Flexural vibration bandgaps in local resonance beam with a novel two-degree-of-freedom local resonance system

In this paper, an elastic metamaterial beam with a novel two-degree-of-freedom local resonator is investigated theoretically, and the dispersion relation is calculated by using transfer matrix (TM). In order to confirm the existence of band gaps, the transmission spectrum of flexural wave are also studied by using finite element method. The formation mechanism of the flexural vibration bandgaps (FVBGs) are further analyzed by studying the displacement fields of the eigenmodes at the band-gap edges. At last, the evolution of the dispersion relations with the increasing of the distance from the one side rubber to the center of the local resonance mass are discussed in detail, and the effects of the outside diameter of the Cu ring and the equivalent stiffness k of the rubbers on the FVBGs are also investigated. Through the above analysis, we can draw the following conclusions, due to the unequal of the torques provided by the two rubbers, two different rotational vibrations of local resonance mass with two different local resonance frequencies are introduced in the local resonance system, thus the elastic metamaterial beam shows two FVBGs at low frequencies. The theoretical results are in good agreement with the numerical results. The magnitude of torques introduced in the local resonance system can obviously affect the locations of the FVBGs. With the asymmetry decreasing, the frequency region of the first FVBG moves to the higher value, while that of the second FVBG tends to the lower value, and when the two torques are equal, the two FVBGs coupled into one wider gap. For the elastic metamaterial beam with heavy resonance mass and weak rubbers is appropriate to obtain a lower band gap, and the total width of the FVBGs becomes wider. However, it does just the opposite under the condition of the case with light Cu ring and strong rubbers, but the total width of the band gaps also becomes wider. The propagation properties of the flexural wave in the designed local resonance beam can potentially be used to control and insulate vibration at low-frequency range and the unequal distances between the two rubbers to the center of the local resonance mass provide possibilities to expand the application of the beams for the vibration control in the needed frequency ranges.