CN105022921A - Fuzzy system reliability design method in combination with modal interval algorithm - Google Patents
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Abstract
The present invention relates to a fuzzy system reliability design method in combination with a modal interval algorithm. The method comprises the following steps of: 1. defining operations of fuzzy numbers; 2. judging multiple events and carrying out derivation on the multiple events and the sub-events of the multiple events; 3. judging whether the monotonicity of the multiple events is consistent with the monotonicity of the sub-events of the multiple events, and separately performing modal interval analysis by using coercive optimal and partial coercive optimal theories; 4. performing calculation according to a modal interval operation rule; 5. performing calculation to obtain an interval reliability index, security possibility degree and invalidation possibility degree of a system for reliability design of the system; 6. determining whether the reliability design of the system is performed in a serial, parallel or serial-parallel manner, and if so, performing the next step, or otherwise, ending; 7. performing modal interval analysis in the same method as the steps 2 to 4 so as to obtain the accurate security possibility degree and invalidation possibility degree of the system; and 8. determining the design reliability of the fuzzy system according to an actual requirement. According to the method, the correctness of the system reliability design is improved and the calculation amount is reduced.
Description
Technical Field
The invention relates to the technical field of structural reliability design, in particular to a fuzzy system reliability design method combined with a modal interval algorithm.
Background
In 1965, the release of "fuzzy sets" marked the emergence of fuzzy theory. Thereafter, fuzzy mathematics has been rapidly developed and widely used in various engineering fields. Unlike traditional mathematical algorithms, fuzzy numbers have their own algorithms of addition, subtraction, multiplication, division, etc. and their corresponding simplified processes.
The structure reliability assessment is necessary for practical engineering, and the safe operation of the structure can be effectively guaranteed. Structural system reliability is generally defined as: under specified conditions and circumstances of use, the structural system is capable of effectively withstanding loads and the ability to withstand the environment for normal operation over a given service life. The reliability of the structural system can be represented by a reliability index, a safety probability (safety probability), or a failure probability (failure probability), which are consistent. Modern structural systems are more and more complex, are easily influenced by various factors, and are difficult to realize accurate expression by using mathematical or mechanical methods. Practice shows that the fuzzy theory can objectively describe the problem of the complex system and obtain a model basically consistent with the actual engineering. Due to the limitation of basic assumptions of the conventional reliability theory, for the fuzzy problem, if the conventional reliability theory is still used, the calculation result is inconsistent with the actual result. For this reason, reliability analysis must be combined with fuzzy mathematics to build a fuzzy reliability analysis model of the structure.
From the expression theorem and the decomposition theorem of the fuzzy number, the fuzzy number corresponds to the interval set, and the algorithm is also established on the interval algorithm. However, the interval operation process is prone to interval expansion, which may result in erroneous reliability estimation during the reliability design of the fuzzy system. Meanwhile, when the interval reliability of the series, parallel and series-parallel systems is calculated, the calculation result is unreasonable because the systems generally contain a plurality of units or a plurality of failure modes.
At present, an improved interval truncation method, a combination method, an optimization method and the like can be adopted to avoid the interval expansion phenomenon. However, the truncation criterion of the improved interval truncation method is not easy to determine, and the calculation result is greatly influenced by the criterion. The combination method and the optimization method have the problems of large calculation amount, application limitation and the like, so that a method more suitable for fuzzy reliability estimation needs to be found. The modal interval algorithm is established on the basis of interval operation, and the result is subjected to semantic interpretation by defining logic predicates, so that accurate parameter interval estimation can be obtained. In view of this, the invention provides a fuzzy system reliability design method combined with a modal interval algorithm.
Disclosure of Invention
The invention aims to provide a fuzzy system reliability design method combined with a modal interval algorithm, which not only improves the correctness of system reliability design, but also reduces the calculation amount.
