基于连续Markov过程首超时间概率分析的结构动力可靠性研究
|
摘要
对时间连续状态连续的Markov过程的首次超越时间Tf概率分布问题进行了研究。令过程在任意时刻t发生微增量?t,通过分析?t微时段内的概率变化关系寻求Markov过程的转移概率分布函数与其首超时间概率分布之间的内在联系,最终得到了首超时间概率密度函数的解析显式表达式。由于在分析推导过程中没有采用苛刻的数学假定,因此该文结果在本质上属精确解。由于在常见随机荷载白噪声激励下线性体系的反应具有Markov性,而对于反应不是Markov过程的情形有不少实用的方法可将其转化为一维Markov过程,因此该文的研究结果具有一般性。
The paper studies the probability distribution of the first-passage time Tf of Markov process,whose times and states are both continuous.Considering minute time increment at time t,the paper examines the probability variation in the minute time-interval to find the internal relation between transition probability distribution function and first-passage times probability distribution.Finally the explicit expression of probability density function of first-passage time is obtained.The responses of linear system have Markov property under common random loads-flat noise,and the non-Markov responses process can be converted to one-dimensional Markov process by quite a few existing practical methods,so this study has generality.
The paper studies the probability distribution of the first-passage time Tf of Markov process,whose times and states are both continuous.Considering minute time increment at time t,the paper examines the probability variation in the minute time-interval to find the internal relation between transition probability distribution function and first-passage times probability distribution.Finally the explicit expression of probability density function of first-passage time is obtained.The responses of linear system have Markov property under common random loads-flat noise,and the non-Markov responses process can be converted to one-dimensional Markov process by quite a few existing practical methods,so this study has generality.
引文
[1]Coleman J J.Reliability of aircraft structures in resistingchance failure[J].Operations Research.,1959,7(5):639―645.
[2]Corotis R B,Vanmarcke E H,Cornell C A.First passageof nonstationry random processes[J].Journal ofEngineering Mechanics Division,1972,98:401―415.
[3]Langley R S.A firstpassage approximation for normalstationary random processes[J].Journal of SoundVibration,1988,122(2):261―275.
[4]Liu Ning,Liu Guangting.Time-dependent reliabilityassessment for mass concrete structures[J].StructuralSafety,1999(21):23―43.
[5]Naess A.Crossing rate statistics quadratictransformations of Gaussian processes[J].ProbabilityEngineering Mechanics,2001(16):209―217.
[6]Madsen P H,Krenk S.An integral equation method forthe first-passage problem in random vibration[J].Journalof Applied Mechanics,ASME,1984(51):674―679.
[7]Engelund S,Rachwitz R.Approximations of first-passage times foe differentiable processes based onhigher-order threshold crossing[J].ProbabilityEngineering Mechanics,1995(10):53―60.
[8]吴斌,欧进萍,张纪刚.结构动力可靠度的重要抽样法[J].计算力学学报,2001,18(4):478―482.Wu Bin,Ou Jinping,Zhang Jigang.Importance samplingtechniques in dynamical structural reliability[J].ChineseJournal of Computational Mechanics,2001,18(4):478―482.(in Chinese)
[9]Ditlevsen O,Madsen H O.Structural reliability methods[M].Chichester:John Wiley&Sons Ltd.,2004.
[10]Papoulis A,PillaiS U.Probability,random variables andstochastic processes[M].New York:McGraw-HillCompanies,Inc.,2002.
[11]李桂青,曹宏,李秋胜.结构动力可靠性理论及其应用[M].北京:地震出版社,1993.Li Guiqing,Cao Hong,Li Qiusheng.The theory and itsapplication on the structure’s dynamic reliability[M].Beijing:Earthquake Press,1993.(in Chinese)
[12]欧进萍,王光远.结构随机振动[M].北京:高等教育出版社,1998.Ou Jinping,Wang Guangyuan.Random vibration ofstructures[M].Beijing:Higher Education Press,1998.(in Chinese)
[13]Ariaratnam S T,Pi H N.On the first-passage time forenvelope crossing for a linear oscillator[J].InternationalJournal of Contol,1973,18(1):89―96.
[14]方丹萍,张惠侨,范祖尧.用矩阵连分法求解非平稳的FPK方程[J].上海交通大学学报,1989,23(4):78―89.Fang Danping,Zhang Huiqiao,Fan Zuyao.Solution toFPK equation by matrix continued fraction method[J].Journal of Shanghai Jiaotong University,1989,23(4):78―89.(in Chinese)
[2]Corotis R B,Vanmarcke E H,Cornell C A.First passageof nonstationry random processes[J].Journal ofEngineering Mechanics Division,1972,98:401―415.
[3]Langley R S.A firstpassage approximation for normalstationary random processes[J].Journal of SoundVibration,1988,122(2):261―275.
[4]Liu Ning,Liu Guangting.Time-dependent reliabilityassessment for mass concrete structures[J].StructuralSafety,1999(21):23―43.
[5]Naess A.Crossing rate statistics quadratictransformations of Gaussian processes[J].ProbabilityEngineering Mechanics,2001(16):209―217.
[6]Madsen P H,Krenk S.An integral equation method forthe first-passage problem in random vibration[J].Journalof Applied Mechanics,ASME,1984(51):674―679.
[7]Engelund S,Rachwitz R.Approximations of first-passage times foe differentiable processes based onhigher-order threshold crossing[J].ProbabilityEngineering Mechanics,1995(10):53―60.
[8]吴斌,欧进萍,张纪刚.结构动力可靠度的重要抽样法[J].计算力学学报,2001,18(4):478―482.Wu Bin,Ou Jinping,Zhang Jigang.Importance samplingtechniques in dynamical structural reliability[J].ChineseJournal of Computational Mechanics,2001,18(4):478―482.(in Chinese)
[9]Ditlevsen O,Madsen H O.Structural reliability methods[M].Chichester:John Wiley&Sons Ltd.,2004.
[10]Papoulis A,PillaiS U.Probability,random variables andstochastic processes[M].New York:McGraw-HillCompanies,Inc.,2002.
[11]李桂青,曹宏,李秋胜.结构动力可靠性理论及其应用[M].北京:地震出版社,1993.Li Guiqing,Cao Hong,Li Qiusheng.The theory and itsapplication on the structure’s dynamic reliability[M].Beijing:Earthquake Press,1993.(in Chinese)
[12]欧进萍,王光远.结构随机振动[M].北京:高等教育出版社,1998.Ou Jinping,Wang Guangyuan.Random vibration ofstructures[M].Beijing:Higher Education Press,1998.(in Chinese)
[13]Ariaratnam S T,Pi H N.On the first-passage time forenvelope crossing for a linear oscillator[J].InternationalJournal of Contol,1973,18(1):89―96.
[14]方丹萍,张惠侨,范祖尧.用矩阵连分法求解非平稳的FPK方程[J].上海交通大学学报,1989,23(4):78―89.Fang Danping,Zhang Huiqiao,Fan Zuyao.Solution toFPK equation by matrix continued fraction method[J].Journal of Shanghai Jiaotong University,1989,23(4):78―89.(in Chinese)
版权所有：© 2023 中国地质图书馆 中国地质调查局地学文献中心