CN114565232A - Storage availability evaluation method for missile engine - Google Patents

Storage availability evaluation method for missile engine Download PDF

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CN114565232A
CN114565232A CN202210118327.3A CN202210118327A CN114565232A CN 114565232 A CN114565232 A CN 114565232A CN 202210118327 A CN202210118327 A CN 202210118327A CN 114565232 A CN114565232 A CN 114565232A
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张朝先
王苗苗
杨力
陈懿
焦华宾
王子健
彭涛
马小兵
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Abstract

The invention provides a missile engine storage availability evaluation method, namely a missile engine storage availability evaluation method based on a secondary sampling inspection scheme and historical sampling inspection data, which comprises the following implementation steps of: firstly, parameter estimation is carried out according to historical sampling inspection data; secondly, calculating the probability of the missile engine passing through the second-level sampling inspection; thirdly, iteratively calculating the instantaneous availability evaluation value of the engine; fourthly, calculating an average availability evaluation value of the engine; the storage reliability and the storage availability of the missile engine are evaluated under the condition of no need of additional experiments, the estimated value is derived from actual data, the method has authenticity, accuracy and stability, the iteration method is simple, the requirement on historical data is low, the operability is high, the average availability of the missile engine can be evaluated, and the instantaneous availability of the missile engine can be evaluated.

Description

Storage availability evaluation method for missile engine
Technical Field
The invention provides a missile engine storage availability evaluation method, namely a missile engine storage availability evaluation method based on a secondary sampling inspection scheme and historical sampling inspection data, in particular to a missile engine storage availability evaluation method based on the secondary sampling inspection scheme and the historical sampling inspection data, and belongs to the field of missile engine storage availability evaluation.
Background
The missile is in a storage state in most of the life cycle of the missile, wherein a missile engine is a key component of the missile, and whether the engine stored for a long time can meet the mission requirement or not can be quantified through data. The availability is an important index for evaluating the readiness integrity of equipment, because the equipment reliability, maintainability, testability, supportability and other design characteristics are comprehensively evaluated, and the missile availability evaluation aims at determining factors and rules influencing the missile availability and establishing a scientific and reasonable missile availability evaluation method.
Mathematical models of the degree of availability can be roughly divided into two types, namely probability models and statistical models; the probability model is used for deducing the reliability quantity index related to the service life of the system from the system structure and the information related to the service life distribution, repair time distribution and the like of the components, and further can discuss the optimal design, use and maintenance strategies and the like of the system; the statistical model estimates and tests the service life, reliability, availability index and the like of the component or the system based on statistical data.
For the missile engine, because secondary sampling inspection is required periodically to check hidden faults and maintain the missile engine during storage, the main factors influencing the usability of the missile engine are hidden fault time and unavailable time caused by maintenance; therefore, the main method for evaluating the availability of the missile engine is to analyze and process data obtained by sampling inspection and then adopt a specific method for evaluation according to the characteristics of the sampling inspection data.
Under the background, how to scientifically evaluate the usability of the missile engine during storage has very important significance for improving the readiness integrity of equipment and saving the maintenance and guarantee cost.
Disclosure of Invention
(1) The purpose of the invention is as follows: the existing method for evaluating the storage availability of the missile engine is to calculate the proportion of the time of the missile engine in an available state in given time to the total time; while this method can assess the average availability of a missile engine, it cannot assess the instantaneous availability of a missile engine at any one time; however, for weapons such as missiles and the like, the instantaneous availability index is more important, so the invention provides a missile engine storage availability evaluation method, namely a missile engine storage availability evaluation method based on a secondary sampling scheme and historical sampling data, namely a missile engine storage availability evaluation method based on a periodic secondary sampling scheme and historical sampling data of an engine.
(2) The technical scheme is as follows:
the basic assumptions made by the present invention are as follows:
Supposing 1 that the storage failure time of the missile engine follows Weibull distribution according to a plurality of literature conclusions and experimental results, the probability density of the storage failure of the engine at the time t is as follows:
Figure BDA0003497381900000021
in the formula: t represents the accumulated working time of the engine after leaving the factory; (t) represents the probability density of engine failure at time t; eta > 0, representing the proportionality coefficient of the Weibull distribution; m > 0, representing the shape factor of the Weibull distribution;
the corresponding cumulative distribution function of storage failure is:
Figure BDA0003497381900000022
in the formula: t represents the accumulated working time of the engine after leaving the factory; f (t) represents the probability of engine failure at time t; eta > 0, representing the proportionality coefficient of the Weibull distribution; m > 0, representing the shape factor of the Weibull distribution;
thus, the storage reliability function of a missile engine is:
Figure BDA0003497381900000023
in the formula: t represents the accumulated working time of the engine after leaving the factory; r (t) represents the reliability of the engine at time t; eta > 0, representing the proportionality coefficient of the Weibull distribution; m > 0, representing the shape factor of the Weibull distribution;
assuming that 2 missile engines carry out secondary sampling maintenance once every delta T time and the engines are in an unavailable state during sampling inspection, N engines are used in a certain sampling inspection, a samples are extracted in the first sampling inspection, the number of allowable sampling inspection samples is b, and the time spent in each sampling inspection is T dAll samples which are randomly inspected are maintained, and whether the random inspection is passed or not is judged according to the following rules:
if the number of unqualified samples for the first sampling inspection is less than b, the sampling inspection is considered to pass;
if the number of unqualified samples of the first sampling inspection is equal to b, performing second sampling inspection, extracting a from the rest N-a engines, if unqualified samples exist, determining that the sampling inspection does not pass, performing factory repair on all the engines, and if unqualified samples do not exist, determining that the sampling inspection passes;
