Non-stationary response of some first-order non-linear systems associated with the seismic sliding of rigid structures

The present investigation focuses on the determination of the statistical properties, transition probability function, mean, and variance, of the response v(t) of the first-order system governed by the non-linear stochastic differential equation dv dt+f{hook}(v)=D(t)W(t), where f{hook}(v) is a piecewise constant. D(t) is an arbitrary deterministic modulation function and W(t) is a stationary Gaussian white noise. An exact solution of the Fokker-Plank equation associated with the process v(t) is first derived in terms of a set of unknown boundary conditions. It is then shown that these functions of time satisfy a system of linear Voltera integral equations of the second kind, which are readily solved numerically and sometimes analytically. Finally, these concepts are applied to the seismic sliding of rigid structures and examples of application are presented.