Insights of Nonlinear Vibration of Oscillators Linked with Two-Degrees-of-Freedom

Damped coupled harmonic nonlinear oscillators are essential to model several physical systems, including electrical circuits, mechanical structures, and definite biological systems. Therefore, the current work aims to examine the honest non-perturbative approach (NPA) to get periodic solutions for damped and conservative coupled mass-spring systems with linear and nonlinear stiffness that exhibit nonlinear free vibrations. The NPA is mainly based on the He’s frequency formula (HFF). Four practical models of two-degree-of-freedom (TDOF) oscillation systems are demonstrated to display the accuracy, effectiveness, and applicability of the proposed approach. Furthermore, the study intends to achieve approximate solutions for small amplitude parametric factors, without observing the restrictions imposed by traditional perturbation approaches. The method is also extended to reveal ideal solutions for systems of nonlinear coupled oscillators. The iterative approximations of the solutions of parametric nonlinear fluctuations necessitate a quick estimation of the frequency-amplitude relationship. The produced parametric equation is validated using the Mathematica Software (MS), and it exhibits excellent agreement with the numerical solution (NS) of the original system. The achieved responses of the examined models, besides the solutions of two limited cases, are graphically represented. These graphs show the temporal behaviors of these models and reveal a good impact of the acted parameters. The time histories of these solutions show consistent decay behavior, implying the stability of the results. Moreover, the related phase plane curves are displayed in various plots that have spiral curves directed inward at one single point. The behavior of the limited cases has periodicity forms with the variation of several acted parameters. Therefore, the corresponding curves of the phase plane have the forms of symmetric closed curves, i.e., limit cycles are produced. The present methodology seems to be straightforward, promising, powerful, and attractive. It can be employed in several kinds of multi degrees of freedom in dynamical systems.