Missile engine storage availability evaluation method

Abstract

The application provides a missile engine storage availability evaluation method, namely a missile engine storage availability evaluation method based on a two-stage sampling scheme and historical sampling data, and the implementation steps are as follows: 1, parameter estimation according to historical sampling data; 2, calculation of the two-stage sampling probability of the missile engine; 3, iterative calculation of the instantaneous availability evaluation value of the engine; 4, calculation of the average availability evaluation value of the engine; the application realizes the evaluation of the storage reliability and storage availability of the missile engine without additional experiments, the evaluation value is derived from actual data, has authenticity, accuracy and stability, the iterative method is simple, the requirement for historical data is low, the operability is strong, and not only the average availability of the missile engine can be evaluated, but also the instantaneous availability of the missile engine can be evaluated.

Description

Technical Field

[0001] This invention provides a method for assessing the availability of missile engine storage, specifically a method for assessing the availability of missile engine storage based on a two-stage sampling inspection scheme and historical sampling inspection data. It particularly relates to a method for assessing the availability of missile engine storage based on a two-stage sampling inspection scheme and historical sampling inspection data, and belongs to the field of missile engine storage availability assessment. Background Technology

[0002] Missiles spend most of their lifespan in storage. The missile engine is a critical component, and whether a long-term stored engine can still meet mission requirements needs to be quantified through data. Availability is a crucial indicator for evaluating equipment readiness, as it comprehensively assesses design characteristics such as reliability, maintainability, testability, and supportability. The goal of missile availability assessment is to identify the factors and patterns affecting missile availability and to establish a scientifically sound and reasonable missile availability assessment methodology.

[0003] Mathematical models of availability can be broadly classified into two categories: probabilistic models and statistical models. Probabilistic models infer quantitative indicators of reliability related to system lifespan based on information such as system structure, component lifespan distribution, and repair time distribution. Further, they can be used to discuss the optimal design, usage, and maintenance strategies of the system. Statistical models estimate and test the lifespan, reliability, and availability indicators of components or systems based on statistical data.

[0004] For missile engines, since they need to undergo periodic secondary inspections during storage to check for hidden faults and perform maintenance, the main factors affecting the availability of missile engines are the duration of hidden faults and the downtime caused by maintenance. Therefore, the main method for assessing the availability of missile engines is to analyze and process the data obtained from the inspections, and then use specific methods to evaluate the data based on its characteristics.

[0005] Against this backdrop, scientifically assessing the availability of missile engines during storage is of great significance for improving equipment readiness and saving maintenance costs. Summary of the Invention

[0006] (1) Purpose of the present invention: The current method for assessing the storage availability of missile engines is to calculate the proportion of time during which the missile engine is in an available state within a given time period. Although this method can assess the average availability of missile engines, it cannot assess the instantaneous availability of missile engines at any given moment. However, for missiles and other weapons, the instantaneous availability index is more important. Therefore, the present invention provides a method for assessing the storage availability of missile engines, namely, a method for assessing the storage availability of missile engines based on a two-stage sampling inspection scheme and historical sampling inspection data. This method can use the two-stage sampling inspection scheme and historical sampling inspection data to assess the instantaneous availability of missile engines and further assess the average availability.

[0007] (2) Technical solution:

[0008] The basic assumptions proposed in this invention are as follows:

[0009] Assumption 1: Based on extensive literature review and experimental results, the storage failure time of a missile engine follows a Weibull distribution. At time t, the probability density of engine storage failure is:

[0010]

[0011] In the formula: t represents the cumulative working time of the engine after it leaves the factory; f(t) represents the failure probability density of the engine at time t; η>0 represents the proportionality coefficient of the Weibull distribution; m>0 represents the shape coefficient of the Weibull distribution;

[0012] The corresponding cumulative distribution function of storage failure is:

[0013]

[0014] In the formula: t represents the cumulative working time of the engine after it leaves the factory; F(t) represents the failure probability of the engine at time t; η>0 represents the proportional coefficient of the Weibull distribution; m>0 represents the shape coefficient of the Weibull distribution;

[0015] Therefore, the storage reliability function of the missile engine is:

[0016]

[0017] In the formula: t represents the cumulative working time of the engine after it leaves the factory; R(t) represents the reliability of the engine at time t; η>0 represents the proportionality coefficient of the Weibull distribution; m>0 represents the shape coefficient of the Weibull distribution;

[0018] Assume that missile engines undergo a secondary sampling maintenance every ΔT time interval, and the engines are unavailable during the sampling period. There are N engines in a given sampling period. The first sampling draws 'a' samples, the allowed number of defective samples is 'b', and the time spent on each sampling is 't'. d All samples that are randomly selected for inspection will be repaired. The following rules will be used to determine whether the inspection passes:

[0019] ①If the number of non-compliant samples in the first random inspection is less than b, the random inspection is considered to have passed;

[0020] ② If the number of non-conforming samples in the first sampling inspection is equal to b, then a second sampling inspection is required. A new batch of a engines will be drawn from the remaining Na engines. If any of these engines are non-conforming, the sampling inspection is considered to have failed and all engines need to be returned to the factory for repair. If no non-conforming samples are drawn, the sampling inspection is considered to have passed.

[0021] ③ If the number of non-compliant samples in the first random inspection is greater than b, the random inspection is considered to have failed and all engines need to be returned to the factory for repair.

[0022] Assumption 3: Since each random inspection only covers a small portion of all engines, it is assumed that the probability of passing each inspection is independent.

