Galerkin scheme based determination of first-passage probability of nonlinear system response

An approximate analytical approach for determining the first-passage probability of the response of a class of lightly damped nonlinear oscillators to broadband random excitations is presented. Markovian approximations of both the response amplitude envelope and the response energy envelope are considered. This approach leads to a backward Kolmogorov equation, which governs the evolution of the survival probability of the system. This equation is solved approximately by employing a Galerkin scheme. A convenient set of confluent hypergeometric functions is used as an orthogonal basis for this scheme. The reliability of the derived analytical solution is demonstrated by comparisons with data derived by pertinent Monte Carlo simulations.

[1]  E. J. Allen,et al.  Numerical solution of first passage problems using an approximate Chapman-Kolmogorov relation , 1998 .

[2]  A. Kovaleva.  An exact solution of the first-exit time problem for a class of structural systems , 2009 .

[3]  R. L. Stratonovich,et al.  Topics in the theory of random noise , 1967 .

[4]  R. Val Ery Roy.  Asymptotic Analysis of First-passage Problems , 1994 .

[5]  B. Øksendal.  Stochastic differential equations : an introduction with applications , 1987 .

[6]  Masanobu Shinozuka,et al.  Probability of Structural Failure Under Random Loading , 1964 .

[7]  The First Passage Time Problem , 1977 .

[8]  H. Haus.  Topics in the theory of random noise, vol. I , 1964 .

[9]  D. Williams.  STOCHASTIC DIFFERENTIAL EQUATIONS: THEORY AND APPLICATIONS , 1976 .

[10]  W. Zhu.  Stochastic Averaging Methods in Random Vibration , 1988 .

[11]  P. Spanos,et al.  Random vibration and statistical linearization , 1990 .

[12]  E. Vanmarcke.  On the Distribution of the First-Passage Time for Normal Stationary Random Processes , 1975 .

[13]  Yu-Kweng Michael Lin.  Probabilistic Theory of Structural Dynamics , 1976 .

[14]  P. Spanos.  Survival probability of non-linear oscillators subjected to broad-band random disturbances , 1982 .

[15]  Arvid Naess,et al.  Reliability of Strongly Nonlinear Single Degree of Freedom Dynamic Systems by the Path Integration Method , 2008 .

[16]  G. Cai,et al.  Some thoughts on averaging techniques in stochastic dynamics , 2000 .

[17]  L. Socha,et al.  Linearization in Analysis of Nonlinear Stochastic Systems: Recent Results—Part I: Theory , 2005 .

[18]  N. C. Nigam.  Introduction to Random Vibrations , 1983 .

[19]  J. Beck,et al.  Important sampling in high dimensions , 2003 .

[20]  J. Roberts,et al.  First-passage probabilities for randomly excited systems: Diffusion methods , 1986 .

[21]  P. Spanos,et al.  Nonstationary Response Envelope Probability Densities of Nonlinear Oscillators , 2007 .

[22]  P. Spanos,et al.  Nonstationary Solution in Nonlinear Random Vibration , 1988 .

[23]  K. Phoon,et al.  Implementation of Karhunen-Loeve expansion for simulation using a wavelet-Galerkin scheme , 2002 .

[24]  Y. K. Lin,et al.  Some observations on the stochastic averaging method , 1986 .

[25]  Helmut J. Pradlwarter,et al.  Evolution of probability densities in the phase space for reliability analysis of non-linear structures , 2009 .

[26]  Pol D. Spanos,et al.  A generalization to stochastic averaging in random vibration , 1992 .

[27]  An importance sampling procedure for estimating failure probabilities of non-linear dynamic systems subjected to random noise , 2007 .

[28]  Pol D. Spanos,et al.  An approximate approach for nonlinear system response determination under evolutionary stochastic excitation , 2009 .

[29]  P. Spanos,et al.  An analytical Wiener path integral technique for non-stationary response determination of nonlinear oscillators , 2012 .

[30]  W. D. Iwan,et al.  Response Envelope Statistics for Nonlinear Oscillators With Random Excitation , 1978 .

[31]  N. Bogolyubov,et al.  Asymptotic Methods in the Theory of Nonlinear Oscillations , 1961 .

[32]  Pol D. Spanos,et al.  On the determination of the power spectrum of randomly excited oscillators via stochastic averaging: an alternative perspective , 2009 .

[33]  J. Beck,et al.  First excursion probabilities for linear systems by very efficient importance sampling , 2001 .

[34]  G. Cai,et al.  Reliability of Nonlinear Structural Frame Under Seismic Excitation , 1998 .

[35]  Roger Ghanem,et al.  Adaptive polynomial chaos expansions applied to statistics of extremes in nonlinear random vibration , 1998 .

[36]  G. Schuëller,et al.  Equivalent linearization and Monte Carlo simulation in stochastic dynamics , 2003 .

[37]  Arvid Naess,et al.  Extreme Response of Nonlinear Structures With Low Damping Subjected to Stochastic Loading , 1999 .

[38]  P. Spanos,et al.  Stochastic averaging: An approximate method of solving random vibration problems , 1986 .

[39]  N. Bershad,et al.  Random differential equations in science and engineering , 1975, Proceedings of the IEEE.

[40]  J. J. Coleman.  Reliability of Aircraft Structures in Resisting Chance Failure , 1959 .

[41]  Siu-Kui Au,et al.  Importance sampling for elasto-plastic systems using adapted process with deterministic control , 2009 .