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research-article

Approximate solution of the Fokker-Planck equation for a multi-degree of freedom frictionally damped bladed disk under random excitation

[+] Author and Article Information
Alwin Förster

Institute of Dynamics and Vibration Research, Leibniz Universität Hannover, 30167 Hannover, Germany
foerster@ids.uni-hannover.de

Lars Panning-von Scheidt

Institute of Dynamics and Vibration Research, Leibniz Universität Hannover, 30167 Hannover, Germany
panning@ids.uni-hannover.de

Jörg Wallaschek

Institute of Dynamics and Vibration Research, Leibniz Universität Hannover, 30167 Hannover, Germany
wallascheck@ids.uni-hannover.de

1Corresponding author.

ASME doi:10.1115/1.4040740 History: Received June 22, 2018; Revised June 28, 2018

Abstract

Bladed Disks are subjected to different types of excitations, which cannot in any case be described in a deterministic manner. Fuzzy factors, such as slightly varying airflow or density fluctuation, can lead to an uncertain excitation in terms of amplitude and frequency, which has to be described by random variables. The computation of frictionally damped blades under random excitation becomes highly complex due to the presence of nonlinearities. Only a few publications are dedicated to this particular problem. Most of these deal with systems of only one or two degrees of freedom and use computational expensive methods, like finite element method (FEM) or finite differences method (FDM), to solve the determining differential equation. The stochastic stationary response of a mechanical system is characterized by the joint probability density function (JPDF), which is driven by the Fokker-Planck equation (FPE). Exact stationary solutions of the FPE only exist for a few classes of mechanical systems. This paper presents the application of a semi-analytical Galerkin-type method to a frictionally damped bladed disk under influence of gaussian white noise (GWN) excitation in order to calculate its stationary response. One of the main difficulties is the selection of a proper initial approximate solution, which is applicable as a weighting function. Comparing the presented results with those from the FDM, Monte-Carlo Simulation (MCS) as well as analytical solutions proves the applicability of the methodology.

Copyright (c) 2018 by ASME
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