if the number of unqualified samples of the first random inspection is more than b, the random inspection is considered to be failed, and all engines need to be returned to the factory for maintenance;
suppose 3, since the engines maintained by each sampling check only occupy a small part of all the engines, the passing probabilities of each sampling check are assumed to be independent;
it is known that the missile engine experiences the total time at t
Figure BDA0003497381900000031
Second-level sampling examination (
Figure BDA0003497381900000039
Lower rounded symbols) comprising n first spot checks and e (n) second spot checks;
the method provided by the invention mainly comprises the following steps: performing parameter estimation according to historical sampling data, calculating the probability of the missile engine passing through secondary sampling, iteratively calculating the instantaneous availability evaluation value of the engine, and calculating the average availability evaluation value of the engine;
Based on the assumptions and the thinking, the invention provides a method for evaluating the storage availability of the missile engine, namely a method for evaluating the storage availability of the missile engine based on a secondary sampling inspection scheme and historical sampling inspection data, namely a method for evaluating the storage availability of the missile engine based on the secondary sampling inspection data, which is realized by the following four steps:
the method comprises the following steps: estimating parameters according to historical spot check data;
given that the missile engine totally carries out n times of secondary sampling inspection, the obtained sampling inspection data comprises: age t of spot examinationiTotal number of spot checks aiNumber of false spot checks biI ═ 1, 2, …, n; firstly, evaluating the storage reliability of the corresponding missile engine during the spot check according to the proportion of the number of unqualified spot checks in the spot check number:
Figure BDA0003497381900000032
in the formula: t is tiIndicating a spot check age of the engine; r (t)i) Indicating that the engine is at tiThe reliability of the time; a isiRepresents tiThe total number of spot checks at that moment; biRepresents tiThe number of unqualified spot checks at the moment;
since the storage failure of the missile engine follows Weibull distribution, the corresponding reliability function is
Figure BDA0003497381900000033
The following mathematical transformation is made to the reliability function:
Figure BDA0003497381900000034
Figure BDA0003497381900000035
in the formula: t represents the accumulated working time of the engine after leaving the factory; r (t) represents the reliability of the engine at time t; eta > 0, representing the proportionality coefficient of the Weibull distribution; m > 0, representing the shape factor of the Weibull distribution;
Consider this equation as being represented by the coordinates (x)i，yi) Linear function of composition: y isi＝cxi+ d, wherein
Figure BDA0003497381900000036
c＝m，xi＝ln ti，d＝-m lnη；
The sampling data is correspondingly transformed and then substituted into the equation, and the least square regression fitting method is utilized to obtain the scale parameter estimation value
Figure BDA0003497381900000038
And shape parameter estimation
Figure BDA0003497381900000037
Step two: calculating the probability of the missile engine passing the secondary sampling inspection;
by Ppass(i Δ T) represents the probability of the engine passing the ith secondary spot check, and there are two cases of passing the spot check according to the secondary spot check rule in hypothesis 2: firstly, the engine is qualified by first sampling inspection; the second sampling inspection of the engine is qualified; calculating the probabilities of the two conditions respectively, and adding the probabilities to obtain the probability of the engine passing the spot check;
the unqualified sample number X is determined in the ith secondary sampling inspection processiObeying a binomial distribution XiB (c, F (i delta T)), wherein F (i delta T) represents the probability that the engine sampling inspection is unqualified (namely the storage failure occurs) at the moment;
thus the number of defective products XiThe probability of j is
Figure BDA0003497381900000041
In the formula: xiThe number of unqualified samples in the ith secondary sampling inspection process is shown; p (X)iJ) is the number of defective products XiA probability of j; a is the number of sampling samples; delta T is the interval time of two secondary sampling tests; f (i delta T) represents the failure probability of the missile engine at the time of i delta T; r (i delta T) represents the reliability of the missile engine at the moment i delta T;
So that the probability P that the engine passes the spot check at the ith secondary spot checkpass(i Δ T) is:
Figure BDA0003497381900000042
in the formula: ppass(i Δ T) represents the probability of the engine passing the spot check at the ith secondary spot check; a is the number of sampling samples; b represents the number of allowable sampling sample miseligibility; delta T is the interval time of two secondary sampling tests; t is tdRepresents the time spent in each spot check; f (i Δ T) represents the probability of engine failure at time i Δ T; r (i Δ T) represents the reliability of the engine at time i Δ T; r (i Delta T + T)d) Indicating that the engine is at i Δ T + TdThe reliability of the time;
step three: iteratively calculating an instantaneous availability evaluation value of the engine;
defining the state function of the missile engine at any moment t as:
Figure BDA0003497381900000043
the probability that the engine is in a usable state at time t is a0(t) P (s (t) 1), the probability cannot be directly calculated because the engine may have been previously maintained, but a structure a may be iteratively calculated0(t) an expression;
by T1The time of the first secondary sampling maintenance completion time of the engine is represented; at time T, according to T1And t, there are two situations in the engine, namely: phi T1The engine is not maintained when the engine is not maintained; ② T1< T, the engine has undergone maintenance; a. the 0(t) consists of the two cases above, namely:
A0(t)＝P(S(t)＝1，T1≥t)+P(S(t)＝1，T1＜t) (8)
in the formula: t represents the accumulated working time of the engine after leaving the factory; t is a unit of1The time of the first secondary sampling maintenance completion time of the engine is represented; a. the0(t) represents the probability that the engine is in a usable state at time t; p(s) (T) 1, T1T) represents the probability that the engine is in a usable state at time t and has not undergone maintenance; p(s) (T) 1, T1< t) represents the probability that the engine was in an available state at time t and underwent maintenance;
the probabilities of the two cases in the available states are calculated as follows:
①T1not less than t, the engine has not undergone maintenance
Instantaneous availability of the engine in this case:
P(S(t)＝1，T1≥t)＝R(t)P(T1≥t) (9)
in the formula: t represents the accumulated working time of the engine after leaving the factory; t is1The time of the first secondary sampling maintenance completion time of the engine is represented; p(s) (T) 1, T1T) represents the probability that the engine is in a usable state at time t and has not undergone maintenance; r (t) represents the reliability of the engine at time t; p (T)1T) represents the probability that the engine has not undergone maintenance at time t;
because the probability of the engine being pumped in each sampling inspection is
Figure BDA0003497381900000051
Therefore, the probability that the engine has not undergone secondary spot check maintenance at time t is:
Figure BDA0003497381900000052
