Maclaurin-Based Harmonic Balance Method for Solving Nonlinear Oscillators
Abstract
This article introduces a Maclaurin-based harmonic balance method (MBHBM) for solving strongly nonlinear oscillators. In the proposed formulation, the nonlinear restoring force is systematically expanded by using a Maclaurin series, enabling the transformation of the governing nonlinear differential equation into a hierarchy of polynomial expressions. The method is applied to Duffing-type oscillators to assess its capability in capturing complex nonlinear systems. The resulting approximate solutions exhibit excellent agreement with numerical solutions and those obtained using the classical harmonic balance method across a broad range of oscillation amplitudes. Comparative analyses reveal that the Maclaurin-based harmonic balance method (MBHBM) provides higher accuracy with lower computational complexity, demonstrating its effectiveness and efficiency. The method exhibits reduced errors, especially in high-amplitude cases where the classical harmonic balance method loses accuracy. Owing to its high accuracy, computational efficiency, and straightforward implementation, the proposed method constitutes a robust and effective analytical framework for a wide class of strongly nonlinear oscillatory systems.
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