Frequency-Amplitude Relationship in Nonlinear Oscillators with Irrational Nonlinearities

This study investigates the frequency-amplitude relationship in nonlinear oscillators with irrational nonlinearities, which pose significant challenges to traditional analytical methods such as the homotopy perturbation method and variational iteration method. By employing a simplified frequency formulation, we analyze the impact of irrational nonlinearities on oscillation dynamics and derive a frequency-amplitude relationship in a single-step calculation. For small amplitudes, our results are consistent with those obtained by the homotopy perturbation method, validating the proposed approach. This research provides new insights into the complex dynamics of nonlinear oscillators and offers practical implications for fields such as physics, engineering, and applied mathematics. Additionally, it contributes to the design and optimization of MEMS systems and suggests future research directions, including the incorporation of damping and external forces for a more comprehensive understanding of system behavior.