Harvesting energy in nonlinear systems through time-delay mechanisms

Energy harvesting (EH) from ambient vibrations is a promising approach for powering self sustained devices. Nonlinear oscillators with time delay feedback have attracted attention for their ability to broaden operating ranges and improve efficiency. Conventional linear harvesters and even nonlinear devices often suffer from instabilities, hysteresis, and limited bandwidth. Understanding how periodic and quasi periodic (QP) vibrations contribute to EH in systems with dual time delays remains an open challenge. Prior studies have shown that time delay in mechanical subsystems can induce large amplitude QP oscillations, while piezoelectric coupling with delay can enhance power output. However, most analyses treat mechanical and electrical delays separately. The combined influence of distinct delays in both mechanical and electrical components has not been systematically investigated, leaving unclear how dual delay mechanisms affect stability and harvested power. This study models an EH system as a delayed Duffing–van der Pol oscillator coupled with a delayed piezoelectric circuit. Using perturbation methods and numerical simulations, we derive periodic and QP solutions near delay induced parametric resonance and quantify harvested power. Results show that small mechanical delay amplitudes favor periodic vibrations, while larger amplitudes destabilize them, shifting energy extraction to QP vibrations with superior performance. Electrical delay further enhances harvested power across specific parameter ranges. These findings advance the theoretical foundation of nonlinear EH, highlighting QP vibrations as a viable strategy for efficient broadband energy extraction. The work provides design guidelines for delay controlled harvesters and suggests future extensions to experimental validation and multi degree of freedom systems.