Flexural waves and deflection mode shapes of periodic and disordered beams

Abstract

This paper deals with free flexural wave motion and natural deflection mode shapes of simply supported infinite uniform periodic beams consisting of repeating units that are identical finite beams having equal and unequal span lengths. Governing equations for the natural frequencies and that for the wave propagation constant have first been set up in terms of the receptances of the individual beam elements. The equations are then applied to compute the natural frequencies and deflection mode shapes together with the propagation constants for some specific disordered periodic beams that include four-span and eight-span periodic and disordered beams. Relationship between the natural frequencies of the symmetric finite repeating beam units and the bounding frequencies of the propagation and attenuation zones has been studied. The bounding frequencies are always identified with the resonance frequencies of such beams with their extreme ends either simply supported or clamped. The combinations of the natural frequencies, with which the bounding frequencies of the propagation and attenuation zones can be identified, have been pointed out. Studies of the deflection mode shapes of symmetrically and unsymmetrically disordered four-span beams have shown that the characteristic free waves in an infinite periodic system consisting of disordered repeating beams (beams of unequal span lengths) do not always decay at the natural frequencies. The conditions under which the waves at the natural frequencies of the repeating beam units decay have been studied and explained.