[0023] It is known that the missile engine has experienced a total of [time t] times. Secondary sampling inspection ( (where n is the floor symbol), which includes n first samplings and e(n) second samplings;

[0024] The method proposed in this invention mainly includes four parts: estimating parameters based on historical sampling data, calculating the probability of the missile engine passing the second-level sampling inspection, iteratively calculating the instantaneous availability assessment value of the engine, and calculating the average availability assessment value of the engine.

[0025] Based on the above assumptions and ideas, this invention provides a method for assessing the availability of missile engine storage, specifically a method for assessing the availability of missile engine storage based on a two-stage sampling inspection scheme and historical sampling inspection data. This method is implemented through the following four steps:

[0026] Step 1: Estimate parameters based on historical sampling data;

[0027] It is known that the missile engine underwent a total of n secondary random inspections, and the obtained inspection data includes: the inspection age t. i Total number of samples a i Number of non-compliant samples (b) i, i = 1, 2, ..., n; First, the storage reliability of the missile engine at the time of sampling is assessed by the proportion of the number of unqualified samples to the total number of samples:

[0028]

[0029] In the formula: t i Indicates the engine's age at random inspection; R(t) i ) indicates that the engine is at t i Reliability at any given moment; a i Indicates t i The total number of random inspections at any given time; b i Indicates t i The number of non-compliant items in random inspections at any given time;

[0030] Since the storage failure of missile engines follows a Weibull distribution, the corresponding reliability function is: The reliability function is transformed mathematically as follows:

[0031]

[0032]

[0033] In the formula: t represents the cumulative working time of the engine after it leaves the factory; R(t) represents the reliability of the engine at time t; η>0 represents the proportionality coefficient of the Weibull distribution; m>0 represents the shape coefficient of the Weibull distribution;

[0034] View this equation as a function of coordinates (x) i y i A linear function composed of y i =cx i +d, where c = m, x i =ln t i d = -m lnη;

[0035] After transforming the sampled data and substituting it into the above equation, the scale parameter can be estimated using the least squares regression fitting method. and shape parameter estimates

[0036] Step 2: Calculate the probability of the missile engine passing the secondary random inspection;

[0037] Use P pass (iΔT) represents the probability that the engine passes the second-level inspection in the i-th time. According to the second-level inspection rules in Assumption 2, there are two situations in which the engine passes the inspection: ① the engine passes the first inspection; ② the engine passes the second inspection. The probability of the engine passing the inspection is obtained by calculating the probabilities of the two situations separately and adding them together.

[0038] Since the number of unqualified samples X in the i-th secondary sampling inspection... i X follows a binomial distribution i ~B(c, F(iΔT)), where F(iΔT) represents the probability that the engine fails the random inspection at this time (i.e., storage failure has occurred);

[0039] Therefore, the number of defective items X i The probability of =j is

[0040] In the formula: X i P(X) represents the number of non-compliant samples in the i-th secondary sampling inspection. i =j) represents the number of non-conforming items X i =j; a is the number of samples to be sampled; ΔT is the interval between two secondary sampling inspections; F(iΔT) represents the failure probability of the missile engine at time iΔT; R(iΔT) represents the reliability of the missile engine at time iΔT;

[0041] Therefore, the probability P of the engine passing the second-level inspection in the i-th time is... pass (iΔT) is:

[0042]

[0043] In the formula: P pass (iΔT) represents the probability that the engine passes the second-level sampling inspection in the i-th time; a is the number of samples to be inspected; b is the allowed number of samples to fail; ΔT is the time interval between two second-level sampling inspections; t d The time spent on each sampling inspection is represented by F(iΔT); the failure probability of the engine at time iΔT is represented by F(iΔT); the reliability of the engine at time iΔT is represented by R(iΔT+t). d ) indicates that the engine is at iΔT+t d Reliability at any given moment;

[0044] Step 3: Iteratively calculate the instantaneous availability assessment value of the engine;

[0045] Define the state function of the missile engine at any time t as:

[0046]

[0047] The probability that the engine is in a usable state at time t is A0(t) = P(S(t) = 1). Since the engine may have been maintained before, the above probability cannot be calculated directly, but the expression of A0(t) can be constructed by iterative calculation.

[0048] Let T1 represent the time when the engine's first secondary sampling maintenance is completed. At time t, based on the relationship between T1 and t, there are two possibilities for the engine: ① T1 ≥ t, the engine has not undergone maintenance; ② T1 < T, the engine has undergone maintenance. A0(t) consists of these two possibilities, namely:

[0049] A0(t)=P(S(t)=1,T1≥t)+P(S(t)=1,T1<t) (8)

[0050] In the formula: t represents the cumulative working time of the engine after leaving the factory; T1 represents the time when the engine's first secondary sampling maintenance is completed; A0(t) represents the probability that the engine is in a usable state at time t; P(S(t)=1,T1≥t) 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 is in a usable state at time t and has undergone maintenance.

[0051] The probabilities of being in an available state in these two scenarios are calculated below:

[0052] ①T1≥t, the engine has not undergone maintenance.

[0053] The instantaneous availability of the engine in this situation:

[0054] P(S(t)=1,T1≥t)=R(t)P(T1≥t) (9)

[0055] In the formula: t represents the cumulative working time of the engine after leaving the factory; T1 represents the time when the engine's first secondary sampling maintenance is completed; P(S(t)=1,T1≥t) 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(T1≥t) represents the probability that the engine has not undergone maintenance at time t;

[0056] Since the probability of an engine being selected in each random inspection is... Therefore, the probability that the engine has not undergone secondary sampling maintenance at time t is:

[0057]

[0058] In the formula: P(T1≥t) represents the probability that the engine has not undergone maintenance at time t; ΔT is the interval between the two secondary sampling inspections; p represents the probability that the engine is selected in each inspection; e(n) represents the average number of the second sampling inspection; n represents the total number of secondary sampling inspections the engine has undergone by time t; P pass (iΔT) represents the probability that the engine passes the second-level inspection in the i-th time;

[0059] The average number of sampling inspections in the second round, e(n), is:

[0060] In the formula: e(n) represents the average number of the second sampling inspection; n represents the total number of secondary sampling inspections the engine has undergone up to time t; a is the number of samples inspected; b represents the number of unqualified samples; ΔT is the interval between two secondary sampling inspections; F(iΔT) represents the failure probability of the engine at time iΔT; R(iΔT) represents the reliability of the engine at time iΔT.

