Tunable low-frequency torsional-wave band gaps in a meta-shaft

The location of a locally resonant band gap is related to the resonant frequency of the local resonator, and thus one could obtain a band gap in a very low-frequency region by designing a resonator with an ultra-low resonant frequency. In this paper, a novel torsional resonator is proposed by introducing a negative stiffness mechanism composed of five pairs of cams and rollers to partially neutralize the positive stiffness of a rubber ring. By attaching this resonator onto an elastic shaft, a meta-shaft structure for attenuating low-frequency torsional wave is devised. The static analysis for the local resonator and the dynamic analysis for the meta-shaft are conducted. The band structure is revealed by the transfer matrix method, which are validated by numerical simulations with the assistance of the Galerkin method. The theoretical results show that the location of the band gap can be effectively shifted from the high frequency region to a lower one by tuning down the net stiffness of the resonator. In addition, the effects of structural parameters on the band gap and torsional wave attenuation are also evaluated, which indicate that a thick rubber ring, a large number of unit cells and a small-amplitude excitation are beneficial to open a broad and deep band gap at low frequencies. This could be a potential solution to the challenge of low-frequency torsional wave attenuation.