Stochastic response of the Volterra integral hysteresis model via a gaussian basis function solution

This paper investigates the random response of hysteretic nonlinear oscillators described by the Volterra integral model, which captures material memory effects via a convolution kernel under Gaussian white noise excitation. The Gaussian basis function (GBF) approximation technique is utilized to estimate the system's steady-state probability density function (PDF). The approximately stationary probability density function is represented as a weighted sum of Gaussian basis functions in this method. The optimal weighting coefficients are determined by minimizing the residual of the stationary Fokker-Planck-Kolmogorov (FPK) equation using the least squares method with Lagrange multipliers, yielding the steady-state joint probability density function. The computed results show excellent agreement with Monte Carlo simulations (MCS), confirming the accuracy and reliability of the Gaussian basis function method and demonstrating the influences of various parameters on the hysteresis loop and the system response.