Pade-type solutions to nonlinear stochastic dynamics
Roy, Romain Valery
Spanos, Pol D.
Doctor of Philosophy thesis
A novel method of analysis for nonlinear stochastic dynamical systems under Gaussian white noise excitation is developed. The system response is Markovian and its probability density function (p.d.f.) is governed by the Fokker-Planck-Kolmogorov (FPK) equation. Of interest is the prediction of statistics of the response. For this purpose, the FPK equation is not solved but is used in a variety of approaches to derive exact analytical representations of the response statistics. One procedure involves the derivation of infinite hierarchies of equations governing the statistics. Another procedure exploits a formal series expansion of the transition p.d.f. All unknowns are expressed in the form of perturbation expansions of a system parameter, or in power series of the variable of interest. These series-type solutions are then recast in various approximations of the Pade-type. Results are obtained for stationary and nonstationary moments, correlation functions, power spectral densities, and Wiener kernels for second-order systems with analytical nonlinearities, and additive/multiplicative excitations. They are validated with exact solutions or Monte Carlo simulations. The pivotal point of this dissertation is that series-type representation after proper transformations can yield quite reliable global solutions by exploiting the local information contained in the first few series coefficients.