Random vibration of beam under moving loads

The stochastic response of a beam excited by moving loads with random amplitude is presented. The force arrivals at the beam are assumed to constitute a Poisson process of events. The input process is considered as a filtered Poisson process, as the response of a linear undamped oscillator excited by a superposition of two Poisson white noise processes. By means of the extension of the Ito's differential rule to the case of delta correlated input processes, the cumulant differential equations of every order of the response process are obtained. These equations are first-order linear differential equations with constant forcing function and a closed-form solution of the nonstationary stochastic response is provided. A numerical application of a bridge subjected to traffic flow has been performed by means of the proposed approach. The results are provided in terms of cumulants up to the fourth order and a comparison with those obtained by means of a Monte Carlo simulation is presented.