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Abstract： This paper suggests a procedure to estimatefirst excursion probabilities for non-linear dynamical systems subjected to Gaussian excitation. The approach is based on the mean up-crossing rate and importance sampling method. Firstly， by using of Poisson assumption and Rice formula， the equivalent linear system is carried out. The linearization principle is that non-linear and linear systems have the same up-crossing rate for a specified threshold. Secondly， an importance sampling technique is used in order to estimate excursion probabilities for the equivalent linear system. The variance of the failure probability estimates， the number of samples and the computational time are reduced significantly compared with direct Monte Carlo simulations.
Key words： First excursion probability； Importance sampling； Mean upcrossing rate
1. INTRODUCTION
Thefirst excursion probability is one of the most basic reliability measures in structural reliability assessment of dynamical systems. So far， no analytical solution has been obtained even in the case of a linear system. Pioneered by Rice （1945） [1，2]， a class of numerical solution methods based on Fokker-plank equation has been developed， such as path integration （1976） [3]， cell-mapping （1993） [4] and stochastic averaging （2003） [5，6]， however these methods increase in complexity at least exponentially with the state space dimension. Over the past decade， Monte Carlo simulations （MCS） with variance reduction technique offered a robust methodology well suited for solving such reliability problems.
For the linear dynamical systems， S. K. Au （2001） developed an extremely powerful methodology by investigating analytically the failure region of linear systems and constructing an efficient importance sampling density [7]. Ka-VengYuen （2005） also presented a simulation method using simple additive rules of probability [8]. Lambros （2006） proposed the domain decomposition method [9]. K. M. Zuev （2011） proposed the Horseracing Simulation algorithm [10]. For the non-linear dynamical systems， H. J. Pradlwater （2004） suggested a procedure based on the so called“averaged excursion probabilityflow”[11]. A. I. Olsen （2007） put forward a two step iterative method [12].
This paper proposes a robust reliability methodology. The approach is focused on out crossing theory and importance sampling technique. By used of Poisson assumption and Rice formula， the equivalent linear system is obtained. Importance sampling density （ISD） function is a weighted sum of the probability density function. 2. THE PROCEDURE
In this section， numerical results forfirst passage probability of Duffing oscillators will be presented. The main idea is to estimate the excursion probability using a linear version of the non-linear Duffing oscillators. Recently， it has been shown that it is the most efficient linearization method for estimating failure probability， the mean up-crossing characteristics are far more important than the characteristics of the mean square response [12]. So it is reasonable to assume that both the nonlinear system and the equivalent linear system have the same up-crossing rate for a specified threshold.
Consider a non-linear dynamic system subjected to Gaussian white noise excitation， Duffing oscillators. Where w（t） is a Gaussian white noise process with spectral intensity G0.
Based on Duhamel’s integral， the input output relationship of the equivalent linear system can be generally written as：
In this section， numerical examples are presented， which demonstrate that the efficiency of the proposed procedure by comparing with the crude MCS. The parameters of Duffing system are chosen：β= 0.05，γ= 0.03， *0= 1.0（rad/s）， T = 15s，？t = 0.05， b（t） = kσ0， nt= 301. According to Equation （4）， the mean up crossing rate is obtained as Figures 1 and 2， respectively， which has the different specified threshold and the different nonlinearity parameterε. The impulse response function of the equivalent linear system is given as：
3. CONCLUSIONS
Based on the mean up-crossing rate and importance sampling technique， a method has been proposed for solving thefirst excursion problem of non-linear system. Simulation results show the procedure to be correct and effective by comparing with the crude Monte Carlo method. The number of samples and the computational time are reduced significantly compared with crude MCS.
For estimating thefirst failure probability， due to simplicity and computational efficiency， equivalent linearization method has become a standard tool of stochastic structural dynamics. When the mean up crossing rate cannot be evaluated analytically， some numerical extrapolation can be implemented to assess it.
REFERENCES
[1] Rice， O. C. （1944）. Mathematical analysis of random noise. Bell Syst. Tech. J.， 23， 282-332.
