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

- 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.

Automated summary

This paper analyzes the parametric Gaylord's equation using the non-perturbative method and obtains the approximate solution and amplitude-frequency equation. The resonance frequency is obtained without perturbation, and the accuracy of the analytic solution is validated by comparing it with numerical data.