Abstract
Technical systems are subjected to a variety of excitations that cannot generally be described in deterministic ways. External disturbances like wind gusts or road roughness as well as uncertainties in system parameters can be described by random variables, with statistical parameters identified through measurements, for instance.
For general systems the statistical characteristics such as the probability density function (pdf) may be difficult to calculate. In addition to numerical simulation methods (Monte Carlo Simulations, MCS) there are differential equations for the pdf that can be solved to obtain such characteristics, most prominently the Fokker–Planck equation (FPE).
A variety of different approaches for solving FPEs for nonlinear systems have been investigated in the last decades. Most of these are limited to considerably low dimensions to avoid high numerical costs due to the “curse of dimension”. Problems of higher dimension, such as d=6, have been solved only rarely.
In this paper we present results for stationary pdfs of nonlinear mechanical systems with dimensions up to d=10 using a Galerkin method, which expands approximative solutions (weighting functions) into orthogonal polynomials.
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References
Iourtchenko, D.V., Mo, E., Naess, A.: Response probability density functions of strongly nonlinear systems by the path integration method. Int. J. Non-Linear Mech. 41, 693–705 (2006)
Bergman, L.A., Masud, A.: Application of multi-scale finite element methods to the solution of the Fokker–Planck equation. Comput. Methods Appl. Mech. Eng. 194, 1513–1526 (2005)
Kumar, P., Narayanan, S.: Solution of the Fokker–Planck equation by finite element and finite difference methods for nonlinear systems. Sadhana 31(4), 445–461 (2006)
Naess, A., Moe, V.: Efficient path integration methods for nonlinear dynamic systems. Probab. Eng. Mech. 15, 221–231 (2000)
Yu, J.S., Lin, Y.K.: Numerical path integration of a non-homogeneous Markov process. Int. J. Non-Linear Mech. 39, 1493–1500 (2004)
Tombuyses, B., Aldemir, T.: Continuous cell-to-cell mapping. J. Sound Vib. 202(3), 395–415 (1997)
Feuersänger, C.: Sparse grid methods for higher dimensional approximation, pp. 63–101. Dissertation, Institut für Numerische Simulation, Universität Bonn (2010)
Kumar, P., Chakravorty, S., Singla, P., Junkins, J.: The partition of unity finite element approach with HP-refinement for the stationary Fokker–Planck equation. J. Sound Vib. 327, 144–162 (2009)
von Wagner, U.: Zur Berechnung stationärer Verteilungsdichten nichtlinearer stochastisch erregter Systeme, pp. 20–31. VDI, Düsseldorf (1999)
von Wagner, U., Wedig, W.V.: On the calculation of stationary solutions of multi-dimensional Fokker–Planck equations by orthogonal functions. Nonlinear Dyn. 21, 289–306 (2000)
von Wagner, U.: On double crater-like probability density functions of a duffing oscillator subjected to harmonic and stochastic excitation. Nonlinear Dyn. 28, 343–355 (2002)
von Wagner, U.: On nonlinear stochastic dynamics of quarter car models. Int. J. Non-Linear Mech. 39, 753–765 (2004)
Arnold, L.: Stochastische Differentialgleichungen, pp. 141–146. Oldenbourg, München (1973)
Golub, G.H., Van Loan, C.F.: Matrix Computations, 3rd edn. Johns Hopkins University Press, Baltimore (1996)
Ammon, D.: Modellbildung und Systementwicklung in der Fahrzeugdynamik, pp. 74–80. Teubner, Stuttgart (1997)
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Martens, W., von Wagner, U. & Mehrmann, V. Calculation of high-dimensional probability density functions of stochastically excited nonlinear mechanical systems. Nonlinear Dyn 67, 2089–2099 (2012). https://doi.org/10.1007/s11071-011-0131-2
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DOI: https://doi.org/10.1007/s11071-011-0131-2