Analytical Modeling of Nonlinear Oscillators Using the Energy Balance Method: Applications to Relativistic and Duffing Systems

Accurate determination of the frequency–amplitude relationship in strongly nonlinear oscillators remains a critical challenge in engineering dynamics, particularly when classical linearization and traditional perturbation methods fail due to the absence of small parameters. This limitation complicates the analysis and design of systems where nonlinear effects dominate, such as mechanical vibrations, structural dynamics, and nonlinear control applications. This study investigates the applicability and effectiveness of the Energy Balance Method (EBM) for analytical modeling of relativistic and Duffing-type oscillators. The method provides reliable and computationally efficient expressions for system frequencies across a wide range of oscillation amplitudes without relying on restrictive perturbation assumptions. The formulation assumes conservative systems undergoing periodic motion, limiting its application to undamped oscillators; within these constraints, EBM offers a simple and robust solution strategy. The approach constructs an energy balance over a single oscillation period using a single-term trial function, requiring only one iteration to obtain the frequency–amplitude relationship. The main novelty lies in demonstrating that such a minimal framework delivers highly accurate results even in strongly nonlinear regimes. Quantitative comparisons with exact solutions and established analytical methods—including the Harmonic Balance Method, Variational Iteration Method, Homotopy Perturbation Method, and the method of multiple scales—show excellent agreement, with negligible relative errors over a broad amplitude range. These results confirm that EBM achieves comparable or superior accuracy with significantly reduced computational effort. Future work may extend the method to damped, forced, and multi-degree-of-freedom systems, further broadening its applicability to complex engineering problems.