[0061] therefore:

[0062]

[0063] To simplify the expression, let:

[0064]

[0065] Let represent the probability that the engine was not maintained in the first n secondary sampling inspections. Then, equation (11) can be expressed as:

[0066] P(S(t)=1,T1≥t)=R(t)G(n) (13)

[0067] In the formula: t represents the cumulative working time of the engine since it left the factory; T1 represents the time when the engine's first secondary sampling inspection and maintenance is completed; P(S(t)=1,T1≥t) 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 secondary sampling inspections the engine has undergone up to time t; a is the number of sampling samples; represents the allowable number of unqualified sampling samples; G(n) represents the probability that the engine has not undergone maintenance within n secondary sampling inspections; p represents the probability that the engine is selected in each inspection; ΔT is the interval between two secondary sampling inspections; F(iΔT) represents the failure probability of the engine at time iΔT; R(iΔT) represents the reliability of the engine at time iΔT; P pass (iΔT) represents the probability that the engine passes the second-level inspection in the i-th time;

[0068] ②T1 < t, the engine has undergone maintenance.

[0069] Assuming the engine undergoes its first maintenance during the nth secondary inspection, the time T1 at which it is returned to the factory and transported back to the field can have three possible scenarios:

[0070] Case 1: T1 = nΔT + t d This indicates that the engine was not repaired in the first n-1 secondary sampling inspections, and was selected in the first sampling inspection of the nth secondary sampling inspection.

[0071] Therefore, the probability of scenario 1 occurring is:

[0072] P(T1=nΔT+t d )=G(n-1)p (14)

[0073] In the formula: T1 represents the time when the engine's first secondary inspection and maintenance is completed; td represents the time spent on each inspection; ΔT is the interval between two secondary inspections; n represents the total number of secondary inspections the engine has undergone up to time t; P(T1=nΔT+t) d ) represents the probability of scenario 1 occurring; G(n-1) represents the probability that the engine has not undergone maintenance within n-1 secondary sampling inspections; p represents the probability that the engine is selected in each sampling inspection.

[0074] Case 2: T1 = nΔT + 2t d ;

[0075] ① The engine was not repaired in the first n-1 secondary sampling inspections. It was not selected in the first sampling inspection of the nth secondary sampling inspection. However, the number of unqualified samples in the first sampling inspection reached the critical value b, so a second sampling inspection was carried out. The engine was selected in the second sampling inspection.

[0076] The probability of this happening is:

[0077]

[0078] In the formula: T1 represents the time when the engine's first secondary sampling inspection and maintenance is completed; t d This represents the time spent on each sampling inspection; ΔT is the interval between two secondary sampling inspections; n represents the total number of secondary sampling inspections the engine has undergone up to time t; P1(T1=nΔT+2t) d ) represents the probability of situation ① occurring; G(n-1) represents the probability that the engine has not undergone maintenance within n-1 secondary sampling inspections; p represents the probability that the engine is selected in each sampling inspection; a is the number of sampling samples; b represents the allowed number of defective sampling samples; F(nΔT) represents the failure probability of the engine at time nΔT; R(nΔT) represents the reliability of the engine at time nΔT.

[0079] ② The engine was not repaired in the first n-1 secondary sampling inspections. It was not selected in the first sampling inspection of the nth secondary sampling inspection, but the number of unqualified samples in the first sampling inspection exceeded the critical value b. The entire batch of engines was returned to the factory for maintenance.

[0080] The probability of this happening is:

[0081]

[0082] In the formula: T1 represents the time when the engine's first secondary sampling inspection and maintenance is completed; t dThis represents the time spent on each sampling inspection; ΔT is the interval between two secondary sampling inspections; n represents the total number of secondary sampling inspections the engine has undergone up to time t; P2(T1=nΔT+2t) d ) represents the probability of scenario ② occurring; G(n-1) represents the probability that the engine has not undergone maintenance within n-1 secondary sampling inspections; p represents the probability that the engine is selected in each sampling inspection; a is the number of sampling samples; b represents the allowed number of defective sampling samples; F(nΔT) represents the failure probability of the engine at time nΔT; R(nΔT) represents the reliability of the engine at time nΔT.

[0083] Therefore, the probability of scenario 2 occurring is:

[0084]

[0085] In the formula: T1 represents the time when the engine's first secondary sampling inspection and maintenance is completed; t d This represents the time spent on each sampling inspection; ΔT is the interval between two secondary sampling inspections; n represents the total number of secondary sampling inspections the engine has undergone up to time t; P(T1=nΔT+2t) d ) represents the probability of scenario 2 occurring; G(n-1) represents the probability that the engine has not undergone maintenance within n-1 secondary sampling inspections; p represents the probability that the engine is selected in each sampling inspection; a is the number of sampling samples; b represents the allowed number of defective sampling samples; F(nΔT) represents the failure probability of the engine at time nΔT; R(nΔT) represents the reliability of the engine at time nΔT.

