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Research Papers

Approximate Solution of the Fokker–Planck Equation for a Multidegree of Freedom Frictionally Damped Bladed Disk Under Random Excitation

[+] Author and Article Information
Alwin Förster

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

Lars Panning-von Scheidt

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

Jörg Wallaschek

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

1Corresponding author.

Manuscript received June 22, 2018; final manuscript received June 28, 2018; published online September 14, 2018. Editor: Jerzy T. Sawicki.

J. Eng. Gas Turbines Power 141(1), 011004 (Sep 14, 2018) (8 pages) Paper No: GTP-18-1279; doi: 10.1115/1.4040740 History: Received June 22, 2018; Revised June 28, 2018

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 (DOFs) and use computational expensive methods, like finite element method 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.

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References

Risken, H. , and Haken, H. , 1989, The Fokker–Planck Equation, Vol. 18, Springer, Berlin.
Sinha, A. , 1987, “Friction Damping of Random Vibration in Gas Turbine Engine Airfoils,” Int. J. Turbo Jet Engines, 7, pp. 95–102.
Cha, D. , and Sinha, A. , 2003, “Computation of the Optimal Normal Load of a Friction Damper Under Different Types of Excitation,” ASME J. Eng. Gas Turbines Power, 125(4), pp. 1042–1049. [CrossRef]
Cha, D. , and Sinha, A. , 2006, “Statistics of Responses of a Mistuned and Frictionally Damped Bladed Disk Assembly Subjected to White Noise and Narrow Band Excitations,” Probab. Eng. Mech., 21(4), pp. 384–396. [CrossRef]
Kumar, P. , and Narayanan, S. , 2009, “Nonlinear Stochastic Dynamics, Chaos, and Reliability Analysis for a Single Degree of Freedom Model of a Rotor Blade,” ASME J. Eng. Gas Turbines Power, 131(1), p. 012506. [CrossRef]
Kumar, P. , and Narayanan, S. , 2010, “Response Statistics and Reliability Analysis of a Mistuned and Frictionally Damped Bladed Disk Assembly Subjected to White Noise Excitation,” ASME Paper No. GT2010-22736.
Narayanan, S. , and Kumar, P. , 2012, “Numerical Solutions of Fokker–Planck Equation of Nonlinear Systems Subjected to Random and Harmonic Excitations,” Probab. Eng. Mech., 27(1), pp. 35–46. [CrossRef]
Wedig, W. V. , 1989, “Generalized Hermite Analysis of Nonlinear Stochastic Systems,” Struct. Saf., 6(2–4), pp. 153–160. [CrossRef]
Wedig , W. , and Kree , P. , eds., 1995, Probabilistic Methods in Applied Physics (Lecture Notes in Physics, Vol. 451), Springer, Berlin.
Wedig , W. V., von Wagner , U., Spanos , P. D. , and Brebbia , C. A. , eds., 1999, Extended Laguerre Polynomials for Nonlinear Stochastic Systems, Springer, Dordrecht, The Netherlands.
von Wagner, U. , 1999, Zur Berechnung Stationärer Verteilungsdichten Nichtlinearer Stochastisch Erregter Systeme (Fortschritt-Berichte VDI Reihe 11, Schwingungstechnik, Vol. 274), VDI-Verl., Düsseldorf, Germany.
von Wagner, U. , and Wedig, W. V. , 2000, “On the Calculation of Stationary Solutions of Multi-Dimensional Fokker–Planck Equations by Orthogonal Functions,” Nonlinear Dyn., 21(3), pp. 289–306. [CrossRef]
von Wagner, U. , 2002, “On Double Crater-Like Probability Density Functions of a Duffing Oscillator Subjected to Harmonic and Stochastic Excitation,” Xenophon's Spartan Constitution, M. Lipka , ed., Vol. 28, De Gruyter, Berlin, pp. 343–355.
Martens, W. , vonWagner, U. , and Mehrmann, V. , 2012, “Calculation of High-Dimensional Probability Density Functions of Stochastically Excited Nonlinear Mechanical Systems,” Nonlinear Dyn., 67(3), pp. 2089–2099. [CrossRef]
Martens, W. , and von Wagner, U. , 2011, “Calculation of Probability Density Functions for Nonlinear Vibration Systems,” PAMM, 11(1), pp. 923–926. [CrossRef]
Martens, W. , 2014, On the Solution of the Fokker–Planck Equation for Multi-Dimensional Nonlinear Mechanical Systems (Berichte aus dem Maschinenbau), 1. aufl., Shaker, Aachen.
Lentz, L. , and Martens, W. , 2014, “On the Solution of the Fokker–Planck Equation on Infinite Domains Using Problem-Specific Orthonormal Basis Functions in a Galerkin-Type Method,” PAMM, 14(1), pp. 767–768. [CrossRef]
Wang, R. , and Zhang, Z. , 2000, “Exact Stationary Solutions of the Fokker-Planck Equation for Nonlinear Oscillators Under Stochastic Parametric and External Excitations,” Nonlinearity, 13(3), pp. 907–920. [CrossRef]

Figures

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Fig. 5

Four-DOF frictionally damped bladed disk model PDF of the first state space variable (displacement)

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Fig. 6

Four-DOF frictionally damped bladed disk model PDF of the second state space variable (velocity)

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Fig. 4

Frictionally damped one-dimensional system with improved weighting functions

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Fig. 1

PDF for Duffing-oscillator

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Fig. 2

Frictionally damped bladed disk model

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Fig. 3

Frictionally damped one-dimensional system with Duffing-weighting functions

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Fig. 8

Four-DOF frictionally damped bladed disk model two-dimensional JPDF's calculated by MCS

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Fig. 9

Fifty-DOF frictionally damped bladed disk model PDF of the first state space variable (displacement)

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Fig. 10

Fifty-DOF frictionally damped bladed disk model PDF of the second state space variable (velocity)

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Fig. 7

Four-DOF frictionally damped bladed disk model two-dimensional JPDF's

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