In the formula: p (T)1T) represents the probability that the engine has not undergone maintenance at time t; delta T is the interval time of two secondary sampling tests; p represents the probability of the engine being pumped at each spot check; e (n) represents the average number of second sampling tests; n represents the total number of times of secondary sampling inspection of the engine by the time t; ppass(i Δ T) represents the probability of the engine passing the spot check at the ith secondary spot check;
wherein, the average times e (n) of the second sampling examination are:
Figure BDA0003497381900000053
in the formula: e (n) represents the average number of second sampling tests; n represents the total number of times of secondary sampling inspection of the engine by the time t; a is the number of sampling samples; b represents the number of false sampling; delta T is the interval time of two secondary sampling tests; f (i Δ T) represents the probability of engine failure at time i Δ T; r (i Δ T) represents the reliability of the engine at time i Δ T;
thus:
Figure BDA0003497381900000054
to simplify the expression, let:
Figure BDA0003497381900000061
representing the probability that the engine has not been serviced for the first n secondary spot checks, equation (11) can be expressed as:
P(S(t)＝1，T1≥t)＝R(t)G(n) (13)
in the formula: t represents the accumulated working time of the engine after leaving the factory; t is1The time of the first secondary sampling maintenance completion time of the engine is represented; p(s) (T) 1, T1T) represents the probability that the engine is in a usable state at time t and has not undergone maintenance; r (t) represents the reliability of the engine at time t; n represents the total number of times of secondary sampling inspection of the engine by the time t; a is the number of sampling samples; indicating the number of allowable miseligibility of the sampling sample; g (n) represents the probability that the engine has not undergone maintenance within n secondary spot checks; p represents the probability of the engine being pumped at each spot check; delta T is the interval time of two secondary sampling tests; f (i Δ T) represents the probability of engine failure at time i Δ T; r (i Δ T) represents the reliability of the engine at time i Δ T; p pass(i Δ T) represents the probability of the engine passing the spot check at the ith secondary spot check;
②T1t, the engine has undergone maintenance
Suppose that the engine is maintained for the first time in the nth secondary sampling inspection, and the time T1 when the engine is returned to the factory and transported back to the outfield has three conditions:
case 1: t is a unit of1＝nΔT+td(ii) a The engine is not maintained in the first n-1 secondary sampling inspection, and the first sampling inspection of the nth secondary sampling inspection is performed;
the probability of occurrence of case 1 is therefore:
P(T1＝nΔT+td)＝G(n-1)p (14)
in the formula: t is1The time of the first secondary sampling maintenance completion time of the engine is represented; td represents the time taken for each spot check; delta T is the interval time of two secondary sampling tests; n represents the total number of times of secondary sampling inspection of the engine by the time t; p (T)1＝nΔT+td) Represents the probability of occurrence of case 1; g (n-1) represents the probability that the engine has not undergone maintenance within n-1 secondary spot checks; p represents the probability of the engine being pumped at each spot check;
case 2: t is1＝nΔT+2td；
Firstly, the engine is not maintained in the first n-1 secondary sampling inspection, the first sampling inspection of the nth secondary sampling inspection is not performed with sampling, but the number of unqualified samples in the first sampling inspection reaches a critical value b, the second sampling inspection is performed, and the engine is performed with sampling in the second sampling inspection;
The probability of this occurring is:
Figure BDA0003497381900000062
in the formula: t is1The time of the first secondary sampling maintenance completion time of the engine is represented; t is tdRepresents the time spent in each spot check; delta T is the interval time of two secondary sampling tests; n represents the total number of times of secondary sampling inspection of the engine by the time t; p1(T1＝nΔT+2td) Representing the probability of occurrence of the situation (i); g (n-1) represents the probability that the engine has not undergone maintenance within n-1 secondary spot checks; p represents the probability of the engine being pumped at each spot check; a is the number of sampling samples; b represents the number of allowable sampling sample miseligibility; f (n Δ T) represents the probability of engine failure at time n Δ T; r (n Δ T) represents the reliability of the engine at time n Δ T;
the engine is not maintained in the first n-1 secondary sampling inspection, the first sampling inspection of the nth secondary sampling inspection is not performed, but the number of unqualified samples in the first sampling inspection exceeds a critical value b, and the whole batch of engines are returned to the factory for maintenance;
the probability of this occurrence is:
Figure BDA0003497381900000071
in the formula: t is1The time of the first secondary sampling maintenance completion time of the engine is represented; t is tdRepresents the time spent in each spot check; delta T is the interval time of two secondary sampling tests; n represents the total number of times of secondary sampling inspection of the engine by the time t; p 2(T1＝nΔT+2td) Representing the probability of occurrence of the case two; g (n-1) represents the probability that the engine has not undergone maintenance within n-1 secondary spot checks; p represents the probability of the engine being pumped at each spot check; a is the number of sampling samples; b represents the number of allowable sampling sample miseligibility; f (n Δ T) represents the probability of engine failure at time n Δ T; r (n Δ T) represents the reliability of the engine at time n Δ T;
the probability of occurrence of case 2 is therefore:
Figure BDA0003497381900000072
in the formula: t is1The time of the first secondary sampling maintenance completion time of the engine is represented; t is tdRepresents the time spent in each spot check; delta T is the interval time of two secondary sampling tests; n represents the total number of times of secondary sampling inspection of the engine by the time t; p (T)1＝nΔT+2td) Represents the probability of occurrence of case 2; g (n-1) represents the probability that the engine has not undergone maintenance within n-1 secondary spot checks; p represents the probability of the engine being pumped at each spot check; a is the number of sampling samples; b represents the number of allowable sampling sample miseligibility; f (n Δ T) represents the probability of engine failure at time n Δ T; r (n Δ T) represents the reliability of the engine at time n Δ T;
case 3: t is1＝nΔT+3td(ii) a The engine is not maintained in the first n-1 secondary spot checks, the first and second spot checks of the nth secondary spot check do not have spot checks, but the number of unqualified samples in the first spot check of other engines reaches a critical value b, the second spot check is carried out, and then the second spot check does not pass;
The probability of occurrence of case 3 is therefore:
Figure BDA0003497381900000081
in the formula: t is1The time of the first secondary sampling maintenance completion time of the engine is represented; t is tdRepresents the cost of each spot checkThe time of (d); delta T is the interval time of two secondary sampling tests; n represents the total number of times of secondary sampling inspection of the engine by the time t; p ═ T1＝nΔT+3td) Represents the probability of occurrence of case 3; g (n-1) represents the probability that the engine has not undergone maintenance within n-1 secondary spot checks; p represents the probability of the engine being pumped at each spot check; a is the number of sampling samples; b represents the number of allowable sampling sample miseligibility; f (n Δ T) represents the probability of engine failure at time n Δ T; r (n Δ T) represents the reliability of the engine at time n Δ T; r (n.DELTA.T + T)d) Indicating that the engine is at n Δ T + TdThe reliability of the time;
