Two-DOF Auto-Parametric Dynamical System Stability and Bifurcation Analysis with Piezoelectric and Electromagnetic Devices and Feedback Control

Abstract Abstract Aim This work analyzes a two-degree-of-freedom (DOF) auto-parametric system for concurrent vibration control and enhanced energy harvesting at resonance. The main objective is to achieve simultaneous vibration mitigation and energy extraction by integrating mechanical and electrical subsystems through electromechanical coupling. Methods The investigated system consists of a damped oscillator with an attached pendulum and two distinct energy harvesting (EH) devices: a piezoelectric harvester and an electromagnetic harvester. The primary structure, composed of a vertical damped oscillator coupled with a suspended pendulum, is equipped with a negative-velocity-feedback (NVF) controller to reduce unwanted vibrations that impair system performance, especially under resonance conditions. The governing equations of motion are derived using Lagrange’s equations (LE) and solved approximately by the perturbation method, namely the multiple scales method (MSM). Validation of the obtained approximate solutions (AS)is performed through comparison with the computed numerical solutions (NS) via the Runge–Kutta fourth order (RK-4) method. Results After classifying the resonance cases, the analysis focuses on the worst-case scenario, namely the primary external resonance. Under this condition, the influence of various system parameters is thoroughly examined. The system’s stable and unstable responses are analyzed using frequency responses (FR), time histories, Poincaré maps (PM), and bifurcation diagrams. The results show that the NVF controller effectively reduces the amplitude of unwanted vibrations while maintaining stable system behavior. Both the piezoelectric and electromagnetic harvesters successfully extract electrical energy from the system’s oscillations. The comparison between analytical and numerical results demonstrates excellent agreement, confirming the reliability of the derived solutions. Applications This dynamical model offers a dual advantage by harvesting energy from vibrating motion through both electromagnetic and piezoelectric transducers, which convert mechanical vibration into electrical power via magnetic induction and material strain, respectively, while actively reducing unwanted oscillations using an NVF controller. It is particularly useful in systems operating near resonance, such as in vehicles, buildings, and industrial machinery. The model enhances both energy efficiency and structural reliability, making it highly suitable for smart infrastructure and self-powered monitoring applications.