An explicit closed-form solution for transverse and longitudinal vibration of beam with multi-directional elastic constraints under an arbitrary moving load

Abstract

Beams with elastic constraints are widely used in dynamic systems in engineering. A general explicit solution is presented here for the vibration of simple span beam with transverse, rotational and axial elastic boundary constraints due to an arbitrary moving load. The Euler-Bernoulli beam theory is adopted, in which the boundary constraints are treated as multi-directional boundary springs. After the modal analyses, the explicit closed-form solutions of transverse and axial vibration of the beam under a constant, sinusoidal and cosinoidal moving loads are obtained, respectively. And the vibration of a beam subjected to an arbitrary moving load is derived by the superposition of Fourier series. The current analytical solution is exact and can be applied in multiple engineering fields to obtain accurate structural vibrations. In numerical examples, the effects of the boundary springs on the natural frequencies, modes, deflection, bending moment and boundary reaction of the beam are studied in details. The effects of the number of terms in Fourier series of arbitrary moving load are also discussed.