Free and Forced Vibrations of Periodic Multispan Beams

Abstract

In this study, the following two topics are considered for uniform multispan beams of both finite and infinite lengths with rigid transversal and elastic rotational constraints at each support: (a) free vibration and the associated frequencies and mode shapes; (b) forced vibration under a convected harmonic loading. The concept of wave propagation in periodic structures of Brillouin is utilized to investigate the wave motion at periodic supports of a multispan beam. A dispersion equation and its asymptotic form is obtained to determine the natural frequencies. For the special case of zero rotational spring stiffness, an explicit asymptotic expression for the natural frequency is also given. New expressions for the mode shapes are obtained in the complex form for multispan beams of both finite and infinite lengths. The orthogonality conditions of the mode shapes for two cases are formulated. The exact responses of both finite and infinite span beams under a convected harmonic loading are obtained. Thus, the position and the value of each peak in the harmonic response function can be determined precisely, as well as the occurrence of the so-called coincidence phenomenon, when the response is greatly enhanced.