Propagation and attenuation characteristics of free flexural waves in multi-stepped periodic beams by the method of reverberation-ray matrix

Abstract

This paper aims to present better physical understanding of flexural wavenumber spectral behaviors of multi-stepped periodic beams by the method of reverberation-ray matrix (MRRM) based on the Timoshenko beam theory. According to the Bloch theorem, the formulation of MRRM is presented and the complex flexural wavenumbers of multi-stepped periodic beams are solved by numerical techniques. The validation of MRRM in analyzing flexural wave behaviors of multi-stepped periodic beams is verified by comparison with the analytical wavenumber spectrum of a homogeneous beam. Numerical examples are provided to illustrate the general wavenumber spectral characteristics of an infinite periodic beam with alternate variation of cross-sections. The effects of the material properties and geometrical parameters of the beam segments are evaluated. The most remarkable finding inspired from the wavenumber spectral curves is that the real wavenumber of the multi-stepped periodic beam does not vary with frequency in the attenuation bands, which indicates that the wavelength of the flexural wave is constant in the attenuation bands. Highlights A multi-stepped periodic beam model with arbitrary finite number of beam segments is established. Complex wavenumber spectrum analysis is performed to reveal the propagation and attenuation behaviors of multi-stepped periodic beams. Similarity and comparative studies on the propagation and attenuation characteristics of beams of a uniform cross-section and of variable cross-sections are carried out to provide better physical understanding. An innovative finding that the wavelength of the flexural wave is constant in the attenuation bands is achieved.