Analysis and Stability Assessment of the Vibratory Motion of a Magnetic Mechanical System Near Resonance

This article examines a novel magnetic dynamical system consisting of a connected linearly damped transverse tuned absorber to a linearly damped spring pendulum (SP), where its suspension point moving along a Lissajous trajectory. The motion is considered under the influence of a magnetic field, a harmonic moment at the suspension point, and a harmonic force on the spring’s radial direction. The controlling equations of motion (EOM) are derived using the second kind of Lagrange’s equations (LEs) and are analytically solved up to a higher order of approximation via the traditional perturbation approach known by multiple-scales method (MSM) to achieve new results. Solvability criteria are obtained in view of the removal of secular terms, and therefore the system’s modulation equations (ME) are achieved according to the examined resonance cases. The Routh-Hurwitz criterion (CRH) is employed as a well-established approach for assessing the stability of linearized systems through the analysis of the characteristic polynomial’s coefficients. This approach is particularly suited to the model under investigation, where the stability regions were identified and examined in relation to the solutions at the steady-state scenario. Graphical representations, including time histories plots, curves of resonance response, and stability regions, are provided to elucidate how various physical parameters affect the system’s behavior. Furthermore, phase portrait diagrams have been drawn which, are powerful tools for gaining deep insights into the behavior and properties of dynamical systems. The numerical solutions (NS) for the EOM are obtained using fourth-order Runge-Kutta algorithms (4RKA) and then compared with the achieved analytic approximate ones. This comparison highlights the strong consistency between them and confirms the accuracy of the applied perturbation technique. This work holds significance for its potential applications in both theoretical physics and engineering, including the analysis of flexible arm robotics control, vibrational dynamics of flexible arms, pump and compressor systems, rotor dynamics, transportation equipment, and shipboard crane operations.