Three-dimensional energy channeling in the unit-cell model coupled to a spherical rotator I: bidirectional energy channeling

This work is the first one in a two part series devoted to the analysis of the complex nonlinear mechanism of three-dimensional energy channeling emerging in a locally resonant three-dimensional, single-cell unit. The system under consideration comprises of an external mass subjected to a three-dimensional linear local potential with an internal spherical rotator. In the present study we focus on the analysis of the regimes of three-dimensional, bidirectional energy transport realized in the limit of low-energy excitations. Unlike the previously reported studies, this system under consideration exhibits rich nonlinear phenomena concerning the dynamics and the bifurcation structure of highly non-stationary regimes. Thus, in the considered limit we unveil analytically the two distinct families of non-stationary regimes corresponding to the in-plane as well as the out-of-plane bidirectional energy channeling. This phenomenon of bidirectional energy channeling is manifested by the three-dimensional, recurrent transformation of general in-plane oscillations of the external element to the orthogonally reoriented in-plane and out-of-plane ones. This three-dimensional energy flow is fully controlled by the internal spherical rotator coupled to the external mass. Here we also show that the regimes corresponding to the bidirectional energy channeling as well as spontaneous energy locking reported in the previously considered planar cases can be generalized analytically to the three-dimensional case. To this end we use a regular multi-scale analysis which enables to characterize and predict the intrinsic mechanisms governing the highly non-stationary regimes of the three-dimensional energy flow. Numerical simulations are found to be in extremely good correspondence with the analysis.