In order to achieve the purpose, the technical scheme of the invention is as follows: a fuzzy system reliability design method combined with a modal interval algorithm comprises the following steps:
step S1, defining fuzzy number operation by decomposition theorem;
step S2, judging multiple event variables in the fuzzy number arithmetic formula, and respectively performing derivation on multiple events and sub-events thereof;
step S3, judging whether the monotonicity of the multiple events and the sub-events thereof is consistent, if so, performing modal interval analysis by using a forced optimal theory, otherwise, performing modal interval analysis by using a part of the forced optimal theory;
step S4, calculating the analyzed formula according to a modal interval algorithm to obtain a reasonable result interval;
step S5, calculating the middle point and the radius of the intervals under different levels of the truncated sets to obtain the interval reliability index, the safety possibility and the failure possibility of the system, and using the interval reliability index, the safety possibility and the failure possibility for the reliability design of the system;
step S6, determining whether to design the system reliability according to the serial, parallel or series-parallel mode, if yes, turning to step S7, otherwise, ending;
step S7, for the situation of calculating the system reliability according to the serial, parallel or series-parallel connection mode, judging the multiple event variables in the serial, parallel or series-parallel connection reliability calculation formula, and respectively performing derivation on the multiple events and the sub-events thereof;
step S8, judging whether the monotonicity of the multiple events and the sub-events thereof is consistent, if so, performing modal interval analysis by using a forced optimal theory, otherwise, performing modal interval analysis by using a part of forced optimal theory, and calculating the analyzed formula according to a modal interval algorithm to obtain the safety possibility and the failure possibility of an accurate serial, parallel or series-parallel system;
and step S9, determining the design reliability of the fuzzy system according to actual requirements.
Further, it is provided withpFor a fuzzy set in the real number domain R,λis a threshold or confidence level, ifAnd ispIs/are as followsλCutting setp λ Is a finite interval contained in R, then definepIs a fuzzy number on R; setting fuzzy numberWhereinAs the fuzzy number set, in step S1, the fuzzy number operation is defined by the decomposition theorem as:
(1)
wherein,representing a multivariate fuzzy function, U represents a union of multiple sets,λis at [0,1 ]]A real number between which a value is taken,is shown to takeλ∈[0,1]A union of all fuzzy sets;λthe subscripts indicate the truncation set,pn λ representing fuzzy numberspnIs/are as followsλHorizontal truncated set for converting fuzzy number into real number interval for interval operation；λ() To representλThe product of the cross product of the set in brackets is used for converting the real number interval after operation into a fuzzy number so as to obtain a membership function;
the membership functions obtained from the expression theorem are:
(2)
wherein the V-shaped represents taking the supreme boundary,is shown in satisfaction ofTaking the supremum boundary under the condition,zthe value in the fuzzy function value domain after fuzzy mapping;
assuming a closed interval set of the real number domain R, and the existence of quantifier E and global quantifier U, the modal interval is defined as:
(3)
in the formulaRepresenting a classical interval, and representing the mode by QX epsilon (E, U), namely one interval corresponds to two modes; is provided with=[a, b]A is less than or equal to b; when in useQX=EWhen, X = [ a, b =]The form of the interval is consistent with that of the classical interval and is defined as a standard interval; when in useQX=UWhen, X = [ b, a =]The method is a form specific to a modal interval and is defined as an unnormalized interval, and the unnormalized interval is used for inhibiting the expansion of the interval in the operation process; in the modal interval algorithm, the interval is not normalizedIs realized by a Dual operator Dual, namely: dual ([ a, b)])=[b,a](ii) a If the function isfThe variable in (1) is converted into an interval form variable, then the functionfAccordingly, becomes a range function, and is recorded asfR(ii) a In-situ calculation formulafRWhen a variable appears more than once, the variable is defined as a multiple event variable, and events corresponding to different positions are defined as sub-events of the multiple events; then, in step S2, the multiple events and their sub-events are differentiated, respectively.
