结构与系统的动态可靠性研究
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摘要
本学位论文以结构与系统的动态可靠性为研究对象，分别建立了随机参数结构动态可靠性模型，区间参数结构动态可靠性模型，模糊参数结构动态可靠性模型，可修复k/n表决系统动态可靠性模型，多状态可修复k/n表决系统动态可靠性模型。对结构与系统的动态可靠性进行了计算、预测、分析。其主要内容如下：
1.随机参数结构动态可靠性
将随机载荷的加载形式分为动载荷累加，等幅且服从参数为t的泊松分布，和载荷随时间变化且不服从任何分布的三种情况，利用应力-强度干涉理论和随机过程理论分别建立了结构强度退化下的结构动态可靠性预测的三种模型。由该模型可获得结构的可靠度随时间变化的规律。由一次随机载荷的概率密度函数，求得多次随机载荷的概率密度函数。在考虑结构承受多次随机载荷作用下结构强度退化和不退化的情况下，根据应力-强度干涉理论和概率密度演化方法，建立了结构的动态可靠度预测模型。应用向前差分方法求解该模型中的概率密度演化微分方程，获得任意时刻结构强度裕度的概率密度函数，进而再利用积分方法获得结构动态可靠度的预测结果。对于目前对结构的可靠性研究仅考虑结构所受的一种载荷的情况，从结构所受的4种随机载荷共同作用于结构的实际情况出发，运用Turkstra方法对应力进行了三种不同的组合，分别求得三种组合的应力作为结构所承受的应力。然后根据应力-强度干涉理论，对结构分为强度不退化，强度退化，分别建立了结构在共同载荷作用下的动态可靠性模型，将在三种组合应力下求得结构动态可靠性指标最低者确定为结构的最终动态可靠性指标。每个模型均用算例进行了验证，通过算例表明，这些方法是实用易行的。
2.区间参数结构动态可靠性
研究了当载荷区间变量随时间变化且结构强度区间变量随时间退化情况下的结构动态非概率可靠性问题，给出了结构强度退化的变化区间及其均值、离差的计算表达式，根据应力-强度干涉理论和区间理论建立了在结构强度随时间退化时，而结构所承受的载荷分别为：阶梯型、等幅交变型和任意时变型三种情况下的结构动态非概率可靠性预测模型。针对多次区间载荷结构随机强度退化和不退化的混合模型的情况，首先将多次区间载荷用等效区间载荷进行了代替，然后把该等效载荷看做均匀分布的随机变量，对均匀分布的随机变量和结构的随机强度进行了当量正态化，根据应力-强度干涉理论，用一次二阶矩法求出了结构在多次区间载荷下的结构的随机强度退化时的结构动态可靠度。算例表明了这些方法的合理性，实用性，精确性。
3.模糊参数结构动态可靠性
针对实际工程问题中结构承受的多次模糊载荷与结构模糊强度的情况，利用应力-强度干涉理论对结构承受的多次模糊载荷与模糊强度随时间变化的情况进行了分析，分别建立了作用于结构的多次模糊载荷随时间变化且结构模糊强度随时间不退化和退化时的结构的动态模糊可靠性预测模型。对于实际工程问题中结构承受的多次随机载荷与结构模糊强度的情况，利用应力-强度干涉理论对结构承受的随机载荷与模糊强度随时间变化的情况进行了分析，分别建立了作用于结构的多次随机载荷下结构模糊强度随时间不退化和退化时的结构的动态随机-模糊可靠性预测模型。最后算例对模型分别进行了验证，说明了模型的可行性与合理性。
4.可修复的k/n表决系统动态可靠性
根据强度-应力干涉理论，给出了k/n表决系统的单元在多次随机作用下、且单元抗力退化情况下的动态可靠性指标计算模型，由可靠性指标求得单元的动态失效概率。基于单元的动态失效概率，给出了在多次随机外部作用下，单元失效数目变化的概率，再由可修复k/n系统所具有的Markov性质，给出了在多次随机外部作用下的可修复k/n系统的转移概率，由转移概率获得概率密度矩阵，通过求解微分方程组计算出系统的动态可靠度。针对目前k/n表决系统中k的设置根据人为经验给出的情况，文中首先利用应力-强度干涉理论及系统元件承受的作用概率分布，给出了单个元件的失效概率，再由可修复k/n表决系统工作时元件的可靠与失效状态的转移情况，应用概率微分方程，建立了可修复k/n表决系统的动态可靠性预测模型，并给出了该模型的解。对可修复k/n表决系统中最少工作单元数目k的设置给出了理论依据。最后通过对算例的分析，表明了所建模型的准确性与可行性。
5.多状态可修复k/n表决系统的动态可靠性
针对多状态可修复k/n表决系统的复杂性，首先对系统的元件的多样性利用离散时间的马尔科夫链和半马尔科夫链进行了分析，给出了状态变化，在状态逗留时间的概率分布计算公式；然后给出了元件在状态变化，状态寿命变化的一步概率转移矩阵，最后根据对元件的分析，导出了系统的可靠度与可用度的预测模型。针对多状态发动机系统，对发动机系统进行了离散量化处理，使发动机的输出功率成为一个状态离散时间连续的马尔科夫随机过程；根据发动机输出功率的变化规律，建立了多状态马尔科夫模型，根据观察得到的数据，对多状态的马尔科夫模型的转移强度的计算给出了算法；利用本文方法计算了发动机单元随时间变化的故障率、输出功率的亏欠期望。从而为动态的预测和监控发动机正常工作可靠性提供了一个新方法。
The reliability from time response of the structure and system were studied in thedegree paper. The model of the structural random reliability from time response, themodel of the structural interval reliability from time response, the model of thestructural fuzzy reliability from time response, the model of the reparable k-out-of-nsystem reliability from time response, the model of the multi-state reparable k-out-of-nsystem reliability from time response are established, respectively. The dynamicreliability of the structure and system are computed, predicted, analyzed. The maincontents are as follows.
1. The structural random reliability from time response.
Stochastic loads are classified three types: dynamic load accumulation, amplitudeand obey the Poisson distribution whose parameter is t, load changing with time andnot subject to any distribution. As these conditions, their reliability from time responsemodels are established under structural strength constant and structural strengthdegenerating by using stress-strength interference theory and stochastic process theory;it shows that the structural reliability is gradually reduced with time, but not unchanged.The