CN108563888A - A method of calculating parallel system reliability - Google Patents
A method of calculating parallel system reliability Download PDFInfo
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Abstract
The present invention provides a kind of method calculating parallel system reliability, including：Step 1：Determine the reliability index and designcheck point of each element；Step 2：According to the result of calculation of step 1, the designcheck point for choosing the maximum element of reliability index is sampling center, extracts normal distribution random vectorSample valueStep 3：Calculate the failure probability P of parallel systemfs, step 4：The reliable guideline of parallel system is calculated according to the failure probability of calculatings.Easy to operate (relative to method of descent) of the invention, it is ensured that higher computational accuracy (relative to the boundary estimation technique), and computational efficiency is higher (relative to direct sampling method), it is stronger to the applicability of practical problem analysis.
Description
Technical field
The present invention relates to System Reliability Analysis field more particularly to a kind of methods calculating parallel system reliability.
Background technology
The reliability for the system being made of several elements depends on the reliability of each element, such as a truss structure system
Reliability be dependent on composition trussing each rod piece reliability.According to the pass between thrashing and component failure
System, can be divided into two kinds of fundamental types of train and parallel system.Train refers to that any element of composition system fails then
The system of thrashing；And all elements that parallel system refers to composition system system that system just fails when all failing.It is assumed that
System is made of m element, and the power function of i-th of element is gi(X1,X2,…,Xn), i=1,2 ..., m, X1,X2,…XnFor
Stochastic variable.The then failure probability p of parallel systemfsIt can be expressed as：
The reliability β of parallel systemsFor：
βs=-Φ-1(Pfs) (2)
In formula, Φ-1() is the inverse function of Standard Normal Distribution.
Because only that parallel system just fails when all elements all fail, so the failure probability of parallel system is than single member
The failure probability of part is low, otherwise the reliability of parallel system is higher than the reliability of discrete component.The reliability of parallel system is not only
Depending on the reliability of discrete component, correlation is influenced also between by element, therefore the reliability calculating of parallel system is one
Complicated nonlinear problem.Especially when the failure probability of element is relatively low, the failure probability of parallel system is very low, in this situation
The lower accurate failure probability (or reliability) for calculating parallel system is more difficult.Precise and high efficiency there is no to calculate complicated taken in conjunction at present
The method of the reliability of system.
Generally pass through the failure probability p of parallel systemfsIt is converted to the reliability β of parallel systems.Calculate parallel system
Failure probability pfsCommon method have it is following several：
(1) method of descent
In formula, Φm() is m dimension standardized normal distributions, and m is the parts number for forming parallel system；β and ρ is respectively each member
Correlation matrix between the reliability index vector sum element of part.
The precision of method of descent is preferable, but due to that will iteratively solve designcheck point in each step reduction process, works as member
Number of packages amount is more, and when the nonlinear degree of element function function is higher, iterative calculation may not restrain.
(2) the boundary estimation technique
Consider that each element is perfectly correlated or is completely independent, it can be deduced that the general compass of parallel system failure probability
For：
In formula, m is the parts number for forming parallel system；pfiFor the failure probability of i-th of element.
The boundary estimation technique calculates simply, but the compass obtained is wider, can not determine parallel system failure probability
Exact value.
(3) direct sampling method
The basic principle that direct sampling method (also known as Monte Carlo Method) calculates parallel system failure probability is as follows：
Assuming that a parallel system is made of m element, the power function of each element isI=1,2 ..., m,
InIt is n dimension random vectors.It randomly selectsSample valueSubstitute into each element
Power function calculates corresponding power function valueWhen the power function value of all elements is both less than 0, thrashing.
If the indicative function that one value of construction is 0 and 1, i.e.,：
In formula,Indicate that the power function value of all elements is both less than 0.
Then the failure probability of parallel system is：
In formula, N is sampling number；For the power function for the m element that jth time sampling obtains
Value.
Direct sampling method calculates simplicity, when sampling number N is sufficiently large, can accurately determine the failure probability of parallel system.
But the disadvantage is that when the failure probability very little of parallel system, very big sampling number N is needed just to can guarantee enough computational accuracies,
Calculating will be dramatically increased to take.
Invention content
It is an object of the invention to solve the problems of the above-mentioned prior art, the weight for calculating parallel system reliability is provided
Sampling, this method is wanted to can guarantee higher computational accuracy, compared with direct sampling method, and greatly improve computational efficiency.
A method of parallel system reliability is calculated, is included the following steps：
Step 1：Determine the reliability index and designcheck point of each element；
Step 2：According to the result of calculation of step 1, the designcheck point for choosing the maximum element of reliability index is sampling
Normal distribution random vector is extracted at centerSample value
Step 3：Calculate the failure probability P of parallel systemfs,
In formula, N is sampling number；For the power function for the m element that kth time sampling obtains
Value；I () is the indicative function that value is 0 and 1；It is normal distribution random vectorJoint
Probability density function；It is by formulaThe joint probability density function of the random number of generation, i.e.,：
Wherein, σjFor XjStandard deviation；U is standardized normal distribution random number；
Step 4：The reliable guideline of parallel system is calculated according to the failure probability of calculatings；
βs=-Φ-1(Pfs)
In formula, Φ-1() is the inverse function of Standard Normal Distribution.
