Some Issues On A Double Shock Oscillator Under White Noise By Fokker-Planck Equation

The study of primitive models can help to qualitatively understand the dynamics of real nonlinear structures. In this framework favouring analytical developments, the paper focuses on the here-called double shock oscillator (DSO): a 2-DOF model made of a simple shock oscillator carrying a linear oscillator, and excited by white noise(s). DSO drastically reduces some configurations of the nuclear engineering involving a loose support (rigid stops with gaps) and severe excitation. Fokker-Planck-Kolmogorov equation (FPK) is candidate for such an analysis. This partial differential equation governs the probability density function (PDF) of diffusion stochastic processes, and applies to DSO in its 4-D phase space. It even has closed-form solutions for the stationary (time independent) PDF, but under very restrictive conditions concerning the damping distribution linked to the white noise excitation. So, the paper explores a way for computing the PDF for “more physical” damping arrangements. The FPK stationary solution is step-by-step approached from a previous analytical solution. For this, the time-evolutive PDF is decomposed into its Fourier components taken as generalized variables and whose number is limited. The transfer between the physical and Fourier spaces is made by FFT. The main difficulties arise from the physical convergence duration (transient response given the initial damping change), and the singularity of the matrix whose product by the Fourier components supplies their time-derivatives. The method, which includes a second order scheme, periodicity changes and renormalization processes, is validated from time-history Monte-Carlo simulation, as well as analytical results in the linear case (without stops).