DR-PDEE-based efficient stochastic dynamical response analysis for high-dimensional nonlinear systems subject to multiplicative non-white excitation

Multiplicative stochastic excitation plays an indispensable role in modeling complex nonlinear dynamical systems across multiple fields of science and engineering.In particular, random noise in physical systems is typically not ideal white noise.The coupling between excitation and system responses in multiplicative problems, combined with the non-white nature of the excitation, leads to significant challenges for probabilistic response analysis, especially in high-dimensional dynamical systems.To address these challenges, an efficient method based on the dimension-reduced probability density evolution equation (DR-PDEE) is proposed for analyzing the response probability density of such systems.In this approach, the multiplicative non-white noise excitation is simulated by a linear analog filter, and the augmented system, which combines the original dynamical system with the linear filter, becomes a higher-dimensional system under additive white noise.Subsequently, by combining the quantity of interest with one filter output that possesses a nonzero intrinsic diffusion function, and solving the two-dimensional DR-PDEE for the joint process, the probability density function (PDF) solution of the quantity of interest can be directly obtained.Notably, due to the introduction of the analog filter, the corresponding intrinsic diffusion function in the DR-PDEE for the joint process becomes a known constant.Therefore, in this method, only the intrinsic drift functions of the DR-PDEE need to be determined, which can be reliably estimated through a small number of deterministic analyses.A numerical example is illustrated to validate the proposed method, demonstrating that the DR-PDEE achieves PDF solutions with high tail accuracy by only a small number of deterministic analyses.