A review of the frequency-amplitude formula for nonlinear oscillators and its advancements

The frequency-amplitude relationship is pivotal in understanding oscillatory systems, dictating their dynamic behaviors and responses. This paper delves into the intricacies of the frequency amplitude formula, elucidating its foundational role in both linear and nonlinear oscillations. Through comprehensive analysis, we highlight the formula's significance in predicting system behaviors, especially in environments with varying amplitudes. Our findings underscore the formula's potential as a robust tool for enhanced system characterization, offering profound implications for diverse applications across scientific and engineering domains. This study delves deep into the formulation of the frequency-amplitude relationship, extending its application beyond the conventional cubic powers of the restoring force. We introduce three novel and equivalent formulations of the generalized frequency amplitude, breaking traditional boundaries by not confining the restoring force to just odd functions. The formula to determine the frequency of the singular oscillator has been set forth. Our approach, characterized by its simplicity, offers a robust tool for analyzing high nonlinearity vibrations, ushering in a richer, multidimensional perspective to vibration analysis.