Inspection of Some Extremely Nonlinear Oscillators Using an Inventive Approach

Abstract

Abstract Purpose A group of classical oscillators of high nonlinearity, which cannot be completely analyzed, is addressed by introducing a novel technique. The main objective of the current investigation is to utilize the generalized He’s frequency formula (HFF) in studying the analytical explanations of specific types of extremely nonlinear oscillators. This interest arises from the growing fascination in the realm of nonlinear oscillators. Regarding several engineering and scientific fields, together with three particular situations, a generic example is presented. Methods Compared to prior perturbation approaches utilized in this field, the new strategy is straightforward and requires less processing and timing. This ground-breaking tactic, which converts the nonlinear ordinary differential equation (ODE) into a linear one, is referred to as the non-perturbative approach (NPA), as an innovative approach. A new frequency that is comparable to a linear ODE, like in a simple harmonic situation, is produced in the procedure. When evaluating the physiologically significant specialized instances, the outcome from this straightforward approach not only exhibits a strong agreement with the numerical findings but also demonstrates that it is more accurate than the outcomes from other well-known approximate methodologies. An extensive description of the NPA is presented to ensure the maximum benefits. Results The theoretical findings are confirmed by conducting a numerical analysis with the aid of Mathematica Software (MS). The numerical solution (NS) and the theoretical responses demonstrated remarkable congruity. Conventional perturbation techniques typically use Taylor expansion to enlarge restoring forces, thereby reducing problem complexity. However, this weakness disappears with the NPA. Additionally, stability analysis of the problem alongside the NPA becomes feasible, unlike with prior conventional methodologies. Conclusion The NPA emerges as a more responsible resource when examining the NS for oscillators with significant nonlinearity. Its exceptional versatility in addressing various nonlinear problems underscores the NPA as a valuable benefit in the fields of engineering and applied science.