Response of Gaussian color noise excited oscillators with inertia nonlinearity based on the radial basis function neural network method

Given the widespread utilization of oscillators featuring inertial nonlinear terms across engineering disciplines and the omnipresence of noise excitations, comprehending their response to noise excitation holds paramount importance. In contrast to Gaussian white noise, colored noise stands out as a more adept mathematical model for simulating ambient noise scenarios. The widely used stochastic averaging method is somewhat ineffective in dealing with the response of oscillators with inertial nonlinearity under Gaussian colored noise excitation, as it cannot display the saddle shaped characteristics of the steady-state probability density function (PDF) caused by inertial nonlinearity. To overcome the shortcomings, this paper adopts a semi-analytical method called a radial basis function neural network method (RBFNN) to address this problem. This method represents the solution of the Fokker-Plank-Kolmogorov (FPK) equation of the system as the sum of a series of weighted radial basis functions, and obtains the optimal weight coefficients through the Lagrange multiplier method. This article investigates two examples, one of which has inertial nonlinearity and the other has multiple potential wells. The effects of inertial nonlinearity coefficients and the delay coefficients of colored noise on response are explored. The mean square errors between Monte Carlo (MC) and RBF are presented, which indicates the RBF results perfectly agree with the MC predictions. •We published a paper titled “Stochastic averaging on a nonlinear oscillator with coordinate dependent mass excited by Gaussian white noises” in 2021. In this paper, we studied an improved stochastic averaging method that can deal with nonlinear oscillators with coordinate-dependent mass (also called as longitudinal inertia in some literature) as well as stiffness nonlinearity. After that, two limitations of this method have been found by the further study.•The presented method is powerless on the multi-stability systems.•For Gaussian colored noise excited systems with strong inertia nonlinearity, the presented method will lose its accuracy dramatically.•In this paper, we applied a radial basis function neural network (RBFNN) algorithm to study the same topic. This new approach can deal with systems with strong inertia nonlinearity as well as multi-stability successfully. Furthermore, this method has much less time consuming than the Mento Carlo method. The influence of the inertia nonlinearity and the col