基于区域分解的结构动力学系统首次穿越失效
|
摘要
提出了高斯白噪声激励的线性及非线性结构动力学系统的首次穿越失效概率的估计方法.对于线性结构动力学系统,失效区域被分解为互斥的基本失效域之和,每个基本失效域可用其设计点完全描述,并以正态分布代替卡方分布估计失效概率中的参数.对于非线性结构动力学系统,基于Rice穿越理论,将非线性方程转化为与之具有相同平均上穿率的线性化方程,然后利用文中方法对等效线性化方程估计首穿失效概率.最后给出了线性及非线性结构动力学系统的数值例子,并将所提方法与蒙特卡罗法及重要样本法相比较,模拟结果显示了方法的正确性与有效性.
The first passage problems of linear and nonlinear dynamical systems excited by Gauss white noise are considered.For linear dynamical system,the failure domain can be described as a union of mutually exclusive events,and every event is completely described by a local design point.The paper uses standard Gaussian distribution instead of chi-square distribution to estimate the parameter of first passage probability.For nonlinear dynamical system,the equivalent linear system is carried out based on the out-crossing theory.The linearization principle is that nonlinear and linear systems have the same up-crossing rate for a specified threshold.Finally the paper gives two examples.The results show that the method of the paper suggested is correct and effective by comparing with the Monte Carlo method.
The first passage problems of linear and nonlinear dynamical systems excited by Gauss white noise are considered.For linear dynamical system,the failure domain can be described as a union of mutually exclusive events,and every event is completely described by a local design point.The paper uses standard Gaussian distribution instead of chi-square distribution to estimate the parameter of first passage probability.For nonlinear dynamical system,the equivalent linear system is carried out based on the out-crossing theory.The linearization principle is that nonlinear and linear systems have the same up-crossing rate for a specified threshold.Finally the paper gives two examples.The results show that the method of the paper suggested is correct and effective by comparing with the Monte Carlo method.
引文
1 Proppe C,Pradlwarter HJ,Schueller GI.Equivalent linearization andMonte Carlo simulation in stochastic dynamics.Probability Engi-neering Mechanics,2003,18(3):1-15
2 Spencer BF,Bergman LA.On the numerical solution of the Fokker-Planck equation for nonlinear stochastic systems.Nonlinear Dynam,1993(4):357-372
3 Li W,Xu W.Stochastic optimal control of first passage failure forcoupled Duffng-Van der Pol system under Gaussian white noise ex-citations.Chaos Solitons&Fractal,2005,25(5):1221-1228
4 Li W,Xu W.First passage problem for strong nonlinear stochasticdynamical system.Chaos Solitons&Fractal,2006,28(2):414-421
5徐伟,孙春艳,孙建桥等.胞映射方法的研究和进展.力学进展,2013,43(1):91-100(Xu Wei,Sun Chunyan,Sun Jianqiao,et al.Development and study on cell mapping methods.Advances in Me-chanics,2013,43(1):91-100(in Chinese))
6徐伟,李伟,靳艳飞等.耦合Duffng-van der Pol系统的首次穿越问题.力学学报,2005,37(5):620-625(Xu Wei,Li Wei,Jin Yanfei,et al.First-passage problem for coupled Duffng-van der Pol sys-tems.Chinese Jounal of Theoretical and Applied Mechanics,2005,37(5):620-625(in Chinese))
7任丽梅,徐伟,肖玉柱等.基于重要样本法的结构动力学系统的首次穿越.力学学报,2012,44(3):648-652(Ren Limei,Xu Wei,XiaoYuzhu,et al.First excursion probabilities of dynamical systems byimportance sampling.Chinese Jounal of Theoretical and AppliedMechanics,2012,44(3):648-652(in Chinese))
8 Au SK,Beck JL.First excursion probability for linear systems byvery effcient importance sampling.Probability Engineering Me-chanics,2001,16(3):193-207
9 Juu O,Hiroaki T.Importance sampling for stochastic systems un-der stationary noise having a specified power spectrum.ProbabilityEngineering Mechanics,2009,24(4):537-544
10 Pradlwarter HJ,Schueller GI.Assessment of low probability eventsof dynamical systems by controlled Monte Carlo simulation.Prob-ability Engineering Mechanics,1999,14(3):213-227
11 Olsen AI,Naess A.An importance sampling procedure for estimat-ing failure probabilities of Non-linear dynamic systems subjected torandom noise.J Non-linear Mechanics,2007(42):848-863
12 Katafygiotis LS,Cheung SH.Domain decomposr-tion method forcalculating the failure probability of linear dynamic systems sub-jected to Gaussian stochastic loads.Journal of Engineering Me-chanics,2006(20):475-486
