Calculation of high-dimensional probability density functions of stochastically excited nonlinear mechanical systems
Calculation of high-dimensional probability density functions of stochastically excited nonlinear mechanical systems
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DOI：
10.1007/s11071-011-0131-2
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发表时间：
2012-02
影响因子：
5.6
通讯作者：
Wolfram Martens;U. Wagner;V. Mehrmann
中科院分区：
文献类型：
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作者：
Wolfram Martens;U. Wagner;V. Mehrmann
技术系统受到各种激励，这些激励通常不能用确定性的方式来描述。外部干扰（如阵风或路面不平度）以及系统参数的不确定性可以用随机变量来描述，统计参数可以通过测量来确定。对于一般系统，统计特性（如概率密度函数（pdf））可能很难计算。除了数值模拟方法（蒙特卡罗模拟，MCS），还有微分方程的PDF，可以解决，以获得这样的特性，最突出的福克-普朗克方程（FPE）。各种不同的方法来解决非线性系统的FPE在过去的几十年中已经研究。大多数这些被限制到相当低的维度，以避免由于“维数灾难”而导致的高数值成本。高维问题（如d =6）很少得到解决，本文利用Galerkin方法，将近似解（权函数）展开为正交多项式，给出了维数tod=10的非线性力学系统的定常pdf的结果。
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 asd=6, have been solved only rarely.In this paper we present results for stationary pdfs of nonlinear mechanical systems with dimensions up tod=10 using a Galerkin method, which expands approximative solutions (weighting functions) into orthogonal polynomials.