Band-gap analysis of a novel lattice with a hierarchical periodicity using the spectral element method

Inspired by the hierarchical structures of butterfly wing surfaces, a new kind of lattice structures with a two-order hierarchical periodicity is proposed and designed, and the band-gap properties are investigated by the spectral element method (SEM). The equations of motion of the whole structure are established considering the macro and micro periodicities of the system. The efficiency of the SEM is exploited in the modeling process and validated by comparing the results with that of the finite element method (FEM). Based on the highly accurate results in the frequency domain, the dynamic behaviors of the proposed two-order hierarchical structures are analyzed. An original and interesting finding is the existence of the distinct macro and micro stop-bands in the given frequency domain. The mechanisms for these two types of band-gaps are also explored. Finally, the relations between the hierarchical periodicities and the different types of the stop-bands are investigated by analyzing the parametrical influences.