基于失效域重构和重要抽样法的结构动力学系统首穿失效概率
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摘要
对于线性动力学系统,重构系统失效域,利用基本失效域概率构造重要抽样密度函数,提出了基于重要抽样技术的首穿失效概率估计方法;对于非线性动力学系统,构建等效线性系统,线性化原理为线性与非线性系统对安全域边界具有相同的平均上穿率.最后给出Gauss(高斯)白噪声激励的线性与非线性系统的数值算例,并与Monte-Carlo(蒙特-卡洛)方法及区域分解方法比较,结果显示该文方法是正确有效的.
For linear dynamical systems, the system failure domain was reconstructed, an important sampling density function was built with the probability of the basic failure domain and the important sampling simulation method was employed. For the non-linear dynamical system, the equivalent linear system was constructed according to the principle that they have the same mean high level crossing rate for the specified threshold. Two numerical examples were given to demonstrate the accuracy and efficiency of the proposed method.
For linear dynamical systems, the system failure domain was reconstructed, an important sampling density function was built with the probability of the basic failure domain and the important sampling simulation method was employed. For the non-linear dynamical system, the equivalent linear system was constructed according to the principle that they have the same mean high level crossing rate for the specified threshold. Two numerical examples were given to demonstrate the accuracy and efficiency of the proposed method.
引文
[1] RICE O C. Mathematical analysis of random noise[J]. Bell System Technology, 1944, 23(3): 282-332.
[2] RICE O C. Mathematical analysis of random noise[J]. Bell System Technology, 1945, 24(3): 46-156.
[3] ROBERTS J B. First passage probability for nonlinear oscillators[J]. Journal of Engineering Mechanics, 1976, 102: 851-866.
[4] SPENCER B F, BERGMAN L A. On the numerical solution of the Fokker-Planck equation for nonlinear stochastic systems[J]. Nonlinear Dynamics, 1993, 4(4): 357-372.
[5] WEI L, WEI X. Stochastic optimal control of first-passage failure for coupled Duffing-van der Pol system under Gaussian white noise excitations[J]. Chaos, Solitons & Fractal, 2005, 25(5): 1221-1228.
[6] WEI L, WEI X. First passage problem for strong nonlinear stochastic dynamical system[J]. Chaos, Solitons & Fractal, 2006, 28(2): 414-421.
[7] PRADLWARTERH J, SCHUELLER G I. Assessment of low probability events of dynamical systems by controlled Monte Carlo simulation[J]. Probability Engineering Mechanics, 2009, 14(3): 213-227.
[8] PROPPE C, PRADLWARTER H J, SCHUELLER G. Equivalent linearization and Monte Carlo simulation in stochastic dynamics[J]. Probability Engineering Mechanics, 2003, 18(1): 1-15.
[9] JUU O, HIROAKI T. Importance sampling for stochastic systems under stationary noise having a specified power spectrum[J]. Probability Engineering Mechanics, 2009, 24(4): 537-544.
[10] NEWTON N J. Variance reduction for simulate diffusions[J]. SIAM Journal on Applied Mathematics, 1994, 54(6): 1780-1805.
[11] AU K L, BECK J L. First excursion probability for linear systems by very efficient importancesampling[J]. Probability Engineering Mechanics, 2001, 16(3): 193-207.
[12] KA-VENG Y, LAMBROS S K. An efficient simulation method for reliability analysis of linear dynamical systems using simple additive rules of probability[J]. Probability Engineering Mechanics, 2005, 20(1): 109-114.
[13] LAMBROS K, SAIHUNG H. Domain decomposition method for calculating the failure probability of linear dynamic systems subjected to Gaussian stochastic loads[J]. Journal of Engineering Mechanics, 2006, 20: 475-486.
[14] 何军. 非平稳随机激励下系统首次穿越概率的近似解法[J]. 应用数学和力学, 2009, 30(2): 245-252.(HE Jun. Approximation for the first passage probability of systems under non-stationary random excitation[J]. Applied Mathematics and Mechanics, 2009, 30(2): 245-252.(in Chinese))
[15] ZUEV K M, KATAFYGIOTIS L S. The horseracing simulation algorithm for evaluation of small failure probabilities[J]. Probability Engineering Mechanics, 2011, 26(2): 157-164.