models are simple and easy computing, the change regular of the structuralreliability from time response can be obtained. The probability density evolutionmethod of the structural reliability from the time response prediction models underseveral times random loads are researched. Firstly, the probability density of severaltimes stochastic loads can be obtained according to the probability density of a timestochastic load. The prediction models of the structural reliability from the timeresponse under several times stochastic loads with the strength degeneration andwithout degeneration over time are established by using the stress-strength interferencetheory and probability density evolution method. The differential equation in the modelscan be solved by using the forward differential method. The probability densityfunctions of the structural strength redundancy in any time can be obtained. Thestructural prediction reliability from time response can be obtained by using integralmethod. For the structural reliability is only considered a load for structural influence incurrently research, the4types common loads that act on the structure have beencombined3different combinations by Turkstra method, the loads of the threecombinations were believed as the structural stress. The models of the structuralreliability from time response without structural strength degeneration and with degeneration over time have been established by using the stress-strength interferencetheory. The minimum reliability index from time response of the three reliabilityindexes from time response which was obtained in the models is determined thestructural reliability index from time response. It is shown that models are practicable,feasible by using example.
2. The structural interval reliability from time response.
The structural non-probabilistic reliability from time response under the intervalvariable of loads changing over time and the interval variable of structural strengthdegrading over time is researched. The formulas for the change interval of structuralstrength degradation and its mean value and deviation are shown. According to thestress-strength interference theory and the interval theory, the structural reliability fromtime response prediction models are established under the structural strengthdegradation over time while the loads of the structure bearing are classified three casesloads: ladder type, constant amplitude alternating type and random changing over timetype. For the hybrid model of several times interval loads and structural stochasticstrength with degradation and without degradation over time are established. Firstly,several times interval loads are substituted by the equivalent interval load. Secondly, theequivalent interval load is thought as the random variable who obeys the uniformdistributed. The random variables of the uniform distribution and structural randomstrength are done with the equivalent normal method. Then the structuralinterval-random hybrid reliability from time response under several times interval loadsand structural random strength degeneration and without degeneration over time iscomputed by using the first secondly order moment method according to thestress-strength interference theory. It is shown that the models are reasonable andpracticable by using examples.
