Stochastic bifurcation analysis and performance optimization of a concave X-shaped structure quasi-zero-stiffness vibration isolator and energy harvesting system with time delay

This paper proposes a concave X-shaped quasi-zero-stiffness vibration isolation and energy harvesting (CXS-QZS-VIEH) system. Unlike conventional single-function isolators, this compact design achieves simultaneous vibration isolation and energy harvesting within a limited space. A stochastic time-delayed electromechanical coupling dynamic model is established, incorporating Gaussian white noise excitation and time-delayed feedback control to reflect real-world operating conditions. In contrast to existing studies that overlook the influence of time delay on the characteristic frequency, the delay-dependent critical frequency is derived for accurate Hopf bifurcation analysis, ensuring theoretical consistency. Through stochastic averaging and bifurcation analysis, it is revealed that an appropriate introduction of time delay can simultaneously enhance both vibration isolation and energy harvesting performance under stochastic excitation, which is further validated by numerical simulations. A comprehensive performance index (CPI) based on the Hill function is proposed to systematically optimize the trade-off between vibration isolation and energy harvesting. This study provides theoretical support and generalizable design guidance for vibration isolation and energy harvesting systems in stochastic environments.