Analytical prediction of the dynamics of beams under traveling loads and external resonance phenomena

The dynamics of slender elastic structures under a generic number of moving loads is addressed in this paper. The problem is solved analytically, assuming the structures are representable via an Euler–Bernoulli beam, the moving loads are equally spaced forces traveling at constant velocity, and the (generic) number of such forces lying on the beam at any time is always the same. The obtained solution is based on a linear map, which transforms the system state at the time at which one of the forces crosses a beam end, to the system state at the time at which the subsequent force crosses the same end. The reiteration of the linear map provides the complete time response of the system. The solution technique described and employed in the paper allows the continuous-time problem to be turned into a discrete-time problem and can in principle also be adopted to study nonlinear dynamic problems. Moreover, it provides analytical expressions for the system state variables and makes it simple and straightforward the analytical detection of the velocities of the traveling forces that can produce a divergent dynamics. It is shown, among the rest, that such velocities (referred to here as divergence velocities) form a subset, yet infinite in dimension, of the set of critical velocities of the system, they correspond to specific external resonance conditions, and are determinable analytically, in closed form. Numerical examples are also reported and corroborate the analytical findings.