3. The structural fuzzy reliability from time response.
For the several times fuzzy loads and fuzzy structural strength in the practiceengineer, the fuzzy loads and the fuzzy strength are analyzed. The predictionfuzzy-fuzzy models of the structural fuzzy reliability from time response are establishedby using the stress-strength interference theory when the several times fuzzy loadschanges over time and the structural strength changes without degeneration and withdegeneration. For the several times random loads and fuzzy structural strength in thepractice engineer, the loads and the fuzzy strength are analyzed. The predictionrandom-fuzzy hybrid models of the structural fuzzy reliability from time response areestablished by using the stress-strength interference theory when the several times random loads changes over time and the structural strength changes withoutdegeneration and with degeneration. Finally, it is shown that the models are feasible andreasonable by examples. Further it is pointed out that the reliability of probability is aspecial case of fuzzy reliability.
4. The reparable k-out-of-n system reliability from time response.
The model of unit reliability from time response of repairable k/n (G) system withunit strength degradation under repeated random shocks has been developed accordingto the stress-strength interference theory. The unit failure number is obtained based onthe unit failure probability which can be computed from the unit reliability from timeresponse. Then, the transfer probability function of the repairable k/n (G) system isgiven by its Markov property. Once the transfer probability function has been obtained,the probability density matrix and the steady-state probabilities of the system can beretrieved. Finally, the reliability from time response of the repairable k/n (G) system isobtained by solving the differential equations. Anovel reliability-based model of k/n (G)system has been developed. Unit failure probability is first calculated based on theapplied load and strength distributions according to the stress-strength interferencetheory. Then, a reliability prediction model from time response for the whole repairablek/n (G) system is established using the probabilistic differential equations. Finally, theresulting differential equations are solved and the value of k can be determined precisely.It is illustrated that the proposed methods are practicable, feasible and gives reasonableprediction which conforms to the engineering practice.
5. The reparable multi-state k-out-of-n system reliability from time response.
For the complexity of multi-state repairable k-out-of-n system, firstly, the diversityof the components of the system are analyzed by using discrete time Markov chains andsemi-Markov chain, the probability distribution computing formula of the change ofstate and the sojourn time in the state are obtained. Secondly, the first order transitionprobability matrix of the change of state and lifetime in the state areobtained. Finally,the reliability and availability prediction model of the system are obtained according tothe analysis of the components. For the multi-state engine system, Firstly, the enginesystem is discrete and quantization processed, the output power of the engine is becomea state discrete time continue Markov random process; Secondly, the multi-state Markovmodel is established, according to the observed data, the algorithm of transitionintensities is obtained; Finally, the proposed method is used to calculate the forcedoutage rate and the expected engine capacity deficiency of the engine changing overtime. For forecasting and monitoring the reliability from time response of the engine to work properly is supplied a new method. It is shown that the methods are practical,effective by using the engineering numerical examples.
1.随机参数结构动态可靠性
将随机载荷的加载形式分为动载荷累加，等幅且服从参数为t的泊松分布，和载荷随时间变化且不服从任何分布的三种情况，利用应力-强度干涉理论和随机过程理论分别建立了结构强度退化下的结构动态可靠性预测的三种模型。由该模型可获得结构的可靠度随时间变化的规律。由一次随机载荷的概率密度函数，求得多次随机载荷的概率密度函数。在考虑结构承受多次随机载荷作用下结构强度退化和不退化的情况下，根据应力-强度干涉理论和概率密度演化方法，建立了结构的动态可靠度预测模型。应用向前差分方法求解该模型中的概率密度演化微分方程，获得任意时刻结构强度裕度的概率密度函数，进而再利用积分方法获得结构动态可靠度的预测结果。对于目前对结构的可靠性研究仅考虑结构所受的一种载荷的情况，从结构所受的4种随机载荷共同作用于结构的实际情况出发，运用Turkstra方法对应力进行了三种不同的组合，分别求得三种组合的应力作为结构所承受的应力。然后根据应力-强度干涉理论，对结构分为强度不退化，强度退化，分别建立了结构在共同载荷作用下的动态可靠性模型，将在三种组合应力下求得结构动态可靠性指标最低者确定为结构的最终动态可靠性指标。每个模型均用算例进行了验证，通过算例表明，这些方法是实用易行的。
