Multiple bandgap formation in a locally resonant linear metamaterial beam: Theory and experiments

Abstract

This study presents a strategy of vibration suppression of a beam when multiple resonant frequencies of the structure are excited. The methodology operates on the ability of local resonators attached to the beam to create frequency ranges corresponding to locally resonant bandgaps in which the vibration of the beam is attenuated. This study discusses the mechanism of multiple bandgap formation by first deriving the equations of motion of the metastructure based on Hamilton’s principle and then utilizing a modal analysis approach to obtain analytical expressions for the edge frequencies of the created bandgaps. As a consequence of this assumption, the lattice constant of the resulting metastructure is much smaller than the operating flexural wavelength of the host beam. These edge frequencies are derived by assuming an infinite number of resonators tuned to different resonant frequencies of the beam at which a bandgap is desired to be centered at. Parametric studies on the steady state dynamic response of the beam, however, reveal that only a finite number of resonators is required to create these bandgaps and that their bandwidths largely depend on the ratio of the mass of the resonators to that of the beam. The proposed analytical approach is used to demonstrate bandgap formation at the first and second resonant frequencies of a cantilever beam both numerically using a commercial finite element solver as well as experimentally. Numerical modal analysis results of the metastructure compare well with experimentally measured modal analysis results and the steady-state response of the tip displacement of the structure clearly demonstrates the creation of two bandgaps both numerically as well as experimentally.