Periodic solution of the parametric Gaylord's oscillator with a non-perturbative approach

Department of Mathematics, Faculty of Education, Ain Shams University - Roxy, Cairo, Egypt

^(a) E-mail: yusryeldib52@hotmail.com; yusryeldib@edu.asu.edu.eg (corresponding author)
^(b) E-mail: nasser522000@gmail.com; nasersaleh@edu.asu.edu.eg

Received: 18 August 2022
Accepted: 16 November 2022

Abstract

The vibration of a regular rigid bar without sliding over a solid annular surface of a specified radius can be considered by a parametric Gaylord's oscillator. The governing equation was the result of a strong nonlinear oscillation without having a natural frequency. The present work is concerned with obtaining the approximate solution and amplitude-frequency equation of the parametric Gaylord's equation via an easier process. The non-perturbative approach was applied twice to analyze the present oscillator. Two steps are used, the first is to transform Gaylord's oscillator to the parametric pendulum equation having a natural frequency. The second step is to establish the amplitude-frequency relationship which was taken out in terms of the Bessel functions. A periodic analytic solution is obtained, in the presence or without the parametric force. The frequency at the resonance case is established without a perturbation for the first time. The stability condition is established and discussed graphically. The analytic solution was also validated by comparing it with its corresponding numerical data which showed a very good agreement. In a word, by dissection of the behavior of strong nonlinearity oscillators, the non-perturbative technique is characterized by its ease and simplicity along with high accuracy when compared to other perturbative methods.