Nonlinear free vibration of conical beams using He's frequency formula: Educational implications

This study presents an analytical solution for the nonlinear free vibration of conical beams using He's frequency formula. This solution aims to efficiently address the challenges posed by their axially varying cross-sectional dimensions and associated nonlinear mechanical behaviors. Conical beams, with their customized stiffness and mass distribution, find wide application in aerospace, civil engineering, and micro-electromechanical systems (MEMS), where precise vibration analysis is imperative for ensuring structural stability and performance. The frequency formula, rooted in residual minimization, is employed to derive the frequency-amplitude relationship of conical beams. This method avoids complex iterative procedures and reduces computational complexity compared to traditional methods like Aboodh Transform-based variational iteration method (ATVIM) or homotopy perturbation. The validation of the method against ATVIM and numerical solutions (4th-order Runge–Kutta method) confirms its accuracy, with close agreement across moderate to large amplitudes and a frequency relative error of less than 5%. Beyond its practical utility in engineering design—enabling rapid parametric analysis for resonance avoidance—the study also highlights educational implications, as the conical beam case study bridges abstract nonlinear dynamics theory with real-world applications, aiding students in understanding frequency-amplitude coupling and method selection. This work demonstrates that He's frequency formula offers a robust, accessible framework for analyzing conical beam vibrations, linking theoretical nonlinear dynamics, engineering practice, and educational value.