[0086] Case 3: T1 = nΔT + 3t d The engine was not repaired in the first n-1 secondary sampling inspections. It was not selected in the first and second sampling inspections of the nth secondary sampling inspection. However, the number of unqualified samples of other engines in the first sampling inspection reached the critical value b, so a second sampling inspection was required. Then, the second sampling inspection also failed.

[0087] Therefore, the probability of scenario 3 occurring is:

[0088]

[0089] In the formula: T1 represents the time when the engine's first secondary sampling inspection and maintenance is completed; t d This represents the time spent on each sampling inspection; ΔT is the interval between two secondary sampling inspections; n represents the total number of secondary sampling inspections the engine has undergone up to time t; P = (T1 = nΔT + 3t) d) represents the probability of scenario 3 occurring; G(n-1) represents the probability that the engine has not undergone maintenance within n-1 secondary sampling inspections; p represents the probability that the engine is selected in each inspection; a is the number of samples inspected; b represents the allowable number of defective samples; F(nΔT) represents the probability of engine failure at time nΔT; R(nΔT) represents the reliability of the engine at time nΔT; R(nΔT+t) represents the reliability of the engine at time nΔT. d ) indicates that the engine is at nΔT+t d Reliability at any given moment;

[0090] Because repairs take time, the time between engine repairs and the next secondary spot check is no longer ΔT, but ΔT-t. d ΔT-2t d ΔT-3t d One of them depends on the time required for the last maintenance, so it is necessary to calculate the probability that the engine is in an available state under this maintenance mode;

[0091] Let A0(k,t) represent the engine's first maintenance cycle as ΔT-kt. d The probability of being in a usable state at time t, where k = 1, 2, 3, can be calculated using the following iterative equation:

[0092]

[0093] In the formula: This indicates that the first maintenance occurs after the start of the i-th maintenance. d Time; t d This represents the time spent on each sampling inspection; ΔT is the interval between two secondary sampling inspections; A0(k,t) represents the engine's first maintenance cycle as ΔT-kt. d The probability that the device will be in an available state at time t; This represents the probability that the engine is in a usable state at time t and has not undergone maintenance; n represents the total number of secondary inspections the engine has undergone by time t. This indicates that the first maintenance occurs after the start of the i-th maintenance. d The probability at any given moment;

[0094] Therefore, the iterative formula for the probability that the missile engine is in a usable state at any time t is:

[0095]

[0096] In the formula: This indicates that the first maintenance occurs after the start of the i-th maintenance. d Time; ΔT is the interval between two secondary sampling inspections; t dA represents the time spent on each sampling inspection; A0(t) represents the probability that the engine is in a usable state at time t. G(n) represents the probability that the engine is in an available state at time t without having undergone maintenance; n represents the total number of secondary inspections the engine has undergone by time t; G(n) represents the probability that the engine has not undergone maintenance within n secondary inspections; R(t) represents the reliability of the engine at time t. This indicates that the first maintenance occurred after the start of the [number]th maintenance. d The probability at any given moment;

[0097] Step 4: Calculate the engine's average availability assessment value;

[0098] Since the engine is specified to be unavailable during the sampling inspection period, and the probability of the engine being available (A0(t) calculated in the previous step) needs to be transformed to obtain the engine availability A(t) under the corresponding regulations.

[0099] At any sampling maintenance time nΔT, n=1,2,…, the sampling has the following possible scenarios:

[0100] ①If the first random inspection passes, the instantaneous availability of the engine during this maintenance cycle is:

[0101]

[0102] In the formula: A(t) represents the instantaneous availability of the engine at time t; ΔT is the interval between two secondary sampling inspections; t d A0(t) represents the time spent on each sampling inspection; A0(t) represents the probability that the engine is in a usable state at time t; n represents the total number of secondary sampling inspections the engine has undergone by time t.

[0103] The corresponding probability of occurrence is:

[0104]

[0105] In the formula: P1 is the probability of scenario ① occurring; ΔT is the interval between two secondary sampling inspections; A0(nΔT) represents the probability that the engine is in a usable state at time nΔT; n represents the total number of secondary sampling inspections the engine undergoes up to time t.

[0106] ② If the first random inspection fails; or if the first random inspection is critical but the second random inspection passes, then the instantaneous availability of the engine during this maintenance cycle is:

[0107]

[0108] In the formula: A(t) represents the instantaneous availability of the engine at time t; ΔT is the interval between two secondary sampling inspections; td A0(t) represents the time spent on each sampling inspection; A0(t) represents the probability that the engine is in a usable state at time t; n represents the total number of secondary sampling inspections the engine has undergone by time t.

[0109] The corresponding probability of occurrence is:

[0110]

[0111] In the formula: P2 is the probability of scenario ② occurring; ΔT is the interval between two secondary sampling inspections; A0(nΔT) represents the probability that the engine is in a usable state at time nΔT; n represents the total number of secondary sampling inspections the engine undergoes up to time t; a is the number of samples inspected; b represents the allowed number of non-compliant samples.

[0112] ③ If the first random inspection is at the critical point and the second random inspection fails, then the instantaneous availability of the engine during this maintenance cycle is:

[0113]

[0114] In the formula: A(t) represents the instantaneous availability of the engine at time t; ΔT is the interval between two secondary sampling inspections; t d A0(t) represents the time spent on each sampling inspection; A0(t) represents the probability that the engine is in a usable state at time t; n represents the total number of secondary sampling inspections the engine has undergone by time t.

[0115] The corresponding probability of occurrence is: P3 = 1 - P1 - P2;

[0116] In summary, the instantaneous availability assessment model for missile engines is as follows:

[0117]

[0118] In the formula: A(t) represents the instantaneous availability of the engine at time t; P1 is the probability of scenario ① occurring; P2 is the probability of scenario ② occurring; ΔT is the interval between the two secondary sampling inspections; t d At represents the time spent on each sampling inspection; A0(t) represents the probability that the engine is in an available state at time t; n represents the total number of secondary sampling inspections the engine has undergone by time t.