Further, in step S3, the constrained optimal theory is defined as: let X be the interval vector,fRis defined inAnd is completely monotonic for all multiple events; let XD be an expansion vector of X, that is, each sub-event of multiple-event XD is an independent sub-event in XD; however, if the monotonic trend of any independent sub-event in XD is opposite to the monotonic trend of corresponding multiple events, the sub-event is changed into dual form, and the interval calculated by the dual formfR(XD) is precise;
the definition part forces the optimal theory to be: let X be the interval vector,fRis defined inAnd is completely monotonic for only a partial subset Y of multiple events; let XD be an expansion vector of X, i.e., each sub-event of each multiple event in the fully monotonic subset is contained in XD; if the monotone trend of any independent sub-event in the XD is opposite to the global monotone trend of the corresponding multiple events, the sub-event is changed into a dual form; the remaining multiple event canonical interval vector ApConverting all sub-events except one sub-event into dual form to obtain sub-vector Ap', so as to convert X into XDT, the calculation formula being as in formula (4), the interval thus calculatedfR(XD) is approximate;
(4)
wherein,XDTkthen it means to divide bykAll but one sub-event is transformed into a dual form.
Further, in step S5, an expression theorem is used to find an accurate membership function of the fuzzy number:
obtaining the interval reliability index of the system based on the interval values obtained under different level cut setsSafety likelihood thetaspAnd the failure probability thetafp，Andrespectively representing horizontal cut setsλThe midpoint and radius of the lower interval.
Further, steps S1-S5 are designed for reliability of a single failure mode or a single unit of the system, and the reliability interval form of a single failure mode or unit is shown in formula (5):
(5)
wherein,is shown asiThe security possibilities of the individual units are,、respectively representiThe upper and lower limit of the security possibility of each unit,is shown asiThe probability of failure of an individual cell,andrespectively representiUpper and lower limit of failure possibility of each unit;
for a complex system with multiple failure modes or consisting of multiple units, the reliability calculation of the system is summarized as the reliability calculation of a series system, a parallel system and a series-parallel system, wherein the reliability of the series system is as follows:
(6)
the reliability of the parallel system is:
(7)
wherein,representing a number of quantities multiplied together.
Compared with the prior art, the invention has the beneficial effects that: 1) the modal interval algorithm has a good theoretical basis, is convenient to operate in problem analysis, and can solve the interval expansion problem to a great extent; 2) in the fuzzy number operation process, the modal interval algorithm can avoid interval expansion, so that a reasonable result interval is obtained, and an accurate membership function is obtained; 3) the system reliability is calculated by using the interval solutions under different levels of the truncated sets, and the ambiguity of the intermediate calculation process is reserved; 4) the reliability of the fuzzy system is described by adopting an interval reliability index, the calculated amount can be effectively reduced, the interval safety possibility and the failure possibility are introduced, the interference condition is considered, and the method is suitable for the actual engineering problem; 5) the reliability of the series, parallel and series-parallel systems is calculated by adopting a modal interval algorithm, so that the accurate reliability of the complex system can be obtained, and the correctness of the reliability design of the system is ensured.
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FIG. 1 is a flow chart of an implementation of an embodiment of the present invention.
Detailed Description
The method uses the modal interval algorithm to replace the classical interval algorithm in the fuzzy number operation, solves the problem of interval expansion which is easy to occur in the interval calculation process, and obtains accurate reliability index, safety possibility and failure possibility of the system interval by the interval values under different levels of truncation. The invention not only effectively reduces the calculated amount, but also keeps the ambiguity of the intermediate calculation process, and the obtained result is more accurate and has engineering practicability. In addition, when the interval reliability of the series, parallel and series-parallel systems is calculated, the modal interval algorithm can avoid interval expansion, and the accurate safety possibility and failure possibility of a typical system are obtained.
The invention is described in further detail below with reference to the figures and the embodiments.
The method for designing the reliability of the fuzzy system by combining the modal interval algorithm comprises the following steps as shown in figure 1:
step S1 defines the fuzzy number operation by the decomposition theorem.
