Modeling, analysis and control of parametrically coupled electromechanical oscillators

This work investigates a parametrically coupled electromechanical system of van der Pol and Duffing oscillators. Such a model finds applications in the development of sensors and transducers for the purpose of measurements and actuation. First, the model was developed from the basic components of electrical and mechanical systems, and the governing equations were derived from an energy-based principle. The system was solved analytically by multiple scales method to validate the numerical solutions. Its stability was examined by Floquet theory, and further analysis reveals the presence of Hopf and Neimark-Sacker bifurcations. Other phenomena, such as period doubling and quasiperiodic routes to chaos, were discovered. Additionally, the system exhibits multistability due to the presence of coexisting attractors and synchronization at parametric resonances. Finally, the chaotic system was stabilized by designing a nonlinear feedback controller with time-varying coefficients, bringing its trajectories to the desired equilibrium at the origin for vibration suppression and the desired limit cycle for periodic solutions, respectively. The controlled outputs show a stable invariant closed curve with its Floquet multipliers falling within a unit circle.