Abstract Summary
This work addresses the dynamical response of slender structures crossed by systems of equally spaced transverse forces which travel at constant velocity along the main length of the structure. The study of the dynamic behavior of elastic structures acted upon by moving loads is a topic of great importance in civil and industrial engineering and has been the subject of many investigations over the years. Different external action models of various complexity have also been proposed in the literature, but the model of the moving force, despite its simplicity, represents a good schematization when the inertia of the traveling mass is negligible with respect to the mass of the entire structure, which is the case considered in this study. Specifically, we investigate the dynamical response of a slender elastic structure schematized as an Euler-Bernoulli beam, which is crossed by a system of equally spaced forces travelling at constant velocity, with the distance between two adjacent forces equal to an integer submultiple of the beam length. This problem represents a natural generalization of a problem recently studied by Luongo and Piccardo (Continuum Mech. Thermodyn., 28, 603-616, 2016) and is solved analytically in the present work. Starting from the case of the single moving force, the effect of the aforementioned train of forces is analytically accounted for via a linear map, which transforms the system state (that is, system position and velocity in modal coordinates) at the time instant tk (time at which one of the equally spaced travelling forces, say Fn, leaves the beam from one of its ends) to the system state at the subsequent time instant tk+1 (time at which the preceding force Fn-1 of the train of forces leaves the beam from the same end). The reiteration of this linear map provides the discrete-time response of the beam. The analytical solution obtained in this work enables the study of the dynamic behavior of the system and, in particular, highlights the presence of critical values of the train velocity for which instability phenomena by response accretion may occur. Comparisons with solutions obtained by resorting to purely numerical methods confirm the accuracy of the proposed analytical solution.
