Nonlinear dynamic analysis of a punctual charge in the electric field of a charged ring via modified frequency–amplitude formulation

In this paper, two types of frequency amplitude formulation method are initially utilized to obtain frequency–amplitude relationship of nonlinear vibration of a punctual charge in the electric field of a charged ring. In order to obtain the nonlinear natural frequency of the considered system, Reng-Gui and Geng-Cai modified methods are implemented. A table is also prepared to provide a brief review of recent development of nonlinear differential equations. The correctness of the obtained results is compared with those obtained from harmonic balance method (HBM) and energy balance method (EBM). A numerical simulation is carried out to investigate the accuracy of the used methods. In accordance with it, the relative errors of the employed approaches are numerically and analytically found based on the exact numerical solutions. It is exposed that the exerted approaches are very reliable and applicable for solving the nonlinear differential equations.