[16] VALDEBENITO M A, JENSEN H A, LABARCA A A. Estimation of first excursion probabilities for uncertain stochastic linear systems subject to Gaussian load[J]. Computers and Structures, 2014, 138: 36-48.
[17] WANG J, KATAFYGIOTIS L S, FENG Z Q. An efficient simulation method for the first excursion problem of linear structures subjected to stochastic wind loads[J]. Computers & Structures, 2016, 166: 75-84.
[18] PRADLWARTER H J, SCHUELLER G J. Excursion probabilities of non-linear systems[J]. International Journal of Non-Linear Mechanics, 2004, 39(9): 1447-1452.
[19] HE J, GONG J H. Estimate of small first passage probabilities of nonlinear random vibration systems by using tail approximation of extreme distributions[J]. Structural Safety, 2016, 60: 28-36.
[20] OLSEN I A, NAESS A. An importance sampling procedure for estimating failure probabilities of non-linear dynamic systems subjected to random noise[J]. International Journal of Non-Linear Mechanics, 2007, 42(6): 848-853.
[21] 任丽梅, 徐伟, 肖玉柱, 等. 基于重要样本法的结构力学系统的首次穿越[J]. 力学学报, 2012, 44(3): 648-652.(REN Limei, XU Wei, XIAO Yuzhu, et al. First excursion probabilities of dynamical systems by importance sampling[J]. Chinese Journal of Theoretical and Applied Mechanics, 2012, 44(3): 648-652.(in Chinese))
[22] KOO H, KIUREGHIAN A D, FUJIMURA K. Design-point excitation for non-linear random vibrations[J]. Probabilistic Engineering Mechanics, 2005, 20(2): 136-147.
[23] AU S K. Critical excitation of sdof elasto-plastic systems[J]. Journal of Sound and Vibration, 2006, 296(4/5): 714-733.
[24] VALDEBENITO M A, PRADLWARTER H J. The role of the design point for calculating failure probabilities in view of dimensionality and structural nonlinearities[J]. Structural Safety, 2010, 32(2): 101-111.
[25] TAKEWAKI I. Critical excitation for elastic-plastic structures via statistical equivalent linearization[J]. Probability Engineering Mechanics, 2002, 17(1): 73-84.
[26] SOONG T T, GRIGORIU M. Random Vibration of Mechancial and Structural Systems[M]. Englewood Cliffs, NJ: Prentice-Hall Inc,1997.
[27] OKSENDAL B. Stochastic Differential Equations: an Introduction With Application[M]. Berlin: Springer, 1998.
[2] RICE O C. Mathematical analysis of random noise[J]. Bell System Technology, 1945, 24(3): 46-156.
[3] ROBERTS J B. First passage probability for nonlinear oscillators[J]. Journal of Engineering Mechanics, 1976, 102: 851-866.
[4] SPENCER B F, BERGMAN L A. On the numerical solution of the Fokker-Planck equation for nonlinear stochastic systems[J]. Nonlinear Dynamics, 1993, 4(4): 357-372.
[5] WEI L, WEI X. Stochastic optimal control of first-passage failure for coupled Duffing-van der Pol system under Gaussian white noise excitations[J]. Chaos, Solitons & Fractal, 2005, 25(5): 1221-1228.
[6] WEI L, WEI X. First passage problem for strong nonlinear stochastic dynamical system[J]. Chaos, Solitons & Fractal, 2006, 28(2): 414-421.
[7] PRADLWARTERH J, SCHUELLER G I. Assessment of low probability events of dynamical systems by controlled Monte Carlo simulation[J]. Probability Engineering Mechanics, 2009, 14(3): 213-227.
[8] PROPPE C, PRADLWARTER H J, SCHUELLER G. Equivalent linearization and Monte Carlo simulation in stochastic dynamics[J]. Probability Engineering Mechanics, 2003, 18(1): 1-15.
