Calculation of probability density functions of mechanical systems under stochastic excitation

by solving the Fokker-Planck Equation

(Berechnung von Verteilungsdichtefunktionen stochastisch angeregter mechanischer Systeme durch Lösung der Fokker-Planck-Gleichung)

project numbers: ME 790/22-1, ME 790/22-2 and WA 1427/9-1, WA 1427/9-2

Technische Universität Berlin DFG-Forschungszentrum

duration: 2009 - 2012
project leaders: Volker Mehrmann, Utz von Wagner
Department of Mathematics, Technical University of Berlin,
Strasse des 17. Juni 136, 10623 Berlin, Germany
Tel: +49 (0)30 - 314 25 736 / - 314 21 264
email: mehrmann(at)
Department of Mechanics, Technical University of Berlin,
Einsteinufer 5-7, 10587 Berlin, Germany
Tel: +49 (0)30 - 314 21 169 / - 314 22 922
email: utz.vonwagner(at)
research assistant: Wolfram Martens
Department of Mechanics, Technical University of Berlin,
Einsteinufer 5-7, 10587 Berlin, Germany
Tel: +49 (0)30 - 314 22 876
email: wolfram.martens(at)
student assistant: Ina Thies
support: Deutsche Forschungsgemeinschaft


Technical systems are subjected to a variety of external excitations that cannot generally be described in deterministic ways. Road surface roughness, wind gusts or oceanic excitations of marine structures are all examples of mechanical excitations that can be described by statistical means only. System responses to such excitations are described by statistical parameters such as the probability density function (pdf), which can be obtained by solving partial differential equations such as the Fokker-Planck Equation (FPE). Exact solutions to the FPE are available only for a small class of problems, and for a wide range of non-linear problems there is a need for numerical FPE-solutions. Most numerical methods are restricted to considerably low dimensions due to high numerical efforts, so that the relevance to most technical applications is limited.
At TU Berlin's Chair of Mechatronics and Machine Dynamics, in cooperation with the Numerical Mathematics Research Group, a method is investigated which allows the efficient solution of comparably high-dimensional FPEs. It is based on the expansion of approximative pdf solutions by orthogonal polynomials, customized to specific problems providing good convergence properties.
The project is funded by Deutsche Forschungsgemeinschaft (DFG).




last modified July 7th, 2011