Geometric nonlinear dynamics of a quasi-zero stiffness isolator integrated with an energy harvester: Monostable, perfect zero-linear stiffness, and bistable oscillation modes

Achieving effective vibration isolation across a broad frequency range while simultaneously harvesting energy from vibrations remains a key challenge in engineering systems. This study examines the nonlinear dynamics and vibration isolation performance of an oblique-type spring quasi-zero stiffness (QZS) isolator integrated with a piezoelectric energy harvester. The coupled system is modeled as a strongly nonlinear oscillator linked to a first-order differential equation governing the harvester's response. The QZS isolator's behavior is characterized by two geometric nonlinearity parameters, the stiffness ratio of oblique to vertical springs ($\rho$) and the ratio of the oblique spring's maximum horizontal compression to its free length ($\lambda$). Closed-form expressions for $\rho$ and $\lambda$ are derived to determine the conditions for monostable, bistable, and perfect zero-linear stiffness operation. The system's response is analyzed using the harmonic balance method, with bifurcation diagrams illustrating oscillation amplitudes, harvested voltage, and displacement transmissibility under different excitation conditions. Key findings indicate that for $\rho <\lambda /\left(2-2\lambda \right)$, the system functions as a monostable QZS isolator, where reducing $\lambda$ and/or increasing $\rho$ suppresses resonant peaks, creating a semi-full-band isolator. When $\rho =\lambda /\left(2-2\lambda \right)$, the system achieves full-band vibration isolation with perfect zero-linear stiffness, enhanced by high pre-compression of the oblique springs. For $\rho >\lambda /\left(2-2\lambda \right)$, the system transitions to a bistable regime, improving energy harvesting but reducing isolation efficiency. Additionally, the piezoelectric harvester not only facilitates energy conversion but also introduces active damping, effectively mitigating resonant peaks and stabilizing the system under strong base excitations while minimally affecting high-frequency displacement transmissibility. This work provides a comprehensive understanding of the oscillation modes in oblique-type QZS systems and offers design insights for optimizing geometric parameters to achieve full-band or semi-full-band isolation and efficient energy harvesting.