because the maintenance requires time, after the engine is maintained, the time from the next secondary sampling inspection time is not delta T any more, but delta T-Td，ΔT-2td，ΔT-3tdDepending on the time required for the last repair, it is therefore necessary to determine the probability that the engine in the maintenance mode is in a usable state;
note A0(k, T) denotes the first maintenance period of the engine as Δ T-ktdProbability of being in usable state later at time t, where k is 1, 2, 3, a0The calculation of (k, t) may construct the following iterative equation:
Figure BDA0003497381900000082
In the formula:
Figure BDA0003497381900000083
jt indicating that the first repair occurred after the start of the ith maintenancedTime of day; t is tdRepresents the time spent for each spot check; delta T is the interval time of two secondary sampling tests; a. the0(k, T) denotes the first maintenance period of the engine as Δ T-ktdProbability of being in an available state later at time t;
Figure BDA0003497381900000084
representing the probability that the engine is in a usable state at time t and has not undergone maintenance; n represents that the engine experiences two stages in total by time tThe number of sampling tests;
Figure BDA0003497381900000085
jt indicating that the first repair occurred after the start of the ith maintenancedA probability of a time of day;
the iterative algorithm for the probability that the missile engine is in a usable state at any one time t is thus:
Figure BDA0003497381900000086
in the formula:
Figure BDA0003497381900000088
jt indicating that the first repair occurred after the start of the ith maintenancedTime of day; delta T is the interval time of two secondary sampling tests; t is tdRepresents the time spent in each spot check; a. the0(t) represents the probability that the engine is in a usable state at time t;
Figure BDA0003497381900000087
representing the probability that the engine is in a usable state at time t and has not undergone maintenance; n represents the total number of times of secondary sampling inspection of the engine by the time t; g (n) represents the probability that the engine has not undergone maintenance within n secondary spot checks; r (t) represents the reliability of the engine at time t;
Figure BDA0003497381900000091
jt indicating that first service occurred after the first maintenance was started dA probability of a time of day;
step four: calculating an average availability evaluation value of the engine;
a calculated in the last step since the engine is specified to be in an unavailable state during the spot check0(t) probability of engine being in a usable state, therefore, it is necessary to assign A to0(t) converting to obtain the engine availability A (t) and
Figure BDA0003497381900000092
at any spot check maintenance time n Δ T, n is 1, 2, …, there is a possibility that the spot check will be:
firstly, the engine passes the sampling inspection, and the instantaneous availability of the engine in the maintenance period is as follows:
Figure BDA0003497381900000093
in the formula: a (t) represents the instantaneous availability of the engine at time t; delta T is the interval time of two secondary sampling tests; t is tdRepresents the time spent in each spot check; a. the0(t) represents the probability that the engine is in a usable state at time t; n represents the total number of times of secondary sampling inspection of the engine by the time t;
the corresponding probability of occurrence is:
Figure BDA0003497381900000094
in the formula: p1The probability of occurrence of the case (1); delta T is the interval time of two secondary sampling tests; a. the0(n Δ T) represents a probability that the engine is in a usable state at time n Δ T; n represents the total number of times of secondary sampling inspection of the engine by the time t;
the first sampling inspection is not passed; or the first sampling inspection is critical, and the second sampling inspection is passed, the instantaneous availability of the engine in the maintenance period is as follows:
Figure BDA0003497381900000095
In the formula: a (t) represents the instantaneous availability of the engine at time t; delta T is the interval time of two secondary sampling tests; t is tdRepresents the time spent in each spot check; a. the0(t) represents the probability that the engine is in a usable state at time t; n represents the total number of times of secondary sampling inspection of the engine by the time t;
the corresponding probability of occurrence is:
Figure BDA0003497381900000096
in the formula: p2Case 2 probability of occurrence; delta T is the interval time of two secondary sampling tests; a. the0(n Δ T) represents a probability that the engine is in a usable state at time n Δ T; n represents the total number of times of secondary sampling inspection of the engine by the time t; a is the number of sampling samples; b represents the number of allowable sampling sample miseligibility;
the first sampling inspection is critical, and the second sampling inspection does not pass, so that the instantaneous availability of the engine in the maintenance period is as follows:
Figure BDA0003497381900000101
in the formula: a (t) represents the instantaneous availability of the engine at time t; delta T is the interval time of two secondary sampling tests; t is tdRepresents the time spent in each spot check; a. the0(t) represents the probability that the engine is in a usable state at time t; n represents the total number of times of secondary sampling inspection of the engine by the time t;
the corresponding probability of occurrence is: p3＝1-P1-P2；
In conclusion, the transient availability evaluation model of the missile engine is as follows:
Figure BDA0003497381900000102
In the formula: a (t) represents the instantaneous availability of the engine at time t; p is1Is the probability of occurrence of case 1; p is2Probability of occurrence of case two; delta T is the interval time of two secondary sampling tests; t is tdRepresents the time spent in each spot check; a. the0(t) represents a probability that the engine is in a usable state at the time; n represents the total number of times of secondary sampling inspection of the engine by the time t;
due to instantaneous availability A (t) and average availability
Figure BDA0003497381900000103
There is a functional relationship between:
Figure BDA0003497381900000104
in the formula: a (t) represents the instantaneous availability of the engine at time t; t is the accumulated working age of the engine (including the working age after returning to the factory for maintenance);
therefore, after the instantaneous availability evaluation model is obtained, the instantaneous availability evaluation model can be converted into an average availability evaluation model, and the estimated value of the scale parameter of Weibull distribution is obtained
Figure BDA0003497381900000105
And shape parameter estimation
Figure BDA0003497381900000106
Substituting equations (26) and (27) to obtain the estimated values of the instantaneous availability and the average availability of the missile engine;
through the steps, the invention realizes modeling of the instantaneous availability and the average availability of the missile engine according to the specific implementation method of the secondary sampling inspection scheme and the Weibull distribution reliability model of the missile engine, and solves the problem of how to scientifically evaluate the average availability and the instantaneous availability of the missile engine during storage.
