Enhancing damping performance below the bandgap in metamaterial beams with geometric nonlinearity and bistable attachments via nonlinear energy transfer

Nonlinear metamaterials embedded with bistable resonators enable efficient energy exchange between acoustic and optical branches via 1:n internal resonances. Recognising that the low-frequency acoustic branch is typically low-damping while the optical one near the bandgap exhibits higher losses, tuning their frequency relationship promotes nonlinear coupling, offering substantial and robust passive damping below the bandgap. To validate the feasibility of this pathway, a theoretical investigation is carried out on a clamped–clamped metamaterial beam with bistable attachment, which is realised by employing a dual pre-curved beam structure to design an easily achievable micro-bistable resonator, then periodically embedding it into the host beam. The geometric nonlinearity of the host structure, modelled using the von Kármán beam theory, is considered to capture the coupling effects accurately. The dynamics of the nonlinear beam are reduced using nonlinear normal modes, computed via the direct parametrisation method for invariant manifolds, which are then assembled with the dynamic equations governing the internal resonators. This strategy provides an accurate reduced-order model of the coupled system, achieving a significant speed-up in the computing time. The nonlinear vibrations of the coupled system are then analysed through linear dispersion spectrum and nonlinear frequency response curves, demonstrating significant damping improvements below the bandgap. Reductions of up to 30 dB and 10 dB are observed in the first and second resonant peaks, respectively, benefiting from the rich nonlinear energy transfer phenomena, enabled by the combination of geometric nonlinearity and the bistable characteristics of the resonators. The parameter analysis provides optimisation guidelines for the parameters related to the bistable configurations of the dual pre-curved beam, where the linear frequency of the bistable resonators, which governs the coupling between the low-damping and the high-damping modes, should remain between one and three times the frequency of the target mode to achieve effective damping enhancement.