Dynamic response of a beam to the train of moving forces driven by an Erlang renewal process

Dynamic response of a beam to a random train of moving forces moving with the same velocity is considered. Unlike a widely used Poisson process model, a more adequate Erlang renewal process is used as a process driving the train of forces. Normal-mode approach is used to convert the problem into that of a renewal driven train of general pulses. Consequently the modal responses are the filtered renewal processes and are expressed as integrals with respect to the response to a single pulse (passage of a force) and to the increments of the counting renewal process. The expressions for the mean values and cross-correlations of modal responses are obtained as single and double integrals, respectively. The results are presented in terms of the renewal density of the underlying Erlang counting process. Mean value and variance of the mid-span deflection of the beam are determined by numerical evaluation of the pertinent integrals. Numerical analysis is carried out for the Erlang processes with integer parameter $k=2$, $k=3$ and 4 and, for comparison, also for a Poisson process. Different traffic conditions such as the velocity and the mean arrival rate of vehicles are taken into account.