Non-reciprocity and selective wave amplification in nonlinear inertia-induced autoparametric periodic structures

Abstract

Inspired by the spatiotemporally modulated metamaterials, we propose a one-dimensional autoparametric periodic structure induced by nonlinear inertia that is realized simply by constructing an array of pendula coupled to local oscillators. The mechanism of autoparametric modulation is revealed using a unit cell. The evolution equations for a superlattice describing the global dynamics of the wave-vibration coupling are obtained analytically by combining the Floquet-Bloch theorem and the multiple-time scales perturbation technique. The band structure exists corresponding to the fixed point in the phase space, wherein large amplitude local vibrations with distributed traveling phases modulate the inertial properties of the host pendula propagating with a small amplitude signal wave by playing the role of a pump wave. Non-reciprocal dispersion relations are demonstrated both by numerical results and perturbation results with two-order accuracy. Selective wave amplification with two-mode directional emission is observed in a two-port system which is composed of a sandwich composite medium. The present research based on the proposed autoparametric structure paves a way towards building tunable wave manipulators, vibration energy migrators and mechanical amplifiers.