Frequency–Amplitude Relationship in Damped and Forced Nonlinear Oscillators with Irrational Nonlinearities

This paper undertakes an exhaustive investigation of nonlinear oscillators subject to damping and external excitation, with a particular emphasis on systems exhibiting irrational nonlinearities. Nonlinear oscillators play a foundational role in the modeling of a broad array of natural and engineering systems. These systems exhibit behaviors that are significantly different from linear systems. These behaviors include amplitude-dependent frequencies, subharmonic and superharmonic responses, bifurcations, and chaotic motions. The frequency-amplitude relationship, which is central to this research, is of significant importance in various fields, including vibration control, energy harvesting, and the study of biological rhythms. It is important to note that this relationship is subject to variation in accordance with amplitude changes. In this study, the frequency formulation is employed to meticulously analyze the response characteristics of damped and forced nonlinear oscillators. This analysis effectively validates the efficacy of frequency formulation in capturing the periodic behavior of these systems. The research findings not only validate the established results under optimal conditions but also extend the analytical scope to encompass the more intricate and nuanced dynamic phenomena encountered in real-world scenarios. The derivation of the frequency-amplitude relationship unveils the underlying mechanism through which damping and external forces influence the system's dynamic response, thereby facilitating a more profound comprehension of the behavior of nonlinear oscillation systems.