The curious dynamics of a free electromechanical system

Abstract This paper presents the modal analysis of a simple electromechanical system composed of two interacting subsystems, one mechanical and one electromagnetic, and shows how its dynamics differs fundamentally from that of purely mechanical systems. Classical mechanical oscillators exhibit motion governed by the exchange between kinetic and potential energies, whereas the system analyzed here oscillates solely through the exchange between mechanical kinetic and magnetic kinetic energies, since no elements store mechanical or electrical potential energy. Its governing equations involve only two matrices, M and G . Although traditionally referred as inertia and gyroscopic matrices, their physical meaning in this electromechanical context differs substantially: M incorporates both mechanical and electromagnetic inertial contributions, and G does not act as a conventional gyroscopic matrix but instead couples the subsystems and their energy exchange. To the best of our knowledge, this is the first modal analysis of a system composed solely of inertia and gyroscopic matrices. The computed natural frequencies and modes are intrinsically hybrid, involving both mechanical and electromagnetic variables. Hybrid modal coordinates are introduced, and resonance characteristics and frequency response functions are examined. An energetic analysis further reveals that the hybrid natural frequency corresponds to the rate at which energy is exchanged between the mechanical and electromagnetic subsystems.