Physically Driven Exact Dimension Reduction of a Class of Nonlinear Multidimensional Systems Subjected to Additive White Noise

Abstract

Stochastic response analysis of nonlinear high-dimensional systems subjected to random excitations is a challenging issue in various engineering fields. The newly developed globally-evolution–based generalized probability evolution equation (GE-GDEE), as a partial differential equation governing the joint probability density of quantities of interest, rather than that of the whole state vector as in the Fokker-Planck-Kolmogorov (FPK) equation, provided a powerful tool to obtain probability density of responses of nonlinear high-dimensional systems. The dimension of GE-GDEE can be reduced with great flexibility, mostly to one or two in engineering applications. In this method, the equivalent drift coefficient serves as the ensemble driving force of probability densities, and thus its construction is critical to the accuracy of the solution. For this purpose, in the present paper, the exact stationary equivalent drift coefficient of the GE-GDEE is derived for a class of additive white-noise excited nonlinear multi-degree-of-freedom (MDOF) systems that achieve energy equipartition in their stationary stage. Based on the deduced explicit expression, a physical interpretation is provided to intuitively reveal the relationship between the equivalent drift coefficient and the inherent physical properties of systems. This leads to the physically driven approach for the determination of equivalent drift coefficient by imposing corresponding constraints on the quantities of interest. The proposed approach is verified by applying the GE-GDEE to the stochastic response analysis of two nonlinear MDOF systems excited by Gaussian white noise. The numerical results demonstrate that utilizing pertinent physical properties of a dynamical system yields a good estimation of the equivalent drift coefficient. In contrast, the method of regression based solely on sample data causes larger errors in assessing the equivalent drift coefficient and the probability density functions of the system responses. Also, the resulting probability density functions (PDFs) verify the high accuracy of the GE-GDEE provided the equivalent drift coefficient is constructed with high accuracy.