Nonlinear Dynamics

, Volume 92, Issue 2, pp 613–626 | Cite as

Nonstationary solutions of nonlinear dynamical systems excited by Gaussian white noise

  • Siu-Siu Guo
Original Paper


Nonlinear dynamical systems to random excitations may fail long before stationarity is achieved. Transient state has to be taken into account. A novel approximate technique for determining nonstationary probability density function (PDF) of randomly excited nonlinear Oscillator is developed. Specifically, it expresses the PDF approximation in terms of polynomial functions with time-dependent coefficients. By applying statistical linearization and weighted residual method, residual error of the FPK equation associated with approximated solution is reduced to a series of nonlinear first-order ordinary differential equations, which can be solved by the numerical method. Finally, a class of nonlinear vibrating systems with additive excitations or/and parametric excitations are considered. The obtained PDF has tail regions of logarithmic form, which are important for reliability and failure analysis, and agrees very well with the simulated ones. In particular, the computational time spent by the proposed procedure is a very small fraction of the one taken by the MCS method. This technique can be used as a convenient tool for assessing the accuracy of alternative, more general, approximate solution methods.


Stochastic processes Random vibration Fokker–Planck–Kolmogorov (FPK) equation Monte Carlo simulation Nonstationary solutions 



This research was jointly supported by the National Natural Science Foundation of China under Grant No. 11502187.


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© Springer Science+Business Media B.V., part of Springer Nature 2018

Authors and Affiliations

  1. 1.School of Civil EngineeringXi’an University of Architecture and TechnologyXi’anChina

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