On the solution of the Fokker-Planck-Equation for multi-dimensional nonlinear mechanical systems

Dynamical physical systems are commonly described by means of mathematical analysis of the underlying system state variables. While a lot of technical applications are sufficiently well described by deterministic differential equations, there are situations where the response of a dynamic system under the influence of random excitations is of specific interest. The modeling of such systems leads to the analysis of stochastic processes, the trajectories of which can be found as the solutions of stochastic differential equations. In addition to the numerical simulation of such processes, probability theory provides analytical tools for the computation of probability density functions of the system states. In the case of Itô-diffusion processes the Kolmogorov-Forward- or Fokker-Planckequation is of fundamental importance. Exact solutions of the Fokker-Planck-equation exist only for very limited classes of problems, so that a variety of numerical solution methods have been investigated in the literature. In the context of random mechanical systems a fundamental problem lies in the fact that the dimension of the Fokker-Planck-equation increases along with the number of degrees of freedom of the underlying system. This thesis is about a solution method which allows for the analysis of comparably high-dimensional mechanical systems. In a lot of cases, approximate solutions can be found, for example through semi-analytical considerations of the underlying system. These are then used as the basis of an efficient Galerkin-scheme for the generation of improved solutions of the Fokker-Planck-equation. Following a thorough investigation of the underlying mathematical relations, the thesis presents results for a variety of nonlinear mechanical systems, with specific focus on highdimensional problems. Different systems from the field of mechanical energy harvesting are investigated in order to demonstrate the relevance of the presented method in the context of technical applications.