(3) The advantages and the effects are as follows:
aiming at the missile engine which is returned to a factory regularly for secondary sampling inspection, the invention carries out modeling on the instantaneous availability and the average availability of the missile engine according to a specific implementation method of a secondary sampling inspection scheme and a Weibull distribution reliability model of the missile engine, thereby realizing a method for evaluating the storage reliability and the storage availability of the missile engine without additional experiments;
regression fitting is carried out on Weibull distribution reliability model parameters obeyed by the missile engine according to historical sampling data of the missile engine to obtain estimated values of storage reliability and storage availability of the missile engine, the estimated values are derived from actual data, and authenticity, accuracy and stability are achieved;
the iteration method is simple, has low requirement on historical data and has strong operability;
compared with the original evaluation method for obtaining the average availability according to the available time ratio, the method provided by the invention not only can evaluate the average availability of the missile engine, but also can evaluate the instantaneous availability of the missile engine;
the method of the invention is scientific, has good manufacturability and has wide popularization and application value.
Drawings
FIG. 1 is a flow chart of the method of the present invention.
Detailed Description
The invention will now be further illustrated with reference to an example, see fig. 1.
Given that the number of missiles in a batch is 100, the secondary sampling inspection scheme is as follows: the sampling period delta T is 4 years, the sampling number a is 8 every time, the allowable sample misqualification number b is 1, and the maintenance time T is every timedDay 90. Meanwhile, the historical spot check data of the missiles are shown in the table 1.
TABLE 1 historical spot check data for missile engines
Age of spot check 2 3 4 5 6 7 8
Total number of spot checks 1 4 5 13 29 48 34
Number of misapprovals 0 1 0 3 6 20 11
The invention discloses a missile engine storage availability evaluation method, namely a missile engine storage availability evaluation method based on a secondary sampling inspection scheme and historical sampling inspection data, which is shown in figure 1 and is realized by the following four steps:
the method comprises the following steps: estimating parameters according to historical sampling inspection data;
substituting the data of table 1 into equation (4) results in table 2:
TABLE 2 missile Engine History storage reliability
Figure BDA0003497381900000111
Figure BDA0003497381900000121
Observing the data in the table 2, the data of the storage reliability in the 3 rd year can be found to have contingency and larger deviation, so the data are removed, and then reliability model parameter fitting is carried out on the rest data to obtain a scale parameter estimation value of Weibull distribution obeyed in the storage failure process of the missile engine
Figure BDA0003497381900000122
And shape parameter estimation values
Figure BDA0003497381900000123
Comprises the following steps:
Figure BDA0003497381900000124
step two: calculating the probability of the missile engine passing the secondary sampling inspection;
assuming that the life of the missile engine is 12 years, since the secondary sampling inspection is performed every 4 years, the total of 3 secondary sampling inspections are performed in the life cycle, and the parameters of the secondary lottery scheme and the reliability parameter model estimation value are substituted into equation (6) to obtain table 3:
TABLE 3 probability of missile engine passing through two-stage spot check
Number of sampling tests 1 2 3
Probability of passing of spot check 0.3909 0.0263 0.0008
Step three: iteratively calculating an instantaneous availability evaluation value of the engine;
substituting the data of table 3 and the estimated values of the weibull distribution parameters into equation (20) yields table 4:
TABLE 4 estimated instantaneous reliability of missile engines
Age of storage 1 2 3 4 5 6
Instantaneous availability 0.9821 0.9487 0.9026 0.8557 0.9035 0.8597
Age of storage 7 8 9 10 11 12
Instantaneous availability 0.8086 0.7596 0.8597 0.8134 0.7682 0.7186
It can be seen that the missile engines in 5 th and 9 th years are improved in instantaneous availability due to the fact that secondary spot check return maintenance is carried out in 4 th and 8 th years;
step four: calculating an average availability evaluation value of the engine;
substituting the data of table 4 into equation (27) yields an average availability estimate for the missile engine:
Figure BDA0003497381900000125
calculating to obtain the average availability of 0.8437 in the life cycle of the missile engine;
the result shows that the method can realize the evaluation of the instantaneous availability of the missile engine by utilizing a secondary sampling scheme and historical sampling data, and further evaluate the average availability, thereby achieving the expected purpose;
In summary, the invention provides a missile engine storage availability evaluation method based on a regular secondary sampling inspection scheme and historical sampling inspection data of an engine.

Claims (1)

1. A method for evaluating the storage availability of a missile engine provides the following conditions:
condition 1: the storage failure time of the missile engine follows Weibull distribution, and at the moment t, the probability density of the storage failure of the engine is as follows:
Figure FDA0003497381890000011
in the formula: t represents the accumulated working time of the engine after leaving the factory; (t) represents the probability density of engine failure at time t; eta > 0, representing the proportionality coefficient of the Weibull distribution; m > 0, representing the shape factor of the Weibull distribution;
the corresponding cumulative distribution function of storage failure is:
Figure FDA0003497381890000012
In the formula: t represents the accumulated working time of the engine after leaving the factory; f (t) represents the failure probability of the engine at the time t; eta > 0, representing the proportionality coefficient of the Weibull distribution; m > 0, representing the shape factor of the Weibull distribution;
thus, the storage reliability function of a missile engine is:
Figure FDA0003497381890000013
in the formula: t represents the accumulated working time of the engine after leaving the factory; r (t) represents the reliability of the engine at time t; eta > 0, representing the proportionality coefficient of the Weibull distribution; m > 0, representing the shape factor of the Weibull distribution;
condition 2: the missile engine carries out secondary sampling maintenance once every delta T time, the engines are in an unavailable state during sampling, N engines are used in a certain sampling, a samples are extracted in the first sampling, the number of allowable sampling samples is b, and the time spent in each sampling is TdAll samples to be spot-checked are maintained, and whether the spot check is passed or not is judged according to the following rules:
if the number of unqualified samples of the first sampling inspection is less than b, the sampling inspection is considered to pass;
if the number of unqualified samples of the first sampling inspection is equal to b, performing second sampling inspection, extracting a from the rest N-a engines, if unqualified samples exist, determining that the sampling inspection does not pass, performing factory repair on all the engines, and if unqualified samples do not exist, determining that the sampling inspection passes;
