Analytical Model for Two Coupled Magneto-Mechanical Cantilevers With Small Perturbations

Works of numerous scholars have predominantly concentrated on considering the entire system, comprising a cantilever beam (or pendulum) and a magnet, exposed to the influence of a fixed permanent magnet. Moreover, these models largely depended on equivalent representations with lumped parameters for their analysis. This paper addresses these limitations by introducing a novel magneto-mechanical oscillator model. In this paper, the dynamics and kinematics of two magneto-mechanical coupled oscillators are analytical investigated. The system consists of two vertical cantilevers, which are affixed to a horizontal base and separated by a specific distance. These beams, characterized by their elasticity, exhibit magnetic coupling due to the inclusion of magnets at their tips. To model the system's behavior, the Galerkin method is used to derive a first-order approximate solution, assuming small oscillations. The influence of system parameters on the motion modes of the cantilever beams is systematically studied. It is shown that symmetric initial positions result in simple motion modes-overdamped, critically damped, or underdamped. Conversely, non-symmetric initial positions yield more intricate superimposed modes, combining these simple modes in pairs. These results offer a theoretical foundation for a comprehensive understanding of the magneto-mechanical oscillator's behavior. Furthermore, the potential phase regions for various motion modes are defined by analyzing a range of system parameters, thereby providing valuable insights for a deeper comprehension of the motion characteristics inherent to the magneto-mechanical oscillator.