非线性随机动力系统的概率密度演化分析
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摘要
阐述了基于概率密度演化理论进行多自由度结构非线性随机动力反应分析的基本思想。采用随机过程的正交分解或物理系统建模的思想,实现随机激励的随机函数表述。对由此获得的随机状态方程采用概率密度演化理论求解,可以获得随机动力系统反应的概率密度函数及其演化。以某剪切型框架结构的非线性随机地震反应分析为例,说明了所发展方法的可行性和有效性。
The basic idea of stochastic response analysis of multi-degree-of-freedom nonlinear structures is outlined.Employing the orthogonal decomposition or physical modeling,the stochastic excitation can be represented by random combination of a set of deterministic functions.The probability density evolution theory can then be applied to the resulted stochastic dynamical system and the instantaneous probability density function and its evolution can thus be obtained.Stochastic response analysis of a shear frame subjected to random ground motions is exemplified,showing the feasibility and validity of the proposed method.
The basic idea of stochastic response analysis of multi-degree-of-freedom nonlinear structures is outlined.Employing the orthogonal decomposition or physical modeling,the stochastic excitation can be represented by random combination of a set of deterministic functions.The probability density evolution theory can then be applied to the resulted stochastic dynamical system and the instantaneous probability density function and its evolution can thus be obtained.Stochastic response analysis of a shear frame subjected to random ground motions is exemplified,showing the feasibility and validity of the proposed method.
引文
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[21]李杰.工程结构随机动力激励的物理模型.随机振动理论与应用新进展[C].上海:同济大学出版社,2009.(LI Jie.Physical stochastic models for the dy-namic excitations of engineering structures.Ad-vances in Theory and Applications of Random Vibra-tion[C].Shanghai:Tongji University Press,2009.(in Chinese))
[22]LI J,CHEN J B.Stochastic Dynamics of Structures[M].John Wiley&Sons,2009.
[2]朱位秋.随机振动[M].北京:科学出版社,1998.(ZHU Wei-qiu.Random Vibration[M].Beijing:Science Press,1998.(in Chinese))
[3]LUTES L D,SARKANI S.Random Vibrations:A-nalysis of Structural and Mechanical Systems[M].Elsevier,Amsterdam,2004.
[4]林家浩,张亚辉.随机振动的虚拟激励法[M].北京:科学出版社,2004.(LIN Jian-hao,ZHANG Ya-hui.Psuedo-Excitation Method in Random Vibration[M].Beijing:Science Press,2004.(in Chinese))
[5]SCHU LLER G I(Ed.).A state-of-the-art reporton computational stochastic mechanics[J].Probabi-listic Engineering Mechanics,1997,12(4):197-321.
[6]ZHU W Q.Nonlinear stochastic dynamics and con-trol in Hamiltonian formulation[J].Applied Me-chanics Reviews,2006,59:230-248.
[7]李杰,陈建兵.随机结构非线性动力响应的概率密度演化方法[J].力学学报,2003,35(6):716-722.(LI Jie,CHEN Jian-bing.The probability density e-volution method for analysis of dynamic nonlinear re-sponse of stochastic structures[J].Acta MechanicaSinica,2003,35(6):716-722.(in Chinese))
[8]LI J,CHEN J B.The principle of preservation ofprobability and the generalized density evolution e-quation[J].Structural Safety,2008,30:65-77.
[9]CHEN J B,LI J.A note on the principle of preserva-tion of probability and probability density evolutionequation[J].Probabilistic Engineering Mechanics,2009,24(1):27-42.
[10]CHEN J B,LI J.Dynamic response and reliability a-nalysis of nonlinear stochastic structures[J].Proba-bilistic Engineering Mechanics,2005,20(1):33-44.
[11]LI J,CHEN J B,FAN W L.The equivalent ex-treme-value event and evaluation of the structuralsystem reliability[J].Structural Safety,2007,29:112-131.
[12]李杰,刘章军.基于标准正交基的随机过程展开法[J].同济大学学报(自然科学版),2006,34(10):1279-1283.(LI Jie,LIU Zhang-jun.Decompositionof stochastic processes based on standard orthogonalbasis[J].Journal of Tongji University,2006,34(10):1279-1283.(in Chinese))
[13]李杰,艾晓秋.基于物理的随机地震动模型研究[J].地震工程与工程振动,2006,26(5):21-26.(LIJie,AI Xiao-qiu.Study on random model of earth-quake ground motion based on physical process[J].Earthquake Engineering and Engineering Vibra-tion,2006,26(5):21-26.(in Chinese))
[14]GARDINER C W.Handbook of Stochastic Methodsfor Physics(2nd Ed.)[M].Chemistry and the Nat-ural Sciences.Springer,Berlin,1983.
[15]李杰.随机结构系统———分析与建模[M].北京:科学出版社,1996.(LI Jie.Stochastic StructuralSystem:Analysis and Modeling[M].Beijing:Sci-ence Press,1996.(in Chinese))
[16]LOEVE M.Probability Theory[M].Springer-Ver-lag,Berlin,1977.
[17]ZHANG L L,LI J,PENG Y B.Dynamic responseand reliability analysis of tall buildings subject towind-loading[J].Journal of Wind Engineering andIndustrial Aerodynamics,2008,96:25-40.
[18]CHEN J B,GHANEM R,LI J.Partition of theprobability-assigned space in probability density evo-lution analysis of nonlinear stochastic structures[J].Probabilistic Engineering Mechanics,2009,24(1):51-59.
[19]WEN Y K.Method for random vibration of hysteret-ic systems[J].Journal of the Engineering Mechan-ics Division,1976,102(2):249-263.
[20]BABER T T,NOORI M N.Random vibration of de-grading,pinching systems[J].Journal of Engineer-ing Mechanics,1985,111(8):1010-1027.
[21]李杰.工程结构随机动力激励的物理模型.随机振动理论与应用新进展[C].上海:同济大学出版社,2009.(LI Jie.Physical stochastic models for the dy-namic excitations of engineering structures.Ad-vances in Theory and Applications of Random Vibra-tion[C].Shanghai:Tongji University Press,2009.(in Chinese))
[22]LI J,CHEN J B.Stochastic Dynamics of Structures[M].John Wiley&Sons,2009.
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