Further, method as described above, the step 1 include the following steps：
Step 1：The initial value of the reliability index of selection elementAnd designcheck pointInitial value, can useI is the serial number of element, and j is the serial number of stochastic variable；
Step 2：The power function of computing element at designcheck point to the partial derivative of stochastic variable, i.e.,K is iterations；
Step 3:Meter sensitivity coefficient, i.e.,：
Step 4：The designcheck point after kth time iteration is determined as the following formula, and k >=0 is：
Step 5：IfAndε is given allowable error, stops changing
Generation, otherwise：
As k=0, directly assume one it is new
Work as k>When 0, determine as the following formulaI.e.：
K=k+1 turns to step 2 and continues iteration.
Beneficial effects of the present invention are：
Easy to operate (relative to method of descent) of the invention, it is ensured that higher computational accuracy (relative to the boundary estimation technique),
And computational efficiency is higher (relative to direct sampling method), it is stronger to the applicability of practical problem analysis.
Description of the drawings
Fig. 1 is the parallel system reliability calculating result figure sampled using the method for the present invention.
Specific implementation mode
To make the object, technical solutions and advantages of the present invention clearer, the technical solution below in the present invention carries out clear
Chu is fully described by, it is clear that described embodiments are some of the embodiments of the present invention, instead of all the embodiments.It is based on
Embodiment in the present invention, those of ordinary skill in the art are obtained every other without creative efforts
Embodiment shall fall within the protection scope of the present invention.
The present invention is achieved through the following technical solutions：
The applicable elements of the method for the invention are：1. the power function for forming the element of parallel system is stochastic variable
Explicit function；2. power function can be to stochastic variable derivation；3. stochastic variable Normal Distribution；4. stochastic variable is mutually only
It is vertical.
When the conditions are satisfied, the reliability that following steps calculate parallel system can be used：
(1) reliability index and designcheck point of each element are determined.
It is assumed that parallel system is made of m element, the power function of i-th of element isI=1,2 ..., m, whereinIt is n dimension normal distribution random vectors, mean value and standard deviation are respectivelyWith
The reliability index and designcheck point of each element are determined using following steps：
1) initial value of the reliability index of selection elementAnd designcheck pointInitial value, can useI is the serial number of element, and j is the serial number of stochastic variable；
2) power function of computing element at designcheck point to the partial derivative of stochastic variable, i.e.,k
For iterations；
3) meter sensitivity coefficient, i.e.,：
4) designcheck point after kth (k >=0) secondary iteration is determined by formula (2), i.e.,：
If 5)And(ε is given allowable error), stops iteration.
Otherwise：1. as k=0, directly assume one it is new2. working as k>When 0, determined by formula (3)I.e.：
K=k+1 turns to step 2) and continues iteration.
(2) according to the result of calculation of step (1), the designcheck point for choosing the maximum element of reliability index is in sampling
The heart extracts normal distribution random vectorSample valueAssuming that i-th element
Reliability index is maximum, and designcheck point is Then random vectorIn j-th
Stochastic variable XjThe sample value of (j=1,2 ..., n) can generate by the following method：
In formula, σjFor XjStandard deviation；U is standardized normal distribution random number, can be generated as the following formula：
In formula, r1And r2For two uniform random numbers on (0,1) section.
(3) failure probability of parallel system is calculated.
In formula, N is sampling number；For the power function for the m element that kth time sampling obtains
Value；I () is the indicative function that value is 0 and 1, i.e.,：
In formula,Indicate that the power function value of all elements is both less than 0.
It is normal distribution random vectorJoint probability density function, i.e.,：
It is the joint probability density function of the random number generated by formula (4), i.e.,：
(4) reliability index of parallel system is calculated.
The reliable guideline of parallel systemsIt is calculated as follows：
βs=-Φ-1(Pfs) (10)
In formula, Φ-1() is the inverse function of Standard Normal Distribution.
Embodiment：
Assuming that a parallel system is made of 4 elements, the power function of each element is：
X in formula (11)1、X2、X3、X4For mutually independent normally distributed random variable, mean μ1=μ2=μ3=μ4=0,
Standard deviation sigma1=σ2=σ3=σ4=1.
The reliability of above-mentioned parallel system is calculated using the method for the invention.Include the following steps：
(1) reliability index and designcheck point of each element are determined.
1) initial value of the reliability index of each elementThe initial value of designcheck point2,3,4, j=1,
2,3,4, i be the serial number of element, and j is the serial number of stochastic variable.
2) power function of each element is calculated at designcheck point to the partial derivative of stochastic variable, i.e.,：
It is above it is various in k be iterations.
3) meter sensitivity coefficient.
By formula
It can be obtained with formula (12)~formula (19)：
α11=1, α22=1, α31=0.19996, α33=0.97980
α41=0.40022, α42=0.90050, α43=-0.08004, α44=0.15008
4) designcheck point after kth (k >=0) secondary iteration is determined by formula (2), i.e.,：
If 5)AndStop iteration.
