Analytical analysis of nonlinear internal resonance bandgap of pendulum-type metamaterial

In this paper, a pendulum-type metamaterial (PTM) is designed with a pendulum bob hinged at the primary mass. Considering the effect of geometric nonlinearity, the nonlinear dynamic equations of PTM unit cell are presented with the aid of the Bloch theorem. The analytical formula of dispersion equation is deduced to discuss the nonlinear internal resonance bandgap using the harmonic balance method. The obtained bandgap of the metamaterial is in good agreement with the numerical simulation result. The nonlinear geometric influence of pendulum on PTM bandwidth is significant. The bandgaps under different internal resonances are substantially different from each other due to the nonlinear effects. The upper boundaries of the frequency bandgap under 1:1/2 and 1:1/3 internal resonance rise nonlinearly to higher than those under linear and 1:1 internal resonance conditions. It shows that the impact of 1:1/2 and 1:1/3 internal resonance on the bandgap could be more obvious as the strong nonlinearity is taken into consideration.