Energy transfer and localization in a forced cyclic chain of oscillators with vibro-impact nonlinear energy sinks

We theoretically investigate the strongly nonlinear dynamics, inter-modal targeted energy transfer and energy localization in an elastically coupled cyclic chain of oscillators with vibro-impact nonlinear energy sinks (VI-NESs) under symmetric harmonic standing or traveling wave forcing. Each identical sector of the chain consists of a single linear oscillator hosting a VI-NES, which is a small mass that is freely placed inside a cavity of the oscillator. We show that the VI-NESs are able to synchronize to the global standing or traveling wave response of the structure in the form of 1:1 resonance captures with the oscillators in each sector. In addition, localized states at higher amplitudes can be found where the VI-NESs synchronize to the motion of their host oscillators in only a subset of all sectors. We derive an analytical model to predict the frequency-amplitude curves of these synchronized solutions and study their local asymptotic stability analytically and their practical stability numerically. We show that the globally synchronized response can experience a modulation instability which gives rise to traveling beat waves. High and practically stable localized amplitudes only arise for sufficiently low excitation wavenumbers and weak inter-sector coupling strengths. However, even the largest practically stable amplitudes show a significant reduction of the vibration level compared to the corresponding linear resonant responses. Hence, a robust high performance of the VI-NESs is observed for all excitation wavenumbers and inter-sector coupling strengths.