Nonlinear vibration of quasi-zero stiffness structure with piezoelectric harvester and RL-load: intra-well and inter-well oscillation modes under 1:1 internal resonance

This study explores the nonlinear dynamics of a quasi-zero stiffness (QZS) vibration isolator coupled with a piezoelectric energy harvester connected to an RL-resonant circuit. The model of the system is formulated with the Lagrangian mechanics, representing a two-degree-of-freedom nonlinear electromechanical system subject to harmonic base excitation under a 1:1 internal resonance condition. The model is normalized, and the conditions dictating monostable and bistable oscillation modes are identified. The bifurcation characteristics of the coupled system are analyzed in both oscillation modes by means of harmonic balance and continuation methods. The vibration isolation performance, with and without the coupled harvester, is evaluated in terms of displacement transmissibility to assess its dual functionalities for vibration isolation and energy harvesting. Analytical results demonstrate that integrating a piezoelectric harvester into a monostable QZS isolator under 1:1 internal resonance does not compromise its vibration isolation capability while enabling efficient energy harvesting at extremely low-frequency base excitation. Furthermore, the system’s response under strong base excitation is investigated exclusively for energy harvesting in both monostable and bistable modes, leading to optimal structural parameter design. The conditions for intra-well and inter-well periodic oscillation modes, as well as chaotic responses, are analyzed analytically and validated numerically through stability charts, basins of attraction, bifurcation diagrams, time histories, and Poincaré maps. This work provides a comprehensive understanding of the oscillation dynamics of QZS isolators and offers valuable insights for optimizing their geometric parameters to function as high-performance vibration isolators and/or energy harvesters.