Subharmonic resonance of geometrical nonlinear structure in 2-D periodic elastic system for mechanical wave filtering

A dynamical analysis is conducted on a representative volume element of a periodic continuous structure. It is proven that the internal part attached to the frame is not constrained to periodic propagation conditions according to a finite element formulation. Hence a periodic structure in 2-D plane with a dynamic absorber in each element can be well designed to block the propagation of elastic waves in specific frequency domains, known as band gap around prescribed frequencies. To overcome the limit of linear vibrating internal structures we introduce an inclusion with geometrical nonlinearities which exhibits a nonlinear localized oscillation since the excitation frequency falls in the neighborhood of one-third of the main resonance frequency, known as subharmonic resonance. A combined configuration is taken as an example including clamp-clamp beams in rectangular periodic frame. According to the band structures and derivation, the designed structure can filter out waves at frequencies lower than fundamental resonance frequency of internal structure, which means the mechanism based on subharmonic resonance may enhance the functions for mechanical wave filtering and further energy harvesting.