Nonlinear Dynamics

, Volume 95, Issue 1, pp 293–308 | Cite as

Reliability function determination of nonlinear oscillators under evolutionary stochastic excitation via a Galerkin projection technique

  • H. Vanvinckenroye
  • I. A. Kougioumtzoglou
  • V. DenoëlEmail author
Original Paper


An approximate semi-analytical technique is developed for determining the response first-passage time probability density function of a class of lightly damped nonlinear oscillators subject to evolutionary stochastic excitation. Specifically, relying on a Markovian approximation of the response energy envelope, and on a stochastic averaging treatment, yields a backward Kolmogorov equation governing the evolution in time of the oscillator reliability function. Next, the backward Kolmogorov equation is solved by employing an appropriate orthogonal basis in conjunction with a Galerkin projection scheme. It is noted that the technique can account for arbitrary evolutionary excitation forms, even of the non-separable type. The special case of an undamped oscillator, for which relevant analytical results exist in the literature, is also included and studied in detail. Further, Markovian approximation of the potential energy envelope is considered as well. In comparison with the conventional amplitude-based energy envelope formulation, the intermediate step of linearizing the nonlinear stiffness element is circumvented, thus reducing the overall approximation degree of the technique. An additional significant advantage of the potential energy envelope formulation relates to the fact that its degree of accuracy appears rather insensitive to the nonlinearity magnitude (at least in the considered examples). Pertinent Monte Carlo simulation data are included in the numerical examples as well for assessing the accuracy of the technique.


Galerkin scheme Reliability assessment First-passage time Nonlinear oscillator Evolutionary excitation Backward Kolmogorov equation 



H. Vanvinckenroye was supported by the National Fund for Scientific Research of Belgium. This research has been conducted during a visit of H. Vanvinckenroye at Columbia University

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.


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Copyright information

© Springer Nature B.V. 2018

Authors and Affiliations

  1. 1.Structural Engineering Division, Faculty of Applied SciencesUniversity of LiègeLiègeBelgium
  2. 2.FRIA (F.R.S.-F.N.R.S)National Fund for Scientific ResearchBrusselsBelgium
  3. 3.Department of Civil Engineering and Engineering MechanicsColumbia UniversityNew YorkUSA

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