[2] Rice， O. C. （1945）. Mathematical analysis of random noise. Bell Syst. Tech. J.， 24， 46-156.
[3] Roberts， J. B. （1976）. First passage probability for nonlinear oscillators. J. Engineering Mechanics， Asce.， 102， 851-866. [4] Spencer， B. F.， & Bergman， L. A. （1993）. On the numerical solution of the Fokker-Planck equation for nonlinear stochastic systems. Nonlinear Dynamics， 4， 357-372.
[5] Li， W.， & Xu， W. （2005）. Stochastic optimal control offirst passage failure for coupled Duffing-Van derPol system under Gaussian white noise excitations. Chaos Solitons & Fractals Chaos， 25（5）， 1221-1228.
[6] Lei， Y.， & Xu， W. （2006）. First passage problem for strong nonlinear stochastic dynamical system. Chaos Solitons & Fractals Chaos， 28（2）， 414-421.
[7] Au， S. K.， & Beck， J. L. （2001）. First excursion probabilityfor linear systems by very efficient importance Sampling. Probabilistic Engineering Mechanics， 16（3）， 193-207.
[8] Ka-Vang， Y.， & Katafygiotis， L. S. （2005）. An efficient simulation method for reliability analysis of linear dynamical systems using simple additive rules of probability. Probabilistic Engineering Mechanics， 20， 109-114.
[9] Katafygiotis， L. S.， & Cheung， S. H. （2006）. Domain decomposrtion method for calculating the failure probability of linear dynamic systems subjected to Gaussian stochastic loads. Journal of Engineering Mechanics， 20， 475-476.
[10] Zuev， K. M.， & Katafygiotis， L. S. （2011）. The horseracing simulation algorithm for evaluation of small failure probabilities. Probabilistic Engineering Mechanics， 26（2）， 157-164.
[11] Pradlwarter， H. J.， & Schueller， G. I. （2004）. Excursion probabilities of nonlinear systems. J. Non-linear Mech.， 39， 1447-1452.
[12] Olsen， A. I.， & Naess， A. （2007）. An importance sampling procedure for estimating failure probabilities of non-linear dynamic systems subjected to random noise. J. Non-linear Mech.， 42， 848-853.
[13] Soong， T. T.， & Grigoriu， M. （1997）. Random vibration of mechancial and structural systems. NJ： Prentice-Hall. Inc. Englewood Cliffs.
Key words： First excursion probability； Importance sampling； Mean upcrossing rate
1. INTRODUCTION
Thefirst excursion probability is one of the most basic reliability measures in structural reliability assessment of dynamical systems. So far， no analytical solution has been obtained even in the case of a linear system. Pioneered by Rice （1945） [1，2]， a class of numerical solution methods based on Fokker-plank equation has been developed， such as path integration （1976） [3]， cell-mapping （1993） [4] and stochastic averaging （2003） [5，6]， however these methods increase in complexity at least exponentially with the state space dimension. Over the past decade， Monte Carlo simulations （MCS） with variance reduction technique offered a robust methodology well suited for solving such reliability problems.
For the linear dynamical systems， S. K. Au （2001） developed an extremely powerful methodology by investigating analytically the failure region of linear systems and constructing an efficient importance sampling density [7]. Ka-VengYuen （2005） also presented a simulation method using simple additive rules of probability [8]. Lambros （2006） proposed the domain decomposition method [9]. K. M. Zuev （2011） proposed the Horseracing Simulation algorithm [10]. For the non-linear dynamical systems， H. J. Pradlwater （2004） suggested a procedure based on the so called“averaged excursion probabilityflow”[11]. A. I. Olsen （2007） put forward a two step iterative method [12].
This paper proposes a robust reliability methodology. The approach is focused on out crossing theory and importance sampling technique. By used of Poisson assumption and Rice formula， the equivalent linear system is obtained. Importance sampling density （ISD） function is a weighted sum of the probability density function. 2. THE PROCEDURE
In this section， numerical results forfirst passage probability of Duffing oscillators will be presented. The main idea is to estimate the excursion probability using a linear version of the non-linear Duffing oscillators. Recently， it has been shown that it is the most efficient linearization method for estimating failure probability， the mean up-crossing characteristics are far more important than the characteristics of the mean square response [12]. So it is reasonable to assume that both the nonlinear system and the equivalent linear system have the same up-crossing rate for a specified threshold.