Step S2, determining multiple event variables in the fuzzy number arithmetic formula, and respectively deriving multiple events and their sub-events.
Is provided withpFor a fuzzy set in the real number domain R,λis a threshold or confidence level, ifAnd ispIs/are as followsλCutting setp λ Is a finite interval contained in R, then definepIs a fuzzy number on R. Setting fuzzy numberWhereinAs the fuzzy number set, in step S1, the fuzzy number operation is defined by the decomposition theorem as:
(1)
wherein,representing a multivariate fuzzy function, U represents a union of multiple sets,λis at [0,1 ]]A real number between which a value is taken,is shown to takeλ∈[0,1]A union of all fuzzy sets;λthe subscripts indicate the truncation set,pn λ representing fuzzy numberspnIs/are as followsλThe horizontal truncated set is used for converting the fuzzy number into a real number interval so as to carry out interval operation;λ() To representλTruncation with bracketed set for converting the computed real number interval to fuzzyAnd counting to obtain the membership functions.
The membership functions obtained from the expression theorem are:
(2)
wherein the V-shaped represents taking the supreme boundary,is shown in satisfaction ofTaking the supremum boundary under the condition,zthe values in the fuzzy function value domain after fuzzy mapping.
In the formula (1), the parentheses areThe operation of (2) is established on the basis of interval operation, and the interval expansion problem can also be generated, so that the final solved membership function of the formula (2) is inaccurate.
The modal interval algorithm is an algorithm which is established on the basis of a classical interval and can prevent the interval from expanding. Assuming a closed interval set of real number domains R, and the presence of quantifiers e (existantial quantifiers) and global quantifiers u (universal quantifiers), the modal interval is defined as:
(3)
in the formulaRepresenting a classical interval, and representing the mode by QX epsilon (E, U), namely one interval corresponds to two modes; is provided with=[a, b]A is less than or equal to b; when in useQX=EWhen, X = [ a, b =]The form of the interval is consistent with that of the classical interval and is defined as a standard interval; when in useQX=UWhen, X = [ b, a =]The method is a form specific to a modal interval and is defined as an unnormalized interval, and the unnormalized interval plays a role in inhibiting the expansion of the interval in the operation process; in the modal interval algorithm, the generation of the unnormalized interval is realized by a Dual operator Dual, namely: dual ([ a, b)])=[b,a](ii) a If the function isfThe variable in (1) is converted into an interval form variable, then the functionfAccordingly, becomes a range function, and is recorded asfR(ii) a In-situ calculation formulafRWhen a variable appears more than once, the variable is defined as a multiple event variable, and events corresponding to different positions are defined as sub-events of the multiple events; then, in step S2, the multiple events and their sub-events are differentiated, respectively.
And step S3, judging whether the monotonicity of the multiple events and the sub-events thereof is consistent, if so, performing modal interval analysis by using a forced optimal theory, otherwise, performing modal interval analysis by using a part of the forced optimal theory.
Wherein, the forced optimal theory is defined as: let X be the interval vector,fRis defined inAnd is completely monotonic for all multiple events; let XD be an expansion vector of X, that is, each sub-event of multiple-event XD is an independent sub-event in XD; however, if the monotonic trend of any independent sub-event in XD is opposite to the monotonic trend of corresponding multiple events, the sub-event is changed into dual form, and the interval calculated by the dual formfR(XD) is precise.
The definition part forces the optimal theory to be: let X be the interval vector,fRis defined inAnd is completely monotonic for only a partial subset Y of multiple events; let XD be an expansion vector of X, i.e., each sub-event of each multiple event in the fully monotonic subset is contained in XD; but in XDIf the monotone trend of any independent sub-event is opposite to the global monotone trend of the corresponding multiple events, the sub-event is changed into a dual form; the remaining multiple event canonical interval vector ApConverting all sub-events except one sub-event into dual form to obtain sub-vector Ap', so as to convert X into XDT, the calculation formula being as in formula (4), the interval thus calculatedfR(XD) is approximate.
