 momenttum wheel unit sample, (G) system of momenttum wheel 3/4 is run 100 hours
Remaining life afterwards is predicted.This 9 samples out-of-service time that life test obtains is 96900,100300 respectively,
100800,122600,103300,103400,105400,151300 and 162400 (hours).
This 9 out-of-service time data are arranged as 96900,100300,100800,103300,103400 according to ascending order,
105400,122600,151300 and 162400.Then, each corresponding failure probability of out-of-service time data is calculated according to formula (3) to estimate
Evaluation is respectively as follows: 0.0745,0.1809,0.2872,0.3936,0.5,0.6064,0.7128,0.8191 and 0.9255.
The point estimation for determining the distribution parameter m and η in momenttum wheel service life according to formula (7) isWith
Finally, by point estimationWithSubstituting into formula (14), prediction obtains momenttum wheel 3/4 afterwards
(G) remaining life of system after 100 hours is 110215.13 hours.
The remaining life of k/n (G) system is pre- when obeying Weibull distribution by the above cell life proposed by the invention
Survey method can be directed to k/n (G) system, when its cell life obeys Weibull distribution, pass through the service life of analytical unit sample
Data predict the remaining life of system.In conclusion when cell life proposed by the invention obeys Weibull distribution
The method for predicting residual useful life of k/n (G) system is easily operated, and result is accurate.
The method of the present invention is not limited to be applied to momenttum wheel k/n (G) system of momenttum wheel composition, and the present invention is for other k/n
(G) system also adapts to extensively.
Contain the explanation of the preferred embodiment of the present invention above, this be for the technical characteristic that the present invention will be described in detail, and
Be not intended to for summary of the invention being limited in concrete form described in embodiment, according to the present invention content purport carry out other
Modifications and variations are also protected by this patent.The purport of the content of present invention is to be defined by the claims, rather than by embodiment
Specific descriptions are defined.
Claims (7)
- The method for predicting residual useful life of k/n (G) system when 1. cell life obeys Weibull distribution, which is characterized in that including such as Lower step:(1) sample for randomly selecting N number of k/n (G) system component units carries out life test, obtains the out-of-service time of each sample, Wherein N >=2；(2) estimate the failure probability of the out-of-service time of each sample；(3) point estimation of k/n (G) system component units service life distribution parameter is calculated based on horizontal error function；(4) remaining life of k/n (G) system is predicted according to the point estimation of k/n (G) system component units service life distribution parameter.
- The predicting residual useful life side of k/n (G) system when 2. cell life according to claim 1 obeys Weibull distribution Method, which is characterized in that in step (2), remember that the out-of-service time of the N number of sample obtained through life test is respectively t1,…,tNIf t1 ≤…≤tN, i is referred to as tiOrder, according to formula (3) estimate tiFailure probability:
- The predicting residual useful life side of k/n (G) system when 3. cell life according to claim 2 obeys Weibull distribution Method, which is characterized in that the implementation method of step (3) is as follows:(3.1) distribution function of Weibull distribution is linearized；(3.2) horizontal error function of the point estimation about Weibull distribution parameters is established:(3.3) point estimation of Weibull distribution parameters is calculated.
- The predicting residual useful life side of k/n (G) system when 4. cell life according to claim 3 obeys Weibull distribution Method, which is characterized in that in step (3.1), according to the probability density function of Weibull distribution, the distribution letter of Weibull distribution can be obtained Number isWherein t > 0 is the service life of product, and m > 0 and η > 0 are respectively the form parameter and scale parameter of Weibull distribution；Logarithm operation is taken to the distribution function of Weibull distribution twice, formula (4) can be converted toLn (- ln [1-F (t)])=mlnt-mln η (5).
- The predicting residual useful life side of k/n (G) system when 5. cell life according to claim 4 obeys Weibull distribution Method, which is characterized in that in step (3.2), the thought based on fitting of distribution constructs horizontal error functionWherein It is tiFailure probability, xi=lnti, m and η are Weibull point The form parameter and scale parameter of cloth.
- The predicting residual useful life side of k/n (G) system when 6. cell life according to claim 5 obeys Weibull distribution Method, which is characterized in that in step (3.3), according to the necessity condition of function minimum it is found that error function S is most in formula (6) Hour, error function S is 0 about the first-order partial derivative of m and η, i.e.,It can be obtained by abbreviationTo acquire the point estimation of k/n (G) system component units service life distribution parameter m and ηWith
- The predicting residual useful life side of k/n (G) system when 7. cell life according to claim 6 obeys Weibull distribution Method, which is characterized in that in step (4), the predicting residual useful life value after k/n (G) system τ moment isWhereinFor incomplete gamma functions.
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