Otherwise：1. as k=0, directly assume one it is new2. working as k>When 0, determined by formula (3)I.e.：
K=k+1 turns to step 2) and continues iteration.
The reliability index and designcheck point of each element determined using above-mentioned calculating step are shown in Table 1.
(2) according to the result of calculation of step (1), the designcheck point for choosing the maximum element of reliability index (element 3) is
Normal distribution random vector is extracted in sampling center using formula (20)Sample value
In formula, σjFor XjStandard deviation；U is standardized normal distribution random number, is generated using formula (5)：
In formula, r1And r2For two uniform random numbers on (0,1) section.
(3) formula (21) is used to calculate the failure probability of parallel system.
In formula, N is sampling number；For the function letter for 4 elements that kth time sampling obtains
Numerical value；I () is the indicative function that value is 0 and 1, i.e.,：
In formula,Indicate that the power function value of all elements is both less than 0.
It is the joint probability density function of the random number generated by formula (20), i.e.,：
(4) reliability index of parallel system is calculated.
The reliable guideline of parallel systemsIt is calculated as follows：
βs=-Φ-1(Pfs) (10)
In formula, Φ-1() is the inverse function of Standard Normal Distribution.
1 × 10 is sampled respectively using the method for the invention4It is secondary, 5 × 104It is secondary, 10 × 104It is secondary, 50 × 104It is secondary, 100 ×
104It is secondary, 500 × 104It is secondary and 1000 × 104The results are shown in Figure 1 for secondary parallel system reliability calculating.As seen from Figure 1,
The reliability index of parallel system is in sampling 50 × 104Start to restrain after secondary, sampling 100 × 104Secondary reliability index is βs=
5.22.And direct sampling method needs sampling 25000 × 104Same result of calculation could be obtained more than secondary, computational efficiency is far not
Method as described herein.
The reliability index and designcheck point of 1 each element of table
Finally it should be noted that：The above embodiments are merely illustrative of the technical solutions of the present invention, rather than its limitations；Although
Present invention has been described in detail with reference to the aforementioned embodiments, it will be understood by those of ordinary skill in the art that：It still may be used
With technical scheme described in the above embodiments is modified or equivalent replacement of some of the technical features；
And these modifications or replacements, various embodiments of the present invention technical solution that it does not separate the essence of the corresponding technical solution spirit and
Range.
Claims (2)
1. a kind of method calculating parallel system reliability, which is characterized in that include the following steps：
Step 1：Determine the reliability index and designcheck point of each element；
Step 2：According to the result of calculation of step 1, the designcheck point for choosing the maximum element of reliability index is sampling center,
Extract normal distribution random vectorSample value
Step 3：Calculate the failure probability P of parallel systemfs,
In formula, N is sampling number；For the power function value for the m element that kth time sampling obtains；I
() is the indicative function that value is 0 and 1；It is normal distribution random vectorJoint probability
Density function；It is by formulaThe joint probability density function of the random number of generation, i.e.,：
Wherein, σjFor XjStandard deviation；U is standardized normal distribution random number；
Step 4：The reliable guideline of parallel system is calculated according to the failure probability of calculatings；
βs=-Φ-1(Pfs)
In formula, Φ-1() is the inverse function of Standard Normal Distribution.
2. according to the method described in claim 1, it is characterized in that, the step 1 includes the following steps：
Step 1：The initial value of the reliability index of selection elementAnd designcheck pointInitial value, can useI=1,2 ..., m, j=1,2 ..., n, i are the serial number of element, and j is the serial number of stochastic variable；
Step 2：The power function of computing element at designcheck point to the partial derivative of stochastic variable, i.e.,k
For iterations；
Step 3:Meter sensitivity coefficient, i.e.,：
Step 4：The designcheck point after kth time iteration is determined as the following formula, and k >=0 is：
Step 5：IfAndε is given allowable error, stops iteration,
Otherwise：
As k=0, directly assume one it is new
Work as k>When 0, determine as the following formulaI.e.：
K=k+1 turns to step 2 and continues iteration.
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Citations (2)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| CN101520652A (en) * | 2009-03-03 | 2009-09-02 | 华中科技大学 | Method for evaluating service reliability of numerical control equipment |
| CN107239645A (en) * | 2017-08-10 | 2017-10-10 | 西南交通大学 | Without backfill arch open cut tunnel structure probability Reliability design method under rock-fall impact |
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Patent Citations (2)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| CN101520652A (en) * | 2009-03-03 | 2009-09-02 | 华中科技大学 | Method for evaluating service reliability of numerical control equipment |
| CN107239645A (en) * | 2017-08-10 | 2017-10-10 | 西南交通大学 | Without backfill arch open cut tunnel structure probability Reliability design method under rock-fall impact |
Non-Patent Citations (2)
| Title |
|---|
| 孙宁: "结构可靠度计算方法分析及在系留机构中的应用", 《中国优秀硕士学位论文全文数据库 工程科技II辑》 * |
| 张峰: "基于模拟退火的并联系统失效概率的计算方法", 《机械强度》 * |
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Application publication date: 20180921 |