13 Yuen KV,Katafygiotis LS.An effcient simulation method reliabilityanalysis of linear dynamical systems using simple additive rules ofprobability.Probability Engineering Mechanics,2005,20(2):109-114
14 Zuev KM,Katafygiotis LS.The Horseracing Simulation algorithmfor evaluation of small failure probabilities.Probability EngineeringMechanics,2011,26(2):157-164
15 Der Kiureghian A.The geometry of random vibrations and solu-tions by FORM and SORM.Probability Engineering Mechanics,2000,15(2):81-90
16 Valdebenito MA,Pradlwarter HJ,Schueller GI.The role of the de-sign point for calculating failure probabilities in view of dimension-ality and structural nonlinearities.Structural Safety,2010(32):101-111
17 Oksendal B.Stochastic Differential Equations:An Introduction withApplication.5th edn.Berlin:Springer,1998
18 Soong TT,Grigoriu M.Random Vibration of Mechancial and Struc-tural Systems.Englewood Cliffs,NJ:Prentice-Hall,1997
2 Spencer BF,Bergman LA.On the numerical solution of the Fokker-Planck equation for nonlinear stochastic systems.Nonlinear Dynam,1993(4):357-372
3 Li W,Xu W.Stochastic optimal control of first passage failure forcoupled Duffng-Van der Pol system under Gaussian white noise ex-citations.Chaos Solitons&Fractal,2005,25(5):1221-1228
4 Li W,Xu W.First passage problem for strong nonlinear stochasticdynamical system.Chaos Solitons&Fractal,2006,28(2):414-421
5徐伟,孙春艳,孙建桥等.胞映射方法的研究和进展.力学进展,2013,43(1):91-100(Xu Wei,Sun Chunyan,Sun Jianqiao,et al.Development and study on cell mapping methods.Advances in Me-chanics,2013,43(1):91-100(in Chinese))
6徐伟,李伟,靳艳飞等.耦合Duffng-van der Pol系统的首次穿越问题.力学学报,2005,37(5):620-625(Xu Wei,Li Wei,Jin Yanfei,et al.First-passage problem for coupled Duffng-van der Pol sys-tems.Chinese Jounal of Theoretical and Applied Mechanics,2005,37(5):620-625(in Chinese))
7任丽梅,徐伟,肖玉柱等.基于重要样本法的结构动力学系统的首次穿越.力学学报,2012,44(3):648-652(Ren Limei,Xu Wei,XiaoYuzhu,et al.First excursion probabilities of dynamical systems byimportance sampling.Chinese Jounal of Theoretical and AppliedMechanics,2012,44(3):648-652(in Chinese))
8 Au SK,Beck JL.First excursion probability for linear systems byvery effcient importance sampling.Probability Engineering Me-chanics,2001,16(3):193-207
9 Juu O,Hiroaki T.Importance sampling for stochastic systems un-der stationary noise having a specified power spectrum.ProbabilityEngineering Mechanics,2009,24(4):537-544
10 Pradlwarter HJ,Schueller GI.Assessment of low probability eventsof dynamical systems by controlled Monte Carlo simulation.Prob-ability Engineering Mechanics,1999,14(3):213-227
11 Olsen AI,Naess A.An importance sampling procedure for estimat-ing failure probabilities of Non-linear dynamic systems subjected torandom noise.J Non-linear Mechanics,2007(42):848-863
12 Katafygiotis LS,Cheung SH.Domain decomposr-tion method forcalculating the failure probability of linear dynamic systems sub-jected to Gaussian stochastic loads.Journal of Engineering Me-chanics,2006(20):475-486
13 Yuen KV,Katafygiotis LS.An effcient simulation method reliabilityanalysis of linear dynamical systems using simple additive rules ofprobability.Probability Engineering Mechanics,2005,20(2):109-114
14 Zuev KM,Katafygiotis LS.The Horseracing Simulation algorithmfor evaluation of small failure probabilities.Probability EngineeringMechanics,2011,26(2):157-164
15 Der Kiureghian A.The geometry of random vibrations and solu-tions by FORM and SORM.Probability Engineering Mechanics,2000,15(2):81-90
16 Valdebenito MA,Pradlwarter HJ,Schueller GI.The role of the de-sign point for calculating failure probabilities in view of dimension-ality and structural nonlinearities.Structural Safety,2010(32):101-111
17 Oksendal B.Stochastic Differential Equations:An Introduction withApplication.5th edn.Berlin:Springer,1998
18 Soong TT,Grigoriu M.Random Vibration of Mechancial and Struc-tural Systems.Englewood Cliffs,NJ:Prentice-Hall,1997