Faster simulation methods for the nonstationary random vibrations of nonlinear MDOF systems

In this paper semi-analytical forward-difference Monte Carlo simulation procedures are proposed for the determination of the lower order statistics and the Joint Probability Density Function (JPDF) of the stochastic response of geometrically nonlinear multi-degree-of-freedom structural systems subject to nonstationary Gaussian white noise excitation, as an alternative to conventional direct simulation methods. These alternative simulation procedures rely on an assumption of local Gaussianity during each time step. This assumption is tantamount to various linearizations of the equations of motion. All of the proposed procedures yield the exact results as the time step goes to zero. The proposed procedures are based on analytical convolutions of the excitation process, hereby, reducing the generation of stochastic processes and numerical integration to the generation of random vectors only. Such a treatment offers higher rates of convergence, faster speed and higher accuracy. These procedures are compared to the direct Monte Carlo simulation procedure, which uses a fourth order Runge-Kutta scheme with the white noise process approximated by a broad band Ruiz-Penzien broken line process. The comparisons show that the so-called Ermark-Allen algorithm developed for simulation applications in molecular dynamics is the most favourable procedure for MDOF structural systems.