If the number of unqualified samples of the first random inspection is more than b, the random inspection is considered to be failed, and all engines need to be returned to the factory for maintenance;
condition 3: the engines maintained by each sampling inspection only account for a small part of all the engines, so that the passing probability of each sampling inspection is independent;
it is known that the missile engine experiences the total time at t
Figure FDA0003497381890000021
The secondary and secondary sampling inspection is carried out,
Figure FDA0003497381890000022
is a lower rounded symbol, which comprises n times of first sampling inspection and e (n) times of second sampling inspection;
the method is characterized in that: the method is realized by the following four steps:
the method comprises the following steps: estimating parameters according to historical sampling inspection data;
it is known thatThe missile engine is subjected to secondary sampling inspection for n times totally, and the obtained sampling inspection data comprise: age of spot examination tiTotal number of spot checks aiNumber of false positive of spot check biI ═ 1, 2, …, n; firstly, evaluating the storage reliability of the corresponding missile engine during the spot check according to the proportion of the number of unqualified spot checks in the spot check number:
Figure FDA0003497381890000023
in the formula: t is tiIndicating a spot check age of the engine; r (t)i) Indicating that the engine is at tiThe reliability of the time; a isiRepresents tiThe total number of spot checks at that moment; biRepresents tiThe number of unqualified spot checks at the moment;
since the storage failure of the missile engine follows Weibull distribution, the corresponding reliability function is
Figure FDA0003497381890000024
The following mathematical transformation is made to the reliability function:
Figure FDA0003497381890000025
Figure FDA0003497381890000029
in the formula: t represents the accumulated working time of the engine after leaving the factory; r (t) represents the reliability of the engine at time t; eta > 0, representing the proportionality coefficient of the Weibull distribution; m > 0, representing the shape factor of the Weibull distribution;
consider this equation as being represented by the coordinates (x)i，yi) Linear function of composition: y isi＝cxi+d，
Wherein the content of the first and second substances,
Figure FDA0003497381890000026
c＝m，xi=lnti，D=-mlnη；
the sampling data is correspondingly transformed and then substituted into the equation, and the least square regression fitting method is utilized to estimate the scale parameter
Figure FDA0003497381890000027
And shape parameter estimation
Figure FDA0003497381890000028
Step two: calculating the probability of the missile engine passing the secondary sampling inspection;
by Ppass(i Δ T) represents the probability of the engine passing the ith secondary spot check, and there are two cases of passing the spot check according to the secondary spot check rule in condition 2: firstly, the engine is qualified by first sampling inspection; the second sampling inspection of the engine is qualified; respectively calculating the probabilities of the two conditions, and adding the probabilities to obtain the probability of the engine passing the spot check;
the unqualified sample number X is determined in the ith secondary sampling inspection processiObeying a binomial distribution XiB (c, F (i Δ T)), wherein F (i Δ T) represents the probability that the engine spot check is not qualified at the moment;
thus the number of defective products X iA probability of j being
Figure FDA0003497381890000031
In the formula: xiThe number of unqualified samples in the ith secondary sampling inspection process is shown; p (X)iJ) is the number of defective products XiJ is a probability; a is the number of sampling inspection samples; delta T is the interval time of two secondary sampling tests; f (i delta T) represents the failure probability of the missile engine at the time of i delta T; r (i delta T) represents the reliability of the missile engine at the moment i delta T;
therefore, the probability P that the engine passes the spot check in the ith secondary spot checkpass(i Δ T) is:
Figure FDA0003497381890000032
in the formula: ppass(i Δ T) represents the probability of the engine passing the spot check at the ith secondary spot check; a is the number of sampling samples; b represents the number of allowable sampling sample miseligibility; delta T is the interval time of two secondary sampling tests; t is tdRepresents the time spent in each spot check; f (i Δ T) represents the probability of engine failure at time i Δ T; r (i Δ T) represents the reliability of the engine at time i Δ T; r (i Delta T + T)d) Indicating that the engine is at i Δ T + TdThe reliability of the time;
step three: iteratively calculating an instantaneous availability evaluation value of the engine;
defining the state function of the missile engine at any moment t as:
Figure FDA0003497381890000033
the probability that the engine is in the enabled state at time t is A0(t) P (s (t) 1), the probability cannot be directly calculated because the engine has been maintained before, but the structure a can be iteratively calculated 0(t) an expression;
by T1The time of the first secondary sampling maintenance completion time of the engine is represented; at time T, according to T1And t, there are two situations in the engine, namely: phi T1The engine is not maintained when the engine is not maintained; ② T1< t, the engine has undergone maintenance; a. the0(t) consists of the two cases above, namely:
A0(t)＝P(S(t)＝1，T1≥t)+P(S(t)＝1，T1＜t) (8)
in the formula: t represents the accumulated working time of the engine after leaving the factory; t is1The time of the first secondary sampling maintenance completion time of the engine is represented; a. the0(t) represents the probability that the engine is in an available state at time t; p(s) (T) 1, T1T) indicates that the engine is in the enabled state at time t and has not been experiencedA probability of over-maintenance; p(s) (T) 1, T1< t) represents the probability that the engine is in an enabled state and has undergone maintenance at time t;
the probabilities of the two cases being in the usable state are calculated as follows:
①T1not less than t, the engine has not undergone maintenance
Instantaneous availability of the engine in this case:
P(S(t)＝1，T1≥t)＝R(t)P(T1≥t) (9)
in the formula: t represents the accumulated working time of the engine after leaving the factory; t is1The time of the first secondary sampling maintenance completion time of the engine is represented; p(s) (T) 1, T1T) represents the probability that the engine is in an available state at the moment t and has not undergone maintenance; r (t) represents the reliability of the engine at time t; p (T) 1T) represents the probability that the engine has not undergone maintenance at the moment t;
because the probability that the engine is pumped in each sampling inspection is
Figure FDA0003497381890000041
Therefore, the probability that the engine has not undergone secondary spot check maintenance at time t is:
Figure FDA0003497381890000042
in the formula: p (T)1T) represents the probability that the engine has not undergone maintenance at time t; delta T is the interval time of two secondary sampling tests; p represents the probability of the engine being pumped at each spot check; e (n) represents the average number of second sampling tests; n represents the total number of times of secondary sampling inspection of the engine by the time t; ppass(i Δ T) represents the probability of the engine passing the spot check at the ith secondary spot check;
