Emergence of Chaotic Behavior in Locally Resonant Nonlinear Metastructures

In this study, nonlinear dynamics of a metastructure composed of a cantilever beam with nonlinear resonators are investigated. Local resonators exhibit cubic stiffness nonlinearity which introduces complex nonlinear system dynamics and leads to rich vibrational phenomena such as nonlinear resonances, bifurcations, and chaotic motion of the overall metastructure. Understanding this dynamic behavior is crucial for predicting the response of nonlinear systems and designing effective vibration attenuation strategies utilizing nonlinear metastructures. This paper investigates complex dynamical behavior arising from the coupling between the host beam and the periodic array of nonlinear resonators. The transmissibility along the metastructure beam is studied for various excitation levels. For low excitation amplitudes, the system shows a linear-like behavior. The resonators create an anti-resonance in the frequency response and a bandgap is observed, similar to the linear metastructures. However, increasing the excitation force amplitude leads to a stronger nonlinear response where the jump phenomenon is observed in the metastructure response due to the nonlinearity of the resonators. In addition, new peaks start to form within the bandgap frequencies where the nonlinearity leads to non-periodic solutions. Through careful examination of attractors in Poincare sections, and Lyapunov exponents analyses, the birth of chaotic solutions is observed. Furthermore, increasing the forcing amplitude results in the merging of this chaotic solution with a periodic solution that leads to a drastic increase in the response amplitude. The findings of this study present potential risks of using nonlinear resonators for vibration attenuation and inform design guidelines for nonlinear metastructures.