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Previously, classical boundary conditions (i.e. simple, fixed and cantilevered) have been adopted for the beams under the moving loads in deriving the dynamic response. In reality, most beams cannot be simply classified as the ones with classical supports, but are elastically restrained. For theoretical completeness, a general theory will be developed in this note for the damped beams with elastic restraints modeled by vertical and rotational springs subjected to a moving damped sprung mass. Essential to the present theory is the solution of the transcendental equation for the frequency by the bi-section method. The solution obtained can also be applied to the classical boundary conditions (i.e. simple, fixed and cantilevered) under a moving sprung mass. The reliability of the present theory, along with the bi-section method for solving the frequency and mode shape, is validated by comparison with the solution obtained by the finite element method (FEM) for various damping ratios, mass ratios and running speeds of the sprung mass.

In this paper, the internal and external cancellation phenomena for damped beams subjected to multi-moving loads are investigated in detail. To start, the theory for the vibration of a simply supported beam is revisited by including the effect of damping. For the first time, a simple expression is derived for the free vibration of the damped beam under multi-moving loads. Based on the concept of local minimum, two cancellation conditions are identified. One is the internal cancellation, which relates to the inherent property of the beam and is conventionally known. The other is the newly formulated external cancellation that relates to the number and spacing of moving loads. For comparison, both the resonant condition and the optimal criterion for span length of the bridge are also briefed. By comparing with the classical solution, the present simple expression for the free vibration of the beam is firstly validated. Then the factors affecting the cancellation are investigated against various load cases and damping levels. The results show that external cancellation occurs more frequently due to the increase in the number and spacing of the moving loads. The damping of the beam has a leaking effect on cancellation, in that nonzero vibration may occur, but it is also quickly damped out by damping itself.