Improved pseudo-force approach for Monte Carlo Simulation of non-linear fractional oscillators under stochastic excitation

This paper presents a step-by-step procedure for the numerical integration of the fractional differential equation governing the response of single-degree-of-freedom (SDOF) non-linear systems endowed with fractional derivatives subjected to stochastic excitation. The procedure, labeled improved pseudo-force method (IPFM), is developed by extending a step-by-step integration scheme proposed by the second author for the numerical solution of classical differential equations. The IPFM relies on the following main steps: i) to use the Grunwald- Letnikov (GL) approximation of the fractional derivative; ii) to treat terms depending on the unknown values of the response, which result from the GL approximation as well as from the non-linear restoring forces, as pseudo-forces; iii) to handle non-linearities by performing iterations at each time step. The IPFM provides accurate solutions by using time steps of larger size compared to classical step-by-step integration schemes. In this paper, the IPFM is applied within the framework of classical Monte Carlo Simulation (MCS) to evaluate the time domain dynamic response of non-linear fractional systems subjected to the generic sample of a stochastic excitation.