(4)
Wherein,XDTkthen it means to divide bykAll but one sub-event is transformed into a dual form.
And step S4, calculating the analyzed formula according to a modal interval algorithm to obtain a reasonable result interval.
Step S5, calculating the middle point and radius of the interval under different levels of the truncated set to obtain the interval reliability index, the safety possibility and the failure possibility of the system, and using the interval reliability index, the safety possibility and the failure possibility for the reliability design of the system, namely using the expression theorem in the fuzzy theory to obtain the accurate membership function of the fuzzy number:
obtaining the interval reliability index of the system based on the interval values obtained under different level cut setsSafety likelihood thetaspAnd the failure probability thetafpWherein the security likelihood thetaspSimilar to the reliability indicator in the probabilistic approach,andrespectively representing horizontal cut setsλThe midpoint and radius of the lower interval.
Steps S1-S5 are designed for reliability of a single failure mode or a single unit of the system, and the reliability interval form of the single failure mode or unit is shown in formula (5):
(5)
wherein,is shown asiThe security possibilities of the individual units are,、respectively representiThe upper and lower limit of the security possibility of each unit,is shown asiThe probability of failure of an individual cell,andrespectively representiThe upper and lower limit of the failure possibility of each unit.
For a complex system with multiple failure modes or consisting of multiple units, the reliability calculation of the system is summarized as the reliability calculation of a series system, a parallel system and a series-parallel system, wherein the reliability of the series system is as follows:
(6)
the reliability of the parallel system is:
(7)
wherein,representing a number of quantities multiplied together.
In the calculation process of the reliability of the complex system, the problem of interval expansion is also generated, so modal interval analysis should be performed.
And step S6, determining whether to design the system reliability according to a serial connection, parallel connection or series-parallel connection mode, if so, turning to step S7, and if not, ending the operation.
And step S7, judging multiple event variables in the serial, parallel or serial-parallel reliability calculation formula for the situation of calculating the system reliability in a serial, parallel or serial-parallel mode, and respectively performing derivation on multiple events and sub-events thereof.
And step S8, judging whether the monotonicity of the multiple events and the sub-events thereof is consistent, if so, performing modal interval analysis by using a forced optimal theory, otherwise, performing modal interval analysis by using a part of forced optimal theory, and calculating the analyzed formula according to a modal interval algorithm to obtain the safety possibility and the failure possibility of the accurate serial, parallel or series-parallel system.
And step S9, determining the design reliability of the fuzzy system according to actual requirements.
The structural reliability analysis model mainly relates to stress and strength, wherein the stress is related to factors such as external load, structural size, material performance and support, and the factors are always provided with fuzzy characteristics, so that the stress is also provided with fuzziness. The material strength is a complex mechanical quantity, and fuzzy factors such as manufacturers, sample quality conditions, test piece sizes, processing methods and the like can influence the material strength. In order to meet specific conditions as much as possible, a fuzzy mathematical method is adopted, and subjective fuzzy variables are adopted for processing according to experience and judgment of designers. Therefore, the modal interval algorithm is adopted to avoid the problem of interval expansion in the fuzzy operation process, and the method plays an important role in the reliability design of the fuzzy system.
The reliability of the fuzzy system is calculated by using a modal interval algorithm, namely, the modal interval algorithm is adopted to replace a classical interval algorithm in the fuzzy number operation process, and the interval expansion is inhibited by monotonicity analysis of multiple events and sub-events thereof and by using a forced optimal theory or a part of forced optimal theory, so that a reasonable result interval is obtained. By the interval estimation values under different level cut sets, the interval reliability index of the system can be calculatedSafety possibility and failure possibility, and is used for the reliability design of the fuzzy system. For the situation that the reliability of the complex system is calculated according to the serial, parallel and parallel connection modes, modal interval analysis needs to be carried out on the serial, parallel and parallel connection reliability calculation formulas, accurate system safety and failure possibility are obtained, and finally the design reliability of the fuzzy system is determined according to actual requirements.