[0119] Due to the instantaneous availability A(t) and average availability There is a functional relationship between them:

[0120]

[0121] In the formula: A(t) represents the instantaneous availability of the engine at time t; T is the cumulative working age of the engine (including the working age after returning to the factory for maintenance);

[0122] Therefore, after obtaining the instantaneous availability assessment model, it can be transformed into an average availability assessment model, using the Weibull distribution's scale parameter estimates. and shape parameter estimates Substituting into equations (26) and (27) yields the evaluation values ​​of the instantaneous availability and average availability of the missile engine;

[0123] Through the above steps, this invention realizes the modeling of the instantaneous availability and average availability of missile engines based on the specific implementation method of the two-level sampling inspection scheme and the Weibull distribution reliability model of missile engines, thus solving the problem of how to scientifically evaluate the average availability and instantaneous availability of missile engines during storage.

[0124] (3) Advantages and benefits:

[0125] ①This invention targets missile engines that undergo periodic return-to-factory secondary sampling inspections. Based on the specific implementation method of the secondary sampling inspection scheme and the Weibull distribution reliability model of the missile engine, it models the instantaneous availability and average availability of the missile engine, realizing a method for evaluating the storage reliability and storage availability of the missile engine without the need for additional experiments.

[0126] ②The present invention also performs regression fitting on the parameters of the Weibull distribution reliability model that the missile engine follows based on historical sampling data of the missile engine, and obtains the estimated values ​​of the missile engine storage reliability and storage availability. The estimated values ​​are derived from actual data and have authenticity, accuracy and stability.

[0127] ③ The iterative method of this invention is simple, has low requirements for historical data, and is highly operable;

[0128] ④ Compared with the original assessment method that obtains the average availability based on the proportion of available time, the method proposed in this invention can not only assess the average availability of missile engines, but also assess the instantaneous availability of missile engines.

[0129] ⑤ The method described in this invention is scientific, has good processability, and has broad application value. Attached Figure Description

[0130] Figure 1 This is a flowchart of the method described in this invention. Detailed Implementation

[0131] See Figure 1 The present invention will be further described below with reference to examples.

[0132] Given that a batch of missiles contains 100 missiles, the secondary sampling inspection plan is as follows: the sampling period ΔT = 4 years, the number of samples sampled each time is a = 8, the allowable number of defective samples is b = 1, and the maintenance time t each time is... d =90 days. Meanwhile, historical inspection data for this batch of missiles is shown in Table 1.

[0133] Table 1 Historical Inspection Data of Missile Engines

[0134] Total number of samples inspected 1 4 5 13 29 48 34 Number of non-conforming items 0 1 0 3 6 20 11

[0135] This invention discloses a method for assessing the availability of missile engine storage, specifically a method based on a two-stage sampling inspection scheme and historical sampling inspection data. (See attached document.) Figure 1 As shown, this can be achieved through the following four steps:

[0136] Step 1: Estimate parameters based on historical sampling data;

[0137] Substituting the data from Table 1 into equation (4) and processing it, we obtain Table 2:

[0138] Table 2 Historical Storage Reliability of Missile Engines

[0139]

[0140]

[0141] Observing the data in Table 2, it can be found that the storage reliability data in the third year has randomness and large bias, so it is removed. Then, the remaining data is fitted with reliability model parameters to obtain the estimated values ​​of the scale parameters of the Weibull distribution that the missile engine storage failure process follows. and shape parameter estimates for:

[0142] Step 2: Calculate the probability of the missile engine passing the secondary random inspection;

[0143] Assuming the missile engine has a lifespan of 12 years, and since a secondary inspection is conducted every 4 years, there are a total of 3 secondary inspections during the lifespan. Substituting the parameters of the secondary inspection scheme and the reliability parameter model estimates into equation (6) yields Table 3:

[0144] Table 3. Probability of missile engines passing secondary random inspection

[0145] Probability of passing random inspection 0.3909 0.0263 0.0008

[0146] Step 3: Iteratively calculate the instantaneous availability assessment value of the engine;

[0147] Substituting the data from Table 3 and the estimated values ​​of the Weibull distribution parameters into equation (20) yields Table 4:

[0148] Table 4. Instantaneous reliability estimates of missile engines

[0149] Instantaneous availability 0.9821 0.9487 0.9026 0.8557 0.9035 0.8597 Storage Age 7 8 9 10 11 12 Instantaneous availability 0.8086 0.7596 0.8597 0.8134 0.7682 0.7186

[0150] It can be seen that due to the secondary random inspection and return to the factory for maintenance in the 4th and 8th years, the instantaneous availability of the missile engine in the 5th and 9th years was improved compared to the previous year.

[0151] Step 4: Calculate the engine's average availability assessment value;

[0152] Substituting the data from Table 4 into equation (27), we obtain the estimated average availability of the missile engine:

[0153]

[0154] The calculated average availability of the missile engine over its entire life cycle is 0.8437.

[0155] The results show that the method of the present invention can be used to evaluate the instantaneous availability of missile engines using a two-stage sampling inspection scheme and historical sampling inspection data, and further evaluate the average availability, thus achieving the expected purpose.

[0156] In summary, this invention provides a missile engine storage availability assessment method based on a periodic two-stage sampling inspection scheme and historical sampling inspection data. This method first establishes a missile engine storage reliability assessment model, then obtains estimated values ​​of the reliability assessment model parameters based on historical sampling inspection data, then calculates the pass probability of each sampling inspection according to the missile engine's two-stage sampling inspection scheme, and further calculates the probability that the missile engine is in an available state. The instantaneous availability estimate of the missile engine is obtained through iterative calculation, and finally, the average availability estimate of the missile engine is obtained through averaging.