wherein, the average times e (n) of the second sampling examination are:
Figure FDA0003497381890000043
in the formula: e (n) represents the average number of second sampling tests; n represents the total number of times of secondary sampling inspection of the engine by the time t; a is the number of sampling samples; b represents the number of false sampling; delta T is the interval time of two secondary sampling tests; f (i Δ T) represents the probability of engine failure at time i Δ T; r (i Δ T) represents the reliability of the engine at time i Δ T;
thus:
Figure FDA0003497381890000044
to simplify the expression, let:
Figure FDA0003497381890000045
representing the probability that the engine was not serviced in the first n secondary spot checks, equation (11) can be expressed as:
P(S(t)＝1，T1≥t)＝R(t)G(n) (13)
in the formula: t represents the accumulated working time of the engine after leaving the factory; t is 1The time of finishing the first secondary sampling inspection maintenance of the engine is represented; p(s) (T) 1, T1T) represents the probability that the engine is in an available state at the moment t and has not undergone maintenance; r (t) represents the reliability of the engine at time t; n represents the total number of times of secondary sampling inspection of the engine by the time t; a is the number of sampling samples; b represents the number of allowable sampling sample miseligibility; g (n) represents the probability that the engine has not undergone maintenance within n secondary spot checks; p represents the probability of the engine being pumped at each spot check; delta T is the interval time of two secondary sampling tests; f (i Δ T) represents the probability of engine failure at time i Δ T; r (i Δ T) represents the reliability of the engine at time i Δ T; ppass(iAT) indicating the probability that the engine passed the spot check at the ith secondary spot check;
②T1< t, the engine has undergone maintenance
The engine is maintained for the first time in the nth secondary sampling inspection, and is returned to the factory for completion and transportationTime T of external field1There are three cases respectively:
case 1: t is1＝nΔT+td(ii) a The engine is not maintained in the first n-1 secondary sampling inspection, and the first sampling inspection of the nth secondary sampling inspection is performed;
the probability of occurrence of case 1 is therefore:
P(T1＝nΔT+td)＝G(n-1)p (14)
in the formula: t is1The time of the first secondary sampling maintenance completion time of the engine is represented; t is t dRepresents the time spent for each spot check; delta T is the interval time of two secondary sampling tests; n represents the total number of times of secondary sampling inspection of the engine by the time t; p (T)1＝nΔT+td) Represents the probability of occurrence of case 1; g (n-1) represents the probability that the engine has not undergone maintenance within n-1 secondary spot checks; p represents the probability of the engine being pumped out at each spot check;
case 2: t is1＝nΔT+2td；
Firstly, the engine is not maintained in the first n-1 secondary sampling inspection, the first sampling inspection of the nth secondary sampling inspection is not performed with sampling, but the number of unqualified samples in the first sampling inspection reaches a critical value b, the second sampling inspection is performed, and the engine is performed with sampling in the second sampling inspection;
the probability of this occurrence is:
Figure FDA0003497381890000051
in the formula: t is1The time of the first secondary sampling maintenance completion time of the engine is represented; t is tdRepresents the time spent in each spot check; delta T is the interval time of two secondary sampling tests; n represents the total number of times of secondary sampling inspection of the engine by the time t; p1(T1＝nΔT+2td) Representing the probability of occurrence of the situation (i); g (n-1) represents the probability that the engine has not undergone maintenance within n-1 secondary spot checks; p represents the probability of the engine being pumped at each spot check; a is the number of sampling samples; b denotes the allowed spot check The number of samples that fail; f (n Δ T) represents the probability of engine failure at time n Δ T; r (n Δ T) represents the reliability of the engine at the time n Δ T;
the engine is not maintained in the first n-1 secondary sampling inspection, the first sampling inspection of the nth secondary sampling inspection is not performed, but the number of unqualified samples in the first sampling inspection exceeds a critical value b, and the whole batch of engines are returned to the factory for maintenance;
the probability of this occurring is:
Figure FDA0003497381890000061
in the formula: t is1The time of the first secondary sampling maintenance completion time of the engine is represented; t is tdRepresents the time spent in each spot check; delta T is the interval time of two secondary sampling tests; n represents the total number of times of secondary sampling inspection of the engine by the time t; p2(T1＝nΔT+2td) Representing the probability of occurrence of case two; g (n-1) represents the probability that the engine has not undergone maintenance within n-1 secondary spot checks; p represents the probability of the engine being pumped at each spot check; a is the number of sampling samples; b represents the number of allowable sampling sample miseligibility; f (n Δ T) represents the probability of engine failure at time n Δ T; r (n Δ T) represents the reliability of the engine at time n Δ T;
the probability of occurrence of case 2 is therefore:
Figure FDA0003497381890000062
in the formula: t is1The time of the first secondary sampling maintenance completion time of the engine is represented; t is t dRepresents the time spent for each spot check; delta T is the interval time of two secondary sampling tests; n represents the total number of times of secondary sampling inspection of the engine by the time t; p (T)1＝nΔT+2td) Represents the probability of occurrence of case 2; g (n-1) represents the probability that the engine has not undergone maintenance within n-1 secondary spot checks; p represents the probability of the engine being pumped at each spot check; a isThe number of sampling samples is selected; b represents the number of allowable sampling sample miseligibility; f (n Δ T) represents the probability of engine failure at time n Δ T; r (n Δ T) represents the reliability of the engine at time n Δ T;
case 3: t is1＝nΔT+3td(ii) a The engine is not maintained in the first n-1 secondary spot checks, the first and second spot checks of the nth secondary spot check do not have spot checks, but the number of unqualified samples in the first spot check of other engines reaches a critical value b, the second spot check is carried out, and then the second spot check does not pass;
the probability of occurrence of case 3 is therefore:
Figure FDA0003497381890000063
in the formula: t is1The time of the first secondary sampling maintenance completion time of the engine is represented; t is tdRepresents the time spent in each spot check; delta T is the interval time of two secondary sampling tests; n represents the total number of times of secondary sampling inspection of the engine by the time t; p (T)1＝nΔT+3td) Represents the probability of occurrence of case 3; g (n-1) represents the probability that the engine has not undergone maintenance within n-1 secondary spot checks; p represents the probability of the engine being pumped at each spot check; a is the number of sampling samples; b represents the number of allowable sampling sample miseligibility; f (n Δ T) represents the probability of engine failure at time n Δ T; r (n Δ T) represents the reliability of the engine at time n Δ T; r (n.DELTA.T + T) d) Indicating that the engine is at n Δ T + TdThe reliability of the time;
