Scrutinizing highly nonlinear oscillators using He’s frequency formula

Abstract

Highly nonlinear oscillators are examined in their capacity to simulate intricate systems in engineering, physics, biology, and finance, as well as their diverse behavior, rendering them essential in the development of resilient systems and technological advancement. Therefore, the fundamental purpose of the current work is to analyze He’s frequency formula (HFF) to get theoretical explanations of many types of very nonlinear oscillators. We investigate, in both analytical and computational, the relationship between elastic forces and the solution of a specific oscillator. This oscillator exhibits significant nonlinear damping. It is assumed that the required quantity of trigonometric functions matches the solution of a strong nonlinear ordinary differential equation (ODE) that explains the motion. The novel approach definitely takes less processing time and is less complex than the traditional perturbation methods that were widely used in this field. This novel method, which is essentially giving a linearization of the nonlinear ODE, is known as the non-perturbative approach (NPA). This procedure produces a new frequency that is similar to a linear ODE, much as in a fundamental harmonic scenario. Readers will benefit from an in-depth account of the NPA. The theoretical findings are validated by numerical examination using Mathematical Software (MS). The theoretical and numerical solution (NS) tests yielded fairly similar findings. It is a well-established principle that classical perturbation methods trust on Taylor expansions to approximate restoring forces, therefore simplifying the current situation. When the NPA is used, this vulnerability does not present. Furthermore, the NPA enables a thorough assessment of the problems’ stability analysis, which was a not possible using prior conventional methodology. Consequently, the NPA is a more appropriate responsibility tool for examining approximations in extremely nonlinear oscillators in MS.