A method for considering a distributed spring constant for studying the flexural vibration of an Euler-beam with lightweight multistage local resonators

Abstract

For the traditional locally resonant beams there always attached the one-stage local resonator and result that the lower band gap the heavier the scattering ring. In order to resolve this problem, the flexural vibration band gap in an Euler beam with periodically arranged lightweight multistage local resonators was theoretically investigated using the transfer matrix method based on discretization of lumped mass. The present method considered a distributed spring constant, which showed fast convergence with less computational requirements. A finite element method was then employed to calculate the frequency response function of a finite sample simultaneously, which demonstrated that the results calculated using the proposed method were closer to the simulation results than those obtained using the traditional transfer matrix method. The study found that, under the same additional mass, the lightweight multistage structure had much lower beginning frequency than one-stage structure, and the total width of the gaps was basically the same. In addition, a simplified model of the beginning frequency of gaps was proposed, and the effect of scattering density on the model precision was further explored numerically. The results show that the lower scattering density, the more important the role of the rubber mass and the higher precision of the simplified model.