Exploring dynamics, stability, and bifurcation in a coupled damped oscillator with piezoelectric energy harvester

In this work, we analyze and investigate the mathematical modeling of a dynamical system connected to a piezoelectric device. Piezoelectric transducers are established as highly effective energy harvesting (EH) devices, often employed in real-world mechanical systems. The structure of the dynamical model contains a damped Duffing oscillator acting as the major component, which is connected to an unstretched pendulum and simultaneously to the piezoelectric harvester. To derive the governing equations of motion (EOM), Lagrange’s equations are applied based on the system’s generalized coordinates, in which they are analytically solved using the multiple scales (MS) perturbation method up to the second approximation. Moreover, the solutions achieved are compared with numerical ones to increase transparency and demonstrate the accuracy of the approximate solutions. In addition, comprehensive graphical representations have been produced to investigate the nonlinear stability of the modulation equations. Phase portraits, bifurcation diagrams, and Lyapunov spectra are displayed to depict various system behaviors, alongside Poincaré maps for deeper understanding. The various stability ranges are also explored and discussed. In conclusion, mechanical vibrations are transformed into electricity by the piezoelectric transducer connected to the dynamic model, which has widespread applications and includes crystal oscillators, medical ultrasound, gas igniters, and displacement transducers.