Reduction of nonlinear dynamical systems excited by random loads

Abstract

The presented research deals with methods for the study of complex interactions between noise and nonlinearities in dynamical systems. Practical applications of this research are beginning to appear across the spectrum of mechanics; for example vibration absorbers, ship dynamics, energy harvesting, and variable speed machining processes. In the presented approach perturbed Hamiltonian systems are considered, which are damped and excited by an absolutely regular non-white Gaussian process. Analytical and semi-analytical solutions to the corresponding nonlinear stochastic differential equations (SDE) are determined. The approach is based on a limit theorem by Khashminskii, from which a class of methods has been derived known as stochastic averaging. In the presented approach Homoclinic orbits, which divide the phase space of the corresponding Hamiltonian flow into different regions, are mapped to a graph in the one degree of freedom case. For two degree of freedom systems this procedure results in a mapping to a book, and so on for higher degrees of freedom. From the drift and diffusion of the resulting averaged process, probability density functions and first passage times are obtained.