An innovative methodology in analyzing certain pendulum oscillators

Abstract

Pendulum oscillators study harmonic motion, energy conservation, and nonlinear dynamics, providing insights into mechanical vibrations, wave phenomena, weather patterns, and quantum mechanics, with real-world applications in engineering, seismology, and clock mechanisms. The present study addresses three distinct issues related to SPs; a charged magnetic spherical simple pendulum (SP), and a SP composed of heavy cylinders that roll freely in a horizontal plane, and a nonlinear model depicting the motion of a damped SP in a fluid flow. The SPs are analyzed via an innovative technology known as the non-perturbative approach (NPA), which is based on He's frequency formula (HFF). This advanced approach linearizes a nonlinear ordinary differential equation (ODE), enabling more straightforward analysis and solution. As-well known, implementing the NPA has several advantages, chief among them the removal of the constraints associated with managing Taylor expansions. Consequently, there have been no augmentations to the current restorative forces. Secondly, the novel method enables us to assess the stability criteria of the system away from the traditional perturbation techniques. The numerical comparison of nonlinear ODEs into linear ones using Mathematica Software (MS) is conducted to validate this innovative method. An analysis of the two responses demonstrates a strong concordance, underscoring the necessity of precision of the methodology. Furthermore, to demonstrate the influence of the components on motion behavior, the time history of the calculated solution and the corresponding phase plane plots are accumulated. The use of multiple phase portraits aims to explore stability and instability near equilibrium points by examining the interaction between expanded and cyclotron frequencies, modulated by the magnetic field, for varying azimuthal angular velocities.