The periodically addressed steady-state response of a nonlinear pendulum like oscillator is investigated in this study using the classical Lindstedt–Poincare technique. The oscillator is obtained by a systematic transformation of the nonlinear differential equation u ̈+(1+u ̇^2 )u=0, which cannot be easily handled in its original form because of its high nonlinearity. The system can be simplified to a corresponding pendulum like nonlinear oscillator of the form x ̈+tanx=0, along with consistently transformed initial conditions. This change does not alter any of the basic dynamical characteristics, but drastically simplifies the analytical treatment. Nonlinear oscillatory systems are common in physics and engineering, and the equations governing their behavior do not often have analytical solutions. Many traditional perturbation techniques can give secular terms that are increasing with time, and provide non-uniform and physically unrealistic approximations. In order to solve this problem, the Lindstedt-Poincare method is used to remove the secular terms by adding a perturbation expansion to the system frequency, and this ensures the uniformly valid periodic solutions. The Lindstedt-Poincare method is used to compute approximate analytical expressions of the solution and frequency of the transformed oscillator. The accuracy and effectiveness of the proposed approach are demonstrated through comparisons with numerical solutions obtained by the fourth-order Runge–Kutta method, as well as with results from the harmonic balance method, frequency–amplitude formulation, and the parameter expansion method. The comparisons show excellent agreement, confirming the reliability and robustness of the present analysis for nonlinear oscillatory systems.
This paper has provided an analytical framework of studying a highly nonlinear oscillatory system by rewriting the original governing equation into a corresponding pendulum like equation. This work simplifies the mathematical treatment of the system and still retains the basic dynamical properties of the system which allows the application of classical analytical methods. Lindstedt-Poincare method successfully has been used to obtain approximate periodic solutions and the corresponding frequency-amplitude relationships. The approach removes the secular terms, and provides uniformly sound solutions over long durations of time. This characteristic renders it especially useful in the analysis of nonlinear oscillators with the amplitude-dependent behavior.
The validity and strength of the proposed method have been rigorously tested by comparing the results with numerical solutions to systems found using the fourth-order Runge-Kutta method, as well as with the results of other established analytic techniques, such as the harmonic balance technique, frequency-amplitude technique, and parameter expansion technique. The findings indicate that there is an excellent agreement of small and high amplitudes, and that the current method is significantly more accurate than the compared methods at higher amplitudes. Specifically, even in the highly nonlinear regime, the error growth is relatively small, which evidently demonstrates the reliability and stability of the approach.
In general, the current analysis proves that the combination of an appropriate transformation and the Lindstedt-Poincare method is a potent and effective tool to investigate nonlinear oscillatory systems. The methodology that has been developed in this work can be applied to a wider range of nonlinear problems that can be used to provide valuable insights into the dynamic behavior and frequency characteristics of these types of nonlinear problems.
Conflict of Interest
No conflict-of-interest present among the authors.
Data Availability
No datasets were generated or analyzed during the current study.
Funding Statement
The authors declare that no funds, grants, or other support were received during the preparation of this manuscript.
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https://doi.org/10.1016/j.ijnonlinmec.2024.104923
