Interaction between Anderson and nonlinear localization in a forced and mistuned cyclic chain of oscillators with vibro-impact nonlinear energy sinks

Abstract

Random mistuning is widely known to lead to strong localization of vibrations in weakly-coupled, rotationally periodic structures composed of a finite number of sectors, which is known as Anderson localization. In the case of forced vibrations, this is typically associated with spatially confined large amplitude magnifications compared to the tuned system’s resonant response. In addition to Anderson localization, strong localization effects can also occur in tuned but nonlinear rotationally periodic structures. In this work, we investigate the interaction between Anderson localization and nonlinear localization in a mistuned and harmonically forced cyclic chain of oscillators which incorporates internal strongly nonlinear devices for vibration mitigation, called vibro-impact nonlinear energy sinks (VI-NESs). A VI-NES consists of a small mass that is placed inside a cavity in each sector’s primary mass where it undergoes dissipative (inelastic) impacts with the cavity walls. The tuned version of the studied system was recently shown to exhibit strong nonlinear localization effects due to Chimera-like steady-state vibrations in which sustained 1:1 resonance captures between the VI-NESs and their host oscillators occur in only a subset of all sectors. In the present work, we show both analytically and numerically that Anderson localization can constructively interact with the system’s nonlinearly localized solutions by increasing both their amplitude and localization level, as well as by creating solution branches in parameter regions where the nonlinearly localized solution would not exist in the tuned configuration. Despite this strong interaction, the VI-NESs are still capable of effectively mitigating the vibrations of the chain. The implications of the findings of this work for real rotationally periodic engineering systems with VI-NESs are discussed.