Abstract
Most existing research on periodic beams concerns bending waves in mono-coupled and bi-coupled periodic mono-component beams with the unit cell containing only one beam segment, and very few works on bi-coupled periodic multi-component beams with the unit cell containing more than one beam segments study the bending waves in structures with only binary unit cells. This paper presents the method of reverberation-ray matrix (MRRM) as an alternative theoretical method for analyzing the dispersion characteristics of bending waves with the wavelength greater than the size of the cross-sections of all components in bi-coupled periodic multi-component beams. The formulation of MRRM is proposed in detail with its numerically well-conditioned property being emphasized, which is validated through comparison of the results obtained with the counterpart results by other methods for exemplified bi-coupled periodic beams. Numerical examples are also provided to illustrate the comprehensive dispersion curves represented as the relations between any two among three in frequency, wavenumber (wavelength) and phase-velocity for summarizing the general features of the dispersion characteristics of bending waves in bi-coupled periodic multi-component beams. The effects of the geometrical and material parameters of constituent beams and the unit-cell configuration on the band structures are also demonstrated by numerical examples. The most innovative finding indicated from the dispersion curves is that the frequencies corresponding to the Brillouin zone boundary may not be the demarcation between the pass-band and stop-band for bending waves in bi-coupled periodic multi-component beams.
Highlights
A periodic structure consists of repeated unit cells which are joined together end-to-end/side-by-side [1]
Since the beam structure is a basic component in engineering and the bending wave is one of the main vibration forms and noise origins in engineering structures [4], the study of bending waves in periodic beams provides a method for vibration control and noise reduction in engineering structures, e.g., railway trains and tracks
For bending waves in a uniform beam attached with both translational/rotational spring supports and mass-spring resonators and with/without lumped mass: Lin and McDaniel [42] analyzed the FRF of a finite periodic beam by transfer matrix method (TMM) based on Euler–Bernoulli beam theory (EBT); based on Winkler’s foundation model and EBT, Yu et al [43] studied the complex band structures and FRF of periodic beams on elastic foundations attached with a locally-resonant mass-spring system; Raghavan and Phani [44] derived by receptance method (RM) the closed-form expressions for the location and width of sub-Bragg bandgaps, obtained the rigid body modes of the unit cell setting the bounding frequencies for local resonance bandgaps, and validated these results by experiments
Summary
A periodic structure consists of repeated unit cells which are joined together end-to-end/side-by-side [1]. For bending waves in a uniform beam attached with both translational/rotational spring supports and mass-spring resonators and with/without lumped mass: Lin and McDaniel [42] analyzed the FRF of a finite periodic beam by TMM based on EBT; based on Winkler’s foundation model and EBT, Yu et al [43] studied the complex band structures and FRF of periodic beams on elastic foundations attached with a locally-resonant mass-spring system; Raghavan and Phani [44] derived by RM the closed-form expressions for the location and width of sub-Bragg bandgaps, obtained the rigid body modes of the unit cell setting the bounding frequencies for local resonance bandgaps, and validated these results by experiments.
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