Advances in nonlinear random vibration analysis via FPK equation, GDEE, and PDIE

In recent years, nonlinear random vibration analysis of engineering structures subjected to complex environmental excitations has attracted increasing attention from researchers. However, the inherent coupling of nonlinearity and randomness in multi-degree-of-freedom systems within these complex environments presents a significant challenge. Historically, random excitations have often been simplified to white noise, thereby transforming stochastic dynamical systems into Markov systems whose transition probability density function satisfies the well-known Fokker–Planck–Kolmogorov (FPK) equation and its equivalent Chapman–Kolmogorov–Smoluchowski equation. Building upon this framework, researchers have developed a series of theoretical and analytical methods for nonlinear stochastic systems. Nevertheless, the computational complexity of solving the FPK equation restricts the dimensionality of solvable systems. To overcome this limitation, novel equations characterizing probability density propagation and evolution have been derived on the basis of the principle of probability conservation, namely, the generalized density evolution equation. Also, the probability density integral equation is further developed. Corresponding random vibration analysis methods have also been developed. This paper provides a comprehensive overview of the fundamental principles of these methods and the associated recent advances in nonlinear random vibration analysis.