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 W. Martens ( ) · U. to d = 10 using a Galerkin method, which expands approximative solutions (weighting functions) into orthogonal polynomials.
Procedures are given for solving the equations of motion of infinite and half-infinite chains of linear springmass oscillators with nearest-neighbor coupling. Arbitrary initial displacements and initial velocities may be prescribed for any finite number of the masses in the chains, and external forces may be applied to any finite number of them. The solutions appear as integrals with integrands involving orthogonal polynomials generated by three-term recurrence relations whose coefficients are determined from the equations of motion. Tables of the polynomials needed for solving the equations of motion of a number of special chains are included.
Solving the Fokker-Planck-Equation for multidimensional nonlinear systems is a great challenge in the field of stochastic dynamics. As for many mechanical systems a general idea about the shape of stationary solutions for the probability density function is known, it seems promising to use an approach that contains this knowledge. This is done using a Galerkin-method which applies approximate solutions as weighting functions for the expansion of orthogonal polynomials, e.g. generalized Hermite polynomials [1]. As examples, nonlinear oscillators containing cubical restoring (Duffing oscillators) and cubical damping elements are considered. The method is applied to the two-dimensional problem of a single-degree-of-freedom oscillator and consecutively extended up to dimension ten. Results for probability density functions are presented and compared with results from Monte Carlo simulations.
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