Electromagnetic and structural dynamic coupling in a two-degree-of-freedom system

Induced current damping is widely used to mitigate vibrations in mechanical systems. Conventional formulations of electromagnetic force matrices often assume constant, linear, or frequency-dependent coefficients. In a recent study, the authors observed that when a cantilever beam subjected to bending or torsional vibrations is exposed to an external magnetic field in multiple directions, an unexpected coupling between vibration modes occurs. This phenomenon suggests that existing models may be insufficient to capture the complete dynamic behaviour resulting from induced currents. Consequently, this Rapid Communication investigates this coupling effect using a simplified two-degree-of-freedom system under an external magnetic field. The effect of this magnetic field implies a coupling between the two degrees of freedom of the system, characterised by a symmetric matrix containing both dissipative and coupling terms. A frequency response analysis reveals a particular behaviour at resonance when the two degrees of freedom are coupled, which occurs only when the magnetic field acts simultaneously in both directions. In this case, increasing the magnetic field magnitude mitigates one vibration mode while amplifying the other. In the time domain, the analysis of the mechanical energy evolution of each degree of freedom reveals how energy is transferred between the two degrees of freedom as the system dissipates energy. These findings highlight the necessity of explicitly incorporating magnetic field orientation and magnitude into electromagnetic force matrices to accurately predict the coupling effects. The results serve as a foundation supporting future studies on more complex multi-degree-of-freedom systems and their practical implications in vibration control.