because the maintenance requires time, after the engine is maintained, the time from the next secondary sampling inspection time is not delta T any more but delta T-Td，ΔT-2td，ΔT-3tdDepending on the time required for the last maintenance, it is therefore necessary to determine the probability that the engine in the maintenance mode is in an available state;
note a0(k, T) denotes the first maintenance period of the engine as Δ T-ktdThen at time t, the probability of being in an available state, where k is 1, 2, 3, a0(k，t)The calculation process of (a) can construct the following iterative equation:
Figure FDA0003497381890000071
in the formula:
Figure FDA0003497381890000072
jt indicating that the first repair occurred after the start of the ith maintenancedTime of day; t is tdRepresents the time spent in each spot check; delta T is the interval time of two secondary sampling tests; a. the0(k, T) denotes the first maintenance period of the engine as Δ T-ktdProbability of being in an available state at the later time t;
Figure FDA0003497381890000073
representing the probability that the engine is in a usable state at time t and has not undergone maintenance; n represents the total number of times of secondary sampling inspection of the engine by the time t;
Figure FDA0003497381890000074
jt indicating that the first repair occurred after the start of the ith maintenancedA probability of a time of day;
therefore, the iterative calculation of the probability that the missile engine is in the usable state at any time t is as follows:
Figure FDA0003497381890000075
in the formula:
Figure FDA0003497381890000076
Jt indicating that the first repair occurred after the start of the ith maintenancedTime of day; delta T is the interval time of two secondary sampling tests; t is tdRepresents the time spent for each spot check; a. the0(t) represents the probability that the engine is in an available state at time t;
Figure FDA0003497381890000077
representing the probability that the engine is in a usable state at time t and has not undergone maintenance; n represents the total number of times of secondary sampling inspection of the engine by the time t; g (n) represents the probability that the engine has not undergone maintenance within n secondary spot checks; r (t) represents the reliability of the engine at time t;
Figure FDA0003497381890000078
jt indicating that the first repair occurred after the start of the ith maintenancedA probability of a time of day;
step four: calculating an average availability evaluation value of the engine;
a calculated in the last step due to the fact that the engine is specified to be in an unavailable state during the sampling inspection0(t) probability of engine being in usable state, therefore, it is necessary to set A to0(t) converting to obtain the engine availability A (t) and
Figure FDA0003497381890000079
at any spot check maintenance time n Δ T, n is 1, 2, …, there is a possibility that the spot check will be:
firstly, the engine passes the sampling inspection, and the instantaneous availability of the engine in the maintenance period is as follows:
Figure FDA0003497381890000081
in the formula: a (t) represents the instantaneous availability of the engine at time t; delta T is the interval time of two secondary sampling tests; t is t dRepresents the time spent for each spot check; a. the0(t) represents a probability that the engine is in an available state at time t; n represents the total number of times of secondary sampling inspection of the engine by the time t;
the corresponding probability of occurrence is:
Figure FDA0003497381890000082
in the formula: p is1The probability of occurrence of the case (1); delta T is the interval time of two secondary sampling tests; a. the0(n Δ T) represents a probability that the engine is in an available state at the time of n Δ T; n represents the total number of times of secondary sampling inspection of the engine by the time t;
the first sampling inspection is not passed; or the first sampling inspection is critical, and the second sampling inspection is passed, the instantaneous availability of the engine in the maintenance period is as follows:
Figure FDA0003497381890000083
in the formula: a (t) represents the instantaneous availability of the engine at time t; delta T is the interval time of two secondary sampling tests; t is tdRepresents the time spent in each spot check; a. the0(t) represents the probability that the engine is in an available state at time t; n represents the total number of times of secondary sampling inspection of the engine by the time t;
the corresponding probability of occurrence is:
Figure FDA0003497381890000084
in the formula: p2Case 2 probability of occurrence; delta T is the interval time of two secondary sampling tests; a. the0(n Δ T) represents a probability that the engine is in an available state at the time of n Δ T; n represents the total number of times of secondary sampling inspection of the engine by the time t; a is the number of sampling samples; b represents the number of allowable sampling sample miseligibility;
The first sampling inspection is critical, the second sampling inspection does not pass, and the instantaneous availability of the engine in the maintenance period is as follows:
Figure FDA0003497381890000085
in the formula: a (t) represents the instantaneous timing of the engine at time tThe use degree; delta T is the interval time of two secondary sampling tests; t is tdRepresents the time spent in each spot check; a. the0(t) represents the probability that the engine is in an available state at time t; n represents the total number of times of secondary sampling inspection of the engine by the time t;
the corresponding probability of occurrence is: p3＝1-P1-P2；
In conclusion, the transient availability evaluation model of the missile engine is as follows:
Figure FDA0003497381890000091
in the formula: a (t) represents the instantaneous availability of the engine at time t; p1The probability of occurrence of the case (1); p2Case 2 probability of occurrence; delta T is the interval time of two secondary sampling tests; t is tdRepresents the time spent in each spot check; a. the0(t) represents the probability that the engine is in an available state at time t; n represents the total number of times of secondary sampling inspection of the engine by the time t;
due to instantaneous availability A (t) and average availability
Figure FDA0003497381890000092
There is a functional relationship between:
Figure FDA0003497381890000093
in the formula: a (t) represents the instantaneous availability of the engine at time t; t is the accumulated working age of the engine;
therefore, after the instantaneous availability evaluation model is obtained, the instantaneous availability evaluation model can be converted into an average availability evaluation model, and the estimated value of the scale parameter of Weibull distribution is obtained
Figure FDA0003497381890000094
And shape parameter estimation
Figure FDA0003497381890000095
The estimated values of the instantaneous availability and the average availability of the missile engine can be obtained by substituting equations (26) and (27).
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Cited By (1)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN115292969A (en) * 2022-10-08 2022-11-04 天津七一二通信广播股份有限公司 Equipment outfield reliability evaluation method and system based on delivery and repair data

Cited By (1)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN115292969A (en) * 2022-10-08 2022-11-04 天津七一二通信广播股份有限公司 Equipment outfield reliability evaluation method and system based on delivery and repair data

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