Nonlinear oscillations of a lumped system with series spring, piezoelectric device, and feedback controller

This paper examines the behavior of a mechanical system with a lumped- mass comprising two nonlinear springs arranged in series and combined with a piezoelectric device. External harmonic excitations, as well as linear and nonlinear damping, are considered. The main system employs a negative velocity feedback (NVF) controller to reduce undesired effects vibrations, particularly under resonance conditions, thereby enhancing the system's efficiency. The system is described by differential and algebraic equations, forming a dynamic model governed by differential-algebraic equations (DAE). A nearly analytical technique is further applied to resolve the initial value problem of the DAE. Applying the Lagrange's equations (LE), the regulating equations of motion (EOM) are derived. The approximate solutions (AS) to third-order are obtained subsequently in the framework of the multiple-scales method (MSM). The AS's accuracy is confirmed by comparing it to the numerical solution (NS) obtained via Runge-Kutta fourth-order algorithms (RK- 4). Examining the resonance cases, along with the criteria of solvability, leads to the derivation of the modulation equations (ME). Graphical representations of the solutions' time histories and frequency response curves are presented using Wolfram Mathematica 9 and MATLAB- 23 software, providing a thorough visualization of the results. In addition, bifurcation diagrams and Poincaré maps (PMs) are graphed to illustrate the different behavioral modes of the system. Conversely, piezoelectric transducers are linked to the dynamic model to transform vibrational motion into electrical energy. This technology represents one of the many energy harvesting (EH) solutions widely utilized across commercial, aerospace, industrial, medical sectors, and automotive. A graphical analysis illustrating the time courses of solutions with and without control is presented. Additionally, resonance frequency curves are plotted to assess stability/instability and evaluate the solutions at steady-state.