Claims

1. A method for assessing the storage availability of missile engines, with the following conditions: Condition 1: The storage failure time of the missile engine follows a Weibull distribution. At time t, the probability density of engine storage failure is: (1) In the formula: This indicates the engine's cumulative operating time since it left the factory; Indicates the engine is in The failure probability density at time step; , representing the proportionality coefficient of the Weibull distribution; , representing the shape coefficient of the Weibull distribution; The corresponding cumulative distribution function of storage failure is: (2) In the formula: This indicates the engine's cumulative operating time since it left the factory; Indicates the engine is in The probability of failure at any given time; , representing the proportionality coefficient of the Weibull distribution; , representing the shape coefficient of the Weibull distribution; Therefore, the storage reliability function of the missile engine is: (3) In the formula: This indicates the engine's cumulative operating time since it left the factory; Indicates the engine is in Reliability at any given moment; , representing the proportionality coefficient of the Weibull distribution; , representing the shape coefficient of the Weibull distribution; Condition 2: Missile engine every A secondary random inspection and maintenance is conducted periodically, and the engine is unusable during the inspection period. During one particular inspection, the engine has a total of [number missing] [units missing]. One, drawn during the first random inspection. The number of samples allowed to be non-compliant is [number]. The time spent on each random inspection is All samples that are randomly selected for inspection will be repaired. The following rules will be used to determine whether the inspection passes: If the number of non-compliant samples in the first random inspection is less than If so, the random inspection is considered passed; If the number of non-compliant samples in the first random inspection equals Then a second sampling inspection is required, from the remaining... Extract from the engine again If any of the samples are found to be substandard, the inspection is considered to have failed and all engines must be returned to the factory for repair. If no samples are found to be substandard, the inspection is considered to have passed. If the number of non-compliant samples in the first random inspection is greater than If the inspection fails, all engines will need to be returned to the factory for repair. Condition 3: Since each random inspection and maintenance only covers a small portion of all engines, the probability of passing each inspection is independent. Known to The missile engine has undergone a total of Secondary sampling inspection. The floor function is the floor function, where, include The first random inspection and Second and second random inspections; The method is characterized by the following four steps: Step 1: Estimate parameters based on historical sampling data; It is known that the missile engine has undergone a total of [number] tests. The secondary sampling inspection yielded the following data: the age of the sampled individuals. Total number of samples inspected Number of unqualified samples , First, the storage reliability of the missile engine at the time of the sampling inspection is assessed by the proportion of non-conforming items to the total number of samples inspected. (4) In the formula: Indicates the age of the engine sampled for inspection; Indicates the engine is in Reliability at any given moment; express The total number of random inspections at any given time; express The number of non-compliant items in the random inspection at any given time; Since the storage failure of missile engines follows a Weibull distribution, the corresponding reliability function is: The reliability function is then subjected to the following mathematical transformation: (5) In the formula: This indicates the engine's cumulative operating time since it left the factory; Indicates the engine is in Reliability at any given moment; , representing the proportionality coefficient of the Weibull distribution; , representing the shape coefficient of the Weibull distribution; View this equation as a coordinate system The resulting linear function: , in, , , , ; After transforming the sampled data accordingly, we input it into the above equation and use the least squares regression fitting method to obtain the estimated values ​​of the scale parameters. and the estimated value of the shape parameter m; Step 2: Calculate the probability of the missile engine passing the secondary random inspection; use Indicates the engine in the The probability of passing the second-level sampling inspection is calculated based on the second-level sampling inspection rules in condition 2. There are two possibilities for passing the inspection: the engine passes the first sampling inspection; or the engine passes the second sampling inspection. The probability of passing the second sampling inspection is obtained by calculating the probabilities of the two scenarios separately and then adding them together. Due to the During the secondary sampling inspection, the number of unqualified samples Follows binomial distribution ,in This indicates the probability that the engine will fail the random inspection at this time; Therefore, the number of defective products The probability is ; In the formula: For the first Number of unqualified samples during the secondary sampling inspection; Number of non-conforming products The probability of; This refers to the number of samples collected during the random inspection. The interval between two secondary sampling inspections; Indicates that the missile engine is The probability of failure at any given time; Indicates that the missile engine is Reliability at any given moment; Therefore, the engine is in the... The probability of passing the secondary sampling inspection for: (6) In the formula: Indicates the engine in the The probability of passing the secondary sampling inspection; This refers to the number of samples collected during the random inspection. Indicates the permissible number of non-compliant samples in the random inspection; The interval between two secondary sampling inspections; This indicates the time spent on each random inspection; Indicates the engine is in The probability of failure at any given time; Indicates the engine is in Reliability at any given moment; Indicates the engine is in Reliability at any given moment; Step 3: Iteratively calculate the instantaneous availability assessment value of the engine; Define the missile engine at any given moment The state function is: (7) Then the engine is The probability of being in a usable state at all times is Since the engine had previously undergone maintenance, the aforementioned probabilities could not be calculated directly, but could be constructed through iterative calculations. The expression; by This indicates the time when the engine's first secondary random inspection and maintenance was completed; At any time, according to and Regarding the relationship, there are two situations for the engine: The engine has not undergone maintenance; The engine has already undergone maintenance; It consists of the two situations mentioned above, namely: (8) In the formula: This indicates the engine's cumulative operating time since it left the factory; This indicates the time when the engine's first secondary spot check and maintenance was completed. Indicates the engine is in The probability that the device is always in a usable state; Indicates the engine is in The probability that it is always in an available state and has not undergone maintenance; Indicates the engine is in The probability that it is always in a usable state and has undergone maintenance; The probabilities of being in a usable state in these two