The above are preferred embodiments of the present invention, and all changes made according to the technical scheme of the present invention that produce functional effects do not exceed the scope of the technical scheme of the present invention belong to the protection scope of the present invention.
Claims (5)
1. A fuzzy system reliability design method combined with a modal interval algorithm is characterized by comprising the following steps:
step S1, defining fuzzy number operation by decomposition theorem;
step S2, judging multiple event variables in the fuzzy number arithmetic formula, and respectively performing derivation on multiple events and sub-events thereof;
step S3, judging whether the monotonicity of the multiple events and the sub-events thereof is consistent, if so, performing modal interval analysis by using a forced optimal theory, otherwise, performing modal interval analysis by using a part of the forced optimal theory;
step S4, calculating the analyzed formula according to a modal interval algorithm to obtain a reasonable result interval;
step S5, calculating the middle point and the radius of the intervals under different levels of the truncated sets to obtain the interval reliability index, the safety possibility and the failure possibility of the system, and using the interval reliability index, the safety possibility and the failure possibility for the reliability design of the system;
step S6, determining whether to design the system reliability according to the serial, parallel or series-parallel mode, if yes, turning to step S7, otherwise, ending;
step S7, for the situation of calculating the system reliability according to the serial, parallel or series-parallel connection mode, judging the multiple event variables in the serial, parallel or series-parallel connection reliability calculation formula, and respectively performing derivation on the multiple events and the sub-events thereof;
step S8, judging whether the monotonicity of the multiple events and the sub-events thereof is consistent, if so, performing modal interval analysis by using a forced optimal theory, otherwise, performing modal interval analysis by using a part of forced optimal theory, and calculating the analyzed formula according to a modal interval algorithm to obtain the safety possibility and the failure possibility of an accurate serial, parallel or series-parallel system;
and step S9, determining the design reliability of the fuzzy system according to actual requirements.
2. The method of claim 1, wherein the fuzzy system reliability design method in combination with modal interval algorithm is further characterized bypFor a fuzzy set in the real number domain R,λis a threshold or confidence level, ifAnd ispIs/are as followsλCutting setp λ Is a finite interval contained in R, then definepIs a fuzzy number on R; setting fuzzy numberWhereinAs the fuzzy number set, in step S1, the fuzzy number operation is defined by the decomposition theorem as:
(1)
wherein,representing a multivariate fuzzy function, U represents a union of multiple sets,λis at [0,1 ]]A real number between which a value is taken,is shown to takeλ∈[0,1]A union of all fuzzy sets;λthe subscripts indicate the truncation set,pn λ representing fuzzy numberspnIs/are as followsλThe horizontal truncated set is used for converting the fuzzy number into a real number interval so as to carry out interval operation;λ() To representλThe product of the cross product of the set in brackets is used for converting the real number interval after operation into a fuzzy number so as to obtain a membership function;
the membership functions obtained from the expression theorem are:
(2)
wherein the V-shaped represents taking the supreme boundary,is shown in satisfaction ofTaking the supremum boundary under the condition,zthe value in the fuzzy function value domain after fuzzy mapping;
assuming a closed interval set of the real number domain R, and the existence of quantifier E and global quantifier U, the modal interval is defined as:
(3)
in the formulaRepresenting a classical interval, and representing the mode by QX epsilon (E, U), namely one interval corresponds to two modes; is provided with=[a, b]A is less than or equal to b; when in useQX=EWhen, X = [ a, b =]The form of the interval is consistent with that of the classical interval and is defined as a standard interval; when in useQX=UWhen, X = [ b, a =]The method is a form specific to a modal interval and is defined as an unnormalized interval, and the unnormalized interval is used for inhibiting the expansion of the interval in the operation process; in the modal interval algorithm, the generation of the unnormalized interval is realized by a Dual operator Dual, namely: dual ([ a, b)])=[b,a](ii) a If the function isfThe variable in (1) is converted into an interval form variable, then the functionfAccordingly, becomes a range function, and is recorded asfR(ii) a In-situ calculation formulafRWhen a variable appears more than once, the variable is defined as a multiple event variable, and events corresponding to different positions are defined as sub-events of the multiple events; then, in step S2, the multiple events and their sub-events are differentiated, respectively.
