An approximate technique for the probability density function of the response of a linear oscillator to Erlang renewal random impulse processes

In this paper, the effect of non Gaussian, non Poissonian impulsive process on linear structural systems response is considered. The particular impulsive process herein considered is the Erlang renewal process, useful for traffic load modelling. In (Iwankievicz, 2005) it was shown that, to take advantage from the well established Poisson differential stochastic calculus, the original Erlang impulse process can be exactly converted into a Poisson driven process with the aid of a jump process regarded as an auxiliary variable. Characterizing the reponse process by a chain of k Markov states and making us of the Chapman- Kolmogorov equation, the evolutionary k equations of the response joint probability density function were derived. The first of this set of equations is integro-differential while all others are partial differential. These equations are transformed to first-order partial differential equations and an approximate solution technique is devised by considering the evolution of the response during small time intervals. Using the method of characteristic, as shown in (Vasta and Luongo, 2004), is then possible to numerically integrate these equations, and an explicit solution can be found. It is shown that numerical integration of these equations confirm the consistency of the theory for different parameters value of the system and of the excitation process, as well as a good agreement between Markov state modeling and direct computational approach.