2.区间参数结构动态可靠性
研究了当载荷区间变量随时间变化且结构强度区间变量随时间退化情况下的结构动态非概率可靠性问题，给出了结构强度退化的变化区间及其均值、离差的计算表达式，根据应力-强度干涉理论和区间理论建立了在结构强度随时间退化时，而结构所承受的载荷分别为：阶梯型、等幅交变型和任意时变型三种情况下的结构动态非概率可靠性预测模型。针对多次区间载荷结构随机强度退化和不退化的混合模型的情况，首先将多次区间载荷用等效区间载荷进行了代替，然后把该等效载荷看做均匀分布的随机变量，对均匀分布的随机变量和结构的随机强度进行了当量正态化，根据应力-强度干涉理论，用一次二阶矩法求出了结构在多次区间载荷下的结构的随机强度退化时的结构动态可靠度。算例表明了这些方法的合理性，实用性，精确性。
3.模糊参数结构动态可靠性
针对实际工程问题中结构承受的多次模糊载荷与结构模糊强度的情况，利用应力-强度干涉理论对结构承受的多次模糊载荷与模糊强度随时间变化的情况进行了分析，分别建立了作用于结构的多次模糊载荷随时间变化且结构模糊强度随时间不退化和退化时的结构的动态模糊可靠性预测模型。对于实际工程问题中结构承受的多次随机载荷与结构模糊强度的情况，利用应力-强度干涉理论对结构承受的随机载荷与模糊强度随时间变化的情况进行了分析，分别建立了作用于结构的多次随机载荷下结构模糊强度随时间不退化和退化时的结构的动态随机-模糊可靠性预测模型。最后算例对模型分别进行了验证，说明了模型的可行性与合理性。
4.可修复的k/n表决系统动态可靠性
根据强度-应力干涉理论，给出了k/n表决系统的单元在多次随机作用下、且单元抗力退化情况下的动态可靠性指标计算模型，由可靠性指标求得单元的动态失效概率。基于单元的动态失效概率，给出了在多次随机外部作用下，单元失效数目变化的概率，再由可修复k/n系统所具有的Markov性质，给出了在多次随机外部作用下的可修复k/n系统的转移概率，由转移概率获得概率密度矩阵，通过求解微分方程组计算出系统的动态可靠度。针对目前k/n表决系统中k的设置根据人为经验给出的情况，文中首先利用应力-强度干涉理论及系统元件承受的作用概率分布，给出了单个元件的失效概率，再由可修复k/n表决系统工作时元件的可靠与失效状态的转移情况，应用概率微分方程，建立了可修复k/n表决系统的动态可靠性预测模型，并给出了该模型的解。对可修复k/n表决系统中最少工作单元数目k的设置给出了理论依据。最后通过对算例的分析，表明了所建模型的准确性与可行性。
5.多状态可修复k/n表决系统的动态可靠性
针对多状态可修复k/n表决系统的复杂性，首先对系统的元件的多样性利用离散时间的马尔科夫链和半马尔科夫链进行了分析，给出了状态变化，在状态逗留时间的概率分布计算公式；然后给出了元件在状态变化，状态寿命变化的一步概率转移矩阵，最后根据对元件的分析，导出了系统的可靠度与可用度的预测模型。针对多状态发动机系统，对发动机系统进行了离散量化处理，使发动机的输出功率成为一个状态离散时间连续的马尔科夫随机过程；根据发动机输出功率的变化规律，建立了多状态马尔科夫模型，根据观察得到的数据，对多状态的马尔科夫模型的转移强度的计算给出了算法；利用本文方法计算了发动机单元随时间变化的故障率、输出功率的亏欠期望。从而为动态的预测和监控发动机正常工作可靠性提供了一个新方法。
The reliability from time response of the structure and system were studied in thedegree paper. The model of the structural random reliability from time response, themodel of the structural interval reliability from time response, the model of thestructural fuzzy reliability from time response, the model of the reparable k-out-of-nsystem reliability from time response, the model of the multi-state reparable k-out-of-nsystem reliability from time response are established, respectively. The dynamicreliability of the structure and system are computed, predicted, analyzed. The maincontents are as follows.
1. The structural random reliability from time response.
Stochastic loads are classified three types: dynamic load accumulation, amplitudeand obey the Poisson distribution whose parameter is t, load changing with time andnot subject to any distribution. As these conditions, their reliability from time responsemodels are established under structural strength constant and structural strengthdegenerating by using stress-strength interference theory and stochastic process theory;it shows that the structural reliability is gradually reduced with time, but not unchanged.The models are simple and easy computing, the change regular of the structuralreliability from time response can be obtained. The probability density evolutionmethod of the structural reliability from the time response prediction models underseveral times random loads are researched. Firstly, the probability density of severaltimes stochastic loads can be obtained according to the probability density of a timestochastic load. The prediction models of the structural reliability from the timeresponse under several times stochastic loads with the strength degeneration andwithout degeneration over time are established by using the stress-strength interferencetheory and probability density evolution method. The differential equation in the modelscan be solved by using the forward differential method. The probability densityfunctions of the structural strength redundancy in any time can be obtained. Thestructural prediction reliability from time response can be obtained by using integralmethod. For the structural reliability is only considered a load for structural influence incurrently research, the4types common loads that act on the structure have beencombined3different combinations by Turkstra method, the loads of the threecombinations were believed as the structural stress. The models of the structuralreliability from time response without structural strength degeneration and with degeneration over time have been established by using the stress-strength interferencetheory. The minimum reliability index from time response of the three reliabilityindexes from time response which was obtained in the models is determined thestructural reliability index from time response. It is shown that models are practicable,feasible by using example.