3. The fuzzy system reliability design method in combination with modal interval algorithm of claim 2, wherein in step S3, the constrained optimal theory is defined as: let X be the interval vector,fRis defined inAnd is completely monotonic for all multiple events; let XD be an expansion vector of X, that is, each sub-event of multiple-event XD is an independent sub-event in XD; but if the single trend of any independent sub-event in XD and corresponding multiple event listIf the adjustment trend is opposite, the sub-events are changed into dual form, and the interval calculated by the dual formfR(XD) is precise;
the definition part forces the optimal theory to be: let X be the interval vector,fRis defined inAnd is completely monotonic for only a partial subset Y of multiple events; let XD be an expansion vector of X, i.e., each sub-event of each multiple event in the fully monotonic subset is contained in XD; if the monotone trend of any independent sub-event in the XD is opposite to the global monotone trend of the corresponding multiple events, the sub-event is changed into a dual form; the remaining multiple event canonical interval vector ApConverting all sub-events except one sub-event into dual form to obtain sub-vector Ap', so as to convert X into XDT, the calculation formula being as in formula (4), the interval thus calculatedfR(XD) is approximate;
(4)
wherein,XDTkthen it means to divide bykAll but one sub-event is transformed into a dual form.
4. The method of claim 3, wherein in step S5, the exact membership function of the fuzzy number is obtained by using the expression theorem:
obtaining the interval reliability index of the system based on the interval values obtained under different level cut setsSafety likelihood thetaspAnd the failure probability thetafp，Andrespectively representing horizontal cut setsλThe midpoint and radius of the lower interval.
5. The fuzzy system reliability design method in combination with modal interval algorithm as claimed in claim 4, wherein the steps S1-S5 are reliability design for a failure mode or a unit of the system, and the reliability interval form of the single failure mode or unit is shown as formula (5):
(5)
wherein,is shown asiThe security possibilities of the individual units are,、respectively representiThe upper and lower limit of the security possibility of each unit,is shown asiThe probability of failure of an individual cell,andrespectively representiUpper and lower limit of failure possibility of each unit;
for a complex system with multiple failure modes or consisting of multiple units, the reliability calculation of the system is summarized as the reliability calculation of a series system, a parallel system and a series-parallel system, wherein the reliability of the series system is as follows:
(6)
the reliability of the parallel system is:
(7)
wherein,representing a number of quantities multiplied together.
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| CN105447271A (en) * | 2015-12-17 | 2016-03-30 | 北京航空航天大学 | Design method of strength reliability of turbine leaf disc structure |
| CN105447271B (en) * | 2015-12-17 | 2018-05-15 | 北京航空航天大学 | A kind of turbine blade Structural Strength Reliability design method |
| CN105825004A (en) * | 2016-03-15 | 2016-08-03 | 万毅 | Reserve parallel connection method of automotive electronic equipment |
| CN110739687A (en) * | 2019-10-24 | 2020-01-31 | 福州大学 | electric power system distribution robust scheduling method considering wind power high-order uncertainty |
| CN110739687B (en) * | 2019-10-24 | 2021-06-01 | 福州大学 | Power system distribution robust scheduling method considering wind power high-order uncertainty |
| CN110929453A (en) * | 2019-11-18 | 2020-03-27 | 西安电子科技大学 | Copula function failure correlation system-based dynamic fuzzy reliability analysis method |
| CN110990921A (en) * | 2019-11-27 | 2020-04-10 | 上海工程技术大学 | Improved interval truncation method for cantilever beam safety design |
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