cases are calculated below: The engine has not undergone maintenance. In this case, the instantaneous availability of the engine is: (9) In the formula: This indicates the engine's cumulative operating time since it left the factory; This indicates the time when the engine's first secondary spot check and maintenance was completed. Indicates the engine is in The probability that it is always in an available state and has not undergone maintenance; Indicates the engine is in Reliability at any given moment; Indicates the engine is in The probability that a device has never undergone maintenance. Since the probability of an engine being selected in each random inspection is... Therefore, the engine is The probability that a device has never undergone secondary sampling inspection and maintenance is: (10) In the formula: Indicates the engine is in The probability that a device has never undergone maintenance. The interval between two secondary sampling inspections; This indicates the probability that an engine will be selected during each random inspection. This indicates the average number of times the second sampling inspection was conducted. Indicated to The total number of times the engine underwent secondary random inspections; Indicates the engine in the The probability of passing the secondary sampling inspection; Among them, the average number of the second sampling inspection for: ; In the formula: This indicates the average number of times the second sampling inspection was conducted. Indicated to The total number of times the engine underwent secondary random inspections; This refers to the number of samples collected during the random inspection. This indicates the number of items that failed the random inspection. The interval between two secondary sampling inspections; Indicates the engine is in The probability of failure at any given time; Indicates the engine is in Reliability at any given moment; therefore: (11) To simplify the expression, let: (12) Indicates that the engine is in the front. If the probability of not performing maintenance in the second-level sampling is given, then equation (11) can be expressed as: (13) In the formula: This indicates the engine's cumulative operating time since it left the factory; This indicates the time when the engine's first secondary spot check and maintenance was completed. Indicates the engine is in The probability that it is always in an available state and has not undergone maintenance; Indicates the engine is in Reliability at any given moment; Indicated to The total number of times the engine underwent secondary random inspections; This refers to the number of samples collected during the random inspection. Indicates the permissible number of non-compliant samples in the random inspection; Indicates the engine is in The probability that the sampled item has not undergone maintenance within the second-level sampling inspection; This indicates the probability that an engine will be selected during each random inspection. The interval between two secondary sampling inspections; Indicates the engine is in The probability of failure at any given time; Indicates the engine is in Reliability at any given moment; Indicates the engine in the The probability of passing the secondary sampling inspection; The engine has undergone maintenance. The engine in The first maintenance was performed during the second-level random inspection, and the item was returned to the factory and transported back to the field. There are three possible scenarios: Scenario 1: This indicates that the engine is in the front. The second-level random inspection did not involve repairs, the first It was selected in the first sampling of the secondary sampling inspection; Therefore, the probability of scenario 1 occurring is: (14) In the formula: This indicates the time when the engine's first secondary spot check and maintenance was completed. This indicates the time spent on each random inspection; The interval between two secondary sampling inspections; Indicated to The total number of times the engine underwent secondary random inspections; This represents the probability of scenario 1 occurring; Indicates the engine is in The probability that the sampled item has not undergone maintenance within the second-level sampling inspection; This indicates the probability that an engine will be selected during each random inspection. Scenario 2: ; The engine is in the front. The second-level random inspection did not involve repairs, the first The sample was not selected in the first sampling of the secondary sampling inspection, but the number of unqualified samples in the first sampling inspection reached the critical value. A second random inspection was conducted, and the engine was selected during the second inspection. The probability of this happening is: (15) In the formula: This indicates the time when the engine's first secondary spot check and maintenance was completed. This indicates the time spent on each random inspection; The interval between two secondary sampling inspections; Indicated to The total number of times the engine underwent secondary random inspections; Indicates the probability of the situation occurring; Indicates the engine is in The probability that the sampled item has not undergone maintenance within the second-level sampling inspection; This indicates the probability that an engine will be selected during each random inspection. This refers to the number of samples collected during the random inspection. Indicates the permissible number of non-compliant samples in the random inspection; Indicates the engine is in The probability of failure at any given time; Indicates the engine is in Reliability at any given moment; The engine is in the front. The second-level random inspection did not involve repairs, the first The sample was not selected in the first sampling of the secondary sampling inspection, but the number of unqualified samples in the first sampling inspection exceeded the threshold. The entire batch of engines was sent back to the factory for maintenance. The probability of this happening is: (16) In the formula: This indicates the time when the engine's first secondary spot check and maintenance was completed. This indicates the time spent on each random inspection; The interval between two secondary sampling inspections; Indicated to The total number of times the engine underwent secondary random inspections; Indicates the probability of the situation occurring; Indicates the engine is in The probability that the sampled item has not undergone maintenance within the second-level sampling inspection; This indicates the probability that an engine will be selected during each random inspection. This refers to the number of samples collected during the random inspection. Indicates the permissible number of non-compliant samples in the random inspection; Indicates the engine is in The probability of failure at any given time; Indicates the engine is in Reliability at any given moment; Therefore, the probability of scenario 2 occurring is: (17) In the formula: This indicates the time when the engine's first secondary spot check and maintenance was completed. This indicates the time spent on each random inspection; The interval between two secondary sampling inspections; Indicated to The total number of times the engine underwent secondary random inspections; This indicates the probability of scenario 2 occurring; Indicates the engine is in The probability that the sampled item has not undergone maintenance within the second-level sampling inspection; This indicates the probability that an engine will be selected during each random inspection. This refers to the number of samples collected during the random inspection. Indicates the permissible number of non-compliant samples in the random inspection; Indicates the engine is in The probability of failure at any given time; Indicates the engine is in Reliability at any given moment; Scenario 3: