Vibration Attenuation of Cantilever Beam Using Electromagnetic Delayed Feedback

This study presents a combined analytical and numerical investigation of a single-input single-output delayed feedback controller designed to suppress vibrations in a cantilever beam actuated by an electromagnet positioned beneath its free end. An analytical model was developed using the linearized equation of motion for a transversely excited cantilever beam, incorporating an electromagnetic actuation force term, and solved numerically. In parallel, a finite element model coupling solid mechanics and magnetic fields was implemented in COMSOL Multiphysics. To capture the nonlinear variation of the magnetic force with beam deflection, a moving mesh technique was employed. Continuous delayed feedback was applied within the finite element framework using built-in operators that access solutions from previous time steps. Stability regions and normalized damping values were obtained from both models and compared to assess the validity of the analytical formulation. The results show that optimal controller gains and delays can be identified, with the delay remaining nearly invariant while the optimal gain exhibited sensitivity to the initial vibration amplitude. Unlike conventional feedback control, the electromagnet-based delayed feedback controller demonstrated amplitude-dependent behavior but achieved significant suppression of the first vibration mode.