2. The structural interval reliability from time response.
The structural non-probabilistic reliability from time response under the intervalvariable of loads changing over time and the interval variable of structural strengthdegrading over time is researched. The formulas for the change interval of structuralstrength degradation and its mean value and deviation are shown. According to thestress-strength interference theory and the interval theory, the structural reliability fromtime response prediction models are established under the structural strengthdegradation over time while the loads of the structure bearing are classified three casesloads: ladder type, constant amplitude alternating type and random changing over timetype. For the hybrid model of several times interval loads and structural stochasticstrength with degradation and without degradation over time are established. Firstly,several times interval loads are substituted by the equivalent interval load. Secondly, theequivalent interval load is thought as the random variable who obeys the uniformdistributed. The random variables of the uniform distribution and structural randomstrength are done with the equivalent normal method. Then the structuralinterval-random hybrid reliability from time response under several times interval loadsand structural random strength degeneration and without degeneration over time iscomputed by using the first secondly order moment method according to thestress-strength interference theory. It is shown that the models are reasonable andpracticable by using examples.
3. The structural fuzzy reliability from time response.
For the several times fuzzy loads and fuzzy structural strength in the practiceengineer, the fuzzy loads and the fuzzy strength are analyzed. The predictionfuzzy-fuzzy models of the structural fuzzy reliability from time response are establishedby using the stress-strength interference theory when the several times fuzzy loadschanges over time and the structural strength changes without degeneration and withdegeneration. For the several times random loads and fuzzy structural strength in thepractice engineer, the loads and the fuzzy strength are analyzed. The predictionrandom-fuzzy hybrid models of the structural fuzzy reliability from time response areestablished by using the stress-strength interference theory when the several times random loads changes over time and the structural strength changes withoutdegeneration and with degeneration. Finally, it is shown that the models are feasible andreasonable by examples. Further it is pointed out that the reliability of probability is aspecial case of fuzzy reliability.
4. The reparable k-out-of-n system reliability from time response.
The model of unit reliability from time response of repairable k/n (G) system withunit strength degradation under repeated random shocks has been developed accordingto the stress-strength interference theory. The unit failure number is obtained based onthe unit failure probability which can be computed from the unit reliability from timeresponse. Then, the transfer probability function of the repairable k/n (G) system isgiven by its Markov property. Once the transfer probability function has been obtained,the probability density matrix and the steady-state probabilities of the system can beretrieved. Finally, the reliability from time response of the repairable k/n (G) system isobtained by solving the differential equations. Anovel reliability-based model of k/n (G)system has been developed. Unit failure probability is first calculated based on theapplied load and strength distributions according to the stress-strength interferencetheory. Then, a reliability prediction model from time response for the whole repairablek/n (G) system is established using the probabilistic differential equations. Finally, theresulting differential equations are solved and the value of k can be determined precisely.It is illustrated that the proposed methods are practicable, feasible and gives reasonableprediction which conforms to the engineering practice.
5. The reparable multi-state k-out-of-n system reliability from time response.
For the complexity of multi-state repairable k-out-of-n system, firstly, the diversityof the components of the system are analyzed by using discrete time Markov chains andsemi-Markov chain, the probability distribution computing formula of the change ofstate and the sojourn time in the state are obtained. Secondly, the first order transitionprobability matrix of the change of state and lifetime in the state areobtained. Finally,the reliability and availability prediction model of the system are obtained according tothe analysis of the components. For the multi-state engine system, Firstly, the enginesystem is discrete and quantization processed, the output power of the engine is becomea state discrete time continue Markov random process; Secondly, the multi-state Markovmodel is established, according to the observed data, the algorithm of transitionintensities is obtained; Finally, the proposed method is used to calculate the forcedoutage rate and the expected engine capacity deficiency of the engine changing overtime. For forecasting and monitoring the reliability from time response of the engine to work properly is supplied a new method. It is shown that the methods are practical,effective by using the engineering numerical examples.
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