[9] JUU O, HIROAKI T. Importance sampling for stochastic systems under stationary noise having a specified power spectrum[J]. Probability Engineering Mechanics, 2009, 24(4): 537-544.
[10] NEWTON N J. Variance reduction for simulate diffusions[J]. SIAM Journal on Applied Mathematics, 1994, 54(6): 1780-1805.
[11] AU K L, BECK J L. First excursion probability for linear systems by very efficient importancesampling[J]. Probability Engineering Mechanics, 2001, 16(3): 193-207.
[12] KA-VENG Y, LAMBROS S K. An efficient simulation method for reliability analysis of linear dynamical systems using simple additive rules of probability[J]. Probability Engineering Mechanics, 2005, 20(1): 109-114.
[13] LAMBROS K, SAIHUNG H. Domain decomposition method for calculating the failure probability of linear dynamic systems subjected to Gaussian stochastic loads[J]. Journal of Engineering Mechanics, 2006, 20: 475-486.
[14] 何军. 非平稳随机激励下系统首次穿越概率的近似解法[J]. 应用数学和力学, 2009, 30(2): 245-252.(HE Jun. Approximation for the first passage probability of systems under non-stationary random excitation[J]. Applied Mathematics and Mechanics, 2009, 30(2): 245-252.(in Chinese))
[15] ZUEV K M, KATAFYGIOTIS L S. The horseracing simulation algorithm for evaluation of small failure probabilities[J]. Probability Engineering Mechanics, 2011, 26(2): 157-164.
[16] VALDEBENITO M A, JENSEN H A, LABARCA A A. Estimation of first excursion probabilities for uncertain stochastic linear systems subject to Gaussian load[J]. Computers and Structures, 2014, 138: 36-48.
[17] WANG J, KATAFYGIOTIS L S, FENG Z Q. An efficient simulation method for the first excursion problem of linear structures subjected to stochastic wind loads[J]. Computers & Structures, 2016, 166: 75-84.
[18] PRADLWARTER H J, SCHUELLER G J. Excursion probabilities of non-linear systems[J]. International Journal of Non-Linear Mechanics, 2004, 39(9): 1447-1452.
[19] HE J, GONG J H. Estimate of small first passage probabilities of nonlinear random vibration systems by using tail approximation of extreme distributions[J]. Structural Safety, 2016, 60: 28-36.
[20] OLSEN I A, NAESS A. An importance sampling procedure for estimating failure probabilities of non-linear dynamic systems subjected to random noise[J]. International Journal of Non-Linear Mechanics, 2007, 42(6): 848-853.
[21] 任丽梅, 徐伟, 肖玉柱, 等. 基于重要样本法的结构力学系统的首次穿越[J]. 力学学报, 2012, 44(3): 648-652.(REN Limei, XU Wei, XIAO Yuzhu, et al. First excursion probabilities of dynamical systems by importance sampling[J]. Chinese Journal of Theoretical and Applied Mechanics, 2012, 44(3): 648-652.(in Chinese))
[22] KOO H, KIUREGHIAN A D, FUJIMURA K. Design-point excitation for non-linear random vibrations[J]. Probabilistic Engineering Mechanics, 2005, 20(2): 136-147.
[23] AU S K. Critical excitation of sdof elasto-plastic systems[J]. Journal of Sound and Vibration, 2006, 296(4/5): 714-733.
[24] VALDEBENITO M A, PRADLWARTER H J. The role of the design point for calculating failure probabilities in view of dimensionality and structural nonlinearities[J]. Structural Safety, 2010, 32(2): 101-111.
[25] TAKEWAKI I. Critical excitation for elastic-plastic structures via statistical equivalent linearization[J]. Probability Engineering Mechanics, 2002, 17(1): 73-84.
[26] SOONG T T, GRIGORIU M. Random Vibration of Mechancial and Structural Systems[M]. Englewood Cliffs, NJ: Prentice-Hall Inc,1997.
[27] OKSENDAL B. Stochastic Differential Equations: an Introduction With Application[M]. Berlin: Springer, 1998.
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