Bifurcation structure of the special class of nonstationary regimes emerging in the 2D inertially coupled, unit-cell model: Analytical study

Abstract

Present work is devoted to the analytical investigation of the bifurcation structure of special class of nonstationary low-energy regimes emerging in the locally resonant unit-cell model. System under consideration comprises an outer mass with internal rotator and subject to the 2D, nonlinear local potential. These regimes are characterized by the slow, purely rotational motion of the rotator synchronized with the periodic energy beats between the axial and the lateral vibrations of the outer element. Thus the angular speed of the rotator and the beating frequency of the outer element satisfy the 1:2 resonance condition. In the present study these regimes are referred to as regimes of synchronous nonlinear beats (RSNB). Using the regular muti-scale analysis in the limit of low energy excitation we derive the slow-flow model. To showcase the evolution of RSNBs we used the special Poincaré map technique applied on the slow-flow model. Results of the Poincaré sections unveiled some interesting local bifurcations undergone by these regimes. Further analysis of the slow-flow model enabled us to describe the RSNBs analytically as well as exposed their entire bifurcation structure. The bifurcation analysis has shown the coexistence of several branches of RSNBs corresponding to the regimes of weak and strong, two-dimensional, recurrent energy channeling. We substantiate the results of the analytical study with numerical simulations of the full model and find them to be in the very good agreement.