The stability analysis of a dynamical system equipped with a piezoelectric energy harvester device near resonance

In this work, the influence of a piezoelectric device on the planar motion of a two-degrees-of-freedom (DOF) dynamical system is examined. This system comprises a nonlinear damping spring pendulum (SP), whose pivot point moves on a Lissajous curve at a constant angular velocity in an anticlockwise direction amidst the influence of external forces and moments. The second kind of Lagrange’s equations is used to derive the governing system of equations of motion (EOM), which is transformed into a dimensionless form. Up to the third approximation, the novel asymptotic solutions of this system are obtained using the multiple scales technique (MST). The numerical solutions of the EOM are obtained applying the fourth-order Runge–Kutta method. The comparison between both solutions, the asymptotic solutions and the numerical ones reveals great agreement between them, which in turn indicates the accuracy of the used perturbation technique. The solvability criteria are established once the generated secular terms are eliminated, and all resonance cases are subsequently categorized. In light of the system’s adjusted phases, the modulation equations for two of these cases are simultaneously investigated. Graphically, numerous plots depicting the time histories of the achieved solutions are examined, and the nonlinear stability analyses of the modulation equations are investigated in accordance with the steady-state solutions. The outputs of the energy harvesting (EH) device are scrutinized to investigate and analyze the influences of damping coefficients, excitation amplitudes, and different frequencies. Multiple regions of stability and instability are delineated within the frequency response curves (FRC), illustrating the impact of varying parameters on the dynamic behavior of the system. The obtained results are considered significant due to their practical applications in our lives. It can be utilized for medical applications, charging electrical gadgets, powering sensors, smartphones, and other gadgets. It also powers keyless entry systems, patient monitors, airbag sensors, fish and depth finders, and audible alarms like smoke alarms.