Physics of rack-and-pinion-inspired metamaterials with rotational resonators for broadband vibration suppression

In the present work, a new design of elastic metamaterial is proposed on the basis of rack-and-pinion mechanism. The design process includes a fundamental model with single resonator and a developed model with multiple rotational resonators. The system of the fundamental model consists of an array of cells acting as the rack; each cell includes a pinion disk which is connected to the cell with a linear spring. In the developed model, a core mass is embedded inside the pinion and connected to it with a torsional spring. The main objective of the proposed model is to obtain a wider band gap in the single resonator model and establish broadband vibration suppression by tuning local resonators in the multiple resonator model. Dynamic behavior of the system is investigated by extracting governing equations of motion for a unit cell and obtaining dispersion relation in order to find the impacts of controlling parameters on the band gaps starting and ending frequencies. Negative effective mass is also presented to express the models in a monatomic lattice model and explore the effect of negative mass on band gap frequencies. To verify the integrity of numerical analyses, 3D models of both systems are created and analyzed through MSC-ADAMS software. Attention is focused on deriving the closed-form expression for the band gap starting/ending frequencies yielding an overall insight for designing vibration attenuation mechanisms.