Consider a non-linear dynamic system subjected to Gaussian white noise excitation， Duffing oscillators. Where w（t） is a Gaussian white noise process with spectral intensity G0.
Based on Duhamel’s integral， the input output relationship of the equivalent linear system can be generally written as：
In this section， numerical examples are presented， which demonstrate that the efficiency of the proposed procedure by comparing with the crude MCS. The parameters of Duffing system are chosen：β= 0.05，γ= 0.03， *0= 1.0（rad/s）， T = 15s，？t = 0.05， b（t） = kσ0， nt= 301. According to Equation （4）， the mean up crossing rate is obtained as Figures 1 and 2， respectively， which has the different specified threshold and the different nonlinearity parameterε. The impulse response function of the equivalent linear system is given as：
3. CONCLUSIONS
Based on the mean up-crossing rate and importance sampling technique， a method has been proposed for solving thefirst excursion problem of non-linear system. Simulation results show the procedure to be correct and effective by comparing with the crude Monte Carlo method. The number of samples and the computational time are reduced significantly compared with crude MCS.
For estimating thefirst failure probability， due to simplicity and computational efficiency， equivalent linearization method has become a standard tool of stochastic structural dynamics. When the mean up crossing rate cannot be evaluated analytically， some numerical extrapolation can be implemented to assess it.
REFERENCES
[1] Rice， O. C. （1944）. Mathematical analysis of random noise. Bell Syst. Tech. J.， 23， 282-332.
[2] Rice， O. C. （1945）. Mathematical analysis of random noise. Bell Syst. Tech. J.， 24， 46-156.
[3] Roberts， J. B. （1976）. First passage probability for nonlinear oscillators. J. Engineering Mechanics， Asce.， 102， 851-866. [4] Spencer， B. F.， & Bergman， L. A. （1993）. On the numerical solution of the Fokker-Planck equation for nonlinear stochastic systems. Nonlinear Dynamics， 4， 357-372.
[5] Li， W.， & Xu， W. （2005）. Stochastic optimal control offirst passage failure for coupled Duffing-Van derPol system under Gaussian white noise excitations. Chaos Solitons & Fractals Chaos， 25（5）， 1221-1228.
[6] Lei， Y.， & Xu， W. （2006）. First passage problem for strong nonlinear stochastic dynamical system. Chaos Solitons & Fractals Chaos， 28（2）， 414-421.
[7] Au， S. K.， & Beck， J. L. （2001）. First excursion probabilityfor linear systems by very efficient importance Sampling. Probabilistic Engineering Mechanics， 16（3）， 193-207.
[8] Ka-Vang， Y.， & Katafygiotis， L. S. （2005）. An efficient simulation method for reliability analysis of linear dynamical systems using simple additive rules of probability. Probabilistic Engineering Mechanics， 20， 109-114.
[9] Katafygiotis， L. S.， & Cheung， S. H. （2006）. Domain decomposrtion method for calculating the failure probability of linear dynamic systems subjected to Gaussian stochastic loads. Journal of Engineering Mechanics， 20， 475-476.
[10] Zuev， K. M.， & Katafygiotis， L. S. （2011）. The horseracing simulation algorithm for evaluation of small failure probabilities. Probabilistic Engineering Mechanics， 26（2）， 157-164.
[11] Pradlwarter， H. J.， & Schueller， G. I. （2004）. Excursion probabilities of nonlinear systems. J. Non-linear Mech.， 39， 1447-1452.
[12] Olsen， A. I.， & Naess， A. （2007）. An importance sampling procedure for estimating failure probabilities of non-linear dynamic systems subjected to random noise. J. Non-linear Mech.， 42， 848-853.
[13] Soong， T. T.， & Grigoriu， M. （1997）. Random vibration of mechancial and structural systems. NJ： Prentice-Hall. Inc. Englewood Cliffs.