The engine is in the front. The second-level random inspection did not involve repairs, the first The first and second rounds of the secondary sampling inspection were not selected, but the number of non-compliant samples from other engines in the first round of sampling inspection reached the critical value. A second random inspection was required, but the second inspection also failed. Therefore, the probability of scenario 3 occurring is: ；(18) In the formula: This indicates the time when the engine's first secondary spot check and maintenance was completed. This indicates the time spent on each random inspection; The interval between two secondary sampling inspections; Indicated to The total number of times the engine underwent secondary random inspections; This indicates the probability of scenario 3 occurring; Indicates the engine is in The probability that the sampled item has not undergone maintenance within the second-level sampling inspection; This indicates the probability that an engine will be selected during each random inspection. This refers to the number of samples collected during the random inspection. Indicates the permissible number of non-compliant samples in the random inspection; Indicates the engine is in The probability of failure at any given time; Indicates the engine is in Reliability at any given moment; Indicates the engine is in Reliability at any given moment; Because repairs take time, once the engine is repaired, the time until the next secondary spot check is no longer... , but , , One of them depends on the time required for the last maintenance, so it is necessary to calculate the probability that the engine is in a usable state under this maintenance mode; remember This indicates the engine's first maintenance interval is... Later The probability of being in a usable state at any given time, where, , The calculation process can construct the following iterative equation: (19) In the formula: This indicates that the first repair occurred on the [date / time]. After the start of the next maintenance time; This indicates the time spent on each random inspection; The interval between two secondary sampling inspections; This indicates the engine's first maintenance interval is... Later The probability that the device is always in a usable state; Indicates the engine is in The probability that it is always in an available state and has not undergone maintenance; Indicated to The total number of times the engine underwent secondary random inspections; This indicates that the first repair occurred on the [date / time]. After the start of the next maintenance The probability at any given moment; Therefore, the missile engine at any given moment The iterative formula for the probability of being in a usable state is: (20) In the formula: This indicates that the first repair occurred on the [date / time]. After the start of the next maintenance time; The interval between two secondary sampling inspections; This indicates the time spent on each random inspection; Indicates the engine is in The probability that the device is always in a usable state; Indicates the engine is in The probability that it is always in an available state and has not undergone maintenance; Indicated to The total number of times the engine underwent secondary random inspections; Indicates the engine is in The probability that the sampled item has not undergone maintenance within the second-level sampling inspection; Indicates the engine is in Reliability at any given moment; This indicates that the first repair occurred on the [date / time]. After the start of the next maintenance The probability at any given moment; Step 4: Calculate the engine's average availability assessment value; Because regulations stipulate that the engine must be in an unusable state during the random inspection period, and the previous calculation... The probability that the engine is in a usable state, therefore it is necessary to... The conversion is performed to obtain the engine availability under the corresponding regulations. and ; At any random inspection and maintenance time , The following situations may occur during random inspections: If the first random inspection passes, the instantaneous availability of the engine during this maintenance cycle is: (21) In the formula: Indicates the engine is in Instantaneous availability at any given moment; The interval between two secondary sampling inspections; This indicates the time spent on each random inspection; Indicates the engine is in The probability that the device is always in a usable state; Indicated to The total number of times the engine underwent secondary random inspections; The corresponding probability of occurrence is: (22) In the formula: The probability of the situation occurring; The interval between two secondary sampling inspections; Indicates the engine is in The probability that the device is always in a usable state; Indicated to The total number of times the engine underwent secondary random inspections; If the first random inspection fails; or if the first random inspection is critical but the second random inspection passes, then the instantaneous availability of the engine during this maintenance cycle is: (23) In the formula: Indicates the engine is in Instantaneous availability at any given moment; The interval between two secondary sampling inspections; This indicates the time spent on each random inspection; Indicates the engine is in The probability that the device is always in a usable state; Indicated to The total number of times the engine underwent secondary random inspections; The corresponding probability of occurrence is: (24) In the formula: The probability of the situation occurring; The interval between two secondary sampling inspections; Indicates the engine is in The probability that the device is always in a usable state; Indicated to The total number of times the engine underwent secondary random inspections; This refers to the number of samples collected during the random inspection. Indicates the permissible number of non-compliant samples in the random inspection; If the first random inspection is near the critical level and the second random inspection fails, then the instantaneous availability of the engine during this maintenance cycle is: (25) In the formula: Indicates the engine is in Instantaneous availability at any given moment; The interval between two secondary sampling inspections; This indicates the time spent on each random inspection; Indicates the engine is in The probability that the device is always in a usable state; Indicated to The total number of times the engine underwent secondary random inspections; The corresponding probability of occurrence is: ; In summary, the instantaneous availability assessment model for missile engines is as follows: (26) In the formula: Indicates the engine is in Instantaneous availability at any given moment; The probability of the situation occurring; The probability of the situation occurring; The interval between two secondary sampling inspections; This indicates the time spent on each random inspection; Indicates the engine is in The probability that the device is always in a usable state; Indicated to The total number of times the engine underwent secondary random inspections; Due to instantaneous availability and average availability There is a functional relationship between them: (27) In the formula: Indicates the engine is in Instantaneous availability at any given moment; The cumulative service life of the engine; Therefore, once the instantaneous availability assessment model is obtained, it can be transformed into an average availability assessment model, using the Weibull distribution's scale parameter estimates. Substituting the estimated value of the shape parameter m into equations (26) and (27) yields the evaluation values ​​of the instantaneous availability and average availability of the missile engine.

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